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Global AnalysisDifferential Forms in Analysis,Geometry and Physics

Ilka AgricolaThomas Friedrich

Graduate Studiesin MathematicsVolume 52

American Mathematical Society

Global Analysis

Global AnalysisDifferential Forms in Analysis,Geometry and Physics


Translated byAndreas Nestke

Graduate Studiesin MathematicsVolume 52

American Mathematical SocietyProvidence, Rhode Island

Editorial BoardWalter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 53-01;Secondary 57-01, 58-01, 22-01, 74-01, 78-01, 80-01, 35-01.

This book was originally published in German by Friedr. Vieweg & Sohn Verlagsge-sellschaft mbH, D-65189 Wiesbaden, Germany, as "Ilka Agricola and Thomas Friedrich:Globale Analysis. 1. Auflage (1st edition)", F}iedr. Vieweg & Sohn VerlagsgesellschaftmbH, Braunschweig/Wiesbaden, 2001

Translated from the German by Andreas Nestke

Library of Congress Cataloging-in-Publication DataAgricola, I1ka, 1973-

(Globale Analysis. English)Global analysis : differential forms in analysis, geometry, and physics / Ilka Agricola, Thomas

Ftiedrich ; translated by Andreas Nestke.p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 52)

Includes bibliographical references and index.ISBN 0-8218-2951-3 (alk. paper)1. Differential forms. 2.Mathematical physics. 1. Friedrich, Thomas, 1949 ll. Title.

111. Series.

QA381.A4713 2002514'.74- dc2l 2002027681

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2002 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.

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Preface

This book is intended to introduce the reader into the world of differentialforms and, at the same time, to cover those topics from analysis, differ-ential geometry, and mathematical physics to which forms are particularlyrelevant. It is based on several graduate courses on analysis and differen-tial geometry given by the second author at Humboldt University in Berlinsince the beginning of the eighties. From 1998 to 2000 the authors taughtboth courses jointly for students of mathematics and physics, and seized theopportunity to work out a self-contained exposition of the foundations ofdifferential forms and their applications. In the classes accompanying thecourse, special emphasis was put on the exercices, a selection of which thereader will find at the end of each chapter. Approximately the first half ofthe book covers material which would be compulsary for any mathematicsstudent finishing the first part of his/her university education in Germany.The book can either accompany a course or be used in the preparation ofseminars.

We only suppose as much knowledge of mathematics as the reader would ac-quire in one year studying mathematics or any other natural science. Fromlinear algebra, basic facts on multilinear forms are needed, which we brieflyrecall in the first chapter. The reader is supposed to have a more extensiveknowledge of calculus. Here, the reader should be familiar with differentialcalculus for functions of several variables in euclidean space R", the Rie-mann integral and, in particular, the transformation rule for the integral,as well as the existence and uniqueness theorem for solutions of ordinarydifferential equations. It is a reader with these prerequisites that we have inmind and whom we will accompany into the world of vector analysis, Pfaf flan

v

vi Preface

systems, the differential geometry of curves and surfaces in euclidean space,Lie groups and homogeneous spaces, symplectic geometry and mechanics,statistical mechanics and thermodynamics and, eventually, electrodynamics.

In Chapter 2 we develop the differential and integral calculus for differentialforms defined on open sets in euclidean space. The central result is Stokes'formula turning the integral of the exterior derivative of a differential formover a singular chain into an integral of the form itself over the bound-ary of the chain. This is in fact a far-reaching generalization of the maintheorem of differential and integral calculus: differentiation and integrationare mutually inverse operations. At the end of a long historical develop-ment mathematicians reached the insight that a series of important integralformulas in vector analysis can be obtained by specialization from Stokes'formula. We will show this in the second chapter and derive in this wayGreen's first and second formula, Stokes' classical formula, and Cauchy'sintegral formula for complex differentiable functions. Furthermore, we willdeduce Brouwer's fixed point theorem from Stokes' formula and the Weier-strass approximation theorem.

In Chapter 3 we restrict the possible integration domains to "smooth"chains. On these objects, called manifolds, a differential calculus; for func-tions and forms can be developed. Though we only treat submanifolds ofeuclidean space, this section is formulated in a way to hold for every Rie-mannian manifold. We discuss the concept of orientation of a ma.nifold, itsvolume form, the divergence of a vector field as well as the gradient and theLaplacian for functions. We then deduce from Stokes' formula the remainingclassical integral formulas of Riemannian geometry (Gauss-Ostrogradski for-mula, Green's first and second formula) as well as the Hairy Sphere theorem,for which we decided to stick to its more vivid German name, `'Hedgehogtheorem". A separate section on the Lie derivative of a differential formleads us to the interpretation of the divergence of a vector field as a mea-sure for the volume distortion of its flow. We use the integral formulas tosolve the Dirichlet problem for the Laplace equation on the ball in euclideanspace and to study the properties of harmonic functions on R". For thesewe prove, among other things, the maximum principle and Liouville's the-orem. Finally we discuss the Laplacian acting on forms over a Riemannianmanifold, as well as the Hodge decomposition of a differential form. Thisis a generalization of the splitting of a vector field with compact support inR" into the sum of a gradient field and a divergence-free vector field, goingback to Helmholtz. In the final chapter we prove Helmholtz' theorem withinthe framework of electrodynamics.

Preface vii

Apart from Stokes' theorem, the integrability criterion of Frobenius is one ofthe fundamental results in the theory of differential forms. A geometric dis-tribution (Pfafflan system) is defined by choosing a k-dimensional subspacein each tangent space of an n-dimensional manifold in a smooth way. Ageometric distribution can alternatively be described as the zero set of a setof linearly independent 1-forms. What one is looking for then is an answerto the question of whether there exists a k-dimensional submanifold suchthat, at each point, the tangent space coincides with the value of the givengeometric distribution. Frobenius' theorem gives a complete solution to thisquestion and provides a basic tool for the integration of certain systems offirst order partial differential equations. Therefore, Chapter 4 is devoted toa self-contained and purely analytical proof of this key result, which, more-over, will be needed in the sections on surfaces, symplectic geometry, andcompletely integrable systems.

Chapter 5 treats the differential geometry of curves and surfaces in euclideanspace. We discuss the curvature and the torsion of a curve, Frenet's formu-las, and prove the fundamental theorem of the theory of curves. We thenturn to some special types of curves and conclude this section by a proof ofFenchel's inequality. This states that the total curvature of a closed spacecurve is at least 27r. Surface theory is treated in Cartan's language of mov-ing frames. First we describe the structural equations of a surface, and thenwe prove the fundamental theorem of surface theory by applying Frobenius'theorem. The latter is formulated with respect to a frame adapted to thesurface and the resulting 1-forms. Next we start the tensorial description ofsurfaces. The first and second fundamental forms of a surface as well as therelations between them as expressed in the Gauss and the Codazzi-Mainardiequations are the central concepts here. We reformulate the fundamentaltheorem in this tensorial description of surface theory. Numerous examples(surfaces of revolution, general graphs and, in particular, reliefs, i.e. thegraph of the modulus of an analytic function, as well as the graphs of theirreal and imaginary parts) illustrate the differential-geometric treatment ofsurfaces in euclidean space. We study the normal map of a surface and arethus lead to its Gaussian curvature, which by Gauss' Theorema Egregiumbelongs to the inner geometry. Using Stokes' theorem, we prove the Gauss-Bonnet formula and an analogous integral formula for the mean curvatureof a compact oriented surface, going back to Steiner and Minkowski. An im-portant class of surfaces are minimal surfaces. Their normal map is alwaysconformal, and this observation leads to the so-called Weierstrass formu-las. These describe the minimal surface locally by a pair of holomorphicfunctions. Then we turn to the study of geodesic curves on surfaces, the in-tegration of the geodesic flow using first integrals as well as the investigation

of maps between surfaces. Chapter 5 closes with an outlook on the geometryof pseudo-Riemannian manifolds of higher dimension. In particular, we lookat Einstein spaces, as well as spaces of constant curvature.

Symmetries play a fundamental role in geometry and physics. Chapter 6contains an introduction into the theory of Lie groups and homogeneousspaces. We discuss the basic properties of a Lie group, its Lie algebra, andthe exponential map. Then we concentrate on proving the fact that everyclosed subgroup of a Lie group is a Lie group itself, and define the structureof a manifold on the quotient space. Many known manifolds arise as homo-geneous spaces in this way. With regard to later applications in mechanics,we study the adjoint representation of a Lie group.

Apart from Riemannian geometry, symplectic geometry is one of the es-sential pillars of differential geometry, and it is particularly relevant to theHamiltonian formulation of mechanics. Examples of symplectic manifoldsarise as cotangent bundles of arbitrary manifolds or as orbits of the coadjointrepresentation of a Lie group. We study this topic in Chapter 7. First weprove the Darboux theorem stating that all symplectic manifolds are locallyequivalent. Then we turn to Noether's theorem and interpret it in terms ofthe moment map for Hamiltonian actions of Lie groups on symplectic man-ifolds. Completely integrable Hamiltonian systems are carefully discussed.Using Frobenius' theorem, we demonstrate an algorithm for finding the ac-tion and angle coordinates directly from the first integrals of the Hamiltonfunction. In 7.5, we sketch the formulations of mechanics according toNewton, Lagrange, and Hamilton. In particular, we once again return toNoether's theorem within the framework of Lagrangian mechanics, whichwill be applied, among others, to integrate the geodesic flow of a pseudo-Riemannian manifold. Among the exercises of Chapter 7, the reader willfind some of the best known mechanical systems.

In statistical mechanics, particles are described by their position probabilityin space. Therefore one is interested in the evolution of statistical states ofa Hamiltonian system. In Chapter 8 we introduce the energy and informa-tion entropy for statistical equilibrium states. Then we characterize Gibbsstates as those of maximal information entropy for fixed energy, and provethat the microcanonical ensemble realizes the maximum entropy among allstates with fixed support. By means of the Gibbs states, we assign a ther-modynamical system in equilibrium to a Hamiltonian system with auxiliaryparameters satisfying the postulates of thermodynamics. We discuss the

Preface ix

role of pressure and free energy. A series of examples, like the ideal gas,solid bodies, and cycles, conclude Chapter 8.

Chapter 9 is devoted to electrodynamics. Starting from the Maxwell equa-tions, formulated both for the electromagnetic field strengths and for thedual 1-forms, we first deal with the static electromagnetic field. We provethe solution formula. for the inhomogeneous Laplace equation in three-spaceand obtain, apart from a description of the electric and the magnetic fieldin the static case, at the same time a proof for Helmholtz' theorem as men-tioned before. Next we turn to the vacuum electromagnetic field. Here weprove the solution formula for the Cauchy problem of the wave equation indimensions two and three. The chapter ends with a relativistic formulationof the Maxwell equations in Minkowski space, a discussion of the Lorentzgroup. the Maxwell stress tensor and a thorough treatment of the Lorentzforce.

