Geometric Theory of Foliations

César Camacho

Alcides Lins Neto

Geometric Theory of

Foliations

Translated by Sue E. Goodman

BIRKH.kUSER Boston • Basel • Stuttgart

Library of Congress Cataloging in Publication Data

Camacho, Cesar, 1943- Geometric theory of foliations.

Translation of: Teoria geometrica das folheacoes. Bibliography: p. Includes index. 1. Foliations (Mathematics) 2. Geometry, Differential.

I. Lins Neto, Alcides, 1947- . II. Title. QA613.62.C3613 1985 514'.73 84-10978 ISBN 0-8176-3139-9

CIP-Kurztitelaufnahme der Deutsch,en Bibliothek

Camacho, César: Geometric theory of foliations/César Camacho; Alcides Lins Neto. Transi. by Sue E. Goodman - Boston; Basel; Stuttgart: Birichauser, 1985.

Einheitssacht.: Theoria geométrica das folheagôes <dt.> ISBN 3-7643-3139-9

NE: Lins Neto, Alcides: 27

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner.

© BirkhAuser Boston, Inc., 1985 ISBN 0-8176-3139-9 ISBN 3-7643-3139-9

CONTENTS

Introduction 1

Chapter I — Differentiable Manifolds

§1. Differentiable manifolds 5 §2.The derivative 13 §3. Immersions and submersions 14 §4. Submanifolds 16 §5. Regular values 17 §6. Transversality 17 §7. Partitions of unity 18

Chapter II — Foliations

§1. Foliations 21 §2. The leaves 31 §3. Distinguished maps 32 §4. Plane fields and foliations 35 §5. Orientation 36 §6. Orientable double coverings 37 §7. Orientable and transversely orientable foliations 38 Notes to Chapter II 41

Chapter III — The Topology of the Leaves

§1. The space of leaves 47 §2. Transverse uniformity 48 §3. Closed leaves 51 §4. Minimal sets of foliations 52 Notes to Chapter III 53

Chapter IV — Holonomy and the Stability Theorems

§1. Holonomy of a leaf 62 §2. Determination of the germ of a foliation in a 67

neighborhood of a leaf by the the holonomy of the leaf

§3. Global trivialization lemma 69 §4. The local stability theorem 70 §5. Global stability theorem. Transversely orientable case 72 §6. Global stability theorem. General case 78 Notes to Chapter IV 80

Chapter V — Fiber Bundles and Foliations

§1. Fiber bundles 87 §2. Foliations transverse to the fibers of a fiber bundle 91 §3. The holonomy of 5. 93 §4. Suspension of a representation 93 §5. Existence of germs of foliations 100 §6. Sacksteder's Example 102 Notes to Chapter V 106

Chapter VI — Analytic Foliations of Codimension One

§1. An outline of the proof of Theorem 1 116 §2. Singularities of maps f: lit" --• JR 118 §3. Haefliger's construction 121 §4. Foliations with singularities on 13 2 123 §5. The proof of Haefliger's theorem 127

Chapter VII — Novikov's Theorem

§1. Sketch of the proof 131 §2. Vanishing cycles 133 §3. Simple vanishing cycles 138 §4. Existence of a compact leaf 140 §5. Existence of a Reeb component 149 §6. Other results of Novikov 152 §7. The non-orientable case 157

Chapter VIII — Topological Aspects of the Theory of Group Actions

§1. Elementary properties 159 §2. The theorem on the rank of S 3 163 §3. Generalization of the rank theorem 165

§4. The Poincaré-Bendixson theorem for actions of IR2 168 §5. Actions of the group of affine transformations 171

of the line

Appendix — Frobenius' Theorem

§1. Vector fields and the Lie bracket 175 §2. Frobenius' theorem 182 §3. Plane fields defined by differential forms 184

Exercises 189

Bibliography

199

Index 203

INTRODUCTION

Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book.

The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painlevé saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field.

The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 1930's: "Does there exist on the Euclidean sphere S3 a completely integrable vector field, that is, a field X such that X - curl X -= O?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the sphere S 3 a two-dimensional foliation?"

This question was answered affirmatively by Reeb in his thesis, where he presents an example of a foliation of S 3 with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu-mulate asymptotically on the compact leaf. Further, the foliation is C. Also in the work are proved the stability theorems, one of which, valid for any dimen-sion, states that if a leaf is compact and has finite fundamental group then it has a neighborhood consisting of compact leaves with finite fundamental group. Reeb's thesis motivated the research of other mathematicians, among whom was A. Haefliger, who proved in his thesis in 1958, that there exist no analytic two-dimensional foliations on S 3 . In fact Haefliger's theorem is true in higher dimensions.

The example of Reeb and others, which were constructed later, posed the following question, folkloric in the midst of mathematics: "Is it true that every

2 Geometric Theory of Foliations

foliation of dimension two on S 3 has a compact leaf?" This question was answered affirmatively by S. P. Novikov in 1965, using in part the methods introduced by Haefliger in his thesis. In fact Novikov's theorem is much stronger. It says that on any three-dimensional, compact simply connected manifold, there exists a compact leaf homeomorphic to the two-dimensional torus, bounding a solid torus, where the leaves are homeomorphic to two-dimensional planes which accumulate on the compact leaf, in the same way as in the Reeb foliation of S 3 .

One presumes that the question initially proposed by Hopf, was motivated by the intuition that there must exist nonhomotopic invariants which would serve to classify three-dimensional manifolds. In fact this question did not suc-ceed in this objective, since any three-dimensional manifold does admit a two-dimensional foliation. However, a refinement proposed by J. Milnor with the same motivation, had better results. In effect, Milnor defined the rank of a manifold as the maximum number of pairwise commutative vector fields, linearly independent at each point, which it is possible to construct on the mani-fold. This concept translates naturally in terms of foliations associated to actions of the group Et'. The problem proposed by Milnor was to calculate the rank of S. This problem was solved by E. Lima in 1963 by showing that the rank of a compact, simply connected, three-dimensional manifold is one. Later H. Rosenberg, R. Roussarie and D. Weil classified the compact three-dimensional manifolds of rank two.

In this book we intend to present to the reader, in a systematic manner, the sequence of results mentioned above. The later development of the theory of foliations, has accelerated, especially in the last ten years. We hope that this book motivates the reading of works not treated here. Some of these are listed in the bibliography.

We wish to express here our appreciation to Airton Medeiros and Roberto Mendes for various suggestions and especially to Paulo Sad for his collaboration in the reading and criticism of the text.

Rio de Janeiro, May 1979

César Camacho Alcides Lins Neto

Addendum to the English edition

This book is a translation of TEORIA GEOMÉTRICA DAS FOLHEAÇÔES, no. 12 of the Series Projeto Euclides, published by IMPA — CNPq (BRAZIL). In this translation the arguments of some theorems were improved and an ap-

Introduction 3

pendix about elementary properties of the fundamental group was suppressed. We wish to acknowledge Sue Goodman for the excellent work of translation.

Rio de Janeiro, March 1984

César Camacho Alcides Lins Neto

I. DIFFERENTIABLE MANIFOLDS

In this chapter, we state the basics of the theory of differentiable manifolds and maps with the intention of establishing the principal theorems and notation which will be used in the book.

§1. Differentiable manifolds

Just as topological spaces form the natural domain of continuous functions, differentiable manifolds are the natural domain of differentiable maps. in order to better understand the definition of manifold, we begin by recalling some aspects of differential calculus.

A map f:U —• 111" from an open set U C Br to Rn is differentiable at X E U if there is a linear transformation T: lie --■ Rn which approximates f in a neighborhood of x in the following sense:

f(x + v) = f(x) + T• y + R(y)

and

(. R y) Jim =0

y --• 0 lvi

for all sufficiently small y E Rm. The map T, when it exists, is unique. It is called the derivative of f at x and

is denoted Df (x).


Df(x) • y

0

Figure 1

The derivative has the following geometric interpretation. Given V E lit m we take a differentiable curve o: Udefined on an open interval I IR con-taining 0 E JR such that a ( 0 ) = x and c' (0) = v. Then

f(a(t)) f (x) d f(a(t)) Df(x) - = dt t 0

i = I

Fixing the canonical basis of III', we define the partial derivatives off at x by ( äf/ax) ( x) = Df(x) e i . In this manner we have that for each vector u = E a,e, in

Df(x) y = E aiDf(x) e, = E -af (x) a i . ax,

We say that f is of class C' on U when all the partial derivatives ( af/ax,)(x)

are continuous as functions of x E U. Proceeding inductively on r, f is of class CT when all the partial derivatives of f are of class C'' U. When f is of class C" for all r EN, we say f is of class Cœ.

The differentiability of the composition of two maps is determined by the chain rule which says that if f: U Cr and f(U) C V then gof: U Ile is of class C' and D(g of) (x) = Dg(f(x)) Df(x).

A map of class C', r 1,f: U V = f(U) between open sets U,V in III' is called a Cr diffeomorphism if f possesses an inverse f -1 : V-- • U of class Cr. in particular, a diffeomorphism is a homeomorphism. Moreover, for each x E U, Df(x) : IR m is an isomorphism and (Df(x)) -1 = = Df-1 (f(x)).

Let f: W IR m , W g IR m be an open set. We say that f is a local diffeomorphism when for each p E W there exists a neighborhood U c W of p such that f I U: U f( U) c JR tm is a diffeomorphism.

1= 1

Ile and g: — • 1R k are of class

Differentiable Manifolds 7

The notion of differentiability, which until now was associated with maps defined on open sets of Euclidean spaces, will be extended next to maps defined on certain topological spaces locally homeomorphic to R m .

With this objective we define a local chart or a system of coordinates on a topological space M as a pair ( U, v ) where U is an open set in M and v : U — ■ Er is a homeomorphism from U to an open set yo ( U) in Ilen. An atlas a of dimension ni is a collection of local charts whose domains cover M and such that if ( U, (if, ) , (- 0,) E a and U n (7 0 then the map 7p so (U n /7) — 7,-0( U CI Ci) is a Cr diffeomorphism between open sets of R m . The diffeomorphisms Y-:, . v ' as above are called changes of coordinates.

Figure 2

The concept of differentiability can now be extended to maps between topo-logical spaces which possess an atlas of class C', r a- 1.

Indeed, let M and N be topological spaces and suppose that a and 63 are atlases of class Cr on M and N respectively. We say that f : M --s. N is dif-ferentiable of class Ck , k r, if f is continuous and for each x E M there exist local charts ( U, sp) E a and ( V, ik ) E 63 with x E U and f(x) E V such that

of. 40 -1 : v (u n f -I (V)) C R m -.– 1k (V) c lir

is of class Ck . Since the changes of variable are C' diffeomorphisms, r _._. k, this definition

is independent of the local charts chosen. It is clear that in the Euclidean spaces R m , if we consider the single local chart ( IV , identity map), the notion of differentiability given above coincides with the usual one.

An atlas Ci. of class C' on M is called maximal when it contains all the local charts ( 0,7",o ) whose changes of coordinates with elements ( U, v ) E a


;76 4o -1 : (U n ej) ;o(U n (j) (*)

are CT diffeomorphisms. The advantage of considering a maximal atlas a is that in this case the domains of local charts of a form a topological basis of M. On the other hand, each atlas a is contained in a unique maximal atlas Ci. Indeed, a is defined as the union of all the local charts ( o ) such that if ( U, so ) E a and un then the changes of coordinates ( * ) are of class Cr.

A maximal atlas of dimension m and class Cr on M is also called a differen-

tiable structure of dimension m and class Cr on M. It is now clear that a continuous map f : M N between spaces M and N with Cr differentiable structures is of class Ck , k r, if and only if for each x E M there exist local

- . charts ( U, so), (V,11/) on M and N such that x E U, f(U) C V and V, o f o . so (U) V/ (V) c 1R" is of class ck.

A differentiable manifold of class Cr and dimension m is a Hausdorff topo-logical space M, with a countable basis, provided with a differentiable struc-ture of dimension m and class cr.

To denote a manifold M of dimension ni we will at times use the notation Mr"

Unless otherwise specified, all manifolds considered hereafter will be of class C oe

A map f : M N of class Cr,r.1, is a diffeomorphism when it possesses an inverse f -1 : N M of class cr. In this case we say that the manifolds are diffeomorphic and we write M = N.

Example 1. The sphere Sm.

The sphere Sm defined by

= ix = (X 1 x„ E V"' I4 + + 4, 1 = H ,

with the topology induced from IR'"' possesses a Coe manifold structure of dimension m.

Indeed, let Uric Sm, a = -±1, 1 i m + 1, be the set U7 = [xEs m 10-x,> 0). The local charts (pc:: U'l I1V"1 defined by so 7(x 1 ,...,xi _,,xi ,xi+ , = = (x, x i _,,x, + , x„, + ,) cover S r" and the changes of coordinates are C.

Example 2. Real projective spaces.

Let F", n 1, be the set of lines of IR " which pass through the origin. can be considered as the quotient space of IR"' — 101 with the equivalence relation which identifies two vectors u, 0 if u = tv, where t E 1R — [0). We consider V' with the quotient topology, that is, U C F" is open if r -I (U) is open in 1R" — SO], where r : IR" — to) F" is the quotient map which associates to each element u E E"' — 101 its equivalence class r [u] The space F" with this topology is compact and Hausdorff and it is called the real projective space of dimension n.


Let us see how one constructs a C atlas on 11)". The equivalence class [x i ,..., x„ 1 1 of the vector (x 1 ,..., x 1 ) can be thought of as being the line in 11" +1 which passes through the origin and through the point ( x 1 x„ 1 ) O. For each i E f i n 1), let U, = [ [x i ,..., x, +11 E IP" I xi 0). We define so, by v i [x i x, Xn+1] = (i/X i )(X i ,•••, Xi-i9Xi+ 1 ,...2 Xn+i)•

It is easily shown that Ui is an open subset of IP", that (p i is a homeo-morphism and that U7_+1 = 13 ". Moreover, if i,j E fi n + 1 ) with i < j, we have that U, n 0 and

is given by the expression 1 1

(Pi ° _

(Y1 ,•••, Yrt) = (Y1 ,•••% Y1-1, Yi+1 ,•••, Y n )•

Then (Ui , so i ) I i = 1 n + 1) is a C° atlas on rn.

Example 3. Classification of manifolds.

A natural problem in the theory of differentiable manifolds is the problem of classification. We will say that two manifolds M and N are equivalent if they are homeomorphic. The problem then consists of determining explicitly a repre-sentative of each equivalence class.

For example it is very easy to show that if M has dimension 1 and is con-nected, then M is homeomorphic to S' or to JR (see exercise 4). In the case of dimension 2 the problem is already complicated, since its solution implies in particular a classification, modulo homeomorphisms, of all open subsets of 11;1 2 . However if we restrict to compact connected manifolds of dimension 2 a classification is possible. One proves ([32] and [61]) that every compact con-nected surface is homeomorphic to the quotient of a polygon by an equivalence relation that pairwise identifies the sides of the polygon according to the follow-ing rules.

ORIENTABLE SURFACES

a

4.

sphere

a

torus

Figure 3

projective plane Klein bottle

Figure 4

a 2

nonorientable surface of genus g

-+

1

6

Figure 5

4

7


NONORIENTABLE SURFACES

a a 1

In figures 3 and 4 the sides with the same letter are identified to preserve the sense of the arrows.

In the sequence of figures below we illustrate how a surface of genus 2 can be obtained by identifications from an octagon.


In figure 5, we first identify sides a, followed by sides c. Observing that the vertices of the octagon are all identified to the same point, in the fourth figure we identify the common endpoints of sides b and d that in the third figure form the boundary of a hole. In the fifth figure we identify the sides d. In the sixth figure we change the position of the figure, deforming it a little. Finally in the seventh figure we see the final position of the curves a, b, c, d on the surface of genus two.

Example 4. Manifolds of dimension 3.

The problem of classification of compact manifolds of dimension 3 is still open. Even a classification of those that are simply connected is not known. However some useful properties of manifolds of dimension 3 can be obtained in an elemen-tary manner. For example one can show that any compact connected manifold of dimension 3 can be decomposed into the union of two handlebodies bounded by a surface of genus g in IR 3 .

This can be shown using the fact that such manifolds can be triangulated, i.e., decomposed into a union of tetrahedra such that any two of them intersect along faces, edges or vertices [51]. The union of all the edges and vertices form a connected set of dimension one which will be the "skeleton" of one of the two handlebodies Tt . The complement of T1 is a handlebody T2 whose skeleton is formed by the edges and vertices of a "triangulation" dual to the first. The dual triangulation is obtained in the following manner. In the center of each tetrahedron put a vertex. The vertices that are in adjacent tetrahedra are joined by an edge which cuts the common face in one point. The faces are taken so that they intercept each edge of the first triangulation in only one point. In this manner we obtain a decomposition of the manifold into polyhedra (not necessarily tetrahedra). For more details see [51].

Example 5. Non-Hausdorff manifolds.

In the definition of manifold it was required that every differentiable manifold is a Hausdorff space. Nevertheless there exist spaces that satisfy all the axioms of the definition of manifold except that of Hausdorff. Such spaces are called non-Hausdorff manifolds and they occur naturally in certain discussions. Let us look at an example.

Let f : 1R 2 —• IR be defined by f( x,y ) = a (x 2 )eY where o: 1R --• JR is a function class C such that a ( t) = 1 if t E ( — c, c ) , a ( 1 ) = 0 and a' (t) < 0 for t > E.

It is easy to see that the level curves of f, f - ' ( t), t fixed, have the follow-ing form:


Figure 6

Each connected component of a level curve of f is called a leaf of f. Let M be the quotient space of R2 for the equivalence relation which identifies points on the same leaf of f. Let r : R2 —I. M be the projection map for the quotient. The space M with the quotient topology is not Hausdorff since the points a = ( 1 , t) and b = 7r ( – 1 ,t), tE IR, are points of ramification, that is, they do not admit disjoint neighborhoods of M. However if it is possible to define a C atlas on M of dimension 1.

In fact, the sets U, = tir (x,y)EMix < 11 and U2 = 17r(X,Y) E M x> are open in M = U1 U U2. Define 40, : U, JR by ço, ( 7r (x,y )) = f(x,y),

E = 1,2. It is easy to show that 4c 1 and 40 2 are homeomorphisms, that 40, ( U l n (J2 ) = (0, + 00 ) and that so o w' is the identity on (0, + 00 ). Hence ( ),(U2 ,,,o 2 )1 is a C' atlas for M.

Observe that M is homeomorphic to the quotient space of two disjoint copies of IR for the equivalence relation which identifies points with the same negative coordinate.

Example 6. A manifold structure for covering spaces.

Let X and Ybe locally path connected topological spaces, and Y be connected. We say that a continuous, surjective map Ir : X Y is a covering map if it satisfies the following properties:

(a) For each y E Y there exists a connected neighborhood U of y such that r (U) = --1 ( , ) V, where for each x E 7r - (y) the restriction r V, : V, U is a homeomorphism.

(b) If x, ,x2 E 7r - (y ) and x, x-, then v V,, = 0.

In particular the set 7r - '(y) is discrete and has no accumulation points. The neighborhood U is called a distinguished neighborhood of y and the set 7r (y ) the fiber over y.


We will consider the following two situations which occur at various times throughout the text:

(A) Suppose that Y is a differentiable manifold. In this case it is possible to define a differentiable manifold structure on X so that dim (X) =

dim( Y) and 7r is a local Cœ diffeomorphism. Indeed let x E X and let y = 7r ( x ) . Consider neighborhoods V of x and U of y such that 71 1V:V U is a homeomorphism. Since Y is a manifold, let w : W be a local chart such that yE Wc U. Take V1 = 7r ( W) n U, which is open in X. Then the composition lk

= o ( 7r ) : w ( W) is a homeomorphism. Also the collection of charts ( u'), constructed as above, is a Cœ atlas of dimension n on X, as can be easily shown. This structure is called the structure co-induced by 7r. It is the unique structure which makes ir a local Cœ diffeomorphism.

(B) In the case that X is a differentiable manifold (but not necessarily Y), with one more natural hypothesis, we can induce on Y a Cr differentiable manifold structure by means of 7r. With this structure 7r will be a CT local diffeomorphism.

Hypothesis. Suppose that given y E Y there exists a distinguished neighborhood U of y with the following property: if x ,x2 E Ir (y) and V, and V2 are neighborhoods of x , and x2 such that 7 rl V,: V, U ( i = 1,2) is a homeomorphism on U then ( 70 V2 ) ( 7r 1 V, ): V2 is a Cr diffeomorphism (r 1).

Construction of the atlas on Y. Given y E Y, we fix U, a distinguished neigh-borhood as in the hypothesis. Let x E 7r -1 (y ) and take V a neighborhood of x such that 71 I V : V U is a homeomorphism. Since X is a differentiable manifold, there exists a local chart so : W E n , with xE Wc V. Let

= so I W) : 7r ( W) 111n. Then 1,t is a homeomorphism. By using the hypothesis it is possible to prove that the collection of all the maps ( W, ) constructed as above, constitutes a CT atlas of Y. We leave the details to the reader.

A specific example is the following. Let M be the quotient space of R 2 for the equivalence relation — where (x,y) ( x ',y ' ) if and only if (x — x' ,y — y' ) E Z 2• Let ir : JR 2 M be the quotient projection. It is easy to verify that 71 is a covering map and satisfies the hypothesis of B. Hence M, with the structure induced by r, is a Cœ differentiable manifold of dimension 2. It is possible to show that M is diffeomorphic to the torus T 2 = S' x S 1 .

P. The derivative

We introduce now the notion of derivative of a differentiable map between two manifolds, taking as a model the geometric interpretation of the derivative


of a differentiable map in Iten given in the beginning. Fixing x E M, we consider the set C.,„ ( M) of all Cc° curves, a : ( c, )

M with f > 0 and a ( 0 ) = x. Given a local chart so : U x E U, the curves a u ( t ) = so -1 ( (x) + tu ), u E fir, are in C, ( M) . So Cx ( M)

0. We introduce the following equivalence relation on Cx (M) : a - )3 if for any local chart ( U, so), x E U, we have (d(yo o a) / dt) (t) 11= 0 = = (d (so (3) / dt ) ( t) I = . Observe that if ( V4') is another local chart with x E V, then ‘t/ o a = ( V/o so -1 )o (c o a), so by the chain rule, the relation - is independent of the chart chosen.

The quotient of CA M) by - is called the tangent space to M at x and is denoted TI M. The space Tx M has a natural structure of a real vector space of dimension m. Indeed, if [a] denotes the equivalence class of a curve a E C(M), for u = [a] , u = [0] and X E IR, we define u + u = = [o-1 (woa + o (3)] and Xu = [so -1 (X • so o a)1, where so : U ffen is a local chart such that so (x) = O. It is easy to show that these definitions are independent of the chart chosen and satisfy the axioms for a vector space. It is also true that if so : U IR'n is a local chart with so (x) = 0 and e ,

e,,4 is a basis for 1R'n , then the set I [so ( te 1 )] ,..., [so - ( te )11 is a

basis for TIM, te, being the curve t 1-4 te„t E IR. Therefore TM is a vector space of dimension m. When I e em is the canonical basis for It m we also use the notation [so - ( te.,)] = a/ ax„ i = 1 ,..., m.

Every differentiable map f : M N induces at each point xEMa linear map Df (x) : Tx (M) --• Ty (N) , y = f(x) , in the following manner. Given u = [a] E TM we take Df (x) • u = [f o a] , the equivalence class of the curve fo a in TN. It is easy to see that Df (x) is well-defined and linear. It is called the derivative of f at x.

A differentiable manifold can be thought of as being a topological space that is locally Euclidean. Taking into account this pictorial image, a tangent vector to Mat x is, essentially, the linear approximation of a differentiable curve which goes through x. The derivative of a differentiable function f : M Nat x E M is a linear approximation of the function in a neighborhood of x. In practice, in order to work with differentiable functions, one should always have this in-tuitive idea in mind.

§3. Immersions and submersions

Given a CT map (r a. 1), f: M ---• N, we say f is an immersion if for every X E M, Df (x) : TM TEN, y = f (x) , is injective. We say f is a submersion if for every x E M, Df (x) : T r M T y N is surjective.

An immersion f : M N is called an imbedding iff : M - • f (M) C N is a homeomorphism, when one gives f (M) the topology induced by N. In par-ticular, if f : M --• N is an imbedding, then f is a one-to-one immersion; however the converse is false, that is, there are one-to-one immersions which


are not imbeddings, as is illustrated by the immersion in figure 7.

f

Figure 7

The following theorems follow from the inverse function theorem in R m .

Theorem (Local form of immersions). Let f: Min ---• Ar be of class CT (r _>... 1). Suppose Df ( p) : TM --• TI N, q = f (p), is injective. Then there exist local charts so : U —* R in i p E U, and V/ : V --0 R n , q E V and a de-composition R n = Dr x 1Rn - "1 such that f (U) C V and f is expressed in the charts (U, so), (V, i) as V, . f . yo -1 (x) = (x, 0) if x E (p (U). In other words f is locally equivalent to the linear immersion x 1-4 (x, O).

N . R2

I

4 ) p(U)

-1 Iliofokio

ir.

Figure 8


Theorem (Local form of submersions). Let f: M"1 --+ N" be of class «(r 1). Suppose Df( p) : TM Tg N, q = f( p), is surjective. Then there exist local charts so : U 111' " , p E U, ‘1, : V , q E V and a decomposi-tion lle" = IR" x II' such that f(U) C V and f is expressed in the charts (U,0), (V,I,G) as 1,1, of o so -1 (x,y) = x. In other words, f is locally equiv-alent to the projection (x,y) 1-0 x.

N = S I

(P)

f

Figure 9

§4. Submanifolds

A subset N c M'" is called a submanifold of M of dimension n and class (r 1) if for every p E N there exists a C local chart, (Clop), with

(U) = V x W where 0 E y c Ftn, OEWc lit m- n and V, W are Euclidean balls such that 40 (N n U) = V x O. In this situation we also say that the codimension of N is m — n = dim (M) — dim (N).

It follows that a submanifold N of M is, in particular, a C. manifold. For example S" is a Cc° submanifold of dimension n of IR"

From the theorem giving the local form of immersions one can obtain the following consequence.

Theorem. Let f: M N be a Cr map (r 1) such that Df(p) : TM TN, q = f( p), is injective. Then there is a neighborhood U of p in M

such that f(U) is a C submanifold of N of dimension equal to dim (M).


Corollary. If f : M N is a CT imbedding then f (M) is a Cr submanifold of N of dimension dim ( M).

When f : M N is an immersion we say that f( M) is an immersed submanifold in N. The example in figure 7 shows that an immersed submanifold may not be a submanifold, even when f is one-to-one.

§5. Regular values

Let f : M N be a Cr map (r 1). When p E M is such that Df ( p) : TM 7 .47 N, q = f( p), is surjective, we say p is a regular point off. When q E N is such that f ( q) = 0 or f (q) consists entirely of regular points, we say that q is a regular value off. A point q E N which is not a regular value of f is called a critical value off.

The following result is an immediate consequence of the local form of submersions.

Theorem. Let f: M N be of class CT (r 1). If q E N is a regular value

of f and f ( q) 0, then f ( q) is a C submanifold of M of codimension

equal to dim ( N).

An example of the above situation is S" = f I () where f(x, x„,) = I 2

= La X.

An important theorem in the theory of differentiable functions is the following ([ 35 ] ).

Theorem (Sard). Let f: M N be a Ce map. Then the set of critical

values off has Lebesgue measure O. In particular the set of regular values of f is dense in N.

§6. Transversality

Let f : M N be a CT map ( r 1) and S C Na submanifold of N. We say f is transverse to S at x E M if y = f ( x) S or if y = f(x) E S and the following condition is satisfied:

T,N = TS + Df(x) (TM).

When f is transverse to S at every point of M we say f is transverse to S. A local characterization of transversality is the following.

Theorem. Let f : M Nn be a Cr map ( r 1 ) and Ss c N a Cr submanifold of N. Let p E M and q = f ( p) E S. A necessary and sufficient condi-

p n-s

1r2 -Ili- 0


tion for f to be transverse to S at p is that there exist a local chart it : V

—• IR A , q E V, a neighborhood U of p in M and a decomposition R n = Rs x EV' such that f (U) C V, (s n v) c 1W x tot and the map 72 o 1,t f U R n is a submersion, where 7r 2 : 1W x wi n - s -n-s 1K is the projection on the second factor.

Figure 10

Theorem. Letf:M—. Nbeof class C` (r a- 1) and S 5 a Cr submanifold of N. Iff is transverse to S and f (S) 0 then f -1 ( S) is a CT submanifold of M, with the codimension of f' (S) equal to the codimension of S.

When S 5 , Al m c Nn are two Cr submanifolds of N, we say that S meets M transversally at x E S if the imbedding i : S N, j (y) = y, is transverse to M at x. This condition is equivalent to TN = TM + TS in case x E S fl M.

For more information about the theory of transversality we refer the reader to [1] or [261.

§7. Partitions of unity

Consider a countable open cover ( Un ) of a manifold M. We say this cover is locally finite if every point of M has a neighborhood which meets only a finite number of Un's.

Differentiable - Manifolds 19

Definition. A partition of unity subordinate to the cover ( Un ) is a collec-tion of nonnegative functions ( w n ) such that

(1)The support of jo„ is contained in Un . Recall that the support of a func-tion is the closure of the set of points where the function is nonzero.

(2)E ço n ( p) = 1 for every p E M.

Theorem. Every countable, locally finite, open cover of a manifold admits a partition of unity subordinate to it.

The proof can be found in [27]. A Riemannian metric on a manifold M is a map that associates to each

point p E M an inner product < , > defined on the tangent space to M at p. The metric is said to be Cr if for each point p E M there is a system of coor-dinates ô = ( x, ) : U with p E U such that the maps g y ( q) =

a a < — (q),— (q)) are of class Cr for all i,j = 1 ,..., m. ax, axi

It is easy to see that on any manifold there exists a C Riemannian metric. In fact, take a countable, locally finite cover ( (J,, ) of open coordinate neighborhoods of a manifold M. On each Un , fix a system of coordinates ( x 1 ,..., xm ) :

and define the following metric < , > ( q), q E U,„ by the values on a a the basis elements of Tg (M) : ( — (q), — (q)) = .5,j where = 0 if i j

ax, axi and 6, „ = 1. Now let (40,,) be a partition of unity subordinate to ( (J,,). The metric on M is defined at each point p E M by the expression < , > (p) =

II. FOLIATIONS

We introduce in this chapter the notion of a foliation and the more elementary properties which will be used in the rest of the book. We will also see various examples illustrating the concept.

§1. Foliations

A foliation of dimension n of a differentiable manifold Mm is, roughly speak-ing, a decomposition of M into connected submanifolds of dimension n called leaves, which locally stack up like the subsets of fir = lle x JR"' with the second coordinate constant.

The simplest example of a foliation of dimension n is the foliation of Ilm = = En x 111'n where the leaves are n-planes of the form litn x Ici with c E Br -".

The diffeomorphisms h: U C ER' --• V C Ile which preserve the leaves of this foliation locally have the following form

(*) h(X,Y) = (h1(x,Y),h2(Y)), (x,Y) E IRn X IR m- n

len-n

h

Figure 1


Definition. Let M be a Coe manifold of dimension ni. A Cr foliation of dimension n of M is a Cr atlas a on M which is maximal with the following properties:

(a) If ( U, so ) E 5 then so ( U) = U, x U2 c Rn x IR"'" where U, and U2

are open disks in R n and IR' respectively. (b) If ( U, so ) and ( V, lit ) E 5 are such that U n y 0 then the change of

coordinates map 1/, . ,.,e, - I : (p (U r) V) ---• V/ (U r) V) is of the form (.), that is, 1,1' c so -' ( x,Y ) = ( h 1 (x,Y),h2(Y)). We say that M is foliated by a , or that a is a foliated structure of dimension n and class Cr on M.

In figure 2, we illustrate the local aspect of a two-dimensional manifold foliated by a one-dimensional foliation.

114000-1

Figure 2

Remark I. When we say M is a C manifold which has an atlas 5 as above, we are implicitly saying that M has an atlas a whose changes of coordinates are C'; however, if ( U, so ) E a and ( V,1,t ) E a and U n y 0 then so . tk - ' and '4 . so - I are C'. The only relation between 5 and a is that the mixed change of variables, as above, are Cr. As is clear from the definition, the atlas 5 is very special (due to condition ( . )) and, as we will see further in §4, not all manifolds of dimension n7 possess a foliation of dimension n, where

Foliations 23

O < n < m. We observe that if A = I(U,op,) i E /1 is an atlas, not maximal, whose

local charts satisfy (a) and (b), then there is a unique maximal atlas which con-tains A and satisfies (a) and (b). This atlas is defined as the set of all charts so : U litn such that if U n u, for some i E I, then h = so o ça» : so,(U ("1 U,) ça (U fl (J,) is a Cr diffeomorphism of the form ( ). Al-though the condition of the atlas being maximal is not necessary in the definition, it is convenient since in that case the set of all domains of local charts form a basis for the topology on M.

From now on we consider only foliations of class Cr, r 1. The charts (U, ça) E if will be called foliation charts.

Let if be a cr foliation of dimension n, 0 < n < m, of a manifold Ar Consider a local chart ( U, so ) of if such that so ( U) = U1 x U2 C 112" x 1R tm ". The sets of the form so ( U1 x ci ), c E U2 are called plaques of U, or else plaques of 5. Fixing c E U2, the map f so -1 / U1 x Ici : U1 x Ici U is a Cr imbedding, so the plaques are connected n-dimensional Cr submanifolds of M. Further if a and 0 are plaques of U then a (1 fi = 0 or a = 0.

A path of plaques of if is a sequence a l a k of plaques of if such that aj CI a• , 0 for all j E 11 k — l. Since M is covered by plaques of a, we can define on M the following equivalence relation: "pRg if there exists a path of plaques a ak with p E a ,,gEoc k ". The equivalence classes of the relation R are called leaves of 5.

From the definition it follows that a leaf of if is a subset of M connected by paths. Indeed, if F is a leaf of if and p,g E F, then there is a path of plaques a 1 Œk such that p E cx, and q E a k . Since the plaques a j are connected by paths and ai n a„, 0, it is immediate that a U U ak C F is connected by paths, so there is a continuous path in F connecting p to g.

Another important fact is that every leaf F of if has the structure of a CT

manifold of dimension n naturally induced by charts of 5. Before proving this, we will look at some examples of foliations.

Example 1. Foliations defined by submersions.

Let f: N" be a Cr submersion. From the local form of submersions, we have that given p E M and g = f(p) E N there exist local charts (U, ça) on M, (Voii) on N such that p E U, g E V. U) = Ul x U2 C fir " X ER" and Vi ( V) = V2 D U2 and ik f : U, x U2 U2 coincides with the projection (x,y) y.


ir2

Figure 3

i

Figure 4

( Ub ) Sob Ub X F

/ P I

Ur,

Foliations 25

Therefore it is clear that the local charts ( U,,,o) define a CT foliated manifold structure where the leaves are the connected components of the level sets

c E N. Let us look at a specific example. Let f : R 3 ---• JR be the submersion defined by f(x 1 ,x 2 ,x 3 ) = a (r2 )e x3 where r = + x; and

: 111 JR is a Coe function such that a ( 1) = 0, a (0) = 1 and if t > 0 then a' (t) < O. Let 5 be the foliation of IR 3 whose leaves are the connected components of the submanifolds f -1 (c), c E IR.

The leaves of 5 in the interior of the solid cylinder C = f (x 1 ,x2 ,x3 ) + x :5 1) are all homeomorphic to IR 2 and can be parametrized by

(x 1 ,x2 ) E D 2 I—. (x i ,x2 ,log (c/a (r 2 ))), where c > O. The boundary of C, ac = (x1 ,x 2,x 3 ) x; + x = 11 is also a leaf. Outside C the leaves are all homeomorphic to cylinders (see fig. 4).

Example 2. Fibrations.

The fibers of a fiber space (E,7r,B,F) define a foliation on E whose leaves are diffeomorphic to the connected components of F. A fibered space (E, consists of differentiable manifolds E, B, F and a submersion r : E B such that for every b E B there is an open neighborhood Ub of b and a diffeomorphism

: 7r- ( ) 11h x F which makes the following diagram commute:

In the above diagram P 1 is the projection onto the first factor. The fibers of the fibration are the submanifolds ( b ), b E B.

An important example of this situation is given by the following theorem, whose proof the reader can find in [27].

Theorem (On Tubular Neighborhoods). Let N C M be a Cr submanifold with r 1. There is an open neighborhood T (N) D Nand a Cr submersion, 71- : T (N) N, such that 7r(q) = q for all q E N. If the codimension of N is k. then T(N) can be obtained in such a way that ( T(N),r,N, il k ) is a fibered space.

Example 3. The Reeb foliation of S 3 , [46].

The following example plays an important role in the development of folia-tion theory.

Consider the submersion of Example I, f: D 2 xR —• 1R given by f( x i ,x2 ,x 3 )

= a(r)-e'3 , where now a(r) is the function ce(r) = exp ( - exp ( 1/1 - r2 ) ).

The foliation defined by f has for leaves the graphs of the functions x 3 =

Figure 5


= exp ( 1 /1 — r 2 ) + b, b E IR and it extends to a C foliation of 1R 3 whose leaves in the exterior of D2 x JR are the cylinders x; + x; = r2 ,r > 1.

On D 2 x [0,11 we identify the boundary points in the following manner: ( x i ,x2 ,0) (y1 'Y2' 1 ) if and only if ( x, ,x2 ) = (y, ,y 2 ). The quotient manifold D2 x [0,1 ] / .= is diffeomorphic to D 2 x s' and, since the foliation defined on D 2 x JR is invariant by translations along the x3 -axis (that is, these translations take leaves to leaves), it induces a Cc° foliation, (R on D2 x S'. It is called the (orientable) Reeb foliation of D 2 x S'. In this foliation the boundary a (D 2 x S') = s' x s' is a leaf. Moreover, the other leaves are homeomorphic to 1R 2 and they accumulate only on the boundary (see Fig. 5).

It is easy to see that this foliation is not defined by a submersion. If, instead of the above identification, we consider on D2 x [0,1 ] the relation (x, ,x 2 ,0) —

(YI,Y2,1) if and only if (x,,x 2 ) = (Y1, — y2 ) then the quotient D 2 x [0,11/ — will be a non-orientable three-dimensional manifold, K3 , whose boundary is dif-feomorphic to the Klein bottle. Since this identification preserves the foliation of D 2 x IR, this induces a foliation CR of K 3 called the non-orientable Reeb foliation of K 3 . The leaves of (R in the interior of K3 are all homeomorphic to IR' and the boundary of K 3 is a leaf.

From two Reeb foliations of D2 x s' we can get a C' foliation of S 3 in the following way. The sphere S 3 = (x,,x 2 ,x 3 ,x 4 ) E JR X ,2 = I can be considered as a union of two solid tori T, = D2 x S', j = 1,2 identified along the boundary by a diffeomorphism which takes meridians of aTI to parallels of a T2 and vice versa. The solid torus T, can be defined by the equations

E' ,,, = 1 and 4 + x 22 :5_ 1 / 2 and the solid torus T2 by the equations

Foliations 27

E = x: = 1 and 4 + x22 1 / 2. Another way of seeing this decomposition of s' into two solid tori is the

following. Consider the stereographic projection 7r : S3 – P —0. IR 3 where P = ( 0,0,0, 1 ) and 71 (x) is the point of intersection with the plane x4 = 0 of a ray containing P and x. Since 71 is a diffeomorphism, S 3 can be thought of as the union of 1R 3 and the point P at infinity.

Consider now in the x i x3-plane the region S, bounded by the circles of radius 1 centered at p = ( 2,0), q = ( – 2,0) and S2 =-7 111 2 - S 1 .

Figure 6

Taking this figure into JR by rotating about the x3 -axis, we generate by the region S 1 a solid torus S; • Let S2' = IR 3 - S 1' and note that 7r I (Si) U P is also a solid torus. Indeed consider the circles in the x 1 x3 -plane with center on the x3-axis and which pass through the points p and q. Let ea be the connected component of one of these circles in S2 which intersects the x3 -axis in the point a. Letting fo, = (X 1 ,0) 14 9), we get a foliation of S2 whose leaves are the curves fea l , a E IR U co J . By rotating the x / x3-plane about the x3 -axis, we see that the curves P a (a E IR) generate disks D, and fo, generates a cylinder D. Each Da , a E JR U too ), meets the torus as; along a parallel.

It is now clear that by adding to IR 3 the point P of the stereographic pro-jection, the x3 -axis together with P defines the axis of a solid torus T2 which is foliated by disks 7r I (Da ), a E lit plus the disk Dp = 7r ( D ) U P.

If we denote by T1 the solid torus 71" I ( S ;) we get that S 3 = T1 U T2 and the parallels of aT1 coincide with the meridians of ô T2 and vice-versa.

Returning to the construction of the foliation of S 3 , consider the foliation that results from glueing two Reeb foliations of T1 and T2 where aT, = aT, is a leaf. We obtain in this way a foliation of S 3 of codimension one called the Reeb foliation of S3 . This foliation is C' and has one leaf homeomorphic to T2 . All the other leaves are homeomorphic to IR 2 and accumulate on the compact leaf.


Example 4. Vector fields without singularities.

A vector field on a manifold M is a map that associates to each point p E M

a vector of the tangent space to Mat p. An integral curve of X through p E M is a curve -y : (a,b) —0 M, with -y (0) = p,suchthatX(-y(t)) = -y ' ( t) for any t E ( a, b ) . Thus integral curves are, locally, solutions of the differential equation dx/dt = X(x). The existence and uniqueness theorem for ordinary differential equations guarantees that under certain differentiability conditions on X, for instance if X is CT, r 1, there passes through any p E M an integral curve of X (of class Cr) which is unique in the sense that any two integral curves through the same point necessarily agree in the intersection of their domains of definition. Moreover, these integrals define a local flow at any point p E M, i.e. there is a neighborhood Up of p and an interval ( a, ) such that for any q E Up the integral curve through q,.-yq ( t) is defined for all t E (a,(3) and the map (local flow) : ( a, (3 ) x Up M given by ,p ( t,q) = = -yq (t) is of class Cr. Clearly, so ( 0,q) = q for any q E M and (p (s, so (t, q)) = = So (s + t, q) provided that ,p ( t,q) E Up and s, t,s + t E ( a, (3). Let X be a vector field on M without singular points. Let j: B"7- M, i (0) = p, be an imbedding of a small in - 1 disc centered at 0 E IR'n , transverse to X everywhere. Since X(p) 0, the map 4) : x (o103) M given by

(x, t ) = So( t,i (x)) hasmaximalrankat(0,O) E B (a, 0 ) . Thus, by the inverse mapping theorem there is a neighborhood V C M of p such that 4) -1 / V is a diffeomorphism onto a product neighborhood B'"1-1 x ( a ' , ' )

C B" x ( a, ) of ( 0,0 ). This map will be a local chart for the one dimen-sional foliation on M defined by the integral curves of X.

Example 5. Actions of Lie groups.

A Lie group is a group G which has a C differentiable manifold structure such that the map G x G G, (x,y) xy is Coe . This last condition is equivalent to saying that the maps (x,y) 1-4 xy and x x -1 are C. An immersed C' submanifold H C G that is also a subgroup of G is called a Lie subgroup of G.

Examples of Lie groups

(1) The additive group WV. (2) The group C* = C - 101, with the multiplication of complex numbers.

The circle S' = z EC! z = 11 is a Lie subgroup of C*. (3) The torus T" = S' x x S I (n times) with the multiplication

z l z „ ) w i ,•••, w„ ) = w1 • zp? wn) •

Foliations 29

(4) The group GL (n, IR ) of all real n x n nonsingular matrices is a Lie group of dimension n 2 . This can be seen by considering each element A = = ( au ) E GL (n, ) as a point of R n2 . SO GL (n, ) can be taken to be an open submanifold of IR'`.

(5) The orthogonal group 0 (n, ) which consists of n x n real matrices A

such that AA' = I is a compact Lie subgroup of GL(n, IR ) . In fact, if S ( n, IR ) denotes the space of symmetric matrices, the map f: GL (n, )

S(n,111), f(X) = XX I is well-defined and is C. Moreover O (n , IR ) = f -1 (I). One can verify that I is a regular value of f. Con-sequently, O ( n, ) is a submanifold of GL (n,R) . If a ,..., a,, are the rows of A E (.9 ( n, IR ) , the condition AA' = I implies lunr = n. So 0 (n, ) is compact.

A Cr action of a Lie group G on a manifold M is a map so : G X M M such that so (e,x) = x and ( g g2 ,x) = (g ,(p ( g 2 ,x)) for any g i ,g2 E G

and x E M. The orbit of a point x E M for the action so is the subset 0„ ( v)) = = fso (g, x) E Mig E G]. The isotropy group of x E M is the subgroup G,( ç0) = = (g E G I so (g,x) = x). It is clear that G, (c) is closed in G. The map 0„ : G —• Mgiven by ;Lx (g) = so (g,x) induces the map Ip x : G/G,((p) M, Vi x (k) = 0,(g) where k = g • Gx ((p). Since g 1-1 g 2 E G(so) if and only if

(g 1 ,x) = th(g1 ) = th(g2 ) = so (g2 ,x) , we conclude that Tx is well-defined and injective. Further it can be shown that G/G,(,,o) has a differentiable struc-ture and that ;// is an injective immersion whose image is ( so ) . (See Chapter VIII).

We say that so : G x M M is a foliated action if for every x E M the tangent space to the orbit of so passing through x has fixed dimension k. When k is the dimension of G we say so is locally free.

Proposition 1. The orbits of a foliated action define the leaves of a foliation.

Proof. Let 90 : G x M ---• M be a foliated action whose orbits have dimension k 1. Fixing xo E M, let E c Te G be a subspace complementary to the tangent space of G,0 ((p) and i, : B k G, j 1 ( ) = e, an imbedding of a k-disk tangent to E at e. If i2 : Bi t?" M, 1 2 (0) = xo , denotes the imbedding of a small transverse section to the orbit 0,0 ( so) of xo , we can define the map cI) : B k x Br?' k Aim given by 4) = ,p(i l ,i2 ). Since D(I) ( 0,0 ) : IR k x fitm -k T o M is an isomorphism, there is a neighborhood U of xo 'c such that (1) : U B k x is a diffeomorphism taking the orbit of

i2 (x ) E U fl 2 (B rn-k ) to an open subset of the surface B k x [xi . This defines a local chart of the foliation by orbits of ça.

r A, 0 A = I where A I =

0 A 2

sin s I and A,

cos S

cos t sin t 1

- sin t cos t j

cos S

- sin s


B'n - k

Figure 7

Remark 2. Given a Lie group G and a Lie subgroup H C G, the map H x G G, (h, g) 1-0 g h defines an action of H on G. The isotropy group of each element g E G is the identity. So this action is locally free and the orbits define a foliation of G whose leaves are all homeomorphic to H. The leaves of this foliation are imbedded if and only if His closed in G. In fact, if H is not imbedded, there exists an h E Hand V a neighborhood of h such that V n H contains a countably infinite number of path-connected components. Let E be a transverse section to the orbits of the action passing through h. Then E n H contains a countably infinite number of points. On the other hand, as is easy to see, the set of accumulation points of E (1 H contains E n H,

so EnHCE is perfect. From general topology we know that a perfect set is not countable, thus EnH-EnH is not empty. Thus H is not closed.

Conversely, if His not closed there is a sequence h, in H such that h,, g H. Hence there exists an open set V containing g such that V (-1 H has an infinite number of path-connected components. Let U = g-1 - V. We have that lEUnH, the sequence g„ h„--1 • h„, is in FÏ , converges to 1 and if m n then g„., and g,, are in distinct path-connected components of H n u. So H is not imbedded.

A specific example of this can be obtained by considering G = 0( 4,1R ) and H the subgroup generated by matrices A of the form

It is easy to see that H is isomorphic to T2 .

Remark 3. An action ,p : IR x M M (IR the additive group of real numbers) is also called a flow on M. A flow ço on M satisfies the following properties:

(a) (0,x) = x forallx E M

Foliations 31

(b) so(s + t,x) = so(s,so(t,x)), s,t E IR, x E M.

To a Cr flow 0 on M(r 1) it is possible to associate a Cr -1 vector field by the formula: X (x) = (do(t,x)/dt) . The field X so defined is such that t 1-4 0(t,x) is the integral of X which passes through x. In fact, from this definition we have dso(t,x)/dt = (d0 (s + t,x)/ds) I s= 0 — = (d (0 (s, so (t,x))/ ds) I s=0 = X (0 (t, x)). Conversely, if X is a Cr vector field on M whose integrais are defined on 112, there exists a unique Cr flow,

: IRxM—> M, such that t 1--+ ( t,x) is the integral of X with initial condition so (0,x) = x. ([20]).

Proposition 1 in the case of flows is called the flow-box theorem. ••-••-

§2. The leaves

Each leaf F of a Cr foliation g has a Cr differentiable manifold structure induced by the charts of a. This structure, called the intrinsic structure of F,

is constructed in the following manner. Given p E F, let ( U, 0) be a chart of • such that p E U and so (U) = Ul X U2 C IR' where U1 and U2 are open balls in fi n and RS respectively. Let a be the plaque of U which contains p. Setting so = 01,02 ) where so : U , 0 2 : U IRs, we define -s-o : a by = sod . It is clear that : a —■ U1 c IR' is a homeomorphism since io (a) U x fa) for some a E U2. Now we prove that

= f( a,sO)laCF is plaque of U with ( U, so ) E

is a Cr atlas of dimension n for F. It is enough to show that if ( a, ), (0, -1,T) are in and a n then a n (3) and V/ ( a rl 0) are open in JR' and (70 1,t : t(a CI (3 )

(a (1 0) is a Cr diffeomorphism. First we show that a fl f3 is open in a and in 0. Let (U,0), (V, ) be in a such that‘7, = so, j ;:i; = 4„ l o . From condition (*) in §1, so o V/ : (U n v) —. 0 (u n v) is given by • o (x,Y) = (x , Y),h2(Y)) E 1W X iRs with (x,y) E X Es .

In particular since a CI # 0 we have (*.) so . (x,b) = (h i (x,b),h 2 (b)) = (17 1 (x,b),a) .

