Recent Titles in This Series

123 M. A. Akivis and B. A. Rosenfeld, Elie Car;tan (1869-1951), 1993 122 Zhang Guan-Hou, Theory of entire and meromorphic functions: Deficient and

asymptotic values and singular directions, 1993 121 I. B. Fesenko and S. V. Vostokov, Local fields and their extensions: A constructive

approach, 1993 120 Takeyuki Hida and Masuyuki Hltsuda, Gaussian processes, 199-3 119 M. V. Karasev and V. P. Maslov, Nonlinear Poisson brackets. Geometry and

quantization, 1993 118 Kenkichi lwasawa, Algebraic functions, 1993 117 Boris Zilber, Uncountably categorical theories, 1993 116 G. M. Fel'dman, Arithmetic of probability distributions, and characterization problems

on abelian groups, 1993 115 Nikolai V. Ivanov, Subgroups of Teichmiiller modular groups, 1992 114 Seize Ito, Diffusion equations, 1992 113 Michail Zhitomirskii, Typical singularities of differential I-forms and Pfaffian

equations, 1992 112 S. A. Lomov, Introduction to the general theory of singular perturbations, 1992 111 Simon Gindikin, Tube domains and the Cauchy problem, 1992 110 B. V. Shabat, Introduction to complex analysis Part II. Functions of several variables,

1992 109 Isao Miyadera, Nonlinear semigroups, 1992 108 Takeo Yokonuma, Tensor spaces and exterior algebra, 1992 107 B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selected problems

in real analysis, 1992 106 G.-C. Wen, Conformal mappings and boundary value problems, 1992 105 D. R. Yafaev, Mathematical scattering theory : General theory, 1992 104 R. L. Dobrushin, R. Kotecky, and S. Shlosman, Wulff construction: A global shape from

local interaction, 1992 103 A. K. Tsikh, Multidimensional residues and their applications, 1992 102 A. M. II'in, Matching of asymptotic expansions of solutions of boundary value

problems, 1992 101 Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Dong Zhen-xi, Qualitative

theory of differential equations, 1992 100 V. L. Popov, Groups, generators, syzygies, and orbits in invariant theory, 1992

99 Norio Shimakura, Partial differential operators of elliptic type, 1992 98 V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and

applications, 1992 97 ltiro Tamura, Topology of foliations: An introduction, 1992 96 A. I. Markushevich, Introduction to the classical theory of Abelian functions, 1 992 95 Guangchang Dong, Nonlinear partial differential equations of second order, 1991 94 Yu. S. II'yashenko, Finiteness theorems for limit cycles, 1991 93 A. T. Fomenko and A. A. Tuzhilin, Elements of the geometry and topology of minimal

surfaces in three-dimensional space, 1991 92 E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, 1991 91 Mamoru Mimura and Hirosi Toda, Topology of Lie groups, I and II, 1991 90 S. L. Sobolev, Some applications of functional analysis in mathematical physics, third

edition, 1991

(Continued in the back of this publication)

Elie Cartan

(1869-1951)

ELIE CARTAN April 9, 1 869-May 6, 1 95 1

Translations of

MATHEMATICAL MONOGRAPHS

Volume 123

Elie Cartan

(1869-1951) M.A. Akivis B. A. Rosenfeld

9JIH KAPTAH (1869-1951) M.A. AKHBHC

E. A. PoseaclieJihA Translated by V. V. Goldberg from an original Russian manuscript

Translation edited by Simeon Ivanov

1991 Mathematics Subject Classification. Primary 01A70; Secondary 01A60, 01A55.

ABSTRACT. The scientific biography of one of the greatest mathematicians of the 20th cen­tury, Elie Cartan (1869-1951), is presented, as well as the development of Cartan's ideas by mathematicians of the following generations.

Photo credits: p. iv-Centre National de la Recherche Scientifique; pp. 2, 3, 9, 10, 17, 19, 25, 27, 28, 29-Henri Cartan; p. 31-Department of Geometry, Kazan University, Tatarstan, Russia

Library of Congress Cataloging-in-Publication Data

Akivis, M. A. (Maks Aizikovich) [Elie Kartan (1869-1951). English] Elie Cartan (1869-1951)/M. A. Akivis, B. A. Rosenfeld; [translated from the Russian by

V. V. Goldberg; translation edited by Simeon Ivanov]. p. cm.-(Translations of mathematical monographs, ISSN 0065-9282; v. 123) Includes bibliographical references. ISBN 0-8218-4587-X (acid-free) 1. Cartan, Elie, 1869-1951. 2. Mathematicians-France-Biography. 3. Lie groups.

4. Geometry, Differential. I. Rozenfel1 d, B. A. (Boris Abramovich) II. Title. III. Series QA29.C355A6613 1993 93-6932 516.31761092-dc20 CIP

Copyright © 1993 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America

The paper used in this book is acid-free and falls within the �idelines established to ensure permanence and durability. �

Information on Copying and Reprinting can be found at the back of this volume. This publication was typeset using AMS-TEX,

the American Mathematical Society's TEX macro system. 10 9 8 7 6 5 4 3 2 I 97 96 95 94 93

Contents

Preface

Chapter 1 . The Life and Work of E. Cartan § 1 . 1 . Parents' home § 1 .2. Student at a school and a lycee § 1 . 3 . University student § 1 .4. Doctor of Science § 1 . 5 . Professor § 1 . 6 . Academician § 1 . 7 . The Cartan family § 1 . 8 . Cartan and the mathematicians of the world

Chapter 2. Lie Groups and Algebras §2 . 1 . Groups §2.2 . Lie groups and Lie algebras §2. 3 . Killing's paper §2.4. Cartan's thesis §2 .5 . Roots of the classical simple Lie groups §2 .6 . Isomorphisms of complex simple Lie groups §2 . 7 . Roots of exceptional complex simple Lie groups §2 .8 . The Cartan matrices §2 .9 . The Weyl groups

§2. 1 0. The Weyl affine groups §2. 1 1 . Associative and alternative algebras §2. 1 2 . Cartan's works on algebras §2. 1 3 . Linear representations of simple Lie groups §2. 1 4. Real simple Lie groups §2. 1 5 . Isomorphisms of real simple Lie groups §2 . 1 6 . Reductive and quasireductive Lie groups §2. 1 7 . Simple Chevalley groups §2. 1 8 . Quasigroups and loops

vii

XI

1 2 4 6 8

1 7 24 27

33 33 37 42 45 46 5 1 5 1 53 55 60 63 67 69 73 78 82 84 85

viii CONTENTS

Chapter 3. Projective Spaces and Projective Metrics 87 § 3 . 1 . Real spaces 8 7 §3 .2 . Complex spaces 93 §3 .3 . Quaternion spaces 95 § 3 .4. Octave planes 96 §3 .5. Degenerate geometries 97 §3 .6 . Equivalent geometries 1 0 1 § 3 . 7 . Multidimensional generalizations o f the Hesse transfer

principle 1 07 §3 .8 . Fundamental elements 1 09 §3 .9 . The duality and triality principles 1 1 3

§3 . 1 0. Spaces over algebras with zero divisors 1 1 6 § 3 . 1 1 . Spaces over tensor products of algebras 1 1 8 §3 . 1 2 . Degenerate geometries over algebras 1 2 1 §3 . 1 3 . Finite geometries 1 23

Chapter 4. Lie Pseudogroups and Pfaffi.an Equations 1 25 §4. 1 . Lie pseudogroups 1 25 §4.2. The Kac-Moody algebras 1 27 §4. 3 . Pfaffi.an equations 1 29 §4.4. Completely integrable Pfaffi.an systems 1 30 §4. 5 . Pfaffi.an systems in involution 1 32 §4.6 . The algebra of exterior forms 1 34 §4. 7 . Application of the theory of systems in involution 1 3 5 §4. 8 . Multiple integrals, integral invariants, and integral

geometry 1 36 §4.9 . Differential forms and the Betti numbers 1 39

§4. 1 0. New methods in the theory of partial differential equations 1 42

Chapter 5 . The Method of Moving Frames and Differential Geometry 1 45

§5. 1 . Moving trihedra of Frenet and Darboux 1 45 §5.2. Moving tetrahedra and pentaspheres of Demoulin 1 47 §5.3 . Cartan's moving frames 1 48 § 5 .4. The derivational formulas 1 50 §5.5 . The structure equations 1 52 §5.6 . Applications of the method of moving frames 1 53 § 5 .7 . Some geometric examples 1 54 §5.8 . Multidimensional manifolds in Euclidean space 1 58 §5 .9 . Minimal manifolds 1 60

§5 . 1 0. "Isotropic surfaces" 1 62 § 5 . 1 1 . Deformation and projective theory of multidimensional

manifolds 1 66

CONTENTS ix

§ 5 . 1 2 . Invariant normalization of manifolds 1 70 § 5 . 1 3 . "Pseudo-conformal geometry of hypersurfaces" 1 74

Chapter 6. Riemannian Manifolds. Symmetric Spaces 1 77 §6. 1 . Riemannian manifolds 1 77 §6.2 . Pseudo-Riemannian manifolds 1 8 1 §6 .3 . Parallel displacement of vectors 1 8 1 §6 .4. Riemannian geometry in an orthogonal frame 1 83 §6 .5 . The problem of embedding a Riemannian manifold into a

Euclidean space 1 84 §6 .6 . Riemannian manifolds satisfying "the axiom of plane" 1 8 5 §6 .7 . Symmetric Riemannian spaces 1 86 §6 .8 . Hermitian spaces as symmetric spaces 1 9 1 §6 .9 . Elements of symmetry 1 93

§6. 1 0. The isotropy groups and orbits 1 96 §6. 1 1 . Absolutes of symmetric spaces 1 98 §6 . 1 2. Geometry of the Cartan subgroups 1 99 §6. 1 3. The Cartan submanifolds of symmetric spaces 200 §6 . 1 4. Antipodal manifolds of symmetric spaces 201 §6 . 1 5. Orthogonal systems of functions on symmetric spaces 202 §6 . 1 6. Unitary representations of noncompact Lie groups 204 §6 . 1 7. The topology of symmetric spaces 207 §6 . 1 8 . Homological algebra 209

Chapter 7. Generalized Spaces 2 1 1 §7 . 1 . "Affine connections" and Weyl's "metric manifolds" 2 1 1 §7 .2. Spaces with affine connection 2 1 2 §7 .3 . Spaces with a Euclidean, isotropic, and metric connection 2 1 5 §7 .4. Affine connections in Lie groups and symmetric spaces

with an affine connection 2 1 6 §7 .5 . Spaces with a projective connection 2 1 9 §7 .6. Spaces with a conformal connection 220 §7 .7 . Spaces with a symplectic connection 22 1 §7 .8 . The relativity theory and the unified field theory 222 §7 .9 . Finsler spaces 223

§7 . 1 0. Metric spaces based on the notion of area 225 §7. 1 1 . Generalized spaces over algebras 226 §7 . 1 2. The equivalence problem and G-structures 228 §7 . 1 3 . Multidimensional webs 23 1

Conclusion 235

Dates of Cartan's Life and Activities 239

List of Publications of Elie Cartan 241

x CONTENTS

Appendix A. Rapport sur les Travaux de M. Cartan, by H. Poincare 263

Appendix B. Sur une degenerescence de la geometrie euclidienne, by E. Cartan 273

Appendix C. Allocution de M. Elie Cartan 275

Appendix D. The Influence of France in the Development of Mathematics 28 1

Bibliography 303

Preface

The year 1 989 marked the 1 20th birthday of Elie Cartan ( 1 869- 1 9 5 1 ) , one of the greatest mathematicians of the 20th century, and 1 9 9 1 marked the 40th anniversary of his death. The publication of this book is timed to these two dates. The book is written by two geometers working in two dif­ferent branches of geometry whose foundations were created by Cartan. The mathematical heritage of Cartan is very wide, and there is no possibility of describing all mathematical discoveries made by him, at least not in a book of relatively modest size. Because of this, the authors pose for themselves a much more modest problem-to describe and evaluate only the most impor­tant of these discoveries. Of course, the authors are only able to describe in detail Cartan's results connected with those branches of geometry in which the authors are experts.

The book consists of seven chapters. In Chapter 1 the outline of E. Car­tan's life is given, and in Chapters 2-7 his main achievements are described, namely, in the theory of Lie groups and algebras; in applications of these the­ories to geometry; in the theory of Lie pseudogroups; in the theory of Pfaffian differential equations and its application to geometry by means of Cartan's method of moving frames; in the geometry of Riemannian manifolds; and, in particular, in the theory of symmetric spaces created by Cartan; in the theory of spaces of affine connection and other generalized spaces. In the same chap­ters the main routes of the development of Cartan's ideas by mathematicians of the following generations are given. At the end of the book a chronology of the main events of E. Cartan's life and a list of his works are presented. The references to Cartan's works are given by numerals without Cartan's name, and the other references by first letters of the names of the authors, with nu­merals added for multiple references. The appendices contain H. Poincare's reference on Cartan's work ( 1 9 1 2 ) ; Cartan's paper On a degeneracy of Eu­clidean geometry, which was omitted in his muvre Completes; his speech at the meeting in the Sorbonne on the occasion of his 70th birthday ( 1 939 ) ; and his lecture, The influence of France in the development of Mathematics ( 1 940) . Chapters 1 -3 and 6 were written by B. A. Rosenfeld, Chapters 5 and 7 were written by M. A. Akivis, and Chapter 4 was written by both authors.

xi

xii PREFACE

The authors express their cordial gratitude to Henri Cartan, a son of E. Car­tan, who himself is one of the greatest mathematicians of this century, for providing numerous facts for a biography of his father and for pictures fur­nished by him.

Moscow, Russia M. A. Akivis

University Park, PA, U.S.A. B. A. Rosenfeld

CHAPTER 1

The Life and Work of E. Cartan

§1.1. Parents' home

Elie Joseph Cartan was born on April 9, 1 869, in the village of Dolomieu located between Lyons and Grenoble in the Departement !sere in the south­eastern part of France. The !sere river, after which the Departement was named, has a very fast current, and several hydroelectric power plants are now located along it. They supply the industrial district of Grenoble, the center of the Departement !sere, with electric power. The first hydroelectric power plant on this river was built by Aristide Berges ( 1 833- 1 904) , the owner of a paper mill in Lancey, in 1 869, the year of Cartan's birth.

The Departement !sere is in the central part of the historic French province Dauphine, which was a patrimonial estate of a dauphin, the eldest son of the king (the crown prince) . Dauphine stretched from the Alps to the Rhone, the left tributary of which is the !sere. Originally the capital of Dauphine was the town of Vienne, which is located on the RhOne just south of Lyon. Later on, the capital was transferred to Grenoble.

During Cartan's childhood, Dolomieu had about 2 ,500 inhabitants. Pre­sently the population is about 1 ,600. Long ago the village was a center of silkworm breeding and silk spinning. Figure 1 . 1 (next page) shows Square Champ-de-Mars (Martial Field) in Dolomieu (presently Place Elie Cartan) and the house where Cartan spent his childhood (he lived there from 1 872 to 1 879) . Cartan's family home is the second from the right. Dolomieu was also the home of the famous geologist Deodat (Dieudonne) Guy Silvain Gratet de Dolomieu ( 1 750- 1 80 1 ) , one of the sons of Fran�ois de Gratet, Marquis de Dolomieu. Deodat Dolomieu was an academician and a participant in the famous Egyptian campaign of Napoleon. He immortalized his own name and the name of his home village through his discovery of the mineral dolomite.

Cartan's ancestors were peasants. His great-grandfather Benoit Cartan ( 1 779- 1 854) was a farmer. Cartan's grandfather, whose first name was also Benoit ( 1 80 1 - 1 854) , was a miller. Cartan's father Joseph ( 1 837- 1 9 1 7) was born in the village of Saint Victor de Morestel, which is 1 3 kilometers from Dolomieu. After he married Anne Cottaz ( 1 84 1 - 1 927) the family settled in Dolomieu, where Anne had lived. Joseph Cartan was the village blacksmith. Cartan recalled that his childhood had passed under "blows of the anvil,

2 I. THE LIFE AND WORK OF E. CARTAN

DOLOM!EU -- Le Champ de Mars

FIGURE 1 . 1

which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel" [ 1 89, p. 5 1 ) .

Figure 1 .2 shows a picture of Cartan's parents, taken approximately in 1 890. Cartan recollected later that his parents were "unpretentious peasants who during their long lives demonstrated to their children an example of joyful accomplished work and courageous acceptance of burdens" [ 1 89, p. 5 1 ) .

Elie was the second oldest of the four Cartan children. His elder sister Jeanne-Marie ( 1 867- 1 93 1 ) was a dressmaker, and his younger brother Leon ( 1 872- 1 956) became a blacksmith, working in his father's smithy. Cartan's younger sister Anna ( 1 878- 1 923) , not without the influence of her brother, graduated from L'Ecole Normale Superieure (the Superior Normal School) for girls and taught mathematics at different lycees (state secondary schools) for girls. She was the author of two textbooks for these lycees: Arithmetic and Geometry, for first-year students, and Geometry, for second-year students. Both textbooks were reprinted many times.

§ 1 .2. Student at a school and a lycee

Elie Cartan began his education in an elementary school in Dolomieu. He later spoke very warmly of his teachers, M. Collomb and especially M. Dupuis, who gave one hundred boys a primary education, the importance of which Cartan could appraise at its true worth only much later. Elie was the

§ 1 .2. STUDENT AT A SCHOOL AND A LYCEE 3

FIGURE 1 .2

best student in the school. M. Dupuis recollected: "Elie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, and this was combined with an excellent memory. There was no question that could be a problem for him: he understood everything that was taught in class even before the teacher finished his explanation." Cartan remembered that in the school he "could, without a moment's· hesitation, list all subprefectures in each Departement" of France as well as the grammatical fine points "of the rules of past participles" [ 1 89, p. 52) .

Elie Cartan was of small stature and did not possess the physical strength of his father and brother. That he became one of the most famous scientists of France was due to the fact that the school where he was studying was visited by Antonin Dubost ( 1 844- 1 92 1 ) . Dubost was a remarkable person­ality in many respects. He was a republican journalist during the empire of Napoleon Ill. After France became the Third Republic, he became a prefect of the Departement Orne, which is to the west of Paris. Later he moved to the Departement Isere and was its representative in 1 880- 1 897. During this period he was the Minister of Justice in the cabinet of Grenobler Casimir Perier. In 1 897 Dubost was elected to the French Senate and was the Pres­ident of the Senate from 1 906 to 1 920. Cartan described him as having "a

4 I. THE LIFE AND WORK OF E. CARTAN

strong optimism, based on a strong belief in progress, in the power of in­tellect and in the hope of discovering truth and doing good". Later Cartan noted: "His visit changed my whole life" [ 1 89, p. 52) . Impressed by the unusual abilities of Cartan, Dubost recommended that he participate in a contest for a scholarship in a lycee. Cartan prepared for this contest under the supervision of M. Dupuis.

At that time in France there were two kinds of secondary schools: colleges, belonging to local self-governments, and lycees, belonging to the Ministry of Public Education. (After restoration of royal power in France in 1 6 1 5 the lycees were renamed "royal colleges"; the name "lycee" was returned to them only after the 1 848 Revolution. The 1 959 reform renamed the colleges "municipal lycees". ) Young Cartan passed the contest exams in Grenoble, the main city of the Departement Isere. He remembered that he "passed these competitions, which turned out to be not so difficult, without particular nervousness" [ 1 89, p. 52] . The brilliant success of Elie in this contest was a source of special pride of M. Dupuis, who supervised his preparation for the contest. Thanks to the continuing support of M. Dubost, who retained a fatherly interest in Cartan's scientific career and achievements throughout his life, Cartan received a full scholarship in the College of Vienne (Vienne is the ancient capital of the province Dauphine) . Elie was 1 0 years old at that time.

Cartan spent the next ten years in colleges and lycees far from home. His first five years ( 1 880- 1 885) were at the College of Vienne. After this his scholarship was transferred to the Lycee of Grenoble, where he was a student from 1 885- 1 887. The teaching in colleges and lycees at that time to a consid­erable extent consisted of a medieval curricula of "trivial" and "quadrivial" sciences. The first group, the so-called trivium (three-path) , were formed by Grammar, Rhetoric, and Philosophy, and the second group, the quadrivium (four-path) , was formed by Mathematical Sciences. Originally they had been comprised of Arithmetic, Geometry, Astronomy and Music. Cartan com­pleted the study of the trivium in the Grenoble college (after passing Rhetoric and Philosophy) . To study mathematical sciences, in 1 887 he moved to Paris, to the Janson-de-Sailly Lycee ("Grand Lycee") , where he was a student un­til 1 888 . With special warmth Cartan remembered two professors from this lycee: Salomon Bloch, who taught "elementary mathematics'', and E. Lacour, who taught "special mathematics". One of his classmates in this lycee was Jean-Baptiste Perrin ( 1 870- 1 942) , who later became one of the most famous physicists in France. A close friendship between Cartan and Perrin, which began during these years, continued throughout their lives.

§1 .3. University student

After graduation from the Lycee Janson-de-Sailly, Cartan decided to be­come a mathematician. At that time in Paris there were three educational institutions with mathematical majors: the Sorbonne (University of Paris) ,

§ 1 .3 . UNIVERSITY STUDENT 5

which was founded by Robert de Sorbon in 1 253 ; l'Ecole Polytechnique (the Polytechnic School) ; and l'Ecole Normale Superieure (the Superior Normal School) . The latter two had been founded during the French Revolution. The Polytechnic School, where one would study for three years (later changed to two years) , gave a mathematical and general technical education, after which one was supposed to study a specialization in practical higher technical insti­tutions. L'Ecole Normale Superieure, where, according to the Convent's de­cision, "the art of teaching, not science itself' should be taught, was a higher pedagogical educational institution in which one would study for three years.

Cartan chose l'Ecole Normale Superieure and enrolled in 1 888 . Of the professors whose lectures he attended in this school and the Sorbonne, Cartan thought most highly of "a mathematical giant, Henri Poincare, whose lectures were flying over our heads" [ 1 89, p. 54] . Poincare ( 1 854- 1 9 1 2) , about whom Cartan wrote that "there was no branch of mathematics which was not under his influence" [ 1 89, p. 54], was a mathematician, physicist, astronomer, and philosopher who created in 1 883 the theory of automorphic functions, which is closely connected with group theory and hyperbolic geometry. He attracted Cartan's attention to geometric applications of group theory.

Listing professors who influenced him, Cartan indicated Charles Hermite ( 1 822- 1 90 1 ), a specialist in analysis, algebra, and number theory, who in­troduced "Hermitian forms" for problems in number theory-forms which play an important role in geometry; Jules Tannery ( 1 848- 1 9 1 0) , one of the founders of French set theory; Gaston Darboux ( 1 842- 1 9 1 7) , one of the founders of the method of moving frames, who is also known by his work in the theory of differential equations; Paul Appell ( 1 8 55- 1 930) , a special­ist in analysis and mechanics; Emile Picard ( 1 8 56- 1 94 1 ) , a specialist in the theory of differential equations who widely used geometric and group theory methods in his work; and Edouard Goursat ( 1 858- 1 936) , a specialist in the theory of differential equations, who also was interested in transformation groups. (In 1 889 Goursat wrote a paper on finite groups of motions of a four-dimensional Euclidean space that are generated by reflections . )

L'Ecole Normale Superieure at that time was closely connected with the Norwegian mathematician Sophus Lie ( 1 842- 1 899) , who from 1 886 to 1 889 was head of the Department of Geometry in Leipzig University. In 1 888-1 889, upon the recommendation of Tannery and Darboux, several French mathematicians, including Ernest Vessiot ( 1 865- 1 952) and Arthur Tresse ( 1 868- 1 958) , studied under Lie in Leipzig. Picard was also very much inter­ested in Lie's papers. After Vessiot returned to Paris, he and Picard published papers on applications of continuous groups to the problem of integrability of differential equations. These papers were a further development of Lie's re­search. In the investigations of Lie, Picard, and Vessiot, the so-called solvable or integrable Lie groups played a special role. This gave rise to the problem of listing all so-called simple Lie groups, since the presence of simple sub­groups in a group indicates that it is nonsolvable. Cartan's interest in these

6 I. THE LIFE AND WORK OF E. CARTAN

problems, to a considerable extent, can be explained by the influence of his classmate Tresse.

After graduation from l'Ecole Normale Superieure in 1 89 1 , Cartan was drafted into the French army, where he served one year and attained the rank of sergeant.

§1.4. Doctor of Science

While Elie Cartan served in the army, his friend Arthur Tresse was a student of Sophus Lie at Leipzig University. When Tresse returned from Leipzig, he informed Cartan that W. K.illing's paper, The structure of the fi­nite continuous groups of transformations [K.il2], had been published in the Leipzig journal Mathematische Annalen, in 1 888- 1 890. In this paper impor­tant results on the classification of simple Lie groups were obtained.

Tresse also told Cartan that, after publication, this paper was found to contain incorrect statements concerning nilpotent groups ("groups of zero rank") , and that the mathematician F. Engel from Leipzig, who was working jointly with Klein and Lie, assigned the task of correcting K.illing's inaccu­racies to his student Carl Arthur Umlauf ( 1 866-?) . Umlauf accomplished the mathematical objective assigned to him and defended his doctoral dis­sertation, On the structure of the finite continuous groups of transformations, especially groups of zero rank [Um] ( 1 89 1 ) . Tresse advised Cartan to inves­tigate whether the main part of K.illing's paper also contained inaccuracies. From this came the subject of Cartan's thesis.

Cartan worked on this subject for two years ( 1 892- 1 894) in Paris. As an excellent student of l'Ecole Normale Superieure, he was a recipient of the grant ("bourse") of the Peccot Foundation, founded in 1 885 to support talented young scientists of l'Ecole Normale Superieure. (The Peccot Foun­dation is still in existence. )

Following Tresse's advice, Cartan studied the Killing paper and became convinced that the principal parts of this work were correct and that the new method, which was used by Killing and which was based on the study of "roots" of simple Lie groups, is an exceptionally powerful method for studying this kind of group. Simultaneously, Cartan discovered a number of inaccuracies and incomplete statements. A rigorous classification of simple Lie groups constituted the main part of Cartan's doctoral dissertation.

In 1 892, at the invitation of Darboux and Tannery, Lie came to Paris and spent six months there. However, the main purpose of Lie's visit to Paris was to meet Cartan. (This information was given by his son Henri in a letter to one of the authors of this book.) Lie and Cartan had discussions on several occasions. Cartan recollected that Lie was interested "with a great good will in the research of young French mathematicians" [20 l , Engl. tr. , p. 265] and that at that time Lie "could often be seen with them around the table at the Cafe de la Source, on the Boulevard Saint-Michel; it was not unusual for the white marble table top to be covered with formulas in pencil, which the

§ 1 .4. DOCTOR OF SCIENCE 7

master had written to illustrate the exposition of his ideas" [20 1 , Engl. tr. , p. 265] . In the same article Cartan gave his impression of Lie's personality: "Sophus Lie was of tall stature and had t_he classic Nordic appearance. A full blond beard framed his face and his gray-blue eyes sparkled behind the eyeglasses. He gave the impression of unusual physical strength. One always immediately felt at ease with him, certain beforehand of his sincerity and his loyalty." He also evaluated Lie's influence on mathematics: "Posterity will see in him only the genius who created the theory of transformation groups, and we French shall never be able to forget the ties, which bind us to him and which make his memory dear to us. " [20 1 , Engl. tr. , p. 267] .

In 1 893 Cartan published his first scientific papers-two notes, The struc­ture of simple finite continuous groups [ 1 ] and The structure of finite continu­ous groups [2]-in Comptes Rendus des Seances de l'Academie des Sciences (Paris) . They were presented for publication by Picard. In these notes Car­tan's results on simple Lie groups were presented briefly. The details were given in the paper The structure of finite groups of transformations [3 ] , pub­lished in German in Mitteilungen (Communications) of University of Leipzig and recommended for publication by Lie. These results comprised Cartan's doctoral dissertation, The structure of finite continuous groups of transforma­tions, which he defended in 1 894 in the Faculty of Sciences in the Sorbonne, and which was published as a book [5 ] .

From 1 894 to 1 896 Cartan published a few more papers on the theory of simple Lie groups: the notes On reduction of the group structure to its canonical form [4] ( 1 894) and On certain algebraic groups [8] ( 1 895 ) , and the paper On reduction of the structure of a finite and continuous group of transformations to its canonical form [9] ( 1 896) . In 1 894 two papers by Cartan [6] , [7] were published in which he gave a new proof of Bertrand's theorem concerning permutation theory. Cartan's proof was based on the properties of complete permutation groups. In 1 896 Cartan's first paper on integral invariants, The principle of duality and certain multiple integrals in tangential and line spaces [ 1 Q] , was published.

Also between 1 894 and 1 896 Cartan was a lecturer at the University of Montpellier, one of the oldest scientific centers in France. Then, during the years 1 896 through 1 903, he was a lecturer in the Faculty of Sciences at the University of Lyons. At this time he continued his intense scientific work: in 1 897 his two notes, On systems of complex numbers [ 1 1 ] and On real systems of complex numbers [ 1 2] , and, in 1 898, his paper, Bilinear groups and systems of complex numbers [ 1 3] , were published. Following French tradition, by systems of complex numbers Cartan meant associative algebras, also called systems of hypercomplex numbers. In these articles, which are also connected to the direction of the Lie school, many notions arising in the theory of Lie groups were generalized for associative algebras. In particular, Cartan gave a classification of both the complex and real simple associative algebras.

8 I. THE LIFE AND WORK OF E. CARTAN

Cartan's ·reflections on differential forms, which he dealt with in his pa­pers on Lie groups and in his paper on integral invariants, brought him to the so-called Pfaff problem-the theory of integration of the Pfaffi.an equa­tions, which are equivalent to a system of partial differential equations. In 1 899 he published his first paper, On certain differential expressions and the Pfaff problem [ 1 4] , on this topic, which was followed by the papers On some quadratures, whose differential element contains arbitrary functions [ 1 5 ] , On the integration of the system of exact equations [ 1 6] , and On the integration of certain Pfaffian systems of character two [ 1 7] ( 1 90 1 ) ; two notes, On the integration of completely integrable differential systems [ 1 8] , [ 1 8a], and the note On the equivalence of differential systems [ 1 9] ( 1 902) .

In 1 903, while in Lyons, Cartan married Marie-Louise Bianconi ( 1 880-1 950) , whose father Pierre-Louis Bianconi ( 1 845- 1 929) , a Corsican by birth, had been Professor of Chemistry in Chambery and was, at that time, "in­specteur d' Academie" in Lyons.

§ 1 .5. Professor

In 1 903 Cartan became a professor in the Faculty of Sciences at the Univer­sity of Nancy. Nancy is the capital of the Departement Meurthe-et-Moselle in the part of Lorraine that was not ceded to Germany after the 1 870- 1 87 1 war. He worked in Nancy until 1 909. In Nancy, Cartan also taught at the Institute of Electrical Engineering and Applied Mechanics. While in Nancy, Cartan's sons Henri ( 1 904) and Jean ( 1 906) were born. Figure 1 . 3 is a 1 904 portrait of Cartan.

After publishing the note On the structure of infinite groups [20] in 1 902, Cartan published two long papers, On the structure of infinite groups of trans­formations [2 1 ], [22], in 1 904- 1 905 . They were followed by the note Sim­ple continuous infinite groups of transformations [23] ( 1 907) and the pa­per The subgroups of continuous groups of transformations [26] ( 1 908) . In these articles Cartan studied the structure of infinite-dimensional analogues of Lie groups. While for Lie groups Cartan used the name "finite continuous groups", for their infinite-dimensional analogues he used the name "infinite continuous groups". Now they are called "Lie pseudogroups". While classi­cal Lie group theory was connected with the theory of systems of ordinary differential equations, the theory of Lie pseudogroups turned out to be con­nected with the theory of systems of partial differential equations and with the theory of systems of Pfaffi.an equations, which are equivalent to the lat­ter. In these articles the foundations of the method of moving frames and of Cartan's method of exterior forms were laid. Later these methods played a very important role in the development of differential geometry. In 1 908, in the French edition of Encyclopaedia of Mathematica/ Sciences, Cartan published the article Complex numbers. This article was Cartan's extended French translation of the paper The theory of usual and higher complex num­bers by Eduard Study ( 1 862- 1 930) [Stu 1 ] , from the German edition of this

§ 1 . 5 . PROFESSOR 9

FIGURE 1 .3

Encyclopaedia. Cartan's translation was four times as long as the original Study paper.

In 1 907- 1 908 Cartan also published two geometric notes under the same title, On the definition of the area of a part of a curved surface [24], [25] . In 1 909 Cartan moved his family to Paris. In Paris he worked as a lecturer in the Faculty of Sciences in the Sorbonne and in 1 9 1 2 became Professor, based on the reference he received from Poincare [Poi6] . Appendix A contains the English translation of this reference, which was not included in Poincare's <Euvres [Poi] . In 1 909 Cartan built a house in his home village Dolomieu (Figure 1 .4, next page) , where he regularly spent his vacations. In Dolomieu Cartan continued his scientific research but sometimes went to the family smithy and helped his father and brother to blow blacksmith bellows (Figure 1 .5 shows a 1 932 picture of Cartan working in his garden) .

I n 1 9 1 0, i n the note On isotropic developable surfaces and the method of moving frames [29] and in the paper The structure of continuous groups of transformations and the method of a moving trihedron [3 1 ], Cartan for the first time connected the theory of Lie groups and the theory of Pfaffian equations with the method of moving frames. This method later became the basic method in the geometric work of Cartan.

10 I. THE LIFE AND WORK OF E. CARTAN

FIGURE 1 .4

FIGURE 1 . 5

§ 1 .5 . PROFESSOR II

In the same year, Cartan's paper The Pfaffian systems with five variables and partial differential equations of second order [ 30] was published. In 1 9 1 1 his papers Variational calculus and certain families of curves [32] and On systems of partial differential equations of second order with one unknown function and three independent variables in involution [33 ] appeared, and in 1 9 1 2 he published two notes, On the characteristics of certain systems of par­tial differential equations [34] and On groups of contact transformations and new kinematics [35 ] .

In 1 9 1 3 Cartan returned to the theory of simple Lie groups and published an important paper titled Projective groups, under which no plane manifold is invariant [37] . In the same year, his Notes on the addition of forces [36] appeared.

In 1 9 1 4, in the paper Real simple finite continuous groups [38] , Cartan solved the problem of classification of real simple Lie groups. This problem is similar to that which he solved in his thesis for complex Lie groups. In the paper Real continuous projective groups, under which no plane manifold is invariant [39] ( 1 9 1 4) , he constructed the theory of linear representations of these groups. Also in 1 9 1 4 his notes On the integration of certain systems of differential equations [40] and On certain natural families of curves [4 1 ] , the paper On the absolute equivalence of certain systems of differential equations and on certain families of curves [42] , and the popular paper Theory of groups [43] appeared.

In 1 9 1 5 , when Cartan was 46 years old, he was drafted into the Army and served at the rank of sergeant (the rank that he attained in 1 892) in a hospital set up in the building of l'Ecole Normale Superieure. While he served in this position, until the end of World War I, Cartan continued his mathematical studies.

In 1 9 1 5 Cartan's papers, On the integration of certain indefinite systems of differential equations [44] and On Backlund transformations [45] , appeared. In the same year, Cartan wrote an extended French translation, Theory of continuous groups and geometry [46], of Gino Fano's ( 1 87 1 - 1 952) paper Continuous geometric groups. Group theory as a geometric principle of classi­fication [Fan] for the French edition of Encyclopaedia of Mathematica/ Sci­ences (from the German edition of this Encyclopaedia) . However, after the beginning of World War I, the French edition of this Encyclopaedia, which was in the process of publication in Leipzig and Paris, was discontinued, and in 1 9 1 4 only 2 1 pages of the paper [ 46] were published. The complete text of this paper, taken from proofs of 1 9 1 5 , was published only after Cartan's death in his (Euvres Completes [207] .

From 1 9 1 6 to 1 9 1 8 Cartan studied the theory of deformation of hypersur­faces. At that time he published the papers The deformation of hypersurfaces in the real Euclidean space of n dimensions [47] ( 1 9 1 6) , The deformation of hypersurfaces in the real conformal space of n > 5 dimensions [48] ( 1 9 1 7) , and On certain hypersurfaces in the real conformal space of five dimensions

12 I. THE LIFE AND WORK OF E. CARTAN

[49) ( 1 9 1 8) . In 1 9 1 8 Cartan published four notes on three-dimensional man­ifolds of n-dimensional Euclidean space: in the note [ 50) the general theory of three-dimensional surfaces of this space is constructed; in the note [50a] Cartan gives the theory of developable surfaces, i .e . , surfaces with vanishing curvature; in [50b] he develops the theory of surfaces of constant negative curvature, which he calls "Beltrami manifolds"; and in [50c] Cartan presents the theory of surfaces of constant curvature, which he calls there "Riemann manifolds". In 1 9 1 9- 1 920 Cartan published the paper On the manifolds of constant curvature of a Euclidean and non-Euclidean space [ 5 1 ) , [52) , where the results of notes [50]-[50c] are generalized to p-dimensional surfaces of both Euclidean and non-Euclidean spaces. Also appearing in 1 920 were the notes, On the projective deformation of surfaces [53 ) and On the projective applicability of surfaces [ 5 3a] , the paper On the projective deformation of sur­faces [54) , and Cartan's lecture On the general problem of deformation ( 55 ) given at the International Congress of Mathematicians in Strasbourg.

In 1 922 Cartan's articles on Einstein's gravitation theory appeared: the paper On the equations of gravitation of Einstein [56] and the note On a geometric definition of Einstein 's energy tensor (57) . The research in general relativity theory and the attempts to create a unitary* field theory brought Cartan to his theory of " generalized spaces"; in the same year he published the notes On a generalization of the notion of Riemannian curvature [58 ) , On generalized spaces and relativity theory [59) , On generalized conformal spaces and the optical universe (60), On the structure equations of generalized spaces and the analytic expression of the Einstein tensor [6 1 ) , and On a fundamental theorem of Wey! in the theory of metric spaces (62) . In 1 922 Cartan published the paper On small oscillations of a fluid mass [63] and the book Lectures on integral invariants [64), in which he summarized his research on integral invariants and gave applications of this theory to mechanics.

The results of the note [62) were given in detail in the paper On a fun­damental theorem of Wey! [65) ( 1 923 ) . Departing from the ideas of Weyl, who constructed a generalization of Riemannian geometry to create one of the first "unitary field theories", Cartan came up with the notions of spaces with Euclidean connection, metric connection, affine connection, and, later, with conformal and projective connection. The space with Euclidean con­nection differs from the Riemannian manifold in the way that the space with metric connection differs from the Weyl space-namely, by the presence of torsion. The geometry of spaces with affine, Euclidean, or metric connection was presented in the paper On manifolds with an affine connection and gen­eral relativity theory, which appeared in three parts: [66] ( 1 923) , [69] ( 1 924) , and [80] ( 1 925) . In the same paper Cartan considered a space with degen­erate Euclidean geometry, which at present is called "the isotropic space"; the generalized isotropic space is called today "the space with an isotropic

*Editor's note. Or unified.

§ l.S. PROFESSOR 1 3

connection". The geometry of spaces with conformal connection was pre­sented in the paper [68] ( 1 923 ) , and the geometry of spaces with projective connection was presented in the paper [70.] ( 1 924) .

In 1 923 Cartan also published the paper Non-analytic functions and singu­lar solutions of first order differential equations [67] . In 1 924 he published the paper Recent generalizations of the notion of a space [7 1 ]; delivered the talks Relativity theory and generalized spaces [72] at the International Philosoph­ical Congress in Naples and Group theory and recent research in differential geometry [73] at the International Congress of Mathematicians in Toronto; and published the notes On differential forms in geometry [74], On the affine connection on surfaces [76] , and On the projective connection on surfaces [77] .

From 1 924 to 1 940 Cartan held the University chair in the Department of Higher Geometry at the Faculty of Sciences at the Sorbonne. From 1 9 1 7 to 1 936 Cartan lived with his family in the village Le Chesnay near Versailles, and in 1 936 he rented an apartment in a multistory house at 95 Boulevard Jourdan in the southern part of Paris, near the square Porte d'Orleans. Cartan lived in this apartment until his death. At present his son Henri and his wife Nicole Weiss live in this apartment.

In Figure 1 .6 (next page) a letter by Cartan is reproduced; it is his letter to a young mathematician, Olga Taussky (b. 1 906) , written in Le Chesnay on the stationary of the Department of Higher Geometry at the Faculty of Sciences of the Sorbonne. In 1 936 Taussky, who worked at that time in Austria and who later moved to England and eventually to the U.S.A., wrote a letter to Cartan in which she explained her results on the theory of division algebras. In his response Cartan wrote: "Mademoiselle, thank you for your letter. Your proof, which relates to the systems of hypercomplex numbers without zero divisors, is very simple and elegant. It would be a great pleasure for me to meet you personally in Oslo in July."

In 1 925 , Cartan published his book Geometry of Riemannian manifolds [84] and gave the talk Holonomy groups of generalized spaces and topology [77] at a session of the Association for the· Development of Science in Greno­ble. He also published the following papers: Irreducible tensors and simple and semisimple linear groups [8 1 ] , on the theory of linear representations of simple and semisimple Lie groups; The duality principle and the theory of simple and semisimple groups [82] , where he considered analogues of the duality principle in projective geometry based on the bilateral symmetry of the systems of simple roots of some simple Lie groups, and also the "triality principle'', based on the trilateral symmetry of one of those systems; On the motions depending on two parameters [83 ] ; and also Note on generation of forced oscillations [78] , written jointly with his son Henri.

In 1 926 Cartan delivered the talk Applications of Riemannian manifolds and Topology [85] at a session of the Association for the Development of Science in Lyons, and published the notes On certain differential systems, in which the unknowns are Pfaffian forms [86] , and On Riemannian manifolds,

1 4 I. THE LIFE AND WORK OF E. CARTAN

PACULTE DES SCIENCES lfJJ 6

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FIGURE 1 . 6

in which parallel translation preserves the curvature [87] , where for the first time Cartan considered an important class of Riemannian manifolds that he later named "symmetric Riemannian spaces". He also published the papers Holonomy groups of generalized spaces [88] , On spheres of three-dimensional Riemannian manifolds [89] , and The axiom of plane and metric differential geometry [90] (the latter in the collection of articles "In Memoriam N. I . Lobachevsky" in Kazan) , and two notes written jointly with J. A. Schouten: On the geometry of the group-manifold of simple and semi-simple groups [9 1 ], and On Riemannian geometries admitting an absolute parallelism [92] ( in English and Dutch, published in Proceedings of the Amsterdam Academy of Science) . In 1 926 and 1 927 his papers On a remarkable class of Riemannian manifolds [93] , [94] appeared. In this two-paper series Cartan gave a detailed treatment of the geometry of symmetric Riemannian spaces.

In 1 927, in the paper The geometry of transformation groups [ 1 0 1 ] , Cartan constructed the theory of symmetric spaces with affine connection. In the

§ 1 .5. PROFESSOR 1 5

same year, h e published the notes On geodesic lines of spaces of simple groups [96] , On the topology of real simple continuous groups [97] , On the geodesic deviation and some related problems [98] , On certain remarkable Riemannian forms of geometries with a simple fundamental group [99] , and On Rieman­nian forms of geometries with a simple fundamental group [ 1 00] ; the papers The geometry of simple groups [ 1 03] , and Group theory and geometry [ 1 05] ; and the important paper On certain remarkable Riemannian forms of geome­tries with a simple fundamental group [ 1 07] , with the same title as the note [99] . These works were devoted to various aspects of the geometry of sym­metric Riemannian spaces. In the same year, the note On curves with zero torsion and developable surfaces in Riemannian manifolds [95] and the pa­pers On certain arithmetic cycles [ 1 02] and On the possibility of imbedding a Riemannian manifold into an Euclidean space [ 1 04] appeared. In the Bul­letin of Kazan Physics-Mathematics Society Cartan published his report on the Schouten memoir The Erlangen program and the theory of parallel trans­lation. New point of view on foundations of geometry [ 1 06], which was de­voted to the geometry of "generalized spaces". In Mathematichesky Sbornik, Moscow, he published the paper On a problem of the calculus of variations in plane projective geometry [ 1 08] . From 1 926 to 1 927 at the Sorbonne, Cartan delivered a series of lectures, the notes of which were published in 1 960 in Russian translation (translated by S. P. Finikov) under the title Riemannian geometry in an orthogonal frame [ 1 08a] .

In 1 928 Cartan's Lectures on the geometry of Riemannian manifolds [ 1 1 4] appeared, in addition to his notes On complete orthogonal systems of func­tions in certain closed Riemannian manifolds [ 1 09] (Cartan's term "closed" means "compact") ; On closed Riemannian manifolds admitting a transitive continuous transformation group [ 1 1 O], On the Betti numbers of spaces of closed groups [ 1 1 1 ] (where the algebraic topology of compact Lie groups was reduced to the algebraic theory of Lie algebras) ; and the complement [ 1 1 3] to the memoir The geometry of simple groups [ 1 03] . This complement was de­voted to finite groups of the Euclidean space generated by reflections. In the same year, Cartan's lecture at the International Congress of Mathematicians in Toronto, entitled On the ordinary stability of Jacobi ellipsoids [ 1 1 2] , was published. This lecture was devoted to the development of the well-known Poincare research on the stable forms of a rotating fluid mass. In 1 928 he also delivered the talk On imaginary orthogonal substitutions [ 1 1 5] at a ses­sion of the Association for the Development of Science in La Rochelle. In the same year, Cartan gave the talks On a geometric representation of non­holonomic material systems [ 1 1 9] and On closed spaces· admitting a transitive finite continuous group [ 1 20] (i .e . , on compact spaces admitting a transitive Lie transformation group) at the International Congress of Mathematicians in Bologna.

In 1 929 Cartan published the papers On the determination of a complete orthogonal system of functions on a closed symmetric Riemannian space [ 1 1 7]

1 6 I . THE LIFE AND WORK OF E. CARTAN

and On the integral invariants of certain closed homogeneous spaces and topo­logical properties of these spaces [ 1 1 8] . He also published the paper Closed and open simple groups and Riemannian geometry [ 1 1 6] , in which he presented a classification of noncompact simple Lie groups using his theory of symmetric Riemannian spaces. (This method is much simpler than the method he used in [38] . )

In 1 930 the book The theory of.finite continuous groups and Analysis situs [ 1 28] (analysis situs is the old name for topology) appeared, in addition to his notes Linear representations of the group of rotations of the sphere [ 1 2 1 ] ; The linear representations of closed simple and semisimple groups [ 1 22]; two notes entitled The third fundamental Lie theorem [ 1 23] , [ 1 23a]; the note A historic note on the notion of absolute parallelism [ 1 24] (devoted to the application of this notion to general relativity theory) ; and the papers On linear representations of closed groups [ 1 25] and On an equivalence problem and the theory of generalized metric spaces [ 1 26] . In the same year, Cartan took part in the First Congress of Mathematicians of the U .S.S.R. in Kharkov and gave the talk Projective geometry and Riemannian geometry [ 1 2 7] . On his way from Kharkov, Cartan made a stop in Moscow and delivered the course of lectures The method of moving frames, the theory of.finite continuous groups and generalized spaces [ 1 44] at the Institute of Mathematics and Mechanics in Moscow University. This course was published in Russian in 1 933 and 1 962 (translated by S. P. Finikov) and in French in 1 935 .

In 1 9 3 1 the book Lectures on complex projective geometry [ 1 34] appeared. In this work a detailed investigation is given of symmetric spaces, whose fundamental groups are the group of projective transformations of three­dimensional complex projective space or its subgroups. In the same year, Cartan published the paper Absolute parallelism and unitary field theory [ 1 30] , devoted to results he had obtained in 1 920 and rediscovered by Einstein in 1 928. Also in 1 93 1 , the expository paper Euclidean geometry and Rieman­nian geometry [ 1 29] and the papers On the theory of systems in involution and its application to relativity theory [ 1 3 1 ] and On the evolvents of a ruled surface ( 1 32] appeared, and Cartan gave the talk The fundamental group of the geometry of oriented spheres [ 1 33 ] .

In 1 9 3 1 Cartan also published a survey of his mathematical works [ 1 87] , which was later republished with supplements in a collection of his articles [204] and in the complete collections of his papers [207] , [209].

During the twenty years after Cartan defended his doctoral dissertation, his ideas were not developed further by other mathematicians. The situation changed in the beginning of the 1 920s when Hermann Weyl ( 1 885- 1 955 ) became interested in Cartan's works. In 1 924- 1 925 Weyl obtained impor­tant results in the theory of simple Lie groups. These results were developed further in 1 933 by Bart el Leendert van der Waerden (b. 1 903) . On the other hand, Cartan's papers on the geometry of "generalized spaces" were closely connected with the papers of Weyl and Jan Arnoldus Schouten ( 1 883- 1 97 1 )

§ 1 .6. ACADEMICIAN 1 7

FIGURE 1 . 7

on the geometry of spaces with affine connection, which appeared respec­tively in 1 9 1 8 and 1 92 1 . That Cartan was isolated during the two decades after receiving his doctoral degree is due to his extreme modesty and to the fact that in this period the center of attention of French mathematicians was in set theory and function theory. In the 1 930s the mathematical commu­nity in different countries recognized the scientific importance of the direc­tions of Cartan's research. Cartan was elected a Foreign Member of several Academies of Sciences: Polish Academy in Cracow ( 1 92 1 ) , Norway Academy in Oslo ( 1 926) , and the famous National Academy dei Lincei ("of lynxes") in Rome ( 1 927) . Finally in 1 9 3 1 Cartan was elected a Member of the Paris Academy of Sciences. In Figure 1 . 7 a portrait of Cartan taken in 1 9 3 1 is reproduced.

§ 1 .6. Academician

After being elected as a Member of the Paris Academy of Sciences, Cartan remained a modest man. He continued his intensive research.

In 1 932 Cartan published the papers On the group of the hyperspherical geometry [ 1 35 ) and On the pseudo-conformal geometry of hypersurfaces of the

1 8 I. THE LIFE AND WORK OF E. CARTAN

space of two complex variables (which appeared in two parts [ 1 36] , [ 1 36a] ) ; and gave two lectures, On the pseudo-conformal equivalence of two hypersur­faces of the space of two complex variables [ 1 39] and Symmetric Riemannian spaces [ 1 38] , at the International Congress of Mathematicians in Zurich. The first four of these were devoted to the geometry of real hypersurfaces of the two-dimensional complex space with respect to analytic transformations of this space, which form a Lie pseudogroup. In the same year, in the mathe­matical journal of the University of Belgrade, Yugoslavia, Cartan published the paper On the topological properties of complex quadrics [ 1 37] , in which he studied globally one of the most important symmetric Riemannian spaces.

In 1 933 the book Metric spaces based on the notion of area [ 1 40] appeared, and in Moscow a translation of his course of lectures [ 1 44] was published. In the same year, there appeared the notes Newton 's kinematics and spaces with Euclidean connection [ 1 40a] and Fins/er spaces [ 1 4 1 ] , and two notes on Finsler spaces: the letter to the Indian geometer Damodar Dharmanand Kosambi ( 1 907- 1 966) [ 1 4 l a] and the note on the paper of the Polish geome­ter Stanislaw Gol� ( 1 902-?) [ l 40b ] .

On October 22 , 1 933 , in Nimes, Cartan gave a speech in memory of one of his teachers, G. Darboux [ 1 88 ] , during festivities accompanying the unveiling of a bust of the scientist.

In 1 934 the book Fins/er Spaces [ 1 42] , two notes [ 1 42a] , [ 1 42b] concerning A. Weil's communications, and the note Tensor calculus in projective geom­etry [ 1 43] appeared. In the same year, Cartan wrote the manuscript The unitary (field) theory of Einstein-Mayer [ 1 43a] , which was published only in the CEuvres Completes of his works [207], and gave three talks at the Inter­national Conference on Tensor Differential Geometry in Moscow, U.S.S.R: Fins/er spaces [ 1 52], Spaces with projective connection [ 1 53] , and The topol­ogy of closed ( i .e . , compact) spaces [ 1 54] . Figure 1 . 8 shows Cartan's arrival in Moscow (Cartan is on the left in the first row; on the right in the same row is the Chairman of the Conference, V. F. Kagan) .

I n 1 936 the French text o f Cartan's lectures [ 1 44] appeared, i n addition to his paper Homogeneous bounded domains of the space of n complex variables [ 1 45] , the notes on the communications of L. S. Pontryagin on the Betti num­bers of compact simple groups [ 1 45a] and of G. Bouligand (b. 1 889) [ 1 46] ; and the paper Projective tensor calculus [ 1 47] in Matematichesky Sbornik (Moscow) . Cartan also gave the talk On a degeneracy of Euclidean geom­etry [ 1 47a] at a session of the Association for the Development of Science in Nantes, in which he considered the two-dimensional isotropic geometry. The text of this talk, which was not included in the CEuvres Completes ( [207, 209] ) , is reproduced in Appendix B of this book.

In 1 936 the papers The topology of spaces representing Lie groups [ 1 50] and The geometry of the integral J F(x , y , y' , y" ) dx [ 1 48] , and the note On the fields of uniform acceleration in restricted relativity (theory) [ 1 49] ap­peared. In the same year, Cartan delivered the lecture The role of group theory

§ 1 .6. ACADEMICIAN 1 9

FIGURE 1 . 8

in the evolution of modern geometry [ 1 5 1 ] at the International Congress of Mathematicians in Oslo.

In 1 937 the following works were published: the books Lectures on the theory of spaces with a projective connection [ 1 55 ] and The theory of.finite con­tinuous groups and differential geometry considered by the method of moving frames [ 1 57] ; the talks [ 1 52]-[ 1 54] at the International Conference on Ten­sor Differential Geometry in Moscow, U.S.S.R. , in Proceedings of the Vector and Tensor Analysis Seminar at Moscow University ( in French and Rus­sian) ; the papers Extension of tensor calculus to non-affine geometries [ 1 56] ; the talk The role of analytic geometry in the evolution of geometry [ 1 58 ] at the International Philosophical Congress; the papers Groups [ 1 59] , Geometry and groups [ 1 60], and Riemannian geometry and its generalizations [ 1 6 1 ] in French Encyclopaedia; and the talks The problems of equivalence [ 1 6 1 a] and The structure of infinite groups [ 1 6 1 b] in Proceedings of French Mathematical Seminar.

In 1 938 the book Lectures on the theory of spinors [ 1 64] was published. This book was devoted to the linear representations of the group of orthog­onal matrices, which were discovered by Cartan as far back as 1 9 1 3 . In the 1 930's they were named the spinor representations because of their appli­cations in physics, which are connected with the spin of an electron. In the same year the papers Linear representations of Lie groups [ 1 62], Galois theory and its generalizations [ 1 65] , and Families of isoparametric surfaces in spaces of constant curvature [ 1 66] , and his note Generalized spaces and integration

20 I. THE LIFE AND WORK OF E. CARTAN

of certain classes of differential equations [ 1 63] were published. In 1 939 the papers On remarkable families of isoparametric surfaces in the

spherical spaces [ 1 67] and The absolute differential calculus in light of recent problems in Riemannian geometry [ 1 69] appeared. He also gave the talk On certain remarkable families of hypersurfaces [ 1 68] at the Mathematical Congress in Liege, Belgium.

On May 1 8, 1 939, at the Sorbonne, a celebration in honor of Cartan's 70th birthday was held. The chairman of the meeting was the well-known physi­cian and biologist, Academician Gustave Roussy ( 1 87 4- 1 948) , the rector of the Sorbonne, who opened the meeting. One of Cartan's teachers, Emile Pi­card, who was at that time the permanent secretary of the French Academy of Sciences, gave a short description of Cartan's works in the theory of Lie groups and the theory of differential equations, Riemannian geometry, and the theory of "generalized spaces". Picard stressed that Cartan was not only "a pure mathematician, an artist and a poet in the world of numbers and forms", but also that he was dealing with problems of physics, connected with relativity theory, and had written a book on spinors.

In his greetings, the Dean of the Faculty of Sciences of the Sorbonne, the famous geodesist Charles Maurain ( 1 87 1 - 1 967) recollected all the universi­ties of the world where Cartan had worked or delivered talks or courses. One of the founders of the method of moving frames, the Belgian Academician Alphonse Demoulin ( 1 869- 1 94 7) , greeted Cartan on behalf of the scientists of the entire world. Without mentioning his own name, Demoulin told that in 1 904 "one of Darboux's students" generalized the Darboux method of a moving trihedron for non-Euclidean spaces and noted the further stages of its development, which brought Cartan in 1 9 1 0 to the general formulation of the method of moving frames.

Arthur Tresse, one of Cartan's former schoolmates in l'Ecole Normale Superieure and the honorary general inspector of secondary schools, greeted Cartan on behalf of his schoolmates and told how the student Cartan deliv­ered lectures to his schoolmates in !'Ecole Normale Superieure. Tresse also greeted the successors of the scientific "dynasty" of E. Cartan, the mathemati­cians Henri and Helene Cartan, and the physicist Louis Cartan. He spoke warmly about the composer Jean Cartan, E. Cartan's deceased son.

The famous physicist and director of the School of Physics and Chemistry, Academician Paul Langevin ( 1 872- 1 946) , described the works of Cartan re­lated to physics. Georges Bruhat ( 1 887- 1 944) , a physicist and the deputy director of l'Ecole Normale Superieure, noted the many connections Cartan had with l'Ecole Normale Superieure. Professor of the Sorbonne, mathemati­cian, and Academician Gaston Julia ( 1 893- 1 965) recalled how he listened to Cartan's lectures in l'Ecole Normale Superieure and how he again met Cartan in a hospital, which was set up at the same school during the war. Julia, a young officer, was seriously wounded in the face and was undergoing a reha­bilitation in this hospital after a series of successive plastic surgeries in the

§ 1 .6. ACADEMICIAN 2 1

hospital Val-de-Grace, where his nose was reconstructed. The President of the French Mathematical Society, Antoine Joseph Henri

Vergne ( 1 879- 1 943) , greeted Cartan as an active member of the Society. Professor of Mathematics at the University of Nancy, Jean Dieudonne (b. 1 906) , saluted Cartan on behalf of young mathematicians. Cartan himself gave a speech at this meeting. In his speech he gave his recollections of his entry into science, from which we have previously quoted. He also replied to each of the speakers who had greeted him. The speeches at this celebration meeting were published in a book [Ju], and Cartan's speech can be found in Appendix C to this book. At the conclusion of this celebration meeting, the orchestra, under conductor Charles Munch ( 1 89 1 - 1 968) , performed the composition To the memory of Dante, written by Jean Cartan.

On the date of this celebration the collection of selected Cartan papers, Selecta [204] , was published. It includes his works [37] , [70], [ 1 1 8 ] , [ 1 50], [ 1 6 1 a] , and [ 1 62], as well as Cartan's survey of his own works [ 1 87] and the list of his mathematical works.

Cartan retired as Professor of the Sorbonne in 1 940, after 30 years of service in this university. While working in the Sorbonne, Cartan also was Professor of Mathematics at the School of Industrial Physics and Chemistry in Paris.

In 1 940 the papers On a theorem of J. A. Schouten and W. van der Kulk [ 1 70], On the linear quaternion groups [ 1 7 1 ], and On families of isoparamet­ric hypersurfaces in the spherical spaces of five and nine dimensions [ 1 72] appeared. In the same year in Moscow, U.S.S.R. , in the collection of articles devoted to the memory of the Soviet Academician D. A. Grave ( 1 863- 1 939) , Cartan's paper On a class of surfaces similar to the surfaces R and the sur­faces of Jonas [ 1 80] was published. (This paper was published in France in 1 944.)

In 1 940, in the Yugoslavian journal Saturn, the Serbian translation of Car­tan's lecture The influence of France in the development of Mathematics [ 1 9 1 ] , delivered during his visit to Belgrade in February of 1 940, appeared. In 1 94 1 this translation was published as a separate booklet. The English translation of this lecture from Serbian, compared with its French text, is given in Ap­pendix D. The introduction to this book was written by the famous Serbian mathematician Mihailo Petrovic ( 1 868- 1 943) , who was Cartan's schoolmate in l'Ecole Normale Superieure. Cartan started this lecture from the works of F. Viete ( 1 540- 1 603) and finished the section on Viete with the following words: "I should tell you that for quite some time Viete was in contact with one of your (i .e . , Yugoslavian) first mathematicians, Marino Ghetaldi (Marin Getaldic) ( 1 5 56- 1 626) , who was born in Dubrovnik and who, in Paris, in the year 1 600, published one of Viete's last works." [ 1 9 1 , p. 6] . Later Cartan considered works of R. Descartes, B. Pascal, P. Fermat, A. C. Clairaut, J. B. D'Alembert, J. L. Lagrange, P. S. Laplace, A. M. Legendre, G. Monge, J. B. Fourier, A. L. Cauchy, J. V. Poncelet, E. Galois, Ch. Hermite, G. Darboux,

22 I . THE LIFE AND WORK OF E. CARTAN

and H. Poincare, and in passing he mentioned the names of other famous French mathematicians.

The last of those mentioned by Cartan was Jacques Herbrand ( 1 908- 1 93 1 ) , who defended his thesis i n 1 930 and was tragically killed the following sum­mer in an accident in the mountains. His thesis was related to proof theory. He also worked in the theory of fields of classes. His last paper, published in 1 93 1 , was written jointly with young Claude Chevalley ( 1 909- 1 984) . Car­tan said of Herbrand's works that "his works, mercilessly interrupted by his early death, were announcing of a great mathematician, perhaps similar to Evariste Galois". We present here the last paragraph of the lecture, where Cartan expressed his general view on mathematics: "More than any other sci­ence, mathematics develops through a sequence of successive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of the problems and entities considered. Carried to an extreme, this proce­dure justifies the well-known joke according to which a mathematician is a scientist who neither knows what he is talking about or whether whatever he is indeed talking about exists or not. French mathematicians, however, never enjoyed distancing themselves from reality; they do know that, although needed, logic is by no means crucial. In mathematical activity, like in any other type of human activity, one should find a balance of values: there is no doubt that it is important to think correctly, but it is even more important to formulate the right problems. In that respect, one can freely say that French mathematicians not only always knew what they were talking about, but also had the right intuition to select the most fundamental problems, those whose solutions produced the strongest influence on the overall development of sci­ence."

In 1 942, the paper On pairs of applicable surfaces with preservation of prin­ciple curvatures [ 1 76) appeared. In this year Cartan also wrote the paper The isotropic surfaces of a quadric in a seven-dimensional space [ 1 77) , which is still unpublished. H. Cartan sent us the manuscript of this paper. We will consider this paper in Chapters 3 and 5. In the same year, Cartan wrote the obituary of the Italian geometer Tullio Levi-Civita ( 1 87 3- 1 94 1 ) and the paper A centenary: Sophus Lie [20 1 ) , on the occasion of the l OOth birthday of Lie. In the latter he recalled his meetings with the founder of the theory of Lie groups during Lie's visit to Paris. This paper was only published in 1 948.

In 1 943 the papers On a class of Wey! spaces [ 1 78) and Surfaces admitting a given second fundamental form [ 1 79) and the obituary of the mathematician Georges Giraud ( 1 889- 1 943) [ 1 93] were published. In 1 944, the paper [ 1 80), published in the U.S.S.R. in 1 940, was published in France.

In 1 945 Cartan published Exterior differential systems and their geomet­ric applications [ 1 8 1 ] and the paper On a problem of projective differential geometry [ 1 82) . In the same year, in Moscow, U.S.S.R. , he participated in celebrations on the occasion of the 220th anniversary of the founding of the

§ 1 .6. ACADEMICIAN 23

Academy of Sciences of the U.S.S.R. In 1 946 a .new edition of Lectures on the geometry of Riemannian man­

ifolds [ 1 83] was published. Cartan included in this edition the topics that he originally intended to include in the second volume of this book. In particular, the study of Riemannian manifolds by means of moving frames (published in Russian translation in [ 1 08a] ) was included. In the same year, Cartan published the paper Some remarks on the 28 double tangents of a plane quartic and the 27 straight lines of a cubic surface [ 1 84] .

In the first half of 1 946, when the President of the Paris Academy of Sciences was sick, Cartan replaced him and chaired the weekly meetings of the Academy. During these meetings Cartan informed the audience about French and foreign members of the Academy who had passed away. These communications by Cartan were brief but detailed obituaries of eminent sci­entists. During this time Cartan delivered obituaries of the following French Academicians: the head of French geodesic service General Georges Per­rier ( 1 872- 1 946) [ 1 94] ; the metallurgist Leon Alexandre Guillet ( 1 873- 1 946) [ 1 96] ; the bacteriologist Louis Martin ( 1 864- 1 946) [ 1 98] ; the famous physi­cist Langevin [ 1 99] ; and the two foreign members of the Academy: the Amer­ican biologist Thomas Hunt Morgan ( 1 866- 1 945) [ 1 95] , the founder of the study of genes as carriers of heredity and their localization in chromosomes, and the American pathology anatomist and biologist Simon Flexner ( 1 863-1 946) [ 1 97] . In the same year, Cartan wrote an article on the occasion of the 80th birthday of his old friend E. Vessiot [200] and a note on the occasion of the 200th birthday of Gaspard Monge ( 1 746- 1 8 1 8) [ 1 98a] . These publi­cations show that Cartan was very familiar with the status of many sciences, including some that are rather far from mathematics.

In 1 94 7 Cartan published the paper A real anallagmatic space of n dimen­sions [ 1 85 ] on the geometry of an n-dimensional conformal space, which, following the old French tradition, he named "anallagmatic space", and the short book The group theory [ 1 85a] .

In 1 948, in the collection of articles Great currents of mathematical thought, which was prepared for publication by Fran!(ois Le Lionnais ( 1 90 1 - 1 984) during World War II, the paper [20 1 ] was published. In the same year, Car­tan published the 30-page book [202] under the same title-Gaspard Monge: his life and work-as his earlier note [ 1 98a] . In this book Cartan published for the first time a series of Monge's letters. This was the reason why the historian of science Rene Taton (b. 1 9 1 5 ) referred many times to this book in his research Scientific works of Gaspard Monge [Ta] .

In 1 949 Cartan published his last two papers: Two theorems of real anallag­matic space of n dimensions [ 1 86] , relating to the n-dimensional conformal geometry, and The life and works of Georges Perrier [204] ; [ 1 94] is a short obituary of Perrier.

After Cartan retired in 1 940, he spent the last years of his life teaching mathematics at the Ecole Normale Superieure for girls.

24 I . THE LIFE AND WORK OF E. CARTAN

Elie Cartan died in Paris on May 6, 1 95 1 , after a long illness. Immediately after Cartan's death, in the years 1 952- 1 955 , a facsimile edi­

tion of his papers [207] was published. It consisted of three parts, and each part appeared in two volumes. In Part I, a list of publications by Cartan, his survey [ 1 87] of his own scientific works, and his papers on the theory of Lie groups and the theory of symmetric spaces were reproduced. In Part II, Car­tan's papers on algebra, theory of Lie pseudogroups, and theory of systems of differential equations were included. Part III contains Cartan's papers on differential geometry and some other areas. In particular, in Part III, for the first time, a complete text [ 46] of his extended translation of the paper [Fa] of Fano and the paper [ 1 43a] on a unitary field theory of Einstein-Mayer were published. In 1 984 a new edition [209] of Cartan's papers was released. In this edition Parts I and II of the 1 952- 1 955 edition were each placed in one volume; at the end of the second volume of Part III, the papers of Shiing-shen Chern and Claude Chevalley [ChC] and of J. H. C. Whitehead [Wh] were added, in which analyses of Cartan's mathematical work were presented.

Cartan's best-known students are the French mathematicians Andre Lich­nerowicz (b. 1 9 1 5) and Charles Ehresmann ( 1 905- 1 979) . Andre Weil (b. 1 906) was also greatly influenced by Cartan. He dedicated his book Integra­tion on topological groups and its applications [Wel] to Cartan.

In addition to the papers [ChC] and [Wh] on the life and research of Cartan, the articles of Dieudonne [Die] , Hodge [Hod] , and Saltykov [Sal] , and the articles in the memorial collection [ECR] published by Roumanian mathematicians on the occasion of Cartan's l OOth birthday, and the research of Hawkins [Haw l ]-[Haw3], are also worthy of note.

§1.7. The Cartan family

Elie Cartan and his wife Marie-Louise had four children: the mathemati­cian Henri, the composer Jean, the physicist Louis, and daughter Helene, who, like her father and eldest brother, became a mathematician. Figure 1 . 9 shows a 1 928 picture of the Cartan family: in the first row from left to right are Louis, Helene, and Jean, and in the second row from left to right are Elie Cartan, Henri Cartan, and Marie-Louise Bianconi-Cartan.

Henri Cartan (b. 1 904) , the eldest son of E. Cartan, became one of the most prominent contemporary mathematicians. He graduated in l 926 from l'Ecole Normale Superieure, the same school from which his father grad­uated. From 1 928 to 1 929 he taught in the Lycee Malherbe in Caen, the center of the Departement Calvados in Normandie. From 1 929 to 1 93 1 he was a lecturer in the Faculty of Sciences at the University of Lille. From 1 93 1 to 1 935 he was a lecturer and from 1 936 to 1 940 a professor in the Faculty of Sciences at the University of Strasbourg. From 1 940 to l 949 he

§ 1 .7. THE CARTAN FAMILY 25

FIGURE 1 .9

was a lecturer in the Faculty of Sciences at the Sorbonne, except for the pe­riod 1 945- 1 947 when he again worked in Strasbourg. From 1 949 to 1 969 he worked as a professor in the Faculty of Sciences at the Sorbonne. Between 1 940 and 1 965 he also taught in l'Ecole Normale Superieure, and from 1 969 to 1 975 he was a professor of the Faculty of Sciences at the University of Orsay, a southern suburb of Paris (this university was later renamed the Uni­versity of Paris-Sud) . Since 1 975 , H. Cartan has been a professor emeritus of this university. In 1 93 5 H. Cartan, with Chevalley, Jean Frederic Delsarte ( 1 903- 1 968) , Jean Dieudonne, and Andre Weil (b. 1 906) organized a group which wrote the mathematical encyclopaedia Elements of Mathematics un­der the pseudonym Nicolas Bourbaki [Bou] . H. Cartan worked in this group until 1 954, when he was 50 years old. This collective work exceptionally influenced the development of mathematics throughout the entire world. In 1 965 H. Cartan was elected as a corresponding member of the Paris Academy of Sciences, and in 1 974 he became a member of this Academy. From 1 967- 1 970 H. Cartan was the President of the International Mathematical Union. In 1 980 he and Andrei N. Kolmogorov ( 1 903- 1 987) were the recip­ients of a very prestigious Wolf Prize in Mathematics. H. Cartan is a foreign

26 I . THE LIFE AND WORK OF E. CARTAN

member of many Academies of Sciences, including the London Royal Soci­ety and the National Academy of Sciences, U.S.A. He is also a honoris causa Doctor of Sciences of many universities. He is the author of several well­known books: The elementary theory of analytic functions [CaH2], Homolog­ical algebra Uointly with S. Eilenberg) [CaE] , and Differential calculus and differential forms [CaH3] . These books have been translated into many lan­guages. He also is the author of numerous papers in the theory of ana­lytic functions, algebraic topology, homological algebra, and potential theory [CaH l ] . He has five children: Jean (b. 1 936) is an engineer, Frarn,;oise (b. 1 939) is a teacher of English, Etienne (b. 1 94 1 ) is a teacher of mathemat­ics, Mireille (b. 1 946) is an expert in ecology, and Suzanne (b. 1 9 5 1 ) is a management expert.

Jean Cartan ( 1 906- 1 932) was a student of Paul Dukas ( 1 865- 1 935) in the Paris Conservatory from 1 925- 1 93 1 . After graduation from the Conserva­tory, J. Cartan was a composer: he is the author of two string quartets, a sonatina for flute and clarinet, a composition for choir and orchestra includ­ing words from the Lord's Prayer, and a composition for orchestra, To the memory of Dante, mentioned earlier. J . Cartan died of tuberculosis at the age of 25 .

Louis Cartan ( 1 909- 1 943) was a talented physicist who specialized in atomic energy. He was a student of Maurice de Broglie ( 1 875- 1 960) . He worked in the X-ray physics laboratory in Paris, and after that became a pro­fessor of the Faculty of Sciences at the University of Poi tiers. He authored the book Mass spectrography. Isotopes and their masses [CaL] , and, jointly with Jean Thibaud and Paul Comparat, the book Some actual technical questions in nuclear physics. Method oftrochoid: positive electrons. Mass spectrography: isotopes. Counters of particles with linear acceleration. Geiger 's and Muller 's counters [TCC] . During World War II L. Cartan was an active participant in the Resistance in Poitiers. In 1 942 he was arrested by the police of the Vichy government and was handed over to the German occupation forces. In February 1 943 he was taken to Germany, and in December 1 943 he was decapitated. The poor parents learned of Louis's death only in May 1 945 . At present three of Louis's children are alive: Annette (b. 1 936) is a teacher of English, Isabelle (b. 1 938) is a teacher of mathematics, and Pierre (b. 1 940) is a financier.

The youngest Cartan child, daughter Helene ( 1 9 1 7- 1 952) , was a math­ematician. She graduated from the Ecole Normale Superieure, as had her father and brother. She taught in several lycees and authored several mathe­matical papers.

Figure 1 . 1 0 shows the grave of E. Cartan, his wife, and their two children in a cemetery in Dolomieu. On the vertical tombstone there is the inscription

The CARTAN FAMILY

The inscription on the left half of the horizontal tombstone reads:

§ 1 .8 . CARTAN AND THE MATHEMATICIANS OF THE WORLD

FIGURE 1 . 1 0

Jean CARTAN, December 1 , 1 906 - March 26, 1 932

Marie-Louise BIANCONI, the spouse of Elie CARTAN February 1 8 , 1 880 - May 2 1 , 1 950

Elie CARTAN, April 9, 1 869 - May 6, 1 9 5 1

The right half of the same horizontal tombstone reads

Helene CARTAN, October 1 2, 1 9 1 7 - June 7, 1 952

§1.8. Cartan and the mathematicians of the world

27

Elie Cartan visited many countries and was connected by friendship with many mathematicians. In 1 920, 1 924, 1 928, 1 932, c;tnd 1 936 he partici­pated in the International Congresses of Mathematicians held in Strasbourg, Toronto, Bologna, Zi.irich, and Oslo. In 1 939 he participated in the Mathe­matical Congress in Liege. In 1 940, in Belgrade, he delivered the lecture on the role of French mathematicians in the history of mathematics.

Cartan greatly influenced mathematicians of many countries. Among Ger­man mathematicians, Ernst August Weiss ( 1 900- 1 942) , a student of Eduard

28 I. THE LIFE AND WORK OF E. CARTAN

FIGURE 1 . 1 1

Study, was most influenced by Cartan; Weiss spent two semesters with Car­tan and developed further Cartan's idea on the "triality principle". Many papers of Shiing-shen Chern (b. 1 9 1 1 ) , a student of Wilhelm Blaschke, also reflected Cartan's influence.

In April and May of 1 93 1 Cartan made a trip to Romania and Poland. In Romania he delivered a series of lectures in Cluj , Bucharest, Ia�i (Yassy) , and Cernauti (Chernovcy, now in the U.S.S.R. ) . In the same year, Cartan was elected an honorary member of the Romanian Academy of Sciences in Bucharest. In 1 934 Cartan was made a corresponding member of the Royal Society of Sciences in Liege, Belgium; in 1 937 he was elected a foreign member of the Amsterdam Academy of Sciences (Netherlands) . In 1 949 he became a foreign member of the National Academy of Sciences of the U.S.A. and a member of the National Academy of Forty in Rome. Cartan was also elected an honoris causa Doctor of Sciences at Harvard University ( 1 936) and the Universities of Liege ( 1 934) , Brussel and Louvain ( 1 947) , and Bucharest and Pisa ( 1 948) .

Cartan corresponded with many scientists. However, although many of his letters have been preserved, only his correspondences with A. Einstein [2 1 0) and the Romanian geometers Gheorghe Titeica ( 1 873- 1 939) , Alexan­dro Pantazi ( 1 873- 1 939) , and Gheorghe Vranceanu ( 1 900- 1 979) [2 1 1 ) have been published.

Figure 1 . 1 1 shows a group of participants at the Congress in Zurich. From left to right in this picture are Ferdinand Gonseth ( 1 890-?) , Elie Cartan,

§ 1 .8. CARTAN AND THE MATHEMATICIANS OF THE WORLD 29

FIGURE 1 . 1 2

FIGURE 1 . 1 3

Gustave Juvet ( 1 896- 1 936) , Gaston Julia, Mrs. Julia, and Mrs. Gonseth. Figure 1 . 1 2 shows a group of participants at the Congress in Oslo. From left to right in this picture are George David Birkhoff ( 1 884- 1 944), Elie Cartan, and Constantine Caratheodory ( 1 8 7 3- 1 9 50) . Figure l . l 3 is a picture of a group of mathematicians in Paris at the beginning of 1 935 . In the first row from left to right in this picture are: Emil Artin ( 1 892- 1 962) , Gaston Julia, Francesco Severi ( 1 879- 1 96 1 ), and Elie Cartan.

30 I. THE LIFE AND WORK OF E. CARTAN

Cartan had a close friendship with many Soviet geometers. Being in Paris in 1 926- 1 927, Serge P. Finikov ( 1 883- 1 964) attended the course of lectures delivered by Cartan. Later Finikov founded a Soviet differential-geometric school that dealt with applications of the method of exterior forms and the method of moving frames. From 1 927 to 1 928 in the Sorbonne, Georgi N. Nikoladze ( 1 888- 1 93 1 ) , under Cartan's supervision, prepared and de­fended his doctoral dissertation On continuous families of geometric figures. Before 1 9 1 7 Nikoladze worked as an engineer-metallurgist in the factories of Donbass. From 1 9 1 9 he taught mathematics at the University of Tbilisi. After his return to Tbilisi, Nikoladze became a professor at the University of Tbilisi and founded the Georgian geometric school. Cartan also was on friendly terms with Veniamin F. Kagan ( 1 869- 1 953 ) , the founder of the So­viet tensor differential-geometric school.

We have already mentioned Cartan's publications in Moscow and Kazan. In 1 937 , in the VIII International Lobachevsky competition, the Lobachev­skian prize was awarded to Cartan for his work in geometry. Cartan visited the U.S.S.R. three times: in 1 930 he participated in the First All-Union Mathematical Congress in Kharkov and later delivered a series of lectures at Moscow University; in 1 934 he participated in the International Conference on Tensor Differential Geometry in Moscow; and in 1 945 he was present during the celebration of the 220th anniversary of the Academy of Sciences of the U.S.S.R.

Ten books and collections of papers by Cartan appeared in Russian transla­tions in the U.S.S.R. In 1 933 the translation of the course of lectures [ 1 44] de­livered by Cartan in 1 930 appeared in Moscow (translated by S. P. Finikov) . In 1 936 in Moscow the translation of the book [ 1 1 4] under the title Geometry of Riemannian manifolds was published (translated by G. N. Berman; edited by A. Lopshits) . In 1 93 7 Cartan's lectures [ 1 52]-[ 1 54] at the International Conference on Tensor Differential Geometry were published in Proceedings of the Vector and Tensor Analysis Seminar. In 1 939 a collection [205] of Cartan's papers [88] , [ 1 05] , and [ 1 40] was published in Kazan (translated by P. A. Shirokov and B. L. Laptev) . In 1 940 these translations were re­published in a collection, The VIII International Lobachevsky Competition . In the same year, the Russian translation, titled The integral invariants, of the book [64] was published in Moscow (translated by G. N. Berman; edited by V. V. Stepanov) . In 1 947 the Russian translation, titled The theory of spinors, of the book [ 1 64] was published in Moscow (translated by P. A. Shirokov) . In 1 949 a collection of Cartan's papers [93], [94], [ 1 0 1 ] , [ I 03] , [ 1 1 6] , and [ 1 28] , titled Geometry of Lie groups and symmetric spaces [206], was published in Moscow (translated by B. A. Rosenfeld; edited by P. K. Rashevsky) . In 1 960, 1 962, and 1 963 the Russian translations of Cartan's books [ 1 08a] , [ 1 44], [ 1 57] , and [ 1 8 1 ] were published in Moscow (translated by S. P. Finikov) . In 1 962 a collection of Cartan's papers [66] , [68]-[70], and [80] , titled Spaces with affine, projective and conformal connection [208], was

§ 1 .8 . CARTAN AND THE MATHEMATICIANS OF THE WORLD 3 1

Courtesy of Department of Geometry, Kazan University, Tatarstan, Russia

FIGURE 1 . 1 4

published in Kazan (translated by P. A. Shirokov, V. G. Kopp, B. L. Laptev, and others; edited by P. A. Shirokov) .

Figure 1 . 1 4 shows a meeting of Cartan (left) with the mathematicians from Kazan: Petr A. Shirokov ( 1 895- 1 944) (center) and Nikolai G. Chebotarev ( 1 894- 1 947) during one of Cartan's visits to Moscow.

The method of exterior differential forms was developed by Finikov in the book The Cartan method of exterior differential forms in differential ge­ometry [Fin] . This method was applied to solutions of a very large number of problems in differential geometry by Finikov and his numerous students and followers in Moscow, Kiev, Vil' nius, Tomsk, and other cities of the U.S.S.R. Also, further development in theory of Riemannian manifolds and spaces with affine connection, particularly symmetric spaces, was achieved in papers of Kagan, Shirokov, and other geometers from Moscow, Kazan, Saratov, Penza, and other cities.

During Cartan's first two visits to Moscow, the authors of this book were high school students. During his third visit to Moscow in May of 1 945, the authors were serving in the Soviet Army. At that time B. A. Rosenfeld's military unit was located near Moscow, and he had the good fortune to see Cartan and discuss with him his own results and plans.

The scientific activities of M. A. Aki vis in the field of differential geometry, which started a few years after World War II ended, also were very closely connected with the development of Cartan's ideas.

CHAPTER 2

Lie Groups and Algebras

§2.1 . Groups

The 1 870s, when Elie Cartan was a lad taking his first steps in his father's blacksmith shop and in the elementary school of Dolomieu, were critical years in the history of France as well as in world history and in the history of mathematics. In 1 870, after its defeat in the Franco-Prussian war, the Second Empire of France fell, and France again became a republic. In the 1 870s a new period of world history began-the Industrial Revolution. At that time a new period in the history of mathematics also began. Two great discoveries, made in the first half of the 1 9th century, were understood: the discovery of group theory by Evariste Galois and the discovery of non-Euclidean geometry by Nikolai I. Lobachevsky.

The mathematical implications of these apparently unrelated discoveries, which were arrived at independently, were very closely related. Before Galois it was believed that only one arithmetic of real and complex numbers was conceivable. Galois showed that there are many different arithmetics defined by different groups and fields. Before Lobachevsky, it was believed that only one geometry, namely Euclidean geometry, was conceivable. Lobachevsky discovered a new geometry, which was as much noncontradictory as Eu­clidean geometry but quite different from it. The discoveries of Galois and Lobachevsky were the principal manifestations of creations of new "alge­bras" and "geometries" in the 1 9th century. Along with Galois groups and fields, a series of new numerical systems was discovered at that time. Later, this series was named "hypercomplex numbers" and "algebras". Along with Lobachevskian geometry, during the 1 9th century, other geometries, differ­ent from classical Euclidean geometry, were also discovered: affine, projec­tive, multidimensional geometries, and finally the Riemannian geometries­geometries of curved spaces. Group and algebra theories as well as non­Euclidean and other geometries discovered at that time played an important role in Cartan's mathematical research. In the mid- 1 8708 another important discovery was made: the set theory of Georg Cantor ( 1 845- 1 9 1 8) . This the­ory and the theory of functions of a real variable, which is closely connected with set theory, became the main areas of research of French mathemati­cians at the end of the 1 9th century and the beginning of the 20th century.

33

34 2. LIE GROUPS AND ALGEBRAS

Originally these two theories were not reflected in Cartan's work. Group theory was created by the young Evariste Galois ( 1 8 1 1 - 1 832) , who

was killed in a duel. However, in his short lifetime he published a few works, and, on the night before the fatal duel, he wrote a summary of his main discoveries. This was later published by a friend. Galois was a student at the same Ecole Normale Superieure where Cartan later studied. Galois made his discovery while trying to determine the solvability by radicals of algebraic equations. If one is given an algebraic equation

(2. 1 )

with rational coefficients, real or complex, then the values of x in this equa­tion which make it an identity are called the roots of the equation. In the case of quadratic equations ( n = 2) , the roots x1 and x2 are expressed in terms of the coefficients a0 , a 1 , and a2 by commonly known formulas, found at the beginning of the 9th century by Muhammed al-Khwarizmi (circa 783-850) . These formulas involve quadratic radicals. In the 1 6th century Niccolo Tartaglia (circa 1 500- 1 557 ) and Girolamo Cardano ( 1 50 1 - 1 576) found the "Cardano formula'', through which the roots of a cubic equation ( n = 3) are expressed in terms of the coefficients a0 , a 1 , a2 , and a3 ; the "Cardano for­mula" involves cubic radicals. Cardano's student, Luigi Ferrari ( 1 522- 1 565 ) , solved a similar problem for n = 4 . For a few centuries mathematicians tried to find a formula expressing the roots of equation (2. 1 ) for n 2:: 5 , in terms of the coefficients of this equation. However, this problem was solved only for the simplest particular cases of this equation, for example, for "binomial equations" xn = a (one root of this equation is expressed by the radical x = efO. and others are the products of this radical and powers of the com­plex number e = e2ni/n = cos 2: + i sin 2: ) • In 1 829 Niels Henrik Abel ( 1 802- 1 829) , in his Demonstration of the impossibility of the algebraic reso­lution of general equations surpassing fourth degree [Ab], distinguished a class of equations solvable by radicals, and this class was wider than the binomial equations. In the paper Memoir on conditions of solvability of equations by radicals [Gal], written before his duel, Galois gave a complete solution to the problem. The Galois solution is based on the notion of groups which he introduced and which was implicitly contained in the paper Reflections on so­lution of equations [Lag l ] by Joseph Louis Lagrange ( 1 736- 1 8 1 3) and in the paper Arithmetic investigations [Gau] by Carl Friedrich Gauss ( 1 777- 1 855 ) . In many branches of mathematics one can find such operations on objects, which assigns to each pair of objects of a set an object from the same set. Examples of such operations are: the addition of numbers, vectors, or matri­ces; the multiplication of numbers or matrices; and the successive realization of transformations. At the very beginning of human civilization, the concept of the natural number, which includes the sets of different objects consisting of the same number of objects, and later the arithmetic of integers and the algebra of rational, real, and complex numbers, were introduced. In a similar

§2. 1 . GROUPS 35

way, the theory, including very diverse arithmetic, algebraic, and geometric operations, appeared next. The term "group" was introduced by Galois, who, in his Memoir on the conditions of solvabi/ity of equations by radicals, wrote about substitutions: "If in such a group there are substitutions S and T , then there is the certainty of there being the substitution ST" (see [Gal, p. 47] or [Ro8, p. 328] ) . Note that Galois used the term "group" in a wider sense than we do. In the famous letter to his friend written on the eve of his fatal duel, Galois wrote: "When a group G contains another group H , the group G can be decomposed into groups" (see [Gal, p. 1 73] or [Ro8, p. 329] ) , where these "groups" are right cosets of G with respect to H . (In the English translation of this letter in [Sm, p. 279] the word "sets" was used instead of "cosets".) At present a group consisting of elements a , b , c , . . . is defined as a set of elements such that

1 ° . To each two elements a and b there corresponds an element c = a o b .

2° . (a o b) o c = a o (b o c) for any a , b , and c . 3° . There exists a "neutral element" e such that e o a = a o e = a for

every a . 4° . For each element a there exists a "complementary element" a such

that a o a = a o a = e .

If within a group the following property holds:

5° . a o b = b o a for every two elements a and b (the group opera�ion is commutative) ,

the group i s called commutative or abelian. In the case of integers, rational, real, and complex numbers, and the opera­

tion of addition, the "neutral element" is 0 and the "complementary element" for a number a is the number -a . For the last three classes of numbers without 0 and the operation of multiplication, the "neutral element" is 1 and the "complementary element" for a number a is its reciprocal a- 1

• In both cases the property 5° is satisfied.