We are grateful to Ms. Heike Pahlisch for her extensive work on the prepara-tion of the manuscript and the illustrations of the German edition. We alsothank the students in our courses from 1998 to 2000 for numerous commentsleading to additions and improvements in the manuscript. In particular,Dipl.-Math. Uli Kriihmer pointed out corrections in many chapters. Notleast our thanks are due to M. A. Claudia Frank for her thourough readingand correcting of the German manuscript with regard to language.

The English version at hand does not differ by much from the original Ger-man edition. Besides small corrections and additions, we included a detaileddiscussion of the Lorentz force and related topics in Chapter 9. Finally, wethank Dr. Andreas Nestke for his careful translation.

Berlin, November 2000 and May 2002


Contents

Preface v

Chapter 1. Elements of Multilinear AlgebraExercises

1

8

Chapter 2. Differential Forms in R" 112.1. Vector Fields and Differential Forms 112.2. Closed and Exact Differential Forms 182.3. Gradient, Divergence and Curl 232.4. Singular Cubes and Chains 262.5. Integration of Differential Forms and Stokes' Theorem 302.6. The Classical Formulas of Green and Stokes 352.7. Complex Differential Forms and Holomorphic Functions 362.8. Brouwer's Fixed Point Theorem 38Exercises 43

Chapter 3. Vector Analysis on Manifolds 473.1. Submanifolds of lR' 473.2. Differential Calculus on Manifolds 543.3. Differential Forms on Manifolds 673.4. Orientable Manifolds 693.5. Integration of Differential Forms over Manifolds 763.6. Stokes' Theorem for Manifolds 793.7. The Hedgehog Theorem (Hairy Sphere Theorem) 81

xi

xii Contents

3.8. The Classical Integral Formulas 823.9. The Lie Derivative and the Interpretation of the Divergence 873.10. Harmonic Functions 943.11. The Laplacian on Differential Forms 100Exercises 105

Chapter 4. Pfaffian Systems 1114.1. Geometric Distributions 1114.2. The Proof of Frobenius' Theorem 1164.3. Some Applications of Frobenius' Theorem 120Exercises 126

Chapter 5. Curves and Surfaces in Euclidean 3-Space 1295.1. Curves in Euclidean 3-Space 1295.2. The Structural Equations of a Surface 1415.3. The First and Second Fundamental Forms of a Surface 1475.4. Gaussian and Mean Curvature 1555.5. Curves on Surfaces and Geodesic Lines 1725.6. Maps between Surfaces 1805.7. Higher-Dimensional Riemannian Manifolds 183Exercises 198

Chapter 6. Lie Groups and Homogeneous Spaces 2076.1. Lie Groups and Lie Algebras 2076.2. Closed Subgroups and Homogeneous Spaces 2156.3. The Adjoint Representation 221Exercises 226

Chapter 7. Symplectic Geometry and Mechanics 2297.1. Symplectic Manifolds 2297.2. The Darboux Theorem 2367.3. First Integrals and the Moment Map 2387.4. Completely Integrable Hamiltonian Systems 2417.5. Formulations of Mechanics 252Exercises 264

Chapter 8. Elements of Statistical Mechanics and Thermodynam: .es 2718.1. Statistical States of a Hamiltonian System 271

Contents xiii

8.2. Thermodynamical Systems in Equilibrium 283Exercises 292

Chapter 9. Elements of Electrodynamics 2959.1. The Maxwell Equations 2959.2. The Static Electromagnetic Field 2999.3. Electromagnetic Waves 3049.4. The Relativistic Formulation of the Maxwell Equations 3119.5. The Lorentz Force 317

Exercises 325

Bibliography 333

Symbols 337

Index 339

Chapter 1

Elements ofMultilinear Algebra

Consider an n-dimensional vector space V over the field K of real or complexnumbers. Its dual space V' consists of all linear maps from V to K. Moregenerally, a multilinear and antisymmetric map,

wk: Vx...xV-+K.depending on k vectors from the vector space V, is called an exterior (mul-tilinear) form of degree k. The antisymmetry of wk means that, for allk vectors vl, ... , vk from V and any permutation a E Sk of the numbers{ 1, ... , k}, the following equation holds:

wk(VQ(1), ....Va(k)) = sgn(a) i.Jk(VI, -..,Vk).Here sgn(a) denotes the sign of the permutation a. In particular, wk changessign under a transposition of the indices i and j:

wk(i1i ...,Vi, ...,Vj, ...,Vk) = -wk(v1, ...,v ....,VI.....Vk).The vector space of all exterior k-forms will be denoted by Ak( V*). Fur-thermore, we will use the conventions A(V') = K and A1(V*) = V.

Fixing an arbitrary basis e1, ... , e in the n-dimensional vector space V, wesee that each exterior k-form wk is uniquely determined by its values on allk-tuples of the form el,,... 441 where the indices are always supposed tobe strictly ordered, I = (i1 < ... < ik). On the other hand, a k-form can bedefined by arbitrarily prescribing its values on all ordered k-tuples of basisvectors and extending it to all k-tuples of vectors in an antisymmetric and

I

2 1. Elements of 1llultilinear Algebra

multilinear way. The number of different k-tuples of n elements is equal to(k) = k. Thus we concludeTheorem 1. If k > n, Ak(V*) consists only of the zero map. For k < nthe dimension of the vector space Ak(V*) is equal to

dim (nk(V`))_ (k)

Exterior forms can by multiplied, and the product is again an exterior form.

Definition 1. Let wk E Ak(V*) and 7/ E A1(V') be two exterior forms ofdegrees k and 1. respectively. Then the exterior product wk n 171 is defined asa (k + 1)-form by the formula

1wknll!(t'1....L'k+l) = Esgn(v)wk(v,(1),...Vo(k))11l(z'o(k+l)....vn(k+l))k!l!

oESk+j

Obviously. wk n,1 is a multilinear and antisymmetric map acting on (k + 1)vectors, i. e. of degeree k+l. The following theorem summarizes the algebraicrules governing computations involving the exterior multiplication of forms.

Theorem 2. The exterior product has the following properties:(1) (wi +w2)AY/=w1 nrft+w2n(2)(3) (awk)A1)l=wkA(ar/)=a(wkAr/)foranyaEIfs:(4) (wkAgl)A m=wkA(r11A ');(5) wk A 111 = (-1)k'711 n wk.

Proof. Only the last two formulas require a proof, in which we shall omitsome of the upper indices for better readability. First

(k+l)!m! (wk A11l) A/'"(v1, ...,vk+l+m)1: sgn(a)(wk A 711)(v,(1), ... , v,(k+1))1f (vn(k+1+1).... , vs(k+l+m))

f7E Sk+l+m

Decompose the permutation group Sk+l+m into residue classes with respectto the subgroup Sk+l C Sk+1+m formed by all permutations acting as theidentity on the last m indices {k + 1 + 1, ..., k + l + m}. Each residueclass R thus consists of all permutations a E Sk+l+m with fixed valuesQ(k + I + 1), ..., v(k + I + m). Fixing any permutation ao E R. all theremaining elements a E R are parametrized by the elements in Sk+l:

or = a0 0 7r, 7r E Sk+l

1. Elements of Multilinear Algebra 3

Hence,

E sgn(o)(wk A rll)(Va(l),- - - , Vo(k+l))11m(Va(k+1+l), - - - , Va(k+l+m))

oER

rsgn(oo)sgn(ir)(w A r/)(Vaoo,r(1), --Vooo,,(k+())1L(Vao(k+t+1), ..Vao(k+t+m)),rESk+i

= sgn(ao)(k + l)!(w A 17)(vao(1), ... , Voo(k+1))1J(VCo(k+1+1), ... , Vao(k+1+m))-Using now the definition of the exterior product wk A 711, this leads to theformula

k! 1! E(k + l)! a Rsgn(a)(w A i))(va(1), vo(k+l+m))= sgn(o)w(va(1),

... va(k))1T(t'a(k+1) ... V0(k+l))1-(Va(k+1+1), - - -Va(k+l+m))-aER

In order to compute the sum over the whole group Sk+l+m, we sum overall the residue classes R and, simplifying the scalar factors, we obtain theequation

(k! l! m!) - (wk A ill) A pm (v1, ... , tk+l+m )E sgn(a)w(va(1), ... Vo(k))17(Vo(k+1), - - Vo(k+l))1(Va(k+1+1), . Vo(k+l+m))-

Sk+I+m

This shows the associativity of the exterior multiplication of forms. The lastformula (5) is proved analogously. 0Definition 2. The exterior algebra A(V') of the vector space V is formedby the sum of all exterior forms

A(V*) = Ek=0

endowed with the exterior multiplication A of forms as multiplication.

Next we will construct an explicit basis of the vector spaces Ak(V*). Todo so, start from any basis el, ... e of V and denote by a, ... , o thedual basis of the dual space V' _ A' (V*). For an ordered k-tuple of indices(k-index for short) I = (i1 < ... < ik) let of denote the k-form defined bythe formula

of := ail A ... A o;k .Obviously, for a fixed k-index J = (j1 < ... < jk),

J 0 if 196 J,ol(ejl, --.,eik) t 1 if I = J.In particular, the k-forms of are linearly independent in Ak(V'). For di-mensional reasons, this immediately implies

4 1. Elements of Multilinear Algebra

Theorem 3. Let e1, ... , en be a basis of the vector space V, and denote byol.... , o its dual basis in the dual space V. Then the forms o,,1 = (i1 0.

Obviously, there are precisely two equivalence classes.Definition 3. An orientation of the real vector space V is the choice of oneof the two equivalence classes in the set B(V) of all bases in V.In the plane R2, an orientation can be understood as a sense of rotation:

Figure 1el el

Figure 2

Figure 1 illustrates R2 with the orientation determined by the basis (el, e2),whereas Figure 2 depicts the plane R2 equipped with the orientation (e2, el).The transition matrix between both bases is the matrix

0 1A __ [0 0with negative determinant, det(A) = -1. Hence the bases (er, e2) and(e2, el) represent the two possible orientations of R2.Example 1. Let (Vg) be a real vector space with fixed orientation. Wechoose a basis el, ..., e,, in V in such a way that the matrix M(g) has thediagonal format of Theorem 4, and (el, ... , e,) is positively oriented. Thenthe formula

g(vl,el), ...,g(vn,el)dV(vl, ...,vn) := det

g(vi, en), ... , g(vn, en)defines an n-form dV E An(V') independent of the choice of the particularbasis e1, ... , e with the stated properties. This form dV is called the volumeform of the oriented vector space with the non-degenerate scalar product g.By means of the dual basis al, ..., an, the volume form is represented as

dV = (-1)Q al A...Aa,,.Here q denotes the index of the scalar product g. The length of the volumeform is, by definition,

g(dV, dV) = (-1)9

1. Elements of Multilinear Algebra

The volume form of the coordinate space R" with the euclidean scalar pro-duct coincides with the determinant:

dR"(vl, ...,vn) = det(vi, ...,vn), v, E R".In particular, the determinant turns out to be an n-form on the space R".

