Job #: 111368
Author Name: Conlon
Title of Book: Differentiable Manifolds
ISBN #: 9780817647667
Modern Birkh~user Classics
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Differentiable Manifolds Second Edition
Lawrence Conlon
R e p rint of the 2001 S e c o n d Edition
Birkh~iuser Boston 9 Basel ~ Berlin
Lawrence Conlon Department of Mathematics W a s h i n g t o n University St. Louis, M O 6 3 1 3 0 - 4 8 9 9 U.S.A.
Originally p u b l i s h e d in the series B i r k h d u s e r A d v a n c e d Texts
ISBN-13:978-0-8176-4766-7 DOI: 10.1007/978-0-8176-4767-4
e-ISBN-13:978-0-8176-4767-4
Library of Congress Control Number: 2007940493 Mathematics Subject Classification (2000): 57R19, 57R22, 57R25, 57R30, 57R45, 57R35, 57R55, 53A05, 53B05, 53B20, 53C05, 53C10, 53C15, 53C22, 53C29, 22E15 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh~iuser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design by Alex Gerasev. Printed on acid-free paper. 987654321 www. birkhauser, com
Lawrence Conlon
Differentiable Manifolds Second Edition
Birkh~iuser
Boston
9 Basel ~ Berlin
Lawrence Conlon Department of Mathematics Washington University St. Louis, MO 63130-4899 U.S.A.
Library of Congress Cataloging-in-Publication Data Conlon, Lawrence, 1933Differentiable manifolds / Lawrence Conlon.-2nd ed. p. cm.- (Birkh~iuseradvanced texts) Includes bibliographical references and index. ISBN 0-8176-4134-3 (alk. paper)-ISBN 3-7643-4134-3 (alk. paper) 1. Differentiable manifolds. I. Title. II. Series. QA614.3.C66 2001 516.'6-dc21 2001025140
AMS Subject Classifications: 57R19, 57R22, 57R25, 57R30, 57R35, 57R45, 57R50, 57R55, 53A05, 53B05, 53B20, 53C05, 53C10, 53C15, 53C22, 53C29, 22E15 Printed on acid-free paper. @2001 Birkh~iuserBoston, 2nd Edition 9 Birkh~iuser Boston, 1st Edition
Birkh~user ~ |
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh~iuser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4134-3 ISBN 3-7643-4134-3
SPIN 10722989
Typeset by the author in lATEX. Printed and bound by Hamilton Printing Company, Rensselaer, NY. Printed in the United States of America. 987654321
This book is dedicated to my wife Jackie, with much love
Contents Preface to the Second Edition Acknowledgments
xi xiii
Chapter 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.
1. Topological Manifolds Locally Euclidean Spaces Topological Manifolds Quotient Constructions and 2-Manifolds Partitions of Unity Imbeddings and Immersions Manifolds with Boundary Covering Spaces and the Fundamental Group
17 20 22 26
Chapter 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.
2. The Local Theory of Smooth Functions Differentiability Classes Tangent Vectors Smooth Maps and their Differentials Diffeomorphisms and Maps of Constant Rank Smooth Submanifolds of Euclidean Space Constructions of Smooth Functions Smooth Vector Fields Local Flows Critical Points and Critical Values
41 41 42 50 54 58 62 65 71 80
1 1
3 6
Chapter 3. The Global Theory of Smooth Functions 3.1. Smooth Manifolds and Mappings 3.2. Diffeomorphic Structures 3.3. The Tangent Bundle 3.4. Cocycles and Geometric Structures 3.5. Global Constructions of Smooth Functions 3.6. Smooth Manifolds with Boundary 3.7. Smooth Submanifolds 3.8. Smooth Homotopy and Smooth Approximations 3.9. Degree Theory Modulo 2* 3.10.Morse Functions*
87 87 93 94 98 104 107 110 116 119 124
Chapter 4.1. 4.2. 4.3. 4.4. 4.5.
131 131 136 142 145 150
4. Flows and Foliations Complete Vector Fields The Gradient Flow and Morse Functions* The Lie Bracket Commuting Flows Foliations
viii
CONTENTS
Chapter 5.1. 5.2. 5.3. 5.4.
5. Lie Groups and Lie Algebras Basic Definitions and Facts Lie Subgroups and Subalgebras Closed Subgroups* Homogeneous Spaces*
161 161 170 173 178
Chapter 6.1. 6.2. 6.3. 6.4. 6.5.
6. Covectors and 1-Forms Dual Bundles The space of 1-forms Line Integrals The First Cohomology Space Degree Theory on S 1.
183 183 185 190 195 202
Chapter 7.1. 7.2. 7.3. 7.4. 7.5.
7. Multilinear Algebra and Tensors Tensor Algebra Exterior Algebra Symmetric Algebra Multilinear Bundle Theory The Module of Sections
209 209 217 226 227 230
Chapter 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.
8. Integration of Forms and de Rham Cohomology The Exterior Derivative Stokes' Theorem and Singular Homology The Poincar6 Lemma Exact Sequences Mayer-Vietoris Sequences Computations of Cohomology Degree Theory* Poincar5 Duality* The de Rham Theorem*
239 239 245 258 264 267 271 274 276 281
Chapter 9.1. 9.2. 9.3.
9. Forms and Foliations The Frobenius Theorem Revisited The Normal Bundle and Transversality Closed, Nonsingular 1-forms*
289 289 293 296
Chapter 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7.
10. Riemannian Geometry Connections Riemannian Manifolds Gauss Curvature Complete Riemannian Manifolds Geodesic Convexity The Cartan Structure Equations Riemannian Homogeneous Spaces*
303 304 311 315 322 334 337 342
Chapter 11.1. 11.2. 11.3. 11.4.
11. Principal Bundles* The Frame Bundle Principal G-Bundles Cocycles and Reductions Frame Bundles and the Equations of Structure
347 347 351 354 357
CONTENTS
ix
Appendix A.
Construction of the Universal Covering
369
Appendix B.
The Inverse Function Theorem
373
Appendix C. Ordinary Differential Equations C.1. Existence and uniqueness of solutions C.2. A digression concerning Banach spaces C.3. Smooth dependence on initial conditions C.4. The Linear Case
379 379 382 383 385
Appendix D. The de Rham Cohomology Theorem D.1. Cech cohomology D.2. The de Rham-Cech complex D.3. Singular Cohomology
387 387 391 397
Bibliography
403
Index
405
P r e f a c e to t h e S e c o n d E d i t i o n In revising this book for a second edition, I have added a significant amount of new material, dropping the subtitle "A first course". It is hoped that this will make the book more useful as a reference while still allowing it to be used as the basis of a first course on differentiable manifolds. In such a course, one should omit some or all of the material marked with an asterisk. More information about these optional topics will be given below. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Mastery of the central topics of this book should prepare students for advanced courses and seminars in differential topology and geometry. There are certain basic themes of which the student should be aware. The first concerns the role of differentiation as a process of linear approximation of nonlinear problems. The well-understood methods of linear algebra are then applied to the resulting linear problem and, where possible, tile results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i.e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, resulting from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above. It is expected that the student has been exposed to the above processes in the setting of Euclidean spaces, at least in low dimensions. This is what, we will refer to as local calculus, characterized by explicit computations in a fixed coordinate system. The concept, of a "differentiable manifold" provides the setting for global calculus, characterized (where possible) by coordinate-free procedures. Where (as is often the case) coordinate-free procedures are not feasible, we will be forced to use local coordinates t h a t vary from region to region of the manifold. When theorems are proven in this way, it becomes necessary to show independence of the choice of coordinates. The way in which these local reference frames fit together globally can be extremely complicated, giving rise to problems of a topological nature. In the global theory, geometric topology and, sometimes, algebraic topology become essential features. These themes of linearization, (re)integration , and global versus local will be emphasized repeatedly. Although a certain familiarity with the local theory is presupposed, we will try to reformulate t h a t theory in a more organized and conceptual way that will make it easier to treat the global theory. Thus, this book will incorporate a modern treatment of the elements of multivariable calculus. Fundamental to the global theory of differentiable manifolds is the concept of a vector bundle. As the global theory is developed, the tangent bundle, the cotangent
xii
PREFACE TO THE SECOND EDITION
bundle and various tensor bundles will play increasingly important roles, as will the related notions of infinitesimal G-structures and integrable G-structures. For conceptual simplicity, all manifolds, functions, bundles, vector fields, Lie groups, homogeneous spaces, etc., will be smooth of class C ~ It is possible to adapt the treatment to smoothness of class C k, 1 < k < oc, but the technical problems that arise are distracting and the usefulness of this level of generality is limited. On the other hand, in much of the literature, the study of Lie groups and homogeneous spaces is carried out in the real analytic (C ~) category. In these treatments, it is customary to note that C a groups can be proven to be analytic, hence that no generality is lost. It seems to the author, however, that nothing would be gained by this approach and that the ideal of keeping this book as self-contained as possible would be compromised. The optional topics (sections, subsections and one chapter, with titles terminating in an asterisk) can safely be omitted without creating serious gaps in the overall presentation. One topic that is new to this edition, covering spaces and the fundamental group, is not starred and should not be omitted unless the students have seen it in some prior course. Some of the optional topics fall into subgroupings, any one of which can be included without dependence on the others. Thus, Subsection 2.9.B and Sections 3.10 and 4.2 constitute a brief introduction to Morse theory, one of the most useful tools in differential topology. Similarly, Sections 3.9, 6.5, and 8.7 constitute an introduction to degree theory, together with some classical topological applications, but in this case any one of these three sections can be treated without serious logical dependence on the others. Apart from minor revisions, this treatment of degree theory is not new to this edition. In Subsection 1.6.B, we classify 1-manifolds. This intuitively plausible result needed is only in the optional Section 3.9. Also easily omitted is the brief Subsection 1.6.A, this being an extended remark on cobordism theory. New to this edition is an optional introductory treatment of Whitney's imbedding theorems (Subsection 3.7.C). We prove only the "easy" Whitney theorem, while stating carefully the general theorem. Imbeddings of manifolds in Euclidean space will be used only in treating some other optional topics, namely, the smoothing of continuous maps and homotopies (Subsection 3.8.B) and the existence of Morse functions (Section 3.10). In Chapter 5, an introduction to Lie theory, adequate for a first course on manifolds, requires only the first two sections. Accordingly, Sections 5.3 (the closed subgroup theorem and related topics) and 5.4 (homogeneous spaces) are optional. Certain topics in de Rham theory, Sections 8.8 (Poincar5 duality) and 8.9 (a version of the de Rham theorem), can be omitted, as can the treatment of foliations defined by closed 1-forms (Section 9.3). Also easily omitted is the brief treatment of Riemannian homogeneous and symmetric spaces (Section 10.7). Finally, Chapter 11, on principal bundles and their role in geometry, gathers together and slightly expands on topics treated in various parts of the first edition and can be reserved to introduce a more advanced course or seminar. There are some significant changes in the appendices also. The original Appendix A has been replaced by one that gives the construction of the universal covering space. The former Appendix D (Sard's theorem) has been moved to the main body of the text. The current Appendix D (formerly Appendix E) has been expanded to include a proof of the de Rham theorem for singular as well as Cech cohomology.
Acknowledgments I am grateful to the late Robby Gardner and his students at Chapel Hill who "beta tested" eight chapters of a preliminary version of the first edition of my book in an intensive, one-semester graduate course. Their many suggestions were most helpful in the final revisions. Others whose input was helpful include Geoffrey Mess, Gary Jensen, Alberto Candel, Nicola Arcozzi and Tony Nielsen. I particularly want to thank Filippo De Mari, whose beautiful class notes, written when he was one of my students in an earlier version of this course, were immensely useful in subsequent revisions and first suggested to me the idea of writing a book. Finally, my students in the academic year 1999-2000 have offered much helpful input toward the final version of this edition.
CHAPTER 1
Topological Manifolds This chapter pertains to the global theory of manifolds. See also [3, Chapter I] and [41, Chapter 1].
1.1. Locally Euclidean Spaces Classical analysis is carried out in Euclidean space, the operations being defined by local formulas. One might hope, therefore, to extend this classical theory to all topological spaces that are locally Euclidean. While this is not generally possible without further restrictions on the spaces, the locally Euclidean condition is fundamental.
Definition 1.1.1. A topological space X is locally Euclidean if, for every x E X, 3 n > 0 (an integer), an open neighborhood U C X of x, an open subset W C ]I~n and a homeomorphism ~ : U --~ W. If we can show that n is uniquely determined by x, we will write n = d(x) and call this the local dimension of X at x.
Example 1.1.2. Any open subset X c_ ~ n is a locally Euclidean space that is also Hausdorff and 2nd countable. We will see that the local dimension is n at every xCX.
Example 1.1.3. Let X = ll~ U (*}, where * is a single point and sqcup denotes disjoint union. Topologize this set so that a basis of open subsets V C X consists of the following: 9 I f * ~ V, then V is open as a subset of ~. 9 If * E V, then 0 ~ V and there is an open neighborhood W C R of 0 such that V = (W \ {0}) U {*}. This space is locally Euclidean and 2nd countable. It is not Hausdorff since every open neighborhood of * meets every open neighborhood of 0. In this case, the local dimension is everywhere 1, even at . and at 0. E x a m p l e 1.1.4. In [41, Appendix A], there is described a bizarre space called the long line. It is connected, Hausdorff, and locally Euclidean with d(x) =- 1, but it is not 2nd countable. In fact, the long line contains an uncountable family of disjoint, open intervals.
Exercise 1.1.5. Prove that each connected component of a locally Euclidean space X is an open subset of X. E x e r c i s e 1.1.6. Prove that a connected, locally Euclidean space X is path connected.
2
1. T O P O L O G I C A L
MANIFOLDS
E x e r c i s e 1.1.7. Give an example of a connected, 2nd countable, Hausdorff space t h a t is not path connected. Recall t h a t a regular space X is one in which any proper closed subset C C X and point x C X \ C can be separated by disjoint, open neighborhoods of each. E x e r c i s e 1.1.8. Prove that a locally compact (in particular, a locally Euclidean), Hausdorff space must be regular. By a theorem of Urysohn [8, p. 195], 2nd countable regular spaces are metrizable. Thus, manifolds are metrizable and questions of continuity, closure, compactness etc. involving manifolds can be reduced to corresponding questions of sequential continuity, sequential closure, sequential compactness, etc. The following difficult result, known as L. E. J. Brouwer's theorem on invariance of domain, is needed in order to show that local dimension is always well defined on locally Euclidean spaces. The proof will not be given. It is best carried out by the methods of algebraic topology [10, p. 303], [39, p. 199], [13, p. 110]. For a more classical proof, see [20, pp. 95-96]. In the theory of smooth manifolds, differential calculus reduces the appropriate analogue of this theorem to elementary linear algebra. T h e o r e m 1.1.9 (Invariance of domain). I f U C_ IRn is open and f : U --~ ~'~ is continuous and one-to-one, then f ( U ) is open in IRn. C o r o l l a r y 1.1.10. I f U C_ ~ n and V C_ I~m are open subsets such that U is horneornorphic to V , then n = m. Proof. Assume that rn ~ n, say, m < n. Define i : R m ~ lI~~ by i(x 1 . . . . ,x m)
:
(X 1 , . . . , x m , 0 , . . . , 0 ) . n--m
This m a p is continuous and one-to-one and i(N m) is not open in N ~ and does not even contain a subset that is open in N n. By assumption, there is a homeomorphism ~o : U ~ V, so the composition
f :U ~
V ~d~U~n
is continuous and one-to-one. Also, while U C_ R ~ is open, we see t h a t f ( U ) = i ( V ) C i(N m) cannot be open in R n. This contradicts Theorem 1.1.9. [] C o r o l l a r y 1.1.11. I f X is locally Euclidean, then the local dimension is a welldefined, locally constant function d : X --* Z + . Proof. Let x C X and suppose that there are open neighborhoods U and V of x in X, together with open subsets U C_ R n, V C N m, and homeomorphisms
~:U--*U
r Since V R U is open in X, it follows that
~ ( v n g ) c_ ~c_
Rn
are inclusions of open subsets and, similarly, t h a t r or
:r
n U) is open in R m. But
n v ) -~ ~ ( u n v )
is a homeomorphism, so m = n by the previous corollary. A l l assertions follow.
[]
1.2. T O P O L O G I C A L MANIFOLDS
3
C o r o l l a r y 1.1.12. If X is a connected, locally Euclidean space, then the local dimension d : X ~ Z + is a constant called the dimension of X . E x e r c i s e 1.1.13. Let X and Y be connected, locally Euclidean spaces of the same dimension. If f : X --~ Y is bijective and continuous, prove that f is a homeomorphism. 1.2. T o p o l o g i c a l M a n i f o l d s Some authors designate by the term "manifold" an arbitrary locally Euclidean space. It is more common, however, to require more. D e f i n i t i o n 1.2.1. A topological space X is a manifold of dimension n (an nmanifold) if (1) X is locally Euclidean and d(x) - n = dim X; (2) X is Hausdorff; (3) X is 2nd countable. Of the three examples in the previous section, only the open subsets of Nn were manifolds. L e m m a 1.2.2. I f X is a compact, connected, metrizable space that is locally Euclidean, then X is an n-manifold, for some n c Z +. Proof. Indeed, X is Hausdorff because it is metrizable. It is 2nd countable because it is locally Euclidean and compact. Being locally Euclidean and connected, X has constant local dimension. [] Here are some examples of manifolds. E x a m p l e 1.2.3. The n-sphere S n = {v E IR~+1 I Ilvll = 1} is an n-manifold. One way to see that it is locally Euclidean is by stereographic projection. Let p+ = ( 0 , . . . , 0 , 1 ) , p_ = ( 0 , . . . , 0 , - 1 )
be the north and south poles of S n, respectively. Then the stereographic projections 7r+ : S ~ \ {p+ } --+ IR~' 7r_ : S n \
{p_}
--+ II{ n
onto the subspace R~= {(xl,...,xn,0)} are homeomorphisms and {S ~ \ {p_}, S ~ \ {p+}} is an open cover of S n. (For a pictorial definition of ~+, see Figure 1.2.1.) Since S ~ is compact and metrizable, it is an n-manifold. E x e r c i s e 1.2.4. In Example 1.2.3, write down formulas for the stereographic projections 7r• and prove carefully that they are homeomorphisms. E x a m p l e 1.2.5. If N is an n-manifold and M is an m-manifold, then N x M is an (n + m)-manifold. Indeed, if (x,y) E N x M, let U be a neighborhood of x in N homeomorphic to an open subset of IRn and V a neighborhood of y in M homeomorphie to an open subset of R m. Then, the neighborhood U x V C_ N x M of (x, y) is homeomorphic to an open subset of IRn x R m = R n+m. Since M and N are Hausdorff and 2nd countable, so is N x M.