Since 1,G(0 n = Ik(u n vn 0) = tk(u n n x Its') and ( fl = so (u n V) rl ( IR" al ), from ( **) we have

O(0 n = so. ;/, -1 (k(0 n U)) = 0 V i (‘G(U n v) n (E n X tbI))

C ço(Un v) n (1Rn X a ) = so(a n v)

i.e.,finucan v. Analogously, ce VcOn U, so a V = = f3 fl U. This proves the claim.


Since and ' are homeomorphisms we get that ‘7, ( a n ) and IT (a n 0) — are open in 112n. The map (7-0 : 1,// ( a fl f3) ( a fl f3) is Cr since

(7, o i t (x) = h i (x,b) if x E ' (a n ,3). Similarly 4, is CT , hence o TT/ ' is a Cr diffeomorphism. This defines the intrinsic structure of F. It can be shown that the leaves have a countable basis. (See exercise 6).

We note that the topology of F associated to the atlas GI defined above is such that the set of all the plaques a of a with a C F constitute a basis of open sets of F. This topology, in general, does not coincide with the one induced naturally by the topology of M. The reason is that the leaf F can eventually meet the domain U of a chart ( ) E a in a sequence of plaques ( a, ) „IN

which accumulate on a plaque a c F, i.e., any neighborhood of a contains an infinite number of plaques of F and, hence F is not locally connected in the topology induced by M, while in the intrinsic topology, F is a manifold and hence locally connected. In Chapter III we will see some examples where this occurs.

We consider now the canonical inclusion j: F M, i( p) = p, and F with its intrinsic manifold structure. It is easy to verify that i is a one-to-one Cr

immersion. When i is an imbedding we say that F is an imbedded leaf. This occurs if and only if the intrinsic topology of F coincides with the topology induced by that of M. We summarize the above in the following:

Theorem 1. Let M be a manifold foliated by a C r n-dimensional foliation a. Each leaf F of 5 has the structure of a Cr manifold of dimension n such that

the domains of the local charts are plagues of 5 . The map j: F M defined

by i ( p) = p is a Cr one-to-one immersion, where on F we take the intrinsic

manifold structure. Further F is a Cr submanifold of M if and only if i is

an imbedding.

S3. Distinguished maps

As one would expect, the transverse structure plays an important role in the study of foliations. This structure is emphasized in the following alternate definition of foliation.

Definition. A Cr ( r 1) codimension s foliation of M is defined by a maximal collection of pairs ( U„f), i E I, where the U,'s are open subsets of M

and the f,'s, f, : U, NV, are submersions satisfying

( 1) U , E , U, = M

( 2 ) if Ui n 0, there exists a local C' diffeomorphism g of Rs such that f, = g of j on U, (.1 Uj .

The f,'s are called distinguished maps of .

In this definition the plaques of in U, are the connected components of the sets ( c), CE 1R 3 .

We verify that this definition is equivalent to the definition of 51. To do this we need the following lemma.

Foliations 33

Lemma 1. Let 'S be a foliation of a manifold M. There is a cover e = U, i E J of M by domains of local charts of 5. such that if U fl Uj 0

then U1 U Uj is contained in the domain of a local chart of

Proof. We consider a cover of M by compact sets K K„ c int (K„). For each n E N we fix a cover of K, by domains of foliation charts of g ,

V,n1 i = f ,..., k n i. Let 6, > 0 be the Lebesgue number of this cover with respect to some fixed metric on M. We can assume that the sequence (o ,, ) is decreasing.

It is now sufficient to take a cover of K, by domains of foliation charts, Ufl j =- I,..., such that the diameters of the U2's are less than 6,,/2 for

all j = 1 ,..., fn . It is clear that if Un, n 0 then U7 U U; C V n for some •

Suppose now that M has an atlas g which defines a codimension s foliation, according to the definition in §1. Consider a cover C = U , i E /I of M by domains of local charts of g , as in Lemma 1.

Given an open set U, E e, i E I, we have defined so : U rRn x Ile on g such that io,(U,) x U', where U 11 and U 2 are open balls in JR' and 1R respectively. Let /3 2 : 111' x IR IRS be the projection on the second factor. Then f, = P2 ° lits is a submersion and for every c E U12, A -1 (c) is a plaque of U„ If U, C.1 0, U, U Uj is contained in the domain of a chart ( ) E a with q, ( V) = V I X V2 . Hence, letting a and 0 be plaques of U, and Uf with a n o we have that a U 0 is contained in a plaque -y of V. Then 0 fl U, c a, which implies that f,(0 n U,) contains a unique point. This implies that given y E Uj ), f o f/' (y) contains a

unique point g 1 (y). We leave it to the reader to verify that g,j :f.,(U, n Uj )

f,(U, fl Uj ) is a Cr diffeomorphism. Suppose now that there exists a col-lection of pairs ( U „ f ,) , i E I, satisfying the definition of this section. Since for every i E I, f,: U, IR' is a Cr submersion, from the theorem of the local form of submersions, given p E U„ there exist V 1 and v, open balls of

and IR respectively, with V, C f, (U,), and a Cr local chart so : V V,

X V2 with V c U„ such that f, o so : V 1 x V 2 — * V2 is the projection on the second factor. The set of all the charts ( V, ,,n) constructed in this manner is a Cr atlas of a foliation of M.

In fact, if ( V, so) and ( W, ;t ) are two charts as above such that V (1 W 0,

then V c U, and W c U., with i,j E I. On the other hand, so - : (V fl W)

(,o(V n 14/) can be written as so o (x,y) = (h,(x,y),h 2 (x,y)), (x,y) E

111" x IR S . So h-_,(.v,y) = p 2 1,1, (x,y) = f, I I (x,y)

= f, o (x,y) = of o x , y = g„ ( y ) only depends on y as we wished.

Definition. Let N be a manifold. We say that g : N M is transverse to

when g is transverse to all the leaves of , i.e., if for every p E N we have


Dg(p) • Tp (N) + Tg (5) = Tg M q = g( p)

where by Tg ( if) we mean the tangent space to the leaf of if which passes through q.

Theorem 2. Let if be a Cr foliation (r a- 1) of M and g : N M a Cr map.

Then g is transverse to if if and only if for each distinguished map (U,f) of if the composition f . g : (U) IR s is a submersion.

Proof. Suppose g is transverse to if and let ( U,f) be a distinguished map of 5. IfpEU and q = g(p), we have Tg M = Tg (5) + Dg(p) • Tp (N).

Applying Df(q): Tg M --•-• Rs to both sides and using the chain rule, we get

IRs = Df( q) • Tg M Df( q) - T g (5) + D(f o g) p • 1,N.

Since f is constant on the plaque of if in U which passes through q, we have Df (q) • T,( if) = LOI, so D(f o g) p • TN = I11 5 , i.e., f . g is a submersion.

Conversely, consider p E N and a distinguished map (U,f) of if such that q = g(p) E U. From the hypothesis, we have that Df(q) Dg(p) Tp N = = Rs , i.e., the restriction Df(q) j Dg ( p) - Tp N is onto 111 5 and so Dg ( p) • Tp N contains a subspace E of dimension s such that Df(q)E:E llts is an isomorphism. Since Tg (5) = (Df(q)) -1 (0) it follows that E n Tg () =

= [01, so Tg M = EC ) Tg (5 ) , since dim (E) = cod ( T ( )). Hence Tg M =

= Dg ( p) • TN + Tg ( ) , as desired. •

As an immediate consequence of this theorem we have the following

Theorem 3. Let if be a Cr foliation on M and g : N M a Cr map transverse to if. Then there exists a unique Cr foliation g* (5) on N of codimension cod (5) whose leaves are the connected components of the sets g - ( F), F a leaf of if

Proof. Let (U„f„gg ) f =g ,1 off, i,j E / be a system of distinguished

maps of 5. Since f, . g = g 11 o f o g on g -1 (U,) n g -1 (VI ), it is evident that the collection (g -1 (U,),f, o g,g 0 )1i,j E /1 is a system of distinguished Cr

maps of a foliation g* (5) on N of codimension cod ( a ) . On the other hand, if F is a leaf of if and a is a plaque of F (-1 U, for some i E I, then the connected components of g - (a) are plaques of g* (5) in g - ( U,) and this implies that g -1 (F) is a union of leaves of g* (5 ) . Since ( F) F is a leaf of 51 is a cover of N, it follows that the leaves of g* (5) are the connected components of the sets g (F), F a leaf of 5. •

In particular when g is a submersion g* if is well-defined for any foliation if of M.

Foliations 35

§4. Plane fields and foliations

A field of k-planes on a manifold M is a map P which associates to each point qEMa vector subspace of dimension k of Tq M.

A field of 1-planes is also called a line field. For example, if X is a vector field without singularities on M, we can define a line field P on M by letting P(q) = 111 • X (q), the subspace of dimension I of Tq M generated by X (q).

Conversely, if P is a line field on M, we can define a vector field without singularities on M by choosing at each point q E Ma nonzero vector in P ( q ) . One says that such a vector field is tangent to P. We say that a line field on M is of class Cr when for each q E M there is a Cr vector field X defined in a neighborhood V of q such that P(x) = 111 • X (x) for every x E V. In general, a continuous line field on M does not have a continuous tangent vector field. The figure below illustrates some examples in 11I 2 101.

Figure 8

Analogous to the case of line fields, we say that a k-plane field P on M is of class Cr if for every q E M there exist k Cr vector fields X 1 ,X 2

defined in a neighborhood V of q such that for every x E V, (X' (x) (x)1

is a basis for P ( x ). A relevant fact is the following.

Proposition 2. Every CT (r 1) k-dimensional foliation 5 on M defines a

C'' k-plane field on M which will be denoted T . .

Proof. Let P(x) = Tj3 be the subspace of TM tangent at x to the leaf of a which passes through x. Given xo E M, let (t], w). X0 E U, be a local chart of . It is easy to see that for each x E U, P(x) is the subspace generated by vectors (a/ax,)(x) = (D,p(x)) -1 e ,) , i = 1 k where e, is the

vector of the canonical basis of Ile'. Since so is CT, the fields 8/ax,,

I = 1 k, are CT' , and hence P is Cr.- 1 . •

In particular, if M does not admit a continuous k-plane field, then M does not possess a k-dimensional foliation. For example, the sphere S 5 does not


have a continuous 2-plane field (see 1531 page 142); so, there do not exist folia-tions of dimension 2 on S s . A natural question then is the following. Given a k-plane field P on M, under what conditions does there exist a k-dimensional foliation a such that, for each q E M, Tq g = P (q)?

This question is answered by the theorem of Frobenius whose statement follows. Its proof is in the appendix.

Definition. One says that a plane field P is involutive if, given two vector fields X and Y such that, for each q E M, X (q) and Y(q) E P (q), then [X,Y1(q) E P(q). (For the definition of the Lie bracket I , 1, see the appendix).

Theorem (of Frobenius). Let P be a Cr k-plane field ( r 1) on M. If P is involutive then there exists a Cr foliation of dimension k on M such that Tq ( ) = P(q) for every q E M. Conversely, if a is a Cr (r ?_- 2) foliation and P is the tangent plane field to then P is involutive.

We say also that an involutive plane field is completely integrable. In particular, if k = I, P is always completely integrable. In this case the

theorem reduces to the existence and uniqueness theorem for ordinary differen-tial equations.

S5. Orientation

Given a vector space E of dimension n 1, we say that two ordered bases of E, = un I and (B' = v,r ), define the same orientation on E if the matrix for the change of basis, A = (au ),, defined by y 1

a q ui , has positive determinant. If B is the set of ordered bases of E, the relation "CB and (B' define the same orientation on E" is an equivalence relation on B, which has two equivalence classes, called the orientations of E.

Let P be a continuous k-plane field on M. We will say that P is orientable if for each x E M it is possible to choose an orientation 0 ( x) on P(x) such that the map x 0 (x) is continuous in the following sense. We consider a cover of M by open sets ( U,) i,, such that for each j E I, the restriction P Ili is defined by k continuous vector fields X' ,..., X k . For each x E li1 , the bases (B(x) = IX' (x) X k (x) j and CB' (x) = X 1 (x),X 2 (x) Xi (x)I define two distinct orientations of P (x), say 0,*(x) and 0,( x). We say that the choice of 0 is continuous if 0 : U, = 0,- for every i and if u, n then 0: = 0 in the intersection.

If k = dim ( M) and P (x) = T,M we say that M is orientable. For example, a line field P on M is orientable if and only if there exists a

continuous vector field X on M such that, for all x E M, P (X) is the subspace generated by X ( x ). In figure 8 we see some examples of line fields that are not orientable.

Foliations 37

§6. Orientable double coverings

The orientable double covering of a k-plane field P is defined in the follow-ing way: Let /1-4 = 1 (x, 0 ) x E M and 0 is one of the orientations of P ( x ) and 7r : A71 M be the projection Ir (x, ) = x. For each x E A7/ , (x) = = 1 (x, ), (x, -Oft where 0 is one of the orientations of P(x) and - 0 the other. On /1-4: consider the topology whose basis of open sets is constructed as follows. Given (x 0 ,0 0 ) E -A;1, let U be a neighborhood of xo on M where there is defined a continuous orientation 0 of PI U. We can assume (9 (x0 ) = 0 0 • We define then a neighborhood V of (x0 ,( 0 ) as V = 1(x,0 (x)) IxE U With this topology, 7r : /t7/ M is a two-sheeted covering (that is, iv 1(X) is a set of two elements for every x E M). Since iv is a local homeomorphism we can define a C' differentiable structure on -A7/, compatible with the topology co-induced by 7r and such that r is a C' local diffeomorphism (see example 6 of Chapter I).

The orientable double covering of P is by definition the k-plane field 7r* ( P) given by 7r* (P) = D7r(x) -I /3 (7r (x)).

Example. Consider on the torus T2 the non-orientable field P tangent to the foliation S sketched in Figure 9a.

Figure 9a

Figure 9b


In this figure, T2 is the square [0, 1] x 10, 1] with sides identified according to the arrows. The vertical sides constitute a unique leaf of 5 . The double cover-ing of P is still defined on T2 and looks like figure 9b. In this figure, T2 is the rectangle [0,2] x 10, 11 with the sides identified.

Theorem 4. Suppose M is connected and let P be a continuous k-plane field on M. Let ( /17/, 7r, 7r* (P)) be the double covering of P. Then

(a) 7r* (P) is orientable. ( b ) P is connected if and only if P is not orientable.

Proof. (a) By definition 71- * (P)(x,(e') = (D7 ( x, O )) • ( P (x)), if

(x, O) E /t7/. We define an orientation 0* for 7r* ( P) by saying that the ordered basis { (x, O) Xk (x,O)j of 7 * (P) (x,0) is in 0* (x) if the basis [D7r ( x, () - X' (x,0) D7r (x, O) • X k (x, O) is in O. One easily verifies that 0* is a continuous orientation of 7 * ( P).

(b) Suppose /a is connected. Given x0 E M, consider a continuous path : [0,1] /t--/ such that -y*(0)= ( x0 , 0 0 ) and -y*(1) = (x0 , - 00 ),

where 0 0 is a fixed orientation on P( xo ) and — 0 0 is the opposite orientation. Let -y = 7r o 7*. Then 7 is a closed path in M with -y ( ) = -y ( 1 ) = x o . Suppose P were orientable with continuous orientation O. We can suppose that O ( xo ) = O. However, in this case the path 5",' (t) = -y (0,0 (-y (1))) satisfies 7r o (t) = -y ( t ) for every t E [0,1] and T(0) = (x0 ,0 ) = -y * ( 0 ). Therefore it is easy to see that we must have 7y (t) = -y* (t) for every t E [ 0,1]

and hence -y* ( 1) = (x0 ,0 0 ) , a contradiction. Conversely, suppose /a is disconnected. If t , is a connected component of

we must have 7r ( m, since -n- is a local diffeomorphism and M is connected. This implies that /11-// has exactly two connected components /17/ 1 and XI- 2

In fact, if there were another connected component A-1- 3 , for every x E M, 7r (x) would contain at least three points, which does not happen. It is also true that for every x E M, if ( x, 0 ) E tva l then (x, 0) E M2. Then, 7r 1 iI is bijective and so h = 7r il7/ 1 /17/ i —••• M is a diffeomorphism. Since 7F * ( P) is orientable, it follows that the restriction 7r * (P) I 4 1 is orientable and we can define a continuous orientation on P by h -1 : M 1 - I , as was done in part (a) of the proof. •

Corollary. If M is simply connected then every k-plane field is orientable (1 k dim (M)). In particular, M is orientable.

§7. Orientable and transversely orientable foliations

In the theory of foliations, more useful than the notion of orientability is that of transverse orientability.

Foliations 39

Let P be a k-plane field on M. We say that I is a field complementary to P or transverse to P if for every x E M we have P(x) + I3 (x) = Titi and P(x) fl 15 (x) = 101. It is a plane field of codimension k. If P is CT it is possible to define a complementary CT field as follows. Fix on M a Rieman-nian metric < , ) . Let P' (x) = Iv E T,,A1 <u,v> x = 0 for all u E P (x) . It is clear that P I is a plane field complementary to P. We leave it to the reader to verify that P I is Cr.

Definition. Let P be a continuous k-plane field. We say that P is tranversely orientable if there exists a field complementary to P which is continuous and orientable.

Proposition 3. If P is tranversely orientable, any continuous plane field com-plementary to P is orientable.

Proof. It is enough to show the following: let P' be a plane field orthogonal to P and let is be a plane field complementary to P. Then 13-1- is orientable if and only if /3 is orientable. For each q E M let A q : Tg M P I (q) be the orthogonal projection. Given u E Tg M, we can write u = + u2 where

E P(q) and u 2 = A g (u) E (q) are unique. If X is a continuous vector field on M, it is easy to see that Y( q) = A g ( X (q)) is also a continuous field. Moreover if F C TM is a plane complementary to P(q) then the restric tion A g j F: F P I (q) is an isomorphism. It follows that A g ; 1-5 (q)

: 13 (q) P" (q) induces a bijection A q* between the orientations of P (q) and the orientations of 13 (q) : if 0 is an orientation of P (q) and u 1 E O then A q* ) is the orientation determined by the ordered

basis IA 4- I ( U l ) ,..., A q 1 ( 14, * k ) 1. Therefore 0 is a continuous orientation of (P" if and only if A* (0), defined by A* (0)(q) = A:( (9 ) is a continuous orientation of /5. •

Theorem 5. Let P be a Cr k-plane field on M. The following properties are true:

(a) if P is orientable and transversely orientable, then M is orientable,

(b) if M is orientable then P is orientable if and only if it is transversely

orientable.

Proof. (a) Let O be a continuous orientation of P and 0 a continuous orientation of P i . We define a continuous orientation -(-9- of M in the following way: given q E M consider fu, uk E (q) and l 11 A +1 ,..., u E (q). Define

( q) to be the equivalence class of the ordered basis u, u,7 1. It is easily verified that (79 is a continuous orientation of M.

(b) Suppose M and P are orientable. Let (79 and 0 be continuous orientations of M and P respectively. We define a continuous orientation of P" in the


following manner: given q E M, let uk ) E 0 (q). We say luk +1 ,..., E O (q) if l u 1 un ) E (79 (q). It is easily shown that 0 is a continuous orientation of P'. If M and P -1- are orientable, one proves similarly that P is orientable. •

Definition. A Cr (r 1) foliation 5 is orientable if the plane field tangent to 5 is orientable. Similarly, 5 is transversely orientable if the plane field tangent to 5 is transversely orientable. We denote the plane field tangent to S by T3:.

If 5 is not orientable, we can consider the orientable double cover of T3:, 7r* ( PS ), where 7r : /t7/ M is as before. Since 7r is a local diffeomorphism, we can define the foliation 7r* (3:) on A. Evidently T( 71-* (3:)) = 7r* ( 77).

So, 7r* ( 5) is orientable. Analogously, if 5 is not transversely orientable, we can consider the orien-

table double cover ( )* T5') of Tg where Tg is a plane field com-plementary to TtJ and 7r -`- : )17/ —• M is the covering projection. The foliation ( )* ( is transversely orientable. This construction is used in the proofs of many theorems in the theory of foliations in order to reduce the arguments to the case that 5 is orientable and transversely orientable.

The transverse orientability of a foliation 5 of co-dimension k and class Cr, r 1, can be expressed in terms of a collection of distinguished maps, D =

: U, IR k ) with U , E , U, = M. Recall that if U, n 0 then there exists a diffeomorphism gii : f1 (U1 rl U1 ) f,(U, fl Uj ) such that = g, o fj • We say that D is a coherent system of distinguished maps when, for every i, j E I, gu :J. (U1 n uj ) --• f, ( U, fl U.,) preserves the canon-ical orientation of IR k . In other words, if te 2 ,..., ek ) is the canonical basis of IR k then for every i, j E I and every q E U, fl L/ ) the basis IDgu (q) • e l ,..., Dgu (q) • ek ) has the same orientation as ) e l ,..., ed. This is equivalent to saying that det (Dgu (q)) > 0 for every q E fj (U, n (if ).

Theorem 6. Let 5 be a Cr (r a- 1) foliation. Then 5 is transversely orientable if and only if 5 possesses a coherent system of distinguished maps.

Proof. Suppose a is transversely orientable. Let -1- be a continuous orientation of P', the plane field orthogonal to 5. Let D = ( U„fi ,gy ) I i,j E I I be a system of distinguished maps of 5. From D we are going to construct a coherent system D = (72i , -/„T,, Li E J ) for 5. For each i E land each q E U„ let A u = Dfi (q) 1111.1 (q) : (q) 1R". Since P" (q) is orthogonal to 5 at q, Alq is an isomorphism for every i E I and every q E U. Let

u, ) E 0' (q). In the case that the basisIA,' ( u, ) A ((1 (1-4 )1 has the same orientation as the canonical basis of IR A , we define f, (q) = f(q). In the other case we define f,(q) = R ( ( q )) where R : IR k is the reflection R (x, ,..., xi ) = ( x1 ,x2 ,..., xi ). Since is a continuous orientation, we have that if:4 (q) = f,(q) for some q E U, then f, = f, on U, (since

Foliations 41

Ui is connected). Analogously if f (q) = R (fi (q)) for some q E U„ then = R o fi . The transition diffeomorphisms are defined by

(i) 170. = gu in the case that f, = f, and fi = fi (ii) = R in the case that ji = _I.; and J,,

= R

(iii) = R o g in the case that ji = R of1 and 7,== (iv) = R o gij o R in the case that ji = R o fi and ji = R

One easily verifies that jri = o fi for any i j such that Ui fl Ui 0. Moreover if A‘q = Dji (q) IP (q), i E I, A (II takes a basis of P -1- (q) in

(q) to a basis of IR k with the same orientation as the canonical basis. Then, if x = Irk ,j (x) = o (Alq ) - I takes the canonical basis of IR k to a basis with the same orientation, as we wished.

Conversely, if 5 has a coherent system of distinguished maps D = = U1 IR k i E /I, consider on P 1 the orientation (91 induced by A qi = DfIP' and by the canonical orientation of tie : uk 1 E (q) if and only if {A ig (u 1 ) A'q (uk ) has the orientation of the canonical basis of IR k . One easily verifies that 0 is a continuous orientation of P'. •

Notes to Chapter II

(1) Suspension of diffeomorphisms

A specific example of a foliation of dimension one, induced by a vector field without singularities, is obtained by the suspension of a C. diffeomorphism (r 1)f:N -1. N defined as follows.

On the product N x R consider the equivalence relation - defined by (x, t) (x ',t') if and only if t t' =nEZ and x' = f n (X) ( f n is defined by f° = id, f n = = f f (n times) if n a- 1 and f n = f -I f I (In! times ) if n - 1). So, if g(x,t) = (f (x),1 + 1). then g is a CI- diffeornorphism of N x Wi and (x, t)

( x t ' ) if and only if (x',t ' ) = (x,t) for some n E Z. Let M = N x Ft/ - be the quotient space, with the quotient topology. and 7r : N x 111 M the projection of the equivalence relation.

Given y = 7(x,t) E Mjet V = N x (t - 1/2,t + 1/2) and U = r(V). It is easily verified that ir I (U) = U„(z V,„ where V, = gn (V) and 7r : Vn U is a homeomorphism for every n E Z . One concludes then that Ir : N X 1R M is a covering map. On the other hand, for every n E Z , one has that g n = (7r! V) -I o 7r I V : V Va is a Cr diffeomorphism. Hence it is possible to induce on M a e. manifold structure such that 7r is a local e. diffeomorphism and dim ( M) = dim (N) + 1 (see example 6 in Chapter I).

On N x R, consider the foliation whose leaves are the lines lxl x Il, x E N, which is tangent to the vector field X° ( x,r) = (0,1) on N x R. This foliation and the field X° , are invariant under the diffeornorphism g, that is, g* ( 0 ) = and g* ( X ° ) = (dg) -1 (X ° o g) = X° ). Under these conditions, it is easy to see that there exists a foliation and a field X on M such that ao 7r * ( ) and X° = 7r* (X). The integral


curves of X are the leaves of a . The foliation a is called the suspension of the diffeomor-

phism f and the field X, the svspension field off.

Let us see the relation that exists between f and F. Let No = 7r (N X 101). Then No is an imbedded subvariety of M, diffeomorphic to N by h 7rI N x 101 No . Given y = 7 (x, O) No , let F be the leaf of 5 which passes through y. Then Fy can be parametrized by -y (t) = lr(x,t), which is the orbit of X with initial conditions -y ( 0 ) = y. This orbit returns to No at t = 1 in the point 7r (x, I ) = 7 (1(x) ,0) = = h of o h ( y), so we can define a map of "first return," : No --• No , by g = h ofo h -I . The map g is a C r diffeomorphism of No conjugate to f (g = h o f o h -1 ).

This map is also known as the Poincaré map of the section No and of the field X. The topological and algebraic properties of f translate to analogous properties of 5. For example, if x E N is a periodic point of f of period k (that is, f f (x) = x if and only if

= j • k for some j E Z ) then the leaf of 5 containing y = h( x) is a closed curve that cuts No k times. If x is not a periodic point of f, then the leaf of a through h(x) is not compact and cuts No in the set Ih(f n (X)) n E71. If the orbit of x for f, lf" (x) In EZ1, is dense in N, the leaf of 5 through h (x) is dense in M.

As a specific example of this situation, consider the diffeomorphism fcT : S I —• S I

defined by fa ( z) e2iria z. This diffeomorphism is a rotation of angle 27ra. The

suspension of fa is a foliation defined on M = T2 , which we denote by Ç. If a = = p/q E Q, where p and q are relatively prime, then all the points of s' are periodic for 4,, which means that the leaves of 1F all homeomorphic to S I and they cut No = 7r ( S 1 x 101) q times. If a fl) all the leaves of ac, are dense in T 2 (see exercise 13).

Identification by f

N= S 1 iT

o

Figure 10

The method of suspension will be generalized in Chapter V. It is used, as we will see.

to construct examples of foliations exhibiting specific recurrence phenomena.

Foliations 43

(2) The Hopf fibration

Let S2n = kz 1 z,,) EC" I E in. =1 1 Zi 1 2 = 11, n 2. Given an n-tuple of integers k = (1( 1 k,,), where ki 0 for all 1 = 1 n, we can associate an action Wk : SI x s2n -' s2n - letting

k 1 k

4pk (u,z 1 ,••., z n ) = (u z1 ,•-•, U zn) •

It is easy to verify that the vector field Xk defined on S 2" I

'V IM ! z r,) = (iki z i ikn z„)

is tangent to the orbits of Wk and hence (pk is locally free, since Xk ( z) 0 for every z E S2n - 1 . The leaves of the foliation ff k , induced by yn k , are all honneomorphic to S I . In the case that k i = k2 = • • = ki = 1, the foliation ff k is known as the Hopffibration.

In this case, for every z E S2" -1 the isotropy group of z is 11) and, for this reason, it is possible to prove that the orbit space of wk has a differentiable manifold structure. This manifold is diffeomorphic to the complex projective space CP" -- , which is defined as being the quotient space of C" - 101 by the equivalence relation which identifies two points z, w E C" - 101 if there is a X E C* such that z = X w. The projective space 4:CPn- I

has real dimension 2n - 2, and since the projection for this equivalence 7r : S 2" I

CP n I is a submersion, = r*( ,;)), where a. 0 is the foliation of CPI I by points. In the case n = 2, CP' l is diffeomorphic to S 2 and the projection 7r can be defined,

for example, by the expression:

7r(z 1 ,Z2) = 2 (Iz i l 2 — 1Z2 1 , 2Re(z i 2 ), 2im(Zii2)) E S

2 .

(3) Holomorphic vector fields

A holomorphic flow is a holomorphic action 4) : C x M M where M is a complex

manifold (locally diffeomorphic to C m by a holomorphic diffeomorphism). The flow 4)

induces a holomorphic field Z on M defined by Z ( p) = (d(1) (T, p) /dT) 17- ,--0 • If Z (p) = 0 then 43 ( T, p) = p for every T E C, or, the orbit of p is the point p. In the

case Z(p) 0, the orbit of 4) through p has complex dimension 1 (2 real).

Consider for example the flow on t" defined by

(T,z) = exp(T • A) • z, TE C, z E C" ,

where A ED(n ,C ), the set of complex n x n matrices. The field induced by (I) is the linear field z 1-4 A z. If A is nonsingular, 4) defines

a foliation of two real dimensions in C" - 101. Let us look at the case where A = diag (X 1 ,..., X ,, ), where X, V IR X J if i j. In this

case, the eigenspaces E,, relative to the eigenvalues X„. are invariant under the field

A, (A (E,) = E.,). So, for i = 1 n. E, - 101 is an orbit of 4) homeomorphic to

111 x S'. The orbits of 4) through points outside E l U... U L , . are all homeomorphic to

C. Let K. ( A ) c C be the closed convex hull of the set IX, 3Q (A) is the smallest.

polygonal convex set which contains lX I ,..., X,,1. If 0 H (A) it is possible to prove

that there exists an ro E C - such that Jim, ( t r0 , z) = 0 for all z EC".

This means that there exists a sphere S of real dimension 2n - 1, transverse to the tra-


jectories of the real "subflow" Xi ( z) = 4)(t • ro ,z). So the intersection of orbits of 4) with S define a foliation 5 of (real) dimension 1 on S. The intersection of each eigenspace Ei with S is a closed curve. Any other leaf of 5 is homeomorphic to U. For example, if n = 2, 5 has two leaves homeomorphic to SI and ainy other has as its limit set the union of these two.

In the case that A = diag (1 ,..., 1), the identity on C n , one sees that the foliation 5 is the Hopf fibration of S' 21.1- 1 .

Another example of a foliation generated by a holomorphic differential equation is the following.

Consider the holomorphic differential equation

(1) y" = f(x,y,y') x E C

where the derivatives of y are taken with respect to the parameter x. Setting z = y ' , equation (1) is transformed to the system

(2) y' = z

z' f(x,y,z)

on C 3 . This equation induces a holomorphic vector field Z on C 3 given by

(3) Z(x,y,z) = (1,Z,f(x,Y,Z))

whose integral surfaces contain the solutions of (2). The field Z can be considered as a plane field of real dimension 2 on C 3 , completely integrable and transverse to the fibers x = constant.

(4) Turbulization of a foliation

Let 5 be a codimension one foliation of a three manifold M. Suppose there exists an imbedding

: S 1 x D2 M

such that:

(1) ' S 1 x 1x1 is transverse to 5 for each x E D 2

(ii) v(19 X D2 ) is contained in a leaf for all 0 E S I .

In this case it is possible to modify 5 in the solid torus (S' x D2 ) without changing the foliation on the outside, as follows:

In the solid cylinder 111 x D 2 consider the C' foliation defined by the cylinder 2

X + = 1/2 plus the graphs of the functions x3 = ( 2 + x,) + b, h E IR, on the points (x i ,x2 ) with 4 + x; 1/2.

Foliations 45

X3

Figure 11

Here a (r) = 0 in a neighborhood of r = 1 and (cra(r)/dra ) — 00 when r 1/2 for all n O. Since this foliation is invariant under translation along the

i x3 -axis, it defines a folaition of S' x D- which n turn induces via 40 a foliation ' of M with one more compact leaf. In the region yo (SI X D1/2), the foliation a has a Reeb

component, where D ] /2 is the disk of radius 1 / 2 .

We say ' is obtained from S by turbulization.

(5) Foliations of three-manifolds

We will sketch here the proof of the theorem of Lickorish [25], Novikov-Zieschang

1 401: every compact, orientable manifold of dimension three admits a foliation of codimen-

sion one. The proof of [40] is based on the following result on the topology of three-manifolds.

Theorem (H. Wallace [58]). Let M be a compact, connected, orientable manifold of dimen-sion three. There exist imbedded, disjoint solid tori T, in M and 7 7 in S 3 , i = 1 n, such that M — U i= , 7-, and S 3

— U := 1 T f are diffeomorphic

One shows then that the tori TI can be taken transverse to a Reeb foliation of S. Using the turbulization process, we obtain a C ce foliation of .53 — U where each a Ti'

in the boundary is a leaf diffeomorphic to T2 . By the theorem of Wallace this induces a foliation of M U n, = T, where the a T1 's are leaves. This foliation is completed to a foliation of M by putting a Reeb foliation in each solid torus T,.

Remark. Using Alexander decomposition, B. Lawson ( [24] ) proved in 1970 the existence

of codimension-one foliations on the spheres S 2À * 3 , k = 1,2,3. This result was gener-alized to any odd-dimensional sphere by Durfee [9] and Tamura [54].


Later, and with different methods, W. Thurston (1551 , [56]) proved that any plane field of codimension one on a compact orientable manifold of dimension n is homotopic to a completely integrable plane field, that is, tangent to a foliation.

In the case n = 3 this result was already known [60]. In particular, one has that any compact manifold with zero Euler characteristic admits a foliation of codimension one.

(6) Geodesic flows

Let M be a compact Riemannian manifold. A unit vector y E TM determines a unique geodesic -y passing through p in the direction of v. We denote by ( p, u) the point of M at distance t from p measured along y in the direction of v. Since the vector (clyos ( p,v) ds) I s , has length I , the map 4), : T 1 M --• T I M given by 95 ( ( p,v) =

= ((pt (p,v), ((Loos ( p,v)/ ds) I s , i ) is well-defined on the fiber space T I M of unit

tangent vectors to M. (See example 3, chapter V). Setting 4), ( p, u) = p, — u) for t <0, we obtain a flow : IR x T I M T M, given by 4) ( t, ( p, v ) ) = This is called the geodesic flow of M.

On manifolds of negative sectional curvature, the geodesic flows has the following property. There is a decomposition of the tangent fiber space T( T' M) invariant under D4)t (t E IR ) :

T(T I M) = E.5 C) @- X

where 0 denotes the Whitney sum and X the space generated by the velocity vector of the geodesic flow, and there are positive constants a, b, X > 0 such that

[/--)43. 1 (P,v) • Ell ae -xl ; Itl if t > 0 and E E s

:I D4)t ( v ) I 5 be xi IV' if t < 0 and E E" .

The plane fields generated by E li C) X and E 5 Ci X are completely integrable, i.e., they define two foliations which intersect transversely along orbits of 4). The study of these foliations, taking into account the dynamical properties of the system, proves that the set of periodic orbits of cl) is dense in T i M, that is, given ( p,v) E T 1 Mand f > 0 there exists ( p ',v') E T i M, eclose to ( p, u) such that the geodesic which passes through p' in the direction of u' is closed. This result is due to Anosov [2]. We also recommend [3] and [4].

III. THE TOPOLOGY OF THE LEAVES

We saw in the previous chapter that the leaves of a Cr foliation inherit a Cr differentiable manifold structure immersed in the ambient manifold. In this chapter we will study the topological properties of these immersions, giving special emphasis to the asymptotic properties of the leaves.

51. The space of leaves

Let Min be a foliated manifold with a foliation 5 of dimension n < m. The space of leaves of 5 , M/ , is the quotient space of M under the equivalence relation R which identifies two points of M if they are on the same leaf of F. From the definition of leaf it is clear that this relation coincides with the rela-tion R defined in §1 of Chapter II. On M/3 take the quotient topology. The topology of M /5 is in general very complicated, possibly being non-Hausdorff as in the case of the Reeb foliation of S 3 , or of the foliation of 1R 2 as in figure 6 of Chapter I.

Let A C M. The saturation of A in is by definition the set a. (A) =

=1xEMIxRy for some y E A. If Ir :M M/5 is the projection for the quotient, we have F (A ) = 7r - ' ( 7r ( A ) ) = u „A F,, where by F, we denote the leaf of containing x.

Theorem I. The projection 7 is an open map, or. the saturation 5 (A ) of an open subset A of M is open.

Proof. Let p E (A) and F the leaf of through p . Then F (1 A 0, and ifqEA fl F there is a path of plaques a ,..., a k such that q E c and p E


Suppose that each a is a plaque of Uf with ( U„, (pi ) E 5 and let 4 o (U i ) =--

= U; x U,"where U„' and U]' are open disks in 1R" and IR'' respectively, j = 1 k. Suppose that for some j E 1 1 , kJ there is an x E ai which has an open neighborhood V C 5 (A) fl U. Since ça ,.: U1 U.; x U; is a homeomorphism, (pi (V) is open in U„' x U,", so if r2 : U; x U," --0 U.; is the projection onto the second factor, r 2- ' (ç j2 ( V)) is open in x Uj". On the other hand W = ça» ( (so. ( V))) is open and a„ c W C (A) and so a, is in the interior of 5(A). It suffices now to note that since A is open, al is in the interior of 5 (A), so a fl a2 is also in the interior of 5 (A) and from the previous argument, there is a neighborhood W of a2 such that a, c W C (A). Repeating this process k — 1 times, one proves inductively that ak is in the interior of 5 (A) and hence that 5 (A) is open. •

In general the projection r is not closed. When this occurs, each leaf F of 5 is a closed subset of M, since F = ( 71- (x)) where x E F. The foliation in figure 6 of Chapter I is an example where r is not closed, although each leaf of 5 is a closed subset of Ile.

Definition. We say that a set A C M is invariant or saturated in 5 when the saturation of A in 5 is A, that is, r -1 (r (A)) = A.

Theorem 2. Let A C M be an invariant subset in a. Then its interior A, its closure A and its boundary aA are also invariant in T.

Proof. The set A is characterized as being the largest open set contained

in A, that is, if B is an open set such that Â c Bc A then B A. Since

ir is open we have that 7r - ( 7r (;k)) = B is open. Further, À C B C A,

since A is invariant, hence A B = 71- I ('r( )). Observe now that if

A is invariant then M — A is also, so int (M — A) is. On the other hand

int (M — A)=M — A, or, M — A is invariant and hence A is invariant.

Since aA = A — A, it follows that aA is also invariant.

§2. Transverse uniformity

Let E be a submanifold of M. We say that E is transverse to 5 when E is

transverse to every leaf of 5 that it meets. When dim ( ) + dim ( ) =

= dim (M) we say E is a transverse section of

Given p E M there is always a transverse section of 5 passing through p.

In fact it suffices to consider a local chart ( Uop ) E with p E U, ça(U) =-

= U, X U2 C IR" x 111 5 , ça(p) = (c,,c2 ) and take E = ça '(c 1 x U2 ).

Since 1c 1 1 x U2 is transverse to the plaques U1 x c E U, it is clear that

E is a transverse section of 5.

The Topology of the Leaves 49

Theorem 3 (Transverse uniformity of Let F be a leaf of 5 V. Given q 1 ,q2 E F, there exist transverse sections of 5, E l and E2, with q, E E, (i = 1,2) and a Cr diffeomorphism, f: E, E2 such that for any leaf F' of 5 , one has f(F' n E l ) = F' fl E2.

Proof. Let q 1 ,q2 E F. Consider a path of plaques a l ak joining q, and q2 , that is (4, 1 E a l and q 2 E ak . Suppose that, for each j, ai is a plaque in ( E T. By Lemma 1 in Chapter II we can suppose that if U, n u, 0 then U, U U, is contained in some open set, the domain of a local chart of 3. Let (p1 (U.1 ) UI x LP2 c itn x W. Fix points p; E cif (-1 a , 1:sj5_k-1, po = q, and Pk = q2 . For each j = 1 k, set 4,, i ( pi ) = (xj ,yi ) and consider the transverse disk Dj = ,p,--1 ({xj 1 x (JD. For j = 0, let sol ( Po) = = ( x0 ,y0 ) and Do = 1-1 ( ixo l x U1).

Figure 1

Since for each j = 0 k — 1, Uj U , is contained in a chart of there is a disk Bi of dimension s with pi E B C Dj n U, such that

each plaque of 1.1.1 , 1 cuts Bj at most one time. We can then define an in-jective map 4: B , calling fj (p), the point of intersection of the plaque of Uvf , which passes through p with DJ ,. 1 . So we have that f1 ( pi ) =

p, *1 and that fj : Bj f1 (BI ) C Dj _ i is a Cr diffeomorphism. From the construction it is immediate that, for every leaf F' of g , f, (F' fl B1 )

F' fl fi (Bi ) if j = 0,1 ,..., k — 1. Finally take a disk E, such that po E E, C B0 n (B 1 ) n ( f (... f k 1 ( (Di) )) and define f: E, -* DA. by f( p) = f, ( ( . f, ( fo ( p)) )). It is clear tffat f is a diffeomorphism on E z = f( E 1 ) C DA and that for every leaf F' of 'S we have f( F' n E 1 ) = F' fl E2, as we wanted.


Let us see some consequences of this theorem.

Theorem 4. Let be a foliation of M, Fa leaf of and E a transverse section of such that E fl F 0. We have three possibilities:

(1) E n F is discrete and in this case F is an imbedded leaf. (2) The closure of E fl F in E contains an open set.

This occurs if and only if the closure has non-empty interior and int () = — aT is an open set which contains F. In this case we say F is locally dense.

(3) E n F is a perfect set (i.e., without isolated points) with empty interior. In this case we say F is an exceptional leaf.

Proof. Suppose pEE nF and q E F, p. By Theorem 3 there exist transverse disks E, and E2 with p E E, C E, q E E2 and a diffeomorphism f: E 1 E2 such that, for every leaf F' of 5, f(F' n E 1 ) = F' fl E 2 . In particular f(F fl E 1 ) = F n E 2 , therefore F n E l is homeomorphic to F n E2 and F (1 E l is homeomorphic to F n E2 (by f). In the case q E E we can take E2 C E. From Theorem 3 we have the following possibilities:

(1) p is an isolated point of E n F. In this case E fl F consists entirely of isolated points, i.e., is discrete,

(2)p is an interior point of E n F in E. In this case all the points of E fl F are interior points of E fl F ,

(3) E n F has empty interior in E and E fl F is not discrete.

Consider alternative (1). In this case, if q E F and ( U, yo) is a foliation chart of 5 with q E U and ,oc, (U) = U1 X U2 then Ds = ye) -1 (ix1 X U2) is a trans-verse disk to 5, so F n Ds is a discrete set which contains q. Therefore, there is a smaller disk 02 C U2 such that D = ,p -1 ([xl x 02 ) cuts F only in q, or

1 U1 X 02 : U 1 X 02 M is a C' imbedding such that F n cul x '2) contains only one plaque of U; hence Fis a C submanifold of M. Conversely, if F is a CT submanifold of M one proves easily that F fl E is discrete.

Let us look at alternative (2). Let q E F and ( U, (p) E 3:, q E U, with ,p (U) = =U1 X U2, as above. Let Ds = yo -1 (1x1 X U2 ) . Then q is an interior point of F n Ds, so there is a disk 02 C £12 such that D = I ( Ix] X 02 ) C F n D5 . Therefore F contains the open set A = tp -1 (U1 x 02 ). On the other hand,

is invariant under 5 so contains the saturation of A, (A), which is open, by Theorem 2. Hence, F c 5(A) C T- and 5(A) = — (3F- as we wanted.

Alternative (3) is the complement of the other two and this concludes the proof. III

As a consequence of this theorem, the immersed submanifold F, sketched in the figure below, is not a leaf of any foliation defined on an open set of 1R2.


Figure 2

The reason is that a point x, as in the figure, is an accumulation point of points of E fl F, where E is a transverse segment passing through x, while the same does not happen with the point y.

§3. Closed leaves

Theorem 5. Let F be a leaf of a foliation on M. The following statements are equivalent:

(a) F is a closed leaf, (b) if (U, 40) is a foliation chart of such that is compact then U fl F con-

tains a finite number of plaques of U, (c) the immersion 1: F M, 1(x) = x, is proper (*). In particular, if F is

closed then F is imbedded.

Proof. Let F be a closed leaf of 3 and (U, (p) a foliation chart of with (U) = U1 x U2 c 111" x llts such that U fl F 0. Let DcDc U2 be a

disk such that E = ça (x x D) is a transverse section with E c U and E n F 0. Since the plaques of U fl F are open and disjoint in the intrinsic topology of F, U fl F contains at most a countable number of plaques and therefore E n F is countable. Since F is closed, Fn E = Fn is countable. However, we know from general topology that a perfect set is uncountable, hence F necessarily satisfies alternative (1) of Theorem 4, so, E n F is discrete and hence finite since E is compact. We conclude therefore that F fl ç '(U1 x D)

I

(•) A continuous map f : X Y is proper if for every compact K c Y, f (K) is compact.


contains a finite number of plaques of U. Taking U compact, this implies that F fl U contains a finite number of plaques. Conversely, suppose that for each foliation chart ( U, yo) of 5 with U compact, U fl F contains a finite number of plaques of U. In this case, if p n , n E N, is a sequence in F with lim n pn =

= p E M, we take a foliation chart (U, ç') of a such that p E U and U n F contains a finite number of plaques. Then there exists an no and a plaque Cr of U such that p,, E Cr for all n no . So p E a and hence p E F, i.e., F is a closed leaf of 5.

We will show that (a) implies (c). If F is a closed leaf of 5, then i F M is an imbedding and the intrinsic topology of F coincide with the topology in-duced by the topology of M. Thus if K C M is compact, K n F i (K) is compact in F, so the immersion i F M is proper.

We will now prove that (c) implies (b). If i F M is proper, given a foliation chart ( U, 4o) of a such that U is compact, i -1 (U) is compact in F; so, U n F contains a finite number of plaques, which shows that Fis a closed leaf.

§4. Minimal sets of foliations

Let M be a manifold with a foliation 5. A subset p C M is minimal if it satisfies the following properties:

(a) p is closed, nonempty and invariant, (h) if p' C y is a closed, invariant subset then either p' = 0 or y' = p.

Examples.

(1) Every closed leaf of if is a minimal subset. (2) The only minimal set of the Reeb foliation of .S3 is the compact leaf

homeomorphic to T2 . (3) Let 5a, be the foliation of T2 obtained by a suspension of the rotation

fa : —1. , fa (z) = z. If Cr is irrational, all the leaves of a c,

are dense in T2 and, therefore T2 is the only minimal set of 5a .

Theorem 6. Every foliation of a compact manifold has a minimal set.

Proof. Let M be a compact manifold and 5 a foliation of M. Denote by e the collection of nonempty compact subsets of M invariant in 5. It is clear that C is nonempty. If F is a leaf of if then F E C. Consider on C the partial order-ing induced by et inclusion. Given a sequence p i D y2 D ... of elements of C, it is clear that p = nr_ p, is nonempty, invariant, compact and p c p for every i E IN. By Zorn's lemma, e contains minimal elements. A minimal ele-ment of C is obviously a minimal subset of a. •


Theorem 7. Let p be a minimal subset of The following properties hold:

(a) Every leaf of contained in it is dense in y.

(b) If M is connected and y has nonempty interior then pi = M. (c) Let p = cod 5 and E a p-dimensional disk transverse to a such that

12 n E 0. If 14 is not a closed leaf then y (1 E is a perfect set. (d) Moreover, if p = 1, aE n y 0 and pi has empty interior but is not a

closed leaf, then 14 n E is homeomorphic to a Cantor set. In this case we say p is an exceptional minimal set.

Proof.

(a) Let F be a leaf of 5 contained in it. We have then that F is closed, F 0 and F c 14, since 12 is invariant. As y is minimal, F = y.

(b) Since p has nonempty interior, there exists an open subset A C p. By (a), all leaves of it meet A , so (A) = p. Since 5 (A ) is open in M, pi is closed and open in M, which is connected, so y = M.

(c) Given x E E fl p. , let F be the leaf of 5 through x. Since ji is not a closed leaf, there exists a leaf F' F, contained in p. Taking a local chart of 5 con-taining x and using (a), one concludes that x is an accumulation point of F' fl E.

(d) By (c), p. n E is a compact, perfect subset with empty interior in E. The proof is reduced to the following lemma, which can be found in [12 ] , p. 123. 11111

Lemma. Let K C IR be a compact perfect subset with empty interior. Then there exists a homeomorphism h : K e where C is the Cantor set in [0,1 ] .

Notes to Chapter III

(1) Foliated structures of the plane.

In Example 5 of Chapter 1 we saw a foliation of 1R 2 whose leaf space is a non-Hausdorff manifold of dimension one, equivalent to the manifold obtained by taking two

copies of the real line and identifying points with the same negative coordinate. One can say that this space is obtained by branching over at the point 0 E IR, a

ray in the positive direction. If we repeat this process at all the rationals, we get a manifold called a simple feather for which IR represents the shaft and the branched rays the barbs.

A double feather is obtained by substituting each barb by a simple feather. If now this process is taken to the limit substituting all the barbs by simple feathers one gets a com-

plete feather. In this space the nonseparable points form a dense set. It is possible to show that there exist foliations of 111 2 whose leaf spaces are simple

feathers. This process is sketched in figure 3.

Taking, in the figure, the curve L as axis, the process is repeated by substituting a simple feather for a barb. Substituting simple feathers for all the barbs by this process,


one obtains a foliation of 1R 2 whose leaf space is a complete feather. This is called Wazewski's example [59]. The following theorems are known about foliations of the plane.

Theorem (Kaplan [22]). To each continuous foliation of the plane it is possible to associate a continuous real-valued function defined on R 2 which does not have either a local max-imum or a local minimum and is constant on each leaf

Theorem (Kamke [21 1). If 5 is a C' foliation and U is bounded then there exists a Coe

map f: U JR which is constant on the leaves of'/ U and such that df 0 at each

point of U.

Theorem (Wazewski [59]). There is a C° foliation of 1R 2 such that each C 1 function

on IFt 2, constant on the leaves, is constant.