Now we introduce the *-operator (Hodge operator) assigning a k-form toevery (n - k)-form. Consider the given real vector space V together with ascalar product g and a fixed orientation. For each k-form Sc E AV'), therule

An-k(V*) E) gn-k_ Wk A r)n-k E An(V')

determines alinear mapping from An-k(V*) to the 1-dimensional spaceAn(V*). The volume form dV is a basis vector in An(V*), and the vectorspace An-k(V*) is equipped with a non-degenerate scalar product. Hence,there exists exactly one (n - k)-form-to be denoted by *wk-such that

Wk A,fin-k = g(*Wke71n-k)dV

holds for all (n - k)-forms 17n-k. Summarizing we defined a linear mapping*: Ak(V*) -A n-k(V*),

the so-called Hodge operator.

Example 2. As above, let e1, ... , en be an orthonormal basis (g;i = fb;i)of the vector space V representing the fixed orientation, and let al, ... , anbe the dual basis. For an ordered k-index I = (i1 < ... < ik), denote byJ = (j1 < ... < in-k) the "complementary" index, i. e. the ordered (n - k)-index containing the numbers 11, ... , n}\{il, ... , ik}. The equations....g(*aj,oj)dV = alAa = sgn l I . Jn o'1A...Aan = (-1)sgn 1 I .Jn dVand immediately imply the formula

l...*oj = (-1)Isgn n(I J gijiI ...gin-yin-4a

The following theorem assembles the main properties of the Hodge *-operatorfor a scalar product of arbitrary signature.Theorem 5.

(1) For each k -form Sc, the twofold application of the Hodge operatoris given by * * wk = (-1)k(n-k)+gwk;

(2) for any k -forms wk, r7k, the following relations hold for the exteriorand the scalar product:

g(*wk, *rlk) = (-1)gg(Wk, rlk), Wk A *,nk = (-1)gg(Wk, rlk)dV .


Proof. To prove the first formula we first compute

= (-1)Qsgn (i...n)9.,... gi..-kin-k * oJ2y l...n l...n\

_ (-1) 5 1 J 9,0, ...9il-kie-ksgn (I J I g";, ...y;k,koi_

(-1)9(-1)kin-kioi. /The other formulas are consequences of this first one and the definition ofthe *-operator. We have

(*wk, *rik)dV = wk A *I?k = (-1)k(n-k) * flk Awk=

(-1)k(n-k)9(** r)k,wk)dV = (-1)99(r/k,wk)dV ,

and this implies the remaining identities. 0Example 3. Let n = 2k be an even number. Then the Hodge operatormaps the vector space Ak (V *) to itself,

*:Ak(V*)-Ak(V*).Moreover,

**talk = (-1)k+gWk,and hence, in case k + q = 0 mod 2, the Hodge operator has the eigenvalues 1.

Exercises

1. Let a,, ..., an be a basis of V*, let wl = E aj o;, ril = F_ bi a; be twoarbitrary elements from V*, and let 2 = c;j o; A aj be a 2-form.a) Compute wI A ill and, in the case n = 3, explain in which sense the

exterior product generalizes the vector product;b) compute wl A 2 and discuss in case n = 3 the relation to the scalar

product.

2. Prove that each 2-form w2 E A2(V*) can be represented asw2 = a, A a2 + ... + a2r_i A a2r

for a certain basis a,, - .. , on of V*-

Prove, moreover, that the number r isindependent of the choice of the basis and is characterized by the condition

(w2)r 76 (w2)r+1= 0 .

Exercises

3. Prove that k linear forms ol, ... , ok on V' are linearly independent ifand only if

olA...nok # 0-

4 (Cartan's Lemma). Let o1i ... , ok be linearly independent linear formsfrom V' and let l, ... k be arbitrary elements of V. If o1 A pi + ... +ok A Ilk = 0 holds, then the forms pi are linear combinations of the off,

k

J=1

Moreover, ail = aji.

5. Let e1, .... e,a be a basis of the vector space V and o1, ... , o the cor-responding dual basis. Then the following formula holds for every k-formWk:

n

o,A(eiJ Wk) =i=1

6. For a given form 0 96 wk E Ak(V*), define the subspace M(Wk) C V' by1tf(wk) = {171 E A'(V*) : ql AWk = O } .

Prove the following statements.a) The inequality dim M(wk) < k holds:b) the equality dim M(wk) = k holds if and only if there exist k linear

forms Cl, .... ok such that wk = of A ... A ok.Forms of the kind wk = of A ... A ok are called decomposable.

7. Prove the following statements.a) A 2-form w2 = F_i.j wig oiAaj is decomposable (see the previous exercise)

if and only ifWijWpg - WipWjg + WigWjp = 0;

b) a 3-form w3 = Eij k Wijk of A of A ok is decomposable if and only ifWickWpgr - WijPWkgr + wi.jq Wkpr - wilt wA-M = 0-

8. The Hodge operator of a 4-dimensional vector space V maps the spaceof 2-forms into itself,

* : A2(V')-A2(V'),


If the index of the scalar product is even, q = 0, 2, then the Hodge operatordecomposes the real vector space A2(V') into the eigenspaces

A2 (V') = {w2 E A2(V') : * W2 = f w2In the case of an odd index, q = 1, 3, the complexification A2(V') $ Canalogously decomposes into the i-eigenspaces. Compute the dimensionof the eigenspaces in both cases, and determine a basis of eigenforms.

Chapter ,2

Differential Formsin '

2.1. Vector Fields and Differential FormsVectors in euclidean space R" can be understood as "free vectors" or as"place-bound vectors" at a point in R". In the first case, we simply considerR" as a euclidean vector space. The second point of view is based on theconcept of R" as a set or a metric space whose elements are called the pointsof space. The "place-bound vectors" then form vector spaces of their own,each one consisting of all those attached to a particular point. For example,vectors at different points cannot be added. This second concept of a vectorin R" leads to the notion of the tangent space at a point in R".

Definition 1. Let p E R". The tangent space to R" at the point p is definedto be the set

TpR" :_ {(p, v) : v E R" } .An addition and a multiplication by scalars \ E R are introduced in thisset by (p, v) + (p, w) := (p, v + w) and A (p, v) :_ (p, \ v), respectively.These operations endow each TpR" with the structure of a real n-dimensionalvector space.

The ordinary differential of a smooth map f from the space R" to the spaceR'" at the point p, D fp, can now be interpreted as a linear map f.,p betweencorresponding tangent spaces.

Definition 2. Let U C R" be open and f : U R' be a differentiablemap. For each point p in the set U, the linear map f.,p : TpR" -. Tf(p) Rm

11

12 2. Differential Forms in 1R"

is defined by the formula

(f (p), Dff(v))Definition 3. A vector field defined on an open subset U of IR" assigns avector V(p) E TDR" in the corresponding tangent space to each point p E U.If e1.... , e is the standard basis in euclidean space lR". the vector fielddetermined by p (p,ei) is usually denoted by 8/8x', i.e. (8/8x')(p) :=(p, e,). (Compare Exercise 5 for an explanation of this notation.) Obviously,every other vector field defined on U can be represented in the form

V (P) = V, (P) iix-l0 (P) + ... + V" (p) 81" (P)with certain functions V1..... V" defined on U. The vector field V is calleddifferentiable of class Ck if all its component functions V1, .... V" have con-tinuous partial derivatives up to order k. The set of all vector fields of classCk is, on the one hand, a real vector space and, at the same time, a moduleover the ring Ck(U) of all real-valued Ck-functions on U. Graphically, avector field can be depicted by drawing at each point p the correspondingsecond component v of the vector from the tangent space. Consider, e.g.,on the plane R2 with coordinates x, y the vector field

V(x, y) =The following figure depicts this vector field:

yA

ttt

ttTf

Each of the tangent spaces Tp R" is a real vector space. Hence we canconsider the dual space, 71R' := (TpR, "")'. as well as its exterior powers,

/"(R") := "k(T;R")

2.1. Vector Fields and Differential Forms 13

An element wk of the space Ap(R") is thus an antisymmetric multilinearmap with k arguments on the tangent space TplR":

wk : Tp1R" x ... x TpR" -p R.In analogy with the notion of vector field, we will now introduce the notionof differential form.

Definition 4. A k -form on the open subset U of lR" assigns to each pointp E U an element wk(p) E Ap(1R")First we want to consider some examples of differential forms.

Example 1. Let f : U R be a smooth real-valued function, let p E U bea fixed point, and let D fp : IR" -, IR be its differential at the point p. Thenthe formula

df (p)(p, v) := Dfp(v)obviously defines a 1-form df on the set U.Example 2. A fixed basis el, ... , e" in IR" determines n coordinate func-tions x1, ... , x" and hence their differentials dx', ..., dx". Thus for a tan-gent vector (p, v) E TpR" we have the identity

dx`(p)(p,v) = v`,where v` denotes the i-th component of v with respect to the basis eI, ... , e".In addition, the 1-forms dxl (p), .... dx"(p) form a basis of the vector spaceA' (R") = T R". Arbitrary exterior products dx" A ... A dx'k as well astheir linear combinations with functions as scalar coefficients lead to furtherexamples of k-forms. Conversely, each k-form wk on U can be representedas

k = 11 ikw - wjl....,tkdx A ...ndx;1

14 2. Differential Forms in R"

In fact, at the point p E U the following equality holds for the vector (p. v):n of

df(p)(p,v) = Dff(v) _ (P)v'axiReplacing the vector components v' by dx'(p)(p, v) and omitting the argu-ment (p, v), we arrive at the stated formula

n 9fdf (p) _ (P) . dx' (p)

The exterior product of multilinear forms can easily be extended to differ-ential forms, defining for two forms wk, 7771 on U a (k + l)-form by

(wk An`)(P) := wk(P) A7l(p)The rules known from the first chapter remain valid without change:

(1) (wk+k)A7j' = wk AY/ +kA7/';(2) (f wk) A = f wk A i/;(3) wk A ii = (-1)kl7! A wk.

In particular, the lastproperty

implies that the exterior product of a formof odd degree with itself always vanishes, e.g., dx` A dx' = 0. Forms of evendegree do not in general have this property: A straightforward calculationshows that for the forms on R4 below the following relations hold:(dx' A dx2 + dx3 A dx4) A (dx' A dx2 + dx3 A dx4) = 2 dx' A dx2 A dx3 A dx4 .For conceptual as well as computational reasons, it is important that differ-ential forms can be "pulled back" by maps.Definition 5. Let f : Ul --y U2 be a differentiable map between two opensubsets Ul C R" and U2 C R, and let, moreover, wk be a k-form on U2.Then a k-form f' (wk) (the pullback or induced form) on Ul is defined by

f,(wk)((P,Vi),...,(p,vk)) := wk(fa,P(P,vl),...,f..P(P,vk))The form f * (wk) obeys the following rules.Theorem 1. Let f : Ul -, U2 be a differentiable map between the open setsUi C R" and U2 C R' with component functions f'. Then

f`(dx`) _ d,,," 8xiLet, moreover, differential forms on U2 and a function g : U2 -p R be given.Then

(1) f*(wi +w2) = f*(wk)+f*(w2);


(2) f' (g . wk) = (g f) f *(wk);(3) f*(wk Arl1) = f'(wk) A f*(r/).