4
1. TOPOLOGICAL MANIFOLDS
F i g u r e 1.2.1. Stereographic projection from p+ E x a m p l e 1.2.6. The n-torus T n-- S 1
X
S1
X
...
X
S1
n factors
is an n-dimensional manifold. Indeed, by Example 1.2.3, S 1 is a 1-manifold and Example 1.2.5, applied successively, then implies that T n is an n-manifold. E x a m p l e 1.2.'7. A vector w E ~ + 1 is defined to be tangent to S n at v E S n if w J_ v. This conforms to naive geometric intuition and can be seen to conform to the general definition of tangent vectors to differentiable manifolds that we will give later. In order to keep track of the point of tangency, we will denote this tangent vector by (v, w) E R ~+1 x •n+l. Thus, the set of all tangent vectors to S ~ is T ( S n) : { ( v , w ) e ]1:~n + l >< ]l~n + l I IlVll ~- 1 , W J- V}.
This space is topologized as a subspace of I~~+1 • R ~+1 . We also define the continuous map p : T ( S n) ---+ S n by p ( v , w ) = v. Thus, p assigns to each tangent vector its point of tangency. For each v0 E S ~, consider
Tvo(SD = {(v0,w) 9 T ( S D } = p - l ( v 0 ) , the set of all vectors tangent to S ~ at v0. This is an n-dimensional vector space under the operations r . (vo, w) = (vo, r " w)
(vo,w,) + (vo,w2) = (vo,Wl + w2). This structure, p : T ( S ~) ---* S '~, is called the tangent bundle of S n. The space T ( S ~) is the total space of the bundle, the space S ~ is the base space of the bundle, and p is the bundle projection. By a common abuse of terminology, the total space is often referred to as the tangent bundle itself. In Exercises 1.2.10 and 1.2.11, you are going to prove that T ( S '~) is a 2n-manifold. A couple of definitions are needed first. D e f i n i t i o n 1.2.8. If U c S n is an open subset, then T ( S n ) I U = T ( U ) is the space p - l ( U ) . The tangent bundle of U is given by Pu : T ( U ) -+ U, where Pu denotes the restriction pIT(U).
1.2. T O P O L O G I C A L MANIFOLDS
5
D e f i n i t i o n 1.2.9. If U C_ S n is open, a vector field on U is a continuous m a p s : U --+ T ( U ) such t h a t Pu o s = idg. E x e r c i s e 1.2.10. Given v0 E S n, show t h a t there is an open neighborhood U C S n of v0 and vector fields si : U --~ T ( U ) , 1 < i < n, such t h a t
{Sl(V), s2(v),., sn(v)} is a basis of the vector space Tv(Sn), V v C U. E x e r c i s e 1.2.11. Let U C S n be as in the previous exercise. Using t h a t exercise, c o n s t r u c t a continuous bijection ~o : U x IR~ --+ T ( U ) and prove t h a t ~ is a homeom o r p h i s m . (This is not very deep. You do not, for instance, need T h e o r e m 1.1.9.) Using this, prove t h a t T ( S n) is a 2n-manifold. Prove also that, for each v E U, the formula qov(w) = ~ ( v , w ) defines an isomorphism ~Ov : IR'~ --* T v ( S n) of vector spaces. A t h o r o u g h u n d e r s t a n d i n g of the tangent bundle of S n eluded topologists for several decades. For instance, it was long unknown w h a t is the m a x i m u m number r(n) of vector fields si : S ~ ~ T(S'~), 1 < i < r(n), t h a t are everywhere linearly independent. T h a t is, we require that, for each v C S n, the vectors { s l ( v ) , . . . ,sr(~)(v)} be linearly independent in T , ( S n) and t h a t no set of r(n) + 1 fields has this property. T h e problem of c o m p u t i n g r(n) was known as the "vector field problem for spheres". Definition
1.2.12. T h e sphere S ~ is parallelizable if r(n) = n.
This brings us to a striking example of global versus local properties. If S ~ is parallelizable, Exercise 1.2.11 implies t h a t T ( S n) ~ S n x IRn. For general n, this s a m e exercise implies t h a t the tangent bundle T ( S ~) is locally a Cartesian p r o d u c t of an open set U C S ~ w i t h IR~, but it is only 91obally such a p r o d u c t when S ~ is parallelizable. Not every sphere is parallelizable. For instance, it has long been known t h a t r(2n) = 0. This means t h a t every vector field on S 2n is somewhere zero. In the case of S 2, this is s o m e t i m e s stated facetiously as "you c a n ' t comb the hair on a c o c o n u t " . It was also known for some t i m e t h a t S 1, S 3, and S 7 are parallelizable. T h e following was finally proven in the late 1950s [4], [27]. Theorem
1 . 2 . 1 3 (R. B o t t and J. Milnor, M. Kervaire). The sphere S ~ is paral-
lelizable if and only if n = O, 1, 3, or 7. T h e case n = 0 is tile trivial fact t h a t the 0-sphere S o = {4-1} c IR a d m i t s 0 i n d e p e n d e n t fields. T h e r e is an interesting relationship between T h e o r e m 1.2.13 and the problem of defining a bilinear multiplication on IR'~ w i t h o u t divisors of zero. Such a multiplication is a bilinear m a p # : IR~ x R ~ --+ IR~, w r i t t e n #(v, w) = vw, such t h a t v w = 0 ~ v = 0 or w = 0. Theorem
1.2.14.
and only if n = 0 , 1 , 3 ,
There is a multiplication on IRn+l without divisors of zero if or 7.
Indeed, R 1 = IR, R 2 = C, and IR4 = ]HI (the quaternions). T h e nmltiplication on ]Rs is given by the Cayley numbers, a nonassociative division algebra whose elements are ordered pairs (x,y) of quaternions [43, pp. 108-109]. This proves the "if" in T h e o r e m 1.2.14.
6
1. T O P O L O G I C A L M A N I F O L D S
E x e r c i s e 1.2.15. If ]~n+l admits a multiplication without divisors of zero, prove that S n is parallelizable. In light of Theorem 1.2.13, this gives the "only if" part of Theorem 1.2.14. The full solution to the vector field problem for spheres was given by F. Adams in the early 1960s [1], culminating a long history of research on that problem by several algebraic topologists. We state Adams' result. Define the function p(n), n > 1, by requiring that S n-1 admit p(n) - 1 everywhere linearly independent vector fields, but not p(n) such fields (thus, r(n) = p(n + 1) - 1). Write each natural number n uniquely as n = (2r + 1)2 c+4d, where r, c, d are nonnegative integers and c <_ 3. This uses the unique factorization theorem and the division algorithm mod 4. T h e o r e m 1.2.16 (F. Adams). p(n) = 2 c + 8d. Remark that p(odd) = 1, since c = d = 0. This gives the classical result that every vector field on an even dimensional sphere is somewhere zero. The easier part of Adams' theorem is that S n-1 does admit at least 2c + 8 d - 1 independent vector fields. The original proof, using only linear algebra, was given by Radon and Hurwitz and, in 1942, an improved proof was given by Eckmann [9]. The harder part of the theorem, that there are at most 2c + 8d - 1 such fields, is much more advanced. E x e r c i s e 1.2.17. Using Theorem 1.2.16, show that p(n) = n if and only if n = 1, 2, 4, or 8. This gives back Theorem 1.2.13. E x e r c i s e 1.2.18. Verify that p(2) = 2 without using Theorem 1.2.16.
1.3. Q u o t i e n t C o n s t r u c t i o n s a n d 2 - M a n i f o l d s We continue the project of constructing manifolds. In this section, examples will be constructed using the quotient topology. Before giving a careful definition of quotient spaces, we look at some intuitive examples of 2-manifolds constructed in this way. Consider the square D = [0, 1] • [0, 1]. In Figure 1.3.1, the arrows indicate that the opposite sides of D are to be glued together so that the vertical edges are glued bottom to top and the horizontal edges from leR to right. When the first pair of edges are glued, the result is a cylinder. When the second pair are glued, the cylinder becomes the 2-torus T 2 = S 1 • S 1. In Figure 1.3.2, the horizontal edges of D are identified in the same sense, b u t the left and right are glued together in opposite senses. If the left and right are identified first, the result is a "MSbius strip". It is not very easy, then, to picture the rest of the identification. If the top and bottom are identified first, the result is a cylinder whose boundary circles are then to be identified with an orientationreversing "flip". The resulting 2-manifold is called the "Klein bottle" K 2 and can be pictured only in 1~3 if one allows the surface to intersect itself. Figure 1.3.2 is an a t t e m p t at such a picture, the self-intersection occurring along a circle. This circle of intersection corresponds to the vertical line g and the circle c in Figure 1.3.3. In order to view K 2 without self intersection, it is necessary to situate it in a 4-dimensional framework. This is not as psychologically hopeless as it might seem.
1.3. Q U O T I E N T CONSTRUCTIONS
7
F i g u r e 1.3.1. The 2-torus
F i g u r e 1.3.2. The Klein bottle with self-intersection
One can, for example, color K 2 by shades of grey, varying continuously over the Klein bottle, in such a way that the circle c in Figure 1.3.3 has no points with the same shade as any point in the line ,e. By continuously assigning numbers from 0 to 1 (lightest to darkest) to these shades, one introduces a fourth "dimension" and the shaded Klein bottle has no self intersections. What was the circle of intersection is now two disjoint shaded circles ~ and c. One can think of the shaded Klein bottle as a topologically imbedded Klein bottle in ]R4. In Figure 1.3.4, we identify each pair of opposite sides of D with a reverse of orientation. The resulting 2-manifold is called the projective plane p2 and, once again, it cannot be imbedded in IR3. The topologist (but not the geometer) can view D as the unit disk D 2 = {v E IR2 ] Ilvll _< 1} in such a way that the gluing
8
1. T O P O L O G I C A L M A N I F O L D S
F i g u r e 1.3.3. Using variable shading to imbed K 2 in ll~4
F i g u r e 1.3.4. The projective plane
identifies antipodal pairs of boundary points. T h a t is, if [Ivll = 1, then v and - v are identified. Another way to think of p2 is to start with S 2 C IRa and to identify the antipodal pairs { w , - w } . To see that this also yields p2 first carry out the identification for the antipodal pairs not lying on the equatorial circle z = 0. The resulting disk has the equatorial circle as boundary and the remaining identifications give the previous description of p2. One also considers surfaces with edges. These are not manifolds in the sense we have defined, but they are manifolds with boundary in the sense to be defined in Section 1.6. For example, the disk D 2 is such a surface, as is the cylinder (also called the annulus) S 1 x [0, 1]. Of particular interest is the Mi~biusstrip DI 2, already referred to in the construction of the Klein bottle. It is the result of gluing one pair
1.3. Q U O T I E N T CONSTRUCTIONS
9
F i g u r e 1.3.5. The MSbius strip 9Jt2 of opposite edges of a rectangle with an orientation-reversing flip, but leaving the other two edges alone (Figure 1.3.5). This nonorientable surface has just one edge (a circle). D e f i n i t i o n 1.3.1. Let M1 and M2 be 2-manifolds and let Di C Mi be imbedded disks, i = 1, 2. If Mi has edges, require that Di be disjoint from the edges. Let M~ = Mi \ int(Di), i = 1,2, and glue these together by a homeomorphism of OM~ = OD1 to OM~ = OD2. The resulting 2-manifold M1 # M2 is called the connected sum of/I//1 and M2. When this definition has been put on a rigorous footing, it can be shown that the connected sum is well defined up to homeomorphism. Thus, strictly speaking, the symbols /I//1, M2, and M I # M 2 should denote homeomorphism classes of 2-manifolds. With this understanding, the connected sum can be seen to be commutative, MI # M2 = M2 # MI, associative, (M1 # M2) # M3 = M1 # (M2 # M3), (hence parentheses can be dropped) and to admit (the homeomorphism class of) S 2 as a 2-sided identity
MI # S 2 =M1 = S2 # M1. In this way, the set of (homeomorphism classes of) surfaces becomes an abelian semigroup. Of particular interest is the subsemigroup of compact, connected surfaces. E x e r c i s e 1.3.2. You are to give an intuitively clear "proof" that the Klein bottle is the connected sum of two projective planes. Proceed as follows. (1) Remove the interior of an imbedded disk from p2 and show that the resulting surface M is the MSbius strip. (Hint. Use the description of p2 as the result of antipodal identifications on $2.) Thus, p 2 # p ~ is obtained by gluing
10
1. T O P O L O G I C A L M A N I F O L D S
F i g u r e 1.3.6. The two-holed torus T 2 # T 2 together two copies of the MSbius strip by a homeomorphism between their boundary circles. (2) Consider the square with edge identifications in Figure 1.3.2 and use two horizontal lines to divide this square into three congruent rectangles. Show that one of these is really a M5bius strip, as is the union of the other two, showing that K 2 is also obtained by gluing together two copies of the MSbius strip by a homeomorphism between their boundary circles. E x a m p l e 1.3.3. The connected sum T 2 # T 2 of two copies of the torus is the two-holed torus pictured in Figure 1.3.6. It is possible to view this surface as the result of gluing together pairs of edges of an octagon, as we now sketch. Consider the irregular pentagon, as pictured in Figure 1.3.7, and glue together each pair of parallel edges as indicated there. Remark that this identifies all five vertices to a single point and that the resulting surface is just the complement in T 2 of the interior of an imbedded disk. Now glue two copies of this surface together as indicated in Figure 1.3.8 to obtain T 2 # T 2. A topologically equivalent picture in Figure 1.3.9 shows that the two-holed torus is obtained from an octagon with edges identified pairwise as indicated there. By an inductive procedure, one can continue this process, showing that the g-holed torus, produced as a connected sum of g copies of T 2, is obtained by pairwise identifications of edges of a 4g-gon. E x e r c i s e 1.3.4. Give an intuitively plausible proof that p2 ~ T 2 = p2 ~ K 2 = p2 # p2 # p2. Of course, the second equality is by Exercise 1.3.2. For the first, proceed as follows. (1) Remove the interiors of two disjoint, imbedded disks in a rectangle [0, 1] • [0, 1]. There are essentially two distinct ways to attach a cylinder S 1 • [0, 1], identifying its boundary circles respectively to the two boundary circles left by the excised disks. Show that one of these is homeomorphic to the surface obtained by removing the interior of a disk from T 2, the other to the surface obtained by the corresponding operation on K 2. (2) Show that, if one identifies one pair of opposite edges of the square with an orientation-reversing flip, both of the surfaces obtained in (1) become homeomorphic.
1.3. Q U O T I E N T CONSTRUCTIONS
11
F i g u r e 1.3.7. Identify the directed edges with the same labels.
F i g u r e 1.3.8. Glue two copies together along their free edges (3) Show t h a t the surface obtained in (2) can be viewed both as the connected sum 9)l2 # T 2 and as 9Jl:2 ~r K 2. (4) Since the MSbius strip is obtained by deleting an open disk from p2 (part (1) of Exercise 1.3.2), conclude to the desired homeomorphism.
Remark. Exercise 1.3.4 shows, in particular, t h a t the semigroup of compact surfaces does not have cancellation. It is not a group. This exercise and constructions as in Example 1.3.3 are involved in the proof of the following classification theorem for compact surfaces. A proof of this theorem, in the spirit of our current discussion, can be found in [26, Chapter 1]. T h e o r e m 1.3.5. The semigroup of compact, connected 2-manifolds with no edges is generated by T 2 and p2. Indeed, every element other than the identity S 2 can be uniquely written as a connected sum of finitely many copies of T 2 or of finitely many copies of p2.
12
1. T O P O L O G I C A L M A N I F O L D S
F i g u r e 1.3.9. Identify the directed edges with the same labels. The number g of copies of T 2 or of p2 in the unique connected sum representation of a compact, connected surface M % S 2 is called the genus of M. We agree t h a t S 2 has genus 0. An important tool in the proof of Theorem 1.3.5 is a triangulation of a compact surface M. Triangulations are useful for many purposes, so we give a brief discussion. Let A c ~2 be the convex hull of the points v0 = (0,0), vl = (1,0), and v2 = (0, 1). T h a t is, A={(x,y)
Ix, y > 0 a n d x + y - < l } ,
a closed, triangular region in the plane having vertices {v0, vl, v2} and edges e0 = { ( x , y e z x ) I x + y =
1},
el = {(0, y) e zx}, e2 = {(x, 0) e zx}. The triangle A C IR2 is called the standard 2-simplex. pieces A~, A 2 , . . . , A~, together with homeomorphisms 9~i : A --+ Ai,
One decomposes M into
l
in such a way that any two of the triangles Ai, A j are either disjoint, have in common just one vertex ~oi(ve) = qoj(vk), or have in common just one edge ~i (ee) = qoj (ek). Such a decomposition is called a triangulation of M. 1.3.6 (T. Rad6). Every compact surface M admits a triangulation. Equivalently, M can be const~cted, up to homeomorphism, by taking finitely many copies of the standard 2-simplex A and gluing them together appropriately along edges.