The addition of numbers: a + b = c , vectors: a + b = c , and matrices: A + B = C and the multiplication of numbers: ab = c are commutative. Numbers, vectors, and matrices with these operations form commutative groups. The simplest example of a noncommutative group is the group of permutations

of n elements, i .e . , substitutions of each element a; of the upper row by the corresponding element b; of the lower row, where the elements b1 , b2 , • • • ,

bn of the lower row are the same elements a 1 , a2 , • • • , an of the upper row

36 2. LIE GROUPS AND ALGEBRAS

but arranged in another order. Here the group operation has the form

(a l a2 . . . an) x (b1 bi bn) = (a1 a2 an) bl bi . . . bn cl C2 . . . en cl C2 en

'

the role of the neutral element is played by the identity permutation

(a a · · · a ) I 2 n a a · · · a ' I 2 n

and the permutation inverse to a permutation

(a a · · · a ) I 2 n b b . . . b I 2 n

is

The multiplication of nonsingular matrices: AB = C is also noncommuta­tive. In the group of nonsingular matrices, where the operation is the matrix multiplication, the neutral element is the identity matrix I and the comple­mentary element for a matrix A is its inverse matrix A- 1

•

A subset H of a group G is said to be a subgroup if H itself is a group with respect to multiplication in G . If H is a subgroup of a group G , then the products aH and Ha of the elements of this subgroup and an arbitrary element a of G from the left and the right are called a left and right coset of the subgroup H . If every right coset of a subgroup H is also a left coset, then the subgroup H is said to be invariant or normal (or a normal divisor) . In this case multiplication of cosets can be defined, and the cosets with this multiplication form a group. This group is called a quotient group (or factor group) of the group G by its invariant subgroup H and is denoted by G / H .

Simple groups play a special role in group theory. A group G is simple if it does not have invariant subgroups except the group G itself and the subgroup consisting of the neutral element of G only. In the case where in G there is a sequence of subgroups G = G0 , G1 , G2 , • • • , Gk = e such that each subgroup Gi+ I is an invariant subgroup of G; and each quotient group G;+ 1 /G; is abelian, the group G is called solvable.

Galois introduced the notion of the group which is now called the Galois group of an algebraic equation. This group is the group of automorphisms of a field which is such an extension of the field F (to which belong the coefficients of the equation defined by the roots of this equation) that leaves its subfield F invariant. This group is a finite group which in general can be represented as a permutation group of roots of this equation . The Galois criterion of solvability of the algebraic equation (2 . 1 ) by radicals is that the Galois group of this equation is solvable. In the case of the binomial equation xn

= a this group is cyclic. In the case of equations that were considered by Abel, this group is the general commutative (abelian) group. (This explains the origin of the name "abelian". ) These two groups are examples of solvable groups.

§2.2. LIE GROUPS AND LIE ALGEBRAS 37

Besides the notion of a group, Galois introduced the concept of a field. A field is a commutative additive group, and its elements, excluding 0, form a multiplicative group, where multiplication is distributive with respect to addition. If the multiplicative group of a field is commutative, the field is called commutative.

Examples of commutative fields are: the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, and the field F

P of remainders modulo a prime integer p ( i .e . , the numbers 0 , I , 2 , . . . , p - I , where the sum or the product is the remainder resulting from the division of the sum or the product of the corresponding numbers by p) . The field F

P consists of p elements. Galois also constructed more general finite fields: Galois fields F q , where q is a positive integer power pk

of a prime number p . In the same way as the field C consists of elements a + bi , where a and b are elements of R and i is the "imaginary unit'', i .e . , a root of the equation x2

+ I = 0 , the field F q consists of elements of the form a; + Eo: io:ao: , where i 1 , i2 , • • • , ik- I are "Galois imaginaries"-roots of an irreducible polynomial of degree k with coefficients from F

P. A similar

extension of fields determined by algebraic equations plays an important role in Galois theory.

The Galois memoir on solvability of algebraic equations by radicals, which was originally published by his friend in an obscure publication, was repub­lished in 1 846 by Joseph Liouville ( 1 809- 1 882) in the Journal de Mathema­tiques Pures et Appliquees, of which Liouville was the editor. Galois's ideas were recognized only after Camile Jordan ( 1 838- 1 922) in 1 865 and 1 869 published his comments on Galois's memoir and in 1 870 released a funda­mental Treatise on permutations and algebraic equations [Jo i ] , in which he presented the theory of the permutation group, the Galois theory, and its application to the problem of solvability of algebraic equations by radicals.

§2.2. Lie groups and Lie algebras

In 1 870, not long before the Franco-Prussian war, two friends, Sophus Lie and the young German mathematician Felix Klein ( 1 849- 1 925) , came to France. In Paris the friends attended the lectures of Darboux, had discus­sions with Jordan, and carefully studied his recently published book [Jo i ] . Al­though Jordan's book was mainly devoted to discrete and even finite groups, Lie and Klein, whose first papers were in geometry, were interested in con­tinuous groups and their importance for geometry.

Examples of continuous groups are the following group transformations of geometric spaces: the groups of motions of a Euclidean and a non-Euclidean space, the groups of rotations about a point in these spaces, the group of translations and the group of similarities of a Euclidean space, the group of affine transformations, and the group of collineations ( i .e . , projective trans­formations) . It is well known that the groups of rotations and translations

38 2. LIE GROUPS AND ALGEBRAS

are subgroups of the group of motions of a Euclidean space (moreover, the group of translations is an invariant subgroup) , the group of motions is a subgroup of the group of similarities, the group of similarities is a subgroup of the group of affine transformations, and the latter is a subgroup of the group of collineations. In 1 8 7 1 , while constructing his famous interpreta­tion of Lobachevskian geometry, Klein proved that the group of motions of Lobachevskian space is also a subgroup of the group of collineations. In 1 8 72 he arrived at his "Erlangen Program" [Kie] . According to this program, ev­ery geometry is defined by a group of transformations, and the goal of every geometry is to study invariants of this group.

Sophus Lie chose another way. As early as in his geometric paper On com­plexes, in particular, on complexes of straight lines and spheres ( 1 872) [Lie l ] , which was written i n Paris and was very highly regarded by Cartan (in his paper on Lie, Cartan wrote: "It was in Paris that Sophus Lie made one of his most beautiful discoveries, the famous transformation which bears his name and which establishes an unforeseen relation between lines and spheres in space on the one hand and between asymptotic lines and lines of curvature of surfaces on the other" [20 1 , Engl. tr. , p. 263] ) , Lie connected geometric trans­formations with differential equations ("Lie transformations", which Cartan mentioned in the above quotation, are imaginary transformations sending straight lines into spheres and sending asymptotic lines of surf aces into their curvature lines) . Cartan wrote further: "But the theory of transformation groups itself, its technique, has not been created and nothing indicated the path to be followed for that creation. Sophus Lie devoted himself to this work from 1 873 on and by intense labor rapidly managed to construct the funda­mental theorems from which he quickly deduced very many consequences. In 1 882, upon reading a paper of the French mathematician Halphen, So­phus Lie realized that his earlier research enabled him to see in perspective the problem considered by Halphen" [20 1 , Engl. tr. , pp. 264-265] . The paper by Georges Halphen ( 1 844- 1 889) mentioned by Cartan is the memoir Reduction of a linear differential equation to integrable forms ( 1 884) [Hal] ; it was written earlier and in 1 88 1 received an award from the Paris Academy of Sciences. The problem considered by Halphen is the problem of integrability of differential equations by quadratures, i .e . , the expression of the solutions of these equations in terms of integrals of known functions. In the paper Classification and integration of ordinary differential equations admitting a group of transformations [Lie2] ( 1 883- 1 884) , Lie considered the problem of integrability by quadratures of differential equations as an analogue of the problem of solvability by radicals of algebraic equations and tried to solve this problem by the Galois method. And, in fact, the Lie criterion of solvabil­ity of differential equations by quadratures proved to be similar to the Galois criterion: with each differential equation, a continuous group "admitted by this equation" is connected, and the Lie criterion is that this group must be solvable. In this connection Lie, according to Cartan, "felt the necessity of

§2.2. LIE GROUPS AND LIE ALGEBRAS 39

expounding in one great didactic work the results of his earlier researches, particularly those dealing with group theory. Thanks to the devoted collabo­ration of a young German mathematician; Friedrich Engel ( 1 86 1 - 1 94 1 ), the projected work was written and published after nine years' labor; it appeared successively in three volumes between 1 888 and 1 893" [20 1 , Engl . tr. , p. 265] . In particular, and as in the case of finite groups, with which Galois dealt, Lie had to study properties of simple and solvable continuous groups.

First of all, in his Theory oJtransJormations groups ( 1 888- 1 893) [LiE] , Lie considered a wide class of continuous groups whose elements depend on a finite number of real or complex parameters. Lie himself called such groups "finite continuous groups" or, since he always presented these groups in the form of transformation groups, "finite transformation groups". At present, these groups are called Lie groups.

Lie considered transformations of the form

(2.2) I i Ji ( I n I r x = x , . . . , x ; a , . . . , a ) ,

where the xi and 'xi are coordinates of a transformable point and a trans­formed point and the a0 are parameters of the group. Our notation differs from the notation used by Lie and Cartan: in their time all indices were written as subscripts, but we write them as superscripts to be able to use tensor notation. In addition, the parameters a0 , b0 , and c0 defining two transformations and their product (the result of their successive realization) are connected by the relations

(2 .3 ) a a ( I r b ' b' ) c = </J a , . . . , a ; , . . . , .

The parameters a0 are chosen such that the values a0 = 0 correspond to h "d • � · I i i · i Ji ( I n 0 0) t e 1 entity trans1ormation x = x , 1 .e . , x = x , . . . , x ; , . . . , .

Next, Lie considered the "infinitesimal transformations'', i .e . , transforma­tions infinitesimally close to the identity transformation. They can be written in the form 'x i = xi + (8 //8a0)da0 (from here on, we shall adhere to the summation convention: whenever the same index symbol appears in a term of an algebraic equation both as a subscript and a superscript, the expression should be summed up over the range of that index) . If we denote da0 = e0d t and 8//8a0 = e! ' then the transformation (2 .2) can be written in the form

(2 .4) I i i a ):i d x = x + e .. t + · · · . a

If F(x 1 , x2 , . . . , xn ) is an arbitrary differentiable function, then dF (8F /8xi)dxi . But, by (2.4) , dxi = e0e!dt . Therefore,

(2. 5 ) dF = eaei 8F . . dt 0 8x'

40 2. LIE GROUPS AND ALGEBRAS

The last expression is a linear combination of the expressions XaF = e!aF /oxi . Thus to an infinitesimal transformation of a Lie group there

corresponds an operator X = ea Xa = eae!a /oxi , which is a linear combi­

nation of the basis operators x()t = e!a / axi . In particular, for the group Tn of translations

(2 .6) I i i i X = X + a

of the Euclidean space Rn we have r = n and e! = o//oaa = J! ( i .e . , 1 for i = a and 0 for i "I a) and x()t = J!a /{)xi ' i .e . , xi = {){)xi .

For the group On of rotations

(2 .7) I i i i x = uix

of the Euclidean space Rn , where U = ( u� ) are orthogonal matrices of order

n , we have r = n(n - 1 ) /2 . If, in a neighborhood of the identity element of the group On , we represent the matrix (u� ) in the form t5J + a� + · · · , where

a� are infinitesimals of the first order and the dots denote higher degree terms, and substitute these expressions into the condition of orthogonality Ek u;u� = Jii , then we obtain

"" k k k k L..)t5i + ai + . . . ) (Ji + ai + . . . ) = Jii " k

It follows from the last equation that the matrix a� is skew-symmetric: a� = -af . The elements of the matrix (a� ) with i < j can be taken as parameters of the group On . Since

I i i i i i i x = uix = x + aix + · · · ,

the infinitesimal operators Xa of the group O(n) can be written in the form

xij = xi{) j{)xi - xia ;axi . For the group of motions

(2 .8 )

of the space Rn we have r = n(n + 1 )/2 , and the operators Xa of this group

are Xi = {)/{)xi and Xii = xi{) /{) xi - xi{) /{) xi . The groups Tn and On

are subgroups of this group; moreover, T;, is an invariant subgroup. For the operators Xa of a transformation group, an operation of transition

from operators Xa and X P to their "Poisson bracket" is defined as

(2 .9)

§2.2. LIE GROUPS AND LIE ALGEBRAS 4 1

This new operation is anticommutative:

(2. 1 0)

and satisfies the "Jacobi identity":

(2 . 1 1 )

In addition, the bracket [XaXp] is a linear combination of the operators x,. :

(2. 1 2)

where the numbers c�P are constants, which are called the structure constants of the group. Lie called the set of operators X = ea Xa with the operation (2.9) the "infinitesimal group" and proved that if an "infinitesimal group" is given, then it completely defines the group of transformations (2 .2) in a neighborhood of the identity transformation 'x i = xi , and the functions (2.2) are solutions of a certain system of differential equations.

At present, Lie groups are considered independently from their "represen­tations" in the form of a group of transformations of a certain space. They are considered as manifolds with a group structure in a neighborhood of the identity element in which coordinates aa are introduced. In this case, instead of the operators X = ea Xa the tangent vectors to this manifold with coordi­nates ea = daa /dt are considered. To lines a( t) and b(t ) emanating from the group identity element (one-parameter subgroups a( t 1 + t2 ) = a( t 1 )a ( t2 ) are usually taken) , there correspond tangent vectors e = {ea } and f = {]} and to their product a ( t)b( t ) there corresponds the sum e + f of the vectors. To the product a(t )b ( t)a- 1 ( t )b- 1 ( t) there corresponds the commutator [efJ , which is anticommutative:

(2 . 1 3 ) [efJ = - [fe]

and satisfies the Jacobi identity

(2 . 1 4) [e[fg]] + [f[ge]] + [g[efJ] = 0.

These properties are similar to properties (2 . 1 0) and (2. 1 1 ) of the opera­tors Xa . In particular, instead of considering the group Tn as the group of translations (2 .6) , one considers it as a group of vectors a with respect to addition. Similarly instead of considering the group On as the group of rota­tions (2 . 7 ) , one considers it as a group of orthogonal matrices, and the group of motions (2 .8) is considered as a group consisting of orthogonal matrices U and vectors a with multiplication defined by ( U , a) ( V , b) = ( U V , a + Uh) .

The "infinitesimal group" is a vector space with the operation [efJ , which may be considered as vector multiplication. At present, a vector space with multiplication is called an "algebra". Because of this, Hermann Weyl ( 1 885-1 955 ) in his paper The structure and representations of continuous groups

42 2. LIE GROUPS AND ALGEBRAS

( 1 935 ) [Wey4] suggested replacing the term "infinitesimal group" with the term "Lie algebra" which is universally accepted nowadays. In the case in which a Lie group is a multiplicative matrix group, the corresponding Lie algebra consists of the matrices A = (d U /dt)0 , where the derivative of the function U(t) is taken at the identity element of the group, which corre­sponds to the value t = 0 of the parameter. The commutator [AB] of two matrices A and B is connected with their usual product by the relation

[AB ] = AB - BA.

When a Lie group is a group of vectors a with respect to addition, the corresponding Lie algebra consists of the vectors e = (da/dt)0 , where the derivative of the function a is taken at the identity element of the group, and the commutator [ef] of any two vectors e and f is equal to 0. In the case of the group On of orthogonal matrices, the corresponding Lie group

consists of skew-symmetric matrices A = (a� ) , a{ = -a� .

§2.3. Killing's paper

The paper by W. Killing, which determined the subject of Cartan's thesis, was published in the journal Mathematische Annalen under the title Continu­ous .finite transformation groups [Kil2] . Wilhelm Killing ( 1 847- 1 923) , a stu­dent of Karl WeierstraB ( 1 8 1 5- 1 897) , was very familiar with the WeierstraB theory of elementary divisors and normal form of matrices. In his doctoral dissertation, which was defended in Berlin in 1 872, he successfully applied these theories to the investigation of mutual disposition of two quadrics ( sur­faces of second order) in a projective space. (This problem is equivalent to the problem of classification of quadrics in a non-Euclidean space.) Based on the recommendation of WeierstraB, Killing became Professor of Mathematics in the Catholic Lyceum Hosianum in the city of Braunsberg in Eastern Prussia (now this city, which is located in Olsztyn wojewodstwo in Poland, has re­turned to its original name Braniewo) . The lyceum was a college for training Roman Catholic clergy, founded in 1 565 by Polish bishop Stanislaus Hosius ( 1 504- 1 579) . When Killing became a professor of this college, he joined the holy order of tertiaries (the biography of Killing written by P. Oellers [Oel] has the subtitle "The university professor in tertiary cloth") .

In Braunsberg, Killing continued his mathematical research. Following WeierstraB's advice, he studied the problem of space forms building on the work in the well-known papers of Klein and William Kington Clifford ( 1 8 34-1 879) . This problem brought Killing to consider infinitesimally small mo­tions. In 1 884, Killing published in Braunsberg the program titled Extension of the notion of space [Kil l ] , in which he, independently of Lie, arrived at the notions of Lie group and Lie algebra and posed the problem of classification of real simple Lie groups. Killing sent this program to Klein . In turn, Klein

§2.3. KILLING'S PAPER 43

informed Killing that his friend Sophus Lie was studying similar problems in Christiania (at the beginning of the 20th century its ancient name Oslo was returned to this city) and gave Lie's address to Killing. Upon Killing's re­quest, Lie sent preprints of his own papers to Killing. When Killing found out that Lie had not studied the problem of classification of simple Lie groups, he returned these reprints to Lie. Lie was offended by this rather fast return of reprints, and his relations with Killing were spoiled for good.

Killing started to correspond with Engel, a colleague of Klein and Lie. Engel helped him publish his paper [Kil2] in Mathematische Annalen. After a wrong statement on the "groups of zero rank" was discovered in this pa­per, Engel advised Killing to assign the correction of this mistake to one of Killing's students. After Killing replied that in Braunsberg he did not have any students in mathematics, Engel assigned this task to his own student, Umlauf, who in 1 89 1 successfully defended his dissertation on this subject (there is no further information on Umlaurs subsequent life and works) .

In the paper [Kil2] Killing did not solve the problem of classification of real simple Lie groups but solved the simpler problem of classification of complex groups of this type. In this paper he applied the theory of eigenvalues of matrices with which he was very familiar and showed that in addition to the four infinite series of groups which were discovered by Lie, there are five more "exceptional" simple groups of dimensions 1 4, 52, 78 , 1 33 , and 248. His paper [Kil2] is a very important event in the development of mathematics. Albert John Coleman (b. 1 9 1 8) [Col2] even considers it as "the greatest mathematical paper of all time".

The four infinite series of complex simple Lie groups discovered by Lie are: the group of collineations of a complex projective space CP

n ; the group

of motions of a complex non-Euclidean space csn

, i .e . , a subgroup of the previous group which leaves fixed a quadric aiJx

i xj

= O(aij = aji ) ; and the "group of a linear complex" (Komplex-Gruppe) , i .e . , a subgroup of the group of collineations of the space CP2n- I , which leaves fixed a linear complex of straight lines, i .e . , a set of straight lines whose Plucker coordinates /j =

xiyj - xj/ satisfy the equation aiJ/j = 0 (aij = -aji ) . Killing named the

groups of the first series "the system A'', the groups of the second series for even n "the system B" and for odd n "the system D'', and the groups of the third series "the system C". The groups of "system A" are locally isomorphic to the groups CSLn+ i of complex unimodular matrices of order n + l . The groups of "system B" are locally isomorphic to the groups C02n+ 1 of complex orthogonal matrices of order 2n + l . The groups of "system C" are locally isomorphic to the groups CSp2n of complex symplectic matrices of order 2n . The groups of "system D" are locally isomorphic to the groups C02n of complex orthogonal matrices of order 2n .

The term "symplectic" was introduced by Weyl in his book [Wey5] . In his lectures [Wey4] he translated the term "Komplex-Gruppe" as "complex

44 2. LIE GROUPS AND ALGEBRAS

group". However, these words also denote any Lie group with complex pa­rameters. Because of this, he suggested the groups of the third series be called "symplectic groups". The word "symplectic" originates from the Greek word "symplektikos'', which has the same meaning as the Latin word "complexus"­complex. The complex dimensions of the groups CSLn+ I , C02n+ I , CSp2n , and C02n are equal to n(n + 2) , n (2n + 1 ) , n (2n + 1 ) , and n (2n - 1 ) , respectively.

Killing considered the problem on eigenvalues of a linear operator gener­ated in the "infinitesimal group" by its fixed element ea Xa . Its action on an arbitrary element X = ;..a Xa has the form

(2 . 1 5 )

and the eigenvectors are defined by the equation

[ea Xa , X] = wX ,

which can be written in coordinate form as

(2 . 1 6) ea cy ;..P - wJ..Y ap - .

The eigenvalues of this operator are roots of the equation

(2 . 1 7) Ll(w) = det(eaC�p - wo; ) = 0 ,

which is called the characteristic equation of a Lie group. This equation can be written

r r- 1 r-2 r- 1 r (2 . 1 8 ) OJ - l/f1 0J + l/f20J - · . . + (- 1 ) l/f,_ 1 w + (- l ) l/f, = 0 ,

where the I/I a are homogeneous functions of the parameters ea :

(2 . 1 9)

P a I ( Y o y o ) a p 1/11 = Cape ' 1/12 = 2 CayCpo - CpoCay e e '

1 (c;o c;o c�o) a P Y 1/13 = -3, det cl!.6 cp6 c/J.C e e e , . . . . . o e { cYC cy, cy,

Killing defined as the group rank the number of functionally indepen­dent coefficients 'Ila of the characteristic equation. For the groups CSLn+ I , C02n+ I , CSp2n and C02n the rank is equal to n . Killing showed that for

a simple group the same operator ;..P Xp is an eigenvector for all matrices

of linear transformations (2 . 1 5 ) corresponding to the operators ea Xa from the subgroup of zero rank which contains the infinitesimal transformation of general type. However, Killing's proof of this fact was invalid. From this statement, which was proved later, Killing showed that for simple groups the

§2.4. CARTAN'S THESIS 45

characteristic equation can be written in the form

a/ IT(w - a(h) ) = 0 ,

where the roots a (h ) are linear functions of infinitesimal transformations h from the subgroup of zero rank. Next, considering different possible combi­nations of these roots, Killing gave the classification of complex simple Lie groups.

Killing denoted complex simple Lie groups by Roman numerals equal to the group rank and by one of the capital letters A , B , C , D , E , and F . He found the isomorphisms of the simple groups IA , IB , and IC of dimen­sion three as well as of the simple groups HB and HC and of the simple groups IHA and HID of dimension 1 5 and proved that the group HD is not simple and consists of two groups IA . For the exceptional groups that he discovered he used the notation IC (since the group of series C of rank 2 is isomorphic to the group HB) , IVE , VIE , VHE , VHIE , and IV F and proved that the dimensions of these groups are equal to 1 4, 52, 78 , 1 33 , 248, and 52 respectively. (Thus, Killing assumed that there are two nonisomor­phic complex simple groups of rank four and dimension 52 . ) Killing called groups which are composed of a few simple groups semisimp/e groups.

§2.4. Cartan's thesis

As early as in his note The structure of simple finite continuous groups [ 1 ] ( 1 893) , Cartan, noting "exceptionally important results" of the Killing paper, indicated: "Unfortunately, in the considerations which led Mr. Killing to these results, the rigor is missing. Therefore, it is desirable to perform this research again, indicating which of Killing's theorems are inaccurate and proving those of his theorems that are C<?rrect" [ 1 , p. 784-785] . This work was performed by Cartan in his thesis [5 ] .

Cartan's research was concerned with those Lie groups which, following Killing, he called "semisimple groups". However, he defined these groups as the groups not possessing a solvable invariant subgroup. The groups satisfying this definition are semisimple in the sense of Killing's definition. Note that all noncommutative simple Lie groups are semisimple and that commutative simple Lie groups, namely, the one-parameter group of translations and the group 02 (which can be considered as the group ID) , are not semisimple. Cartan showed that, when the coefficient 1/11 (e) = 0 , the form 2 1/f2 (e) has the form

( t5 JI a p

- 21/12 e ) = CaJICPt5e e .

When the form 1/11 (e) is not zero, the expression on the right-hand side of this equation can be written in the form

46 2. LIE GROUPS AND ALGEBRAS

(2 .20) 2 t5 l' a P qJ (e) = Vii - 2 1/12 = c0,.cp6e e ,

The condition that a Lie group be semisimple is the nondegeneracy of the form qJ (e) , and the condition that a Lie group be solvable is the vanishing of this form (the vanishing of this form for commutative groups is obvious since in this case c�P = 0) . Since the forms (2. 1 9) were introduced by Killing and the value of the form qJ (e) in deciding whether a Lie group is semisimple or solvable was discovered by Cartan, this form is called the "Killing-Cartan form". Since for semisimple groups the form qJ (e) is a nondegenerate quadratic form (the metric in the Lie algebra in which the square of the length of the vector e is equal to the value of this form for this vector) , in the Lie algebra of a complex semisimple Lie group, this form defines the metric of a complex Euclidean space CR' . Moreover, since a Lie algebra can be considered as the tangent space to a Lie group at its identity element, this form defines the metric of the complex Riemannian manifold CV' in the complex semisimple group itself. At present this Riemannian metric in semisimple Lie groups is called the Cartan metric.

Cartan gave a rigorous proof of the fact that the "subgroup of zero rank" of a semisimple Lie group is commutative and can be considered as a set of group elements that commute with a general element ("regular element") of the group. Because of this fact, at present this subgroup is called the Cartan subgroup of a semisimple Lie group, and the subalgebra of the Lie algebra corresponding to this subgroup is called the Cartan suba/gebra.

Cartan slightly changed Killing's notations of simple Lie groups: he sug­gested that groups in the classes A , B , C , and D of rank n be denoted by An , Bn , Cn , and Dn , respectively, and the groups VIE , VIIE , and VIIIE by E6 , E7 , and Es . In addition, he proved that the group IVE is isomor­phic to the group IV F and suggested that these two groups be denoted by F4 and the group IIC by G2 •

Thomas Hawkins (b. 1 938 ) in [Haw3] made a thorough comparison of the Cartan thesis with the Killing paper [Kil2] . He noted all instances where Cartan corrected errors or omissions of Killing. In particular, he noted that while considering the group Es , Cartan, who was a skilled and intrepid calcu­lator, checked the Jacobi identities for all e;s) = 2 ' 5 1 1 ' 496 combinations of the basis elements of the Lie algebra of this group taken three at a time, and Killing did not accomplish this.

§2.5. Roots of the classical simple Lie groups

We see that Killing gave to the word "root'', already heavily used in math­ematics, one more very important meaning. The word "root" first appeared in the works of medieval Arab mathematicians. They translated the San­skrit word "pada", whose meaning is the base of a wall or the root of a tree,

§2.5 . ROOTS OF THE CLASSICAL SIMPLE LIE GROUPS 47

as "j idhr'', whose meaning is the root. The Indians used the word pada as a translation of the Greek word "basis'', which was used by Pythagoreans for bases "square numbers" (they represented these numbers in the form of squares) . The Arabs began to use the word "j idhr" not only for nota­tion of roots of numbers, i .e . , roots of the equations xn = a , but also for notation of roots of any algebraic equation and notation of unknown quan­tities. European mathematicians who wrote in Latin began to use the Latin translation of this word, "radix", and mathematicians who wrote in Ger­man, French, and English used the words "Wurzel", "racine", and "root", respectively, with the same meaning. This application of botanical terms in mathematics inspired one of the founders of projective geometry, Gerard Desargues ( 1 593- 1 662), to use such terms as "trunk", "branch", "shoot", "tree", "stump", and "involution"-the twisted form of young leaves. Only the latter term was widely used later and in significantly wider meaning than in Desargues's works. Recently in mathematics the term "tree" has been used in the sense of a connected graph without cycles and the term "forest" in the sense of a disconnected graph without cycles, i .e . , a set of "trees".

The botanical term "root" of Killing was added to the system of terms similar to the Desargues system by Hans Freudenthal ( 1 905- 1 990) and H. de Vries (b. 1 932) in the book Linear Lie groups [FdV] ( 1 969) . In this book, Freudenthal and de Vries used the word "trunk" for the Cartan subgroup of a Lie group and the Cartan subalgebra of a Lie algebra, the word "branches" for eigenvectors corresponding to the roots, and the word "nodes" for the commutators of branches corresponding to opposite roots.

For the group CSLn+ I , the Cartan subgroup consists of those diagonal matrices (ea op) , for which ITa e

a = 1 , and the corresponding Cartan subal­gebra consists of those diagonal matrices (ha op) , for which Ea h

a = 0 . The eigenvectors of the linear transformation x --+ [h , x ] are the matrices Eap having 1 in the intersection of the ath row and the Pth column and zeros in all other places. If we denote h = EY h

y En , then

[h , Eap ] = L hy EyyEap - L hy EapEyy = (ha - hp)Eap ·

y y

Thus, the eigenvalue corresponding to the eigenvector E ap of the linear trans­formation x --+ [h , x ] is a linear form on the Cartan subalgebra, whose value on a vector h of this subalgebra is ha - hp . Because of this property, we will denote this linear form by wa - wp .

In the case of the group C02n , which consists of the matrices preserving

the quadratic form EA xA x2n-A- I (here and further A. = 0 , 1 , . . . , n - 1 ) ,

the Cartan subgroup consists of the diagonal matrices (e"op ) , for which

e2n-A- I = (i) - 1 , and the Cartan subalgebra consists of the diagonal ma­trices (ha op) , for which h2n-A- I = -hA . As in the case of the groups

48 2. LIE GROUPS AND ALGEBRAS

CSLn+ l , it follows that the eigenvalues of the linear transformation x -+

[h , x] corresponding to the eigenvectors EAµ , EA , 2n-µ- I , E2n-A- I , µ , and A µ A µ A µ d A µ E2n-A- I , 2n-µ- I are w - w , w + w , -w - w , an -w + w , respec-

tively. For the group co2n+ I ' which consists of the matrices preserving the qua­

dratic form EA xAx2n-A + (xn ) 2 , the Cartan subgroup consists of the diagonal

matrices (e0c5p) , for which e2n-A = (i) - 1 , en = 1 , and the Cartan subalge­

bra consists of the diagonal matrices (h0c5p ) , for which h2n-A = -hA , hn = 0 . It follows from this that the eigenvalues of the linear transformation x -+

[h , x] corresponding to the eigenvectors EAµ • EA , 2n-µ , E2n-A , µ , E2n-A , 2n-µ , E E E d E A µ A µ A µ A , n ' 2n-A , n ' n , A ' an n , 2n-A are W - W • W + W • -W - W •

A µ A A A d A ' l -w + w , w , -w , -w , an w , respective y. In the case of the group CSp2n , which consists of the matrices preserving

the bilinear form EA(xAy2n-A- I - /x2n-A- I ) , the Cartan subgroup con­

sists of the diagonal matrices (e0c5p) , for which e2n-A- I = (i)- 1 , and the Cartan subalgebra consists of the diagonal matrices (h0c5';) , for which

h2n-A- I = -hA . It follows that the eigenvalues of the linear transformation x -+ [h , x] corresponding to the eigenvectors EAµ , EA , 2n-µ- I , E2n-A- I , µ , E2n-A- 1 , 2n-µ- 1 (A. =I µ) • EA , 2n-A- I • and E2n-A- l , A are WA - Wµ , WA + wµ , -WA - wµ , -WA + wµ , 2wA , and -2o/ , respectively.

Thus, in the case of simple groups .An , Bn , en , and Dn , the roots of their characteristic equations can be written as:

(2 .2 1 ) An : Bn : en : Dn :

WA - wµ (EA WA = 0) '

±WA ± Wµ ' ±WA ' ±WA ± Wµ ±2WA ' ' ±WA ± Wµ .

Killing noticed that all these roots are linear combinations with integer coefficients of a certain number of forms composing a basis, that these coef­ficients can take only the values ± 1 , ±2 , and ±3 , and that the number of forms in this basis is equal to the group rank. We can also find a basis formed by the roots of a semisimple group. For the simple groups An , Bn , en , and D n , the roots composing the latter basis are:

(2.22) An : Bn : en : Dn :

0 I a l = w - w ' 1 2 a l = w - w ' I 2 a l = w - w ' I 2 a l = w - w ' n- 1 n an = w + (J) •

1 2 a2 = w - w ' . . . 2 3 a2 = w - w ' . . .

2 3 a2 = w - w ' . . . 2 3 a2 = w - w ' . . .

n- 1 n ' an = W - W ; n- 1 n n ' an- I = w - W ' an = W ; n- 1 n 2 n , an- 1 = W - (J) , an = W ; n- 1 n • an- I = (J) - W

a)

§2.5 . ROOTS OF THE CLASSICAL SIMPLE LIE GROUPS

82

b)

FIGURE 2. 1

C2 2cil2

-20Y­c)

49

D2

d)

The Cartan-Killing theory was significantly simplified by Weyl in the paper Theory of representations of semisimple continuous groups by linear transfor­mations ( 1 925) [Wey3] . Developing the results of Weyl's paper, van der Waerden in his paper The classification of simple Lie groups [Wae] ( 1 933) introduced a very visual representation of the roots of a simple Lie group by vectors of the Euclidean space Rn . The possibility of such a representation follows · from the fact that the Cartan metric in a complex simple Lie group determines the metric of the complex Euclidean space R' in the Lie algebra of this group, and the Cartan subalgebra is an n-dimensional plane in this space; i .e . , this algebra is the space CRn . Since the roots of a simple Lie group are linear forms in the Cartan subalgebra, they can be represented by the vectors of the space CRn , and since all the roots are linear combina­tions with integer coefficients of n linearly independent forms, these roots can also be represented by vectors of the real Euclidean space Rn . In this representation, a root aA. a/ is represented by a vector a with coordinates aA. . Figure 2 . 1 shows such systems for the groups A2 , B2 , C2 , and D2 •

A further simplification of the classification of complex simple Lie groups was made by Eugene B. Dynkin (b. in 1 924) in his paper [Dy 1 ] of 1 946 under the same title as the van der Waerden paper [Wae] (see also [Dyn2] ) . The paper [Dyn 1 ] was written in 1 944 when the author was 19 years old; Dynkin followed the advice of Gel' fand, whose seminar he participated at . that time.

Dynkin introduced the notion of "simple roots" of semisimple Lie groups. If we write the roots as linear combinations aA. a/ , A. = 1 , 2 , . . . , n or A. = 0 , 1 , 2 , . . . , n , with integer or rational coefficients aA. , we will say

that aA. a/ > 0 if the first nonzero coefficient aA. is positive and that a root

a = aA.a/ is greater than a root b = bA.a/ if the difference a - b is positive. A root is called simple if it is positive and cannot be represented as the sum of other positive roots. Any positive root can be represented as the sum of simple positive roots with positive coefficients. The Cartan-Killing basis roots considered above are simple roots in Dynkin's sense.

50

<X1 a) An

<X1 b) Bn

c) en

d) Dn

2. LIE GROUPS AND ALGEBRAS

<Xi <X3

<X2 <X3

FIGURE 2.2

- - - --0--0

a.n-1 an - - - -Cl > 0

Dynkin introduced a very simple representation of systems of simple roots in the form of graphs in which simple roots are represented by the graph dots. These dots are not joined if the corresponding vectors are orthogonal, they are joined by a line if the angle between vectors is 1 20° , and they are joined by a double line if the angle between vectors is 1 3 5° . In his papers [Dyn l ] and [Dyn2] , Dynkin indicated the lengths of vectors representing the roots by special marks next to the corresponding dots. Later, in the 1 950s, he represented the longer vectors by black dots and the shorter vectors by white dots. Lev S. Pontryagin ( 1 908- 1 988) in his book Topological groups [Pon2] , used the Dynkin graphs* , but he did not show the lengths of vectors. Jacques Tits (b. 1 930) , in the paper On certain classes of homogeneous spaces of Lie groups [Ti l ] ( 1 955 ) , suggested, in the case when the vectors representing the roots have different lengths, putting the sign > in the direction of the dot representing the vector of smaller length. At present, the majority of mathematicians use the Dynkin graphs in the form suggested by Tits, and the white and black dots are used in the modification of the Dynkin graphs suggested by Ichiro Satake (b. 1 927) for another purpose. Nevertheless, Joseph A. Wolf (b. 1 936 ) , who in the first editions of his book Spaces of constant curvature [Wo2] used the the Dynkin graphs in the Tits form, in the last edition of this book, returned to the form of these graphs used by Dynkin in the 1 950s. Dynkin himself called his graphs "schemes of angles". Tits in the paper [Ti l ] called them the "Schlafli figures". Wolf, in the book [Wo2], used the term the "Schlafli-Dynkin diagram". Because of a similarity of the Dynkin graphs with the Coxeter diagrams for groups generated by reflections (we will discuss these groups later) , these graphs are sometimes called "Coxeter-Dynkin graphs". Figure 2 .2 shows the Dynkin graphs in the Tits form for the complex simple groups An , Bn , Cn , and Dn .

•Editor 's note. Or diagrams. Same for Coxeter and Satare graphs.

§2.7 . ROOTS OF EXCEPTIONAL COMPLEX SIMPLE LIE GROUPS 5 1

§2.6. Isomorphisms of complex simple Lie groups

Killing noted the isomorphisms between some complex simple Lie groups: the isomorphism of the groups A 1 , B1 ; and C1 , the isomorphism of the groups B2 and C2 , and the isomorphism of the groups A3 and D3 in addtion to the fact that the group D2 is not simple and is isomorphic to the direct product of two groups A 1 • Since Killing actually considered not Lie groups but their Lie algebras, the isomorphism of groups stated by him is in reality a local isomorphism.

In the case when two simple groups are isomorphic or locally isomorphic, the vector systems representing their roots or the Dynkin graphs of these groups are similar. For the groups A 1 , BL , and C1 , the vector systems consist of two opposite vectors a and -a and the Dynkin graphs consist of one point alone (Figure 2. 3a) ; for the groups B2 and C2 , the vector systems have the form shown in Figures 2. 1 b and 2. 1 c, and the Dynkin graphs have the form shown in Figure 2. 3c; and for the groups A3 and D3 , the Dynkin graphs have the form shown in Figure 2 . 3d.

In the case of a semisimple group, which is a direct product of a few simple groups, the vector systems of root systems consist of a few systems of vectors for simple groups located in mutually orthogonal subspaces, and the Dynkin graphs consist of a few Dynkin graphs for simple groups. An example of the latter group is the group D2 , which is isomorphic to the direct product of two groups A 1 = B1 = C1 (Figures 2. l d and 2 .3b) .

al al � a1 a.z a3 0 B2 0 �� 0 A3 0-----0---0 D2

0

o � C2 �� D3 �

a) b) c) d)

FIGURE 2. 3

§2.7. Roots of exceptional complex simple Lie groups

The complete systems of roots of characteristic equations of the simple Lie groups in the five "exceptional classes" G2 , F4 , E6 , E1 , and £8 can be written as: (2.23 )

(Ji - (J/ ' ±al ,

± Ei(al - 3a/) , i , j = O , 1 , 2 ;

±Q i - ±a/ , ! (±w1 ± w2 ± w3 ± w4) ,

i , j = l , 2 , 3 , 4 ; ±./'J.wo ' ! ( f wo + !wi - wi - wk - w' ) '

i , j , k , 1 = 1 , 2 , . . . , 6 ;

S2 2. LIE GROUPS AND ALGEBRAS

FIGURE 2.4

i j c I "' i h j k I ) E1 : w - w • ± 2 L..., ; w - w - w - w - w ' h , i , j , k , l = O , 1 , . . . , 7 ;

Es : ±w; ± wi ' ±C t L; w; - wi) ' ±C t L; w; - wi - wk - wt) ' i , j , k , I = 1 , 2 , . . . , 8 .

The systems of simple roots for these groups have the form:

I 2 a l = w - w ' 2 3 a l = w - w '

I 2 a l = w - w ' s 6 as = w - w ' I 2 a l = w - w ' s 6 as = w - w , 2 3 a l = w - w ' 7 s a6 = w - w ,

a2 = wo + w1 - 2w2 ; 3 4 4 a2 = w - w ' a3 = w '

I ( I 2 3 4 ) a4 = 2 w - w - w - w ; 2 3 a2 = w - w ' 3 4 4 s a3 = w - w ' a4 = w - w '

a6 = {/-wo - t Cw1 + w2 + w3 - w4 - ws - w6 ) ; 2 3 3 4 4 s a2 = w - w ' a3 = w - w ' a4 = w - w '

a6 = t (wO _ W I _ ())2 _ (JJ3 + ())4 + (J)S + ())6 + W1 ) ; 3 4 4 s s 6 a2 = w - w ' a3 = w - w ' a4 = w - w 6 7 as = w - w '

a7 = t cw l - ())2 - (JJ3 - ())4 + (J)s

+ ())6 + ())7 + ws ) ' 7 s as = w + w . For these groups there are also the vector systems of root systems and the Dynkin graphs: Figure 2.4 represents the vector systems of root systems for the group G2 , and Figure 2 . 5 the Dynkin graphs for all five exceptional simple Lie groups.

In the case of the group G2 , the angle between vectors representing simple roots is 1 50° : in this case the corresponding dots of a diagram are joined by a triple line. Van der Waerden in his paper [Wae] showed that the vec­tors representing a root system of a simple Lie group can form only the angles 90° , 60° , 45° , 30° , 1 20° , 1 3 5° , 1 50° , and 1 80° ; the lengths of non­orthogonal vectors are in no way related to each other, the lengths of vectors forming the angles 45° and 1 35° are related by b2 = 2a2 , and the lengths of vectors forming the angles 30° and 1 50° are related by b2 = 3a2 • These

§2.8 . THE CARTAN MATRICES 53

a) c) E6

~ d) e) E8

<X.1 <Xi <X.3 <X.4 <X.5 a6 �

FIGURE 2 .5

results give a rather simple method of classification of simple Lie groups. The method of classification, used by Killing and Cartan and based on com­putation of determinants, is much more complicated.

Since in Cartan's thesis Lie groups were considered as transformation groups, in this work he also gave a representation of the exceptional sim­ple Lie groups in the form of transformation groups: these groups are rep­resented there as certain subgroups of projective transformations. In what follows, we will present simpler geometric realizations of these groups based on Cartan's later results.

§2.8. The Cartan matrices

In the calculations of Killing and Cartan, the integers aii appeared often. These integers can be defined with the help of the inner products of the vectors representing simple roots a; as follows:

2a; , ai (2 .25) a . . = . I) (aj ' O!;) Cartan called these numbers the "fundamental integers". Possibly this is

the reason why at present these matrices are called "Cartan matrices". For­mula (2 .25) shows that all diagonal entries of the Cartan matrices are equal to two, and all nondiagonal entries of these matrices are nonpositive, and, in general, these matrices are not symmetric. For the groups An , Bn , Cn , and D n these matrices have the form:

(2.26)

( 2 - 1 0 · · · 0 OJ - 1 2 - 1 . . . 0 0

· ·�· · · ·::ii · · · · · ·� · · · '. ·:· : · · · · · ·� · · · ·�·i '

0 0 0 . . . - 1 2 ( 2 - I 0 . . · 0 OJ - I 2 - 1 . . · 0 0

· · ·6· · . . ::ii · · · · · ·� · · · : ·:· : · · · · · ·� · · · ·�i '

0 0 0 . . . - 2 2

( 2 - 1 0 . . · 0 OJ - 1 2 - 1 . . . 0 0

· ·� · · · ·::ii · · · · · ·� · · · : ·:· : . . · · · ·� · · · ·�·� '

0 0 0 . . . - 1 2 (:L . . �L . . :. � � . . . . . . � . . . . . . ?. . . . � . . . . . . ?J 0 0 . . . - 1 - 1 2 - 1 0 0 . . . 0 - 1 2 0 0 0 . . . 0 - 1 0 2

54 2. LIE GROUPS AND ALGEBRAS

and for the groups G2 , F4 , E6 , E7 and E8 , the Cartan matrices are:

(2 .27)

- 1) (-i 2 ' 0

0

-� -� �) (J - I 2 - I ' 0 0 - I 2 0

0

J ] =i J !J ' 0 0 - I 2 - I 0 0 0 - I 2

2 0 - I 0 0 0 0 2 0 - 1 0 0

- I 0 2 - I 0 0 0 - I - 1 2 - 1 0 0 0 0 - I 2 - I 0 0 0 0 - 1 2 0 0 0 0 0 - 1

0 0 0 0

' 0

- I 2

2 0 - I 0 0 0 0 0 0 2 0 - I 0 0 0 0

- I 0 2 - I 0 0 0 0 0 - I - 1 2 - I 0 0 0 0 0 0 - 1 2 - I 0 0 0 0 0 0 - 1 2 - I 0 0 0 0 0 0 - I 2 - I 0 0 0 0 0 0 - 1 2

Note that for the groups An , the determinants of the Cartan matrices are equal to n + 1 , for the groups Bn and en they are equal to 2, for the groups Dn they are equal to 4, for the groups G2 , F4 and E8 they are equal to 1 , for the groups E6 they are equal to 3, and for the groups E7 they are equal to 2. These numbers, which are called the connection indices, determine important algebraic and topological properties of these groups.

In the theory of simple Lie groups, the inverse matrices of the Cartan matrices play an important role. For the groups An , Bn , en and Dn , these matrices have the following form:

(2 .28) n n- 1 n-2 3 2 I

n+ I n- 1 n+ I 2n-2 n+ I 2n-2 nt l n4 1 ni l n+ I n-2 n+ I 2n-4 n+ I 2n-6 n9 1 nt l nj l .'!t. � . . • • . '?t. ! . . . . . !'.t.1 . • • • • • . • • • • • • �t. ! . . . . . !!t.1 • • . • • • '!t.1• 2 nt l n+ I

4 6 2n-4 2n-2 ni l nj l n+ I n- 1 n+ I n- 1 n+ I n+ I n+ I n+ I 1 1 1 1 1 1 2 2 . . . 2 2 1 2 3 . . . 3 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 n - 1 n - 1 I 1 3 n- 1 n 2 2 -2- 2

I 2 I

1 1 2 2 2 3

n- 1 n+ I n n+ I

I 2 3

I 2 1

I 2 I

1 1 2 2

1 1 2 2 2 3

1 2 3

1 3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 n - 1 2 3 . . . n - 1

n- 1 -2-n- I -2-I t 2

2 3 3 2 3 2

n - 2 n-2 n-2 -2- -2-n-2 n n-2 -2- 4 -4-n-2 n-2 n -2- -4- 4

§2.9. THE WEYL GROUPS 55

and for the groups G2 , F4 , E6 , E7 and E8 , these matrices have the form:

(2.29)

2 2 3 4 2 ! 4 6 3 4 6 8 4 6 8 1 2 3 t 6 9 2 3 4 6 1 ! 2 3

3 9 2 6 9

2 1 3 ! 4 2 6 3

Ii 5 � 5 4 2 5 2 2 !

� 1 i 2 1 2 2 3 i 2 130 4 2 3 4 6

4 3 2 8 3 4

2 3 1 4 3 2

4 2 8 4 1 0 5 l l 2 � l 3 3 3 3

4 5 7 1 0 8 6 5 8 10 1 5 1 2 9 7 1 0 1 4 20 1 6 1 2

1 0 1 5 20 3 0 24 1 8 8 1 2 1 6 24 20 1 5 6 9 1 2 1 8 1 5 1 2 4 6 8 1 2 1 0 8 2 3 4 6 5 4

4 2 6 3 8 4

1 2 6 1 0 5 8 4 6 3 3 2

If we denote by 1C; the vectors 2aJ(a; , a; ) and by 1C; the vectors of the

basis dual to the basis (1C; ) (i .e. the inner products (7r; , 1Cj) are equal to of ) , then the Cartan integers are equal to the coordinates of the vectors a; in the basis (7r ; ) , i .e.

(2. 30) i aj = aij1C ,

and the coordinates of the vectors 1C; in the basis (a;) are equal to the

entries Aij of the matrix A- 1 = (Aij ) which is the inverse matrix of the Cartan matrix, i .e. ,

(2. 3 1 ) i ij 1C = A ar The integer multiples of the vectors a; define the root lattice of a simple

Lie group, and the integer multiples of the vectors 7C ; define the weight lattice of this group. These lattices are discrete additive groups of vectors; the first of these groups is a subgroup of the second one, and the order of the quotient group of the second group by the first one is equal to the connection index of the Lie group.

§2.9. The Weyl groups

Killing in the paper [Kil2] and Cartan in his thesis considered transforma­tions of the root systems of complex simple Lie groups. Since these roots are roots of the characteristic equation (2. 1 8) of the group, these transformations can be considered as elements of the Galois group of this equation. This was why Cartan called the group of these transformations the "Galois group of

56 2. LIE GROUPS AND ALGEBRAS

the Lie group". Both Killing and Cartan connected an involutive substitution S0 of the system of roots with every root a , considered products of these substitutions, and wrote these substitutions and their products in the form of linear transformations where the linear transformations corresponding to the substitutions S0 have the form of reflections.

Weyl in his paper [Wey3] showed that the transformations corresponding to the substitutions S0 can be written in the form

(2. 32) J:' _ J:

_ 2(� , a )

.. - .. a , (a , a )

and, in the metric of the Euclidean space Rn in the Cartan subalgebra, these transformations are reflections in hyperplanes of Rn orthogonal to the vec­tors a . Moreover, the linear transformations corresponding to the products of the substitutions S0 are rotations of the space Rn , i .e. the matrices of these transformations are orthogonal matrices of the group O

n . The prod­

ucts of n reflections S0 corresponding to simple roots are especially im­portant. At present, these transformations are called the Coxeter transforma­tions. For any order of the factors S0 , the eigenvalues of the matrices of these transformations have the form eM; , where the numbers M; have the form 2na; / h , and the numbers a; are integers called the exponents of a simple Lie group, and the number h is the Coxeter integer of this group. The latter number is connected with the rank n of the group and its dimension r by the relation

(2 .33) r - n h = - . n

In the paper [Wey3] , Weyl also considered the group of rotations of the space Rn generated by these reflections. He used for this group the name "group (S) ". At present, this group is called the Wey! group.

If at the common initial point of the vectors a; (simple roots of a Lie group) , we construct the hyperplanes H; orthogonal to these vectors, then connected sets of points of the space Rn not belonging to the hyperplanes H; are called open Wey! chambers, and their closures are called closed Wey! chambers. The Weyl chambers have the form of cones with vertices at the common initial point of the vectors a; and with faces that are faces of an n-faced angle. The Weyl chambers are the fundamental domains of the Weyl group.

The exponents a; of simple Lie groups are also called the exponents a; of its Weyl group. These integers belong to the interval 1 � a; < h where h is the Coxeter integer (2 .33 ) . All integers of this interval that are relatively prime with h are integers a; = h - ah- i+ l . These integers are equal to:

§2.9. THE WEYL GROUPS

FIGURE 2 .6

(2 . 34) for the groups An : for the groups Bn and Cn : for the groups Dn (n is even) : for the groups Dn (n is odd) :

1 , 2 , 3 , . . . , n - 1 , n (h = n + l ) ; 1 , 3 , 5 , . . . , 2n - 1 (h = 2n) ; 1 , 3 , 5 , . . . , n - 2 , n - 1 , . . . , 2n - 3 ; 1 , 3 , 5 , . . . . , n - 3 , n - 1 , n - 1 ,

(2 . 35)

for the group G2 : for the group F4 : for the group E6 : for the group E7 : for the group E8 :

n + 1 , . . . , 2n - 3 (h = 2n - 2) ;

1 , 5 (h = 6) ; 1 , 5 ' 7 ' 1 1 (h = 1 2) ; 1 , 4 , 5 , 7 , 8 , 1 1 (h = 1 2) ; 1 ' 5 ' 7 ' 9 ' 1 1 ' 1 3 ' 1 7 (h = 1 8 ) ; 1 ' 7 ' 1 1 ' 1 3 ' 1 7 ' 1 9 ' 23 ' 29 (h = 30) .

57

As we will see later, the exponents of simple Lie groups play an important role in the most unexpected questions of the theory of simple Lie groups such as the topology of real simple Lie groups and the theory of finite groups which are the analogues of simple Lie groups.

Influenced by the Weyl paper [Wey3] , in 1 925 Cartan returned to the the­ory of roots of simple Lie groups and showed in the paper The duality princi­ple and the theory of simple and semisimple groups [82] that for all simple Lie groups, except the groups An , Dn and E6 , their Weyl groups coincide with the Galois groups, and the Weyl groups of the excluded groups are invariant subgroups of their Galois groups. Moreover, he proved that, for the groups An , Dn (n =f. 4) , and E6 , the quotient group of the Galois group by the Weyl group is isomorphic to the multiplicative group { 1 , - 1 } , and, for the group D 4 , it is isomorphic to the general group of permutations of three elements. This is connected with the fact that the Dynkin graphs of the groups An , Dn (n =f. 4) , and E6 possess bilateral symmetry and also with the "duality prin­ciple" of the spaces where the groups An , Dn , and E6 act, and with the fact that the Dynkin graph of the group D4 (Figure 2.6) possesses the trilateral symmetry and with the "triality principle" in the space where the group D 4 acts (the latter principle was introduced by Cartan in the paper [82] ) .

The finite groups of reflections of the space Rn generated by reflections in hyperplanes of this space were studied by Harold Scott MacDonald Coxeter (b. 1 907) in the paper Discrete groups generated by reflections [Cox l ] ( 1 934) where he gave a complete classification of these groups and characterized them by means of graphs whose structure is very close to the Dynkin graphs which appeared ten years later. The vertices of the Coxeter graphs represent

58 2. LIE GROUPS AND ALGEBRAS

(3" - I ] 0----0----0-- . . . -0----0--0

[311 - 2• 4] 0----0----0-- . . . -0----0--0 4

13n - 3. I . I ] 0----0----0-- . . . � [n] 0--0 n

[3, 5] 0---0---0 5 [3, 4, 3] � [3 , 3, 5] 0--0--------0--

FIGURE 2 .7

hyperplanes of the space R" , reflections in which generate the group. The vertices are not joined if corresponding hyperplanes are orthogonal. They are joined by a line without a numerical mark if the angle between these hyperplanes is 60° and with the mark n if this angle is 1 80° /n . Figure 2. 7 shows the Coxeter graphs of finite groups generated by reflections. If the Coxeter graphs of these groups consist of lines without marks, the groups are denoted by [3" ] if the graph consists of n lines and does not have branches and by [31

• m

• " ] if the graph consists of three branches having I , m , and n lines, respectively. If the Coxeter graph consists of a few lines with marks k , I , and m (the absence of mark is counted as the mark 3) and does not have branches, the group is denoted by [k , I , m] . If the graph consists of graphs of different types, the group notation consists of the notation of the corresponding groups.

The group [3"- 1 ] is the group of symmetries of the regular n-dimensional simplex of the space R" . The group [3"-2 , 4] is the group of symmetries of the n-dimensional cube of the same space. The group [3"- 3

• 1

• 1 ] is the group

of symmetries of the n-dimensional "semicube", i .e . , the convex polytope obtained from the n-dimensional cube by selection of one vertex on each edge and rejection of the other vertex of this edge. The group [n] is the

§2.9. THE WEYL GROUPS 59

group of symmetries of a regular n-angle. The group [3 , 5] is the group of symmetries of an icosahedron and a dodecahedron. The group [3 , 4 , 3] is the group of symmetries of a regular polytope of the space R4 with 24 faces whose vertices are 1 6 vertices of the four-dimensional cube and the reflections of its center of symmetry in its eight faces. The group [3 , 3 , 5] is the group of symmetries of regular polytopes of the same space R4 with 600 and 1 20 faces. The group [32 ' 2 ' 1 ] is the group of symmetries of a cubic surface with 27 rectilinear generators in the projective space P3 • The group [33 ' 2 ' 1 ] is the group of symmetries of a quartic (a curve of the fourth order) with 28 double tangents in the plane P2 • The group [ 34 ' 2 ' 1 ] is a subgroup of the group of permutations of a set of 1 20 elements which was called by Jordan "the first hypoabelian group".

The Weyl group of the complex simple group An is isomorphic to the group [3n- t ] of symmetries of the regular n-dimensional simplex. The Weyl groups of the complex simple groups Bn and Cn are isomorphic to the group [3n-2 , 4] of symmetries of the n-dimensional cube. The Weyl group of the complex simple group Dn is isomorphic to the group [3n-3 • 1 · 1 1 of symmetries of the n-dimensional "semicube". The Weyl groups of the complex exceptional simple groups G2 , F4 , £6 , E7 , and £8 are isomorphic to the groups [6] , [3 , 4 , 3] , [32 ' 2 ' 1 ] , [33 ' 2 ' 1 ] , and [34 ' 2 ' 1 ] , respectively. Isomorphism of the Galois groups of the characteristic equations of the last three Lie groups and three last finite groups was shown by Cartan as far back as 1 894 in his note On reduction of the group structure to its canonical form [ 4] and was proved in the paper On reduction of the structure of a finite and continuous group to its canonical form [9] ( 1 896) (for the group £6 , Cartan made this result more precise in the paper [82] ) . Commenting on the solution of the characteristic equation for the "groups of type E" of rank l , Cartan wrote in [9] : "For the latter ones, if l = 6 , it is reduced to an equation of the same nature as the equation defined by 27 generators of the cubic surface; if l = 7 , it is reduced to an equation of the same nature as the equation defined by 28 double tangents to the curve of the fourth order; and finally, if l = 8 , it is reduced to the equation of 1 20th degree whose group is the first hypoabelian group of 1 20 letters" [9, p. 57] .

Cartan returned to these groups in one of his last works, Some remarks on 28 double tangents of a plane quartic and 27 lines of a cubic surface [ 1 84] ( 1 946) , where he, using the term "Galois group of a configuration" for the group of transformations (for the two cases which he considered, these trans­formations are collineations) keeping this configuration fixed, formulated the following theorem: "The Galois group of the characteristic equation of the simple Lie group of rank 7 and order 1 33 is isomorphic to the Galois group of the configuration of 28 double tangents of a plane quartic without dou­ble points. The Galois group of the characteristic equation of the simple Lie group of rank 6 and order 78 is isomorphic to the Galois group of the

60 2. LIE GROUPS AND ALGEBRAS

configuration of 27 lines of a cubic surface without double points" [ 1 84, pp. 1 -2] .

Comparison of Figures 2 .3 and 2.5 with Figure 2 .7 shows that the Dynkin graphs of complex simple Lie groups differ from the Coxeter graphs of the Weyl groups of these groups only by presence of inequality signs. The Coxeter graphs were first applied to the theory of simple Lie groups by Coxeter himself in the paper [Cox2] with the same title as [Cox l ] . This paper [Cox2] was published as Appendix to the Weyl paper [Wey4] ( 1 935 ) . Later, in the paper Groups of reflections and enumeration of semisimp/e Lie rings [Wit] ( 1 94 1 ) , Ernest Witt (b. 1 9 1 4) applied the Coxeter graphs to classification of simple Lie algebras.

§2.10. The Weyl affine groups

In the paper [Cox l ] Coxeter found also all infinite discrete groups of mo­tions of the space Rn generated by reflections in hyperplanes of this space. These groups are also described by graphs similar to the graphs of the finite groups of this type. Figure 2 .8 shows the notation and the Coxeter graphs

[3 1 , I , 11 - 5, I , I ]

[oo]

[3, 6]

[3 , 3, 4, 3]

[32, 2, 2]

0--------0----- . . . --0------0----0 4 4

� - - · � � - - · �

0-----0 00

0---0----0 6

0-----0-----0--4

FIGURE 2 .8

§2. 1 0. THE WEYL AFFINE GROUPS 6 1

of these groups. The group [ oo] is the group of motions of the Euclidean line R1 generated by reflections of this line in two of its points or the group of motions of the space Rn generated by reflections of this space in two of its parallel hyperplanes. The groups crJ are the groups of symmetries of polyhedral angles. The group [4 , 3n- I , 4] is the group of symmetries of honeycombs of the space Rn formed by tessellation of this space with n-dimensional cubes. The group [3 , 6] is the group of symmetries of honey­combs of the plane R2 formed by tessellation of this plane with equilateral triangles or regular hexagons. The group [3 , 3 , 4 , 3] is the group of sym­metries of honeycombs of the space R4

formed by tessellation of this space with regular polytopes.

The affine Weyl group of a complex simple Lie group is defined as the infinite discrete group of motions of the space Rn determined by the vec­tors of the root system in the Cartan subalgebra of the Lie algebra of this group, and the metric of this space is induced by the Cartan metric in the Lie group. To find the affine Weyl group, Coxeter supplemented the reflec­tions in hyperplanes generating the Weyl group by the reflection in one more hyperplane passing through the common initial point of the vectors of the root system, the terminal point of one of these vectors and orthogonal to this vector. As this vector, Coxeter took the vector representing the minimal root in the order of roots which was later defined by Dynkin. The Dynkin graphs, supplemented by one more dot representing the root which is opposite to the maximal root, are called the augmented Dynkin graphs. It turned out that these graphs are very useful in solving many problems related to simple Lie groups. Figure 2 .9 (next page) shows the extended Dynkin graphs for simple Lie groups. If the minimal root has the form µ = L; mp.; , where a; are simple roots, then on these graphs, the dots representing the roots a; are marked by the number m; and the new dot is marked by the number 1 . (Note that the Dynkin graph itself can be obtained from the extended one by deleting any dot marked 1 . ) Actually,_ the extended Dynkin graphs were considered by Dynkin himself in the paper [Dyn2] as "impossible graphs" (see also the book by Pontryagin [Pon2] ) .

Comparison o f Figures 2 . 8 and 2 . 9 shows that the extended Dynkin graphs of complex simple Lie groups differ from the Coxeter graphs of the affine Weyl groups of these groups only by the presence of the inequality signs, i .e . , in just the same way as the Dynkin graphs of complex simple Lie groups differ from the Coxeter graphs of the Weyl groups of these groups.

Note that the connection index of simple Lie groups is exactly equal to 1 , 2, 3, 4, and n + 1 in the cases when the extended Dynkin graphs of these groups do not possess the symmetry or possess the bilateral, trilateral, quadrilateral symmetry, or the symmetry of order n + 1 , respectively. The connection index of simple Lie groups is equal to the numbers of marks I in its extended Dynkin graph.

62 2. LIE GROUPS AND ALGEBRAS

a) AO> n

b) B( I ) � � n

c) c O > � . . . -o-------a:::¢: n

d) v <o r � n

e) o Ol 2 �

f) F C i l 4 �

g) E ( I ) 6

h)

i)

FIGURE 2 .9

As for the usual Weyl groups, for the affine Weyl groups of simple Lie groups, it is also possible to define the fundamental domains which are the simplices whose n faces coincide with the faces of the Weyl chamber. Some­times, these domains are called the Wey! alcoves. The fundamental domains of the affine Weyl groups of simple Lie groups were first considered by Weyl himself in his work [Wey3] . However, their relations were not indicated. Weyl used this notion to prove that the connection group (the Poincare group) of compact real semisimple groups is finite (for simple Lie groups without center, the order of this finite group is equal to the connection index of the Lie group, i .e . , the determinant of its Cartan matrix) . Weyl used the finiteness of these groups for proving the complete reducibility of linear representations of complex semisimple Lie groups.

Shortly after the publication of the Weyl paper [Wey3], in the paper The geometry of simple groups [ 1 03] ( 1 927) , Cartan described the fundamental domains of usual and affine Weyl groups, and, in the addendum [ 1 1 3) ( 1 928) to this paper, he proved that any irreducible finite group generated by reflec­tions of the space Rn in its hyperplanes possesses a fundamental domain

§2. 1 1 . ASSOCIATIVE AND ALTERNATIVE ALGEBRAS 63

which cuts a spherical complex on a hypersphere with center at the point of intersection of the hyperplanes. In the same paper, Cartan proved the uniqueness of the maximal and minimal roots relative to an arbitrary system of roots.

§2.1 1 . Associative and alternative algebras

As we have already noted, in the 1 9th century, along with the groups and fields defined by Galois, a number of new numerical systems were introduced. These numerical systems, which are generalizations of the field of complex numbers, originally were also called systems of complex numbers. Later on, in order to distinguish them from the usual complex numbers, mathe­maticians began to call them systems of hypercomp/ex numbers or associative algebras.

Along with associative algebras, i .e . , vector spaces where an associative multiplication of vectors is defined which is distributive with respect to their addition and commutes with multiplication of vectors by numbers, more general algebras were considered. These new algebras differ from associative algebras by the fact that multiplication of their elements is not associative. The Lie algebras which we discuss in this chapter are nonassociative algebras.

If in a vector space a basis {e; } is given (for algebras, the elements e; are often called the "units" of an algebra) , multiplication of elements of an algebra is defined by the formula

(2 .36)

Formulas (2. 1 2) are a particular case of formulas (2 .36) . They differ in that the operators X°' in formulas (2. 1 2) play the role of the vectors e; , and the commutators [Xo:Xp] play the role of the products eiej .

For an arbitrary algebra, the numbers < are also called its "structure constants". For Lie algebras, multiplication is neither commutative nor as­sociative. For these algebras, these properties are replaced by the property of anticommutativity (2 . 1 3 ) and the Jacobi identity (2 . 1 4) . While multiplica­tion of elements of Lie algebras is written in the form c = [ab] , multiplication in associative algebras is written in the form c = ab . Multiplication of ele­ments of nonassociative algebras with the "alternativity" property (any two elements of an alternative algebra generate an associative algebra) is written in the same form as in associative algebras.

The appearance of algebras was closely connected to the appearance of vectors. The simplest algebra is the field C of complex numbers with the units 1 and i , i2 = - 1 . In the works of Leonhard Euler ( 1 707- 1 783) , Jean Le Rond D'Alembert ( 1 7 1 7- 1 783) , Gauss, and Augustin Louis Cauchy ( 1 789- 1 857) , the geometric interpretation of complex numbers was estab­lished in the "plane of a complex variable" with addition according to the

64 2. LIE GROUPS AND ALGEBRAS

parallelogram rule and multiplication according to which the moduli of com­plex numbers are multiplied and their arguments are added. Following this, in the first half of the 1 9th century, attempts were made to "generalize the complex numbers for the space'', i .e . , to construct a number system with three units. These algebras were constructed by Augustus de Morgan ( 1 806- 1 87 1 ) and Charles Graves ( 1 8 1 0- 1 860) . However, in all such algebras there were "divisors of zero", i .e . , elements a and b which are themselves different from zero but whose product is zero. The best known among these algebras is the algebra of "triplets" which is isomorphic to the direct sum of the fields R and C , i .e . , to a set of pairs (a , a) of numbers where a is a real number and a is a complex number and where addition and multiplication are defined by the following formulas: (a , a) + (b , P) = (a + b , a + ft) , (a , a)(b , ft ) = (ab , aft) .

In 1 844 William Rowan Hamilton ( 1 805- 1 865) discovered the algebra of quaternions, which is the algebra with four units 1 , i , j , k , i2 = / = - 1 , ij = -j i = k . This algebra was of significantly greater importance both for algebra and geometry. Hamilton called expressions of the form xi + yj + zk "vectors" and viewed quaternions of general type as sums of scalars (real numbers) and vectors. The algebra of quaternions, which is denoted by H after Hamilton, is a noncommutative field (skew field) . As in the field C , in the field H , a transition to the conjugate element a -+ a (which is multiplication of the quaternion units i , j and k by - 1 ) is defined satisfying the property:

(2 .37) aft = /ia.

The product aa , which is equal to the sum of squares of coordinates of the quaternion, is called the square of the modulus l a l of the quaternion. The modulus l a l in the fields H and C possesses the property

(2 .38) laft l = l a l l ft l .

Two algebras important for both algebra and geometry are connected with the name of Arthur Cayley ( 1 82 1 - 1 895 ) . The first of these algebras is the algebra 0 of octaves with eight units 1 , i , j , k , I , p , q , r , i2 = / , 12 = - 1 , ij = ji = k , ii = -I i = p , kp = -pk = q , jp = -pj = r . Octaves are often called "Cayley numbers" or "Graves-Cayley numbers" since almost at the same time as Cayley, they were discovered by John Thomas Graves ( 1 806- 1 870), brother of Charles Graves. The algebra 0 , like the algebra H , is a skew field, and its multiplicaion is not associative but alternative in the sense indicated above. As in the fields C and H , in the field 0 , the transi­tion to the conjugate element a -+ a (which is multiplication of the octave units i , j , k , I , p , q , and r by - 1 ) is defined satisfying property (2 .37) in addition to the modulus J a l whose square l a l

2 = aa is equal to the sum of squares of coordinates of the octave and which satisfies property (2 . 38 ) . We

§2. 1 1 . ASSOCIATIVE AND ALTERNATIVE ALGEBRAS 65

will see below that the field 0 is closely connected with exceptional simple Lie groups.

The second algebra discovered by Cayley is the algebra of matrices; it appeared in his Memoir on the theory of matrices [Cay l ] ( 1 858 ) . The al­gebra Rn of real matrices of order n consists of square arrays A = (a;) which are added and multiplied according to the rules: A + B = (au + bu )

and AB = ('L,i aiibik ) . The algebra Rn has n2 units Eu discussed ear­lier. The algebras en , and Hn of complex and quaternion matrices can be defined in the same manner. The algebras Rn , en , and Hn possess divi­sors of zero. In the algebras Rn and en , such elements are matrices with zero determinant. In the algebra Hn , such elements are matrices with zero "semideterminant"-a real number equal to the determinant of a real or com­plex matrix representing the given quaternion matrix in the algebras e2n and R4n containing a subalgebra isomorphic to Hn . If we introduce the notion of tensor product A ® B of algebras A and B with bases {e; } and {fa} as an algebra with the basis {e;fa} (eJo: = fae; ) , then the algebras en and Hn can be defined as the tensor products Rn ® e and Rn ® H .

The founder of multidimensional algebra and geometry Hermann Grass­mann ( 1 809- 1 877) in 1 844 in the work The science of linear extension [Gra l ] , having defined the n-dimensional linear space, also introduced the "exte­rior product" of vectors of this space. Later, in his geometrical works, Cartan often used this notion. At present, the exterior product of vec­tors x 1 , x2 , • • • , xk is written in the form x1 /\ x2 /\ • • • /\ xk . It is un­changed by an even substitution of the vectors x 1 , x2 , • • • , xk , multiplied by - 1 for an odd substitution of them, and equal to zero when the vec­tors x1 , x2 , • • • , xk are linearly dependent (in particular, x /\ x = 0) . The vectors x1 , x2 , • • • , xn , along with all their possible exterior products X; /\

I X; /\ · · · /\ X; , i 1 < i2 < · · · < ik , form a basis of an algebra with 2n units.

2 k A modification of the Grassmann algebra is the algebra with 2n units

constructed by Clifford in the paper Applications of Grassmann 's extensive algebra [Cl2] ( 1 878 ) . He wrote the units of his algebra in the form I , e1 , e2 ,

2 . . . , en and e,. ,. ,. = e1 e2 . . • ek where e,. = - 1 and the products e,. ,· ,· are I 2 ' " k I 2 " ' k

not changed if their indices undergo an even substitution and are multiplied by - 1 if the indices undergo an odd substitution (i .e . , products of distinct factors behave in the same way as exterior Grassmann products) .

I f we denote the Clifford algebra with n units e; by the symbol Kn+ l , then the algebra K1 coincides with the field R of real numbers, the algebra K2 with the field e of complex numbers, and the algebra K3 with the field H of quaternions. Other algebras Kn+ 1 are generalizations of the field H of quaternions but in a direction other than that of the field 0 of octaves. Namely, all algebras Kn+ l are associative, and the algebra K4 is isomorphic to the direct sum H EEl H of two fields H , the algebra K5 is isomorphic to the algebra H2 of quaternion matrices of second order, the algebra K6

66 2. LIE GROUPS AND ALGEBRAS

is isomorphic to the algebra e4 of complex matrices of fourth order, and the algebra K7 is isomorphic to the algebra R8 of real matrices of eighth order. Clifford showed that the algebras Kn+ I are isomorphic to the following algebras: Ksm+ I = R24m , K8m+2 = e24,,. , Ksm+J = H24m , K8m+4 = H24,. EB H24m , Ksm+S = H24m+t , K8m+6 = e24m+2 , K8m+? = R24m+3 , KS(m+ I ) = R24m+3 $ R24m+3 .

In 1 872, in the paper A preliminary sketch of biquaternions [Cl l ] , Clifford introduced two modifications of the algebra e now known, respectively, as the algebra of split complex numbers and the algebra of dual numbers. These algebras are denoted by 'e and 0e , respectively. The algebra 'e has the units 1 , e , e

2 = 1 and the algebra 0e has the units 1 , e , e2 = 0 . If in

the algebra 'e we take the basis e+ = ( 1 + e)/2 , e_ = ( 1 - e )/2 for which e! = e + , e=_ = e _ , e + e _ = 0 , we see that this algebra is isomorphic to the direct sum R EB R of two fields R . Next, Clifford extended the notion of biquaternions (complex quaternions) introduced by Hamilton, to split com­plex and dual quaternions. He called the Hamilton biquaternions hyperbolic, and he called split complex and dual quaternions elliptic and parabolic bi­quaternions, respectively. The algebras of hyperbolic, elliptic, and parabolic biquaternions are the tensor products H © e , H © 'e , and H © 0 e .

The general notion of the associative algebra whose particular cases are the algebras Rn , en , and Hn and similarly defined algebras 1 en and 0en as well as the algebras Kn , was introduced by Benjamin Peirce ( 1 809- 1 880) in his posthumously published paper Linear associative algebras [Pe] ( 1 88 1 ) . Peirce introduced the notion of nilpotent element one of the powers of which is equal to zero (the "dual unit" e of the algebra 0e is an example of such an element) and the notion of "idempotent element" for which a2 = a (the split complex numbers ( 1 ± e)/2 are examples of such elements) . He used these notions for classification of algebras of small dimensions.

In 1 883- 1 885 several papers by outstanding mathematicians on the the­ory of algebras appeared. In 1 883 Weierstra8 wrote a letter to H. Schwartz, a fragment of which was published in 1 884 in the form of the note To the theory of complex quantities formed by n principal units [Wei] . In the same year, Poincare's note On complex numbers [Poi2] appeared, and in 1 885 the paper [Ded] by Richard Dedekind ( 1 83 1 - 1 9 1 6) , with the same title as Weier­stra8's note [Wei], was published. Poincare studied the relation between al­gebras (which Poincare called "systems of complex numbers", as WeierstraB and Dedekind had done earlier) and continuous groups, namely, "bilinear groups", i .e . , groups of transformations (2.2) for which the functions / are linear in both the variables xk

and the variables a0 • Weierstra8 showed that any commutative associative algebra is isomorphic to the direct sum of a few fields R and e . He called the elements of associative algebras "complex quantities" and the basis elements (units) of these algebras "principal units".

Dedekind's paper was devoted to finite algebraic extensions of the field

§2. 1 2. CARTAN'S WORKS ON ALGEBRAS 67

Q of rational numbers. These extensions are similar to the Galois fields F q discussed earlier which are finite algebraic extensions of the fields EP of residue classes. Both extensions can be considered as algebras over the fields Q and R , respectively.

In 1 892 Theodor Molien ( 1 86 1 - 1 94 1 ) defended in Dorpat (now Tartu) his doctoral dissertation On systems of higher complex numbers. In 1 893 this dissertation was published in Leipzig (see [Mol]) in the same journal Math­ematische Annalen in which the Killing paper [Kil2] had appeared. Molien was in close contact with German algebraists. In his dissertation, he gener­alized the notions of simplicity and semisimplicity used by Killing for Lie algebras to associative algebras. He also found a criterion of semisimplicity of a complex algebra in the form of nondegeileracy of the quadratic form

(2 .39)

similar to form (2 .20) , i .e . , reducibility of this form to the sum of squares of all coordinates a;

. Molien's dissertation dealt with complex algebras, and its main result is that any simple complex algebra is isomorphic to the algebra Cn and any semisimple complex algebra is isomorphic to the direct sum of such algebras.

§2.12. Cartan's works on algebras

Cartan's works on classification of simple and semisimple associative alge­bras were a natural development of his works on classification of simple and semisimple Lie groups and algebras. Two of his notes On systems of complex numbers [ 1 1 ] and On real systems of complex numbers [ 1 2] ( 1 897) and the extensive paper Bilinear groups and systems of complex numbers [ 1 3] ( 1 898) were devoted to this problem.

Following Poincare, in the latter paper Cartan used the term "systems of complex numbers" for algebras and considered "bilinear groups" of transfor­mations connected with algebras. In addition to solving problems of clas­sification of simple and semisimple algebras, Cartan revised the notions of simplicity and semisimplicity of algebras and introduced the notion of an "invariant subsystem of the system of complex numbers'', which is similar to the notion of an invariant subgroup of a group and is the most impor­tant particular case of an invariant subsystem-"pseudonull invariant sub­system". An "invariant system" is a subalgebra remaining invariant under multiplication by an arbitrary element of the algebra from the right or from the left. At present, such subalgebras are called ideals of an algebra. The term "ideal" was originated from the term "ideal prime factors" introduced by Ernst Kummer ( 1 8 1 0- 1 893) in his theory of algebraic integers. In the ring Z of regular integers an ideal (defined for a ring in the same manner as for an algebra since an associative algebra is a ring with respect to addition and multiplication) consists of numbers that are multiples of an integer. This

68 2. LIE GROUPS AND ALGEBRAS

was the reason, when Kummer encountered ideals in the rings of algebraic integers, he considered them as sets of numbers that are multiples of "ideal factors". The Cartan term "pseudonull invariant subsystem" arose from the word "pseudonull" which Cartan used for nilpotent elements of algebras. At present, such subalgebras are called radicals. An algebra is called simple if it does not contain ideals different from the algebra itself and zero. An algebra is called semisimple if it does not contain a radical.

In his paper, Cartan proved that any complex or real algebra is a direct sum of a semisimple algebra and a "pseudonull subsystem'', i .e . , a radical, and that any semisimple algebra is isomorphic to a direct sum of simple algebras and a "pseudonull subsystem". Moreover, we saw earlier that in 1 892 Molien proved that any complex simple algebra is isomorphic to the algebra en of complex matrices. For the algebra e2 , which is isomorphic to the algebra of complex quaternions, Cartan used the term the "algebra of quaternions"; for the algebra e3 , following James Joseph Sylvester ( 1 8 1 4- 1 897 ) , he used the term the "algebra of nonions"; and for the algebra en he used the term the "algebra of n2-ions".

Next, Cartan proved that any real simple algebra is isomorphic to either the algebra Rn of real matrices or the algebra en of complex matrices or the algebra Hn of quaternion matrices. In the first note mentioned above, Cartan announced results related to complex matrices and in the second one to real matrices. In particular, it follows from Cartan's results that all Clifford algebras are simple or semisimple.

In 1 898 , in the German Encyclopaedia of Mathematical Sciences, the sur­vey paper The theory of usual and higher complex numbers [Stu 1 ] by Study appeared where the development of the theory of algebras in the 1 9th century was summarized. Following Molien, Study used for algebras the name "sys­tems of higher complex numbers". In this paper, by analogy with split com­plex and dual numbers, Study defined the algebra 'H of split quaternions with the units l , i , e , f having the properties i2 = - 1 , e2

= l , ie = e i = f . This algebra is isomorphic to the algebra R2 of real matrices of second order. In this paper, Study also defined the algebra 0H of semiquaternions with the units 1 , i , e , 17 having the properties i2 = - 1 , e2 = 0 , ie = -ei = Yf . Semi­quaternions are often called "Study's quaternions". As in the fields e and H , in the algebras 'e , 0 e , 'H , and 0H , then the transition to the conjugate element satisfying property (2 . 30) and the modulus l n l satisfying the prop­erty (2 .3 1 ) can be defined. The difference is that while in 'e and 'H the products nO: are algebraic sums of squares of all coordinates, in °e and 0H they are sums of squares of coordinates in 1 and i only. In the same manner, if in the definition of an alternative skew field 0 of octaves, one replaces the unit / by the units e and e with the same properties as in the algebras 'e , 0e , 'H and 0H , then the algebra 'o of split octaves and the algebra 00 of semi-octaves will be obtained where properties (2 .37 ) and (2 . 38 ) also hold.

§2. 1 3. LINEAR REPRESENTATIONS OF SIMPLE LIE GROUPS 69

Study's paper was translated into French and significantly revised by Car­tan. This revised translation was published under the title Complex numbers [27] in the French edition of Encyclopaedia of Mathematica/ Sciences ( 1 908) . While Study's original paper was 34 pages long, Cartan's extended translation was 1 40 pages long.

After presenting the theory of Clifford algebras Kn along Study's lines, Cartan added that "it is possible to consider more general systems" in which some squares of e; are equal to - 1 and some are equal to + 1 . At present, the algebras which differ from the algebras Kn , by the fact that for I of its units e7 = + 1 and for the remaining n - I - 1 units e7 = - 1 , are denoted by K� . Cartan noted that "all these systems are simple or semisimple" and indicated the structure of the algebras Kn and K� in the following way. After introducing the number h = 1 - E; e7 , he indicated that K� = R2.- 1 if h = 1 (mod 8) , K� = C2.- 1 if h = 2 (mod 8) , K� = H2.- 1 if h = 3 (mod 8) ' Kn , [ = R2n- I E9 R2n- I if h = 0 (mod 8) ' and Kn , / = "2·- · E9 "2·- · if h = 4 (mod 8) . Cartan denoted the algebras R2,, - 1 , C2n - 1 , and H2,, - 1 by Sm ' csm ' and QSm ' respectively, and the direct sums R211 - I E9 R211 - I and H2.- 1 E9 H2.- 1 by 2Sm and 2QSm , respectively. He concluded his supplement by saying that "these systems are reducible if h is a multiple of 4" [27, p. 464]. Note that the algebra K� coincides with the algebra 'C and that the algebras K� and K� coincide with the algebra 'H .

§2.13. Linear representations of simple Lie groups

In his paper Projective groups, under which no plane manifold is invariant [3 7], published in 1 9 1 3 , Cartan constructed the theory of linear represen­tations of complex simple Lie groups. This theory is the foundation of a number of mathematical theories that have important applications to mod­ern physics.

A linear representation of a group G is a homomorphic mapping of this group into a subgroup of the group GLN of real matrices of order N or the group CGLN of complex matrices of order N . A linear representation is said to be reducible if in a linear space of representation, i .e . , in a linear space, whose matrices of linear transformations form a representation of the group, there is a subspace which is invariant under these transformations. A linear representation <p is said to be completely reducible if the linear space of representation decomposes into a direct sum of invariant subspaces. In these invariant subspaces, representations rp 1 , rp2 , • • • , <pk of the group G occur. In this case, the representation <p is called the direct sum of these represen­tations and is denoted by rp 1 E9 rp2 E9 · · · E9 <pk . The title of the Cartan paper indicates that he considered irreducible linear representations. He used the term "projective groups" for groups of linear transformations since matrices of linear representations can also be considered as matrices of collineations of projective spaces. First, Cartan showed that all linear representations of

70 2. LIE GROUPS AND ALGEBRAS

semisimple Lie groups are completely reducible. It follows from this that the study of general linear representations of these groups is reduced to their irreducible linear representations.

A linear representation of a Lie group G induces a linear representation of the Lie algebra of this group. One type of linear representation of simple Lie groups was already considered in Cartan's thesis. In the same manner as in that case, one can show that for any irreducible linear representation of a simple Lie group the linear transformations X --+ [H X] , where X and H are the matrices representing an arbitrary element x of the Lie algebra and an element h of the Cartan subalgebra of this algebra, have the same eigenvectors, and the corresponding eigenvalues are linear combinations with rational (no longer integer) coefficients of the basis roots; as these roots, one can take "simple roots". These linear combinations are called the weights of the linear representations. Since for any two weights, as for any linear combinations of simple roots, the notion "greater than" can be defined, it is possible to distinguish the maximal weight among all weights of a linear rep­resentation. This maximal weight is called the dominant weight of this linear representation. Cartan showed that a linear representation of a semisimple Lie group is completely determined by its dominant weight.

If two linear representations rp and If/ of a group G are given in M- and N-dimensional spaces with vector coordinates x i and l' , i = 1 , 2 , . . . , M , o: = 1 , 2 , . . . , N , then the products x i y°' also undergo linear trans­formations forming a linear representation of the group G in an (MN)­dimensional space with vector coordinates i°' . This representation is called the Kronecker product of representations rp and I/I and is now denoted by rp ® I/I . Cartan showed that if the dominant weights of representations rp and I/I are the forms w1 and w2 , then the dominant weight of the representation rp ® I/I is the form w1 + w2 • For a linear representation rp of a group G in M-dimensional space, it is possible to define the kth exterior power rp lk l

_a linear representation of the group G in the (�) -dimensional space of skew­symmetric tensors ai1 i2 · · · ik . Cartan showed that if the dominant weight of a representation rp is a form w1 and its following weights in decreasing order are the forms w2 , w3 , • • • , wk , . . . , then the dominant weight of the kth exterior power rplkJ is the sum of the forms w1 + w2 + · · · + wk .

Cartan also showed that all linear representations of a complex simple Lie group G are Kronecker products of exterior powers of several basic representations whose number is equal to the rank of the group, and each of these basic representations corresponds to a certain simple root of the group G . The groups of matrices corresponding to these basic representations were called the "fundamental groups" by Cartan. Since the term "fundamental group" has several different meanings (later Cartan used this term for the transitive group of transformations of a homogeneous space) , we will apply Cartan's term "fundamental" not to the groups but to their representations,

§2. 1 3 . LINEAR REPRESENTATIONS OF SIMPLE LIE GROUPS 7 1

i .e. , we will call fandamental representations of a simple Lie group those of its representations from which it is possible to obtain all its representations. The dominant weights of these fundamental representations are called the fundamental weights. These weights are linear combinations (2 . 3 1 ) of simple roots ai whose coefficients are the entries of the inverse matrix A- 1 of the Cartan matrix of a simple Lie group. Thus, to each fundamental weight ni , there corresponds a simple root ai . On the other hand, the dominant weight of any linear representation of a simple Lie group is a linear combination with integer coefficients of the fundamental weights, i .e . , of the points of the weight lattice of this group (and, therefore, they are linear combinations with rational coefficients of simple roots) . If the dominant weight of a linear representation of a simple Lie group is a linear combination mini , this linear representation is represented by the Dynkin graph where next to each dot a; , the integer mi is written.

In particular, for a simple Lie group in the class An , fundamental rep­resentations are its representation rp 1 by matrices of order n + 1 from the group CSLn+ I and the exterior powers <pk = <p�

kJ , k = 2 , 3 , . . . , n , of this

representation by matrices of order (nt° 1 ) . For a simple Lie group in the class Bn , fundamental representations are its representation rp 1 by matrices Of order 2n + l from the group C02n+ I , the exterior powers <pk = <p�

k] , k =

2 , 3 , . . . , n - 1 , of this representation by matrices of order (2"tc+ 1 ) , and a representation lfli by the matrices of order 2n , which later received the name "spinor representation". For a simple Lie group in the class en , fundamental representations are its representation rp 1 by matrices of order 2n from the group CSy2n and irreducible representations <pk of order (2kn) - (k2!!.2) from

the exterior powers rp�kJ of the representation rp 1 , k = 2 , 3 , . . . , n . For

a simple Lie group in the class Dn , fundamental representations are its rep­resentation rp 1 by matrices of order 2n from the group C02n , the exterior powers <pk = rp�

kJ , k = 2 , 3 , . . . , n - 2 , of this representation by matrices

of order (2kn) , and representations by the ·matrices of order 2n- I . Similarly

to the representation lfli of the group Bn , the representations lfli and lf/2 presently are called spinor representations.

We will call the dominant weights of linear representations <pk of the groups An , Bn , en the forms n

k and the dominant weights of linear rep­

resentations If/ and lfli , lf/2 of the groups Bn and en the forms nn and n- 1 n · l TC , TC , respective y. The roots ai of the adjoint representation x ---+ [ax ] were considered

in Killing's paper [Kil2] and in Cartan's thesis; the maximal roots coincide with the dominant weights of these representations. For the groups An , they . 2 are n' + nn ; for the groups Bn and Dn , they are TC ; and for the groups en , they are 2ni . Note that the dots of the Dynkin graphs marked by the

72 2. LIE GROUPS AND ALGEBRAS

numbers and 2 which correspond to these representations coincide with those dots to which the additional dots of the extended Dynkin graphs are attached. The dominant weights of adjoint representations of the exceptional simple Lie groups have the same property.

Cartan's book Lectures on the theory of spinors [ 1 64], devoted to the spinor representations 'Iii and 1/12 , was written in 1 938 when it was discovered that similar representations of the group ol of pseudo-orthogonal matrices (the Lorentz transformations defining the transitions from one inertial coordinate system to another in the space-time of special relativity) are closely connected with electron spin discovered in the 1 930s. The vectors of spaces of these representations are called the spinors.