Y Y

1

1

trivial foliation insertion of a barb at y = 1

1

1/2

insertion at y = —1 2

Figure 3

These theorems were proved again by Haefliger and Reeb in [16] and [47], by means of an analysis of leaf spaces of foliations of the plane. They observed that the leaf space of a C r foliation of R 2 is. in general, a non-Hausdorff manifold, of dimension one, simply connected and of class C r . Kaplan's theorem follows from this and from the fact that on a simply connected, one-dimensional manifold (Hausdorff or not), there always exists a continuous strictly monotonic function. Kamke's theorem follows from the fact that the leaf space of 'S U has a C manifold structure. Finally, Wazewski's theorem follows from the fact that, on a complete feather, there exists a C

structure such that any C I function defined on it is necessarily constant.


(2) Minimal sets of foliations of codimension 1.

The study of minimal sets began with H. Poincaré who studied the behavior of orbits of analytic diffeomorphisms of the circle. The concept of rotation number, which he introduced, can be defined by the limit p ( f) = ( lim n 0, (7e? (X) — x)/ n) (mod 1 ) , where 7: R R is a diffeomorphism which covers f, that is, w (7( x) ) = f(71 - (x))

where w (x) = e2wix . One shows that the limit always exists and is independent of X. Poincaré proved that p ( f) is rational if and only if f has a periodic point. In terms of

foliations, this means that a foliation 5. of T2 has no compact leaves if and only if it is equivalent to the foliation defined by the suspension of a diffeomorphism of S i with ir-rational rotation number.

In 1916 Bold ([4]) observed, without constructing them that there exist foliations of T2 having exceptional minimal sets. In 1924 Kneser ([23]) constructed examples of such foliations of class C° .

The theory of foliations on T2 has as one of its highest points the theorems of A. Denjoy ([7 1 and [8 1 ). In 1932, he constructed examples of C I foliations on T2 with exceptional minimal sets. Further he proved that if 5 is a C2 foliation of T2 and is a minimal set of then M = T2 or p. is a compact leaf ( S I ) . In particular if does not have compact leaves, Denjoy's theorem implies that a is topologically equivalent ( ' ) to a foliation obtained by a suspension of an irrational rotation of the circle (see note

in Chapter II). In note 3 of this chapter we construct Denjoy's example. Later results on Cr diffeomorphisms of the circle can be found in: M. R. Herman, Sur la conjugaison différentiable des diffeomorphismes du cercle b. des rotations. Pub!. IHES no 49 (1979).

The results of Denjoy motivated the study of minimal sets of foliations of codimension one on compact manifolds of dimension a- 3. In Chapter V we will see an example, due to Saksteder ([491, 1964) of a C D:' codimension one foliation of M2 x S I (M2 = surface of genus two) which has an exceptional minimal set.

Later, Raymond showed, using Fuchsian groups, the existence of foliations of S3 with exceptional minimal sets.

(3) Denjoy's example.

Denjoy's example is obtained by suspending a C I diffeomorphism f : S I which has an exceptional minimal set. A subset C S 1 is called minimal for f if it is closed, nonempty, f(p) = II and for all C M with the above three properties we have pc' = M.

A minimal set is called exceptional when it is perfect and has empty interior, i.e., homeomorphic to a subset of the Cantor set on S I .

The construction of a diffeomorphism with an exceptional minimal set will be done in the following manner. Given an irrational rotation R a (e2T1t ) = e271(l+Q) , cut the circle S I at all the points of an orbit {On n E Z) of R In each cut insert a segment J,7 of length en where 2., , _ = L < on . We obtain in this manner a new circle longer than the first. The set S t — U J„ = it will be homeomorphic to the Cantor set. The diffeomorphism f will be constructed in such a way that PA ' will be conjugate to

( • ) We say two foliations F and ' on M are topologically equivalent when there exists a homeomorphism h : M M which takes leaves of .3 to leaves of ' .


1?,/ (S 1 — (O n i n E Z)) where ti ' = 1.4 — u ncz J„. From this fact one sees that every orbit of f in j will be dense in A.

In order to construct f formally, we construct a C 1 increasing diffeomorphism, 7 —• 1R such that 7( t) = t + ,,o(t) where so is periodic of period I. As it is known,

such a map 7 induces a C 1 diffeomorphism f : S I --• S I which can be defined by the formula

f( e2ril.(1)

(A) Construction of the minimal set pt.

Fix a E (0,1) — () and yo E (0,1). Given n E the number yo tux is not an integer so yo + na E (m,m +1) for some m E Z. Set y, = yo + na — m E (0,1).

Since a is irrational the sequence IYn1 nE Z is dense in [0,1] and moreover y„ y, if n tn. Let if„I ncz be a sequence of positive numbers satisfying the following properties: (1) E f, = L < 00, (2) E n e /f -n= 1 and (3) 1/3 < f /f n +1 n < 5/3 for all n E Z. For example the sequence en = ( n 2 + I) 1 satisfies these properties.

From the sequences j y„I„ E and we will construct a set K C (0,1 + L), z IfninEZ homeomorphic to the Cantor set and which will essentially be the minimal set of Denjoy's example. The idea for the construction of K is, basically, that for each n EZ one cuts the interval [0,1] at the point y, and inserts an interval of length g„. Formally we proceed in the following manner.

Consider the two sequences a, =y,. + E e, and On = an + fn . Let 4, = = It is clear that for every n E Z one has 0 < a, < te, < 1 + L, so

C (0,1 + L). Moreover one can verify that if y, < y, then On < an, and hence n i, = 0. Set K = [0,1 + Ll U „ E In . It is clear that K is compact.

Lemma 1. K is perfect and has empty interior.

Proof. With this goal we define two auxiliary functions. Define h - and h + : [0,11 10 ,1 Ll by IC (y) = y + E 0<yr<y fi and h + ( y) =y + E 0<1< t.

that the series E „en converges, one easily proves that h - and h + are monotonically

increasing that h - is lower semi-continuous (i.e. S<y y h (s) = h- (y)) and that

h + is upper semi-continuous (i.e., s>y liMs y h + (s) = h+ (y)). Moreover h - ( y, ) = an and h + ( ) = 0,, n E Z, and if y n E ZJ then h - (y) = h + ( y).

K has empty interior. Suppose [a, b] C K is such that a, b E a K. Since a E aK, there is a sequence 10,,,I kE _z such that O n, < < a for all k and lim k = a. Since

1, n= 0 if n In we have that On, < a flk < < a so lim k oe nA = a.

Since the sequence(a flA jkaN is increasing, the corresponding sequence ly„,1 kEIN is also

increasing, so lim k yn, = y for some y E [0,1] and hence a = lim an , = lim k h - (y) = h- ( y). Analogously one concludes that b = h (y' ) for some

y' E [0,1]. In the case y = y, for some n we have a = an so a = b since

n I, = 0. In the other case y y, n E Zlandsoh - (y) = h + ( y) = a. Hence,

h . (y)=ab=h+ (y') and thus we conclude that y y' because IC. is increasing.

On the other hand if y < y' there exists y, E ( y,y' ) since 1 yn l nEEN is dense in [0,1]

so a = h+ (y) < h + (y,) < h + (y' ) = b or else [a, b) n 0, a contradiction. We

conclude that y = y' so a = b and hence K has empty interior.

( On — an ) 2 [ i3 n an

6 r n - I — I 1 6(1 + L) 2 en-1 1


K is perfect. Let x E K. In the case x = an for some n E Z , let fyn,I kEiN be a sequence such that yn* <y, for all k E N and limk,yflk= yn . We have then that an = = Lim k c„ h - (y nk ) = lirnk 0, an, and therefore an is an accumulation point of K. In the case x = On for some n E Z , with an analogous argument, one proves that

= urn k tink where gn, > gn for all k E N and therefore f3, is an accumulation point of K. On the other hand if x fan ,13„ In E j, it is easy to see that x

= lim k an* for some sequence fu nk )", since K has empty interior. This proves that K is perfect. III

Now set X = K - cin ,(3,7 1n E ZI and Y = [0,1] - Lyn In EZ). Define h : Y Xby h(y) = h + (y) = h (y), yE Y.

Lemma 2. 77 is a homeomorphism onto ?.

Proof. 7 is continuous since h - is lower semicontinuous and h ir is upper semi- continuous. On the other hand h is increasing since h - is. The inverse of h is given by

= x — E

x E ;17 ,

0<cv,<x 0<,6i<x

which, as one can verify directly, is continuous on 51('

We consider now the map ik (x) = (1 + L) -1 x. The homeomorphism 1,G takes K C [0,1 + L] to the set 1,t, (K) C [0,11, which is homeomorphic to the Cantor set. Set 71 = 1,G ( K) + Z. By an abuse of notation we designate 1,1, ( in ) by In , 1Y (an ) by an and 1,t ) by On . With this notation we have [0,1) - = 1.) nErN

(B) Construction of 7:

Lemma 3. There exists a continuous function yo : 111 1R satisfying the following properties:

(a) so is positive and periodic of period I.

(b) so 1,

(c) so ( T ) dr =[3nI Otn , = (1 + L)

Proof. It is enough to define yn on U ne z 4. Using this we extend so to 112, letting t + in) = so(t) if m E Z and so! 1. We then define so (t) = 1 + X, ( - t) •

(t a, ) for t E In where

Condition (c) is immediately verified. We verify that so is positive and continuous. For

t E 1,, we see that

3 Is^( 1 ) = I Xn(On 1 ) (I — an ) ( 13 p? —

, = —2

since 1/3 < f < 5/3. So so is positive. On the other hand

'x..

X n

en+1 1

< I In


sup fç(t) - 11 = -3

tEI 2

so Jim,, cr, (sup,an ( - 11) = 0, since cn en + I en = 1. This implies that so is continuous on Ti n [o, 1] and thus continuous on 1R. •

Now let c = (1 + L) + E 0<yi < c, ed where a is the irrational number used in definition of the sequence 1 yn I nE . Define 7: JR 111 by 7(x) -- c + .f ok ( 7) dr. It is clear that f is a C 1 diffeomorphism.

Lemma 4. The following properties are true:

(a) 7(t) = t + 0 (t) where 0 is periodic of period 1, (b) for ea,ch n E Z such that y n + a < 1, one has that 1(an ) = an+ f and 7(0n ) =

13,1+1, (c) for each n E Z such that yn + a> 1, one hcts that 7(an ) = 1 + a,,+1 and 7(On

= 1 + 13n+1.

Proof. Let m be the Lebesgue measure on R. We have then that

7( an C = 0 it) ( ) Kn 4,0 Ch Ln (T) Ch.

where K,, = i n [0, ci„] and L,, = u 0<ai<an /J • Since S' ( r)dr = ( 1 + L) -1

one sees that I L,, (r)dr = (1 + L) -1 ei, i . On the other hand

= dT = III( n [o,otn i) = an - (1 + L ) ' E 0<aj <an

since T4 n [0,a,,] = f0,an ] U <an ./j and m (l) = (1 + L) -1 f l . Recalling that E o<yecyn E gi and that after the change of scale 0, an = .( 1 + L) -1 •

(Yn E0<Yj < Yn fi ) we have finally

7( an ) = ( 1 + L) ( yn + a + E f, + E .

Suppose y,, + a < 1. In this case, from the definition of the sequence YJ JJEz , one sees that y,, + a = yr,„ i and _zy + a = yj+ for every j such that y 5_ y,,.

o< v.,< v„ E and therefore 7( a„ ) = We have then that =

= ( 1 + L) I Din + 1 E 0, „< , f» = un-:- On the other hand 7030 =

7( an ) + in ( )dr = ± an+ , — = On+ 1 • In an analogous manner one proves that f( a,,) = 1 + an , 1 and 7030 = 1 + O n _ 1 , in the case y„ + a > 1. We leave the details to the reader. In order to prove (a) it is enough to observe that so is periodic of period 1 and that

en +1 1

.,r_1(0,11,40(r)dr + f s 1.1

4,o(r)dr = (p(r)dr = - 0 U

nE Z

T)CIT =


00

= (1 + L) 1 + (1 + L) E ej

Therefore 7( t + ) - t) = 1 for all t E H, sol( t ) t + 0 (1) where 0

of period I. Now let p = ! t E a, = e2r1trn bn e271-0, .1n /e2rit t e2ri( ) e2ril(t) .

phism of S I . Further p. c S I is homeomorphic to the Cantor set and ki = S I - It is clear that f : S I S I is well-defined and is a C I

is periodic

E In ) and diffeomor-

U nEz in .

Lemma 5. p is a minimal set of f.

Proof. Conditions (b) and (c) of the preceding lemma imply that f( Jn ) = J„ for all n EZ and hence f(p) = S i U f( )

= u , ti is invariant under f. In order to nE Z

prove that p is a minimal set for f it is enough to prove that the orbit of O under f, 0 (0) = (fn (0) inEZ 1, is dense in for all 0 E

1 st case. O = ak (or bk ) for some k E Z . In this case O (ak ) = an I n EZI which is dense in p since S I - = u nEZ Jn , p. is homeomorphic to the Cantor set and for all n E Z, an is an endpoint of in .

2nd case - general case. Consider the rotation by angle 22-a, R O,(e 2") = e2Ki(l+a)

For each n E Z set On = e2" . It is easy to see that O n = R ( 00 ) for all nE Z. Con-sider also the homeomorphism : c [0,1 + L]. This map induces a homeomorphism

h : Y = (S 1 - !nEZI) —• S I - I an ,bn It E Z1) = X setting h(e2tnt ) = 271(1+ L) = e ii7(t) ,t E P. Observe that Y is invariant under Ro, and Xis invariant under

f. The proof resumes now to show that the following diagram commutes

R. Y

h

Indeed, the fact that all the orbits of R„, are dense implies that all the orbits of f in X are dense in X, so in p. The commutativity of the diagram is equivalent to showing that 7. (ii( y )) - y + a) E Z for all y E Y. This can be verified directly using the formulas given above. We leave the details to the reader.

IV. HOLONOMY AND THE STABILITY THEOREMS

In this chapter g denotes a foliation of codimension n and class Cr, r 1,

of a manifold M". Our objective is to study the behavior of the leaves near a fixed compact leaf F. By the transverse uniformity of g it is sufficient to study the first returns of leaves to a small transverse section E of dimension n passing through a point p E F. For each closed path 7 in F passing through p, these returns can be expressed by a local CT diffeomorphism of E, 4, with

( p) = p and where for x E E sufficiently near p, f,, (x) is the first return "over 7" of the leaf of g which passes through x.

If 7 is another closed path in F through p, homotopic to -y (with fixed end-points) then it happens that f,, and 4, coincide on a common neighborhood of p E E, that is, A, and j; have the same germ at p E E . So it follows that the map -y 1-4 f, induces a homomorphism

-n- ( F, p ) --• G(E,p)

from the fundamental group of Fat p (the group of homotopy classes of closed paths) and the group of germs of Cr diffeomorphisms of E at p. This represen-tation is called the holonomy of F at p. This notion was introduced by Ehres-mann in [10] .

It is clear that if 7i- ( F, p) is small the recurrence of leaves near F will be small. The Theorem of Local Stability expresses this in the following manner: if F is a compact leaf of g with finite fundamental group, then there exists a neighborhood V of F, saturated by leaves of 5:, such that every leaf in V is compact with finite fundamental group.


The Global Stability Theorem offers the following global version: if cod if = 1, M is compact and connected and there is a compact leaf with finite fundamental group then all the leaves are compact and have finite fundamental group. These theorems are due to G. Reeb [46].

1. Holonomy of a leaf

Let 7 :1 = [0,1] F be a continuous path and E0 , E l small transverse sections to if of dimension n passing through po = -y ( 0 ) and pi = 7(1)

respectively. We will define a local map between E2 and E I "along" the leaves of a, "over" the path 7 taking po to p i .

Figure 1

According to Lemma 1 of Chapter II, there exists a sequence of local charts ( U1 ) k,, o and a partition of [0, 1] , 0 = to < < tk = 1 such that (a) if U, fl U then U, U Ui is contained in a local chart of 5. and (b) -y ( [ t„ t,,]) C U, for all 0 i k. We will say then that ( U1 ),k_ o is a chain subordinated to

For each 0 < I < k + 1 fix a transverse section to • , D( c u,_, n U„ homeomorphic to an n-dimensional disk, passing through -y ( t, ) . Also set D(0) = Eo and D(1) = E. Then for each x E D(t1 ) sufficiently near 'y ( t, ) the plaque of U, passing through x meets D(t, ±1 ) in a unique point f, (x).

The domain of the map f, contains a disk D . D(t,) containing -y ( t, ) .

Holonomy and the Stability Theorems 63

So it is clear that the composition

fy = fk ° fk-i °•-•° fo

is well-defined in a neighborhood of Po E Eo . We will call fy the holonomy map associated to y.

Lemma 1. The map fy is independent of the disks (D (t.)) chosen and of the chain subordinated to y, that is, if .7.), and f are two holonomy maps associated to the same path then L and f coincide on the intersection of the two domains.

Proof. First fix the chain subordinated to -y, and suppose that (D(0),k,_ and ( B ( t,)) 1 are two collections of disks as above with -y ( t, ) E D(t1 ) fl B(t,) for every i. Given x E D(t,) near -y( t, ) we can define )3, (x) E B(t,) as the inter-section of the plaque of Ui which passes through x and B ( r). We have then defined a homeomorphism 3 : b, --• h, between open sets b, c D(t,) and fi, c B(t,) containing -y ( t,) . Let f : D: D(t,,,) and g, : B: B(t,, 1 )

be as before. Since U, U C V, a local chart of , each plaque of V meets each transverse disk in at most one point; then, the following diagram commutes

f,

g,


Restricting the maps !,, g, and (.3i , 0 i k, to smaller domains, we have that

= fk ° lk _ ° • • ° fo

= 07,+I 1 ° gk ° 0k ° ° gk-1 °•••° 01 ° 0Ï 1 ° g0 ° 00 gy

since 00 and 0", are identity maps. With an analogous argument one proves thatf, is independent of the partition 0----, to < t, < < tk+ , 1. Now we take ( U,) /L o and ( 0, two chains subordinated to -y. If for every I E tO , , k C U some j, it is immediate that f„ and f; coincide on the intersection of their domains. The argument is similar to the one above. In case this does not occur, by Lemma 1 of Chapter II, we can construct a chain subordinated to -y, ( U1") /,`„: 0 which refines 1 U. , that is, such that for every i E 10 k "1, U," c U, (1 U: for some j E 10 ,..., ki and

E 10 k '1. This implies that f = f.," and f = f„" on the intersection of their domains, so f = f;.

Remark 1. The following facts follow from the definition of fo„ :

(a) f,, ( Po ) = Pi , (b) if -y 1 (t) = -y(1 — t) then fr = ( ) (c) if is of class Cr (r I). The intermediate maps f, (0 i k) are cr

sofy is Cr. Sincef has an inverse, also Cr, f.), is a C r diffeomorphism, (d) if ;y- is obtained from 7 by a small deformation in the leaf, with endpoints

fixed, then f = f, in a neighborhood of po E E. One proves this ob-

serving that, for :)," near -y, there is a chain that is simultaneously sub-ordinated to -y and 7y,

(e) if Ao c U0 and A, C Uk are other transverse disks to 5. , centered at

Po and Pi, the projections along the plaques of U0 and Uk define Cr diffeomorphisms, soo : A0 E0 and 4,17 1 : E 1 . The new holonomy

transformation, Ao D A i satisfies

g,(x) = ,p,- ' of o wo (x), x E Ao .

In particular, if -y is a closed path, E0 = E, and -10 = , then g and f, are conjugate, that is, g = o where so = yoo =

Definition 1. Let X, Y be topological spaces and x E X. On the set of maps

D y -4 Y, where V is a neighborhood of x, we introduce the equivalence relation R : fRg if there is a neighborhood W of x such thatf W = g W. The equivalence class of f is called the germ of f at x.

When X = Y the set G(X,x) of germs of local homeomorphisms which leave x fixed is a group with the multiplication germ ( f ) o germ ( g ) = germ ( f o g)


where the domain of f o g is the intersection of the domain of g with g-, (domain of f).

Theorem 1. Let 7, : I M, i = 0,1 be paths contained in a leaf F of 5' such that -y, (0) = po , y ( 1 ) = pl , i = 0,1. Let E 0 , E I be transverse sections

fyi to F at Po, Pi; E0 D Di E I holonomy maps associated to -y, and the germ of Ai at po .

(1) If -yo -y i rel ( 0, 1 ) then = (2) If p o = 131 and Eo = E i then the transformation -y 1— P >

homomorphism ( I induces a

43 : ( F, po )

from the fundamental group of F at po to the group of germs of Cr dif feomorphisms of E 0 which leave p o fixed.

Proof. Let H : I x F be a homotopy between -yo and , i.e., H(t,0) = -y o (t), H(t,l) = y 1 (t) and H(0,$) = Po , H(1,$) = p l for all s, E I.

For each path 'y,: I —0 F, y s ( t) = H(t,$), there is a chain subordinated to -ys . By continuity, each chain is also subordinated to any 7, with s' near s. Consequently, we can find a collection of chains ( C,)',7_ 1 and a partition of I: O = s0 < .5 1 < < s„, = 1 such that for any i = 1 m, C, is subordinate to all the paths -ys , s, _ 1 s s,. This implies by (d) of Remark 2 that germ ( ) = germ ( ) for all i = 1 ,..., m. So tp,0 = For this reason, When Po = p l thé map 4): [7] is well-defined. Finally, if Cx ,C, are chains subordinated to X - and I then Ci., U Ç is a chain subordinated to the product X - of the paths X and pt,. This says that

upt • [ _ = 40, (p ), - = ( ( [ xl

and so 43 is a homomorphism between the groups 7 1 ( F, po ) and G (E0 , Po). •

Definition 2. The subgroup Hot ( F, po ) = ( ( F, po )) of G (E0 , po ) is called the holonomy group of F at po . Given po ,p, E F, any path a : F with a (0) = po and a ( 1 ) = p 1 induces an isomorphism

a* Hol ( F, po ) Hol ( F, Pi)

where a* ( (11121) = (pa 43[A] o ‘,0 In this way we can speak of the holonomy group of F as any group isomorphic to Hol ( F, po ). To simplify nota-tion, we will at times identify an element of the holonomy group with a par-ticular representation.


Example I. In the Reeb foliation of S3 the fundamental group of the com-pact leaf is isomorphic to AZ 0 Z. Its holonomy group can be represented by two C diffeomorphisms f, g : 1R, f (0) = g ( 0 ) = 0 such that

f(x) = xforx Oandf(x) < xifx < 0; g(x) = xifx 5_ Oandg(x) < x

for x > 0 (see figure 3).

Figure 3

Remark 2. The holonomy of a leaf F can also be defined using a fibration transverse to F. With this goal in mind we first prove the following lemma.

Lemma 2. Let F be a leaf of 5 and KC Fa compact subset. There exist neighborhoods U 3 W of K, U open in M and W open in F and a Cr retrac-tion : U --• W such that for any x E W, 7r -l (x) is transverse to 5 / U.

Proof. Since K C F is compact there exists an imbedded surface W formed by a finite union of plaques of , such that K c W. Let 7r : W be a CT tubular neighborhood of W. Since each fiber 7r -1 (y), y E W, meets W transversally and at y only, it is clear that if x E 7r - ( y ) is sufficiently near y, then 71- - ( y ) meets the leaf of a through x transversally at x. We can then obtain a neighborhood U -a/ such that for all y E U fl F, r ( y ) meets g/ U transversally. •

Now let -y : 1 —0- F be a continuous curve with -y ( 0) --. po and -y ( 1 ) = p 1 . Let U be a neighborhood of -y ( I) in M, W a neighborhood of -y (I) in F and r : U Was in Lemma 2. Given x E r ( po ) sufficiently near po , there is a unique path -T : I —0 M, contained in the leaf Fi of g which passes through x such that 5; (0) = x and ir ("it ( t) ) = -y (t) for all t E I. The path -1 is a lift of -y to the leaf Fi along the fibers of 7r. The map f r (x) = -T ( 1 ) is well-defined for x near po and coincides with the holonomy map associated to y (see figure 4).


f(x)

Figure 4

52. Determination of the germ of a foliation in a neighborhood of a leaf by the holonomy of the leaf

Let a be a foliation of a manifold M, F a leaf of a and Po a point of F as above. The holonomy of F induces a local action of 7r 1 ( F,p0 ) on Lo defined as follows: to each IT] E ir 1 ( F, p0 ) and x E E0 sufficiently near Po , we associate the point f. , ( x ) E E0 . When F is compact, this action characterizes the folia-tion in a neighborhood of F. More precisely, let (M, 3:), (M' ,a be folia-tions of codimension n and class Cr, r 1, and F C M, F' c M' be compact leaves. We say the holonomies of F and F' are Cs conjugate (s r) when there exist transverse sections to F and F', E0 and E I:;, po E E0 n F, p (!, E E6 fl F' and a homeomorphism h : F U E0 F' U E6 such that h ( po ) = /36, h I F and h 1E0 are Cs diffeomorphisms (a homeomorphism if s = 0) and for each ['y] E 71- 1 ( F, po ), one has

o fy h -1 (x') = fh.,(x' )

for every x ' E E6 sufficiently near 136.

Theorem 2. Let F and F' be compact leaves of g and V' respectively. The

holonomies of F and F' are C5 conjugate if and only if there exist neighbor-hoods V D F, V' D F' and a C5 diffeomorphism H: V V', H(F) = F', taking leaves of g / V to leaves of 5 '/ V' (homeomorphism ifs = O). In this

case we say g and V' are locally equivalent on F and F' and H is a local

equivalence.

Proof. Suppose first that there exists the local equivalence H. Fix po E F and let E0 and (; = H(E0 ) be transverse sections passing through po and 1,6 = = H( po ). Let [ -y] E 7r 1 (F, po ) and C = (Ll1 ) ki=0 be a chain subordinated to -y. The open sets U,' = H(U1 ) define a chain C' = ( U,'),k=0 subordinated to

H o -y. Thus it follows that the holonomy transformations E0 D Wo E0


and E (Ç D Ej, satisfy H oo = fif.7 (x' ) if x' E E6 is sufficiently near pj,. Then the holonomies of F and F' are conjugate.

Suppose now that the holonomies of F and F' are conjugate. Since F and F' are compact, there exist, by Lemma 2, neighborhoods Vof F, V' of F' and retractions 7r : V F and 7r : V' F' with fibers transverse to 5 and 3:' respectively. Let h : F U Eo F' U E (; be a homeomorphism which conjugates the holonomies of F and F'. We can suppose without loss of generality that Eo ( Po ) and E6 = (ii' ) ( h Po)). Given p E F,

P Po, let -y : F be a curve joining po to p. Let f y : Dp0 C 7r -1 (Po) (p) and .1;; ( ..„ ) : D1(P0) ( 7r ' ) ( h( p)) be the holonomy transfor-

mations of -y and h . For x E f. (D,,) we define H(x) = A (7) (h(f),_,(x))) (see figure 5).

H

Figure 5

This definition is independent of the path -y chosen. In fact, if i : F is another path with tt ( 0 ) = po and pc ( 1 ) = p, we have by hypothesis that h -1 th( „, - 1) o h, consequently A(0 o h , = fl; (1) . h of , = H.

"e Now we see the domain of H. Given p E F let Up be a coordinate neighbor-

hood of 5 such that, for every y E Up fl F, (y ) meets all the plaques of U,, and up n F contains only one plaque. Fixing a curve -y which goes from po top, we can suppose that for every z E Up , the plaque of Up through z cuts ii ( p ) at a point z, in the domain of Let U; (p) , be a coordinate neighborhood of 5 such that every plaque of Uh' (p) , cuts ( ) -1 (h (p)) in a point of the domain off' (1,')• Reducing, if necessary, Up and U,; (p) we can also sup- h - pose that h(f_c1 (r -1 ( p) fl Up )) = f ,)(( 7r ' ) ( h( p)) (") U14 ) ) and h (Up n F) = u,; (p) fl F'. With these properties H is defined on Up and H( U) = UI; (p) . Since F is compact we can obtain a neighborhood of F, U = U U U9, , where H is defined.

From the construction, it is immediate that H : U H(U) = U' is a C5 diffeomorphism (or homeomorphism if s = 0) and that H takes leaves of 5/ U to leaves of 5 ' /U'.


§3. Global trivialization lemma

Denote by a a foliation of class Cr, r 1, and codimension n on a manifold

Lemma 3. Let 7 : I Al be a continuous, simple (injective) path whose image

is contained in a leaf F of 5. There is a neighborhood V D 7 (I) and a Cr

diffeomorphism

h : Dr" x V

such that h* if is a foliation whose leaves are the surfaces P-1 (y) where

P : Drm " x Dn Dri is the projection P(x,y) = y.

Proof. Let ( ( i ) ii =0 be a cover of 'y (I) by coordinate neighborhoods such that if Ui n then U1 (1 U a coordinate neighborhood of a. Let r : U = U i,`=0 U, W C F be a retraction as in Lemma 2. Since 7 has no self-intersections we can suppose that for each i, U, meets at most U,_ 1 and UN and U0 n uk = 0. Let Po = 7( 0 ), PI = 7( 1 ) and E0 = E l = ( p, ) . It is clear that each leaf of 5 / U cuts Eo at most once. The

f, holonomy map Eo D Do —0' E i is well-defined on a disk Do with po E Do . Let V be the saturation of Do by leaves of 5/ U k, =0 U, and f: V Do the map which to each x E V associates the intersection of the leaf a/ v passing through x with Do . Since a is , f is also Cr. One easily verifies that f is a submersion.

Let A c F be the leaf of a/ V containing 7. It is clear that by reducing A if necessary we can suppose that A is homeomorphic to the disk Dr" - " by a diffeomorphism kl : Dr" -- " A. Let k2 : Do be a diffeomorphism such that k2 (0) = po . If if is Cr we can take k i and k2 as Cr diffeomorphisms. Define h : D"'" x —• V letting h(x,y) = 7r -1 (k 1 (x)) fl f -1 (k2 (y)). Then h is a homeomorphism which takes the surface P -1 (y) = Drtm - " x y E D", in the leaf (k2 (y)) of V. If if is Cr, r 1, it is easy to see that h is a Cr diffeomorphism, whose inverse is given by h ( p) = ( k 1 (7r (p)),

k2 l (f(P))). When cod (if) = 1, this result can be improved, as is shown in the follow-

ing lemma which is used in later chapters.

Lemma 4. Let F be a leaf of a transversally oriented foliation on M of codimen-

sion 1, and fo : K F a continuous map homoto pic to a constant, K com-

pact and path-connected. There is a continuous family of maps f : K M,

t E 1 = [0,11, such that f, (K) is contained in a leaf F,. For x E K fixed, the curve t f,. (X) is normal to


Proof. Let 01 be a normal vector field to 5 and denote by DI, (x) the parametrized orbit of x. We can suppose that f is homotopic to the constant map C : K F, C(x) = yo where yo = f0 (x0 ) for xo E K. Given a path

: I K,7(0) = xo , ( 1 ) = x, the path 7y- : I F, f0 o 'y induces a holonomy map

J D V J YO Y Yo E V, Y = fo(X),

where Jy denotes an integral segment of M. For t sufficiently small, define f1 (x) = DI, (y0 )). This definition is independent of 7. In fact, let

:l —wKbe another path, ti ( 0 ) = xo , ( 1) = x and H : K F the homotopy between fo and C, that is, H(x,0) = fo (x), H(x,1) = yo for all x E K. Then the map 1-71 : Ix F, = H(-y p -1 (s),u) is a homotopy between the path (s) and the constant path E'(s) = yo . So '.1% = and the germs of g.7, and g4 at yo coincide.

By the compactness of K, there is an interval [0,E] such that f,(x),t E [0,E] is well-defined for all x E K. Then it suffices to parametrize the integral curve JYo in order to obtain the same for all t E [0,1].

§4. The local stability theorem ([10],[461)

Theorem 3. Let be a c" codimension n foliation of a manifold M and F a

compact leaf with finite holonomy group. There exists a neighborhood U of F, saturated in a, in which all the leaves are compact with finite holonomy groups.

Further, we can define a retraction 7r : U F such that, for every leaf

F' C U, : F' F is a covering with a finite number of sheets and,

for each y E F, 7r -1 (y) is honteomorphic to a disk of dimension n and is

tranverse to . The neighborhood U can be taken to be arbitrarily small.

Proof. Since F is compact, there is a covering ( U, ) of F by coordinate neighborhoods, where Û is compact, and diminishing the U t 's if necessary we can by Lemma 2 define a retraction 7r : U /:=0 U, F of class C' such that for any x E F, (x) is transverse to 5 / U /,`=0 U,. In each U, consider D, = 71- -1 (x,) where x, E U, (1 F. By hypothesis there exist closed paths 'y1 : I = eyj ( 1 ) = xo , j = 1 ,..., ni, and corresponding holonomy

maps I, : C Do —• Do such that Hol ( F,x0 ) = 1f0 = l,f1 ,..., f,. Con- sequently, if W = n , the domains of all the maps contain W. Set V= U 0 U,.

We will now show that there exists W0 C W, a neighborhood of xo , such that, if y E W0 , then the leaf ,Fy of the restriction 5 I V which passes through y does not meet the boundary of V. This will imply in particular that if Fy is a leaf of 5 through y then Fi, = fy . Moreover, if 5 ( WO is the saturation of Wo in 5 then 5 ( W0 ) c V. Observe first that if y E W then f.„ ("1


C Ifi (y) J = O ,..., m . In fact, if there is a z E fry n D0 such that z (y), j = m, there exists a curve rif" : I — • P. such that (0) =

= y, ( 1 ) = z and (I) c V. Since 7r (y) = 7r (z) = x0 , the curve 'y =

= 71 0 .:y" induces an element f E Hol ( F, x0 ) such that j, (y) = z j m, which is absurd.

The global trivialization lemma shows that for all i = 0 k there is a neighborhood W, of x0 in D0 such that, for all y E W , fry (1 D, contains at most m + 1 points. Let 1/17, w n w, n n Wk c D, and for y E Wc;, fry n Di = {qO, j,y) ! 1 j 5_ Pi } where fi 5_ m + 1. Let P( 1, j'y) be the plaque of Ui through q(i, j,y).

Let d be a metric on M. We have d(aV,F) = b > O. Let b,(y)

= max i , i d(xi ,P(i,j,y)), for y E 1,Vc; and 1 i k.

For i = 0 we have

lim max d(x0 ,q(0,j,y)) 5_ lim max d(x0 ,fi (y)) = O. y.x 0 1 y--.x 0 05.j5m

Then lim y (50 (y ) = O. Analogously lim y S, (y) = 0 for 1 k. XO

We conclude therefore that there is a neighborhood W0 C 141(; of x0 , such that for all y E W0 and i = O ,..., k,S i (y) 5_ b/2. We have then that for all y E W0 and i = 0 k the plaques of fry n u, do not meet av, SO Fy C V. The above argument also shows that for all y E W0 the leaf Fy = ry is compact. In fact, for all y E Wo , F1, is covered by a finite number of plaques P(i,j,y),

0 5_ 15k,15_j5_ L, m+1; so F. is compact. In particular the saturation U = i (W0 ) c V is a neighborhood of F and 7r = 71 U: U F is a retraction such that 7r F' :F' F is a covering map with a finite number of sheets for each leaf F' C U.

We have yet to prove that the holonomy group of the leaves F' C U is finite. It suffices to note that for all j = O ,..., ni, w, w, is defined and therefore Ho! ( F' ) since every closed curve in F' is the lift of a closed curve in F.

Corollary. Let g be a C', codimension n foliation of a manifold M and F a

compact leaf with finite fundamental group. Then, there exists a neighborhood

U of F, saturated by , in which all the leaves are compact with finite fundamen-

tal group.

Proof. First, F has finite holonomy. So it follows from Theorem 3 that there exists a neighborhood U of F saturated by 'S such that if F' C U is a leaf then 71 F' : F' F is a covering. Since (71 F').: 7r (F' ) 7r, F) is injec-tive, we have that 7r 1 ( F ' ) is finite.


§5. Global stability theorem. Transversely orientable case

In this section 5 will denote a codimension one, transversely orientable, CT ( r 1) foliation on a compact connected manifold M.

Theorem 4 (global stability). Suppose TF has a compact leaf F with finite fun-damental group. Then all the leaves are diffeomorphic to F. Moreover there is a C r submersion f: M S' such that the leaves of are the sets f -1 (0), 0 E S I

Since F has finite holonomy group, we begin by studying the properties of these groups.

Lemma 5. Let Horn ( IR , 0 ) be the group of germs at 0 E JR of hcrmeomorphisms which leave 0 E JR fixed. Let G be a finite subgroup of Horn ( , 0 ) . Then G has at most two elements. If G has two elements, then one of them reverses orienta-tion and the other is the identity.

Proof. Let f E G. We claim that f 2 = 1, the identity of Hom ( ,0 ). Con-sider a representation of f, which we will denote by f also. We have then that f: (a,b) (c,d) where a < 0 < b, c < 0 < d and f(0) = O.

Suppose initially that f (0 ,b) C (O, + ). If there exists a y E (0, b) such that f(y) < y, we have that for all k > 0, 0 < f k (y ) < f" (y) < < f(y) < y so the points of the sequence ff k (y) k E NI are distinct points and therefore f k f if k j. This contradicts the hypothesis of G being finite. Analogously we cannot have f (y) > y. We conclude therefore that f(y) = y for all y E ( 0, b ) . By an analogous argument f(y) = y if y E (a, 0) and therefore f = I. In the case that f(0,b) C ( co,0), that is, f reverses orientation, we have f2 ( 0, b) C (0, +cc), so f2 = 1. We con-clude therefore that every element of G has order 2. In particular G is abelian.

Suppose there exists f,g E G such that f,g 1. Let ( a, b) be the intersection of the domains of f and g with the image of g. Since f,g 1, we have g(0,b) C (a,0) and f g(0,b) C (0, +co); so fog= I and hence f = =

g'. Since g 2 = 1, we have f = g = g, as desired.

Corollary. Let F be a compact leaf of a codimension 1 foliation 06 such that r ( F ) is finite. Then Hol ( F,x0 ), xo E F, contains at most two elements. If is transversely orientable, then Hol ( F, xo ) = 11j.

Proof. By Lemma 5, it suffices to prove that Ho! ( F,x0 ) = f 11, in the case that 6 is transversely orientable. In this case, there is a vector field X on M

that is nonzero and transverse to 6. Given x E F, take as a transverse section to 6 through x a small segment E x of the orbit of X through x. So, if


f: E C Ex„ Ex° is an element of Hol ( F,x0 ), we have f(x0 ) = x0 and by the above lemma, f = 1 if and only if f preserves the orientation of E xo , that is, if and only if Df (x0 ) - X (x0 ) = X, X(x0 ) where X, > 0 (in fact X, = 1). Let 'y : [0,1] F be a closed curve such that f is the element of Hol ( F,x0 ) corresponding to [7]. For each t E [0,11, there is a holonomy map ft :E; C Eo E,, ( ,) and we have Dft (x0 ) X(x 0 ) X(t)X(7 (I)) where X: [0,1] 1R is a continuous nonvanishing function. Since X (0 ) = 1, we have X, = X ( 1 ) > 0 as desired.

Lemma 6. Let 5 be a codimension one, CT, (r 1) foliation. Suppose F is

a compact leaf of such that # (Hol ( F,x0 ) ) = 1. Then there exists an open neighborhood V( F) of F in M, saturated by 65, and a Cr diffeomorphism h : ( — 1,1 )xF--• V(F) such that the leaves of in V( F) are the sets of the type h ([t} x F), t E ( - 1,1 ), letting F = h( 10] x F). In particular V( F) — F has two connected components.

Proof. By the local stability theorem there is a neighborhood V = V (F) of F, saturated by 0, and a retraction 7r : V F such that, for every x E F,

(X) is a segment transverse to 0. Moreover if F' C V is a leaf of 0 then 1 #(F' n 71- -1 (x)) #(Hol(F,x0 )) = 1, so #(F' f") r' (x)) = 1. Consider a Cr. parametrization a : ( —1,1) —• 7r -- I (x0 ) such that a (0) = x0 and a ' ( t) 0 for all t E ( - 1,1 ). Denote by F, the leaf of 0 in V which cuts r (x0 ) at a (t). The map t 1—■ F, is evidently one-to-one. Consider now the map g: V ( — 1,1) x F defined by g(y) = (g,(y),r(y)) where g, (y) is the unique element of ( — 1,1 ) such that y E Fg1 (y)0 If F, is as above, we have g ( F,) = id X F, because 7r (F,) = F. Further, g is one-to-one because if g(y) = g(y') we have t = g (y) = ,g, (y ' ) so y and y ' are in the same leaf F, of 0 and since r F,: F, F is a diffeomorphism, we have y = y' because r (y ) = r (y ' ) . Let h : ( — 1,1 ) X F V be the inverse of g. We have then that h(t,x) is the unique point of F, fl 7r - I (x). One easily verifies that h and g are Cr and hence are diffeomorphisms. •

Corollary. Let F be a leaf of 0 such that # ( Hol ( F, x0 )) = 1 and let V(F) be a neighborhood of F as in Lemma 6. If e is a segment transverse to 0 such

that C V( F) then for every leaf F' C V( F) we have # (e fl F') 1. If the

endpoints of C are in a v then # (f fl F' ) = 1 for every leaf F' C V( F).

Proof. This follows immediately from the preceding lemma.

Proof of Theorem 4. Let U be the set of points x in M such that the leaf F, of 'S through x is compact and has finite fundamental group. By hypothesis U 0 and by Lemma 6, U is open. Let -0 be a connected component of U.

We will show that a0 = 0 and, since M is connected, we will have r./ = = U = M.


Since U is saturated by g , C/ and ari are also saturated by F. Suppose by contradiction that 0. We prove first that all leaves contained in a0 are compact.

In fact, suppose by contradiction that I' C U is a non compact leaf. Since M is compact, I' has an accumulation point p E M F. Let J be a compact segment transversal to g with p E J. Clearly f n J 0 and since p f, it follows that f n J is infinite. We can suppose that the extremities of J are p and a point of P. Moreover, since 1' C (30, from the transverse uniformity of a (Theorem 3, Chapter III), it follows that any transverse segment which intersects f, intersects U. In particular -0 n J = U nEB j , where for each n E B, J an open segment in J and J fl J„, = 0 if n tn. Let us prove that B is infinite. Since f is a segment we can order Jand consider a monotone sequence lx„1„ E p4 in f n J, such that lim n xn = p. For each n7 E B, clearly

C (xx t ) for some n E IN, where (xn ,xn+ , ) is the segment of J between xn and xn , . Now, ( x,,x, ) n C./ 0, which implies that there exists m E B such that Jn, C (xn _ i ,x„,) . Since there is an infinite number of such segments, it follows that B is infinite.

Figure 6

Now we prove that for each n E B, sat ( ) = Cr. Clearly sat ( C C/ and

from Lemma 6, it follows that sat ( ) is open in O. Since CI is connected it is sufficient to prove that sat ( ./n ) is closed in U. Let p,1 014 be a sequence in sat ( Jr ) such that lim „ pn E U. Since Fp C CI, by Lemma 6, there exists a neighborhood V(Fp ) = V C U , saturatld by a and such that 5/ V is equivalent to a product foliation ( — 1,1 ) x FI!, . Since pi E V forj suf-ficiently big, it follows that ./„ n V 0. Now, the eX*tremeties of can not


be in V, therefore J„ intersects all the leaves of V, hence pco E sat (J„). This implies that sat (J,) = CI for each n E B.

Now, let F' c CI be a leaf of 5:. Since sat ( J„) = -0 it follows that for each n E B, F' n J„ 0. On the other hand, F' and J are compact and so F' (-1 J is finite, which implies that B is finite, a contradiction. Therefore f is com-pact. The same argument implies that for any compact segment I which is transverse to 5:, then I n 0. contains a finite number of open intervals.

Now, let us fix a tubular neighborhood W of f C 30, such that aw is com-pact, and a retraction r : W such that for any p E f, (p) is a seg-ment transverse to 5. Let us prove that there exists a leaf F' of g such that F' c W fl U. Suppose by contradiction that for any leaf F' C U such that F' n w 0 then F' fl aw 0. Let J = (x), where x E f is fixed. Since f c aU , x is accumulated by points of .1 n U and so J n CY contains a monotone sequence tx, „ Eri such that lim x„ = x. For each n E N, let Y,, E n aw. Since aw is compact, by considering a subsequence if necessary, we can suppose that lim„_. yn = y E aw. Set S„ = sat (x„,x) and S = n S,, where (x,,,x) is the segment of J between x,, and x. Since S„ is compact, connected, saturated and S„ 1 C S„ for each n E N, we get that S is also compact, connected and saturated. Moreover for all n E IN, 1x,„),,, E S„, which implies that tx,y1 E S. Let us consider the leaf Fy . Clearly Fy intersects the segment [x,, ,x) for all n 1 and since Fy Fx , this implies that Fy n [x, ,x] is infinite. On the other hand, y = lira n 0, y,, , yn E U , and so y E at 7, which implies that Fy is compact. Therefore F. fl [x1 ,x] is finite, a contradiction. It follows that there exists a leaf F' C 1.7 n w.

Now F' is compact and r /F 1 : F' P is a local diffeomorphism, which implies that r/F' : F' Pis a covering map. Since F' is compact and has finite fundamental group the same is true for f, hence f c t7 and a -O = as we wish.

We will now prove that all the leaves of 5 are diffeomorphic. Given F', a leaf of g, let

UF , = Ix E M I Fx is diffeomorphic to F' .

By Lemma 6, UF , is open in M for every F'. For the same reason M — UF ,

is also open in M. Since M is connected, we have M = UF , as desired. To conclude the proof we prove that there is a submersion f: M

such that the leaves of g are the surfaces f - (0), 0 E S'. For this it is enough to show that there exists a closed curve -y : M transverse to g and such that -y (S 1 ) meets each leaf of g in exactly one point. In fact, this being the case, it is enough to define f(p) = -y - 1 ( Fp n -y ( S i ) ) . The existence of -y

follows from the following two lemmas.

Lemma 7. Let if be a codimension one foliation defined on a compact 'manifold M. Then


(a) There is a curve 7 : M transverse to if, (b) If if is transversely orientable, -7 : -+ M is transverse to if and 7y- (S i )

(1 F0 0 where Fo is a compact leaf of if, then there is a : S' M transverse to if such that 7(S' ) fl F0 contains only one point.

Proof. (a) Suppose initially that if is transversely orientable. Let X be a vector field transverse to if. It suffices to get an integral curve of X which cuts a leaf F of if twice. In fact, suppose that a : IR M is an integral curve of X such that a ( t i ) and a ( t2 ) E F where t l < 12 . If a ( t l ) = a (12 ) the curve a is a closed path in X and we are done. If a ( t, ) p i p2 = a ( t2 ) let .5 : [0,1] F be a simple curve such that (0 ) = p i and +5 ( 1 ) = p2 . By the global trivialization lemma there is a neighborhood V of 6, (I) and a dif-feomorphism h : x ( - E,€) V such that h* (5) = 5* is the foliation whose leaves are of the form Dm x tj, t E --e,E) where Pi ,p2 E h (Dm-I x [0 1 ) C F. Let X* = h* (X) be the vector field X* (q) = = (Dh(q)) -1 (X (h(q)). This vector field is transverse to 5* so we can assume that its last component in D"1- x ( - f,E) is positive. Let S I and 02 be the segments of a [t, ,t2 ] (1 V which contain pl and p2 respectively. Since t, < 12 we have that h -1 (S I ) C D' - ' x [0,E) and h - ' (02 ) C Dm -1 x

( - E, 0 1 . It suffices now to define 'y: [t, ,12 ] M by modifying the curve i ,t2] as in the figure below.

Figure 7

In the figure, a is modified in intervals of the form [t i ,t, + 6] and [t2 - 0 ,12 1 so that 7 [ t i + 0 ,12 - (5], -y I [1 1 ,t, + 0] and -y [1 2 - (5,t2 ] are transverse to

-y(t i ) = -y(12 ) and 7' (t 1 ) (t2 ).

We prove now that given an integral curve a of X, not closed, then a meets


a certain leaf of if, an infinite number of times. Let a : JR M be such a

curve and p an accumulation point of the sequence la (n)I Let U be a

foliation box of if containing p. Then there exists a sequence nk co such

that a ( nk ) E U. The segments of a (IR ) n U which contain a ( nk ) necessarily

meet the plaque of U which contains p for k sufficiently large. This proves our

claim. If if is not transversely orientable consider a line field L transverse to if and

take the double covering r : /17/ M of L. Since )17/ is compact the above case

applies and we obtain a curve ;y: : S' transverse to 7r* (5 ) . The curve

7r o = y is transverse to if.

Figure 8

(b) Suppose if is transversely orientable, 7y : S 1 M is transverse to if and 'V (S I ) fl F0 0 where Fo is compact. Then 5% ( S' ) fl Fo is finite. In the case that ;y" ( S' ) n F0 contains more than one point we choose 0 1 ,02 E S I such that 51' ( 0 I ( 0, ) E F0 and T(0 1 ,02 ) n Fo 0 where ( 0 1 ,02 ) is one of the segments of S' [0 1 ,02 ]. One can now repeat the first argument, since if is transversely orientable. •

Lemma 8. Let if be a transversely orientable, codimension one foliation of a compact, connected manifold M. Let : S' M be a closed curve trans-

verse to if such that every leaf in sat ( y (S I )) is compact and has finite fun-damental group. Then sat( 7 (S I )) = M. Moreover, the number of points of intersection of the leaves of if with 1, (S') is constant.

Proof. We will show that W = sat ( ( S' ) ) is open and closed in M and hence W = M.

W is open. Given x E W, F, fl -y (S I ) 0. If y E F n -y (S I ) we consider a simple curve pt : [0, 1 ] F„ such that kc ( 0 ) = x and ti ( 1 ) y. Using the


global trivialization lemma it is easy to show that there is a neighborhood A

of p, [0 , 1] such that for every z E A , F fl -y ( S i ) 0 as desired.

W is closed. Given x E -y ( S 1 ), by Lemma 6 there exist a neighborhood V(Fr ) of Fx , saturated by if and a diffeomorphism h: ( - 1,1) x V(Fr ) such that the leaves of if in V( Fr ) are the sets of the form h (It) x Fr ), t E ( — 1,1). Denote by V, the compact neighborhood of F„, h([— 1/ 2,1 / 2] x Fr). It is clear that 17, fl y ( S' ) contains an open interval which contains

x, which we will call I. We have then U I = -y S i ) and since -y (S I ) is com-pact there is a finite cover of y ( S' ) by intervals 14,1 1; =1 . It is easily verified that W is the finite union K. U U l' compact sets, so it is compact hence closed, from which it follows that W = M.