Theorem
This theorem is intuitive but nontrivial [31, pages 58-64]. The first proof was given by T. Rad6 [36]. The standard 2-simplex is oriented by a choice of ordering of its three vertices. Two orderings give equivalent orientations if they differ by an even permutation. Thus, up to equivalence, there are two orientations of A. Such an orientation
1.3. Q U O T I E N T C O N S T R U C T I O N S
13
F i g u r e 1.3.10. A triangulation of S 2
induces a direction along the edges of A, leading to the terminology "clockwise orientation" and "counterclockwise" orientation. The standard orientation of A, given by the ordering (Vo, Vl, v2), is the counterclockwise orientation. Note that the homeomorphism a : A --+ A, defined by a(x, y) = (y, x) is orientation reversing. Triangulations give us a way of defining the notion of "orientability" for a compact surface. The idea is that a triangle Ai = g)i(A) has the standard orientation (~oi(vo),~oi(Vl),~oi(v2)). Two triangles Ai and Aj that have a common edge e are coherently oriented if their standard orientations induce opposite directions along e. A little thought should convince the reader that this is the natural notion if we are to picture the orientations of both Ai and Aj as counterclockwise. If Ai and Aj have no edge in common, they are also said to be coherently oriented. Given a triangulation of a connected surface M, one can attempt to make all the orientations coherent as follows. Starting with one triangle, say A1, look at any triangle, say A2, having an edge in common with A1. If they are coherently oriented, well and good. If not, replace ~o2 with ~o2 o a, making them coherently oriented. Continuing in this way either orients the triangulation or leads to conflicting orientations on at least one simplex. In the first case, the triangulation is said to be orientable and, in the second, to be nonorientable. If M has more than one component, each must be treated separately and the triangulation of M is orientable if and only if the triangulation of each component is orientable. It can be proven that some triangulation of M is orientable if and only if every triangulation of M is orientable. D e f i n i t i o n 1.3.7. We say that M is orientable if some, hence every, triangulation of M is orientable.
14
1. T O P O L O G I C A L M A N I F O L D S
E x a m p l e 1.3.8. The sphere S 2 is orientable. Indeed, define the standard 3-simplex A a C R a to be the convex hull of the set of points {v0 = (0,0,0),vl = (1,0,0),v2 = (0, 1,0),v3 = (0,0, 1)}. T h a t is, A a={(x,y,z) lx,y,z>0andx+y+z< 1}. Any three of the vertices {vo,vl,v2,va} correspond to a triangular face of A a. These four triangular faces unite to form a surface homeomorphic to S 2, defining thereby a triangulation of S 2 (Figure 1.3.10). Orienting these faces by (vl,v2, v3), (vo, v2, vl), (va, v2, vo), and (v3, vo, Vl), respectively, gives coherent orientations to all the triangles. An interesting invariant of surfaces is the Euler characteristic. If the compact surface M has a triangulation T, let V denote the number of points of M that are vertices of the triangulation, E the number of arcs in M that are edges of the triangulation, and F the number of triangular faces. T h e o r e m 1.3.9. The number F - E + V depends only on M, not on the choice of triangulation 9". This number is called the Euler characteristic of M and is denoted X(M). This theorem has many proofs. One method, useful only for smoothly imbedded surfaces in IRa, is to use the Gauss-Bonnet theorem, which asserts that the integral of the Gauss curvature over a is 27r)((M) ([15, p. 111], [34, pp. 380-382], [44, pp. 237-239]). E x a m p l e 1.3.10. In the triangulation of S 2 given in Figure 1.3.10, we have V = 4, E = 6, and F = 4, so x ( S 2) = 2. E x e r c i s e 1.3.11. Give an intuitive but completely convincing proof t h a t V - E + F = 2 for every triangulation of S 2. E x e r c i s e 1.3.12. Consider a subdivision of S 2 into n-gons, where n _> 3 is fixed, such t h a t any two of these n-gons meet, if at all, only along a common edge or a common vertex. Assume further that there is a fixed integer m _> 3 such that exactly rn edges meet at each vertex. Using the fact that X(S 2) = 2, prove that the number of n-gons in this subdivision must be 4, 6, 8, 12 or 20. This proves the result, known from antiquity, that there are only five regular polyhedra (the "Platonic solids"). W h a t are the values of n and m for these regular polyhedra? E x e r c i s e 1.3.13. Produce triangulations of the surfaces T 2, /(2, and p2. Use these to prove that T 2 is orientable and that K 2 and p2 are not. Use Theorem 1.3.5 to determine all orientable, compact, connected surfaces and all nonorientable ones. E x e r c i s e 1.3.14. For compact, connected surfaces M1 and M2, prove t h a t x(M1 # M2) = X(M1) + x(M2) - 2.
Use this equation, Theorem 1.3.5 and Exercise 1.3.13 to compute the Euler characteristics of all compact, connected surfaces without edges. All compact, connected surfaces with edges are obtained from those without edges by removing the interiors of finitely many disjoint, imbedded disks. The resulting surface, obtained (say) from M , is said to have the same genus as M and is orientable if and only if M is orientable. Give a formula for the Euler characteristic of this surface in terms of )/(M) and the number of disks removed.
1.3. QUOTIENT
CONSTRUCTIONS
15
Our discussion of the topology of 2-manifolds belongs to tile "cut and paste" brand of topology. To put such constructions on a rigorous footing, it is necessary to introduce quotient spaces. Let X be a topological space and let ~ be an equivalence relation on X. That is, the relation satisfies the following three conditions: 9 x~x,
VxCX;
9 x~y~y~x; 9 x~yandy~z~x~z. Such a relation partitions X into a collection {X~}~ev~ of disjoint subspaces, the equivalence classes of ~. D e f i n i t i o n 1.3.15. The set {X~}ac~ of equivalence classes of X is called the quotient space of X modulo ~ and is denoted by X/,.,. The surjection 7r : X - - ~ X / ~
is the map that assigns to each x C X its equivalence class r=(x) E X / ~ . A subset U C X/~,, is said to be open if and only if rc-l(U) is open in X. It is trivial to check that these open sets constitute a topology on the quotient space X / ~ . This is the quotient topology, characterized as the largest (i.e., the finest) topology on X / ~ relative to which the canonical projection 7r : X - +
X /~
is continuous.
Remark. Here and throughout this book, the term "canonical" will appear. While it has a precise meaning in category theory, our use will be less formal. It always indicates some sort of independence of arbitrary choices, the word "canon" being the Latin word for "law". For the theologically inclined, the term might be read as "God-given". E x a m p l e 1.3.16. Let X = ]R x {0, 1}, topologized as the disjoint union of two copies of IR. Define an equivalence relation on X by setting
if a n d o n l y if e i t h e r ~ = Z a n d x = y, or ~ r
a n d x = y r 0. T h u s { ( 0 , 0 ) } a n d
{(0, 1)} each constitute a distinct equivalence class, but all other classes are pairs {(x, 0), (x, 1)} where x # 0. The quotient space X / ~ is the non-Hausdorff, locally Euclidean space given in Example 1.1.3. As the above example shows, even if X is Hausdorff, the quotient X / ~ may fail to be Hausdorff. L e m m a 1.3.17. If the space X is compact, so is X / ~ .
Proof. The continuous image 7r(X) = X / ~ of a compact space X is compact.
[]
L e m m a 1.3.18. If the space X is connected, so is X / ~ .
Proof. The continuous image 7r(X) = X / ~ of a connected space X is connected.
[]
D e f i n i t i o n 1.3.19. A map f : X --+ Y respects an equivalence relation ~ on X if x ~ y ~ f(x) = f(y). In this case, the induced map
f :X/~--+Y is well defined by /(re(x)) = f(z).
16
1. T O P O L O G I C A L M A N I F O L D S
L e m m a 1.3.20. Let X and Y be topological spaces, let ~ be an equivalence relation
on X , and let f : X ~ Y be a map respecting this equivalence relation. Then f is continuous if and only if f is continuous. Pro@ Consider the commutative diagram X
ir
.x/~
Y Since f = f o 7r, continuity of f implies continuity of f. For the converse, assume t h a t f is continuous, hence that f - l ( U ) is open in X whenever U is open in Y. But this implies, via the commutative diagram, that
~-l(f-I(u)) = f-~(u) is an open subset of X. By the definition of the quotient topology, f - 1 (U) is open in X / N , so f is continuous. [] E x a m p l e 1.3.21. Let X = [0, 1] and let Y = o~ C C. Define
f:X--*Y by f(t) = e 2'~it, a continuous surjection. It is not quite one-to-one since f(1) = f(0). On X , define the equivalence relation x ~ y by requiring either that x = y or that { x , y } = {0, 1}. Clearly, f respects this relation. Denote X / ~ by [0, 1]/{0, 1} and remark t h a t f : [0,1]/{0,1} ~ S 1 is bijective. By Lemma 1.3.20, f is also continuous. But [0, 1]/{0, 1} is compact by Lemma 1.3.17 and S 1 is Hausdorff. A one-to-one, continuous map from a compact space onto a Hausdorff space is a homeomorphism, so f gives a canonical homeomorphism [0, 1]/{0, 1} ~ S 1. Intuitively, we have glued together the two ends of the interval [0, 1] to obtain a circle. E x a m p l e 1.3.22. Consider the map p : S 1 x[0,1]---*D 2 defined by viewing S 1 C D 2 C C and writing
p(z,t) = (1 - t)z. This is one-to-one on S 1 x [0, 1) and collapses S 1 x {1} to the single point 0 C C. Arguing as in the previous example, we see that the quotient space (S a x [0, 1 ] ) / ( S 1 x {0}) is canonically homeomorphic to D 2. Intuitively, we have collapsed the top of the cylinder S 1 x [0, 1] to a point, obtaining a cone that can then be flattened to a disk. Generally, if A C_ X, one can define the equivalence relation ~ A by writing x ~ A Y if and only if either x = y or x, y E A. The quotient space X / N A is thought of as the result of crushing A to a single point in X and will be denoted by X / A . Some care should be taken in using this notation. If X = G is a topological group and H C G is a subgroup, then G / H denotes the space of left cosets of H, not the space obtained by collapsing H alone to a point. The context should make clear which interpretation is intended. Remark that G / H is also a set of equivalence
1.4. P A R T I T I O N S
OF
UNITY
17
classes, t h e relation being gl ~ g2 if and only if 91--1 92 E H . Thus, G / ' H can be given the quotient topology and the continuous m a p 7r : G --~ G / H is 1r(9 ) = g H .
Definition 1.3.23. If G is a topological group and H _C G is a subgroup, the q u o t i e n t space G / H is called the (left) coset space of G rood H. M a n y i m p o r t a n t topological groups are also manifolds. These are called Lie 9roups and are enormously i m p o r t a n t t h r o u g h o u t m a t h e m a t i c s and in m a t h e m a t i cal physics. We close this section with some exercises using quotient constructions to produce manifolds of higher dimension. E x e r c i s e 1.3.24. Let D n = {v E IRn I llvll <_ 1}, the unit n-disk (also called the closed n-ball) w i t h b o u n d a r y OD n = S n-1. Prove t h a t D ' ~ / S ~ - I is h o m e o m o r p h i c to S ". (We proved the case n = 1 in E x a m p l e 1.3.21.) E x e r c i s e 1.3.25. View IRn as an abelian group under vector addition. This is a topological group. T h e integer lattice
z ~ = {(~1,-~,
, - < ) I - ~ c z,
1 < i < n}
is a (normal) s u b g r o u p of R ~. Let T ~ = R ~ / Z n be the coset space (actually, a Lie group). P r o v e t h a t this space is homeonlorphic to S I x S I x ... x S I n factors
our definition of the n-torus in E x a m p l e 1.2.6. Use this to show t h a t the surface c o n s t r u c t e d in Figure 1.3.1 is, indeed, T 2. E x e r c i s e 1.3.26. Define an equivalence relation on S n C IRn+l by writing v ~ w if and only if v = :kw. T h e quotient space p n = s n / ~ is called projective n-space. (This is one of the ways t h a t we defined the projective plane p2.) T h e canonical p r o j e c t i o n 7r : S ~ -~ P~ is just re(v) = {:t:v}. Define U~ C P'~, 1 < i < n + 1, by setting
u~ = { ~ ( ~ l ,
, ~ n + ~ ) l ~ r 0}
Prove
(1) Ui is open i n / o n .
(2) (a)
{ U 1 , . . . , Un+l} covers p n T h e r e is a h o m e o m o r p h i s m g)i : Ui ---+tRn. (4) p n is compact, connected, and Hausdorff, hence is an n-manifold.
1.4. Partitions of U n i t y P a r t i t i o n s of unity play a crucial role in manifold theory. Their existence is a consequence of the fact t h a t manifolds are p a r a c o m p a c t , hi this section, we establish these facts. For further information a b o u t p a r a c o m p a c t spaces, the reader is referred to [8, pp. 162-169].
Definition 1.4.1. A family C = { C ~ } ~
of subsets of X is locally finite if each x E X admits an open neighborhood Wx such t h a t W~ N Ca r ~ for only finitely m a n y indices a E 9.1. T h e following exercise will be useful.
18
1. T O P O L O G I C A L M A N I F O L D S
E x e r c i s e 1.4.2. If e = {Ca}ae9~ is a locally finite family of closed subsets of X , prove that U~e~ c a is a closed subset of X. D e f i n i t i o n 1.4.3. Let 12 = {U~}~eg~ and V = {Vz}/~em be open covers of a space X. We say that ~ is a refinement of II if there is a function i : ff~ --+ 91 such t h a t D e f i n i t i o n 1.4.4. A Hausdorff space X is paracompact if it is regular and if every open cover of X admits a locally finite refinement. Actually, it is redundant to require regularity, but it will shorten some argurnents. Theorem
1.4.5. Every locally compact, 2nd countable Hausdorff space X is para-
compact. Proof. By Exercise 1.1.8, X is regular. Since the theorem is evident for compact Hausdorff spaces, we assume that X is not compact. Consequently, there is a countable, increasing nest K 1 C K 2 C "'" C K r
C ...
of compact subsets of X such that K~Cint(K~+l),
1
X = 0 int(Kr). v=l
Indeed, let {Wi}i~l be a countable base of the topology of X such t h a t each W i is compact. We set K1 = W1 and, assuming inductively that K j has been defined, 1 < j _< r, we let 2 denote the least integer such that g K~C_ U W i i=1
and
set
g+r Kr+l = Uwi. 1=1
The required properties are easily checked. Let 11 = {U~}ae~ be an open cover of X. We select a refinement as follows. We can choose finitely many 1/i = U~ E 12, 1 < i < 21, that cover the compact set t(1. Extend this by { U ~ } ~ e l + 1 to an open cover of K2. Since X is Hausdorff, the compact set K1 is closed, so ~ = U~ x K 1 is open, 21+1 < i < 22, and {V~}~=I is an open cover o f / ( 2 . We have arranged that K1 does not meet V~, i > 21. Proceeding inductively, we obtain a refinement V = {V/}i~I of II with the property t h a t K r meets only finitely many elements of V, Vr > 1. Given x E X, choose r >_ 1 such t h a t x E int(K~), a neighborhood of x that meets only finitely many elements of V. [] C o r o l l a r y 1.4.6. Every manifold is paracompact. This corollary is the main reason that we required manifolds to be 2nd countable.
1.4. P A R T I T I O N S
OF
UNITY
19
D e f i n i t i o n 1.4.7. Let 1 / = {Ua}~E~ be an open cover of a space X. A partition of unity, subordinate to 1/, is a collection ~ = {s of continuous functions , ~ : X -* [0, 1] such that (1) supp(A~) C U~, V a E 9.1 (where the support supp(A~) is the closure of the subset of X on which ~ ~ 0); (2) for each x E X, there is a neighborhood Wx of x such that A~II/Vx ~ 0 for only finitely many indices a E 91; (3) the sum ~ 9 ~ ~ , well defined and continuous by the above, is tile constant function 1. Our goal is to prove t h a t open covers of manifolds admit subordinate partitions of unity. In fact, we will prove this for all paracornpact spaces. L e m m a 1.4.8. I f X is paraco,npact and ~d = {Ua}~e~ is an open cover of X , there is a locally finite refinement V = {Vo}~e~, indexed on the saute set 91, such that V ~ C U~, V a E 91. This will be called a precise refinement. Proof. Indeed, paracompact spaces are regular, so it is possible to find a refinement W = {W~}~en with j : ~ + 9.1 such that W~ C Uj(~), Vr~ E J~. Passing to a locally finite refinement of W gives a locally finite refinement V' = {V~}~Em of 1/ with --I
--/
associated map i : % --+ 91 such that V~ C Ui(~), g/3 E ~B. Remark that {V~}~Em is a locally finite family of closed subsets of X. For each a E 91, let ~B~ = i - l ( a ) , a possibly empty subset of indices in %. Remark that %~ N ~ = 0 if a r ~/. Setting V~ = U~Em.~ V~ (possibly enlpty), V a E 91, gives the desired refinement V. Indeed, the local finiteness of V follows fl'om that of V' and
by Exercise 1.4.2.
[]
Recall that a topological space X is said to be normal if, whenever A, B C X are closed, disjoint subsets, there is an open set U D A such that U/-1 B = 0. L e m m a 1.4.9. I f X is paracompact, it is normal. Proof. Let A, B C X be closed, disjoint subsets. The space X being regular, there is a family {Ua}aca of open subsets of X, covering A and such that Ua C X \ B, V a E 9.1. The space X being paracompact, the open cover of X, obtained by adjoining X \ A to {U~}~c~l , has a locally finite refinement. Thus, we lose no generality in assuming that {Ua}~E~ is a locally finite family of open sets, hence t h a t {U~}aE~ is also locally finite. Set U = U~Ea U~, an open neighborhood of A. By Exercise 1.4.2, U = U~E~ U~ and this set does not meet B. []
In the construction of partitions of unity, we will use the following well-known property of normal spaces [8, pp. 146-147]. We do not give the proof here since, ultimately, our interest is in the C ~ version that we will prove later (Corollary 3.5.5). T h e o r e m 1.4.10 (Urysohn's lemma). If X is a normal space and i r A and B are closed, disjoint subsets, then there is a continuous function f : X --+ [0,1] such that fI A = 1 and s u p p ( f ) C X \ B.
20
1. T O P O L O G I C A L M A N I F O L D S
The existence of partitions of unity can now be established. 1.4.11. If X is a paracompact space and ~[ = {Ua}ae~ is an open cover of X , then there exists a partition of unity subordinate to 1/.
Theorem
Proof. Use Lemma 1.4.8 to choose a precise refinement V = {V~}~E~ of 11 and a precise refinement W = {Wa}~E~ of V. For each a E 9/, use Theorem 1.4.10 to define a continuous function 7~ : X ~ [0, 1] such that %IW~ = 1, supp(%) C V~ C U~. The local finiteness of ~2 implies that {supp(v~)}~e~ is also locally finite and {V~}~e~ satisfies properties (1) and (2) in Definition 1.4.7. It is also clear that "7 = E ' T a
< cx~
c~EN
is continuous and nowhere O. Therefore,
is a partition of unity subordinate to 1/.