Before describing spinor representations of the groups COn , we describe similar representations of real groups On of orthogonal matrices. We noted earlier that in 1 878 , in the paper [Cl2], Clifford defined the algebras Kn with 2n- I units. In 1 886, Rudolf Lipschitz ( 1 832- 1 903) , in his dissertation Research on the sums of squares [Lip] , discovered an important connection between these algebras and the groups On . The simplest way to describe this connection is the following consideration. In the algebra Kn , as well as in the algebra H of quaternions which is its particular case, one can define an "involution" a --+ a with the properties: a = a ' a + p = a + 7i ' and ap = a/i . If we write an element a of this algebra in the form a = E ai 1 ·

· · ik ei · · · i , I k

then the element a has the form E ai 1 . . . ik ei i · · · i . Then the coefficient of k k- 1 I 1 in the product aa is equal to the sum of squares of all coordinates of the element a . If we call this coefficient the square of the modulus la l and take as the distance between elements a and P the modulus I P - a l of their

difference, then in the algebra Kn the metric of the Euclidean space R 211 - I

will be defined. Next, note that the algebra Kn is isomorphic to a subalgebra of the algebra Kn+ ! generated by the units with even numbers of indices. Consider now the following transformation of the algebra Kn+ I :

(2 .40) ' J! - 1 )! .,, = a .,, a ,

where a is an element of the algebra Kn , represented as a linear combination of units of the algebra Kn+ i with even numbers of indices, and e is an element of the algebra Kn+ ! of the form e = xi ei , and assume that the element a is such that the element I e of the algebra Kn+ I is also of the form 'x iei . Then the elements a form a group which is homomorphic to the group On , and the kernel of this homomorphism is a subgroup of this group consisting of the elements 1 and - 1 .

The coordinates a , aii , . . . , ai 1 i2 . . · ik of elements a of the algebra Kn with even numbers of indices satisfy the condition Li i . . . ,·k (a i 1 iz · · · ik ) 2 = 1

I 2 ·

§2. 1 4. REAL SIMPLE LIE GROUPS 73

and the equations

(2 .4 1 )

where (2k - 1 ) ! ! = 1 • 3 · 5 · · · · • (2k - 1 ) and [ ] is the alternation symbol. The surface (2 .4 1 ) in the projective space PN , where N = 2n

-I - 1 (those

equations first appeared in the work [Lip] of Lipschitz) , is called the Lips­chitzian and is denoted by nn . At present, we say that this group doubly covers the group On and call this group the spinor group of the group On .

Thus, the spinor group of the group On is a subgroup of the group of invert­ible elements of the algebra Kn . But from the structure of the algebra Kn discovered by Clifford it follows that the result of complexification of the al­gebra K2k+ I , i .e . , the tensor product K2k+ I © C , is isomorphic to the algebra C2k of complex matrices of order 2

k , and the result of complexification of

the algebra K2k , i .e . , the tensor product K2k © C , is isomorphic to the direct sum C2k - • E9 C2k - I (in explicit form, this result was first obtained by Richard Brauer ( 1 90 1 - 1 977) and Weyl in their paper Spinors in n dimensions [BrW] in 1 935 ) . It follows from this that the order of the matrices of the spinor representation I/Ii of the group C02k+ I is equal to 2

k and that the orders

of the matrices of the spinor representations lf/1 and lf/2 of the group C02k 1 2

k-I are equa to .

§2.14. Real simple Lie groups

Cartan found the classification of real simple associative algebras imme­diately after finding the classification of complex simple associative algebras. However, he was able to solve the similar problem for real simple Lie groups only more than 20 years after solving this problem for complex simple Lie groups in his thesis. This happened in 1 9 1 4 shortly after Cartan constructed the theory of linear representations of complex Lie groups in the paper Real simple finite continuous groups [38] . As in his thesis, in this paper Cartan characterized real simple Lie groups by the nondegeneracy of their "Killing­Cartan form" (2 .20) , which he denoted in this paper by lfl(e) . However, in contrast to complex simple Lie groups for which this form can always be reduced to a sum of squares, for real simple Lie groups, if the group is com­pact, this form can be reduced to the sum of negative squares whose number is equal to the group dimension, and if the group is noncompact, it can be reduced to the sum of a certain number of positive and a certain number of negative squares. Cartan characterized real simple Lie groups by an integer o called the character which is equal to the difference between the number of positive and negative squares in the canonical form of the form lfl(e) .

74 2. LIE GROUPS AND ALGEBRAS

The Cartan metric in real simple and semisimple Lie groups is defined by the metric in their Lie algebras in which the square of the length of the vector e is equal not to l/f(e) but to -1/f (e) . Therefore, the square of the linear element in the Cartan metric in simple and semisimple real Lie groups is equal to

(2 .42)

and this form is positive de�nite for compact groups and indefinite for noncompact groups. Thus, compact real simple Lie groups in their Cartan metrics are real Riemannian manifolds V' , and noncompact real simple Lie groups in their Cartan metrics are real pseudo-Riemannian manifolds V/ . The character o of a noncompact real simple Lie group is connected with its dimension r and the index I (the number of negative squares in the canonical form of the form (2 .27)) of the pseudo-Riemannian Cartan metric of this group by the relation o = 21 - r , and the character o of a compact real simple Lie group is equal to the product of its dimension and the number - 1 .

In 1 929, after constructing the theory of symmetric Riemannian spaces, he returned to the problem of classification of noncompact real simple Lie groups and solved it by much simpler methods in the paper Closed and open simple groups and Riemannian geometry [ 1 1 6] (here "closed groups" and "open groups" are compact and noncompact Lie groups, respectively) . In the introduction to this paper, Cartan wrote that based on the geometric theory constructed by him, "it will now be possible to reduce significantly the calculations which I have performed. Thus, the extensive memoir in which I determined all real forms of simple groups can now be reduced to twenty from the original 90 pages" [ 1 1 6 , p. 2] . By the "extensive memoir" Cartan had in mind his work [38 ] .

If a complex simple Lie group is a subgroup of the group of CGLN of complex matrices of order N , the compact real group having this group as its complexification is the intersection of this group and the group of complex unitary matrices of the same order. This intersection is called the "unitary restriction" of this group. Instead of discussing Lie groups defined up to a local isomorphism, it is more convenient to discuss the Lie algebras of these groups.

Essentially, Cartan's reference to the theory of symmetric Riemannian spaces was that in the theory he had found all involutive automorphisms of the Lie algebras of compact simple Lie groups. If in the Lie algebra G of a Lie group G , an involutive automorphism is given, i .e. , an automorphism A --+ AJ of this algebra such that (AJ )J = A , then in the Lie algebra G one can take a basis { e1 } whose elements e..., remain invariant under the automorphism (e0f = e .. and the elements e; are multiplied by - 1 (e{ = -e; ) . Moreover, the Lie algebra is decomposed into the direct sum

§2. 1 4. REAL SIMPLE LIE GROUPS 75

(2.43 )

of two linear subspaces with the bases { e,,J and { e; } . Decomposition (2 .43) is often called the "Cartan decomposition". In the

basis e1 , the structure equations (2. 1 2) of the algebra G have the form

(2.44)

from which we can see that the subspace H is a subalgebra of the Lie algebra G .

To each Cartan decomposition, there corresponds a new Lie algebra whose basis can be obtained from the basis e1 of the Lie algebra G by multipli­cation of the basis elements ei by the imaginary unit i . If we denote the products iei by the same letters e; , then the structure equations (2 . 1 2) of the new Lie algebra have the form

(2.45) [e e . ] = ci .e . , a I C<I J

This new algebra is denoted by

(2.46) 1G = H E9 iE.

The Lie group 'G defined by the new algebra 'G obviously has the same complexification CG as the group G . However, the group ' G is no longer compact. Since the number of the base vectors e0 with positive inner square is equal to dim H and the number of the base vectors ie ; with negative inner square is equal to dim E , the character t5 of the noncompact group ' G is equal to the difference dim E - dim H . Thus, we can find all noncom pact groups 'G with the same complexification CG as the given compact group G . At present, the transition from the Lie algebra (2 .43) to the Lie algebra (2.46) and from the corresponding compact group G to the noncompact group 'G is called the Cartan algorithm.

Note that the Cartan algorithm can be applied not only to Lie algebras but also to the associative and nonassociative algebras discussed above. In partic­ular, applying this algorithm to the field C and its involutive automorphism a --+ a , we obtain the algebra 'C of split complex numbers, and applying this algorithm to the field H of quaternions and its involutive automorphism a --+ i- 1 ai , we obtain the algebra 'H of split quaternions. Next, applying this algorithm to the alternative skew field 0 of octaves and its involutive automorphism, under which the units i and j are not changed and the unit I is multiplied by - 1 , we obtain the alternative algebra '0 of split octaves. Finally, applying this algorithm to the algebra Kn and its involutive auto­morphism under which the units e; are not changed and the units e., are multiplied by - 1 , we obtain the algebra K� .

76 2. LIE GROUPS AND ALGEBRAS

The compact real form of the group CSLn+ I is the group csun+ I of complex unimodular unitary matrices. The Lie algebra of this group consists of complex skew-Hermitian matrices (a; ) , i .e . , complex matrices satisfying

the condition aJ = -a{ . This condition can be obtained from the unitarity

condition UUT = I . Moreover, for the Lie algebra of the group csun+ I ' the condition Tr U = 0 , which is obtained in a similar manner from the unimodularity condition of matrices U .

If we denote by E1 the diagonal matrix with diagonal entries e 1 from which l entries e0 are equal to - 1 and the other entries e; are equal to 1 ,

and by J the matrix (-� �) of order 2n , where I is the identity matrix

of order n , then all involutive automorphisms of the Lie algebra of the group CSUn+ I Can be written in the form:

(2 .47)

(2 .48) A � A ,

(2.49) A � -JAJ.

Compact real groups in the classes Bn and Dn are the groups 02n+ I and 02n of real orthogonal matrices whose Lie algebras are the algebras of real skew-symmetric matrices of the same orders. All involutive automorphisms of the Lie algebra of the groups 02n+ I and 02n can be written in the form (2 .47) , and for the group 02n it can be also written in the form

(2 . 50) A � -JAJ.

On many occasions, Cartan considered a compact group in the class en as the intersection of the groups CSp2n and CSU2n . However, in the paper On certain remarkable Riemannian forms of geometries with a simple funda­mental group [ 1 07] ( 1 927) , which will be discussed in more detail in Chapter 6, Cartan indicated that a compact group in the class en can be represented by quaternion unitary matrices of order n , i .e . , by unimodular quaternion matrices "keeping invariant a quaternion positive definite Hermitian form" xj aijx

; where "the quaternions aij satisfy the condition aij = li;/' [ 1 07, p.

392] . We will denote this group in the class en by the symbol HUn (the uni­modularity of matrices of this group follows from their unitarity) . At present, a compact group in the class en is represented only in this way. Chevalley was the first to represent systematically a compact group in the class en by matrices of the group HUn . He did this in his book The theory of Lie groups [Chv l ] ( 1 946) . The Lie algebra of the group HUn consists of quaternion skew-Hermitian matrices of the same order, i .e . , of quaternion matrices (aJ ) satisfying the condition aJ = -a{ . All involutive automorphisms of this Lie

§2. 1 4. REAL SIMPLE LIE GROUPS 77

algebra have the form (2 .47) and

(2 . 5 1 ) A -+ - iA i.

Applying the Cartan algorithm to the group cun+ I and the involutive automorphisms (2 .47) , (2 .48) , and (2 .49) of its Lie algebra, we obtain re­spectively: the group CSU�+ ! of complex unimodular matrices satisfying the condition

(2 . 52)

the group 'CSUn+ I of split complex unimodular unitary matrices which is isomorphic to the group SLn+ I of real unimodular matrices, and a group iso­morphic to the group HSL(n+ l l/2 of quaternion unimodular matrices. The

characters of the groups CS Un+ I ' csu�+ I ' 'csun+ I = SLn+ I ' and HSLn+ I are equal to -n(n + 2) , 4/(n - I + 1 ) - n (n + 2) , n , and -n - 2 , respec­tively.

Applying the Cartan algorithm to the group 02n+ l and the involutive au-

tomorphism (2.47) of its Lie algebra, we obtain the group o;n+ i , t of real matrices satisfying the pseudo-orthogonality condition:

T (2 . 53 ) UE1U = E1 •

The characters of the groups 02n+ l and o;n+ I are equal to -n (2n + 1 ) and 2/(2n - I + 1 ) - n (2n + 1 ) , respectively.

Applying the Cartan algorithm to the group HUn and the involutive au-tomorphisms (2 .47) and (2. 5 1 ) of its Lie algebra, we obtain the group HU� of quaternion matrices satisfying the pseudo-unitarity condition (2 .52) and the group 'HUn of antiquaternion unitary matrices which is isomorphic to the group Sp2n of real symplectic matrices, respectively. The characters of the groups HUn , HU� , and 'HUn = Sp2n are equal to -n (2n + 1 ) , 8/ (n - I) - n (2n + 1 ) , and n , respective�y.

Applying the Cartan algorithm to the group 02n and the involutive au­tomorphisms (2 .47) and (2 . 50) of its Lie algebra, we obtain respectively: the group o;n of real pseudo-orthogonal matrices and the group HSqn of quaternion symplectic matrices, i .e. , quaternion matrices satisfying the con­dition

(2 . 54) U iUT = ii.

The characters of the groups 02n , o;n , and HSqn are equal to -n(2n - 1 ) , 21(2n - I) - n (2n - 1 ) , and -n , respectively.

In the same papers, Cartan also found all real simple Lie groups in the exceptional classes. He showed that there are two simple Lie groups in the class G2 with the characters - 1 4 and 2, three simple Lie groups in the class F4 with the characters -52 , -20 , and 4, five simple Lie groups in

7S 2. LIE GROUPS AND ALGEBRAS

the class E6 with the characters - 78 , -26 , - 1 4 , 2 , and 6, four simple Lie groups in the class E1 with the characters - 1 33 , -25 , - 5 , and 7, and three simple Lie groups in the class Es with the characters -248 , -24 , and 8 . In particular, Cartan showed that a compact simple group in the class G2 is isomorphic to the group of automorphisms of the alternative skew field 0 of octaves. Applying the Cartan algorithm to this group and the unique involutive automorphism of its Lie algebra, we obtain a noncompact Lie group in the same class which is isomorphic to the group of automorphisms of the alternative algebra '0 of anti-octaves.

In 1 9 1 4, in the paper [38 ] , Cartan found all real simple Lie groups, sub­sequently. In the same year, in the paper Real continuous projective groups, under which no plane manifold is invariant [39] , he constructed all irreducible linear representations of these groups.

For real simple Lie groups, simple roots of the Lie algebras can be real or imaginary. Thus, the system of simple roots of these groups can be rep­resented by the Dynkin graphs where real simple roots are represented by white dots, imaginary simple roots are represented by black dots, and pairs of imaginary conjugate simple roots are represented by white dots joined by curved double arrows. These graphs are called the Satake graphs because Satake in the paper On representations and compactifications of symmetric Riemannian spaces [Sat] ( 1 960) used them for characterization of symmet­ric spaces with compact simple fundamental groups. (As noncompact real simple Lie groups, these spaces correspond to involutive automorphisms of compact simple Lie groups with the same complex forms. )

The Satake graphs can be also defined for compact real simple Lie groups: these graphs coincide with the Dynkin graphs, but all dots of these graphs are black. For noncompact real simple Lie groups, all simple roots of which are real (such groups are called split or anticompact) , the Satake graphs coincide with the Dynkin graphs, and all dots of these graphs are white. Figure 2. 1 0 represents the Satake graphs for noncompact simple Lie groups in the classes An , Bn , Cn , and Dn , and Figure 2. 1 1 (see page 80) represents the Satake graphs for non-compact simple Lie groups in the classes G2 , F4 , E6 , E1 , and Es . Here, the noncompact simple Lie groups are denoted by the same symbols which Cartan used for symmetric spaces with compact simple fun­damental groups defined by the same involutive automorphisms.

§2.lS. Isomorphisms of real simple Lie groups

There are isomorphic groups from different classes among real simple Lie groups. All these isomorphic groups were found by Cartan in the paper [38 ] where he wrote (he designated the rank of the group by 1 ) :

"I. The real groups of the type (A) (/ = 1 ) have the characters J = I or J = -3 and the same ones have groups of the type (B) [and (C) ] .

1 ° J = 1 . There are isomorphisms between:

a) AI

b) All

c) Allla

d) Alllb

§2. 1 5 . ISOMORPHISMS OF REAL SIMPLE LIE GROUPS

- -�

H-t--�-� � a(n-1)(2

CX. a-1 CX.-2 <Xi•+l l/2

k) Cllb

I) Dia

m) Dlb

n) Die

CX.- 1 a1 "2 a, � 0----0---0- - - � � �

...

� CX.- 1

79

e) AIV Q • • - - • • Q a1 a, � al "2 a, a._, a•-1 a. o) Dll 0--•1----il�t- - -�3 a_"---.

a1 � a.1 al+I a!!:"2 �n I) Bia Q---0- - - -o----+ - - -�

� � � _ _ a.a:.2 a� h) Bll 0---0----0- �

i) CI 6------6---a- - -� a1 "2 a, <Xii "21+ 1 �

j) Clla -------.o----e- - - --0---... - - �

...

an-I a1 a, a, a.-�

p) Dllla � - - -e �

a.

FIGURE 2. 1 0

• The special homogeneous linear group in two real vari­ables;

• The linear group of two complex variables x1 , x2 of the Hermitian form x1x1 - x2x2 ;

• The linear group of the real quadratic form x� + xi - xi .

2° o = -3 . There are isomorphisms between: • The linear group of the Hermitian form x1x1 + x2x2 ; • The group X' = AX of one quaternion variable [X] and

one quaternion parameter [A] ; • The linear group of the real quadratic form x� + xi + xi .

II. The real semisimple groups, which are obtained from complex groups formed by subgroups of rank 1 of the type (A) , have the characters o = 2 , 0 , -2 , and -6 , and the same ones have groups of the type (D) .

80

d)

f)

2 . LIE GROUPS AND ALGEBRAS

al � � a4 as

•) Ell� El � al � � a4 as

a6

Elll � � a � � a4 as I aJ a4 as

g) ElV a1 Ia6

• 0 a1 �

a6

FIGURE 2. 1 1

1 ° o = 1 . There is an isomorphism between: . • The linear group formed by the special linear group of

two variables x1 , x2 and by the special linear group of the variables x3 , x4 ;

• The linear group of the quadratic form x� + xi - xi - x; . 2° o = 0 . There is an isomorphism between:

• The special linear group of two complex variables Xi , x2 ; • The linear group of the quadratic form x� + xi + xi - x; .

3° o = -2 . There is an isomorphism between: • The linear group formed by the special linear group of

two real variables x1 , x2 and by the linear group of the Hermitian form x3x3 + x4x4 ;

• The linear group of the quadratic form Xi X2 + x3x4 and of the Hermitian form Xi Xi - x2x2 + x3x3 - x4x4 of four complex variables.

4° o = - 6 . There is an isomorphism between: • The linear group formed by the linear group of the Her­

mitian form Xi x1 + x2x2 and by the linear group of the Hermitian form x3x3 + x4x4 ;

• The linear group of the real quadratic form x� + xi + 2 2 X3 + X4 .

§2. 1 5 . ISOMORPHISMS OF REAL SIMPLE LIE GROUPS 8 1

III. The groups of the type (B) and (C) (/ = 2) can have the char­acters o = 2 , -2 , and - 1 0 .

1 ° o = 2 . There is an isomorphism between: • The linear group of the real quadratic form x� + xJ +

2 2 2 X3 - X4 - X5 ; • The linear group of the real skew bilinear form [x1 x2] +

[X3X4] · 2° o = -2 . There is an isomorphism between:

• The linear group of the real quadratic form x� + xJ + 2 2 2 X3 + X4 - X5 ;

• The linear group of the skew bilinear form [x1 x2 ] + [x3x4] and the Hermitian form x1 x1 + x2x2 + x3x3 - x4X4 of four complex variables x1 , x2 , x3 , and x4 .

3° o = - 1 0 . There is an isomorphism between: • The linear group of the real quadratic form x� + xJ +

2 2 2 X3 + x4 + xs ; • The linear group of the skew bilinear form [x1 x2 ] + [x3x4]

and of the Hermitian form x1 x1 + x2x2 + x3x3 + x4x4 . IV. The real groups of type (A) or of type (D) (/ = 3) can have the characters o = 3 , 1 , - 3 , and - 1 5 .

1 ° o = 3 . There is an isomorphism between: • The special linear group of four real variables (projective

group in the space) ; • The linear group of the real quadratic form x� + xJ +

2 2 2 2 X3 - X4 - X5 - X6 •

2° o = 1 . There is an isomorphism between: • The linear group of the Hermitian form x1x1 + x2x2 -

X3X3 - X4X4 ; • The linear group of the real quadratic form x� + xJ +

2 2 2 2 X3 + X4 - X5 - X6 •

3° o = -3 . There is an isomorphism between: • The linear group of the Hermitian form x1 X1 + x2x2 +

X3X3 - X4X4 ; • The linear group of of the quadratic form x1 x2 + x3x4 +

x5x6 and of the Hermitian form x1x1 - x2x2 + x3.x3 -X4X4 + X5X5 - X6X6 ·

4° o = - 5 . There is an isomorphism between: • The group of X' = AX + BY , Y' = C X + DY of quater­

nion variables X , Y and parameters A , B , C , D ; • The linear group of the real quadratic form x� + xJ +

2 2 2 2 X3 + X4 + X5 - X6 •

5° o = - 1 5 . There is an isomorphism between:

82

a) AI = Bil 0 al

g)

i)

k)

2. LIE GROUPS AND ALGEBRAS

h)

j)

a.4 FIGURE 2. 1 2

e) BI CI

• The linear group of the Hermitian form x1 x1 + x2x2 + X3X3 + X4X4 ;

• The linear group of of the real quadratic form x� + xi + xi + x; + xi + x; . " (38 , pp. 353-355]

These isomorphisms are clearly seen on the Sa take graphs of the real simple groups (Figure 2. 1 2) analogous to the Dynkin graphs on Figure 2 .3 ; here, as in Figure 2 .3 , to the Satake graphs of isomorphic simple groups, the graphs of isomorphic semisimple groups in the class D2 and the direct product of two simple groups in the class A 1 = B1 = C1 are added (as we will see, in one case a noncompact group in the class D2 is a simple group which is isomorphic to a complex simple group in the class A 1 = B1 = C1 ) .

For real noncompact simple Lie groups, in addition to isomorphisms anal­ogous to isomorphisms between complex simple Lie groups, there is one more isomorphism between two noncompact simple Lie groups in the class D 4 , namely, between groups of the types DI and Diii . This isomorphism is a consequence of the triality principle in the spaces with fundamental groups in the class D 4 • The Sa take graphs of these groups are represented in Figure 2. 1 2k.

§2.16. Reductive and quasireductive Lie groups

Semisimple Lie groups are particular cases of reductive Lie groups, whose Lie algebras are the direct sums of simple Lie groups without the requirement of noncommutativity of these simple groups. The term "reductive groups" was introduced by Armand Borel (b. 1 923) and Tits in their paper Reductive

§2. 1 6. REDUCTIVE AND QUASIREDUCTIVE LIE GROUPS 83

groups [BoT] ( 1 965 ) . All commutative groups and all compact Lie groups, as well as the group GLn of all nonsingular matrices of order n , are reductive groups.

If a reductive group G is a group of automorphisms of a commutative group R , the semidirect product G x R , i .e . , the set of pairs (g , r) , g E G , r E R with multiplication

(2. 55 )

where g1 r2 i s the result of application of the automorphism g1 to the el­ement r2 , is called the quasireductive group. The most important example of a quasireductive group is the group of affine transformations in the affine space En .

An important class of quasireductive Lie groups, which are closely con­nected with Cartan's work, are quasisimple Lie-groups-semidirect products G x R defined above where G is a semisimple Lie group. The Lie algebras of these groups can be obtained from the Lie algebras of semisimple Lie groups, represented in the form (2 .43) , by transition to the Lie algebra

(2 .56) 0G = H + eE ,

where e is a dual unit of the algebra ° C of dual numbers. The transition from the Lie algebra (2 .43) to the Lie algebra (2. 56 ) , which is similar to the Cartan algorithm transferring the Lie algebra (2 .43) to the Lie algebra (2 .46) , is called the quasiCartan algorithm. Examples of quasisimple Lie groups are the groups of motions of the Euclidean space Rn and the pseudo-Euclidean spaces R7 (if n = 4 and l = 1 , this is the nonhomogeneous Lorentz group which is important in theoretical physics) . The general definition of qua­sisimple Lie groups was formulated by Katsumi Nomizu (b. 1 924) in the paper Invariant affine connections on homogeneous spaces [No, p. 50] ( 1 954) and by Marcel Berger (b. 1 927) in the paper Non-compact symmetric spaces [Beg3, p. 93] ( 1 957) (the main content of their works will be discussed in Chapters 6 and 7 ) . For a Lie group 0 G obtained from a simple Lie group by the quasiCartan algorithm, Gel 'fand (who called simple Lie groups obtained from one another by the Cartan algorithm "dual groups in the sense of Car­tan") and his coauthors used the name "trial group in the sense of Cartan" (see [BGN] ( 1 956) ) .

If in the definition of a quasireductive group we change a reductive group G for a quasireductive group, we obtain a biquasireductive Lie group. Tri­quasireductive and r-quasireductive groups as well as biquasisimple, tri­quasisimple, and r-quasisimple groups can be defined in a similar manner. Cartan considered biquasisimple groups in the papers On manifolds with an affine connection and the general relativity theory [66] ( 1 922) and On a de­generacy of Euclidean geometry [ 1 47a] ( 1 935 ) .

Transitions from reductive Lie groups to quasireductive groups and from (r - 1 )-quasireductive groups to r-quasireductive groups are particular cases

84 2. LIE GROUPS AND ALGEBRAS

of contraction of Lie groups which were defined in connection with problems of theoretical physics by the famous physicist Eugene Paul Wigner (b. 1 902) and his student Erdal Inonu (b. 1 926) , a son of a President of Turkey (Ismet Inoni.i) and presently ( 1 99 1 ) himself a politician of that country, in the paper On the contraction of groups and their representations [IW] ( 1 9 5 3) (see also the book Contractions and analytic prolongations of classical groups. An analytic approach [Gro] ( 1 990) by Nikolai A. Gromov) .

§2.17. Simple Chevalley groups

In 1 954, in the paper On certain simple groups [Chv3], Chevalley defined analogues of simple Lie groups over fields that are different from the fields R and C , namely, over the Galois fields Fq , the fields QP of p-adic numbers, and the fields Q( a , . . . ) of algebraic numbers. These groups are defined as the groups of automorphisms of the Lie algebras over the corresponding fields with the same integer structure constants as the Lie algebra having the same name over the fields R and C . The analogues of simple Lie groups defined in this manner are called algebraic groups or Chevalley groups. Com­plete classification of simple Chevalley groups was given by Tits in the paper Classification of algebraic semisimple groups [Ti5] ( 1 965 ) .

Simple Chevalley groups over the Galois fields F q or their quotient groups by their centers are finite simple groups. These groups are denoted in the same way as the corresponding complex simple Lie groups. The finite groups An , Bn , Cn , and Dn are the groups FqSLn+ I , Fq02n+ I , FqSp2n , and Fq02n or the quotient groups of these groups by their centers. If the Galois groups of the characteristic equations of simple Lie groups do not coincide with their Weyl groups, there are also "2-twisted" simple groups A�2l , D�2l , E�2l and the "3-twisted" simple group D�3l . The simple Chevalley groups are char­acterized by the same Dynkin graphs as the corresponding Lie groups. The 2-twisted simple Chevalley groups are characterized by the Satake graphs rep­resented in Figures 2. 1 Od, 2. 1 Om, and 2. 1 1 e. The 3-twisted simple Chevalley group is characterized by the Satake graph represented in Figure 2. 1 3 .

The orders (the number of elements) of the finite simple groups corre­sponding to the finite Chevalley groups can be expressed by the single for­mula:

(2 .57) �qN IT (qa;+ I _ I ) , i

where N is the number of positive roots of the corresponding compact simple Lie group, u is the number of the elements of the center of a finite simple Chevalley group, and the numbers a; coincide with the exponents (2. 34) and (2. 3 5 ) of corresponding simple Lie groups. The orders of the twisted finite

§2 . 1 8. QUASIGROUPS AND LOOPS

FIGURE 2. 1 3

Chevalley groups A(2) D(2) E(2) and D(3l are equal respectively to n ' n ' 6 ' 4

(2. 58 )

tqN I]; (q i+ I - (- 1 ) i+ l ) ' tqN (qn + l ) IJ; (q2i _ l ) ' tqN (q l 2 - l ) (q9 + l ) (q8 - l ) (q

6 - l ) (q 5 + l ) (q2 - 1 ) '

tqN (q8 + q4 + l ) (q6 - l ) (q2 - 1 )

(see Tits's talk [Ti4, pp. 2 1 3-2 1 4] ) .

85

Note that while only one noncommutative division algebra, namely, the field H , can be defined over the field R , it is possible to define different division algebras of dimension m2 over the fields Q

P and Q(a , . . . ) (when

a field is extended to an algebraically closed field, these algebras become algebras of matrices of order m over this field) . Thus, for the fields QP and Q(a , . . . ) , there are Chevalley groups of the class An whose Satake graphs differ from the Satake graphs of the groups All (Figure 2. 1 Ob) by the fact that each black point of the graph is replaced by m - 1 black points.

§2.18. Quasigroups and loops

Recently, in geometry as well as in theoretical physics, algebraic systems with nonassociative operations have become of greater importance. First of all, quasigroups and loops should be named among such systems. A quasi­group is a set Q where a binary operation is defined which to any two el­ements x and y assigns a third one, z , x o y = z , and is invertible with respect to each of the factors on the left-hand side of this equation. In general, this operation is not associative. If there is a two-sided unit in a quasigroup, such a quasigroup is called a loop. A group is a particular case of a loop-it is a loop whose binary operation is associative. Weakening in different ways the associativity condition, one can obtain the most important classes of loops: the Moufang loops defined by Ruth Moufang ( 1 905- 1 977) , who also introduced the term "quasigroup", the Bo! loops defined by Gerrit Bol ( 1 906- 1 987) , and the monoassociative loops.

If a set of elements of a loop is a manifold and its algebraic operation can be expressed by differentiable or analytic functions, a loop is called smooth or analytic, respectively. Such a loop is a nonassociative analogue of a Lie group, and it is possible to construct an analogue of the Lie algebra for such a loop. For analytic Moufang loops, such algebras were defined by Anatolii

86 2. LIE GROUPS AND ALGEBRAS

I. Mal'cev ( 1 909- 1 967) in the paper Analytic loops [Mal] ( 1 955 ) . At present, these algebras are called Ma( cev algebras. For the Bol loops, such algebras were constructed by Lev V. Sabinin (b. 1 932) and Pavel 0. Mikheev in the paper On analytic Bo/ loops [SM] ( 1 982) . In contrast to Lie algebras and Mal 'cev algebras, they have not only a binary but also a ternary operation. For general loops, such algebras were constructed by Maks A. Akivis (b. 1 923) in the paper The local algebras of a three-dimensional three-web [Ak7] ( 1 976) (see the end of Chapter 7 of this book on the connection of quasigroups and webs) .

It is well known that a Lie algebra completely defines a local Lie group. A similar theorem for Moufang loops was proved by Evgenii N. Kuz'min (b. 1 938) in 1 97 1 and for Bol loops by Sabinin and Mikheev in 1 982. For general loops and quasigroups, a theorem of this kind is not valid. However, there exist certain classes of quasigroups defined by a certain number of constants. In 1 985 , Alexander M. Shelekhov (b. 1 942) proved that mono-associative loops form one of these classes. However, there is no complete description of local algebras connected with such loops.

Moufang loops are the closest to Lie groups. At present, the theory of Moufang loops has been extensively developed. In particular, the theory of simple smooth Moufang loops which is similar to the theory of simple Lie groups was constructed by A. S. Sagle in the paper Simple Mal ' cev algebras over fields of characteristic zero [Sag] ( 1 962) . In this paper, Sagle showed that simple smooth Moufang loops are hyperspheres l a d = l of alternative algebras 0 , '0 and 0 ® C of octaves, solit octaves, and complex octaves.

The current status of the theory of quasigroups and loops is described in the book Quasigroups and loops: Theory and applications [CPS] ( 1 990) edited by Orin Chein (b. 1 943) , Hala 0. Pflugfelder and Jonathan D. H. Smith (b. 1 949) (see also the textbook Quasigroups and loops: Introduction [Pfl] ( 1 990) by H. 0. Pflugfelder) . In the book [CPS] we note the chapters Local differ­entiable quasigroups and webs [Glb2] by Vladislav V. Goldberg (b. 1 936) , Quasigroups and differential geometry [MS] by Sabinin and Mikheev, and Topological and analytic loops [HS] by Karl H. Hofmann and Karl Stram­bach.

CHAPTER 3

Projective Spaces and Projective Metrics

§3.1 . Real spaces

In the titles of papers [37] and [39] Cartan called linear representations of simple Lie groups "projective groups", i .e . , groups of projective transforma­tions (collineations) of projective spaces CPn and Pn . We saw earlier that complex simple groups in the classes An , Bn , Cn , and Dn are represented by the groups of collineations of the spaces CPn , the groups of motions of the non-Euclidean spaces CS2n , the groups of symplectic transformations of the symplectic spaces csy2n- t , and the groups of motions of the non-Euclidean

Cs2n- 1 . l spaces , respective y.

Real simple Lie groups admit similar geometric interpretations in the real forms of these spaces: in the real projective space Pn and in the real non­Euclidean spaces-the elliptic spaces S2n and S2n- t , the hyperbolic spaces s;n and s:n- I , and the symplectic space sy2n- I . All these spaces are de­fined in the same way as the spaces CPn ' cs2n ' cs2n- I and csy2n- I were defined. In addition, note that for the spaces SN the left-hand side of the quadric equation aiix

i xi = 0 is a positive definite quadratic form, i .e . , this

equation can be reduced to the form l::; (xi )2 = 0 , and for the spaces s{

the left-hand side of the quadric equation is a nondegenerate form of index I , i .e . , this equation can be reduced to the form - Ea(x0/ + l::; (x

i ) 2 = 0 , where the number of negative terms is equal to / . The spaces defined in this manner, where the classical groups can be interpreted, are called the classical spaces. Cartan's book Lectures on complex projective geometry [ 1 34] ( 1 93 1 ) was devoted to geometries of many of these spaces.

Historically, the first of these spaces which is different from the Euclidean space was the hyperbolic Lobachevsky space S� . The geometry of the space S� was discovered by Nikolai I. Lobachevsky ( 1 792- 1 856 ) , who presented it for the first time in his paper On the principles of geometry [Lob 1 ] ( 1 829) ; by Janos Bolyai ( 1 802- 1 860) , who presented his discovery in the form of an Appendix [Boy] to the book of his father in 1 832; and by Gauss, who arrived at the same geometry before Lobachevsky but did not publish his discovery during his lifetime.

87

88 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

We have already mentioned that hyperbolic geometry was widely recog­nized by mathematicians only in the 1 870s, the years of Cartan's childhood. This recognition was made possible by a series of important discoveries of the 1 9th century. In the middle of this century the treatment of projective properties of figures became independent projective geometry, which, in the book Geometry of position [Sta] ( 1 847) , Christian von Staudt ( 1 798- 1 867) freed from definitions connected with Euclidean geometry. In 1 8 59 , Cayley showed, in A sixth memoir upon quantics [Cay2] , that the Euclidean plane can be considered as the projective plane where, in addition, a line ("the line at infinity of the Euclidean plane") and a pair of imaginary conjugate points on it (the pair of "cyclic points" in which this line intersects all circles) are given, and the "elliptic plane" , i .e . , a sphere with antipodal points being identified, can be considered as the same projective plane where, in addition, an imaginary conic is given. Cayley thus exclaimed: "Metrical geometry is thus a part of descriptive geometry, and descriptive geometry is all geometry, and reciprocally. " [Cay2] ( see [Cay, vol. 2, p. 592] ) .

At the same time, i n the papers of August Ferdinand Mobius ( 1 790- 1 866) , the treatment of affine properties of figures became affine geometry, and the treatment of circular transformations in the plane generated by inversions with respect to circles became conformal geometry (also called Mobius ge­ometry and inversive geometry). At the time that the geometries of projec­tive, affine, and conformal planes were created, the geometry of the three­dimensional projective, affine, and conformal spaces arose. After the pub­lication of Grassmann's paper The science of linear extension [Gra] ( 1 844 ) , the geometries of the multidimensional Euclidean space Rn , the multidimen­sional hyperbolic space s� ' the multidimensional elliptic space sn ' the mul­tidimensional projective space pn , the multidimensional affine space En , and multidimensional conformal space en appeared.

The recognition of Lobachevsky's hyperbolic geometry came in 1 868, when Eugenio Beltrami ( 1 83 5- 1 900) constructed an interpretation of the hyper­bolic plane S� in a circle of the Euclidean plane, and in 1 870, when Klein showed that the plane S� can be realized as a part of the projective plane P2 bounded by a real conic. In Klein's interpretation, the motions of the plane S� are represented by projective transformations preserving the conic (if this conic is a circle, Klein's interpretation coincides with Beltrami's) . The mul­tidimensional spaces sn and S� can be realized as the projective space Pn , where an imaginary quadric is given, and as a part of the space Pn bounded by an oval quadric, respectively. The motions of the spaces sn and S� are represented by projective transformations of the space pn preserving these quadrics.

In 1 882, Poincare proposed another interpretation of the Lobachevskian plane in a circle of the Euclidean plane. In this interpretation, motions of the Lobachevskian plane are represented by circular transformations preserving

§3. 1 . REAL SPACES 89

the circumference of the circle. (In this interpretation, a circle can be replaced by a half-plane. ) Similarly, the space S� can be represented as the interior of a hypersphere in the space en , and, in .this interpretation, the motions of the space s� are presented by conformal transformations of the space en preserving this hypersphere.

Along with the Lobachevskian space S� represented as the interior of an oval quadric of the space Pn , one can also consider the exterior of this quadric which is called the ideal domain of the space S� (in this case, the space defined by Lobachevsky is called the proper domain of the space S� ) . The space S� , considered a s a set of the proper and the ideal domain and the oval quadric which divides these domains, is a particular case of the hyperbolic space SF-the space pn in which a quadric of index l is given whose equation can be reduced to the form E; 8; (x

; )2 = 0 , where i = 0 , 1 , . . . , n , 8; = - 1 for i < l , and 8; = 1 for i ;:::: I . This quadric is called the absolute of the space s; , and motions of this space are projective transformations of the space pn preserving the absolute.

If in the definition of the space s; we replace the quadric a;1x; x1 = 0

by a linear complex aiJpiJ = 0 (aiJ = -a1; ) of straight lines, where piJ =

xiyJ - /x1 are the Plucker coordinates of a straight line XY (joining the points X(x; ) and Y(/) ) , we obtain the space Syn . In the 1 9th century, this space was called "the space of linear complex". Later, after Weyl proposed calling the "group of linear complex" the symplectic group, this space was given the name symplectic space. In the space Syn , the "null-system" U; = a;1x

1 (aiJ = -a1; ) sending each point X(x; ) into the hyperplane u;x; = 0

passing through this point plays the role which the polar transformations U; = aiJx

1 (a;1 = a1; ) play in the spaces s; . The linear complex aiJpiJ = 0

is called the absolute linear complex of this space and consists of isotropic straight lines which are transformed into the (n - 2)-planes passing through these lines. Since the null-systems are nondegenerate (their determinants det(a;) are different from zero) only for odd values of n , the dimensions of symplectic spaces are odd.

After Einstein's special relativity was discovered ( 1 905) , the notion of a pseudo-Euclidean space R7 appeared. A pseudo-Euclidean space R7 is an affine space En in which an inner product of vectors xy is defined that can be reduced to the form xy = E; 8;X

i/ , i = I , . . . , n , and the numbers 8; have the same values as they have for.quadrics of index l + I . The space-time of the special relativity is the space R� .

If we extend the space R7 by adding the point at infinity which, under inversions in hyperspheres of this space, corresponds to the centers of these hyperspheres, and by adding the "ideal points" corresponding, under these in­versions, to the points whose distances from these centers are equal to zero, we obtain the pseudo-conformal space e; . As we saw for the conformal

90 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

space en , conformal transformations of the space c; are its transforma­tions that are generated by inversions in its hyperspheres.

In 1 887, Poincare proposed another model of the Lobachevskian plane on a two-sheeted hyperboloid with identified antipodal points. As the distance between two points of the hyperboloid Poincare took a number which is proportional to the cross ratio of the rays going to these points and of two asymptotes lying in the plane of these two rays.

From the modern point of view, this definition of distances on a hyper­boloid can be formulated in the following manner. If we write the equation of a two-sheeted hyperboloid in the form L::; e; (x; )2 = -q2 and introduce in the space the metric of the space R: with inner product xy = L:; e;x; / , the hyperboloid will be a sphere of pure imaginary radius in the space R: , and the Poincare metric on this hyperboloid will coincide with the metric of this sphere. The models of Beltrami-Klein and Poincare in circles are closely connected with the last Poincare model. Namely, if we project the upper half of a sphere of pure imaginary radius qi of the space R: from its center onto a tangent plane to that sphere (which is a Euclidean plane) , we obtain the Beltrami-Klein model (Figure 3 . 1 ) , and if we project the same upper half from its "south pole" onto its equatorial plane (which is also a Eu­clidean plane) , we obtain the Poincare model in a circle (Figure 3 .2) , which essentially is a stereographic projection of the sphere onto this plane. The interpretation of the hyperbolic plane S� on a sphere of imaginary radius explains an important similarity of hyperbolic geometry and usual spherical geometry. As was noticed by Lobachevsky himself, trigonometric formu­las in the hyperbolic plane can be obtained from formulas of the spherical trigonometry if we consider the lengths of the sides of a triangle to be pure imaginary (i .e . , these formulas are trigonometric formulas on a sphere of imaginary radius) , and motions of the plane S� are rotations of the space R: . The fact that the space-time of the special relativity can be considered as the space Ri and the group of Lorentz transformations (which is important in this theory) is isomorphic to the group of rotations of the space Ri is the cause for the deep connections between hyperbolic geometry and the special relativity (in particular, the formula for addition of velocities in the latter theory is equivalent to the law of cosines in hyperbolic geometry) .

The elliptic space Sn can also be realized as the geometry on a hypersphere of the space Rn+ I with identified antipodal points and as the geometry on the hyperplane at infinity of the space Rn+ I . The imaginary quadric of this hyperplane in which it intersects all hyperspheres of the space Rn+

1 plays the

role of the absolute of the space Sn . Similarly, the hyperbolic space s; can be interpreted as a geometry on the hyperplane at infinity of the space R�+ i , and each of the domains in which the space s; is divided by its absolute can be interpreted as a hypersphere of real or pure imaginary radius with

§3. 1 . REAL SPACES 9 1

FIGURE 3 . 1

FIGURE 3 .2

identified antipodal points in the space R7+ ' . If the radius of a hypersphere, on which the space sn or one of the domains of the space s; is interpreted, is equal to r or q i , then the distance w between the points X(x; ) and Y (/) of the space Sn or one of the domains of the space s; is respectively defined by the following formulas:

(3 . 1 )

(3 .2)

2 (J) o::::: . xi/)2 cos - __ ,.,...,, �-�

r - E; (x; )2 E; (/ )2 '

92

( 3 . 3 )

3. PROJECTIVE SPACES AND PROJECTIVE METRICS

i i 2 h2 (J) (L i B;X y )

cos q = L; B; (Xi ) 2 L; B; (Yi ) 2 .

The numbers 1 / r2 and - 1 / q2 are called the curvatures of the spaces sn and s; , respectively.

Note that in the space Sn of curvature 1 /r2 the area of a triangle ABC with angles A , B , and C is equal to

( 3 .4) 2 S = r (A + B + C - n)

and the area of such a triangle in the space s; of curvature - 1 / q2 is equal to

( 3 . 5 ) 2 S = q (n - A - B - C) ,

where the angles A , B , and C in formulas ( 3 .4) and ( 3 . 5 ) are measured in radians.

The spaces Rn , En , Pn , Sn , S� , and en were considered by Klein in his Erlangen program [Kle] ( 1 872) . In this program, Klein also formulated cer­tain "transfer principles" which enable one to interpret one space within an­other. These principles are based on the isomorphisms of the groups of trans­formations of these spaces (Cartan called geometries of such spaces equivalent geometries) and on the interpretation of the space en on the absolute of the space s�+ t . The latter interpretation is based on the fact that if one takes the angle between two hyperspheres as the distance between them (if the hy­perspheres are tangent to one another, this angle is equal to zero, and if the hyperspheres have no real intersection, this angle is imaginary) , then the set of hyperspheres of the space en is isometric to the ideal domain of the space s�+ I of curvature 1 . Furthermore, the points of the space en that can be considered as the hyperspheres of radius zero are represented by the points of the absolute of the space s�+ t , imaginary hyperspheres are represented by the points of the proper domain of the space s�+ t , and the conformal transformations of the space en are represented by motions of the space s�+ t . Similarly, one can prove that the space C1n is realized on the absolute of the space s;:/ , and the conformal transformations of the space c; are represented by motions of the space S'/:/ .

The groups of rotations of the spaces Rn and R7 , the subgroups of affine transformations of the space En keeping fixed one point of this space and preserving the volumes of parallelotopes (such transformations are called centro-affine) , the groups of motions of the spaces sn and s; , the group of projective transformations of the space Pn , the group of conformal trans­formations of the space en , and the group of symplectic transformations of the space sy2n- t (the groups of projective transformations of this space preserving its absolute linear complex) are all simple Lie groups except the

§3.2. COMPLEX SPACES 93 groups of rotations of the spaces R4

and R� and the group of motions of the spaces S3 and si ' which are semisimple groups.

Properties of the corresponding Lie groups were substantially used by Car­tan in his works on differential geometry of the spaces Rn , En , pn , and en .

The extended Cartan translation [ 46] of the Fano paper [Fa] ( 1 907) , pub­lished in the German edition of Encyclopaedia of Mathematical Sciences, was devoted to the connections between Lie groups and various spaces.

Cartan's translation [46] had the title Theory of continuous groups and ge­ometry. We have already mentioned that in 1 9 1 4 only 2 1 pages of this paper were published; because of the beginning of World War I, the rest of paper was not published at that time. The complete text of the paper was published only after Cartan's death in his CEuvres Completes [207] . (As was the case with Cartan's extended translation of Study's paper [Stu l ] on complex num­bers and their generalizations, Cartan's translation contains many additions; Fano's paper had 1 00 pages, and Cartan's translation had 1 34 pages) .

§3.2. Complex spaces

In the 1 9th century, along with real spaces, complex spaces obtained by the transition from real coordinates to complex ones were the subject of study. The complex projective space CPn was especially widely used in algebraic geometry. In addition, the complex Euclidean space CRn , the complex non­Euclidean space CSn , and occasionally, the complex conformal space cen realized on the absolute of the space csn+ i , and the complex symplectic

Cs 2n- I 'd d space y were cons1 ere . We have already noted that the group of projective transformations of the

space CPn is a complex simple Lie group in the class An , and complex sim­ple Lie groups in the classes Bn , en , and Dn are represented by subgroups of groups of projective transformations of complex projective spaces. When Cartan gave the title "Projective groups, under which no plane manifold is invariant" to his paper [37] on linear rep�esentations of complex simple Lie groups, he meant exactly this latter representation.

We have already mentioned that the group of motions of the space CS2n is a complex simple group in the class Bn ; the group of symplectic transfor­mations Of the space CSy2n- I is a complex simple group in the class en ; and the group of motions of the space cs2n- I is a complex simple group in the class Dn (for n = 2 , the latter group is a semisimple group locally isomorphic to the direct product of two complex simple groups B1 ) .

In his extended translation of Fano's paper, along with complex spaces that were the subject of study in the 1 9th century, Cartan considered some new complex spaces by means of which certain real simple Lie groups can be realized. First of all, he introduced the Hermitian elliptic space CSn and the Hermitian hyperbolic space cs; , the geometries of which were first studied by Guido G. Fubini ( 1 879- 1 943) in the paper On definite metrics of

94 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

a Hermitian form [Fub2] ( 1 903) , and by Study in the paper Shortest paths in the complex domain [Stu4] ( 1 905) . The space CSn can be defined as the space CPn in which a positive definite real metric is given with the distance w between points X and Y defined by the formula

(3 .6) 2 w L:i xii · L:i /xi

cos - = . . . . ' r L:i x'x' L:i y 'y' which differs from formula ( 3 . 1 ) by substitution of the Hermitian forms · · · 2 -n L:i x'x' for the quadratic forms L:i (x' ) . The space CS1 is defined as the space CPn with the distance w between points X and Y defined by the formula

2 w L: · e .x ii · L: · e./xi (3 .7 ) COS - = I I . . I I . . ' r L:i eix'x' L:i eiy 'y'

which differs from formula (3 .2 ) by substitution of the Hermitian forms . . . 2 L:i eix 'x' for the quadratic forms L:i ei (x' ) . The groups of motions of the spaces CSn and CS7 are represented by

matrices of a compact group in the class An (the matrix group CS Un+ I ) and of a noncompact group in the same class (the matrix group CSU�+ 1 ) .

Cartan considered in detail the geometry of the spaces CS3 and CS� in his above-mentioned Lectures on complex projective geometry [ 1 34] ( 1 93 1 ) . He showed that the line CS1

is isometric to a sphere of radius � of the space R3 (while the complex projective line CP 1 can be considered as the extended complex plane, the complex Hermitian elliptic line CS1

can be considered as the Riemannian sphere) .

In the above-mentioned paper [Stu4] , Study also defined the complex Her­mitian Euclidean space CRn as the space CEn in which a Hermitian inner product is defined, and this product can be reduced to the form L:i x

ii ; the Hermitian inner square L:i x

ixi is defined as the square of the modu­lus !x i of a vector x = {x i } , and the distance between the points X and Y is defined as the modulus IY - xi of the vector y - x . The space CRn , which is also called the unitary space, is often used in linear algebra. This space is isometric to the Euclidean space R2n • The group of motions of the space CRn consists of transformations (2 . 8 ) where U = (u� ) are ma­trices of the group C Un and the products of these transformations and the transformation I xi = xi . Substituting in the definition of the space CRn the inner product L:i eix

ii for the inner product L:i xii , we obtain the

complex Hermitian pseudo-Euclidean space CR7 , which is isometric to the space R;7 . The group of motions of this space consists of transformations (2. 8 ) where U = (u� ) are matrices of the group CU� and the products of

these transformations and the transformation 'x i = xi .

§3. 3. QUATERNION SPACES 95

§3.3. Quaternion spaces

Among Cartan's supplements to Fano's paper [Fa] , we must mention the introduction of the quaternion projective space HPn , which can be obtained from the complex projective space CPn with the substitution of quaternion projective coordinates (defined up to multiplication by a quaternion factor from the right) for complex projective coordinates. The group of projective transformations of the space HPn , i .e . , the group HSLn+ I of quaternion unimodular matrices, is one of the noncompact groups in the class A2n+ I .

In the paper On certain remarkable Riemannian forms of geometries with a simple fundamental groups [ 1 07] ( 1 927) , Cartan used the representation of a compact simple Lie group in the class Cn by the group HUn of quaternion unitary matrices. It follows from this representation that the latter group can be considered as the group of motions of the quaternion Hermitian Euclidean space HSn- I , i .e . , the quaternion space HPn- I in which a real metric is de­fined by the same formula (3 .6) as in the space CSn- i . Similarly, one can use formula (3 . 7) to define a real metric of the quaternion Hermitian hyper-

-n- 1 -I bolic space HS1 • In the same manner as the isometry of the line CS to a sphere of radius r /2 in the space R3 was proved, one can prove the isom­etry of the line HS1

to a hypersphere of radius r /2 in the space R5 • The

space HSn was defined in the paper Symmetric spaces and their geometric applications [Ro l ] by Boris A. Rosenfeld (b. 1 9 1 7) , which was published as a supplement to the collection of his translations of Cartan papers titled Ge­ometry of Lie groups and symmetric spaces [206] ( 1 949) . In the same paper, Rosenfeld proved that the space CSn is isometric to a paratactical congru­ence of straight lines of the space s2n+ 1 if one takes as the distance between lines of the congruence their unique stationary distance and that the space HSn is isometric to a para tactic congruence of the space cs2n+ I . Further­more, in this paper Rosenfeld defined the Hermitian hyperbolic spaces cs; and Hs; . This material was presented in detail in Chapter VI of Rosenfeld's book Non-Euclidean geometries [Ro3] ( 1 955 ) (see also [Ro7] ) . The complex and quaternion Hermitian elliptic and hyperbolic spaces CSn , cs; , HSn , and Hs; can be defined as the space CPn or the space HPn whose fun­damental groups are subgroups of the group of projective transformations preserving the Hermitian hyperquadric xj aijx

i = 0 (aij = aji ) , and there­fore commuting with the polarity relative to this hyperquadric defined by the formula U; = Xj aij .

As Hilbert showed in his book Foundations of geometry [Hil] ( 1 899) , in spaces over nonassociative fields the Desargues configuration does not take place, and this is the reason that the geometry of such spaces (the plane OP2 is one of them) is a non-Desarguesian geometry. In the same book Hilbert proved that, if n > 2 , the Desargues configuration follows from the incidence axioms of the projective geometry, and this is the reason that a

96 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

non-Desarguesian geometry is impossible in a space of dimension greater than two.

In the paper Alternative fields and the theorem on complete quadrilateral (D9) [Mou] ( 1 933 ) , Moufang proved that, in a projective plane over an al­ternative skew field, the configurational theorem on complete quadrilateral holds. Using this paper of Moufang, Guy Hirsch (b. 1 9 1 5 ) , in his paper The projective geometry and the topology of fiber spaces [Hir] ( 1 949), defined the plane OP2 by topological methods and proved that the straight lines of this plane are homeomorphic to eight-dimensional spheres. In the paper The projective plane of octaves and spheres as homogeneous spaces [Bor l ] ( 1 950) , A. Borel defined the plane OS2 as the plane OP2 (defined by Hirsh) with the metric of the symmetric space V 1 6 • In the paper Octaves, exceptional groups and the geometry of octaves [Frd l ] ( 1 9 5 1 - 1 985 ) , Freudenthal defined the planes OP2 and OS2 algebraically by means of the octave coordinates in these planes.

Reducing the group of projective transformations of the space HPn to its subgroup consisting of projective transformations commuting with the "null-system" u; = xi aii (aii = -aii ) , we obtain the quaternion Hermitian symplectic space HSyn defined by Ludmila V. Rumyantseva (b. 1 937) in the paper Quaternion symplectic geometry [Ru] ( 1 963 ) . In the space HSyn , one can always choose a coordinate system in which the "absolute null-system" of this space will be of the form uk = xk i . (The complex Hermitian symplectic spaces defined in a similar manner coincide with the spaces csn and cs; since multiplication of a skew-Hermitian matrix ( aii ) by i gives a symmetric Hermitian matrix.)

Note that, as was shown by David Hilbert ( 1 862- 1 943) in his book Foun­dations of geometry [Hil] ( 1 899) , in spaces over noncommutative skew fields the Pappus-Pascal configuration does not take place, and this is the reason the geometry of such spaces (the space H pn is one of them) is a non-Pascalian geometry.

§3.4. Octave planes

A compact simple group in the class G2 , which is the group of automor­phisms of the alternative skew field 0 of octaves, is the transitive group of rotations of the six-dimensional sphere that is the intersection of the hyper­sphere l a l = 1 of octaves of modulus one with the hyperplane a = -a . Furthermore, the metric of the space R8 is introduced into the field 0 of octaves: in this metric the distance between two octaves a and p is the modulus I P - a l of their difference. Identifying antipodal points of this hy­persphere, we obtain the elliptic space S6 whose fundamental group is the group indicated above. Such a space is called the group of motions of the G-elliptic space and is denoted by Sg6 •

§3 .5 . DEGENERATE GEOMETRIES 97

A noncompact simple group in the class G2 which is the group of auto­morphisms of the algebra '0 of split octaves admits a similar representation as the group of rotations of the six-dimensional sphere of the space R� and the G-hyperbolic space S g: . A. Borel, in the paper Octave projective plane and spheres as homogeneous spaces [Bor] ( 1 950) , and Freudenthal, in the pa­per Octaves, exceptional groups and octave geometry [Frd l ] ( 1 95 1 ) , showed that a compact simple group in the class F4 is the group of motions of the

Hermitian elliptic plane OS2 which is defined in the same way as the planes -2 -2 CS and HS , and that one of the noncompact simple groups in the class

E6 is the group of projective transformations of the projective plane OP2 •

In contrast to the planes CP2 and HP2 , the plane OP2 cannot be defined by means of arbitrary triplets x0 , x 1 , and x2 of elements of the field 0 defined up to multiplication by an arbitrary element a of this field since nonassociativity of the field 0 implies (x;a )P =/; x; (ap ) . However, points of the plane OP2 can be defined by means of triplets x0 , x 1 , and x2 from an associative subfield of the field 0 defined up to multiplication by an arbitrary element of this associative subfield. Freudenthal defined points of the planes OP2 and OS2 by means of Hermitian symmetric octave matrices . . . . . . k k . . x'1 = x'1 satisfying the condition x'1 x1 = x' xn , from which it follows that all octaves xiJ belong to an associative subfield and thus it is possible to find three octaves x; in this skew subfield such that xij = xixj .

Note that in the paper Two-point homogeneous spaces [Wan] ( 1 952) , Wan Hsien Chung showed that any compact metric space in which, for any two equidistant pairs of points, there is a motion transforming one of these pairs into the other, is isomorphic to a hypersphere of the space R

n+ t , the spaces n -n -n -2 S , CS , HS and the plane OS . In a series of papers under the common title The relations between E7 and E8 to the octave plane [Frd2] ( 1 954- 1 963) , Freudenthal also defined the

octave Hermitian symplectic space osy 5 . However, since it is impossible to define projective spaces of dimension higher than two over nonassociative

- 5 5 fields, the space OSy cannot be defined as the space OP with the funda-mental group being a subgroup of the group of projective transformations of

- 5 2 this space. The space OSy can only be defined as a set of planes OP that are the analogues of two-dimensional isotropic planes of the space sy 5 .

§3.5. Degenerate geometries

In Chapter 2, we defined the quasi-Cartan algorithm transforming a Lie group G with the Lie algebra G = H EB E into a Lie group 0 G with the Lie algebra 0G = H EB eE where e is the dl,lal unit of the algebra

0c of dual numbers a + be (e2 = 0) . If G is a semisimple Lie group, the Lie group 0 G is called a quasisimple group. Application of this algorithm r times to a semisimple Lie group leads to an r-quasisimple Lie group. The quasi-Cartan

98 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

algorithm can also be applied to both associative algebras and the alternative algebras: if such an algebra A possesses an involutive automorphism and can be represented as the direct sum A = B EB C , where B is a subalgebra consisting of the elements that are invariant under this automorphism and C is a linear subspace consisting of the elements that are anti-invariant under this automorphism (i .e . , under this automorphism, they are multiplied by - 1 ) , then the quasi-Cartan algorithm transfers this algebra A into the algebra 0c = B EB eC . The algebra 0c itself is obtained in this manner from the field C and the algebra 'C of split complex numbers which have undergone the involutive automorphism a --+ a . Applying this algorithm to the algebras H and 'H and their automorphism a --+ iai- 1 , we obtain the algebra 0H of semiquaternions, and, applying this algorithm to the algebras 0 and '0 and their automorphism, under which the field H is an invariant subalgebra, we obtain the alternative algebra 0o of semioctaves.

In Chapter I of the memoir On manifolds with an affine connection and general relativity theory [66] ( 1 923) devoted to relativity theory, Cartan con­sidered the transformations of the space and time coordinates of classical Galilei-Newton mechanics which he wrote in the form:

( 3 . 8 )

x' = a1 x + b1y + c1 z + g1 t + h 1 , y' = a2x + h2Y + C2Z + g2 t + h2 ' z' = a3x + b3y + c3 z + g3 t + h3 , t' = t + h ,

where the matrix with entries ai , bi , and ci is an orthogonal matrix of order three. Cartan assumed that x , y , z , and t are the coordinates of a point of a four-dimensional space whose fundamental group is the group of transformations

(3 .9 )

x' = a 1 x + a2y + a3 z + h 1 , y' = b1 x + b2y + b3 z + h2 , z' = c1 x + c2y + c3 :z + h3 , t' = g1 x + g2y + g3 z + t + h .

This space coincides with an isotropic hyperplane of the pseudo-Euclidean space Ri . The hyperplane is tangent to the isotropic cone of Ri . This is the reason that the space Ri is called the isotropic space and is denoted by In . The rotations of this spaces (motions ( 3 .9) with h 1 = h2 = h3 = 0) transform the vector {O , 0 , 0 , 1 } into itself, and the direction of the axis Ot is invariant under these rotations. The space Ri can be considered as the affine space E4 , in whose hyperplane at infinity a point ( in the direction of the axis Ot) and an imaginary hypercone of second order (defining in this hyperplane the geometry of the co-Euclidean space * R3 , dual to the space R3 ) are defined. The group of motions (3 .9 ) of this space is a biquasisimple Lie group which can be obtained by applying the quasi-Cartan algorithm to

§3 .S . DEGENERATE GEOMETRIES 99

the groups of motions of the spaces R4 and R1 . Later, the space /4 was considered by Karl Strubecker ( 1 904- 1 99 1 ) in the paper Differential geometry of isotropic space [Str] ( 1 94 1 ) . Alexander P. Kotelnikov ( 1 865- 1 944) in the paper The principle of relativity and the Lobachevskian geometry [KotA3] ( 1 926) considered the space with the group of motions ( 3 . 8 ) . He thought that this space is the space-time of Galilei-Newton mechanics. Since this space is the affine space E4 , with the geometry of the space R3 in its hyperplane at infinity, it is called the Galilean space and is denoted by r4

• The group of motions of this space is also a biquasisimple Lie group. In the same chapter of the memoir [66] , Cartan considered a four-dimensional manifold whose tangent spaces are the spaces /4 • Such a manifold can be called the "space with an isotropic connection".

Cartan's note On a degeneracy of Euclidean geometry [ 1 47a] ( 1 935 ) was devoted to a two-dimensional isotropic geometry. In it, Cartan considered the geometry of the "isotropic plane" /2 , i .e . , a plane of the space R: tan­gent to an isotropic cone. The geometry of such a plane coincides with the geometry of the Galilean plane r2 . Cartan's note was the exposition of his talk at a session of the French Association for the Development of Science in Nantes. Despite the fact that similar talks by Cartan at other sessions of this association were included in Cartan's <Euvres Completes [207] and [209], this note is missing from them. The note began as follows: "The geometry in an isotropic plane differs deeply from the geometry of the classical plane: in this plane, the lines, that in a non-isotropic plane play the role of circles, are parabolas tangent to the line at infinity at the same point" [ 1 47a, p. 1 28] . Cartan wrote the motions of this plane in the following form:

{x' = x + a , y' = ex + hy + b.

If h = 1 , these transformations are analogues of transformations ( 3 .9) for the two-dimensional case. If h =F 1 , these transformations are analogues of similarities of the plane R2

•

The "isotropic plane" is the affine plane E2 , in whose line at infinity a point is defined. The latter can be considered as a result of coincidence of the imaginary cyclic points of the plane R2 or the real cyclic points of the plane R� . (While in the planes R2 and R� circles are conics passing through the cyclic points of these planes, in the plane /2 obtained by the passage to the limit from these planes, the role of circles is played by conics tangent to the line at infinity at the point of coincidence of cyclic points, i .e . , parabolas with diameters directed toward this point . )

The group of motions of the plane r2 is a biquasi-simple Lie group ob­tained by the quasi-Cartan algorithm from the groups of motions of the planes R2 and R� . In the note [ 1 47a], a number of problems on an isotropic plane were solved.

100 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

The title of the note [ 1 47a] shows that Cartan considered the passage from the geometry of the Euclidean plane to that of the plane /2 as a "degeneracy of Euclidean geometry". Note that this degeneracy is not the only possibil­ity. The idea of more general "degenerate geometries" obtained from the non-Euclidean spaces sn and S� was suggested in Klein's lectures on non­Euclidean geometry ( 1 9 1 0) , and the complete enumeration of all such geome­tries was given by Duncan Maclaren Young Sommerville ( 1 879- 1 934) in the paper The classification of geometries with projective metrics [Som] ( 1 9 1 0) . A year later, i n the paper Euclidean kinematics and non-Euclidean geome­try [Bla l ] ( 1 9 1 1 ) , Wilhelm Blaschke ( 1 885-1 962) considered an important case of a degenerate elliptic geometry-the geometry of the quasielliptic space S1 • 3 • In 1 9 1 2, in the paper Construction of the entire geometry on the basis of the projective axioms alone [Mil] , Ch. Muntz arrived at the same geometries as Sommerville. Blaschke defined the metric in the quasielliptic space S1

• 3

as an analogue of the Cartan metric for the group of motions of the Euclidean plane R2 • Blaschke also introduced the term "quasielliptic space". In 1 966 Freudenthal analogously proposed the term "quasisimple group". The latter term was the cause of the terms "quasisimple algebra", " r-quasisimple group and algebra" and "quasi-Cartan algorithm".

The quasielliptic spaces Sm · n whose groups of motions are represented by matrices of the quasi-simple group On+I can be defined as the space Pn in which a degenerate imaginary quadric is given. This quadric is a cone of second order with a plane (n - m - 1 ) -dimensional vertex. This cone is called the absolute cone, and its equation can be reduced to the form Ea(x

a) 2 = 0 , a = 0 , 1 , . . . , m . In addition, in the plane vertex of this cone (this vertex is called the absolute plane, and its equation has the form xa = 0), a nondegenerate imaginary quadric is given. This quadric is called the absolute quadric, and its equation can be reduced to the form Eu(x

u ) 2 = 0 , u = m + 1 , . . . , n . If the projective coordinates x; of the points of this space are normalized by the condition Ea(x

a) 2 = 1 , the distance w between the points X(x; ) and Y(/) is defined by the relation cos w = Ea xaya . If the line X Y intersects the absolute plane and w = 0 , then the distance between the points X and Y is defined as the number I/I determined by the relation: 1/12 = Eu(Y

u - xu)2 • It is easy to see that the space s0 · n , whose absolute cone is the hyperplane x0 = 0 taken twice, coincides with the Euclidean space Rn . {In this case the absolute quadric coincides with that imaginary quadric in which all hyperspheres of this space intersect one another, and if n = 2 , it coincides with the cyclic points of the plane R2 . ) The space sn- l , n coincides with the co-Euclidean space *Rn corresponding to the space Rn according to the principle of duality of the space Pn . {In this case the absolute cone is an imaginary cone with a point vertex, and the role of the absolute quadric is played by the vertex of this cone taken twice. ) The absolute cone of the space S1 , n degenerates into a pair of imaginary

§3.6. EQUIVALENT GEOMETRIES 1 0 1

conjugate hyperplanes. In particular, for the Blaschke quasielliptic space S

1 ' 3 , the absolute consists of a pair of imaginary conjugate planes, their intersection line, and a pair of imaginary conjugate points on this line. (The Blaschke metric in the group of motions of the plane R2 is defined as follows: the distance w between two motions A and B is defined as the angle of rotation of the motion BA- 1 if this motion is a rotation about a point; if this motion is a translation and w = 0 , the distance I/I is defined as the length of the vector of translation.)

If we substitute the space sm1 • n-m0- 1 for the space sn-m0- 1 in the abso­lute plane of the space sm0 , n ' we obtain the biquasielliptic space sm0 , m 1 , n

with a biquasisimple group of motions. In particular, the space So , n- l , n

is the isotropic space In considered by Cartan in the paper [66] for n = 4 and in the paper [ 1 47a] for n = 2 . The space s0 • I , n = rn was considered by A. P. Kotelnikov in the paper [KotA3] mentioned above. Note that in the paper Projective geometry of the Galilean space-time [Sil] ( 1 925 ) , Ludwik Silberstein ( 1 872- 1 942) considered not the space /4 or the space r4 but the space R� . Repeating this procedure a few times, we obtain the r-quasi­el/iptic Space smo , m l , . . . , mr- I , n WhOSe group Of motiOnS iS an r-qUaSiSimple Lie group. A particular case of this space is the flag space s0 • I , . . . , n- t , n = Fn whose absolute is a "flag" consisting of m-dimensional planes of all dimen­sions m from 0 to n - 1 each contained in all p-dimensional planes (p > m) of this flag. (The flag plane F2 also coincides with the plane /2 considered by Cartan. )

Substituting in the definition of the space S111 • n a real cone of index /0 and a real quadric of index /1 for an imaginary absolute cone and an imagi­nary quadric, we obtain the quasihyperbolic space s; 1 whose particular cases

0 I are the pseudo-Euclidean space R7 and its dual space * R7 . The r-quasi-hyperbolic spaces S��;

1�'. . '. "i;

, m,_ 1, n are defined in a similar manner. The

groups of motions of the spaces sm. n and s,m ·t can be obtained by ap-

0 • I plying the quasi-Cartan algorithm to the groups of motions of the spaces sn and s; , and the grOUPS Of motiOnS Of the SpaCeS smo ' m I ' . . . ' mr- I ' n and

S��i1�'. . '."i; • m,_ 1 , n can be obtained from the same groups by applying the quasi­

Cartan algorithm r times. The general theory of the spaces sm , n

, S1m ,1n , sm0 , m , , . . . , m,_ 1 , n , and 0 ' I

S�� ;1 �'. . '."i; ' m,_ p n was presented in the paper Projective metrics [YRY] ( 1 964) by Isaac M. Yaglom ( 1 92 1 - 1 988 ) , Rosenfeld, and Evgeniya U. Yasinskaya (b. 1 929) and in Rosenfeld's book Non-Euclidean spaces [Ro7] ( 1 969) (see also his book A history of Non-Euclidean geometry [Ro8] ( 1 988 ) ) .

§3.6. Equivalent geometries

Chapter 2 of Fano's paper [Fa] was entitled "Relationships of different geometries from group-theoretical point of view". In Cartan's extended

1 02 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

translation [ 46] of this paper, this chapter received the shorter title "Equiva­lent geometries". This is the term Cartan used for geometries of spaces with isomorphic fundamental groups. In Klein's "Erlangen program'', the repre­sentation of objects of one geometry by geometric objects of another was called the "transfer principle". The first of these principles was the "Hesse transfer principle" suggested by Otto Hesse ( 1 8 1 1 - 1 874) in the paper On a transfer principle [Hes] ( 1 866) which gave the name to these principles. The Hesse transfer principle is based on the stereographic projection of a conic in the projective plane P2 onto a line P 1 in this plane and on the isomorphism between the group of projective transformations of the plane P2 preserving a conic and the group of projective transformations of the line P1 •

Since the first of these groups can be considered as the group of motions of the plane s� ' any geometric object of the plane s� is represented by

a certain geometric object of the line P1 • In particular, straight lines of the plane S� are represented by pairs of points of the line P 1 • Thus the Hesse transfer principle is based on the isomorphism of the simple groups A 1 and B1 • Another transfer principle-the "Pli.icker transfer principle" suggested by Julius Pliicker ( 1 802- 1 868) in his paper New geometry of space based on considering a straight line as a spatial element [Plii] ( 1 868)-is well known. This principle is based on the representation of the straight lines of the space P3 by the points of the space P5 whose projective coordinates are the Pliicker coordinates piJ = xi y1 - x1 yi , i , j = 0 , 1 , 2 , 3 , where xi and / are the projective coordinates of the points X and Y of the line XY . Since the coordinates piJ are connected by the quadratic relation p0 1 p23 +

p02p3 1 + p03p 1 2 = 0 , the straight lines of the space P3 are represented by the points of the quadric of index 3 of the space P5 • Moreover, the group of projective transformations of the space P3 is isomorphic to the group of projective transformations of the space P5 preserving this quadric. Thus, the Pliicker transfer principle is based on the isomorphism of the simple Lie groups A3 and D3 •

In his paper [Fa] , Fano gave only a few geometric interpretations known at the beginning of the 20th century. In the section Equivalent geometries of the extended translation [46] of Fano's paper, Cartan gave geometric in­terpretations of all isomorphisms from real simple Lie groups in the paper [38] .

The groups indicated by Cartan as the groups A 1 = B 1 = C1 (c5 = 1 ) are the group of collineations of the line P 1 and the groups of motions of the plane S� and of the line cs: . The isomorphism of the first two

groups defines the Hesse transfer of P 1 on the absolute conic of S� . The

isomorphism of the last two groups defines the Poincare interpretation of S� in the complex plane.

The groups A 1 = B1 = C1 (c5 = -3) are the groups of motions of the

§3.6. EQUIVALENT GEOMETRIES 1 03

line CS1 ' the quaternion group IA I = 1 ' and the group of motions of the plane Si (or the group of rotations of a sphere in the Euclidean space R3 ) . The isomorphism o f the first and the third groups defines the metric o f the Riemannian sphere in the complex plane. The isomorphism of the last two groups defines the representation X' = AX A- 1 , IA I = 1 , of the group of ro­tations of a sphere in the space R3

• Cartan formulated these representations as follows:

" (a) The projective geometry of the real line is equivalent to the hyperbolic non-Euclidean geometry of the plane and to the hyperbolic Hermitian geometry of the line'', and " (a' ) The elliptic non-Euclidean geometry of the plane is equivalent to the elliptic Hermitian geometry of the line" [ 46, p. 1 8 34 ] .

The groups Di = A 1 x A 1 (o = 2) are the groups of motions of the space

si and the direct product of two groups of collineations of the line P1 • This

isomorphism defines the interpretation of the manifold of straight lines of the space si by pairs of points of two lines P 1 and by points of two planes

S� , and also the interpretation of the line 'CP 1 over the algebra 'C of split complex numbers a + be , ei = + 1 , on the absolute ruled quadric of the

3 space Si . The group Di (o = 0) is not semisimple but simple. It is isomorphic not

to the direct product of two real groups A 1 but to the complex group A 1 • Being the group of collineations of the line CP 1 , the group Di ( o = 0) is the

group of motions of the space S� . The isomorphism of these groups defines the Kotelnikov-Study transfer of the manifold of straight lines of the space S� on a sphere of the complex Euclidean space CR3

•

The groups Di = A 1 x A 1 (o = -2) are the group of symplectic transfor­

mations of the line HSy 1 and the direct product of the group of collineations of the line P1 (which is isomorphic to the group of motions of the plane S�) and the group of motions of the plane Si . This isomorphism defines the in­terpretation of the line HSy I by the points of the planes Si and s� .

The groups Di = A 1 x A 1 (o = -6) are the group of motions of the space

S3 and the direct product of two groups of motions of the line CS1

or of two planes Si . This isomorphism defines the Fubini-Study transfer of the manifold of straight lines of the space Si on the points of two spheres of the Euclidean space R3 and the Kotelnikov-Study transfer of the same manifold on a sphere of the split complex Euclidean space 'CR

3 •

Cartan formulated these interpretations as follows:

" (b) The real projective geometry of a real ruled quadric of the space E3 (Cartan denoted any n-dimensional space by En ) is equivalent to the union of the real projective geometries of two lines'',

1 04 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

" (b' ) The hyperbolic non-Euclidean geometry of the space £3 (or the real projective geometry of a real non-ruled quadric) is equivalent to the projective geometry of the complex line", and " (b") The elliptic non-Euclidean geometry of the space £3 is equiv­alent to the elliptic non-Euclidean geometry of two planes, or to the elliptic Hermitian geometry of two lines, or to the Euclidean geome­try of two spheres" [46, pp. 1 08- 1 1 0] .

Cartan omitted the case (b111 ) of equivalence of "the projective geometry of a quadric x1x2 + x3x4 = 0 and of a hyperquadric (Hermitian quadric) x1x1 + x2:X2 + x3x3 + x4x4 = 0 of the space £3 and the geometry of the union of two non-Euclidean planes-one elliptic and one hyperbolic".

The groups B2 = C2 (o = 2) are the group of symplectic transforma­

tions of the space Sy3 • This isomorphism defines the interpretation of the

manifold of straight lines of the space sy3 in the space s� .

The groups B2 = C2 (o = -2) are the groups of motions of the space si and of the line HS : . This isomorphism defines the Poincare interpretation

of the space si in the quaternion 4-space. The groups B2 = C2 ( o = - 1 0) are the groups of motions of the space

S4 and of the line HS1

• This isomorphism defines the isometry of the line

HS1 and a hypersphere in the space R5

•

Cartan formulated these interpretations as follows:

" (c) The real projective geometry of the quadric z� + z; + zi - z� = 0 of the space £4 or the geometry of cycles of the plane (Lie's "higher geometry" of oriented circles) is equivalent to the real projective ge­ometry of a linear complex'', " (c' ) The real hyperbolic non-Euclidean geometry of the space £4 or the real conformal geometry of the space £3 is equivalent to the projective geometry of the linear complex p12 + p34 = 0 and of the hyperquadric x1x1 + x2:X2 + x3x3 - x4x4 = 0 of the space £3", and " (c" ) The real elliptic non-Euclidean geometry of the space £4 is equivalent to the projective geometry of the linear complex p1 2 + p34 = 0 and of the hyperquadric x1x1 + x2:X2 + x3x3 + x4x4 = O" [46, pp. 1 1 0- 1 1 1 ] .

The groups A3 = D3 (o = 3) are the group of collineations of the space

P3 and the group of motions of the space S� . This isomorphism defines

the Plucker transfer of the manifold of straight lines of the space P3 on the

absolute quadric of the space s� . The groups A3 = D3 (o = 1 ) are the groups of motions of the spaces

CS� and Si . This isomorphism defines the interpretation of the manifold of straight lines of one of these spaces in the manifold of lines of the other.

The groups A3 = D3 (o = -3) are the group of motions of the space

cs� and the group of symplectic transformations of the plane HS/ . This

§3.6. EQUIVALENT GEOMETRIES I OS

isomorphism defines the interpretation of the manifold of straight lines of -3 - 2

the space CS 1 in the plane HSy The groups A3 = D3 (c:5 = -5 ) are the ·group of motions of the space S�

and the group of collineations of the line HP1 • This isomorphism defines

the interpretation of the line HP1 in the absolute quadric of the space s� .

The groups A3 = D3 (c:5 = - 1 5) are the group of motions of the spaces

CS3 and SS . This isomorphism defines the interpretation of the manifold

of straight lines of one of these spaces in the manifold of lines of the other. Cartan formulated these interpretations as follows:

" (d) The real projective geometry of the quadric zt + z� + zi - z; ­z� - z� = 0 of the space Es is equivalent to the general real projec­tive geometry of the space E3 '', " (d') The real projective geometry of the quadric zt + z� + zi + z; - z� - z� = 0 , or the geometry of oriented spheres of the space E3 , is equivalent to the Hermitian geometry of the hyperquadric X1X1 + X2X2 - X3X3 - X4X4 = O", " (d") The real hyperbolic non-Euclidean geometry of the space Es , or the real conformal geometry of the space E4 , is equivalent to the projective geometry of the quaternion line", " (d"' ) The real elliptic non-Euclidean geometry of the space Es is equivalent to the elliptic Hermitian geometry of the E3", and " (d"") The hyperbolic Hermitian geometry of the space E3 is equiv­alent to the projective geometry of the fundamental quadric x1 x2 + x3x4 + xsx6 = 0 and of the fundamental hyperquadric x1x1 + x2:X2 + X3X3 - X4X4 + XsXs - X6X6 = 0 of the space Es " [46, pp. 1 1 1 - 1 1 2] .

Note that instead of the quaternion spaces HSn , Hs; , and HSyn (the

first of which appeared only in [ 1 07] ( 1 927) ) , in his translation of Fano's paper, Cartan used complex (2n + 1 )-dimensional spaces with a quadric or a linear complex and with a "hyperquadiic" (Hermitian quadric) .

The interpretation of the complex projective line CP1 in the absolute of

the space S� was formulated by Klein in his "Erlangen program". The in­

terpretation of the manifold of straight lines of the space S3 in a pair of

spheres of the space R3 was proposed by Fubini in his dissertation Clifford

parallelism in elliptic spaces [Fu l ] ( 1 900) and by Study in the paper On non­Euclidean and line geometry [Stu2] ( 1 902) . Similar interpretations of the manifold of straight lines of the space si in a sphere of the split complex space 'CR3

and of the manifold of straight lines of the space S� in a sphere

of the space CR3 were proposed by A. Kotelnikov in his doctoral thesis Pro­

jective theory of vectors [KotA2] ( 1 899) and by Study in his book Geometry of Dynames [Stu3] ( 1 903) . Cartan devoted a special section in his Lectures on complex projective geometry [ 1 34] to the interpretation of the quaternion

1 06 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

line HP1 in the absolute of the space S� which was considered in detail in

Study's paper An analogue of the theory of linear transformations of a com­plex variable [Stu5] ( 1 923- 1 924) . In the paper [ 1 34) , Cartan also considered the interpretation of the spaces S5 and si as manifolds of paratactic con-

-3 -3 gruences of lines in the spaces CS and CS 2 •

Note also that in the same way that the conformal space C3 is interpreted on the absolute of the space S� , the group of the Lie "higher geometry of spheres" (defined in Lie's paper [Lie l ] ( 1 872) on a line and spherical com­plexes) which is the group of transformations of the manifold of oriented spheres of the space R3 (points and planes are considered as spheres of zero and infinite radius) , is isomorphic to the group of motions of the space si , and the manifold of spheres of this geometry is interpreted in the absolute of the space si (Lie's imaginary transformation of the manifold of lines of

the space P3 in the manifold of spheres of R3

is based on the imaginary transformation of the absolutes of the spaces s; and si ) .

The interpretation o f the manifold of straight lines of the space CS� i n the

same type of manifold of straight lines of the space si forms the foundation of the "twistor program" of Roger Penrose (b. 1 93 1 ) presented in his paper Twistor theory, its aims and achievements [Pen] ( 1 97 5 ) . The twistors are the spinors of the group of motions of the space si . They are vectors of the

space C4 representing the points of the absolute of the space CS� . Thus,

the points of this absolute represent rectilinear generators of the absolute of the space si , and similarly, the points of the absolute of the space si are

represented by rectilinear generators of the absolute of the space CS� . But

the absolute of the space si represents the pseudoconformal space c� which

is obtained as the extension of the space R� -the space-time of the special

relativity. Thus, the points of the absolute of the space si can be considered as the space-time points of the Universe of special relativity. This explains the title The complex Universe of Roger Penrose of the paper [Gi] ( 1 983) by Semen G. Gindikin (b. 1 937) which is devoted to this interpretation.

In addition to the "transfer principles" based on the isomorphisms of sim­ple Lie groups, in the papers [Fa] and [ 46) , Fano and Cartan also gave "trans­fer principles" based on the isomorphisms of quasi-simple Lie groups which are obtained by passage to the limit from simple groups. The first of these

principles was the interpretation of the elliptic Hermitian line °CS1 over the algebra 0c of dual numbers a + be , e2 = 0 , in the manifold of straight lines of the Euclidean plane R2

presented in detail by I . M. Yaglom in his book Complex numbers in geometry [Ya2] ( 1 968 ) . The second principle was the interpretation of the dual projective line °CP 1 in the geometry of Laguerre transformations in the real plane, i .e . , the geometry of the manifold of ori­ented circles of the plane R� in which those nonpoint transformations are

§3 .7 . GENERALIZATIONS OF THE HESSE TRANSFER PRINCIPLE 1 07

considered that transfer circles into circles and straight lines into straight lines and preserve the tangential distances between circles (the segments of the common tangents between the points of tangency) . If we take the tangential distance as the distance between circles, this manifold becomes isometric to the pseudo-Euclidean space R� , and the group of Laguerre transformations is isomorphic to the group of motions of this space. Finally, the interpreta­tion of the dual projective plane °CP2 in the manifold of straight lines of the space R3 was considered. In this interpretation, the motions of the space R3 are represented by the motions of dual elliptic plane 0cs2

• The first of the above-mentioned interpretations is based on the isomorphism of quasi­simple groups which are obtained by passage to the limit from the isomorphic complex groups, and the second and the third interpretations are based on the isomorphism of quasi-simple groups which are obtained by passage to the limit from the complex groups D2 and B 1 x B 1 • The latter interpretation was studied in detail by A. Kotelnikov in his master's thesis Twist calculus and some of its applications to geometry and mechanics [KotA l ] ( 1 895 ) and by Study in his Geometry of Dynames [Stu3] ( 1 903 ) .

§3.7. Multidimensional generalizations of the Hesse transfer principle

After presenting a few "transfer principles" based on isomorphisms of sim­ple Lie groups, Fano formulated a few generalizations of the "Hesse transfer principle". First, he gave the generalization of the Hesse principle suggested by Wilhelm Franz Meyer ( 1 8 56- 1 934) in the book Apolarity and rational curves [Me] ( 1 883 ) . According to this generalization, the points of the pro­jective line P1 are represented by the points of the "unicursal normal curve" of the space Pn , i .e . , by the algebraic curve defined by the parametric equa­tions

( 3 . 1 0) i t i x = ' i = 0 , 1 , . . . , n ,

where t i is the ith power of the parameter t . The group of projective transformations of the space Pn preserving this

curve is isomorphic to the group of projective transformations of the line P1

• For n = 2 , this interpretation coincides with the Hesse transfer prin­Ciple. Fano also gave another generalization of the Hesse transfer principle according to which the conics aijx

i xj = 0 of the plane P2 are represented

by the points of the space P5 with coordinates au . In this representation, the degenerated conics that are decomposed into pairs of straight lines are represented by the points of the algebraic hypersurface det(ai) = 0 of this space, and the degenerated conics that are twice taken straight lines are rep­resented by the points of the two-dimensional algebraic surface of fourth order

(3 . 1 1 )

1 08 3 . PROJECTIVE SPACES AND PROJECTIVE METRICS

in the same space. This surface was studied by Cayley in 1 868 and by Guiseppe Veronese ( 1 854- 1 9 1 7) in the paper The two-dimensional normal smooth surface of fourth order in a .five-dimensional space and its projections onto a plane and onto the usual space [Ver] ( 1 884) . Veronese's term "normal smooth surface" (superficie omaloide normale) stresses that he considered this surface as a generalization of the unicursal normal curve. The same mapping, with the reference to Veronese, was considered by Corrado Segre ( 1 863- 1 924) in the paper Geometry of conic sections in the plane and on its representation in the form of complexes of straight lines [SeC l ] ( 1 885 ) . In this title C. Segre emphasized the analogy between the representation of conic sec­tions of the plane P2 in the space P5 and the representation in the same space of linear complexes of the space P3 which follows from the "Plucker transfer principle": the linear complexes of the space P3 are defined by the equation au/j = 0 in the Plucker coordinates, and thus, in the space P5 , they are represented by cross-sections of the quadric of index three by hy­perplanes and also by the points of the space P5 that are the poles of these hyperplanes with respect to the quadric. Fano indicated that this representa­tion can be generalized to the representation of quadrics of the space pn in the form of points of the space PN where N = ( (n + l ) (n + 2)/2) - 1 , and the quadrics that are twice taken hyperplanes are represented by the points of the surf ace ( 3 . 1 1 ) of the space PN which, at present, is also called the Veronese hypersurface or the Veronesian and is denoted by Yn .

Fano mentioned one more generalization of the Hesse transfer principle under which a set of points of several projective spaces of different dimen­sions is represented by the points of an algebraic surface in a certain projec­tive space PN . He indicated that the simplest cases of this representation were considered by C. Segre in the paper Varieties representing pairs of points of two planes or spaces [SeC2] ( 1 89 1 ) . In this paper, Segre considered the rep­resentation of two pairs of points of two planes P2

or two spaces P3 in the form of the points of the algebraic surface

(3 . 1 2) iet i a z = X Y

of the space P8 or P1 5 • At present, surface ( 3 . 1 2) in the space pmn+m+n whose points represent the pairs of points of two spaces Pm and Pn , where in general m =f n , is called the Segre surface or the Segrean and is denoted by l:m , n ·

When Cartan translated this section of Fano's paper [Fa] , in his transla-tion [ 46] he added the section "The Hesse principle applied for generation of the projective groups that are isomorphic to a given projective group". In this section, he considered the Segreans ( 3 . 1 2) of general type in the space pmn+m+n and indicated that the subgroup of the group of projective trans­formations of this space preserving the Segrean is the Kronecker product of the groups of projective transformations of the spaces Pm and Pn .

§3.8 . FUNDAMENTAL ELEMENTS 1 09

The Veronesians and the Segreans as well as the surfaces that are obtained by their projections onto planes (the "quasi-Veronesians" and the "quasi­Segreans") have important applications in differential geometry. We will see in Chapter 5 that the Veronesians and the quasi-Veronesians are the indica­trices of curvature of p-dimensional manifolds in the space Rn , and that they also appear in the theory of p-dimensional manifolds in the space Pn . We will see in Chapter 6 that the Veronesians define the absolutes of one type of symmetric Riemannian spaces and the Segreans are the local absolutes of several types of symmetric Riemannian spaces. We will also encounter in Chapter 7 an application of the Segreans to the theory of multidimensional webs. Application of the Segreans and the quasi-Segreans to different prob­lems of differential geometry is discussed in. the two papers of Rosenfeld, M. A. Polovceva, T. I . Yuchtina, et al. : The Segreans and the quasi-Segreans and their application to the geometry of straight lines and planes [RPRY] ( 1 989) and The metric and symplectic Segreans and quasi-Segreans [RKSY] ( 1 989) .