Now let k(x) = #(F, fl -y (S I )) , X E (S I ). We want to prove that k(x) is constant. For this we consider the set = ix E (S i ) I# (7 (S i ) n Fx) = = j). We claim that -yi is open in -y ( S' ) for every j E N. In fact, given x E let V( F) be a neighborhood as in Lemma 6. Then 7 (S I ) n V( F) has a finite number of components fl f,. For all i = 1 r the endpoints of 4 are contained in V( F) so by the corollary of Lemma 6, # F) = 1 for every leaf F C V(Fr ). In particular # (P, r) Fr) = 1, i = 1 r so r = j. Moreover for every leaf F c V( Fr), # (e(V) fl F) = j so 7; is open in 7 (S 1 ). We have then -y (S i ) = U , ,y; so (S 1 ) = yk for some k E N, since 7 ( S' ) is connected.

§6. Global stability theorem. General case

Theorem 5. Let if be a C', codimension one foliation of a compact connected manifold M and F a compact leaf of a with finite fundamental group. Then all the leaves are compact with finite fundamental group.

Proof. By Theorem 4 we can suppose if is not transversely orientable. Let P be a line field transverse to the plane field of codimension one which is tangent to the leaves of if. Let 7r : /t7/ M be the orientable double covering of P.

Since P is not orientable, if/i is connected. Consider on /17/ the foliation 5* = w* ( 5 ) . Since Ir takes leaves of if to leaves of a, 5* has two types of leaves:

(1) leaves F* such that w F* : P -n- ( F*) is a diffeomorphism and (2) leaves F* such that r F* : F* ir ( F*) is a two-sheeted covering.

In either case, the leaf r ( P) is compact and has finite fundamental group. III

Example 2. We will now construct a foliation if on a three-dimensional manifold which has two leaves, Fl and F2, diffeomorphic to the projective plane. The other leaves of if are diffeomorphic to S2.


First consider A-1 = S2 x [I ,21, foliated trivially by S 2 x 10, t E [1,2].

2 x2

Figure 9

On Mconsider the equivalence relation — such that ( x,1) — (x' ,t' ) if and only if x = x' , t = t' or x = — x ' when t = t' = 1 or when t = t' =2. Taking M = 1171 / — , we have that S 2 x [11 and S2 x 121 are transformed to two projective planes F1 and F2. The other leaves S 2 x 1t1 are taken to spheres; with this we get a foliation 5 on M with the desired properties.

Example 3 ([46]). We will see here that the global stability theorem of Reeb does not generalize to foliations of codimension greater than one.

Let Mn = Sn -2 x S i x S' and designate a point p E Ar by p = (x, O) where x = (x, ,..., xn _, ) E IRn -1 , 4 +...+ xn2 _, = 1, and ço and 0 are angular coordinates defined modulo 2w.

The differential forms

77 1 = c10, 71 2 = ( ( 1 - sin0) 2 + .)d)do + sin0 dx,

are linearly independent at every point and define on Mn a plane field r of codimension two by the following equation:

r(p) = Iv E 7p M1 (v) = 71 2 (v) =-- O.

Since dn, A in A n, . d71 2 A n i A 77 2 = 0, the plane field r is completely in-tegrable, that is, it is tangent to a foliation 5 of codimension two on M" (see the appendix).

It is easy to verify that the surfaces given by sin 0 = 0 and ço = constant are leaves of this foliation, homeomorphic to Sn - 2 , and consequently simply connected if n > 3. Nevertheless the surfaces defined by sin 0 = 1 and it, = = wo + 1 /x, are non-compact leaves of V.


Remark 3. Reeb's stability theorem was generalized by W. Thurston [57] in the following way. Let F be a compact leaf of a transversely oriented codimen-sion one, C' foliation g . If ( F,1R ) = 0 then the holonomy of 5 is trivial. Applying Lemma 6 we can conclude that g I V is a product foliation where V

is a small neighborhood of F.

Remark 4. Reeb's stability theorems (§ 4 , 5,6) hold even when the foliation is C° . In the C' case the proof simplifies and illustrates better the techniques of the theory of foliations. For these reasons we restricted ourselves to the C' case.

Notes to Chapter IV

(1) Foliations induced by closed 1-forms.

Every Cr, closed 1-form w (r 1) defined on a manifold M induces a codimension

one, Cr foliation 5 ( ) on M – sing ( ) where sing ( ) = fp E M wr = 0). A system of distinguished maps for 5 ( w ) can be defined in the following manner.

Let Po E M – sing ( ) . Consider a neighborhood D of Po , homeomorphic to a disk and suet that wp o 0 for every p E D. Since D is simply connected, by Poincaré's lemma (see [52]), there is a Cr function f : D —* 1R such that df = w. This function is a submersion, since df( p) = wi,, 0 0 for every p E D. One easily verifies that the set of all pairs (D,f), constructed as above, constitutes a system of distinguished maps and hence a foliation 5 (w).

One particularly interesting fact about such foliations is that the holonomy of any leaf is trivial. This follows from Stokes' theorem, as we will see in the following. Although the result also holds for C I forms, we are going to suppose that co is C2 , in order to simplify the argument. Let F be a leaf of 5 ( w ) and y : S I F be a closed curve in F. We can suppose that -y is a C2 immersion. By Remark 2 of this chapter the holonomy of -y can be defined using a one-dimensional fibration along -y and transverse to 5 (w).

More precisely, there is a C2 immersion h:S I x (– E, E) —* A/ such that h ( 0,0) = -y ( 0 ) for every 0 ES' and the curve t h(0,t) is transverse to 5 ( ) . The inter-

section of the leaves of 5 ( w ) near F with h (S' x ( – E, E)) define the holonomy of y. The map h can be defined, for example, by taking h (0,t) = X (y (0)) where X is the C2 vector field defined by wp ( y) = <X( p), t;) p for every p E M – sing (w) and

E TrM, letting ( , ) be a Riemannian metric on M. This vector field X is also called the gradient of w with respect to < , ) and one easily verifies that it is normal to the leaves of 5 (co ).

Let w* = h* (6)) be the form co-induced by co and h (see [52]), defined by atom (y) = = Wh (8 a ) ( dh om • y) . Then w' is a closed C I 1-form on S I x ( – , ) and the foliation defined by w', 3:', is exactly the foliation h* (5 (w)) . One sees also that S I x [01 is a leaf of 5* and that 5* is transverse to the lines f01 x ( – E, €), O E . Fix a transverse line fo = 100 1 x ( – E, E ) and let g : ty, C Pp eo be a representative of the holonomy of S I x 10j. Then g is conjugate to the holonomy map of 7, f E Hol ( F). This follows from the fact that 5' = (5 ( 0))) . It suffices to show that g = identity.


Suppose on the contrary that there is a q = (00 1 t) E f6 such that g(q) = (00 ,1 1 ) = = qi q. In this case the leaf L q of if which passes through q meets fo for the first time at q 1 . Let a be the simple closed curve of S I x E) formed by the segments qq, of eo and q i q of Lq as in Figure 9.

Figure 10

The curves cr and S I x (01 define a region RCS x( --€,€). Applying Stokes' theorem to the form 0.7* in the region R. we have

co = cico* = 0 x lolua *

and hence Ç = s i (.0*. Since S I x f01 is an integral curve of co*, we have

\ 0). = o . „

Therefore

0 = = a)* + qq, t l tr i

Since el is transverse to g, it is easy to see that 0.)* 0, vA.lich is a contradiction. Hence we must have q = q l , i.e., the holonomy of is trivial. The following converse of the above-mentioned fact was proved by Sacksteder in 1501.

Theorem. Let if be a C 2 codimension one foliation defined on a manzfold M. Suppose the holonomy of every lea( of S is trivial. Then S is topologically equivalent to a foliation defined by a closed 1-form. We say that two foliations S and if' of M are topologically equivalent when there is a homeomorphism of M which takes leaves of S to leaves of S ' .


(2) Singular points of completely integrable 1-forms.

A differentiable 1-form co on IR" of class Cr is written

• E a (x)dx,

where the a,(x) are Cr. functions. The form ce is called completely integrable when w A dw = 0 (see the appendix).

A singularity of co is a point p where all the a,(x) vanish simultaneously. In what follows we will show that in certain cases it is possible to determine the topological struc-ture of the leaves of co near a singularity using the stability theorems of Reeb.

For simplicity we suppose a, (0) = 0 for i = 1 n. Let bu = (8a) ( )/ (axj ), 1.-5 i,j 5_ n. Then we can write

R i (x) ai (x) = E R,(x), urn

Then,

E bu xi dx, + R, R = E R,(x)dx, I.) = 1 = 1

The form co l = i.j , t bij xi dxi is called the linear part of w at O.

Lemma. The form w i is completely integrable.

Proof. From the relation co A du) = 0 we have that

0=condco.w i ncko l +w i AdR +RAdw l +RAdR.

From the above expression we see that co t A dw i is the linear part of w A do.). So co t A dco t = O. •

Proposition. Let w be C I and completely integrable and 0 E a singularity of w. We have two possibilities:

(1) dw = O. hi this case B = (b,» 1< ,, j„ is symmetric.

(2) dco x , 0 O. In this case B has rank 2.

Proof. Since dR = 0, one has that &el x=0 = dwi . So, dch., , =0 ( bii — bu )dxi A dXj . Clearly, if cho I x=o = 0 then B is symmetric.

Suppose now that dw i O. We claim that dw i = 01 A 0 2 where 0 1 and 0, are linearly independent 1-forms which do not vary with the point (i.e. 0i ,., = 00 , x E R n ). In fact, set du), =E i< , cu dx, A dxj , bu . Since ch.! ' 0, we can suppose that c 12

0, for example. Set 0 1 = 1 /c12 E rf= c li dxj and 02 = E jn=,c2dx»c1, = c22 = 0 ). Taking the product we get

1=1 X - - • o

EC12 ( CI, C2 .1 - Ci i ) dx, A dXj .

t<,/

(1) 01 A 02


On the other hand the relation w i A dw = 0 implies that dco l A dw1 = O. The coefficient

of dxi A dX) A d.,VA. A dx,, in dul l A do.' is 2( cu cke + cki cie + c)k cif ) so taking k = 1 and

P = 2 we get

Cl2Cii C11Ci2 Cj1 C,2 = O.

Comparing (ii) with (i), we see that e i A 02 = dw i , which proves the claim.

Take now fi (x) = —1 E and f2 ( x) = E c x

C12 j=1

We have then that (L i = df1 A df2 SO d (w i - f 1 df2 ) = O. By Poincares lemma ([52]), there is a g : 1Fi n 111 of degree 2 such that w i = f df2 + dg. Take a linear change

of variable y = A (x) defined by

yi = (X), _)I2 f2 (X) , yk = xk for k 3.

In this system of coordinates w i is written as A .(w i ) = y1 dy2 + dh, h = g o A -1 ,A. =

(A I )* . Using now the relation w i t dco i = 0 we get dy i A dy2 A dh = 0, a relation that is

equivalent to ah/ayk = 0 for k 3. Therefore h depends only on y i and y2 , i.e.,

A (w i ) = —ah

dy i + (Yi + ) dy2 = (e11 y 1 + e12 y2 ) dy + (e21 y 1 + e22 y2 ) dy 2 ay ]

Then the matrix B of A (w ) has rank 2. Thus it is possible to deduce that B has rank 2. We leave the final details to the reader. •

Examples

2 2 x32 (1) Let co = xi dx i + x2 dx2 - x3 dx3 = df where f = 1/2( 4 + x2 - 4 ). The leaves of 6.) are the connected components of the level surfaces of f 111 3 - 101

and are given in figure 11 below.

Figure 11


(2) The form 77 = x2 dx, — 2x 1 dx2 in R 3 satisfies II A d = O. Further dn —3dx, A dX2 # O. The set of singularities of n is the x3-axis. The leaves of n are

products of the x3-axis with the orbits of the vector field X = 2x 1 (i9/3x 1 ) + x2 (a/ax2 ) (see figure 12).

Figure 12

Study of centers in IR", n 3.

Let LLi be a completely integrable 1-form in JR' with a singularity at O. Let Le i =

= E ki xi dx, be the linear part of Lc at O.

Definition. We say that 0 is a simple or nondegenerate singularity for Lo when

det ( ) O. This definition does not depend on the coordinate system chosen. Further, the simple

singularities are isolated. By the previous proposition if n a- 3 and 0 E 1R n is a simple singularity then neces-

sarily du) I x=0 = dw i = O. Consequently, Le i = df where f is a quadratic function. By means of a linear change of variable we can suppose f is written as the sum of squares. Thus

Definition. The singularity 0 E IR n of Le is called a center when it is positive definite

or negative definite, that is, when w i is written Le i = ± d( + + ) in some linear system of coordinates.

In this case the leaves of w i are the spheres x + x ( 2 . Our objective here is to show that if L,J is C 2 then its leaves are also diffeomorphic to spheres.

Theorem (Reeb [46 ] ). Let w be a completely integrable. C 2 form defined in a neighbor-

hood of O E 1R'1 , n 3. Suppose that 0 is a center ofw. Thell there is a neighborhood of

0 in which all the nonsingular lea ces of w are diffeomorphic to the sphere S" .


Proof. Since 0 is a center of w, by a linear change of coordinates we can suppose that w i = df, f( x) = 1/2 E 7.. 4-

Consider the map 1,t : 1R x Sn -1 defined by (p, u) = p • u where u = = ( u l ) E S'21 and p E IR.

Observe that the restrictions ik I (0,00 ) x - 1 and %,/, ( — 00,0) x - 1 are diffeomorphisms on le — 101. Therefore if F is a leaf of ik* ( w ) contained in (O, c0 ) x r -1 , (F) will be a leaf of w in IR" — 101. We will now prove that all the leaves of

* ( w ) in a neighborhood of 101 x Sn-1 are diffeomorphic to S n- 1 and this implies that all the nonsingular leaves of w in a neighborhood of 0 E R" are diffeomorphic to S r' 1 . To do this, set

17

(J) = df + E R i (x)dx i , dR 1 (0) = 0, R 1 (0) = 0 .

We have then

(.)

(w) = d(f 04) + E R,(p • u)d(p • u,) .

7=1 2i Since f o ‘li(p u) = 1/2 p2 u 1 /- p2, one sees that d(f o %,1,) = pdp. As R . of class C2 , R. (0) = 0 and dR, (0) = 0 one obtains, R i (p • u) = pSi (p,u) where s, is C I on JR x Sm- 1 and lim, 0 Si ( p,u ) = O. From (*) above we have

(W) P (dP 71), = E u,S,)dp + p E Si du, . i=I 1=1

Consequently the form (7) defined on IR x S n -1 by "6") = 1/p i,t* ( w ) for p 0 and 7.,; = dp

for p = Ois C 1 , integrable and has the same integral surfaces as Vi* ( w) on (JR — 101) X S"

1 . Observe now that 01 x Sn _i is an integral surface of, so, since n 3, by Reeb's

stability theorem, there is a neighborhood V of [0] x -1 such that all the integral sur-

faces of(.7) in V are diffeomorphic to S" - 1 . This concludes the proof. • In the case that w is analytic, the above theorem can be strengthened.

Theorem (Reeb). Let w be an analytic, integrable 1-form defined in a neighborhood of

0 E 111 n , n a 3. Suppose that w o = 0 and that w has nondegenerate linear part w ) =

= df. Then there exists an analytic diffeamorphisrn h from a neighborhood U of 0 to a neighborhood V of0 and an analytic function g : U11 1 such that g(0) = 1, Dh (0)

is the identity and h* (w) = gdf.

The proof of this theorem can be found in 146].

Other results about the existence of first integrals of integrable forms can be found in [37] and [31].

For the study of singularities of integrable forms we suggest the following references:

1 331, [30] and [5].

V. FIBER BUNDLES AND FOLIATIONS

The notion of holonomy of a leaf introduced in the previous chapter is essen-tially of local character. It is defined by a group of germs of diffeomorphisms of

a transverse section to a leaf, with a fixed point. In certain circumstances, however, it is possible to associate to the foliation a group of diffeomorphisms of a global transverse section, containing in a certain well-defined sense the holonomy of each leaf. This is the case of foliations whose leaves meet transversely all the fibers of a fiber bundle E. The importance of these folia-tions is in the fact that they are characterized by their holonomy, in this case given by a representation yo :ir 1 ( B ) Diff ( F) of the fundamental group of the base of E to the group of diffeomorphisms of the fiber of E. In this man-ner properties of so translate to properties of the foliation. For example, the action 40 has exceptional minimal sets if and only if the same occurs for the foliation. Sacksteder's example, of a C codimension one foliation with an ex-ceptional minimal set, is a typical case of what we will see in this chapter.

A necessary and sufficient condition for the existence of a foliation transverse to the fibers of a fiber bundle is that it has discrete structure group. Once show-ing this, We obtain the following result due to Ehresmann [111: let Nbe a com-pact C 2 submanifold of M. A necessary and sufficient condition in order to exist a foliation of a neighborhood of N for which N is one of the leaves is that the normal bundle to N be equivalent to one that has discrete structure group.

As an elementary application of this theory we will prove a classic result about the solutions of the differential equation of Ricatti (see note 1).

51. Fiber bundles

A fiber bundle consists of differentiable manifolds E,B,F and a differentiable map 7r : E —• B. Further E has a local product structure defined by a cover

(U, n x F 43y

(U1 flUi ) x F


( U, ) 1E ,, of B and by diffeomorphisms : 7r - (U ) U, x F, i E J, which make diagrams of the type below commutative

7r - 1 ( U, )

ça ` x F

that is, r = p o ço„ where p is the projection on the first factor. In particular r is a submersion on B and E is locally diffeomorphic to the product of an open set in B and F.

The space E is called the total space, B the base and F the fiber of the bundle. The map 7r is called the projection. If x E B, the submanifold Fx = 7r - '( x)

F is called the fiber of E over x. Usually a fiber bundle is denoted by (E,7r,B,F). If there is no danger of confusion we will denote it simply by E. Moreover we will assume that E is C', that is, all the manifolds and maps in-volved in the definition of a fiber bundle are assumed to be C' .

Given i, j E J such that U, n u, 0, we can define gy : U, r") U./ Diff ( F) in such a way that the map cky , which makes the following diagram commute,

is written 4'„, ( x,y ) = ( x, ( x,y ) ) . For each x E U, n Up the map : F F defined by y —0- h u ( x,y ) is clearly a diffeomorphism. We will use

the notation g,1 (x) = h.

When F and the fibers are vector spaces and all the gy ( x ) are linear automorphisms of F, we say that E is a vector bundle. We say that the fiber bundle E has discrete structure group if, for any i and j, the map x is locally constant.

When E has discrete structure group the foliations of 7r -1 (U,) given by submersions

7r (U ,) U, x F F, q(x,y) =y

define distinguished maps of a foliation whose leaves are transverse to the fibers of E. The plaques of the foliation in ir --1 (U,) are the sets (q o so,) -1 (y), y E F.

Fiber Bundles and Foliations 89

Example 1. The tangent bundle.

Let M be a Coe, n-dimensional differentiable manifold. For each p E M, let Tp M be the space of tangent vectors to M at p, and

TM = f (p,v p ) p E M, vp E Tp /V/1 .

If 7r: TM M is given by 7r ( p, v p ) = p then ( TM, 7r,M,111' ) is a Coe vector bundle. In fact, given a local chart x: U Dr of M, introduce a local chart

(s) : (U) U X 111"

for TM in the following manner:

a (p,x(p)) = Y(P, E ) = (Ace' an) = (P,Dx(P) X(P)) ax,

When x : U 112", y : V IR", u n v and X(p) = E al

aiax,. E 3/ay -1, we have that

° (13,a 1 an ) = (P03 1 On )

vl ay'

oi= Ld ak ax

This permits us to define on TM both a differentiable manifold structure by (*) and a vector bundle structure by (*) and (**).

A section of a fiber space is a map o: B E such that r o a = identity. A section of the tangent bundle is a vector field on M. So, a vector field X,

is, in effect, a map which to each p E M associates a vector X ( p) E Tp M.

Example 2. The normal bundle.

Let < , > be a Riemannian metric on Al m and N n C M in a submanifold of M.

Given p E M, let 7, N 1 C Tp M be the subspace of normal vectors to Tp N; we define 1) (V) = ( p, v p ) p E N, v p E Tp l\i ± ) and

7r : v(N) N, 7r(p,vp ) = p .

with

(.*) k= I

Then ( P(N), 7r, ) is a vector bundle.


In fact, since N is a submanifold, for each p E N there exists a local map X: U IR'" on M,pEU, such that qE Un N if and only if x q ) has its last ni – n coordinates zero. These coordinate functions x"' x"1 define vector fields X,, = grad x"' X„, = grad xm, which form a basis for Tq N'. The vector field grad f is defined by the relation Df(p) • u = < grad f(p),u> p .

So, we can define local charts Y : 7r -1 (U f") N) (U ("1 N) x 112 1" -n by

(q,X(q)) = (q, E am ) i=n+r

whose changes of coordinates are: if y : V IR"I is as above and u n then

(.) y ,..., am ) = (q,On+ , ,..., Om ) for q E U (1 V ,

where X(q) = E i=n oe, E 7=n+113jY» Y , = grad y l ,n+1<j<m

and

= i=n+1 ay,

Thus a vector bundle structure is determined on 1, (N) by (.).

Example 3. The unit tangent bundle.

Let < , > be a Riemannian metric on 1W" and 7.1 M = 1(p3 vp ) p E M, vp E Tp M,Iy p i =

Denote by 7r: T I M M the map 7r ( p,vp ) = p. Then T' M ,M,S n-1 ) is a fiber space whose fiber is the unit sphere s" -1 c IR" Tp M.

In fact, let x : U 112" be a local chart of M, where x = (xi x,,). Let

< —a >. The matrix G = (01 ., 1„ is positive definite and

ax, symmetric so it has a unique square root A x = (a01 ,,, j , (i.e., A which is symmetric and positive definite. We define local charts for TI M by setting

( : 7r -1 (U) u X sti '

(p,A,(Dx( p) v))•

If y = , u, 3/x, is a unit vector, we have A v (Dx(p) • y) = E,, av1 e1 -= = 1,,ai ei where 1 e i ,..., I is the canonical basis of IR", so the norm of A x (Dx(p) y) in the usual inner product is Ei cx j2 = id.k yia,‘Jdj!kyk


EI , k v i gvk = 1, or, A x (Dx(p) • y) E If y: V IR" is another local chart on M with I/ fl 0 then

Y Y -1 (Ace' an ) = (P,01 )

where E, = E, /3; =1 and i3k = Et( j,k E a" —a-Yi ake) letting (ai,7)1<j,k<n

y axk x f? a) )4

A'. Since B = ( bid 1<„, where bd = E j. k ax , takes unit vectors a

,xk

to unit vectors, B is orthogonal (that is, B = ). We say then that T' M

is a fiber bundle with structure group (9 ( n ), the group of orthogonal n x n

matrices.

Example 4. Let M be a compact connected manifold and g a transversely orientable, codimension one foliation of M which has a compact leaf F with finite fundamental group. Then the leaves of g are fibers of a fiber bundle with base S', fiber F and discrete structure group.

In fact, by the global stability theorem all the leaves of g are diffeomorphic to F. Further, there is a submersion f: M S I such that the leaves of g are the sets of the form (0), 0 E S I . In this case, ( M, f,S' ,F) is a fiber bundle, as can easily be seen.

S2. Foliations transverse to the fibers of a fiber bundle

Let ir : E B be the projection of a fiber bundle with fiber F. We say a foliation a of E is transverse to the fibers when it satisfies the following properties:

(a) for every p E E the leaf L g which passes through p is transverse to the fiber F,(p) and dim ( ) + dim ( F) = dim ( E).

(b) For every leaf L of g, 7i- IL:L B is a covering map. It follows from the definition that for every p E E one has

TrE = T(L)

An important observation due to Ehresmann is the following.

Proposition 1. Suppose that the fiber F is compact. In this case (a) implies (b).

Proof. Let L be a leaf of g. Since 7r : E B is a submersion, L is transverse to the fibers and dim ( L ) = dim ( B), it follows that 7riL:L B is a local diffeomorphism. To prove that ir L : L B is a covering it suffices to show that, for every b E B, there is a disk U C B with b E U such that for each x E U

the fiber 7 - 1 (x) cuts each leaf of the restricted foliation g ( U) in ex-actly one point.


In fact, the connected components of L fl 7r - 1 ( U ) are leaves of I 7r -1 ( U). If V is one such connected component, and if x E U then 7r - 1 (x) fl V contains exactly one point, so 7rIV:V---• U is one-to-one and onto and hence is a dif-feomorphism, which shows that 7rIL:L B is a covering.

In order to construct U we show first that, given z E r -1 (b) there is a folia-tion box W. of S such that, for every x E it ( ) , 7r -I (x) cuts each plaque of W, in exactly one point.

Figure 1

Let L., be the leaf of g through z E 7r - 1 (b). Since z is an isolated point of

7r - (b) n L : in the intrinsic topology of L z , there is a disk zED C L z such that for every y E D, the fiber it I ( 7r (y ) ) cuts D exactly in the point y. This implies that the map h: it - I ( 7r ( D)) D such that h (y) = 7r - (it (y ) ) f") D is a submersion and h (D) = D. On the other hand, D could have been chosen so that D is contained in a plaque of a foliation box 1-V: of It is easy to see now that it is possible to construct a foliation box W. of such that W, c 17V- n 7r - I ( 7r ( D)) and the plaques of W, cut each fiber 7r -1(X) in exactly one point, if xE it (D). We leave the details to the reader.

Since 7r - (b) is compact, we can take a finite cover ( W, = W, ) k, = , of 7r - ( b ) . For each i E k I , 7r ( W, ) is a neighborhood of b in B so fl 1 r(W) is a neighborhood of b in B. Let U c n 4, = 1 7r ( W, ) be a disk con-taining b. 1ff is a leaf of a 1 7r I ( U ) , then J is contained in some plaque P of W, for some i E 11 ,..., kl. Thus we can conclude that it HI: .1 U is in-jective. On the other hand, from the injectivity, it follows that 7r( J) =

r(P n 7 -1 (0) = it (P) fl U ---- U. Therefore it I J : ../ U is a diffeomorphism, as desired.


§3. The holonomy of a

When a is a Cr (r a- 1) foliation transverse to the fibers of E, there is a representation

: (B,b) Diffr (F) = Diff r ( 7r -1 (b))

of the fundamental group 71- 1 ( B,b) in the group of Cr diffeomorphisms of F, Diffr (F), called the holonomy of 5, which we define as follows.

Let a : / = [0,1] B, a ( ) = a ( 1 ) = b and y E 7r -1 (b). Since 7r I Ly : Ly B is a covering, there exists a unique path ay : I Ly such that ay (0) = y and 7r o ay = a -1 . If we identify (b) with F, we can define a map soc, : F F by soa (y) = y ( 1 ) .

Since the endpoint of the curve ay only depends on the homotopy class of a-1 , it follows that so (y) only depends on the homotopy class of a- and we can write tpa = ,p[a] . Further so [Œi _ = Opial ) -1 and if [0] E 7r1 (B) ,

yofro som . It is easy to verify that p is Cr. Since ,p [od has an inverse that is also C`, is a Cr diffeomorphism. So, (,,9 : 71) (B,b 0 ) Difr(F) given by ( [a]) = so" is the desired group homomorphism.

§4. Suspension of a representation

In note 1 of Chapter II we saw the simplest example of a foliation transverse to the fibers of a fiber bundle. It is that of a suspension of a diffeomorphism f : F F to a foliation of dimension one of the quotient space E = F x [0,11/ — where — is the equivalence relation which identifies ( 0, y) with ( 1,f - I (y) ). This foliation is the one induced on E by the trivial foliation of F x [0,1] with leaves ix) x [0,1], x E F. The space E is fibered over S' with discrete structure group since we can take as local charts of E the quo-tients O, F of U = (c,1 — €) X F, V = ( [0,2e) U ( 1 — 2€, 11) x F.

Then C/ n 17 has two connected components W12, W21, and grI2 : W, 2 12

Diff (F), g2, W21 Diff (F) are given by gi2(x) = identity and g2 , (x) = f

A natural question is the following: how does the above construction generalize to the case where we have a representation so : G --+ Diffr ( F) where G is a group? As we have seen, G must be the fundamental group of the base B. More specifically the problem is the following: given two manifolds F, B and a representation ço : 7 1 (B) Diffr (F), to find a manifold E ( ço) fibered over B with fiber F, and a foliation if ( ço ) transverse to the fibers of E such that the holonomy of if (cc) is cc. Before giving the general construc-tion, let us see an example.

Example 5". Let B = T2 and F = \Ve have that 7r 1 (T- ) = Z 2 so a


representation cp : ,Z 2 •—• Diffr ( S' ) is determined by two diffeomorphisms of S', namely ! = (p ( 1,0) and g = ,p(0,1). These diffeomorphisms are com-mutative since fo g = so(1,0) . so (0 ,1) . (I, ( (1 OD) + (0,1)) = (p(0,I) . ,p (1,0 = g of. Suppose for simplicity that f preserves the orientation of S' but not g. We will now construct the suspension of ,,o. On /2 x S' consider the foliation a, whose leaves are /2 x 101, 0 £ S I .

Figure 2 — The foliation 3 ,

We define E (ço) by identifying the points of the boundary of /2 X s ' accord-ing to the following equivalence relation: ( x,y, 0 ) and (x',y',0') on /2 x S' are equivalent if

(1) x = 0, x' = 1, y = y' and 0 = f (0' ) or (2) x = x' , y = 0, y' = 1 and 6 = g (0' ).

Making identification (1) we obtain a manifold diffeomorphic to I x S' x S' = I x T2 since f preserves orientation. Evidently 5 1 induces a foliation a 2 transverse to the boundary whose holonomy is generated by f.

E(,0)

Figure 3 — The foliation S2


In making identification (2) on I x T2 we obtain a nonorientable manifold E (v) fibered over T2 with fiber s'. Since g commutes with f, the leaves of

2• I 1 0J x T2 are taken to the leaves of 52 1 1 1 I X T2 by identification (2). So 52 induces a foliation if (so) on E(,,o). Since S, is transverse to the curves Ix) x fyi x S', (x,y) E /2 , if (ç o ) is transverse to the fibers of E(so). The leaves of E( ) are homeomorphic to 1.1 2 , 1R x S 1 or T2 and in any case the projection of the fibers restricted to each leaf is a covering.

Theorem 1 (Suspension of a representation). Let B and F be connected manifolds and ç>: 7 1 (B, bo ) ---■ Diff (F) a representation. Then there exist a fiber bundle (E (w), 7,B, F) and a foliation a ( sci ) transverse to the fibers of E ((to) whose holonomy is v. The fiber bundle E(v) has discrete structure group.

Proof. Let P : b —• B be the universal covering of B. The representation so induces an action

Y-o : r i (B, bo ) --• Diffr Ch x F)

given by iTo ( [a]) ( -6,f) = ([al - i; , cp ([a])-1 - f) . In the above definition, [a] - -6 denotes the image of ii by the deck transfor-

mation associated to [a]. Recall that a deck transformation is a diffeomorphism g : h --• h such that Pog=P. The deck transformation associated to [a] is defined in the following manner. Fix -60 E -ij such that P ( To ) = /3,0 . Given T3 E n consider a curve -.7 : [0,1] —• /I such that "-y- ( 0 ) -,-- I; and 52 ( 1) = = Lo . Let -y = P c• "I and S = y * a -1 * -y -1 . Then [a] - -6 = -6. ( 1 ) where 75 is the unique lift of 6 such that Z ( 0 ) = T) .

[c ]si; '------------4 ""s"---1

5 i; .----1.----- 1 - g o

P b

Figure 4

For the details see [29]. - P

The set of deck transformations B —1. B with the operation of composition

is a group naturally isomorphic to 7, (B, b 0 ), since h is the universal cover-ing of B. We are identifying [a] with the automorphism associated to it by this isomorphism. In Example 5 the action •,b is defined by

Ço(m,n) • ( x,y,0) . (x + m,y + n,f -m . g' n (0)) .


The action Zro satisfies the following property. For every -6 E h there is a connected neighborhood V of b. in b such that if g E 7r 1 (B, bo ) and g 1

then rp'(g)(V x F) (1 (V x F) = 0. In fact, let V be a neighborhood of E• such that P: V P(V) is a diffeomorphism. For every g E (B, b 0 ), g 1, g-t/nV=0 since if there were ay E g( V) (1 V, we would have that y = g(x) with x E V, so P(x) = P(y) and P would not be injective on V. We have then that 'Zro(g)(V x F) (1 (V x F) = (g V X so(g) -1 F)

(1 (V x F) = 0. We will now describe the construction of E(,,o) and ff ( ) .

Introduce on b x F the equivalence relation which identifies two points when they are in the same trajectory of -yr, that is, (L, y) y' ) if there exists a g E 7r 1 ( B,b 0 ) such that ' ( g) (L, y) = , y ' ) . In Example 5 this equivalence corresponds to the identifications (1) and (2). Using the above-mentioned property we prove that the quotient space of b x F by this equivalence rela-tion is a differentiable manifold which we denote by E(so).

From the expression for -(70 it follows that it preserves the fibers of h Pi

x F B, P I (b,y) = L. We can then define a map 7r : E(so) B that makes the following diagram commute:

Q x F

NPI (*) E(so)

71- NN,

In the above diagram Q is the projection of the equivalence relation. Observe also that the leaves, h x I», of the product foliation b x F are

preserved by . So, this foliation will induce on the quotient space E(so) a folia-tion transverse to the fibers of 7r, /hich we will denote a ( ça).

We go now to the proofs. E (so) is a fiber bundle with base B, fiber F and projection let V C b be

an open set such that if g E B, bo ),g then g(V) n V= 0. From this property it follows that QIVxF:VxF Q(V x F) = 7r - (P(V)) is bijective. On the other hand, QiVx F is continuous and open, as is easy to see, so Q1VxF is a homeomorphism. It follows then that Viv = = ( Q V x F) : 7r - 1 (P(V)) --•• V x F is a homeomorphism. We will show that the set of such defines on E(tio) a Cr differentiable structure.

Let us prove first that if U,V C h are two open sets as above with P(U) n p( v) then

;4.1,4 1 : (13-1 (P(U)) n x F— (U n P -1 (P(V)) x F

is given by ik t o = (g) where g E 1 B , b o ) is fixed in each connected component of ( (P(U)) (1 V) x F. To simplify the argument we suppose that ( P - '(P(U)) (1 V) x Fis connected, which is the same as saying that P(U) r") P( V) is connected.


Fix ( -6 1 ,f, ) E (P-1 (P(U)) rl V) x F. Since kt 1, 1 = Q on V x F and

4= (QIUx F) -1 we have that (62,f2) = 0 ;Li.' ) is equivalent

to ( -6 1 ,f, ), so ( -62,f2) = ;70(g) ( -6 1 ,f 1 ) for some g E 7r 1 (B,b 0 ). Take now

(b,f) E (P-1 (P(U)) n v) x F and set ktu kt /Y (6,f) = ( -6',f'). In this

case (T),f) is equivalent to ( 73 ,f')andb' E un P-1 (P(V)). We have then

that = g' - b and f' = q7(g') -1 •f for some g' E 1- 1 (B, b0 ). On the other

hand P-1 (P(V)) = U hEfft(B,b0) h(V), where lh ( V) h E (B,b0)1 is the

set of connected components of /3- (P(V)). Since -62 = g bl EU r) g

• V 0 and U n P-1 (P(V)) is connected, we have U (P(V)) =

= U g • V, so T = g ' • t; = g • with E E V. Therefore "C.E (g -1 • g' • V)

n V so g -1 • g' = 1 and then g = g'. It follows therefore that Vi u

° ' (,f) = ;0- (g)(6,f) for every (b,f) E (P-1 (P(U)) fl V) x F,

or, Vi u o = We can now introduce on E(v) a Cr. manifold structure, composing each

1,tv as above with local charts t : W 1R k where Wc VxF is an open set, obtaining in this manner a Cr atlas

g• VflV=ø if

g 1 and ( W,E ) is a local chart on V x F) .

The fact that 1,t u e = Z-p(g) on each connected component of its domain implies that the change of coordinate maps are Cr.

The composition of 1,t v with P x idF yields a diffeomorphism ;T v : 7r (P(V)) P(V) x F. The set ( V, ;-t y ) I g • y (1 V = 0 if g 1 ) induces on

E(so) a structure of a fiber bundle with base B, fiber F and projection 7, as is easy to see from the definition of §1. Observe that E(‘,0) with this structure has discrete structure group.

The foliation g ( tp) is transverse to the fibers of E (so). A set of foliation boxes of a(,0) can be defined using the C diffeomorphisms 1,t v : 7r -1 (P(V))

V X F defined before. The plaques of 5 ( ) on 7 -1 (P(V)) are the sets of the form kt,Y ( V x if)), JE F. Since 1,t u 14 1 = (g) for some g E 7r, (B, b0 ) and rp preserves the leaves -I-3 x (f) of the product foliation of 13 x F, it follows that 5 ( ye' ) is well-defined.

Observe that from the construction it also follows that 5 ) is transverse to the fibers of E( so). In fact, the charts ( V, Ov) are both local charts for E ( (p ) as well as for the foliation 5 (so), so the fibers of E (47) in ( V x F) =

= 7r -1 (P(V)) are represented in V x F by submanifolds of the form vl x F

and the plaques of 5 ( 40) are represented by submanifolds of the form V x tfl;

hence the condition

(a) TE(p) = T(L) 3


from the definition of section 2 is immediate. On the other hand the map 7r /7r -1 ( P ( V)) is represented in V x F by

P o P1 / V x F, where P1 is the first projection. Therefore the restriction of in to each plaque of 5 (so)/ roe' (P ( V)) is represented by P P1 : V x [fi

P ( V), which is a homeomorphism. This implies that for any leaf L of if (p), rIL:L B is a covering, as we wanted.

The holon,omy of if (0 is so. Take bo as base point. The fiber 7r - (b 0 ) is naturally diffeomorphic to 1 -60 ) x F by the diffeomorphism

Q/(1 -60 ) x F) : lbo i x F

where P (To ) = bo . Identify the points of F with those of 7r -1 (130 ) by the dif-feomorphism 0 ( f) = Q 0 ,f) . Now consider g E 7r 1 (B, b0 ). Let ei : [0,1] —• :8- be a curve such that ei (0) = bo and & ( 1 ) = g • bo . Let a -1 =

P o• From the construction it is clear that [a] = g and ert is a lift of a -which satisfies ei (0) = To . We will see now that the image of f E F by the holonomy transformation associated to g is (p (g) f (taking into account the identification of F with 7 -1 (b0 ) by O). First, note that the leaf L of if (p) which passes through Q ("60 ,f) E 7r -1 (b0 ) is Q (T3 x (f)). Fur-ther, Q/ii x If I : x [f] L is a covering map, where on L we take the intrinsic C' manifold structure. These facts follow from the construction of if ( ‘,0 ) and from the commutativity of the diagram (o). Let ei" : [0,11 L be the covering of a that & ( 0) = Q 0 , f) . The image of ( -60 ,f) by the holonomy map associated with [a] will be ire ( 1). On the other hand the lift of re to fl x [f ) by Q/. x ff) which begins at (,f) is the curve t ( t),f) , as is easy to see. The endpoint of the curve is (& (1) , f) = (g i30 ,f) which is identified in E (p) with the point (60 , (g)

f) E C-60 j x F. We have then that ei ( 1) = Q (6 0 ,4p (g) • f) which is iden-tified by 0 with (,o(g) • f. This proves the theorem.

Definition. We say that two representations (p : 7r 1 (B, b0 ) Diff ( F) and : in 1 (B, AD ) Diff ( F ' ) are C5-conjugate if there is a Cs diffeomorphism

(if s 1) or homeomorphism (if s = 0) h : F F' such that, for every [a] E 7r 1 ( B, too ) , we have 4,0 [a] = h I ° (P' ( [a] ) o h.

Theorem 2 (Uniqueness of the suspension). Let ço and ço' be C5-conjugate representations as above. There exists a C5 diffeomorphism H : E ( so)

E ' ) (homeomorphisrn if s = 0) such that

(a) in' o H = in and consequently H takes fibers of E ( ) to fibers of E (( to ' ). (b) H takes leaves of 5 ((p) to leaves of 5' (40 ' ).

Proof. Let Ti : 1-3xF--■ T3xF' be the CS homeomorphism defined


by ñ( , f) = ( -6,h ( f)) , where h conjugates so and so ' . It is immediate that 1":1 preserves the fibers of the fibration ET» x F and also takes the leaf /3 x f J of the product foliation to the leaf LI x th (f)j. Further, for every g E r i ( B, b0 ) we have iT (iTo (g)) = ço' (g) o where i-„o' and iTo ' are as in the construction of Theorem 1.

Therefore Ti takes equivalent points of /3 x F to equivalent points of X F' . Thus it follows that there exists a CS ( s 1) diffeomorphism (or homeomorphism if s = 0)H:E(so) E( so') such that the diagram below commutes.

x F x F '

IQ' E(so)-1-11■ E(4p' )

It is immediate that H takes fibers of E(yo) to fibers of E (so ' ) and leaves of 3( cc) to leaves of ( so ) . •

Note. Theorem 2 can be proved in a little more general context. Let B, B' and F, F' be connected manifolds. Suppose that so : 7r 1 ( B, 1,0 ) Diffr(F)

and so ' : 7r I ( B' ,13(;) ( F' ) are conjugate representations in the following sense: there exist diffeomorphisms g: B B' and h : F F' such that g ( bo ) = b c; and for every [a] E r i B , bo ) we have

([a]) = h -1 (p' (g.([a])) h

where g. : 71- 1 (B,b0 ) 7r 1 ( B ' , N) is the isomorphism induced by g between fundamental groups. Then there exists a diffeomorphism H: E(yo) E (so' ) which takes fibers of E (so) to fibers of E (so' ) and leaves of F ( so ) to leaves of 5 ((p' ).

Suppose now that ( E, 7r, B,F) is a fiber bundle and that 3 is a foliation on E transverse to the fibers. Let so be the holonomy of a (with respect to the fiber 7r - ' (bp), to o E B). How can one relate the foliation ( so ) on ( E(so),7r„„B,F) with the foliation The answer is given in the following theorem.

Theorem 3. Let S be a Cr(r 1) foliation transverse to the fibers of

(E,r,B,F) whose holonomy in the fiber 7. -1 ( bo ), bo E B, is ,p : 7r ( B,b 0 )

Diffr( F) . Then there exists a Cr diffeomorphism H: E E( cc) which

takes leaves of 5. to leaves of () and such that 7r, o H = 7r. In particular,

H takes fibers of E to fibers of E

We leave the proof to the reader ( see exercise 38 ) .

Definition. We say that the fiber bundles ( E, 7T, B, F) and ( E ',7r' ,B, F) are


( Cr) equivalent if there exists a Cr diffeomorphism H: E E' such that 7r ' o H = 7r . In the case of vector bundles we require also that H takes fibers to fibers linearly.

As a consequence of Theorem 3, we have the following.

Theorem 4. Let (E, 7r, B, F) be a fiber bundle. Then there exists a foliation on E transverse to the fibers if and only if (E,7r,B,F) is equivalent to a fiber bundle with discrete structure group.

Proof. Suppose there is a foliation a transverse to the fibers of E. Let ço : 7r 1 ( B , b o ) Diffr ( F) be the holonomy of 5. As we saw (E,r ,B, F) is equivalent to ( E((p),r,,B,F) which by Theorem 1 has discrete structure group.

Let us consider the converse. We can suppose that (E,Ii-,B,F) has discrete structure group. In this case we define the foliation 1V using the local trivialization for the fiber bundle. Let I U„,1,Es where U iEs B and for every i E S,

U, is open in B and 1,G, : 7r - ( ) U, x F is a chart of the fiber bundle. Since the fiber bundle has discrete structure group, the change of variables

: (u, n (.J; ) x F (U, n (.J,) x F

is written in the form

° (b,f) = (b,4) u (f))

that is, the second component cl) does not depend on b. So the charts : -1 ( U,• ) U, x F induce a foliation 5 on E whose plaques on -yr U, )

are of the form 1,G,-1 (U, x f D. f E F. It follows immediately from the con-struction that if L is a leaf of 5 then ir 1L:L B is a covering and hence 5 is transverse to the fibers of E. •

We will now see some applications.

§5. Existence of germs of foliations

In this section we will study the following problem. Let N C M be a sub-

manifold imbedded in M. Under what conditions does there exist a foliation

5 defined on a neighborhood of N and such that N is a leaf of 5?

Let p ( N) = 1( p, v ) E TMlp E N and u E Tv I be the normal bundle of

N with respect to a fixed Riemannian metric < , ) on M. Then the zero section

of (N), No = ( p , E y(N)1 is diffeomorphic to N and by the tubular

neighborhood theorem [27], y ( N) is diffeomorphic to a neighborhood U of N in M by a diffeomorphism f: i ( N) U such that f (No ) = N. We see then

that the existence of a foliation defined in a neighborhood of N which has N

as a leaf is equivalent to the existence of a foliation in a neighborhood V of

No in v(N) which has No as a leaf.


Suppose that y (N) is equivalent to a vector bundle E which has a discrete structure group. The equivalence in this case must be a diffeomorphism H: y (N) E which takes fibers to fibers linearly; in particular taking the zero section No of y ( N) to the zero section of E. By Theorem 4 there is a folia-tion W on E transverse to the fibers, so g = H* (W) is a foliaton on y (N) transverse to the fibers. The local charts of W, as we saw in Theorem 4 are also local charts of the fiber bundle E and since they are linear in the fibers the zero section of E must be a leaf of g which implies that No is a leaf of W.

Conversely, suppose there exists a foliation i defined in a neighborhood of No in P (N) such that No is a leaf of

Consider a cover ( U, ) iEs of No by foliation boxes of a, where there are defined submersions j; : U 111P , p = m — n, such that the plaques of a U, are of the form A-1 (x), X E RP and f; (N0 (.1 = O. Consider also the changes of coordinates hki : fi (Ui fl Uk ) fk (Ui fl Uk ) such that fk = = hki

R P RP

Using the distinguished maps f„ we will redefine the structure of the normal bundle so that in the new structure p(N) has discrete structure group. This is the same as defining an equivalence between fibrations.

Given q E No fl U„ consider the linear map L i (q) = df1 (q)1 7q N4- : Tq N I 111'. Since No is a leaf of for every i E S and q E U„ L1 (q) is an

isomorphism. Fix a basis (e 1 ,..., ep ) of IRr. Then the vectors ,((q)

= (L,(q)) (ei ), j = 1 p, form a basis of Tq N6`. We can now define local charts Yc i : (N0 fl 1.1,) (No fl U,) x IRP

Tc, ( g, X ( q )) = (q,L,(q) • X (q)) = (q, E ai ed J = 1

where X (q) = aj xJ,( q ) E


If U, n Uk fl Nc, 0, the change of coordinates will be .-V-k 5C7 1 ( q , y) = (q,gk,(q) - y) where gk,(q) = Lk (q) o L ,' (q) = Dh (0 ) . Therefore,

the new structure group of y (N) is discrete. We have then the following.

Theorem 5. Let N C M be an imbedded C 2 submanifold of M. There is a neighborhood U i N and a foliation a on U such that N is one of the leaves of if if and only if the normal bundle of N has some fibered structure with discrete

structure group.

Note that the hypothesis of C2 in the statement of Theorem 5 permits us to use the tubular neighborhood theorem of [27].

§6. Sacksteder's example

As was mentioned in note 2 of Chapter III, a codimension one Cr (r 2) foliation of T2 does not have exceptional minimal sets (Denjoy's theorem). In this section we describe a codimension one Coe foliation of a compact three manifold which has an exceptional minimal set. The example is due to Sack-steder ( [491) and shows that the original version of Denjoy's theorem is not valid in dimensions larger than 2. Sacksteder, in 1964, proved a generaliza-tion of Denjoy's theorem for foliations of codimension one and class Cr (r 2). For more details see note 2 of this chapter.

Let V2 be an orientable compact surface of genus two (two-holed torus). This surface can be represented by the region bounded by an octagon as in figure 5 Chapter I, with the sides with the same letter identified according to the arrows.

The fundamental group of V2 is a non-abelian group generated by closed curves a,b,c, and d so that the unique nontrivial relation between these curves is aba- 13 - I cdc -I d -I = I (see [29]).

Let G be the subgroup of 7r, ( V2 ) generated by a and c. The subgroup G is free, since all relations in 7r 1 ( V2 ) involving only a and c are trivial. Conse-quently, given two diffeomorphisms f, g E Difr ( S' ) we can define a homo-morphism so : G Diff' ( S' ) setting so ( a) = f and so ( c ) = g. The homo-morphism so induces, in turn, a homomorphism so : r i ( V2 ) —• Dar° ( S' ) defined on the generators of r i ( V2 ) by (io, b) = tp (d) = 1, ço (a) = f and 41' (c) = g. Observe that so is well-defined, since

ço(aba - `13 -1 =f of l og. =

Let ) be the suspension of so. The fiber bundle E ((p) , where if (p) is defined, is homeomorphic to V2 X S ' , if we choose f and g preserving orien-tation. Moreover, by Theorem 7 of Chapter III, if ( so ) has an exceptional minimal set if and only if the action ço of the group G on S' defined by f and g has a minimal set homeomorphic to the Cantor set. In what follows we con-struct such an action.


We consider here S I as IR/Z . So, a diffeomorphism f of S I can be thought of as a diffeomorphism of IR satisfying the property f(x + k ) = f(x) + k for every x E JR and k E Z. Define f: IR --* IR by f( x) = x + 1 / 3 and g: PR JR such that g (x + k) = g(x) + k ifx E IR and k E .7 and the graph of g in the interval [0,1] has the following form:

Figure 6

The diffeomorphism g is defined in such a way that g(x) = x/3 if x E [0,1/2], g(x) = 3x — 5/3 if x E [2/3,5/6], g(1) = 1, g' (1) = 1/3, g (k) (1) = 0 if k 2 and g is C on [0,1]. The diffeomorphism f cor-responds to a rotation of angle 2t-/3 on S' and its expression on IR/Z is f(x(mod 1)) = (x + 1/3)(mod 1).

Let K be the subset of [0,1] defined by K = n 7, 0 K, where for every j E IN, KJ is a union of closed intervals, defined inductively in the following manner:

K0 = [0,1/6] U [1/3,1/2] U [2/3,5/6] =

[0,1] — (1/6,1/3) U (1/2,2/3) U (5/6,1) .