[]
C o r o l l a r y 1.4.12. Every open cover of a manifold admits a subordinate partition
of unity. Partitions of unity will be needed for Riemann integration and Riemannian geometry on (smooth) manifolds. For these and similar applications we will need smooth partitions of unity, a notion that we are not yet ready to define. In the next section, we will use the existence of continuous partitions of unity to prove a topological imbedding theorem for manifolds. E x e r c i s e 1.4.13. Let M be a manifold, U _c M an open subset, K C U a set that is closed in M, and let f : U --~ R be continuous. Prove that the restriction fI K extends to a continuous function f : M ~ IlL E x e r c i s e 1.4.14. Let K C S n be a closed subset, U D K an open neighborhood of K, v a vector field defined on U. Prove that v l K extends to a vector field on all of S ~.
1.5. Imbeddings and Immersions We will prove that compact manifolds can always be imbedded in Euclidean spaces of suitably large dimensions.
Definition 1.5.1. Let N and M be topological manifolds of respective dimensions n < m. A topological imbedding of N in M is a continuous map i : N --~ M that carries N homeomorphically onto its image i(N).
Definition 1.5.2. If N and M are as above, a topological immersion of N into M is a continuous map i : N ~ M such that, for each x E N, there is an open neighborhood W of x in N such that i i W : W --~ M is a topological imbedding.
1.5. I M B E D D I N G S
AND IMMERSIONS
21
F i g u r e 1.5.1. The topologist's sine curve For example, Figure 1.3.2 depicts an immersion of the Klein bottle K 2 into IRa. Every i m b e d d i n g is also a n immersion, b u t even one-to-one immersions can fail to be imbeddings. The immersion of IR in IR2, pictured in Figure 1.5.1, is one-to-one, b u t is not a homeonlorphism onto its image. D e f i n i t i o n 1.5.3. If M is a manifold and X c M is a subspace, we say t h a t X is a submanifold if there is a manifold N and an imbedding i : N ~-~ M such t h a t X = i(N). D e f i n i t i o n 1.5.4. The image of a one-to-one immersion i : N --+ M is called an immersed submanifold of M. Some authors use the term "submanifold" to include immersed submanifolds. From the point of view of a topologist, this seems dangerously misleading. E x e r c i s e 1.5.5. Define f : S 2 -~ IR4 by the formula f ( x , y, z) = (yz, x z , x y , x 2 + 2y 2 + 3z2).
Prove t h a t f passes to a well-defined, topological imbedding f : p2 ~_~ ii{4. (It is known t h a t p2 c a n n o t be imbedded in IRa.) E x e r c i s e 1.5.6. Let g : $2 -~ Ra be defined by
F i n d six points P l , . . . , P G C P2 such that 9 :P2 \ {Pl,...,P6} ~IR3 is a topological immersion. T h e m a p p i n g ~ of p2 itself into It~3 is known as Steiner's surface. It is simply the imbedding f into II{4 followed by projection onto a threedimensional subspace of It{4. Prove that 9 does not restrict to an i m b e d d i n g of any neighborhood of pi, 1 < i < 6. (It is known t h a t p o f c a n n o t be an immersion for any linear surjection p : 11{4 ~ IRa.)
22
1. T O P O L O G I C A L
MANIFOLDS
1.5.7. If M is a compact n-manifold, then there is an integer k > n and an imbedding i : M ~-+ IRk.
Theorem
Proof. Since M is compact, there is a finite open cover II = {Uj}}'=I of M and a collection of h o m e o m o r p h i s m s ~oj : Uj --* Wj C_ ]Rn, 1 < j < r. Let ~ = {AJ}5=l be a p a r t i t i o n of unity subordinate to ~d. We will take k = r(n + 1) and c o n s t r u c t an i m b e d d i n g i : M ~ IRk. Define i:M---+iRn• ... x ]~n XIRX ... X IR r fact . . . . .
f•tors
by
i(~) = ( ~ ( ~ ) ~ 1 ( ~ ) , . . . , ~(x)~,.(x), ~1(~),..., ~,.(x)). Here we m a k e the convention t h a t 0.p3 (x) = () 9 IR~, even when ~ j (x) is undefined. Since supp(Aj) C Uj and d o m ( ~ j ) = Uj, the expression )~j(x)~oj(x) is identically (~ n e a r the set-theoretic b o u n d a r y of Uj and on all of M \ Uj. This implies t h a t t h e m a p i : M --+ IRa is continuous. Since M is c o m p a c t and i(M) is Hausdorff, we only need to prove t h a t i is one-to-one in order to prove t h a t i is a h o m e o m o r p h i s m onto its image. Let x , y 9 M and suppose t h a t i(x) = i(y). Since ~ is a p a r t i t i o n of unity, there is a value of j such t h a t h i (x) ~ 0. B u t the (nr + j ) t h coordinates of i(x) and i(y) are Ay(x) = Aj(y), so x , y 9 supp(/~i) C Uj. Also, )U(x)~oj(x) = )U(y)~oj(y), so ~j (x) = ~j (y). Since ~oj : Uj ~ IRn is one-to-one, it follows t h a t x = y. [] T h e i m b e d d i n g dimension k = r(n + 1) given by this t h e o r e m for c o m p a c t nmanifolds is often much too generous. For example, Exercise 1.3.26 gives a covering of p 2 by r = 3 open sets h o m e o m o r p h i c to ]R2, so the theorem guarantees only t h a t P~ can be i m b e d d e d in ]R9. In fact, it is possible to imbed p2 into ]R4, as you showed in Exercise 1.5.5. Generally, if an n-manifold is differentiable (Definition 3.1.6), it can be proven t h a t M imbeds in ]R2~+1 ( T h e o r e m 3.7.12). This result, due to H. W h i t n e y , is best possible in the sense t h a t there exist n-manifolds t h a t cannot be i m b e d d e d in IR2n.
1.6. Manifolds with B o u n d a r y Manifolds are m o d e l e d locally on Euclidean n-space. S o m e t h i n g like the closed n-ball D n = {v E IR~ I I[v[I < 1} fails to be a manifold because a point on the b o u n d a r y 0 D n = S ~-1 does not have a neighborhood h o m e o m o r p h i c to an open subset of IR~. It does, however, have a neighborhood h o m e o m o r p h i c to an open subset of E u c l i d e a n half-space. Definition
1.6.1. T h e Euclidean half-space of dimension n is ]HIn =
x n) e R n l x 1 <_ 0}
{(xl,...,
and t h e b o u n d a r y of IHD is (~]I n :
((xl
...,
X n) 9 IRn I X I ~- 0 } .
T h e interior of H ~ is i n t ( H ~) = H ~ \ 0]HI~ = { ( x l , . . . , x n) 9 IR~ ] x 1 < 0}. R e m a r k t h a t 0IR n is canonically identified with IR~-I by suppressing t h e coordin a t e x 1 = 0 and renumbering the remaining coordinates as yi = x i+1, 1 < i < n - 1 .
1.6. M A N I F O L D S W I T H B O U N D A R Y
23
D e f i n i t i o n 1.6.2. A topological space X is a n n-manifold with b o u n d a r y if (1) for each x E X, there is an open neighborhood U~ of x in X, an open subset W~ C N n, a n d a homeomorphism ~ : U~ ~ W~; (2) X is Hausdorff; (3) X is 2nd countable. D e f i n i t i o n 1.6.3. Let X be a manifold with boundary. We say that x C X is a b o u n d a r y point if a suitable choice of ~ : Ux + W , , as above, carries x to a poiut ~(x) E c0IHIn. T h e interior points x E X are those such that a suitable homeomorphisnl ~ : Ux --* W~ carries x to a point ~(x) E int(Nn). The b o u n d a r y O X is the set of all the b o u n d a r y points of X a n d the interior i n t ( X ) is the set of all the interior points of X. It is clear from the definition t h a t every point of X is either a b o u n d a r y point or an interior point. T h a t is, X = 0 X U i n t ( X ) . It is not immediately evident, however, t h a t a point c a n n o t be both an interior point and a b o u n d a r y point. E x e r c i s e 1.6.4. Use T h e o r e m 1.1.9 to prove that O X A i n t ( X ) = 0. Also prove t h a t O X is an (n - 1)-manifold and that i n t ( X ) is an n-manifold. E x a m p l e 1.6.5. T h e closed, unit n-ball D n is a manifold with b o u n d a r y OD ~ = S ~-1. The interior is the open ball B ~ = {v E R n ] Ilvll < 1}. To see this rigorously, recall t h a t the orthogonal group O(n) is the group of all n x n matrices A over IR such t h a t A T = A - I a n d t h a t the s t a n d a r d m a t r i x action of O(n) on II~~ restricts to a group of homeomorphisms of D ~ onto itself. If el E S n-1 is the column vector with first entry 1 a n d r e m a i n i n g entries 0, t h e n A e l is the first c o l u m n of the m a t r i x A E O(n). Since every unit vector v E S n-1 occurs as the first c o l u m n of some m a t r i x A E O(n), it follows t h a t v E S "-1 is carried by A -~ E O(n) to q . Thus, an a r b i t r a r y point of S ~-1 will be a b o u n d a r y point of D n if and only if el is a b o u n d a r y point. Let U C D ~ be the open neighborhood of el defined by the condition x 1 > 0. T h e n ~ : U -~ N ~, defined by
carries the u n i t vectors in U into c~II~'* and the other points of U into int(IHI"). Furthermore, ~ has continuous inverse ~ given by the formula ~(yl,y2,...,y~)=
(yl + x / 1 _
(y2)2 . . . . .
(y~)2,y2 ...,yn).
We have proven t h a t S n - 1 C_ OD n. Since the open ball B n = D n ".. S n - 1 is a n open subset of II~~, it is clear that B n C_ i n t ( D n ) . By Exercise 1.6.4, all assertions follow. E x e r c i s e 1.6.6. If O M = 0 and ON r (0, show t h a t N x M is a manifold with boundary O(N x M) = ON x M.
E x a m p l e 1.6.7. By Exercise 1.6.6, the n-torus T n is the b o u n d a r y of the solid torus D 2 x T n - l , a compact (n + 1)-manifold with boundary. E x a m p l e 1.6.8. If W1 a n d W2 are compact 3-manifolds with nonempty, connected boundaries, let Di C O W i be a closed, imbedded 2-ball (a disk), i = 1, 2. By fixing a h o m e o m o r p h i s m f : D1 --~ D2, we set up an equivalence relation ~ f on the disjoint u n i o n W1 U W2 by defining x ~ f y if a n d only if either x = y, y = f ( x ) , or
24
1. T O P O L O G I C A L M A N I F O L D S
or z = f ( y ) . The quotient space is denoted by W1 tOy W2 and can be proven to be a compact 3-manifold with boundary the connected sum OWl # OW2. This is intuitively obvious, and we will not a t t e m p t a rigorous proof. It follows that, if M and N are compact, connected 2-manifolds that bound suitable compact 3manifolds, then the connected sum M # N is also the boundary of some compact 3-manifold, By Examples 1.6.5 and 1.6.7, S 2 and T 2 are such boundaries, hence the classification of compact surfaces (Theorem 1.3.5) implies that every compact, connected, orientable surface bounds a compact 3-manifold. W h a t about the compact, connected, nonorientable surfaces? It is known that the Klein bottle is such a b o u n d a r y K 2 = OW. Intuitively, we construct W from the solid cylinder D 2 x [0, 1] by gluing D 2 x {1} to D 2 x {0} by an orientation-reversing flip about a diameter. Since p 2 # p2 = K2, it follows that the connected sum of an even number of projective planes is the boundary of a compact 3-manifold. By Theorem 1.3.5, every compact, connected, nonorientable surface is a connected sum of finitely many copies of p2. Thus, the only compact, connected surfaces that might fail to bound a compact 3-manifold are connected sums of an odd number of projective planes. In fact, these do fail to bound. We will not prove this but, in Exercise 1.6.12, you will reduce the assertion to the special case that p2 does not bound. 1 . 6 . A . C o b o r d i s m * . Example 1.6.8 suggests an interesting topological problem. Given a compact n-manifold (connected or not), is it the boundary of some compact (n + 1)-manifold? A closely related problem is that of classifying compact n-manifolds up to cobordism, an equivalence relation t h a t was first defined by H. Poincar6 [35, Section 5]. D e f i n i t i o n 1.6.9. Let M and N be compact n-manifolds with empty boundary. We say t h a t M and N are cobordant, and write M ~ o N, if there is a compact (n + 1)-manifold W such that OW is homeomorphic to the disjoint union M U N. The cobordism class of M is denoted by [M]o. The set of cobordism classes of compact n-manifolds is denoted by 91n. The fact that cobordism is an equivalence relation is elementary (Exercise 1.6.10) as is the fact that homeomorphic manifolds are cobordant. If we make the standard convention t h a t the empty set is a manifold of every dimension, the cobordism class [@]o E 91n is precisely the set of compact n-manifolds that bound. In Exercise 1.6.11, you will show that disjoint union well defines an abelian group structure on 9In under which every element is its own inverse. It is a nontrivial fact t h a t this group is finitely generated, hence 91n ~ Z2 O Z2 | "'" O Z2. It is rather obvious that 910 = Z2, the nontrivial element being [point]o. We are about to see that, up to homeomorphism, the only compact, connected, boundaryless 1manifold, is S 1 = OD2, so 911 = 0. In Exercise 1.6.12, you will show that 912 = Z2, generated by [P2]o. E x e r c i s e 1.6.10. Prove that cobordism is an equivalence relation on the set of all compact n-manifolds (ignore Zermelo Frankel set-theoretic scruples about the meaning of "set of all compact n-manifolds"). Prove that homeomorphic manifolds are cobordant. E x e r c i s e 1.6.11. Show that the operation [M]o+[N]o = [MLIN]o is a well-defined binary operation on 91~. Prove that this operation makes 91n into an abelian group with 0 element the class [0]o and -[Nlo = IN]o, g [N] E 91~.
1.6. M A N I F O L D S
WITH
BOUNDARY
25
E x e r c i s e 1.6.12. Using Example 1.6.8 and assuming t h a t p2 is not a boundary, prove t h a t 912 = Z2, the nontrivial element being [P2]o. If M and N are compact, connected surfaces, prove t h a t [M]o + [NJo = [iF[# N]a. 1 . 6 . B . C l a s s i f i c a t i o n o f 1 - m a n i f o l d s * . We sketch the classification of compact l-manifolds, possibly with boundary. It is enough to classify the connected ones. T h e result is "intuitively evident" and will be needed only in the proof of L e m m a 3.9.4, so this subsection can be omitted in a first reading. T h e o r e m 1.6.13. I f N is a compact, cormected 1-manifold, then N is homeomorphic either to S 1 or to [0, 1]. The following exercise is a critical step in tile proof of this theorem. E x e r c i s e 1.6.14. Let N be as in Theorem 1.6.13. We say t h a t V C_ N is an open interval in the 1-manifold N if it is an open subset that is homeomorphic to an open interval in 1R. You are to prove that there is a maximal open interval in N, proceeding as follows. (1) Let ~ be tile family of open intervals in N, partially ordered by inclusion. Let V = { V ~ } a ~ be an infinite, linearly ordered subset of 1. Prove t h a t there is a sequential nest
v~cV,~,c...cv,~c... in V such that, for each a C 9,1, there is a n integer k > 0 with V~ C Vc~k. (Hint: N is 2nd countable.) (2) For { V.~k}~=l as in part (1), prove t h a t it is possible to choose the homeomorphisms hk of V~k to open intervals in N so that h k + ~ l V ~ = hk, Vlc > 1. oc Show t h a t these assemble to define a homeomorphism h of ~ k = l Vak onto an open interval in ]R. (3) Using the above, show thai; the partially ordered set J is inductive, hence contains a m a x i m a l element (Zorn's lemma). Proof of Theorem 1.6.13. Let U C N be a maximal open interval and fix a homeo m o r p h i s m f : (0, 1) --* U. Here we use the well-known fact t h a t any two open intervals in ]R are homeomorphic. Since N is a compact Hausdorff space, there is a strictly increasing subsequence {a~:}~=l C_ (0, 1) such t h a t ak T 1 and lira f ( a k ) = x+ C N \ U
/c~oo
exists. It follows easily that, for every monotonic sequence xk T 1 in (0, 1), lira f ( x k ) = x+. Similarly, there is a unique x_ C N \ U with lira f(Yk) = x _
k~oc
whenever yk ~ 0. Therefore, U = U U {x+, x_ } is the closure of U in N, a compact, connected subset. Define 9 : [0, 1] --* U to be the extension of f by g ( 0 ) = z _ a n d .q(1) = x + ,
clearly a continuous map. We consider the cases x+ = x_ and x+ r z _ . If x+ = z _ , 9 induces a one-to-one, continuous m a p ~ : [0, 1]/{0, 1} --~ U.
26
1. T O P O L O G I C A L
MANIFOLDS
Since S ~ ~ [0, 1]/{0, 1} is compact and N is Hausdorff, this defines an imbedding S 1 ~ N of a compact, connected, boundaryless 1-manifold into a connected 1-manifold. By an application of Theorem 1.1.9, the image of this imbedding is an open subset of N. Being compact, this subset is also closed in N and, being nonempty, it is all of N. This proves that N is homeomorphic to S 1. If x+ r x_, it follows that g : [0, 1] -* N is one-to-one, hence is a topological imbedding. If x+ c ON, let W C N be an open neighborhood of x+ and find 0 < 5 < 1 such that 9(1 - 5, 1] = J C_ W. We can assume that there is a homeomorphism h : W --+ ( - 0 % 0] such that h(x+) = 0, so h(J) is a connected subset of ( - o o , 0] containing 0 and having more than one point. Using the standard fact t h a t the only connected subsets of IR are the intervals, we conclude t h a t h(J) is a nondegenerate interval. By an application of Theorem 1.1.9, one sees t h a t h(J) has the form ( - e , 0] for suitable e > 0, from which it follows that J is open in the topology of N. By Theorem 1.1.9, g(0, 1) is also open in N, so g(0, 1] is open. Similarly, if x_ c ON, g[0, 1) is an open subset of N. Thus, if x+ and x_ both belong to ON, the image of g is open and, being compact, is also closed in N. By connectivity, this image is all of N, proving t h a t N is homeomorphic to [0, 1]. Suppose, therefore, that x+ c int(N) and deduce a contradiction. The same argument will show that x_ r int(N). Let h : V --+ IR be a homeomorphism of an open neighborhood V of x+ in N onto an open subset of 1R and let J denote the connected component of 9(0, 1] a V containing x+. Since x+ is the limit of a sequence in g(0, 1), J does not degenerate to a single point, hence h(J) is a halfopen interval with h(x+) as its endpoint. Rechoosing h, if necessary, assume t h a t h(J) = ( - 1 , @ For a small value of e > 0, there is an open set W C V such t h a t h(W) = ( - 1 , e) and J = g(0,1] n W . It follows easily that W Ug(0,1) = W U U is homeomorphic to an open interval in R, contradicting the maximality of U. [] The following innocuous corollary will be quite important for our treatment of rood 2 degree theory in Section 3.9. C o r o l l a r y 1.6.15. Every compact, 1-dimensional manifold A~ has OM equal to a finite set of points with an even number of elements. 1.7. C o v e r i n g S p a c e s a n d t h e F u n d a m e n t a l
Group
Covering spaces play a fundamental role, not only in manifold theory, b u t throughout topology. In this section, there will be no reason to restrict our attention to manifolds, everything being true for a quite large class of topological spaces. Accordingly, we fix only the following hypothesis for the entire section. Hypothesis.