§3.8. Fundamental elements

Geometric objects of the spaces CPn , CS2n , csy2n- I , and CS2n- l are connected with the fundamental linear representations of complex Lie groups in the the classes An , Bn , Cn , and Dn . These objects help to give geometric interpretations to these groups. Probably, because of the connection of these geometric objects with the "fundamental groups" of Cartan, Tits, who further developed the theory of these groups, called these objects the fundamental elements.

The linear representation <p 1 of a complex simple Lie group in the class An that can be considered as the group of projective transformations of the complex projective space CPn is a representation of this group in the space CPn itself, i .e . , the coordinates of vectors of this representation coincide with the projective coordinates of points of the space CPn . The coordinates of vectors of the linear representation <pk of this group coincide with the Grassmann coordinates

( 3 . 1 3)

of (k - 1 )-dimensional planes (for k = 2 , these coordinates coincide with the Pliicker coordinates /i of straight lines) of the space CPn . Thus, the fundamental elements of the space CPn are points, straight lines, and m­dimensional planes ( m = 2 , 3 , . . . , n - 1 ) .

Grassmann coordinates ( 3 . 1 3) satisfy the equations

(3 . 1 4)

In the projective space PN where N = (��11 ) - 1 , surface (3 . 1 4) represents the manifold of m-dimensional planes of a complex or real n-dimensional

1 1 0 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

projective space. This manifold is called the Grassmann manifold and is de­noted by G n , m • This is the reason that surface ( 3 . 1 4) is called the Grassman­nian and is denoted by rn , m . We will see in Chapter 6 that the Grassmanni­ans (as the Segreans) are the local absolutes of several symmetric Riemannian spaces.

The linear representation <p 1 of complex simple Lie groups in the classes Bn and Dn that can be considered as the groups of motions of the com­

plex elliptic spaces CS2n and CS2n- I are representations of these groups in the spaces CS2n and CS2n- I themselves; i .e . , the coordinates of vectors of these representations coincide with the projective coordinates of points of the spaces CS2n and CS2n- I . The coordinates of vectors of the linear repre­sentations <pk of these groups also coincide with the Grassmann coordinates

/0;1 . . . ;k- 1 (for k = 2 , with the Plucker coordinates /i of straight lines) of the spaces cs2n and cs2n- I • As to the spinor representation '1'1 of the group Bn and 1{11 and 1{12 of the group Dn , in his Lectures on the theory of spinors [ 1 64] ( 1 938) , Cartan showed that the coordinates of vectors of these representations, the so-called spinors, can be considered as the coordinates of the isotropic spaces of maximal dimension of the complex Euclidean spaces CR2n+ I and CR2n whose groups of rotations are complex simple Lie groups of the classes B n and D n • But the isotropic subspaces of maximal dimen­

sion of the spaces CR2n+ I and CR2n cut on the hyperplanes at infinity of the spaces CR2n+ 1 and CR2n the plane generators of maximal dimensions of these absolutes. These plane generators form one family for the space S2n and two families for the space s2n- I . Thus, the fundamental elements of the

Cs2n d CS2n- I d" h . . h spaces an correspon mg to t e spmor representations are t e plane generators of maximal dimensions of absolutes of these spaces. Thus, as the fundamental elements of the spaces CS2n and CS2n- I correspond to the linear representations <p 1 , <p2 , and <pk , one must consider not arbi­trary points, straight lines, and (k - 1 ) -dimensional planes of these spaces but only the points and rectilinear and plane generators of the absolutes of these spaces. Therefore, the fundamental elements of the spaces CS2n and cs2n- I are the points and the rectilinear and m-dimensional plane genera­tors of the absolutes of these spaces ( m = 2 , 3 , . . . , n - 1 for the group Bn and m = 2 , 3 , . . . , n - 3 , n - 1 for the group Dn ) .

The linear representation of a complex simple Lie group in the class Cn that can be considered as the group of symplectic transformations of the spaces CSy2n- I is a representation of this group in the space CSy2n- I it­self, i .e . , the coordinates of vectors of this representation coincide with the projective coordinates of points of the space csy2n- i . The coordinates of vectors of the linear representations <pk of this group coincide with the Grass­mann coordinates of (k - ! )-dimensional isotropic planes (for k = 2 , with the Plucker coordinates of isotropic lines) of the space csy2n- l , i .e . , such straight lines and planes that lie entirely in those (2n - k - 1 )-dimensional

§3 . 8. FUNDAMENTAL ELEMENTS I l l

planes that correspond them relative to the null-system of the space csy2n- I . The values of the dimensions of the spaces of the representations <pk , which are equal to (�Z) - (k2!!,2) , are explained by the fact that the Grassmann co­

ordinates p;0; . . . . ;k- • = k !x1;0xi • · · · X ;k- i l of the (k - ! )-dimensional isotropic planes are connected by the linear relations a; ; p

;0 ; • . . . ;k - • = 0 whose par-. . 0 I

ticular case is the equation aiip'1 = 0 of a linear complex of straight lines

in the space CP3 • Thus, the fundamental elements of the space csy2n- I are its points (the absolute null-system of this space maps each point of the space into a hyperplane passing through this point) , isotropic lines, and m­dimensional planes ( m = 2 , 3 , . . . , n - 1 ) .

Since to each simple root of a simple Lie group there corresponds a fun­damental linear representation of this group, to this root there corresponds also a fundamental element of the corresponding classical space. Thus, the sets of the Dynkin graphs shown in Figures 2 .2 and 2 . 5 not only repre­sent simple roots of simple Lie groups but also the fundamental elements of the corresponding spaces. In particular, to the graph dots represent­ing simple roots of a simple Lie group in the class An , there correspond points, straight lines, and m-dimensional planes ( m = 2 , 3 , . . . , n - 1 ) of the space CPn ; to the graph dots representing simple roots of a simple Lie group in the class Bn , there correspond points, rectilinear generators, and m-dimensional plane ( m = 2 , 3 , . . . , n - 1 ) generators of the absolute of the space CS2n ; to the graph dots representing simple roots of a simple Lie group in the class Cn , there correspond points and isotropic and m­

dimensional ( m = 2 , 3 , . . . , n - 3 , n - 1 ) planes of the Space CSy2n- I ; and to the graph dots representing simple roots of a simple Lie group in the class Dn , there correspond points and rectilinear and m-dimensional ( m = 2 , 3 , . . . , n - 3 , n - 1 and n - 1 ) plane generators of the absolute of the space CS2n- I (here, the two numbers n - 1 are related to two families of plane generators of maximal dimension; in this case, (n - 2)-dimensional plane generators are not fundamental elements of the space CS2n- I - they are determined by two plane generators of maximal dimension belonging to different families) .

Similarly, starting from linear representations of real simple Lie groups, we can find the fundamental elements of the real spaces Pn , Sn , s; , and

S 2n- I d d 2n - I h 1 . y ; we un erstan the latter space as the space P w ere a mear 1 ij 0 f . h l ' . . F h pn sin S in-I comp ex aiip = o stra1g t mes is given. or t e spaces , n , y ,

and s;n- I , whose fundamental elements are defined in the same way as the fundamental elements of the corresponding complex spaces CPn , CS2n , csy2n-l , and cs2n- I , all these elements are real. For the spaces s2n and Sin- I 11 h 1 . . .

d I." h s2n d s2n- I , a t ese e ements are imagmary, an , 1or t e spaces / an / for 0 < I < n , there are both real and imaginary elements.

1 1 2 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

In the lecture On the general problem of deformation [ 55 ] ( 1 920) presented at the International Congress of Mathematicians in Strasbourg, Cartan noted that the manifold of straight lines of the projective space P3

can be mapped by means of an imaginary transformation onto the conformal space C4

• This transformation maps the group of projective transformations of the space P3

to the group of conformal transformations of the space C4 , which allows us

to apply the theory of deformations of surfaces of the space C4 to the theory

of deformations of families of straight lines of the space P3 • The latter

Cartan conclusion was based on the "Pliicker transfer principle" by means of which the manifold of straight lines of the space P3

is represented by the absolute of the space sj , and the imaginary transformation mentioned

above is a transformation of the space sj into the space s� whose absolute

represents the space C4 • Similar to the imaginary transformation used by

Lie in his paper [Lie l ] ( 1 872) , here Cartan essentially introduced the idea of pseudo-conformal space ct and its interpretation on the absolute of the

spaces s�+ l and s;:,.1 ; in this interpretation, conformal transformations

of the space C1n are represented by motions of the space S'/:i1 • Since the

points of the absolutes of the spaces s;:/ are the fundamental elements of these spaces, the points of the spaces en and cln are also the fundamental elements of these spaces.

The geometry of fundamental elements was further developed in the the­sis On the topology of certain homogeneous spaces [Eh l ] ( 1 934) of Cartan's student Ehresmann. In his thesis, Ehresmann found topological invariants of many manifolds of fundamental elements. In the book Unitary representa­tions of the classical groups [GN2] ( 1 950) , Israel M. Gel' fand (b. 1 9 1 3) and Mark A. Naimark ( 1 909- 1 980) used many of these manifolds for construc­tion of linear representations of noncompact Lie groups by means of uni­tary operators of the Hilbert space. Stationary subgroups of "fundamental elements" were studied by Vladimir V. Morozov ( 1 9 1 0- 1 975) in his unpub­lished dissertation On nonsemi-simple maximal subgroups of simple groups (Kazan, 1 943) . These groups are maximal nonsemisimple subgroups of sim­ple Lie groups.

The theory of "fundamental elements" was substantially developed in the papers of Tits who, along with these "fundamental elements", considered more general geometrical elements whose stationary subgroups are parabolic subgroups of simple Lie groups (i .e . , these subgroups contain a maximal solv­able subgroup of these groups which is called the Borel subgroup) . At present, these geometric elements are called the parabolic elements. (In the paper Figures of simplicity and semi-simplicity [Ro6] ( 1 963 ) , Rosenfeld called the "fundamental elements" the "figures of simplicity'', and he called the more general parabolic elements the "figures of semisimplicity". ) Tits considered the "parabolic elements" in the papers On certain classes of homogeneous spaces of Lie groups [Ti l ] ( 1 95 5 ) and On the geometry of R-spaces [Ti3]

§3.9. THE DUALITY AND TRIALITY PRINCIPLES 1 1 3

( 1 957 ) . The R-spaces here stand for the manifolds of parabolic elements. At present, these spaces are called the parabolic spaces. According to Tits, two fundamental elements are incident if the intersection of their stationary subgroups is a parabolic subgroup.

For the space CPn , parabolic elements are "flags" consisting of planes of different dimensions enclosed one inside the other (straight lines and points are considered to be 1 - and 0-dimensional planes) .

For the spaces csn and csy2n- I ' the parabolic elements are the "flags" consisting of the plane generators of the absolute or of isotropic planes. This is the reason that the parabolic spaces are also called "flag manifolds" (see, e.g., the paper by Wolf, The action of a real semi-simple group on a complex flag manifold [Wo l ] ( 1 969) ) . A general survey of the geometry of parabolic spaces has been presented in the paper Parabolic spaces [RZT] ( 1 990) by Rosenfeld, Mikhail P. Zamakhovsky (b. 1 942) , and Tamara A. Timoshenko (Stepashko) (b. 1 949) .

Later on, departing from the geometry of " R-spaces", Tits constructed a more general geometry of buildings (immeubles) of "spherical types" (see his book Buildings of spherical types and finite B N-pairs [Ti6] ( 1 980)) and also a geometry of buildings of Euclidean (affine) types corresponding to analogues of usual and affine Weyl groups (see [Ti7] and [Ti8] ) .

§3.9. The duality and triality principles

In the paper The duality principle and the theory of simple and semisimple groups [82] ( 1 925) which we discussed earlier, Cartan posed the question of transformations of simple and semisimple Lie groups "preserving the group structure" and of representations of these transformations in spaces for which these Lie groups are groups of transformations. In this paper, Cartan proved that the only simple Lie groups whose Weyl group is a subgroup of the Galois group of the characteristic equation of the group are groups in the classes An , Dn , and E6 , and for them the quotient groups of the Galois groups by the Weyl groups are isomorphic to the multiplicative group { 1 , - 1 } and, for the group D 4 , to the group of permutations of three elements. The presence of these quotient groups explains the symmetries of the Dynkin graphs discussed earlier and sh�wn in Figures 2 .3 , 2 .4, and 2 .6 . These graphs have the bilateral symmetry for the classes An and E6 in the vertical axis of symmetry, for the class Dn (n =/: 4) in the horizontal axis of symmetry, and the trilateral symmetry for the class D 4 • Cartan connected these symmetries with the transformations of the spaces under which the fundamental elements of these spaces corresponding to simple roots of the groups are transformed into the fundamental elements corresponding to other simple roots of these groups.

For the first of these symmetries, points of the projective space CPn cor­respond to hyperplanes of these spaces, straight lines correspond to (n - 2)­dimensional planes, and m-dimensional planes correspond to ( n - m - l )-

1 1 4 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

dimensional planes. This correspondence is expressed by the classical duality principle of the space CPn , and the latter was the reason for the title of the Cartan paper [82) . This duality can be realized by means of the correlation U; = aijxi which maps the point X(x;) into the hyperplane U;X

; = 0 with the tangential coordinates u; .

For the second of these symmetries, ( n - 1 ) -dimensional plane generators of the absolute of the space cs2n- I from one family correspond to ( n - 1 ) ­dimensional plane generators of the same absolute from another family. In the space cs2n- l ' there are transformations interchanging the plane gener­ators of these two families. Cartan considered the correspondence between plane generators of these two families as an analogue of the duality principle of the space CPn .

There is also the duality principle in the real space Pn , and there is the correspondence between plane generators of two families of generators in the

S2n- I d S2n- I spaces an n •

We mentioned earlier the Freudenthal interpretation of one of the non­compact groups in the class E6 in the form of the group of projective trans-

formations of the octave plane OP2 • It follows from this interpretation that

the complex simple group E6 admits the interpretation in the form of the

projective plane (0 ® C)P2 over the algebra of complex octaves (the tensor product of the algebras 0 and C) . This is the reason that the third symme­try indicated by Cartan coincides with the duality principle of the projective planes OP2 and (0 ® C)P2

•

As to the trilateral symmetry of the graph of groups in the class D 4 , Car­tan connected it with the isomorphism of stationary subgroups of the points of the absolute of the space CS 7 and of the three-dimensional plane genera­tors from different families of this absolute and also with coincidence of the fundamental linear representations <p 1 , I/Ii , and I/Ii (the matrices of all of these three representations are matrices of order eight) . In this connection, Cartan wrote: "We can say that the duality principle of projective geometry is replaced here by the triality principle" [82, p. 373) .

·

By this "triality principle'', the points of the absolute of the space CS7 can be replaced by three-dimensional plane generators of the absolute from both families of these plane generators. But the points and three-dimensional plane generators of the absolute of the space CS 7 are the fundamental ele­ments of this space corresponding to three simple roots of the group of its motions, and these elements are transferred one into another by the rotation of the graph of simple roots of this group through 1 20° (see Figure 2 .6 ) . Similar triality principles hold in the real spaces S7 and sX . Note that, in the same paper, Cartan connected this "triality principle" with the algebra of octaves: if one introduces the metric of the space R8 in the algebra 0 of octaves taking as the distance between two octaves a and p the modulus

I P - al of their difference, then the metric of the space CR8 arises in the

§3 .9. THE DUALITY AND TRIALITY PRINCIPLES 1 1 5

algebra 0 ® C . Thus, any point of the absolute of the space CS 7 can be represented as a complex octave of zero modulus. But the three-dimensional plane generators of this absolute can be also represented by octaves of zero modulus since the equations o:e = 0 and ea = 0 (where 0: and e are com­plex octaves of zero moduli and the complex octave o: represents a point of the absolute) define the three-dimensional plane generators of different fam­ilies of the absolute of the space CS 7 • On the other hand, if the complex octaves o: and P of zero moduli are octaves such that their product o:P = y is not zero, then this product is a nonzero complex octave of zero modu­lus. If the complex octaves o: and p represent two three-dimensional plane generators from different families of the absolute, then the complex octave y represents their unique common point. If the complex octaves o: and p represent a three-dimensional plane generator from one of the families and a point of the absolute, then the complex octave y represents a three­dimensional plane generator from another family which intersects the first generator at this point of the absolute. Cartan's research on the connection between the triality principle in the space S 7 and octaves was continued by E. A. Weiss in the paper Octaves, Engel 's complex and the triality principle [Wes] ( 1 938 ) .

In the joint paper On Riemannian geometries admitting an absolute paral­lelism [92] ( 1 926) , Cartan and Schouten considered a similar triality principle in the real space S7 which is obtained by identifying antipodal points on the hypersphere lo: ! = 1 in the algebra 0 with the metric of the space R8

• They ' defined in this space two continuous families of transformations of this hy­persphere mapping the OCtave e intO the OCtaVe I e and the OCtaVe 1'f intO the octave ' 17 • These transformations are called the absolute parallelisms, and there are (+)-parallelisms

( 3 . 1 5 ) 17 ( 'C10:) = 17 (c!- 1

0:)

and ( - )-parallelisms

( 3 . 1 6) I - 1 I - 1 (o: c! ) 17 = (o:c! ) 17 ,

each of which depends on seven parameters. They noted that "the points of S7 and the ( + )- and ( - )-parallelisms can be considered as elements of S7"and that "thus we have triality in S7 by means of which we can define the distance between two ( + )- or two ( - )-parallelisms, etc. " [92, p. 944 ] . Parallelisms (3 . 1 5 ) and ( 3 . 1 6) are defined in such a way that the segments e 1'f and I e' 1'f are Considered tO be parallel tO One another.

Cartan also considered the triality principle in the space CR8 , which is equivalent to that in the space CS7 , in his Lectures on the theory of spinors [ 1 64] where he formulated this principle for isotropic vectors of the space CR8 representing the points of the absolute of the space CS 7 and for "semi­spinors" of first and second kinds defining the four-dimensional isotropic

1 1 6 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

planes of the space CR8 representing the three-dimensional plane generators of the absolute of the space CS 7 • Cartan did not mention octaves here but he mentioned the Brioschi formulas which express the product of two sums of eight squares in the form of a similar sum and are equivalent to the octave identity: l aP I = i a l lP I ·

Finally, in the unpublished manuscript Isotropic surfaces of a quadric in a seven-dimensional space [ 1 77) , Cartan, without mentioning the triality prin­ciple, in fact considered an application of this principle in the space sX . In

this manuscript, he considered the real projective space P7 and the quadric .Q(x) = x0 x 7 + x 1 x6 + x2 x5 + x3 x4 = 0 in this space. This quadric is the ab­solute of the sX . The collineations ("homographies") , preserving this quadric

(i .e . , the motions of the space sx ) . were called the "absolute homographies" by Cartan. The hyperquadric considered by Cartan has two families of three­dimensional plane generators. Cartan called these generators the "generator spaces". Cartan proved that, in general, two "spaces" from different fami­lies always intersect each other at a point, and if they have one more com­mon point, they have a two-dimensional intersection. Cartan called these 2-planes of intersections "isotropic planes". Through any isotropic plane, there passes a unique pair of "generator spaces" from different families. In general, two "generator spaces" of one family do not have common points. If they have a common point, then they have a common "isotropic line". The main goal of this manuscript was the study of "isotropic surfaces'', i .e . , two-dimensional surfaces all of whose tangent planes are isotropic planes. Cartan's terms "isotropic line", "isotropic curve" (a curve whose tangents are isotropic lines) , and "isotropic surface" indicate that he considered a hy­perquadric of the space P 7 as the pseudo-conformal space c: although he never used this term. Cartan applied the equations of "generator spaces" which are equivalent to the equations ac; = 0 and c;a = 0 , where a and c; are elements of the alternative algebra '0 of split octaves satisfying the conditions .Q(c;) = 1 c; 1 2 = 0 and .Q{a) = la l 2 = 0 . In the manusript, Car­tan also gave the formula (equivalent to the formula aP = y) for three split octaves, one of which represents a point of the absolute and the other two representing two "generator spaces" from different families.

§3.10. Spaces over algebras with zero divisors

We have already discussed many times the spaces over the field C of com­plex numbers considered by Cartan, namely, the spaces CPn , CEn , csn , csn

' csy2n- l ' and others as well as the analogues of some of these spaces over the algebra 'C of split numbers and the algebra

0c of dual numbers, over the skew field H of quaternions, the alternative skew field 0 of oc­taves, and some other algebras. Many of these spaces are used for geometric interpretations of simple Lie groups.

§3. 1 0. SPACES OVER ALGEBRAS WITH ZERO DIVISORS 1 1 7

The projective line R2P1

over the algebra R2 of real matrices of second order was first considered by Niccolo Spampinato ( 1 892- 1 97 1 ) in the paper On geometry of the line space considered as a hypercomplex S1 [Spa] ( 1 934) . In the paper The manifold S5 of lines considered as the hypercomplex S2 con­nected with a regular complex algebra of order 4 [Cab] ( 1 936) , his student Carmela Carbonaro considered the projective plane R2P

2 over the same alge­

bra. Spampinato and Carbonaro also studied interpretations of the projective line R2P

1 and the projective plane R2P

2 in the form of the manifolds of

straight lines of the real projective spaces P3 and P5 and indicated that the

results obtained by them can be generalized to higher dimensions. For this generalization, one must define the n-dimensional projective space Rm+ 1 Pn over the algebra Rm+ I of real matrices of order m + 1 . Each point of this

space is defined by n + 1 coordinates xi that are the matrices (x'// ) defined

up to multiplication xi -t xi a by a nonsingular matrix (aj; ) of the same

algebra. In this representation, each point x(xi ) is represented by an m­dimensional plane of the real space pmn+m+n defined by the points x P with

coordinates x// . (When the coordinates xi are multiplied by the matrix

(aj; ) , the points Xp are replaced by their linear combinations Yy = Xpa� . ) An attempt to construct a general theory of spaces over rings with zero

divisors was made by Dan Barbilian ( 1 89 5- 1 96 1 ) (who is also known as the poet "Ion Barbu") in the paper The axiomatics of projective plane ring geometries [Bab] ( 1 940- 1 94 1 ) . Because of this, the spaces over rings and algebras with zero divisors are often called the "Barbilian spaces". Barbilian was first to notice that, although in general through two points of these spaces there passes a unique straight line, there are pairs of points in these spaces through which there passes more than one straight line. Barbilian called the points in the first case the "points in clear position" and the points in the second case the "points in spectral position". At present, the pairs of points through which there passes more than one straight line are called the adjacent points, and the pairs of straight lines having more than one common point are called the adjacent lines.

In the paper Symmetric spaces and their geometric applications [Ro 1 ] ( 1 949) , Rosenfeld defined the spaces I CPn and I csn over the algebra I c of split numbers and the spaces 'HPn and 'HSn over the algebra 'H of split quaternions (by the isomorphism 'H = R2 , the space 'HPn coincides with R2P

n ) . He proved that the spaces 'csn and 'HSn admit the interpretation in the form of the manifolds of "0-pairs" (a 0-pair is a point and a hyper­plane) of the space pn and in the form of the manifolds of straight lines of the space sy2n+ I . He also proved that the groups of motions of the spaces 'csn and 'HSn are isomorphic to the fundamental groups of the spaces Pn and sy2n+ i , respectively.

In the interpretation of the space R2Pn

in the form of the manifold of

1 1 8 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

straight lines of the space P2n+ i , adjacent points are represented by inter­secting lines and adjacent lines by the manifolds of straight lines in three­dimensional planes belonging to a four-dimensional plane, and therefore these 3-planes have a common 2-plane. The terms "adjacent points" and "adjacent lines" were introduced by Wilhelm Klingenberg (b. 1 924) in the paper Projective and affine planes with adjacent elements [Kli] ( 1 954) . In 1 957 , in the papers Projective spaces over algebras [Jav l ] and Non-Euclidean geometries over algebras [Jav2], Maqsud A. Javadov ( 1 902- 1 972) defined the spaces Rm+ 1 pn and the Hermitian elliptic spaces Rm+ 1 Sn over the al­gebra Rm+ I and found the interpretations of these spaces in the manifolds

of m-dimensional planes in the spaces pmn+m+n and smn+m+n . A detailed exposition of the geometry of projective and non-Euclidean spaces over al­gebras is given in Chapter VI of Rosenfeld's book Non-Euclidean geometries [Ro3] ( 1 955 ) (see also his book [Ro7] ) .

§3.1 1 . Spaces over tensor products of algebras

Projective, Hermitian elliptic and Hermitian hyperbolic spaces were also defined over the tensor products of algebras C , H , and 0 and their ana­logues. The Hermitian elliptic spaces over the tensor products A © B are -n denoted by (A © B)S , and the Hermitian hyperbolic spaces over the same -n algebras are denoted by (A © B)S1 (these notations are explained by the fact that in the definitions of these spaces, the involution a --+ � is used in the tensor product A © B , and this involution consists of the involution a --+ a in the algebra A and the same involution a --+ a in the algebra B) . Note that the tensor product C © C-the elements of this algebra were called the bicomplex numbers by Cartan-is isomorphic to the direct sum C E9 C and -n the space (C © C)S admits the interpretation in the form of the pair of spaces CSn . The tensor product H © C-the elements of this algebra were called the biquaternions by Hamilton, the inventor of the algebra H-is iso­morphic to the algebra C2 of complex matrices of second order, and the -n space (H © C)S admits the interpretation in the form of the manifold of

straight lines of the space CS2n+ 1 • The tensor product H © H is isomorphic -n

to the algebra R4 of real matrices of fourth order, and the space (H © H)S

admits the interpretation in the form of the manifold of three-dimensional planes of the space s4n+3 •

In the papers [Ro2] ( 1 954) and [Ro4] ( 1 956) , Rosenfeld defined the Her--2 -2 - 2

mitian elliptic planes (0 © C)S , (0 © H)S , and (0 © O)S and showed that the groups of motions of these elliptic planes are compact simple Lie groups in the classes E6 , E1 , and E8 •

Later he also defined the Hermitian hyperbolic planes over the same tensor products and the Hermitian elliptic planes (which can be obtained from the

§3. 1 1 . SPACES OVER TENSOR PRODUCTS OF ALGEBRAS 1 1 9

Hermitian elliptic planes over the tensor products 0 ® C , 0 ® H , and 0 ® 0 by replacing one or both factors in these tensor products by the corresponding algebras 'c , 'H , and 'O) and showed that the groups of motions of these planes are noncompact simple Lie groups in the same classes. In the earlier mentioned paper [Frd2] on the connection between the simple Lie groups E7 and Es with the octave plane, Freudenthal noticed that the groups of motions of two-dimensional Hermitian elliptic planes over the fields R , C , H , and 0 , the groups of projective transformations of two-dimensional projective planes over the same fields, and the groups of symplectic transformations of five-dimensional symplectic spaces over the same fields can be represented as the first three rows of the following "magic square":

Bi A2 C3 F4

(3 . 1 7) A2 A2 x A2 A s E6 C3 As D6 E1 F4 E6 E1 Es

This was the reason that Freudenthal used the term metasymplectic geome­tries for the geometries whose fundamental groups are the groups indicated in the fourth row of the square ( 3 . 1 7) . We will denote the spaces corresponding to these geometries over the fields R , C , H , and 0 by Ms , CM s , HM s , and OMs , respectively. These spaces are sets of so-called symplecta which in turn can be considered as sets of two-dimensional isotropic planes of the symplectic spaces sy5 ' CSys ' HSys ' and osy5 . Thus, the main geo­metric objects of metasymplectic geometries are symplecta, two-dimensional projective planes, projective lines of these planes, and points of these lines and planes. In the paper Metasymplectic geometries as geometries on the ab­solutes of Hermitian planes [RoS] ( 1 983 ) , Rosenfeld and Stepashko showed that the Freudenthal metasymplectic geometries are represented in the ab-

2 -2 -2 solutes of Hermitian elliptic planes 'OS , (' 0 ® C)S , (' 0 ® H)S , and

-2 ('o ® O)S and that the similar metasymplectic geometries 1CMs , 'HMs , and 'OM s are represented in the absolutes of the Hermitian elliptic planes

-2 -2 -2 (10 ® 1C)S , ('0 ® 1H)S , and (0 ® 10)S . The Freudenthal "magic square" represents also the geometric interpretations of compact Lie groups forming this square. In this case all the groups of this square are the groups of mo­tions of the Hermitian elliptic planes over the fields R , C , H , and 0 and their tensor products.

Table 3 . 1 (next page) is the table of the real simple Lie groups (up to a local isomorphism) . In the first column of this table, the class of the sim­ple Lie group is indicated, in the second column the character of the real simple group, in the third column the spaces for which these groups are the fundamental groups, and in the fourth column the number of the figure rep­resenting the Dynkin graph of a compact group or the Satake graph of a noncompact group.

1 20

An

Bn

en

Dn

G2

F4

E6

E1

Ea

3. PROJECTIVE SPACES AND PROJECTIVE METRICS

J = -n(n + 2) J = n

J = -n - 2 J = 4/(n - I + 1)

- n (n + 2) J = 2n - n2

J = -n(2n + 1) J = 2/(2n - I + 1)

- n(2n + l) J = 3n - 2n2

J = -n(2n + 1 ) J = 2n J = 8/(n - /)

- n(2n + l )

J = -n(2n - 1) J = 2/(2n - /)

- n (2n - 1) J = Sn - 2n2 - 2 J = -n

J = -14 J = 2

J = -52 J = -20 J = 4

J = -78

J = -26

J = -14

J = 2

6 = 6

J = -133

J = -25

J = - S

J = 7

J = -248

J = -24

J = 8

-n :::::.(n - 1 )/2 CS = (H ® C)S pn = 'csn

-(n - 1 )/2 HP(n - 1 i12 = (H ® 'qs csn

I

cs7 s2n

s2n I

s2n I

HSn - 1

syr- 1 = 'HSn - 1 -n - 1 HS1

s2n- I = 'HSyn - 1 S2n- I

I

S2n- I I

HSyn- 1

Sg6

Sgf os2

os2 I

'os2 = Ms :::::.2

(O ® C)S -2

OP2 = (0 ® 'C)S1 :::::.2

(0 ® C)S 1 -2

('O ® C)S = CMs -2 'op2 = c'o ® 'qs1 = 'cMs

:::::.2 (O ® H)S

:::::.2 0Sp5 = (0 ® 'H)S

-2 I :::::.2 (0 ® H)S1 = ( 0 ® H)S = HM s '0Sp5 = (10 ® 'H)S2

= 'HMs :::::.2

(O ® O)S :::::.2

(O ® 'O)S = 0Ms :::::.2 :::::.2 I ('O ® 'O)S = (0 ® 0)S 1 = OMs

TABLE 3. 1

Fig. 2.2 (a) Fig. 2. 1 0 (a) Fig. 2. 1 0 (b) Fig. 2. 1 0 (c, d)

Fig. 2. l O (e)

Fig. 2.2 (b) Fig. 2. 1 0 (f, g)

Fig. 2. 1 0 (h)

Fig. 2 .2 (c)

Fig. 2. I O (i) Fig. 2. 1 0 (j, k)

Fig. 2.2 (d) Fig. 2. 1 0 ( I , m, n)

Fig. 2. 1 0 (o)

Fig. 2. 1 0 (p, q)

Fig. 2 .S (a) Fig. 2. 1 1 (a) Fig. 2 .S (b) Fig. 2. 1 1 (c) Fig. 2. 1 1 (b)

Fig. 2.S (c) Fig. 2. 1 1 (g)

Fig. 2. 1 1 (f)

Fig. 2. 1 1 (e)

Fig. 2. 1 1 (d) Fig. 2.6 (d) Fig. 2. 1 1 (j)

Fig. 2. 1 1 (i)

Fig. 2. 1 1 (h)

Fig. 2.S (e)

Fig. 2. 1 1 ( I ) Fig. 2. 1 1 (k)

Table 3.2 gives the table of isomorphisms of real simple Lie groups (up to a local isomorphism) . The construction of Table 3 .2 is similar to that of Table 3 . 1 .

§3. 1 2. DEGENERATE GEOMETRIES OVER ALGEBRAS

A 1 = B1 = C1 0 = - 3

0 = - 1

D2 = B1 x B1 0 = -6 0 = 0 0 = 2 0 = 6

B2 = C2 o .= - 1 0 0 = -2

0 = 2

A3 = D3 0 = - 1 5

0 = - 5

o = l 0 = 3

0 = - 3

D4 0 = -4

cs1 = S2

cs: = P 1 = 'CS1 = S�

s3 = s2 x s2 = 'cs2 s3 = cs2

� 2 2 I 2 s2 = s1 x s1 = cs1 I 2 2 HSy = S x S1 S4 = HS1 4 -I SI = HSI

s1 = Sy3 = 'HS1

s5 = cs3 S� = HP1

5 -3 S2 = CS2 si = p3 = 'cs3

- 2 -3 Hsy = cs1 - 3 7 HSy = S2

TABLE 3 .2

Fig. 2 .3 (a)

Fig. 2. 1 2 (a)

Fig. 2.3 (b)

Fig. 2. 1 2 (b) Fig. 2. 1 2 (c) Fig. 2. 1 2 (d)

Fig. 3 (c)

Fig. 2. 1 2 (f)

Fig. 2. 1 2 ( e)

Fig. 2.3 (d)

Fig. 2. 1 1 (b)

Fig. 2. 1 2 (h)

Fig. 2. 1 2 U)

Fig. 2 . 1 2 (i)

Fig. 2 . 1 2 (k)

§3.12. Degenerate geometries over algebras

1 2 1

Applying to the groups o f motions o f the spaces CSn and cs; quasi-Cartan algorithms similar to the algorithm by means of which from the groups of motions of the spaces sn and s; we obtained the groups of motions of the quasielliptic spaces sm ' n ' the quasihyper­bolic spaces S'('/ , the r-quasielliptic spaces smo · m1 • . . . . m,_ , • n , the r -quasi-

o I hyperbolic spaces s:"t0 :.� · · · · · •

m,_ , , n , we obtain the complex Hermitian, quasi-o I r

elliptic, quasihyperbo/ic, r-quasiel/iptic and r-quasihyperbolic spaces csr/ n ' 0 I CSm0 ' m , ' . . . ' m,_ , ' n and csr0{'..� ' ' . • • ' m,_ , ' n . In like manner, from the spaces

I r

HSn and Hs; we obtain the spaces HSm ' n , HSr/ n , HSm0 • m, •

· · · · m,_ , • n 0 1

d H-Sm0 , m 1 , • • • , m,_ 1 , n an / 1 . . . 1 , etc. 0 I r

On the other hand, applying to the groups of motions of the spaces CSn and HSn

the quasi-Cartan algorithms corresponding to the involutive auto­morphisms a --. aaa where a is the motion 'xi = xi and 'x i = ixi i- 1 , we obtain the groups of motions of the Hermitian elliptic spaces °CSn and 0HSn over the algebra °C of dual numbers and the algebra 0H of semi­quaternions. The spaces, similar to those we have defined over the fields C and H , can be also defined over the algebras °C and 0H .

The quasi-Cartan algorithm applied to Lie groups is a particular case of the "contraction of Lie groups" defined by Wigner and Inonu in the earlier mentioned paper [IW] ( 1 953 ) . A complete classification of all quasisimple

1 22 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

Lie groups and of their geometric interpretations was given by Rosenfeld and Ludmila M. Karpova (b. 1 934) in the paper Flag groups and contraction of Lie groups [RK] ( 1 966) . (In this paper, the authors called quasisimple Lie groups "flag groups".)

The quasisimple Lie groups obtained by the method indicated above from compact simple groups are obtained by the same method from noncompact simple groups enumerated by Cartan. Because of this, we will denote these groups by the same Cartan symbols which we used for notation of noncom­pact simple Lie groups.

In particular, the quasisimple Lie groups AI , All , Alli , and AIV are o -n o ::::.(n- 1 )/2 -m n the groups of motions of the spaces CS , (H ® C)S , CS ' , and

CRn , respectively. The quasisimple Lie groups BI and Bil are the groups of motions of the spaces sm , 2n (for m > 0) and R2n , respectively. The quasisimple Lie groups CI and CII are the groups of motions of the spaces o -n- 1 -m n- 1 -n- 1 HS , HS ' ( HR for m = 0) , respectively. The quasisimple Lie groups DI , DII , and Diii are the groups of motions of the spaces Sm ' 2n- I (for m > 0) , R2n- I , and 0HSyn- I , respectively. The quasisimple Lie group G is the group of motions of the space S g2 ' 6 • The quasisimple Lie groups

0 -2 FI and Fil are the groups of motions of the planes OS over the algebra 00 of semioctaves (obtained by the quasi-Cartan algorithm from the field 0)

-2 and OR , respectively, etc. As we have for simple Lie groups, to the isomorphic or locally isomor­

phic quasisimple Lie groups there correspond "equivalent geometries" whose fundamental groups are such groups.

Tables 3 . 3 and 3 .4 are the tables of real quasisimple Lie groups that are ob­tained by the quasi-Cartan algorithm from compact simple Lie groups (up to a local automorphism) and the isomorphisms of real quasisimple Lie groups. As in Tables 3 . 1 and 3 .2, Tables 3 . 3 and 3 .4 also indicate the spaces whose fundamental groups are the corresponding groups.

-(n- 1 ) /2 I I 0c'Sn , (H ® 0C)S , CS - ,n

s/- 1 , 2n OHSn- 1 , HS/- 1 , n - I /- 1 s2n- I ' OHSyn- l

sg2 · 6

OR2 0os2 ' -2 - 2 -2

(0 ® 0C)S , (O ® C)R , (00 ® C)S -2 - 2 -2

(0 ® 0H)S , (O ® H)R , (00 ® H)S -2 - 2

(0 ® 00)S , (O ® O)R

TABLE 3 . 3

A 1 = B1 = C1 D2 = B1 x B1

B2 = C2 A3 = D3

§3. 1 3. FINITE GEOMETRIES

CR1 = °CS1 = R2 R3 = °CS2 ' 1S 1 ' 3 = 'CR2 ' 0HSy 1 = S2 x R2

R4 = HR 1 , s 1 • 4 = 0Hs1

Rs = (H ® OC)Sl ' s l , 5 = csl , 3 ' S2 ' 5 = °CS3 , HSy2 = CR3

OHSy3 = Sl , 7

TABLE 3 .4

§3 .13 . Finite geometries

1 23

The simple Chevalley groups also admit geometric interpretations in spaces constructed over the corresponding fields and over the algebras built over these fields.

The finite Chevalley groups admit similar interpretations in spaces similar to those which were considered in this chapter but constructed over the finite fields Fq . In particular, groups in the class An can be interpreted as the groups of collineations of the projective space F qpn , groups in the class

Bn as the group of motions of the non-Euclidean space FqS2n , groups in the class en as the group of symplectic transformations of the symplectic

space F qsy2n- I , groups in the class D n as the group of motions of the non­

Euclidean space F qs2n- I , groups in the class A�2> as the group of motions

of the Hermitian space F q2Sn , and groups in the classes D�2> and D�3> as

the groups of motions of the non-Euclidean spaces FqS�2�- I and Fq3sr3>.

The Chevalley groups whose Satake graphs are shown in Figure 2. 1 3 ad­mit geometric interpretations in the form of groups of collineations of pro­jective spaces over division algebras constructed over corresponding fields. The geometry of the projective line P 1 and the plane S� , whose group of

collineations is isomorphic to that of the line P1 over the field F2 of 2-adic numbers, were considered by Jean-Pierre Serre (b. 1 926) in the paper Trees, amalgams, SL2 [Se l ) ( 1 977) . The geometric interpretations of arbitrary Chevalley groups are particular cases of the Tits "buildings" whose theory, as we have already indicated, is presented in his book [Ti5] and in his papers [Ti6] and [Ti7] .

CHAPTER 4

Lie Pseudogroups and Pfaffian Equations

§4.1 . Lie pseudogroups

After solving in his thesis the problem of the structure of usual (finite­dimensional) Lie groups which Cartan called "finite continuous groups'', Car­tan posed the similar problem for "infinite continuous groups'', i .e . , for in­finite-dimensional analogues of Lie groups. The following papers by Cartan were devoted to this problem: the two-part paper On the structure of infinite groups of transformations [2 1 ], [22] ( 1 904 ), Simple continuous infinite groups of transformations [23] , (28] ( 1 907 and 1 909) , and Subgroups of continuous groups of transformations [26] ( 1 908) .

While the finite-dimensional Lie groups are connected with the theory of ordinary differential equations, their infinite-dimensional analogues are re­lated to the theory of partial differential equations. Cartan started to study the latter as far back as 1 899.

At present, infinite-dimensional analogues of Lie groups are called Lie pseudogroups. The Lie pseudogroup considered by Cartan is a set of trans­formations of a space that contains the identical transformation (playing the role of the neutral element) and possesses the property that the result of suc­cessive realization of two transformations of this set (when this is possible) belongs to the same set. However, in contrast to usual Lie groups of trans­formations, in this case the successive re�lizations of transformations is not always possible: each such transformation is given by functions defined in certain domains, and the domain of one of the transformations may not have common points with the domain to which another transformation maps its domain. This explains the fact that this set of transformations is not a group and is the reason it is called "pseudogroup".

In the papers mentioned above, Cartan considered manifolds whose points are defined by complex coordinates and assumed that the transformations which he studied were given by analytic functions of these coordinates. As in the case of the finite-dimensional Lie groups, Cartan considered only "in­finitesimal transformations". This explains why he did not encounter the cases when for two transformations the result of their successive realization cannot be found. Because of this, Cartan used the term "groups" for sets of such transformations.

1 25

1 26 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

As he did in the case of the finite-dimensional Lie groups, Cartan consid­ered only those transformations of manifolds for which there is no subdi­vision of manifolds into the classes transposed by the transformations un­der consideration. Such groups and pseudogroups of transformations are called primitive groups and pseudogroups. Cartan showed that every infinite­dimensional primitive pseudogroup of complex analytic transformations be­longs to one of the following six classes:

1 ° . The pseudogroup of all analytic transformations of n complex vari­ables.

2° . The pseudogroup of all analytic transformations of n complex vari­ables with a constant Jacobian ( i .e . , transformations that multiply all volumes by the same complex number) .

3° . The pseudogroup of all analytic transformations of n complex vari­

ables whose Jacobian is equal to one ( i .e. , transformations that preserve vol­umes) .

4 ° . The pseudogroup of all analytic transformations of 2n � 4 complex variables that preserve the double integral

( 4. 1 )

5° . The pseudogroup of all analytic transformations of 2n � 4 complex variables that multiply the double integral ( 4. 1 ) by a complex function.

6° . The pseudogroup of all analytic transformations of 2n + 1 complex variables that multiply the form d z0 + E7= 1 z

; d zn+i by a complex function. The pseudogroup 4° is called the symp/ectic pseudogroup since its transfor­

mations preserve the exterior form E7= 1 d/ /\ dzn+i , and the latter defines the "symplectic geometry" in the hyperplanes at infinity of the tangent spaces CE2n to the manifold under consideration. This pseudogroup is also called the Hamiltonian pseudogroup since the mechanical system with generalized coordinates q;

and generalized impulses P; , whose motion is described by the Hamiltonian equations, can be viewed as a space where the exterior closed differential form w = dq; /\ dp; is given ( dw = 0) . The pseudogroup 6° is called the contact pseudogroup since in this case 2n + l complex coordinates

b . d 2 2 d' 0 I n 0 I n can e v1ewe as n + coor mates z , z , . . . , z , w , w , . . . , w , connected by the relation Q(z0 , z 1 , • • • , zn , w0 , w 1 , • • • , wn ) = 0 which establishes the correspondence between the points z (z0 , z 1 , • • • , zn ) of an n-dimensional space and hyperplanes of the space with coordinates w0 , w 1 , . . . , wn . Such transformations are called contact transformations (or trans­formations of tangency) . The theory of contact transformations was devel­oped by Lie.

Cartan showed that the pseudogroups l 0 , 3° , 4° , and 6° are simple pseu­dogroups, or, in his terms, they are "simple infinite continuous groups'', and the pseudogroups 2° and 5° are "invariant subgroups" of the pseudogroups

§4.2. THE KAC-MOODY ALGEBRAS 1 27

3° and 4° . He called the classes 1 ° , 3° , 4° , and 6° the "four large classes of simple infinite continuous groups" and considered them to be analogous to the "four large classes of simple finite continuous groups"-the infinite series of simple finite Lie groups.

There are similar classes of pseudogroups for primitive pseudogroups of real transformations defined by analytic functions of real variables.

In his extended French translation [46] of Fano's paper [Fa] Cartan de­voted a few sections to the real Lie pseudogroups. Along with Lie's and Cartan's research described by Fano, Cartan included in [46] some results which were obtained after the publication of [Fa] . We note one pseudogroup from pseudogroups considered in [46], namely, the pseudogroup of transfor­mations of the set of straight lines of the space R3 under which the normal congruences (the congruences of normals to a surface) are transformed into the same kind of congruences. Cartan noted the importance of these trans­formations for optics.

The infinite-dimensional pseudogroups were applied to geometry in the book of Oswald Veblen ( 1 880- 1 960) and John H. C. Whitehead ( 1 904- 1 960) , The foundations of differential geometry [VW] ( 1 932) , since the transforma­tions of coordinates of a differential-geometric manifold form precisely such a pseudogroup. In this connection the series of papers by Victor V. Wagner ( 1 908- 1 97 1 ) is very interesting. We note his papers On the theory of pseu­dogroups of transformations [Wag 1 ] ( 1 950) and Algebraic theory of differential groups [Wag2] ( 1 9 5 1 ) .

§4.2. The Kac-Moody algebras

At present, several types of infinite-dimensional generalizations of Lie al­gebras and groups are being studied. Victor G. Kac (b. 1 943) in his book Infinite-dimensional groups with applications [Kac2] ( 1 985 ) indicated that, although the general theory of infinite-dimensional theory of Lie algebras and groups had not yet been constructed, "there are, however, four classes of infinite-dimensional Lie groups and algebras that underwent more or less intensive study. There are, first of all , the . . . Lie algebras of vector fields and the corresponding groups of diffeomorphisms of a manifold. Starting with the work of Gel'fand-Fuks . . . , there emerged an important direction having many geometric applications, which is the homology theory of infinite­dimensional Lie algebras of vector fields on a finite-dimensional manifold. There is also a rather large number of works which study and classify various classes of representations of the groups of diffeomorphisms of a manifold . . . The second class consists of Lie groups (respectively Lie algebras) of smooth mappings of a given manifold into a finite-dimensional Lie group (resp. Lie algebra) . In other words, this is a group (resp. Lie algebra) of matrices over some function algebra but viewed over the .base field. (The physicists refer to certain central extensions of these Lie algebras as current algebras. ) . . . The third class consists of the classical Lie groups and algebras of operators

1 28

e) v e2i 6

4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

FIGURE 4. 1 FIGURE 4.2

in a Hilbert or Banach space. There is a rather large number of scattered results in this area . . . Finally the fourth class of infinite-dimensional Lie algebras is the class of so-called Kac-Moody algebras" [Kac2, pp. ix-x] . The Lie pseudogroups belong to the first of these classes. Here, Kac mentioned the paper by Gel'fand and Dmitry B. Fuks (b. 1 939) , The cohomology of the Lie algebra of tangent vector fields on a smooth manifold ( 1 969- 1 970) [Gel, vol . 3 , pp. 290-306 and 323-329] (see also the book Cohomology of infinite-dimensional Lie algebras [Fuk] ( 1 986) ) .

The Kac-Moody algebras introduced b y Kac i n the paper Simple graded Lie algebras of finite growth [Kac l ] ( 1 968) and by Robert V. Moody (b. 1 94 1 ) in the paper A new class of Lie algebras [Moo] ( 1 968) are closest in their properties to the Lie algebras of the simple Lie groups. Kac's book [Kac2] is also devoted to the theory of these algebras. As for the Lie algebras of simple Lie groups, for the Kac-Moody algebras, the following notions can be defined: the root systems, the Weyl groups which in this case are infinite discrete groups generated by reflections, and also the systems of simple roots and the Dynkin graphs. The Kac-Moody algebras are divided into three types: nontwisted algebras, 2-twisted algebras, and 3-twisted algebras. These names and notation of these algebras are given by analogy with the names and notation of the Chevalley groups mentioned earlier. The Dynkin graphs of the nontwisted Kac-Moody algebras coincide with the augmented Dynkin graphs (Fig. 2 . 8 ) . Figures 4. 1 and 4.2 show the Dynkin graphs of the 2- and 3-twisted Kac-Moody algebras, respectively [Kac2, pp. 44-45) .

The congruence of the Dynkin graphs of the nontwisted Kac-Moody alge­bras with the augmented Dynkin graphs of the simple Lie groups is connected with the isomorphisms of the Weyl groups of the nontwisted Kac-Moody al­gebras and the affine Weyl groups of the simple Lie groups. Note that the . (2) (2) (2) (2) twisted Kac-Moody algebras are denoted by Kac as A2 , A2n , A2n- I , Dn+ I , E�2l , and D�3l , and by Moody (if one transfers on top the low indices 2 and 3 which Moody places after the low index preceded by a comma) as

§4. 3 . PFAFFIAN EQUATIONS 1 29 (2) (2) (2) (2) (2) (3) . .

A 1 , BCn , Cn , Bn , F4 , and G2 [Moo, p. 229] , respectively. This no-tation corresponds to the notation of those Lie groups whose Dynkin graphs are obtained by removal of one vertex of the Dynkin graph of the twisted Kac-Moody algebras.

§4.3. Pfaffian equations

Cartan's first work on the theory of partial differential equations was his paper On certain differential expressions and the Pfaff problem [ 1 4] ( 1 899) , which was followed by the papers On some quadratures, whose differential el­ement contains arbitrary functions [ 1 5] ( 1 90 1 ) , On the integration of systems of exact equations [ 1 6] ( 1 90 1 ) , On the integration of certain Pfaffian systems of character two [ 1 7] ( 1 90 1 ) , On the integration of completely integrable dif­ferential systems [ 1 8] ( 1 902) , and On the equivalence of differential systems [ 1 9] ( 1 902) .

In the first of these papers Cartan showed that every system of partial differential equations is equivalent to a system of differential equations:

()°' Q d i (4.2) = a; (x) x = 0 , a = l , 2 , . . . , s ,

a so-called system of Pfaffian equations. For example, the Laplace equation

(4. 3) a 2z a 2z -2 + -2 = 0 , ax ay

which is one of the fundamental equations of mathematical physics, with the help of substitution a z /ax = u , a z /a y = v , can be reduced to the first-order system of partial differential equations:

au av au av (4·4) ax = ay ' ay = - ax '

the so-called system of Cauchy-Riemann equations which the real and imag­inary parts of an analytic function w = u + iv = f(x + iy) of a complex variable x + iy satisfy. The latter system is equivalent to the following sys­tem of differential equations:

(4. 5) 0 1 = du - pdx - qdy = O , 2 () = dv + qdx - pdy = 0 ,

where p = au/ox = av /oy , q = 8u/8y = -av /ox. The appropriateness of transition from systems of partial differential equations to Pfaffian equa­tions is explained by the fact that equations ( 4.2) are invariant with respect to an arbitrary change of both dependent and independent variables, while in a system of partial differential equations the choice of independent variables is predetermined.

The Pfaffian equations are named after the mathematician and astronomer Johann Friedrich Pfaff ( 1 765- 1 825) who considered such equations in 1 8 1 4-1 8 1 5 . The term "Pfaffian equations" was introduced by Carl Gustav Jacob Jacobi ( 1 804- 1 8 5 1 ) who named the problem of integration of such equation

1 30 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

the "Pfaff problem". Papers of Feodor Deahna ( 1 8 1 5- 1 84 1 ) , August Leopold Crelle ( 1 780- 1 855 ) , Jacobi, Ferdinand Georg Frobenius ( 1 849- 1 9 1 7) , Lie, and Darboux were devoted to the investigation of this problem. Appearing in Cartan's research in 1 899, the Pfaffian equations were the subject of his investigations and then the tool of research in many of his papers during his entire life. After finding in his papers of 1 899- 1 902 a new approach to the investigation of such systems, Cartan used them widely both in his geomet­ric papers and in his papers on the theory of Lie groups and mathematical physics.

System ( 4.2) of differential Pfaffian equations permits the following geo­metric interpretation. If one considers the variables to be the coordinates of points of an n-dimensional manifold xn , then the differentials dxi can be considered as coordinates of the vector dx belonging to the tangent linear space Tx (Xn ) of the manifold xn at its point x . If system (4.2) contains s linearly independent equations, s < n , then it defines a linear subspace Ah (x) of dimension h = n - s in the space Tx (Xn ) . If the rank of system ( 4 .2) remains constant and equal to s in the whole manifold xn , then this system defines a subspace Ah (x) of the space Tx (Xn ) at each point x of

the manifold xn . The set of subspaces ll.h (x) of the tangent spaces Tx (Xn ) taken at each point x of the manifold xn is said to be a distribution Ah .

An integral manifold of Pfaffian system ( 4.2) is a smooth submanifold vk

of the manifold xn such that, at each of its points, it is tangent to the subspace ll.h (x) determined at the point x by system (4.2) . The di­mension k of the integral manifold Vk

cannot exceed the dimension h of Ah (x) . System (4.2) always has one-dimensional integral manifolds-the in­tegral curves, and in finding them, the system ( 4 .2) is reduced to a system of ordinary differential equations.

§4.4. Completely integrable Pfaflian systems

Pfaffian system ( 4 .2) . is said to be completely integrable if it has integral manifolds vh of maximal dimension h , and through every point of the manifold xn there passes a unique integral manifold vh , i .e . , the integral manifolds vh of a completely integrable system ( 4.2) form a foliation in the manifold xn .

.

Conditions for the system ( 4 .2) to be completely integrable were found by Frobenius. In order to write down these conditions, one must construct the bilinear covariants of the system. Let x be a point of the manifold xn and d1 x = {d1 x

; } , d2x = {d2x; } be two tangent vectors to this manifold at the

point x . Denote by (}"'(d1 ) = a� (x)d1 xi and (}"' (d2 ) = a�(x)d2x

i the values of these forms on these vectors. Differentiate the first of these expressions along the vector d1 x and the second one along the vector d2x . Then, the

§4.4. COMPLETELY INTEGRABLE PFAFFIAN SYSTEMS

bilinear Frobenius covariant is the difference of these two differentials:

(4 .6) difJ a (d1 ) - d1 ()a (d2 ) = d2a� (x)d1 x; - d1 a� (x)d2x;

1 3 1

(on the right-hand side of expression ( 4.6) the terms containing the mixed differentials d2d1 x

; and d1 d2x; cancel) . Initially Cartan called the left-hand

side of expression ( 4.6) the exterior derivative of the form ()a and denoted it by ( ()a ) ' . Later he started to call this expression the exterior differential of the form ()a and denoted it by d ()a . Cartan called the right-hand side of expression (4.6) the exterior product of the forms da� and dx; . He initially denoted the exterior product of the forms w1 and w2 by the symbol w1 w2 and later by the symbol [w 1 w2] . At present, this product is denoted by w1 /\ w2 , and relation ( 4.6) can be written in the form:

(4. 7)

On any integral manifold vk of system ( 4 .2) , equations ( 4.2) are satisfied

as well as the equations

(4 .8)

obtained by exterior differentiation of system ( 4.2) . By ( 4. 7) , equations ( 4.8) can be written in the form (oa� /8xi)dx; /\ dxi = 0 . But since dx; /\ dxi = -dxi /\ dx; , one can also write these equations in the form:

(4.9) ({)a� - {)a! ) dx; /\ dxi = 0. 8x1 ox'

These equations impose conditions on the coordinates of any two vectors d1 x and d2x tangent to an integral manifold. If the vectors d1 x and d2x satisfy equations ( 4. 9) , then we say that they are in involution relative to the system of exterior forms ( 4 .8 ) .

Exterior differentiation of equations ( 4 .8 ) leads to identities since

d(d()a ) = d ({)a� ) /\ dx; /\ dxi = ( 8� a� k ) dx; /\ dxi /\ dxk

= 0 8 x1 8 x1 8 x

by the symmetry of the second mixed derivatives. Thus, the system of equa­tions ( 4.2) and ( 4 .8 ) is closed with respect to the operation of exterior differ­entiation.

If the system of equations ( 4.2) is completely integrable, the integral man­ifolds of this system are of dimension h = n - s , and equations ( 4 .8 ) must not impose any new relations on the coordinates of the tangent vectors in addition to the relations imposed by equations ( 4.2) . It is easy to see that this condition can be written in the form

(4. 1 0) 0: ' p = 1 ' 2 ' . . . ' s .

1 32 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

In his paper of 1 877 Frobenius proved that condition ( 4. 1 0) is not only necessary but also sufficient for the complete integrability of system ( 4.2) .

§4.S. Pfaffian systems in involution

In the papers On integration of systems of exact equations [ 1 6] ( 1 90 1 ) and On the structure of infinite groups of transformations [2 1 ] , [22] ( 1 904- 1 905) , Cartan constructed the theory of systems of Pfaffian equations that are not completely integrable. Following Lie, who used the term "involutive systems of equations'', Cartan said the system of Pfaffian equations ( 4.2) were in in­volution if at least one two-dimensional integral manifold V2 passes through each integral curve V1 of this system, at least one three-dimensional integral manifold V3 passes through each of its integral manifolds V2 , etc. , and finally at least one integral manifold VP passes through each of its integral manifolds vp- t •

Cartan found necessary and sufficient conditions for the system of Pfaffian equations ( 4.2) to be in involution. For this, he considered p-dimensional elements consisting of a point x of the manifold xn and a p-dimensional subspace EP of the tangent space Tx (Xn ) to the manifold xn at the point x . This element is called the integral element and is denoted by IP if all its vectors satisfy the system of Pfaffian equations ( 4 .2) , i .e . , IP belongs to !:ih , and if, in addition, any two vectors of this integral element are in involution relative to the system of equations (4. 8 ) . It is obvious that if system (4 .2) is in involution, then a two-dimensional integral manifold I2 passes through each of its one-dimensional integral elements I 1 , a three-dimensional inte­gral element I3 passes through the integral element I2 , etc. , and finally an integral element IP passes through the integral element Ip- I . This sequence of enclosed integral elements I 1 , I2 , • • • , IP is called an integral chain. An integral chain is said to be regular if each of its integral elements is in general position, i .e . , no more integral elements Ik pass through an element Ik- t than through any neighboring (k - 1 )-dimensional integral element. Cartan proved that a necessary and sufficient condition for the system of Pfaffian equations ( 4.2) to be in involution is the existence of a regular chain of inte­gral elements I 1 , I2 , • • • , IP for each point x of the manifold xn .

When one is constructing an integral chain, there comes a time when there is no integral element of dimension g + 1 passing through an integral element of dimension g . In this case system ( 4.2) is in involution for all p $ g , but it does not have this property for p > g . The number g is called the genre of system (4.2) .

Cartan proved the existence theorem for solution of system ( 4 .2) of genre g . Let IP be an integral element of system ( 4.2) of dimension p $ g at a point x of the manifold xn ; then there exists an infinite set of

§4.5 . PFAFFIAN SYSTEMS IN INVOLUTION 1 33

p-dimensional integral manifolds passing through a manifold vp- I and tan­gent to the element IP at the point x0

, and for p = g there exists only one such manifold. The proof of this theorem is based on the classical Cauchy­Kowalevskaya theorem on the existence of a solution of a system of partial differential equations. Since the Cauchy-Kowalewskaya theorem is valid only in the class of analytic functions, the Cartan theorem is also valid only in the case when all coefficients of equations ( 4 .2) are analytic functions, and the desired integral manifolds are analytic manifolds.

The Cartan theory not only gives the answer to the question of the existence of integral manifolds of system ( 4.2) but also establishes arithmetic tests under which there exist integral manifolds of a certain dimension p (p � g) of system ( 4.2) and indicates an arbitrariness with which these manifolds exist.

These tests were formulated by Cartan in his paper On the structure of in.finite groups of transformations [2 1 ] , [22] ( 1 904- 1 905 ) . Here, applying the notion of the "character of a system of Pfaffian equations" introduced by Eduard von Weber in the paper On the theory of invariants of a system of Pfaffia.n equations [Web] ( 1 898) , Cartan determined the system of characters of a system of Pfaffian equations (the von Weber character was the first of Cartan's characters) , and using these characters, he established necessary and sufficient conditions for existence of a solution of a system of Pfaffian equations. We now show in more detail how Cartan established these tests.

Suppose one is looking for p-dimensional integral manifolds of the system of Pfaffian equations ( 4.2) where p � g . System ( 4 .8 ) of exterior differentials of this system can be reduced to the form

( 4. 1 1 ) o (J i ()j 2

o (J i (ju o (ju ()v O aij /\ + aiu /\ + auv /\ = ,

where (J i , i = 1 , . . . , p , are Pfaffian forms that are independent on an integral manifold, and (Ju , u = 1 , . . . , q , are the remaining characteristic forms of system ( 4.2) . Let r; be the rank of the system of linear equations which is obtained from ( 4. 1 1 ) when one constructs an ( i + l )-dimensional integral element. The characters s; , l � i � p , of system ( 4.2) are the differences r; - r;_ 1 (note that s1 + s2 + · · · + sp- I � q) , and the character sP = q - (s1 + s2 + · · · + sP_ 1 ) . The number Q = s1 + 2s2 + · · · + psP , which at present is called the "Cartan number", is equal to the number of parameters on which a p-dimensional integral element depends. The characteristic forms (Ju can be expressed from system ( 4. 1 1 ) in the form of linear combinations of the basis forms (); : (Ju = b; () ; . If we denote by N the number of independent coefficients in these decompositions, then the Cartan test for the involutivity of system ( 4 .2) is expressed by the relation: N = Q . Moreover, if the last nonvanishing character is sm , then the solution of system ( 4.2) depends on sm functions of m real variables.

1 34 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

§4.6. The algebra of exterior forms

We have already mentioned the operation of exterior differentiation of a linear form and the operation of exterior multiplication of two such forms which were introduced by Cartan. These operations are particular cases of more general operations applied by Cartan not only to linear forms but also to differential forms

( 4. 1 2)

called exterior forms of degree p . Here a; ; . . . ; is the tensor which is skew-1 2 p

symmetric in all its indices, i .e . , it changes the sign with any odd substitution of indices and preserves the sign with any even substitution of indices, and /\ is the symbol of exterior multiplication which also indicates that the form ( 4. 1 2) changes the sign with any odd substitution of the differentials and preserves the sign with any even substitution of the differentials. For exterior forms, the operation of exterior multiplication:

(4. 1 3 )

w /\ w = (a . . . dx;1 /\ dx;2 A . . . /\ dx;P ) I 2 1 1 12 · · · 1P . . . /\ (b . . . dx1 1 A dxli /\ · · · /\ dx1q )

l 1 Ji · · ·Jq = a . . . b . . . dxi1 dxi1 /\ dx;2 /\ • • • 1 1 12 " ".° IP J 1 Ji · ·:Jq . .

/\dx'P /\ dx1 1 /\ dxli /\ · · · A dx1q

and the operation of exterior differentiation:

( 4. 1 4)

are defined. Moreover, if w 1 and w2 are differential forms of degrees p and q re­

spectively, then the exterior differential of the product w 1 /\ w2 is equal to

( 4. 1 5 )

We have seen that Frobenius used the operations (4. 1 3 ) and (4. 1 4) . We also encountered a particular case of the rule:

( 4. 1 6) d(dw) = 0 ,

which essentially was known to Poincare and thus frequently called the Poincare theorem.

An exterior differential form w is called closed if d w = 0 and exact if there exists a differential form () such that w = d () . By the Poincare theorem, every exact differential form is closed, although not every closed differential form is exact.

§4.7. APPLICATION OF THE THEORY OF SYSTEMS IN INVOLUTION 1 35

Cartan often used the property that if the equation ()i /\ a/ = 0 holds

where Oi and a/ are Pfaffian forms and the forms a/ are linearly indepen­

dent, then the forms Oi are linear combinations of the forms a/ , and the coefficients bu of these linear combinations are symmetric:

( 4. 1 7) ()i = bua/ , bu = b1 i .

At present this statement is called the Cartan lemma. The exterior forms constitute an algebra with respect to their addition and

multiplication. This algebra coincides with the Grassmann algebra.

§4.7. Application of the theory of systems in involution

In many of his investigations Cartan applied the theory of systems of Pfaf­fian equations in involution which he created. In the paper The Pfaffian sys­tems with five variables and partial differential equations of second order [30] ( 1 9 1 0) , this theory was applied to the investigation of a system of two partial differential equations of second order-a problem investigated by Edouard Goursat ( 1 858- 1 936) . Using this theory, in this paper Cartan investigated a system of Pfaffian equations with five variables to which these two equa­tions can be reduced, solved the equivalence problem relative to admissi­ble transformations for two such systems, and gave a detailed classification of the systems of this type. In the paper On systems of partial differential equations of second order with one unknown function and three independent variables in involution [33] ( 1 9 1 1 ) , Cartan investigated the systems indicated in the title that can be reduced to a system of four Pfaffian equations. In the paper On Backlund transformations [ 45] ( 1 9 1 5 ) , the theory of systems in involution was applied to the study of Backlund transformations by means of which the known solutions of a system of partial differential equations can be transformed into certain new solutions of this system. In the paper On the theory of systems in involution and its application to relativity theory [ 1 3 1 ] ( 1 9 3 1 ) , this theory was applied to the investigation of equations to which certain problems of general relativity can be reduced. Most applications of the theory of systems in involution are related to differential geometry of sub­manifolds of various homogeneous spaces, which we will consider in Chapter 5�

In 1 934 the theory of systems in involution constructed by Cartan for Pfaffian equations was generalized for systems consisting not of only Pfaf­fian equations but also of exterior differential equations of different orders by Erich Kahler (b. 1 906) in his book Introduction to the theory of systems of differential equations [Kah2] . In the book Exterior differential systems and their geometric applications [ 1 8 1 ] ( 1 945) , Cartan presented a systematic ex­position of both his own theory of solution of Pfaffian equations and Kahler's theory. Following Kahler, in this book Cartan changed his original term "ex­terior derivative" to the presently accepted term "exterior differential" and

1 36 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

his original notation w' of this operation to the presently accepted notation dw .

The books Geometric theory of partial differential equations [Ra2] ( 1 94 7) by Petr K. Rashevskii ( 1 907- 1 983) and Cartan 's method of exterior forms in differential geometry [Fin] ( 1 948) by Finikov are devoted to the original expositions of the Cartan theory.

§4.8. Multiple integrals, integral invariants, and integral geometry

The calculus of exterior forms created by Cartan turned out to be very useful in the theory of multiple integrals as well as in the theory of integral invariants and in integral geometry, which are both connected with the theory of multiple integrals.

While the simple Riemann integral is invariant under a change of vari­ables, the double integral f fv f(x , y) dx dy , under the change of variables x = x( u , v ) , y = y( u , v ) , will be transformed according to the following formula:

(4 . 1 8 ) /Lf(x , y) dx dy = jfv, f(x(u , v ) , y (u , v ) )J (u , v ) du dv ,

where J(u , v ) = l �;j�: �;j�� ' is the Jacobian of the functions x =

x(u , v ) , y = y(u , v ) with respect to the variables u and v , and D' is the domain of the variables u and v (the functions x = x( u , v ) , y = y( u , v ) are assumed to be differentiable, and the Jacobian J ( u , v ) is assumed to be nonvanishing in the domain D') . Formula ( 4. 1 8) shows that a double integral is not invariant under a change of variables, i .e . , the right-hand side of this formula cannot be obtained by a simple substitution of the differentials dx = (8x/8u)du + (8x/8v )dv and dy = (8y/8u)du + (8y/8v )dv into its left-hand side. The same is true for triple and other multiple integrals. However, the expression of a double integral can be made invariant if we write it in the form:

( 4. 1 9) J k f(x , y) dx /\ dy ,

i .e. , use the exterior multiplication dx /\ dy in its integrand since dx /\ dy = J(u , v )du /\ dv . After this, formula (4 . 1 8 ) can be written in the form:

(4 .20) JL f(x , y) dx /\ dy = J fv, f(x(u , v ) , y(u , v ) ) dx(u , v ) /\ dy(u , v ) .

Similarly, fo r a surface integral J fs (P dy dz + Q dz dx + R dx dy) to be

invariant under a change of variables, we should write it in the form

( 4 .2 1 ) Jfs(P dy /\ dz + Q dz /\ dx + R dx /\ dy) ,

§4.8. MULTIPLE INTEGRALS AND INTEGRAL GEOMETRY 1 37

i .e . , in the form of an integral of an exterior form. Integrals ( 4. 1 9) and ( 4 .2 1 ) are particular cases of the integral

(4.22)

along a p-dimensional submanifold VP of an n-dimensional manifold xn . Expression ( 4.22) remains invariant under any differentiable transformation of coordinates in the manifold xn . The classical formulas of Green, Gauss, and Stokes are particular cases of the general formula:

(4 .23)

where a VP is the boundary of the submanifold VP , w is an exterior dif­ferential form of degree p - 1 , and d w is the exterior differential of the form w which is an exterior form of degree p on the closed manifold-the closure of the manifold VP . For the Green formula, p = 2 , V2 is a plane domain, a V2 is its boundary, the form w is w = Pdx + Qdy , and

dw = ( �� - ��) dx t\ dy.

For the Gauss formula, p = 3 , V3 is a domain of a three-dimensional space, a V2 is a surface boundary of this domain, the form (J) is:

and

w = Pdy t\ dz + Qdz t\ dx + Rdx t\ dy ,

(aP aQ aR ) dw = ax + ay + a z dx t\ dy t\ dz.

For the Stokes formula, p = 2 , V2 is a domain on a two-dimensional sur­face, a V2 is its boundary, the form w is w = Pdx + Qdy + Rd z , and

dw = (aQ - aP ) dx t\ dy + (aR - aQ ) dy t\ dz + (aP - aR ) dz t\ dx. ax ay ay az az ax Formula ( 4 .23) is called the generalized Stokes formula.

Cartan systematically presented the theory of integral invariants fa his book Lectures on integral invariants [64] ( 1 922) where, applying the method of exterior forms� he completed the construction of this theory created by Poincare.

Suppose a system of ordinary differential equations

(4.24) dx; ; 1 2 n dt = P (x , x , . . . , x , t) , i = l , 2 , . . . , n ,

is given, where P; (x 1 , x2 , • • • , xn , t) are differentiable functions. An in­tegral invariant of this system is an integral f vP w along a submanifold VP of dimension p < n on which the parameter t has a constant value, and

1 38 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

this value is not changed when the points of the submanifold VP move along integral curves of system ( 4.24) . An integral invariant is called absolute if the property of invariance holds for any domain of integration, and it is called relative if this property holds only for closed domains.

Applications of this theory to mechanics are most important. The funda­mental differential equations of mechanics can be written in the form of the Hamiltonian equations:

(4 .25) dp . 8H -' = dt -{)qi '

where q i are the generalized Lagrange coordinates of a system, pi are gener­

alized momenta, and H = H(pi , / , t) is a Hamiltonian function. A relative

integral invariant of this system is the integral f8 vP p; d q; , and its absolute

invariant is the integral f fvp dp; /\ dqi . By the generalized Stokes theorem, we have the following relation:

(4 .26)

In Cartan's book these and some other integral invariants of mechanics are investigated in detail. The general theory, which was developed during this investigation, was applied to the three-body problem, to light propaga­tion in a homogeneous medium, and to other problems of mechanics and mathematical physics.

As far back as 1 896, in his paper The principle of duality and certain mul­tiple integrals in tangential and line spaces [ 1 0) , Cartan considered multiple integrals on families of straight lines and planes of the space R3

. These inte­grals are integral invariants relative to the groups of motions of the spaces R2

and R3 • Such an invariant for a one-parameter family of straight lines inter­

secting a given closed curve is the "perimeter" which is proportional to the curve length. Cartan also defined an integral invariant for a two-parameter family of straight lines in the space (a rectilinear congruence) . This integral vanishes if a congruence is normal ( i .e . , it is a congruence of normals to a surface) . By means of this invariant, one can prove very simply the classical theorem of Etienne Malus ( 1 775- 1 8 1 2) , which states that a normal congru­ence remains normal after any number of reflections and refractions. This Cartan paper initiated a branch of geometry which is at present called in­tegral geometry. Before this paper, problems from this branch of geometry were considered in probability theory. Such problems include, for example, the problem of throwing a disk, a square plate, and a needle, which were solved by Georges Louis Buff on ( 1 707- 1 788) in his Essay of moral arith­metic [Buf] ( 1 777) , and the "Crofton formulas" found by Morgan William Crofton ( 1 826- 1 9 1 5 ) in his paper On the theory of local probability [Cro]

§4.9. DIFFERENTIAL FORMS AND THE BETTI NUMBERS 1 39

( 1 868) . Cartan was the first to solve problems of this type as pure geometric problems.

Integral geometry was significantly developed in the 1 930s. The essen­tial role in this development was played by the invariant measure in the Lie groups used by H. Weyl and later by Cartan himself in their research on the theory of simple Lie groups. This measure allows one to define invariant measures in the manifolds of different geometric objects in the spaces whose transformation groups are the groups indicated above. The term "integral ge­ometry'', by analogy with the term "differential geometry", was suggested by Blaschke in his books Integral geometry I. Determination of density for linear subspaces in En [Bla4, vol. 2, pp. 2 1 9-238] ( 1 935 ) and Lectures on integral geometry [Bla5] ( 1 936) . Following the book Integral geometry I, Blaschke and his students and co-workers (Boyan Petkantschin, 0. Varga, Luis An­tonio Santalo (b. 1 9 1 1 ) , Wu Tayen, Hildegard Rohde, and others) wrote a long series of papers under the general heading Integral geometry [Bla l ] . Altogether there were 33 papers in this series. They were related to integral geometry in the Euclidean, non-Euclidean, affine, projective, and Hermitian spaces. These and many other investigations in integral geometry were sum­marized by Santalo in his books Introduction to integral geometry [San 1 ] ( 1 953 ) and Integral geometry and geometric probability [San2] ( 1 976) . We note also Chern's paper On integral geometry in Klein spaces [Chr l ] ( 1 942) , where the author introduced a general method for solving problems of this type based on integration in Lie groups. New directions in integral geometry were found by Rashevskii in the paper Polymetric geometry [Ra 1 ] ( 1 94 1 ) (pa­pers by Boris V. Lesovoi ( 1 9 1 6- 1 942) , Measure of area in a two-parameter family of curves on a surface [Les] ( 1 948) , and I. M. Yaglom, Tangential met­ric in a two-parametric family of curves on a surface [Ya l ] ( 1 949) , are also related to these directions) and in the book Integral geometry and representa­tions theory [GGV] ( 1 962) by Gel'fand, Mark I. Graev (b. 1 922), and Naum Ya. Vilenkin ( 1 920- 1 99 1 ) , where a series of problems of integral geometry connected with the theory of representations of noncompact Lie groups by linear operators in function spaces was solved (see also the book Groups and geometric analysis [Hel2] ( 1 984) by Sigurdur Helgason (b. 1 927) ) .

§4.9. Differential forms and the Betti numbers

In the paper On the integral invariants of certain closed homogeneous spaces and topological properties of these spaces [ 1 1 8] ( 1 929) , Cartan considered integral invariants that are integrals of exterior invariant forms on compact homogeneous spaces and that are invariant relative to transformations of these homogeneous spaces. He showed how to use these invariants to define important topological invariants of these spaces-the so-called Betti numbers.

The term "topology", i .e. , the geometric discipline that studies the invari­ants of one-to-one continuous transformations whose inverses are also con­tinuous, came from the term Analysis situs or Geometria situs. This term was

1 40 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

introduced by Gotfried Wilhelm Leibniz ( 1 646- 1 7 1 6) , who, in 1 679, in his well-known letter to C. Huygens, expressed the idea that in addition to alge­bra, "we need still another analysis which is distinctly geometric or linear and which will express the situation ( situm) directly as algebra expresses the mag­nitude". Under the influence of this idea of Leibniz, Euler in his "problem on the seven Konigsberg bridges" used the term "geometry of position" (geome­tria situs) in the sense of what we now call topology. Afterwards the term the "geometry of position" (geometrie de situation, geometrie de position, Geometrie der Lage) was used in the sense of the theory of chess problems by A. T. Vandermonde ( 1 735- 1 796) and in the sense of projective geometry by Lazare Carnot ( 1 753- 1 823) , Theodor Reye ( 1 838- 1 9 1 9) , and von Staudt. Grassmann created a vector calculus in a multidimensional space also under the influence of this idea. This term in the sense of topology was used by Gauss, and Bernhard Riemann ( 1 826- 1 866) in Theory of Abelian functions [Rie l ] ( 1 857 ) gave to this term in the same sense the name Analysis situs­the "analysis of position". This term was used by Poincare for the title of his fundamental memoir on combinatorial topology.

The term "topology" appeared in 1 84 7, as the translation of the Latin term of Leibniz into Greek, in the paper of Gauss's student Johann Bene­dict Listing ( 1 808- 1 882) , Preliminary studies in topology [Lis] . However, this term was accepted only in the 20th century. Originally Cartan used the Riemann and Poincare term and, in spite of the fact that in the titles of his papers [97] ( 1 927) and [ 1 1 8] ( 1 929) the terms "topology" and "topological properties" appeared, in the title of the book [ 1 28] ( 1 930) he again used the term "Analysis situs". In Theory of Abelian functions [Rie l ] , Riemann considered multivalent surf aces which represent multivalued functions of a complex variable and are defined by algebraic equations F (x , y) = 0 con­necting the complex variables x and y . At present these surfaces are called Riemannian surfaces. He subdivided such surfaces into simply connected surfaces (divided into two parts by any cut) , doubly connected surfaces (the cuts that do not divide them into two parts make them simply connected surfaces) , triply connected surfaces · (the cuts make them doubly connected) , etc. , and to each closed two-sided surface he put in correspondence the "or­der of connection" determined by the number of cuts that are necessary to make the surface simply connected. In the case of closed two-sided surfaces, this number of cuts is always even and if one denotes this number by 2p , then the "order of connection" is equal to 2p + l (for a sphere, p = 0 , for a torus, p = 1 , and for a "sphere with p handles'' , it is equal to p) . At present, the number p for Riemannian surfaces defined by the equation F (x , y) = 0 is called the genus of a plane algebraic curve F (x , y) = 0 . For a polyhedron with N0 vertices, N1 edges and N2 faces, the number p is connected with the Euler characteristic x = N0 - N1 + N2 by the relation x = 2 - 2p . In his Fragments related to Analysis situs, published posthu­mously, Riemann suggested a multidimensional generalization of his "orders

§4.9. DIFFERENTIAL FORMS AND THE BETTI NUMBERS 1 4 1

of connection" defined by his friend Enrico Betti ( 1 823- 1 892) in his paper On spaces of arbitrary numbers of dimensions [Bet] ( 1 8 7 1 ) . Betti introduced the orders of connection of the mth type. in various dimensions.

The theory outlined by Riemann and Betti was developed by Poincare in his memoir Analysis situs [Poi4], which we already mentioned above. In this memoir Poincare introduced the notion of homeomorphism of manifolds that are curves or surfaces in a multidimensional space (actually this space is an affine space En ) and the Betti numbers of these manifolds coinciding with the "orders of connection" of Betti. Poincare defined these numbers as follows. To each p-dimensional manifold yP , he put in correspondence the (p - 1 ) -dimensional manifold a VP and the boundary of VP , and he called the manifold VP homological to 0 if this manifold itself is the boundary Of a (p + 1 )-dimensional manifold yP+ I : yP = 8 Vp+ I . If yP = 8 Vp+ I , then the boundary of yP is equal to 0, i .e . , a yP = 0 . Distinguishing the positive and negative orientation of manifolds, Poincare defined multiplica­tion of manifolds by integers where multiplication by - 1 means change of orientation. Poincare also defined the sum of manifolds, their linear combi­nations with integer coefficients, and the linear independence of these linear combinations. If a manifold vn carries Pm - 1 and only Pm - 1 linearly independent closed m-dimensional manifolds, Poincare said that the "order of connection" of the manifold Vm relative to the m-dimensional manifolds is equal to Pm . The numbers p1 , p2 , • • • , Pn- I defined in this way, and coinciding with the Betti "orders of connection'', Poincare called the Betti numbers. Poincare also defined the commutative groups that are quotient groups of groups of all closed linear combinations of submanifolds of the given manifold (at present they are called cycles) by the subgroup of this group consisting of all linear combinations homological to 0. These groups are the direct sums of a certain number of free cyclic groups Z (which are isomorphic to the additive group Z of integers) and a few finite cyclic groups Z1 . • The number of free cyclic summands of this group is equal to Pm - 1 , i .e. , one less than the Betti number defined by Poincare (at present, the num­bers Pm - 1 themselves are called "Betti numbers" and are denoted by Pm ) , and the orders t; of finite cyclic summands Z1 . of these groups are called the "torsion coefficients". Since these groups ar� closely connected with the Betti numbers, Poincare called these groups the Betti groups.

In the earlier mentioned paper, On the integral invariants of certain closed homogeneous spaces and topological properties of these spaces [ 1 1 8] ( 1 929) , Cartan, developing Poincare's idea on the importance of integrals of exact differentials for topology (which Poincare expressed in his Analysis situs) , showed that the Betti numbers of compact topological spaces can be cal­culated as the number of linearly independent integrals of the exact differ­ential forms of order p . In this paper Cartan introduced the polynomials 'E;P/-i , whose coefficients are the Betti numbers, and suggested calling them the Poincare polynomials.

1 42 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

The essence of the connection of integrals of differential forms with the Betti numbers established by Cartan was explained by Georges de Rham ( 1 903- 1 990) in the paper On the Analysis situs of manifolds of n dimension [Rh] ( 1 93 1 ) , where he defined the so-called de Rham cohomology groups. These groups are the quotient groups of the groups of closed differential forms of order p relative to their subgroups consisting of the exact differential forms. De Rham also established the isomorphism of these groups and the "Betti groups".

In the memoir Analysis situs Poincare also defined the noncommutative group consisting of closed paths on a manifold that are defined up to a con­tinuous transformation of these paths into one another. This group is called the connection group of a manifold, or the Poincare group or fundamental group. The study of this group forms the basis of the homotopy theory of manifolds.

§4.10. New methods in the theory of partial differential equations

The theory of partial differential equations which, in the beginning of the 20th century, was developed in different directions by Vessiot and Cartan, underwent new developments during the last decades due to the synthesis of their methods and some new methods of contemporary mathematics. Among these new methods, homological algebra should be especially noted. Homo­logical algebra has grown, to a great extent, from Cartan's papers in homology theory of compact simple Lie groups and symmetric Riemannian spaces. In this connection, we first note the following papers of H. L. Goldschmidt: Existence theorems for analytic partial differential equations [Gls l ] ( 1 962) , Prolongations of linear partial differential equations [Gls2] ( 1 965 ) , Integra­bility criteria for systems of non-linear partial differential equations [Gls3] ( 1 969) , and On the structure of the Lie equations [Gls4] ( 1 972) , the thesis of Daniel G. Quillen, Formal properties of over-determined systems of linear par­tial differential equations [Qu] ( 1 964) ; and the paper Over-determined systems of linear partial differential equations [Spe 1 ] ( 1 965) by Donald C. Spencer (b. 1 9 1 2) . For investigation of systems of partial differential equations Spencer and A. K. Kumbera developed a special technique in the papers Deformation of structures of manifolds defined by transitive continuous pseudogroups [Spe2] ( 1 962- 1 965) and Lie equations: general theory [KuSp] ( 1 972) .

The theory of "contraction" of Lie algebras and groups (Inonu and Wigner) mentioned in Chapter 2 was also generalized for Lie pseudogroups by D. S. Rim in the paper Deformation of transitive Lie algebras [Rim] ( 1 966) and by William Stephen Piper (b. 1 940) in the paper Algebraic deformation theory [Pip] ( 1 967) . We also note the paper The classification of irreducible com­plex algebras of infinite type [GuQS] ( 1 967) , by Victor Guillemin (b. 1 93 7 ) , Quillen, and Shlomo Sternberg, where a new simpler proof was given for Cartan's theorem on classification of irreducible Lie pseudogroups.

§4. 1 0. NEW METHODS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 1 43

The systematic presentations of new methods in the theory of partial dif­ferential equations that are developments of Cartan's methods are given in the book Exterior differential systems [BCG] ( 1 990) by R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, and the book Systems of partial differential equations and Lie pseudogroups [Porn 1 ] ( 1 978) by Jean Fran<;ois Pommaret (b. 1 945) (see also his books Differential Galois theory [Pom2] ( 1 983) and Lie pseudogroups and mechanics [Pom3] ( 1 988 ) ) .

CHAPTER 5

The Method of Moving Frames and Differential Geometry

§5.1 . Moving trihedra of Frenet and Darboux

Numerous papers by Cartan and his successors on differential geometry of classical spaces are based on the application of the method of moving frames. This method is connected with the theory of finite continuous groups and the theory of systems of Pfaffian equations in involution, both developed by Cartan.

Cartan indicated that he adopted the method of moving frames from Dar­boux, who used it in his classical Lectures on the general theory of surfaces [Da] ( 1 887) under the name of the method of moving trihedrons. In reality, this method was first used by Martin Bartels ( 1 769- 1 836) , a professor of the University of Dorpat (now Tartu in Estonia) . He is best known as a teacher of young Gauss and, later, while at the University of Kazan, of Lobachevsky. To each point of a space curve, Bartels associated a trihedron, which at present we call the "Frenet trihedron", and obtained formulas that are equivalent to the Frenet formulas. These formulas were published by his student Carl Ed­uard Senff ( 1 8 1 0- 1 849) in the book Principal theorems of the theory of curves and surfaces [Snf] ( 1 83 1 ) . He indicated that these formulas were obtained by Bartels. The moving trihedron related to the rotating globe was also used by another Bartels student-Petr I. Kotelnikov (the father of A. P. Kotel­nikov mentioned earlier) in Presentation of analytical formulas determining the perturbation of the rotational motion of the Earth [KoP] ( 1 832) . Later the Frenet formulas appeared in Joseph Serret's ( 1 8 1 9- 1 855 ) On some formulas related to the theory of curves of double curvature [Srt] ( 1 8 5 1 ) and in Jean Frederic Frenet's paper ( 1 8 1 6- 1 900) On certain properties of curves of double curvature [Fm] ( 1 852) . However, Frenet's thesis, where these formulas were given, appeared in 1 84 7. The axes of the Frenet trihedron are directed along the tangent to a curve, its principal normal (the straight line that is orthog­onal to the tangent and located in the osculating plane) , and the binormal (the perpendicular to the osculating plane of a curve) . If we denote the unit vectors parallel to these axes by e 1 , e2 , and e3 , the Frenet formulas can be written in the form

1 45

1 46 S. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

( 5 . 1 )

where s is the arc length of the curve, k is its curvature, and K is its torsion. Note that Frenet found only six formulas equivalent to the first two formulas of ( 5 . 1 ) and that Serret discovered all nine formulas equivalent to all formulas ( 5 . 1 ) .

The Frenet formulas were generalized for an n-dimensional space Rn by Camille Jordan ( 1 838- 1 922) in his work On the theory of curves in a space of n dimensions [Jo2] ( 1 87 4 ) . With every point of a curve in a space Rn , Jordan associated an n-hedron whose axes are directed along the tangent line to the curve, the straight line in the osculating 2-plane of the curve orthogonal to the tangent line and, in the same way, the straight line in the osculating ( i + 1 ) ­

plane of the curve orthogonal to the osculating i-plane, and finally the normal line is orthogonal to the osculating hyperplane of the curve. If we denote the unit vectors parallel to these axes by e 1 , e2 , • • • , en , the generalized Frenet formulas can be written in the form

(5 .2 )

de . -d ' = -k. l e . I + k.e . , . . . ' S I - I - I I

den = -k e ds n- 1 n- 1 '

where s is the arc length of the curve and k1 , k2 , • • • , kn- I are its 1 st, 2nd, . . . , (n - l )th curvatures.