Furthermore, in the case n = m, the following additional equality holds:

Proof. The definition of f' implies

f*(dxt)(P)(P,v) = dx'(f(P))(f(P), Dfp(v))But this is precisely the i-th component of Dfp(v). Hence

4(p)dx'(p)(p,v).f(dx')(P)(P, v) = at (P)zai=1 i=1

The three rules immediately follow from the definitions and will thereforenot be proved here. To derive the last equation, we use the second rule andthe identity just proved:

A...ndxn) = (guf)f*(dxl)n...A fs(dx")= (g f)

afdxJI] fn n

A ... A 8xi^it=1 in=1nn afl (9f

_ (g f) E aTi1 " ' 5; dxit n ... A dxinit, ....i.,=1

Since the exterior square of a 1-form always vanishes, in this sum onlythose terms remain whose n-index (j1.... ,j) is a permutation of (1, ..., n).Using the antisymmetry of the multiplication in the exterior algebra, weobtain the formula

(gf)Esg11(1 J n)BxJi'...'8xj. dx1 n...ndx"Jand thus the determinant of the differential D f . 0Recall that a Ci-function f on U can be considered as a 0-form, so thedifferential d is also a map turning a 0-form f into a 1-form df. Applyingthis map d in a suitable way to the coefficient functions of a differentialform, the exterior derivative can be extended to act on k-forms in general.

Definition 6. Let wk be a k-form on the open set U,k E it A ikwit, .ik dx ... A dx

it

16 2. Differential Forms in IR'

We define its exterior derivative dwk by the formulaCO = E d(wi1....,ik) dxi' A ... A dxik

i 1


Now we will show that dd = 0. Applying the derivative twice yields theexpression

n dx n dxiddwk = EE 013x 3I 3=1 o=1

(a2`a,'\ dx3 A dx A dxl.

I a

18 2. Differential Forms in Rn

and the 1-form on R3 defined in the coordinates x, y, z byw' = ydx+xdy+xyzdz.

Compute f'(w') as follows. First,f*(wl) = (yof)f'(dx)+(xof)f'(dy)+(xyzof)f*(dz)

= v3 f' (dx) + u2 f' (dy) + u3v4 f' (dz) .Then we use the fact that the exterior derivative commutes with the inducedmap:

f'(dx) = d(f'x) = d(xof) = d(u2) = 2u du.Similarly,

f*(dy) = 3v2dt,, f'(dz) = vdu+udv,from which the result follows:

f* W) = (2uv3+u3v5)du+(3u2v2+u4v4)dv.2.2. Closed and Exact Differential FormsFrom the theory of the Riemann integral, it is well-known that every con-tinuous function on R has a primitive function. In the language of forms,this can be expressed by saying that for every 1-form ' = g(x) dx with acontinuous coefficient function g : R - R, there exists a function f such thatdf = p'. If, in addition, g is differentiable, then certainly d' = 0, sinceeach 2-form on R vanishes. Now we want to pose the analogous question ofwhether any differential form has a "primitive form":

Let pk E fli (U) be a k-form. What are the conditionsguaranteeing the existence of a (k-1)-form gk-' E ft2 ' (U)whose exterior derivative coincides with k, dgk-1 = pk

The equality dd = 0 immediately implies that dpk = 0 is a necessary con-dition, but, in general, it is not sufficient for the solvability of the equationdq ' =,[k.Definition 7.

(1) A k-form wk E f1k(U) is called closed if dwk = 0;(2) a k-form wk E fZi (U) is called exact if there exists a (k - 1)-form

nk-' E f4-'(U) such that de-' = wkThe property dd = 0 states that each exact form is closed.

Example 6. Consider on the open set U = JR2-10) the winding formi -y xW = x2+y2dx +x2+y2dy

2.2. Closed and Exact Differential Forms 19

and calculate its derivative dwi:-dy y(2x dx + 2y dy)1 r dx x(2x dx + 2y dy) l y

Ax2 + y2 + (x2 + y2)2 J dx + [x2 + y2 (x2 + y2)2 A d

- y2+x2-2y2+x2+y2-2x2dx Ad = 0.(x2 + y2)2 yHence, wl is closed; later we will see that wl is not exact (Example 10). Theintegral of the winding form along a closed curve surrounding zero measureshow often it "turns around" the origin (Exercises 1 and 2).In algebraic topology, it is common to describe the difference between closedand exact forms by the so-called de Rham cohomology. The vector spacesof "cycles" Zk(U) and "boundaries" Bk(U) are defined by

Zk(U) {wk E !1k (U) wk is closed},Bk(U) {WA E S1;(U) WA is exact} .

Thus Bk(U) is a subspace of Zk(U), and the k-th de Rham cohomology ofU is defined as the quotient space

HDR(U) := Zk(U) / Bk(U).The winding form, e.g., is a non-trivial element in H)DR(R2 - {0}) # 0, andlater (Exercise 14) we will show that

HDR(II 2 - {0}) = R.The k-th de Rham cohomology only depends on the topological shape of theset U. For example, the de Rham cohomology of a convex set vanishes. Forslightly more general sets this is the contents of Poincare's lemma, whichwe are going to discuss now.

Definition 8. A subset U of IRn is called star-shaped (or a star-region if itis open in addition) if there exists a point po E U with the property thatfor every second point x E U the segment joining po with x is completelycontained in U. Obviously, star-regions are path-connected.

20 2. Differential Forms in Rn

Theorem 3 (Poincare's Lemma). Let U be a star-shaped open set in lid".Then

HDR(U) = 0for every k = 1, ... , n.In other words: For each closed k form wk E 1l (U) there exists a (k - 1)-form rik-1 E Qk-1 such that drik-' = wk.

Proof. In order to show this, we assign to every k-form wk = wi dx' a(k - 1)-form P(wk) satisfying the identity

wk = P(dwk) + dP(wk) .For closed forms the first term vanishes, which proves the assertion. Theform P(wk) is defined as follows:

P(wk) _k rj'tk_1wI..ik(tx)dt] _

(-1)-1I x dxi' A... Adxio A... A

1j< ...

2.2. Closed and Exact Differential Forms 21

We apply the operator P to this result:

[oma

P(dwk) = E L [ f't tk &i1...ik (tx)dt] xj dxi' A ... A dxikaxii1

22 2. Differential Forms in IR"

where the functions f, g, h : R3 R still have to be determined. Theexterior derivative is easily computed:

d'l' [ax ay] dx A dy + [82 az] dx A dz + I ah - ag] dy A dz.yHence the functions have to satisfy the conditions

ag of Oh - Of2 and

ah - OgX Y, 2xOX ay y' az 49Z y anay Oz

Integrating, e.g., the first two with respect to x yieldss

dx.g = 2 x2y + ,l dx, h = 2xy + , J49Z

Inserting the result into the last condition, /

2x = 2x + / Oy dx - J a N dx,we see that it is satisfied for function f; in particular, we may choosef = 0. Then g = x22 y, h = 2xy, and hence, using

171=

21x2ydy+2xydz,

we obtain a solution, as is easily checked. The integration method computesthe "primitive form" by means of the map P(w2) introduced in the proof ofPoincare's lemma. In the example, this turns out to be the sum of 6 terms,namely

1

P(w2) = + Cf

t(tx)(ty)dt J xdy - (J1 t(tx)(ty)dt) ydx

(1o / ro l

+21

t(tx)dt J ydz - 2 (J 1 t(tx)dt I zdy

+2 Uo t(ty)dt I x dz - 2 (jI t(ty)dt I z dx.Each of the summands can easily be computed:

2- 1 2 I 2 2 2 2P(w) = 4x ydy- 4xy2dx+ 2xydz - 2xzdy+ 2xydz - 2yzdx1 2 2 2 2 4(4xy + 2yz dx + 4x y- 2xz dy + 2xydz.

Which of these two methods leads to a solution more quickly depends onthe particular situation; since an integration has to be carried out in anycase, it might not be possible to find an explicit elementary solution (just.as not every continuous function is elementary integrable).

2.3. Gradient, Divergence and Curl 23

2.3. Gradient, Divergence and CurlEach tangent space TpRn of the coordinate space is an oriented, euclideanvector space, and hence there is the volume form

dRn(p) E Ap(R")as well as the Hodge operator

A (Rn) ---, /\p-k(Rn).This allows us to associate with every differential form wk of degree k onRn a corresponding (n - k)-form *wk defined by applying the *-operatorpointwise, i.e., at each point p E Rn to wk(p). Consider an orthonormalbasis e1, . . . , en of the space Rn and the corresponding coordinate functionsx 1, ... , x". For a k-form wk = F w1dxr expressed in these coordinates theassociated form is thus determined by the following formula:

*Wksgn l...

Jn

wt dxj.

Here, J = (jl < ... < jn_k) is the complementary multi-index to I = (i1 (l) j 1 a(ck o I(j,o))-

a(ck o Ik 1))/j=1

k k-I

= E j:(-1)i+j [ckI(j o)I(+.oj - C'(j,o)I(i 1) - C'(j,1)I(i:)j=1 i=1 LIn order to be able to transform this expression, we introduce two additionalsummation indices a,,3 = 0, 1, and rewrite it as

k k-1a(ack)

_

(-1)i+j+a+0 l(ck o I(j.a) oj=1 i=1 a=o,1

0=o,1

For j < i and a,,3 = 0,1 we havek k-1 1 k-2 k 1 i-1 i k-21U-) 01 (i.li) (x , ... , x ) = I(j,a) (x , ... , x , fl, x , ... , x )

= (x1, ... , xj-1,Similarly, one rewrites I(i+1 p) o 1(j,a) as1(i+1.0) IU*(x1, ... , xk-2) = I(i+1,0)(x1, ... , xj-1, a, x1, ... , xk-2)

1 1 i-1 i k-2

Together these yield the identity

I(j,a) I(i,8) = I( i+1,0) I(j,)In order to apply this, we split the sum appearing in the expression forO(Ock) into those terms for which j < i and the remaining ones:

a(ack) = L.r Lam(-1)i+j+a+Qck o Ik o Ik-11


In a similar way as the de Rham cohomology, we can define the k-th cubichomology group of a set U C Rn:

Hk''(U) := ker (8 : Ck(U) --' Ck-1(U)) / im (0: Ck+,(U) ~ Ck(U))

2.5. Integration of Differential Forms and Stokes' TheoremConsider a singular k-cube ck : [0, 1]k _ U C R" of class C', as well as ak-fonn wk on U. Then the induced differential form (ck)'wk is defined onthe unit cube [0,1]k, and, as such a multiple of dxl A A dx"

(ck)`wk = f (x) dxl A ... A dx'for some function f : [0, 1]k - R. We can thus define the integral of wk overthe singular cube ck as follows:

Definition 17. Setwk f(x)

ck [01Jk

and extend this definition linearly to any k-chain sk = >, lj in U byrwk

1kWk

Example 9. For k = 1 a singular k-cube is simply a parametrized Cl-curve c : (0, 1] -i U C R'. In this case, the integral of the 1-form wl =p1dyl +... + p"dy" is called the line integnzl of wl along c. If cl, ..., c" arethe component functions of c, the pullback of the form is written as

c'w'(t) = pl(c(t)). d dtt) dt + ... + pn(c(t)) - dcndtt) dt,and hence, in this situation, we obtain the following general formula for theline integral:

Jw' =

J 1 pi(c(t)).dctt)1 dt.