All spaces are locally path-connected.
Note t h a t local path-connectedness implies that connected spaces are pathconnected. Note also that we do not require the Hausdorff property. There are actually useful applications of covering space theory to non-Hausdorff 1-manifolds. 1.7.A. The basics of covering spaces. D e f i n i t i o n 1.7.1. Let p : Y --+ X be a continuous map. An open, connected subspace U C_ X is said to be evenly covered by p, if each connected component of p-1 (U) is carried homeomorphically by p onto U.
1.7. C O V E R I N G
SPACES AND THE FUNDAMENTAL
GROUP
27
D e f i n i t i o n 1.7.2. A continuous m a p p : Y --~ X is a covering map if X is connected a n d each point x C X has a connected neighborhood t h a t is evenly covered by p. T h e triple (Y, p, X ) is called a covering space of X. In practice, one usually abuses this terminology, referring to Y itself as the covering space. Note t h a t we do not require Y to be connected. However, we will be mostly interested in connected covering spaces. Of considerable i m p o r t a n c e are automorphisms of covering spaces, defined precisely as follows. D e f i n i t i o n 1.7.3. Let p : Y -+ X be a covering map. A covering transformation, also k n o w n as a deck transformation or an automorphism, is a h o m e o m o r p h i s m h : Y -+ Y such t h a t p o h = p. T h a t is, the following diagram commutes: Y
h
~ Y
X L e m m a 1.7.4. The set F of covering transformations associated to a covering map p : Y ---+X forms a group under composition, called the covering group.
Proof. Indeed, if hi and h2 are covering transformations, p o ( h i o h 2 ) = (p o h i ) o h2 = p o h~ = p
Furthermore,
poh =pop=
( p o h ) o h -1 = p o h -1.
Finally, it is clear t h a t idv is a covering transformation, so F is a group. Example
[]
1.7.5. T h e m a p p : IR --+ S 1, defined by p(t) = e 27tit,
is a covering map. Indeed, if z0 = e 27tit~ C S 1, p carries the compact interval [to - 1/4, t0 + 1/4] one-to-one, hence homeomorphieally, onto a compact arc in S 1 c o n t a i n i n g z0 in its interior U. T h e n p - l ( U ) is the disjoint union of open intervals (to + n - 1/4, t0 + n + 1/4), as n ranges over the set Z of all integers. Evidently, each of these intervals is a connected c o m p o n e n t of p-1 (U) and is carried by p homeomorphically onto U. Finally, p(t) = p(s) if and only if s = t + m, for some m E Z. Thus, h : IR --~ II{ is a covering t r a n s f o r m a t i o n if and only if h(t) = t + m t where rat E 2~, - o o < t < oo. By continuity, mt depends continuously on t. B u t Z is a discrete space a n d R is connected, so ra t = ra is constant. The group of covering transformations is the group of translations by integers, hence is canonically isomorphic to the additive group Z. E x e r c i s e 1.7.6. Let G be a connected, locally path-connected, topological group. Let H C G be a closed, discrete subgroup. (Recall that a subspace is discrete if, in the relative topology, each of its points is open.) Prove t h a t the coset projection
p: G ~ G/H is a covering space (where G / H has the quotient topology). Show that the group F of covering t r a n s f o r m a t i o n s consists of right translations
g E G~-~ gh
28
1. T O P O L O G I C A L M A N I F O L D S
by elements h E H. More precisely, prove that
~o : h E H--+ qOh E F, ~h(g) =
gh -1,
defines an isomorphism of tile group H to the group P. (The need for h -1 rather than h is due to the possible noncommutativity of these groups. W i t h this definition, one has that ~ghlh2 ~hl o ~9h2. For commutative groups, left and right translation are equivalent and the inverse could be omitted.) :
Note that Example 1.7.5 is a special case of Exercise 1.7.6. More generally, the projection p : R ~-~
Rn/Z ~ = T n
is a covering map with covering group isomorphic to the integer lattice Z n. E x a m p l e 1.7.7. The quotient map p : S ~ -~ P~, defined as in Exercise 1.3.26, is a covering map. The group of covering transformations is generated by the antipodal interchange map, hence is Z2. Definition
1.7.8.
If
y,
X'
f
f
~y
~X
is a commutative diagram of continuous maps, where p~ and p are covering maps, we say t h a t f i s a lift of f to the covering spaces. In the case that X ~ = X and f = i d x , such a lift is called a homomorphism of covering spaces. A homomorphism of covering spaces that is also a homeomorphism is called an isomorphism of covering spaces. Of course, an automorphism of covering spaces, as defined earlier, is an isomorphism. L e m m a 1.7.9. If f is a lift o f f , as in the preceding definition, and if the covering
space Y~ is connected, then "f is completely determined by f and by the value of f at a single point. Proof, Let f and f" be lifts of f which agree at some point y C Y'. Let x = p'(y) and note t h a t f ( x ) = p('f(y)) = p('f(y)). Denote this point by z. Let V' be an evenly covered neighborhood of x and let U be an evenly covered neighborhood of z such that f ( U ' ) cc_ U. This is possible since f is continuous. Let V ~ be the component of (p')-l(U') that contains y and V the component of p - l ( u ) that contains f ( y ) = f'(y). By the definition of "lift", the diagrams V' ~P'I U'
Y )V Ip , U
1.7. COVERING SPACES AND THE FUNDAMENTALGROUP
29
and V'
r
P'I
,V 1~
U'
~U f exist and commute. But p~ and p are one-to-one on the components V I and V, so the maps y and f must agree on the open subset V ~ C_ Yq This proves that the set of points o n which f = f is open in Y~. On the other hand, the set of points on which f and f do not agree is also open. Indeed, if f ( y ) r f ( y ) , then the component V1 of p - l ( u ) containing f ( y ) is disjoint from the component V2 containing ]'(y). Since V' is connected and y E V', we conclude that f ( V ' ) C 1/1 and "f(V') C V2. T h a t is, f a n d f'disagree at every point of the open set V'. The fact t h a t Y~ is connected and that the functions agree at some point implies that
1=_?.
[]
C o r o l l a r y 1.7.10. Let p : Y ~ X be a coverin 9 map, assume that Y is connected, and let y E Y . Then a coverin9 transformation h is uniquely determined by the point h(y). Indeed, a covering transformation is a lift of id : X --~ X, where we take Y~ = Y and p~ = p. Another important type of lift is one for which the covering p~ : Y~ ~ X ~ is the trivial covering id : X ~ ~ X ~. In this case, the continuous lift fits into a commutative triangle Y
X'
~
, X f
L e m m a 1.7.11 (Path-lifting property). Let p : Y --~ X be a coverin9 space, let x c X and y C p - l ( x ) , and let cr : [a,b] --* X be a continuous path with ~(a) = x. Then there is a unique lift ~ : [a, b] --~ Y such that ~(a) = y. Proof. For each t C [0, 1], let Ut denote an evenly covered neighborhood of or(t). Then {~r-l(Ut)}o
30
1. TOPOLOGICAL MANIFOLDS
Since each x C X has an evenly-covered neighborhood U, it is clear t h a t distinct points of the "fiber" p-1 (x) lie in distinct components of p-1 (U), so the fiber is a discrete space. Covering transformations must map p - l ( x ) one-to-one onto itself, hence Corollary 1.7.10 requires that the covering group P permute this set in such a way that only the identity element of F has a fixed point. A permutation group with this property is said to act "simply". The most useful covering spaces are those in which F must also be transitive. (This means that, for each pair of points y , z E p - l ( x ) , there is h r P such that h(y) = z.) In this case, we say t h a t F permutes the fiber "simply transitively". E x e r c i s e 1.7.12. Use the path-lifting property to prove that the group F of coyering transformations is transitive on one fiber if and only if it is transitive on every fiber. (You only need X to be connected, not Y.) D e f i n i t i o n 1.7.13. A covering space p : Y -+ X is said to be regular if the group F of covering transformations permutes each fiber p-1 (x) simply transitively. Thus, for regular coverings, F can be put in one-to-one correspondence with p - l ( x ) by selecting a "basepoint" Y0 C p-1 (x) and setting up the correspondence h @ F ~ h(yo) e p - l ( x ) .
Another important lifting property for covering spaces is the homotopy-liftin 9 property. We need a definition. D e f i n i t i o n 1.7.14. Let f0 and fz be continuous maps of a space Z into a space W. A homotopy between these maps is a continuous map
H:Zx[O,
1]~W
such t h a t fo(z) = H(z,O),
Vz e z,
fl(z)=H(z,
VzeZ.
1),
If C C Z and folC ~ f~lC, we say that H is a (relative) homotopy mod C if, in addition, H ( z , t ) = fo(z), V z C C and O < t < l. If there is a homotopy between f0 and f l , we say that f0 is homotopic to f l and write f0 ~ f l . If this is a homotopy rood C, we write f0 ~ c f l . We think of a homotopy H as a continuous deformation of the map f0 to the map f l through intervening maps A(z) : H(z,t),
0 < t < 1
E x e r c i s e 1.7.15. Prove that homotopy and homotopy mod C are equivalence relations. Also prove that, if f ~ g (respectively, f H a g), then u o f ~ u o g (respectively, u o f ~"c u o g), whenever these compositions are defined. The equivalence classes under this relation are called homotopy classes. E x e r c i s e 1.7.16. Consider a commutative triangle Y
Z ~
,X
1.7. C O V E R I N G S P A C E S A N D T H E F U N D A M E N T A L G R O U P
31
of continuous maps, where p is a covering map. If H : Z x [0, 1] --+ X is a h o m o t o p y such t h a t H I Z x {0} = f , prove t h a t there is a unique lift H : Z x [0, 1] --+ Y such t h a t H I Z x {0} = f . (Hint. H l { z } x [0, 11 is a path, for each z E Z.) This is the homotopy-lifting property for covering spaces. A particularly i m p o r t a n t case of relative homotopy will be t h a t in which the m a p s are paths ai:[a,b]-~W, i=0,1, a n d C = O[a, b] = {a, b}. T h e n the curves have the same endpoints a0(a) = Crl(a) = z, ~0(b) = ~ l ( b ) = v
and the h o m o t o p y mod {a, b} deforms the one curve to the other while keeping the e n d p o i n t s z a n d y fixed. Since this situation arises so often, we will use the n o t a t i o n or0 ""o (71 for "homotopy rood tile endpoints". In this case, the homotopy-lifting property implies the following. L e m m a 1.7.17. If p : Y -~ X is a covering map, if ~ri : [a,b] --~ X are paths with the same endpoints, i = 0, 1, and if cro ~ o or1, then lifts of these paths starting at the same point must also terminate at the same point.
Proof. Indeed, let a~(a) = w, (7~(b) = v, let H be a homotopy rood the endpoints, a n d let ~i be lifts s t a r t i n g at a point w t E p - l ( w ) , i = 0, 1. These exist by the path-lifting property. If v ~ E p - l ( v ) is tile terminaI point of ~0, we must prove t h a t it is also tile t e r m i n a l point of ~1. By the homotopy-lifting property, there is a u n i q u e lift ~r of the homotopy that agrees with ~0 along [a, b] x {0}. Being a lift, ~r must carry the interval {a} x [0, 1] into p - l ( w ) and {b} x [0, 1] into p - l ( v ) . Since these fibers are discrete and H is continuous, the images of these intervals must be the respective singletons {w'} and {v'}. Thus, H restricts to [a, b] x {1} to define a lift of (71 s t a r t i n g at w', hence equal to K1. This lift terminates at the point vq [] Usually, we will parametrize paths on the u n i t interval [0, 1]. If ~ a n d r are two such p a t h s such t h a t ~(1) = r(O), we can join them at this c o m m o n point to produce a p a t h (7 9r joining or(O) to r(1) and parametrized on [0, 1] as follows: = ( a . T)(t)
c~(2t), [ ~ ( 2 t -- 1),
1 0 < t < ~,
1
Intuitively, one runs along (7 at twice the original speed, t h e n along T at twice the original speed. Since or(l) = 7(0), the resulting p a t h is continuous. E x e r c i s e 1 . 7 . 1 8 . Suppose that or(l) = r(0), as above, and that (7' a n d r ' are similar p a t h s such t h a t cr ~ o (7' and r ~ o r ' . Prove t h a t a . r ~ o (7' 9r ' . 1.7.B. Simple connectivity. X , these being paths
Of particular interest are loops at a point z0 E o : [0, 1] -~ X
such t h a t ~(0) = a(1) - z0. We will also let z0 denote the constant loop ~ = Zo. D e f i n i t i o n 1 . 7 . 1 9 . A topological space X is simply connected if the following conditions hold: (1) X is path-connected;
32
i. TOPOLOGICAL
MANIFOLDS
(2) there is a point x0 E X such that every loop a at x0 satisfies a ~ o x0. A (not necessarily connected) space X is locally simply connected if each neighborhood of each point x E X contains a simply connected open neighborhood of X.
E x a m p l e 1.7.20. The open unit ball
B '~ = {x e ~n I Ilxll < 1} and the closed unit disk
D ~ = {x c R n l l l x l l < 1} are both simply connected. Indeed, every convex subset C c_C_II~'~ is simply connected. To see this, choose an arbitrary point x0 C C and consider a loop a based at x0. For each fixed t e [0, 1], consider the line segment {sxo + (1 - s)a(t)}, 0 < s < 1. By convexity, this segment lies in C. Also, if a(t) = xo, the segment is the constant p a t h x0. Thus, we define a continuous homotopy H : [0, 1] x [0, 1] -~ C
by •(t, s) = s~0 + (1 - s)~(t), noting that this is a homotopy a No x0. Since manifolds are locally homeomorphic to B ~, manifolds are locally simply connected. It will be convenient to have various equivalent characterizations of simple connectivity. L e m m a 1.7.21. The following properties o f a path-connected space X are equivalent:
(1) E v e r y c o n t i n u o u s m a p f : S 1 ~ X is h o m o t o p i c to a c o n s t a n t map. (2) E v e r y c o n t i n u o u s m a p f : S 1 ~ X extends to a c o n t i n u o u s m a p F : D 2 X.
(3) I f or and ~- are paths in X havin9 the s a m e initial point and the s a m e t e r m i n a l point, t h e n cr ~ o r .
(4) I f a is a loop at an arbitrary point x E X , then a ~ o x. (5) X is s i m p l y connected. Proof. By Example 1.3.22, it is immediate that (1) ~
(2). For the implication (2) ~ (3), one needs the fact that [0, 1] x [0, 1] is homeomorphie to D 2. All closed rectangular domains in N 2 are homeomorphic via suitable affine transformations. Thus, we can replace the rectangle [0, 1] x [0, l] with [-1, 1] x [ - 1 , 1]. The unit disk is inscribed in this rectangle and a suitable radial transformation gives a homeomorphism. The elementary details are left to the reader. Thus, if a and T are paths parametrized on [0, 1], both joining x0 to x l , we obtain a map on the b o u n d a r y of [0, 1] x [0, 1] by using cr along the b o t t o m edge, r along the top, the constant m a p x0 along the left edge and Xl along the right. By (2), this map extends over [0, 1] x [0, 1] and this extension is the desired homotopy a ~ o T. Evidently, (4) is just a special case of (3), so (3) =~ (4). Likewise, (4) =~ (5) is immediate. For the implication (5) ~ (1), we need the path-connectivity of X. We assume t h a t there is a point x0 C X such that every loop at x0 is homotopic rood the boundary to the constant loop x0. Let f : S 1 ~ X be continuous. In Exercise 1.7.22, you will be asked to show that path-connectivity implies that f ~ f0, where f0(1) = x0. By example 1.3.21,this map can be viewed as a loop at x0, hence, by (5), it is homotopic mod the boundary to x0. This homotopy is constantly equal to x0 along
1.7. COVERING SPACES AND THE FUNDAMENTAL GROUP
33
F i g u r e 1.7.1. Projection of the square onto the union of three boundary segments
the two vertical edges and the top edge of the boundary of [0, 1] x [0, 1]. Identifying (0, t) with (1, t), 0 < t < 1, we view the homotopy as a map on S 1 X [0, 1]. Then, appealing to Example 1.3.22, we view this as a map of the disk, obtaining the desired extension of f. [] E x e r c i s e 1.7.22. Given a continuous map f : S 1 --~ X and assuming that X is path-connected, let x0 E X and construct a homotopy f ~ f0 such that f0(1) = x0. (Hint. Think about the projection of [0, 1] x [0, 1] onto part of its boundary indicated in Figure 1.7.1.) 1.7.C. T h e u n i v e r s a l c o v e r i n g . Reasonably nice spaces admit a particularly useful type of covering space, defined by a "universal" property. D e f i n i t i o n 1.7.23. A covering map 7r : X -~ X, is said to be universal if X is connected and, for any covering map p : Y -~ X such that Y is connected, there is a continuous map ~ : J~ -~ Y that is a lift of 7r. The covering space X is called a universal covering space of X. E x e r c i s e 1.7.24. Prove that ~ is a covering map, hence that every connected covering space is intermediate between X and the universal covering space X. At this point it will be useful to start working in the category of pointed spaces. D e f i n i t i o n 1.7.25. A pointed space is a pair (Z, z0), where z0 E Z is a fixed choice of basepoint and Z is path-connected. A map
f : (z, z0) -~ (w, ~0) is a continuous map of Z into W such that f(zo) = wo. These are called basepointpreserving maps.