In the theory of surfaces of the space R3 , the moving trihedrons were first used by Albert Ribaucour ( 1 845- 1 893) in his Investigation of elassoides or surfaces of zero mean curvature [Rib] ( 1 882) . (Ribaucour's "elassoides" are now called minimal surfaces; he called the application of the method of moving frames to the theory of surfaces the method of "perimorphie". )

The method of moving frames was systematically applied to the theory of surfaces by Darboux in_ his Lectures on the general theory of surfaces [Da] . For studying curves on surf aces, Darboux considered trihedra whose vectors e 1 and e3 are parallel to the tangent line to the curve and to the normal line to the surface, and for studying the surfaces themselves, he considered trihedra whose vectors e 1 and e2 are parallel to the tangent lines to the curvature lines of the surface, i .e . , parallel to the two principal directions, and the vector e3 is parallel to the normal line to the surface. Darboux considered the derivatives of the vectors of the first frame relative to the arc length of a curve on the surface. These derivatives have the form

§5 .2. MOVING TETRAHEDRA AND PENTASPHERES OF DEMOULIN 1 47

where kg is the geodesic curvature of the curve (if kg = 0 , the curve is a geodesic line) , k

n is the normal curvature of the surface along the given

curve, and Kg is the geodesic torsion of the curve on the surface. In the second case Darboux considered the derivatives of the vectors of

the frame relative to the arc lengths of the curvature lines. The coefficients of the decompositions of these derivatives with respect to the vectors of the trihedron are the principal curvatures of the surface and the geodesic curvatures and the geodesic torsions of its curvature lines.

§5.2. Moving tetrahedra and pentaspheres of Demoulin

For spaces different from the Euclidean �pace, the method of moving frames was generalized by the Belgian geometer Demoulin in his papers On the application of a moving tetrahedron of reference to the Cayley geometry [Dem l ] ( 1 904) and Principles of the anal/agmatic and line geometry [Dem2] ( 1 905) . In the first of these papers, Demoulin considered non-Euclidean spaces with nondegenerate absolutes. Actually he considered only the geom­etry of the elliptic space S3 but indicated that the same theory is applicable to any "Cayley space'', i .e . , to any space sJ . With any point of a curve or

a surface of the space S3 Demoulin associated a moving tetrahedron which is an autopolar tetrahedron with respect to the absolute of the space. For a curve, Demoulin placed one of the vertices of the tetrahedron at the point of the curve and directed the edges of the tetrahedron emanating from this vertex along the tangent line to the curve, its principal normal and binormal. In the case of a surface, Demoulin also placed one of the vertices of the tetra­hedron at the point of the surface and directed the edges of the tetrahedron emanating from this vertex along the principal directions of the surface and its normal. Demoulin considered the derivatives of coordinates of the ver­tices of the moving tetrahedron of the curve relative to the length of the curve and partial derivatives of the vertices of the moving tetrahedron relative to the lengths of its curvature lines and obtained formulas similar to the Frenet and Darboux formulas.

In the second paper, Demoulin considered the conformal space C3 , and with every point of a curve or a surface, he associated a moving pentasphere, Le. a system of five mutually orthogonal spheres defining a system of pen­taspherical coordinates in this space. The Darboux transfer maps these five spheres onto the vertices of an autopolar simplex of the space s: whose abso­

lute represents the space C3 • Demoulin also considered the manifold of the straight lines of the projective space P3 , and with each rectilinear generator of a ruled surface or with each line of a congruence, he associated six linear complexes that are pairwise in involution. The Pliicker transfer maps these six complexes into points of the space sg , and these points are the vertices of a simplex which is autopolar with respect to the absolute of this space. This absolute represents the manifold of straight lines of the space P3 • However,

1 48 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

Demoulin did not notice that one of the spheres of the pentasphere which he considered must be imaginary, and actually, instead of the spaces S� and

S� , he considered the elliptic spaces S4 and S5 •

Also, in the year 1 905, Ernest J. Wilczynsky ( 1 876- 1 932) , in the paper General projective theory of space curves [Wil] , constructed the theory of curves of the projective space P3 applying the moving tetrahedron of this space, and E. Vessiot, in the paper On minimal curves [Yes] , applied the moving trihedron for study of imaginary isotropic curves of the space R3 •

At that time these curves were called "minimal curves". These curves have zero arc length, and, because of this, the usual Frenet formulas are not valid for them.

Finally, in the same year, 1 905, Emile Cotton ( 1 872- 1 950) published the paper Generalization of the theory of moving frame [Cot] , where he introduced the concept of generalization of the method of moving frames for arbitrary spaces that possess transformation groups.

§5.3. Cartan's moving frames

Developing the ideas of Darboux and Cotton, in 1 9 1 0 Cartan published first the short note On isotropic developable surfaces and the method of moving frames [29] and later the paper The structure of continuous groups of transfor­mations and the method of a moving trihedron [3 1 ] . In the first note Cartan applied the method of moving trihedrons to the theory of imaginary devel­opable surfaces of the space R3 whose rectilinear generators are isotropic straight lines. In the second paper he connected the "method of a moving system of reference'', which later received the name moving frame, with the structure of Lie groups and the theory of Pfaffian equations.

With every homogeneous space xn where a transformation group G acts, one can associate a family of frames R0 with the property that the group G acts simply transitively on it, i .e . , each pair of frames defines a unique transformation S of this group that sends the first frame into the second one.

For example, in the Euclidean space Rn , the systems of orthogonal unit vectors e; , e;e1 = oiJ , with the origin at an arbitrary point x of the space can be chosen as these frames. Since, in any orthogonal coordinate system, the coordinates of the vectors of such a frame are elements of an orthogonal matrix belonging to the group On of dimension n(n - 1 ) /2 and the origins of these frames are determined by n coordinates, the frames {x , e; } of the space Rn depend on the same number, n(n - 1 ) /2 + n = n(n + 1 ) /2 , of real parameters as the group of motions of the space Rn . The frames in the pseudo-Euclidean spaces R� can be chosen in a similar way, but in this case the orthonormality condition for the vectors of a frame has the form e;e1 = e;o;1 , where e0 = - 1 , a = 1 , . . . , / , eu = 1 , u = l + l , . . . , n . In any orthogonal coordinate system, the coordinates of the vectors of such a

§5 .3 . CARTAN'S MOVING FRAMES 1 49

frame are elements of a pseudo-orthogonal matrix belonging to the group O� , the frames of the space R7 depend on the same number, n (n - 1 ) /2 + n = n (n + 1 ) /2 , of real parameters as the group of motions of the space R7 .

In the affine space En , one can take the systems of linearly independent vectors e; with the initial point at an arbitrary point x of the space as the family of moving frames. Since in any affine coordinate system, the coordinates of the vectors of such a frame depend on n2 parameters, the frames of the space En depend on the same number, n2 + n , of parameters as the group of affine transformations of the space En .

As a model of the projective space Pn Cartan considered the linear space L n+ I in which collinear vectors are assumed to be equivalent. To each one­dimensional subspace of the space L n+ 1 , there corresponds a "geometric point" of the space Pn , and Cartan called each vector of this subspace an "analytic point". The operations of addition and multiplication by real num­bers, typical for vectors, are applied to these "analytic points". Cartan em­phasized that the "analytic points" x and A.x determine the same "geometric point" x of the space Pn . Because of this, to define a projective frame in the space Pn , one should take n + 1 points e; , i = 0 , 1 , . . . , n , of gen­eral position and a unit point e . The vectors representing these points in the space Ln+ I are connected by the relation e = I:; e; and are defined up to a common real factor. It follows from this that a projective frame of the space Pn depends on the same number, n (n + 2) , of parameters as the group SLn+ I of unimodular matrices which is locally isomorphic to the group of projective transformations of the space pn .

The non-Euclidean spaces sn and s; can be considered to be the pro­jective space Pn where an absolute is given as a nondegenerate quadric Q(x , x) = 0 whose equation does not contain or contains exactly I neg­ative squares, respectively. In these spaces, a frame is formed by points e; that are vertices of an autopolar simplex with respect to the absolute and nor­malized in such a way that Q(e; , ej ) = e;<J ;j , where e0 = - 1 , a < I , eu = 1 , u ;::: I . The orthonormal frames in these spaces depend on the same num­ber, n (n + 1 ) /2 , of parameters as the groups On+I of orthogonal matrices

and O�+ 1 of pseudo-orthogonal matrices which are locally isomorphic to the groups of motions of these spaces.

The conformal space en can be represented in the form of an oval quadric Q(x , x) = 0 in the projective space pn+ t . Thus, the group of conformal transformations of the space en coincides with the group of motions of the Space S�+ I and is locally isomorphic to the group of pseudo-orthogonal

matrices 0�+2 • Such a representation is determined by the Darboux transfer

which maps the points of the space pn+ t that are outside of the quadric Q onto real hyperspheres of the space en , the points that are inside of the quadric Q onto imaginary hyperspheres of the space en , and the points of the hyperquadric Q itself onto the points of the space en .

1 50 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

As a frame in the space s�+ i , one can take a system of n + 2 points that are vertices of an autopolar simplex with respect to the absolute, but in such a frame one point is always inside the absolute and the remaining points are outside of it; in the space en , to such a frame there corresponds a frame consisting of n + 2 mutually orthogonal hyperspheres one of which is imaginary. This kind of frame is inconvenient, and this was the reason why Cartan chose, in the space en , a conformal frame consisting of two points e0 and en+ l and n mutually orthogonal real hyperspheres passing through these two points. In this frame, the equation of the absolute of the space s�+ 1 has the form

( 5 .4) Q(x , x) = :�:)xi )2 + 2x0xn+ I = 0 , i = 1 , . . . , n .

The conformal transformations of the space en are represented by linear transformations of coordinates preserving equation ( 5 .4) .

§5.4. The derivational formulas

The derivational formulas are the formulas that determine the transi­tion from a frame Ra of a given homogeneous space xn to an infinites­imally close frame Ra+da . To find these formulas, we fix a frame R0 and denote by Sa the transformation mapping the frame R0 onto the frame Ra , Ra = SaRo . Then, the transition from the frame Ra to the frame

Ra+da is defined by the transformation Sa+das; ' . Since Sas; 1 = I , this transformation is in a neighborhood of the identity I of the group of ad­missible transformations of frames. Thus, it can be written in the form Sa+das; 1 = I + Sw + o(da) . Cartan called the transformation Sw the in­finitesimal transformation of a frame of the homogeneous space under consid­eration. Using this transformation, the derivational formulas can be written as dRa = SwRa . Now we can say that the transformations Sw belong to the Lie algebra G of the group G of transformations of the homogeneous space xn . Denote by w" , u = 1 , . . . , r , the coordinates of the transformation S w in the algebra G . These coordinates are invariant forms of the Lie group G .

In the affine space En the frame Ra consists of a point x and vectors e; , and the frame Ra+da consists of a point x + dx and vectors e; + de; . Thus, in this space, the derivational formulas can be written as

( 5 . 5 )

where wi and w{ are differential forms, depending on parameters a (that determine the position of the frame) and their differentials d a . Since the group of transformations of frames of the space En is the ( n2 + n )-parameter . . 2 group, the forms w' and w: , whose number is also equal to n + n , are linearly independent.

§5 .4. THE DERIVATIONAL FORMULAS 1 5 1

In the spaces Rn and R7 , the derivational formulas have the same form

( 5 . 5 ) , but now the forms w{ are not linearly independent. By differentiat­ing the relations e;e1 = oiJ , we find that in the space Rn these forms are connected by the relations

( 5 .6)

Similarly, by differentiating the relations e;e1 = e;0;1 , we find that in the space R7 these forms are connected by the relations

( 5 .7 )

The derivational formulas for the frames R� = { e; } of the space Pn can be written as

( 5 . 8 ) i , j = 0 , 1 , . . . , n .

Since now the vectors e; allow multiplication by a common factor, the fam­ily of frames can be reduced by imposing the condition of equality for the volumes of the parallelepipeds [e0 , e 1 , • • • , en ] constructed on these vectors. From this condition we obtain the relation

( 5 .9) 0 I n Wo + W1 + . . . + wn = 0 '

connecting the forms w� . Relation ( 5 .9 ) distinguishes the unimodular group SLn+ I in the general linear group GLn+ I .

The derivational formulas in the spaces Sn and s; have the form ( 5 . 8 ) , but now the forms w� are connected by relations similar to relations ( 5 .6) and ( 5 . 7 ) . These relations follow from the fact that the corresponding frames consist of vertices of simplices that are autopolar with respect to the absolute of the space.

The derivational formulas in the spaces en also have the form ( 5 . 8 ) where i , j = 0 , 1 , . . . , n + 1 . However, since in the Cartan frame the equation of the absolute has the form ( 5 .4) , the forms w� are connected by the relations

( 5 . 1 0)

i , j = 1 , . . . , n .

Suppose further that, i n a homogeneous space xn with an r-parameter group G of motions, there is given a smooth family I. of frames depending on p :::; r parameters. On this family, the forms wu defining the infinitesimal displacements of frames also depend on p parameters and their differentials. Cartan noted that if there are two families I. and I.' of frames such that I.' = SI. where S is a fixed transformation of the group G , the forms wu and ' wu defining the infinitesimal displacements of frames in these families coincide. Conversely, if two families I. and I.' of frames in a homogeneous space xn depend on the same number, p :::; r , of parameters and under

1 52 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

an appropriate bij ective correspondence between frames of these families we have 'al = wu , then these families can be superposed by a transformation of the group G . This theorem is important when one studies submanifolds of homogeneous spaces by means of the method of moving frames.

§5.5. The structure equations

Invariant forms wu of the group G of transformations of a homogeneous space xn satisfy the structure equations

( 5 . 1 1 ) d u u v w (J) = cvw (J) /\ (J) ' U , V , W = l , . . . , r ,

which are equivalent to equations (2 . 1 2) . For the groups of transformations of the classical homogeneous spaces, these structure equations can be ob­tained from derivational formulas ( 5 . 5 ) and ( 5 . 8 ) .

Taking exterior differentials of equations ( 5 . 5 ) and equating to zero the coefficients of the linearly independent vectors e; , we obtain the structure equations of the spaces En , Rn

, and R7 :

( 5 . 1 2) i k i d (J) = (J) /\ (J)k '

where in the space Rn the forms w{ satisfy relations ( 5 . 6 ) and in the spaces R7 they satisfy relations ( 5 . 7 ) .

Similarly, exterior differentiation of equations ( 5 . 8 ) leads to the structure equations of the spaces pn ' sn ' s; ' and en :

( 5 . 1 3 )

where for the spaces pn ' sn ' s; ' i ' j ' k = 0 ' 1 ' . . . ' n ' and for the space en ' i ' j ' k = 0 , 1 , . . . ' n + 1 , and, in addition, in the space pn the forms w{ satisfy relations ( 5 .9 ) , in the spaces Sn and s; they satisfy relations ( 5 .6 ) and ( 5 . 7 ) , and in the space en they satisfy relations ( 5 . 1 0) .

The structure equations of a homogeneous space xn are the conditions of complete integrability of its derivational formulas. From this follows the important theorem which Cartan noted in all his works devoted to the method of moving frames: Let the forms wu , u = 1 , . . . , r , be given; suppose that they depend on p , p � r , parameters and their differentials and satisfy the structure equations of a homogeneous space xn ; then they define in this space a p-parameter family I: of frames uniquely, up to a transformation S of the group G . This theorem is a generalization of the theorem on determination of a curve in the space R3 by its curvature and torsion and the 0. Bonnet theorem on determination of a surface in the space R3 by its first and second fundamental forms. As we will see, the Codazzi and Gauss equations, which the coefficients of these forms must satisfy, follow from the structure equations of the space R3

•

§5 .6 . APPLICATIONS OF THE METHOD OF MOVING FRAMES 1 53

§S.6. Applications of the method of moving frames

Cartan applied the method of moving frames to the study of submanifolds in various homogeneous spaces. We will give the general scheme of investiga­tion of a submanifold VP in a homogeneous space xn indicated by Cartan. With every point x of a submanifold VP there is associated a family l:x of frames subject to only one condition: the point x belongs to all frames of the family l:x . Such frames are called "frames of order zero". These frames

depend on p principal parameters u 1 , • • • , uP , on which the point x of the submanifold VP depends, and on r - n secondary parameters whose num­ber is equal to the difference between the dimension r of the group G and the dimension n of the space xn . The whole family l: of frames of order zero is a fiber bundle whose base is the submanifold VP and the fibers are the families :I:x • The number of secondary parameters can be reduced if one replaces the frames of order zero by the frames of order one whose elements are connected in a certain way with the first-order differential neighborhood of the point x of the submanifold VP . The frames of order one form a fiber subbundle l:( l ) of the fiber bundle :I: which has the same base VP . Further, families of frames of orders two, three, etc. , are constructed whose elements are chosen by means of the corresponding differential neighborhood of the point x of the submanifold VP . This procedure is called the specialization of frames. There are two possibilities when we follow this procedure.

In the process of specialization we exhaust all the secondary parameters and, for some number k , the family l:(k) of frames will depend only on p principal parameters. Such a family of frames is called canonical. In this case, all differential forms in the derivational formulas are linear com­binations of the differentials of the principal parameters. The coefficients of these combinations are invariants defining the submanifold VP up to a transformation of the fundamental group of the space.

The second possibility is that the process of specialization of frames stops before reaching the end, i .e . , on a certain step l:(k+ l ) = :I:(k) , but not all secondary parameters will be exhausted. Then, the submanifold VP admits a certain group of transformations into itself.

For instance, for a curve in the Euclidean plane R2 , the family :I: of frames of order zero depends on one principal and one secondary parameter -the angle of rotation of the orthonormal pair of vectors e 1 and e2 relative

to a point x of the curve. When we construct the family l:1 of frames of order one, we take the vector e 1 to coincide with the tangept to the curve. This family is canonical since it depends only on the unique principal param­eter. The frames constructed are the Frenet frames for a plane curve. The family of canonical frames for a curve in the space R3 can be constructed in a similar manner. In this case, the canonical frame is determined by the tangent line and the principal normal to the curve, and this canonical frame is a frame of order two. For a curve in the space Rn , the Frenet frame is

1 54 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

a frame of order n - 1 . The nonvanishing differential forms in the deriva­tional formulas for the Frenet frames have the form w;+ i = -w;+ , = -k;ds , and the quantities k; form a complete system of invariants defining a curve in the space Rn up to a motion.

The Darboux frames for a hypersurface in the space Rn are also canonical frames. These frames are formed by the vector en parallel to the normal to the hypersurface and the vectors e 1 , e2 , • • • , en- I parallel to its principal directions. These frames are frames of order two.

§S.7. Some geometric examples

Cartan noted that simple geometric considerations do not always lead to the construction of a canonical frame. In such cases the construction may be conducted purely analytically by means of the structure equations of the space. We show how this can be done for an isotropic curve of the space CR3

• These curves were considered by E. Vessiot in 1 905 . Cartan consid­ered them in the book The theory of finite continuous groups and differential geometry considered by the method of moving frames [ 1 5 7] and in his lectures on The method of moving frames, the theory of finite continuous groups and generalized spaces [ 1 44] which he delivered in Moscow in 1 930.

An isotropic curve x = x( t) in the space CR3 is said to be a curve each tangent vector x' of which is isotropic, i .e . , (x' )2 = 0 . The latter equation

implies x' x" = 0 . The arc length of such a curve is equal to zero, and the

normal and tangent planes coincide. Thus, it is impossible to construct the Frenet frame for such a curve. For studying an isotropic curve, Cartan used the cyclic frames in the space CR3 whose vectors satisfy the relations

( 5 . 1 4)

Only three out of the nine forms w{ determining the infinitesimal displace­ments of this frame are independent. Differentiating equations ( 5 . 1 4) and using equations ( 5 . 5 ) , we easily find that they are connected by the relations

While constructing a canonical frame, we save one step by immediately associating with the curve the frames of order one. For this, we place the origin of a frame at the point x of the curve and take its isotropic tangent vector (x)' as the vector e 1 • Since now we have dx = w 1

e 1 , on the curve the following equations hold:

( 5 . 1 6) 2 3 Q) = Q) = 0.

The form w 1 is called the basis form. It contains the differential of the parameter t defining the location of a point x on the curve. If we apply

§5 .7. SOME GEOMETRIC EXAMPLES 1 5 5

exterior differentiation to equations ( 5 . 1 6) with the help of the structure equa­tions ( 5 . 1 1 ) , then, by ( 5 . 1 5) and ( 5 . 1 6) , we obtain only one exterior quadratic equation w1 /\ wi = 0 . This equation implies that

( 5 . 1 7) 2 I w1 = pw .

The form wi is principal since it vanishes when the point x is fixed. More­over, there will be only two nonvanishing independent forms on the curve, namely, the forms w : and wi . They determine the admissible transfor­mations of frames of order one. Thus, the family of frames of order one depends on one principal and two secondary parameters.

For further specialization of frames, we apply exterior differentiation to equation ( 5 . 1 7) . This gives

from which it follows that

( 5 . 1 8) I I dp - 2pw1 = -2qw .

If we fix a point x on the curve, then w1 = 0 , and equation ( 5 . 1 8 ) takes the form

( 5 . 1 9) I t5p - 2pn 1 = 0 ,

where t5 denotes differentiation with respect to secondary parameters and

n : = w : I . w 1 =0

In equation ( 5 . 1 9) we distinguish two cases. If p = 0 for all points of the curve x = x{t) , then further specialization is impossible, and the family of frames of order two coincides with the family of frames of order one. Since in this case equation ( 5 . 1 6) implies that wi = 0 , it follows from equations ( 5 . 5 ) that

I I dx = w e 1 , de 1 = w 1 e 1 • It follows from this that, in the case p = 0 , a curve x = x{t ) is an isotropic straight line.

If p =f. 0 , equation ( 5 . 1 9) can be written in the form

I t5 ln p - 2n 1 = 0.

It is easy to check that dn : = 0 if w1 = 0 . Thus, the secondary form n: is

a total differential: n : = rJ ln rp . Substituting this value and integrating the

previous equation, we obtain p = Crp2 • Here rp is a secondary parameter which determines the magnitude of the vector e 1 • By an appropriate choice of this parameter, we can reduce the quantity p to + 1 or - 1 . Let us take

1 56 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

the first case. In this case, equations ( 5 . 1 7) and ( 5 . 1 8 ) take the form

( 5 . 20) I I w1 = qw .

These equations define the family of frames of order two associated with an isotropic curve.

To construct a family of frames of order three, we take exterior differentials of the second of equations ( 5 .20) . As a result, we obtain the equation

from which we find that

( 5 .2 1 )

I I (dq + w2 ) /\ W = 0 ,

I I dq + w2 = kw .

If we fix the point x on the curve, we obtain I <5q + 1t2 = 0.

Here again the form n� is a total differential: n� = -<5 VJ . This implies

q = VJ + C.

It follows from this that by an appropriate choice of the secondary parameter VJ the quantity q can be reduced to 0. Now the second equation in ( 5 . 20) and equation ( 5 .2 1 ) can be written in the form

( 5 .22) I I w2 = kW .

These equations show that all secondary forms are already expressed in terms of the basis form w

1 • Therefore, the frame of order three is canonical.

Note that, by previous formulas, dw1

= 0 . Thus, the form w1 is a

total differential: w1

= da . The parameter a is called the pseudoarc of an isotropic curve x = x{t) . It was introduced by E. Vessiot in the paper [Yes] mentioned above. The quantity k in the second equation of ( 5 .22) is an invariant which is called the pseudocurvature of an isotropic curve. By previous relations, the Frenet formulas for an isotropic curve have the form

dx de 1 de2 k de3 (5 .23 ) da = el ' da = e2 ' da = e , - e3 ' da = ke2 .

Two isotropic curves coincide up to a motion of the space CR3 if for both curves, the pseudocurvature k is the same function of the pseudoarc a .

The method of moving frames can be applied to the study of manifolds with any generating element. As an example, in the book ( 1 57) , Cartan con­sidered ruled surfaces of the space R3

• Starting with the family of orthonor­mal frames {x , e 1 , e2 , e3 } of order zero, where the point x belongs to a generator I of the ruled surface and the vector e 1 is directed along this generator, Cartan arrived at a canonical frame whose origin is located at the central point of the generator I , the vector e3 is directed along the common

§ 5.7 . SOME GEOMETRIC EXAMPLES 1 57

perpendicular of two infinitesimally close generators, and e2 = e3 x e 1 . The derivational formulas for the family of canonical frames have the form

( 5 .24) dx da = ae 1 + ke3 '

The last three of formulas ( 5 .24) are the Darboux formulas for a curve on the sphere described by the terminal point of the vector e 1 • This curve is the spherical image of the ruled surface under consideration. The parameter a coincides with the arc length of this spherical image since da = lde 1 I , and the invariant b represents its geodesic curvature. The invariant k is the distribution parameter of the ruled surface which is equal to the limit of the ratio of the shortest distance between its two rectilinear generators and the angle between them when one of these generators approaches another. This invariant is determined by the first-order differential neighborhood of the generator l , and the invariants a and b are determined by its second-order differential neighborhood. If three arbitrary functions k = k(a) , a = a(a) and b = b (a) are given, then there exists a unique ruled surface for which these functions are the corresponding invariants. If k = 0 , a ruled surface is developable.

Cartan also showed how the method of construction of a canonical moving frame which he developed can be applied to nonmetric geometries. In his book [ 1 44], he considered the theory of plane curves in affine geometry, and in the book [ 1 5 7], he considered the theory of plape curves in projective geometry. In both cases, starting with the family of frames of order zero, he constructed the family of canonical frames, found the derivational formulas for this family, and gave the geometric characterization to the invariants in these formulas. In addition, he found some special cases for which the construction of the canonical frame is impossible.

Thus, the Cartan books [ 1 44] and [ 1 57] contain not only the general the­ory of the method of moving frames but also its applications to a series of concrete geometric problems.

Among similar problems which Cartan solved in his other papers, we note the analogue of the Frenet formulas which Cartan obtained in his paper On a degeneracy of Euclidean geometry [ 1 47a] ( 1 935 ) for the isotropic plane 12 • In this paper, with every point x of a curve in the plane /2 , Cartan associated the frame consisting of the unit tangent vector e 1 and the unit vector e2 which is parallel to the isotropic straight lines of this plane and wrote the analogue of the Frenet formulas in the form

( 5 .25) dx ds = e 1 '

de2 = 0 ds ·

1 58 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

§5.8. Multidimensional manifolds in Euclidean space

In the book Riemannian geometry in an orthonormal frame [ 1 08a] , Cartan considered certain special topics of the theory of p-dimensional manifolds VP in n-dimensional Riemannian manifolds vn and, in particular, in the Euclidean space Rn . We consider in more detail how Cartan constructed this theory.

With every point x of a p-dimensional manifold VP of the space Rn , we associate a family of orthonormal frames whose vectors e; , i = 1 , . . . , p , are located in the space Tx ( VP ) tangent to the manifold VP , and the vectors ea , a = p + 1 , . . . , n , belonging to its normal space Nx ( VP ) . Then, the manifold VP is determined by the following system of Pfaffian equations:

( 5 . 26) wa = 0 ,

and the forms w; are linearly independent on the manifold VP . The square

of the linear element of the manifold VP has the form ds2 = L; (w; ) 2 • This

form is called the first fundamental form of the manifold VP . Exterior differentiation of equations ( 5 .26) by means of the structure equa­

tions ( 5 . 1 2) leads to the equations

( 5 .27) d a i a O (J) = (J) I\ W; = .

Applying Cartan's lemma to this equation, we find that

( 5 .28 ) a ba j (J) i = ij(J) '

The coefficients b� are the coordinates of the vectors bii = b�ea . From derivational formulas ( 5 . 5 ) it follows that on the manifold VP we

have

( 5 .29) d2 (d i j i ) i a x = (J) + (J) (J)j e; + (J) (J)i ea .

Therefore, the quadratic forms

( 5 . 30) a i a ba i j <p = (J) (J) i = ij(J) (J)

define the deviation of the manifold VP from its tangent space Tx ( VP ) . The vector-valued quadratic form <p = rpaea = biiw

; wi is called the second fundamental form of the manifold VP .

Consider a curve x = x(s) on the manifold VP given by a vectorial func­tion of its arc length s . The vector �; = a = a;

e; is its unit tangent vector.

Since a; = w; / ds , with the help of ( 5 . 30) , we obtain for this curve

( 5 . 3 1 ) d x da da i w1 ; 1 2 ( i i ) ds2

= ds = ds + a ds e; + biia a .

§5 .8. MULTIDIMENSIONAL MANIFOLDS IN EUCLIDEAN SPACE 1 59

The vector d2x/ds

2 is the vector of curvature of the curve x = x(s) . For­

mula ( 5 . 3 1 ) gives its decomposition into the tangent and normal components. Because of this the vector of normal curvature of this curve has the form

( 5 . 32)

It follows from this that the vector kn depends only on the direction of the tangent line to the curve x = x(s) .

The linear span of the set of vectors kn coincides with the linear span of the system of normal vectors bij and determines the principal normal N; ( VP ) of the manifold VP . Its dimension is equal to p1 :::; min{n - p , p (p + 1 )/2} . The direct sum of the principal normal and the tangent space Tx ( VP ) is the first osculating space r; ( vP ) of the manifold

vP at the point x . Its dimension is dim r; ( VP ) = p + P 1 . If we change the tangent vector a in the tangent space Tx ( VP ) and p 1 ?. p ,

the terminal point of the vector kn describes a (p - 1 ) -dimensional algebraic

surface in the principal normal N; ( vP ) . This surface is called the indicatrix of curvature. If p 1 < p , the terminal point of this vector describes a closed

domain in N; ( vP ) which is called the domain of curvature. As an example, following Cartan, we consider a two-dimensional surface

V2 in the space Rn . In this case, p 1 :::; 3 , a = e 1 cos () + e2 sin () , and

formula ( 5 . 32) takes the form

( 5 . 33 ) kn = b 1 1 cos

2 () + 2b 1 2 cos () sin () + b22 sin

2 () = ! (b 1 1 + b22 ) + b 1 2 sin 20 + ! (b 1 1 - b22 ) cos 20 .

We can see from this that if p 1 = 2 or 3 , the terminal point of the vector kn describes an ellipse in the normal N; ( V2 ) with the center determined by the vector ! (b 1 1 + b22 ) and the vectors b 1 2 and ! (b 1 1 - b22 ) parallel to its conjugate diameters. Cartan called this ellipse the ellipse of curvature of the surface V2 • If p 1 = 1 , then the terminal point of the vector kn describes a

segment in the one-dimensional normal N; ( V2 ) which is called the segment of curvature. Its ends correspond to the principal directions of the surface v2 .

A p-dimensional manifold VP of the space Rn depends on n - p arbi­trary functions of p real variables. As these functions, in the general case one can take the functions expressing the coordinates x" of a point x of this manifold in terms of the coordinates xi taken as independent variables. In the general case the indicatrices of curvature of p-dimensional manifolds are the Veronesians and the quasi- Veronesians.

We will now illustrate the application of the Cartan test by investigating '11.e system of Pfaffian equations ( 5 . 26) which determines a manifold VP in the space Rn . The character s 1 of this system is equal to the number of linearly independent exterior quadratic equations ( 5 .27) obtained as a

1 60 5 . THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

result of exterior differentiation of system ( 5 .26) ; i .e . , it is equal to n - p : s 1 = n - p . The ranks r; of the system of linear equations determining the (i + l )th integral elements are r; = i (n - p) . Thus, the remaining characters s; = r; - r;_ 1 are also equal to n - p . The sum of the characters s1 + Si + · · · + sP = q is equal to the number p(n - p) of independent forms w� . The Cartan number

p(p + 1 ) Q = s1 + 2s

i + · . . + psP = ( 1 + 2 + · · · + p) (n - p ) = 2 (n - p)

coincides with the number N of independent coefficients b� . Since Q = N , by Cartan's test, system ( 5 .26) is in involution, and its solution depends on sP = p(n - p) functions of p real variables. This corresponds to the arbitrariness of the existence of a manifold VP in the space Rn which we indicated above.

§5.9. Minimal manifolds

In the book Riemannian geometry in an orthonormal frame [ 1 08a] Car­tan also considered minimal surfaces vi in the Euclidean space R4

• The condition for vi to be minimal is that the variation of surface area be equal to zero. This condition has the form o J w1 /\ wi = 0 . It follows from this condition that the vector ! (b 1 1 + b

ii) = 0 , i .e . , the center of the ellipse of

curvature of a minimal surface coincides with its point x . In the case of a manifold VP in the space Rn , Enrico Bompiani ( 1 889-

1 975 ) , one of the founders of multidimensional differential geometry, called the vector b = E; b;; the vector of mean curvature. If p > 2 , the equation b = 0 also characterizes the minimal manifolds.

However, the minimal surfaces vi of the space R4 are also remarkable by the fact that they carry a complex structure (this is not true for minimal manifolds VP for p > 2) . Namely, each minimal surface vi in the space R4 can be viewed as a real interpretation of an analytical curve y in a two­dimensional complex plane endowed with the metric of a complex Hermitian

-i I i plane CR where the length of the vector z = { z , z } is equal to i z l = J z 1 z I + zi zi .

Generalizing this property of minimal surfaces vi in the space R4 , we can arrive at the notion of strongly minimal manifolds VP in the Euclidean space Rin . For this, we consider the complex Hermitian space CRn and construct its real interpretation Rin . We take the vectors of a unitary or--n thonormal frame of the space CR as the vectors eik- I of a frame in the

space Rin , and as the vectors eik

we take the products of the vectors of

the same frame in the space CRn by i . Then, in addition to the conditions

w� = -w� , the differential forms w� in the equations of the infinitesimal

§S .9 . MINIMAL MANIFOLDS 1 6 1

displacements of the frame in the space R2n are also connected by the con­ditions:

( 5 . 34) 2/- 1 2/ W2k- 1 = W2k •

2/- 1 2/ W2k = -w2k- 1 • k , I = 1 , 2 , . . . , n .

Note that the number of relations ( 5 . 34) , whose form remind us the Cauchy­Riemann conditions (4 .4) , is equal to the difference n (2n + 1 ) - n(n + 2) of the dimensions of the group of motions of the spaces R2n and CRn .

Next, we consider a ( 2p )-dimensional manifold V2P of the space R2n

representing an analytic manifold C VP of complex dimension p in the space CRn . With every point of this manifold, we associate an orthonormal frame consisting of vectors e2;_ 1 , e2; , i = 1 , 2 , . . . , p , lying in the tangent space

Tx ( V2P ) , and vectors e20_ 1 , e20 , o: � p + 1 , . . . , n , lying in the normal

space Nx( V2P ) . Then, the manifold V2P is defined by the following system of Pfaffian equations:

( 5 . 35 ) 2a- I 2a Q W = W = .

Exterior differentiation of equations ( 5 . 3 5) leads to the following exterior quadratic equations:

( 5 . 36) 2i- I A 2a- I + 2i A 2a- I Q w " w2;_ 1 w " w2i = , 2i- I 2a 2i 2a

W A w2i- I + W A <.v2; = 0.

Since the forms w2i- I and w

2i are linearly independent on the manifold V2P , application of Cartan's lemma gives

( 5 . 37) 2a- I b2a- I 2j- I b2a- I 2j w2i- 1 = 2;- 1 , 2j- 1 w + 2;- 1 , 2jw

2a- I b2a- I 2j- I + b2o- I 2j W2; = 2i , 2i- l w 2i , 2iw •

2a b2a 2j- I b2a 2j w2i- 1 = 2;- 1 , 2j- 1 w + 2;- 1 , 2jw

2a b2a 2j- I b2a 2j W2 · = 2 · 2 ·- 1 <.v + 2 · 2 ·<.v • I I , J I , J

where b11 = bJ1 . By conditions ( 5 . 34) , we also find the following relations

between the coefficients b11 : ( 5 . 38)

b2a- I b2a- I b2a b2a b2n- I b2a- 1 2i- l , 2j- I = 2j- l , 2i- I = 2i , 2j- I = 2j- l , 2i = - 2j , 2i = - 2i , 2j '

b2a b2a. b2a- I b2a- I b2n b2a 2i , 2j = 2j , 2i = 2j , 2i- I = 2i- l , 2j = - 2j- l , 2i- I = - 2i- l , 2j- I '

One consequence of these relations is that

( 5 . 39) �(b2o- I b2a- I �(b2a b2a ) L..,, 2i- l , 2i- I + 2i , 2; ) = L..,, · 2i- l , 2 i- I + 2i , 2 i '

i .e. , the vector of mean curvature of a strongly minimal manifold is equal to zero, and this manifold is minimal in the usual sense. A p-dimensional

1 62 5 . THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

analytic manifold CVP in the space CRn is defined by n - p analytic func­

tions of p complex variables. But to define each of these functions, it is sufficient to take two functions of p real variables. Thus, a strongly minimal manifold V2P in the Euclidean space R2n

depends on 2(n - p) functions of p real variables. This result can also be obtained by applying the Cartan test to the system of Pfaffian equations ( 5 . 3 5 ) . The character s1 of this system is equal to the number of linearly independent exterior quadratic equations (5 . 36 ) , i .e. , s1 = 2(n - p) . The ranks ri of the systems of linear equations that determine the ( i + 1 )-dimensional integral elements are ri = 2i(n - p) . This implies that the remaining characters si = ri - r;_ 1 are also equal to 2(n - p) : si = 2(n - p) . The sum of characters s1 + s2 + · · · + sP = q is

equal to the number 2p(n - p) of independent forms w���/ and w��- I . The Cartan number

Q = s1 + 2s2 + · · · + psP = p (p + l ) (n - p)

coincides with the number N of independent coefficients b1J . Since Q = N , system ( 5 . 35 ) is in involution and its solution depends on sP = 2(n - p) functions of p real variables.

§S.10. "Isotropic surfaces"

The bulk of Cartan's unpublished paper Isotropic surfaces of a hyperquadric in seven-dimensional space [ 1 77] is devoted to the differential geometry of "isotropic surfaces'', i .e . , two-dimensional surfaces on the absolute of the hy­perbolic space SJ which can be considered as the pseudoconformal space

C� . Cartan assumed that all tangent two-dimensional planes of these sur­

faces are plane generators of the absolute of the space sJ , and hence through each of its two-dimensional planes there passes one three-dimensional plane generator of this absolute of the first and second family. While in the con­formal space en

Cartan used a frame consisting only of two points of the conformal space and a few mutually orthogonal hyperspheres, in the space C� he used a frame consisting of points represented by such points of the absolute that the straight lines joining pairs of these points are mutually polar with respect to the absolute. If the equation of the absolute is

( 5 .40) 0 7 1 6 2 5 3 4 Q(x) = x x + x x + x x + x x = 0 ,

Cartan takes as the points of the frame analytic points A0 , A 1 , • • • , A7 for which the quadratic form ( 5 .40) vanishes: Q(Ai ) = 0 , i = 0 , 1 , . . . , 7 , and the bilinear form Q(Ai , Aj ) obtained by the polarization of the quadratic form ( 5 .40) is equal to 1 if i + j = 7 and 0 in all other cases. This means that the straight lines AiA1_ ; are mutually polar with respect to the absolute. Note that in the manuscript of this paper, Cartan denoted analytic points not only by capital Latin letters, as he did in most of his works, but also by small Latin letters with arrows --+ over them; moreover, he also called the forms

§S. 1 0. "ISOTROPIC SURFACES" 1 63

!l(x) and O(x , y) the "inner square" of an analytic point and the "inner product" of two analytic points; i .e . , in fact, he considered analytic points as vectors of the pseudo-Euclidean space R: . The derivational formulas of this frame have the form (5 . 8 ) , and the structure equations have the form ( 5 . 1 3) . Differentiating the equations A;Aj = 1 , i + j = 7 , Cartan obtained the relations

( 5 .4 1 ) 7-j 7- i W; + (J)j = 0 ,

which show that the matrix ( w� ) is skew-symmetric with respect to the sec­ondary diagonal, and in particular, all the entries of this diagonal are equal to 0.

With every point of an isotropic surface, Cartan associated the frames whose point A0 coincides with this point of the surface, the points A 1 and A2 belong to the tangent isotropic plane to the surface at this point, the point A3 lies in that "generating space" (three-dimensional plane generator) of the first family which passes through the tangent isotropic plane, the point A4 lies in the "generating space" of the second family, the points A5 and A6 lie outside of the "generating spaces" mentioned above, in the hyperplane which is tangent to the absolute at the point A0 , and the point A7 lies outside of the latter hyperplane. Thus, the Pfaffian equations of the isotropic surface have the form

( 5 .42)

(the analogous equation w� = 0 is a consequence of relations ( 5 .4 1 ) ) . By ex­terior differentiation of equations ( 5 .42) Cartan found the following exterior quadratic equations:

( 5 .43)

( 5 .44)

(5 .45)

( 5 .46)

Since equations ( 5 .4 1 ) imply that w� = w� , it follows from (5 .45) and (5 .46) that

(5 .47)

Exterior differentiation of equations ( 5 .47) leads to the relations:

1 64 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

which, by ( 5 .4 1 ) , are reduced to one relation:

( 5 .48)

Thus, the closed system of differential equations defining an isotropic sur­face V2

of the space c; consists of Pfaffian equations ( 5 .42) and ( 5 .4 7) and exterior quadratic equations ( 5 .43) , ( 5 .44) , and ( 5 .48) . Investigation of this system by Cartan's test shows that since new forms w� , w� , w� , and w� enter into the quadratic equations, i .e . , their number q = 4 , and the number of independent exterior quadratic equations is s 1 = 3 , the second Cartan character is s2 = q - s 1 = 1 and the Cartan number is Q = s 1 + 2s2 = 5 .

Applying Cartan's lemma to relations ( 5 .42) and ( 5 .43) , Cartan obtained the equations:

( 5 .49) 3 I b 2 w 1 = aw0 + w0 , 4 I I b' 2 WI = a Wo + Wo ,

Substituting expansions ( 5 .49) into equation ( 5 .48) , he found the relation:

( 5 . 50) ac' + ca' - 2bb' = 0.

Therefore, the degree of freedom of the most general integral element of an isotropic surface V2

is equal to N = 6 - l = 5 , and the system of Pfaffian equations ( 5 .4 1 ) and ( 5 .47) defining this surface is in involution, and, since s2 = 1 , an isotropic surface V2

in the space c; depends on one function of two real variables.

Since the differential of the analytic point A0 tangent to the surface V2

is equal to dA0 = w�A0 + w�A 1 + w�A2 , the second differential of this point

modulo the tangent plane to the surface V2 is equal to d2 A0 = w�w� A,.. , i =

1 , 2 , a = 3 , 4 , . . . , 7 . But by equations ( 5 . 42) and ( 5 .47) , modulo the same tangent plane, we have

( 5 . 5 1 )

It follows from this that the osculating plane T� ( V2) of the isotropic surface 0

V2 is determined by the points A0 , A 1 , A2 , A3 , A4 , and thus it is four-

dimensional. (The osculating plane of the general surface V2 in the space of

dimension exceeding five is five-dimensional . ) This plane coincides with the four-dimensional plane which is determined by those two "generating spaces" of the quadric that pass through the isotropic tangent plane to the isotropic surface V2

• This plane is a polar plane of the tangent plane relative to the quadric. The coefficients of the points A3 and A4 in expression ( 5. 5 1 ) are the

3 · 3 4 · 4 2 second fundamental forms <I> = w�w; and <I> = w�w; of the surface V . Substituting relations ( 5 .49) into these expressions, we obtain the following

§ 5 . 1 0. "ISOTROPIC SURFACES" 1 65

form for the second fundamental forms of the surface V2 : ( 5 . 52)

3 1 2 1 2 2 2 <I> = a ( w0) + 2bw0w0 + c( w0) , 4 I 1 2 / ) 2 I 2 2 <I> = a ( w0) + 2b w0w0 + c ( w0) .

Cartan called the nets <1>3 = 0 and <1>4 = 0 the nets (I) and (II) . Next, he considered the general case when, at any point of the surface, the equations <1>3 = 0 and <1>4 = 0 do not have common roots (the pairs of points defined by these equations on the line A 1A2 have no common point) and found on

the surface V2 a net (Ill) which is "harmonic for the nets (I) and (II) '', i .e . , a net of lines on this surface such that if the points A 1 and A2 belong to

the tangents to the lines of this net then the forms <1>3 and <1>4 can both be simultaneously reduced to algebraic sums of squares. In this case b = b' = 0 , a = c = a' = -c' = 1 . At present, the net (III) is called the conjugate net of the surface V2

• In this case

( 5 . 53 )

and the forms ( 5 . 52) become

( 5 . 54)

The tangents to each family of lines of the net (Ill) form a two-parameter family of straight lines. For the general two-parameter family of straight lines in the spaces pN , N � 5 , the limiting positions of three-dimensional planes, passing through a straight line of the family and infinitesimally close straight lines tending to it, do not coincide but belong to a certain five-dimensional plane (called the tangent plane of the family) . Unlike the general case, in the case of the two families of tangents considered above, these three-dimensional planes coincide. Therefore, the neighborhoods of lines of these families have structure similar to that of straight lines of the congruences of straight lines of the space P3 , and each of these straight lines have two foci-the points of intersection of this line with infinitesimally close straight lines. The latter infinitesimally close lines define two developable ruled surfaces of this family passing through each of its lines. Such two-parametric families of straight lines are called focal families. One of the foci of each of these straight lines is the point A0 • We place the points A 1 and A2 into the second focus of the tangents from the first and second family, respectively, and choose the points A3 and A4 in such a way that the points A0 , A 1 , A2 , A3 + A4 and A0 , A 1 , A2 , A3 - A4 define three-dimensional tangent planes of the straight lines of the first and second family, respectively. The foci A 1 and A2 gener­

ate two-dimensional surfaces ' V2 and " V2 obtained from the surface V2 by Laplace transforms. These surfaces also belong to the absolute of the space s; , but they are not isotropic. Thus, they can be considered as nonisotropic

1 66 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

surfaces of the space e; . Cartan introduced a system of curvilinear coor­

dinates u and v on the surface V2 , whose coordinate lines are the curves of the net (III) and whose differentials are the forms w� and w� , and con­structed a canonical frame.

Cartan also considered special classes of isotropic surfaces. In the case when the foci A 1 and A2 generate not surfaces but lines, Cartan expressed all points A; in terms of analytic points U and V depending on the variable u alone and the variable v alone, respectively. These expressions of the points of the canonical frame are:

( 5 . 5 5) I I A0 = U + V , A 1 = U , A2 = V ,

A - l U"' A - - l V"' 5 - 2 ' 6 - 2 ' Completing the construction of the canonical frame, Cartan found the

complete system of invariants defining an isotropic surface V2 on the abso­lute of the space s; up to a motion of this space, i .e . , a surface V2 of the

space e; up to a conformal transformation of this space.

§5.1 1 . Deformation and projective theory of multidimensional manifolds

While in his books Cartan considered problems of differential geometry using mainly examples of curves and surfaces in spaces of lower dimensions, in his theoretical papers on differential geometry he studied problems of the theory of multidimensional manifolds in the spaces Rn , Sn , pn , and en . The first paper of this type was his paper The deformation of hypersurfaces in the real Euclidean space of n dimensions [47) ( 1 9 1 6) . By the well-known theorem of Richard Beez ( 1 827- 1 902) , in general, the hypersurfaces in the space Rn , n > 3 , are not deformable; i .e . , if n > 3 , any pair of hypersurfaces of the general type can be superposed by means of a motion. In the paper indicated above, Cartan investigated such hypersurfaces in the space Rn that admit a nontrivial deformation, i .e. such a deformation that leaves them isometric, but such that they cannot be superposed by means of a motion. After solving this problem in the space Rn , in the paper The deformation of hypersurfaces in the real conformal space of n > 5 dimensions [ 48) ( 1 9 1 7) , Cartan considered a similar problem in the conformal space en . Later, applying the notion of projective deformation of surfaces introduced in 1 9 1 6 by Fubini, in the paper On the projective deformation of surfaces [54) ( 1 920), Cartan solved the problem of projective deformation. In the same year in his lecture On the general problem of deformation [55 ) , Cartan also defined the projective deformation of the congruences and complexes of straight lines in the projective space P3 and noted that, with the help of Plucker transfer, this problem can be reduced to the problem of conformal deformation of surfaces of dimensions two and three in the space e4

• In 1 9 1 9- 1 920 Cartan returned to the problem of deformation of manifolds in the space Rn ; in the

§5 . 1 1 . PROJECTIVE THEORY OF MULTIDIMENSIONAL MANIFOLDS 1 67

paper On the manifolds of constant curvature of Euclidean and non-Euclidean space [ 5 1 ) , [52) , he solved similar problems in n-dimensional non-Euclidean spaces. In all these papers Cartan systematically used the method of moving frames in the spaces Rn , pn , Sn , S? , and en .

During the study of deformation of manifolds in n-dimensional Euclidean and non-Euclidean spaces, it was detected that in this theory the projective properties of manifolds, i .e . , properties that are invariant under projective transformations of the space, play an esssential role. This was one reason Cartan studied the geometry of a manifold VP in the projective space Pn in Chapter 4 "Manifolds of p dimensions in the projective space of n dimen­sions. Osculating planes. Asymptotic linear systems" of his two-part paper [ 5 1 ) , [52) .

With a manifold VP he associated a moving point frame whose point e0 coincides with a varying point x of the manifold VP , whose points e; , i = 1 , . . . , p , are in its tangent space Tx ( VP ) and whose points ea , a = p + 1 , . . . , n , are outside of this space. If we denote the Pfaffian forms w� in derivational formulas ( 5 . 8 ) by w

k, then the Pfaffian equations defining

the manifold VP in Pn will have the same form ( 5 .26) as in the space Rn , their exterior differentials will have the form ( 5 .27) , and the application of Cartan's lemma will again give relations ( 5 .28 ) . Cartan called the quadratic forms ( 5 . 30) asymptotic forms of the manifold VP • The linear system of these forms is projectively invariant and does not depend on the metric prop­erties of the manifold VP even if the latter belongs to the space Rn . Simi­larly the first osculating space r; ( VP ) defined for the manifolds VP in the Euclidean space Rn has also an invariant meaning. If one places the points e; , i 1 = p + 1 , . . . , p + p1 , into this osculating space, then the asymptotic

I

forms </>P+Pi become linearly independent, and the forms </>A. , A. > p + p 1 , are identically equal to zero. After this Cartan defined the asymptotic and conjugate directions on the manifold VP .

In the same manner, Cartan introduced the osculating spaces r; ( VP ) , k > 1 , and linear systems of asymptotic differential forms of higher orders and established relations among them. Then he posed the problem of projective classification of multidimensional manifolds according to the structure of their linear systems of asymptotic differential forms of certain order.

As the first example, Cartan considered the manifolds on which the asymp­totic forms of first order can be reduced to the form

,1..P+ i = (w

i)2 '

. l 'I' l = ' . . . , p ; l = 0 , A. > 2p .

At present, these manifolds are called Cartan manifolds. They carry a con­jugate net, and their osculating spaces r; ( VP ) have dimension 2p . Cartan investigated the system of Pfaffian equations defining such manifolds and proved that they exist and depend on s2 = p(p - 1 ) functions of two vari­ables.

1 68 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

Next, Cartan considered tangentially degenerate manifolds VP whose tan­gent spaces Tx ( VP ) depend on q < p parameters. The number q is called the rank of the tangentially degenerate manifold VP . The asymp­totic quadratic forms of such a manifold can be expressed in terms of q linearly independent forms ol and have the form

,1,.a ba a b 'fJ = ab (J) OJ ' a , b = l , . . . , q .

Later on, it was proved that a tangentially degenerate manifold VP is foliated into a q-parameter family of (p - q )-dimensional planes along each of which the tangent space Tx ( VP ) is the same.

Cartan considered two classes of tangentially degenerate manifolds. For the first class, the asymptotic forms can be reduced to the form

If q > 1 , such manifolds depend on s2 = q ( q - 1 ) functions of two variables. If q = 1 , they are envelopes of a one-parametric family of p-dimensional planes and depend on s 1 = n - l functions of one variable. The second clas5 is characterized by the fact that, among the forms </>a , there is the maximal possible number q(q + 1 ) /2 of linearly independent forms. In this case, if q > 1 , the manifold VP is a cone with a (p - q - 1 )-dimensional vertex and (p - q )-dimensional plane generators.

These results of Cartan were further developed by many geometers. Akivis in the paper On multidimensional surfaces carrying a net of conjugate lines [Ak3] ( 1 96 1 ) and Vyacheslav T. Bazylev ( 1 9 1 9- 1 989) in the paper On a class of multidimensional surfaces [Baz] ( 1 96 1 ) considered the manifolds VP in the space Pn for which all asymptotic quadratic forms can be reduced to sums of squares. As the Cartan manifolds discussed above, such manifolds carry a conjugate net, but their osculating spaces r; ( VP ) have dimension not exceeding 2p . For these manifolds, conditions of holonomicity of their conjugate net were found (for the Cartan manifolds it is always holonomic) as well as conditions under which the manifold VP belongs to its osculating space T; (VP ) .

Manifolds VP c Pn that carry a conjugate system with multidimensional components were considered by Valery V. Ryzhkov (b. 1 920) in the pa­per Conjugate systems on multidimensional surfaces [Ry 1 ] ( 1 95 8) and by Akivis in the paper On the structure of two-component conjugate systems [Ak6] ( 1 966) .

Tangentially degenerate manifolds were studied in detail by Akivis in the papers Focal images of a surface of rank r [Ak2] ( 1 9 57) , and On a class of tangentially degenerate manifolds [Ak5] ( 1 962) , by S. I. Savelyev in the paper A surface with plane generators along which the tangent plane is fixed [Sav] ( 1 957) and by Ryzhkov in the paper On tangentially degenerate surfaces [Ry2] ( 1 960) . In particular, Akivis studied the structure of focal images of

§S . 1 1 . PROJECTIVE THEORY OF MULTIDIMENSIONAL MANIFOLDS 1 69

a (p - q)-dimensional plane generator of a tangentially degenerate mani­fold of dimension p and rank q . Further, in the paper Multidimensional strongly parabolic surfaces [Ak9] ( 1 987 ) , he showed that the structure of reg­ular strongly parabolic manifolds in Euclidean and non-Euclidean spaces is connected with focal properties of the tangentially degenerate manifolds.

In the paper On n-dimensional surfaces with asymptotic fields of p-direc­tions [Lu] ( 1 959) , Ulo G. Lumiste (b. 1 929) showed that, in the general case, such surfaces possess an (n - p )-parameter family of p-dimensional plane generators. In the same paper, he considered manifolds with a complete sys­tem of asymptotic directions. He evaluated the dimension of their osculating space and described their structure.

Phillip A. Griffiths (b. 1 938) and Joseph Harris (b. 1 95 1 ) devoted their paper Algebraic geometry and local differential geometry [GrH] ( 1 979) to the study of the projective structure of multidimensional manifolds. In this pa­per, they applied methods of algebraic geometry for studying linear systems of asymptotic differential forms of the manifold VP introduced by Cartan. The main goal of the paper was the study of manifolds whose projective struc­ture is not general. Griffiths and Harris again studied tangentially degenerate manifolds (they called them manifolds with degenerate Gauss mappings) , and next they studied manifolds with degenerate dual varieties, manifolds with degenerate Chern forms, manifolds with degenerate secant varieties, etc. Lin­ear systems of asymptotic differential forms of such manifolds have a special structure and their algebraic-geometric analysis allowed Griffiths and Harris to study not only local but also global structure of such manifolds.

Finally, in 1 988 Akivis and Polovtseva created a new procedure in the problem of projective classification of multidimensional manifolds (see the abstract of Akivis's lecture On projective differential geometry of submani­folds [Ak l O] ( 1 988 ) ) . Let PN; ( vP ) = r; ( vP ) fTx ( VP ) be a "projective nor­mal" of the manifold VP which is a projective space of dimension p1 - 1 , and let PTx (VP ) be the projectivization of its tangent space. Consider the

mapping b : PTx (VP ) --+ PN; ( vP ) defined by the formula ya = b�xixj . This mapping can be represented as a superposition b = P o v where v : PTx ( VP ) --+ Pm , m = p(p + 1 ) /2 - 1 is the Veronese mapping ( 3 . 1 1 ) , and

p : Pm --+ PN; ( VP ) is the linear mapping ya = b�xij . Arbitrary projective transformations of the space P Tx ( VP ) induce projective transformations of the space Pm that preserve the Veronesian W1 , defined by parametric equa­tions ( 3 . 1 1 ) , and also the algebraic manifolds Wk defined by the equations

rank(x ij ) = k , k = 2 , . . . , p - 1 (if k = 1 , we obtain again the manifold W1 ) . These manifolds form a filtration f : W1 c W2 c · · · c Uj,_ 1 c Pm . The kernel K = ker p of the mapping p which is a subspace of the space pm of dimension m - p1 - 1 . To the points of intersection K n »'t there correspond the asymptotic directions on VP , and to the points of intersec­tion K n w2 there correspond the pair of conjugate directions on VP . The

i 70 5 . THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

projective structure of the linear systems of the asymptotic quadratic forms and, consequently, of the manifold VP , are determined by the location of the kernel K relative to the filtration f in Pm . A similar construction can be carried out for osculating spaces and linear systems of asymptotic differential forms of higher orders.

Polovtseva in her dissertation Projective differential geometry of three­dimensional manifolds [Pol] ( 1 988) applied this method to the study and classification of three-dimensional manifolds V3

in Pn . For V3 , the di­

mension of the osculating space r; ( vP ) is equal to 3 + Pi where Pi � 6 . Since the cases Pi = 1 , 2 were studied in detail earlier, Polovtseva consid­ered in her dissertation only the cases Pi = 3 , 4 , 5 , 6. For Pi = 6 , the

linear systems of the asymptotic quadratic forms of all manifolds V3 be­

long to one class. For this reason, in this case the projective classification of manifolds V3

is determined by the structure of linear system of asymptotic cubic forms which arise in the third-order differential neighborhood of the manifold VP . If Pi = 5 , 4 , or 3, there are 3, 8, or 1 5 classes of the lin­ear systems of the asymptotic quadratic forms, respectively. Each of these classes determines a class of manifolds V3

which is projectively invariant. For each of these classes, Polovtseva investigated the geometric structure of manifolds V3 , indicated the presence, if any, of conjugate pairs and asymp­totic directions in them, and evaluated the dimension of the osculating spaces of orders higher than one. In addition, for the cases where n = 3 + Pi , she proved the existence of manifolds from each of the classes and established their arbitrariness.

§S.12. Invariant normalization of manifolds

The problem of construction of the canonical moving frame, which Cartan considered in some of his papers and monographs, is directly connected with the problem of invariant normalization of manifolds embedded in homoge­neous spaces.

Let VP be a manifold of an n-dimensional Euclidean or non-Euclidean space. Its invariant normalization is a family of normals naturally determined by the geometry of the ambient space. The normals Nx ( VP ) of the manifold VP are completely orthogonal to its tangent spaces Tx ( VP ) and defined in the first-order differential neighborhood of a point x of the manifold VP . The invariant normalization induces in VP the inner geometry, i .e . , the metric, the geodesic lines, the Gaussian curvature, etc. None of these properties of the manifold VP depend on the choice of a system of curvilinear coordinates on VP or on the choice of coordinates in the normals Nx ( VP ) .

For manifolds in spaces with a wider group of transformations (the affine, projective, or conformal or some other spaces) , it is impossible to determine an invariant normalization in the first-order differential neighborhood. For the first time the problem of construction of invariant normalization arose

§5 . 1 2. INVARIANT NORMALIZATION OF MANIFOLDS 1 7 1

in affine differential geometry. I t turned out that for a surface V2 i n the

space E3 , the invariant normal, i .e . , a straight line passing through its point x and not lying in the tangent space Tx ( VP ) , can be determined only in the third-order differential neighborhood. It was first constructed hy Blaschke in his paper On affine geometry. V. Characteristic properties of ellipsoids [Bla2] ( 1 9 1 7) .

I n projective differential geometry the situation i s even more complicated. In this case, to construct an invariant normalization of a surface V2

in the space P3 , one must find its normals of the first and second kinds. The former is defined in the same way as in affine geometry, and the latter is a straight line in the plane Tx ( V2 ) not passing through the point x (see the book Spaces with affine connection [Nor] ( 1 950) by Alexander P. Norden (b. 1 904) ) . The problem of construction of invariant projective normals of a surface V2

in the space P3 was considered by Wilczynski in 1 909, Fubini and Eduard Cech ( 1 893- 1 960) in 1 927, and Finikov in 1 937 . However, in their works the invariant normalization was connected with the choice of a certain coordinate net on the surface.

A normalization of a surface V2 in the conformal space C3

is determined by a family of tangent spheres Cx and normal circles Sx passing through

the point x E V2 • In this case the construction of invariant normalization is

connected with the second-order differential neighborhood and was consid­ered in 1 924 by Blaschke and in 1 948 by Norden.

In 1 953 , Herman F. Laptev ( 1 909- 1 972) in the paper Differential geome­try ofimbedded manifolds [Lap3] developed a general method of differential­geometric investigations of manifolds embedded in homogeneous spaces or spaces with connections. This method is based on the theory of represen­tations of Lie groups and the Cartan method of moving frames. The idea of the method is that during the differential prolongations of the system of equations, defining the manifold VP under consideration in the space xn

,

one constructs a sequence of geometric objects connected with this manifold. This sequence contains complete information on the differential geometry of the manifold VP and is the basis for all geometric constructions related to this manifold. Using this sequence, one can construct an invariant normal­ization of the manifold VP and also other geometric images connected with it. The construction indicated above does not require us to fix a coordinate system on the manifold VP . Because of this, this construction is invariant.

As an example, we consider the application of the Laptev method to the study of the geometry of a hypersurface vn- I in the affine space EN . If we place the origin of a moving frame at a point x of the hypersurface and its vectors e; , i = 1 , . . . , n - 1 , in the tangent space Tx ( vn- I ) , then the equation of the hypersurf ace can be written in the form wn = 0 . After triple prolongation of this system (i .e . , three exterior differentiations followed by the application of Cartan's lemma), we obtain

1 72 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

w� = A.uwj ,

( 5 . 56) 'V A.u + A.uw� = A.ijkwk ,

'V A.ijk + A.ijk(J)� - 3A.(iik)l(J)� = A.ijk/ '

where 'V is constructed according to the rule

k k 'V A.;j = dA.;j - A.;kwj - A.kjwi

and the quantities A.ii , A.ijk , and A.ijkl are symmetric in all indices. These quantities form the fundamental sequence of geometrical objects indicated above. Here and in what follows parentheses mean a cycle of the indices i , j , and k followed by division by 3.

Let us fix a point x on the hypersurface vn- I . Then w;

= 0 , w� = 0 , and the remaining equations of system ( 5 . 56) take the form:

( 5 . 5 7)

( 5 . 58 )

Here, as earlier, the symbol o denotes differentiation relative to the sec­ondary parameters, and n: = w: (o ) , u , v = 1 , . . . , n . The previous equa­tions show that the quantities A.ii form a double covariant symmetric relative

tensor-the asymptotic tensor of the hypersurface vn- I , and the quantities A.ijk do not form a tensor since they depend on the choice of the vector en at the point x . The latter fact follows from equations ( 5 . 58) containing the forms n� defining the displacement of the vector en .

Suppose that det(A.;) =/:- 0 , i .e . , the hypersurface vn- i is not tangentially degenerate. We construct a geometric object whose coordinates depend only on the displacement of the vector en • For this, we first construct the tensor

;.ii which is the inverse tensor of the asymptotic tensor. By ( 5 . 57) , this tensor satisfies the equations

( 5 . 59)

Next, we set 1 ij A.k = --A. ;,. "k ' n - 1 IJ

Differentiating the last equations relative to the secondary parameters and using formulas ( 5 . 57) and ( 5 . 5 8 ) , we find that

§S . 1 2. INVARIANT NORMALIZATION OF MANIFOLDS 1 73

It follows from this that the quantities ;,,i form the desired object. It is easy to check that the vector n = en - ��: J..;e; constructed with the

help of the object ;..i satisfies the equation c5n = n:n , and therefore, its di­rection does not depend on the choice of the vector en at the point x . This direction is internally connected with the geometry of the hypersurface and gives the affine normal of Blaschke. This normal is determined by the third­order differential neighborhood of a point x of the hypersurface vn- I since for its construction we used the quantities )..iik connected with this neigh­borhood. It is possible to prove that this normal is parallel to the diameter of the paraboloid which has a tangency of second order with the hypersurface vn- I at its point x .

The quantities J..iik and ).; allow us to construct an important tensor that is connected with the third-order differential neighborhood of a point x of the hypersurface vn- i . This tensor is defined by the formula

n - 1 b . "k = ). . "k - --- 3).( . .

).k) I} I} n + l IJ

and is called the Darboux tensor. It satisfies the condition biik;..ii = 0 , i .e . ,

it is apolar to the asymptotic tensor ).ii . For a two-dimensional surface in the three-dimensional space this tensor was constructed by Darboux. For a hypersurface this was done by Galina V. Bushmanova (b. 1 9 1 9) and Norden in their paper Projective invariants of a normalized surface [BN] ( 1 948) . The vanishing of this tensor characterizes the hypersurfaces of second order.

The method developed by Laptev was widely used for solving concrete problems in the theory of embedded manifolds. In the paper An invariant construction of the projective differential geometry of a surface [Lap 1 ] ( 1 949) , Laptev himself used this method for study of the geometry of a surface V2

in the space P3

, and later in the paper On fields of geometric objects on imbedded manifolds [Lap2] ( 1 95 1 ) , for study of the geometry of a hypersurface vn- I in the space Pn . For these manifolds, he found the fundamental sequence of geometric objects, considered a family of osculating hyperquadrics, and constructed a few invariant normalizations. Later Akivis solved similar prob­lems, in the paper Invariant construction of the geometry of a hypersurface of a conformal space [Ak l ] ( 1 952) , first for a hypersurface vn- I , and second, in the paper On the conformal differential geometry of multidimensional surfaces [Ak4] ( 1 96 1 ) , for a manifold VP of arbitrary dimension p in the conformal space en . In the paper Invariant constructions on an m-dimensional surface in an n-dimensional affine space [Shv] ( 1 958 ) , Petr I. Shveikin considered the problem of constructing invariant normalization for a manifold VP in the affine space En . In the papers On the geometry of a multidimensional surface in a projective space [Os l ] ( 1 966) and Distributions of m-dimensional linear elements in a space with a projective connection II [Os2] ( 1 97 1 ) Natalia

1 74 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

M. Ostianu (b. 1 922) studied this problem for a manifold VP and a distri­bution of hyperplane and p-dimensional elements in the space Pn and in a space with a projective connection.

Note that if in the beginning of its development the method of moving frames was very often opposed to the tensorial methods which also were of great importance for differential geometry, then after the creation of the Laptev method, it became clear that these two methods can easily be com­bined and can complement one another. The example considered above is a good evidence of this.

§5.13. "Pseudo-conformal geometry of hypersurfaces"

While in his works on differential geometry that we have considered so far Cartan considered manifolds embedded in spaces with Lie groups as their groups of transformations, in his two-part paper On the pseudo-conformal geometry of hypersurfaces of the space of two complex variables [ 1 36, 1 36a] ( 1 932) , Cartan studied the geometry of three-dimensional surfaces of the two­dimensional complex space with analytic transformations of this space, which form a Lie pseudogroup. The term "pseudoconformal geometry" which Car­tan used and which means a generalization of conformal geometry is presently used for the geometry of the pseudoconformal space c: . In Cartan's papers indicated above this term had another meaning: Cartan understood the term "conformal geometry" in the sense of geometry of the pseudogroup of an­alytic transformations of the plane of one complex variable and the term "pseudoconformal geometry" in the sense of geometry of the pseudogroup of analytic transformations of the space of several complex variables. The study of real hypersurfaces of a two-dimensional complex space was initiated by Poincare in the paper Analytic functions of two variables and conformal mapping [Poi5] ( 1 907) . In this paper Poincare proved that such a hyper­surface possesses an infinite set of invariants relative to transformations of this space. Using the analogy with analytic transformations of one complex variable, Poincare himself called these transformations conformal mappings. The term "pseudoconformal mappings" was suggested by Severi . In 1 93 1 Beniamino Segre ( 1 903- 1 973) in the papers On the Poincare problem on pseudo-conformal mappings [SeB 1 ] and Geometric questions associated with functions of two complex variables [SeB2] found new geometric properties of these hypersurfaces. Cartan, who became interested in the "Poincare prob­lem" under the influence of these papers by Segre, gave a classification of real hypersurfaces of the complex plane CE2

(which can be considered as the space £4

with complex structure) according to the groups of "pseudo­conformal mappings" admitted by these hypersurfaces.

These Cartan papers were substantially developed in the paper Real hyper­surfaces in complex manifolds [ChM] ( 1 97 4) by Chern and Jiirgen Kurt K. Moser (b. 1 928 ) .

§5 . 1 3. "PSEUDO-CONFORMAL GEOMETRY OF HYPERSURFACES" 1 75

The Laptev method generalizing the Cartan method of moving frames was expanded by Anatoly M. Vasil'ev ( 1 923- 1 987) to spaces where infinite­dimensional Lie groups act in his papers General invariant methods in dif­ferential geometry [Va 1 ] ( 1 9 5 1 ) and Differential algebras and differential­geometric structures [Va2] ( 1 973) and the book Theory of differential­geometric structures [Va3] ( 1 987) . The Vasil' ev method encompasses a wider circle of differential-geometric investigations.

CHAPTER 6

Riemannian Manifolds. Symmetric Spaces

§6.1 . Riemannian manifolds

The Euclidean space Rn , the elliptic space Sn , and the Lobachevskian space S� are particular cases of the Riemannian manifold vn introduced by B. Riemann in his famous lecture On the hypotheses which lie at the foun­dations of geometry [Rie2) ( 1 854) . Cartan contributed much to the geometry of Riemannian manifolds. His books Geometry of Riemannian manifolds [84] ( 1 926) , Lectures on the geometry of Riemannian manifolds [ 1 1 4] ( 1 928) , [ 1 83] ( 1 946), and Riemannian geometry in an orthogonalframe [ 1 08a] ( 1 927) and many of his papers were devoted to this topic.

A Riemannian manifold vn is a manifold whose points are defined by real coordinates x 1 , x2 , • • • , xn , and the transition from these coordinates to another coordinate system is performed with the help of differentiable functions. In addition, the distance ds between infinitesimally close points with coordinates xi and xi + dxi is given by the formula

(6 . 1 ) 2 i j ds = gijdx dx ,

where gij are differentiable functions of the coordinates xi of the points and the quadratic form ( 6 . 1 ) is positive definite.

Integrating the expression ds defined by formula ( 6. 1 ) along a curve in the space vn , we find the arc length of this curve. Comparing different curves joining two points of the space vn , we find geodesics: the curve is geodesic if and only if, for any of its points sufficiently near, the arc of this curve between these two points is the shortest one. On the other hand, the coefficients gij allow us to find the angle <p between the differentials {dxi } and {Jxi } :

(6 .2) g . . dxiaxj

COS <p = I) J gijdxidxj J g;ix;adxj

The volume element of the space vn can be expressed in terms of the de­terminant of the matrix (gi) by the formula

(6 .3 ) dV = vgdx1 /\ dx2 /\ • • • /\ dxn , 1 77

1 78 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

and the volume of a particular domain of the space vn is equal to the integral of expression (6 . 3) over this domain. Volumes of the domains of any m-dimensional surfaces of the space vn and, in particular, the areas of domains of two-dimensional surfaces of this space can be defined in a similar manner. Formula (6 .2) for finding the angles between curves of the space vn and the formula for finding the area of domains of two-dimensional surfaces of this space allow us to define the most important notion of Riemannian geometry, namely, the sectional (Riemannian) curvature of the space vn at a given point and a given two-dimensional direction. For calculation of the sectional curvature at a given point x(xi ) and a given two-dimensional direction defined by the differentials dxi and oxi of the coordinates of this point, one must take two geodesics through this point in the direction of these differentials, join points y and z of these geodesics by the third geodesic, find the area SA of the geodesic triangle xy z obtained, the angles A , B , C of this triangle, and their sum A + B + C . Then, the sectional curvature at the given point in the given two-dimensional direction is the limit of the ratio of the difference A + B + C - n to the area SA of this triangle as the triangle is shrunk to the given point in such a way that its sides remain tangent to the given two-dimensional direction:

(6 .4) K = lim A + B + C - n . A-+O SA

In particular, for the space sn ' by (3 .4 ) , we have SA = r2 (A + B + c - n) ' and the expression under the limit sign is equal to 1 / r2 • This implies that the sectional curvature K of the space sn at all its points and in all two­dimensional directions is equal to the curvature 1 /r2 of this space. For the space S� , by ( 3 . 5 ) , we have SA = q2 (n - A - B - C) , and the expression

under the limit sign is equal to - 1 / q2 . This implies that the sectional cur­

vature K of the space S� at all its points and in all two-dimensional direc­

tions is equal to the curvature - 1 / q2 of this space. Thus, the elliptic space

sn and the Lobachevskian space S� are Riemannian manifolds of constant curvature, positive and negative, respectively. Since, in his lecture of 1 8 54 Riemann paid special attention to the spaces of constant curvature, the space sn is often called the non-Euclidean Riemannian space. The Euclidean space Rn is also a particular case of the space vn . Since in the space Rn the . sum of angles of any triangle is equal to n , the expression under the limit sign in formula (6 .4) is equal to zero. Hence, in this space the curvature K at all its points and all two-dimensional directions is equal to zero, and the Euclidean space Rn is a Riemannian manifold of constant zero curvature.

With any point x of the Riemannian manifold vn there is associated the Euclidean space Tx ( Vn ) tangent to Vn at this point, and the differentials

dxi of coordinates of points of the space vn can be considered as coordi­nates of vectors of the space Tx ( Vn ) tangent to vn at this point. In the first

§6. 1 . RIEMANNIAN MANIFOLDS 1 79

edition of his Lectures on the geometry of Riemannian manifolds [ 1 1 4] with each point of the space vn , Cartan associated the so-called natural frame­the vectorial basis of the space Tx ( Vn ) consisting of the vectors e; that are

tangent to the coordinate lines of a coordinate system {x; } up to infinitesi­mals of higher orders coinciding with the segments joining the points x(x; ) and x' (x i + dx;) . Therefore, up to infinitesimals of higher orders, the co­ordinates of vectors of the space Tx ( Vn ) are equal to the differentials dx; . The vectors e; of the natural frame are often denoted by a /ax; .

Riemannian geometry in the natural frame usually is presented by means of the tensor calculus developed by Levi-Civita in the paper Methods of the absolute differential calculus and their applications [LeC] ( 1 90 1 ) .

Under the coordinate transformations x;' = /1 (x 1 , x2 , • • • , xn ) , the vec­tors e; of the natural frame undergo a transformation according to the fol­lowing rule:

(6 .5 ) ax; e ., = e . --, . 1 ' ax; Under the same transformation, the coordinates of contravariant vectors

a = {a ; } undergo the transformation

(6 .6) ;' ; ax;' a = a --. ,

ax'

where (ax;' fax;' ) is the inverse of the matrix (ax; /ax;' ) , and the coordi­nates of covariant vectors a = {a; } undergo the transformation

(6 .7) ax; a ., = a .--, , 1 ' ax;

· ' · ' · ' 1 1 · · · I with the same matrix as the vectors e; . The tensors T) } � with p con­J 1h · ·Jq

travariant indices and q covariant indices undergo the transformation

(6 .8 )

The differentials dx; of coordinates x; undergo the same transformation as coordinates of a contravariant vector, and the partial derivatives a <p /ax; undergo the same transformation as coordinates of a covariant vector. The quantities gii form a doubly covariant tensor which is called the metric tensor. In the natural frame, the equations of geodesics of the space vn have the form

(6 .9)

1 80 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

where the functions r�k , which are called the Christoffel symbols and do not form a tensor, can be expressed in terms of the tensor gii by the following formula:

(6 . 1 0)

where gii is the inverse tensor of the tensor gii ( gih ghi = oJ ) . The sectional

curvature in a two-dimensional direction defined by the differentials {dx; } and {ox;} can be expressed by the formula

(6 . 1 1 ) · k · I

_ Rij , k1dx'dx ox1ox K - ; k i I ' (gikgi1 - gikgu )dx dx ox ox

where Rii , kl is the Riemann-Christoffel tensor and can be expressed in terms of the tensor gii and its derivatives by the formula

(6 . 1 2) _ (af'Jk ar:k _,1i ....g _,1i ....g ) R . . kl - --. - --. + L 1 �k - 1 . 1 �k' ghl ' I} , OXI OX} 1g J jg I

It follows from formula (6 . 1 2) that the tensor R;j , kl satisfies the relations:

(6 . 1 3 )

(6 . 1 4)

( 6. 1 5 )

R · · kl = -R · · kl = -R · · lk = R · · lk • I} , J I , I} , J I ,

R . . kl = Rkl · · • I} , , I}

R · · kl + R ·k ·1 + Rk · ·1 = 0. I} , J , I I , j Relation (6 . 1 5 ) is called the Ricci identity; it is named after the founder of the tensor analysis Gregorio Ricci-Curbastro ( 1 853- 1 925) who discovered this identity.

Cartan encountered the Riemannian manifolds in his works on the theory of simple Lie groups. In his thesis he proved that the condition for a complex Lie group to be semisimple is the nondegeneracy of quadratic form (2 .20) . In other words, he showed that semisimple complex Lie groups are complex Riemannian manifolds. Similarly, Cartan's result of 1 9 1 4 that the condition for a real compact Lie group to be semisimple is the negative definiteness of the same quadratic form (2 .20) (see [38] ) means that semisimple compact real Lie groups are real Riemannian manifolds if one takes the form -V1(e) as di . Note that the Riemannian metric, defined in this manner in the complex and compact real simple Lie groups and at present called the Cartan metric, is a unique (up to the scale of the Riemannian metric) metric in these groups which is invariant relative to the group operations x --+ ax , x --+ xa , and

- 1 X --+ X At the end of the introduction to his Lectures on the geometry of Rieman­

nian manifolds Cartan wrote: "I was forced to leave aside many important

§6.3 . PARALLEL DISPLACEMENT OF VECTORS 1 8 1

problems. They might compose the content of the next volume where the method of moving frames and its numerous applications will be presented" [ 1 1 4, p. 6). The main part of this material was included in the second edition [ 1 83] of this book.

§6.2. Pseudo-Riemannian manifolds

In 1 9 1 4, in the same paper [38] discussed in the previous section, Cartan actually introduced the so-called pseudo-Riemannian manifolds which differ from the Riemannian manifolds by the fact that the quadratic form (6 . 1 ) , defining the metric of this space, is no longer positive definite; it can be any indefinite nondegenerate form. In this paper, Cartan showed that if one takes the form -VJ(e) as ds2 , then a noncompact real Lie group is a pseudo-Riemannian manifold. At present, a pseudo-Riemannian manifold, whose metric is defined by a nondegenerate quadratic form ( 6. 1 ) of index I , is called a pseudo-Riemannian manifold of index I and is denoted by �n

•

Thus, in the Cartan metric, a noncompact real group Lie of dimension r , whose character is equal to o , is a pseudo-Riemannian manifold �n where the index I is connected with the dimension r and the character o of the group by the relation o = r - 21 .

The tangent pseudo-Euclidean spaces Tx ( �n ) for the space �n play the same role as the tangent Euclidean spaces Tx ( Vn ) for the Riemannian man­ifold vn . The ideal domain of the hyperbolic space S� and both domains of the spaces s; , I > 1 , are pseudo-Riemannian manifolds of constant pos­itive or negative curvature, and the pseudo-Euclidean space R7 is a pseudo­Riemannian manifold of constant zero curvature.

General relativity theory, created by Albert Einstein ( 1 879- 1 955 ) in 1 9 1 6 , played the important role in attracting the interests of mathematicians to the geometry of Riemannian and pseudo-Riemannian manifolds since, according to this theory, the space-time is a pseudo-Riemannian manifold V.4

whose curvature is greater in those places where the density of matter is larger.

§6.3. Parallel displacement of vectors

In 1 9 1 7, shortly after the appearance of the general relativity of Einstein, Levi-Civita introduced one of the most important notions of the Riemannian and pseudo-Riemannian geometries-the parallel displacement of vectors. At the same time as Levi-Civita, Schouten discovered parallel displacement of vectors in the Riemannian geometry. Schouten's colleague Dirk Jan Struik (b. 1 894) recently recalled: "One day in 1 9 1 8 Schouten came bursting into my office waving a paper he had just received from Levi-Civita in Rome. 'He also has my geodesically moving systems,' he said, 'only he calls them parallel . ' The paper had in fact already been published in 1 9 1 7 , but the war had prevented it from arriving sooner. " [Row, p. 1 6] .

1 82 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

In the same year, 1 9 1 7, when the paper of Levi-Ci vita was published, in the paper On the curvature of surfaces and manifolds [Svr] , Severi gave the geometric definition of this notion which Cartan widely used. The essence of Severi's definition is that to each vector a of the tangent space Tx ( vn ) at a point x of the Riemannian manifold vn , one can set in correspondence a certain vector 'a of the tangent space at a point x' infinitesimally close to the point x of the same space. This vector 'a is defined by a mapping of a neighborhood of the point x onto a neighborhood of the point x' that is a result of the sequential reflecting about the point x of the neighborhood of the point x along geodesics emanating from this point and the similar reflection in the point x0 of a neighborhood of the point x0 located on the geodesic xx' , half way between the points x and x' (provided that the neighborhood of the point x0 contains the neighborhoods of the points x and x' ) . If, up to infinitesimals of higher orders, the vector a coincides with the geodesic segment xa , and the mapping indicated above sends the point a of the neighborhood of the point x into the point a' of the neighborhood of the point x' , then up to infinitesimals of higher orders, the vector 'a coincides with the geodesic segment x' a' . If the vector 'a is given at the point x (xi ) of the space vn , then the result of its parallel displacement into the infinitesimally close point x' (xi + dxi) is a vector 'a with coordinates

(6 . 1 6) I i i i kd j a = a + rika x .

Assuming that the scalars are not changed under a parallel displacement, we find that the parallel displacement of covariant vectors a = { ak } is defined by the formula

(6 . 1 7)

i · · · i and the parallel displacement of an arbitrary tensor T. 1 .P i s defined as J i · · ·Jq

(6 . 1 8) I i · · · i T} .P = 11 · · ·Jq

If in the space vn , a vector or tensor field is given, i .e . , at any point of this space a vector or a tensor of a certain type is defined that is a function of this point, then we can define the covariant derivative of this vector or tensor by subtracting from the value of this vector or tensor at the point x' the result of its parallel displacement from the point x into the point x' , dividing this difference by the difference of the coordinates xi + dxi and xi of the points x' and x and taking the limit of the ratio obtained when the point x' tends to the point x . The covariant derivative V ia

i of the vector a = {a; } has

§6.4. RIEMANNIAN GEOMETRY IN AN ORTHOGONAL FRAME

the coordinates

(6 . 1 9) i ( k A �k ) I (

k ) O i " i -l' a x + UA - a x - a ri k

v .a - im . - -. + .ka , l t:.xi -+O !lx1 8 X1 l

1 83

and the covariant derivative V' . T'. 1 . .

. '.p of the tensor r'.1 . . . '.p has the coordi-1 1 1 · · ·lq 1 1 . . . lq nates

(6 .20)

The covariant derivative of any vector or tensor is again a tensor which has one more covariant index than the original vector or tensor. The result of the contraction of the covariant derivative V' j Tj11 .·.· .·J: with the differential

dxj is called the absolute differential and is denoted by DTj11 .·.·.·J: . Comparing formulas (6 .9) and (6 . 1 9) , we see that formula (6 .9) can be

written in the form V/dx;/ds) (dxj /ds) = 0 from which it follows that the geodesic lines of the space vn can be also defined as the lines along which their tangent vectors dx; /ds undergo a parallel displacement. Note also that relation (6 . 1 0) is equivalent to the relation V'kgij = 0 .

In the spaces v,n , parallel displacement and covariant differentiation can be defined in a similar manner.

§6.4. Riemannian geometry in an orthogonal frame

In Cartan's book Riemannian geometry in an orthogonal frame [ 1 08a], which was composed of his lectures of 1 926- 1 92 7 written by Finikov and translated by him into Russian, and in the second edition of the book Lectures on the geometry of Riemannian manifolds [ 1 83] , Cartan introduced a new presentation of Riemannian geometry associating with every point of the space an orthonormal frame {e; } (e;ej = <5u ) instead of the natural frame.

The derivational formulas of an orthonormal frame { e; } have the same form ( 5 . 5) as for an orthonormal frame in the space Rn (the differential dx here is the vector with the coordinates dx\ but in this case the structure equations are more complicated than in the case of the space Rn , namely, they are:

(6 .2 1 ) i k i d (J) = (J) /\ (J)k '

where R{ , kl is the curvature tensor which in this case can no longer be expressed in terms of the tensor gij and its derivatives by formulas ( 6 . 1 2) .

In an orthonormal frame, the sectional curvature K in a two-dimensional direction can again be calculated by the same formula ( 6 . 1 1 ) as in the case

1 84 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

of the natural frame; but, since in this case we have gii = oii , this formula becomes:

(6 .22) i k j 1 K = R . . kla a b b , I} '

where a; and b i

are the coordinates of two unit orthogonal vectors a and b defining this two-dimensional direction.

Comparing formulas (6 .2 1 ) with formula ( 5 . 1 3 ) held in the space of con­stant curvature 1 /r2 , we find that in this space and in an orthonormal frame the curvature tensor R ii , kl has the form

(6 .23)

The application of orthogonal frames enables one to solve many problems of differential geometry in the space vn in the same simple way as in the space Rn .

Contracting the tensor Rii , kl of the space vn in the indices i and 1 , we obtain the Ricci tensor :

(6 .24) ii R .k = R . . klg

J I} '

Since at each point of the space vn two tensors gii and Rii are always

given, in the general case with each point of the space vn , n principal directions in the sense of Ricci are associated, and these directions are the directions of the eigenvectors of the matrices R� = Rkig

ki .

§6.5. The problem of embedding a Riemannian manifold into a Euclidean space

The problem of embedding a Riemannian manifold vn as a surface into a Euclidean space RN of sufficiently large dimension was posed by one of the founders of multidimensional geometry, Ludwig Schlafli ( 1 8 1 4- 1 895) , in his Note on the memoir "On spaces of constant curvature" of Mr. Beltrami [Sel l ] ( 1 87 1 - 1 873 ) . Schlafli's argument was as follows. If the space vn with the metric form di = giidu; dui is embedded into the space RN in

the form of a surface x = x( u 1 , u2 , • • • , un ) , then the coefficients gii are

connected with the partial derivatives X; = 8x/8u; by the relations gii =

X;X i . Since the number of coefficients gii is equal to n ( n + 1 ) /2 , the number of the equations obtained is the same. Schlafli concluded from this that it is possible, at least locally, to embed the space vn into the space RN where N = n(n + l )/2 .

Schlafli's statement on the possibility of local embedding was proved by Maurice Janet ( 1 888- 1 984) in the paper On the possibility of imbedding a given Riemannian manifold into a Euclidean space [Ja] ( 1 926) . Janet's re­sults were revised by Cartan in the paper [ 1 04] ( 1 927) with the same title.

§6.6. RIEMANNIAN MANIFOLDS SATISFYING "THE AXIOM OF PLANE" 1 85

While Janet wrote the problem of embedding the space vn into the space RN in the form of a system of partial differential equations which he in­vestigated using rather complicated methods, Cartan applied his own theory of systems in involution. He wrote the system of equations describing the embedding of the space vn into the space RN in the form of a system of Pf ffi . - i ; - a 0 . 1 N a an equations ro = ro , ro = , z = 1 , . . . , n , a = n + , . . . , , where ro;

are basis forms of the space vn and the forms ii/ are basis forms of a surf ace vn of the space RN onto which the space vn is mapped. The differential prolongation of these systems leads to the equations w� = (J)� •

Applying structure equations (6 .2 1 ) of the space vn and the structure equa­tions of the surface vn of the space RN

' Cartan showed that this system of Pfaffian equations is in involution and its general solution depends on n arbitrary functions of n - 1 real variables. This solution of the problem of embedding the space vn into the space RN in the paper [ 1 04], which was much simpler than Janet's solution in the paper [Ja] , offended Janet. After Cartan's death Janet tried to convince Pommaret that his (own) methods of solution of partial differential equations were better than Cartan's meth­ods, complaining that his methods were undeservedly forgotten. Pommaret wrote in his book Lie pseudogroups and mechanics [Pom3] ( 1 988 ) : "These comments were given privately to us by Maurice Janet, again mathematician and mechanician (sic. ) , who died in January 1 984 at the age of 96." [Pom3, p. 7] .

§6.6. Riemannian manifolds satisfying "the axiom of plane"

In 1 927, Cartan published the paper The axiom of plane and metric dif­ferential geometry [90] . This paper appeared in the collection of articles "In Memoriam N. I. Lobatschevsky" which was published in Kazan, U.S.S.R. , on the occasion of the l OOth anniversary of the discovery of non-Euclidean geometry by Lobachevsky. Figure 6. 1 (next page) reproduces the first page of Cartan's manuscript of this paper, which is kept in the Department of Ge­ometry of the University of Kazan. Cartan called a surface of a Riemannian manifold geodesic at a certain point if this surface coincides with the union of geodesics of the Riemannian manifold, emanating from this point and tangent to a plane element of the space at this point, and he called totally geodesic a surface which is geodesic at each of its points. The requirement that any geodesic surface be totally geodesic at each of its points was called "the axiom of plane" by Cartan. The notion of a totally geodesic surface was introduced by Jacques Hadamard ( 1 865- 1 963) in his paper On linear ele­ments of many dimensions [Had] ( 1 90 1 ) . Hadamard defined these surfaces as surfaces such that each geodesic of them is a geodesic of the space.