Supposing Kj _, = U 211 - 1 [01 -1„ .- I , I ] then Ki is defined from K1 _ 1 by removing from the interval [a" , 13 ',, - I ], its middle third, for every n =

1 ,..., 3 • 2" - '. Doing this we get K1 = U n3 211 P J where for every in = 1 ,..., 3 • 2 and for every n = 1 ,..., 3 - 2

1 a Jn - 1 )

6 • 3

It is clear that K is compact and nonempty. From the construction it is easy to verify that K is homeomorphic to the usual Cantor set, since after the first step the construction of both is the same.


Lemma: Consider K as the subset of S'. Then K is a minimal set of the action generated by f and g.

Proof. First we prove that K is invariant. We will denote by G the subgroup of Diff °°(S' ) generated by f and g. From the construction of the sequence [K JEN , one has f (K i ) = Ki for every j E IN so f (K) = K. From the defini-tion of g and the inductive construction of K1 , we have

g (-1 [0,1/21) = K1 n [0,1/6 ] ( j 1 )

and

g(Ki n [2/3,5/6 ]) = K, fl [1/3,5/6] ( j 1) .

Since Ki c [0,1 / 2] U [2/ 3,5/6], from the above relations we get that

g(K1 ) C Ki+ , U KJ _ 1 c IÇ,.

So g(K) = g(n 7_ , K; ) = n 7_ g(Ki ) C n KJ _ = K. Analogously

g - '(K Cl [0,1/6 ]) = K1 , (") [0,1/2]

and

n [1/2,5/6 ]) n [2/3,5/6] ,

SO

g -1 (K) C K1 , .

Then g - ( K) C K. Therefore g (K) = K. We conclude therefore that K is in-variant under f and g so K is invariant under G.

Since K is compact and invariant under G, K contains a minimal subset We will see next that ti = K. To do this we will prove first that 1 / 3 is in A

and then that the orbit of 1/3 in G, 0 ( 1/ 3 ), is dense in K. We will have then that p. = 0 ( 1 / 3 ) = K since 0 ( 1 / 3 ) is dense in A and A is minimal.

1 / 3 E A. Let x E u. We can suppose, without loss of generality, that x E [0,1 /2], since if x E ( 1/2,1] it is clear that f (x) or f -2 (X) E 10,1/21 (") ti. Consider then the case that x E [0,1/2] (.1 . Since is invariant under G, for every n 0, gn (x) = ( 1 / 3" ) x E ti, so lim n gn (x) = 0 E

Therefore 1/3 = f (0) E 12.0 ( 1 / 3 ) is dense in K. Let/ — K = U ,IN ( an,i3n). Since K is homeomor-

phic to the usual Cantor set, the set B = i3 n E IN] is dense in K. Observe that 1 / 3 E B, since the interval ( 1 /6,1 / 3 ) is one of the intervals of the decom-position of I — K. Next we will prove that 0 ( 1 / 3 ) = B and hence 0 ( I / 3 ) = =K.


If suffices to prove that 1 — K = U h E G h( 1/6,1/3 ). Indeed, if there were the case, for every n E N there would exist an h E G such that h (1 /6, 1 / 3 ) =

( an 43,, ). Since dh/dt > 0 for every h E G we see that h( I / 3 ) = [3,, and therefore ( 1 /3 ) = B. We will prove that I — K = U hEG h ( 1 /6,1 / 3 ). In the first place, it is clear that U h€G h (1/6,1/3) C I — K. To see that I—KcU n(G h( 1 /6,1/3 ) it is sufficient to prove that for every j E IN, 1 — Ki C U hEG h ( 1 /6,1 /3 ). We prove this by induction on j.

For j = 0, we have I — K o = (1/6,1/3) U (1/2,2/3) U (5/6,1) and (1/2,2/3) =f(1/6,1/3), (5/6,1) =f 2 (1/6,1/3). So I — K o C U hEG

h ( 1 /6,1 / 3 ). Suppose by induction that I — K J c U hEG h ( 1/6,1/3) for 0 j — I. We have that / — Ke D / — Ke_i and (I — 14) — (1 — K r _ 1 ) U Ien , where the intervals f„ have length 1/6 • 3. Since (I — K 1 ) — ( I — K _ 1 ) c (0,1/6) U (1/3,1/2) U (2/3,5/6), f 1, given n E 1,2 ,..., 3 • 2" - I I, there exists an i E 0, —1, —21 such that

(11,' ) C (0,1 /6). On the other hand, as we saw, g' (K fl [0,1 /6]) = = K fl [0,1/2] for every j E N, so g -1 ((I — K r ) rl (0,1/6)) = = (1 Kr _ i ) n (0,1 /2) and therefore o f (I) c I - IC1-1. By the induction hypothesis = h( 1 /6,1 /3 ) for some h E G, so I" = = g h(1/6,1/3), and so /", c U I,EG h(1/6,1/3). •

Let us look at a simple way to see the foliation 3 (it') geometrically. Let Es and ei be two closed curves in V2 which do not meet and are homotopic to b and d respectively. Let P2 = V2 -( U j) and ici = P2 X S ' . The figure below represents Al

Identification

Identification by (4, ,g) by (0,f)

Figure 7

In this figure a manifold with boundary -12, x [0,1] is drawn. The mani-

fold :12i should be thought of as P2 x [0,1 ], where we identify the points

of P2 x 0 and P, x 1 with the same first coordinate. Observe that tq is a

3-dimensional manifold such that a1t7/ is made up of four copies of T 2 , x


S', -62 x s' ,J, x s' and c72 X S', where -6, and -62 (resp. J, and J2 ) are the results of lifting -6 (resp. J) from V2. In order to obtain V2 from F2 we need two diffeomorphisms of identification, say fi : -62 -6, and J2 •

Let us construct M from A71. Given the diffeomorphisms of S', f and g define : (15 2 x s') u (J2 x s') ( -6, x s') u (J, x s') by )1, (x,y)

= (0(x),f(Y)) if (x,y) E -62 x S and 0 (x,y) = (6(x),g(y)) if (x,y) E d2 x S'.

Consider on A-1 the equivalence relation - defined by (x,y) (x' ,y') if x = x' and y = y' or (x,y) = 1,1,(x',y/), or (x,y) = (x',y 1 ) in the case ( x',y' ) E ait71. Then M = / - is a compact manifold of dimension 3 and we have defined a submersion 71" : M •—• V2 which makes the following diagram commute:

In this diagram 'w1 is projection on the first factor r i (x,y) = x, 13 is the projection of the identification and p is the natural projection from P2 onto V2 . By construction IT : M 1/2 defines on M a fibered space structure with base V2 and fiber S

Let be the product foliation on it-f, whose leaves are of the form i72 X x E S'. A leaf P2 X tx) meets aft-i in four circles b i x txj, -62 x [xi, J, x fx), and c72 X [xi. Observe that the diffeomorphism of identi-fication 0 takes the circle -62 X X I to the circle -6 x If(x)] and the circle J2 X [xi to the circle J, x [ g (x), or, as we identify the points of aA7f by 0 we automatically glue the boundary of a leaf of In this way induces on M a foliation 5 equivalent to ( ) .

Notes to Chapter V

(1) An application to the study of Ricatti's equation.

The equation of Ricatti in the complex domain is a differential equation of the follow-ing type:

—dy

= a(x)y- + b y + c(x), dx

E C ,

where a, b and c are entire functions, i.e. holomorphic in C .


In this note we will prove that the solutions of the Ricatti equation are meromorphic functions from C to (E. This result was originally proved by Painlevé [41]. Observe that this fact is not true for example for the equation dy/dx = y 3 ,x, y E C. In fact, this equa-tion can be integrated by separation of variables and the solution such that g ( 0) = = yo 0 is y(x) = yo/ ± xf1 - 2xyl, , which is not a function in the usual sense of the word, since it is multi-valued, that is, for each x 1/2y, y ( x) assumes two distinct values.

Before we prove this result we will remember some facts about the theory of mero-morphic functions. Consider the Riemann sphere of radius 1 / 2, S 2 C 1R 3 with a parametrization defined by the stereographic projections

4,01 :C - (0,0, - 1/2) and so2 :C —0- S 2 - (0,0,1/2)1

1 1 (z 1 ,z2, - ( I - ! z i 2 ))

I + IzI2 2

-) ( z = z i + iz2 ) = 1 I , 1 2

so 1 + !Z1 2 (zi, —Z2 ' —2: k I Z — 1 )) .

The changes of parametrizations are given by

1 502 ° (Pi(z ) = ° So2 (z ) = 9 4P2 ° (p, - C 0 .

A meromorphic function f :€ f:C can be considered a holomorphic function 7 : c S 2 . In fact, if z EC is not a singularity of f, set f (z) = (p i o f(z) and if

z is a singularity of f, set f (z) = ( 0,0, -1/2). We claim that f: —• S 2 is holomorphic. Proof: if zo is not a singularity of f then o f is holomorphic in a neighborhood of zo and if f(z 0 ) 0, ,p 2-1 7( z ) = 1/f(z) is also holomorphic in

a neighborhood of zo . In case z o is a singularity of f then zo is a pole of f and therefore

(z) = (z - z o ) k g (Z), where k 1 and g is holomorphic and nonvanishing in a neighborhood of zo . We have then that (p 2 1 )7 (Z) = — Z0) k • (g(z)) 1 , which is holomorphic.

Conversely, if 7 : c s' is a holomorphic function then the function f(z) = -1 -

= ■Pi ° f (z), z EC, is meromorphic.

Consider now the Ricatti equation

dy = a(x)y- + b(x)y + c(x),

dx y E C

The local solutions of this equation are (complex) integral curves of the complex 1-form

on C"-

= dy - (a(x)y 2 b(x)y + c(x))dx.

From the real point of view, ± ke„. where

(2 1 = dy i - Re(F(x,y))dx i + Im(F(x,_,v))dx2

= iz2) =


cc), = dy, — lm(F(.x-,y))dx, Re(F(x,y))dx,

where

F(x,y) = a(x)y 2 + b(x)y + c(x) .

Therefore the plane field defined by u.) = 0 onC 2 has real codimension 2 and complex 1. This plane field is integrable and can be considered as a plane field on C x S 2 ,

transverse to the fibers 1.z1 x S 2 . Indeed, consider the parametrizations of C x S 2 ,

4, 1 :C x€ C x (S 2 -1(0,0,-1/2)1) and \t2 :CxC C x (S 2 — 1(0,0,1/2)1)

defined by tt i (x,y) = (Aw l (Y)) and 1l2(x,,v) = (x,sc2(Y)). On C x (S 2 — 1(0,0, — 1/2)1) the plane field is defined by the complex 1-form

( I ) ( ) , and on C x (S 2 — 1(0,0,1/2)1) the plane field is defined by the complex 1-form Oh I )*(n) where n= —dy — (a(x) + b (x)y + c(x)y 2 )dx. We must verify that this plane field is well-defined. It is enough to show that they coincide on C

x ( S 2 — 1(0,0,1/2), ( 0,0, — 1/2)1). In this region, the plane field defined by ( ‘ti I )* (n) is equivalent to the plane field defined by the form ( ik 2-I )*(n) =

ok 2 r(,) ) on C x (C — 101), which can be expressed

Oh Vi i )*(71) = —d(y 1 ) — (a(x) + b(x)y 1 + c(x)y 2 )dx =

y-2

(dy — (a(x)y + b(x)y + c(x))dx) =

-2 • w

Hence the two complex 1-forms ce and 1,G; (1,t 1 )* (17) differ by the factor y -2 which does not vanish on C x (C — 101), and this proves that the plane field is well-defined on C x S 2 .

On the other hand, since co (x,y) ( + 62 ) = + 62 and n t ,3) (0,;, + 62 ) =

• _• = —y, we have that the plane field is transverse to the fibers 1.z1 x S 2 of the fibration x S 2 ,7r 1 ,C,S 2 ) where ri is projection on the first factor. Let be the foliation of complex codimension 1 defined by the plane field on C x S 2 . Since the fiber is compact, by Proposition 1, given a leaf L off, the map r i I L : L C is a covering. Since C is simply connected, we can conclude that r i L : L C is a diffeomorphism,

so L CC x S 2 is the graph of a function 37: C S 2 . The function y (X) = -1 — = cc, ( y (x) ) is single-valued and rneromorphic, being also a solution to the Ricatti

equation. We can then state the following

Theorem. Let a, b,c :C C he holornorphic functions. The solutions (if the Ricatti

equation

dy = a(x) y2

+ b(x)y + c(x), dx

a re meromorphic functions from C to C.

(2) Sacksteder's Theorem (1501).

In 1965, Sacksteder proved the following theorem.

x,y E C


Theorem. Let 5 be a codimension one, C2 foliation of a compact manifold M m ( m 2).

Suppose that 5 has an exceptional minimal set p.. Then there is a leaf L C p. and a closed

curve y : --• L such that, if f is the germ . of holonarny of y on a transverse segment

E passing through p = y ( 0 ) y (1) then f' ( p) < I. In particular, the holonomy of

L is not trivial.

Denjoy's theorem (see note 2 of Chapter III) is a consequence of the above theorem. In fact, if 1.). is an exceptional minimal set of a foliation 5 of codimension 1 defined on a compact manifold M2 then all the leaves are homeomorphic to IR and hence simply connected; so the holonomy of such leaves is trivial. By Sacksteder's theorem such a foliation cannot be C 2 . In fact, the calculations made by Sacksteder for the proof of the above theorem are similar to those made by Denjoy.

In the example of Sacksteder described in the text, two leaves which satisfy the con-clusion of the theorem are those corresponding to the fixed points 0 and 5/6 of the dif-feomorphism g defined in §6.

(3) Hector's example ([19]).

We will see here an example of a codimension one foliation on R 3 = E2 x IR whose leaves are dense and transverse to the fibers 1x1 x IR, x E R2 . This foliation satisfies condition (a) of the definition of §2 , but not condition (b), that is, 71/, : L R 2 is not a covering, where L is a leaf and r the projection on the first factor.

Consider the solid cylinder D 2 x R c R 3 where D 2 = I (xl ,x2 ) E R 2 + <

and let S I x R = 3 ( D 2 x R2 ). Given a diffeomorphism f : IR 111 such that f(x) = x for every x E (a,b), we are going to define a foliation 5 ( f) on D 2 x R

- (101 x ( 00,a1 U (0) x [6,00 )), as follows. Consider on S I x IR a foliation g(f), the suspension of f, such that for every

z E (a, b) the circle S I x {zl is a leaf of g(f) and the leaves of (f) are transverse to the lines (q) x IR, q E S i .

f(z

z '

f(z) ,_ Z

Figure 8

Define now the leaves of 5 ( f ) saturating the leaves of g ( f) by the flow of the radial field X (x,y,z) = (- x, - y,0 ) inside D 2 x S I , as in figure 8.

So, if t --• (cos t, sin t, z(t)) = y(t) is the parametrization of a leaf g ( f) which passes through a point (x,y,z) E S I x D 2 such that z > b or < a, then 'y r t = (r cos t, rsin 1, z(t)), I E 91,0 < r < 1, is the parametrization of a leaf of 5 (f). The leaves of ( f) in the region D 2 x (a,b) are the disks D 2 x IzI, z E (a,b).


Now consider a C c° diffeomorphism h: D2 x JR D 2 x 1R — (101 x ( —00,a]

U [01 x [b,00)) which satisfies the following properties:

(a) h (10) x (a,b)) = 101 x 111, (b) for every x E D 2 - 101, h(fxI x it) = 1x1 x R,

(c) h is the identity in a product neighborhood of S i x 1R.

We can define a diffeomorphism h as above taking h(x 1 ,x2 ,z) = (x 1 ,x2 ,a(r2 )13 (z)

+ (1 — a (r2 ))z), where r 2 = 4 + 4 and the graphs of a and j(3 are as in figure 9.

Figure 9

It is clear that in the above construction we can have a = 00 or b = + co . Let (f) = h* (5 (f)). It is clear that 5* (f) is a foliation on D 2 x JR whose leaves are

transverse to the fibers 1x1 x 1R, x E D 2 . Further, 5* ( f) coincides with 5 ( f) on a neighborhood of S I x R. The idea now is to glue together various foliations like this defined initially on solid, disjoint cylinders, identifying the boundaries of these cylinders in the following manner.

Suppose that ft : 111 R and f2 : JR are diffeomorphisms with f, ( x ) = x if x E i = 1,2. We can suppose that the leaves of g ( ),i = 1,2, are horizon-tal in the region Vi of S I x NI defined by V, = 1( 1,6,z) E S I x11110, — 7r/2 <0 <0i + 7r/21 (where we are writing the point (X, ,x2 ) E S 1 in polar coordinates and 0, E S i is fixed). Figure 11 illustrates what we want.

Figure 10

3


With this restriction the leaves of 5 (f,) are horizontal in the region Pi = (r,O,Z) E D2 x 111 O < r < 1,0,– 7/2 < 0 < 0, + 7/21.

Since 5* ( ff ) coincides with 5 ( fi ) in a neighborhood A, x 112 of S I x 1R, where A, = ( r,B) 1 – < r < 1), the leaves of 5* (ft ) I ((A, fl V,) x ) are horizontal.

Glue together now the two cylinders by a diffeomorphism g : ( A 1 CI V I ) x IR (A 2 CI V2 ) x JR of the form g(x,z) = (a(x),z) where a : A 1 (--) V A2

fl V2 is a diffeomorphism. Thus we obtain a foliation 5( f1 ,f2 ) as in Figure 11.

Figure 11

The region where 5 (f,, f2 ) is defined is a product of the form U x IR where U is dif-feomorphic to the unit disk D 2 . So we can consider a (f j'2) as being a foliation of D 2 x IR transverse to the fibers (xi x JR, X E D 2 .

Let us see now how a leaf F of 5 (fi ,f2 ) meets a fiber fx0 1 x IR, X0 E D2 . Observe first that in every step of the construction vertical fibers were taken to vertical fibers. If f l and e2 are the axis of the initial cylinders, there are p q E D 2 such that C I is taken to (pi x JR and P2 to fqj x IR, at the end of the process. If we consider 5 (A ,f2 ) restricted to (D 2 – x 114, it is easy to see that, for every leaf F of the new foliation, T-IF:F D 2 – 113,0 is a covering, where r is the projection r (X, z) = = x,x E D2 , Z E IR. We have then that the new foliation satisfies the hypothesis (a) and (b) of the definition of §2.

Given x0 E D 2 - if -y : I —• D 2

- lp,q1 is a continuous curve such that ( 0 ) = -y ( 1 ) = x0 , we can get a holonomy transformation g 171 : x0 1 x 1R 1X0 1

X IR as in Theorem 1. Observe that the fundamental group of D 2 – Ip,q1 is the free group of two generators, a and i3, which can be taken as in the figure below:

Figure 12


It is easy to see that the holonomy transformations g 1c, 1 (or gm 1) and 43] (or 431 - ) are conjugate to fi and f2 respectively, by the same diffeomorphism. We have then that the intersection of the leaf F of g .12) which passes through (xo , z ) with x0 l x JR can be represented by the set

k 1 f i k2 2 0(Z) = if , 1 012 of °f2 e -••° f2j (Z) E Z; .

We now consider diffeomorphisms fk : JR R as before. Using the same process as before, we can inductively define a foliation g (fi fk ) on D 2 x IR transverse to the fibers {xi x 1R, x E D 2 . As before, the intersection of the leaf of

5 fk ) which passes through (xo ,z) with Ixo l x JR can be represented by the set

0(x) = , ll f.,; ( r) I n EZ, E .Z and 4E11 ,..., k11

,=1 n

where H = fi , 0.... ffn ( z ) . Finally observe that since D 2 is diffeomorphic to R2 , we can consider a ( f, fk )

as being a foliation on R 3 = R2 X IR transverse to the fibers (x1 x R, x E 112 2 .

Now we will see that there exist diffeomorphisms ( f1 ),4• =1 such that all the leaves of

5 (fi f4 ) are dense in R 3 .

First fix 95, : R R of class C that (I, (x) = 0 if x S 0, çô (x) = I if x 1 and ,p ' (x) 0 for every x E IR. Define fi ( x) = x + ap (x), where a > 0 is irrational. It is clear that fi is a C œ diffeomorphism and that fl (x) = x if x 0 and

(x) = x + a if x 1. Set 12(x) (x + a). It is clear that 12 is a C and that f2 ( x) = x if x 1. Analogously define f3 (x) = x + yo(x) and

- 14(x) =

1 (x + 1). We claim that if a is irrational, all the leaves of a ( f, ,..., ) are dense in R 3 . In fact, let g 1 (x) =f1 ef2 (x) x + a and g2 (x) =--f3 -f4 (x) =

x + 1. It suffices to prove that, for every z E IR, the set B = (grg 2"( z) m,n EZ;

is dense in R. Indeed, grg 2n ( z ) = z + ma + n and, since a is irrational the set (ma + n m,n E t is dense in 111, so B is dense in IR. •

(4) Foliations on open manifolds.

As was already mentioned not every compact n-dimensional manifold has a codimen-sion k foliation where 0 < k < n. However when the manifold is open (i.e., not compact) the situation changes. The following theorem in this direction is known.

Theorem. Every codimension one plane field defined on an open manifold is hom °topic to a plane field tangent to a codimension one foliation.

This theorem is contained in a more general result due to Gromov [15 ] and Phillips [44] about homotopies of plane fields of arbitrary codimension, of which the following is a sample.

Theorem. Let u be a codimension k plane . field on an open Riemannian manifold. If the normal fibrat ion a of a has discrete structure group then a is homotopic to a com-pletely integrable plane .held.


(5) Foliations on 1R?.

A problem which has been studied recently is that of the characterization of a folia-tion when one imposes strong conditions on the topology of the leaves. A typical result in this direction is the following, due to C.F. Palmeira ([421).

Theorem. Let be a Cr ( r a- 0), codimension one foliation defined on R n ( n a 3). Suppose that all the leaves of S are closed subsets of " and are homeomorphic to a n - I .

Then there is a foliation on R2 and a C r diffeomorphism (homeomorphism if r = 0)

f n IR 2 x En-2 such that the leaves of are of the form f ( F X an 2 ), where F is a leaf of . In other words S is topologically equivalent to a product foliation of a foliation on 1R 2 with fibers 112' 2 .

This theorem can in fact be strengthened, to the following statement ( 1 421): Let and g be two foliations on lR ( n 3 ) whose leaves are closed and homeomorphic to Pe -1 . Then g and g are topologically equivalent if and only if the leaf spaces of and g are homeomorphic. This fact is not true for n = 2.

The above result admits other generalizations which can be found in [42 ] .

VI. ANALYTIC FOLIATIONS OF CODIMENSION ONE

A codimension n foliation g of an in-dimensional manifold is analytic when the change of coordinate maps which define a are analytic local diffeomorphisms of IV. Under these conditions any element of the holonomy of a leaf of g has a representation which is an analytic local diffeomorphism of litn.

For example, the Reeb foliation of S 3 is not analytic. Indeed, the Reeb folia-tion has a compact leaf whose holonomy is represented by the germs of two commuting diffeomorphisms of the line, f and g, with the following characteristics: f( x) = x, g(x) <x if x E (0,00) andf(x) < x,g(x) = x if x E ( — co , 0 ) . These diffeomorphisms are not analytic since an analytic map which coincides with the identity on an interval is the identity. It is natural then to ask if there exists analytic foliations of S 3 . In this chapter we will see that the answer is no. More generally, we will show that no compact manifold with finite fundamental group has an analytic foliation of codimension one. The results of this chapter are due to A. Haefliger ([171) and [181). The obstruc-tion to the existence of analytic foliations of codimension one is contained in the following theorem.

Theorem 1. Let 5 be a codimension one C2 foliation of a manifold M. Sup- pose there exists a closed curve transverse to 5 homoto pic to a point. Then there exist a leaf F of 5 and a closed curve r c F whose germ of holonomy in a segment J transverse to 3, with x o = J fl F. is the identity on one of the compone-nts of J ixo ) but differ from the identity on any neighborhood of x o in J.

Corollary. Let M be a real analytic manifold on which is defined an analytic foliation 5 of codimension one. Every closed curve transverse to 5 represents an element of infinite order in r i (M).

saddle


Proof of the Corollary. Let y S I --• M be a curve transverse to g and [ -y] its homotopy class. Suppose, by contradiction, that [y] has finite order, [ -y = 1. Then the path a: S 1 M defined by a (e''' ) = -y( e ) is homo-topic to a point. This yields a contradiction, since by Theorem 1, there is a path in a leaf of g whose holonomy germ cannot be analytic.

51. An outline of the proof of Theorem 1

In this section we will present a sketch of the proof of Theorem 1. In subse-quent sections this proof will be developed in detail.

Let a : S 1 M be a curve transverse to g homotopic to a point. This means that there is an extension of a to a map A : D 2 --• M. This map is continuous, but by the Stone-Weierstrass approximation theorem, we can assume that A is C (see [26]).

The map A is next approximated by a map g : D2 M transverse to 3:, except at a finite number of points I p i ,..., , where the tangency of g with the leaves of 5 is nondegenerate, that is, of one of the following types:

Figure 1

From this we obtain a foliation with singularities on D 2 , g* ( 5 ) , transverse to the boundary and whose singular set 1 p i ,•-., p,1 is made up of centers and saddles.

Analytic Foliations of Codimension One 117

When pi is a saddle of 5*, if we restrict to a small neighborhood V of pi as in figure 1, 5* I V has four integral manifolds Vi , -y2 , 73 , y4 which accumulate on pi . These leaves are called local separatrices of pi . If 7 is a leaf of 5* such that -y n V contains a local separatrix of pi , we say that -y is a separatrix of pi , and if y n V contains two local separatrices of pi , -y is a self connection of

a saddle. When y is a separatrix of two distinct saddles, we say that -y is a saddle connection and that the two saddles are connected. In the final part of the proof of the theorem we will use the fact that it is possible to obtain g so that no two distinct saddles of 5* are connected.

Observe that if pi and pi are two connected saddles of 5*, then g (pi ) and g (pj ) are contained in the same leaf of 5. When this occurs, we can modify g to obtain a function k : D 2 M, near g, such that k- (Jo and k- (7if ) are in distinct leaves of 5 whenever Pi are saddles of r

(3:). We can then sup-

pose, without loss of generality, that 5* does not have distinct connected saddles, possibly having self-connections (which are indestructible under small perturbations). In the figure below we illustrate an example of what can happen:

Figure 2

Observe that the foliation is locally orientable (even in a neighborhood of a singularity). Since D 2 is simply connected there is a vector field Y on D 2 , with singularities 1p1 pj, whose regular orbits are the leaves of Applying Poincaré-Bendixson theory to the vector field Y, one shows that there is a closed curve F, invariant under Y, and a segment J, transverse to Y, such that it is possible to define a first-return map f, in a neighborhood of x0 = = F n J in J, following the positive orbits of Y, which is the identity on one of the components of — x 0 l, but is not the identity on any neighborhood of xo in J. The image of F under g defines a closed curve g ( ) in a leaf of whose holonomy is conjugate to f. This concludes the proof.


§2. Singularities of maps f : R" JR

A singularity of a differentiable function f: IR" IR is a point p E IR' where df(p) = O. If ( x„) is a coordinate system about p, this means that

af af ax I n

A singularity p of f is nonclegenerate if the symmetric quadratic form

n 2 — 1

H(p) • u ---- D 2 f(p)(u,u) E a (fpx ) (x(p»ut.u., ax,ax,

is nondegenerate, where (u 1 un ) = dx (p) u. Since p is a singularity of f the form H(p) is independent of the system of coordinates chosen. So, if y = ( y, y,) : U --• IR", p E U, is another system of coordinates and dy(p) • u = (v i va ), one easily verifies that

n — 1 (t) E 2 (f y ) E a (f x )

(Y(P))vi - v 2

(x(p))u,- ui . ay,ayi aXi

j I j = I

Associated to the quadratic form H(p) and to the system of coordinates

X: V --• Rn , we have the symmetric matrix ( a2 (!: °,x ) x( p ))) 1 ,,, j„ fax,axi

which is called the Hessian of f at p with respect to x. The form H (p) is nondegenerate if and only if the Hessian matrix of f, in

some system of coordinates, is nondegenerate. These notions extend naturally to differentiable functions defined on mani-

folds. In fact, if f : M --• JR is a differentiable function, a singular point of fis a point p E M such that df(p) = O. A singular point of f is nondegenerate if for some system of coordinates x:UcM p E U, the function f e : x (U) --• IR has a nondegenerate singularity at x(p). In virtue of (.), this definition is independent of the system of coordinates. Moreover the Hessian quadratic form of f at p remains well-defined by

, H(p) • u = 19-( f ° x-1) (x(p)) u, E

ax,ax,

where (u 1 tin ) = dx(p) u.

Let p E M be a singularity of f: M IR. If E is a subspace of Tp M, we say that H( p) is positive (respectively negative) definite on E if H( p ) • u > 0 (resp. H(p) - u < 0) for every u E E — 101. The index of f at p is the largest possible dimension for a subspace E c 7,,M where H(p) is negative definite:

j = 1

I, j = 1

Analytic Foliations of Codimension.One 119

ind ( f,p ) = max dim ( E) 1E is a subspace of Tp M

and H(p) • u < 0 if u E E — 10j) .

The proof of the result below can be found in [36].

Theorem (Morse Lemma). Let p be a nondegenerate singular point off with ind ( f,p) = k. There is a system of coordinates x = (x 1 ,..., xn ) on a neigh-borhood U of p such that

f(q) = f(p) — x(q) xl(q) + (q) +...+ x2n (q) .

In other words, the function f is locally equivalent to the function

2 2 2

In the case n 2 one has three possible canonical forms:

(a) f(x) = f(p) — x — x; (b) f(x) = f(p) + x +

f(x) f(p) — + x.

In the first two cases the level curves of f in a neighborhood of p are dif-feomorphic to circles and p is a local maximum or minimum, respectively. In the third case p is a saddle point of f and the equation f(x) = f(p) defines the four local separatrices of p (see figure 1).

We say that f : M IR is a Morse function when all its singularities are nondegenerate.

Next we will see how one defines the Cr uniform convergence topology on the space of Cr functions of M to 1R, CT ( M,IR ), in the case when M is com-pact. For r = 0 and f E C° ( M,1R), define

jf = sup !f(x)i • xEM

One easily verifies that 11 . 10 defines a norm on C° (M,111). With this norm C° ( M,Ilt ) is a Banach space. For r 1, the CT norm can be defined by means of finite covers of M by systems of coordinates. Since M is compact, consider covers (U,), and ( V,) k, , of M such that for every î E l ,..., kl, one has U, C V, and V, is the domain of a local chart : V, 1Rn. Given f E C'(M,ilfl, define the Cr norm of f on U, by

flir,, = max ( sup I'D' (f ° so, --1 )(so,(P)) pEU,

For each j 1 and each p E U„ D3 ( f ,,c, --1 )( ça,(p)) = L., is a sym-


metric j-linear map from Ilt" x x IW = ( )i to IR, where 11L1 11 = = supl!lLi (u, uj ) , = ..-= = 11.

The Cr norm of f with respect to the cover ( U1 ) ,k= , and ( 1/0 is defined by IlfIlr = max i = k [1f11,,1- One verifies that ll 1 1,. is a norm on Cr (M, IR ) . With this norm this space is Banach.

Let ( U/) 1 , (171') 1 be other covers of M such that for every j E 1 ,..., Li, c V; is the domain of a local chart ;t i : V; 1W. Let i l:be the Cr norm

induced on Cr ( M, 1R ) by the covers ( and ( 1/1') fi=1 . One verifies easily that there exist constants K2> K 1 > 0 such that

1111; d iI K2 11b/- •

Therefore, the Cr uniform convergence topology is well-defined. For more details see [27]. The result stated below proves, in particular, the existence of Morse functions.

Theorem. Let M be a compact manifold, with or without boundary. If r 2 the set of Cr Morse functions is open and dense in Cr (M,1R ).

The proof can be found in [36]. The following lemma will be used in the next section:

Lemma 1. Let U,V and W be open sets of lit" where U C VC C W and W is an open ball. Let h: W JR be Cr, r _>1 2. For every > 0, there is f: W • It which is Cr and such that Ilf - hri r < E, fi(W- V)= =hl(W - V) and fl U is a Morse function.

Proof. By the preceding theorem, given (5 > 0 there is an f0 : W JR

such that fo is Morse and ll fo -hjj < (5. Let cp : JR be C' and such that ço U 1, j (1Rn - V) 0 and 90 ?.: O. Let f = fo + (1 - Oh. We have ( f - h) p ( - h) = so • a. Differentiating ,p • a k times we get D( = E;_ o Cik DJ Op) D (a) , where for every j = 0 k,

is a constant independent of a and of so. We have then

li Dk (40 0110 EjCJ iDiseqo •

lc_ E Icy ; )=0

Letting

max Ecfl fD I0I , Os_klc_r j-0


we get !ID" Op • a )11 0 Ls. a • (5 for 0 Ls. k s r. So, if .5 is sufficiently small, we will have II If — h11, < E.

In an analogous manner one defines the Cr uniform convergence topology for maps between manifolds (see [27]).

S3. Haefliger's construction

In this section if will denote a Cr( r 2) codimension-one foliation defined on a manifold M of dimension 3. If h : D 2 M is a C map, we say that p E D 2 is a point of tangency of h with if if Dh(p) IR 2 c Th(p) g.

Proposition 1. Let A : D 2 M be a Coe map such that the restriction A 1 3D 2 is transverse to if. Then for every E > 0 and every r 2, there exists g : D2 M, Coe , E near A in the cr topology and satisfying the following properties:

(a) g I 3D 2 is transverse to if. (b) For every point of tangency p E D 2 of g with if, there exist a foliation

box U of if with g ( p) E U and a distinguished map : U IR such that p is a nondegenerate singularity of r o g : g (U) IR. In particular there are only a finite number of tangencg points, since they are isolated, and they are contained in the open disk D 2 = fzEIR 2 bld <1).

(c) If T = [pi p 1 is the set of points of tangency of g with if, then g ( p,) and g (pi ) are contained in distinct leaves of if, for every i j. In par-ticular the singular foliation if * = g* (if) has no distinct connected saddles.

Proof. Let Q 1 ,..., Qk be foliation boxes for if whose union covers A ( D 2 ) and 7r, : 42, --• JR distinguished maps of a, i = 1 k. Observe that A meets if transversely at p E A (Q i ) if and only if D( A ) ( p) O. We will use the notation W, = A - (Qi ), 1 = 1 ,..., k.

It is clear that U t W = D2 and since D 2 is compact, there exist covers vi ) ;(= and (U, )',L, of D 2 by open sets such that Li, c V, c V, c W, for

every i = I k. Let b = min i <i< k d (A (x), y) x E -V, and y E aQi l, where d is a

metric on M. If f : D 2 M is such that do ( f,A) < 6 where do ( f,A) = = sup1d( f (x),A (x)) ix E D 2 1, then f c Q, and therefore the maps r i of: JR are well-defined. All the maps we consider from here on are such that do ( f, A ) < b.

For each i = 1 ,..., k consider a foliation chart of a, : Q, ---• IR", such that 40, = cp _ 1 ,7r i ) . The nonempty subsets of Q, defined by ( t ) are the plaques of if in Q,. We have then jo o A = (A; ,..., A i„ 1 ,7 , A). A point p E V, is a point of tangency of A with if if and only if


D(7ri o A)(p) = O. The construction of g satisfying (a) and (b) will be made by inductive modifications of A I U 1 W„ j E (1 ,...,

For j = 1. By Lemma I of §2 there exists A : JR such that AI u1 is Morse fl ( – ) ir 1 o A I (W 1 – V I ) and f1 – ° Ah, < S I . We now define g i : D 2 M setting g i I (D 2 – V 1 ) -= A I (D 2 – V 1 ) and g l W1 ---- ,..., ) . Since R- 1 o A coincides with f1 on W 1 – V I , it is clear that g 1 is well-defined and that 7r 1 g1 I U1 = f 1 U1 is a Morse function. Further, taking (5, sufficiently small, we have cl,.(g 1 ,A) < c/k.

Suppose by induction that a map : D2 — • M, 1 j k – 1, has been constructed so that dr (A,gi ) < k, g1 I (D 2 – U Vi ) = A I (D 2

V,) and for every i = 1 : U1 JR is a Morse function. By Lemma 1, there exists f : JR such that (1/1/J+1 – Vi±1 ) =

= 7r1, ° g1 ( VV.; – , 4_ 1 1 Uji_ i is a Morse function and 114 +1 –

Ir./ -t- 1 ° gj r < 5,/ + I • Then set ( D 2 – Vi , 1 ) = g1 I (D 2 – Vi , 1 ) and

g»,1 I Wi sc j 0 (g ,..., _ 1 ,4_ 1 ) where (g1 gi ) = It is clear that gi+ , is well-defined since f I ( NPV».-1 V.1 -1-1) =

= gf – 1'i+1 ). Moreover, if Ei is sufficiently small we will have dr (g»,,,gf ) < c/ k and also vi 0 gi+1 : U JR a Morse function, i = = I , , j + I, since the set of Morse functions is open and j in, gi+ , – ri gj t. can be taken arbitrarily small (in Ui n Uf+i ). Proceeding induc-tively we get finally gk : D 2 M such that for every i = I k, 0 gk : JR is a Morse function and cl,.(gk ,A) < e. From this we obtain gk satisfying (b). Observing that A aD 2 is transverse to g, we see that gk /aD2 is transverse to g if € is sufficiently small.

Let T = Al be the set of tangencies of gk with Next we will in- dicate how it is possible to modify gk to obtain g such that the set of tangencies of g is T and g satisfies (c). Suppose for example that g ( p i ) and g ( p2 ) are contained in the same leaf F of S and that p i E U,. Set f = gk : V, IR , t = f ( p i ) and fix neighborhoods U C V of p i such that U C Vc Vc V; and p i is the only singularity of f, in V. Let or : V1 IR be a C function such that a 1 U 1, a ( – V) 0 and a a- O. De- fine 1; = + bor where (5 > O. Since dji + WOE, one verifies easily that if ô

is sufficiently small then p i is the only singularity of in V, which is non-degenerate. Moreover the set of singularities of . coincides with that of since (f, – fi ) (V i – V) 7.-2 O. Define then h :D2 M setting gk 1

_1 , ° kg1 en- ) and g, (D 2 — v) gk (D 2 – V). One easily

verifies that T k satisfies (a) and (b) if (5 is sufficiently small and that the set of tangencies of k is T.

Observe now that F fl Qi contains at most a countable number of plaques of S so there exists (5 0 arbitrarily small so that in,' (t + (5) = ir, ( .7i ( p 1 ))

F, or gk ( p i ) and Tk (p,) are in distinct leaves of . Repeating the process


a finite number of times it is possible to obtain g, arbitrarily near A, whose set of tangencies p l ,•-•,Pfl is such that g ( pi ) and g (p1 ) are in distinct leaves of 5 if i j. •

Corollary. Let -y : S M be a C curve transverse to a homotopic to a constant. Then there is a C' map g : D2 ---• M such that g* (g) = 5* is a foliation with singularities, satisfying the following properties:

(a) g 3D 2 is homotopic to -y and g* is transverse to 3D 2 , (b) the singularities of g* are saddles or centers (see §2), (c) 5* has no distinct connected saddles.

Proof. Since -y is homotopic to a constant in M, there is a Cc° map A : D 2

M such that A 1, 3D 2 = -y (see [26]). It suffices now to take g as in Pro-position 1, where dr (g,A) < E, E sufficiently small. IN

S4. Foliations with singularities on D 2

In this section we denote by S* a foliation of D 2 with a finite number of singularities and transverse to the boundary 3r)2 . The set of singularities of 5* will be denoted by T. We say that 5* is Cr locally orientable if for every p E D 2 there exist a neighborhood U of p and a Cr. vector field Y on U such that if qE U— T then Y( q ) 0 and is tangent to the leaf of 5* through q, and if q E T then Y( q ) O. We say that 5* is CT orientable if there is a vector field Y as above, defined on the whole disk D 2 .

Proposition 2. is a foliation with singularities on D 2 , CT locally orien-

table, then g* is Cr orientable.

Proof. For every p E D 2 there is a Cr. vector field Y defined on a neighborhood U of p such that Y is tangent to a* and Y( q ) = 0 if and only if q

is a singularity of 5*. Let U, Uk be a cover of D 2 by such neighborhoods and Y, Yk the corresponding vector fields.

The problem of determining a vector field Y as desired reduces to con-veniently orienting Y, Yk , obtaining a new collection of vector fields

such that for every i E I ,..., k , P, = Yi or V, = - Y, and, fur-ther, if q E (U, n uj ) - T then V ( q) and fi ( q ) have the same orientation.

In fact, suppose for a moment that this is possible. In that case, it suffices to consider a partition of unity ( ço, ) A, = , and define

2 Since ( D - Ui ) -= 0, the vector field f, extends to a Cr vector field


on D 2 setting (ça , Vi ) ( q ) 0 if q U. So Y is well-defined and is C' on D 2 . It is tangent to 5*, since each P, is. Further Y (q) = 0 if and only if q is a singularity of a* since if q is not a singularity of 5*, at least one of the vec-tors <pi • i, (q) is not zero and those that are not zero coincide in direction and sense.

Now we will see how it is possible to obtain vector fields fik as above. First observe that we can assume without loss of generality that the open sets U, U. are connected so that if U, n Ui 0 then U, n u; is connected. On (U n T we can write Yi (q) X,, ( q ) }",(q) where X ,, : (U, fl Ui ) T JR is continuous and does not vanish. So X 0 or X6 < O. If X , > 0 we say Y1 and Yi have the same orientation. If X,i < 0 we say they have opposite orientations.

Fix = Y, and a point p, E U1 . Given j > 1 , fix a curve -y : [0,1] D 2 such that -y ( 0 ) = p l and -y ( 1 ) = pi E Ui . Consider a cover (V) ;11_ 1 of y(1) by neighborhoods so small that V, n Vi 0 if and only if ) = i + 1 and for every i we have 1/, c U, , . Now define V, , , 1 s n, inductively so that Pi = Y, and has the same orientation as V. Note that if f =11, we have f, fi; = ± Then set Pi = Further, if we take a curve 6 : [0,1] D 2 such that 6 ( /) C the definition of fi along 6 will coincide with the definition of V along -y.

It follows from this fact and from D 2 being simply connected that the defini-tion of V is independent of the curve chosen. We leave it to the reader to verify the details. •

Corollary. Let be a Cr (r 2 ) codimension one foliation on M and g : D 2 M a Cr map such that the points of tangency of g with 3: are nondegenerate. in the sense gr (b) of Proposition 1. Let ------ g* (3: ). Then is C' orientable.

Proof. Let Q, Qk be foliation boxes of 5 such and r, : Q, IR be distinguished maps of a on Q. of 5* in Q, are the connected components of level

g = g,. One easily verifies that for every i E 11

3g 3g Y, ( ---L, ) is tangent to the level curves

ax„

5* is Cr - locally orientable, hence Cr - '-orientable.

that U Q, D g (D 2 ) The nonsingular leaves curves of the function

k the vector field

of g,. This proves that

We will now see some properties of the vector field Y constructed above, in the case that 5* satisfies the properties of the corollary of Proposition 1.

Consider first a singularity p of Y, where pE W=g (Q), Q being a distinguished map of 5. Let it : Q fR be a distinguished map of 5. Then p is a nondegenerate singularity of the map f= it og: W IR and the orbits of Y are contained in the level curves of f. By the Morse Lemma there is a diffeomorphism a : U V, where pE VC W and a (0) = p, such that


f ce(x,Y) = f(p) + q(x,y)

X2 2 + y 2 0 _

r where q(x,y) (x 2 + y 2 ) or x 2 — y 2 . In the first two cases p is a center of Y, that is, the nonsingular trajectories of Y in Ware all closed. In the last case p is a saddle of Y, that is, f -1 (f(p)) is made up of two dif-ferentiable curves S i and (52 that meet transversely at p so that S I U •52 — f contains four regular trajectories of Y V, which we call a 1 , a2 , a3 , a4 where a l U a3 U [IA = 6 1 and a2 U a4 U tP62.

The trajectories aj , j 4, are the local separatrices of Y. Two of them, say a, and a3 , are characterized by the property lirn,„ Y, ( q ) = p, q E aj ( i=1 ,3 ) and the other two are characterized by the property lim,, _ Y, ( q ) = p, q E a, (i = 2,4). The trajectories of the first type are called stable and those of the second type unstable.

The fact of g 3D 2 being transverse to 5 implies that Y is transverse to 3D 2 S'. We can suppose that Y points to the inside of D 2 along S', since in the other case we simply substitute Y by — Y.

Given a trajectory -y (that is, -y t Y, (q), q E D 2 ) we define co ( y ) = tx E D 2 x = lim, -y ( ) where 1im 00 I and a (y) lx E D 2 x = lim, 'y (t , 1 ) where lim, t = By the Poincaré-Bendixson theorem (see [201), the set w ( -y ) can only be a

singularity, a closed orbit or a graph of Y. A graph is a connected set r formed by saddles and separatrices of saddles such that if p is a saddle of F then at least a stable and an unstable separatrix of p are contained in F. Analogously, a (y) can only be a singularity, a closed orbit, a graph or a ( -y ) = 0 in the case where -y meets a 13 2 .

Definition. We say that r c D' is a limit cycle of Y if r does not reduce to a singularity and there is an orbit -y of Y such that y n F = 0 and (.0 ( -y ) = r or a ( -y ) F.

From what we saw above, a limit cycle F of Y is a closed orbit or a graph of Y. One also can verify that r is always connected and that D 2 — r contains at least two connected components, one of which contains 3D 2 r n 3D 2 = 0). In the figures below we illustrate some possibilities for in the case that it is a graph:

Figure 3

In the ease we are considering, Y has no connections between distinct saddles and hence the above possibilities are the only ones up to homeomorphisms.


Let E be the set of all limit cycles of F. Given r E E we denote by R ( F ) the union of the closures of the connected components of D 2 — r which do not contain a D 2 . On E we define a partial ordering by

F , 5_ F, R(f 1 ) c R( 1' 2 ) .

Observe that R (r) is invariant under Y, since r is.

Proposition 3. Let (r ) noN be a decreasing sequence of limit cycles and F o,

the boundary of the set fl „ I R ( Fa ). Then F a closed orbit or a graph of Y.

Proof. Without loss of generality we can suppose that all the terms of the sequence ( ro „IN are distinct. Since the number of saddles of Y is finite, Y has a finite number of graphs and hence we can also suppose that all the are closed orbits. Under these conditions closed orbits of Y accumulate upon F c,,. For every n 1, R (r „ ) is invariant under Y, homeomorphic to a closed disk and R ( r„ ) c R ( rn ). It follows therefore that n„, R ( r,,) and F invariant under Y and nonempty. Further roe contains at least one point p such that Y( p ) O. In fact, there are no centers in r o,, since those are not accumulated upon by limit cycles. On the other hand, if F a saddle, it must also contain two separatrices.

Suppose that F cc is not a closed orbit of Y and let p E F oe be a regular point. We claim w ( p) ( * ) is a saddle contained in F. First observe that w ( p) does not contain a center of Y. If w ( p) were not a saddle of Y, w ( p) would contain a point q such that Y( q) 0, since w ( p) is connected.

Observe that since r 3, is not a closed orbit of Y then q '0(p) = IY,(p) t E IR I. The argument now is analogous to that of the Poincaré-Bendixson theorem. Let (5 be a small segment transverse to Y with q E 6. Then 0 ( p) meets

in a monotonic sequence p„ = Y en ( p) where lim t„ = co and lim „_, p„ = q (see figure 4.).

Figure 4

(•) (.4( p) = u -y) and( p) = c (y)where is the orbit of Y such that (0) = p.


Let 77 be a closed curve of D 2 made up of the union of segments [pn ,p„, 1 ] C 6 and = Y, ( p) tn t tn , 1 1 C 0(p) and let be the connected component of D 2 — n which contains q. An orbit which cuts the segment [pn ,p„, 1 ] cannot be closed, since it enters the region and never again leaves, so p„ cannot be an accumulation point of closed orbits of Y, a contradiction since pn E r oe . Hence w( p) is a saddle. Analogously a ( p) is a saddle.

Observe that since a saddle in r oe is an accumulation point of closed orbits of Y, then it has at least one stable and one unstable separatrix contained in r oe (see figure 5). Hence F o, is a graph.

Figure 5

§5. The proof of Haefliger's theorem

Let be a C 2 codimension one foliation on M and suppose y : S I M is a closed curve transverse to 5. , homotopic to a point. By the corollary to Proposition 1 there is a map C g : D 2 M such that g* (a) = a* is a foliation with singularities on D 2. By the corollary to Proposition 2 there is a C' vector field Y, tangent to .S*, whose singularities are the same as those of . This vector field Y is transverse to 3D 2 and on 3D 2 points inward toward D 2 , its singularities are centers or saddles (finite in number) and there do not exist connections between distinct saddles.

Let f, he the set of limit cycles of Y, to which we add the closed orbits and graphs obtained as in Proposition 3 of §4. Observe that if is nonempty since if p E 3D 2 , w(p) is a limit cycle. Moreover by Proposition 3, is such that if (r,,) „N is a decreasing sequence in , then there exists F such that r oe F, for every n E N. The order relation on is analogous to that


already considered on E. By Zorn's lemma has a minimal element r. Such a r bounds a region R(r) whose interior contains no limit cycles, that is, for every p E R(r), 0(p) is a singularity, a periodic orbit or O ( p ) is con-tained in a graph.

Figure 6

Consider now a segment 6, transverse to Y, cutting I' in a point p such that Y ( p) O. Suppose first that F is a closed orbit of Y. In this case it is possible to define a holonomy transformation f: 6 1 b. This transformation is the identity on 6 1 n R (T) but not the identity on any neighborhood of p in 6, since if it were, r would not be in r. Let /2 = g (F) and n = g(6). Then kt is a closed curve in F, the leaf of 5 which contains g ( p), whose holonomy h is given by h(x) = g o f o (g/ 6 1 ) -1 (x). It is clear that h is the identity on one of the components of ij — g ( p), but not the identity on any neighborhood of g(p) in n.