34
1. T O P O L O G I C A L M A N I F O L D S
Remark. The set of all pointed spaces (ignore Zermelo-Frankel scruples about sets that are "too large"), together with all basepoint-preserving maps, is called the category of pointed spaces. The pointed spaces themselves are the objects of the category, the basepoint-preserving maps being called the morphisms of the category. Remark that composition of morphisms, whenever defined, is again a morphism and, for each object (Z, z0), the identity map
id(z,z0) : (Z, z0) --* (Z, z0) is a morphism. (We will often omit the subscript and simply write id.) These are the defining properties of the term "category". Other examples of categories are the set of all groups (the objects) and group homomorphisms (the morphisms), usually called the category of groups, and the set of all vector spaces over a given field K (the objects) and linear maps (the morphisms), called the category of K-vector spaces. We will continue to see examples of categories throughout this book. One can think of a category as a whole mathematical discipline, such as topology, group theory, linear algebra, differential geometry, etc. Typically, one considers "functors" between categories, these being maps that take the objects of one category to the objects of the other and take the morphisms of the first to morphisms of the second, preserving the category structure. Thus, if F : A --* ~ is a functor, we must have (1) F(idA) = idF(A), for each object A E Jt; (2) F ( f o g) = f ( f ) o F(g), whenever f and g are morphisms of A for which f o g is defined. Functors are used to transform problems in one category to analogous problems in another. Shortly, we will introduce a functor, the fundamental group, from the category of pointed spaces to the category of groups. A proof of the Brouwer fixed point theorem will be given to illustrate the usefulness of this functor. Actually, what we have just defined is called a covariant functor. Later in the book, we will have occasion to consider contravariant functors. These reverse all morphism arrows and, consequently, satisfy F ( f o g) = F(9) o F ( f ) .
In the category of pointed spaces, notions such as "covering map" and "covering space" have their obvious meanings. Note, however, that covering transformations are not basepoint-preserving. In this category, the definition of "universal covering map" 7r : (X, .~0) --~ (X, x0) requires that, for any covering map p : (Y, Y0) --* (X, x0), there be a commutative diagram
(x, ~0)
. (z, y0)
(x, x0) According to Lemma 1.7.9, the lift ~ is uniquely determined by the requirement, built into the language of pointed spaces, that ~(x0) = Y0.
1.7. COVERING
SPACES
AND
THE
FUNDAMENTAL
GROUP
35
L e m m a 1.7.26. If Te : (-~,Xo) ~ (X, xo) and "~ : (X,'xo) --+ (X, xo) are both universal covering maps, then there is a unique homeomorphism ~o making the diagram
(2:, ~o ) ..~..~
, (2,,~o) (X, xo)
commutative. Proof. Indeed, the existence of a continuous map ~o, making the diagram commute, is guaranteed by the fact t h a t ~ is a covering map and 7r is a universal covering map. As remarked above, uniqueness is by Lemma 1.7.9. Interchanging the roles of ~ and 7r, we get a unique commutative triangle
(x,~o)~..~
r
,
(x, ~0)
(X,~o) and the composition ~ o r is the unique continuous map making the triangle
(2,.~0) ~ ~ 1 6 2
, (2,~0)
(x,~0) commutative. But the identity map id(2,~o) would also make the triangle commutative, so qo o ~/a = id(2,~o) . Similarly, ~/J o ~ is the identity map on ()(,xo), so ~b and 9) are mutually inverse homeomorphisms. [] By this lemma, we can speak of "the" universal covering space of (X, xo), provided t h a t it exists. C o r o l l a r y 1.7.27. If it exists, the universal covering space is a regular covering.
Proof. Indeed, if~0 and Yo lie in the fiber ~ - l ( x o ) , we can take )~ = )(, ~ = 7r and x0 = Y0 in Lemma 1.7.26. The map 4o, interpreted as a homeonlorphism of )( to itself, is obviously a covering transformation taking x0 to Y0. [] Lemma 1.7.26 is the uniqueness lemma for universal covering spaces. Existence is harder and requires that we further restrict the space X. Hypothesis.
From now on, all spaces will be locally simply connected.
By Example 1.7.20, this hypothesis includes all manifolds. In fact, the hypothesis can be weakened to require only that spaces be "semi-locally simply connected" (see [26, Theorem 10.2 on page 175]). This means that the homotopy a N0 x0 of loops based at xo in a neighborhood of x0 is not itself required to stay in the neighborhood.
36
1. TOPOLOGICAL MANIFOLDS
E x e r c i s e 1.7.28. If p : (II, Yo) --* (X, xo) is a covering, prove that any simply connected, open subset of X is evenly covered. Conclude that the composition of covering maps is a covering map and that ~ in Definition 1.7.23 is also the universal covering map. There is a canonical construction of the universal covering space
~: (z,.~0) -~ (x, x0) in which the points of )( are the H a homotopy classes [a] of paths a in X with a(0) = x0. The basepoint 70 is taken to be Ix0], the class of the constant loop, and the projection map is defined by ~[a] = a(1). The details are a bit involved and could be distracting, so we relegate them to Appendix A, where the following important theorem will be proven. T h e o r e m 1.7.29. If X is path-connected and locally simply connected, it admits a universal covering space. Furthermore, a covering space is universal if and only if it is simply connected.
Remark. By this theorem, together with Lemma 1.7.26, we are now fully justified in speaking of <'the" universal covering space of (X, x0) in the category of (pathconnected and locally simply connected) pointed spaces. It is a common abuse of language to use this phraseology even if no choice of basepoints has been specified, but a little caution is recommended. L e m m a 1.7.30. If p : (Y, yo) ---* (X, xo) is the universal cover of X , then i d : (Y, Yo) ---* (Y, Y0) is the universal cover of Y
Proof. Indeed, by Exercise 1.7.28, commutativity of the diagram
(Y, y0)
id / (Y, Y0) ~
P
P
(X, x0) []
implies the assertion.
1.7.D. T h e f u n d a m e n t a l f i r o u p . The group F of covering transformations of the universal covering space X of X is isomorphic to a group ~rl(X, x0), called the fundamental group of (X, x0), that plays an important role in algebraic topology. Although they are abstractly isomorphic, these groups are subtly different. Since covering transformations, other than the identity, fix no point, they do not really coexist comfortably with the category of pointed spaces. On the other hand, the basepoint x0 plays an essential role in the study of the fundamental group. The choice of this basepoint is also essential for specifying an isomorphism F =~ Irl(X, x0). Let T(X, x0) denote the space of paths a : [0, 1] ~ X issuing from the basepoint xo and let ~(X, xo) C [P(X, xo) denote the subset of loops in X based at xo. These are the paths issuing from xo at time t = 0 and returning to xo at time t = 1. The homotopy relation ~o restricts to an equivalence relation on ~t(X, xo) and we set 71-1(X, x o ) = ~ ( X , xo)/,-~o .
1.7. COVERING SPACES AND THE FUNDAMENTAL GROUP
37
If o , r E f~(X, xo), it should be clear that 0 . r E f~(X, xo). By Exercise 1.7.18, this passes to a well-defined "multiplication" on 71"I ( X , X0): [~][7] ; [~. 7]. This is our candidate for the group operation. Our candidate for the identity element is Ix0], and our candidate for the inverse of [a] E 7rl(X, xo) will be the homotopy class of the loop a - 1 obtained by traversing a backwards. Formally, o - l ( t ) = o(1 - t),
0 < t < 1.
T h e o r e m 1.7.31. With the above operations, 7rl(X, z0) is a group that is isomor-
phic to the group F of covering transformations of the universal covering space 7r : X ~ X . This isomorphism depends only on the choices of basepoints zo E X and x0 E 7r-l(x0). Indeed, let 7r : ()(,20) + (X, xo) be the universal covering space, and let F be the group of covering transformations of )(. Given a E f~(X, xo), let be the unique lift of cr to a path in )( starting at 2o. By Lemma 1.7.17, ~(1) depends only on [a] E 7rl ( X , 2;0). Furthermore, if the lifts ~ and ~ of 7, a E Ft(X, x0) have ~(1) = ~(1), the simple connectivity of Jf implies that ~ ~o ~. Thus, by Exercise 1.7.15,
[0] -- [~ o ~] = [~ o ~ = [7] and we have set up a one-to-one correspondence between the set 71"1 ( X , 2;0) and the set 7r-l(z0). Given [a]E 7rl(X, xo), let po E F be the unique covering transformation that satisfies p~(20) = ~(1). Thus, [a] ~ ~ is a one-to-one correspondence between 7rl (X, x0) and F, canonically determined by the choice of basepoints. Since F is a group, we obtain a group structure on 7rl (X, x0). That this is the "correct" group structure, as described above, is the content of the next lemma. L e m m a 1.7.32. The correspondence cr ~ ~ (1) ~z.r = ~G o p~;
has the following propeT~ties:
(3) ~z0 = id.
Pro@ Let o, ~- E ft(X, Xo) and, as usual, denote the respective lifts to paths issuing from 2o by g and ~. As ~a o ~ is the unique lift of ~- to a path issuing from g(1), it is evident that the lift of o . 7 issuing from 2o is J - ~ = ~" (9)~ o ~). Thus, ~P~.,(2o) : 6 - - - ~ ( i ) : qo~,(~(1)) : ~p~(cp~-(2o)).
Since a covering transformation is uniquely determined by its value at one point, po.~ = p~ o ~ , , proving (1). For (2), remark that the unique lift of a - 1 to a path starting at ~(1) will be a path ending at 2o. Applying the above argument to 7 = a - 1 , we see that ~o.~-~(20) = 2o, implying that ~ o po ~ = id, as desired. Finally, it is evident that ~ o ( 2 0 ) = 20, so q~o = id and (3) is proven. [] The proof of Theorem 1.7.31 is complete. Remark that, if we construct J( as indicated prior to the statement of Theorem 1.7.29, there is a canonical choice of the basepoint 20, so the identification F -~ 7rl(X, x0) can be said to depend only on the choice of basepoint x0. There are situations, however, in which one prefers not to think of X in this way and in which the choice of basepoint 20 should be free.
38
1. T O P O L O G I C A L
MANIFOLDS
E x a m p l e 1.7.33. By Example 1.7.5 and the fact that simply connected covering spaces are universal (Theorem 1.7.29), the universal covering space of the circle is p : IR --~ S 1 (N, being convex, is simply connected) and the group of covering transformations is isomorphic to the infinite cyclic group Z. Thus, 7r1(S 1, x0) ~ Z, where x0 = p(Z). Remark that a loop on S 1, based at x0 and generating 71"1 ( S 1 , x 0 ) , is given by the restriction a = pl[0, 1]. E x e r c i s e 1.7.34. Verify directly, without using covering spaces, t h a t the operation [a][T] = [ a . T] makes 7rl(X, xo) into a group, the inverses and identity element being as described above. This requires constructing a number of homotopies. For example, you need to construct a homotopy
~ ( ~ ~ ) ~ 0 ( ~ ~).~. E x e r c i s e 1.7.35. If X is (path-) connected and x0, xl E X, show t h a t a choice of path a from x0 to Xl can be used to define a group homomorphism 0~ : 7rl (X, x l ) --* 7rl(X, xo) t h a t depends only on [a]. Show t h a t 0o-1 is a two-sided inverse to 0~, hence t h a t the two fundamental groups are isomorphic. Since this isomorphism generally depends on the choice of [a], it is not canonical. At any rate, triviality of the fundamental group at one basepoint implies its triviality at all basepoints. By the definition of the fundamental group, we obtain the following addition to the list of equivalent properties in Lemma 1.7.21 L e m m a 1.7.36. If X is path-connected, it is simply connected if and only if Trl (X, xo)
is trivial, for some (hence every) basepoint xo E X . Consider a continuous, basepoint-preserving map f : (X, x0) ~ (Y, Y0) between pointed spaces. If a E f~(X, x0), then, since f is basepoint-preserving, foc~ E ft(Y, y0). I f a ~ o T, then f o a "~o f o r by Exercise 1.7.15. This enables us to define an induced map
f, : 7cl(X, xo) --* 7rl(Y, y0)
f,([~]) = If o ~]. We use the notation f ~xo g for homotopy rood the singleton {x0}. This is a homotopy through basepoint-preserving maps. The following is a very routine exercise, but important. E x e r c i s e 1.7.37. Prove that f . is well defined and is a group homomorphism. Show that, whenever
(x, xo) ~ (z, yo) Z (z, zo), then
( f o g ) , = f, og,, and t h a t id, = id (where we use "id" for identity maps on any suitable domain). Finally, if f ~xo g, prove t h a t f , = g,.
Remark. These properties are smnmed up by saying that the fundamental group defines a homotopy-invariant, covariant functor from the category of pointed spaces and continuous, basepoint-preserving maps to the category of groups and group homomorphisms. This makes it possible to "paint algebraic pictures" of difficult topological problems. As with all pictures, a great deal of detail is lost (for example,
1.7. COVERING SPACES AND THE FUNDAMENTAL GROUP
39
homotopic maps become indistinguishable), b u t the algebraic problem t h a t appears on this "canvas" is often more manageable. After solving this problem, one t h e n tries to interpret the solution in terms of the original topological problem. T h e following example is a good case in point. E x a m p l e 1.7.38. We indicate how to use properties in Exercise 1.7.37 to prove the Brouwer fixed point theorem. This theorem asserts t h a t every continuous m a p f : D 2 -~ D 9~ has a fixed point. One proceeds by assuming t h a t f has no fixed point a n d deriving a contradiction. For each x E D 2, we assume t h a t x ~ f ( x ) . O u t of the point f ( x ) , draw the unique ray Rx t h a t passes through x a n d let h(x) denote the point of intersection Rx N O D 2. This defines a map h : D 2 -+ OD 2 = S 1 a n d it is geometrically plausible that h is continuous. In fact, with a little care, one can write down an explicit formula for h t h a t makes the continuity evident. Fix a point x0 E oqD2 C D 2 to serve as basepoint for both of these spaces. R e m a r k that, if x E cgD2 = S 1, it is immediate from our definition that h(x) = x. In particular, h is basepoint-preserving, as is the inclusion m a p i : S 1 ~-~ D 2. It should be clear t h a t the diagram ( S 1 , XO)
i
(D 2, x0) h ( S 1 , xo)
is commutative. Then, by Exercise 1.7.37, so is the diagram i. 71-1( S 1 , XO) .~
~'
~l(D2,zo) h. ~I(Sl,xo)
By E x a m p l e 1.7.33, 71-1(S 1, 270) = 77... Also, 71"1 ( O 2, x0) = 0, since D 2 C IR2 is convex, hence simply connected. This gives the commutative diagram i, m.
h.
Z a n d this is t r a n s p a r e n t l y absurd. Prom this contradiction, we conclude that f had a fixed point. Finally, using the f u n d a m e n t a l group, we formulate a very i m p o r t a n t necessary a n d sufficient condition for the existence of lifts.
40
1. TOPOLOGICAL MANIFOLDS
T h e o r e m 1.7.39. Letp : (Y, Yo) ~ (X, Xo) be a covering map and let f : (Z, zo) --* (X, xo) be a continuous, basepoint-preserving map. Then there is a lift f : (Z, zo) --~ (]I, Yo) if and only if f. (~rl(Z, z0)) C_p.(TCl(Y , Yo)).
Proof. If the liR exists, then f.(Trl(Z, zo)) = p. o Y. (Trl(Z, z0)) C P.(~I(Y, Y0)). For the converse, we assume that f. (7q (Z, zo)) C_ p. (rl (]I, Yo)) and construct the lift using the path-lifting property. First, we fix some notation. If a : [0, 1] --* Z is an arbitrary path with a(0) = zo, we will denote by 3 the path f o a in X, starting at xo, and by Y the lift of 3 to a path in Y, starting at Y0. For each z E Z, choose a path az from zo to z. We attempt to define f(z) = az(1). If f is well defined, it will be continuous by an easy argument that can be left to the reader. Evidently, p o f = f and f(zo) = Yo, so we will have constructed the required lift. In order to prove that f is well defined, let 7z be another path from zo to z. Then 7 = az 9T71 is a loop determining [3'] C ~i(Z, z0), ~ is a loop at x0, and f.[3,] = [~]. The lift of ~ is 3, = a z ' (T'z) - 1 . If we can prove that ~ is a loop at Y0, it will follow that ~z(1) = ~z(1), proving that f i s well defined. By our hypothesis and the fact that [~] is in the image of f., we see that this class is also in the image of p.. Thus, there is a loop 3,/ in Y at Y0 such that p o 3,' No ~- By Lemma 1.7.17, it follows that y0 = 3,'(1) = ~(1) and ~ is a loop at Y0. [] The following is perhaps the most frequently used application of Theorem 1.7.39. C o r o l l a r y 1.7.40. Letp : (Y, Yo) ---" (X, x0) be a covering map and let Z be simply connected. Then every continuous, basepoint-preserving map f : (Z, zo) --~ (X, xo) has a unique lift f : (Z, zo) --~ (Y, Yo).
CHAPTER 2
T h e Local T h e o r y of S m o o t h Functions In this chapter, we treat the fundamentals of differential calculus in open subsets of Euclidean spaces. Everything will be set up so as to extend naturally to global differential calculus on smooth manifolds.