In the case of Euclidean and non-Euclidean spaces, geodesics are straight lines and planes are totally geodesic surfaces. This explains the name of this Cartan axiom. It is obvious that the Euclidean and non-Euclidean spaces

1 86 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

Counesy of Depanment of Geometry, Kazan University, Tatarstan, Russia

FIGURE 6. 1

satisfy this axiom. In his paper Cartan proved that if a Riemannian manifold satisfies "the axiom of plane'', then it can be geodesically and conformally mapped onto a Euclidean or non-Euclidean space.

§6.7. Symmetric Riemannian spaces

In the note On Riemannian manifolds in which parallel translation pre­serves the curvature [87] ( 1 926) Cartan remarked on Harry Levy's paper The canonical form ds2 for which the five-index Riemann symbols are annihilated [Lev]. Since Levi called the coordinates Rij , kl of the Riemann tensor the four-index Riemann symbols and the coordinates 'V' hRij , kl of the covariant derivative of the Riemann tensor the five-index Riemann symbols, the spaces singled out by Levi are Riemannian manifolds satisfying the condition:

( 6 .25) 'V' hRij , kl = 0.

Levi established that condition (6 .25) holds for the spaces of constant curvature, i .e . , the spaces Rn , Sn , and s7 and the Cartesian products of these spaces, but he did not find other Riemannian manifolds satisfying this property. Neither Levi nor Cartan knew that the same class of Riemannian manifolds was introduced by Petr A. Shirokov ( 1 89 5- 1 944) in the paper Constant.fields of vectors and tensors of second order in Riemannian manifolds [Sh 1 ] published in 1 925 in Kazan. Shirokov also established that the spaces

§6.7. SYMMETRIC RIEMANNIAN SPACES 1 87

of constant curvature satisfy this condition, and, in addition, unlike Levi, he found the general form of three-dimensional Riemannian manifolds of this type.

In his note [87] Cartan indicated that irreducible spaces of this type "are separated into 1 0 large classes each of which depends on one or two arbi­trary integers, and, in addition, there exist 1 2 special classes corresponding to the exceptional simple groups G". He wrote further: "Among the general classes, besides the spaces of simple groups which were discussed in the note I, I will only indicate the class of the spaces of constant curvature found by Mr. Levi and the class of Hermitian hyperbolic and elliptic spaces." [87 , p. 245] . In the text of the note instead of the word "spaces" at the end of the quotation written above, it is incorrectly written "groups". In this quotation the "note I" means the joint note On the geometry of the group-manifold of simple and semisimple groups [9 1 ] by Cartan and Schouten also published in 1 926 but a little later than the note [87] . In this note [9 1 ] , the authors defined three types of parallel displacements in Lie groups and denoted them by ( - ) , ( +) , and ( o ) . The first two types of these parallel displacements are absolute parallelisms, and the third type in the case of simple and semisimple Lie groups is the parallel displacement of the vectors of the Riemannian or pseudo-Riemannian manifold satisfying property (6 .25 ) . Briefly mentioning this fact in the note [87] , Cartan indicated that the totally geodesic subman­ifolds of the group-manifold of simple and semi-simple Lie groups possess the same property. Since the number 1 0 of "large classes" is the mean of the numbers of types of symmetric Riemannian spaces with the classical funda­mental groups- 1 1 types AI - IV , BI - II , CI - II , DI - Diii and 9 types AI - IV , BDI - II , CI - II , Diii which Cartan used later, and the number 1 2 of "special classes" coincides with the number of types of symmetric Rie­mannian spaces with the special fundamental groups EI - IX , FI - II , and GI , and since Cartan indicated that in simple and semisimple Lie groups one can define the metric of symmetric Riemannian spaces which can take place also on totally geodesic submanifolds of these groups, we see that when Cartan was publishing his note [87] , he already knew most of the results of his theory of symmetric spaces.

Cartan gave a systematic exposition of this theory in his paper On a re­markable class of Riemannian manifolds published in two parts [93] ( 1 926) and [94] ( 1 92 7). Cartan defined these space as the spaces characterized by "the property that the Riemannian curvature of any face is preserved under a parallel displacement, or in more abstract terms, by the property that the covariant derivative of their Riemann-Christoffel tensor is identically equal to zero" [93, p. 2 1 4] , i .e . , by identity (6 .25 ) . In this paper, Cartan called these spaces the "spaces W". However, in his late works he gave them the name "symmetric Riemannian spaces".

In the papers [93]-[94], Cartan proved that condition (6 .25) is equivalent to the fact that the reflection in each point of the space along geodesics is an

1 88 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

isometric transformation (motion) of the space. This very property was the reason that in his subsequent papers Cartan called these spaces the symmetric Riemannian spaces. In addition, Cartan showed that all compact simple and semisimple Lie groups are symmetric Riemannian spaces provided that one introduces in them the Cartan metric which will be in this case a Riemannian metric (in this case the reflection in the identity element of the group has the form x --+ x - I , and the reflection in an element a has the form x --+ ax - I a) . In the same paper, Cartan proved that any irreducible compact symmetric Riemannian space can be realized in the form of a totally geodesic surface in the group of motions of this space which passes through the identity element of this group (if u is the reflection in an arbitrary point of this space and u0 is the reflection in a certain fixed point, then this totally geodesic surface in the group of motions consists of products u0u) .

Cartan also considered the symmetric Riemannian spaces whose groups of motions are noncompact simple and semisimple Lie groups. These spaces can also be realized in the form of a totally geodesic surface in their groups of motions if we introduce in these groups the Cartan metric. However, since for noncompact simple and semisimple Lie groups this form is nondegenerate indefinite, the Cartan metric in the Lie group is a pseudo-Riemannian metric. In this case, the Lie group is a symmetric pseudo-Riemannian space, i .e . , a pseudo-Riemannian manifold satisfying condition (6 .25 ) . For symmetric pseudo-Riemannian spaces, the two properties which were indicated above and which were established by Cartan for symmetric Riemannian spaces also hold.

For the case, when the group of motions of a symmetric Riemannian or pseudo-Riemannian space is a simple Lie group, the Lie algebra of this group admits the "Cartan decomposition" (2 .43) . Moreover, the algebra H is the Lie algebra of the isotropy group of this space (the group of rotations about its point) , and the subspace E can be considered as the tangent space to the totally geodesic surface in the group in which the symmetric space is realized or, equivalently, as the tangent space to the symmetric space. By formulas (2 .36) the subspace E of the Lie algebra G , which can be considered as the tangent space to a symmetric Riemannian space, is closed under the operation [ [X , Y] , Z] . Such spaces are called Lie triple systems.

Since in the Lie algebra of a Lie group the Cartan-Killing form has the form (2.20) , the Riemannian or pseudo-Riemannian Cartan metric in a Lie group is defined by the linear element

(6 .26)

and the Riemann tensor of this symmetric space has the form

(6 .27)

§6.7. SYMMETRIC RIEMANNIAN SPACES 1 89

In the case when a symmetric space corresponds to the Cartan decompo­sition (2 .43) and the structure equations of the group are written in the form (2 .36) , the metric of the symmetric space is defined by the linear element

(6 .28)

and the Riemann tensor of this symmetric space has the form

(6 .29)

Comparing formulas (6.26) and (6.28) with formulas (6.27) and (6 .29) , we see that in both cases the Ricci tensor of the symmetric space is proportional to its metric tensor. It follows from this that in the symmetric Riemannian and pseudo-Riemannian spaces it is impossible to define . the principal direc­tions in the sense of Ricci.

Next, Cartan considered the classification of involutive automorphisms of Lie algebras of compact simple Lie groups which we presented in Chapter 2 for Lie groups in the classes An , Bn , Cn , and Dn and gave the classifica­tion of symmetric Riemannian spaces whose groups of motions are compact simple Lie groups. These symmetric spaces are characterized by the same characters o as noncompact simple Lie groups corresponding to the same involutive automorphisms of compact simple Lie groups. In the case of sym­metric spaces with compact simple groups of motions, these characters have a simple geometric meaning: if G is the group of motions of a symmetric space and the isotropy group is a subgroup H of this group (in this case the space is denoted by G/ H), then the character o of the symmetric space is equal to the difference dim E - dim H between the dimension dim E of the symmetric space and the dimension dim H of its isotropy group. (Since the dimension dim G of the group G is equal to the sum of dimensions dim E + dim H , the character o of the symmetric space is equal to the dif­ference dim G - 2 dim H . )

Cartan also found the isotropy groups of irreducible symmetric Rieman­nian spaces, i .e . , spaces that cannot be represented in the form of Cartesian products of other symmetric spaces. In the case of the symmetric space VN , the isotropy group is the group of rotations of the Euclidean space RN tan­gent to the space VN , i .e . , the group ON or a subgroup of it. Moreover, for the case when the isotropy group is a simple group or a direct product of simple groups, Cartan found the linear representations in which the isotropy group or its direct factors are realized in the group ON (in addition to the noncommutative direct factors, the isotropy group may also contain a rep­resentation rp0 of the commutative simple group D1 = 02-the group of rotations of a circle) . Cartan proved that, for irreducible symmetric Rieman­nian spaces, the isotropy groups of these spaces coincide with their holonomy groups. The latter are the subgroups of the isotropy groups defined by parallel displacements of vectors of these spaces along closed contours. (The require­ment of irreducibility is essential since, for example, for the Euclidean space

1 90 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

An A l n n(n+3) 9' 1 ( 0n+ 1 l 2

All -n - 2 (n - l )(n+2) 9'2 ( Ccn+ 1 )!2 ) 2

AIII 4/(n - I + 1 ) - n(n + 2) 2l(n - I + 1 ) 9' 1 (A1- 1 ) 181 9' 1 (An - I ) 181 9'o

A IV 2n - n2 2n 9' 1 (An - I ) 181 9'o Bn Bl 2l(2n - I + 1 ) - n(2n + 1 ) l (2n - I + 1 ) - 91 1 ( 01 ) 181 9' 1 ( 02n+ 1 - 1 )

Bil 2n - n2 2n 9'1 ( 02n )

en CI n n ( n + 1 ) 2 9' 1 (An - 1 ) 181 9'o

en 8/(n - I) - n (2n + 1 ) 4l(n - I) 9' 1 ( C, ) ® 9' 1 ( Cn - 1 )

Dn DI 21(2n - I) - n(2n - 1 ) l(2n - I) 9' 1 ( 01 ) 181 9' 1 ( 02n - 1 )

Dll (2 - n ) (2n - 1 ) 2n - I 9' 1 ( 02n - 1 )

Diii - n n (n - 1 ) 9'2 (An - 1 l 181 9'o

G2 GI 2 8 3

9' 1 (A 1 ) 181 9' 1 ( A 1 ) F4 Fl 4 2 9'3 ( C3 ) 181 91 i (A 1 )

Fil - 20 1 6 1/11 ( 09 )

E6 El 6 4 9'4 ( C4 )

Ell 2 40 9'3 (A 5 ) 181 9' 1 (A i )

EIII - 1 4 32 1/11 (01 0 ) 181 9'o

E/ V - 26 26 9'1 ( F4 ) E1 E V 7 70 9'4 (A 7 )

E V/ - 5 64 lfl1 ( 01 2 ) ® q.i 1 (A , )

E V/I -25 54 qJ I ( E6 ) 181 9'o Eg E V/// 8 1 28 lfl, ( 0, 6 )

E/X - 24 1 1 2 9' 1 ( E7 ) 181 9' 1 ( A , ) I

TABLE 6 . 1

Rn which is the Cartesian product of n straight lines R 1 , the isotropy group is the group of rotations On while the holonomy group consists of the identity transformation · alone. )

In Table 6. 1 we give Cartan's notation for different types of compact sym­metric Riemannian spaces.

§6.8. HERMITIAN SPACES AS SYMMETRIC SPACES 1 9 1

In the first column of Table 6. 1 we indicate the class of the simple Lie group, in the second the Cartan notation for the type of the symmetric space, in the third and the fourth the character and the dimension of the symmet­ric space, and in the fifth the isotropy group of the symmetric space with indication of the linear representation of each of the direct factors of this group.

In the book Symmetric spaces [Loo] ( 1 969) by Ottmar Loos, the symmet­ric Riemannian spaces were considered as spaces with a certain algebraic structure in which to any pair of points x and y there corresponds a third point z of this space which is the reflection of the point x in the point y . A symmetric space with this operation is a quasigroup, i .e . , it differs from a group by the absence of the identity element and the associativity of the main operation.

While Cartan found all symmetric spaces whose fundamental groups are compact simple Lie groups and those noncompact simple Lie groups for which the stationary groups are compact, Berger in the paper Classification of irreducible homogeneous symmetric spaces [Beg l ] ( 1 955 ) and Anatoly S. Fe­denko (b. 1 929) in the paper Symmetric spaces with simple non-compact fun­damental groups [Fed] ( 1 9 56) found all symmetric spaces with noncom pact simple fundamental groups of infinite sequences. In another paper Structure and classification of symmetric spaces with semi-simple groups of isometries [Beg2] ( 1 955 ) Berger solved the same problem for noncompact exceptional simple Lie groups. Note also that in the paper Non-compact symmetric spaces [Beg3] ( 1 957) , Berger found the isotropy groups of all irreducible symmetric spaces and representations of the direct factors of these groups. This enables one to find the orbits of the isotropy of these spaces.

The current status of the geometry of Riemannian manifolds is described in the books Einstein manifolds [Bes] ( 1 985 ) by Arthur L. Besse, Differential geometry, Lie groups and symmetric spaces [Hel l ] ( 1 978) by Helgason, and Spaces of constant curvature [Wo2] ( 1 984) by Wolf.

§6.8. Hermitian spaces as symmetric spaces

The symmetric Riemannian spaces Bil and DII of dimension 2n and 2n - 1 , respectively, whose groups of motions are compact simple Lie groups in the classes Bn and Dn (the groups of orthogonal matrices 02n+ I and 02n ) , are Riemannian manifolds of constant curvature-the elliptic spaces

S2n and S2n- I . The symmetric Riemannian space AIV of dimension 2n whose group of

motions is a compact simple Lie group in the class An (the group CSUn+ I of complex unimodular matrices is of this type) is the complex Hermitian elliptic space csn

' i .e. , the space cpn where the metric is defined by for­mula ( 3 . 6 ) . We have already mentioned that, as was noted by Cartan in his

expanded translation of the Fano paper, the complex straight line CS 1 of

1 92 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

curvature 1 /r2 is isometric to a sphere of radius r/2 in the space R3 • On the other hand, assuming that the coordinates x;

and / in formula (3 .6 ) are real, we obtain formula ( 3 . 1 ) for finding the distances in the real elliptic space sn of curvature 1 / r2 •

Applying formula ( 6 .29) for computing the curvature tensor to the Rie­mannian manifold v

2n which is isometric to the space csn ' we find that

in the orthonormal frame of V2n , whose vectors e2;_ 1 coincide with the

vectors f; of a unitary-orthonormal frame of the space CSn ( (f; , f) = oii ) and whose vectors e2; coincide with the products if; , the tensor RiJ . kl has the form

(6 . 30)

where eij is the skew-symmetric tensor whose nonvanishing components are e2;_ 1 , 2; = -e2; , 2;_ 1 = 1 . Formula (6 . 30) is similar to the formula for the

coordinates of the tensor Rii , kl of the space V2n which is isometric to

-n the space CS in the natural frame. The latter formula was found by P. A. Shirokov in his posthumously published paper On a certain type of symmetric space [Sh2] ( 1 957 ) .

Substituting expressions (6 . 30) of the coordinates of the tensor R . . k l into I] ' formula (6.22) , we obtain the expression for the sectional curvature K in a two-dimensional area defined by orthogonal unit vectors a and b in the form

(6 . 3 1 )

where a is the "angle of inclination of the two-dimensional element" intro­duced by P. A. Shirokov in the above-mentioned paper which is defined by the relation (a , b) = i cos a . It is easy to check that the values of a and K do not depend on the choice of the vectors a and b in the two-dimensional element and that the angle a is equal to the angle between two complex straight lines of the space CS n tangent to the vectors a and b . If the area element lies in the complex straight line (in this case the two-dimensional direction is called holomorphic) , we have a = 0 and K = 4/r2 ; this can also be seen from the fact that the line CS 1 of curvature 1 / r2 is isometric

to a sphere of radius r /2 in the space R3 . If the two-dimensional element lies in the manifold x; = x; or in the manifold obtained from it by a mo­tion of the space CSn (Cartan called such manifolds "normal space chains") ,

we have a = n/2 and K = 1 /r2 ; this can also be seen from the fact that if x; = x; , / = ·yi , formula (3 .6 ) takes the form (3 . 1 ) . Formula (6 . 3 1 ) shows that the sectional curvature of any area element of the space V

2n iso­metric to the space CSn lies in the interval: 1 /r2 :::; K :::; 4/r2 • Since the sectional curvature of a holomorphic two-dimensional element of the space

§6.9. ELEMENTS OF SYMMETRY 1 93

csn (which is called the holomorphic curvature of this space) is equal to the

constant value 4/r2 , at present, the space CSn is often called the Hermitian space of constant holomorphic sectional curvature.

If l = 1 , the symmetric Riemannian space en of dimension 4(n - 1 ) whose group of motions is a compact simple Lie group in the class en is the

quaternion Hermitian elliptic space HSn- I , i .e . , the space HPn- I where the metric is defined by the same formula (3 .6 ) as for the space csn .

The symmetric Riemannian space Fil of dimension 1 6 whose group of motions is a compact simple Lie group in the class F4 is the octave Hermitian

elliptic plane OS2 defined by A. Borel and Freudental in 1 9 50- 1 9 5 1 . The

sectional curvatures of these spaces V4n and V 1 6 are calculated by the same formula ( 6. 3 1 ) as in the space V2n where a is the "angle of inclination of the two-dimensional element" defined in the same manner as in the space CSn . In the case of holomorphic two-dimensional elements, i .e . , the two­dimensional elements situated in quaternion straight lines of the plane HSn or in octave straight lines of the space OS2

, we have a = 0 and K = 4/r2 • This corresponds to the fact that the straight lines HS 1 and OS1

are isometric to hyperspheres of radius r /2 of the spaces R5 and R9

• Next, if the two-dimensional element lies in the normal real plane chain, we have a = n/2 and K = 1 /r2 •

Note that applying formula ( 5 .29) for computation of the Riemann tensor of a symmetric Riemannian space to the spaces V4n and V 1 6 (which are

isometric to the Hermitian elliptic space HSn and plane OS2) we obtain the

formula:

(6 . 32) 1 R . . kl = -2 (o .ko ·1 - o .10 .k ) + "(e .ke ·1 - e ·1 e .k + 2e . . e kl ) ' I] , I J I J L....,, Otl Ot} Otl Ot} Otlj Ot r °'

where the matrices of the tensors eaik are the matrices of the operators of the complex structure defined by the units i°' of the algebras H and 0 .

The symmetric Riemannian spaces EIII , EIV , and EVIII of dimension 32, 64, and 1 28 whose groups of motions are a compact exceptional simple Lie group in the classes E6 , E1 , and £8 respectively, admit similar inter­pretations in the form of Hermitian elliptic planes over the tensor products O ® C , O ® H , and 0 ® 0 .

§6.9. Elements of symmetry

In the introduction to his paper On a remarkable class of Riemannian manifolds [93] , Cartan wrote: "The new spaces immediately admit a direct geometric definition: they can be represented by geometric figures admitting a simple definition in the ordinary space (of dimension three or higher) " [93 , p. 2 1 7] . At present, these geometrical figures are called elements of symmetry.

1 94 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

To the geometric interpretation of symmetric Riemannian spaces in the form of manifolds of elements of symmetry, Cartan devoted his great paper On certain remarkable Riemannian forms of geometries with a simple funda­mental group [ 1 07] ( 1 927) , which we have already mentioned in Chapter 2 as the paper where an interpretation of a compact simple Lie group in the class

-n- 1 Cn in the form of the group of motions of the space HS first appeared. In Chapters 2 and 3 of this work entitled Classification of spaces of constant neg­ative curvature connected with real non-unitary groups and Spaces of positive curvature connected with real unitary simple groups (in the title of Chapter 3 instead of the word "unitary" the incorrect "nonunitary" was written) , Cartan gave geometric interpretations of noncompact and compact symmetric Rie­mannian spaces with compact isotropy groups, respectively. Cartan called these spaces the spaces of negative and positive curvature, respectively, by analogy with the hyperbolic and elliptic spaces S� and sn which are their particular cases. The terms "unitary" and "nonunitary groups" are connected with the fact that these groups are represented by unitary and pseudo-unitary complex matrices, respectively. For compact symmetric spaces of types AIII and BDI , Cartan wrote that these spaces are spaces or can be defined as spaces of "pairs of plane manifolds of q - 1 and p - 1 dimension mutually polar relative to the form F" [ 1 07, pp. 448 and 45 1 ] ; the form F is the right-hand side of the equation of the absolute in the space csn and the space sn , respectively. In other cases, Cartan indicated the type of sym­metry relative to the corresponding element of symmetry without giving the names of these figures. For the compact symmetric spaces of types AI , Aii , and AIII , these symmetries are the transformations:

(6 .33 ) I i _; x = X ,

(6 . 34) I 2i -i+ l I i+ l 2i x = X x = X '

(6 . 35 ) I a a I u u x = X ' x = -X of the space csn that are the reflection in the normal space chain, the shift for a half-line along the lines of a paratactic congruence, and the reflection in a m-dimensional plane and its polar, respectively. For the compact symmetric space of type BI , the symmetry is the transformation (6 . 35 ) of the space S2n which is the reflection in an m-dimensional plane and its polar. For the compact symmetric spaces of types DI and DII , the symmetries are the transformations ( 6 .35 ) and

(6 .36) I 2i 2i+ l X = -X I 2i+ i 2i X = X , which are the reflection in an m-dimensional plane and its polar and the shift for a half-line along the lines of a paratactic congruence of the space s2n- l

• Finally, for the compact symmetric spaces of types CI and CII , the

§6.9. ELEMENTS OF SYMMETRY

symmetries are the transformations

(6 .37) I i . i . - ! X = lX l . ,

1 95

and (6 .35 ) of the space HSn- t , respectively, which are the reflections in the normal space chain and in an m-dimensional plane and its polar, respectively.

Note that symmetries (6 .33 ) , (6 . 34) , (6 . 35 ) , (6 .36 ) , and (6 .37) correspond to involutive automorphisms (2 .48) , (2 .49) , (2 .4 7) (2 .50) , and (2 . 5 1 ), respec­tively, in the Lie algebras of the groups of motions of symmetric spaces.

Thus, if m = 0 , the compact symmetric Riemannian spaces of types AIV , Bil , DII , and CII coincide with the elliptic spaces CSn , S2n , s2n- t ,

-n- 1 ·

and HS ; if m -::f. 0 , the spaces of types Alli , BI , DI , and CII can be interpreted as the Grassmann manifolds CGRn , m , GR2n , m , GR2n- l , m , and HGRn- l , m of these spaces, the spaces of types AI and CI can be

interpreted as the manifolds of normal space chains of the spaces csn and -n- 1 HS , and the spaces of types All and DII can be interpreted as the manifold of paratactic congruences of straight lines of the spaces csn and s2n- l .

Cartan also noted that a compact exceptional simple group in the class G2 is the group of automorphisms of the alternative field 0 of octaves. If we define a metric of the space Rs in the alternative field 0 by setting the distance between octaves a and p to be equal to the modulus IP - al of their difference, then the group of automorphisms of the alternative field 0 is a transitive group on a six-dimensional sphere which is the intersection of the hypersphere la l = 1 and the hyperplane a = -a . This sphere played an important role in the history of mathematics: in the tangent space to this sphere at its point representing an octave a , there is defined a complex structure transforming the differential d a of the octave a into the product ada , and this structure is nonintegrable since it is impossible to define com­plex coordinates on a sphere. This sphere was the first example of a space with nonintegrable complex structure (at present this structure is called the almost complex structure) . The study of spaces with almost complex stnic­ture started from the study of this sphere. Identifying antipodal points of this sphere, we get the space Sg6 • Elements of symmetry of this space are those planes of the space Sg6 which are cut in it by associative subfields of the field 0 . The latter subfields are isomorphic to the alternative field H . These planes are holomorphic relative to the almost complex structure of the space Sg6 •

We have already indicated that compact exceptional simple Lie groups in the classes F4 , E6 , E1 , and Es are the groups of motions of the elliptic

-2 ::::.2 ::::.2 ::::.2 planes OS , (0 © C)S , (0 © H)S , and (0 © O)S . The compact symmet-ric Riemannian spaces of types Fil , EIII , EVI , and EVIII coincide with these planes. Most of the remaining compact symmetric Riemannian spaces

1 96 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

whose groups of motions are these groups can be interpreted as manifolds of normal plane chains of different kinds of these planes.

Cartan considered noncompact symmetric Riemannian spaces to be of the same type as compact symmetric Riemannian spaces if the isotropy groups of these spaces were isomorphic. If m = 0 , the noncompact symmetric Riemannian spaces of types AIV , Bil , Dll , and CII coincide with proper

-n 2n 2n I -n- 1 domains of the spaces CS 1 , S 1 , S 1 - , and HS 1 ; if m =I- 0 , the non-com pact symmetric Riemannian spaces of types AIII , Bl , DI , and CII can be interpreted as the manifolds of m-dimensional elliptic planes of these spaces, the noncompact symmetric Riemannian spaces of types Al and All can be interpreted as the manifolds of imaginary quadrics of the space pn and imaginary Hermitian quadrics of the space HP(n- I J/2 , and the non­compact symmetric Riemannian spaces of types CI and Diii can be in­terpreted as the manifolds of imaginary quadrics of the spaces sy2n- I and HSyn- I . The noncompact symmetric Riemannian space of type GI can be interpreted as the manifold of holomorphic two-dimensional elliptic planes of the space Sy� ; the noncompact symmetric Riemannian spaces of types Fil , EIII , EVI , and EVIII coincide with proper domains of the hyperbolic

-2 .:::::.2 .:::::.2 .:::::.2 planes OS1 , (0 ® C)S1 , (0 ® H)S1 , and (0 ® O)S 1 ; most of the remain-ing noncompact symmetric Riemannian spaces whose groups of motions are noncompact simple Lie groups of these classes can be interpreted as man­ifolds of normal plane chains of different kinds of elliptic planes over the algebras 'O , O @ 'C , 'O ® C , O @ 'H , 'O ® H , and 10 ® 0 .

§6.10. The isotropy groups and orbits

The isotropy groups of the symmetric Riemannian spaces VN which Car­tan found in the paper [94] are subgroups of the groups O

N of rotations of

the spaces RN tangent to the spaces vN . Because of this, they act in the hyperplanes at infinity of the spaces RN which themselves are the spaces SN- I even if these groups are not transitive in these spaces. They transform certain surfaces of these spaces, the so-called local absolutes, into themselves. The isotropy groups are transitive in the spaces SN- I only in the cases when the space VN is a space of constant curvature, i .e . , in the symmetric spaces BDll .

In the case of the symmetric spaces VN of type BDI(N = (m + l ) · (n - m)) , whose models are the Grassmannians of m-dimensional planes of the spaces sn ' the isotropy groups are isomorphic to the direct products om+ I x on-m ' and the local absolutes are the Segreans �m . n-m- 1 ( 3 . 1 2 ) in

the space s<m+ l ) (n-m)- I . In the case of the symmetric space VN of type AIV , i .e . , the space CSn ,

the isotropy group is isomorphic to the direct product of the group of motions of the space csn- I and the group DI of motions of the line S 1

• In this

§6. 1 0. THE ISOTROPY GROUPS AND ORBITS 1 97 case, the isotropy group transforms into itself a paratactic congruence of straight lines of the space s2n- l which is isometric to the space csn- l (the group of motions of the line S1 is the group of shifts along the lines of the congruence) . In this case, the local absolute is the pair of imaginary conjugate (n - ! )-dimensional plane generators of the absolute of the space s2n- I which is an imaginary focal surface of this congruence.

In the case of the symmetric space VN of type AI (N = n(nt3l ) , whose model is the manifold of the normal space chains isometric to the space Sn , the isotropy group is isomorphic to the group On+ I , and the local absolute consists of the Grassmannian Grn , I and n vertices of the autopolar simplex

of the space sn- l . In the case of the symmetric spaces v4n-4 of type CII (/ = 1 ) , i .e . , the

space HSn- l , the isotropy group is isomorphic to the direct product of the -n- 1 group of motions of the space HS and the group A 1 = B1 = C1 (the group of automorphisms of the field H) . In this case, the isotropy group transforms into itself a paratactic congruence of three-dimensional planes of

the space s4n- s which is isometric to the space HSn- l . In this case, the local absolute is the imaginary Segrean 1:1 . 2n_ 3 (2 . 1 2) which is an imaginary

focal surface of this congruence and lies on the absolute of the space s4n-s . In the case of the symmetric spaces V 1 s , V32 , V64 , and V 1 28 of types

-2 ::::..2 ::::..2 Fii , EIII , EVI , and EVIII , i .e . , the planes OS , (0 ® C)S , (0 ® H)S ,

-2 and (0 ® O)S , the isotropy groups are isomorphic to the spinor group of the group 09 , to the direct product of the spinor group of the group 010 and the group D1 , to the direct product of the spinor group of the group 01 2 and the group A 1 = B1 = C1 , and to the spinor group of the group 016 , respectively. In these cases, the isotropy groups transform into themselves congruences of planes of the spaces S1 s ' S3 1 ' S63 ' and S1 27 which are iso­metric to the Hermitian lines over the same algebras. In these cases, the local absolutes are the imaginary Lipshitzeans .Qs , .Q6 , Q7 , and .Q8 (2 . 34) which are imaginary focal surfaces of these congruences and lie on the absolutes of the spaces S1 s , S3 1 , S63 , and S1 27 . Note that in the case of symmetric

d R. . v2n v.4n-4 v; l 6 v.32 v.64 d u l 28 .

pseu o- iemanman spaces n , 2n_2 , 8 , 1 6 , 32 , an Y 64 , isomet-ric to the Hermitian spaces and planes that can be obtained from the spaces and planes mentioned above by substitution of the algebras 'C , 'H , and '0 for the fields C , H , and 0 , the local absolutes are real pairs of planes, Seg­reans and Lipshitzeans (see the papers [RKoY] and [RB] of B. A. Rosenfeld, T. I . Yuchtina, T. A. Burtseva and others) .

Cartan gave the unitary equation of the local absolute only for irreducible symmetric Riemannian spaces whose isotropy group coincides with their holonomy group; he wrote this equation not in point but in line coordinates:

(6 .38 ) ij kl R

. . kip p = 0 , I] '

1 98 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

where R . . kt is the Riemann tensor of the symmetric space and /i are the I} ' Plucker coordinates of a straight line in the space sN- 1 -the hyperplane at infinity of the space RN tangent to the symmetric space vN ( /i are also the coordinates of a bivector defining a two-dimensional element of the space VN) .

For the simple Lie groups endowed with the Cartan metric, the role of the group of motions is played by the group of transformations g' = agb ( a , b , and g are elements of the given group) which is isomorphic to the direct product of the given group by itself, and the role of the isotropy group is played by the adjoint group, i .e . , the group of transformations g' = aga- 1

which is locally isomorphic to the given group.

§6.1 1 . Absolutes of symmetric spaces

In the paper [ 1 07] , for the symmetric Riemannian space of negative cur­vature of type AI whose model is the manifold of nondegenerate quadrics of the space Pn , Cartan introduced the notion of the absolute of the sym­metric space. Considering this space as the manifold of positive definite quadratic forms, Cartan wrote: "The absolute is formed by degenerate def­inite quadratic forms" and further he wrote that, for n = 2 , N = 5 , "the absolute is formed in a projective space ( aii ) by the part of a cubic manifold obtained by equating to zero the discriminant of the form (aii ) , i .e . , that part which corresponds to the conics decomposing into a pair of imaginary conju­gate straight lines. It contains also the Veronese surface corresponding to the quadratic forms that are perfect squares" [ 1 07 , p. 387] . The term "absolute" is undoubtedly explained by the fact that, in the case n = 1 , N = 2 , i .e . , in the case of pairs of imaginary conjugate points of the projective line P

1 , the

symmetric space is isometric to the hyperbolic plane S� and the manifold of pairs of coinciding points of this line is represented by the absolute of the plane S� . In the general case, we have N = n (n + 3)/2 , and the space is also

represented by a convex domain of the space PN whose point coordinates are the coefficients aii of equations of quadrics; in this case the absolute is an algebraic hypersurface of order N - n which represents the imaginary quadrics (the imaginary cones of second order with real point, line, and plane vertices) . This hypersurface contains the Veronesean ( 3 . 1 1 ) which represents the quadrics decomposing into a pair of coincident hyperplanes.

Similar absolutes can be defined for all symmetric Riemannian spaces with noncompact groups of motions. For the symmetric space of type All whose model is the manifold of nondegenerate Hermitian quadrics, the absolute represents the degenerate imaginary quadrics. For the symmetric spaces of types AIII , BDI , and CII representing the manifolds of m-dimensional

-n 2n 2n I -n- 1 planes of the hyperbolic spaces CS1 , S1 , S1 - , and HS1 , the absolutes represent the planes tangent to the absolute of the hyperbolic space and its

§6. 1 2 . GEOMETRY OF THE CARTAN SUBGROUPS 1 99

plane generators. Similar imaginary absolutes can be defined for symmetric Riemannian spaces with compact groups of motions.

Note that the Veronesean for the symmetric space Al and the similar sur­face /i = xixi for the symmetric space All , as well as submanifolds of the absolutes of the symmetric spaces Alli , BDI , and CII (the latter sub­manifolds represent the parabolic elements which are the plane generators of

the absolutes of the spaces cs; , S�n , S�n- I , and Hs;- i ) possess the prop­erty that the groups of collineations of the corresponding projective spaces preserving these submanifolds are isomorphic to the groups of motions of symmetric spaces.

§6.12. Geometry of the Cartan subgroups

In the paper The geometry of simple groups [ 1 03] ( 1 927) , Cartan studied geometric properties of the most important class of symmetric Riemannian spaces-the group spaces of compact Lie groups. The geometry of the Car­tan subgroups (which were called in [ 1 08] the "subgroups y") was studied in this paper in utmost detail . As are all subgroups, these subgroups are totally geodesic surfaces in Lie groups with their Cartan metric. However, unlike arbitrary subgroups, they have the property that every geodesic in a Lie group with its Cartan metric lies in one of those subgroups, and for a geodesic com­posed of regular elements such a Cartan subgroup is unique: it consists of all elements of the group commuting with elements of this geodesic. Since the Cartan subgroups are commutative, all their structure constants are equal to zero. Since the Cartan metric in compact simple Lie groups is Rieman­nian, the metric of the Cartan subgroups is locally Euclidean. Thus, these subgroups are compact spaces with the Euclidean metric; these spaces are the so-called Clifford forms of Euclidean spaces. (The simplest of such forms is the Clifford quadric in the space S3 , i .e . , the locus of points of the space S3 which are equidistant from a straight line of this space and the polar of this line . ) Cartan studied closed and non closed geodesics of the Cartan subgroups and showed that the shortest closed geodesics are determined by the characteristic equations of these groups. In particular, for the simple Lie groups of rank two, these geodesics are situated as shown in Figure 6.2 (next page) (i .e . , closed geodesics of these subgroups are directed along the vectors representing the roots indicated in Figures 2. 1 and 2.4) .

Further, Cartan considered the Weyl group of a given Lie group and pointed out that in his paper [82] he showed that in some cases (namely, for the groups An , Dn , and E6) , there exist substitutions of roots that do not belong to this group. Next, Cartan defined the "fundamental domain" (D) of the Weyl group bounded by those hyperplanes the reflections in which generate the Weyl group. He also showed that "each infinitesimal transforma­tion of the group y is homologic to one and only one transformation inside this domain " [ 1 03, p. 2 1 5] and that every internal point of the fundamental

200

a)

6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

b) c)

FIGURE 6 .2

d) e)

domain (D) is a set of "homologic infinitesimal transformations" depending on r - n parameters ( r is the dimension of the group and n is its rank) . (At present, this domain is called the Wey! chamber. ) Passing from "infinitesimal transformations" to finite elements, Cartan defined the "net R" consisting of the points with integer coordinates relative to the basis formed by the vectors representing the fundamental system of roots and the affine Weyl group as the group generated by the transformations of the Weyl group and the trans­lations preserving the net R . He showed that this group is also generated by reflections in hyperplanes. Next, Cartan defined the "fundamental polytope (P)" of this group (at present, this polytope is called the Wey! alcove) and proved that every finite element of the "group y" is represented at least by one point of this polytope. It follows from this that, in its Cartan metric, the Cartan subgroup of a compact simple Lie group is isometric to a polytope composed of a few polytopes (P) with the points of the boundary of this polytope being identified.

§6.13. The Cartan submanifolds of symmetric spaces

In the paper [ 1 07] , Cartan constructed a similar theory for symmetric Riemannian spaces. If such a space is represented by a totally geodesic sur­face in a Lie group with its Cartan metric, then the intersection of this totally geodesic surface with the Cartan subgroup is called the Cartan submanifold of the symmetric space. Since the Cartan submanifold is the intersection of two totally geodesic surfaces, this submanifold itself is a totally geodesic surface. From the properties of the Cartan subgroup, it follows that every geodesic of the symmetric Riemannian space lies in one of these submanifolds, and in the general case, a geodesic lies in a unique submanifold, and that the geometry of the Cartan submanifolds is locally Euclidean. The dimension of the Cartan submanifold is called the rank of the symmetric Riemannian space and is equal to the number of metric invariants of a pair of points of this space. Symmetric Riemannian spaces of rank one are the space Sn and the spaces v2n ' y4n ' and V 16 which are isometric to the spaces csn and

-n -2 HS and the plane OS . All geodesics of these spaces are closed and have the same length in each of these spaces.

§6. 1 4. ANTIPODAL MANIFOLDS OF SYMMETRIC SPACES 20 1

In the paper [ 1 07] , Cartan found the ranks, the Cartan submanifolds, and the form of geodesics for all irreducible symmetric Riemannian spaces. In particular, for the symmetric spaces BDI whose models are the Grassmanni­ans of m-dimensional planes of the space sn , the rank is equal to m + 1 if m < n - m - 1 . The stationary distances (the lengths of the common perpen­diculars) between m-dimensional planes can be taken as metric invariants of these spaces. In this case, the Cartan submanifolds are represented by the families of m-dimensional planes intersecting with m + 1 mutually polar straight lines, and geodesics are represented by one-parameter families of planes intersecting with the same m + 1 straight lines and cutting on them proportional segments. (If m = 1 , these families are the families of rulings of ruled helicoids of the space S3 .) These fa�ilies are called m-helicoids. Note that the ranks of symmetric spaces Ell , EVI , and EVIII are equal to 2, 4, and 8 respectively. This corresponds to the fact that the straight -2 -2 -2 lines of the planes (0 ® C)S , (0 ® H)S , and (0 ® O)S are interpreted by the Grassmannians Gr9 , I ' Gr1 1 , 3 , and Gr1 5 , 7 • The term "local abso­lute", which was not used by Cartan, is introduced by analogy with his term "absolute of a symmetric space".

§6.14. Antipodal manifolds of symmetric spaces

In the note On geodesics of spaces of simple groups [96] ( 1 927) , Cartan considered geodesics of simply connected compact simple Lie groups with the Cartan metric, gave a classification of these geodesics, and defined the antipodal points-the points that can be joined by an infinite set of closed geodesics. He also defined the antipodal manifold of a point-the manifold of antipodal points of this point. (For the spinor group of the group 03 , which

is isometric to a hypersphere of the space R4 , the antipodal manifold consists of one point. )

Cartan developed the theory of antipodal manifolds in the paper The geom­etry of simple groups [ 1 03] . He showed that, for a simply connected compact simple Lie group of rank n , each of its points possesses n antipodal mani­folds. This can be explained by the fact that if one characterizes geodesics by the "angular parameters" (i .e . , the coordinates of the tangent vector relative to that basis in the Cartan subgroup through the geodesic which is defined by the root system) , then for geodesics directed towards antipodal points, all nonzero angular parameters are equal to one another. Cartan noted that these geodesics are directed along the edges or the facets of the polyhedron (P) .

Antipodal manifolds can be also defined in symmetric spaces. The number of such manifolds for a point of a symmetric space is equal to the rank of this space. In particular, for the symmetric spaces of rank one, there is only one antipodal manifold: in the case of a hypersphere of the space R11 it is the antipodal point; in the case of the spaces csn and HSn and the plane

202 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

OS2 ' the antipodal manifolds are the polar hyperplanes and the polars of the

points. In these cases the polar images are isometric to the sphere of two, four, and eight dimensions, respectively, and the geodesics joining antipodal points belong to these spheres.

Antipodal manifolds can also be defined in nonsimply connected groups and symmetric spaces. However, in these cases, the geodesics joining the points with points of their antipodal manifolds can be unique, and the an­tipodal manifolds are defined as the manifolds consisting of the midpoints of the closed geodesics of a certain type. For the symmetric spaces represented by the Grassmannians Grn m of m-dimensional planes of the space Sn , the antipodal manifolds are 'represented by the manifolds of m-dimensional planes that lie in the polar plane of the given m-dimensional plane or per­pendicular to this polar plane.

§6.15. Orthogonal systems of functions on symmetric spaces

We saw in Chapter 2 that Cartan's works on classification and theory of linear representations of simple Lie groups were significantly developed in Weyl's paper [Wey3] of 1 925 . Continuing this work, Weyl, in the paper Completeness of primitive representations of closed continuous groups [PeW] ( 1 927) written jointly with F. Peter (the words "closed continuous groups" in the title mean compact Lie groups), showed that all irreducible compact Lie groups (they can always be represented by real orthogonal or complex unitary matrices) can be obtained by means of orthogonal systems of functions given on the group. A representation of the space L2 (G) of functions f(g) with integrable square of their modulus given in a compact Lie group G (this space is an infinite-dimensional Hilbert space) as the direct sum of finite­dimensional spaces of orthogonal or unitary representations of this group was presented in the paper [PeW]. This representation is a generalization of the classical expansion of a periodic function in a Fourier series: assigning such a function with a period 2n is equivalent to assigning a function f( t) in the compact group T = R/(2nZ) ( R and Z are the additive groups of the field R and of the ring Z of integers) , and the expansion of such a function in a Fourier series:

(6 .39) f(t) = �o + L: ak cos kt + L bk sin kt k k

is equivalent to the representation of the Hilbert space L 2 ( T) as the direct sum of a straight line and two-dimensional planes of "vector diagrams" whose vectors represent the harmonics ak cos kt + bk sin kt . Each of these planes can be considered as a complex plane of representation of the group T by the complex numbers eikt

. These representations are called the characters of the group T . Similar characters of any commutative group form a group themselves whose identity element is the representation of all elements of the group by the number 1 (the "unit character") .

§6. 1 5. ORTHOGONAL SYSTEMS OF FUNCTIONS ON SYMMETRIC SPACES 203

For a simple commutative group, the group of characters is isomorphic to the group itself. The group of characters of an infinite discrete commutative group is a compact commutative group, and, conversely, the group of char­acters of any locally compact commutative group (all noncom pact Lie groups are of this type) is a commutative group of the same type. In particular, the group of characters of the group T is isomorphic to the group Z , and con­versely. The group of characters of the group R is isomorphic to the group R itself.

Departing from the paper [PeW] of Peter and Weyl, in the paper On the determination of a complete orthogonal system of fanctions on a closed sym­metric Riemannian space [ 1 1 7] ( 1 929) , Cartan constructed a similar theory for functions defined on a compact symmetric space E that forms a Hilbert space L2 (E) . The spaces L2 (G) and L2 (E) are infinite analogues of the complex Hermitian Euclidean space CRn

where the role of vectors is played by the complex-valued functions f(x) , the role of the inner product of vec­tors is played by the integral

(6 .40) (f , g) = k f(x)g(x) dV ,

where d V is the volume element of the space or the group, and the role of the square of the modulus of a vector is played by the integral

(6.4 1 ) If we denote by ax the point of the symmetric space E which is obtained

as a result of application of an element a of the group of motions of this space to a point x of this space, the transformation

(6.42) T0f(x) = f(ax) is a linear transformation in the space L2 (E) . Since the group considered by Cartan is compact, a linear transformation of this group arising in this manner can be split into finite-dimensional transformations of type:

(6 .43) J;(ax) = af Jj(x) . The sequence of functions J; (x) , J; (x) , . . . , /p (x) defining such trans­

formations was called by Cartan the fundamental sequence of functions. Car­tan showed that the functions of these fundamental sequences can be chosen to be orthogonal; i .e . , they satisfy the conditions (J; , fj) = oiJ . Thus, one can construct an orthogonal sytem of functions on a symmetric space. If we expand an arbitrary function f(x) of the space L 2 (£) with respect to these functions:

p (6 .44) f(x ) = L a;J; (x) ,

i= I

204 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

then the coefficients a; of this expansion possess the property that the series of their squares is convergent to the integral:

(6 .45) p 2 1 2 2 L a; = lf(x) I dV = lfl i= I E

playing the role of the square of the magnitude of a vector of this functional space.

Thus, the functions J;(x) of a fundamental sequence are analogues of the functions cos kt and sin kt determining the Fourier series and forming a system of orthogonal functions on the group T .

Cartan also defined "zonal functions" on symmetric spaces, which are ana­logues of the spherical functions of the space R3

•

§6.16. Unitary representations of noncompact Lie groups

The consideration of systems of functions on homogeneous spaces whose fundamental groups are noncompact real Lie groups led to the theory of unitary representations of noncompact simple and quasi-simple Lie groups, i .e . , representations of these groups by unitary operators in Hilbert spaces L2 (E) of functions on homogeneous spaces whose fundamental groups are given groups. A unitary operator in the space L 2 (E) with inner product (f , g) is a linear operator U of this space which satisfies the condition ( U f , U g) = (f , g) . These operators are infinite analogues of matrices of the group cun .

The first work on such representations was the paper On unitary represen­tations of the inhomogeneous Lorentz group [Wig] ( 1 939) by the physicist Wigner. In this paper, Wigner constructed unitary representations of a qua­sisimple group of motions of the pseudo-Euclidean space R� , i .e . , the space­time of special relativity. Such representations are characterized by one real and one integer parameters. Wigner connected this fact with the phenomenon that the elementary particles in physics also characterized by one continuous parameter-the mass and one discrete parameter-the spin. Wigner con­cluded from this that these representations are important in physics.

In 1 943 Gel' fand and Dmitry A. Raikov ( 1 905- 1 980) published the paper Irreducible unitary representations of locally bicompact groups [GeR] . (From the 1 920s to the 1 940s, in Soviet mathematical literature the word "bicom­pact" had the meaning "compact", and the word "compact" had a different meaning.)

In 1 94 7, in the paper Irreducible unitary representations of the Lorentz group [Bag] by Valentine Bargmann ( 1 908- 1 99 1 ) and in the similarly titled papers [GeN] by Gel'fand and Naimark and [Har l ] by Harish-Chandra, uni­tary representations of the noncompact simple Lorentz group-the group of rotations of the space R�-were found. Note that the paper [Har l ] , the Ph.D. thesis of Harish-Chandra, was written by him when he was a young physicist

§6. 1 6. UNITARY REPRESENTATIONS OF NONCOMPACT LIE GROUPS 205

studying particle physics under the supervision of Paul Adrian Maurice Dirac ( 1 902- 1 984) .

Since the Lorentz group O� is locally isomorphic to the group CS L2 (which is its spinor group) , Gel'fand and Naimark posed the general prob­lem of studying infinite-dimensional unitary representations of all classical complex simple Lie groups considered as noncompact real simple groups (the Satake graphs of such groups consist of two copies of the Dynkin graph of the corresponding complex group, provided that the corresponding dots of these graphs are joined by two-sided arrows) . This theory was presented by Gel' fand and Naimark in their monograph Unitary representations of the classical groups [GeN2] ( 1 950) . Later Gel'fand and Graev solved the con­siderably more difficult similar problem for arbitrary noncompact groups of infinite series. These results were briefly presented in their note Unitary rep­resentations of the real simple groups [GeG] ( 1 952) and, in more detail, in the Graev paper [Grv] ( 1 958 ) under the same title.

Harish-Chandra, who became a famous mathematician, independently constructed the theory of these representations in a series of papers concluded by the paper Representations of semi-simple Lie groups [Har2] ( 1 95 1 - 1 956) . In the papers of Gel' fand and his co-workers as well as in the papers of Harish-Chandra, the Hilbert spaces L 2 (E) of functions in various parabolic spaces whose fundamental groups are the noncompact groups under consid­eration were investigated. For construction of the principal series of unitary representations of a noncompact group G , they considered the Hilbert space L 2 ( G / B) of functions given in the parabolic space G / B defined by the Borel subgroup B . For construction of the "degenerate series'', they considered the Hilbert space L 2 ( G / P) of functions given in the parabolic space G / P de­fined by an arbitrary parabolic subgroup P . (The subgroup B is defined by all positive roots of the group G while an arbitrary parabolic subgroup P is defined by a part of these roots or even by one of them.) For the principal series of unitary representations of the Lorentz group, which is locally iso­morphic to the group of motions of the hyperbolic space S� whose parabolic images are the points of the absolute, the space G / B can be identified with an oval quadric in P3

and with the extended complex plane. Thus, in this case, one can take the space of functions f ( z) of a complex variable as the space L2 (G/B) . (In this case, the group G is the group of linear-fractional transformations of the complex plane. )

The roots and the weights of finite-dimensional representations of semisim­ple Lie groups are characters of the Cartan subgroups of these groups. For the infinite-dimensional representations of these groups, the role of characters is played by certain distributions which are linear functionals of the Hilbert space. These distributions themselves, as characters of a locally compact commutative group, compose locally compact commutative groups which are the direct products of a certain number of infinite discrete groups Z and a

206 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

certain number of the groups R . The elements of the groups Z determine the integer parameters of unitary representations, and the elements of the groups R determine the real parameters of these representations.

For a complex simple Lie group of rank n , the maximal compact subgroup is a compact group of the same class and rank, and the Cartan subgroup is the direct product of n groups T and n groups R . Thus, its group of characters is the direct product of n groups Z and n groups R , and the unitary representations of this group are determined by n integers and n real parameters.

For an arbitrary real noncom pact group, there are a few nonisomorphic Cartan subgroups each of which determines a series of unitary representations of the group. If the Cartan subgroup is the direct product of / groups T and m groups R , then its group of characters is the direct product of / groups Z and m groups R , and the unitary representations of this group are determined by / integers and m real parameters.

In the cases when the Cartan subalgebra is compact, i .e . , it is the direct product of n groups T , its group of characters is the direct product of n groups Z , and there are only the discrete series of unitary representations depending on n integers (the group of characters of the Cartan subgroup of a compact simple Lie group of rank n is isomorphic to the direct product of n groups Z ) .

The theory of unitary representations o f quasi-simple Lie groups i s sim­ilar to the theory of unitary representations of noncompact semisimple Lie groups.

The analogues of the classical spherical functions in symmetric spaces, whose study was initiated by Cartan in the paper [ 1 1 7), are closely connected with the unitary representations of compact and noncompact Lie groups.

We saw earlier that the functions cos kt and sin kt , where k is an integer, determine representations of the group T isomorphic to the group 02 of

rotations of the Euclidean plane R2 • Similarly, the hyperbolic functions cosh pt and sinh pt where p is a real parameter determine representations of the group O� of rotations of the pseudo-Euclidean plane R� . The spherical

functions of the space R3 are the Legendre polynomials Pn (cos 0) where (} is the latitude of a point on a sphere and n is an integer. The spherical functions of the space R� are the Legendre polynomials PP (cos 0) where (} is the analogue of the latitude and p is a real parameter. The analogues of the spherical functions in the plane R2

are the Bessel fanctions J0 (pr) where r is the first polar coordinate of a point in the plane and p is a real parameter. At present, similar theories are constructed for many symmetric spaces with compact and noncompact semisimple and quasisimple fundamental groups (see the paper [Gel l ] by Gel'fand) .

In the book Special functions and the theory of group representations [Vil] ( 1 965 ) , Vilenkin showed that all classical special functions can be considered

§6. 1 7 . THE TOPOWGY OF SYMMETRIC SPACES 207

as elements of matrices of infinite order determined by linear operators of representations of simple and quasisimple Lie groups.

Note that the hypergeometric functions are determined by a representation of the group O� of rotations of the pseudo-Euclidean space R� , the Hanke/­MacDonald functions are determined by a representation of the group of motions of the plane R� , the Gegenbauer polynomials are determined by a representation of the group On of rotations of the space Rn , and the Hermite polynomials are determined by a representation of the group of motions of the space Rn .

§6.17. The topology of symmetric spaces

While in his thesis and in the papers of the 1 890s and the early 1 900s, fol­lowing Lie, Cartan restricted himself to considering only the neighborhoods of the identity element of a Lie group, in his papers of the 1 920s and the 1 930s, he was interested in the topological structure of Lie groups in the large as well as in the topological structure of the compact symmetric spaces closely connected with the simple Lie groups.

In the paper The geometry of simple groups [ 1 03] ( 1 927) , Cartan investi­gated in detail simply connected and nonsimply connected compact simple Lie groups and, for the latter groups, he found the "connection groups"-the homotopy Poincare groups. Cartan showed that these "connection groups" are finite commutative groups isomorphic to the centers of simply connected simple groups of the same type and that these groups are isomorphic to the quotient groups of the weight lattice of the group which Cartan called the "net R" by the root lattice which is its subgroup. (We defined these lat­tices in Chapter 2 . ) Cartan called the orders of the "connection groups" the connection indices of simple Lie groups. (In Chapter 2, we saw that the connection indices, whose name is explained by their relation with the con­nection groups, are equal to the determinants of the Cartan matrices of the corresponding Lie groups. )

In the paper The theory of.finite continuous groups and Analysis situs [ 1 28] ( 1 930) , Cartan showed that a noncompact simple Lie group, and, in particu­lar, a complex simple Lie group, is homeomorphic to the topological product of its maximal compact subgroup and a Euclidean space.

In Chapter 4, we already mentioned the paper On the integral invariants of certain closed homogeneous spaces and topological properties of these spaces [ 1 1 8] ( 1 929) . In this paper, Cartan showed that, if a compact homogeneous space is a symmetric Riemannian space (in particular, the group manifold of a compact simple Lie group with its Cartan metric) , any integral invariant of this space is an integral of an exact differential form.

In the paper On the Betti numbers of spaces of closed groups [ 1 1 1 ] ( 1 928) , Cartan applied the theory of exterior differential forms for computing the Betti groups of simply connected compact ("closed") simple groups in the classes An and Bn , i .e . , the groups CSUn+ I and the spinor group of the

208 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

group 02n+ I . Here, for the first time, Cartan introduced the term Poincare polynomial for the polynomial L:; p/ whose coefficients are the Betti num­bers P; and showed that the Poincare polynomials of simply connected com­pact groups An and Bn are equal to

n (6 .46) II u2i+ l + 1 ) i= l and

n (6 .47) IIu4i- l + 1 ) ' i= l respectively.

The method used by Cartan in this paper was developed by Pontryagin in the paper On the Betti numbers of Lie groups [Pon l ] ( 1 935 ) , where he computed the Poincare polynomials of simply connected compact simple groups of all four infinite series: for the groups An and Bn Pontryagin arrived at the same expressions (6 .46) and (6 .47) as Cartan; for the group Cn Pontryagin found the same expression (6 .47) ; and for the group Dn , the spinor group of the group 02n , he got the polynomial

n- 1 (6 .48) ( t2n- I + l ) II (t4i- l + l ) . i= l

Later the Pontryagin method was also applied to compact exceptional sim­ple Lie groups. In the paper The Betti numbers of exceptional simple Lie groups [BoC] ( 1 955 ) , A. Borel and Chevalley calculated the Betti numbers for all simply connected compact simple Lie groups in the exceptional classes. The Poincare polynomials of simply connected compact simple Lie groups can be expressed by one formula:

n

(6 .49) II ( t2a;+ I + l ) , i= l where the integers a; are the exponents (2 . 34) and (2 .35 ) of simple Lie groups (formulas (6 .46) , (6 .47) , and (6 .48) are particular cases of formula (6 .49) ) . The coincidence of the numbers a; in formula (6 .49) with exponents (2 .34) and (2 . 35 ) was explained by A. J. Coleman in the paper The Betti numbers of the simple Lie groups [Col l ] ( 1 958 ) .

When Cartan found the Poincare polynomials (6 .46) and (6 .47) , he in fact found the exponents for the simple groups An and Bn .

In the paper [ 1 1 8] , mentioned earlier, Cartan found the Poincare polyno­mial of the space CSn (the symmetric space AIV) in the form:

(6 . 50) t2n+2 _ l t2n + t2n-2 + . . . + 12 + l = . t2 - 1

§6. 1 8 . HOMOLOGICAL ALGEBRA 209

He also found the "Clifford form" of the space CS2n+ t which is obtained by identification of its points X(x2; , x2i+ t ) with the points 'X(' x2; , 'x2i+ t ) obtained from the points X by symmetry (6 .34) . This symmetry defines, .

h C-S2n+ t . . . h H-Sn m t e space , a paratact1c congruence, 1sometnc to t e space . In addition, the lines of this congruence are isometric to two-dimensional spheres, which under the Cartan identification become elliptic planes S2 . These facts imply that the quotient of the Poincare polynomial found by Cartan by the Poincare polynomial t2 in a plane S2 is equal to

( 6 .5 1 ) t4n+4 _ l t4n + t4n-4 + . . . + t4 + l = . t4 - 1

This quotient is the Poincare polynomial of the space HSn (a particular case of the symmetric space CII) . Formula (6 . 5 1 ) was used by A. Borel in his paper [Borl ] . In this paper, Borel also found the Poincare polynomial of the

-2 plane OS (the symmetric space Fii) in the form:

t24 1 (6 . 52) t

1 6 + t8 + 1 = __

-

_

t8 - 1 '

analogous to polynomials (6 . 50) and (6 . 5 1 ) . Another generalization of the polynomial (6 .50) was found by Ehresmann

in his thesis [Eh l ] ; it is the Poincare polynomial of the Grassmannian CGrn , m ( i .e . , the symmetric space AIII) and has the form:

( t2n+ I _ l ) ( t2n _ l ) , . . ( t2n-2m- I _ I ) ( t2m+ I _ l ) ( t2m _ l ) , . . ( t2 _ 1 )

(6 . 53 )

In the paper On the topological properties of complex quadrics [ 1 37) ( 1 932) , Cartan considered the topological properties of another symmetric Rieman­nian space-the (2n )-dimensional real space represented by the quadric E; (X

i ) 2 = 0 of the complex projective space cpn+ I . In the space cpn+I ' Cartan introduced the metric of the Hermitian elliptic space CSn+ I . As a result, the quadric in the space cpn+ 1 becomes the Riemannian manifold v2n • Cartan noted that the space v2n is a symmetric space. It is not difficult to see that this space admits the representation in the form of the Grassman­nian Grn+ t , t

(the absolute of the space sn+ t is an imaginary quadric in the

space cpn+ t , and to any straight line of the space sn+ t there corresponds a pair of imaginary conjugate points of this quadric at which the straight line intersects the quadric) . In this paper, Cartan also computed the Betti numbers of this symmetric space.

§6.18. Homological algebra

Cartan's paper [ 1 1 8) , where the topological problem of computing the Betti numbers in a compact Lie group was reduced to the purely algebraic problem

2 1 0 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

in the corresponding Lie algebra, gave birth to a new algebraic discipline whose terminology was adopted from the homology and cohomology theories. For this reason, this discipline got the name "homological algebra". Following the homology theory of Lie algebras constructed by Cartan, the cohomology theory of associative algebras was developed by Gerhard Paul Hochschild (b. 1 9 1 5) in the paper On the cohomology groups of an associative algebra [Hoc] ( 1 945 ) . The cohomology groups of arbitrary groups was constructed by Samuel Eilenberg (b. 1 9 1 3) and Saunders MacLane (b. 1 909) in the paper Cohomology theory in abstract groups [EM] ( 1 94 7) , and the cohomology groups of Lie algebras over an arbitrary commutative ring were defined by Chevalley and Eilenberg in the paper Cohomology theory of Lie groups and Lie algebras [ChE] ( 1 948) .

The cohomology groups of abstract groups can be defined as follows. Let a multiplicative group G and an additive group A (as the latter group the additive group Z of integers or the finite cyclic group Zm often is taken) be given, where the elements of the group G operate on the left of the elements A. of the group A and x(A. 1 + A.2 ) = xA. 1 + xA.2 , x2 (x1 A.) = (x2x1 )A. , 1 · A. = A. . Define an n-dimensional cochain of G over A as a homogeneous function F(x0 , x1 , • • • , xn ) (F (xx0 , xx1 , • • • , xxn ) = xF(x0 , x1 , • • • , xn ) ) with its values in A . Since F1 + F2 is again an n-dimensional cochain, these cochains form an additive group Cn (G , A) . The coboundary of this cochain is an (n + 1 )-dimensional cochain

t5F(x0 , x1 , • • • , xn ) = �:)- l /F(x0 , x1 , • • • , X;_ 1 , X;+ i • • • • , xn ) . i

The coboundaries have the following properties: '5 (F1 + F2) = t5F1 + t5F2 , t5'5F = 0 . The n-dimensional cochains satisfying the condition t5 F = 0 are called the n-dimensional cocycles. The co boundaries are partic­ular cases of cocycles. The n-dimensional cocycles and the n-dimensional coboundaries form commutative groups which are denoted by zn ( G , A) and Bn ( G , A) , respectively. The second of these groups is a subgroup of the first one. The quotient group of the first group by the second one is called the r-dimensional cohomology group Hn ( G , A) of G over A .

If, in this definition, we change the group G to an associative algebra A or to a Lie algebra G , we obtain the cohomology groups Hn (A , A) and Hn (G , A) of these algebras. In both cases, since the functions F(x0 , x1 , • • • , xn ) are homogeneous, they can be written in the form of

multilinear forms: A; ; . . . ; x�0 x:• · · · x�· where A; ; . . . ; is a tensor. In the 0 I n 0 I n

case of Lie algebras, since the operation of commutation in them is skew-symmetric, this tensor is skew-symmetric in all indices and defines an exterior form considered by Cartan.

However, if the Cartan theory was related only to the complex Lie groups, this theory is related to the Lie algebras of arbitrary Lie groups.

All aspects of this theory were presented as a single theory in the book Homological algebra [CaE] ( 1 956) by Henri Cartan and S. Eilenberg.

CHAPTER 7

Generalized Spaces

§7. 1 . "Affine connections" and Weyl's "metric manifolds"

We have already pointed out the importance of Einstein's discovery of gen­eral relativity for the development of Riemannian and pseudo-Riemannian geometry. According to the general theory of relativity, the space-time and the gravitational field of matter are described by means of a four-dimensional pseudo-Riemannian manifold Yi_4 whose curvature is connected with the den­sity of matter. The problem of construction of a unified field theory posed by Einstein has played an exceptional role for the creation of further gener­alizations of the notion of space. Einstein departed from the idea that all physics could be reduced to mechanics and electrodynamics, that the gravi­tational field of matter is already taken into account in the geometry of the space Yi_4 , and that for a description of a unified theory of a physical field it is necessary to construct a more general spatial scheme which would de­scribe not only the gravitational field but also the electromagnetic field. In Einstein's special and general relativity, the electromagnetic field, which was defined in classical electrodynamics by the tension vector E of the electric field and the tension vector H of the magnetic field, is characterized by a single skew-symmetric tensor Fii , i , j = 1 , 2 , 3 , 4 , of the electromagnetic field whose coordinates are connected with the vectors E and H and the speed of light by the relations

F2 1 = -F 1 2 = H3 ,

F 1 3 = -F3 1 = Hi ' F32 = -F23 = HI .

The first attempt to construct a geometry more general than the Rieman­nian or pseudo-Riemannian geometry, which would describe both the grav­itational field and the electromagnetic field, was made by Weyl in his paper Pure infinitesimal geometry [Wey l ] ( 1 9 1 8) . In this paper, Weyl distinguished three types of manifolds: "manifold-place" (situs - Mannigfaltigkeit) , which he identified with the "empty world'', i .e . , with the world without matter; the "affinely connected manifold" (affin zusammenhiingende Mannigfaltigkeit) , which he understood to be a manifold with a parallel displacement of vec­tors and which he called the "world with a gravitational field"; and the "metric

2 1 1

2 1 2 7. GENERALIZED SPACES

manifold'', which he also called the "ether" and which he understood to be the "world with the gravitational field and the electromagnetic fields". Ac­tually, Weyl's "affinely connected manifold" coincides with the space-time of the Einstein theory of general relativity, i .e . , with a pseudo-Riemannian manifold. Weyl's "metric manifold" is a generalization of the Riemannian and pseudo-Riemannian manifolds. If in the spaces vn and �n

a parallel displacement of vectors induces an isometric mapping of the tangent spaces Tx ( Vn ) and Tx ( �n ) onto tangent spaces in infinitesimally near points, in the Weyl "metric manifolds", the mapping of the tangent spaces (which as was the case for the spaces vn

and �n are the Euclidean and pseudo-Euclidean spaces Rn and R7 ) onto the same kind of tangent spaces in infinitesimally near points takes place. However, in the Weyl "metric manifolds'', these mappings are not isometric mappings anymore-they are similarity mappings (i .e . , transformations preserving the angles between vectors and multiplying the linear dimensions by real numbers) .

Weyl expressed the same ideas in his book Space-Time-Matter [Wey2] . The first edition of this book was published in 1 9 1 8 . This edition was followed by a series of new editions (the fifth was in 1 923) . In 1 922, the book was translated into French and later into English. The book became very popular throughout the mathematical world. (In this book the well-known point­vector axiomatics of n-dimensional affine and Euclidean spaces En and Rn was presented. )

Although Einstein was occupied with the construction of a unified field theory for a few decades, neither he nor other physicists were able to con­struct such a theory. On the contrary, as physics was developing, new forms of interactions of matter (the "weak interaction" and the "strong interac­tion") were discovered, and they were not reduced to either mechanical or electromagnetic interactions. Nevertheless, for multidimensional differential geometry, the impetus that it received from physicists who were trying to construct a unified field theory was very helpful.

§7 .2. Spaces with affine connection

The term "affinely connected manifold" introduced by Weyl soon received a wider meaning than that given by Weyl. Namely, it was applied to such spaces An whose tangent spaces Tx (An ) are affine spaces En

and for which the mappings of tangent spaces in infinitesimally near points are defined and are affine mappings of these spaces. Such spaces were defined by Schouten who arrived at them while generalizing the parallel displacement of vectors in a Riemannian manifold which he discovered simultaneously with Levi­Civita. This was the reason that he called the mapping of tangent spaces En of the space An a displacement (Ubertragung) . Schouten defined these spaces in his paper On different kinds of displacements which can be taken as

§7.2. SPACES WITH AFFINE CONNECTION 2 1 3

a basis of differential geometry [Sco l ] ( 1 922) . The Schouten theory was pre­sented in detail in his book Ricci calculus [Sco2] ( 1 924) (as was mentioned above, Ricci calculus is one of the names of the tensor calculus) . Schouten defined the space An as a manifold with coordinates at each point of which functions r�k = rii are given. Under coordinate transformations, these functions are transformed according to the same rule as the Christoff el sym­bols of a Riemannian manifold; however, in general, these functions cannot be expressed in terms of the metric tensor gii by formulas (6 . 1 0) . Thus, in

the general case, in the space An , it is impossible to define the lengths of lines and the angles between lines, but it is possible to define geodesics that are integral curves of differential equations (6 .9 ) . Furthermore, the parame­ter s for which these differential equations preserve their form is defined up to an "affine transformation" s --+ as + b . This is the reason this parameter is called the affine parameter of geodesics.

With each point x of the space An there is associated a tangent space Tx (An ) which is an affine space En similar to the tangent spaces Tx ( Vn ) and Tx ( J-'t) . As was the case for the spaces Tx ( Vn ) and Tx ( V/ ) , in the

space An , the contravariant and covariant vectors ai and a; and the tensors

i · · · i n Ti ,' · ··i: are defined. A parallel displacement in the space A is defined by

means of the functions r�k according to the same formulas ( 6 . 1 9 ) and ( 6 .20)

as for the spaces Tx ( Vn ) and Tx C V/ ) . A somewhat more general definition of a space with an affine connection

was introduced by Cartan in the paper On manifolds with an affine connec­tion and the general relativity theory consisting of three parts [66] ( 1 923) , [69 ] ( 1 924) , and [80] ( 1 925 ) . The book [209a] contains English translations of the papers [66] , [69] , and [80] , and [208] contains Russian translations of Cartan's papers on the spaces with affine, projective, and conformal connec­tions. Explaining the term "affine connection", Cartan wrote in the foreword to the paper [66] that "the expression 'affine connection' is borrowed from H. Weyl (here Cartan made reference to the Weyl book Space-Time-Matter) , although it will be used here in a more general context" [209a, p. 25] .

We will denote the Cartan spaces with an affine connection by the same symbol An which was used for the Schouten space. As Schouten did, Cartan defined an affine transformation of a tangent space Tx (An ) onto a tangent space Tx1 (An ) at an infinitesimally near point, but the Cartan mapping was more general than the Schouten mapping. To define his mapping, Cartan considered a frame {x , e; } in the space Tx (An ) and a frame {x' , e; } in the space Tx, (An ) and defined the principal part of the mapping by the relations:

(O (x) = x + dx , (O (e; ) = e; + de; where dx = wie; and de; = w�ei . These

formulas precisely coincide with derivational formulas ( 5 . 5 ) of a frame in the affine space En ; however, here they are not completely integrable as they were in the space En since the forms w

i and w� in these formulas satisfy

2 1 4 7. GENERALIZED SPACES

the structure equations of a space with an affine connection:

da/ - al /\ a/ + !S; wi /\ wk - k 2 jk ' d j k i 1 Ri k I (J)i = W; /\ Wk + 2 i , kl(J) /\ (J) '

(7 . 1 )

which are more complicated than formulas (6 .2 1 ) . In formulas (7 . 1 ) , sJk is

the torsion tensor and R�kl is the curvature tensor of the space An . Deriva­tional formulas ( 5 . 5 ) admit integration only along a curve x = x ( t) belonging to the space An , and their one-dimensional integrals define "developments" of a space An with an affine connection onto an affine space En .

If the vectors e; form the natural frame in An , i .e . , if e; = a /ax; and

w; = dxi , the forms w{ defining an affine connection on An are expressed

in the form w� = r�kdxk . The coefficients r�k are called the coefficients of affine connection. Unlike the similar Schouten coefficients, they are not assumed to be symmetric: r�k = r�i . Thus, the tensor

(7 .2) ; r; r; sjk = jk - kj arises. This tensor is called the torsion tensor. In addition, in the spaces under consideration, there is also the curvature tensor:

(7 . 3 ) . ari.k ar;., -Ji . -Ji .

R�k' = _1_, - -t + i jkr�, - i j,r�k , ax ax defined by a formula similar to formula (6 . 1 2) for calculating the Riemann tensor of the spaces Tx ( vn ) and Tx ( V/ ) .

If sJk = 0 , the space An is called a torsion-free space, and if R�kl = 0 ,

it is called a curvature-free space. Since in the spaces vn and �n , we have

r�k = r�j , these spaces can be considered as torsion-free spaces with an

affine connection, and the spaces En and E? are both a torsion-free and curvature-free space.

As we found for the spaces vn and �n , the result of a parallel displace-

ment of a vector a = {a; } along a closed contour defined in a neighborhood of a point x of the space An by the differentials dxi and ox; of the co­ordinates differs from the original vector a = {a; } by an increment which is equal to the vector with coordinates R�k1ai dxk ox1

• Up to infinitesimals

of higher order, the vector with coordinates sJkdxi oxk in the tangent space

Tx (An ) is equal to the path between the end of the segment ox; , displaced

in a parallel way from the point x along the segment dx; , and the end of the segment dxi , displaced in a parallel way from the point x along the segment ox; . This gives a geometric meaning to the torsion and curvature tensors of a space An with an affine connection.

§7.3. SPACES WITH A EUCLIDEAN, ISOTROPIC, AND METRIC CONNECTION 2 1 5

§7.3. Spaces with a Euclidean, isotropic, and metric connection

Cartan defined spaces with an affine connection in Chapter II of his paper On manifolds with an affine connection and the generalized relativity the­ory [66] . In Chapter I of this paper entitled Dynamics of continuous media and the notion of affine connection of the space-time, Cartan analyzed the space-time of the theory of general relativity considering it as a space with a "pseudo-Euclidean connection" whose tangent spaces are pseudo-Euclidean spaces R� . In the same chapter, he also considered the space-time of the classical Galilei-Newton mechanics "from the point of view of the Einstein theory", i .e . , he considered this space-time as a space with an "isotropic con­nection" whose tangent spaces are isotropic spaces /4 • Cartan did not intro­duce this notion, but he wrote the transformations of spatial coordinates and time under a passage from one inertial coordinate system of classical me­chanics to another, and these transformations coincide with the coordinate transformations of the space /4 •

After he defined the spaces An with an affine connection in Chapter II of the paper [66], in Chapter III Cartan introduced the spaces with a metric connection and the spaces with a Euclidean connection.

By analogy with Weyl's term "metric manifolds", Cartan used the term spaces with an affine connection for those spaces An whose tangent spaces Tx (An ) are Euclidean spaces Rn in which the group of similarities acts. In

this case, the forms w ; are equal to each other (Cartan denoted these forms

by w), and the forms w{ , i =I j , are connected by relation (6 .3 1 ) . The struc­ture equations of the spaces with a metric connection have the same form (7 . 1 ) as for the general spaces An . The mappings of the tangent spaces Rn

of this space onto the tangent spaces in infinitesimally near points are simi­larity transformations. Cartan called a space with a Euclidean connection a particular case of a space with a metric connection for which the form w is identically zero, i .e . , the case when the mapping of the tangent spaces of this space onto the tangent spaces in infinitesimally near points are isome­tries. The Riemannian manifolds vn are a particular case of spaces with a Euclidean connection for which the torsion tensor S�k is identically equal to zero. The "metric manifolds" of Weyl (at present called spaces with a Weyl connection) are distinguished by the same condition among the spaces with a metric connection. Spaces with a pseudo-Euclidean connection whose particular cases are the pseudo-Riemannian spaces �n can be defined in the same way.

Cartan's memoir On manifolds with an affine connection and the general relativity theory [66], [69], and [80] was preceded by a series of notes de­voted to the attempts to construct a unified field theory. In the note On a generalization of the notion of Riemannian curvature [58] ( 1 922) , Cartan de­fined a space with a Euclidean connection with torsion, and in the paper On generalized spaces and relativity theory [59] ( 1 922) , he defined a space with

2 1 6 7 . GENERALIZED SPACES

a metric connection and suggested characterizing the space-time as a space whose tangent spaces are pseudo-Euclidean spaces R� .

In the paper Recent generalizations of the notion of space [7 1 ] ( 1 924 ) , Cartan gave a simple example o f a space with a Euclidean connection: a sphere on which the parallel displacement of tangent vectors is defined in such a way that an initial vector and its parallel displacement compose equal angles with the meridians passing through their initial points. In this case, geodesics are loxodroms (rhumb lines) . Since this parallel displacement does not depend on the path of displacement, it is an absolute parallelism. Cartan gave the same example in his letter of May 8, 1 929, to Einstein in connection with the fact that Einstein, who, independently of Cartan, arrived at the notion of absolute parallelism in 1 928 (he called it "Fernparallelismus") , tried to apply this notion in his unified field theory. This letter started an intensive correspondence between Cartan and Einstein concerning absolute parallelism. This correspondence was published with English translation in the book [2 1 0) ( 1 979) .

§7 .4. Affine connections in Lie groups and symmetric spaces with an affine connection

Although the notions of a space with an affine connection were initially created by Schouten and Cartan independently, in 1 926 two joint papers of both geometers were published: On the geometry of the group-manifold of simple and semi-simple groups [9 1 ] and On Riemannian geometries admitting an absolute parallelism [92) . Both papers were concerned with Riemannian geometry, but in both cases, one way or another, the geometry of a space with an affine connection was discussed. In the first of these notes, the authors considered three affine connections associated with any Lie group. In this note these connections were called the (+)-connection, ( - )-connection, and ( o )-connection. The authors indicated that for simple and semisimple Lie groups, the latter connection is determined by the Riemannian or pseudo­Riemannian Cartan metric of this group. As we noted in Chapter 3, in the second note the authors considered the absolute parallelisms (3 . 1 5 ) and (3 . 1 6) in the elliptic space S7 and similar absolute parallelisms in an arbitrary simple compact Lie group with the Riemannian Cartan metric.

The theory of three affine connections was presented in more detail by Cartan in his paper The geometry of transformation groups [ 1 0 1 ) ( 1 927) . In this paper Cartan called these connections the "absolute parallelisms of the first and the second type". The parallel displacement of vectors in the first two connections is determined by mappings of neighborhoods of a point a onto a neighborhood of a point b by means of the translations x --+ ( ba - i )x and x --+ x(ba- 1 ) . Since these mappings do not depend on the path joining the elements a and b , the vector obtained as the result of a translation in both connections along a closed contour coincides with the initial vector.

§7.4. AFFINE CONNECTIONS, LIE GROUPS AND SYMMETRIC SPACES 2 1 7

This proves that these connections are curvature-free, i .e . , they define an absolute parallelism. At the same time, each of these connections possesses a torsion, and for the first of these connections the components of the torsion tensor sJk coincide with the structure constants cJk of the Lie group and for the second one they differ from them in the factor - 1 . Both of these connections are invariant under transformations of the group.

The third connection defined by Cartan on a Lie group, which is also invariant under transformations of the group, is torsion-free. It is determined by its geodesics and their affine parameter: the role of geodesics through the identity element of the group is played by one-parameter subgroups, and the role of their affine parameter is played by their canonical parameter t . For the latter parameter the product of elements x(t 1 ) and x ( t2 ) of a subgroup coincides with the element x(t1 + t2 ) , and this parameter is defined up to a real factor. The role of geodesics not passing through the identity element of the group is played by cosets of one-parameter subgroups, and the affine parameter on these cosets is defined up to an affine transformation t -+ at + b . The curvature tensor of this space is expressed in terms of the structure constants cJk of the group .bY the same formula (6 .27) which defines the Riemann tensors of the Riemannian or pseudo-Riemannian Cartan metric in the simple or semisimple Lie groups. This shows that in these groups the Cartan torsion-free affine connection is defined by the invariant Riemannian or pseudo-Riemannian metric (6 .26) of these groups.

The torsion-free affine connection in Lie groups defined by Cartan in the paper The geometry of transformation groups is a particular case of a connec­tion of a symmetric space with an affine connection. In spaces with such an affine connection, the mapping along geodesic lines preserves the affine con­nection, i .e . , this mapping transfers geodesics into geodesics and preserves their affine parameter. As he did for symmetric Riemannian spaces, Cartan showed that condition V hR�kl = 0 analogous to condition (6 .25) is neces­sary and sufficient for a torsion-free space with an affine connection to be a symmetric space. These spaces can be realized in Lie groups in the form of totally geodesic surfaces a0a passing through the identity element of the group and generated by the reflections in the points of these spaces. Here a is a reflection in an arbitrary point of a space with an affine connection and a0 is a reflection in a certain fixed point of this space. As in the case of symmetric Riemannian and pseudo-Riemannian spaces, the Lie algebra of the Lie group generated by reflections in points of a symmetric space with an affine connection admits the "Cartan decomposition" (2 .43) where the subal­gebra H is the Lie algebra of the stationary subgroup of a point of this space (the isotropy group) , and the subspace E can be considered as the tangent space to a totally geodesic surf ace in the group in which the symmetric space with an affine connection is realized or, equivalently, as the tangent space to a symmetric space with an affine connection.

The curvature tensor of a symmetric space with an affine connection is

2 1 8 7. GENERALIZED SPACES

expressed in terms of the structure constants c:v and c;w of the group gen­erated by reflection in points by the same formula ( 6.29) which defines the Riemann tensor in symmetric Riemannian and pseudo-Riemannian spaces.

As in the case of tangent spaces to symmetric Riemannian spaces, tangent spaces to symmetric spaces with an affine connection are closed with respect to the operation [[X , Y] , Z] , and therefore they are triple Lie systems.

As in the case of symmetric Riemannian spaces, in symmetric spaces with an affine connection, the Loos quasigroup (see [Loo]) is defined which as­sociates to any two points x and y of this space the point z that is the reflection of the point x in the point y along geodesics of this affine con­nection.

The Loos quasigroups, which were defined by Loos in symmetric Rie­mannian spaces and symmetric spaces with an affine connection, are smooth quasigroups. The idea of Loos was further developed by A. J. Leger in the paper Generalized symmetric Riemannian spaces [Leg] ( 1 9 5 7) and in the pa­per Affine and Riemannian s-spaces [LeO] ( 1 968) written jointly with Morio Obata (b. 1 926) . The Leger s-spaces which generalize symmetric Rieman­nian spaces and symmetric spaces with an affine connection were also studied by Fedenko in the paper Regu.lar spaces with symmetries [Fe2] ( 1 973) and in the book Spaces with symmetries [Fe3] ( 1 977) . Application of quasigroups and loops to symmetric spaces and their generalizations was first suggested by Mishiko K.ikkawa in the paper On local loops on affine manifolds [Kik] ( 1 964) and was extended to generalizations of spaces with an affine con­nection which differ from the spaces with an affine connection in that they have fewer requirements on the differentiability of functions under consid­eration by Sabinin in the paper Methods of the non-associative algebra in the differential geometry [Sab) ( 1 98 1 ) . In this paper, which is a supplement to Sabinin's translation of the book Foundations of differential geometry [KoN] ( 1 963- 1 969) by Shoshichi Kobayashi (b. 1 932) and Nomizu, these general­izations are called "geoodular structures" (Sabinin used the word "odulus" for a nonassociative analogue of the modulus) .

The theory of symmetric spaces with an affine connection was developed further by Rashevskii in the paper Symmetric spaces with an affine connection with torsion [Ra3] ( 1 959) . In this paper Rashevskii considered spaces with an affine connection in which not only the curvature tensor is covariantly con­stant ('V hR� , kt = 0) but the nonvanishing torsion tensor is also covariantly

constant ('V hsJk = 0) . Rashevskii showed that the fundamental group G of this space and its stationary subgroup H possess the property that the Lie algebra G of the group G admits decomposition (2 .43) into the Lie algebra H of the subgroup H and the subspace E for which, as was the case for a symmetric Cartan space, the commutator of vectors h and e from the spaces H and E belongs to the subspace E , but the commutator of vectors e1 and e2 from the subspace E does not belong to the subalgebra H . At

§1.5 . SPACES WITH A PROJECTIVE CONNECTION 2 1 9

present, these spaces are called the reductive spaces. This term was suggested by Nomizu in the earlier mentioned paper Invariant affine connections on ho­mogeneous spaces [Norn] ( 1 954) . In the same paper, applying formula (6 .27) , Nomizu calculated the curvature tensor of reductive spaces.

§7.5. Spaces with a projective connection

In the paper On manifolds with a projective connection (70] ( 1 924) , Cartan, by analogy with spaces An with an affine connection, defined spaces IIn with a projective connection as n-dimensional manifolds in such a way that each point x of the manifold is associated with a "tangent" space Tx (IIn ) , and the latter space is the space Pn . Moreover, to each pair of infinitesimally close points x and x' of the space IIn there corresponds a projective mapping of the space Tx (IIn ) , and this mapping is an analogue of a parallel displacement of vectors of the space An . The derivational formulas of the projective frames in the spaces IIn have the same form (5 . 8 ) as for the space Pn , but the structure equations of the spaces IIn differ from equations ( 5 . 1 3 ) and have a more complicated form:

(7 .4)

h , i , j = O , 1 , . . . , n , k , l = l , . . . , n , where the tensor A{ kt (an analogue of the curvature tensor of the space An ) is called the tensor of projective curvature. For the case in which the tensor A{ kt vanishes, Cartan called a space IIn a holonomic space. At present, such spaces are called projectively flat spaces. Cartan denoted the exterior quadratic forms A{ kt OJ� A OJ� by 0{ . As in the case of the space Pn , an infinitesimal displacement of frames in the space IIn with a projective connection is determined by the forms OJ{ • Cartan denoted the forms OJ� and Q� by OJ; and Qi , respectively. By analogy with the forms Q; of the

space An , Cartan called the forms gi the torsion forms, and if gi = 0 , he called a space with a projective connection a torsion-free space.

As in the case of the spaces An , in the spaces IIn , geodesics can be defined as curves that preserve their direction under an infinitesimal displacement along the line. However, in contrast to the spaces An , it is impossible to define an affine parameter for geodesics in the spaces IIn .

The role which Riemannian and pseudo-Riemannian manifolds play for the spaces An is played by normal spaces with a projective connection for the spaces IIn . These are torsion-free spaces with a projective connection for which AZ , ii 'I 0 and which are completely determined by the system of

their geodesics. The normal spaces IIn are connected with the problem of geodesic mapping of Riemannian manifolds, i .e . , a mapping of a Riemannian manifold vn onto another Riemannian manifold under which geodesics are

220 7. GENERALIZED SPACES

transferred into geodesics. For the solution of this problem, normal pro­jective connections in Riemannian manifolds defined by their geodesics are constructed. The equivalence of these connections is equivalent to the exis­tence of a mapping of one of these spaces onto the other.

§7 .6. Spaces with a conformal connection

In the paper Spaces with a conformal connection [68] ( 1 923) , also by anal­ogy with the spaces An , Cartan defined the spaces Kn with a conformal connection, i .e. , n-dimensional manifolds each point x of which is associ­ated with a "tangent" space Tx (Kn ) , and the latter space is the space en . Moreover, to each pair of infinitesimally near points x and x' of the space Kn there corresponds a conformal mapping of the space Tx (Kn ) , and this mapping is an analogue of a parallel displacement of vectors in the space An . The derivational formulas of the conformal frames in the spaces Kn have the same form (5 .8 ) as in the space en , and the forms w{ are connected by the same equations ( 5 . 1 0) , but the structure equations of the spaces en

differ from equations ( 5 . 1 3) and have a more complicated form:

(7 . 5 )

h , i , j = 0 , 1 , . . . , n + 1 , k , l = 1 , . . . , n . The tensor A{ kl (an analogue of the curvature tensor of the space An ) is

called the tensor of conformal curvature. In case the tensor A{ kl vanishes, Cartan called a space en a curvature-free space or a holonomic space. At present, such spaces are called conformally flat spaces.

As was the case for the space en , an infinitesimal displacement of frames in the space Kn with a conformal connection is determined by the forms w{ . Cartan denoted the exterior quadratic forms A{ klw� /\ w� by n{ . Cartan

denoted the forms w� and Q� by w;

and Q; , respectively. He also called

the forms gi the torsion forms, and if gi = 0 , he called the space en a torsion-free space. Depending on coincidence of the forms w

; and w{ of

two spaces Kn , there are four types of isomorphisms of these spaces. Among spaces Kn , the normal spaces are also defined: they are torsion­

free spaces with a conformal connection for which AZ , ii = 0 . They play a role similar to that of Riemannian manifolds among spaces with a metric connection. The theory of normal spaces Kn can be applied to the theory of conformal mappings of Riemannian manifolds, i .e . , mappings that preserve the angles between curves in these spaces. The linear elements ds at the corresponding points of such spaces differ by a factor. Cartan considered three-dimensional normal spaces with a conformal connection in detail and also constructed the theory of manifolds embedded into the spaces Kn .

The spaces with a conformal connection appeared under the name "gener­alized conformal spaces" as far back as 1 922 in Cartan's note On generalized

§7.7. SPACES WITH A SYMPLECTIC CONNECTION 22 1

conformal spaces and the optical Universe [60] . It is well known that the conformal transformations of the space-time also play an important role in the theory of special relativity since the Maxwell equations are invariant not only with respect to the Lorentz transformations (the rotations of the space R1) and the Poincare transformations but also with respect to the confor­

mal transformations of the space ci ; the latter space is obtained from the space R1 by adding the point at infinity and the ideal points. In the note [ 60] , Cartan tried to construct a similar conformal theory for the theory of general relativity. Probably, this attempt was the principal stimulus for the construction of the theory of spaces with a conformal connection by Car­tan. Later on, by analogy with spaces with a conformal connection, Cartan constructed the theory of spaces with a projective connection (his paper [68] on spaces with a conformal connection was written one year earlier than the paper [70] on spaces with a projective connection) . Cartan called the four­dimensional space with a conformal connection (actually it was the space Ki with a pseudo-conformal connection but not K4) the "optical Universe" since the rays of light are propagated along isotropic lines of this space (in Ki they are real while in K4

they are imaginary) .