Suppose now that r is a graph of Y. In this case the transformation f can only be defined on an open segment 6 1 c â — (6 fl R(F)) . We can extend f continuously to p so that f(p) = p. On 6 fl int ( R (F ) ) , although the orbits of Y are closed, the "holonomy transformation" in this component does not cor-respond to the curve r (see figure 6). Nevertheless the segment g(S) is transverse to 5 and the element h of the holonomy of a. corresponding to g (r) c F is unilaterally conjugate to f, that is, if x E g ( 6 1 ) we have h ( x ) = = g o fo (g/6 1 ) -1 (x) and hence h is not the identity on any neighborhood of g ( p) in g(6). It is possible to prove however that h is the identity on g(62 ), where 62 = â n R(r). Indeed, let s be the saddle of Y which is con-tained in F. Any regular orbit -y C r is such that w ( y ) = a ( -y ) so r contains one or two regular orbits (see in figure 3 the last two pictures sketched). In the case that r contains only one regular orbit, the other two local separatrices of Y must be contained in R (F ) , since a graph r as in the next figure cannot be a minimal element of If.


6 2

Figure 7

In figure 7, 0, and 02 are not in F. We leave to the reader the verification of the above-mentioned fact. Note

that this implies that the two local separatrices of s which are not in F must be connected, since otherwise there would be a limit cycle contained in the in-terior of R(F).

Therefore the separatrices of s are pairwise connected, forming closed curves U fsi and -y2 U fs) as in figure 6, where = 7 1 U -y 2 U 1s1 or =

= 7 1 u Is] and ô n 0. Consider for example the first case ( F = -y U -y2 U Est ) . Take a segment

(5' transverse to Y such that 6' fl -y2 0. Let 62 = R(r) fl 6, ô = R(F) fl 6' and .5 1' = 6' –

We can then define unilateral holonomy maps f, : 62 (5 2 and f,' : —0- ô. We have f1 (x) = x if x E 43 2 and f i'(x) = x if x E ô , since the orbits of Y in the interior of R (F) are closed. Let A i = g (1, 1 U (51 ) and ,u2 = = g ( -y2 U i s ) ) and h 1 , h 1' be the holonomy maps of A t and which are defined on open segments of g (6) and g ( ' ) .

The restrictions h, I g (ô2 ) and h g (6 ) are conjugate to f, and so h, ( x ) = x if x E g (62 ) and h i' (x) = x if x E g (6;). Since the holonomy map h of g (F) is the composition of those of pt, and p2, one easily sees that h g (62 ) = identity.

The case that = 6 1 U is analogous to the above and we leave it to the reader. This ends the proof of Theorem 1.

Theorem 2. Let M he an analytic compact manifold with finite fundamental group. Then there exists no analytic jbliation of codimension one on M.

Proof. By Lemma 7 of Chapter IV there is a closed curve 7y" : S 1 M transverse to fF. Since M has finite fundamental group, [y ]" = I for some n, so the curve -y ( ) = (eint ) is homotopic to a constant on M and is trans-verse to EF . It now suffices to apply Theorem 1. •

In particular, there are no analytic . foliations of codimension One on S3.

VII. NOVIKOV'S THEOREM

The following theorem, due to Novikov [40], is one of the deepest, most beautiful theorems in foliations.

Theorem. Every C 2 codimension one foliation of a compact three-dimensional manifold with finite fundamental group has a compact leaf

In particular every C 2 codimension one foliation of S 3 has a compact leaf homeomorphic to T 2 . This theorem has an elaborate proof; for this reason we will first give an informal sketch of the more important steps. Later we will see a rigorous proof.

§1. Sketch of the proof

The first step consists of showing the existence of vanishing cycles. A vanishing cycle is a map f0 : S' M contained in a differentiable family f,: S' --• M, t E [0,Ej, E > 0, such that: (1) for every t, f(S 1 ) is contained in a leaf A„ (2) f,: S' --• A, is homotopic to a point (in A,) if and only if t 0 and (3) for every x E S the curve t f, (x) is transverse to a .

Example I. In the Reeb foliation of the solid torus D 2 x S', any meridian of the boundary is a vanishing cycle.

The existence of a vanishing cycle follows, as one would expect, from the techniques of Haefliger, described in Chapter VI. The vanishing cycle is then modified to get a simple cycle in the following way: if 71- ,1, : A, denotes


the universal covering of A„ then the lifts j : A, of f, are imbeddings

for every t O. The rest of the proof of Novikov's theorem consists of proving that the ex-

istence of a simple vanishing cycle implies the compactness of A o . Having a simple vanishing cycle ( f,), (1 , it is possible to prove that there is a family of immersions F,: D2 M such that F,1 S 1 = f, and F1 (D 2 ) c A, for every

t 0 and such that for every x E D 2 , the curve t F, (x) is normal to a (Figure 1).

Figure 1

Next one shows that for any a > 0 there exist t, , t 2 , O < t2 < t i < a, such that (1) 24 11 = (2) F,, (D 2 ) C F1, (D 2 ) and (3) for every t such that o < t2 < t < t one has F, (D2 ) n F, (D2 ) = 0. (Figure 2). Hence it follows that arbitrarily near A 0 there is a region R bounded by the surfaces

= U F,(x) and C2 = F, (D2 ) - F, (D2 ) 1

xES'

'2 <I< (1

such that A 0 fl R = 0. This region has the following property: any simple curve transverse to 5: that enters R is trapped to wander indefinitely without being able to leave R. Since R is arbitrarily near A o it follows that through the leaf A o there passes no closed curve transverse to S. On the other hand, one easily shows that if A is noncompact then there exists a closed curve transverse to S passing through it. So A L, must be compact.

Novikov's Theorem 133

Figure 2

52. Vanishing cycles

Consider a codimension one foliation 5 on a manifold M.

Definition 1. Let A 0 be a leaf of . We say that a closed curve fo : S 1 A o

is a vanishing cycle /./. fo extends to a differentiable map F : [0,d x S 1 M satisfying the following properties:

(a) for every t E [0,d. the curve f1 : M, defined by f (x) = F(t,x)

is contained in a leaf A, (b) for every x E S I the curve t F(t,x) is transverse to ,

(c) for t > 0, the curve f, is homotopic to a constant in the leaf A, and fo

is not homotopic to a constant in A o .

We will call a map F as above a coherent extension of the vanishing cycle fo . In this section we will examine diverse properties of vanishing cycles. We

begin with an existence theorem:


Proposition 1. Let M be a compact manifold of dimension n a- 3 with finite fundamental group and a' a C 2 codimension one foliation of M. Then a has a vanishing cycle.

Proof. Since M is compact and r i (M) is finite by Lemma 7 of Chapter IV, there is a curve a : s' transverse to and homotopic to a point. Under these circumstances, by Haefliger's construction (Chapter IV), there is a C' map g: D 2 M such that the foliation a'* = g* of D 2 is given by the orbits of a vector field Y on D 2 whose singularities are centers or saddles and such that the only saddle connections permitted are self-connections.

By what was seen in §5 of Chapter VI, there exists an open region R of trivial holonomy in Y such that the boundary aR is a curve with nontrivial holonomy (see figure 6 in Chapter VI).

The proof that a. has a vanishing cycle consists of finding a curve F C R which is a periodic orbit or a graph invariant under Y such that (1) g ( F ) is not homotopic to a constant in the leaf A g(r) and (2) F is the boundary of a band C c R, a union of periodic orbits of Y, C = U (E(01] -y„ where g (-y, ) is homo-topic to a constant in the leaf A g(,) .

Since the tangency of g with a at the centers is of parabolic type, all the regular trajectories y of Y, near a center, are closed and have image g ( -y )

homotopic to a point in A g() (see Figure 1 of Chapter VI). It is natural then to look for F starting from the centers of Y in R. Observe that if the number of centers of Y is f 1, then the number of saddles is f – 1. This is a conse-quence of the index theorem (see [35]). The same is true for the closure R of R, that is, if c, ck ( k 1) are the centers of Yin R, then Y has k – 1 saddles in , say s, s . For i = 1 k let V, be the maximal connected subset of R that contains c, and is a union of c, and closed orbits of Y homotopic to a constant in their respective leaves.

The set V, is, by construction, non empty and invariant under Y. Moreover it is open and av, = v vi is a closed connected curve in D 2 which is a union of orbits of Y.

In fact, let -y () C V, . Since all the orbits of Y in R have trivial holonomy, it follows from Theorem 3 of Chapter IV, that there exist a cylindrical neigh-borhood C0 of -yo , Co C R, and a diffeomorphism h : ( - 1,1) x -yo —0 Co , such that for any t E ( — 1 , 1 ) , = h( [ti x -y o ) is a closed orbit of Y. Denote by A, the leaf of fF which contains g ( -y, ) . Let us fix some Riemannian metric < , > in M. Since g (1 ) ) is homotopic to a constant in A o . there exists a dif-ferentiable map fo : D 2 —• A, such that fo ( 3D 2 ) = g ( y 0 ) . It follows from Lemma 4 in Chapter IV that fo extends to a continuous family of maps f,: D 2 M, t E ( f ,E ), such that f, (D 2 ) C A, and for each x E D 2 the curve t f, (X) is normal to . The curve f,(aD 2 ) = 6, is clearly homotopic to a constant in A,. Moreover, for t small, the curves 6., and g ( -y, ) are homotopic in A,, because u , b, = lim 0 g ( -y, ) = g ( "y ( )). Therefore g ( -y, ) is homotopic to a constant in A„ which implies that V, is open.


Since V, is invariant under Y, it follows that av, is either a closed orbit or a graph of Y. If av, is a closed orbit 70 of Y then, g (y o ) is not homotopic to a constant in A g ( .o)• Since 70 C R, it follows that in the interior of -y all the orbits of Y are closed, i.e., -yo belongs to a family of orbits of Y, t -y, „ , such that for t > 0 g (y e ) is homotopic to a constant in A„ and we are done. In the other case, av, = r, is a graph of Y and we have two possibilities. The first is that there is 1 i k such that g ( ro is not homotopic to a constant in A g(r ). In this case g (F,) is a vanishing cycle in A g(ç) . The second possibility is that for all i = 1 k, g (r ) is homotopic to a constant in A g(r ). Since the difference between the number of centers and saddles in —12- is one, it follows that two of these graphs, let us say r, and r2 , contain the same saddle s l as in the following figures.

Figure 3

In this case, V, U V2 is homeomorphic to a closed disk or to two closed disks with one point in common. Consider the graph r' aT7, U 0i 2 = r, u r2 . Since g ( F 1 ) and g(F2 ) are homotopic to constants in A g(ç1) = g(12) = A g(r i ) , it follows that g (F 1 ) is homotopic to a constant in A g(r 1 ) .

So there is a neighborhood U of F' such that U F' consists of closed orbits of Y, homotopic to constants in their respective leaves. This can be verified using a simple extension of the proof of Lemma 4 (Chapter IV).

Let r' = r , where k i :5_ k/2, be all the graphs of Y obtained in the same way as F'. For each i = 1 ,..., k l , let V: be the maximal connected set which contains F , 1 and such that V: — r) consists of centers and of closed orbits of Y homotopic to constants in their respective leaves. By the arguments used above V: is open. Its boundary is a closed curve in R, union of orbits of Y. As before, if there exists 1 5. i k, such that g ( 31/)) is not homotopic to a constant in A„, , then g ( 8 v) is a vanishing cycle in A .

There remains the possibility that for every i = 1 ,..., k l , g (av;) is homotopic to a constant in A,, a , . In this case aV: is a graph for every i = 1 ,..., k i .

Again at least two of the , let us say 0V and ay,' , contain the same saddle.


Figure 4

Let 172 = â V U ô V. Clearly r2 is a graph of Y and g (r 2 ) is homotopic to a constant in Ag(r2 ) . If k2 is the number of graphs as above, we evidently have 1 s k2 ( 1 / 2) k 1 . Moreover r2 has a neighborhood U such that U — r2 con-sists of closed orbits of Y homotopic to constants in their respective leaves. Since aR n aD 2 = 0 and there are only a finite number of singularities of Y in R, proceeding inductively with the above argument we get, after a finite number of steps, a closed orbit or a graph r of Y such that g (F ) is a vanishing cycle in A g(r) . •

Suppose now that 5 is a transversely orientable foliation of a manifold M and X a vector field normal to 5 with respect to a Riemannian metric < , > on M. We say that a curve transverse to 5, y : I M, is positive (respec-tively, negative) if, for every t E I, ( y ' t),X ( -y ( t ) ) > > O (resp. < 0).

Let f0 : s' A o be a vanishing cycle and F: [0,e] x S t M a coherent extension of fo . We say that fo is a positive (resp. negative) vanishing cycle if the curves t F(t,x) are transverse positive (resp. negative). If further, the curves t F(t,x) are orthogonal to the leaves of 5, we say F is a normal extension of fo .

Proposition 2. Let fo : s' A o be a positive vanishing cycle. Then

( i) there is a normal positive coherent extension of A, (ii) ifjo : s' A ° is homotopic to fo in A o , then jo is a positive vanishing

cycle.

Proof. (i). Denote by X, the flow generated by the normal field X. The map f( t,x) = X, ( fo (x)) is well-defined on [0,6] x s' for (5 > 0 suffi-

ciently small. Let us reparametrize now the curves t f ( 1 ,x) by a

1—+ t ( ,x ),x) in such a way that the new map -f(a,x) = P(t(cv,x),x)

is a normal positive coherent extension of f0. Since jo is a positive vanishing

cycle, the positive holonomy of L is the identity. This implies that the integral

curves of the foliation f*( ) of [0,6] x s' near 0 x s' are all simple closed

curves. Fix xo E s' and let S, denote the integral curve of f* ( s) passing


through ( t, x0 ) E 10,61 x s'. Thus it follows that, for a 0 small, S

the curve t ( t, x) in only one point ( t ( a,x),x).

Figure 5

By construction, the map F : [OM X S' M given by F( a, x) =f(t(a,x),x)

satisfies conditions (a) and (b) of the definition of coherent extension. We show now that by reducing more the domain of F we can guarantee that the curves :fa (x) = ( a, x) are homotopic to zero for a > O.

Indeed, let F: 10,F] x s' f,(x) = F(t,x) be a positive coherent extension for L. For simplicity suppose F and that, for t f, (S') and — f,(S 1 ) are in the same leaf. It is clear that for t small the curves f, and 7, are close in the intrinsic topology of A, and therefore are homotopic. Con-sequently, restricting to a domain of the form [0,E1 x S', E Z, we have that T is a normal, positive coherent extension of fo .

(ii) Observe first that there exists E > 0 such that if f0 : s' A o and do ( fo ,f0 ) < E then 70 is a vanishing cycle, where do is the intrinsic metric on A 0 . The proof of this fact can be done considering positive normal exten-sions F: [0,6] x S I M and T: [OA x s' Ai of fo and 70 re-spectively, where F is coherent. The proof that the restriction of F to an interval of the form [0,E] x s' < E ) is coherent reduces, as in (i), to proving that the curves f, and f, are close for small t, in the intrinsic metric of A„

and therefore are homotopic. Suppose now that L and 70 are homotopic in A o and that fo is a vanishing

cycle. Let : [0, 11 x Ao be a homotopy connecting fo and L. By the preceeding observation and by compactness of the interval [0, 11, there exists a positive integer n o such that if ‘,C, k no ( X) = (pk (x) is a positive vanishing cycle, then ( x) is a positive vanishing cycle. By transitivity, we conclude that ion° (x) = J0 (x) is a positive vanishing cycle.


S3. Simple vanishing cycles

From now on we assume that M is a compact orientable three-dimensional manifold and if a C2 foliation of codimension one on M.

Let fo : S' A o be a positive vanishing cycle and F: [0,E] x S' M a positive, normal coherent extension of fo . By transversality theory and by Proposition 2 we can assume, taking an approximation of fo , that (j) f0 is an immersion, (ii) the points of self-intersection of fo are double points and transverse, that is, the sets of the form ./V ( p ) contains at most two points and if p = fo (x l ) = fo ( x2 ) with x, x2 in S I then the vectors f6 (x, ) and

f,1 (x i ) generate the tangent space to A o at p.

With this consideration, it is clear that the set Bo = 1p Efo(S I ) IA -1(P) contains two elements) is finite. Let B, = fp E f, ( S' ) f,-1 (p) contains two elements) and n, = #B, = cardinality of B, if t O.

Since fo ( S' ) is compact and the curves t 1-4 F(t,x) are reparametriza-tions of trajectories of the normal vector field, we can take E > 0 sufficiently small so that if x, ,x2 E S ' , the curves -y; = F([0,e] j --- 1,2 either do not meet or one is contained in the other. It follows therefore that n, Is no for o < t < E.

Given a leaf A of if denote by Â its universal covering and by rA : Â A

the covering projection. Since f,,t > 0, is homotopic to a constant in A,, we can lift f, to a curve] ; : S 2, such that wAr = f, (see [29]). Our lifts always neglect base points, implicitly assuming that they are fixed.

Definition. We say F is a simple coherent extension of fo if the :4 are simple curves in 2, for t > O.

Proposition 3. Let fo : A 0 be a positive vanishing cycle and

F: [0,f] x S' M a positive normal coherent extension of fo . Let f,(x) = = F(t,x), x E S' and A, be the leaf of if that contains f, (S I ). Then there

exists a sequence t,, 0 such that for every n 1, the leaf A in contains a

vanishing cycle g„ : S l A„, which has a simple, positive, normal coherent

extension (it is possible that, for every n 1, we have t = 0).

Proof. Let 7, : S' 2, be a lift of f, and B, = p E f, (S') I p = f,(x l ) =

f, (x) with x, x2 in S' j. As we have already seen, we can take E small enough so that n, = #B, #B0 = k. For t > 0 set h, = jp (S' ) p = (x, ) = j; (x2 ) with x, x2 in S' 1. Clearly 7r- A, (n,) C B, and therefore #fi, 5_ nt k. Suppose Bo = j p, and let x, x,' in S' be such that fo (xi) = fo (x 1') = p„ i = 1 ,..., k. By uniqueness of solutions of the normal vector field, if p E B,, then p = f, (X,) = f, (x i') for some j =

= 1 k. Let K 1 = j E 10,f] fi (x 1')1 and U, = jI E (0,0 1:4(x,) =

(x,')). Clearly K 1 is closed and K i D U,. We will show that U, is open.


Let S' — ix, ,x 1') = a U 13, where a and 1 are open segments and a ( .1 f = = 0. If t E U, we have that:Z(x, ) = , -/s; (xi) and therefore the restriction of f, to the closure a- of a is a closed curve in A, homotopic to a constant in A,. By Lemma 4 of Chapter IV there exists a b > 0 such that for s E ( t b,t + ),

Fe is closed and homotopic to a constant in A s , so f5 a is a closed curve in ;15 , which implies that s E U, and therefore U, is open. We have three

possibilities: (1) There exists (5 > 0 such that [0 ,S] n u1 = 0. In this case, restricting

F to [0,b] x S ' we obtain h, < no = k. We continue the argument with U2 = t E (OM LZ(x2 ) =

(2) There exists b > 0 such that (0,6] c U,. In this case ) for t E (0 05] so f,(x l ) = f, (x,') for t E [OM. We can consider g o = = foi and To = fo as maps of S i on A o . Since fo is not homotopic to a constant in A o , we can suppose, for example, that g o is not homotopic to a constant in A o . Take G = F I [0 ,S] x Ti and g, (x) = = G(t,x). Since :Z(x 1 ) = :11 (x 1') for t E (OM, g, is homot,opic to a con-stant in A, for t E (0,6]. If h , = #[p ,(x) = ,(x' ) = p, x

x' in a- , we have h < h, k. (3) There exists a sequence 7- ml 0 such that 7- 1„, E 171 - U, C K 1 . We can

assume that for every m 1, there exists Em > 0 such that ( + Em

C U1. As in the above case we have f(x,) = f, (x 1 ) for t E [r ,r + Em ].

Since T E K, — U1 , the restrictions fr i 1Fe and fri l 13 are not homotopic

to a constant in A» , and since ( 7-„,1 + Em ] c U, for T t T ml

E m , we have that f, I a and f, I 13 are homotopic to constants in A,. Therefore for every in a- 1, m la- is a positive vanishing cycle on Ai and F [rml + Em ] X Fie is a positive, normal, coherent extension of fr), a- . If hr,n = tif p E ( c7) p = 1;(x) = Ji (x' ) with fx,x' [xi ,x;) and x x' in Fe 1 then h;"< h, k.

In any case we obtained a sequence 7„1 , --+ 0 such that, for every m 1, the leaf A 7 contains a vanishing cycle g,,,, which has a positive, normal coherent extension G 1„, : [T ml + E m l X S I M such that the number of self-intersections of k a lift of g = G m' (t, — ), is h;,,,

min1 k — 1,h, — 1 In the first two cases = 0 for every m I. In the last case we identify the endpoints of a- , obtaining S I .

By an analogous argument we can obtain a sequence T 2,„ T ,!„ such that

the number of self-intersections of is n mini k — 2,h, — 21. Ex- tracting a subsequence T n ( = —0' 0, we get vanishing cycles g,, which

have positive, normal coherent extensions 6,2„ such that the number of self-

intersections of /2„ .1 is h ,, , minlk — 2,n, — 21. Proceeding with this argu-

ment, at the end of at most k — 2 steps, we get a sequence T,„ 0 such that

for every in 1, the leaf A T contains a vanishing cycle satisfying the desired

properties, remembering that at each stage we can always substitute for the

vanishing cycle, which may not be differentiable at a point, a differentiable

vanishing cycle. •


§4. Existence of a compact leaf

The objective of this section is to prove the following theorem.

Theorem 2. Let A 0 be a leaf of a which admits a vanishing cycle with a simple, positive, normal coherent extension F: [0,e] x S I M. Then

(a) A 0 is a compact leaf (b) A o is the boundary of a submanifold V c M such that V = U „ 10,, ) A„ (e) for every t E (0,d, A, — A, = limA, = Ao , (d) for every t E (0,€1, A, is diffeomorphic to 111 2

This theorem will be a consequence of the following propositions.

Proposition 4. If A is a noncompact leaf of a foliation 5. of codimension one of a compact manifold, then for every p E A, there exists a closed curve -y,

transverse to passing through p.

Proof. Since M is compact and A is not compact, lirn A = A — A 0. Con- sider a point xo E limA and let Ube a foliation box of 3, such that xo E U. Let f C U be a segment normal to 'S such that xo E f. Since xo E limA, there is a sequence of distinct points 1 Xn,nEN such that for every n E N , x E f f") A. Con-sider the segment f with an orientation as in the figure below.

Figure 6

Let a : [0,11 A be a differentiable curve without self-intersections,

such that a ( 0 ) = x By the global trivialization lemma

(Chapter IV), there exists a neighborhood V of a( [0,1 ] ) such that V is

equivalent to the product foliation D" - x ti on D" x — p,p). Suppose

first that '5 is transversely orientable. In this case we can assume that the orien-

tation of f is compatible with the normal orientation of 5 V, so f cuts V as

in figure 7.


V T,

-. ■ ..■ --.. 5;

Xn li h

on 4+1 (if

Figure 7

Let bn and (S„ I be connected components of f n V which contain x„ and x„ respectively and 7: [0,11 f be the curve that begins at x,, and ends at x„ . It is clear that aV fl n contains two points, q and q' say, where we can assume q > q' in f (the orientation of f induces a natural order). Since x„ 1 is "below" q and if V is the horizontal foliation on V, there is a curve : 1 0, 11 —• V, transverse to if V, compatible with the positive orientation of if V and such that 7y ( 0 ) = x„ 1 and ( 1 ) = q. We can also assume that the tangent vectors of 37 and "Y at x„ , and q coincide. Hence the curve -y : [0 , 1 ]

M defined by -y ( t ) = -7(2t) if t E [0,1/2] and -y ( I ) = – 1) if t E [1/2,1 J, is a C 1 curve transverse to if and cuts A at x„ .

Let x be an arbitrary point of A. Let 0 : [0,11 A be a differentiable curve without self-intersections and such that 0 ( ) = x„ 1 and 0 ( 1 ) = x. Let W be a neighborhood of 0 ( [0, 11 ) such that a w is equivalent to a horizontal foliation D n- x it) on Dn - I x ( ,E ) . It is clear that -y fl W contains two points q and q' with q > q' , say. We can then define a curve : [0,1] W, transverse to 5 and such that ( ) = q', (1) = q and rt. ( 1 / 2) = x,

assuming without loss of generality that the tangent vectors of 7 and i at q

and q' coincide. Joining the curve ri with the segment of -y contained in the exterior of W, we get a closed curve IA , transverse to a and which cuts A at x (see Figure 8).

A X

Xpri +1

'No

A

Figure 8


Suppose now that is not transversely orientable. In this case the argu-ment is the same, except that we must consider a foliation box V that contains at least three points of the sequence 1.,,c,1 „ E tN as in the following figures.

A

■ ■

A \ i A

.

\

... f _. , xn , _lc _n +1

.,,

•

"cn +2

.....4

J

• 't /

/ /

•

•

/ /

/

Xn tr-

Xn + 1 •

xr/ +2 A

Figure 9

Proposition 5. Let D 2 c 1R 2 be the closed disk of radius 1. There is a dif ferentiable ( • ) immersion H : (0,E] x D 2 M satisfying the following properties:

(a) the curves t H(t,x), x E D 2 , are positive and normal,

(b) for every t E (OA, H(t x D 2 ) C A„ (c) the restriction of H to (0,f] x S' coincides with F (0,E] x S

Proof. First of all, observe that for any leaf A of its universal covering Â is homeomorphic to 111 2 . Indeed, either A IR 2 or Â S 2 . If Â then either A = S2 or A = 1P 2 . In this case, it follows from Theorem 4 of

Chapter IV that a has no vanishing cycle. Therefore Â 1R 2 . In particular Â, IR 2 . Since F is a simple coherent extension of fo , the

curve :4 : st Â, is simple and so by the Jordan curve theorem there exists h, : D2 A, such that ii, 3D 2 Since (S' ) is an embedded simple curve we can suppose that 1 , is a diffeomorphism onto the disk (D 2 ).

It follows that f = 7,4 o extends to an immersion h, = o : D 2 A,.

By Lemma 4 of Chapter IV, h, extends to an immersion H: (€0 ,E] x D 2 M such that (a ' ) the curves t (t,x ) are positive normal and (b ' )

H ( t x ) c A, for every t E (c o , c]. Let P be the restriction of H to

(• ) Observe that (0,c] x D 2 is a manifold with boundary and corners. The notion of differentiability for mappings with these domains can be seen in [39].


to, E1 x S 1 . We have then f(e.,x) = 17,(x) = f, (x) = F( €,x). Since the curves t f(t,x),t I-0 F(t, x) are reparametrizations of trajectories of the normal vector field, by considering a reparametrization of the interval (€0 ,f], we can suppose that i4( t,x) = F(t,x), that is: (c ' ) H (€0 ,c] X s' = F (€0,c] x S 1 .

Assuming fo > 0 we are going to prove that for every x E D 2 lirn

H(t,x) exists. Indeed, the simple curve:40 is the boundary of a disk in Â,0 and therefore using the same argument as before, there exist > 0 and an exten-sion H' : (€0 — ô,e 0 + 5) x D2 Al satisfying (a ' ), (b ' ),(c ').

If co < to < c o + 6, the restriction h,'0 = H' Ito x D 2 is homotopic to a constant in A, 0 and therefore 11 1'0 lifts to : D 2 . By construction 0 0

a (h,'(D 2 ) ) = fto (S I ) a ([1,,(D2 )), since H' = H= F on to x s'. Since 71,0 = lR 2 and ko , iitio are imbeddings, one sees that ii,

0 (D 2 ) = hI (D 2 ).

This implies that for every x E D 2 there exists y E D 2 such that ii:(x)

= 'Trio ( y). So, by the uniqueness of trajectories of the normal field, H ( t, x)

H' ( t,y) for every t E ( E0 , 0 + 5) and therefore H(t,x) =

= Hi( € 0 , y) exists. This argument shows that we can extend H to (€0 —

X D2 since A ) is homotopic to a constant in A, 0 and 4 is a simple curve

in ;I, o . By connectivity we can extend H to (0,E] x D.

Since the construction of H was done using orbits of the normal field, clearly

H is an immersion. •

Proposition 6. Let H: (OA x D 2 M be as in Proposition 5, h, the restriction of H to t x D 2 and D, = h,(D2 ). There exists a decreasing sequence T r,, > 0 such that

(a) A, = A, ,= A for n 1, (h) D.:, D D, for n 1, (c) for every t; 1, there exists a map g, : D 2 such that g, :

g,, (D 2 ) is a diffeomorphism and h,n = h , o g„.

Proof. Let U = x E D 2 ,_. 0 lim i3O H( t,x) exists l. Then U is an open set in D 2 containing s' and U D2 . Indeed, that U is open follows from the fact that the curves t 1-0 H(t,x) are reparametrizations of orbits of the normal field and of the tubular flow theorem for vector fields. Also, U D S i because the restriction of H to (OA x S' coincides with F. Finally if U = D2 , H would extend continuously to 0 x D 2 which would mean that fo is homotopic to a constant in A o , a contradiction.

Consider xo E D 2 - U. Since M is compact, there is a decreasing sequence s„ 0, s„ > 0, such that p,, = H ( s„,xo ) and p„ po E A, a leaf of Let V be a foliation box of po . We can assume that pn E V for every n.

V

Pn Pn +i

I qn qn+1 Po

_


Figure 10

Since the curve 7 (t) = H(t,x0 ) is normal to a, for every n 1, there exists tr, > 0 such that q,, = H(t,„x0 ) E A and the segment of 7 between sr, and t ,, is contained in V (see figure 10). Clearly the sequence tr, is decreasing,

0 and A, = A for every n. This proves (a). We can assume A Ao. We claim that Po E D,n for n sufficiently large. Indeed, if not, since tin

E int (D, ), there exists a sequence bn E ar), such that b,, —• po . On the other hand, 3D , C f, (S'), which converges uniformly to fo (s'). So Po E fo (S')

A which is absurd. This proves the claim. Fix fk, E 7r,74 1 ( Po) and neighborhoods W of Po in A and of »0 in Â such

that rA : w --0 W is a diffeomorphism. Consider a sequence E 11/, 4,? with 71-A (On ) = qn = h, (x0 ). Let i , : D 2 A be the unique lift of h,

such that 'Ti t (x0 ) = a- 1) and set 4 = i , (D2). If m n E IN, then curves h,m (k t ) = f, (S') and h, (S') = ( are disjoint in A and the

curves and (s') are simple and disjoint in Â, and therefore C D, or D, C A n , since these disks have a common point po . Thus we con-clude that D,n, i D, or

Let lim (A) = A – A be the adherence of A. Let dA be the intrinsic metric

on A. Since the sequence of curves h, (S') converge to fo ( s'), it follows that

fo ( ) c lim (A); so lim ,„ c14 ( po ,h, (S i )) = 00. We can then choose a

decreasing subsequence Tk of tr, such that dA ( po ,h,k (S')) is increasing

and tends to infinity. The sequence T,, satisfies (a) and (b) of the lemma. Let

Dr, = h, (D2 ) and b,, = (D 2 ). Observe now that for all n E IN, : D 2 b,, is a diffeomorphism and

that b,, C b„,,. Define g„ : D2 — • n D 2 by gn = (k„. , ) ° hs„. The diffeomorphism g„ evidently satisfies (c). •

Remark. For every tE (OA, there is a sequence r,, —• 0 such that Dr.n

C A, and U „ I Drn = A,. This fact, which is to be proved next, implies that

IE f D = U A t• This observation will be used later.

(0,1 tE(0,(1


We will verify this fact fi rst for the leaf A = A 7 . Let be the metric on Â, co-induced by dA . The fact that dA ( po ,I;rn (S')) co implies that dA h7 (s' )) co where 7rA ( fro ) = Po , so U „ i D,, = Â. It follows therefore that U „ 1 D„ = A.

Consider now po as in the above proof. Then, as we have already seen,

Po E Drn for large n, or, po = H(r,x) for some x E D 2 and T = 7,, fixed. Observe now that po is in the w-limit set of the orbit of the normal field through the point q„ = H (T,x0 ). Since the co-limit set of an orbit is invariant under the flow ([34]), p. 13), we conclude that, if t E (OA, then

= H ( t , X) = lir n H ( r ,x0 ) for some sequence r„ --• O. Using now the same arguments made in the proof of Proposition 6, we can conclude that there exists a sequence r,, --• 0 such that Drn C Drn C A t and P1 E Dr for

n no large. Repeating what was already done for the leaf A 7„ , we conclude that U „, I Dr = A t .

Consider now a manifold with boundary and corners K„ obtained from [r„ ,,r„1 x D 2 by identifying each point of the form ( r„,x ) E r,, x D 2 with ( r„,,g„(x)) E r„, x D2 (see figure 11). Let rn : [7,, ,rn] x D 2 —• K„ be the projection of the equivalence relation.

X

Figure 11

Observe that for every x E D 2 , H ( in 1 ,ga (x)) = h,. = h, (x) = = H(r,,,x). Therefore there exists a map f : K„ M such that H = H,

-Tn . Since H and 71-,, are immersions, H„ is an immersion. This construction contains the main idea of the proof of Novikov's Theorem.

Proof of Theorem 2. Suppose, by contradiction, that A o is not compact. Fix X0 E s' and let qo = j (x0 ). By Proposition 4, there is a closed curve -y


transverse to 'S passing through g o . Modifying the curve y a little we can ob-tain a positive closed curve if" satisfying the following properties:

(i) meets F( [0,f] x S') along the normal segment [1( x0 ) t E (0,a11 where 0 < a E ,

(ii) -1/ does not meet the normal segment If, (x)It E [OA) if x xo .

Reparametrizing -1-, we can assume without loss of generality that 7)% ( t ) = = ft (xo ) for t E [0,a]. Let 0 < + < r„ < a be as in Proposition 6 and H, : K, M be as in the above construction.

For each x E D 2 consider the curve 0, ( t) = ( t,x). By construction H,(13x (t)) = H(1,x) and hence the curve f3x ( t) is mapped by If, to a tra-jectory of the normal vector field. In particular, -T, ( t) = 11(0.,0 (t)) for t E 17n-1-1,Trri•

We are going to lift the part of contained in H(K„) to a curve in K,,

such that 1-1,, = • Define ( 1) = -47,0 (t) if t E [rn , i ,r]. Since (t) is in the interior of H(K) for 7-, < t < 7- + p., we can lift -T, in the interval

17n+11rn

p.],

p. > 0 (see figure 12).

K n

Figure 12

For, since H,, is a local diffeomorphism, we can continue to lift while -;s7 remains in the interior of K. We will only be obliged to stop the lift when --y

touches ax, again. On the other hand, ;y- must again meet axn , since is a closed curve and the lift begins in aKn . So, there are only two possibilities for -7 to meet ax, : (1) In 7 r( [ , r,] X S I ). This is impossible since -7 cuts H ([7,,_ 1 ,7„) X S' ) only along the segment t f (x0 ) = (t), t E [7-„.-1,7„]•


(2) In B = 71-„(7„, x D2) 7r,, 7, X D 2 ). This also is impossible. In fact, since H,(B) C A = A 7 , if .7y- leaves K„ at a point z E B, its tangent vector at this point would be pointing outside of K„, so the tangent vector to 7i at the point il,, ( z ) would be pointing in the negative direction of 5, a contradiction (see figure 12). This proves that A o is a compact leaf.

Consider now the compact, connected regions V H„ (K,,) = H(1 wrn+1 ,7n]

X D 2 ) C M, n 1. Observe in the argument above that the curve 0, ( t) enters the interior of K„ and never again leaves, for every x E D 2 . This corresponds to the fact that the positive integral curves of the normal field, which enter V„, never again leave_ This fact will be used later and implies that

C U D, = V„. Let V = U „ I V,. It is clear that V is connected, compact and its boundary ay D A o . We will show next that ay = A o . Observe first that V = U 0<p,, A„ so V and av are saturated by 5. Indeed, by the remark after Proposition 6, we have that U 0 ,,,, D , =

= U 0< ,,, A„ so

U = U U D,= U D, = U A,. 7-n -sr5_( 0<15_f 0<tc

Hence V = U A,. We claim that ay is connected. Indeed, ay is the set of accumulation points

of U „ avn . This follows from the fact that 1/,, c 1/,„ 1 (n 1) and V = = U V. Suppose by contradiction that ay is disconnected. In this case there exist open disjoint sets W1 , W2 C M such that ay c W 1 U W2 and av n wi 0 (i = 1,2). This implies that there exists n o such that avn n tv, and avn (1 W2 are nonempty for n no , so for n no there exists pn 017„ (W1 U W2 ) , since O Vn is connected. We can assume without loss of generality that p,, 1 E av,,,, — Vi,,, n no . Since M — (W 1 U W2 ) is compact, the sequence 1p,7 1 has an accumulation point p. Since p V„ it is clear that p is an accumulation point of U „ I 31/„, or, pE 3V— (W U W2 ) , a contradiction. So av is connected. Next we will prove that A o is a connected component of av. This will imply that Ao = ay. Let N be a tubular neigh-borhood of A 0 . Since 5 is transversely orientable N is diffeomorphic to A x ( —6,(5), where we can assume that the fibers of N are the trajectories of the normal field to 5. Given g E A o , let 0,7 be the fiber of N which passes through g. It suffices to prove that for every g E A 0 , ay n = tqi , if the length of the fibers is sufficiently small. In fact, this will imply that a V (.1 N = A o , so A 0 will be a connected component of ay. Observe first that it suffices to verify the above fact for some g E A o . In fact, this follows from the fact that ay is invariant under 5 and from transverse uniformity of 3 (Chapter II). On the other hand, given g E f0(S I ) C A o , let W c Nbe a folia-tion box of 5 which contains g. Set -1,i/ = H - I (W). Since the orbits of the normal field in W meet A o , one sees that if (10 ,x0 ) E IT/ then xo E U = = x E D 2 I H (t,x) existsl and therefore lim,. 0 H(t,x0 ) E A. It


follows therefore that the set of accumulation points of U „>, (3v„ fl W) is contained in A o , or, av n w c A o . This proves that 0q n av fq), as desired.

Part (c) of Theorem 2 follows from the fact that av D lim A, D A o for every t and that av = A o .

In order to prove that A„ t > 0, is diffeomorphic to IR 2 , we are going to use the following facts:

(a) Let N be a two-dimensional manifold without boundary, not diffeomor-phic to R 2 , whose universal cover is 1R 2 • Then there is a deck transfor-mation f: 11 2 1R2 which has infinite order (that is f k identity if k 0). Recall that a deck transformation is a diffeomorphism f: 11 2

1R 2 such that 7r o f = Ir , where 7r : 1R 2 N is the covering projection.

(h) If f : 11 2 111 2 is a deck transformation such that f ( x ) = x for some X E lie, then f = identity.

We leave the proof of (a) and (b) as an exercise for the reader. We need the following lemma.

Lemma 1. Let IR 2 = U n> i h„, where for every n, b„ is a disk, b„ C

and 34, n ab„, = 0. Let 7r : 1R 2 N be a covering and suppose that for

every n, 7r ( ah„) is a closed curve in N whose self-intersections are transverse double points. Let m k be the number of self-intersections of A-(34). If the sequence m k is bounded then Ir : 1R 2 N is a diffeomorphism.

Proof. Suppose 7r is not a diffeomorphism. In this case, there exists a deck transformation f: IR 2 R 2 of infinite order. Let zo E b, and set zk =

f k z0). If zk = ze with k e, we would have f k -= identity, contrary to the hypothesis. So the sequence { z k k E Zi is made up of distinct points.

On the other hand, for every k f k (Ai ) contains points outside 13,„ since otherwise, by the Brouwer fixed point theorem, f k would have a fixed point in 4, a contradiction.

For n large, let k„ be such that zo zkn E b„ and zkr, b„. We have that 3h,, n (ah„) contains at least two points, 1 5_ k,„ since z, E

fl f' ( b,,) and h„ n ( J11 2 fi(b n )) 0. Let p,,p; E ah„ n (afin ), 1 j k„. Since 7r o = 7r , 7r ( pj ) and r p;) correspond to points of self- intersection of 315r,) and since these points are all double points, for I s i k„ and i j we have p, pi ,pj', so if m n is the number of self- intersections of 7r ( abr,), then m n k„. Since U h„ = 1R 2 , it follows that Lim _,. cc k = 00 and therefore m„ is not bounded, contrary to hypothesis.


a(3n)

We now prove that A r n = A =1R2 . The proof that A, =1R2 , t 7-,7 is similar.

By the considerations preceding Proposition 3, if n, is the number of self-intersections of the curve f,(S I ) C A„ then n, no . Since is the number of self-intersections of r A ( ab k ) = frk (S 1 ), it follows from the above lemma that A = 1R2 and 7r,,, : Â A is a diffeomorphism. In particular nT, -= 0 for every k. This proves Theorem 2. •

As a consequence, we have the following.

Theorem 1. Let be a transversely orientable, C2 , codimension one foliation

of a compact, three-dimensional manifold with finite fundamental group. Then 5' has a compact leaf

Proof. By Proposition 1 there is a vanishing cycle of 1F. By Proposition 3, has a leaf A o containing a vanishing cycle, which admits a simple, positive,

normal coherent extension. By Theorem 2, the leaf A o is compact. •

55. Existence of a Reeb component

In this section we will present a sketch of the proof that the compact leaf A o , found in Theorem 2, in fact bounds a Reeb component. More precisely, with the notation of Theorem 2:

Theorem 3. The manifold V bounded by A o is diffeomorphic to D 2 x S 1 and i V is topologically equivalent to the Reeb foliation on D 2 x

Proof. By Propositions 5 and 6 there is an immersion H: (0,E1 x 13 2 M, h, (x) = H(t,x). and a sequence 7,, 0 such that (1) h,(D 2 ) c A, for


every t, (2) the curves t 1-4 h, (x) are positive normal, (3) A,., = A ,. , for every n 1, (4) hrn, . (D 2 ) D 11 7„ (D 2 ). Further for every n 1 there is a

map g, : D2 D 2 that is a diffeomorphism onto its image and such that 11 7,, = Iirn+ I ° g, •

As before, let Kn be the manifold with boundary obtained from f X D2 by identifying the points (T,,X) E r x D 2 with (T„i,g„(x)) E

x D 2 and 7r n : [7, +1 ,7n 1 X D 2 —11' Kn be the quotient projection. Also let K„ M be the immersion induced by H and V 11,,(K). The nor-

mal field to a will be denoted by X. Since the set B, = t E [rn± 0-„1 A, = A, , j is discrete (since two distinct curves f, (S') and fs (s') contained in A T, are far apart in the intrinsic metric of AO , we can assume that if t E (r ± ,r,,) then A, A rn , n 1. We will next prove that, with this hypothesis, H„: K, is a diffeomorphism. As a consequence of the lemma of §4 we have that for every I E [0,d, the curve fl (S 1 ) is simple and this implies that the restriction of H to t x D 2 can be taken to be injective (review the proof of Proposition 5). Since A r A, if < < t s rn it follows that H is injective and therefore a diffeomorphism.

On the other hand, since M is orientable, for every n 1, V„ is orientable, so K„ is also_ It follows therefore that the identification diffeomorphism g„: D 2 D 2 , from which we obtain Kn , preserves the orientation of D 2 . We conclude then that Ka and V,, are topological solid tori for every n 1.

We are now going to construct a sequence of two-dimensional tori Tri l such that

(a) T„ meets each leaf of Y transversely in a closed curve which is the

boundary of a disk in this leaf, (h) T„ is the boundary of a solid torus R„ c V = u n vn , (c) the trajectories of the normal field cut T„ in at most one point,

(d) T„ C V,, and, therefore, if n is sufficiently large, every positive trajectory of the normal field through a point of A o cuts T„.

Consider a cylinder C C D,.„ — D7„ such that 0c t (S 1 ) U -y , where -y is a closed curve such that n f,. (S I ) = 0. Saturatin-g C by X in an in-terval [0,p1, i > 0 we obtain a region U of V„ diffeomorphic to C x [0411 where the normal field is represented in C x 10,0 by the field ä/0t = ( 0, 1 ) (see Figure 14). We take p. such that f.„ (S' ) C U. Construct now T ,', by attaching the cylinder


Vn + 1 C /4 7, n 1/,

Figure 14

C1 = — — C such that aci = -y U f, (S l ), to a cylinder C2 transverse to X and to a, C2 C C X [0 ,pd and .5c, = fr (S I ) U -y. (See Fig. 14). It is clear from the construction that T,; = C1 U C; satisfies proper-ties (b), (c) and (d), as desired. Modifying the torus T r; slightly, we can get a torus T to the foliation g (see Fig. 15).

Vn+i

Figure 15


The above figure is a type of "dual" to figure 14. In that figure the elongated part of av„, is contained in the leaf A, and the short part is transverse to if while in figure 15 the elongated part is transverse to if and the short part is contained in A rn . The transverse torus is obtained by pushing C I slightly to the inside of V as shown in figure 15. It follows from the construction that Tn can be gotten arbitrarily near T n' , which is transverse to the normal field, so, we can assume that T is transverse to if and to the normal field.

Now consider p E A o . Let X, be the flow of X. The orbit of X through p enters the region V bounded by A o and never again leaves. Further, since V = U V„, V V,, A o is compact, given (5 > 0 there exists n o E IN

such that for n ?_- no , X ( p ) E int ( V) for every p E A o . This implies that for n sufficiently large and for every p E A 0 , the segment of the orbit X, ( p) t E [0,(5]1 cuts 7", in only one point P(p). This defines a map P:

Ao — 0 T evidently is a diffeomorphism. So A o is diffeomorphic to T2 . In a similar manner it follows that V is diffeomorphic to V. So V is diffeomor-phic to D 2 x S 1 . We leave it as an exercise for the reader to prove that if V is equivalent to the Reeb foliation of D 2 x S'.

Remark. Using Proposition 3 and Theorem 3 one can show that every leaf A o of a which contains a vanishing cycle is the boundary of a Reeb component.

Indeed, let f0 : S' M be a positive normal extension of fo . By Proposi- tion 3, there exists a sequence T„ /-4 0 such that for every m 1, the leaf A 7 contains a positive vanishing cycle g : S 1 — • A 7m , which has a simple, positive, normal, coherent extension. By Theorem 3, A, is compact and is the boundary of a Reeb component of 5, which we denote by vm. We claim that

= A i., and V" = V for arbitrary rtl,f E IN. In fact, since for every m 1, g„, is a positive vanishing cycle which has

a simple coherent extension, it follows that for p E A 7 , V'n entirely contains the positive orbit 0 + ( p ) of the normal field to 5, passing through p at t = O. On the other hand, if e > m, there exists a point p = fr, (x0 ) E A T, such that O (p) fl V- , so v" (1 V' 0. Since the boundaries of V"1 and V' are leaves of if, V" D . By Theorem 2, the leaves of if contained in int ( vï) are diffeomorphic to planes and therefore A, = avm , which is compact, can-not be contained in the interior of V', so A 7, = A r and V' = V'. Hence we see that A, = A o and V' = v' for every ml I so A o is the boundary of a Reeb component of 5.

§6. Other results of Novikov

A natural problem to propose is that of the characterization of three-dimensional manifolds which only admit foliations with Reeb components. By Theorem 2 this problem is equivalent to characterizing compact three-dimensional manifolds such that every foliation has a vanishing cycle. In this direction we have the following theorem due to Novikov 140].


Theorem 4. Let 5 be a transversely orientable, C2 , codimensi on-one foliation of a compact, orientable, three-dimensional manifold M such that 72 (M)

O. Then either 5 has a Reeb component or the leaves of 5 are compact with finite fundamental group.

This applies for example to the manifold S 2 x S 1 . This theorem is a consequence of Theorem 3 and of the following lemma:

Lemma 2. Let 5 be a transversely orientable, C 2 , codimension-one foliation of a compact, orientable, three-dimensional manifold M such that 7r2 (M)

O. If 5 has no vanishing cycles then all the leaves of 5 are compact with finite fundamental group.

Proof. Saying that 72 (M) = 0 is equivalent to saying that every continuous map f: S 2 M is homotopic to a constant.

Let g : S 2 M be a map not homotopic to a constant. By methods we used in Haefliger's construction, g can be approximated by another map which we will again denote by g such that g* (5 ) is a foliation of S 2 defined by a vec-tor field Y whose singularities are centers or saddles and such that Y admits no saddle connections, other than self-connections.

We will next see that the hypothesis that 5 does not have vanishing cycles implies that if -y is a nonsingular trajectory of Y then .y is closed or -y is con-tained in a graph of Y. Moreover, if F is a closed trajectory or graph of Y then g ( r ) (with the orientation given by Y) is homotopic to a constant in the leaf of T that contains it.

Indeed, suppose that Y has a non-closed orbit -y that is not a self-connection of saddles. In this case by the Poincar&Bendixson Theorem, the a or ce-limit set of -y is a closed orbit or a graph formed by a saddle and one or two separatrices. Call it F. The curve g ( F) is not homotopic to a constant in the leaf of T which contains it. We can apply then the same arguments as in §2 in order to obtain a vanishing cycle of T. Since 5 does not have vanishing cycles, an orbit -y as above cannot exist. So, all the nonsingular orbits of Y are closed or are separatrices of saddles forming self-connections. If there exists a closed path or a graph F such that g ( F) is not homotopic to a point in Aon , again by the arguments of §2 , we can obtain a vanishing cycle. So, if F is a closed orbit or graph of Y, g ( F) is homotopic to a constant in the leaf which contains it.

We will prove ahead that there is a map g 1 : M, homotopic to g and such that g i ( S 2 ) c A, for some leaf A of Y. We claim that this implies that A -= P2 or S 2 , therefore it is compact with finite fundamental group and so by Reeb's global stability theorem (Chapter IV) all the leaves of 5 are compact with finite fundamental group. Let us prove the claim.

Let ir : A be the universal covering of A Then Â 111 2 or Â S 2 . It suffices to prove that Â = S 2 . Suppose by contradiction that IR 2 . Since S 2 is simply connected we can lift g i to a map : S 2 Â such that


ir 0 ok i = — 1 . g Since ,:=1.‘ 111 2 , RI is homotopic to a constant. Let P: I x S 2 be a holonomy taking f l to a constant. Then F = 7r o P is a homotopy

taking g, to a constant, which is a contradiction, since g, is homotopic to g which is not homotopic to a constant, by hypothesis. Therefore, all the leaves of are compact with finite fundamental group.

We must still show that there exists g, : S 2 --• Mhomotopic to g and such that g, ( S 2 ) C A for some leaf A of T. The idea, basically, is the following: using that for every closed orbit or graph F of Y, g ( F) is homotopic to a constant in A g(r) , starting from the centers we are going to deform g progressively, obtaining a continuous family g, : S 2 M, t E [0,1], satisfying the follow-ing properties:

(1) For every t E [0, 1 ] there exists a compact region R, C S 2 such that if s > t then Rs R t .