If the 1-form wl is the differential of a smooth function f, then byc*(wl) = c*(df) = d(f o c)

we obtain the following value for the integral over df :

J df =jdt)dt

=

In particular, the line integral of an exact 1-form depends only on the endpoints of the curve and not on its shape. It vanishes for a closed curve.

2.5. Integration of Differential Forms and Stokes' Theorem 31

Example 10. We will use the last remark to show that the winding form,

cal = x2 + 2 dx + x2 x 2 dy,Y2 y2is not exact on R2 - {0}. To this end, consider the parametrization c(t) _(cos 21rt, sin 21rt) of the circle by the interval [0.1]. Then

c'wl = 27r dt and jw1 = 27r.

The integral of a k-form over a singular k-cube is, up to sign. independentof the parametrization of the cube. This is a consequence of a well-knowntransformation rule for higher-dimensional integrals: Let y, : U -' V be adiffeomorphism, and let f : V -+ R be integrable; /then

f(f o )(y) . I det(D(y))I dy J f (x)dx .VTwo parametrizations and 4 : [0, 1]A U of one and the same pointset in U differ by a diffeomorphisin cp : [0,11k [0, 1]k of the unit cube.4 = ci o gyp. The determinant of the differential Dp(x) has constant sign,which will be denoted by e(,p).Theorem 7. In the above situation we haveJk

=f wkcz"

Proof. By Theorem 1 the pullback of the form

( i)"wk = F*(ci)"(wA) = cp`(f dxlA...Adx") = (foip)'4p'(dx'A...ndx")coincides with

(c2)*wk = (f o cp) dxl A ... A dx".Thus the statement follows from the quoted transformation rule for n-dimen-sional integrals. 0In case k = 1, changing the direction of a curve results in a change of signfor the line integral.

Now we prove the essential result of this chapter, Stokes' theorem. This isa far-reaching generalization of the Fundamental Theorem of Calculus andcomprises the classical integral formulas of the 19-th century as special cases.The proof to follow will, however, reveal to the careful reader that the coreof the matter is the fact that integration and differentiation are mutuallyinverse operations.

32 2. Differential Forms in 1R"

Theorem 8 (Stokes' Theorem). Let wk be a differential form defined on theopen subset U C IR", and let sk+l : [0, 1]k+l - U be a (k + 1)-chain. Then

f.9$k+lwk = r d k

Jgk+1

Proof. Since the integral is additive, it suffices to prove the formula forsingular (k+1)-cubes. First we consider the standard cubelk+l : [0, 1]k+lR k+1 and represent the k-form wk on Rk+l as

k+l _

dxk+1wl~ = fi d21 / ... / dxi A ... A

The derivative is theni=1

k+lk = [(_1)1_1L]i dxl A ... A dxk+1i=1 8xi

Hence, by the definition of the integral,f k+1

J dwk axi dxjk+1 i-1 [0,11kOn the other hand, applying the maps IV Ql parametrizing the differentparts of the boundary of the unit cube leads to the formula

Ik+l * k- 1 , X j 1 , a, Xj. . . ., xk) dx1 A . . . A dskf,(x( > w - ...,J.a )from which we conclude that

J (Ii I ` U ) _ f k[0.1[k

/10111k

` J

r flJ J afi (xl,..., J-1, t, xJ, ... , xk)dt dxl ... dxk

o dxj[o.1[k!

adx.dxi[O,1Jk+l

By the definition of the boundary of Ik+1, this implies thatkr+l r kr+1Jk w= L L. f f dfx)'dx = fk

ajk+l J-1 Q-o'1Ik+1 j= l [o.llk+l lk+1li.Q)and hence we have verified Stokes' formula for the standard cube. For anarbitrary singular (k + 1)-cube ck+1 : [0, 1]k+1 U C 1R" we now use the

2.5. Integration of Differential Forms and Stokes' Theorem 33

fact that the exterior derivative commutes with the pullback of forms. Wehave

J dWk = f (ck+ll*( k) d((Ck+l)'Wk)Ck+1 Jk+1 l J jkJ+1

l` J J

j=1 a=0,1Ck+lolk+1U.u)

k+1__ k+1 wk _ f (ck+1 ` Wk

ajk+1 j=1 Q=0.1 +1wk.)

k+1

wk = Jk.

ask+l

We already emphasized that the line integral of an exact 1-form is indepen-dent of the particular shape of the curve. As a first application of Stokes'theorem, we will prove that the line integrals of a closed 1-form along twodifferent curves coincide if there exists a continuously differentiable defor-mation of one curve into the other leaving their initial and end points fixed.This leads to the notion of homotopy, which is fundamental in topology.

Definition 18. Two Cl-curves co, cl : [0,11 U C R" are called homo-topic if there exists a C1-map F : [0,1] x [0,11 - U with the properties

F(t, 0) = co(t), F(0, s) = co(0) = c(0).F(t,l) = cl(t), F(1,s) = co(1) = cl(1).

The map F is called a homotopy between the curves co and cl.

Theorem 9. Let co, cl : [0,1] - U be two homotopic C1-curves, and let w1be a closed 1 -form on U. Then the line integrals along co and cl coincide:

Lw1 = jwll

Proof. Choose any homotopy F : [0,1]2 -> U between co and c1. This mapF is, at the same time, a singular 2-cube, and hence the 2-form dw1 can be

34 2. Differential Forms in IR

integrated over F. By assumption we have dwl = 0, and the correspondingintegral vanishes:

dwl.0=IF

On the other hand, by Stokes' theorem, we have for the right-hand sider dw1 fJ w1 + J

_ J _ IJF F (t,O) (1,s) (t,I) (O,s)

These four integrals are computed using the homotopy property of F. First,F(t, a) = ca(t) immediately implies, for a = 0,1,

FlZt,a)w1= cowl.

Moreover, F(a, s) is independent of s, and henceFl*(a,,)wl = 0.

Lastly, these relations combine to yield the equation1 1

cw0 =J

cowl- J l ,0 0

and by the definition of the integral this concludes the proof. 0We will now generalize these observations concerning line integrals to thehigher-dimensional case. Let sk lj ch- be a singular k-chain withimage set A C IR", and suppose that Oak = 0. The set A can, e.g., be a k--dimensional sphere Sk in R". Consider a smooth map f : A A defined ona neighborhood of A which is homotopic to the identity IdA. This is againsupposed to mean that there is a smooth map defined on a neighborhood Uof A x (0.11 in R"+1

F : A x (0,11 -+ A such that F(a, 0) = f (a) and F(a,1) = a.Then

Fo(skxIdlo,11) :=Eis a singular (k + 1)-chain in R", and, because Oak = 0, the boundary is

OF o (sk x Id10,11) = sk - f o sk.For a k-form wk defined on an open neighborhood of the set A C W', thefollowing holds.

Theorem 10. If Osk = 0 anldf is homotopic to the identity, thenI k=J wkJsk jock

2.6. The Classical Formulas of Green and Stokes 35

Proof. We compute the difference using Stokes' theorem:

f Wk -J

k = f k = f ksk fosk OFo(skxId1o,11) Fo(skxld[O,j])

The (k + 1)-form (sk x Id[o,I))`F*(dwk) vanishes. This follows from theimplicit function theorem together with the assumption that A is the imageof a k-dimensional chain. But this immediately implies the statement. 0

2.6. The Classical Formulas of Green and StokesIn this section, we will discuss the classical two-dimensional special cases ofStokes' formula. Let D C R2 be a subset of R2 which can be representedas the image of a CI-map f defined on the standard cube [0, 112. By ODwe denote the boundary of this set, considered as a singular 1-chain. Thederivative of the 1-form w1 := (x dy - y dx)/2 is the volume form on R2.Hence

fdwlvol(D)2 aD

The formula transforms the calculation of a two-dimensional volume intothe evaluation of a line integral, and this turns out to be a special case ofGreen's formula to be discussed now. Consider the 1-form

wI

and compute its derivative:

dw I = I - -J

dx A dy.

Thus we arrive at Green's first formula.

Theorem 11. Let P(x, y) and Q(x, y) be functions of class CI. Then

I fDIOQ-8PJdxAdy.

D L Ox ay

Suppose now that P(x, y) and Q(x, y) are of class C2, and consider the1-forms

wl P [-LQ dx+ LQ dy] and qI := Q I-5y . dx + 5P dy]The derivative of the difference wI - 771 is easily computed to be

d(w'-1 ) =Applying Stokes' formula leads to Green's second formula.

36 2. Differential Forms in W'

Theorem 12. Let P(x, y) and Q(x, y) be functions of class C2. Then[PO(Q) - QO(P)]dx n dy =f [QLP - P l dx + fPQ - QPl dy.

8DJ L 1

Stokes' formula concerning certain surface integrals is, in a similarly simpleway, a special case of the above general integral formula. In fact, let F C R3be a subset of IIt3, which can be represented as the image of a Cl-map fdefined on the standard cube [0,1]2 (a surface piece). By 8F we denotethe boundary of this set considered as a singular 1-chain in JR3. Consider avector field V defined on an open neighborhood of F. Its curl is defined bythe following condition:

dwy := curl(V) J dllt3 = *wcurl(V)Integrating the 2-form sweurj(V) over the surface piece F, we obtain Stokes'theorem in its classical form.

Theorem 13. Let V be a smooth vector field on a neighborhood of F. Then

J(V1dx1+V2dx2+V3dx3) = / &,;y =

OF F8V2 8V11 1 2 r8V3 8V11 IOV3 8V21 2 3axl - axe I dx Adx +II axl - 5x3 I dx ndx3+ axe - 'ax-3 dx nda .

F

Remark. Using the volume form dF of a regular surface piece yet to bediscussed, the classical Stokes formula can be written more concisely as

Lcun1'1) dF.Here N denotes the normal vector to the surface in R3. We will return tothis in Chapter 3.

2.7. Complex Differential Forms and Holomorphic FunctionsThe complexification of the vector space of all real-valued forms is called thespace of complex-valued forms on an open subset of R". Such a form can besplit into its real and imaginary parts;

wk = wo + i Wkand differentiation as well as integration are defined with respect to thisdecomposition:

d,jk dwo + i dwi, LkjkOLkI .

2.7. Complex Differential Forms and Holomorphic Functions 37

In a similar way we extend the exterior product to complex-valued forms:W k A 771 := (WO A 170 - W I A 911) + i (,oo A 7l1 + W I A 770) .

Then the previous computational rules and Stokes' theorem still hold. Nowwe want to apply complex-valued forms to study holomorphic functions and,to do so, first identify the real vector space R2 with the complex numbersC. From z := x + i y and z := x - i y we obtain the differential formsdz := dz := and dzndz =Let f be a complex-valued function with real and imaginary parts u and v ofclass C1, f (z) = u + i v. Denote by u=, uy, vv, vy the partial derivatives withrespect to the corresponding variables. Then f (z) dz is a complex-valueddifferential form. We compute its differential:

Now let f : U --+ C be a complex-differentiable function defined on an opensubset of C. Elementary complex analysis starts by proving that its real andimaginary, parts are smooth functions in the sense of real analysis (Goursat'stheorem). Furthermore, the Cauchy-Riemann equations hold:

ux=vy and uy= -vx.Theorem 14. If f (z) is a complex-differentiable function, then the 1-formf (z) dz is closed,

d(f (z) dz) = 0.An immediate consequence is Cauchy's theorem.