§7. 7. Spaces with a symplectic connection

The geometry of spaces I,vn with a symplectic connection, which is of­ten called simply "symplectic geometry", has important applications in the theory of differential equations and theoretical mechanics. With each point x of such a space there is associated the tangent space Tx (I,vn ) at whose

hyperplane at infinity the geometry of the space sy2n- I is defined. This is

equivalent to assigning a skew-symmetric tensor gii = -gii or an exterior

differential form w = giidxi /\ dxi at each point of the manifold I,vn . The most important of these spaces are those in which the form w is closed, i .e . , the exterior differential d w of this form is equal to zero. The usage of this space in mechanics is based on the fact that a mechanical system given by generalized coordinates qi and generalized momenta P; can be considered

as a space with the closed exterior differential form w = dq; /\ dp; ; n­dimensional submanifolds of this space whose tangent n-planes cut, on the hyperplanes at infinity of the tangent spaces Tx (I,vn ) , (n - 1 )-dimensional

null-planes of the space sy2n- I are called Lagrangian submanifolds of these

spaces. This name is explained by the fact that a six-dimensional manifold of this type was studied by Lagrange in his Memoir on the theory of varia­tions of elements of planets [Lag2] ( 1 809) . In this paper Lagrange considered perturbations of motions of planets around the sun under the influence of exterior forces. Lagrange took as his point of departure the fact that planets travel around the sun along ellipses with the sun at one of the foci of these

222 7. GENERALIZED SPACES

ellipses, and a perturbed motion of a planet possesses the same property. Be­cause of this, Lagrange described a possible motion of a planet by means of a plane passing through the sun, the major axis of an ellipse and a location of a planet on this ellipse. He considered six "elements of planets" defined in this way as coordinates of a six-dimensional space in which he transferred to the coordinates q 1 , q 2

, q3 , p 1 , p2 , and p3 • In his book Geometric the­ory of partial differential equations [Ra2] ( 1 94 7) , Rashevskii applied spaces with a symplectic connection to the investigation of a wider class of differ­ential equations than the equations of mechanics. He called the spaces with a symplectic connection the "spaces of a linear form of even class". Victor P. Maslov (b. 1 930) widely used the geometry of spaces with a symplectic connection in his book Theory of perturbations and asymptotic methods [Mas] ( 1 965) where the term "Lagrangian submanifolds" was introduced.

§7 .8. The relativity theory and the unified field theory

We have already indicated the exceptional role of Einstein's general rel­ativity in the development of the geometry of Riemannian and pseudo­Riemannian manifolds in the attempts to construct the unified field theory and in the development of the theory of spaces with an affine connection and other generalized Cartan's spaces. Thus, it is natural that a series of Cartan's works was devoted to the problems of relativity theory and unified field theory.

Cartan became interested in the problems of general relativity even before he started to study the theory of generalized spaces. As far back as 1 922, he wrote the paper On the equations of gravitation of Einstein [56], in which he investigated the equations of general relativity by means of his theory of Pfaffian equations in involution. Cartan found a system of Pfaffian equa­tions which is equivalent to Einstein's system of equations, calculated the characters of this system, proved that the system is in involution and its gen­eral solution depends on n (n - 1 ) /2 functions of n real variables (i .e . , in the case of four-dimensional space-time, it depends on six functions of four variables) . In the paper On manifolds with an affine connection and general relativity theory [66] , [69] , and [80] ( 1 923- 1 925) , Cartan first considered the space-time of general relativity and classical Galilei-Newton mechanics. Af­ter an exposition of the geometry of spaces An , spaces with a metric and Euclidean connection, and the theory of curves and surfaces in these spaces, in Chapter V, "The gravitational Universe of Newton and the gravitational Universe of Einstein'', Cartan studied different spaces with an affine con­nection consistent with properties of "the Universe of Newton" and "the Universe of Einstein". He studied these not only from the point of view of description of mechanics of continuous media in these two "Universes", but also from the point of view of description of electromagnetic fields in them.

§7.9. FINSLER SPACES 223

In the paper A historic note on the notion of absolute parallelism [ 1 24] ( 1 930) , Cartan noted that the idea of absolute parallelism, which he intro­duced in one of his papers in 1 922, was rediscovered by Einstein in 1 928 who decided to use it in the foundation of a unitary theory of gravitational and electromagnetic fields. Einstein also defined the tensor piJ of an elec­tromagnetic field in terms of the torsion tensor of this space. The Cartan book Absolute parallelism and unitary field theory [ 1 30] ( 1 9 3 1 ) was devoted to the unitary theory of gravitational and electromagnetic fields based on the notion of absolute parallelism.

In the paper The unitary (field) theory of Einstein-Mayer [ 1 43a] (which was written in 1 934 but published only in Cartan's <Euvres Completes [207] after his death) , Cartan gave a "geometrically intuitive" presentation of unitary field theory constructed by Einstein and Mayer in 1 93 1 . In this presentation, space-time is a totally geodesic surface in a five-dimensional space with a Euclidean connection.

§7.9. Finsler spaces

Another generalization of the Riemannian manifold is the Fins/er space in which a linear element or, using Cartan's words, the distance between two infinitesimally close points x(x; ) and x' (x;

+ dx; ) on a manifold X is defined by the formula

(7 .6 ) ds = F(x1, • • • , xn ; dx

1 , • • • , dxn ) ,

where F is a positive function which is first degree homogeneous with re­spect to dx

1, • • • , dxn . This notion arose in connection with a geometric

interpretation of the variational calculus problem for the integral:

(7 .7 ) J = f12 F(xi, .X

i)dt 1 ,

and was first considered by Paul Finsler ( 1 894- 1 970) in his thesis On curves and surfaces in generalized spaces [Fis] ( 1 9 1 8) . The extremals of this integral are geodesics of the Finsler space.

The simplest space of this kind was defined by Hermann Minkowski ( 1 864- 1 909) in his (posthumously published) work Theory of convex bod­ies, especially the foundation of the notion of a surface [Min] ( 1 9 1 1 ) . The space considered by Minkowski is an affine space En in which a metric is introduced not with the help of a hypersphere (as in the Euclidean space Rn ) but with the help of a closed centrally symmetric convex "gauge surface". Minkowski showed that if one defines the distance between the points X and Y of this space as the ratio of the segment XY to the parallel segment OP enclosed between the center 0 of the gauge surface and the point P of this surf ace, then the triangle inequality X Y + Y Z ;::: X Z holds in this space. The Finsler space is locally the Minkowski space since in each of its tangent spaces Tx (Xn ) , by means of the function F entering under the integral sign

224 7. GENERALIZED SPACES

in equation (7 .7 ) , a gauge surface is defined by the formula F(xi , <!; ) = 1 , where <!; are the coordinates of the tangent vector <! of the space Tx (Xn ) . The Finster geometry was further developed in the paper Generalization of Riemannian line-element [Sy] ( 1 925 ) by John Lighton Synge (b. 1 897) and the papers On parallel displacement in spaces with commonly defined distances [Bew l ] ( 1 926) , On two-dimensional generalized metric spaces [Bew2] ( 1 925) by Ludwig Berwald ( 1 88 3-?) , and other works. In 1 934, in his lecture On the Fins/er and related spaces [Ber3] at the Congress of Mathematicians of Slavic Countries, Berwald replaced the vague term "general metric spaces" by the term "Finsler spaces" commonly used at present. In the paper On the affine foundation of the metric of one variational problem [Win] ( 1 930) , Artur Winternitz ( 1 89 3-?) gave the definition of the Finsler space as a space with a connection whose tangent spaces are Minkowski spaces.

In the book Fins/er spaces [ 1 42] ( 1 934) and in his lecture [ 1 52) under the same title at the International Conference on Tensor Differential Geometry in Moscow, U.S.S.R. , also in 1 934, Cartan developed a new approach for studying Finsler spaces. He indicated that the theory of these spaces can be connected with general problems of equivalence. Such problems arise dur­ing the study of many objects in differential geometry. For example, if we construct the Riemannian geometry, we encounter the problem of finding out whether two differential forms with the same number of variables can be transformed into one another by a change of variables. Each of these differ­ential forms is the metric form of a point space with a Riemannian metric, and the equivalence of two differential forms is reduced to the geometric applicability of these two Riemannian manifolds. As Cartan noted, for the Finster space, the notion of a point space was insufficient. For this space, we are forced to consider spaces of linear elements with a Euclidean connec­tion. A linear element of xn consists of a point x(x; ) of this manifold and a vector x(x;) of the tangent space Tx (Xn ) of this manifold. This space

will be defined if one assigns an expression giidx; dx1 to the square of a

linear element in it, where now gii = K;/xk , xk ) , and the expression of the

absolute differential D<!; of the vector <! = {e; (xk , xk ) } has the form:

(7 .8 )

The condition D<!; = 0 defines the parallel displacement of a vector in the Finsler space. The problem is: among all spaces of linear elements with a Euclidean connection, determine the space that is uniquely defined by the function F (x ; , x;) assigning the distance between two infinitesimally close points in the Finster space. To do this, we set

(7 .9 ) 1 a2 F2 (xk ' xk ) gij = 2 a.x;a.x1 ,

§7 . 1 0. METRIC SPACES BASED ON THE NOTION OF AREA 225

where Cijk = gihC� . Then oxkc;k = 0 and Cijk = Cjik . Under this condition, the parallel displacement preserves the length lc! I of a displaced vector e ' and the square of this length is 1 <! 1 2 = gije

ic;j . Further, Cartan extended the tensor calculus to the Finsler geometry and

studied submanifolds embedded in a Finsler space.

§7.10. Metric spaces based on the notion of area

In the book Metric spaces based on the notion of area [ 1 40] ( 1 933 ) , Cartan introduced another generalization of the notion of a Riemannian manifold; particularly, the basic notion is that the area of a surface given by the equation z = f(x , y) in a three-dimensional space is expressed by the equation:

(7 . 1 0) da = F (x , y , z , :� , �; ) dx dy ,

where the function F depends on coordinates x , y , z of a point and on the quantities p = g� and q = g; . These quantities p and q deter­mine the position of the tangent plane to the surface z = f(x , y) at the point P(x , y , z ) . The surfaces giving the extremum to the integral II du = II F dx dy play the role of geodesics in this geometry.

For construction of such a geometry in an n-dimensional manifold xn , Cartan considered the set of "support elements" consisting of a point x of the manifold xn and an (n - 1 )-dimensional subspace u of the tangent space Tx (Xn ) . If a coordinate system is chosen in the manifold xn , then

the point x is defined by coordinates xi and the subspace u is defined by homogeneous coordinates ui . Cartan defined the square of the distance between two infinitesimally close points x and x' by means of the quadratic form ds2 = gijdxidxj as in the case of the Riemannian geometry, but now the coefficients gij of this form depend not only on the coordinates

xi of the point x but also on the coordinates ui of the subspace u of the tangent space. Next, Cartan defined the absolute differential di; of the support element c; = <!(x , u) by the formula:

(7J 1 )

The quantities C�j and r�j determine a Euclidean connection in the space of support elements.

After this, Cartan showed how to construct a Euclidean connection in the space of support elements in such a way that this connection would be invariantly related to the surface element d <J indicated above and to a more general hypersurface element on an n-dimensional manifold.

The construction of these Euclidean connections was further applied to the solution of the equivalence problem for multiple integrals of the form

226 7. GENERALIZED SPACES

ff F(x , y , z , p , q) dx dy and similar integrals on an n-dimensional mani­fold. A necessary and sufficient condition for two such integrals to be equiv­alent is that the spaces with a Euclidean connection associated with these integrals must be geometrically equivalent; i .e . , there exists a correspondence between these spaces (and therefore a correspondence between their support elements) such that the metric and the Euclidean connection of the first space is transformed by this correspondence into the metric and the Euclidean con­nection of the second space.

Cartan's book [ 1 40] inspired many works devoted to the geometry of mul­tiple integrals. In particular, we mention here the paper Metric spaces of n dimensions based on the notion of area of m-dimensional surfaces [Au] ( 1 9 5 1 ) of Maya V. Aussem (Vasil' eva) (b. 1 926) , in which the author studied the geometry of an m-dimensional integral

(7 . 1 2) J · · · J F(xk , p�) dx 1 /\ dx2

• • • /\ dxm

....__..., m

over an m-dimensional surface x0 = ](xi ) where p� = 8x0 /a xi and the

paper The geometry of the integral f F(x0 , xn , x; , x;P , . . . ) dx 1 /\ dx2 • • •

/\ dxn- I [Ev] ( 1 958 ) by Leonid E. Evtushik (b. 1 93 1 ) , in which the author also considered the geometry of an integral of type (7 . 1 2) but with the func­tion F depending not only on the coordinates xi of the point x and the first order derivatives 8x0 /a xi but also on the derivatives of higher order up to some order p . This forced the author to reconstruct the analytic appa­ratus that had been used previously for studying similar problems; instead of the classical tensorial methods he applied the invariant apparatus of exterior differential calculus, also originated by Cartan.

§7.1 1 . Generalized spaces over algebras

Analogues of Riemannian manifolds and other generalized spaces have also been constructed over commutative algebras-first, over the field C of complex numbers and over the algebras 'C and ° C of split complex and dual numbers. The most important among these spaces is the Hermitian space first defined by P. A. Shirokov in 1 925 in the same paper [Sh 1 ] in which he defined the symmetric spaces. This space was also defined by Schouten in the paper On unitary geometry [Sco3] ( 1 929) .

The points of Hermitian spaces are defined by complex coordinates xi .

The distance ds is defined between the points x(x; ) and x' (x ; + dx; ) of this space, and the square of this distance is given by the formula:

(7 1 3) d 2 d id _j -. s = gij x x ' gij = gji " Thus, an n-dimensional Hermitian space is isometric to a real Rieman­

nian manifold V2n in which an operator J is given, and this operator has

§7. 1 1 . GENERALIZED SPACES OVER ALGEBRAS 227 the property J2 = - 1 and is covariantly constant with respect to an affine connection determined by the metric of the space. With the Hermitian form (7 . 1 3) there is associated the exterior quadratic form

(7 . 1 4)

defining a symplectic connection in the space under consideration. If this form is closed, i .e . , d(J) = 0 , the Hermitian space is called Kiihlerian.

It was named after Kahler who first considered such spaces in his paper On a remarkable Hermitian metric [Kah l ] ( 1 932) . Hermitian spaces are a par­ticular case of spaces with an affine connection.

If at each point of the Riemannian manifold v2n or a differentiable man­ifold x

2n the operator J with the property J2 = - 1 is given, but it is impossible to introduce complex coordinates xi in the space, we say that the space is endowed with an almost complex structure (or a nonintegrable com­plex structure) . As we discussed earlier, historically the first example of an almost complex structure was the six-dimensional sphere which is the inter­section of the hypersphere l o: I = 1 and the hyperplane o: = -a of the algebra 0 of octaves (where the geometry of the space R8 is realized) . The opera­tor J of complex structure considered at each point of this sphere transfers the differential do: into the product o:do: . (We saw in Chapter 3 that on this sphere a transitive subgroup of the group of its rotations is acting and that this subgroup is isomorphic to a compact simple Lie group in the class G2 .) The almost complex structure on this sphere was first discovered by A. Frohlicher in the paper On the differential geometry of complex structures [Fro] ( 1 955 ) .

In complex and almost complex spaces i t i s possible to separate the holo­morphic manifolds whose tangent spaces are invariant under the operator of a complex structure, the antiholomorphic (or "completely real") manifolds whose tangent spaces are transformed into the planes totally orthogonal to them under the operator of a complex structure, and the CR-submanifolds whose tangent spaces are the direct sums of the tangent spaces to holomorphic and antiholomorphic submanifolds. In particular, in the spaces CRn and csn ' the holomorphic submanifolds are complex straight lines and planes of these spaces, and the antiholomorphic submanifolds are their normal space chains. Note that the spaces V2n which are isometric to the spaces CSn are Riemannian manifolds of variable sectional curvature given by formula (6 .3 1 ) and taking on values from 1 /r2 to 4/r2 • But the sectional curva­ture of this space in holomorphic two-dimensional directions is equal to the constant value 4/r2 , which explains the name Hermitian spaces of constant holomorphic sectional curvature for the spaces csn .

If in the definition of complex and almost complex spaces we substitute split complex numbers or dual numbers for complex numbers, we obtain split complex and almost split complex spaces or dual and almost dual spaces,

228 7. GENERALIZED SPACES

respectively. Since the algebra 'C of split complex numbers is isomorphic to the direct sum R E9 R of two fields R , an n-dimensional space over the algebra 'C can be represented in the form of the Cartesian product of two real spaces xn

. The split complex and almost split complex spaces are often called the space-products. The dual and almost dual spaces are also called the contact spaces and the almost contact spaces. Note that the six­dimensional sphere, which is the intersection of the hypersphere !a l = 1 and the hyperplane a = - a in the algebra 'O of split octaves (with the geometry of the space R!) , also forms an almost complex space, and the six-dimensional sphere of imaginary radius, which is the intersection of the hypersphere la l 2 = - 1 and the hyperplane a = -a in the same algebra with the same geometry, forms an almost split complex space.

The most important results on the geometry of generalized complex spaces are given in the second volume of the earlier mentioned monograph Founda­tions of differential geometry [KoN] ( 1 969) by Kobayashi and Nomizu, and the results on the geometry of generalized spaces over more general algebras are given in the book Spaces over algebras [VSS] ( 1 985 ) by Vladimir V. Vish­nevskii (b. 1 929) , Alexander P. Shirokov (b. 1 926) , and V. V. Shurygin.

§7.12. The equivalence problem and G-structures

We have already mentioned the equivalence problem while discussing Car­tan's papers on the theory of Finsler spaces. In fact, this problem is con­nected with all generalized Cartan spaces, and Cartan became interested in this problem as far back as 1 902, long before he started to develop the theory of generalized spaces.

In the general case, the equivalence problem is formulated as follows: let, on the one hand, a system of n linearly independent Pfaffian forms w1 , w2 , • • • , wn with respect to independent variables x 1 , x2 , • • • , xn and

m independent functions y 1 , y2 , . . . , ym of these variables be given, and, on the other hand, let a system of n linearly independent Pfaffian forms QI , Q2 , . . . , Qn With respect to independent variables X 1 , X2 , . . . , Xn

and m independent functions Y 1 , Y2 , • • • , ym of these variables be given. It is required to find out whether there exists a change of variables that sends h f . I 2 m · h f · y I y2 ym d I t e unct10ns y , y , . . . , y mto t e unctions , , . . . , an a -

lows the forms Q1 , Q2 , • • • , Qn to be Obtained from the forms W1 , W2 , • • • , wn by means of a linear substitution from some linear group r , where the coefficients of finite transformations of this group can depend on the func-

• I 2 m t10ns y , y , . . . , y .

In 1 902 Cartan devoted his note On the equivalence of differential systems [ 1 9] to this problem. He also considered this problem in his note On the integration of certain systems of differential equations [ 40] ( 1 9 1 4) , and in the papers On the absolute equivalence of certain systems of differential equations and on certain families of curves [ 42] ( 1 9 1 4 ) , On an equivalence problem and

§7. 1 2. THE EQUIVALENCE PROBLEM AND G -STRUCTURES 229

the theory of generalized metric spaces [ 1 26] ( 1 930) , and The problems of equivalence [ 1 6 1 a] ( 1 937) . The papers The subgroups of continuous groups of transformations [26] ( 1 908) , The Pfaffian systems with five variables and partial differential equations of second order [ 30] ( 1 9 1 0) , and On the pseudo­conformal geometry of hypersurfaces of the space of two complex variables [ 1 36 , 1 36a] ( 1 932) were also mostly devoted to the equivalence problem. In these papers Cartan solved this problem for various groups r .

The equivalence problem is closely connected not only with generalized spaces but also with more general fiber spaces and with G-structures on smooth manifolds.

The spaces that are closest to generalized spaces are fiber spaces whose bases are differentiable manifolds xn , and whose fibers are the sets of all frames {x , e; } in the tangent spaces Tx (Xn ) of the manifold xn which are transformed to one another by transformations of a subgroup G of the group GLn . The subgroup G is called the "structural group" of a fiber space. At present, the fiber spaces are also called G-structures of first order. If we substitute the spaces T;k) (Xn ) of the differentials of kth order for the tangent spaces Tx (Xn ) in the above definition, we get the definition of a G-structure of kth order.

An example of G-structures of first order is the Riemannian manifold vn . It is defined on a manifold xn by means of a positive definite quadratic form ds2 = gudxi dx1 • This form separates the subset of orthonormal frames in the frame manifold of the tangent space Tx (Xn ) , and in this subset this

form is reduced to the form ds2 = L ; ( w i ) 2 and defines the group G = on of orthogonal transformations mapping the set of orthonormal frames into itself. Thus, the Riemannian manifold vn is a G-structure of first order with the structural group G = On . Similarly, a pseudo-Riemannian manifold �n is a G-structure of first order with the structural group G = O� of pseudo­orthonormal transformations of index l .

If the fibers can be identified with a certain group G , or more precisely, if the group G operates (on the right) on the space in such a way that G is simply transitive on these fibers, the fibration is called principal. A connection · in the principal fiber spaces plays an important role in the theory of G­structures. Let Xn be the base of a fibration with fibers Fx = G of dimension r forming the principal fibration xn+r of dimension n + r . Further, let a point y belong to the fiber Fx , and let, in the tangent space Ty (xn+r ) , an n-dimensional subspace HY be chosen in such a way that it has only one common point y with the tangent space V,. to the fiber Fx at the point

y and the space Ty (xn+r ) is the direct sum of the subspaces HY and V,. . These subspaces considered at all points y of the fibration xn+r form the horizontal and vertical distributions, respectively. If the distribution HY is differentiable and invariant under the action of the group G on the fibration xn+r , it is called a connection in the principal fibration.

230 7. GENERALIZED SPACES

Note that the spaces with affine, projective, and conformal connections considered above are particular cases of a connection in the principal fiber space. In these cases, the fibers can be identified with the groups of affine, projective, and conformal transformations in the spaces En , pn , and en , respectively. In particular, for the space An with an affine connection, the horizontal distribution defines the parallel displacement of frames along a curve in the base.

The affine, projective, and conformal connections are G-structures of first order in the corresponding fiber space xn+r . However, usually they are considered as G-structures of higher order on the manifold xn ; for the affine and conformal connections the order of this G-structure is two, and for the projective connection the order is three.

The notion of a G-structure was first formulated by Cartan's student Ehres­mann in his note Fiber spaces of comparable structure [Eh2] ( 1 942) , and the term " G-structure" first appeared in Chern's paper Infinite continuous pseudo-groups [Chr3] ( 1 954) in which these structures were connected with Lie pseudogroups studied by Cartan under the name "infinite continuous groups". Note also another of Chern's papers, The geometry of G-structures [Chr4] ( 1 966) , and the paper On the equivalence problem of certain infinites­imal structures [Lib] ( 1 954) by Paulette Libermann (b. 1 9 1 9) .

The complex and almost complex structures played an important role in the construction of the theory of G-structures. The first of these structures is defined on an n-dimensional complex manifold cxn . Its structural group is the group CGLn . If we take the real interpretation of the space CXn , we

obtain a real manifold X2n . In its tangent space Tx (X2n ) , the group G of

dimension 2n2 (which is the real interpretation of the group CGLn ) acts. The elements g of this group commute with the operator J of the almost complex structure. This operator satisfies the condition J2 = - 1 and cor­responds to the scalar operator ii in the group CG Ln . An almost complex

structure is a G-structure on the real manifold X2n whose structural group is the same as the structural group of the complex manifold cxn . However, in contrast to a complex manifold, in general this structure cannot be ob­tained as a realization of the manifold cxn . The following problem arises in this connection: to find under what condition an almost complex struc­ture becomes a complex structure. The solution of this problem is reduced to finding conditions of complete integrability of two systems of Pfaffian equa­tions that define imaginary conjugate eigenspaces of the operator J in the tangent space Tx (X2n ) of the manifold X2n . This required condition is the

vanishing of a certain operator, NJk , of the third valence (which is called

the Nijenhuis operator) on the manifold x2n • In a similar manner, the structure of an almost-product can be defined on

a manifold xn+m . In this case, the operator J satisfies the condition J2 = I and has real eigenspaces of dimensions m and 2n - m . The elements of the

§7. 1 3. MULTIDIMENSIONAL WEBS 23 1

structural group G commute with this operator J . In particular, if m = n , the operator J defines an almost split complex structure on the manifold x2n .

There is the following problem in the theory of G-structures: given a man­ifold xn

carrying a G-structure, is it possible to define an affine connection on xn

whose parallel displacements preserve the G-structure ? If the answer is positive, the G-structure is called a G-structure of finite type. Otherwise, it is called a G-structure of infinite type. Since the parallel displacements in Riemannian and pseudo-Riemannian manifolds generate a single affine con­nection, the G-structures associated with these spaces are of finite type. On the contrary, in a space with an almost complex structure, it is impossible to find a single affine connection in which the operator J of the almost complex structure will be invariant under a parallel displacement. Moreover, it can be proved that in these spaces it is impossible to find a single affine connec­tion defined even by means of differential prolongations of this G-structure. Thus, this G-structure is a G-structure of infinite type.

There is an extensive bibliography on differential geometry of G-structures. We note here only the book Transformations groups in differential geometry [Ko] ( 1 972) by Kobayashi.

§7 . 13. Multidimensional webs

Another interesting example of G-structures of first order is connected with webs on smooth manifolds formed by a certain number of smooth fo­liations. Web theory was founded by Blaschke at the end of the 1 920s and during the 1 930s. In 66 papers composing the series Topological questions of differential geometry [BlaT] ( 1 928- 1 936) and many other publications, Blaschke and his co-workers considered webs formed by families of curves in the plane and the families of curves and surfaces in three-dimensional space. This explains the title of the series of papers on webs indicated above. In these papers, it was established that web theory is connected with many branches of geometry as well as with some other branches of mathematics and, in particular, with some parts of algebra. These investigations in the web theory were summarized by Blaschke and Bol in their book Geometry of webs [BlaB] ( 1 936) , and later in Blaschke's book Introduction to the geometry ofwebs [Bla6] ( 1 955 ) .

However, a s far back as 1 908, in the paper The subgroups of continuous groups [26], Cartan considered an example in which he posed the problem of the equivalence of two differential equations:

(7 . 1 5 ) dy d Y dx = f(x , y) and dX = F(X , Y)

with respect to transformations of the form

(7 . 1 6) x = X(x) , y = Y(y) .

232 7 . GENERALIZED SPACES

The latter transformations leave invariant the coordinate lines x = a , y = b in the plane xOy as well as the integral lines of equations (7 . 1 5 ) . These three families of lines form a three-web in the plane. Thus, the problem con­sidered by Cartan is equivalent to the problem of classification of curvilinear three-webs in the plane. Cartan distinguished three classes of differential equations of type (7 . 1 5 ) : the equations admitting a three-parameter group of transformations of type (7 . 1 6 ) , the equations admitting a one-parameter group of transformations of type (7 . 1 6) , and the equations not admitting such transformations. To these three classes of differential equations there correspond three classes of curvilinear three-webs in the plane.

In the 1 930s, along with webs in the plane and in three-dimensional space, webs on manifolds of dimension higher than three were studied. First, in 1 935 Bol published the paper On a three-web in a four-dimensional space [Bol] in which he considered a three-web formed on a four-dimensional man­ifold by three two-dimensional foliations. Next, in 1 936, Chern's paper An invariant theory of the three-web of r-dimensional manifolds in R2, [Chr l ]

appeared, in which Chern studied three-webs formed on a manifold R2' by three r-dimensional foliations. During the last 20 years these studies were continued by Akivis, Vasil' ev, Goldberg, and their students and co-workers.

Let us consider, for example, a web W formed on a manifold x2n by three foliations A.a , a = 1 , 2 , 3 , of dimension n . Through any point x of

the manifold X2n there pass three leaves F belonging to these foliations A. . a a Denote by Tx (Fa) the n-dimensional subspaces of the space Tx (X2n ) which are tangent to the leaves Fa passing through the point x . The subgroup of

the group of linear transformations of the space Tx (X2n ) preserving the subspaces Tx (Fa) is the structural group of the G-structure induced on the

manifold X2n by the web W . It is not difficult to show that in this case G = GLn . This G-structure is a structure of finite type since it defines

an affine connection on X2n in which the web leaves are totally geodesic submanifolds of the manifold x2n •

The subspaces Tx (Fa) define in the space Tx (X2n ) an algebraic cone which cuts the Segrean ( 3 . 1 2) in the hyperplane at infinity of this space. This cone is called the Segre cone. Since the Segrean defined by this cone has rectilinear generators and ( n - 1 )-dimensional generators, the cone itself has two-dimensional generators and n-dimensional generators. Linear transfor­mations preserving this Segre cone form a group G which is the direct prod­uct of the groups GLn and SL2 • This group G defines a new G-structure

in the space X2n , and this G-structure is called the almost Grassmann struc­ture AGrn+ l , I . This name is explained by the fact that in the simplest case,

when this G-structure is integrable, the manifold X2n admits a mapping on the Grassmannian Grn+ I , 1 of straight lines of the projective space pn+ l

. In this case, a web W is called Grassmannizable and is defined by a triple of hypersurfaces in the space pn+ 1 •

§7 . 1 3. MULTIDIMENSIONAL WEBS 233

The almost Grassmann structure AGrn+ i , 1 and the G-structure defined by the web W itself are of finite type. However, an affine connection on the structure AGrn+ I , 1 is defined by the differential neighborhood of third order while an affine connection of the G-structure induced by the web W is defined by the differential neighborhood of second order.

Multidimensional three-webs are connected with differentiable quasi­groups: if we map the n-dimensional bases of the foliations A.°' forming a three-web W on a manifold X2n

onto the same n-dimensional mani­fold Q , an algebraic operation arises in Q which defines a smooth local quasigroup. Moreover, to smooth quasigroups and loops there correspond different classes of three-webs which are characterized by some closure con­ditions which are satisfied in these three-webs. In particular, the important classes of three-webs correspond to the Lie groups and the Moufang, Bol, and monoassociative loops.

The theory of multidimensional three-webs is presented in the book Ge­

ometry and algebra of multidimensional three-webs [AS] ( 1 99 1 ) by Akivis and Shelekhov, and the theory of multicodimensional (n + 1 )-webs is presented in the book Theory of multicodimensional (n + 1 ) -webs [Glb l ] ( 1 988) by Goldberg.

Conclusion

As a rule, Cartan built his scientific research on works of his predecessors, developing their ideas so well that other mathematicians often forgot the original works. This was the case in the theory of simple Lie groups with the Killing paper, in the method of moving frames with the Cotton paper, and in the theory of symmetric Riemannian spaces with the Levy paper. It was somewhat different in the ca�e of the theory of generalized spaces, since Cartan continued to work fruitfully with the founders of this theory, Weyl and Schouten. In some of the works of Cartan's predecessors (e.g. , the papers of Cotton and Levy) , only the initial definitions were given for the future theories which were later constructed by Cartan. In other cases, for example in the case of Killing's paper, the important notions of the new theory were introduced and the main results of this theory were formulated, but the rigorous proofs of these results were given only in the famous Cartan thesis [5 ] ; as a result, after the appearance of this thesis, Killing's paper [Kil2] on the structure of groups of continuous transformations was read by almost no one. This explains the enthusiasm that A. J. Coleman had while reading the Killing paper mentioned above and that he expressed in his own paper entitled The greatest mathematical paper of all times [Col2] ( 1 989) . In this paper, Coleman wrote:

"Cartan did give a remarkably elegant and clear exposition of Killing's results. He also made an essential contribution to the logic of the argument by proving that the 'Cartan subalgebra' of a simple Lie algebra is abelian. This property was announced by Killing but his proof was invalid . . . In the last third of Cartan's thesis, many new and important results are based upon and go beyond Killing's work. Personally, following the value scheme of my teacher Claude Chevalley, I rank Cartan and Weyl as the two greatest mathematicians of the first half of the twentieth century. Cartan's work on infinite dimensional Lie algebras, exterior differential calculus, differential geometry, and above all, the representation theory of semisimple Lie algebras was of supreme value. But because one's Ph. D. thesis seems to predetermine one's mathematical life work, perhaps if Cartan had not hit upon the idea of basing his thesis on Killing's epoch-making work he might have ended his

235

236 CONCLUSION

days as a teacher in a provincial lycee and the mathematical world would have never heard of him!" [Col2, p. 30] .

A similar situation occurred with Cartan's works on the theory of Pfaffian equations, which was considered in the books of J. F. Pommaret [Pom l -3] .

Coleman's "prediction" was completely justified by the fate of Cartan's predecessor in the correction of inaccuracies in Killing's results - C. A. Um­lauf, the author of the thesis [Um]; his further life and activities are unknown. However, this was not the case with Cartan. After Cartan gave the classifi­cation of complex simple Lie groups, he created a similar classification of complex and real associative algebras and complex simple Lie pseudogroups. The latter led him to the theory of Pfaffian equations, whose application to differential geometry implied a complete transformation of this discipline and helped Cartan and his followers to solve numerous problems in the dif­ferential geometry of various spaces. Cartan's work on simple Lie groups was followed by his remarkable theory of representations of these groups. Subse­quently Cartan solved the problem of classification of real simple Lie groups. The latter problem was posed by Killing, but the author of "the greatest math­ematical paper of all times" could not solve it himself. Following this, Cartan created the geometries of "generalized spaces" and the theory of symmetric spaces by means of which the problem of classification of real simple Lie groups unexpectedly obtained a new and much more elegant solution.

Cartan's papers eclipsed the papers of many of his predecessors: after the publication of Cartan's papers, practically no one, except the historians of science, read either Killing's or Janet's papers (Janet bitterly complained about this to Pommaret) .

The most spectacular confirmation of the enormous influence that Cartan has had on the development of contemporary mathematics was the creation of the encyclopaedia of mathematical sciences, Elements of mathematics [Bou] of Nicolas Bourbaki. This pseudonym was used by a group of mathemati­cians, among whom leading roles were played by Cartan's son Henri, Andre Weil, Jean Dieudonne, Claude Chevalley, and Jean Frederic Delsarte. The title of this encyclopedia was supposed to indicate that, according to the idea of its authors, this work would play the same role for mathematics of the 20th century as Euclid's Elements played for ancient mathematics. While the first part of this work contained a concise survey of the principal "math­ematical structures" on which algebra, topology, and analysis are based, its second part gave a systematic explanation of the theory of Lie groups and Lie algebras, the bulk of which was Cartan's creation. The authors of Elements of mathematics, who belonged to another generation, often put in the fore­front what Cartan had not. While Cartan considered himself first of all as a geometer and headed the Department of Higher Geometry at the Sorbonne, in the Bourbaki work, geometry was dissolved in algebraical, topological, and analytical "structures". Such pure geometrical structures as the affine, projec­tive, and conformal geometries, considered as sets of points in which some

CONCLUSION 237

specifically geometric subsets (straight lines and planes, circles and spheres) are distinguished, were not included in Elements of mathematics; neither were the spaces with affine, projective, and conformal connections which are based on these geometries and which played so important a role in Car­tan's research. But nevertheless the influence of Cartan's papers penetrated Elements of mathematics. Cartan's works also influenced those mathemati­cians whose research was out of the scheme of Elements of mathematics and who continued to develop different directions of Cartan's research. In our description of Cartan's scientific results, we often mentioned works of math­ematicians from different countries who developed those or other of Cartan's ideas: Weyl, Blaschke, Chern, Freudenthal, Ehresmann, Lichnerowitz, Serre, Tits, Finikov, Rashevsky, Norden, Wagner, Laptev, Vasilyev, and many oth­ers ( including the authors and the translator of this book) .

Another spectacular confirmation of Cartan's influence on many branches of contemporary mathematics was the conference, "The Mathematical Her­itage of Elie Cartan'', which was held in Lyons, France, on June 25-29, 1 984, on the occasion of the 1 l 5th anniversary of Cartan's birth. The conference took place at the University of Lyons, and Henri Cartan and S. S. Chern were the co-chairmen of its Organizing Committee. The participants in the conference made a trip to Dolomieu.

The program of the conference contains the following lectures:

( 1 ) S. S. Chern: Moving frames. (2) J. M. Souriau: On differential forms. ( 3 ) J. Tits: Analogues of great classification theorems of Elie Cartan. ( 4) M. Gromov: Isometric immersions of Riemannian manifolds. ( 5 ) V. Kac: Computing homology of compact Lie groups and their in­

finite-dimensional analogues. (6) V. Guillemin: Some microlocal aspects of integral geometry. (7) B. Kostant: Simple Lie algebras, finite subgroups of SU2 , and the

MacKay correspondence. (8 ) A. Trautman: Optical structures in relativity theories. (9) M. Berger: The Riemannian manifolds as metric spaces.

( 1 0) R. Bryant: The characteristic variety and modern differential geom-etry.

( 1 1 ) Y. Choquet-Bruhat: Causality of supergravities theories. ( 1 2) J. L. Koszul: Schouten-Nijenhuis brackets and cohomology. ( 1 3) C. Feffermann: Conformal geometry. ( 1 4) M. Kuranishi: Cartan connection and CR-structures with nondegen­

erate Levy form. ( 1 5 ) M. Duflo: Noncommutative harmonic analysis and generalized Car­

tan subgroups. ( 1 6) S. Helgason: Fourier analysis on symmetric spaces.

238 CONCLUSION

( 1 7) W. Schmid: Boundary value problems for group invariant differential equations.

( 1 8) G. D. Mostow: Discrete subgroups of Lie groups. ( 1 9) I . Piatetskii-Shapiro: L-functions for automorphic forms. (20) A. Weinstein: Poisson manifolds. (2 1 ) I . M. Singer: Families of Dirac operators with applications to physics. (22) I . M. Gel' fand: New models for representations of reductive groups

and their hidden symmetries.

Following Gel' fand's lecture was the ceremony of his inauguration in the degree of Doctor Honoris Causa of Lyons University.

The lectures of this conference were published in the book [ECM] .

1 869 1 880- 1 885 1 885- 1 887 1 887- 1 888 1 888- 1 89 1 1 89 1 - 1 892 1 892- 1 894 1 892- 1 894

1 894

1 894- 1 896 1 896- 1 903 1 898

1 899 1 903 1 903- 1 909

1 904- 1 905

1 908

1 909- 1 9 1 2 1 9 1 0 1 9 1 2- 1 940

Dates of Cartan's Life and Activities

Born in Dolomieu, France, April 9 Student at the College of Vienne Student at the Lycee of Grenoble Student at the Lycee Janson-de-Sailly in Paris Student at the Superior Normal School in Paris Drafted into the French army; achieved the rank of sergeant Boursier of the Pecaut Foundation Acquaintance with Sophus Lie and discussions with him in Paris In the Sorbonne defended the doctoral thesis The structure of the finite continuous groups of transformations, in which he constructed the theory of simple complex Lie groups Lecturer of mathematics at the University of Montpellier Lecturer of mathematics at the University of Lyons Constructed the theory of complex and real simple algebras in the paper Bilinear groups and systems of complex numbers Published his first paper on the Pfaff problem Married Marie-Louise Bianconi in Lyons Professor of mathematics at the University of Nancy and the Institute of Electrical Engineering and Applied Mechanics Published first papers in the theory of "infinite continuous groups of transformations" (Lie pseudogroups) and the theory of systems of Pfaffian equations in involution Published the paper Complex numbers for the French edition of Encyclopaedia of Mathematical Sciences, which contains a survey and further development of the theory of algebras Lecturer of mathematics in the Sorbonne, Paris Published first papers on the method of moving frames Professor of Differential and Integral Calculus, and, from 1 924, Professor of Higher Geometry in the Sorbonne, Paris; Professor of the Municipal School of Industrial Physics and Chemistry, Paris

239

240

1 9 1 3

1 9 1 4

1 9 1 5- 1 9 1 8

1 9 1 5

1 9 1 6- 1 920

1 922

1 923- 1 925

1 925 1 926

1 926- 1 927

1 927

1 928

1 930

1 9 3 1 1 9 3 1 1 934 1 937

1 938 1 938 1 945

1 946

1 945 1 95 1

DATES OF CARTAN'S LIFE AND ACTIVITIES

Constructed the theory of linear representations of complex simple groups in the paper Projective groups under which no plane manifold is invariant Constructed the theory of real simple Lie groups in the paper Real simple finite continuous groups and constructed linear representations of these groups Drafted into the Army and served at the rank of sergeant in the military hospital of the Superior Normal School Published the paper Theory of continuous groups and geom­etry for the French edition of Encyclopaedia of Mathematica/ Sciences Published papers on the theory of deformation of surfaces in the Euclidean, conformal, and projective spaces Published papers on the theory of gravitation and the book Lectures on integral invariants Published papers on geometry of spaces with affine, projective, and conformal connections Published the book Geometry of Riemannian manifolds Created the theory of symmetric Riemannian spaces in the paper On a remarkable class of Riemannian manifolds Presented lectures in the Sorbonne afterward published under the title Riemannian geometry in an orthonormal frame Created the theory of symmetric spaces with an affine connection in the paper The geometry of transformations groups Published the book Lectures on the geometry of Riemannian manifolds Presented lectures in the Moscow University later published under the title The method of moving frames, the theory of finite continuous groups, and generalized spaces Published the book Lectures on complex projective geometry Elected to the Paris Academy of Sciences Published the book Fins/er spaces Published the books Lectures on the theory of spaces with a projective connection and The theory of finite and continuous groups and differential geometry Published the book Lectures on the theory of spinors Awarded the Lobachevskian prize for geometric works Published the book Exterior differential systems and their geometric applications Published the second largely augmented edition of the book Lectures on the geometry of Riemannian manifolds Member of Bureaux of Longitudes Died in Paris, May 6

List of Publications of Elie Cartan

In the List of Publications of Elie Cartan, we begin by listing chronolog­ically his mathematical works, and then his works in the history of science, his reminiscences, complete collections of his works and collections of his selected papers, and publications of his scientific correspondence.

List of Cartan's mathematical works

The list of Cartan's mathematical works reproduces the lists published in the editions [204] (before 1 939) , [207], and [209]. To our list, we added the translations of Cartan's books, as well as his works which were omitted in the two lists mentioned above. If, in the lists published in the editions [207] and [209], a paper was given under the number with the suffix bis or ter, we list this paper under the same number followed by the letter a, b, or c.

1893

1 . Sur la structure des groupes simples finis et continus, C. R. Acad. Sci. Paris 1 16, 784-786; <Euvres completes: Partie I , Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 99- 1 0 1 .

2. Sur la structure des groupes finis et continus, C. R. Acad. Sci. Paris 1 16, 962-964; <Euvres completes: Partie I , Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 03- 1 05 .

3 . Uber die einfachen Transformationsgruppen, Sitzungsber. Siichs. Ges. Wiss. Leipzig, Mat.-Phys. Kl. 45, 395-420; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 07-1 32.

1894

4. Sur la reduction de la structure d 'un groupe a saforme canonique, C. R. Acad. Sci. Paris 1 19, 639-64 1 ; <Euvres completes: Partie I , Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 33- 1 3 5 .

5 . Sur la structure des groupes de transformations finis et continus, These, Nony, Paris; 2nd ed., Vuibert, Paris, 1 933 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 37-287 .

24 1

242 LIST OF PUBLICATIONS OF ELIE CARTAN

6. Sur un theoreme de M. Bertrand, C. R. Acad. Sci. Paris 1 19, 902; <:Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , p. I .

7. Sur un theoreme de M. Bertrand, Bull. Soc. Math. France 22, 230-234; <:Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 9 55 , pp. 3-7.

1895

8. Sur certains groupes algebriques, C. R. Acad. Sci . Paris 120, 544-548; <:Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 289-292.

1896

9. Sur la reduction a sa forme canonique de la structure d'un groupe de transformations fini et continu, Amer. J. Math. 18, 1 -6 1 ; <:Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 293-353 .

1 0. Le principe de dualite et certaines integrales multiples de l 'espace tan­gentiel et de l 'espace regle, Bull. Soc. Math. France 24, 1 40- 1 77; <:Euvres completes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 9 53 , pp. 265-302.

1897

1 1 . Sur /es systemes de nombres complexes, C. R. Acad. Sci . Paris 124, 1 2 1 7- 1 220; <:Euvres completes: Partie II, Algebre. Formes differen­tielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 9 53 , pp. 1 -4.

1 2. Sur /es systemes reels de nombres complexes, C. R. Acad. Sci. Paris 124, 1 296- 1 297; <:Euvres completes: Partie II, Algebre. Formes dijferen­tie/les, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 5-6.

1898

1 3 . Les groupes bilineaires et /es systemes de nombres complexes, Ann. Fae. Sci. Toulouse 12B, 1 -99; <:Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 7- 1 05 .

1899

1 4. Sur certaines expressions dijferentielles et le probleme de Pfaff, Ann. Sci. Ecole Norm. Sup. 16, 239-332; <:Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes dijferentiels, vols. 1-2, Gauthier-Villars, Paris, 1 953 , pp. 303-396.

1901

1 5 . Sur quelques quadratures dont / 'element dijferentiel contient des fonc­tions arbitraires, Bull. Soc. Math. France 29, 1 1 8- 1 30; <:Euvres com-

LIST OF PUBLICATIONS OF ELIE CARTAN 243

pletes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , 397-409.

1 6 . Sur / 'integration des systemes d 'equations aux differentielles totales, Ann. Sci. Ecole Norm. Sup. 18, 24 1 -3 1 1 ; (Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 4 1 1 -48 1 .

1 7 . Sur / 'integration de certains systemes de Pfaff de caractere deux, Bull. Soc. Math. France 29, 233-303; muvres completes: Partie II, Algebre. Formes dijferentielles, systemes dijferentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 483-553 .

1902

1 8 . Sur / 'integration des systemes differentiels completement integrables. I, C. R. Acad. Sci. Paris 134, 1 4 1 5- 1 4 1 8 ; (Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 555-558 .

1 8a. Sur / 'integration des systemes dijferentiels completement integrables. II, C. R. Acad. Sci. Paris 134, 1 564- 1 566; muvres completes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 559-56 1 .

1 9 Sur / 'equivalence des systemes differentiels, C. R. Acad. Sci. Paris 135, 7 8 1 -783 ; muvres completes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 563-565 .

20. Sur la structure des groupes in.finis, C. R. Acad. Sci. Paris 135, 8 5 1 -853 ; muvres completes: Partie II, Algebre. Formes dijferentielles, sys­temes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 567-569.

1904

2 1 . Sur la structure des groupes in.finis de transformations. I, Ann. Sci. Ecole Norm. Sup. 21 , 1 53-206; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 57 1-624.

1905

22. Sur la structure des groupes in.finis de transformations. II, Ann. Sci. Ecole Norm. Sup. 22, 2 1 9-308; (Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 625-7 1 4.

1907

23. Les groupes de transformations continus, in.finis, simples, C. R. Acad. Sci. Paris 144, 1 094- 1 097; (Euvres completes: Partie II, Algebre. Formes dijferentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 7 1 5-7 1 8 .

244 LIST OF PUBLICATIONS OF ELIE CARTAN

24. Sur la definition de l 'aire d 'une portion de surface courbe. I, C. R. Acad. Sci. Paris 145, 1 403- 1 406; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 9-1 2.

1908

25 . Sur la definition de l 'aire d 'une portion de surface courbe. I I , C. R . Acad. Sci. Paris 146, 1 68 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , p. 1 2 .

26. Les sous-groupes des groupes continus de transformations, Ann. Sci. Ecole Norm. Sup. 25, 57- 1 94; (Euvres completes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 7 1 9-856 .

27 . Nombres complexes, Encyclopedia Math. Sci . , edition fran�aise I 5 , pp. 329-468; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 07-246.

1909

28. Les groupes de transformations continus, in.finis, simples, Ann. Sci. Ecole Norm. Sup. 26, 93- 1 6 1 ; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 857-925 .

1910

29. Sur /es developpables isotropes et la methode du triedre mobile, C. R. Acad. Sci. Paris 151, 9 1 9-92 1 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp . 1 4 1 - 1 43 .

30. Les systemes de Pfaff ii cinq variables et /es equations aux derivees partielles du second ordre, Ann. Sci . Ecole Norm. Sup. 27, 1 09- 1 92; (Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 927- 1 0 1 0.

3 1 . La structure des groupes de transformations continus et la theorie du triedre mobile, Bull. Sci. Math. 34, 250-284; (Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 45- 1 78 .

191 1

32. Le ca/cul des variations et certaines families de courbes, Bull. Soc. Math. France 39, 29-52; (Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 0 1 1 - 1 034

33 . Sur /es systemes en involution d 'equations aux derivees partielles du second ordre ii une fonction inconnue de trois variables independantes, Bull. Soc. Math. France 39, 352-443; (Euvres completes: Partie

LIST OF PUBLICATIONS OF ELIE CARTAN 245

II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 03 5- 1 1 25 .

1912

34. Sur /es caracteristiques de certains systemes d 'equations aux derivees partielles, Soc. Math. France 40, C. R. des seances, p. 1 8 ; muvres completes: Partie II, Algebre. Formes differentielles, systemes differen­tiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 27 .

35 . Sur /es groupes de transformations de contact et la Cinematique nou­velle, Soc. Math. France 40, C. R. des seances, p. 23 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 79 .

1913

36. Remarques sur la composition des forces, Soc. Math. France 41 , C. R. des seances, 58-60; muvres completes: Partie II , Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 247-248 .

37 . Les groupes projectifs qui ne laissent invariante aucune multiplicite plane, Bull. Soc. Math. France 41 , 53-96; Selecta. Jubi/e scientifique de M. Elie Cartan, Gauthier-Villars, Paris, 1 939, pp. 1 37- 1 5 1 ; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 9 52, pp. 3 55-398.

1914

38 . Les groups reels simples finis et continus Ann. Sci. Ecole Norm. Sup. 31 , 263-355 ; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 399-49 1 .

39 . Les groups projectifs continus reels qui ne laissent invariante aucune multiplicite plane, J. Math. Pures Appl. 10, 1 49- 1 86 ; (Euvres com­pletes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 493-530.

40. Sur / 'integration de certains systemes d 'equations differentielles, C. R. Acad. Sci. Paris 158, 326-328; (Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 1 1 29- 1 1 3 1 .

4 1 . Sur certaines families naturelles de courbes, Soc. Math. France 42, C. R. des seances, 1 5- 1 7 ; muvres completes: Partie III, Divers, geo­metrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 8 1 -1 83 .

42. Sur ! 'equivalence absolue de certains systemes d 'equations differentielles et sur certaines families de courbes, Bull. Soc. Math. France 42, 1 2-48; muvres completes: Partie II, Algebre. Formes differentielles, systemes dijferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 33- 1 1 68 .

43 . La theorie des groupes, Revue du Mois 17, 438-468.

246 LIST OF PUBLICATIONS OF ELIE CARTAN

1915

44. Sur / 'integration de certains systemes indetermines d 'equations differen­tielles, J. Reine Angew. Math. 145, 86-9 1 ; (Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 69- 1 1 74.

45. Sur /es transformations de Backlund, Bull. Soc. Math. France 43, 6-24; muvres completes: Partie II, A/gebre. Formes dijferentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 75 - 1 1 93 .

46. La theorie des groupes continus et geometrie (the extended translation from German of Fano's article [Fa] ) , Encyclopedia Math. Sci. III 5, 332-352 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 727- 1 86 1 .

1916

47. La deformation des hypersurfaces dans / 'espace euc/idien reel a n di­mensions, Bull. Soc. Math. France 44, 65-99; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 8 5-2 1 9 .

1917

48. La deformation des hyperfurfaces dans / 'espace conforme reel a n � 5 dimensions, Bull. Soc. Math. France 45, 57- 1 2 1 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 22 1 -285 .

1918

49. Sur certaines hypersurfaces de / 'espace conforme reel a cinq dimen­sions, Bull. Soc. Math. France 46, 84- 1 05 ; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 287-308.

50. Sur /es varietes a 3 dimensions, C. R. Acad. Sci. Paris 167, 357-359; (Euvres completes: Partie III, Divers, geometrie differentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 309-3 1 1 .

50a. Sur /es varietes deve/oppables a trois dimensions, C. R. Acad. Sci. Paris 167, 426-429; muvres completes: Partie III, Divers, geometrie dijferentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 3 1 2-3 1 4.

50b. Sur /es varietes de Beltrami a trois dimensions, C. R. Acad. Sci. Paris, 167, 482-484; muvres completes: Partie III, Divers, geometrie dijferen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 3 1 5-3 1 7 .

50c. Sur /es varietes de Riemann a trois dimensions, C. R. Acad. Sci. Paris 167, 550-55 1 ; muvres completes: Partie III, Divers, geometrie dijferen­tiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 3 1 8-3 1 9 .

LIST OF PUBLICATIONS OF ELIE CARTAN 247

1919

5 1 . Sur /es varietes de courbure constante d 'un espace euc/udien ou non euc/idien, Bull. Soc. Math. France 47, 1 25- 1 60; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 32 1 -359 .

1920

52. Sur /es varietes de courbure constante d 'un espace euc/udien ou non euc/idien, Bull. Soc. Math. France 48, 1 32-208 ; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 360-432.

53. Sur la deformation projective des surfaces, C. R. Acad. Sci. Paris 170, 1 439- 1 44 1 ; muvres completes: Partie III, Divers, geometrie differen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 9 55 , pp. 433-435 .

53a. Sur / 'applicabilite projective des surfaces, C. R. Acad. Sci. Paris 171 , 27-29; muvres completes: Partie III, Divers, geometrie differentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 437-439.

54. Sur la deformation projective des surfaces, Ann. Sci. Ecole Norm. Sup. 37, 259-356; muvres completes: Partie III, Divers, geometrie differentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 44 1 -538 .

55 . Sur le probleme general de la deformation, C. R . Congres Math. In­ternat. (Strasbourg, 1 920), pp. 397-406; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 539-548.

1922

56 . Sur /es equations de la gravitation d 'Einstein, J. Math. Pures Appl. 1 , 1 4 1 -203; (Euvres completes: Partie III, Divers, geometrie dijferentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 549-6 1 1 .

57 . Sur une definition geometrique du tenseur d 'energie d 'Einstein, C. R. Acad. Sci. Paris 174, 437-439; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Vil�1rs, Paris, 1 9 55 , pp. 6 1 3-6 1 5 .

58 . Sur une generalisation de la notion de courbure de Riemann et /es espaces a torsion, C. R. Acad. Sci. Paris 174, 593-595; (Euvres completes: Partie III, Divers, geometrie differentiel/e, vols. 1-2, Gauthier-Villars, Paris, 1 955 , pp. 6 1 6-6 1 8 ; English transl. , Cosmol­ogy and Gravitation (Bologna, 1 979) , NATO Adv. Study Inst. Ser. B. Phys. , vol . 58 , Plenum Press, New York and London, 1 980, pp. 493-496.

59 . Sur /es espaces generalises et la theorie de la re/ativite, C. R. Acad. Sci . Paris 174, 734-736; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 9 55 , pp. 6 1 9-62 1 .

60. Sur /es espaces conformes generalises et / 'Univers optique, C. R. Acad. Sci. Paris 174, 857-859; (Euvres completes: Partie III, Divers, geo-

248 LIST OF PUBLICATIONS OF ELIE CARTAN

metrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 622-624; English transl. , On manifolds with an affine connection and the theory of general relativity, Bibliopolis, Naples, pp. 1 9 5- 1 99.

6 1 . Sur /es equations de structure des espaces generalises et / 'expression analytique du tenseur d 'Einstein, C. R. Acad. Sci. Paris 174, 1 1 04-1 1 06 ; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , Partie III, pp. 625-627 .

62. Sur un theoremefondamental de M. H. Wey/ dans la theorie de l 'espace metrique, C. R. Acad. Sci. Paris 175, 82-85 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 629-632.

63 . Sur /es petites oscillations d 'une masse fluide, Bull. Sci. Math. 46, 3 1 7-352, 356-369; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 3-6 1 .

64. Le�ons sur /es invariants integraux, Paris, Hermann, 2nd ed. , 1 958 , 3rd ed. , 1 97 1 .

1923

65 . Sur un theoreme fondamental de M. H. Wey/, J. Math. Pures Appl . 2 , 1 67- 1 92; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 633-658.

66. Sur /es varietes a connexion affine et la theorie de la relativite generali­see. I, Ann. Sci. Ecole Norm. Sup. 40, 325-4 1 2; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 659-746; English transl., On manifolds with an affine connection and the theory of general relativity, Bibliopolis, Naples, pp. 29- 1 05 .

67. Les fonctions reel/es non analytiques et /es solutions singulieres des equations differentielles du premier ordre, Ann. Polon. Math. 2, 1 -8 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 63-70.

68 . Les espaces a connexion conforme, Ann. Polon. Math. 2, 1 7 1 - 22 1 ; <Euvres completes: Partie Ill, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 747-797.

1924

69. Sur /es varietes a connexion affine et la theorie de la relativite generali­see. II, Ann. Sci. Ecole Norm. Sup. 41 , 1 -25; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 799-823; English transl. , On manifolds with an affine connection and the theory of general relativity, Bibliopolis, Naples, pp. 1 07- 1 27.

70. Sur /es varietes a connexion projective, Bull . Soc. Math. France 52, 205-24 1 ; Selecta. Jubile scienti.fique de M. Elie Cartan, Gauthier­Villars, Paris, 1 939, pp. 1 65-20 I ; <Euvres completes: Partie Ill, Divers,

LIST OF PUBLICATIONS OF ELIE CARTAN 249

geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 825-86 1 .

7 1 . Les recentes generalisations de la notion d 'espace, Bull. Sci. Math. 48, 294-320; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 863-889.

72. La theorie de la relativite et /es espaces generalises, Atti V. Cong. In­ternat . , Filosofia, pp. 427-436.

73 . La theorie des groupes et /es recherches recentes de geometrie differen­tielle, Enseign. Math. 24 ( 1 925 ) , 1 - 1 8 ; Proc. Internat. Math. Congress Toronto 1 ( 1 928) , 8 5-94; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 89 1-904.

.

74. Sur /es formes differentielles en geometrie, C. R. Acad. Sci. Paris 178, 1 82- 1 84; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 905-907.

75 . Sur la connexion affine des surfaces, C. R. Acad. Sci. Paris 178, 292-295 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 209-2 1 2.

76. Sur la connexion affine des surfaces developpables, C. R. Acad. Sci. Paris 178, 449-45 1 ; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 9 1 2-9 1 4.

77 . Sur la connexion projective des surfaces, C. R. Acad. Sci. Paris 178, 750-752; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 2 1 5-2 1 7 .

1925

78 . Note sur la generation des oscillations entretenues (with Henri Cartan) , Ann. Postes, Tel. et Tel. 14, 1 1 96- 1 207; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 7 1 -82.

79. Les groupes d 'holonomie des espaces generalises et /'Analysis situs, Assoc. Avanc. Sciences, 49e session, Grenoble, pp. 47-49; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 9 1 9-920.

80. Sur !es varietes a connexion affine et la theorie de la relativite generali­see, Ann. Sci. Ecole Norm. Sup. 42, 1 7-88; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 92 1-992; English transl. , On manifolds with an affine connec­tion and the theory of general relativity, Bibliopolis, Naples, pp. 1 29-1 93 .

8 1 . Les tenseurs irreductibles et /es groupes lineaires simples et semi-simples, Bull. Sci. Math. 49, 1 30- 1 52; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 5 3 1 -553 .

250 LIST OF PUBLICATIONS OF ELIE CARTAN

82. Le principe de dualite et la theorie des groupes simples et semi-simples, Bull. Sci. Math. 49 , 36 1 -374; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 555-568.

83 . Sur le mouvement a deux parametres, Nouvelles Ann. 1 , 33-37; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 9 55 , pp. 83-87.

84. La geometrie des espaces de Riemann, Memorial Sci. Math . IX, Gauthier-Villars, Paris.

1926

85 . L 'application des espaces de Riemann et /'Analysis situs, Assoc. Avanc. Sciences, 50e session, Lyon, pp. 53-55 ; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 993-995 .

86 . Sur certains systemes differentiels dont /es inconnues sont des formes de Pfaff, C. R. Acad. Sci. Paris 182, 956-958 ; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 95- 1 1 97 .

87 . Sur !es espaces de Riemann dans lesquels le transport par parallelisme conserve la courbure, Rend. Accad. Lincei 31 , 544-547; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 569-572.

88 . Les groupes d'holonomie des espaces generalises, Acta Math. 48, 1-42; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 997- 1 038 .

89 . Sur /es spheres des espaces de Riemann a trois dimensions, J. Math. Pures Appl. 5, 1 - 1 8 ; (Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 039- 1 056.

90. L 'axiome du plan et la geometrie differentielle metrique, in Memoriam of N. I. Lobatschevskii, vol. 2, "Glavnauka'', Kazan, 1 927, pp. 4-1 2; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 057- 1 065 .

9 1 . On the geometry of the group-manifold of simple and semi-simple groups (with J. A. Schouten), Proc. Akad. Wet. Amsterdam 29, 803-8 1 5 ; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 573-585 .

92 . On Riemannian Geometries admitting an absolute parallelism (with J. A. Schouten), Proc. Akad. Wet. Amsterdam 29, 933-946; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 067- 1 080.

93 . Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France 54, 2 1 4-264; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 587-637.

LIST OF PUBLICATIONS OF ELIE CARTAN 25 1

1927

94. Sur une classe remarquable d 'espaces de Riemann, Bull. Soc. Math. France 55 , 1 1 4- 1 34; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 639-659.

95 . Sur /es courbes de torsion nu/le et /es surfaces developpables dans /es espaces de Riemann, C. R. Acad. Sci. Paris 184, 1 38- 1 40; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 08 1 - 1 083 .

96. Sur /es geodesiques des espaces de groupes simples, C. R. Acad. Sci . Paris 184, 862-864; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 66 1 -663 .

97 . Sur la topologie des groupes continus simples reels, C. R. Acad. Sci. Paris 184, 1 036- 1 038 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 664-666.

98 . Sur l 'ecart geodesique et quelques questions connexes, Rend. Accad. Lincei 51 , 609-6 1 3 ; <Euvres completes: Partie Ill, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 085- 1 089.

99. Sur certaines formes riemanniennes remarquables des geometries a groupe fondamental simple, C. R. Acad. Sci. Paris 184, 1 628- 1 630; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 9 52, pp. 667-669.

1 00. Sur /es formes riemanniennes des geometries a groupe fondamental sim­ple, C. R. Acad. Sci. Paris 185, 96-98; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 9 52, pp. 670-672.

1 0 1 . La geometrie des groupes de transformations, J. Math. Pures Appl. 6, 1 - 1 1 9; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 673-79 1 .

1 02. Sur certains cycles arithmetiques, Nouvelles Ann. 2, 33-45; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 89- 1 0 1 .

1 03 . La geometrie des groupes simples, Ann. Mat. 4, 209-256; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 793-840.

104. Sur la possibilite de plonger un espace riemannien donne dans un es­pace euclidien, Ann. Polon. Math. 6, 1 -7; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 09 1 - 1 097.

1 05 . La theorie des groupes et la geometrie, Enseign. Math. 26, 200-225; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 84 1 -866.

1 06 . Rapport sur le memoire de J. A. Schouten intitule "Erlanger programm und Ubertragungslehre. Neue Gesichtspunkte zur Grundlegung der Ge­ometrie", Izv. Kazan Fiz.-Mat. Obshch. 2, 7 1 -76; <Euvres completes:

2S2 LIST OF PUBLICATIONS OF ELIE CARTAN

Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 099- 1 1 04.

1 07. Sur certaines formes riemanniennes remarquables des geometries a groupe fondamental simple, Ann. Sci. Ecole Norm. Sup. 44, 345-467; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 867-989.

1 08. Sur un probleme du calcul des variations en geometrie projective plane, Mat. Sb. 34, 349-364; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 05- 1 1 1 9.

1 08a. Riemannian geometry in an orthogonal frame (Cartan 's 1 926- 1 92 7 lectures in Sorbonne) , Izdat. Moskov. Univ. , Moscow, 1 960. (Rus­sian)

1928

1 09. Sur !es systemes orthogonaux complets de fonctions dans certains es­paces de Riemann clos, C. R. Acad. Sci. Paris 186, 1 594- 1 596; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 99 1-993 .

1 1 0. Sur /es espaces de Riemann clos admettant un groupe continu transitif de deplacements, C. R. Acad. Sci. Paris 186, 1 8 1 7- 1 8 1 9; <Euvres completes: Partie I , Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 995-997.

1 1 1 . Sur /es nombres de Betti des espaces de groupes clos, C. R. Acad. Sci. Paris 187, 1 96- 1 98 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 999- 1 00 1 .

1 1 2. Sur la stabilite ordinaire des ellipsoi"des de Jacobi, Proc. lnternat. Math. Congress Toronto 1, 9- 1 7; <Euvres completes: Partie Ill, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 03- 1 1 1 .

1 1 3 . Complement au memoire "Sur la geometrie des groupes simples", Ann. Mat. Pura Appl. ( 4) 5, 253-260; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 003- 1 0 1 0.

1 1 4. Le(;ons sur la geometrie des espaces de Riemann, Gauthier-Villars, Paris.

1 1 5 . Sur !es substitutions orthogonales imaginaires, Assoc. Avanc. Sci­ences, Congres de La Rochelle, pp. 38-40; <Euvres completes: Par­tie II, Algebre. Formes differentielles, systemes differentie/s, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 249-250.

1929

1 1 6 . Groupes simples clos et ouverts et geometrie riemannienne, J. Math. Pures Appl. 8, 1 -33 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 0 1 1 - 1 043.

1 1 7. Sur la determination d 'un systeme orthogonal comp/et dans un espace de Riemann symetrique clos, Rend. Circ. Mat. Palermo 53, 2 1 7-

LIST OF PUBLICATIONS OF ELIE CARTAN 253

252; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 1 045- 1 080.

1 1 8 . Sur /es invariants integraux de certains espaces homogenes clos et /es proprietes topologiques de ces espaces, Ann. Polon. Math. 8, 1 8 1 -225; Selecta. Jubile scientifique de M. Elie Cartan, Gauthier-Villars, Paris, 1 939, pp. 203-233 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 08 1 - 1 1 25 .

1 1 9 . Sur la representation geometrique des systemes materiels non holo­nomes, Atti Cong. Internat. Mat. (Bologna, 1 928) , vol. 4, pp. 253-26 1 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 3- 1 2 1 .

1 20. Sur /es espaces clos admettant un groupe transitif clos fini et continu, Atti Cong. Internat. Mat. (Bologna, 1 928 ) , vol. 4, pp. 243-252; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 1 1 27- 1 1 36 .

1930

1 2 1 . Les representations lineaires du groupe des rotations de la sphere, C. R. Acad. Sci. Paris 190, 6 1 0-6 1 2; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 1 37- 1 1 39.

1 22 . Les representations lineaires des groupes simples et semi-simples clos, C. R. Acad. Sci. Paris 190, 723-725 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 1 40-1 1 42.

1 23 . Le troisieme theoreme fondamental de Lie. I, C. R. Acad. Sci. Paris 190, 9 1 4-9 1 6; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 1 43- 1 1 45 .

1 23a. Le troisieme theoreme fondamental de Lie. II , C. R. Acad. Sci. Paris 190, 1 005- 1 007; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 1 46- 1 1 48.

1 24. Notice historique sur la notion de parallelisme absolu, Math. Ann. 102, 698-706; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 2 1 - 1 1 29.

1 25 . Sur /es representations lineaires des groupes clos, Comment. Math. Helv. 2, 269-283 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 1 49- 1 1 63 .

1 26 . Sur un probleme d 'equivalence et la theorie des espaces metriques gene­ralises, Mathematica 4, 1 1 4- 1 36 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 3 1- 1 1 53 .

1 27 . Geometrie projective et geometrie riemannienne, Trudy I Vsesoyuz. Matern. S'ezda, Khar'kov, 1 930, 1 79- 1 90; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 5 5- 1 1 66 .

254 LIST OF PUBLICATIONS OF ELIE CARTAN

1 28 . La theorie des groupes finis et continus et I 'Analysis situs, Memorial Sci. Math. XLII, 2nd ed. , Gauthier-Villars, Paris, 1 9 52; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 9 52, pp. 1 1 65- 1 225 .

1931

1 29 . Geometrie euclidienne et geometrie riemannienne, Scientia (Milano) , 393-402.

1 30. Le parallelisme absolu et la theorie unitaire du champ, Rev. Metaph. Morale, pp. 1 3-28 ; Actualites Sci. Indust . , no. 44, Hermann, Paris, 1 932; 2nd ed. , 1 974; muvres completes: Partie III, Divers, geometrie di.fferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 67- 1 1 8 5 .

1 3 1 . Sur la theorie des systemes en involution et ses applications a la Rela­tivite, Bull. Soc. Math. France 59, 88- 1 1 8 ; muvres completes: Partie II, Algebre. Formes di.fferentielles, systemes di.fferentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 1 99- 1 229.

1 32. Sur /es developpantes d 'une surface regle, Bull. Acad. Roumaine 14, 1 67- 1 74; muvres completes: Partie III, Divers, geometrie di.fferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 87- 1 1 94.

1 33 . Le groupe fondamental de la geometrie des spheres orientees reel/es, Assoc. Avanc. Sciences, Nantes, pp. 2 1 -28; muvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 1 95- 1 202.

1 34. Ler;ons sur la geometrie projective complexe, Gauthier-Villars, Paris; 2nd. ed., 1 950.

1932

1 35 . Sur le groupe de la geometrie hyperspherique, Comment. Math. Helv. 4, 1 58- 1 7 1 ; muvres completes: Partie III, Divers, geometrie di.fferen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 9 5 5 , pp. 1 203- 1 2 1 6 .

1 36. Sur la geometrie pseudo-conforme des hypersurfaces de l 'espace de deux variables complexes. I, Ann. Mat. Pura Appl. (4 ) 1 1 , 1 7-90; muvres completes: Partie II, Algebre. Formes di.fferentielles, systemes dijferen­tiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 23 1 - 1 304.

1 36a. Sur la geometrie pseudo-conforme des hypersurfaces de·l 'espace de deux variables complexes. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1 , 333-354; muvres completes: Partie III, Divers, geometrie di.fferentiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 2 1 7- 1 238 .

1 37 . Sur /es proprietes topologiques des quadriques complexes, Puhl. Math. Univ. Belgrade 1 , 55-74; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 227- 1 246.

1 38 . Les espaces riemanniens symetriques, Verb. Internat. Math . Kon­gresses Zurich, vol. I, pp. 1 52- 1 6 1 ; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 247-1 256 .

LIST OF PUBLICATIONS OF ELIE CARTAN 255

1 39 . Sur / 'equivalence pseudo-conforme de deux hypersurfaces de / 'espace de deux variables complexes, Verb. Internat. Math. Kongresses Zurich, vol . II, pp. 54-56; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 1 305- 1 306.

1933

1 40. Les espaces metriquesfondes sur la notion d'aire, Exposes de Geometrie, vol . I, Hermann, Paris.

1 40a. La cinematique newtonienne et /es espaces a connexion euclidienne, Bull. Math. Soc. Sci. Math. R. S. Roumanie 35 ( 1 933 ) , 69-73; muvres completes: Partie III , Divers, geometrie differentielle, vols. 1 -2 , Gauthier-Villars, Paris, 1 955 , pp. 1 239- 1 243 .

1 40b. Observations sur: St. Golq,b. Sur la representation conforme de l 'espace de Fins/er sur l 'espace euclidien, C. R. Acad. Sci. Paris 196, 27-29.

1 4 1 . Sur /es espaces de Fins/er, C. R. Acad. Sci. Paris 196, 582-586; muvres completes: Partie III , Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 245- 1 248.

1 4 1 a. Observations sur le memoire precedent (lettre a D. D. Kosambi) , Math. Z. 37, 6 1 9-622; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 249- 1 252.

1934

1 42. Les espaces de Fins/er, Exposes de Geometrie, vol . II, Hermann, Paris. 1 42a. Remarques au sujet de la Communication de M. Andre Weil, C. R.

Acad. Sci. Paris 198, 1 742- 1 743; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 257- 1 258 .

1 43 . Le ca/cul tensoriel en geometrie projective, C. R. Acad. Sci . Paris 198, 2033-2037; muvres completes: Partie III, Divers, geometrie differen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 253- 1 257 .

1 43a. La theorie unitaire d'Einstein-Mayer, preprint; muvres completes: Par­tie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 863- 1 875 .

1935

1 44. La methode du repere mobile, la theorie des groupes continus et /es espaces generalises, Exposes de Geometrie, vol. V, Hermann, Paris; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2 , Gauthier-Villars, Paris, 1 955 , pp. 1 259- 1 320.

1 45 . Sur /es domaines bornes homogenes de l 'espace de n variables com­plexes, Abh. Math. Sem. Univ. Hamburg 1 1 , 1 1 6- 1 62; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 259- 1 305 .

1 45a. Remarques au sujet d 'une communication de M. L. Pontrjagin sur /es nombres de Betti des groupes de Lie, C. R. Acad. Sci . Paris 200, 1 280-1 28 1 .

256 LIST OF PUBLICATIONS OF ELIE CARTAN

1 46. Observations sur une Note de M. G. Bouligand, C. R. Acad. Sci. Paris 201 , 702; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 32 1 .

1 47 . Le calcul tensoriel projectif, Mat. Sb. 42, 1 3 1 - 1 47; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 323- 1 339 .

1 47a. Sur une degenerescence de la geometrie euclidienne, Assoc. Avanc. Sciences, Nantes, pp. 1 28- 1 30; this book, Appendix B.

1936

1 48 . La geometrie de l 'integrale J F(x , y , y' , y" ) dx , J. Math. Pures Appl. 15, 42-69; muvres completes: Partie Ill, Divers, geometrie differen­tiel/e, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 34 1 - 1 368.

1 49. Sur !es champs d 'acceleration uniforme en Relativite restreinte, C. R. Acad. Sci. Paris 202, 1 1 25- 1 1 28; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 369- 1 372.

1 50. La topo/ogie des espaces representatife des groupes de Lie, Exposes de Geometrie, vol . VIII, Hermann, Paris; Enseign. Math. 35, 1 77-200; Selecta. Jubile scientifique de M. Elie Cartan, Gauthier-Villars, Paris, 1 939, pp. 235-258 ; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 307- 1 3 30.

1 5 1 . Le role de la theorie des groupes de Lie dans ! 'evolution de la geometrie moderne, C. R. Congres Math. Internat. (Oslo) , vol. 1 , pp. 92- 1 03 ; (Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 373- 1 384.

1937

1 52. Les espaces de Fins/er, Trudy Sem. Vektor. Tenzor. Anal. 4, 70-8 1 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 385- 1 396.

1 53 . Les espaces a connexion projective, Trudy Sem. Vektor. Tenzor. Anal. 4, 1 47- 1 59 ; muvres completes: Partie III, Divers, geometrie differen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 397- 1 409.

1 54. La topologie des espaces homogenes clos, Trudy Sem. Vektor. Tenzor. Anal. 4, 388-394; (Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 33 1 - 1 337 .

1 5 5 . Le�ons sur la theorie des espaces a connexion projective, Gauthier­Villars, Paris.

1 56 . L 'extension du ca/cul tensoriel aux geometries non-a/fines, Ann. of Math. (2) 38, 1 - 1 3 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 4 1 1 - 1 423 .

1 57 . La theorie des groupes finis et continus et la geometrie differentielle traitees par la methode du repere mobile, Gauthier-Villars, Paris; 2nd ed. , 1 95 1 .

LIST OF PUBLICATIONS OF ELIE CARTAN 257

1 58 . Le role de la geometrie analytique dans / 'evolution de la geometrie, Travaux du Xlth Congres Internat. Philosophie (Congres Descartes) , vol . VI, Paris, 1 47- 1 53 ; Actualites Sci. Indust . , no. 535 , Hermann, Paris.

1 59 . Les groupes, Encyclopedie Fran�aise, vol. 1 , 3rd part, 1 . 66- 1 - 1. 66-8. 1 60. La geometrie et la theorie des groupes, Encyclopedie Fran�aise, vol. 1 ,

3rd part, 1 .88- 1 2-1.90-2. 1 6 1 . La geometrie riemannienne et ses generalisations, Encyclopedie Fran­

�aise, vol. 1 , 3rd part, I. 90-3 - I. 90-8. 1 6 1 a. Les problemes d'equivalence, Seminaire de Math. expose D, 1 1 jan­

vier 1 937 ; Selecta. Jubile scientifique de M. Elie Cartan, Gauthier­Villars, Paris, 1 939, pp. 1 1 3- 1 36 ; <Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 1 3 1 1 - 1 334.

1 6 1 b. La structure des groupes in.finis, Seminaire de Math. , exposes G et H, l er et 1 5 mars 1 937 , pp. 1 -50; <Euvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier­Villars, Paris, 1 953 , pp. 1 335- 1 384.

1938

1 62. Les representations lineaires des groupes de Lie, J. Math. Pures Appl. 17, 1 - 1 2; Selecta. Jubile scienti.fique de M. Elie Cartan, Gauthier­Villars, Paris, 1 939, pp. 1 53- 1 64; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 339- 1 3 5 1 .

1 63 . Les espaces generalises et / 'integration de certaines classes d'equations differentielles, C. R. Acad. Sci. Paris 206, 1 689- 1 693 ; <Euvres com­pletes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier­Villars, Paris, 1 955 , pp. 1 425- 1 429.

1 64. Le�ons sur la theorie des spineurs. I, II, Exposes de Geometrie, vol. XI, Hermann, Paris; English transl . , Hermann, Paris and MIT Press, 1 966; 2nd ed. , Dover, New York, 1 98 1 .

1 65 . La theorie de Galois et ses generalisations, Comment. Math. Helv. 1 1 , 9-25 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 23- 1 39 .

1 66 . Families de surfaces isoparametriques dans /es espaces a courbure con­stante, Ann. Mat. Pura Appl. ( 4) 27, 1 77- 1 9 1 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Pari� 1 955 , pp. 1 43 1 - 1 445 .

1939

1 67 . Sur des families remarquables d'hypersurfaces isoparametriques dans !es espaces spheriques, Math. Z. 45, 335-367; - <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 447- 1 479.

258 LIST OF PUBLICATIONS OF ELIE CARTAN

1 68 . Sur quelques families remarquables d' hypersurfaces, C. R. Congres Math. de Liege, pp. 30-4 1 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 95 5 , pp. 1 48 1 - 1 492.

1 69 . Le ca/cul differentiel absolu devant /es problemes recents de geometrie riemannienne, Atti Fondaz. Alessandra Volta 9, 443-46 1 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 493- 1 5 1 1 .

1940

1 70. Sur un theoreme de J. A. Schouten et W. van der Kulk, C. R. Acad. Sci. Paris 211 , 2 1 -24; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 95 3, pp. 1 307- 1 3 1 0.

1 7 1 . Sur /es groupes lineaires quaternioniens, Viertelj schr. Naturforsch. Ges. Zilrich, SS, 1 9 1 -203; muvres completes: Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, Gauthier-Villars, Paris, 1 953 , pp. 25 1 -263 .

1 72 . Sur des families d 'hypersurfaces isoparametriques des es paces spheriques a 5 et 9 dimensions, Univ. Nae. Tucuman. Revista A 1 , 5-22; muvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 5 1 3- 1 530.

1941

1 73 . Sur /es surfaces admettant une seconde forme fondamentale donnee, C. R. Acad. Sci. Paris 212, 825-828; muvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 53 1 - 1 534.

1 74. La geometria de las ecuaciones diferenciales de tercer orden, Revista Mat. Hisp.-Amer. ( 1 ) 1 , 3-33 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp . 1 535- 1 565 .

1 75 . La notion d'orientation dans /es differentes geometries, Bull. Soc. Math. France 69, 47-70; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 569- 1 570.

1942

1 76. Sur /es couples de surfaces applicables avec conservation des courbures principales, Bull. Sci. Math. (2) 66, 5 5-85 ; muvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 59 1 - 1 62 1 .

1 77. Les surfaces isotropes d'une quadrique de l 'espace a sept dimensions, preprint.

UST OF PUBLICATIONS OF ELIE CARTAN 259

1943

1 78 . Sur une c/asse d'espaces de Wey!, Ann. Sci . Ecole Norm. Sup. ( 3 ) 60, 1 - 1 6; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 62 1 - 1 636 .

1 79. Les surfaces qui admettent une seconde forme fondamentale donnee, Bull. Sci. Math. (2) 67, 8-32; <Euvres completes: Partie Ill, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 637- 1 66 1 .

1944

1 80. Sur une c/asse de surfaces apparentees aux surfaces R et aux surfaces de Jonas, Bull. Sci. Math. (2) 68, 4 1 -50; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 663- 1 672.

1945

1 8 1 . Les systemes differentiels exterieurs et leurs applications geometriques, Actualites Sci. Indust. , no. 994, Hermann, Paris.

1 82. Sur un probleme de geometrie differentielle projective, Ann. Sci. Ecole Norm. Sup. (3 ) 62, 205-23 1 ; <Euvres completes: Partie III, Divers, geometrie dijferentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 673- 1 699.

1946

1 83 . Ler;ons sur la geometrie des espaces de Riemann, 2nd ed., Gauthier­Villars, Paris; English transl. in Lie Groups, History, Frontiers and Applications, vol. 1 3, Math. Sci. Press, Brookline, MA, 1 983 .

1 84. Quelques remarques sur !es 28 bitangentes d 'une quartique plane et !es 27 droites d 'une surface cubique, Bull. Sci. Math. (2) 70, 42-45 ; <Euvres completes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier­Villars, Paris, 1 952, pp. 1 353- 1 356 .

1947

1 85 . Sur l 'espace anallagmatique reel a n dimensions, Ann. Polon. Math. 20, 266-278; <Euvres completes: Partie III, Divers, geometrie differen­tielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 70 1 - 1 7 1 3 .

l 85a. La theorie des groupes, Alen!ton, Paris.

1949

1 86 . Deux theoremes de geometrie anallagmatique reelle a n dimensions, Ann. Mat. Pura Appl. (4) 28, 1 - 1 2 ; <Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1 -2, Gauthier-Villars, Paris, 1 955 , pp. 1 7 1 5- 1 726.

260 LIST OF PUBLICATIONS OF ELIE CARTAN

List of Cartan's works in the history of science and his reminiscences

1931

1 87. Notices sur /es travaux scientifiques, Paris; Selecta. Jubile scientifique de M. Elie Cartan, Gauthier-Villars, Paris, 1 939, 1 5- 1 1 2; <Euvres com­pletes: Partie I, Groupes de Lie, vols. 1 -2, Gauthier-Villars, Paris, 1 952, pp. 1 - 1 0 1 ; Gauthier-Villars, Paris; 2nd ed. , 1 974.

1937

1 88 . Discours prononce a / 'inauguration d 'un buste eleve a la memoire de Gaston Darboux a Nfmes le dimanche 22 octobre 1 933 , Notices et Discourrs Acad. Sci. Paris 1 924- 1 936 , pp. 437-478.

1939

1 89 . A/locution a la Sorbonne 18 mai 1 939 , in Jubile scientifique de M. Elie Cartan cetebre a la Sorbonne 1 8 mai 1 939, Gauthier-Villars, Paris, 1 939, pp. 5 1 -59; this book, Appendix C.

1941

1 90. Charles Maurain, Jubile de Charles Maurain, Paris, pp. 5- 1 4. 1 9 1 . Le rfJ/e de la France dans le developpement des mathematiques, preprint;

English transl. , this book, Appendix D* .

1942

1 92. Notice sur Tullio Levi-Civita, C. R. Acad. Sci. Paris 215, 233-235 .

1943

1 93 . Notice necrologique sur Georges Giraud, C. R. Acad. Sci. Paris 216, 5 1 6-5 1 8 .

1946

1 94. Notice necrologique sur Antoine-Fram;ois-Jacques-Justin-Georges Perrier, C. R. Acad. Sci. Paris 222, 42 1 -423.

1 95 . Notice necro/ogique sur Thomas Hunt Morgan, C . R. Acad. Sci. Paris 222, 705-706.

1 96 . Notice necrologique sur Leon Alexandre Guillet, C. R. Acad. Sci. Paris 222, 1 1 49- 1 1 5 1 .

1 97 . Notice necro/ogique sur Simon Flexner, C. R. Acad. Sci. Paris 222, 1 265- 1 266.

1 98 . Notice necro/ogique sur Louis Martin, C. R. Acad. Sci . Paris 222, 1 4 1 7- 1 4 1 9.

1 98a. Gaspard Monge. Sa vie, son <Euvre, C. R. Acad. Sci . Paris 223 , 1 049-1 054.

•Added in Proof. The Serbian publication (without the preface) : Saturn, 1 940, No. 4-5 , 8 1 -96, No. 6-7, 1 29- 1 44. The publication of Cartan's French text: Publications de l'lnstitut Mathematique (N.S. ) , Beograd 51 (65) , 1 992, 2-2 I .

LIST OF PUBLICATIONS OF ELIE CARTAN 26 1

1 99. Notice necrologique sur Paul Langevin, C. R. Acad. Sci. Paris 223, 1 069- 1 072.

200. L 'a:uvre scientifique de M. Ernest Vessiot, Bull . Soc. Math. France 75, 1 -8 .

1948

20 1 . Un centenaire: Sophus Lie, Les grands courants de la pensee mathema­tique, Cahiers du Sud, pp. 253-257; 2nd ed. , vol. 1 , Paris, 1 962; English transl. , Great currents of mathematical thought, vol. 1 , Dover, New York, 1 97 1 , pp. 262-267.

202. Gaspard Monge: sa vie, son a:uvre, Alen�on, Paris.

1949

203. La vie et l 'a:uvre de Georges Perrier, Annuaire Bureau des Longitudes, Paris, c 1 -c4.

Collections of Cartan's works

204. Selecta. Jubile scientifique de M. Elie Cartan, Gauthier-Villars, Paris, 1 939.

205. Gruppy golonomii obobshchennykh prostranstv. Teoriya grupp i ge­ometriya. Metricheskie prostranstva osnovannye na ponyatii ploshchadi, Series of Monographs and Studies in Non-Euclidean Geometry, no. 1 , Izdat. Kazan. Univ. , Kazan, 1 939 .

206. Geometriya grupp Lie i simmetricheskie prostranstva, Izdat. Inostr. Literat. , Moscow, 1 949.

207. muvres completes: Partie I, Groupes de Lie, vols. 1 -2, 19 52; Partie II, Algebre. Formes differentielles, systemes differentiels, vols. 1 -2, 1 953 ; Partie III, Divers, geometrie differentielle, vols. 1 -2, 1 955 , Gauthier­Villars, Paris.

208. Prostranstva affinnoi, proyektivnoii konformnoi svyaznosti, Series of Monographs and Studies in Non-Euclidean Geometry, no. 3, Izdat. Kazan. Univ. , Kazan, 1 962.

209. muvres completes: Partie I, Groupes de Lie; Partie II, Algebre. Formes differentielles, systemes differentiels; Partie III, Geometrie differentielle. Divers, vols. 1 -2, C. N. R. S. , Paris, 1 984.

209a. On manifolds with an affine connection and the theory of general rela­tivity, Bibliopolis, Naples, 1 986.

Cartan's scientific correspondence

2 1 0. Elie Cartan-Albert Einstein letters on absolute parallelism 1 929- 1 932, Princeton Univ. Press, Princeton, NJ, 1 979.

2 1 1 . Lettres d 'E. Cartan a G. Tzitzeica, A. Pantazi et G. Vranceanu, Elie Cartan, 1 869- 1 95 1 , Hommage de l'Acad. Republique Socialiste de Roumanie, a !'occasion du centenaire de sa naissance, Editura Acad. R.S.R. , Bucharest, 1 975 , pp. 83- 1 1 6 .

APPENDIX A

Rapport sur les Travaux de M. Cartan

fait a la Faculte des Sciences de l'Universite de Paris

PAR H. POINCARE1

. . . . . Le role preponderant de la theorie des groupes en mathematiques a ete longtemps insoup�onne; ii y a quatre-vingts ans, le nom meme de groupe etait ignore. C'est GALOIS qui, le premier, en a eu une notion claire, mais c'est seulement depuis les travaux de KLEIN et surtout de Lie que l'on a commence a voir qu'il n'y a presque aucune theorie mathematique ou cette notion ne tienue une place importante.

On avait cependant remarque comment se font presque toujours les progres des mathematiques; c'est par generalisation sans doute, mais cette generalisa­tion ne s'exerce pas dans un sens quelconque. On a pu dire que la mathe­matique est l'art de donner le meme nom a des choses ditferentes. Le jour ou on a donne le nom d'addition geometrique a la composition des vecteurs, on a fait un progres serieux, si bien que la theorie des vecteurs se trouvait a moitie faite; on en a fait un autre du meme genre quand on a donne le nom de multiplication a une certaine operation portant sur les quaternions. II est inutile de multiplier les exemples, car toutes les mathematiques y passeraient. Par cette similitude de nom, en etfet, on met en evidence une similitude de fait, une sorte de parallelisme qui aurait pu echapper a l'attention. On n'a plus ensuite qu'a calquer, pour ainsi dire, la theorie nouvelle sur une theorie ancienne deja connue.