Notation. Elements of N n, when thought of as vectors, will be written as column n-tuples. When thought of as points, they will be written as rows. 2.1. Differentiability Classes Let U C_ IRn be an open subset. Let x = (x 1 , . . . , x n) denote the general (variable) point of U and let p = ( p l , . . . ,p~) be a fixed but arbitrary point of U. Let f : U -* ]R be a function and let Lp : U --~ IR be an affine (i.e., inhomogeneous linear) map n
Lp(x) = c + E
bixi
i=1
such that
Lp(p) = f(p). D e f i n i t i o n 2.1.1. If f and Lp are as above and if lira f ( x ) - Lp(x) _ O,
x~,
I I x - pjI
then Lp is called a derivative of f at p. If f admits a derivative at p, then f is said to be differentiable at p. We think of a derivative Lp as a linear approximation of f near p. By the definition, the error involved in replacing f ( x ) by Lp(x) is negligible compared to the distance of x from p, provided that this distance is sufficiently small. It follows from the definition that an affine map is a derivative of itself. The above definition of "derivative" as a linear approximation embodies the real philosophy of differential calculus. As it stands, however, this definition is a bit unsatisfying. The use of the indefinite article (a derivative) raises the issue of uniqueness, while the relationship of the notion of a derivative to the familiar operation of differentiation is also unclear. The following exercise resolves these doubts. E x e r c i s e 2.1.2. If Lp(x) = c + ~-~.~=1 bixi is a derivative of f at p, then
of
b~ = ~x~ (p), 1 < i < n. In particular, if f is differentiable at p, these partial derivatives exist and the derivative Lp is unique.
42
2. LOCAL THEORY
H a v i n g seen t h a t derivatives are given by partial derivatives, we center our a t t e n t i o n on these more familiar operators. D e f i n i t i o n 2.1.3. T h e class of continuous functions f : U -+ IR is d e n o t e d by C~ If r _> 1, the class C~(U) of functions f : U --+ IR t h a t are s m o o t h of order r is specified inductively by requiring t h a t Of/Ox i exist and belong to C ~ - I ( U ) , 1 < i < n. T h e functions t h a t are s m o o t h of order r are also called C r - s m o o t h . E x e r c i s e 2.1.4. Inductively, prove t h a t
C~
D C I ( u ) D . . . D C~-I(U) D C ( U ) D ... .
E x a m p l e s show t h a t these inclusions are all proper. Definition
2.1.5. T h e set of infinitely s m o o t h functions on U is
c
(U) = N c r ( u ) . r>O
It is not u n c o m m o n simply to call C c~ functions "differentiable" or " s m o o t h " . We will be concerned primarily with such functions and will usually refer to t h e m as " s m o o t h " . R e m a r k t h a t the coordinates in U are themselves s m o o t h functions x i : U ~ IR. Thus, q C U has coordinates xi(q), 1 < i < n, and we can write q=
(xl(q),...,xn(q)).
E x e r c i s e 2.1.6. If dim U = 1, prove t h a t the derivative Lp exists if and only if f'(p) exists. If, however, dim U = 2 and p = (0, 0), find a function f : U --* N such t h a t b o t h partial derivatives exist at every point of U, but such t h a t the derivative L(0,0) does not exist. E x e r c i s e 2.1.7. Let
D(U) = { f : U -* R I f is differentiable at x, Vx E U}. Show t h a t C~ D_ D(U) D_ CI(U). P r o d u c e examples to show t h a t b o t h of these inclusions are proper. (Hint: First do this for dim U = 1 and t h e n e x t e n d to a r b i t r a r y dimensions.) E x e r c i s e 2.1.8. Let U c_ N n be open, let f C Cr(U), where 1 < r < oo, and let g : 1R -* R be C r - s m o o t h also. Prove t h a t the composition g o f belongs to Cr(U). 2.2. T a n g e n t
Vectors
We continue to let U C_ ]Rn be a fixed but a r b i t r a r y open set. We fix p C U and describe the t a n g e n t space Tp(U) of U at p. In calculus, it is c u s t o m a r y to t r a n s l a t e a t a n g e n t vector g at p to the origin 0 E NIn, thereby identifying ~7 canonically w i t h an element of IRn. T h a t is, we set Tp(U) = ]R~. This will not do for our purposes since we are trying to set up a local calculus t h a t will make sense on manifolds where, generally, there will be no preferred coordinate system and t r a n s l a t i o n will be meaningless. Also, the custom of representing vectors as directed straight line s e g m e n t s in ]]~n will not do, since a straight line segment in one coordinate system m a y look like a curved line segment in another. Instead of these naive definitions, we will view a t a n g e n t vector as a certain type of operator on functions. This definition will have no dependence on the choice of coordinates.
2.2. TANGENT VECTORS
43
In s t a n d a r d calculus, the vector
defines a directional derivative D~ at p by the formula
D ~ ( f ) = lira f ( p + h ~ ) - f(p) ~ iOf , , h~o h = 2..., a ~ ~p), i=1
where f is an a r b i t r a r y s m o o t h function defined on a n open neighborhood of p. In the n o t a t i o n of differential operators, n
(9
P
i=1
A p p l y i n g this operator to the coordinate functions x i gives
D~(x i) = a i. So the vector ~7 is uniquely determined by its associated directional derivative. A n o t h e r way to o b t a i n this directional derivative is to consider a curve s: (-5,~)-~ U (where e, ~ > 0), w r i t t e n 8(~;) :
(X 1 ( ~ ) , . . . , x n ( t ) ) ,
such t h a t each xi(t) is of class at least C 1 a n d s(O) = p, ~(o) = ~.
T h a t is, ~i(0) = a i, for 1 < i < n. By s t a n d a r d calculus,
D~(f) = lira f ( s ( h ) ) - f(p) h~O h ' an e q u a t i o n t h a t makes sense without explicit reference to coordinates. In other words, although the directional derivative was defined as differentiation at t = 0 along a straight line curve
g(t)=p+t~,
-~
with c o n s t a n t velocity g, it could just as well have been defined via any C 1 curve out of p with initial velocity Y. While the n o t i o n of "straight line" will not have m e a n i n g on general manifolds, the notion of "C 1 curve" will. D e f i n i t i o n 2.2.1. Given p 6 U, C~176 will denote tile set of smooth, real valued functions f with d o m ( f ) an open subset of U and p C d o m ( f ) . D e f i n i t i o n 2.2.2. T h e set of all C 1 curves s : ( - 8 , e) --~ U (where the n u m b e r s 5 > 0 a n d e > 0 d e p e n d on .s) such t h a t s(0) = p is denoted by S(U,p).
44
2. LOCALTHEORY
F i g u r e 2.2.1. Some infinitesimally equivalent curves at p D e f i n i t i o n 2.2.3. If sl,s2 C S(U,p), we say that 81 and s2 are infinitesimally equivalent at p and we write sl ~p s2 if and only if
~f(sl(~)) t = 0 : df(s2(t))t=0' for all
f C C~176
It is easy to check that ~p is an equivalence relation on the set S(U, p). Following Isaac Newton, we think of each equivalence class as an "infinitely short curve", but not as a single point. In fact, we are simply lumping together all curves sharing the same position p and velocity vector ~ at time t : 0 (see Figure 2.2.1). But, while the notion of "velocity vector" will not have an obvious meaning for curves in a manifold, the definition of "infinitesimal equivalence" will be meaningful in that context, allowing us to define the velocity vector as an "infinitesimal curve". D e f i n i t i o n 2.2.4. The infinitesimal equivalence class of s in S(U,p) is denoted by (S)p and is called an infinitesimal curve at p. An infinitesimal curve at p is also called a tangent vector to U at p and the set
Tp(Y) : S(U,p)/~p of all tangent vectors at p is called the tangent space to U at p.
Remark. Once we are sufficiently familiar with these notions, we will replace the notation (S)p with the more standard i(0) and call this the velocity vector of s at time t = 0. The following is immediate by the definition of infinitesimal equivalence. L e m m a 2.2.5.
['or each (S)p E Tp(U), the operator
D(~>.: C~(U,p)
~ ]I{
2.2. T A N G E N T V E C T O R S
45
is well-defined by choosing any representative s E (S)p and setting D(s),,(f) = d f ( s ( t ) ) t = 0 '
for all f E C~(U,p). Conversely, (S}p is uniquely determined by the operator D(s),. Note t h a t D<s),, is a linear operator. That is,
D<s>,"(af + bg) = aD<s>,,(f) + bD(s>,,(g), Vf, gECoo(U,p),
Va, b E R .
Since the whole reason for introducing tangent vectors is to produce linear approximations to nonlinear problems, it will be necessary to exhibit a natural vector space structure on Tp(U). In order to carry this structure over to manifolds, we do not want it to be dependent on the coordinates of ~n. The key lemma for this follows. L e m m a 2.2.6. Let (Sl)p,(S2)p E Tp(U) and a,b E JR. Then there is a unique
infinitesimal curve {S}p such that the associated operators on Coo(U,p) satisfy D<,% =
aD<sl),+ bD<s2),.
Proof. W e are allowed to use the coordinates of ]I{n to prove this assertion. The important point is that the assertion itself is coordinate-free. It is clear, then, that s(t) = asl(t) + bs~(t) - (a + b - 1)p, defined by coordinatewise operations for all values of t sufficiently near 0, is a C 1 curve in U with s(0) = p, and that this curve represents <s}v C Tp(U) such that D(~>,, is the desired operator. By Lemma 2.2.5, {s}~ is uniquely determined by
D<~>,,.
[]
D e f i n i t i o n 2.2.7. Let { S i p , {S2}p E Tp(U) and a, b E R. Then a {Sl)p + b {.s2)p E Tp(U) is defined to be the unique infinitesimal curve {s}p given by Lemma 2.2.6. E x e r c i s e 2.2.8. Prove that the operation of linear combination, as in Definition 2.2.7, makes Tp(U) into an n-dimensional vector space over R. The zero vector is the infinitesimal curve represented by the constant p. If (s)~ E Tp(U), then - (S)p = (S-}p where s - ( t ) = s ( - t ) , defined for all sufficiently small values of t. The operator D<~), described above, does not "see" all of f C C ~176 only the behavior of f in arbitrarily small neighborhoods of p. The proper way to say this is t h a t D(~),, (f) depends only on the "germ" of f at p. There is some usefulness in formalizing this. D e f i n i t i o n 2.2.9. We say that the elements f,g E Coo(U,p) are germinally equivalent at p, and write f -p g, if there is an open neighborhood W of p in U such that W C dora(f) N dora(g) and f t W = 91W. It is clear that - p is an equivalence relation on Coo(U,p). D e f i n i t i o n 2.2.10. The germinal equivalence class [f]p of f E Coo(g,p) is called the germ of f at p. The set Coo(U,p)/=-z, of germs at p is denoted by ~Sv.
46
2. L O C A L
THEORY
D e f i n i t i o n 2.2.11. For each s E S(U,p) the operator
D(s)p : ~)p --> is defined by
D(s>,[f]p
:
~f(s(t))lt:o.
Remark. The discussion so far would have worked equally well if we had fixed an integer k k 1, replaced C~(U,p) with the set Ck(U,p) of C k functions defined in neighborhoods of p, and taken {~p : {~pktO be the germs of these functions. This remark is crucial if one wants to formulate the theory of C k manifolds. There is a purely algebraic characterization of Tp(U) that, though admittedly more formalistic than the one we have given, has its charms. This definition of the tangent space is valid only for the C 0r case (the default). First, recall that an algebra ~ over R is a vector space over ]i{, together with a bilinear map
called multiplication. If (r = r for all (, r E ~, the algebra is said to be commutative. If, for all ~, C,X E ~, (~r : ~(r the algebra is associative. If there is e ~ such that ~( = (~ : (, for all ( 6 ~, then L is called a unity. Now, define algebraic operations on germs as follows. * Scalar multiplication: t[f]p : [tf]p, Vt 9 11{,V [f]p 9 OF. . Addition: [f]p + [g]p = [f[W + g[W]p, V [f]p, [g]p 9 ~gp, where W is an open neighborhood of p in dora(f) A dora(g). 9 Multiplication: [f]p[g]p : [(f]W)(g]W)]p, V If]p, [g]p 9 r where W is again as above. L e m m a 2.2.12. The above operations are well defined and make r
a commutative
and associative algebra over ]~ with unity. The elementary proof of Lemma 2.2.12 is left to the reader. The unique unity, of course, is the germ of the constant function 1. D e f i n i t i o n 2.2.13. The evaluation map ep : {~p ~ ]~ is defined by
ep[f]p
=
f(p).
The following is immediate. L e m m a 2.2.14. The evaluation map ep : ~)p --+ ]~ is a well-defined homomorphism
of algebras. D e f i n i t i o n 2.2.15. A derivative operator (or, simply, a derivative) on ~hp is an ]l{-linear map D : {~p ~ If{ such that
D(ab) : D(a)ep(b) + ep(a)D(b), for all a, b 9 ~p. Temporarily, we will denote the set of all derivatives on Op by T(~p). However, we will see shortly that it is a vector space that is canonically isomorphic to Tp(U). We define algebraic operations on T(C3p).
2 , 2 . TANGENT
VECTORS
47
9 scalar multiplication: (tD)(a) = t(D(a)), Vt C IR and V D C T(Op), Va C Or; 9 addition: (D1 + D2)(a) = Dl(a) + D2(a), VDa,D2 E T(Op), Va C Or. Lemma
2.2.16.
The space T(~Sv) is a vector space over R under the above oper-
ations. Again, the proof will be left to the reader. Example
2 . 2 . 1 7 . Define Di,p : Op --~ IR by
of
Di,p[f]p = ~xi (P). This is a well-defined, R-linear map. Furthermore, by the Leibnitz rule for partial derivatives,
Di,p([f]p[g]p) = ~-~(f g)(p) = ~--fxi(p)g(p)
09 + f(P)-~-~x~(P)
= Di,p([f]p)ep([g}p) + ep([f]p)Di,p([g]p). T h u s Di,p is a derivative, 1 < i < n. E x a m p l e 2 . 2 . 1 8 . If (S>p is an infinitesimal curve, then D<s>, : 0 v ~ derivative. Indeed,
]R is a
n
D<s>~ [f]p = E aiDi,P[f]P' i=1
where
~(o)
=
; a
so the assertion follows from the previous example. It is obvious that
(s>p ~ D<~>,, defines a linear m a p from the space of infinitesimal curves into the space of derivatives of Op. It is also clear t h a t this linear m a p is injective. The fact t h a t it is an isomorphism of vector spaces (Corollary 2.2.22) requires proof. Lemma
2 . 2 . 1 9 . If c is a constant function on U and D E T(Ov), then
D[c]v = O. Proof. Consider first the case c = 1. T h e n D[1}p = D([1]p[1]p)
= D([1]p)%([1]p) + %([1]p)D([1]p) = 2D[1]p, from which it follows t h a t D[1]p = 0. For an arbitrary constant c, D[c]p = ~D[1]p
by linearity.
= 0,
[]
48
X=
2. LOCAL THEORY
In order to get more information on T(qSp), we need a technical lemma. Let ( x l , . . . , X n) a n d p = (xl(p),...,xn(p)).
L e m m a 2.2.20. Let f E C~~
Then there exist functions g l , . . . , g n C C~176
and a neighborhood W c dora(f) A dom(gt) A . . . N dom(gn) of p such that (1) f ( x ) = f(p) + E n = l ( X i - xi(p))gi(x), V x e W ; (2) g~(p) = ~ ( p ) , 1 < i < ~ Proof. Define gi(x) =
/01
( t ( x - p ) +p)dt.
This is clearly a smooth function defined at all points x sufficiently near p. In order to prove (2), consider g~(P) =
J l I Of 7x~ (p) dt
Of fo 1 = Ox i (p) dt
of = Ox~ (p)"
In order to prove (1), consider
f ( x ) - f(p) = fro1
d (f(t(x - p) + p) dt
dt =fo 1 { ~-~"~fxi(tx-p)+p)(xi-xi(p)i=l {~ol ~fxi(t(x- p) +p)dt} (xi-x~(P))
i=l g~(x)(z ~ - ~(p)). i=1
[] T h e o r e m 2.2.21. The set { D l , p , . . . , Dn,p} is a basis of the vector space T(C3p),
Proof. Suppose that n
E
aiDi,p = O.
i=l For the coordinate functions x j, 1 < j <_ n,
Ox j Di,p[xqp = 5 7 ( p )
= ~ij,
the Kronecker delta. Thus,
0=
[XJ]p = a j, \i=1
/
2.2. TANGENT
VECTORS
49
for 1 < j < n. This proves t h a t {Dl,p,..., Dn,p} is a linearly i n d e p e n d e n t subset of T(~Sp). We must prove t h a t it is also a s p a n n i n g set. Let D E T(~Sp). Set a i = D[x~]p, 1 < i < n. Given an arbitrary [f]~ E ~Sp, write n
f(x) = f(p) + ~--]~(xi - xi(p))gi(x) i=1
as in L e m m a 2.2.20. Then, n
D[f]p = D[f(p) + ~-~(z ~ - x~(p))gi(x)]p i=1
= ~ D[(x~ -
~(P))gd~
i=1 n
= ~-~{D[xi]pgi(p)+ (xi(p) -xi(p))D[gi]p} i=1
= ? _ . a -g-S~p~ i=1
/
kC=l
Since [f]p E cSp is arbitrary, it follows t h a t D = ~ i ~ l aiDi,p 9
[]
Remark. Tile above proof would not work for derivatives of the algebra ~ipk of germs o f C k functions, k < oo. The problem is t h a t 9 i E C k - l , 1 < i < n; soD[gi]p is not even defined. In fact, for 0 < k < oo, the space of derivatives of ~Sp k is infinitedimensional [33]. C o r o l l a r y 2.2.22.
The spaces Tp(U) and T( ~p) are canonically isomorphic vector
spaces. Proof. Indeed, the linear injection, defined in E x a m p l e 2.2.18, between the two vector spaces m u s t be an isomorphism since both are n-dimensional. [] We write Tp(U) and T ( e p ) interchangeably, usually preferring Tp(U).
Remark. We can identify this vector space canonically with IRn via
~-~ aiDi,p +-~ i=1
a
O n tile other hand, one should be wary since Tp(U) should not be thought of as identical with Tq(U) when p 54 q. There will be no such canonical identification on manifolds. Let T(U) = [Ixcu T~(U), a disjoint union. There is a one-to-one correspondence T(U) ~ U x IRn given by
~ i=l
aiDi,x ~
x, ~k
a
50
2. L O C A L T H E O R Y
We use this to transfer the topology of U x ]l~n to T(U).