Theorem 15. Let U be an open subset of C, and let -y be a closed curve inU, which is the boundary of a singular 2-cube. Then the integral

0

vanishes for each complex-differentiable function f.

In a similar way we derive Cauchy's integral formula. To do so, we assumethat f (z) is a complex-differentiable function on a neighborhood of the disc

K(zo,M) = {zEC:IIz-zoII Z - zo ax(zo,e) Ilz - zol12

=e2

fK(zo0

(z - zo)f(z) dz.

38 2. Differential Forms in IR"

We compute the derivative of the 1-form (z - zo) f (z) dz:2i -

Thus

J 2i. f(x) dandy.K(zo,M) Z - ZO=

E2 K(zo.e)The mean-value theorem of integral calculus states that there is a numberze in K(zo, e) for which

f (z) dx n dy = f (ze)vol(K(zo, e)) = 7r `2 f (ze)K(zo.e)

If e tends to zero, then f (ze) converges to f (zo), and we arrive at Cauchy'sintegral formula:

Theorem 16.f(zo) = 127ri JaK(zO,jvf) z - zo

This formula is fundamental for the theory of functions. It implies, e.g.,that every function which is complex-differentiable in the neighborhood ofa point can be expanded into a power series (i.e. is an analytic function).2.8. Brouwer's Fixed Point TheoremA fixed point of a map f : X --+ X from a set to itself is defined to be apoint xo which is not moved by f, f (xo) = xo. In topology, several fixedpoint theorems are known. They state that certain continuous maps from ametric space to itself necessarily have at least one fixed point. If, e.g., X isa complete metric space, and f : X -p X is a contracting map, then by theBanach fixed point theorem the map f has at least one fixed point. Thereare, of course, (non-contracting) maps from a complete metric space to itselfwithout fixed points; translations in R" are examples for this. A topologicalspace X is said to have the fixed point property if every continuous mapf : X - X from X to itself has a fixed point. This is obviously a topologicalproperty, i. e., homeomorphic spaces either all have the fixed point propertyor none of them has it. The unit circle X = S' is compact; nevertheless itdoes not have the fixed point property; rotations are continuous maps fromthe unit circle to itself without stationary points. Brouwer's fixed pointtheorem states that the closed n-dimensional ball of radius R,

D"(R) = {xER": IIxii

2.8. Brouwer's Fixed Point Theorem 39

Theorem 17. Every Cl-map f : D" -+ D" from the n-dimensional ball toitself has at least one fired point.

Proof. Suppose that f : D" - D" does not have any fixed points. Thendefine the map F : D" Sn-' from the ball to its boundary by assigningto every point x E D" the point of intersection of the ray from f (x) throughx with the sphere Sn-'. The formula for F,F(x) = x

X - f(x) 2 x - f(x) x - f(x)_ IIxI12 + (X,

IIx - f(x)II 1-(X'

IIx - f(x)II> IIx - f(x)IIshows that F is smooth. Moreover. F acts on the boundary of the ball asthe identity, F(x) = x for all x E S. Let Fl.... , F" be the componentsof F. Differentiating the following relation, which is valid for all x E Dn.

n

E(F'(x))2 = 1,i=1

yields

2 (Fiaa*5) Idxi = 0,i=1 ij=1

and hence for each index jj:Fi(x)8F11'(x)

= 0.11T1

i=1

Therefore, the system of equations

8P (x) = 0Ox)i=ohas a non-trivial solution (al, ... , an) = (Fl (x), ..., F"(x)) j4 (0.... , 0).Hence the determinant of the matrix

det 0

vanishes. Now we apply this observationto the differential formWn-' = F1 A dF2 n ... A dF"

and conclude that its differential vanishes:i \&,n-1 = dFlAdF2A...AdF" = det(aa2 )dx'A...Adx" = 0.

40 2. Differential Forms in 1t"

By Stokes' theorem the integral of the form w' 1 over the boundary Sn-1of the singular cube D" is equal to zero:

0 = do-1 = n-1- JD- jn-1

On the other hand, F acts on the sphere S"-1 as the identity; hencewn-1ISn-I

= x1 dx2 A ... A dxnlSn-i .This implies

r0 = J x1 dx2 A ... A din = J dxl A ... A dx" = vol(D"),n -I Dn

and we arrive at a contradiction. 0Theorem 18 (Brouwer's Theorem). Every continuous map f : D" Dnfrom the n-dimensional ball to itself has at least one fixed point.

Proof. The proof will be reduced to the case of a C1-map by applying theStone-Weierstrass approximation theorem (see, e.g., [Rudin, 19981, Theorem7.32). The ball D" is compact. Consider the ring C(D") of all real-valuedcontinuous functions on it, as well as the subring R of those functions whichare the restriction to D" of a C1-function with strictly larger, open domainof definition. Obviously, the subring R contains the constant functions andseparates points. By the Stone-Weierstrass theorem, it is dense in C(U).Applying this to the components f 1, ... , f" of f, we conclude that for eache > 0 there exists a C1-map

p : D' - R" such that IIf (x) - p(x)II < e for all x E D'.Consider the renormalized map P(x) := p(x)/(1 + s). Since

IIP(x)II - IIf (x)II 5 IIP(x) - f (x)II < e and IIf (x)II < 1,we have IIP(x)II 5 1 + e; hence IIP(x)II 5 1. Therefore, P is a map from theball to itself. Moreover, P can be estimated against f :

IIf (x) - P(x)II 5 IIf (x) - P(x)II + IIP(x) - P(x)II 5 e + IIP(x)II 11- 1 +

< e+(1+e)1+e < 2e.Summarizing, we have proved that for each e > 0 there exists a C1-mapP : D" D" satisfying for every x E D" the estimate

IIf(x)-P(x)II

2.8. Brouwer's Fixed Point Theorem 41

would be strictly positive. Choose for F = /2 a smooth approximation pwith the properties stated above; by Theorem 17 this map then has a fixedpoint xo E Dn, for which in turn

IIf(xo) - Axo)II = Ilf(xo) - xoII < iwould have to hold. But this contradicts the definition of .

Brouwer's fixed point theorem can be viewed as an existence result for real,non-linear systems of equations. We state a possible application, which playsan important role in Galerkin's method.

Theorem 19. Let g1:... , gn : D' (R) -+ R be continuous functions de-fined on the ball Dn(R) of radius R, and suppose that for all points x =(x1, , xn) E S'-'(R) in the sphere Sn-1(R) the following inequality holds:

n9i(x).xi > 0.

i=1Then the system of equations

91(x) = 92(X) _ ... = 9n (x) = 0has at least one solution in D'(R).Proof. We combine the functions to define a map g : Dn(R) --+ Rn, g(x) :=(91(x), .. , 9n(x)). If g(x) 0 0 holds for all points x E Dn(R), we canconsider the map f : Dn(R) -p Sn-1(R),

f (x) := -R g(x)119(x)11

whose image lies in the sphere Sn-1(R). By the fixed point theorem, f hasa fixed point in S"-'(R). Hence there exists a point xo E Sn-1(R) suchthat

xo = -R 9(xo)119(xo)11

This implies R2 II9(xo)II = -R (g(xo), xo), contradicting the assumption ofthe theorem.

In particular, the assumption of the theorem is satisfied for gi(x) = xi +hi(x) if the functions hi : R' -+ R grow more slowly than linear forms,

Ihi(x)I < Ci 11x11"Under this condition the sum

n n

E 9i(x)xi = R2 + c` hi(x)x'i=1 i=1


behaves like R2 on the sphere S"-'(R) and becomes positive for sufficientlylarge radii.

Corollary 1. The system of equationshl(x) = x',

..., h"(x) = x"has at least one solution f o r arbitrary continuous functions bounded by II I'at infinity.

Example 11. The system of equations x = ' 1 + x2 + y'2. y = cos(x + y)has in l 2 at least one solution, e.g.,

x = 1.2758079, y = 0.14722564.Example 12. Consider in 1R2 the following system of equations:

91(x, y) = x + e-(s+y)2 = 0, 92(x, y) = y + e-(I-U)2 = 0.The picture below diaplays the graph of the function g, (x, y) x + 92(x. y) yover the set [-2.2] x [-2,2]. It shows that this function is positive on thecircle S' (2) of radius 2. Hence the above system of equations has at leastone solution in the disc D2(2). A numerical computation of the solutionleads to the values

x = -0.303122, y = -0.789407.

Exercises 43

Exercises

1. Let f : 1R2 - {0} R2 - {0}, f (r, 0) = (r cos 0, r sin 0), be the polarcoordinate map on the "punctured" plane. Prove:a) The winding form satisfies f dx) = d9;b) the radial form x dx + y dy satisfies f ` (x dx + y dy) = r dr.

2. Consider on R2 - {0} the winding form wl = 114.-1., as well as thefollowing family of curves depending on the integer parameter n E Z:

c : [0, 11 - . R2 - {0}, (cos2ant,sin2nrnt).Compute the line integral of the winding form along the curve c,,, and con-clude that the curves c,l are not homotopic in R2 - {O} for different valuesof the parameter n.

3. Compute the exterior derivative of the following differential forms:a) xydxAdy+2xdyAdz+2ydxAdz;b) z2dxAdy+(z2+2y)dxAdz;c) 13 x dx + y2dy + xyz dz;d) e' cos(y) dx - e' sin(y) dy;e) xdyAdz+ydxAdz+zdxAdy.

4. Consider 1R2n with coordinates x1, ... , x2n and the following differentialform of degree 2:

w2 = dx' n dxn+l + dx2 A dxn+2 + ... + dxn n dx2n .Prove:

a) The form w2 is closed;b) the n-th exterior power of w2 is related to the volume form via the

formula

w2 n ... n w2 = (-1)n(n-1)/2n! dxl A ... A dx2n .

5. The subject of this exercise is to explain why it makes sense to denotethe vector fields x -+ (x, e;), defined using the standard basis e1, . . . , e,, of


R", by a/8x'. Writing these vector fields for the moment as E;, every vectorfield can be written in the form

n

V =

For each function f : U R on an open set U of R" we define a newfunction. the derivative of f in the direction of the vector field V, by

(V(f))(x) := (Df=)(V(x))Prove the formula

i=1which by omitting the argument f provides the explanation asked for.

6. Prove the following rules for vector fields on )1t3:a) div(V1 x V2) _ (curl(Vl),V2) - (VI,curl(V2));b) curl(curl(V)) = grad(div(V)) - 0(V), where the Laplacian is to be ap-

plied componentwise to V.

7. Compute the line integral

(x- 2xy)dx + (y2 - 2xy)dyJC2along the curve C = {(x, y) E R2 : x E [-1.1), y = x2}.