II faut s'entendre, toutefois: ii faut donner le meme nom a des choses ditferentes, mais a la condition que ces choses soient ditferentes quant · a la matiere, mais non quant a la forme. A quoi tient ce phenomene mathema­tique si souvent constate? Et d'autre part en quoi consiste cette communaute de forme qui subsiste sous la diversite de la matiere? Elle tient a ce que toute theorie mathematique est, en derniere analyse, l'etude des proprietes d'un groupe d'operations, c'est-a-dire d'un systeme forme par certaines operations

1Acta Mathematica 38 ( 1 9 1 4) , 1 37- 1 45 .

263

264 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE

fondamentales et par toutes les combinaisons qu'on en peut faire. Si, dans une autre theorie, on etudie d'autres operations qui se combinent d'apres les memes lois, on verra naturellement se derouler une suite de theoremes corre­spondant un a un a ceux de la premiere theorie, et les deux theories pourront se developper avec un parallelisme parfait; il suffira d'un artifice de langage, comme ceux dont nous parlions tout a l'heure, pour que ce parallelisme de­vienne manifeste et donne presque l'impression d'une identite complete. On dit alors que les deux groupes d'operations sont isomorphes ou bien qu'ils ont meme structure.

Si alors on depouille la theorie mathematique de se qui n'y apparah que comme un accident, c'est-a-dire de sa matiere, il ne restera que l'essentiel, c'est-a-dire la forme; et cette forme, qui constitue pour ainsi dire le squelette solide de la theorie, ce sera la structure du groupe.

On distinguera parmi les groupes possibles quatre categories principales, sans compter certains groupes etranges ou composites qui ne rentrent dans au­cune categorie, ou qui participent des caracteres de deux ou plusieurs d'entre elles. Ce sont:

I. Les groupes discontinus et finis, ou groupes de Galois; ce sont ceux qui president a la resolution des equations algebriques, a la theorie des permutations, etc . . . . . II. Les groupes discontinus et infinis; ce sont ceux que l'on rencontre dans la theorie des fonctions elliptiques, des fonctions fuchsiennes etc . . . . . III. Les groupes continus et finis ou groupes de LIE proprement dits; ce sont ceux auxquels se rattachent les principales theories geome­triques, telles que la geometrie euclidienne, la geometrie non­euclidienne, la geometrie projective, etc . . . . . IV. Les groupes continus et infinis, beaucoup plus complexes, beau­coup plus rebelles aux efforts du geometrie. Ils sont en connexion naturelle avec la theorie des equations aux derivees partielles.

M. CARTAN a fait faire des progres importants a nos connaissances sur

trois de ces categories, la 1 ere , la 3e , et la 4e . 11 s'est principalement

place au point de vue le plus abstrait de la structure, de la forme pure, independamment de la matiere, c'est-a-dire, dans l'espece, du nombre et du choix des variables independantes.

Groupes continus et finis

Je commencerai par les groupes continus et finis, qui ont ete introduits par LIE dans la science; le savant norvegien a fait commaitre les principes fondamentaux de la theorie, et il a montre en particulier que la structure de ces groupes depend d'un certain nombre de constantes qu'il designe par la lettre c affectee d'un triple indice et entre lesquelles il doit y avoir certaines

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 265

relations. II a enseigne egalement comment on pouvait construire le groupe quand on connaissait ces constantes. Mais ii restait a discuter les diverses manieres de satisfaire aux relations qui doivent avoir lieu entre les constantes c ; on pouvait supposer que les divers types de structure seraient extremement nombreux et extremement varies, de sorte que l'enumeration en serait a peu pres impossible. II ne semble pas eu etre tout a fait ainsi, au moins en ce qui conceme les groupes simples.

La distinction entre les groupes simples et les groupes composes est due a GALOIS et elle est essentielle, puisque les groupes composes peuvent toujours etre construits en partant des groupes simples. II est clair que le premier probleme a resoudre est la construction des groupes simples.

Vers 1 890, KILLING a annonce que tous les groupes simples continus et fi­nis rentrent: soit dans quatre grands types generaux deja signales par LIE, soit dans cinq types particuliers dont les ordres sont respectivement 1 4, 52, 78 , 1 33 , et 248. C'etait Ia un resultat d'une tres haute importance; malheureuse­ment toutes les demonstrations etaient fausses; ii ne restait que des aper�us denues de toute force probante.

II etait reserve a M. CARTAN de transformer ces aper�us en demonstrations rigoureuses; ii suffit d'avoir lu le memoire de KILLING pour comprendre com­bien cette tache etait difficile. La methode repose sur la consideration de l'equation caracteristique, et en particulier de la forme quadratique 1/1, (e) qui est le coefficient de w'-2 dans cette equation; cette consideration per­met de reconnaitre si le groupe integrable, ou de trouver son plus grand sous groupe invariant integrable, ou enfin de reconnaitre si le groupe est simple ou semisimple.

M. CARTAN a donne une maniere de former, dans chaque type, les groupes lineaires simples dont le nombre des variables est aussi petit que possible.

Une des plus importantes applications des groupes de LIE est l'integration des equations differentielles ordinaires ou partielles qui sont inalterees par les transformations d'un groupe. M. CARTAN a applique cette methode au cas des systemes d'equations aux derivees partielles don't l'integrale generate ne depend que de constantes arbitraires. Les operations a faire sont toutes de nature rationnelle ou algebrique.

Groupes discontinus et finis

M. CARTAN a fait faire aussi un progres important a la theorie des groupes de GALOIS, en les rattachant a celle des nombres complexes. On sait qu'on designe par nombres complexes des expressions algebriques susceptibles de subir des operations qui peuvent etre regardees comme des generalisations de l'addition et de la multiplication, et auxquelles on peut appliquer les regles ordinaires du calcul avec cette difference que la multiplication, quoique as­sociative, n'est pas commutative. La plus connu des systemes de nombres

266 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE

complexes a re�u le nom de quaternions et on en a fait des applications nombreuses en Mecanique et en Physique Mathematique.

Ces nombres complexes ont un lien intime avec les groupes de Lie et en particulier avec les groupes lineaires simplement transitifs; il y a, a ce su­jet, un theoreme de M. POINCARE dont M. CART AN a donne une nouvelle demonstration. La theorie des nombres complexes a ete poussee plus loin par M. M. ScHEFFERS et MoLLIEN qui en ont entrepris la classification et ont les premiers mis en evidence !'importance de la distinction entre les systemes a quaternions et les systemes sans quaternions.

M. CARTAN est arrive a resoudre completement le probleme, par une heureuse adaptation des methodes qui lui avaient reussi dans l'etude des groupes de Lie. 11 a pris comme point de depart une equation caracteristique qui n'est pas tout a fait la meme que celle qu'on envisage a propos des groupes de Lie, mais qui se prete a une discussion analogue. M. CARTAN a montre comment on peut construire un systeme quelconque par la combinaison d'un systeme pseudonul et de systemes simples et comment les systemes simples se reduisent aux quaternions generalises; comment enfin les systemes dits de la 2e classe se deduisent facilement de ceux de la 1 ere classe. 11 a etudie aussi le cas oil les coefficients sont des nombres reels.

Ces resultats ne constituent pas, comme on pourrait etre tente de la croire, une simple curiosite mathematique. 11s sont au contraire susceptibles d'appli­cations nombreuses. En particulier, ils se rattachent a la theorie des groupes de GALOIS; il est clair que les lois de la composition des substitutions d'un groupe de GALOIS sont associatives, sans etre commutatives; elles peuvent done etre regardees comme les regles de la multiplication d'un systeme d'unites complexes; et par consequent elles definissent un systeme de nom­bres complexes. Or si on applique a ce systeme le theoreme de M. CARTAN, on retrouve, de la fa�on la plus simple et pour ainsi dire d'un trait de plume, les resultats que M. FROBENIUS avait obtenus par une tout autre voie et qui avaient ete regardes a juste titre comme le plus grand progres que la theorie des groupes de GALOIS eut fait depuis longtemps.

On peut, par cette voie, reconnaitre quels sont les groupes lineaires les plus simples qui sont isomorthes a un groupe de GALOIS donne, ce qui nous conduit au probleme de !'integration algebrique des equations differentielles lineaires. M. PoINCARE a eu !'occasion d'appliquer les principes de M. CAR­T AN a !'integration algebrique d'une equation lineaire.

Groupes continus et infinis

La determination des groupes continus infinis presente beaucoup plus de difficultes que celle des groupes finis et c'est la que M. CARTAN a deploye le plus d'originalite et d'ingeniosite. 11 s'est restreint d'ailleurs a une certaine classe de groupes infinis, la plus importante au point de vue des applica­tions, et celle sur laquelle !'attention de Lie avait surtout ete attiree, je veux

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 267

parler des groupes dont les transformations finies dependent de fonctions ar­bitraires d'un ou de pluseurs parametres, ou, plus generalement, de ceux ou les variables transformees, considerees comme fonctions des variables prim­itives, constituent !'integral general d'un systeme d'equations aux derivees partielles.

M. CARTAN s'est d'ailleurs servi, dans cette etude, de resultats importants qu'il avait obtenus dans des travaux anterieurs relatifs aux equations aux derivees partielles et aux equations de PFAFF, travaux dont nous parlerons plus loin.

La theorie de la structure, telle que LIE l'expose dans l'etude des groupes finis, n'est pas susceptible d'etre immediatement generalisee et etendue aux groupes infinis. M. CARTAN lui substitue done une autre theorie de la struc­ture, equivalente a la premiere en ce qui concerne les groupes finis, mais sus­ceptible de generalisation. Si f est une fonction quelconque des variables x , et si les XJ representent les symboles de Lie, on aura identiquement:

les w; etant des expressions de Pfaff dependant des parametres du groupe et de leurs diff erentielles.

Au lieu de faire jouer le role essentiel aux symboles XJ , comme le faisait LIE, M. CARTAN l'attribue aux expressions de PFAFF w qui sont invariantes par les substitutions du groupe des parametres. Les relations qui definissent la structure se presentent alors sous une autre forme. Au lieu de relations lineaires entre les XJ et leurs crochets, nous aurons des relations lineaires entre les covariants bilineaires des w et des combinaisons bilineaires de ces meme expressons. Le coefficients de ces relations sont les memes dans les deux cas, quoique dans un autre ordre; ce sont les constantes c de LIE.

Sans sortir encore du domaine des groupes finis, M. CARTAN a illustre cette theorie nouvelle en l'appliquant a des exemples concrets, et en particulier au groupe des deplacements de l'espace; il a montre comment elle se rattachait a la theorie classique du triedre mobile de M. DARBoux et comment elle permettait l'etude des invariants differentiels des surfaces et en particulier de ceux de certaines surfaces imaginaires remarquables.

Voyons maintenant comment ces notions peuvent etre etendues aux groupes infinis. La notion d'isomorphisme holoedrique peut etre facilement definie en ce qui concerne les groupes finis, parce que l'on n'a qu'a faire cor­respondre une a une les transformations infinitesimales des deux groupes a comparer. Nous ne pouvons plus employer ce procede lorsque les transfor­mations infinitesimales sont en nombre infini; M. CARTAN donne done une definition differente, quoique equivalente a la premiere dans le cas ou celle-ci a un sens. Un groupe est le prolongement d'un autre quand il transforme les memes variables que cet autre et de la meme maniere et qu'il transforme

268 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE

en meme temps d'autres variables auxiliaires. Par exemple, le groupe des deplacements des points de l'espace aura pour prolongement le groupe des deplacements des droites ou celui des cercles de l'espace. Deux groupes sont alors isomorphes quand deux de leurs prolongements sont semblables.

La theoreme fondamental de LIE peut alors etre etendu aux groupes infinis; on montre que tout groupe infini est isomorphe au groupe qui laisse invari­antes a la fois certaines fonctions U et certaines expressions de PFAFF w et iiJ . Les differentielles totales des U s'experiment lineairement en fonctions des w , les covariants bilineaires des w (mais non ceux des ro) s'expriment bilineairement en fonctions des w et iiJ • Les coefficients de ces relations lineaires ou bilineaires jouent le role des constantes c de LIE. Ce sont des fonctions des invariants U . Ce qui caracterise les groupes transitifs, c'est qu'il n'y a pas d'invariants et par consequent que les coefficients se reduisent a des constantes. Ce qui caracterise les groupes finis, c'est que les expressions n'existent pas.

Les coefficients en question peuvent-ils etre choisis arbitrairement? Non, ils sont assujettis a certaines conditions que M. CARTAN determine et que peuvent etre regardees comme la generalization des conditions de structure de LIE.

Les trois theoremes fondamentaux de LIE se trouvent done etendus aux groupes infinis, de sorte que M. CARTAN a fait pour ces groupes ce que LIE avait fait pour les groupes finis.

Cette analyse a mis en evidence des resultats tout a fait surprenants. Un groupe fini est toujours isomorphe a un groupe transitif, par exemple a celui qu'on appelle son groupe parametrique, et on aurait pu etre tente de croire qu'il en etait de meme pour les groupes infinis, puisqu'au premier abord la demonstration ne semblait mettre en reuvre que la notion generale de groupe. Au contraire, M. CARTAN a montre qu'il existe les groupes infinis qui ne sont isomorphes a aucun groupe transitif.

Ce n'est pas tout: un groupe infini peut etre meriedriquement isomorphe a lui-meme, un groupe infini peut n'admettre aucun sous groupe invariant maximum, etc. , . . . . La notion du prolongement normal permet ensuite a M. CARTAN de determiner tous les groupes isomorphes a un groupe infini donne. Citons un resultat particulier. Les groupes qui ne dependent que de fonctions arbitraires d'un argument, s'ils sont transitifs, sont isomorphes au groupe general d'une variable.

Etant donne un groupe defini par ses equations de structure, M. CARTAN montre qu'on peut determiner les equations de structure de tous ses-groupes par des procede purement algebriques et applique cette methode a des cas particuliers tels que celles du groupe general de deux variables ou i1 retrouve, par une voie nouvelle, quelques sous groupes deja connus et importants par leurs applications.

Si l'on se donne deux systemes differentiels et un groupe, on peut se deman­der s'il y a des transformations du groupe qui transforment un des systemes

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 269

dans l'autre et quelles elles sont; on peut se demander egalement s'il y a dans le groupe des transformations qui n'altereront pas l'un de ces systemes differentiels et qui naturellement formeroil.t un sous-groupe. L'etude de ce sous-groupe a fait egalement l'objet d'un memoire de M. CARTAN .

Enfin M. CARTAN s'est propose en ce qui concerne les groupes infinis, le meme probleme qu'il avait resolu pour les groupes finis, la formation de tous les groupes simples. 11 a montre qu'ici aussi, les groupes simples peuvent se ramener a un nombre restreint de types; ceux qui sont primitifs et d'ou l'on peut deduire tous les groupes transitifs simples se repartissent en six grandes classes; quant aux groupes simples qui ne sont isomorphes a aucun groupe transitif, ils peuvent etre deduits des precedents par des procedes des procedes que M. CARTAN nous fait connahre. ·

Le probleme propose se trouve done entierement resolu.

Equations aux derivt�es partielles

Le probleme de l'integration d'un systeme d'equations aux derivees par­tielles a fait l'objet de travaux nombreux. M. CARTAN s'est place pour l'etudier a un point de vue particulier; il remplace le systeme d'equations aux derivees partielles par le systeme correspondant d'equations de PFAFF, c'est-a-dire d'equations aux differentielles totales.

Dans la theorie des expressions de PFAFF, il y a une notion, introduite par M. M. FROBENIUS et DARBoux, qui joue un role extremement impor­tant, c'est celle du covariant bilineaire; nous avons deja vu apparahre ce covariant a propos de la theorie des groupes infinis. M. CART AN en a donne une interpretation nouvelle a l'aide du calcul de GRASSMANN, et cette in­terpretation l'a conduit a une generalisation. De chaque expression de PFAFF, il deduit une serie d'expressions differentielles qu'il appelle ses derivees; la derivee premiere est la covariant bilineaire; la derivee ne est n + 1 fois lineaire. C'est en cherchant quelle est la premiere de ces derivees qui s'annule identiquement que l'on reconnaitra si, et jusqu'a quel point, il est possi­ble de reduire le nombre des variables independantes sur lesquelles porte l' expression.

Cette consideration a permis a M. CART AN de retrouver sous une forme extremement simple tous les resultats connus relatifs au probleme de PFAFF et un assez grand nombre de resultats entierement nouveaux.

Comment maintenant cela peut-il servir a la resolution d'un systeme d'equations de PFAFF, et surtout a reconnaitre quel est le degre d'arbitraire que comporte l'integrale generale d'un pareil systeme? C'est en se servant de la notion d'involution que M. CARTAN a resolu cette question. Un systeme est dit en involution si, jusqu'a une certaine valeur de m , par toute mul­tiplicite integrale a m dimensions passe une multiplicite integrale a m + l dimensions. M. CARTAN donne une maniere de reconnaitre si un systeme est

270 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE

en involution pour les valeurs de m inf erieures a un nombre donne, et, par la, de savoir combien la solution generale contient de fonctions arbitraires de 1 , de 2, . . . , de n variables.

On retrouve ainsi sous une forme nouvelle la theorie des caracteristiques de CAUCHY, celle des caracteristiques de MONGE, celle des solutions singulieres, etc. , . . . ; on retrouve egalement sous une forme plus simple tous les resultats de M. RIQUIER.

M. CARTAN a applique sa methode a un certain nombre de cas particuliers oil l'integration peut se faire par des equations differentielles ordinaires. II l'a egalement completee en s'aidant de la theorie des groupes qui lui etait si familiere; ii a ainsi reconnu des cas oil l'on peut determiner les invariants d'un systeme de PFAFF, sans en determiner les caracteristiques, c'est-a-dire d'une fa�on rationnelle, et d'autres oil les caracteristiques s'obtiennent sans integration.

Conclusions

On voit que les problemes traites par M. CARTAN sont parmi les plus im­portants, les plus abstraits et les plus generaux don't s'occupent les Mathema­tiques; ainsi que nous l'avons dit, la theorie des groupes est, pour ainsi diew, la Mathematique entiere, depouillee de sa matiere et reduite a une forme pure. Cet extreme degre d'abstraction a sans doute rendu mon expose un peu aride; pour faire apprecier chacun des resultats, ii m'aurait fallu pour ainsi dire lui restituer la matiere dont ii avait ete depouille; mais cette resti­tution peut se faire de mille fa�ons differentes; et c'est cette forme unque que l'on retrouve ainsi sous une foule de vetements divers, que constitue le lien commun entre des theories mathematiques qu'on s'etonne souvent de trouver si voisines.

M. CARTAN en a donne recemment un exemple curieux. On connait l'importance en Physique Mathematiques de ce qu'on a appele le groupe de LORENTZ; c'est sur ce groupe que reposent nos idees nouvelles sur le principe de relativite . et sur Dynamique de l'Electron. D'un autre cote , LAGUERRE a autrefois introduit en geometrie un groupe de transformations qui changent les spheres en spheres. Ces des groupes sont isomorphes, de sorte que mathematiquement ces deux theories, l'une physique, l'autre geome­trique, ne presentent pas de difference essentielle.

Les rapprochements de ce genre se presenteront en foule a ceux qui etudieront avec soin les travaux de LIE et de M. CARTAN. M. CARTAN n'en a pourtant signale qu'un petit nombre, parce que, courant au plus presse, il s'est attache a la forme seulement et ne s'est preoccupe que rarement des diverses matieres dont on la pouvait revetir.

Les resultats les plus importants enonces par M. CARTAN lui appartien­nent bien en propre. En ce qui concerne les groupes de LIE, on n'avait que

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 27 1

des enonces et pas de demonstration; en ce qui concerne les groupes de GA­LOIS, on avait les theoremes de FROBENIUs qui avaient ete rigoureusement demontres, mais par une methode entierement differente; enfin en ce qui concerne les groupes infinis on n'avait rien: pour ces groupes infinis, l'reuvre de M. CARTAN correspond a ce qu'a ete pour les groupes finis l'reuvre de LIE, celle de KILLING, et celle de CARTAN lui-meme.

APPENDIX B

Sur une degenerescence de la geometrie euclidienne

PAR M. ELIE CARTAN

Professeur a la Faculte des Sciences de Paris 1

La geometrie dans un plan isotrope differe profondement de la geometrie plane classique; les lignes qui jouent dans un plan nonisotrope le role des circonferences sont, dans un plan isotrope, des paraboles toutes tangentes en un meme point a la droute de l'infmi. Si l'on prend pour axe des y une parallele a la direction isotrope unique du plan, le groupe de la geometrie euclidienne du plan isotrope est la forme:

( 1 ) {x' = x + a , y' = ex + hy + b ,

l'arc elementaire ds d'une courbe etant reduit a dx . La notion ordinaire de courbure disparait, mais il s'y substitue une pseudoeourbure egale a

7;;g/ , lorsque la courbe est definie par y = f(x) . Le groupe ( 1 ) est un sous-groupe du plus grand groupe affine qui laisse

invariant le point a l'infini dans la direction Oy , a savoir:

(2) {x' = kx +a , y' = ex + hy +b ;

un autre sous-groupe invariant de ce dernier, a savoir le groupe

(3 ) {x' = kx +a ,

y' = ex + y +b ,

peut etre pris comme base d'une geometrie plane a direction isotrope privi­legiee. Dans cette geometrie, qui est en un certain sens une degenerescence de la geometrie euclidienne, on peut definir la longueur d'un vecteur parallele

1 Assoc. Fran�. Avanc. des Sciences, 59e session, Nantes, 1 935 , 1 28- 1 30.

273

274 B. SUR UNE DEGENERESCENCE DE LA GEOMETRIE EUCLIDIENNE, BY E. CARTAN

a la direction isotrope comme etant la difference des ordonnees y' et y de son extremite et de son origine, mais la notion de longueur disparait pour les vecteurs nonisotropes.

La geometrie fondee sur le groupe (3 ) est interessante; on voit tout de suite qu'etant donnee une ligne plane autre qu'une droite, on peut definir d'une maniere intrinseque un element d'arc ds par la formule

(4) di = dxd2y d�dyd2x

= /' (x)dx2 .

Le second membre est en effet le rapport de deux aires, l'aire du par­allelogramme construit sur les deux vecteurs (dx , dy) et (d2x , d2y) , et l'aire du parallelogramtne construit sur les deux vecteurs (dx , dy) et (0 , 1 ) . Cet element d'arc est identiquement nul quand la ligne consideree est une droite. Si l'on attache a chaque point de la ligne deux vecteurs f et N , le premier tangent a la ligne et de composantes �� , ¥s , le second parallele a Oy et de longueur 1 , on a les formules de Frenet generalisees:

dM � df � � dN ( 5 ) ds = T , ds = k T + N , ds

= 0. 2 !"' Le coefficient k = �/'!If = - ! /' g» est la courbure. Les courbes de

courbure nulle sont les paraboles tangentes a la droite de l'infini au point a l'infini sur Oy . La courbure est du reste un invariant pour le groupe general (2) .

Ce quit donne un certain interet a la geometrie precedente, c'est qu'elle se presente d'elle-meme quand on veut chercher des proprietes geometriques intrinsequement attachees a une integrale J F(x , y , y' , y" ) dx , ou F est

une fonction donnee de x , y , y' = * , y" = � ; une propriete est dite intrinseque si elle ne depend pas du choix des coordonnees x , y . Si la fonc­tion F se reduit a H ' le plus grand groupe qui laisse invariante l'integrale

I I ( ' l est precisement le groupe ( 3 ) . Si F est de la forme Y +;/ .:._�· Y , o u A et B sont des fonctions de x , y , on a une geometrie que joue par rapport a la geometrie de groupe (3 ) le meme role que la geometrie riemannienne par rapport a la geometrie euclidienne, avec cette difference que l 'espace doit etre regarde comme engendre non par des points (x , y) mais par des elements lineaires (x , y , y' ) ; l'espace est un espace d'e/ements lineaires a connexion affine, assimilable au voisinage de chaque element lineaire a un plan euclidien isotrope de groupe (3 ) .

Un autre cas particulier interessant est celui de l'integrale J H dx qui est liee a la geometrie affine unimodulaire.

APPENDIX C

Allocution de M. Elie Cartan

A la fin de cette emouvante ceremonie, apres tous les eloges dont YOUS m'avez comble et que j 'ai conscience de n'avoir qu'imparfaitement merites, permettez que ma pensee se reporte vers ceux qui ne sont plus et qui auraient ete si fiers de les entendre. Je pense a mon pere et a ma mere, humbles paysans qui pendant leur longue vie ont donne a leurs enfants l'exemple du travail joyeusement accompli et des charges vaillamment acceptees. C'est au bruit de l'enclume resonnant chaque matin des l'aube que mon enfance a ete bercee, et je vois encore ma mere actionnant le metier du canut, aux instants que lui laissaient libres les soins de ses enfants et les soucis du menage.

En meme temps qu'a mes parents je pense a mes premiers mahres, les instituteurs de l'Ecole primaire de mon village de Dolomieu, M. Collomb, et surtout M. Dupuis; ils donnaient a plus de deux cents gar�ons un enseigne­ment precis dont j'appreciai plus tard la valeur. Je suis oblige d'avouer-et je n'en ai pas honte-que j'etais un excellent eleve; j 'etais capable d'enumerer sans hesitation les sous-prefectures de n'importe quel departement, et aucune subtilite des regles du participe passe ne m'echappait. Un jour un delegue cantonal qui s'appelait Antonin Dubost et qui devait plus tard devenir un des plus hauts personnages de l'Etat vint inspecter l'ecole; cette visite orienta toute ma vie. II fut decide que je me presenterais au concours des bourses des lycees; M. Dupuis dirigea ma preparation avec un devouement aff ectueux que je n'oublierai jamais. Tout cela me valut un beau voyage a Grenoble, ou je subis sans trop d'emoi des epreuves pas trop redoutables. Je fus re�u bril­lamment, ce qui remplit M. Dupuis de fierte et grace a l'appui de M. Dubost, qui s'interessa pendant toute sa vie avec une affection toute paternelle a ma carriere et a mes succes, je fus gratifie d'une bourse complete au College de Vienne.

A l'age de dix ans je quittai done joyeux le foyer paternel, sans me douter que bien peu de jours me suffiraient pour regretter ce que je perdais. II fallut m'adapter a la vie d'internat que je devais mener pendant plus de dix ans. Apres cinq ans de college pendant lesquels je dus mettre les bouchees doubles, ma bourse fut transferee au Lycee de Grenoble ou j 'achevai mes etudes classiques par la rhetorique et la philosophie, puis au Lycee Janson-de­Sailly, qui etait dans toute la fraicheur de sa premiere jeunesse, rayonnant du

275

276 C. ALLOCUTION DE M. ELIE CARTAN

succes que venait d'obtenir Le Dantec re!fu premier a l'Ecole Normale. J'eus a Janson des professeurs remarquables, Salomon Bloch en mathematiques elementaires A, et en mathematiques speciales Emile Lacour dont tu as su, mon cher Tresse, sans l'avoir connu comme professeur, depeindre la noblesse de caractere. C'est dans cette classe que j 'eus comme camarade, avec Eugene Perreau qui devait entrer avec moi a l'Ecole Normale, Jean Perrin, plus jeune que nous, et qui devait devenir une des plus grandes gloires de la science fran!;aise.

C'est avec emotion, mon cher Tresse, que je t'ai entendu evoquer nos annees d'Ecole Normale. Je ne suis pas sur que le recul du temps n'ait pas em­belli le souvenir que tu as garde de moi et du role que j 'aurais joue aupres de mes camarades. Ce que je me rappelle, c'est en effet une camaraderie frater­nelle et une collaboration qui s'est montree surtout assez etroite dans l'annee de preparation a l'agregation. Je vois encore les seances oil le soir, reunis dans une salle quelconque, nous ecoutions l'un de nous exposer la le!;on qu'il devait faire le lendemain. La les critiques etaient libres et franches et com­bien profitables. Je me rappelle particulierement une le!;on sur l'intersection des quadriques qui nous frappa pour la maniere elegante et neuve dont la question etait cOn!;ue; l'auteur de cette le!;On etait Arthur Tresse.

Tu as parle tout a l'heure, mon cher ami, de l'admiration que nous pro­duisaient les cours de M. Emile Picard, qui excellait a nous ouvrir de vastes perspectives dans un domaine encore nouveau pour nous. A l'Ecole meme c'est Jules Tannery qui exer!fa sur nous la plus profonde influence; par une sorte de transposition mysterieuse due a l'ensemble de toute sa personne, a son regard peut-etre, le respect de la rigueur dont il nous montrait la necessite en mathematiques devenait une vertu morale, la franchise, la loyaute le re­spect de soi-meme. Comme on l'a dit deja, Tannery etait notre conscience: c'est pourquoi nous l'aimions, c'est pourquoi nous avons voue a sa memoire un culte fidele.

Nous admirions aussi l'elegance de certaines conferences de Krenigs, la clarte de l'enseignement de Goursat. A la Sorbonne c'etait la limpidite des cours de Mecanique rationnelle d' Appell, l'elegance incomparable des cours de Darboux. Les le!;ons qui nous produisaient l'impression la plus profonde peut-etre etaient celles d'Hermite, dont le visage et les yeux d'une beaute admirable s'illuminaient comme s'il contemplait au sein de la Divinite ce monde eternel des nombres et des formes dont nous parlait tout a l'heureu M. Picard.

Tannery, Goursat, Appell, Darboux, Picard, Hermite, que de grands noms s'offraient a l'admiration de notre jeunesse. Je n'ai pas parle du geant des Mathematiques, Henri Poincare, dont les le!;ons passaient bien au-dessus de nos tetes; il n'est aucune branche des mathematiques modernes qui n'ait subi son empreinte, et vous comprendrez que je garde a sa memoire une particuliere reconnaissance puisque le dernier travail de sa vie si brusquement interrompue a ete un rapport sur mon reuvre scientifique. De cette illustre

C. ALLOCUTION DE M. ELIE CARTAN 277

pleiade de grands mathCmaticiens, vous seul, mon cher Mahre, nous restez; nous admirons toujours votre jeunesse et je me felicite que mon age me donne encore le privilege d'entendre retracer ma carriere scientifique par le mahre admire qui, il y a un demi-siecle, m'initiait a l'Analyse mathematique, presentait mes premieres notes a l' Academie et etait le rapporteur de mon jury de these.

Apres ma these dont le sujet, tu l'as peut-etre oublie, mon cher Tresse, me fut signale par toi a ton retour de Leipzig OU tu avais etc l'eleve de Sophus Lie, je fus nomme mahre de conferences a Montpellier. Je garde le meilleur souvenir des quinze ans que j 'ai passes en province, a Montpellier d'abord, a Lyon, et a Nancy ensuite. Ce furent des annees de meditation dans le calme, et tout ce que j 'ai fait plus tard est contenu en germe dans mes travaux murement medites de cette periode. C'est a Nancy que je commen�ai a me familiariser avec les vastes auditoires. J'avais a y enseigner les elements de l'Analyse aux eleves de l'lnstitut electrotechnique et de Mecanique appliquee. Institut encore jeune, mais deja prospere sous la direction de l'homme au devouement admirable qu'etait Vogt. Cet enseignement m'interessait beau­coup et j 'eus la satisfaction de sentir tout de suite le contact s'etablir avec les eleves. Je me trouvai ainsi prepare a l'enseignement des mathematiques generales qui devait m'etre confie un peu plus tard a la Sorbonne.

C'est un enseignement analogue que je donne a l'Ecole de Physique et de Chimie depuis vingt-neuf ans. Dans la mesure ou je merite les eloges af­fectueux que votre amitie m'a prodigues, mon cher Langevin, je suis tres heureux d'avoir pu YOUS aider a realiser le dessein qui YOUS tient a creur, celui de faire de l'Ecole technique que vous dirigez un veritable etablissement d'enseignement superieur en assurant aux eleves une culture theorique forte­ment organisee. La tache, la encore, m'a ete rendue facile par le courant de sympathie qui n'a cesse d'unir le matre et les eleves, toujours attentifs et desireux d'acquerir les connaissances dont ils reconnaissent eux-memes l'utilite pour leur carriere future. Ce n'est pas sans un vif regret que je quit­terai bientot, cette Ecole a laquelle me rattachent tant de liens; mon depart ne pourra affaiblir les sentiments d'admiration que j 'eprouve pour le savant et l'homme qui la dirige.

Tu as retrace tout a l'heure, mon cher Maurain, en termes qui m'ont par­ticulierement touchC, venant de l'ami, du doyen affectueusement venere de tous ses collegues, ma carriere de prof esseur a la Sorbonne. Cela a toujours ete pour moi une grande joie que d'enseigner; je me suis toujours interesse a ce que j 'enseignais: c'est une condition necessaire et peut-etre suffisante pour interesser ceux qui vous ecoutent. Si ma prochaine mise a la retraite ne me vieillit pas premeturement, il me sera agreable de donner de temps en temps quelques series de le�ons sur des sujets que je n'ai pas encore eu l'occasion d' enseigner.

C'est a l'Ecole Normale que s'est exercee une grande partie de ma carriere de professeur; pendant quelque quatorze ans j 'y ai eu tout mon service. II

278 C. ALLOCUTION DE M. ELIE CARTAN

est Yrai que j 'y comprends les annees de guerre, pendant lesquelles je Yous ai accueilli a plusieurs reprises, mon cher Julia, lorsque grand blesse Yous Yeniez Yous reposer dans notre Yieille Ecole des operations successiYes qu'on etait oblige de Yous faire subir au Val de Grace. II est difficile d'imaginer un auditoire plus interessant que celui l'Ecole Normale; deYant lui on peut aborder tous les problemes et j 'en ai aborde un certain nombre. J'ai ete heureux d'entendre de Yous, mon cher Bruhat, et de Yous, mon cher Julia, !'opinion qu'ont bien Youlu garder de moi mes eleYes. Ce sont maintenant des mahres; un grand nombre enseignent dans les Facultes. L'un d'eux, celui que ses camarades de Janson enYoyaient passer leurs colles chez Cartan, est l'un des plus jeunes membres de I' Academie des Sciences.

Nous, leurs aines, nous aYons la grande joie de Yoir sortir de l'Ecole Nor­male des generations successiYes de brillants mathematiciens; nous sommes assures ainsi qu'elle n'abdique pas le role de pepiniere des mathematiques qu'elle joue depuis longtemps et qui inspira autrefois a Sophus Lie l'idee de lui dedier son grand traite sur la theorie des groupes. Et puisque, par une pensee touchante, le fils de Sophus Lie a Youlu marquer ce Jubile par l'enYoi du buste de son pere, ne serait-il pas naturel que la place de ce buste soit a la bibliotheque des Sciences de l'Ecole Normale? II rappellerait aux promotions successiYes a la fois le grand mathematicien norvegien et les normaliens qui ont ete ses eleYes a Leipzig et ont illustre l'Ecole, les Vessiot, les Tresse, les Drach.

Mon cher Bruhat, Yous aYez parle en termes qui me sont alles au creur de la dynastie normalienne des Cartan. Me permettrez-Yous d'adjoindre aux deux noms d'Henri Cartan et d'Helene Cartan les noms de deux autres nor­maliens qui m'ont ete tres chers? Le premier est celui de mon beau-frere Antoine Bianconi, cacique litteraire de la promotion de 1 903, dont la mort sur le champ de bataille interrompit l'reuYre philosophique qu'il meditait et qui promettait d'etre importante. Le second est celui de ma plus jeune sreur Anna Cartan, dont le succes au concours d'entree a SeYres m'aYait rempli de joyeuse fierte; eleYe elle aussi de Jules Tannery, dont elle ne pouYait parler sans emotion, elle a termine prematurement sa brillante carriere comme pro­f esseur au Lycee annexe de SeYres. II m'est doux de penser qu'elle est un peu presente ici, en Yoyant au milieu de nous la compagne de promotion a qui la liait une tendre affection, ma chere amie Madame la Directrice de l'Ecole de SeYres.

Mon cher Julia, c'est aYec empressement que je me suis associe a Yotre projet de fonder pour les jeunes mathematiciens un cercle d'etudes, Yotre seminaire, oiI ces jeunes gens, traYaillant en collaboration, exposeraient chaque annee une question importante de Mathematiques. Yous nous ayez dit a ce propos que les jeunes sentent; sans peut-etre trop se l'aYouer, le be­soin de s'appuyer sur leurs aines. En entendant tout a l'heure Dieudonne, nous aYons compris combien Yous aYiez raison.

C. ALLOCUTION DE M. ELIE CARTAN 279

Mon cher Dieudonne, les paroles que vous m'avez adressees me touchent au dela de toute expression. Elles montrent que vous avez l'enthousiasme de la jeunesse, vertu que je vous souhaite ·de conserver toute votre vie. Cet enthousiasme ne vous a-t-il pas fait depasser la mesure? J'aurais certes mau­vaise grace a vous contredire, mais je suis assez age pour savoir ne pas tirer de vos eloges un orgueil deplace, sachant tres bien que si j 'ai les qualites que vous m'attribuez, il m'en manque un certain nombre d'autres qui m'auraient permis de rendre plus de services a l'enseignement et a la science; elles ne sont sans doute pas dans ma nature, mais je n'ai peut-etre pas eu assez de ferme volonte pour les acquerir.

Mon cher Demoulin, nous sommes lies par une vieille amitie et de nom­breux souvenirs communs; nous avons ecoute ensemble les mahres dont je rappelais les noms tout a l'heure. Je suis tres sensible aux felicitations que vous m'apportez au nom des savants etrangers. Je remercie particulierement tous ceux d'entre eux, et je les vois ici nombreux, qui ont tenu a assister en personne a cette ceremonie. Leur presence m'est precieuse et l'empressement avec lequel des savants de nombreuses nations etrangeres ont bien voulu s'associer a mon Jubile m'a vivement touche. Dans le monde trouble ou nous vivons, il est indispensable que la collaboration internationale, au moins dans le domaine scientifique, soit maintenue malgre tous les obstacles.

En meme temps qu'aux delegues etrangers, j 'adresse mes remerciements aux amis, aux collegues, aux eleves qui ont bien voulu repondre a l'appel du Comite jubilaire. Je remercie les membres de ce Comite qui ont accepte de donner leur concours a !'organisation de cette fete, et surtout mon collegue et ami Darmois qui, avec l'aide de mon eleve Ehresmann, a pris sur lui la part la plus lourde de cette organisation.

Plusieurs des orateurs precedents, et j 'en suis particulierement touche, ont tenu a associer le nom de la compagne de ma vie a cette commemoration de ma carriere scientifique. Depuis plus de trente-six ans elle est la flamme ardente qui anime le foyer familial. Nos enfants nous ont reserve de grandes joies; la douleur ne nous a pas ete epargnee. Nous n'oublierons jamais l'empressement avec lequel le Comite a tenu a faire sienne la pieuse pensee de rendre presente ici, grace au grand artiste qu'est M. Charles Mi.inch, l'ame de l'enfant disparu dont toi, mon cher Tresse, vous, mon cher Julia, et vous, mon cher Dieudonne, avez SU evoquer la memoire en termes si emouvants. La ceremonie de ce matin, ou vous avez tenu a ne pas dissocier l'homme du professeur et du savant, nous a donne a ma femme et a moi les plus grandes joies qui puissent encore nous etre reservees.

APPENDIX D

The Influence of France in the Development of Mathematics 1

Like any science, mathematics is a common,· international possession; it is

the commonwealth that belongs to all developed nations, the commonwealth to which every nation contributes according to its abilities. It would be un­acceptable if any well-regarded mathematician would decline to pay awed re­spect to the great foreign minds of the past: Galilei from Italy, Newton from England, Euler from Switzerland, Abel from Norway, Leibniz, Gauss, and Riemann from Germany, to mention but the most significant. They opened new routes in different fields of the science that, without them, would not be what it is today. However, I hope to make you realize that the French mathematicians made one of the most noteworthy contributions to the de­velopment of mathematics, and that, when it comes to the number of great mathematical minds, France does not take second place to any other nation. I am honored and pleased to be given this opportunity to talk about this par­ticular subject in front of a friendly audience and in a country tied with my own by many common memories.

In mathematics, as in any other science, there are two kinds of scientists: those who open royal avenues by coming up with new ideas, usually simple ones but nevertheless ones that have not occurred to anyone else; and those who, on the vast land cleared by the first, till their own gardens, often picking tasty fruits, and sometimes collecting magnificent harvests. When it comes to the development of any science, the latter are not simply significant but rather indispensable; however, it is clear that the names of the former are those that are remembered and honored. Those are the people about whom I speak today.

Joseph Bertrand tells us that, at a Fontainebleau reception for the Dutch ambassador, King Henri IV took pleasure in recalling great Frenchmen who, by their achievements in literature and art, exceeded their foreign rivals. "Those I myself admire," said the Dutchman, by training a mathematician

1 This talk was presented by Elie Cartan in the French Institute in Belgrade, Yugoslavia, on February 27, 1 940. The talk was translated from French into Serbian by Milorad B. Protic, published in 1 940 in the Yugoslavian journal Saturn and in 1 94 1 as a separate book with the introduction written by Mihaila Petrovic (see [ 1 90) ) . For this Appendix the lecture was translated from Serbian into English by Dr. Jelena B. Gill, who also wrote all footnotes.

28 1

282 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

whose field was geometry, "but I must notice that, so far, France failed to produce any mathematicians." "Romanus se trompe!" cried Henri IV and, having at once turned to one of the servants, asked that M. de la Bigottiere be brought in. The first great French mathematician, M. de la Bigottiere­whose real name was Fran�ois Viete ( 1 540- 1 603 )-was the founder of mod­em algebra. He was the first to realize that the procedure for solving special numeric equations would be simplified if the operational symbolism-whose beginnings can be traced back to the ancient times-was applied to letters as well; also, he deserves most of the credit for the systematic development of that idea, and he predicted its unbounded expansion. At the end of the six­teenth century, when Galilei and an advanced geometry school brought fame to Italy, it was Fran�ois Viete who secured for France a distinguished place in the process of founding modern mathematics. I s

.hould tell you that, for

quite some time, Viete was in contact with one of your first mathematicians, Marin Getaldic ( 1 566- 1 626) , who was born in Dubrovnik and who, in Paris, in the year 1 600, published one of Viete's last works.

For France, the seventeenth century was particularly glorious. In the his­tory of mathematics, mechanics, and physics, three names from this period especially stand out: Descartes, Pascal, and Fermat.

A philosopher, mathematician, and physicist, Rene Descartes ( 1 596-1 650) is frequently considered the originator of a new era in the history of the human mind. As a physicist, he witnessed a defeat of his attempts to explain the world; however, his idea that all physical phenomena can be expressed in terms of space and motion has retained its attractiveness until the present day, because the founder of the general theory of relativity him­self believed that it may be possible to interpret physics by using geometric terms (it was nothing but the past development of mathematics that enabled Einstein to carry his ideas further than Descartes could have) . Even if we deny him credit for the creation of analytical geometry ( 1 637) , we must not undermine his role in mathematics. It is known that Greek geometers freely used numbers and computations in their thinking, but for them the numbers had not yet completely lost the geometric character they had in hellenistic science; as the words "square" and "cube" stand for both the numbers and the geometric forms it is clear that the common speech of today still shows traces of this double use. Descartes was the first to use abstract numbers systematically to represent geometric forms and to convert geometric reason­ing into computations. In that way he created an extraordinarily powerful tool . To him we must ascribe the growth of geometry that stemmed primar­ily from analytical and differential geometry; he enriched the latter with a general method for finding tangents of algebraically defined curves. Thanks to analytical geometry, mathematicians not only succeeded in understanding a space of any number of dimensions but also learned to think geometrically in such a space. It is possible to say that it is in fact analytical geometry that taught mathematicians to feel comfortable in, for example, a spheric three-

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 283

dimensional space, i .e . , the one that, only recently, physicists started using to explain physical phenomena. All of this represents, although remote, nev­ertheless unquestionable consequences of Descartes's ideas and results. In algebra, it is to him that we owe the rule about the signs. In pure geometry, he should be credited with a theorem that, having been independently discov­ered by Euler, now bears Euler's name. A result of analysis situi , a science unknown at the time, this theorem establishes the relationships between the number of vertices, edges, and sides of a convex polyhedron. Finally, in mechanics, Descartes's principle of conservation of linear momentum pro­vides an illustration of the intuition that required nothing more than a proper refinement to bring about one of the basic principles of classic mechanics.

Even in his early youth, Blaise Pascal ( 1 623- 1 662) , a somewhat strange but extraordinary genius, exhibited an unusual talent for geometry by writing, at the age of sixteen, Traite sur /es sections coniques, a treatise about curves that are most frequently studied as flat conic section and play an important role in Kepler's planetary laws. Pascal used the results of his contemporary Gerald Desargues, who was one of the most significant French geometers and who, alongside Pascal, was a forefather of projective geometry. By taking, in a way similar to Desargues's, the perspective as a starting point, Pascal succeeded in reducing all properties of conic sections to a property that he called "L 'hexagramme mystique": if a hexagon is inscribed into a cone, the three points at which pairs of opposite sides cross each other always lie on a straight line. Even by this result Pascal demonstrated the creative power of an eminent geometer.

As soon as Pascal the forefather of projective geometry established himself, Pascal the founder of mathematical probability took the stage. When his friend Chevalier de Mere asked him a couple of questions concerning a game of chance, Pascal answered them by reducing all possible outcomes to those most basic. Pierre de Fermat, on the other hand, came up with the same answer but in a completely different way. The evolution of the principles of mathematical probability is well illustrated in the letters exchanged between Pascal and Fermat. The scope of this new research did not escape Pascal : "By connecting the exactness of a mathematical approach with the uncertainty of chance," he was known to say, "the new science can rightly be given an astounding name-Geometry of Chance. " From the famous betting proof, it is known to what extent his research and thinking were influenced by his interest in this new geometry. It is also known that this geometry played an instrumental role in the development of modern science, in which entire portions of physics are nothing but chapters of mathematical probability, and many of the laws of physics are nothing but laws of chance.

Pierre de Fermat ( 1 60 1 - 1 665 ) , whom we mentioned earlier, is one of the greatest mathematical geniuses. He became a counselor of the parliament at

2The old name for topology.

284 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

the age of thirty and held that position until his death. Although his voca­tion did not predestine him for mathematical fame, he made sure to devote enough time to his favorite avocation. Fermat is especially famous for his research in arithmetic and number theory. On the margins of a copy of Dio­phantus's work about undefined equations (which was published in 1 6 1 2 by Bache de Meriziac, the author of Problemes plaisants et delectables) he wrote a number of important theorems without proofs; it is a matter of common be­lief that he was in possession of their proofs. The most famous among those theorems is the one frequently called Fermat's Last Theorem-according to which the sum of the nth degrees of two integers cannot equal the nth degree of a third integer for any integer n that is greater than two. This theorem inspired a wealth of results whose authors, in spite of having at their dis­posal modern algebraic results that had been unknown to Fermat, have never been able either to prove or disprove it. It has been believed for a long time that, even if the theorem is wrong in general, it might in fact be wrong only for some values for n ; however, it is by no means known if the number of the values for which it is wrong is finite or infinite. Through the research prompted by this single theorem-conducted in nearly all mathematically developed theories-Fermat influenced the growth of number theory. His contemporaries readily recognized his extraordinary skills in that field. In one of his letters, Pascal wrote that his own results in number theory were surpassed by Fermat's and that his was but to admire them.

The first half of the seventeenth century was an era of strong advancement of integral and differential calculus. With respect to integral calculus (deter­mining areas and volumes, finding centers of gravity) , it is enough to mention Cavalieri3 and de Roberval4 • As Fermat's own research, however, went quite far in this field, we are indebted to him for the classical integration proce­dures. On the other hand, once while trying to fight a tremendous toothache by solving roulette problems, Pascal accidentally discovered a procedure for obtaining integrals of higher powers of trigonometric functions. The names of those whom we have been talking about are found in differential calculus as well (the tangent problem) . By his method "de maximis et minimis'', Fer­mat introduced the notion of an infinitesimally small number. Lagrange and Laplace considered Fermat to be the actual founder of infinitesimal calculus, while Emile Picard5 believed Pascal's works about roulette to represent the beginnings of integral calculus. Originally, Leibniz scribbled his formulae of infinitesimal calculus on a copy of one of Pascal's manuscripts, which, as he himself put it, had suddenly showed him the way.

It would be unfair to conclude the account of these great minds with­out mentioning that, at the age of twenty-eight, Pascal constructed the first

3Bonaventura Cavalieri ( 1 598- 1 64 7 ) . 4Gilles Personne Roberval ( 1 602- 1 675 ) . 5Charles Emile Picard ( 1 856- 1 94 1 ) .

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 285

arithmetic machine, capable of adding and subtracting. Due to his work Traite de l 'equilibre des liqueurs, Pascal can be considered-together with Archimedes-one of the founders of hydrostatics; this is why it comes as no surprise that the barrel he used to check what is today known as Pas­cal's Principle is displayed next to his death mask in the little chapel erected in the churchyard of Port Royal. Finally let me mention the experiments concerning atmospheric pressure, which, it is suspected, he conducted un­der the influence of Mersenne6 , the soul of a small group of philosophers, mathematicians, and physicists that, before the creation of the Academy of Sciences in 1 666, represented the first small but lively academy.

Those were fortunate times when one and the same man could be accom­plished in philosophy, mathematics, and physics, and when a philosopher such as Malebranche 7 could have the extraordinary feeling that colors might be related to the number of vibrations of which light is composed!

II

The second half of the seventeenth and the beginning of the eighteenth century were dominated by Christian Huygens ( 1 629- 1 695) from the Nether­lands, Isaac Newton ( 1 642- 1 727) from England, and Gottfried Wilhelm von Leibniz ( 1 646- 1 7 1 6) from Germany. It should be enough to mention that the last two are credited with the discovery or, rather, the systematization of infinitesimal calculus, while the first is famous for his works in differential geometry, rational and applied mechanics, and especially his works concern­ing the theory of light (in which he originated and developed an undulatory theory as opposed to Newton's particle theory) . In this period, a remarkable scientific revolution was triggered by Newton's proof that stars and objects on Earth move according to the same laws of mechanics, namely, that one and the same law, the law of gravitation, explains the motion of planets, the moon, and comets as well as the existence of Earth's gravity, high and low tide, and so on. It was Newton's genius that created an entirely new science­celestial mechanics. But even if the earliest beginnings of this science did take place in England, it was France that provided a particularly fertile soil for its future development. To realize this, it is enough to recall the names of those whose works contributed the most to its growth: Clairaut, d' Alembert, Euler, Lagrange, Laplace, Gauss, Cauchy, Poisson, Le Verrier, Tisserand, and finally and especially-Henri Poincare.

I pause for a moment on the first of them, Clairaut. The second in a family of twenty-one children, with a father who was a teacher of mathematics, Alexis Claude Clairaut ( 1 7 1 3- 1 765 ) demonstrated talents similar to those of Pascal; however, unlike Pascal, his first works in no way revealed the significance of those that followed. He sent his first announcement to the Academy of Sciences before reaching the age of thirteen, and addressed an

6Marin Mersenne ( 1 588-1 648) . 7Nicolas de Malebranche ( 1 638- 1 7 1 5) .

286 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

article about lines with double curvatures at the age of sixteen. He was eighteen when, against the existing rules, the king named him a member of the Academy of Sciences, the Division of Mechanics. I shall refrain from telling you about his research in the field of pure mathematics in general and about the part connected with solving differential equations in particular-the latter of which should be well known to all who studied differential equations-and focus instead on those results that made him famous. Newton and Huygens came up not only with a theoretical proof that, instead of being a perfect sphere, the earth is a sphere flattened at the poles, but also with a way to calculate the measure of flatness. However, when in 1 70 1 , at the Pyrenees, Cassini8 determined the degree of arc of the Paris meridian, their conclusions came to be questioned. After debates that were occasionally confusing but always lively, in 1 736 the Academy of Sciences decided to launch, under the guidance of de Maupertuis9 , an expedition that would travel to Lapland to determine the degree of the Lapland meridian arc. Working under very hard conditions, which were further complicated by snow and polar night, the team-which included Clairaut as well-came up with a numerical value that was remarkably larger than the one Cassini had obtained in France, hence proving beyond any doubt that the earth is indeed flattened at the poles. Understandably, de Maupertuis won laurels for the success of the expedition: with his head wrapped in a bear skin, his hand pressing against a globe, he posed for a portrait. But Clairaut continued to think about a possible cause of the earth's polar flatness and tried theoretically to determine the shape that a fluid planet would assume under the influence of Newton's attraction. The results of his research were published in 1 743 in La Theorie de la Figure de la Terre, the book that d' Alembert characterized as a classical account of everything that had been done by that time, the account that marked an important date in the history of celestial mechanics. In addition, Clairaut explained the motion of the moon and in so doing contributed to Newton's lunar theory. He summarized his results from this field in Theorie de la Lune, a book published in 1 732, to which, two years later, he added numerical tables, which, as Fontaine had put it, made it possible to find out "every step that the moon makes in the sky". A few years later, by predicting the next return of Halley's comet, Clairaut reached popular recognition and fame. After explaining that the perturbations caused by Saturn would delay the return of Halley's comet for about one hundred days and the influence of Jupiter would delay it for an additional five hundred and eighteen days, he predicted that its next passage through the perihelion would occur around April 1 3 , 1 759 , but cautioned that, due to numerous other factors that he had to neglect, this date might be off by up to one month-indeed, Halley's comet passed through the perihelion on March 1 3 , 1 7 59 . Almost one century

8Jacques Cassini ( 1 677- 1 756) . 9Pierre Louis Moreau de Maupertuis ( 1 698- 1 759) .

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 287

later, by determining the position of an until-then-unknown planet that had been the main cause of the disturbance of Uranus, French astronomer Le Verrier1 0 attained nearly the same glory.

III

The second half of the eighteenth century was dominated by Euler and Lagrange and, in a somewhat lesser degree, by d' Alembert.

Leonhard Euler ( 1 707- 1 783 ) , "the prince of mathematicians", was born in Basel and spent part of his life in St. Petersburg and Berlin. His genius glowed in all areas of mathematics, and his work has had significant and lasting influence. I will always remember the delight I experienced while reading his Introduction to the infinitesimal analysis, the book that was given to me as an award at the end of my final year of gymnasium: it opened a whole new world in front of me, preparing me to understand better the lectures I would attend at the Sorbonne and in l'Ecole Normale.

Jean Le Rond D'Alembert ( 1 7 1 7- 1 783 ) left his trace in many different areas of mathematics. A well-known algebraic theorem that bears his name asserts that the total number of solutions (real and complex) of a rational equation equals the highest degree of the variable. Although d' Alembert's proof of this result was wrong, it should be mentioned that Euler's proof, based on completely different principles, was not without flaws. Only when the famous mathematician Gauss entered the mathematical scene was a cor­rect proof found, and only with Cauchy's appearance was a real and very simple justification of this theorem established. In analysis I shall men­tion only the first correct formulation-which came from d'Alembert-of a partial differential equation describing vibrations of strings. And finally, it is well worth mentioning that, in mechanics, d' Alembert came up with a principle-nowadays known as d'Alembert's principle-which paved the way for Lagrange's analytical mechanics.

Joseph Louis Lagrange ( 1 736- 1 8 1 3) was born in Torino, in a French fam­ily; although, like Euler, he spent a few years in Berlin, in 1 787 he made his permanent home in Paris, entitling France to consider him one of her very own most celebrated minds. He is truly one of the most significant mathematicians of all times. He worked in all fields of mathematics. In the theory of numbers he proved Fermat's theorem for the power four. In alge­bra, through developing a unique method for solving a polynomial equation by reducing it to an equation of a lower degree, he cleared a path for Abel, Gauss, and Galois; in addition, he demonstrated that polynomial equations of the fifth degree cannot be solved in the way used for solving those of the third and fourth degree. In analysis, he gave the method for solving partial differential equations of the first order and came up with the notion of a sin­gular solution. In function theory, he attempted but did not quite succeed in establishing a rigorous foundation for infinitesimal calculus, the area whose

1 0Urbain Jean Joseph Le Verrier ( 1 8 1 1 - 1 877) .

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principles had not yet been developed with desired exactness but whose con­sequences were nevertheless trusted. However, in spite of this lack of full success, his method of considering functions in an abstract way, indepen­dent of their geometric or mechanical meaning, had remarkable influence in preparing the terrain for the modern theory of functions. Lagrange's talent for generalizing became truly obvious in his works concerning the calculus of variations.

The calculus of variations was developed during the eighteenth century, through the works of Bernoulli and Euler, both from Switzerland. Its roots are in some problems of geometry and mechanics, the simplest of which might be the problem of determining the shortest path between two points on the same surface; here, the unknown quantity is not a number but, much more complexly, a line consisting of infinitely many points. De Maupertuis was the one who, by his Principle of Least Action, reduced the problem of determining a trajectory of a particle in a given force field to a problem of maxima and minima, giving special importance to this kind of calculus. It should not be forgotten, however, that by that time Fermat had already reduced the laws of optics to a similar principle, according to which the path chosen by light is the shortest in terms of time. By applying the infinitesimal variation on an unknown line and by showing how that variation can be calculated, Lagrange introduced a general method into a theory in which nearly every problem required a special procedure in order to be solved.

I shall omit Lagrange's work in celestial mechanics and, instead, devote more time to his most significant work, Mecanique Analytique ( 1 788 ) . Galilei, Descartes, Huygens, Leibniz, Newton, and d' Alembert gradually developed all of the grand principles of modern mechanics. But the problem of deter­mining the trajectory of a system governed by given forces was frequently complicated by the necessity to take into account unknown relations between the forces. With ingenious intuition, in the case without friction Lagrange completely removed the difficulty and gave a general procedure for determin­ing equations that would give the trajectory in question: to achieve this it is enough to determine the active force of that system as well as the work of that force for an infinitely small movement of the system. Aside from practical importance, this wonderful creation has remarkable philosophical importance because it completely illuminates everything that is, from the point of view of mechanical properties, important in a system of particles. In this respect, Lagrange's genius is equal to that of Descartes, the creator of analytical geometry.

The so-called Lagrange's equations in Mechanique Analytique represented an analytical model for various mechanical explanations of certain physical theories. From that point of view this work has great philosophical signifi­cance; but, although it is the most important work of the nineteenth century, it created the impression that everything can be explained by the principles of mechanics-an impression as erroneous as Descartes's belief that everything

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can be explained in terms of geometry-which is the reason that, today, it is completely abandoned. Nevertheless, it illustrates the ability of mathematics to provide physicists with the tools they require to carry out their theories.

Some of the extraordinary minds were inclined to see danger in the man­ufacturing of structures (similar to the one created by Lagrange) that offered insights into infinite arrays of phenomena; they feared that such structures might cause a loss of connection with reality. For instance, the great geome­ter Poncelet, known for his works in mechanics, avoided using Lagrange's method and, instead, preferred following to the last detail the influences and interactions of various forces in order to determine, step by step, their actual works. The same type of skepticism prevented Poncelet from using analytical geometry and prompted him, instead, to examine directly relations between various geometric figures by applying principles of classic geometry. With respect to accepting the latest results, there are indeed two kinds of minds, both equally important for the development of science and both found among great French mathematicians.

IV

Visible as early as the end of the eighteenth century, the French superiority in mathematics became especially clear during the French Revolution and at the beginning of the nineteenth century. Among the great names of that era one must include Monge, Laplace, and Legendre.

Pierre Simon de Laplace ( 1 749- 1 82 7) owed his reputation to his research in celestial mechanics, summarized in his charming treatise Exposition du Systeme du Monde. The peculiar result stating that even the finest details of almost all celestial phenomena can be explained evolved into scientific determinism, according to which, in order to be able to determine positions and velocities of cosmic particles at a given time, it is enough to know their positions and velocities at any other time, provided it is known, in addition, which principles regulate the forces-modeled after the forces of Newton's gravitation-that the particles are governed by. For a long time mathemati­cal physics developed according to this result; only recently, electromagnetism and atomic physics succeeded in proving it to be wrong. Still, this result had strong influence on the development of science. A very significant treatise, Theorie Analytique des Probabilities ( 1 8 1 2) , is another one for which we are grateful to Laplace; the most important part of this work deals with the appli­cation of the notion of probability in the theory of least squares, the possibil­ity of which had been indicated by Legendre. While studying the inclination of an ellipsoid, Laplace introduced spherical functions by means of which one can express any function dependent on a point on a sphere. We should not forget Laplace's famous equation which is satisfied by Newton's potential function; this equation is of extraordinary importance in many problems of analysis, geometry, mechanics, and physics.

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Adrien Marie Legendre ( 1 752- 1 833 ) is responsible for the rejuvenation of number theory, previously successfully treated by Euler. Although Euler was the first one to publish the reciprocity law in arithmetic, Legendre explained it clearly and partially proved it; the law is named after Legendre. Gauss was the third mathematician discover this same law, but the first one to construct a correct and complete proof. Legendre's significant work of several years, Sur /es lntegrales Elliptique, a tract in two volumes, was published in 1 825 and 1 826. There he presented a complete study of integrals involving square roots of fourth-degree polynomials and developed different forms that can be given to them. Although with this work Legendre became a forefather of the marvelous theory of elliptical functions, he let Jacobi and Abel take credit for its founding. Finally, let us mention his Elements de Geometrie ( 1 794 ) , a work which had numerous editions and which, in schools of the Anglo-Saxon countries, soon replaced Euclid's theory; in the history of the non-Euclidean geometries, this work had definitive importance.

Gaspard Monge ( 1 746- 1 8 1 8) was one of the best French geometers. There are two reasons why. First, by founding modern projective geometry, he joined the long process of development of perspective, the theory whose prin­ciples had been known to Italian renaissance painters, which Desargues and Pascal applied to the theory of conic sections, and which, following the pre­vious two, the French geometer de la Hire 1 1 expanded to the theory of poles and polars of a circle. Monge systematized projective geometry and enriched it with constructions on surfaces that are not flat. On the other hand, by his treatise Applications de /'Analyse a la Geometrie he gave a substantial boost to differential geometry, the field that was separated from Descartes's analytical geometry by Euler's and Meusnier's significant works concerning the properties of surfaces; it is Monge to whom we are indebted for the no­tion of measure of curvature, as well as for its application in stereometry; it was his idea to characterize a vast family of surfaces by obtaining them as a solution set of a single partial differential equation. He managed to integrate the equation of minimal surfaces, surfaces which have been and still are an object of important research, and which had been obtained first in Plateau's experiments. Monge presented his theories during his lectures at l'Ecole Normale-the school founded in 1 795 as a convent-as well as at l'Ecole Polytechnique (at which Lagrange and Laplace taught as well ) . I am pleased to have a chance to mention Dupin 1 2 , for he was one of the nu­merous students with whom Monge worked; Dupin is known for his work Developpement de Geometrie, in which he introduced the notions of conju­gated tangents and indicatrix at a point of a surface; also, Dupin can be considered a creator of a new branch of geometry.

1 1 Phillipe de la Hire ( 1 640- 1 7 1 8) . 1 2Francois Pierre Charles Dupin ( 1 784- 1 873) .

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 29 1

v

The most remarkable names in France during the first half of the nine­teenth century were those of Fourier, Cauchy, Poncelet, and Galois. Although quite different from each other, they all cleared new paths in science.

Jean Baptiste Joseph Fourier ( 1 768- 1 830) can be considered the founder of mathematical physics. I shall neglect his important results in algebra and instead tell you about Theorie Mathematique de la Chaleur, the work he did not publish until 1 822 but which must have been in his thoughts since at least 1 807. With this work Fourier opened up a new field in mathematical analysis. "Unknown to the ancient geometers, and for the first time used by Descartes for researching curved lines and surfaces," Fourier says, "analytical equa­tions are by no means limited to these general phenomena. Since mathemat­ical analysis determines the most diverse relations and measures time, space, forces and temperature, it is safe to say that it is as wide and rich as Nature itself. It always follows the same paths and gives the same interpretations, in that way certifying about the unity, simplicity and stability of the Universe." It should not be forgotten that, according to Fourier, the richest source of all mathematical discoveries lies in the study of nature. As, for instance, the mathematical theory of heat had a significant influence on the development of pure mathematics, we may say that Fourier's viewpoint was correct. Created by Fourier to help him integrate frequently encountered partial differential equations, the theory of trigonometric series prompted incredibly many arti­cles, all of which were trying to establish a rigorous foundation for this theory as well as to complete and further develop it. The basic ·problem that needed to be solved was determining which functions can be represented in the form of a Fourier series. As even many of Fourier's own examples were peculiar, it did not take much to make the mathematicians truly puzzled, in a way in which a musician would be puzzled upon discovering that, by combining finite or infinite numbers of pure sounds and their various multiples (har­monics) , it is possible to create any disconnected sequence of sounds. These unusual results forced mathematicians to check once more and specify the notion of a function and to start thinking, bit by bit, about the foundations of their own science. This is what brought about unbelievable consequences which have not yet fully presented themselves. Group theory-a field which so frequently failed mathematicians and which caused many paradoxes that, I am afraid, have not yet been successfully resolved-was one of the branches of mathematics that eventually evolved from these efforts; another branch that had its origins in the same efforts is the theory of functions of one real variable, a creation of French mathematics from the end of the nineteenth and the beginning of the twentieth century.

Augustin Cauchy ( 1 789- 1 857) , an extraordinarily fruitful theorist, was successful in all areas of mathematics: number theory, geometry, analysis and celestial mechanics. Unlike Euler, he did not explore series without first

292 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

finding out whether they made sense, that is, whether they were convergent; in that way, we may say, he opened up the era of exactness. Cauchy discov­ered the general rule-later found independently by J. Hadamard 1 3 -which explains how to determine those values of the variable for which a power series is convergent. Creation of a theory of functions with a complex (or imaginary) variable is another of Cauchy's great accomplishments. For more than three centuries, imaginary quantities were a scandal in mathematics. They were encountered for the first time in the sixteenth century, by Italian algebraists, in the formula for the roots of a third-degree equation in the paradoxical case when all of the roots are real. But, once researchers got adjusted to these new quantities and learned how to use them, it was easy to determine important results concerning real numbers, some of which could not have been obtained in any other way. Sometime toward the end of the eighteenth century, the Swiss mathematician Argand explained the secret of imaginary quantities by finding their importance in the possibility of express­ing a vector in a plane whenever one needed to give not only the length of the vector but its orientation as well. When Cauchy started representing a point in the plane by just one imaginary (or, better, complex) quantity instead of two real coordinates, he got the idea of a function with a complex variable, a function which would assign one point in the plane to another point in the plane. In this way Cauchy created a whole new world. The elements of that world are perfectly organized: just as Cuvier1 4 was able to reconstruct a creature from the antediluvial era from just one piece of its skeleton, a mathematician became able to reconstruct one of Cauchy's functions, pro­vided he knew its values at every point of the arc, no matter how small the arc might be. The perfect order in this world, its marvelous harmony, and-with the exclusion of number theory-a long sequence of theorems determining properties of functions and their numerous applications, all leave the most magnificent impression.

As Cauchy created the right conditions for more discoveries than he could have possibly anticipated, the significance of his opus should be measured by the length of the sequence of works concerning functions of a complex vari­able. A single theorem from this sequence, whose beauty is in its simplicity, was nearly enough to immortalize the name of Liou ville 1 5 • Another theorem on the same subject-named after Emile Picard, perhaps the greatest among the living mathematicians-opened vast and until-then hidden horizons, and created a stream of articles that has not yet ceased.

By using a viewpoint different from Cauchy's, the German mathematician Weierstrass also developed a theory of functions of a complex variable. For a long time it had been believed that the viewpoint one chose was irrelevant,

1 3Jacques Salomon Hadamard ( 1 865- 1 963) . 1 4Georges Cuvier ( 1 769- 1 832) , a French naturalist. 1 5Joseph Liouville ( 1 809- 1 882) .

D. THE INFLUENCE OF FRANCE IN THE DEVEWPMENT OF MATHEMATICS 293

but Borel 1 6 demonstrated-in one of his most charming results-that this was not true, and that Cauchy's viewpoint penetrates deeper into the heart of the matter. Borel indeed took out of the plane so much that no circle, regardless of how small, was left intact, and yet inside of what remained he managed to construct a function that, although satisfying all of Cauchy's requirements, did not satisfy Weierstrass's definition, that conditions the ex­istence of a function of a complex variable by the existence of an intact por­tion of the plane. By starting his celebrated collection of monographs about the theory of functions-the collection whose past and present contributors include mathematicians from all countries-Borel himself contributed a lot to the development of functions of a complex variable.

With Jean Victor Poncelet ( 1 788- 1 867) we enter the era of pure geom­etry. Poncelet is considered the founder of projective geometry, the field whose subject is studying those properties of objects that do not change in projections. He is the one who discovered the new and very useful notion of transformations by means of reciprocal polars, the transformations which make it possible to derive one flat figure from another, with a provision that, peculiarly, the sides of the new figure correspond to the vertices of the old one, and vice versa. Frequently, a transformation of this type makes it possible to explore the properties of some figure by reducing them to the easier-to-explore properties of another. Somewhat later, Gergonne

1 7 used this to derive the duality principle, a principle very important in projective geometry. Finally, Poncelet was the one who discovered the continuity principle, according to which if a figure had a certain property, it will retain the same property even after being deformed, provided that the ratios between its various elements were taken into account. By many simple examples Cauchy proved that this principle, as formulated by Poncelet, was wrong; however, if formulated in a slightly different and much more precise way, this principle is in fact correct. Being very helpful, this principle is frequently used. In geometry, Poncelet's influence was remarkable: in Germany, Steiner and Staudt owe the existence of their works to Poncelet; in France, Chasles

1 8 , the first member of the department of higher geometry at the Sorbonne, was the most outstanding representative of modern pure geometry. To Chasles we are indebted for the important historical monument L 'Aper�u historique sur le Developpement de la Geometrie, which led to the correction of a certain number of wrong opinions.

Before ending our discussion of Poncelet, I note that he played an impor­tant role in developing applied mechanics, which he taught for a long time, first in Metz and then at the Sorbonne.

1 6Emile Borel ( 1 87 1 - 1 956) . 1 7Joseph Diez Gergonne ( 1 77 1 - 1 859) . 1 8M ichel Chasles ( 1 793- 1 880).

294 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

Evariste Galois ( 1 8 1 1 - 1 832) is one of the most unusual figures in the history of science. Having twice failed the entrance exam at l'Ecole Poly­technique, in 1 8 3 1 he was accepted to l'Ecole Normale, only to leave it a year later. Taking an active part in politics earned him several months in prison; not quite twenty-one years old, he was killed in a duel triggered by an insignificant quarrel. He had presented his mathematical discoveries in equa­tion theory to the Academy of Sciences in two different announcements, but both of them were later lost; fortunately, he had also published them in sev­eral small articles in Bulletin de Ferussac in 1 830 and also talked about them to his friend Chevalier in a letter written shortly before his death. Some other results, discovered among his papers, were published in 1 846, in Liouville's magazine.

The significance of his work can be explained quickly. Tartaglia, Cardano, and Ferrari, Italian algebraists of the sixteenth century, used the second and third roots to solve equations of the third and fourth degree; however, all efforts to solve equations of higher degrees in the same way were in vain. By showing that some classes of equations can indeed be solved in that same way, Lagrange, Abel, and Gauss contributed a great deal to this problem. Abel first showed, in 1 826, that a general equation of the fifth degree cannot be solved by means of radicals. In that way it became clear that the problem, with which mathematicians had wrestled since the sixteenth century, had not been well formulated. The glory for solving it belongs to Galois, for he showed that each equation determines a certain number of permutations of its roots, the permutations forming a so-called group; although applied to the roots, these permutations do not disrupt their rational interactions (the meaning of the term "rational interactions" needs an additional explanation) . The nature of that group determines the basic properties of the equation, whether it is possible to find its roots or not, and, in a general case, the nature of auxiliary equations whose solving would result in solving the original equation. By starting from his own idea, Galois easily found the results of his predecessors and successfully incorporated them into his own result.

The theory of substitution groups, i .e . , groups of permutations of a certain number of objects, which was founded by Cauchy, demonstrated its full value through Galois's works. Galois improved its important aspects and demon­strated how basic was the role of ordinary groups. Moreover, he enriched number theory by introducing new classes of imaginary quantities (Galois's imaginary numbers) , each of which was tied to a power of a prime number; Galois's name is frequently encountered not only in the theory of equations but also in modern algebra. The letters he sent to his friend Chevalier make it clear that in analysis he had as many important results as in algebra and that his works on Abel integrals were twenty-five years ahead of those of the famous German mathematician Riemann. Although it makes me sad to think how much science lost by Galois's early death, I must also say that, as Emile Picard once put it, "When confronted with such a short and turbulent life,

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one's respect for the extraordinary mind which left so deep trace in science gets even greater. "

It was Galois's theory that made it possible to explain the miracle which allowed imaginary quantities to appear in the formula for solving a third­degree equation with real roots; indeed, it became possible to show that, if an equation has all roots real and if it can be solved by means of radicals, then it can be solved by means of square roots only. By using the same theory, it can also be shown that some of the ancient problems-such as the problem of doubling a cube or the problem of trisecting an angle-cannot be solved with a ruler and a compass. By his significant work Traite des Substitutions, Jordan 1 9 erected a monument in honor of Galois.

Being both simple and profound, Galois's main idea permitted applica­tions in areas other than algebraic equations. Emile Picard and Ernest Ves­siot, for example, considered it highly important in integration of linear dif­ferential equations. It is noteworthy that Drach and Vessiot attempted to extend Galois's theory to solving the most general differential equations but encountered difficulties that could be overcome only if the original theory were altered or if, at least, some of its magnificent simplicity were sacrificed.

The development of science after Galois demonstrated the growth of the importance of groups in the most diverse branches of mathematics and physics. Norwegian mathematician Sophus Lie, the founder of the theory of groups of transformations, introduced them into analysis and geometry. A great admirer of Galois, he dedicated his momentous opus about groups of transformations (in 1 889) to l'Ecole Normale Superieure. Indeed, the most significant results concerning developing, refining, extending, and finding new applications of Galois's theory were made in France. Poincare claimed that the notion of group had already existed in the spirit of geometry; the axiom that two geometric figures are equal to each other if each of them is equal to a third is in fact identical to the statement that there is a group that regulates geometry, more precisely a family of procedures by which one figure turns into another that is equal to the first. It is extraordinarily important that group theory is capable of giving us all concrete, connected meanings that can be given to the expression "equal figures"; as it was shown in 1 872 by the great German mathematician Felix Klein, exactly this implies the existence of infinitely many geometries, each ruled by a special group, as well as by the fact that each geometry can be investigated independently, without resorting to elementary geometry. This framework encompasses projective geometry, the field in which two figures are considered equal if one of them can be obtained from the other by a sequence of projections.

VI

Since Galois's death one century has passed. During that period mathe­matics has developed remarkably; innumerable volumes have been written,

1 9Camille Jordan ( 1 838- 1 922) .

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some of which, I must say, take undeserved space in libraries. Some of the theories, just formulated at the time of Galois, have since been profoundly explored, and some of them have penetrated other areas of mathematics; in a word, as mathematics, like other sciences, has been constantly and dramat­ically changing, it became difficult for a mathematician, no matter who he might be, to have true insight into its current state. There are fewer and fewer minds capable of making significant discoveries in either pure or applied mathematics. It is rare to encounter a genius similar to that of the French­man Andre Ampere ( 1 775- 1 836) , who was also a physicist, the founder of electrodynamics, and a remarkable mathematician (he and Monge share the credit for creating the theory of partial differential equations of the second order) . The Frenchman Gabriel Lame ( 1 795- 1 870) was an analyst, geometer, and the founder of elasticity theory, while the Frenchman Simeon Poisson ( 1 78 1 - 1 840) is famous for his works in analysis and mathematical physics; Augustin Fresnel ( 1 788- 1 827) , the creator of physical optics-whose works had finally ensured, at least until the appearance of quantum physics, a tri­umph of the modular theory of light-can be considered a mathematician as well .

Instead of giving you a long, and likely tedious, list of names, let us focus on just a few of the greatest contemporary French mathematicians, those who were my professors and to whom I am honored and happy to have a chance to pay respect.

Soon after being admitted to l'Ecole Polytechnique, Charles Hermite ( 1 822- 1 90 1 ) wrote to the well-known professor Jacobi-who, along with Abel, was one of the founders of the theory of elliptical functions-and sent him an article about classifying Abel's transcendental functions, the func­tions related to integration of the most general algebraic differentials. Jacobi, who was once, under similar circumstances, kindly received by Legendre, congratulated the young Hermite on his marvelous results. That was only the beginning of regular correspondence between these two great mathematicians. It was Jacobi to whom, at the age of twenty-four, Hermite sent his discover­ies in advanced algebra, the discoveries that ultimately secured him a place among the most prominent geometers. Building on the most famous Gauss's results, he confidently approached the algebraic theory of shapes in their most general form and introduced continuous variables into number theory, a field characterized by discontinuity. The fact that he was the one who introduced quadratic forms with indefinite conjugate terms, today known as Hermite's forms, is the reason that his name is one of the most frequently found in works from quantum physics. In 1 873 , Hermite became famous by discover­ing the transcendentality of e , the base of Neper's logarithm (the existence of transcendentals, the numbers that satisfy no algebraic equation whose coeffi­cients are rational numbers, had first been demonstrated by Joseph Liouville) . As Hermite's result made a strong impression, some expected him to prove transcendentality of n , and thus, consequently, to destroy forever the hope

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that a circle can be squared with a ruler and a compass; however, having found inspiration in Hermite's method and having devised a way to modify it properly, Ferdinand Lindemann, a Germ:an mathematician, came up with a proof instead, securing the honor for himself.

Hermite always left a profound impression on his listeners. "No one will ever forget the sermon-like sound of Hermite's lectures," said the well-known mathematician Painleve

20 , "or the feeling of beauty and revelation that one had to experience while listening to him talk about a marvelous discovery or something that was still waiting to be discovered. His word had the ability to open vast horizons of science; it conveyed affection and respect for high ideals. " Every time I had a chance to listen to Hermite, I had before me an image of quiet and pure joy caused by contemplations about mathematics, joy similar to the one that Beethoven must have felt while feeling his music inside of himself.

Gaston Darboux ( 1 84 7- 1 9 1 7) was an analyst and geometer at the same time. Although he was the initiator of some results in analysis, I shall not talk about that part of his work because it was his work in geometry that brought him recognition. He surely was not one of the geometers who avoided tar­nishing the beauty of geometry by flattering analysis, and neither was he one of the analysts inclined to reduce geometry to calculations without any con­cern for or interest in their geometric meanings. In this respect he followed in Monge's footsteps, connecting fine and well-developed geometric intuition with skilled applications of analysis. All of his methods are extraordinar­ily elegant and perfectly suited for the subject under investigation. While teaching in the department of higher geometry at the Sorbonne, where he succeeded Michel Chasles, he frequently and with reverence spoke about the theory of triple orthogonal systems, with pleasure stressing the importance of Lame's works; not less frequently he spoke about the theory of deformations of planes, the theory which originated in Gauss's Disquisitiones circa Super­ficies Curvas and which, even before Darboux, was a subject of significant works of French mathematicians, among whom Ossian Bonnet certainly de­serves a mention. Finally, Darboux demonstrated the usefulness of a system of local coordinates, i .e . , coordinates connected with the investigated figure rather than independent of it. Thanks to the theory of groups, Elie Cartan further developed this approach and adapted it to the most diverse spaces cre­ated as a consequence of general relativity theory. Darboux had tremendous influence on the development of geometry; of his numerous students and fol­lowers, I shall mention only the well-known Roumanian geometer Tzitzeica, one of the founders of the Mathematical Reviews of the Balkan Union, a man whose recent death is still mourned in the world of science. Classic in its field, Darboux's work Theorie des Surfaces is a splendid monument erected in honor of both analysis and geometry.

20Paul Painleve ( 1 863- 1 933) .

298 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

A story has it that, when a young German mathematician expressed his puzzlement over Lagrange's refusal to recognize Gauss as the greatest German geometer, Lagrange told him, "No, he cannot be the greatest German geome­ter for he is the greatest European geometer! " In the same spirit one could say that Henri Poincare ( 1 854- 1 9 1 2) was not only a great mathematician but mathematics itself. It is impossible to find a branch of mathematics-a branch of physics even-in which he did not leave a trace or which he did not rejuvenate or from which he did not infer a completely new field. After creating Fuchsian functions

2 1 , he used uniform functions with the same pa­rameter to express the coordinates of a point on an algebraic surface, and in that way obtained the result which, before him, was known only for some spe­cial classes of surfaces. He solved the uniformization problem in a way that, at the time, was quite brave. He was a forerunner of the theory of functions with several complex variables. Also, he created the theory of differential equations in a real field; due to that theory, he was then able to restore the methods of celestial mechanics, to study periodic solutions of problems of this field, and to investigate stability problems. In analysis situs, the part of geometry interested only in those properties of objects that are not affected by continuous transformations, Poincare authored several treatises that would become the starting point for nearly all later results in that field. At the Sor­bonne, by lecturing on all areas of mathematical physics, he influenced the ideas triggered by Michelson's experiment

22. With his early death, science

lost one of its most prominent leaders. Translated to many languages, his scientific-philosophical works La Science et l 'Hypothese and La Valeur de la Science are well known to the entire world. In some ways-one of which is well illustrated by Poincare's words, "Thought is only a flash in the middle of a long night, but the flash that means everything"-Poincare can be com­pared with Pascal. It will take a long time to develop all of Poincare's ideas and to explore all of the paths that he had paved by his rich and diverse work.

Finally,! would like to mention Paul Appell and Edouard Goursat-the first of whom is the author of Traite de M ecanique Rationnelle, and the second of Traite de Ca/cul Differentiel et Integral-and also, once again, Emile Pi­card, the last living from that celebrated generation. Two years ago, together with the great German mathematician David Hilbert, Emile Picard received a gold medal from the Mittag-Leffler Institute, and only several weeks ago, at the celebration of the fifty years since Picard was elected a member of the Academy of Sciences, Emile Borel talked about his scientific opus. I already mentioned the famous theorem named after him, as well as those among his works that developed Galois's theory. His work concerning algebraic func­tions with two variables represents the foundation of algebraic geometry, a

2 1 It was Poincare himself who named them this way after the German mathematician Lazarus Fuchs; nowadays, these functions are called automorphic.

22 Also known as the Michelson-Morley experiment.

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 299

branch of geometry especially well developed in Italy. It is true that the viewpoint of Italian geometers from the past century was more clearly de­fined than that of Emile Picard, but, as Emile Borel put it, algebraic geometry would be certainly crippled without Picard's contributions.

VII

The glory of French mathematics created by the greatest results of Hermite, Darboux, Poincare, and Picard has not darkened. Indeed, the flame is as strong as it has ever been. As the time is short, to justify this statement I am forced to limit myself to just a few names.

Gabriel Krenigs was a fine geometer; the elegance of some of his works can be compared with that of Darboux. By creating new transcendentals, Paul Painleve solved a problem that even to Poincare seemed unapproachable; Poincare characterized Painleve's results in analysis by saying: "Mathemat­ics is a well-ordered continent whose countries are united; the work of Paul Painleve is a magnificent island in an ocean." But this judgment is some­what incomplete because Painleve-who, for a long time, taught mechanics at l'Ecole Polytechnique-also remarkably advanced mechanics; besides, his theoretical research prompted development of aviation in such a measure that one may say that, thanks to Painleve, aviation is an exclusively French creation.

The results of Jacques Hadamard were numerous and significant: in arith­metic, he worked on the Riemann's function related to the complicated prob­lem of distribution of prime numbers; in geometry, he researched geodesic lines with opposite curvatures; in analysis, he published works about par­tial differential equations in mathematical physics. Also, he gave a strong stimulus to the calculus of variations and functional analysis, the new sci­ence founded by the Italian mathematician Volterra. Finally, his seminar at College de France, where all foreign mathematicians wished to present their latest results, influenced international collaboration in mathematics. As he is still young, I may say with certainty that his work is far from finished.

The research of functions with complex variables has always been very successful in France. Here I mention Emil Borel; the short-lived analyst Fatou; Paul Montel, famous for his theory concerning families of normal functions; Gaston Julia, known for his works about elevation of rational functions; and so forth.

The theory of functions with real variables is of almost exclusively French origin. Set up by Camille Jordan's Traite d'Analyse (which, like Emile Pi­card's treatise of the same name, had international influence) , founded by the works of Emile Borel, Henri Lebesgue (who defined measure of a set) , Rene Baire (who introduced integrals which today bear his name) , and Denjoy (the creator of the totalization theory) , it introduced unexpected harmony into a field that had been neglected for a long time, testimony to the daring and talent of its creators.

300 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

I cannot but mention Maurice Frechet's theory of abstract spaces, Bouli­gand's infinitesimal geometry, and Elie Cartan's works in analysis and geom­etry, the last of which I am not qualified to judge.

Institut Henri Poincare is born from new French enthusiasm for research in the field of mathematical physics. Emile Borel, the soul of probability theory, started a praise-deserving series of publications in this field, similar to the one in function theory-a series in which Frechet, Paul Levy, and Georges Darmois presented their excellent results. The Department of Theoretical Physics is headed by Louis de Broglie, the creator of wave mechanics, who restored atomic physics and reconciled the undulatory and corpuscular theory of light. I should not forget to mention the Institut of Mechanics, headed by Henri Villat, known for his results in hydrodynamics, who is also editor of the internationally known collection Memorial des Sciences Mathematiques and editor-in-chief of Journal de Mathematiques Pures et Appliquees, a journal which, nearly a century ago, was started by Liouville and which for quite some time was edited by Camille Jordan.

The account of French mathematical activity would be incomplete with­out a mention of l'Ecole Polytechnique and l'Ecole Normale. For more than a century, great French mathematicians have owed their education to one of the two institutions; in the last half-century that marvelous role belonged almost exclusively to l'Ecole Normale, which, even a good fifty years ago, Sophus Lie considered a nursery of French mathematics. Young talents from many countries have been coming here to get the same education as their French colleagues. That is why it is difficult not to consider Georges Tzitze­ica, whom I already mentioned, to be a French mathematician. For the same reason, I am inclined to include among French mathematicians my good friend Mihaila Petrovic, a doyen of Yugoslav mathematics, who is widely recognized for his great originality in inventing the spectral method · in arith­metic, algebra, and analysis, and also for creating general phenomenology, the field which systematically examines the problems of existence of analyt­ical molds that could be used to present simultaneously several apparently different physical theories. I hope that you will not object if I credit his results to the accomplishments which mathematics owes to France.

Thanks to l'Ecole Normale, young mathematicians are ready to replace the older ones. One might say that it is too early to mention names, but some of them are nevertheless already well known. I shall mention only Jacques Herbrand, whose works, mercilessly interrupted by his early death, were announcing a great mathematician, perhaps similar to Evariste Galois.

Ladies and gentlemen, it is time for me to finish this talk, for I have already used a great deal of your kind attention. In conclusion, I would like to make just one remark of general nature.

More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of the problems and entities considered. Car-

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 30 1

ried to an extreme, this procedure justifies the well-known joke according to which a mathematician is a scientist who neither knows what he is talking about or whether whatever he is indeed talking about exists or not. French mathematicians, however, never enjoyed distancing themselves from reality; they do know that, although needed, logic is by no means crucial. In math­ematical activity, like in any other type of human activity, one should find a balance of values: there is no doubt that it is important to think correctly, but it is even more important to formulate the right problems. In that re­spect, one can freely say that French mathematicians not only always knew what they were talking about, but also had the right intuition to select the most fundamental problems, those whose solutions produced the strongest influence on the overall development of science.

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[GeR]

[Gi]

[Glb l ]

[Glb2]

[Gis i ]

[Gls2]

[Gls3]

[Gls4]

[Grv]

[Gra]

[GrH]

[Gro]

[GQS]

[GS l ]

[GS2]

[Had]

[Hal]

[Har]

[Har l ]

[Har2]

[Haw ! ]

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3 1 0

[Kli]

[Ko]

[Ko NJ

[KoP]

[KoA I ]

[KoA2]

[KoA3]

[Ku]

[KuS]

[Lag]

[Lag ! )

[Lag2)

[Lap ! )

[Lap2)

[Lap3)

[Leg)

[LeO]

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[Lev]

[Lib]

[Lie]

[Lie ! )

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[LiE]

[Lip)

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[Loo]

[Lu)

[Mal]

[Mas)

[Me]

[MiS]

[Min]

[Mol]

[Moo]

[Mou]

[Mu]

[Norn]

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[Oe]

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[SeB I ]

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