(2) The connected components of aR, are centers, closed orbits or graphs of Y.

(3) If R R kt , are the connected components of R, then g,(R",) C A", for some leaf A', of ‘S, where g, : R — • Ai, is homotopic to a constant in

(4) g, I (S 2 – R,) = g I (S 2 – R,). (5) R o is a center of Y and R, = S 2 .

In the figure below we see two stages of the deformation in a neighborhood of a center.

Cl

Figure 16

The following lemma implies Lemma 2.

Lemma 3. Let G : [0,a] x S 2 —0 M be continuous and set g,( p) =

= G (1,p). Suppose that the family g,, t E [0,a1, satisfies properties (I) to (4) above, where R„ S 2 . Then we can extend G to [0,a + fl X S 2 , obtaining an extension of the family g„ which satisfies properties (1) to (4).


Proof. The technique of proof is similar to that used in Lemma 4 of Chapter IV and in Proposition 5 of this chapter. The only difference is that we cannot use the normal field to 5 to obtain the extension of g, to a neighborhood of g„(ak). We consider three cases:

(1) Ra has a connected component which is a center. (2) R a has a connected component S a such that asa consists of closed orbits

of Y. (3) For every connected component S of R a , as contains a graph of Y.

In the first case the center of Y has a neighborhood V saturated by Y and such that all the orbits of Y in V — c} are closed, constituting a continuous family t t E [a,a + €1, where Ft is the boundary of a small disk V, such that V, n R c = . In this case, for t E [a, a + g (r,) is the bound-ary of a small disk in A, and we can extend the deformation as shown in figure 17. Here, the image g, (S 2 ) is changed only in a foliation box of 3:.

Figure 17

In the second case, let r' ,...,rk be the connected components of asa . For each i = 1 k, there exists a small tubular neighborhood V' of r' saturated by Y and such that all the orbits of Y in V' are closed. Observe that g„ (F ) g„( rA) are contained in the same leaf A c, of 5 and are homotopic to a constant in this leaf. Since g„ S, is homotopic to a constant in A„, by Lemma 4 of Chapter IV, we can extend the family g, i So to a continuous family g, : S„ M, t E [a, a + 61, such that for every t E [a,a + (5], k",(S„) C A, for some leaf A, of 5. This implies that it is possible to define continuous families t rt„ t E (a, a + El , = . k, such that for every j E f 1 ,...,

and every t E (a,a + il , F C V' — R (, and g (r it ) = g, (r) c A, where 0 < (5 is sufficiently small. Let S, be the connected region of S 2 which con-


tains S„ and whose boundary is F u...0 F set R, = S, U Ra . Then the family R, satisfies properties (1) and (2) of the lemma. The construction of the family g, that extends ga and satisfies (3) and (4) is obtained using arguments analogous to Lemma 4 of Chapter IV, with the only difference being that it is not possible here to use the normal field of F. We leave the details to the reader.

We consider now the third and last case. This is divided into two subcases:

(a) Some connected component of s' — R a contains a center c of Y. (b) S 2 — R contains no centers of Y.

Subcase (a). In this case it is sufficient to add the center c to R a , reducing the problem to the first case.

Subcase (b). In this case we will prove that R a contains a connected compo-nent S properties analogous to those of the second case, that is, if F

r k are the connected components of as„ then there exists a neighborhood V of as, such that V — S a is saturated by Y and the orbits of Y in V — S

closed. Observe that from the construction of g (Chapter VI), we see that if s and

s' are distinct saddles of Y then g (s) and g (s ' ) are in distinct leaves of i.

It follows therefore that if S is a connected component of R a then as contains at most one graph of Y. On the other hand, given a graph F of Y which con-tains two separatrices, S 2 — F has three connected components, say Bi (F), j = 1,2,3. Two of these components, say B 1 (r) and B2 ( F ) , are such that Bi ( F) is homeomorphic to a closed disk, while B3(F ) is not homeomorphic to a disk. Observe that B, ( F ) must contain a center for each i =- 1,2,3.

It suffices to prove the following fact: ( .) There exists a component S R a such that if ra is the graph contained in as, then F„ contains the two separatrices and Bi (F a ) rl S„ 0 for j =- 1,2 or B3 ( ) fl S„ 0.

If S (.) we can define a family of regions S„ t E [a,a + e], satisfying (1) and (2) of Lemma 3. In the case that B3 ( F„ ) (.1 S„ 0, the extension S, of S„ is obtained inside the regions B, ( r„ ) and B2 ( ) and in the other case, inside B3 ( F, ) .

Let S, Sk be connected components of R„. For each j = k, as, contains a saddle point si . Let Fi be the union of the two separatrices of s,, so that FJ is a graph of Y and at least one of its separatrices is contained in as,. Sup-pose S I does not satisfy CO. In this case it is easy to see that Bit (F 1 ) contains a center c S I for some j, = 1,2,3. So Bit ( F 1 ) contains a component of R„ distinct from S, which we can assume to be S2. If S2 does not satisfy (.), by the same argument, for some j, =- 1,2,3, B , ( F,) contains a component of R a

distinct from S, and S,, say S3, with S3 C B ( F, ). Proceeding inductively we obtain finally a component s, of R„ satisfying ( ).


We can now extend the family R,., t E [0,c], obtaining a new family, satis-fying (1) and (2) of Lemma 3. In order to extend the family t g„ the argu-ment is similar to the one used in the second case. •

End of the proof of Lemma 2. As was already said above we begin the defor-mation of g at a center of Y. Using Lemma 3 successively we obtain a homotopy G : (0,a) x S 2 — • M satisfying properties (1) to (4). For each t E [0,a)

there exists a region R, such that g, (R,) C A, for some leaf A, of J and

g, 1 (S 2 – R,) = g I (S 2 – R i ). Observe that U R, = R a is a region of

S 2 whose boundary ak, consists of closed orbits or graphs of Y. Let S be a

connected component of Ra . Let rd F k be the closed orbits or graphs with a separatrix contained in S. Since S C S2 the closed curves F' generate the fundamental group of S. Since g : —• A is homotopic to a constant in A, we can extend g I aS to a continuous map h : S A. By Lemma 4 of Chapter IV, we can assume that there is a continuous family of maps h,: S M, t E [a - 6,a + 6] such that kt = h and h, (S) C A„ t E [a - 6,a + 6]. Set Sr = S n R„ For every t E [a - (5,a), h, S, is homotopic to a constant in A„ so it is homotopic to g, S, by a homotopy Ft : I x (R, n s) A,. Using the (continuous) family of homotopies t F„ we can modify the family h, obtaining a new family k , such that,, _ 6 So,_ 5 — g,, I _ and k et I S = h(t _ 5/2 I Sa_5/2.

Observe that we can obtain the family t h, such that h,1 ascr = g I as,,. The family k t thus defined extends g, S„ t E [0, a – 6] to a family S„ t E [0,ce] which satisfies properties (1) to (3). We leave the details to the reader. Proceeding in a similar manner in the other components of R a , we can finally obtain a family k f : s2 A„ t E [OA, which satisfies property (4) also.

By Lemma 3 and by the above argument, we stop the process only when R tt = S2 • This concludes the proof of Lemma 2.

7. The non -orientable case

Suppose that is a transversely orientable foliation of a compact-three dimen-sional manifold M. Suppose further that has a vanishing cycle which admits a simple, positive, coherent extension. Theorem 2 guarantees the existence of an open set V C M saturated by leaves homeomorphic to IR 2 • Further, V is homeomorphic to the set K,, obtained from [T„ _ ' 1-n ] x D 2 by identifying ( ,.v ) E Tn x D 2 with ( . 1 ,g, ( x ) ) E T,, . X D 2 where g, : D 2 —0- D 2 is a diffeomorphism onto g, ( D 2 ).

In the proof of Theorem 3 we used the orientability of M only to guaran-tee the orientability of V, which has as a consequence that g,, preserves orientation.

When M is not orientable we must consider two cases:


(1) g orientation; then K,, is homeomorphic to D 2 x S' and if 1 V is equivalent to the orientable Reeb foliation of D 2 x S'.

(2) g, reverses orientation; then K,, is homeomorphic to the solid Klein bottle K and if I V is equivalent to the nonorientable Reeb foliation of K.

Since the existence of vanishing cycles, with the hypothesis 7r, (M) finite OF 7r2 (M) 0, is independent of orientation of M, we have the following theorem:

Theorem 5. Let if be a transversely orientable C2 codimensi on-one foliation

of a compact three-dimensional manifold M. If r i (M) is finite then if has a Reeb component (orientable or not). If r2 (M) 0 and if has no Reeb components, then all the leaves of if are

compact with finite fundamental group.

VIII. TOPOLOGICAL ASPECTS OF THE THEORY OF GROUP ACTIONS

In this chapter Mdenotes a Cc° differentiable manifold and G a simply con-nected Lie group.

Our objective here is to show how the theory of foliations can be applied to obtain global informations about the orbit structure of a locally free action of G on M. The central theorem of this chapter is the theorem on the rank of S 3 due to E. Lima, which states that two commutative vector fields on S 3 are necessarily linearly dependent at some point. In other words, there do not exist locally free actions of the group 1R 2 on S 3 . We generalize this theorem to higher dimensions showing that no locally free action of G on a compact simply connected manifold M has vanishing cycles.

§1. Elementary properties

Recall that a Ck action of G on M is a Ck map ,p : G X M M satisfying the properties below:

(1) ço(e,x) = x for every x E M, where e is the identity of G. (2) (p (g lop (g2 ,x)) w(g, g 2 ,x) for any g 1 ,g2 E G and x E M.

Denote by ,pg : M M the map (Pg ( X) (10 (g,X) • From the definition it follows that (Pg_ = (40g) I for every g E G, which proves that ,,og : M

M is a CA diffeomorphism. The orbit of a point x E M (by the action 40) is the subset Ox (.9x ( )

= (g,x) g E GI. The isotropy group of x E M (for the action is the subgroup of G defined by G. = G, (so) = g E GI (g, x) = xi. From the definition it is evident that G. is a closed subgroup of G. Consider the


equivalence relation – on G such that g, g2 if and only if g 1 • g2 E G. Let G/G, be the quotient space of G by – and 7r : G --• G/G, the projection which takes g E G to its equivalence class 7r (g) .

Theorem 1. There is on G/ G„ a unique differentiable structure such that : G —J. G/ G, defines a fibered space with fiber G. In particular if we con-

sider G/ G, with this structure, 71- is a submersion and if G, is discrete then 7r is a covering map.

Proof. Letting k = iv (g) we have 7r -1 () = g • G, = { g • g' Ig' E G„), a set which is diffeomorphic to G, by left translation Lg : G —• G, Lg (g') = = g • g'. Consider now a disk D, imbedded in G, of class C', transverse to G„, with e E D and such that dim ( D) = cod ( G,) . Shrinking D if necessary, we can assume that for every g E D, D is transverse to g • G., in g and more-over that D does not have distinct equivalent points. We have then that 7r -1 ( Ir (D)) = D G„ = fg higE D,h E G,1, so the map

D x G —• 71--1 (7r(D))

given by 1k (g, h) = g • h is a bijection, since it is surjective and if g • h = = g' • h' with g,g' E D and h, h' E G, then g – g' so g = g' and h = h'. Further, since g • G, is transverse to D at g, for every g E D, and dim (D) = = cod ( G,), it follows that 1,/, is a C diffeomorphism. Therefore 7r - ( 7r ( D)) is open in G, so, by definition of the quotient topology, 'w (D) is open in G/G, and iv D : D 7r(D) is a homeomorphism, since it is a continuous, open bijection.

For each g E G, consider the disk Dg = g • D = Lg ( D) . Then g E Dg , L g ID:D -4.• Dg is a diffeomorphism and for every g ' E Dg ,

Dg is transverse to g' • G, at g'. By an argument analogous to the above, we have that 7r (DO is open in G/G, and Ir Dg : Dg --• (Dg ) is a homeo-morphism. Moreover the map Vig : x G, —• 7C 1 ( (Dg ) ) defined by i1' (g' ,h) = g' • h is a Coe diffeomorphism. This proves that Ir : G ---• G/G, defines a fibered space structure. The collection A = 1(7r Dg ) : (

Dg IgE Cl is a C' atlas on G/Gx , which defines a C' manifold structure on G/G„. •

Theorem 2. Let so : G x M --• M be a C k, k 1, action and let x E M. There exists a unique one-to-one C k immersion, 47,„ : G/ G, --• 0, such that ,T9 „o 7r = ça„. If G, is discrete, the map so„ : G ---• 0,, is a covering map.

Proof. Let = 7r (g). Define : G/G„ --• 0, by 7„(k) = (g,x),It is immediate that is well-defined and that it is a bijection. We will show now that is an immersion. Given Dg c G as before, we have that ‘,7„. o (w Dg ) : Dg --• 0, is given by T-p, o ( 7r ; Dg )(g') = so(g',x) = sc.„(g'); so it is CA,

Topological Aspects of the Theory of Group Actions 161

and if u is a tangent vector to Dg at g', D(T,57, o 7r Dg ) u = Dso,(g ' ) • u. On the other hand g ' • G, = {h E G p (h, x) = w(g' ,x)) so; 1 ( ,ox (g' )). One sees therefore that Tg , (g' - = Ix E Tg (G) Ap(g' ) u = 0 1 ; so, if to,p,(g') • u = 0 and u E Tg , (Dg ) then u = 0, since g' • G. and Dg have complementary dimensions. This shows that Ox is a manifold immersed injec-tively in M. If G, is a discrete subgroup of G, 7r : G —0. G/G, is a covering, so so, : G —0. 0, is also a covering, since tpx = (T ox 7. In particular dim (G ) = = dim ( G/ G) = dim ( Ox ). Conversely if dim ( G) = dim ( Ox ), we have dim ( G) = dim ( G/G,), which implies that G, is discrete. •

When for every x E M, dim (Or) = dim G, we say that the action is locally free. The orbits of a locally free action on Mare leaves of a foliation in M (see Proposition 1 in Chapter II).

Let us see what the orbits of an action of IR 2 on a manifold M can be.

Proposition 1. Let G be a closed subgroup of IR 2 . Then G is isomorphic to one of the groups x Ze where k g are integers with 0 k + f s 2. In par-ticular, the orbits of an action of 112 2 are immersions of some of the following manifolds: a point, S' x S', S' x 1R,IR 2 .

Proof. Let G c IR 2 be a closed subgroup. We distinguish two cases:

(1) G is not discrete (2) G is discrete.

Consider case (1). In this case there exists a sequence ( ) , x E G – 101, xn O. Let us write x ( rn ,0,,) in polar coordinates, where 0 :5_ On Is. 27. Let 00 E [0,27r] be an accumulation point of the sequence (0,,). We assert that the line with direction 00 is contained in G. Indeed, fix z = (r,00 ), r 0, and a subsequence ( = Pk) such that Wk 00 . Put pk = r„,. Now, the set Pk Z divides IR in intervals of length Pk and this implies that there exists ink E Z such that 1Mk Pk — ri < pk • This implies that lim k ( m k pk opk ) = = (r,00 ) and so limk--■ cc mkXnk= z. Since G is closed it follows that z E G and so the line IReo = f ( r, 00 ) r E 1111 c G. Now, we have two subcases to consider:

(a) There exists a sequence x,, = (r,„0„) E G – WI such that x,, — 1. 0 and the sequence (O ,, ) has two accumulation points 0 1 ,02 , which define dif-ferent directions in

(b) There exists (30 E [0,71 such that for any sequence x,, = (r,,,0,,) such that x„ 0, the set of accumulation points of (0,,) is contained in

leo,Oo

In subcase (a), G contains two different vector subspaces of dimension 1 of IR 2 and so G = 112 . In subcase (b) . G contains the vector subspace R oo = = r, 00 ) :rE RI, but does not contain any other line which passes through


the origin. We have two possibilities: (i) G = Roo or (ii) G 0 Roo . Let us consider possibility (ii). In this case for any x E G — 11200 , the line x + — 0 =k+ yiyE Roo ) c G. Let f = R ot be the subspace of IR 2 which is perpen-dicular to R oo . Then P n G is a subgroup of IR 2 . Moreover f (1 G is discrete. In fact, ifenG was not discrete then e C G and we were in case (a). Therefore f fl G is discrete and this implies that k' fl G = • Z, where Tc Een G is such that = b, where t5 = min [ ; x E n G — 101). It follows that G = IR oo

C).X•Z1Rx2. Suppose now that G 0 101 is discrete. If there exists x E 1R 2 such that

G = xZ then clearly G Z. If this is not the case, let x, ,x2 E G be such that

Ix' I = G — [011 and lx1 1 = inftxj1xEG — x, IRI. We will prove that G = xiZ + x2Z Z C) Z. For this it suffices to prove that in the parallelogram with vertices 0,x, ,x, + x2 ,x2 there do not exist other elements of G. In fact, in the triangle with vertices 0,x 1 ,x2 there do not exist other elements of G by the definition of x, and x2 , and if there were agE G in the triangle x, ,x, + x2 ,x2 then x, + x2 — g would be in the triangle 0,x, ,x2 .

The rest of the proposition follows from Theorem 2.

Example. Let us see now an example of an action of IR 2 on S 3 . Let D 2 x S' = [ (x,y) E IR 2 X IR 2 jx + x; < 1,y 2, + y; = 11.

Consider the vector fields

X(x 1 ,x2 ,y) = (p(r)xl f3x2) —a

+ ( x, + p(r)x2 ) OX2

a a a a Y ( x, ,x2 Yi 9 Y2 ) ca2 — + coci — Y2 - yi axi 3x2 ayi ay2

where r + x; and p ( r) is a C nonnegative function such that p(r)

= 0 if and only if r = 1, cr p/drn (1) = 0 for all n I and p (r) 1 for r in a neighborhood of zero.

It is easy to see that the flows X, and Y, commute, i.e., X o Y, = Y, ( s, t E 1R) on D 2 x s', so they define a C' action of 1R 2 on D 2 x S having two compact orbits which are 101 x S' and a(D2 x s'). In the case that a

is an irrational number, all the remaining orbits are planes dense in D 2 x S'. When a is rational they are imbedded cylinders. Taking two copies of D 2 x SI and identifying them along the boundary a(D2 x S' ) = S' x S' by means of a diffeomorphism which takes parallels of one to meridians of the other and vice-versa, we obtain examples of actions of IR 2 on S 3 in which the orbits of dimension < 2 are two linked circles


D 2 x{y }

(orbits of X) (orbits of Y)

Figure 1

Other examples of actions can be obtained by taking the suspension of an action so : Z (I) Z --• Diff ( M). As we saw in Chapter V, these are locally free actions of 1R 2 , that is, the dimension of every orbit is two.

§2. The theorem on the rank of S3

We will show in this section that it is impossible to define locally free actions of Et 2 on S 3 . All the actions considered will be Cr, r ?_.- 2.

An important result in the direction of this theorem is the following proposition.

Proposition 2 ([28 ] ). Let X and Y be commuting vector fields on D 2 X s' such that on points of a(D2 x s') they are linearly independent and tangent to a (D' X $ 1 ) . There exists a point of D 2 X s' where X and Y are linearly dependent.

Proof. Let (p : IR 2 x (D 2 x S' ) --• D 2 x s' be the action induced by X and Y. By hypothesis 8 (D 2 x s') is an orbit of so, so there exist r, ,r2 E IR 2 — 101, r, 0 r2 IR, such that the isotropy group of 8D 2 x S', is Z • r, + Z • r2 . If we take r = mi r, + rn2 r2 where m i 0 or m2 0, then the flow sor ( t,x ) = so ( tr,x),x E 0D 2 x s', has all orbits closed with period 1. Now, it is easy to see that there exists ( ni, , f112) E Z2 — f Oh such that the or-bits of (Pr are tangent to the meridians of S' x s', so that any of such orbits is the boundary of a disk in D 2 x S'. After a change of basis in Z• r, + Z

a - r2 we may assume that r = r 1 . Let e l ( x) = -5-i so ( t - r i ,x) I t=0 and e2 (x) =


a = —at 4

(t • r2 ,x) I . Since ço is an 1R 2-action, e, and e2 are linear combina- t=o tions of X and Y. Since the orbits of e l are closed and bound a disk in D 2 x S', we can suppose (after a diffeomorphism) that e l is tangent to the boundary of D 2 x 10 c D 2 x S' and e2 is transverse to D 2 x WI in the points of the boundary s' x (0). Consequently the proposition will be proved with the following lemma.

Figure 2

Lemma 1. Consider the disk D 2 C 1R3 : x 2, + x; lç 1, x3 = 0 and the vector

fields e l (x) = x2 a/ ax, + x1 ô/ 3x 2 , e2 (x) = 0 / 3x3 , x = (xi ,x2 ,0) E 3D 2 . Then for any continuous extension (e i (x),ei(x)) xED 2 of ( ei (x),e2(x)) xon 2 there exists xo E D 2 where el (x0 ) and e2 (x0 ) are linearly dependent.

Proof. First we observe that if there exists an extension (e, (x) , e2 (x)) xED 2 of linearly independent vector fields, then we have an extension by ortho-normal fields. In fact if f2 (x) = e2 (x) a'i (x),e 2 (x)) -e- 1 (x), ë 1 (x) = el(x) f2(x) and ( X) = , the pair re,(x) ,2(X)) would be an ortho-

If2 (x)I normal system.

To prove the lemma, it suffices to show then that there do not exist orthonor-mal extensions of the pair (e, ( x ) , e2 ) ,EaD2 •

Let V2 , 3 be the space of ordered pairs of orthonormal vectors of 112 3 . The set (e l (x),e 2 (x)),_EaD2 can be understood as a path a : 3D 2 V2 , 3, a (x)

(e 1 (x) , e2 (x)). It is sufficient then to show that a is not homotopic to a point.

To prove this we identify V2 , 3 with the real projective space DP 3 , of dimen-sion 3, in the following manner. To each ( f1 ,f, ) E V2 , 3 we associate a vector w = w ( f2) E B (0,r) C 1R 3 of the ball with center 0 E iR 3 and radius r defined as follows: the matrix F, whose columns are f, , f2, f1 x f2 is orthog-onal and has as axis of rotation the subspace f of dimension 1 in 1R 3 . Let P be the plane orthogonal to C. In P the matrix F acts as a rotation of angle

E [0,7r]. For 0 = 0 one verifies that F = I and we set w = O. For 0 < 0 < 7r. we associate the vector w E C with magnitude 0 and the same direction as v x F( v ), where v E P — (01. For 0 = r, u x F(v) = 0, and in this case


the direction of w is not well-defined. This corresponds to identifying antipodal vectors of magnitude on e. We obtain in this way a homeomorphism on the quotient space of B ( 0, 71- ) by the equivalence relation which identifies antipodal points of the boundary, which, as is known, is homeomorphic to 11) 3 . The path a, in this model, is represented by the diameter of B (0 ,7r) , parallel to the x3 -axis and traversed once from the bottom to the top. This path, after iden-tification of its endpoints, is nontrivial in 1P 3 .

Definition. The rank of a manifold is the maximum number of linearly in-dependent commuting vector fields that the manifold admits. This concept was introduced by J. Milnor.

Theorem 3 ([28]). Every compact three-dimensional manifold with finite fun-damental group has rank one. For example, S 3 has rank one.

Proof. As is well-known every manifold of odd dimension has Euler charac-teristic zero. So it admits a non-singular vector field and its rank is 1. Taking a double cover, we can assume that the manifold is orientable. Proceeding by contradiction, suppose that there exists a locally free action of 1R 2 on some compact three-dimensional manifold with finite fundamental group. By Novi-kov's theorem, there exists a compact orbit bounding an imbedded solid torus, where there are defined two linearly independent commuting vector fields, which is impossible by Proposition 2. •

§3. Generalization of the rank theorem

In this section we will see that the theorem on the rank of S 3 generalizes to actions of simply connected groups on compact manifolds with finite fun-damental group.

Proposition 3. Let so : G x M M be a locally free action of a Lie group

G on a manifold M and f:I x I M be a C° map such that for every t E I

the curves f (s, I) is contained in an orbit F, of p. Let x ( t ) = f (0,t ) and

suppose that, for every t E I, X' (t) T, (1) ( Ft ). Then there exists a unique con-

tinuous function .7 : Ix I G sumh that :1(0,0 e, and (p (s,t),x (t)) = = f(s,t) for any (s,t) E I x I.

Proof. By Theorem 2, for every t the map (p oi) : G F, given by (1) (g) = ,,c(g,x(t)) is a covering of F,. By the lifting theorem for paths,

there exists a unique continuous curve :f; : G such that (0) = e and for every s E l one has so, ( , ) (f(s)) = f(s,t). Define s, t ) = ( s ) . It is evident thath 0 , t ) e and ,p (s,t), x(t)) = f(s,t) for any ( s, t) E I x I. Moreover, if h:IxI--•G satisfies the conditions h ( 0, t) = e and


(h (s,t),x(t)) = f(s,t) for every (s,t) E X /, we must have h f, by the uniqueness of the lift of the curve s f(s,t) to the covering sox( , ) : G —• F,. It remains to be proved that ./ is continuous.

Let xo E M, yo = (p(g0 ,x0 ). Let E, xo E E, be a transverse section to the orbits of Since (tog° : M M is a diffeomorphism which preserves the orbits of ço it follows that T = D (g0 ,x0 )(7:g) G x 7x0 E ). So Dso (g0 , x0 ) : Tgo G x TE Ty0 M is an isomorphism. By the inverse function theorem there exist neighborhoods g o E Vc G and xo EDCE such that U = = (it) (V x D) is a neighborhood of y o and vi = V x D is a diffeomorphism of V x D onto U.

Let us prove that :f is continuous at all the points of the type (0,4), to E I.

Fix to E I and let xo = x(t o ), yo = f(0,t o ), go = 1(0, t0 ) = e and consider D, V, U and ik =wiVxD:VxD U as above. Let lo = [0,6) and Jo = ( to — 6,10 + 6) be intervals such that f(Io x Jo ) C U and x ( J0 ) c D. Define g : /0 x Jo V C G by (g(s,t),x(t)) = 1,l/ -1 (f(s,t)). Then g is continuous, g(0,t) = e and so (g(s,t),x(t)) = f(s,t) for every (s, t) E

x J. By uniqueness of the lift of the curve s f(s,t) across v) : G F,, one concludes that g (s, t) = :1(s, t) for (s, t) E 1 x Jo , so .7 is con-

tinuous on ( 0, to ). We prove now that the set A = [ (s, t) EIx /if is continuous at (s,

is open in I x I. Let (so ,to ) E A, xo = x(10 ), yo = f(so ,to ), go = :l(so ,to ) and consider D,V,U, and 1,G = 1VxD as above. Since :f is continuous at (so , to ), there exist open sets 10 ,4 c I such that so E 1 , to E J0 , x( Jo )

C D, _7(10 x J0 ) C V and f(I0 x Jo ) C U. For every (s,t) E I x Jo we have (hs,t),x(t)) = (f(s,t)), which proves that f is continuous on /0 x Jo , so A is open in I x I.

For each t E I, let a( t) = sups E /1 7 is continuous at ( ta) for every E [0,$) j. It suffices to prove now that (a (t),t) E A for every t E I. Fix

to E I, so a (to ), xo = x( t o ), yo = f(s0 ,t0 ) and go = Î(s0 ,1 0 ). Consider D, V, U and ik = I V x D as above. Since s 1-0 7(s, t) is continuous (by construction), there exists 6 > 0 such that if s E [so — 6,s0 ] then 7(s, t0 ) E V and.f is continuous at (s i ,t0 ), s 1 = so — 6. Since A is open there exists 6, > such that the segment f = Isl j x [(to — b i ,to + 51 ) fl I] c A. Set I = = (s, — 5,s0 + 6) n land Jo = ( to — 6 1 , to + 6 1 ) n 1. If 6 and 6 1 are suf-ficiently small, we have f(10 x Jo ) C U, x( Jo ) C D and :7(f) c V. Now define g :10 x Jo V c Gby (g(s,t),x(t)) = 11/ -1 (f(s,t)). However it is clear that g(s l ,t) Tf(s i ,t) for (.5. 1 ,0 E f and that ,p (g (s, t), x( I)) = = f(s,t) for (s, t) E J x Jo . By uniqueness of the lift of the curve s

f(s,t), it follows that g = 7 I x Jo and therefore :f is continuous on ( so , to ) as desired. •

Theorem 4. Let ,p : G X M M be a locally free action of a simply connected Lie group G on a manifold M where dim M dim G + I. Then the folia-


lion of M by orbits of 4, does not have vanishing cycles. In particular, if dim G = = 2 and dimM = 3, if has no Reeb components.

Proof. Suppose by contradiction that there exists a leaf F0 of if which has a vanishing cycle y : I Fo . Then y is a closed curve not homotopic to a constant in F0 and there is an extension f : I x M such that for every s E I, f(s,0) = -y (s) and for every t > 0, t E I, the curve ft (s) = f(s,t) is a closed curve contained in a leaf Ft of if, homotopic to a constant in F. Let : I x G be as in Proposition 3. Now, for t > 0, the curve j; (s ) is homotopic to a constant in F„ so the curve :,,(s) = 7(s, t), which is a lift of f, to G, is necessarily closed, that is, :.f; (0) = :f*", ( 1 ). By continuity we have 1'0 (0) = ( 1 ), or, 1.0 is a closed curve. But 20 is a lift of y to G, so it cannot be closed since G is simply connected and y is not homotopic to a con-stant in F0 , which is a contradiction. We conclude therefore that if has no vanishing cycles. •

Corollary 1. Let G be a simply connected Lie group and M a compact manifold with finite fundamental group such that dim (M) = dim ( G) + I a. 3. Then there is no locally free action of G on M.

Proof. By Proposition 1 of Chapter VII we know that a foliation if of codimen-sion 1 on M necessarily has a vanishing cycle, so it cannot have a locally free action (,o :G><M—M. •

Corollary 2. Let G be a simply connected Lie group and M a manifold such that dim (M) = dim (G) + 1 3. Let (p : G X M M be a locally free action. Then the following properties are true:

(a) if F is an orbit of 4,t, and : F M is the canonical immersion then : lri (F) 7r 1 (M) is injective, that is, if y is a closed curve in F

homotopic to a constant in M then y is homotopic to a constant in F, (b) if y is a curve transverse to the orbits of (p, , then the homotopy class of

y in 7r i (M) has infinite order.

Proof. (b) is immediate from Theorem 4 and from the proof of Proposition 1 of Chapter VII. Let us consider (a). Suppose y is a closed curve in F, hornotopic to a constant in M. Let f: D 2 M be a Cr ( r 2 ) map such that f y and f is in general position with respect to the foliation if

defined by orbits of so. Let X be a vector field on D 2 tangent to the foliation with singularities f* (if ) (see Proposition 2 of Chapter VI). S' = a D 2 is a closed orbit of X. We claim that all the nonclosed orbits of X are separatrices of saddles. Indeed, if X has a nonclosed orbit which is not a separatrix of a saddle, let us say 6, the u-limit set of 6 is a closed orbit or a graph and this implies that there exists a closed curve y in D 2 transverse to the orbits of X.


Figure 3

Repeating the argument of Proposition 1 of Chapter VII on the disk D 21 C D 2 bounded by p., one concludes that has a vanishing cycle, which is not possible. Therefore all the regular orbits are separatrices of saddles. Sup-posing by contradiction that S' = a D 2 is not homotopic to a constant in its leaf, repeating the argument of Proposition 1 of Chapter VII, it is possible to prove that has a vanishing cycle. We leave the details to the reader.

Corollary 3. Let G be a simply connected group and M a (noncompact)

manifold with finite fundamental group such that dim (M) --= dim (G) + 1 3. Let so : G x M M be a locally free action and 5 the foliation whose leaves are the orbits of. If F is a leaf of 5 then r (F) is finite and F is a closed subset of M. In particular every minimal set of a is a leaf and

if M is simply connected then all the leaves are diffeomorphic to G.

Proof. Let F be a leaf of . By (a) of Corollary 2, i* : r i (F) r i (M) is injective so it (F) is finite and #71- 1 (F) 5_ ttr i (M). In particular if M is simply connected, F is also and therefore if x E F the map : G F given by so, (g) = p(g,x) is a diffeomorphism, since it is a covering.

Suppose by contradiction that 'S has a leaf not closed in M. Then there is a sequence xn E F, x„ xo 0 F. Let Q be a foliation box of x0 . The leaf F

cuts Q in an infinite number of plaques, therefore we can construct a closed curve transverse to 3. Since r i ( M) is finite this curve has finite order, which contradicts (b) of Corollary 2. •

§4. The Poincaré-Bendixson Theorem for actions of 111 2

One of the versions of the Poincaré-Bendixson theorem for a flow E on the sphere S 2 or in the plane says that every minimal set of E is an orbit.


In this section we generalize this theorem to locally free actions of R2 on manifolds of dimension three. Other results in this direction can be found in [61, [381 and [451.

Let w:G x M M be an action. A subset of M is invariant (under w) if it is the union of orbits of so.

A minimal set of so is a subset p, c M which is closed, invariant, non-empty and such that no subset properly contained in p. has these three properties.

The theorem below is a special case of Corollary 3 of Theorem 4. We think it is interesting to include this new proof because it is more geometric than the first and also because it contains the original argument of [281 in the proof of the fact that an action of R 2 on S 3 (if it exists) must have an invariant torus.

Theorem 5. Let tp : 1R 2 x M M be a locally free action on a simply connected three-dimensional manifold. Then any minimal set of so is an orbit (in fact by Corollary 1 of Theorem 4, the manifold is not compact).

Proof. Given p E M we define 00, ( so ) n7„ 1 op ( so) — lc, where Ka C K 1 C Op(SO) are compact neighborhoods of p and U 1 K Op( ) • It is clear that Op(S0) Op(S0) U 80p(SO) • Now let 12 C M be a minimal set of ye; and p E j.

Proceeding by contradiction we suppose that ( w). Since kt is minimal we have that p = Op(SO) • So aOp(SO) is non empty and, since it is closed and invariant, 3(.9„((p) = A. This means that if U is a coordinate neighborhood of the foliation g (w) induced by so, with p E U, the orbit Op(SO) has as intersec-tion with U an infinite number of plaques, each being an accumulation point of plaques in 0,, ( w ) .

Thus it follows (Proposition 4, Chapter VII) that there is a closed path a : M passing through p and tranverse to the leaves of g (p). Since M is simply connected, a extends to a map A : D 2 M in general position with respect to the leaves of g ( w ) . The foliation A''‘ (so) of D 2 has singu-larities which are saddles and centers. Further it is transverse to the bound-ary. Let q E 31) 2 be such that A (q) =-- p.

Modifying A slightly if necessary, we can assume that the point q is not con-tained in a separatrix of a saddle. There will exist, then, in the limit set of the point q a limit cycle -y : s' D 2 , inducing a curve 7-y : S l M, = A o y , 5 (S' ) C 0,0 ( C pc with nontrivial holonomy. So 0,0 ( ) is homeomorphic to JR x s' or s' x s'. Since 0,0 ( w) p., 0 xo ( so ) cannot be a torus. Consequently 0(so) is an immersed cylinder with 30,0 ( 40) D 0,0 ( w ). We next show that this is impossible.

Since G ( p) 0 there exist ri ,r2 E 1R 2 , ri 1R1-2 , such that the orbit F through x0 of the flow w,, so, ( t,x) = so ( trI ,x), is periodic of period to and the orbits of W2, W2 ( t, X) = w ( tr2 ,x), for x E r are transverse to F. Since so, and so, are commutative, for every x E 0 ,0 ( so) the orbit of w, through x is periodic of period to.


Let C be a two-dimensional cylinder imbedded in M, transverse to the orbits of the action so with F C C. We next show that arbitrarily near r there exists a closed curve r' c Oxo n C such that r'nr 0 and r' U r is the boundary of a closed cylinder B C Oxo (so). Let V be a cylindrical neighbor-hood of F in C, such that the orbits of (p2 are transverse to V. If E > O is small, the set V, = f (p2 ( t, X) — E < t < E, X E V) is a neighborhood of r in M and the orbits of s02 I V, define a projection r : V, V. Since r is a periodic orbit of so, with period to and F c V,, there exists a neighborhood U C V, of F such that, for every x E U, so l (t,x) E V, for I ti < 2t0 . Since 30xo (w ) D 0(so),

V

Figure 4

there exist sequences x,, E F and sn E JR where is,i 1 —■ 00 , such that sn ) X0 E U. If n is sufficiently large, y, = so 2 (s,,,x,) E U fl 000, so

w ( t, Y, ) E V, for t < 2t0 and since the orbit 0,, of yr, under so, has period to, we have that 0,, C V, . If F,7 = ( O n ) it is clear that r„ c 0(w) and F,, n r = 0. On the other hand, if d is the intrinsic metric of 0 so) d(F,On ) --• co; so, r u F ,, istheboundaryofacylinder B,, c 0 0 ((p) since d(rn ,o„) is bounded because 0,7 c se,2 ( t, X) IX E < (). This proves what we wanted.

Let A,, c e be the annulus bounded by F and F„ . Since lim„, f ,, = F there exists n o such that .4,70 r.) B no = u rno . We have then that T = = A ,, U B,70 is a topological torus imbedded in M.

Since M is simply connected, T is the boundary of an open bounded set a

W c M (see [28 ]) . Let X be the vector field given by X (x) = gD2( t,X) t_o .

We have that X is tangent to T on and X is transverse to T on A no . We can assume that for every x E A no , X(x) points inward toward W. This implies that every orbit of ‘02 which enters W does not leave in positive time, so 0 A n A,, > U F, a contradiction. •


Remark. The above theorem can be used to prove, by contradiction, the rank

theorem of S 3 . Indeed, since S 3 is compact, every action : IR 2 x S 3 has a minimal

set ti. If so were locally free, p, would have to be a compact orbit. So, p. would be diffeomorphic to a torus S l X S ' . Using Alexander's theorem (Proc. Nat. Acad. Sci. (1924) pp. 6-8) according to which for any imbedded torus p., one of the connected components of S 3 — p. is a solid torus, it follows again from Proposition 2 that there is a contradiction. This proof of the rank theorem was given in ([28]), when Novikov's theorem had not yet been proved.

One problem still open about the rank of manifolds is to find out the rank of s2n-t-1

It is well-known that there do not exist 2-dimensional plane fields on S 5 ([53]). On the other hand, the existence of locally free actions of R 2 on S 7 is

unknown. In [48] , H. Rosenberg, R. Roussarie, and D. Weil prove the following result:

a compact orientable three-dimensional manifold has rank 2 if and only if it is a fibration over S with fiber T 2 .

§5. Actions of the group of affine transformations of the line

The Lie group of affine transformations of the line is the set G2 of all affine diffeomorphisms f: JR JR with the operation of composition ( fo g. The affine transformations are those of the form f(x) = ax + b where a,b E IR, a 0, are fixed. Given f,g E G2, f(x) = ax + b, g(x) = = a' x + b', the composition fog(x) = ( aa ' )x + (ab' + b) is also affine so G2 is a two-dimensional Lie group which has two connected components. We consider only the connected component G 2+ = ff(x) = ax + b I a > of affine transformations that preserve the orientation of 114.

The following facts can be easily verified:

(a) G 21- is not abelian, so is not isomorphic to IR 2 , (b) G 2+ is diffeomorphic to IR 2 , so is simply connected, (c) G 2± is isomorphic to the subgroup G - of GL( 2,1R), defined by the set

[ a 131 of matrices I such that b E JR and a > O.

[ O 1

In this section we will prove that the two-dimensional orbits of an action of G 2+ on a manifold 1WI are diffeomorphic to planes or cylinders. Finally, we will see an example of a locally free action of G; on a compact manifold M 3

which has a countably infinite set of cylindrical orbits whose complement con-sists of orbits diffeomorphic to the plane. As we will see, all the orbits of this action are dense in M3 .

Proposition 4. Let so be an action of G - on a manifold M". Then the two-dimensional orbits of ,p are diffeontorphic to planes or cylinders.


Proof. Let p E M and Gp fg E G + ! 4p (g, p) = pl be the isotropy group of p. As we know, the orbit 0 ( p ) is diffeomorphic to the manifold G + Gp ,

obtained as the quotient of G ± by the equivalence relation — on G + which identifies g l and g2 if g 1 g' E Gp . If 0 (p) has dimension 2, it is clear that GI,

is a discrete subgroup of G ± . It suffices then to prove that every discrete

ab subgroup of G ± is of the form H = 1gn I n E Zi where g =

01 fixed. Let us identify G + with the half plane P+ = t(xy) E IRL jx>0).

With this identification, the product on P± is given by (x, y) o (x' ,y')

1-0 (xx' ,xy' + y) and g = (a, b) . First suppose that a 1. In this case the line r = ( 1, y) y E JR I is identified by — with the line r g = 1(a,b + y) = = (1,y) o (a,b) y E IRJ.

The lines r and r o g bound a strip A = 1(x, y) x max ( a, a - ' )1. It is easy to see that the two distinct points contained in the interior of A are not identified by — ; so 0 (p) is diffeomorphic to the quotient of A by — ; that is, the manifold obtained from A be identifying the boundaries r and r o g by the diffeomorphism ( 1,y) E r (a,b + y) E r o g, which is clearly a cylinder.

In the case that a = 1 and b 0 we can make the same argument, substituting the line r by the line s (x, ) j x E JR and the strip A by the strip B = 1(x, y) 0 y I bjx bounded by the lines s and s g =

=

In the case that a = 1 and b = 0, clearly 0 ( p) is diffeomorphic to IR 2 . Con-sider then a discrete subgroup H C G ± . First we are going to prove that H is contained in a line f which passes through ( 1,0). Suppose that (a, b), (c, d) E H where a 1. In this case it is easy to see that (a, b)" = (an ,b (an — 1 ) ( a — 1) -1 ) for every n E Z Therefore

= (a,b) " (c,d) • (a,b) n = ( c, a 1 [bc(1 — a n ) + b(a — 1)1 + da ").

If a > I one sees that lim n h, = (c,(bc — b)/ (a — 1)) = h € II, since h, E H for every n E Z and H is closed. Since H is discrete one concludes that h, = h for every n n o , which implies that (a,b) and (c,d) commute, as can easily be seen using the relation h„,, o h,;-1 = ( 1,0) = 1. On the other

hand, this implies the relation b(c — 1) = d (a — 1) and so (c, d) E f =

= (x, g) j b(x 1) = (a — 1)yi. In the case that a = 1 and c 1 the same argument applies, so we can assume that a = c = 1 and in this case

H C f 1,y)IyEIR). Suppose now thatHcfnP - = f ( x, y) x > 0 and y = k (x — 1)). In this case f fl P- is a subgroup of G . isomorphic to the

additive group IR by the isomorphism f( t) (e l ,k (e t — 1)), as can easily be seen. Therefore f (H) is a discrete subgroup of IR, so H = fg" n E Z1

for some g E H. In the case that H C f = 1(1,y) j y E El, the argument is similar. •

is


Example. First we are going to define a locally free action ;To of G ± on 1R 3 . The manifold M3 will be constructed as the quotient of 1R 3 by an equivalence relation — defined in such a way that ço induces an action so of G + on M.

(A) Construction of o.

Let A = . As can be directly verified the eigenvalues of A are

X and X 1 where 0 < X = (3 — )/2 < 1. Let y = ( v2 ,/i3 ) be an eigenvector of A relative to X. Define ‘,7) : G x 1R 3 R 3 by

(g;(x,z)) (x + 1 log ce, z + e - " • v) -67

0 where in the above expression g = , z E IR 2 and a = In (X) < O.

01 As can be verified directly ço (g' o g; p) = 7, (g ';;-0(g; p)) and ,70 (I, p) = p for any g,g' E G" and p E 1R 3 . Therefore z",0 is an action of G + on IR 3 .

(B) Construction of M.

Consider on 1R 3 = IR x IR2 the equivalence relation — which identifies two points (x, z) and (x ',z') if one of the following conditions are true: (a) x = x' andz — z'EZ 2 or(b)x— x' =kEZ andz'=A k z.

Making the identification (a) we get the manifold JR x T2 . Note that A and A -1 are matrices with integer entries. This implies that if z — z' E Z 2 then A k Z A k e E Z 2 so A induces a diffeomorphism X1 T 2 —•-w T2 . The iden-tification (b) is equivalent to identifying two points (x, p) and (x',p') E JR x T2 when x x' = kEZ and p ' = A k (p). Note that the equivalence relation — is generated by the diffeomorphisms f1 ,f2 and f3 : R 3 R 3 defined by fi (x,z) (x + 1,A -1 z), f2(x,z) = (x,z + e t ) and f3 (x,z) = = ( x, z + e2 ) where e, = () and e2 = (?). This means that two points q and q ' of 111 3 are equivalent if and only if q = h ( q ' ) where h = f ri o f f t3' 1 0 ... 0 f rini, f 2nk f f3k, k E N and m„n„fi E Z for j = 1 ,...,k. Denote by

H the set of all diffeomorphisms of this type. Let M = 1R 3 / — with the quotient topology and 7r : R 3 M the natural

projection of — . It is easily verified that r is a covering map and therefore we can induce on M a C differentiable manifold structure as was done in Chapter I. An atlas of M is the set of all local charts ( 7r ( V), (r V) -I ) where V = (x,z,2,z 1 ) — c < 1/2, iz2 — 3 2 ; < 1/2, 'z 3 /3 , 1 < 1/21, where

(cx,132,(33 ) E IR 3 . The changes of coordinates of this atlas are restrictions of dif-

feomorphisms of H.


(C) Construction of it,

Observe that (74, commutes with the diffeomorphisms f1 ,f2 and f3, that is, for every (g,p) E G + x R 3 and i = 1,2,3 one has rp (g,f,(p)) =

= f,(0(g,p)). We can then define a C action on M setting

= r(rp(g,p)) .

It is easy to verify that so is well -defined by this expression, that is, if r ( p) =

= (p') then 7r(Z,b(g,p)) = 71- (ZMg,p')).

(D) Geometric description of the orbits of v.

First let us see what happens to the orbits of iTo as we make the identification (a). Observe that the orbit 5 (p) of a point p E ill 3 by Zro is the plane that passes through p and is parallel to the subspace of 111 3 generated by the vec-tors ( 1 ,0,0 ) and (O, u2 , v3 ) . In making the identification (a) we do not change the first coordinate and each plane fx1 x 1R 2 is transformed in the quotient to the surface fx) x T 2 where T2 = 1R 2/Z 2 . On the other hand 5 ( p) cuts the plane [xi x 1R2 along the line t p' + tv, p' = (p2 ,p3 ), which has slope 0 = u3 /u2 = ( 1 + \FS) / 2 in this plane. Therefore after identification (a), the orbit ( p) is transformed into an immersed submanifold JR x E, where

is an orbit of the irrational flow of slope 0 in T2 . As we know E is dense in T2 (see exercise 13 of Chapter II), so R x E is dense in JR x T2 . This implies that all the orbits of so are dense in M.

We will now see what occurs after identification (b). The immersed sub-manifold JR x E meets RI x T2 in the line { kj x E (k E Z) and this line will be identified by (b) with the line 10j x A k (). Now, the irrational flow of slope 0 is invariant under A, since u is an eigenvector for A, so A () is an orbit of this flow. In the case that E contains a periodic point q of period k of A (A k (q) = q), the endpoints [O x E and [kj x E of the strip fx) x 0 k, will be identified and we hence have a cylindrical orbit of so.

Conversely, if the endpoints [01 x E and [I/ x E (k > 0) are identified, we have A k (E ) = and this implies that E contains a unique fixed point qo of A k , since A k IE is a contraction, that is, if q and q ' E E then d(A (q),A k (q' ))

5- X ic d (q,q'). With this argument we establish a one-to-one correspondence between cylindrical orbits of so and periodic orbits of A. As is known, A has countably infinite periodic orbits (see [341), so so has countably infinite cylin-drical orbits. The remaining orbits of ço are diffeomorphic to the plane, by the previous proposition.

APPENDIX FROBENIUS' THEOREM

Let if be a k-dimensional, cr( r 1) foliation defined on a manifold M of dimension n. The foliation if induces on Ma Cr - k-dimensional plane field, denoted by Tg (see Chapter II). Here T5' ( p) = if is the subspace of Tp M tangent to the leaf of if which passes through p. A natural question is the following. Let P be a Cr ( r 1) k-plane field on M. Under what conditions does there exist a foliation if such that 77 = P?

As we saw earlier this does not happen in general for an arbitrary k-plane field (k 2 ) . Frobenius' Theorem provides necessary and sufficient condi-tions for such a foliation to exist. With the purpose of proving this theorem, we introduce in §1 of this appendix the notion of Lie bracket between two vec-tor fields, proving its main properties. In §2 we prove Frobenius' Theorem. In §3 we state and proof, without going into details the dual version of Frobenius' Theorem in terms of differential forms.

§1. Vector fields and the Lie bracket

Recall that a Cr (r O) vector field on M is a Cr map X: M TM such that r o X = identity map of M, where r : TM —1. M is the natural projec-tion of the tangent fibration. Denote by X' ( M) the set of all Cr vector fields on M.

Definition. Let X E X ° (M). A C' curve 'y : (a, b) M is called an integral curve ofX if -y' (t) = X(-y(t)) for every t E ( a, b).

On a local chart x: Uc M Rn a vector field X is represented by the

vector field x, (X) defined on x(U) C Rn by the expression

X(p) = x.(X)(p) = dx x _, (p) - X(x -1 (P)) •


Since x: U --• is a C map, the vector field X. has the same class of differentiability as X. A fact that can be easily verified is the following. Let y: (a,b) U C M be a C' curve. Then y is an integral curve of X if and

only if x y: ( a,b) x(U) is an integral curve of X. Putting the above fact together with the theorem on the existence and uniqueness of solutions of ordinary differential equations on Ile, we can state the following result.

Theorem (Existence and uniqueness). Let X be a Cr , r 1, vector field on M. Given Po E M and to E JR there exist E > O and a neighborhood U0 of po in M such that for every p E U0 there exists an integral curve -y of X defined on (to — , to E ) with y (to ) = p. If : M and -y 2 : 12 M are two integral curves of X such that to E I fl 12 and -y i ( to ) = -y2 ( to ) then y i = -y2 on

From the above theorem we can conclude that for each p E M there exist a < 0 < b and an integral curve y: (a, b) M with y(0) = p such that for any other integral curve y i : M such that 0 E /1 and y i (0) = p then /I c (a, b) and = y /I . The interval (a, b) is called the maximal interval of definition of the solution which passes through p; we will denote it by /p .