8. Compute the line-integral

ICsin(y)dx + sin(x)dy,

where C is the segment joining the points (0, x) and (n. 0).

9. Consider on R3 the differential form w2 = y dxAdy. Determine all 1-forms17 1 = p dx + q dy satisfying dal = w2.

10. Prove the following converse to one of the statements of Example 9: Ifwl is a 1-form defined on the open set U, and if the line integral of wl isindependent of the curve, then wl is exact.Hint. Prove this by explicitly constructing a "primitive function". Inphysics, the vector field V corresponding to the 1-form wy is called con-servative, if it does no work along any closed curve -y. 11 wy = 0. Thus, V isconservative if and only if w1, is exact.

n

V(f) _ 08xi (fl

Exercises 45

11. Consider the singular 2-cube, f : [0, 2ir] x [0, 27r] -r S2 C R3 - {0},f (u, v) _ (cos u sin v, sin u sin v, cos v),

as well as the 2-form w2 = (xdyAdz+ydz Adx+zdxndy)/r3 on 1R3 - {0},where r = (x2 + y2 + z2)1/2 denotes the distance from the origin.a) Prove that w2 is closed;b) compute the integral of w2 over f ;c) conclude from the properties just proved that w2 is not exact on 1R3-{0},

and that there is no singular 3-chain c3 in 1R3 - {0} whose boundaryequals f, 0c3 = f.

12. Prove that the integral defines a unique bilinear map

HDR(U) X Hkub(U) R, ([WI], [s']) -- J Wk.gk

13. Let w1 = f (x) dx be a 1-form on the interval [0, 1] with f (0) = f (1).Prove that there exist a real number p and a function g with g(0) = g(1) bymeans of which wl can be written as

1W = pdx+dg.

14 (Continuation of 13). Let 771 be a closed 1-form on R2 - {0} and wlthe winding form. Prove that there exist a number p as well as a functiong : 1R2 - {0} -+ R for which

77 1 = pwl +dg.Consequently, the winding form is the generating element of the first deRham cohomology of R2 - {0}.Hint. Consider the polar coordinate map f from Exercise 1 and its pullbackf `wl. This can be written as f `wl = A(r, O)dr + B(r, B)dO; here B(r, O)dOis a 1-form on [0, 27r] (depending on the parameter r) to which the previousexercise applies.

15. Consider on 1R3 the following exact differential form known from 7:W2 = xydxAdy+2xdyAdz+2ydxAdz,

and the upper half-ball A C S2:A = {(x,y,z)EIR3: x2+y2+z2=1, z>0}.

Prove that the integral of w2 over A vanishes.

46 2. Differential Forms in R

16. Let C be the circle in R2 with the equation x2 + (y - 1)2 = I in itsstandard parametrization. Compute the line integral

f xy2 dy - yx2 dx

a) directly;

b) using Green's formula.

17. Let E be the ellipse with the equation x2/a2 + y2/b2 = 1 (a > 0, b > 0)in its standard parametrization. Compute by means of Cauchy's integralformula the integral

drJE zand obtain from this the value of the integral

12 dtJp a2 cos2(t) + b2 sin2(t)

18. Let 7-1 be an infinite-dimensional Hilbert space, and D = {x E lI IxI I < 11 its unit ball. Does D have the fixed point property?

19. A subset A C X of a metric space X is called a retract of X, if there isa continuous map r : X --. A such that r(a) = a for all points a E A. Provethat if X has the fixed point property, then so does every retract A of X.

zz

20. The set A = { (x, y) E [-1,1]2: xy = 0 } has the fixed point property.

Chapter 3

Vector Analysis onManifolds

3.1. Submanifolds of RnIn Chapter 2 we introduced an integration method for differential forms oversets which can be represented as images of singular chains. These sets, how-ever, may be quite irregular, and it is rather difficult to develop a differentialcalculus for functions defined on them. Further notions like tangent space,vector field, etc., are not available either in their context. Hence we will nowrestrict the possible subsets of Rn to a class for which a differential as wellas an integral calculus can be established in a satisfactory manner. Thesesets are called manifolds, and they are-intuitively speaking-characterizedby the fact that their points can be defined locally in a continuous (differen-tiable) way by finitely many real parameters, that is, locally these sets lookjust like euclidean space. It was the fundamental idea of B. Riemann in hisHabilitationsvortrag (1854) to introduce the notion of a manifold as the newbasic concept of space into geometry. In physics, manifolds occur as config-uration and phase spaces of particle systems as well as in field theory. Theprecise description of what a submanifold of euclidean space is supposed tobe is the content of the following definition.

Definition 1. A subset M of Rn is called a k-dimensional submanifoldwithout boundary if, for each point x E M, there exist open sets x E U C R"and V C R' as well as a diffeomorphism h : U V such that the imageh(U n M) is contained in the subspace IRk C R :

h(UnM) = Vn(IRkx{O}) = {yEV: yk+1- =yn=0}.47

48 3. Vector Analysis on Manifolds

The set U* := UnM together with the map h' := hlu.: U* -- V' := VnRkis called a chart around the point x of the manifold. The sets U* and V'are open subsets of M and Rk, respectively (see next page).

VRn

v n (Rk x {o})

h-1

A family of charts covering the manifold M is called an atlas. The no-tion "diffeomorphism" can be understood in the sense of an arbitrary C'-regularity (1 > 1). Correspondingly, we have manifolds of regularity classCL. For simplicity, we suppose in this chapter that all maps, manifolds, etc.,are of class CO, and for this reason we will simply talk about smooth maps,manifolds, etc. Note that, without any change, all the statements also holdassuming only C2-regularity.

For any two charts (h', U') and (hi, Ur) of the manifold for which the in-tersection u* n u; is not empty, one can ask how they are related. Themap

h'o(hi)-1:

is called the chart transition from one chart to the other. The sets h1(U` nUl) and h' (U' n Ul) are open subsets of the coordinate space Rk, and thetransition function h' o (hi)-1 is obviously a diffeomorphism.

The notion of dimension also needs to be explained for manifolds. If a sub-set of Rk is mapped homeomorphically onto an open subset of the space R,the dimensions of both coincide, k = 1. Under the additional assumption of

3.1. Submanifolds of 1R 49

differentiability, which will always be made here, this is easy to prove: thedifferential of a diffeomorphism at an arbitrary point is a linear isomorphismbetween the tangent spaces T1IRk and TyR', and this immediately impliesk = 1. The corresponding fact for homeomorphisms is a deep topologicalresult going back to Brouwer (1910). In any case, the number k occurringin the definition of a manifold is uniquely determined and will be calledthe dimension of the manifold. Sometimes we will write the dimension of amanifold as an upper index, i.e., denote the manifold also by Mk.

In the first theorem we will prove that, under certain conditions, subsets of1R' defined by equations are submanifolds. This will give rise to plenty ofexamples.

Theorem 1. Let U C lRn be an open subset, and let f : U - Rn-k be asmooth map. Consider the set

M = {x E U : f (x) = 0} .If the differential D f (x) has maximal rank (n - k) at each point x E M ofthe set M, then M is a smooth, k-dimensional submanifold of 1R' withoutboundary.

Proof. The proof is based on a straightforward application of the implicitfunction theorem. If xo E M is a point from M, then there exist an openneighborhood xo E Uxo C U and a diffeomorphism hxo : Uxo -' hxo(Uxo) CRn such that the map f o hzp : hxo (Uxo) - Rn-k is given by the formula

f oh=o (x1, ...,xn) = (xk+1, ...,x').This implies Uxo n M = h=o (hxo (Uxa) n ]R' ), and hence the chart around thepoint xo E M we asked for is constructed. 0Example 1. Every open subset U C Rn is an n-dimensional manifold with-out boundary.Example 2. The sphere Sn = {x E Rn+1 : IIxUI = 1} is an n-dimensionalmanifold. To see this, we consider the function f : Rn+1 -, R defined byf (X)

= I Ix1I2 - 1. Then we have D f (x) = (2x', ..., 2xn+1), and the rank ofthe (1 x n)-matrix D f (x) on Sn is equal to 1. Theorem 1 implies that Sn isan n-dimensional manifold.

Example 3. Consider a smooth map f : Rn ]Rm and its graphG(f) = {(x, f (x)) : x E Rn} C 1Rn+m

G(f) is the zero set of the map 1 : IRn+m = lRn x 1R- 1Rm defined byO (x, y) = f (x) - y. The differential D4 has maximal rank equal to m ateach point. Therefore, G(f) C Rn+m is an n-dimensional manifold.


Example 4. The torus of revolution is the surface in R3 described by theequation (0 < r2 < rl )

( x2 + y2 - rl )2 = r2 - z2 .

A parametrization can be obtained by the formulasx = (r1 + r2 cos cp) cos V,, y = (r1 + r2 cos cp) sin 0, z = r2 sin cp

with parameters 0 < cp, ' < 2ir. The partial derivatives of the functionf(x,y,z) = ( x2+y2-rl)2-r2+z2 are

= 2z,Ox - 2x(1 - 21 ), of - 2y(1 21 2), Ozx-+y2 y VI'X +yand it is obvious that the vector D f does not vanish at any point of thetorus of revolution. Hence Theorem 1 applies and shows that the aboveequation defines a manifold.Example 5. Not every set defined by an equation is a manifold. For ex-ample, consider the set in R2 described by the equation x4 = y2:

Near the point (0, 0) E ]R2 this set is not a manifold. In fact, after havingdeleted this point, any neighborhood of the set splits into four componentsand hence cannot be homeomorphic to an interval. Indeed, the assumptionof Theorem 1 concerning the differential of f (x, y) = x4 - y2 is not satisfied

2at the point (0, 0) E R.

3.1. Submanifolds of R" 51

Example 6. On the other hand, there exist manifolds in R" which cannot bedescribed by systems of equations satisfying the assumptions of Theorem 1.Later we will see that every manifold defined as in Theorem 1 has a particularproperty-it is orientable. An example of a non-orientable manifold is theso-called Mobius strip. One of its parametrizations isx = cos(u)+vcos(u/2)cos(u), y = sin(u)+vcos(u/2)sin(u), z = vsin(u/2)with parameters 0 < u < 2ir, -7r < v < 7r.

Apart from equations, manifolds can also be defined by prescribing theirlocal parametrizations (charts). Let us explain this construction principle.Theorem 2. Let M be a subset of R" and assume that for each point x E Mthere are an open set U, x E U C R", an open set W C Rk, and a smoothmap f : W U such that the following conditions are satisfied:

(1) f(W) = M n U;(2) f is bijective;(3) the differential Df(y) has rank k at each point y E W;(4) f'1 : M n U W is continuous.

Then M is a k-dimensional submanifold without boundary in R.

Proof. For an arbitrary point x E M we choose a map f : W -i U withthe stated properties and denote by y the pre-image of x, f (y) = x. Thedifferential has rank k, and hence we can assume without loss of generalitythat 1r

det 9k 0.