Let V = ( t, p ) E JR xA//tE Ip . Define so : V --• M such that for each fixed p E M, the curve t ( t , p) is the integral curve of X with so ( 0, p ) =

= P.

Definition. The map defined above is called the flow of X. Denote the map p E M (t, p), t E p , by X,. For convenience, we omit the domain of such a map.

From the existence and uniqueness theorem, it follows that V is open in 111 x M and that so has the following property:

(s + t, P ) = t, P = ( s, P

whenever the expressions make sense, that is, are compatible with the domain V. If V = JR x M, the above identities can also be written as = X, X, = X, o X.

The theorem on the differentiable dependence of the solutions of an ordinary differential equation on with respect to initial conditions implies the following:

Theorem. Let X be o Cr( r 1) vector field defined on M. Then : V M is C.

Suppose now that to E IR is such that so ( to ,p) is defined for all p E M. In this case X, : M M is a Cr diffeomorphism. In fact, if ,p ( to ,p) is defined for all p E M, it is easy to see that sc ( to, P) is also, therefore X =

Appendix 177

= X, 0 o X, 0 = the identity on M. Since X and X, 0 are Cr, by the above theorem we conclude that X10 is a Cr diffeomorphism. More generally, if V, is the open set in M, defined by V, = fpEMItElp ) then V, is the domain of X„ where X, ( V, ) = V_,. Therefore X,: V, —• is a Cr diffeomorphism whose inverse is X_, : V_, --• V. If M is compact it is possible to prove that V = 11/ x M and hence V. = M for every t E 112 (see [34]).

Change of variables

Let f : N P be a local Cr (r 1) diffeomorphism. If X is a C1 vector field on P we can define a vector field f* (X) on N by

f* (X)(q) = (Df(q)) - ' • X(f(q)) .

The vector field f* (X) is Cf where e = min (s,r — 1). If f is a diffeomor-phism we can write

f* (X)(q) = (Df(q)) -1 • X(f(q)) = (f(q)) • X(f(q))

In this case the operator f* has an inverse f,, defined by

.f.,(Y)(P) = Df(r i (P)) Y(.1 -1 (P))

The following proposition is a direct consequence of these definitions.

Proposition 1. Let f: N P and g: P M be Cr (r 2 ) diffeomorphisms. The following properties hold:

(a) (g f)* = f* g* and (g o f). = g. o f..

(b) Let X be a C (s 1) vector field defined on P. Let ço : V P and V* — • N be the flows of X and f* (X) respectively. Then V* =

= t , p) E I>< E VI and further f ( so* (t, p)) = = ,,o(t,f(p)) for every (t, p) E V*.

Property (b) above tells us that if we know the integral curves of f* (X)

and the diffeomorphism f, then we know the integral curves of X, and con-versely. In the notation X, (p) = (t, p), (b) is written as f o X7

Lie Bracket

Let X and Y be two vector fields on M. For simplicity we are going to assume that X is C 2 , Y is C 1 and the flow of X is defined on IR x M.

Fix p E M and t E Eft; the vector

u(t) = X7(Y)(p) = DX_ ,(X,(p)) • Y(X,(p))

is tangent to M at p. So t v( t) is a C' curve in 7; M.


Definition. The Lie bracket of X and Y is the vector field [X, Y] defined by by

d EX,Y1(P) = —dt (X7(Y))1-0 •

Note that the definition of [X, Y] ( p) is local, that is, depends only on values of X and Y in a neighborhood of p. Consequently the hypothesis of the flow of X being defined on lR x M is unnecessary.

We will next see that the bracket [X, Y) is defined even when X is c'.

Lemma 1. (Expression of [X, Y] on a local chart). Let .,,c:UCM 111" be a local chart. Denote by a / axi the vector field on

U defined by a/ ax,( q ) = (dx (q)) - - e. fe l ,..., en ) is the canonical basis of 1R'1 . If X and Y are vector fields on M, we can write X =

E a, a/ax„ Y = E ni = bs a/ax,. Then [X, Y] E , c1 3/&, where

17 ab, aa, . (a - c -i E — ) . ax ax

i=1

In particular we can define [X, Y] even when X is C'. Further, if X and Y are Cr, [X, Y] is Cr- 1 .

Proof. We can write x. ( X) = E a, e, = X and x. ( Y) = E , 0,e, = = 37, where a s = a,. and (3 i i = 1 n. With this notation, taking into account Proposition 1, it is easy to see that ( X, )* ( ) = = x. ( (X,)* (Y)) where X and X, are the flows of X and X respectively. At a point q E x(u), this identity can be written as below

(Y,)* (Y)(q) = Dx(p) ((X)* (Y)(p)), p = (q) .

Since p is fixed, Dx(p):Tp Al IR" is a linear transformation and therefore

[X, 11(q) = 1Dx(p) • ((X7)(Y)(p))1, =0

d Dx(p) —dt lX7(Y)(P)11=0 Dx(P) • [X,Yi(P) = x.([X,Y1)(q) .

We can say then that x.[X, Y] = [x.(X),x.(Y)]. It now suffices to calculate ] (q). For this we are going to look at X and 17 as functions k Y :

x(U) IRn. We will also use the notation W, = DX _,(7A7 ,(q)). We will look at t Wt as a curve ( C' ) in oe ( IR" ) , the set of linear transformations from to JR". With this notation we have that Wo = I and that

Appendix 179

[X ,Y](q) = cl [W, • Y (k ,(q})

d

W, (q) + DY (q) - —d (X—,(q)) . dt 1 1=0 dt i=o

On the other hand, it can be verified that W. - DX,(q) = I, therefore

d d — t =

—d

(D),(q)) -D ( X ,(q)) = -DX (q) . d wi l (-0 dt i=o dt f=o

Thus we conclude that

= D7 (q) - X (q) - DX (q) • Y (q)

Therefore the i th component of [Y,T1 ](q) is

ao, ace,

i=. ax,

This concludes the proof. An important consequence of this lemma is the following.

Proposition 2. The following properties hold:

(a) If f:N P is a 2) diffeomorphism and X,Y Er(P),

s 1, then

f* ([X,Y]) = If* (X),f * (Y)]

(b) If X and Y are two vector fields defined on an open set U C IR", then

[X,Y1(q) = DY (q) • X(q) - DX(q) - Y (q), q E V .

(c) [X,Y] = -[Y,X1

(d) [ , ] is bilinear: [aXi + bX2 , Y] = a[XI ,Y1+ b[X2 ,Y ] and bY2 ] = a[X, Y,1 + b[X,Y2]•

The proof is immediate from the lemma.

Commuting flows and actions of IR k .

Recall that a Cr action of a Lie group G on M is a Cr map so:GxM--••M

such that

(a) e,p) = p for every p E M, where e is the identity of G, (b) ço ( g, •g2 , p) = (g lop (g2 ,p)) for any g i ,g2 E G and p E M.

It follows immediately from the definition that for every g E G, the mapp E M


,pg (p) = so(g,p) is a CT diffeomorphism, whose inverse is so _ (p) = (p(g-I ,p).

If G 11e, the additive group, an action so : 1R k X M M is generated by k vector fields on M. Indeed, let v i vk I be a basis for Il k and for each j E 11 k ) set so., ( t, p) = (tvi , p). As is easy to verify, for every j E 11 ,..., k), yoj is a CT flow, to which is associated the vector field X' , of class

CT -1 , defined by X' ( p) = p) . =

It is easy to see that the flow t o

of X" is soi , so we can write

E tivi,p) = o o••• . is I

We say that the vector fields X' ,..., XI are generators of the action so. Since addition in is commutative, we have that

for any i, j E 1 , k ) and s, t E IR.

Proposition 3. Let X' ,..., X" be Cr (r 1) vector fields defined on M. Sup- pose that for every i E 11 k), the flow X; is defined on 1R x M. The following statements are equivalent:

(a) The fields X' ,..., X" are generators of a CT action so : 1R k x M M. (b) For any i, j E 11 ,..., kJ and s, t E 1R, we have X o Xis X o X;. (c) For any i, j E 11 ,..., kl we have [X' = 0.

Proof. (a) (b) has already been done. (b) (c): The condition X; = , o implies that

• (P) = —

d d

(p) 11 t=o —

dt

(X-1 5

° ° X fs (P)) t=0 = (Xis )* (X') (p)dt

as is easy to see. Differentiating the above expression with respect to s at s = 0,

we get [Xl,X1( p) = ( ). p I =

= 0, since X' (p) does not

ds s o depend on s.

(c) (b). The chain rule implies that

d (X )* (X J )(P)) = —

d ((Xi s r (XJ )(P))

ds r - s=c)

d —dri (X) o ((X;) ))(P) s=0 =

On the other hand ( X; )* ( X') = X' so

Appendix 181

(X i )* [XI,X1 = [(X;)* (X'), (X;)* (Xi )] = [X', ()(;)*(XJ)]

and therefore g- ( (xi,)* (XJ)( p)) = 0, or, (X,)* (X -')( p) is independent of dt

t, so (Xl,)*(X')(p) = (p).

By Proposition 1, the flow of (X)* (Xi) is given by

S "—• (X;) —1 0 X i 0 X; = )(i t 0 X is 0 X; .

Thus one sees immediately that ,r o = . x for any s, t E IR. (b) (a): Let (e 1 ,..., ek be the canonical basis of lie. It suffices to define

,( E t,,,p ) = X: , o X(p) . •

Definition. A local action of Il k on M is a map so : V —• M, where V

is an open set of lR k x M containing 101 x M, and so satisfies the prop-erties so (0,p) = p for every p E M, and so ( u + v, p) = ço(u,,,o(v,p)) = = so(v,(p(u,p)), whenever (u,p), (v,p) and (u + v,p) E V.

Corollary 1. Let X' X k be Cr(r a- 1) vector fields defined on M. If [X' ,Xi] = 0 for any i, j E 11 k I then there exists a basis I v 1 vk I of 11 k and a local action : V c IRk x M M such that (tv„ p) = X', ( p) for every (t, p) E IR x M such that both sides are defined.

The proof is similar to that of Proposition 3.

Corollary 2. Let X' X k be Cf (r 1) vector fields defined on M such that [X' = 0 for any i, j E {1 ,..., kJ. Suppose that for every q E M the subspace P(q) of TM, generated by X' (q) X k (q) has dimension k. Then the k-plane field P is tangent to a unique Cr foliation 5 of dimension k.

Proof. We will see how the local charts of J are defined. Let so : V M be the local action given by Corollary 1. Given p E M, fix an imbedded disk Dn -k = D C A/ such that D is compact, p E D and for every q E D we have

(*) Tci M P(q) 0 Tq D

Since D is compact and V is open, it is easy to see that there exists a ball B k

such that 0 E B k C Fe and B k x D C V. Set 1,G = so B k x D. Given ( u, v) E Il k X Tp D, u = E u f e„ it is easy to see that

a 1 v, ( 0,, ) • U E u,X'(p) and (3 2 (0,p) • v = v . i = 1

So from the condition (*) we have that Dlk (0,p) : fft k x TD —•• TM is


an isomorphism. By the inverse function theorem there exist small disks B' B' x D' 1,/, (8' x D') C M is a CT diffeomorphism onto the open set l' ( B' x D') of M. Set U = x D') and = (1,P . Then ( U, ) is a local Cr chart on M and the submanifolds of the form E (B' x L qi), q E D', are the plaques of a CT foliation 5u defined on U and tangent to P.

We claim that 5 u is the unique foliation on U that is tangent to P. Indeed, the leaves of 5u are mapped by to disks of the form B' x lqi, q E D'. On the other hand, the plane field P is mapped by DE to the horizontal plane field

P* (q) = DE(E -1 (q)) • P(E -1 (q)) = IR k x [0), 0E Tg D 1 .

As is easy to see, the foliation of B' x D' whose leaves are B' x qi . q E D', is the only one on B' x D' that is tangent to the plane field q 1-0 x o j. This proves the claim.

The set of all charts ( U, ) constructed as above defines a Cr foliation 5 on M. Indeed, if (C7, -j) is another local chart as above with U n then, by the preceding claim, 5u coincides with 5 on U n C./ and this implies that the change of coordinate map o E ( U fl CI) c BxD -E- (U

x 15 is of the form (x, y) (h 1 (x,y),h 2 (y)). •

§2. Frobenius' theorem

Before stating the theorem we are going to fix some notations. Given a k-plane field P on M we say that the vector field X is tangent to P if X( q ) E P(q) for every q in the domain of X.

A CT ( r 1) k-plane field P is called involutive if given X and Y, C' fields tangent to P, then [X, 1 ] is tangent to P. We say that Pis completely integrable if there exists a Cr foliation of dimension k on M such that T5' P, where T5 is the plane field tangent to 5.

Theorem 1 (Frobenius). Let P be a Cr (r ?: 1) k-plane field defined on M. Then P is completely integrable if and only if it is involutive. Further if either of these conditions hold, the foliation tangent to P is unique.

Proof. Suppose P is involutive. The idea is to use Corollary 2 of Proposition 3. We prove that for every p E M there exist Cr vector fields, X' ,..., X', defined on a neighborhood U of p, such that [X', Xi] = 0 for any i, j E 1 ,..., kl and P(q) is the subspace of Tg M generated by IX' ( q) (q)1 for every q E U. By Corollary 2, there will exist a unique Cr k-dimensional foliation on U, 3:, such that T5 c = P U. In this manner, it will be possible to define a cover U1 } 11 of M be open sets such that for each i E I there will be defined a foliation 5, on U. By uniqueness of such foliations it follows that

Appendix 183

if U, fl LI) 0 then if, t/i fl U.) = 5) 1 Lli n y , . In this way we will have defined a unique foliation if on M such that T5' = P.

Fix then p E M. From the definition of Cr k-plane fields, there exist k C r vector fields, Y` Yk defined on a neighborhood V of P such that P(q) is the subspace generated by fY' (q) Y k (q)J for every q E V. We can assume that V is the domain of a local ( Cm ) chart x: V JR". For each j E [ 1 k I we can write YJ = ,n= , au a/ ax, where la/ ax, ,..., a/ ax n) is the basis associated to the local chart x and au : V —• IR is of class Cr. Consider the matrix A = (au ) . The fact that Y' ( p ) Y k ( p) are linearly in-dependent implies that the matrix A ( p) has rank k, that is, there exists a k x k submatrix B of A such that det ( B ( p)) O. Since the determinant is a continuous function we have that det(B(q)) 0 for every q E U, a neighborhood of p. Permuting the variables (x, xn ) if necessary, we can assume that B = (a,1 ) 1<i, ) , k • In this way the matrix = (to ,j ) 1 , J . 0, is defined and has Cr coefficients on U.

Set xi = Ek,=, b,1 Y, j = 1 k. In the first place, it is clear that [,‘-' (q) X k (q)) is a basis of P (q) for

every q E U. On the other hand,

k n n k

= E bd E a , 1 a/axe = E

i=1 f=1 C=I i=1

For e = j we have that E aii bù = 1 and for 1' j, with 1 e 5_ k, we have that E ki =, a , b ,1 = 0; so

a/ax, Eca/ax,, for j = 1 k . f=k+1

We claim that [X', = 0 for any i, j E f 1 k J. In fact, from Lemma 1 one has that [a/ axr , 8 / âxj = 0 for any r,s E [ 1 ,..., ni so

[Xt, XJ] = f / aXe, / aXil ± [a / aXi, E C a/axm ]

f=k+I rn -= k -r- 1

E 3/8x, ci,„ a/ axm l = E d, a / ax, .

f.rn=k 0-1 r=k+1

Since P is involutive and IX' (q) X k (q)1 is a basis of P (q) for every q E U, we must have that

E d, a / aX, = E f,X s = E f a/ax, ±E g, a/ ax, . 5=1 s=1

From the above relation we can conclude that fl = = f, = 0, so [X I ,X J ] = = 0, as desired.


Suppose now that Pis completely integrable. Let 5 be a foliation such that Ta = P. Consider two cases:

1st case. is Cr (r 2): Fix p E M and a foliation chart for a, so: U Il k X IR" -k . Let X and Y be c' fields tangent to 5 and defined on U.

Since the leaves of 5 U are mapped to plaques IR k x y), y E a n-k , the fields X. = w * (X) and Y so. ( Y) are of the form X .(x, y) = ( f (x, y), 0) and Y. (x, y) = (g (x, y),0) where f,g : 1R k x —• le. Using (b) of Proposition 2, one sees that [X.,Y.](x, y) = (h(x, y),0) where h(x, y) = = f(x,y) ag/ax — g(x,y) a f/ ax. So [X, Y] = so* [X .,Y .1 is tangent to P.

2hd case. a is : Although 5 is C', the fact that P is c' (by hypothesis) implies that the leaves of a are C 2 . Indeed, let Q be a plaque of 5 contained in a coordinate chart of M, X: U IR". Then x(Q) is a submanifold of lit" x which we can assume is the graph of a function lit : D k C Ift k IFin -e. In this way Q is mapped by x onto the submanifold x ( Q) defined by the equation y = (x), x E D k and PI Q is mapped onto the k-plane field 13 along x( Q) defined by 13 (x4 (x)) = = f(u,A,G(x) • u) u E ak i. However -15 is C', so D1,G is C', as is easy to see, therefore V/ and Q are C2 . Since Q is C 2 , given p E Q, there exists a C2 diffeomorphism so : U' iRn = x Efe - k p E U, such that so (U' fl Q) IR k x O J. We can now apply the same argument as in the first case.

Remark. The argument in this theorem proves that if P is a CT involutive plane field on M then the leaves of the foliation associated to P are of class C" although 5 is in general Cr.

Examples.

(1) Every 1-plane field is completely integrable. (2) Define a 2-plane field P on Fe, setting P(x1 ,x2 ,x3 ) = the plane

generated by ( 1,0,0 ) and ((Le -X I , e x ' ) . Then P has no integral surfaces. Indeed, if X 1 (x,,x2 ,x3 ) = ( 1,0,0) and X 2 (x,,x2 ,x3 ) = = E P(x i ,X2 ,x3 ) then [X 1 ,X2 1(x,,x2 ,x3 ) = (0, – e -x1 ,e x1 )

P(xl ,x2 ,x3 ). Therefore P is not involutive at any point (x1 ,x2 ,x3 ) E

§3. Plane fields defined by differential forms

In this section we will see a version of Frobenius' theorem in terms of dif-ferential forms. We assume here that the reader has some basic knowledge of the theory of differential forms, such as the exterior product, exterior derivative, co-induced forms, etc. For more details about the subject we refer the reader to [52].

Appendix 185

Let w 1 w k be Cr (r O) 1-forms defined and linearly independent on an open set U c Mn. For each q E U define

P(q) = 1.) E Tq (M) (.4(v) =...= w(v) = 01 .

Since 0) 41 ,..., w are linearly independent elements of the dual ( Tq M)* , it is clear that P(q) is a subspace of codimension k of Tq M. We claim that the k-

plane field q 1-0 P ( q) is Cr . Indeed, given p E U, let x : V Et" be a local chart with pE Vc U, where x(q) = (x 1 (q)x, 1 (q)), : V IR for j = 1 n. Consider also for each j E 11 ,..., n1 the Cœ 1-form dxf . Then 1dx1 ( p) dx( p)) is a basis of ( Tp M)* , so it is possible to determine indices i • • • , in— k such that {(0 11, W pk , dx, i ( p) dx, n k (p)1 is a basis of ( Tp M)* , this because w are linearly independent. Set, for convenience, dx, = w k+1 dx, k = wn so that 1w pI w1 is a

n -

basis of ( TpM)* • By continuity there exists a neighborhood W of p with W C V, such that 1w p1 w nq 1 is a basis of ( 7M)* for every q E W. Con-sider now vector fields X' ,..., X" defined on W, such that for every q E W the set 1X 1 ( q) X n (q)) is the dual basis to jw w gn1 that is, w qi ( ( q )) = ô , (b,i = 0 if i j and 15,i = 1). It is easy to see now that the

vector fields X' ,..., X' are CT and further that P(q) is generated by the vectors X"' (q) X n (q), q E W. Conversely, given a Cr plane field P of codimen-sion k defined on U c M, it is possible to locally define P as the kernel of k Cr 1-forms. Indeed, given p E U, consider n — k C T vector fields X"' ,..., X' such that P(q) is the subspace of TM generated by X' q) (q) for every qE V c U. By a similar argument to that made in the case of forms, it is possible to define k C. `" vector fields X' ,..., X k on V such that X' ,..., X' are linearly independent on a neighborhood W with pE Wc V. It suffices now to consider the dual basis 1w n1 which consists of CT forms. We have then

P(q) = Iv E Tg Ail w ig (v) =...= u.) /(ci (v) = 01 .

We sum up what was done above in the following:

Proposition 4. A Cr(r 0) codimension k plane field can be defined locally as the kernel of k C r linearly independent 1-forms. Conversely

if 0) 1 w k are Cr linearly independent 1-forms then P(q) = ft E Tq M

(v) w k4 ( y) ----- 01 define a C r codimension k plane field.

Proposition 4 assures that there must be a version of Frobenius' theorem in terms of differential forms. Indeed, this version is given by the following result.

Theorem 2. Let P be a Cr (r 1 ) codimension k plane field defined on an open set U C M by k linearly independent Cr LArms Le' . Then P is completely integrable if and only iffor every j E k I we hare


(.)i ch.o A co I A...A = 0 .

Proof. Suppose that the equalities (*), are true. It suffices to prove that P is involutive. For this we will need two lemmas.

Lemma 2. Let n be a 2-form on U such that n A co l A...A 0.) 1( = O. Given p E U there exist a neighborhood V of p with V C U and k 1-forms a l ,..., ak such that n a l A co l + + ak A co k .

Proof. Since co 1 ,..., co l' are linearly independent, it is possible to define n – k C 1-forms co" I ,..., co" and a neighborhood V of p with V C U, such that for every q E V, (co qi wnq i is a basis of ( Tq M)* We can write n = E 00.4,' A coi. Since n A co l A...A co k = 0 one sees that

So n

E aii o.) A ckri A co l A...A (.0 k = 0, or, a 1 = O if k< i < j . i< j

k E i<k auco i A 6) = E i=1 ai A co where ai = . •

i<j

Lemma 3. Let w be a C. (r 1) 1-form and X and Y two C' vector fields defined on an open set U C Ile. Then

dco(X,Y) =d(co(X)) - Y d (6(Y)) - X+ co([X, Y]).

We denote the function q wq (X(q)) by co ( X).

Proof. Suppose co = E ,n= , a,dx, where a, : U R is of class Cr for i = 1 n and Idx, cix„; is the dual basis of the canonical basis f e, en ). Also set X = E ,n= bi e, and Y = E ,n= ce,. We have then that

(i) LO (X) = E a•bi , w(Y) = E and i= i=

n n

0.4[X, Y1) = E a, E (bf actc —ab,

) = ax_ ax_

ab, = E ai (b; — d ix•

I , j

aa aa (ii) do, = E i<j aif dxi A dx a,- = – so do.)(X, Y) = E

ax, axi <i – bic,).

Appendix 187

By (i) we have that

a (ai bi ) abi act, ) dco(x) • Y = E = E + bi — ax. ax- ax;

and

do.) ( Y) - X = E

0(a,ci ) act 0a; ax. = E bi ( al Tx- c ' )

i,j

So

(iii) cico (X) • Y – clo.)( Y) • X = abi

E ci -07c- — ax- 1 i.j

L – b-c ) .

ax- "

Summing the expression for w([X, 11 ]) from (i) with (iii), we obtain

d. (X) • Y – c1(.4(Y) X + co([X, Y1) = (b,c- – b c ) ax j

J

Thus,

aa, aa, aa• E — (b-c• – b-c.)

E — )(b.c. b•c.) = clo)(X,Y) .11 axi ax. axi i‹;

Suppose then that X and Y are two C' vector fields tangent to P. Since dwi A 6) 1 A...A 0J k = 0, from Lemma 2 we obtain

cloy' = E a, A (.4)

:=1 and therefore

ch.,' (X, Y) = E (ct ),(X)w s (Y) – a li (Y)Lo i (X)) =

since for every i = 1 k we have w` (X) = Lo'( Y) = O. On the other hand from Lemma 3, we obtain

0 = dwf(X, Y) = d(w-i(X)) Y — d(o.)-1 ( Y)) X + (.4.1 ([X, Y1) .

Since w'( X) = ( Y) = 0, one sees that w ([X, Y ] ) = 0 for every j = = 1 k, i.e., [X, Y] is tangent to P.

We leave the converse to the reader. •

Exercises

Chapter I

1. Let f M N be a submersion, where N is connected. (a) Prove that if A is an open subset of M then f (A) is open in N.

(b) Prove that if M is compact then N is also.

2. Let a: la,b] M be a continuous simple (i.e., injective) curve. Suppose that d is a metric on M. Prove that, given E > 0, there exists a cover of aaa,b1), by domains of local charts U1 ,..., U. such that, if U, fl U1 0,

then jE 1, 1, 1 + 1j and the diameter of U, in the metric d is less than e for all E 1 1 ,..., kj.

3. Let M be a connected manifold. Prove that, given p,q E M, there exists a simple C curve a : [0,11 M such that a (0) = p, a ( 1) = q and (da)(t)/dt 0 for every t E [0,1j.

4. Let M be a connected 1-dimensional manifold. Prove that M is diffeomor-phic to IR or to S'. Hint. Use Exercise 3.

5. Let < , > be a Riemannian metric on M. If f : M 111 is a Cr(r 1) function, one defines the gradient of f in the metric < , ) as the unique vector field v f such that dfp (u) = <v f(p),u >, for any p E M and u E Tp M.

(a) Prove that vf is well-defined and is of class C' (b)Prove that p E M is a singularity of vf if and only if it is a critical point

of f. (c) Prove that vf is normal to the level surfaces of f and therefore has

no periodic orbits.


Chapter II

6. Let 5 be a foliation on M and F a leaf of a. Consider on F the intrinsic manifold structure. Prove that F has a countable basis. Hint. Show that F meets a foliation box of 5 in a countable number of plaques.

7. Let 5 be the Reeb foliation of D 2 x S' and Nan n-dimensional manifold. Prove that if f: D 2 xxS 1 is continuous and constant on each leaf of 5 then f is constant.

8. Let S 3 = 1(z i ,z2 ) E C 2 1Z11 2 A- 1z21 2 z= 11. Prove that the map 7r : S 3

S2 defined by

r(z 1 ,z2 ) = (izii 2 2 Re (z, 2 ), 2 Im(z i -z-2 )) ,

is a submersion and that 7r -1 (p) = S' for all p E S 2 . Show that there does not exist a submersion f: S 3 111 2 .

9. Let f : M N be a Cr ( r 1) submersion and a the foliation on M given by level surfaces of f. Prove that if N is orientable then 5 is transversely orientable.

10. Let 5 be a transversely orientable foliation on N and f : M N a map transverse to 5. Prove that f (5) is transversely orientable.

11. Let a and g be two C foliations on M. Suppose that for every p E M one has Tp M = Tp T + Tp s. Prove that there exists a foliation 3C on M, of class CT, such that if F and G are leaves of 5 and g respectively, then the connected components of F (1G are leaves of R. Show that dim ( 3C ) = = dim ( 5 ) + dim (g) - dim M.

12. Let M be a compact manifold of dimension n. Prove that if there exists a foliation of codimension one on M then the Euler characteristic of M is zero. Hint. If there exists a continuous nonsingular vector field on M then the Euler characteristic of M is zero [26].

13. Let a E 1R — Q. (a) Prove that the set lm + na m,n E _AZ is dense in 1R.

(b) Define : s' s' by fc,(z) = e2' • z. Prove that for every z E s' the orbit of z by f,„ O (z) fn,„( z) I n E Z1 is dense in S'.

(c) Let X" be the vector field obtained by the suspension of sfQ. to a com-

pact manifold M 2 (see note 1 of Chapter II). Prove that M 2 is dif-

feomorphic to T2 and that all the orbits of X" are dense.

14. Let P be a CT k-plane field and < , > a ( C' ) Riemannian metric on M.

Define P' by P - (q) = iv E Tq M (u,v), = 0 for every u E P(q)1.

Exercises 191

(a) Prove P' is of class Cr.

(b) If A g : TM P I (q) is the orthogonal projection on (q) and X is a cs (s s r) vector field, prove that the vector field Y, defined by Y( q) = A q ( X( q )) is of class C5 .

15. Let — be the equivalence relation on or"' — 101 defined by z w if and only if there exists X E C — 10 1 such that z = Xw. Let CP(n) be the quotient space of C" — 101 by —. Denote by [z] the equivalence class of z E C n — (01.

(a) Let U, = lizi ,-••, 4+1] ECP(n) 0), 1 s i s n + 1, and (p i :

U, —Do C" defined by so; [z, ,• • ., zn+i ] = z1 1 (z1 zi ,-••, ,•••, Zn + I) • Prove that 42 = ( U„ so, ) I i = 1 ,..., n + 11 is a (holomorphic) atlas for CP(n).

(b) Let a be the foliation of S 2" -1 defined by the Hopf fibration. If M is the quotient space of S 2n -1 by the equivalence relation which identifies the points which are in the same leaf of J, prove that M is homeomor-phic to CP( n — 1).

16. Let f : JR be of class C T (r a- 2 ) and G C IR 2 the graph of f. For each p E G, let fp be the normal line to G at p with p E Pp . (a) Prove that there exists an open neighborhood U of G such that, if Fp

is the connected segment of fp n U that contains p, then Fp fl Fp .

0 implies that p = p'. (b) Prove that the segments Fp , p E G, are the leaves of a C r - 1 foliation

on U. (c) Prove that if f is not Cr +1 then 5 is not Cr.

(d) Give an example of a foliation on IR 2 , of class Cr, but not C" 1 , whose leaves are

Chapter III

17. Prove that all the leaves of the Reeb foliation of S 3 are imbedded.

18. Let f: 1R2 IR be defined by f (x, y) = (1 — x2 )eY . Prove that f is a submersion. Let 5 be the foliation of IR 2 given by level surfaces of f and 7r : IR 2 •—• 111 2 / 5 the projection onto the leaf space of a. Prove that r is not closed.

19. Let 5 be a C r (r a 1) 1-dimensional foliation on 1R 2 and 7r : 1R 2 •—• 31 2 /a be the projection onto the leaf space of 5.

(a) Prove that all the leaves of 5 are closed, but that S has no compact leaves.

(b) Prove that 111 2 /a has a cr 1-dimensional manifold structure (in general non-Hausdorff). Show that 7r is a cr submersion.

(c) Prove that 1Ft 2 / 5 is simply connected.


Hint. Prove that if a : [0,1] --• R 2 /5 is a closed curve then there exists a closed curve a : [0,1] --• R 2 such that a = 7 0 a. (d)Give an example of a foliation if on Ile such that 1R 2 /5 has three

points which are pairwise nonseparable. We say that p and q are nonseparable if, given arbitrary neighborhoods Uand V ofp and q then U n v 0.

20. Given an example of a foliation if of codimension 1 on T3 which has every leaf dense ( T 3 = S 1 x S' x S' ) . Using the method of turbilization (note 4 of Chapter II) modify if, obtaining a foliation if' on T3 with the follow-ing properties: (a) There exists a solid torus D2 x S 1 imbedded in T3 such that D 2 x S 1

is saturated by if' and a '/D 2 x s' is the Reeb foliation. (b)All the leaves of a' are dense in T3 – (D2 x S ' ).

21. Let a be a 1-dimensional C' foliation on T2 . Prove that a has at most one exceptional minimal set. Hint. Suppose that ii is an exceptional minimal set of a. Show that there exists a circle C imbedded in T2 , such that C is transverse to a, c n

0 and C n II' = 0 for every minimal set i ' such that u ' fl ii = 0 (see Prop. 4 of Chapter VII). Using the fact that T 2 – C = R x S I and the Poincaré-Bendixson theorem [34], prove that a has no exceptional minimal sets in T2 – C.

Chapter IV

22. Show with an example that the global stability theorem is not true in R 3 – [0].

23. Let if be a transversely orientable, C (r a- 1) foliation, F a leaf of a and f: E' c E —o- E an element of holonomy of F. Prove that Df ( p) : Tp E --• Tp E preserves the orientation of Tp E, that is, takes a basis of Tp E to another basis with the same orientation.

24. Let if be a C 1 codimension-one foliation on M. If F is a nonclosed leaf of 5, prove that, given p E F, there exists a closed curve transverse to if passing through p. Hint. Use Lemma 3.

25. Let f : M --• N be a Cr. ( r a 1) submersion and if the foliation on M

given by level surfaces of f. Suppose M and N are connected. Let F

C f -1 (q) be a leaf of 5. (a) Prove that, given p E F, there exists a neighborhood V of q and a section

E transverse to a through p, such that f 1E:E ---• V is a Cr

diffeomorphism. (b) Show that the holonomy of F is trivial. (c) Suppose that F is a compact leaf of 5. Let r : V --• F be a Cr tubular

Exercises 193

neighborhood of F. Show that there exists a neighborhood U C V of F, saturated by 5 and such that the intersection of each leaf of g I U with a fiber (q) U contains only one point.

(d) With the notation of (c), define h : U--•Fxf(U) in the following way: h(p) = (7r (p),f(p)). Prove that h is a Cr diffeomorphism which takes the leaves of if contained in U to the sets F x (y), y E f (U).

(e) If all the leaves of if are compact, prove that they are diffeomorphic to F.

26. (a) Let f : JR 1R be a C' diffeomorphism. Suppose that there exists 0 < X < 1 such that f(Xx) = Xf(x) for every x E 1R. Prove that ! is linear.

(b) Let if be a C' (r 1) codimension-one foliation on T2 x JR such that F = T2 x [0) is a leaf of 5. Fix a section E = Ep x ( — e ,E )

transverse to if which we identify to ( — E,e ). Suppose that one of the elements of the holonomy of F, g: E E, is given by g(x) = Xx where 0 < X < 1. Prove that all the elements of holonomy of a are linear.

(c) If if is as in (b), describe if I V where V is a neighborhood of F.

27. Let (.0 1 cok be linearly independent, C' (r1) closed 1-forms defined on M. Let P be the codimension k plane field defined by P(q) =

Ev E Tq M I o. 1 (q) - u = 0, i = 1 ,..., 11. (a) Prove that there exists a unique Cr foliation on M such that 7; if =

= P(q) for every q E M. Show that if is transversely orientable. (b) Prove that the holonomy of any leaf of if is trivial.

28. Let f: M ---1.1R be a Cr (r 1) submersion and if the foliation on M given by level surfaces of f. Suppose that M is connected and that all the leaves of if are compact. (a) Show that there does not exist a closed curve transverse to g. (b) Let F be a leaf of if. Prove that there exists a Cr diffeomorphism

h : F x 1R M such that h* () is the foliation whose leaves are of the form F x It), t E IR.

Hint. For the construction of h use the trajectories of a vector field transverse to if.

29. Let M" (n 3 ) be a compact connected manifold with boundary (see [39]) am = U ,k,_, F, where k 1 and F, is simply connected. Suppose that if is a c' (r 1) transversely orientable codimension 1 foliation on M such that F, Fk are leaves of 5. (a) Prove that k = 2. (b) Prove that there exists a Cr diffeomorphism h : Fi x [0,1] M


such that h* (5) is the foliation whose leaves are F1 x it!, t E [0,11.

(c) Give an example of a codimension 1 foliation 5 defined on a compact manifold M 3 with boundary am _---. s 2 and such that am is a leaf of 5.

30. Let 5 be a Cr transversely orientable codimension-one foliation defined on a connected manifold of dimension n ?._ 2. Suppose that all the leaves of 5 are compact. (a) Show that any two leaves of 5 are diffeomorphic. (b) Suppose that M is compact. Prove that there exists a submersion

f: M --• S i such that the leaves of 5 are the level surfaces f - ' (0) , 0 E S'.

(c) Prove that there does not exist a foliation of codimension 1 on 1lt 3 such that all its leaves are compact.

Chapter V

31. Let 5 be a Cr (r _?... 1) foliation transverse to the fibers of the fibration (E,r,B,F). Prove that any element of holonomy of 5 is Cr.

32. Let ( E,71- ,B, F) and (E' ,r' ,B,F) be two fibered spaces where B and F are connected. Let 5 and 5' be C r (r ?._ 1 ) foliations transverse to the fibers of E and E' respectively. Let H: E --+ E' be a continuous one-to-one map which takes fibers of E to fibers of E' and leaves of 5 to leaves of 5 ' . Prove that His a homeomorphism and that there exists a homeomor-phism h : B --• B such that r' oH=ho r. Suppose now that h and

the restriction of H to some fiber Fo of E are Cr diffeomorphisms. Prove that H is a C r diffeomorphism.

33. Let f : M --• N be a submersion where M is compact and connected. Let p E N be fixed. (a) Prove that (M, f,N,f - ' (p)) is a fibered space. (b) Show by an example that (a) is not true if M is not compact, N being

compact or not.

34. Let ( E, ii- ,B, F) be a fibered space where E is connected and F compact. Suppose 5 is a foliation on E such that for every p E E one has Tp E = = Tp g + Tp ( Fp ) where Fp = 7r - 1 ( 7r ( p)). Prove that if L is a leaf of a then L cuts all the fibers of E.

35. Let f: S 3 --• S 2 be a submersion. Prove that there does not exist a folia-tion 5 of codimension 1 on S 3 whose leaves are transverse to the fibers f -1 (p),p E S 2 . Conclude that there does not exist a fibered space struc-ture with discrete structure group, whose total space is S 3 and base is S 2 .

36. Let E be a fibered space with base s' and compact fiber. Prove that E is equivalent to a libration with discrete structure group.

Hint. Prove that there exists a vector field on E transverse to the fibers.

Exercises 195

37. Let ( E, ir , B, F) and ( E ' ',B ' ,F') be two fibered spaces and if and if' Cr foliations transverse to the fibers of E and E ' , with holonomies w and p' respectively. We say that 9 and p' are Cr-conjugate at bo and b (; if there exist Cr diffeomorphisms h: B —0 B' and f: - ( bo )

—0 (71" 1 ( bd) such that h (AO = b (; and for every [a] E 7r 1 (B, 40 one has that f o o ([«]) = '([h o a]) o f . Prove that, if so and p ' are C'-conjugate, then there exists a diffeomorphism H : E E' taking leaves of a to leaves of if' with 72- ' o H = h o 71-

38. Let (E,71- ,B ,F ) be a fibered space, if a Cr foliation transverse to the fibers of E and w the holonomy of if. Let if (p) be the suspension of to E (so) . Prove that the holonomy of 5 ( 9) is cp.

39. Let if be a C 2 transversely orientable foliation. Let F be a compact leaf of a. Prove that F has a tubular neighborhood diffeomorphic to a product. Hint. Prove that the normal fibration of F is diffeomorphic to a product.

Chapter VI

40. Show that there does not exist a foliation of codimension one on S 3 whose leaves are all homeomorphic to IR 2 .

41. Let if be an analytic, codimension-one, transversely orientable foliation on Mn . Let F be a leaf of if. Suppose that -y : S M is homotopic to a constant and that -y (S I ) C F. Show that the holonomy of -y is trivial. Hint. If the holonomy of y is not trivial, construct a curve : S i M

transverse to a and homotopic to y.

42. Let if be an analytic foliation of codimension one defined on a simply con-nected manifold. (a) Let -y : JR —• M be a curve transverse to if. If F is a leaf of if such

that 7 ( IR)(1F0 prove that y (JR) fl F contains exactly one point. (b) Prove that all the leaves of if are closed. (c) If -y ( 1R) cuts all the leaves of if prove there exists a submersion

f : M —0 R such that if is the foliation whose leaves are the level sur-faces of f.

43. Show that an analytic foliation of codimension one on Ill n (n 2) has no compact leaf.

44. Let S and Ç be two analytic foliations defined on a connected manifold M. Suppose there exists an open set U c M such that alt.J=giu. Prove that if = g. Hint. Use the fact that two analytic functions which coincide on an open set are equal on the intersection of their domains.

45. Let if be the foliation by parallel planes in IR 3 and f: R 3 – 101 S2


X JR defined by f (x) = (4 -1 - x, log114). Set 5* = ( f -I ) * ( 5 I R 3 – 10j).

(a) Prove that 5* is invariant by translations of S 2 x JR of the type (g, t) 1--+ (y,! + to ), to fixed.

(b) Prove that S * induces an analytic foliation on S 2 x S ' which has one leaf F diffeomorphic to T2 and two Reeb components, R I and R2, such that aR, = aR2 = F.

(c) Generalize the above procedure, obtaining an analytic foliation of codimension one on S' x S' which has one leaf diffeomorphic to Sr1-2 x s' and the others diffeomorphic to JR''.

Chapter VII

46. Let Mbe a compact manifold of dimension 3, without boundary, with finite fundamental group. If X is a vector field on M, without singularities and transverse to a C2 codimension-one foliation, then X has nonperiodic orbits and at least one periodic orbit.

47. Let F be a compact leaf of a C 2 codimension-one foliation if on M" (n a 3 ) . Suppose there exists a closed curve -y : S' —■ F such that -y is homotopic to a constant in M but not in F. Prove that if has a vanishing cycle.

48. Give an example of a foliation on S 3 which has a Reeb component R such that S 3 – R is not homeomorphic to a solid torus. Hint. Consider the Reeb foliation of S 3 and a closed transverse curve that is knotted. Turbulize.

49. Let if be a codimension-one foliation on 1R 3 . Prove that (a) if a has a compact leaf then a has a Reeb component, (b) if F is a compact leaf of a then F is diffeomorphic to T 2 .

Hint (for b). Prove that the Euler characteristic of F is zero.

50. Let S be a codimension-one foliation on D 2 x S ' such that T2 = = a (D 2 x S ' ) is the only compact leaf of if. Show that S is topologic-ally equivalent to the Reeb foliation of D 2 x s'. Hint. Prove that the curve S' x 101 C T2 is a vanishing cycle for a and that the leaves of if, contained in the interior of D 2 x S 1 , are diffeomor-phic to 111 2 .

51. Let 7r : R 2 —* M be a covering. Prove that if M R 2 then there exists a covering transformation f: 1R 2 --• R2 which has infinite order (i.e., f k = id k = 0).

Hint. Let a be a simple closed curve in M not homotopic to a constant. Let f be the transformation relative to [a]. If f k = id then [a] k = 1 so, if a is a lift of a 1 = a .,.... a (k times) then a is a simple closed curve in

Exercises 197

IR 2 . Let A be the region of R 2 bounded by a. Then f(A) = A so by

Brouwer's fixed point theorem ([35 ] ), f has a fixed point in A. Conclude

that f = id.

Chapter VIII

52. On IR' consider the additive group structure. Let G C I n be a closed

subgroup. (a) Suppose G is not discrete. Show that G contains a subgroup of dimen-

sion 1. (b) Given vEG–t01, let L = IR • y, the subspace generated by v. 11 H

is a subspace of dimension n – 1 complementary to L, prove that G

is isomorphic to (H n G) e L, if L C G, or (H CI G) e z V, if L 0 G.

(c) Using (a) and (b), prove by induction on n that G is isomorphic to

Rk x Z e where 0 k + < n.

53. Let io be a C 2 action of IR 2 on 114 3 . (a) Prove that all the orbits of ‘,0 are closed subsets of IR 3 homeomorphic

to ne. (b) Suppose there exists a curve -y : IR 3 which cuts all the orbits of

,p transversely. Prove that io is conjugate to the product action ( s,, t, (x, y, z ) ) = (x + s,y + t, z). We say that ço and q, are conjugate

if there exists h : 1R 3 —* R 3 such that h o so (u, p) = 1,1i (u, h ( p)) for any (u,p) E 1R 2 x IR 3 .

(c) Give an example of an action of IR 2 on Ile which is not conjugate to the product action st'.

Hint. Prove that a curve transverse to the orbits of ço cuts each orbit in at most one point. In (b) define h : IR 2 x JR —0 IR 3 by h(u,t) =

= yo(u, -y (t)) . Prove that h is a diffeomorphism and conjugates ç with 1,G .

54. Actions of R 2 linearly induced on S". Let A and B be commuting ( n + 1) X (n + 1) matrices. Define ç : IR 2 x 1Rn÷ —o IR n+ I by

so(s,t;p) = exp (sA + (B) - p .

(a) Prove that ç is an action of 1R 2 on (h) Define ,,-23 : 111 2 x S" —0 S" by

,p(s,t;p) (s,t;p) p E S'' .

ilso(s,t;P)

Show that (7 defines an action of 111 2 on S".

(c) Prove that a necessary and sufficient condition for all the orbits of to have dimension 1 is that A,B and I be linearly dependent, where


I is the (n + 1) X (n + 1) identity matrix. (d) Suppose that A,B and I are linearly independent and that the eigen-

values of A and B are pairwise distinct. For n = 2 and 3 describe the orbits of ;-,0 in terms of the Jordan canonical forms of A and B.

55. Let X and Y be Cd vector fields on S 2 such that [X, Y] = O. Prove that there exists p E S 2 such that X ( p) = Y ( p) = O. Hint. If Y, is the flow of Y and X( p) = 0, prove that X( Y, ( p)) = 0, for every t E E. Next show that the a and 0.)-1imit sets of p for Y consist of singularities of X. Now use the Poincaré-Bendixson Theorem.

56. Let f: S 2 —• S 2 be a diffeomorphism and X a vector field on S 2 . Suppose that f* (X) = X . (a) Prove that f . X, = X, . f for every t E IR, where X, is the flow of X. (b) Show that there exists p E S 2 that is simultaneously a periodic point

of f and a singularity of X. Hint. Every diffeomorphism of S 2 has a periodic point. If f is a dif-feomorphism of 1R 2 without periodic points, then the w-limit set of any point is empty (Brouwer).

57. Prove that the rank of the following manifolds is one: 1133 , S 2 X S i , W 2 X S' and S 2 X IR.

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INDEX

Actions of a Lie group, 28,29,159,161 of the affine group, 171 Atlas of a manifold, 7

Bohl, 55

Centers, 84 Change of coordinates, 7 Classification of manifolds, 9,11 Closed leaves, 51 Codimension, 16 Coherent extension of vanishing cycles,

133 Commuting flows, 179 Completely integrable 1-forms, 82 plane fields, 36,182 Conjugated actions, 98 Covering spaces, 12 Critical value, 17

Deck transformation, 95 Denjoy's example, 55 theorem, 55 Derivative, 5, 14 Diffeomorphism, 6,8

Differentiable function, 6,7 manifold, 5,8 map, 13 _structure, 8 Dimension of a foliation, 21,22 of a manifold, 8 of a submanifold, 16

Distinguished maps, 32

Ehresmann's theorem, 91 Exceptional leaves, 50 minimal sets, 53,55

Fiber, 25, 88 space, 25 bundles, 87

Fibration, 25 Fields of k-planes, 35 Flow, 30, 176 box theorem, 31 Foliated action, 29 Foliation, 22 charts, 23 Foliations defined by a submersion, 23 defined by closed 1-forms, 80 of 3-manifolds, 45 of 1112 , 53 _on open manifolds, 112 on iRn , 113 on spheres, 45 with singularities on D 2, 123

Frobenius' theorem, 36,175,182,185 Fundamental group, 65 Geodesic flows, 46 Germ of a map, 64 Global stability theorem, 72,78 trivialization lemma, 69 Gromov-Phillips theorem, 112 Group of affine transformations, 171

Haefliger, 54 Haefliger's construction, 121 theorem, 115

Hector's example, 109 Hessian, 118 Holomorphic vector fields, 43 Holonomy group, 65

map, 63 of a leaf, 62 of a foliation on a fiber bundle, 93

Hopf fibration, 43

Imbedding, 14 Immersion, 14 Integral curve, 28, 175 Intrinsic structure of a leaf, 31 Invariant set, 48


Involutive plane fields, 36, 182 Isotropy group, 29, 159

Kamke's theorem, 54 Kaplan's theorem, 54 Kueser, 55

Leaf, 23 Lie bracket, 178 group, 28 subgroup, 28 Lima's theorem, 165 Line field, 35 Local action, 181 chart, 7 diffeomorphism, 6 equivalence, 67

flow, 28 form of immersions, 15 form of submersions, 16 stability theorem, 70 Locally dense leaves, 50 free action, 29, 161

Maximal atlas, 7 interval of definition, 177 Minimal sets of actions, 169 of diffeomorphisms, 55 of foliations, 52,55 Morse function, 119

lemma, 119 theorem, 120

Non Hausdorff manifold, 11 Non orientable Reeb foliation, 26 Non orientable surfaces, 10 Normal bundle, 89 extension of vanishing cycle, 136 Novikov's theorem, 131, 153

Orbit of a point, 29, 159 Orientable double coverings, 37 _foliations, 40 k-plane fields, 36

Painlevé, 107 Palmeira's theorem, 113 Partitions of unity, 18 Plane fields, 35 Plaques, 23 Poincaré, 55

Poincaré-Bendixson theorem for De actions, 168

Poincaré map, 42 Positive vanishing cycles, 136 Projective spaces, 8, 43

Rank of manifold, 165 theorem, 163

Reeb's center theorem, 84,85 component, 149

foliation, 25,66 stability theorems, 70, 72 Regular point, 17 value, 17 Ricatti's equation, 106 Riemannian metric, 19

Sacksteder's example, 103 theorems, 81,109 Saddle connection, 117 self-connection, 117 Sard's theorem, 17 Saturated set, 48 Saturation of a set, 47 Section of a fiber bundle, 89 Simple vanishing cycle, 138 Singularity of a function, 118 of an integrable form, 82 Space of leaves, 47 Submanifolds, 16 Submersions, 14 Support of a function, 19 Suspension of diffeomorphisms, 41 of a representation, 93,95

Tangent bundle, 89 space, 14 Thurston's stability theorem, 80 Topological equivalent foliations, 55 Total space, 88 Transversality of manifolds, 18 of maps and foliations, 33 of maps and manifolds, 17 Transverse section, 48 uniformity, 49 Transversely orientable foliations, 40

_k-plane fields, 39 Tubular neighborhood, 25 Turbulization of a foliation, 44

Uniform topology, 119.

Index 205

Uniqueness of the suspension, 98 Unit tangent bundle, 90

Vanishing cycle, 131,133 Vector bundle, 88

_fields, 28

Wasewki's example, 53,54 theorem, 54

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Geometric Theory of Foliations

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