4ak

Consider the map defined by g(a, b) := f (a) + (0, b) with g : W x R"-k -+ R".Then the determinant of the differential of g coincides with the determinantabove, and hence, in particular, it is different from zero. Applying the


inverse function theorem of differential calculus, we obtain two open setsV1 and V2. with (y, 0) E V1 and x E V2 in R", for which g : V1 V2is a diffeomorphism. We invert this map and denote the resulting inversediffeomorphism by h := g-' : V2 -+ V1. By assumption f is continuous,and hence there exists an open set 0 C R" such that

{f(a):(a,O)EV1} = f(w)no.Consider now the sets V2 := V2 nO and V1 = g-'(V2). Then we have

V2nM = V2nOnM = {g(a,0):(a,0)EVII,and thus we obtain

h(V2nAf) = g-1(V2 nM) = V1 n(Rk x 101)Therefore, the condition to be satisfied for each point of a manifold holdsfor A1.

Now we will extend the notion of manifold, taking into account also bound-ary points. We confine ourselves to the case that the boundary itself is asmooth manifold without boundary (no corners or edges). We define thek-dimensional half space Hlik to be the set

Hk= {xERk:xk>O}.

Definition 2. A subset M of R" is called a k-dimensional submanifold(with boundary) if for each point x E M one of the following conditions issatisfied:

(1) There exist open sets U and V, with x E U C R", V C R", and adiffeomorphism h : U -+ V such that

h(U n M) = V n (Rk x {0}) .(2) There exist open sets U and V, with x E U C R", V C R", and a

diffeomorphism h : U -+ V such that

h(UnM) = Vn(Hk X {O})and the k-th component hk of h vanishes at the point x, hk(x) = 0.

Conditions (1) and (2) cannot be satisfied at the same time for one andthe same point x E M, for otherwise there would exist diffeomorphismsh1 :U1 -iV1, h2 : U2 --+ V2 such that

h1(UInM) = VlnRk and h2(U2nM) = V2nlEllk, h2(x)=0.

3.1. Subinanifolds of R" 53

The set hl (U1 n U2) then would be an open subset in IRk mapped diffeo-morphically onto h2(Ul n U2) by the chart transition map h2 o h, 1. Sinceh2(x) = 0, the set h2(U, nU2) would thus contain a point from the boundary8Hk = Rk-4 of the half-space. Consequently, it could not be open in Rk.Altogether this contradicts the inverse function theorem. This observationjustifies the followingDefinition 3. Let M C 1[t" be a manifold. A boundary point of M is apoint x E M for which condition (2) of Definition 2 is satisfied. The set ofall boundary points is denoted by 8M and called the boundary of M.

Theorem 3. Let AEI be a k-dimensional manifold. Then its boundary OMis either empty or a smooth (k-1)-dimensional manifold without boundary,

88M=0.Proof. Fix a boundary point x E OM and choose open sets U C R n. X E U,and V C Ht" with

h(U n m) = V n (Hk x (01).For every other boundary point x' E U n OM the k-th component of h hasto vanish at x` by the preceding observation, hk(x*) = 0. Hence we have

h(U n 8M) = V n (IItk-' x {0}),and thus (h IunaAf, U n OM) is a chart for the boundary 8M.Example 7. The boundary of the Mobius strip is a closed space curve.

Example 8. The (n - 1)-dimensional sphere Sn-' is the boundary of then-dimensional ball D".


3.2. Differential Calculus on ManifoldsWhen a manifold is covered by charts, every chart range is an open subsetof R" and hence a set on which differential calculus is familiar from anal-ysis. In this way it is possible to develop a differential calculus on manifolds.

As in the preceding section, we will now start from a k-dimensional mani-fold Mk and a chart h : U V around a point x and denote by y := h(x)the image of x under this chart map. Then h-1 : V -{ U is smooth and(Dh-')y = (h-1).,y is a linear map between the tangent spaces to R" (com-pare Definition 2, Ch. 2)

(h-1).,y: TyR" - T=R" .Definition 4. The tangent space of the manifold Mk at the point x isdefined to be the image of TVRk under the map (h-1).,y:

Trllfk (h-1).,y(Ty %) C .,R" .The tangent space T?Mk is a k-dimensional vector space, since the differ-ential of the diffeomorphism h-1 is injective. Moreover, we have to checkthat the tangent space of the manifold just defined does not depend on thechoice of the chart. But this is an immediate consequence of the equivalentdescription for the tangent space that is to follow next.

Theorem 4. The tangent space TTMk consists of all vectors (x, v) E T=R"for which there exists a smooth curve y : [0, E) - Mk C R" such that-y(0) = x and y(0) = v.Proof. A vector v = (x, v) E TXMk in the tangent space can be representedas v = (Dh-'),(w) for a certain vector w E Rk. The image under h' ofthe straight line in Rk through y in the direction of the vector w is the curve7(t) we were looking for: the equality

-t(t) = h-1(y + tw)immediately implies -y(O) = h-1(y) = x, and from the chain rule we obtainfor the tangent vector

ddtt) = dt (h-1(y + tw)) l t=o = (Dh-'),(w) = v.The converse is proved analogously.

If the manifold Mk is defined by (n - k) equations, and if, in addition,the differentials of the defining functions are linearly independent, then thetangent space has a simple description.

3.2. Differential Calculus on ?Manifolds 55

Theorem 5. Let fl, ... , fn_k : 1R" , 1R be smooth functions and supposethat

df1 A ... A 0 0 .Then the tangent space TTMk of the manifold

Mk = {xER": f1(x)=...=f,,-k(:r.)=0}consists of all vectors v E T11R" satisfying

df1(v) = ... = dfn-k(v) = 0.In particular, the euclidean gradient fields grad(f1), ... , grad(fk) are per-pendicular to the tangent space of the manifold at each point of :11k.

Proof. Taking a curve r) Alk in 11k and differentiating theequation

f1('r(t)) = ... = f"-k(r(t)) = 0with respect to the curve parameter t yields

df1(i(t)) _ ... = df"-k(i(t)) = 0.The tangent space TA1k is thus the subspace of all those vectors v E T IR"on which all the differentials df1, . df"-k vanish. Comparing the dinlen-sions of these two vector spaces shows that they have to coincide.

The set of all tangent spaces to the manifold is called the tangent bundle ofAlk and denoted by TMk. It is a manifold of dimension 2k. In fact, at leastin the case that A1k is determined by equations, f1 = . . . = f"_k = 0, theset TAlk is in turn defined as a subspace of 1R" x 1R" by the equations

fl (:r) = ... _ .fn-k(x) = 0 and df1(x. v) = ... = df"-k(x, v) = 0.These are 2(n - k) conditions in R2". The corresponding functional determi-nant (Jacobian) does not vanish, since the differentials df1 are linearly inde-pendent. The formula (x, v) := x defines a projection 7r : TMk . Alk onthe tangent bundle of the manifold assigning to every vector its base point.


Example 9. Consider the sphere S' = {x E R"+1 :IIxII = 1}. Thedifferential of the function IIXI12-1 is 2(x1, = xn+1) and hence the tangentspace to the sphere at any point consists of all vectors perpendicular to thispoint:

TS" = {(x, v) E R"+1 x Rn+1: IIxII =1, (x, v) = 01.Definition 5. Let Mk C R' and N' C R' be two manifolds, and letf : Alk - N' be a continuous map. We call f a differentiable map if foreach chart h-1 : V - Mk of the manifold Mk the resulting map f o h-1V -. N' C R' defined on the open subset V C Rk is differentiable.

As in euclidean space, the differential of a smooth map can be introduced asa linear map between tangent spaces. For a tangent vector (x, v) E TZMk wechoose a curve y : [0, e) - Mk with 7(0) = x and y(O) = v. The compositionf o y(t) is a curve in N1 passing through f (x) E N', and its tangent vectordescribes the result of applying the differential of f to the tangent vector(x, v),

f.,x (X, V) := (f(x), dtf o y(0) IThe differential of a smooth map between two manifolds has all the proper-ties which are familiar from euclidean space.

Theorem 6. The differential f.,2 : TXMC -+ Tj(t)NI of a smooth map isa linear map between the tangent spaces, and the differential of the super-position of two smooth maps f and g is equal to the superposition of theirdifferentials,

(g o f).. = 9.,f1=) f*.xThe last formula is the generalized chain rule.

Definition 6. A vector field V on a manifold Mk assigns to every pointx E Alk a vector V(x) E ,,Mk in the corresponding tangent space.

If the map V : Mk - TRn = Rn x R" is smooth, then we will speak of asmooth vector field on the manifold. Vector fields can again be added andmultiplied by functions, so that the vector space of smooth vector fields isa module over the ring Coo(Mk) of all C-functions on Mk.Example 10. The formula V(x) = (x, (x2,-x1,0)) defines a vector fieldon the 2-dimensional sphere (see the figure on the next page).

3.2. Differential Calculus on Manifolds 57

For a chart map h : V Mk C R', which this time and and sometimesalso later will be denoted by h instead of h-1, h.(a/ayi) are vector fieldstangent to Mk defined on the subset h(V) C Mk, and they provide a basisin each tangent space. For simplicity and as long as it is clear to which chartwe refer, these vector fields on the manifold will also be denoted by a/ayi.On the subset h(V) C Mk. every other vector field V can be represented astheir linear combination

k

V(y) _ Vi(y)5iii=1 y

Here V2(y) are functions defined on the set h(V); using the chart map. nowand then they will also be considered as functions on the parameter set V.These functions are called the components of the vector field V with respectto the fixed chart.

Example 11. In euclidean coordinates on 1R2, consider the vector field

V=x15x2-x25x1

depicted on the next page. Introducing in R2 - (0} polar coordinates by theformula

h(r,p) = (r cos V, r sin so), 0 < r < oo, 0 <


Differentiable functions f : Mk -- R can be differentiated with respect to avector field V. At a fixed point x E Mk we choose a curve ry : [0, EJ Mksatisfying the initial conditions y(0) = x and y(0) = V(x). The derivativeoff at the point x in the direction of V(x) is now defined by the formula

V(f)(x) := tf o y(t)It=oThe result is a C30-function V(f) defined on the manifold Mk. In the nexttheorem we stunmarize the properties of this differentiation:

Theorem 7.(1) (V+W)(f) =V(f)+W(f);(2) V(f1 + f2) = V(f1) + V(f2);

(3) V(f1 - f2) = V(fl) - f2 + f1 - V(f2);(4) If the vector field V is represented as V = V'(y)818yi in some

local chart, thenk

V(f) _ Vi(y)8(foh)i=1

Proof. We will prove (4); all the other claims follow immediately. If thepoint x E Mk corresponds to the point y E V under the chain map h : VMk, then -y(t) = h(y + t(V 1(y), ... , Vk(y))) is a curve in Mk satisfying theinitial conditions -t(O) = x and ry' (0) = V(x). The formula for V(f)(x) to beproved then follows from the chain rule:

k

V(f)(x) = d f o9 (y+t(V1(y),...,Vk(y))) _ V'(y)a(f h). 0dt i=1'

3.2. Differential Calculus on Manifolds 59

Next we will discuss the notion of Riemannian metric on a submanifold Mkof euclidean space. The scalar product in R" is denoted by (v, w). Werestrict it to the tangent spaces of the submanifold.

Definition 7. Let M11k

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