Remark. This method of topologizing T(U) does seem to use the coordinates. We will see, however, that the topology on T(U) is actually independent of the choice of coordinates. D e f i n i t i o n 2.2.23. The tangent bundle 7r : T(U) ~ U is defined by 7r
Di, p \i=1
= p. /
Via the canonical identification T(U) = U x R n, 7r is just the standard projection onto the first factor. For each x E U, T~(U) should be thought of as the linear approximation of U at x. This is going to enable us to approximate smooth maps between open subsets of Euclidean spaces by linear maps. E x e r c i s e 2.2.24. Let ~hp C ~hp be the kernel of the evaluation map ep and let ~;* c ~p be the vector subspace spanned by the germs of functions gf, where 9, f E C~176 and g(P) = 0 = f(p). Prove that the quotient space ~5;/qhp* is canonically isomorphic to the vector space dual of Tp(U). In particular, this quotient space is n-dimensional. 2.3. S m o o t h M a p s a n d t h e i r D i f f e r e n t i a l s Let U c N n and V C_ N'~ be open subsets. Consider functions 9 : U --* V and their coordinate representations r ~1,r162 where each qhi : U --* N. D e f i n i t i o n 2.3.1. We say that 9 U --* V is a map of class C k (where 0 < k < oe) if/i)i C Ck(U), 1 < i < m. If 9 is of class C ~ it is called a smooth map. The following lemma is clear by the standard chain rule. L e m m a 2.3.2. Wherever defined, compositions of smooth maps are also smooth. L e m m a 2.3.3. Let 02 : U ---* V be smooth and let p C U. If s C S(U,p), then the
infinitesimal equivalence class of ~2 o s at ~(p) depends only on the infinitesimal equivalence class of s at p Proof. Indeed, let f C C ~ ( V , d2(p)) be arbitrary and note that df(r
t=0
can be interpreted as the derivative of f o ~5 along s at t = 0. This depends only
on (s)p. Definition
[] 2.3.4.
If ~5 : U ~ V is smooth and if p C U, let
dCp = r
: Tp(U) ~ T~(p)(Y)
be defined by
r
(sip = (r o s)~p~,
for arbitrary {S}p C Tp(U). This is called the differential of ~5 at p.
2.3. SMOOTH
MAPS
51
Under the identification of an infinitesimal curve (s); with its associated derivative operator D(s),,, we can write
dq~p(D(s)~,) = Cp.p(D(s)p) = D(r If, for simplicity of notation, we let f E C~176 ~(p)) stand in for its germ [f]r this operator has the form = df(~(s(t)))
~.;(D(~),,)(f) = D(r
= D(,},(f o ~). t=0
T h a t is, to differentiate f by ~.p(D(~)~), one "pulls back" f to the function f o ~ E Coo(U, p) and differentiates t h a t function by D ( , ) . This "pullback" of functions is denoted by ~ * ( f ) = f o ~. It passes to a well-defined map ~ ; : ~5r
--+ qSp,
r
= If o r
It is almost immediate that this is a homomorphism of algebras and one obtains the following9 L e m m a 2.3.5. Under" the identifications T(| = T;(U) andT(r162 the formula for the differential dp,p: T(qSv) ---, T(~5r becomes
~ . ; ( D ) = D o ~;,
= Tr
V D e T(r
In particular, (p.p is a linear map. E x e r c i s e 2.3.6. Relative to the respective bases { D
,
042, 2 I
~
37r IP) kP)
Jr
a~, ~ ~
o~, ~ ( p )
O (p 2 t
0q5 2 /
~ ~
~rJ
oz'~
kP)
~P)
=
9~ m
0 ~ "~
~-(p)
..
0a~ m
(p).
of 9 at p. In particular, if 9 : U -+ ~ and Dp C Tp(U), conclude that, under the canonical identification Tr = ~, 9 .p(Dp) = D ; ( f ) .
Remark. The differentials (I).z, computed at all points x C U, assemble to a mapping r
= d e : T(U) ~ T(V),
called the differential of 4) on U and given by
Here, we have identified T(U) with U x R ~ and T ( V ) with V x IRm
52
2. L O C A L T H E O R Y
C o r o l l a r y 2.3.7. Relative to the identifications T ( U ) = U • ] ~ and T ( V ) = V • N m, the differential dO = O. : T ( U ) --+ T ( V ) is a smooth map from an open subset of ]~2~ to an open subset of R 2m. C o r o l l a r y 2.3.8. Let 02 : U --~ V be of the f o r m 0(x) = L(x) + Yo, where L : N ~ --~ ]Rm is linear and Yo E ] ~ is fixed. Then, denoting by L the matrix of L relative to the standard coordinates of ]~n and ]~m, we have that J0(p)=L,
VpeU.
In this case, d0=(0,
L ) : U x l R n - - * V x R m.
In case 9 = L is itself linear, dL = (L,L). Tr = R m, we can write dLp = L.
If we identify Tp(U) = 1Rn and
T h e o r e m 2.3.9 (The general chain rule). If U C IR'~, V C_ R m, and W C_ ][~q are open subsets and if 9 : U ~ V and 9 : V --* W are smooth, then d(g~ o O)p = d~r
o dOp.
P r o @ Indeed, d(q2 o O)p (S)p = (~ o 9 o s)r162
= d~r
(0 o s>r
= d~r
(Sip). []
Remark. In terms of Jacobian matrices, the general chain rule can be written J (~P o O )(p) = J ~ ( O(p) ) . ZO(p). One can verify this directly by applying the less general chain rule for real-valued functions and the formulas for matrix multiplication. This is the usual proof in multivariable calculus, but the proof via infinitesimal curves is more elegant and more intuitive. E x e r c i s e 2.3.10. Viewing 0,~ in terms of derivatives of germs, give a direct proof of the chain rule (again avoiding Jacobian matrices).
D e f i n i t i o n 2.3.11. If U C_ N ~ and V C_ N'~ are open, a map 9 : U --+ V is a diffeomorphism if it is smooth and bijective and if 9 -1 : V --* U is also smooth. P r o p o s i t i o n 2.3.12. I f O : U ~ V is a diffeomorphism of an open subset of ~ n onto an open subset o f ~ m and if p E U, then
d(O~: Tp(U) ~ T~(~)(V) is a linear isomorphism. In particular, n = m. Proof. Since 0 -1 o 9 = idu is the restriction to U of idRn, a linear map, Corollary 2.3.8 shows that d(O -1 o O)p = idT,(u), for each p E U. By the general chain rule, it follows that the linear map dOp is invertible with inverse d(O-1)(i,(p). [] Remark. Thus the dimension n of an open subset U C_ ~ is a diffeomorphism invariant. We saw earlier that the Brouwer theorem of invariance of domain implies the equality of dimensions, even if 9 were only a homeomorphism. That theorem was very deep, while the proof of Proposition 2.3.12 is quite elementary. This is an example of the technique of reducing nonlinear problems to linear ones via derivatives.
2.3. S M O O T H
MAPS
53
E x a m p l e 2.3.13. Let flJt(n) denote the set of n x n matrices with real entries. This is a vector space over N and, by suitably ordering its entries, we can fix an identification of 9Yt(n) with N n2. An important subset Gl(n) of flYt(n) is the set of nonsingular matrices. These form a group under matrix multiplication, called the general linear group. This is an open subset of 97t(n). Indeed, the determinant function det : 9)I(n) ~ R is a polynomial, hence is smooth. The set N* of nonzero reals is open in N, hence the general linear group Gl(n) = det -1 (St*) is open. If P c 9Y~(n), the left multiple map L p : 9)I(n) -~ 9:~(n) is given by L p ( Y ) = P Y , VY 6 g,R(n). Similarly, the right multiple map is given by R p ( Y ) = Y P , V Y 6 gJt(n). Clearly, both L p and R p are linear transformations. Thus, by Corollary 2.3.8 and the subsequent remark, d ( L p ) z : Ty(gYC(n)) --~ Tpy(gJt(n)),
d ( R p ) y : T y ( ~ ( n ) ) -~ T y p ( ~ ( ~ ) ) are given, via the natural identifications of these tangent spaces with the underlying Euclidean space 9)l(n), by d(Lp)z(A) = PA, d ( R p ) y ( A ) = AP, Y A E ff2(n). If P r Gl(n), then the restrictions of L p and R p to Gl(n) are diffeomorphisms of this open set onto itself. Indeed, the respective inverses are Lp-~ and Rp-1. These diffeomorphisms L p , R p : Gl(n) --~ Gl(n) are called, respectively, the left and right translations by P. The precise sense in which the differential d~p is a linear approximation of near p is given by the following theorem, in which d~by is interpreted as a linear map of ]I~n ~ / ~ m Vy 6 dora(q)). Theorem p6U,
2.3.14. I f U C_ ~n is open and q5 : U --* ]Rm is smooth, then, for each lim (x,y)--(p,p)
~(x)
- q~(y) - d O p y ( x - y )
Ifx - Yfl
=0.
Proof. Using the coordinate representation
we see that it is enough to prove the assertion for maps ~J = f : U -~ N. In Lemma 2.2.20, let p be a variable point y and write
g~(~,y) =
(t(x-y) +y)dt
Of n
f ( x ) - f ( y ) = E ( x i - yi)gi(x,y). i=I
54
2. L O C A L T H E O R Y
Thus, n X i _ yi f ( x ) - f ( y ) = [Ix - yll E ]7-x_-'~] (gi(x,y) i=1
_
n gi(Y,Y))+ E ( xi i=1
n('~,v)
i, _
y
Of,
,
~--gT~v~.
eS~('~-y)
Since (x i - yi)/]]x - y[] is bounded, 1 < i < n, it is clear that lim Rex, y) = 0 (x,y)~(p,p) and the assertion follows.
[]
E x e r c i s e 2.3.15. If f E Coo(U,p) and Lp is the derivative of f at p (Definition 2.1.1), use Theorem 2.3.14 to express Lp in terms of dfp. E x e r c i s e 2.3.16. Let p C S n C 1Rn+l and define
Tp(S n) : {(8)p C Tp(~{n+l)
I 8: (-e,e) ---+]I~n+l has
ira(s)C
sn}.
Prove that Tp(S n) is the linear subspace of ]I~n+l Tp(]I~n+l) consisting of all v / p. This is what we earlier called the tangent space of S n at p (Example 1.2.7). =
In Exercises 2.3.18 and 2.3.19, you will need the following definition. D e f i n i t i o n 2 . 3 . 1 7 . If A c IRn is an arbitrary subset, define Coo(A) to be the set of all functions f : A --+ IR such that f = flA, where f : U --+ IR is a smooth function defined on some open neighborhood U of A. E x e r c i s e 2.3.18. If A = [0, 1] x [0, 1] and I C C~176 let f : U --+ IR be a smooth extension as in Definition 2.3.17. Prove that df(o,o) depends only on f , not on the
choice of f. E x e r c i s e 2 . 3 . 1 9 . I f A = S n, f C COO(sn), p E S n, and f is as in Definition 2.3.17, show by an example that dfp may well depend on the choice of extension f , but prove that dfvlTp(S '~) depends only on f. 2.4. D i f f e o m o r p h i s m s a n d M a p s o f C o n s t a n t R a n k
If 9 : U -~ V is a diffeomorphism between open subsets of N n, then the Jacobian matrix JO(p) is nonsingular, Vp E U. While the converse is not exactly true, a strong version of the converse is true locally. Theorem
2.4.1 (Inverse function theorem). Let 9 : U -~ V be smooth, where
[7, V C_ IR~ are open subsets, and let p E U. If dO~ : T~(U) --* T~(p)(V) is a linear isomorphism, then there is an open neighborhood W~ of p in U such that O[WN is a diffeomorphism of Wp onto an open neighborhood 0(Wp) g O ( p ) in V. This is a remarkable and fundamental result. From a single piece of linear information at one point, it concludes to information in a whole neighborhood of that point. This theorem is often proven in courses in advanced calculus. We will give a proof in Appendix B that works for C k maps, 1 < k < oo, and even works for maps between open subsets of a Banach space. There is a generalization of Theorem 2.4.1, called the "constant rank theorem", that is actually equivalent to the inverse function theorem. Our main goal in this
2.4. MAPS OF CONSTANT RANK
55
section is to prove the constant rank theorem using the inverse function theorem. T h e s t a t e m e n t and proof are greatly facilitated by "smooth changes of coordinates". 2 . 4 . A . D i f f e o m o r p h i s m s as c o o r d i n a t e c h a n g e s . Let W and W be open subsets o f N n and let F : W --* W be a diffeomorphism. If we denote the coordinates of W by x = ( x ~ , . . . x n) and those of W by w = ( w l , . . . , wn), t h e n the coefficient functions F i of F can be denoted by wi(x). T h a t is, F is given by a system of s m o o t h equations W i = wi(xl,...
,3cn),
1 < i < n.
T h e existence of a s m o o t h inverse F -1 can be interpreted as the existence of s m o o t h solutions x i=xi(wl,...,w~), l
F
~W
G
and we can interpret ~ as a new formula for the map 9 relative to the respective coordinate changes F and G in the domain and range of qS. One looks for c o o r d i n a t e changes t h a t make the formula for 9 simpler. E x a m p l e 2.4.2. Let 9 : W --* Z be s m o o t h and let in E W. If we take F : N1n --~ IRn to be t r a n s l a t i o n by - p and G : IR~ --* IRm to be translation by -fl~(p), we can set W = F ( W ) , Z = G ( Z ) , and view F and G as coordinate changes. In the new coordinates, p is replaced by 0 C IRn, 4~(p) is replaced by 0 E R m and the new formula for 9 satisfies ~ ( 0 ) = O. We say t h a t , "by suitable translations in the range and domain of q5, we lose no generality in assuming t h a t p = 0 and qS(p) = 0". E x a m p l e 2 . 4 . 3 . Linear changes of coordinates are frequently useful. We note particularly those linear changes t h a t simply p e r m u t e the order of the coordinates. More precisely, suppose t h a t cr is a p e r m u t a t i o n of {1, 2 , . . . , n}, t h a t T is a permutation of {1, 2 , . . . , m} and t h a t the coordinate changes F : W ~ W and C : Z --* are given by F-I(wl
...,~/Jn) = (I/)o'(1),...,wc,(n)),
c , ( y l , . . . ,ym) = (f(x) . . . . . y,(~)). Thus, if q5 : W -~ Z has coordinate functions gpi, its new formula d) has coordinate functions
~ i ( w I , . . . , w n) = gJ(i)(w~'(O,...,w~'(n)),
l < i < m.
It follows t h a t J ~ is o b t a i n e d from JgP by p e r m u t i n g the rows by 7- and the columns by or. Thus, for example, if the m a t r i x JCpp has rank h, we can assume, after suitable
56
2. LOCAL THEORY
p e r m u t a t i o n s of the coordinates in the d o m a i n and range, t h a t J ~ p has as its upper left k x k block a nonsingular matrix. 2.4.B.
The constant
rank theorem.
D e f i n i t i o n 2.4.4. A s m o o t h m a p r : U ~ V, between open subsets of Euclidean spaces of possibly different dimensions, has constant rank k if the rank of t h e linear m a p d~x : Tx(U) --~ T~(x)(V) is k at every point of U. Equivalently, the J a c o b i a n m a t r i x J ~ has constant rank k on U. Example
2.4.5. Consider the composition IRk x N n - k ~ N
k
i~lRm,
where k < n, k < m, and
7c(xl,.
,xk,yl,...,y~-k)
i(zl,...,
= ( X l , . . . , X k)
x k) = (xl,..., x k, o,..., o). ?n -- k
T h e J a c o b i a n of i o ~r is constantly the m x n m a t r i x having Ik as its u p p e r left k x k corner and 0's elsewhere. T h e rank is constantly k. T h e constant rank theorem asserts that, in a certain precise sense, m a p s of c o n s t a n t rank k locally "look like" the above example. 2 . 4 . 6 ( C o n s t a n t rank theorem). Let U C_ R ~ and V C_ R m be open and let q~ : U ---* V be smooth. Let p E U and suppose that, in some neighborhood of p, 9 has constant rank k. Then there are open neighborhoods W of p in U and Z D_ ep(W) of ~2(p) in V, together with smooth changes of coordinates Theorem
F:W-~W,
G:Z~Z, such that, throughout the neighborhood W of F(p), the new formula ~ for 4~ is @(wl,...,W
n) =
(wl,...,W
k,O,...,O).
Pro@ By E x a m p l e 2.4.2, we make preliminary changes of coordinates so as to assume t h a t p = 0 C IRn and ~5(p) = 0 E IRm. Similarly, by E x a m p l e 2.4.3, we assume t h a t the upper left k x k block ["0~ 1
~176 a ( x l , . . . , x k)
0~ 1-1
ioe~ L-g-~-z~
9
or Ozk J
of J ~ is nonsingular at p = 0. Let x = ( x l , . . . , x ~) and define F : U ~ II~~ by
F(x) = (@l(x),...,q)k(x),xk+l,...,xn). T h e n F ( 0 ) = 0 and
JF =
[
o(x 1...... k) 0
* In-k
]
is a m a t r i x t h a t is nonsingular at p = 0. By T h e o r e m 2.4.1, there is a n e i g h b o r h o o d W of 0 on which F is a d i f f e o m o r p h i s m onto an open set W C R n. Let w =
57
2.4. MAPS O F CONSTANT RANK
( w l , . . . , w n) denote the coordinates of W. q o k + l , . . . , qom, we get the formula oF -l(w)
= (wl,...,w
Then, for suitable s m o o t h functions
k,gpk+l(w),.,,,)9
m(*/))).
Since F - l ( 0 ) = 0 and (I)(0) = 0, we note t h a t the functions qnJ all vanish at the origin. We also note t h a t Ik
0 0qok+~
dq~ . J F -1 = d(d~ o F -1) =
...
o~k+~,
0~ k+t
o~n. ...
COrdn
Since Jq~ has rank k in a neighborhood of 0 and J F -1 is nonsingular on the n e i g h b o r h o o d W = F ( W ) of 0, we can choose W smaller, if necessary, so as to assume t h a t the above m a t r i x has rank k at every point of W. It follows t h a t the lower right block must consist entirely of 0s, hence t h a t the functions ~oY d e p e n d locally only on ( w l , . . . , wk). Thus, choosing W smaller if necessary, we can write
9/(w)=9~J(wl,...,wk),
k + l <_j <_ra.
Let y = ( y l , . . . ,ym) and define
a(v) = (Vl,...,yk,vk+l
_ ~k+~(y, .... ,yk),...,ym
_ ~,~(y~,...,yk)).
This is defined on a suitably small neighborhood of 0 in ]Rm. It is clear t h a t
*
Lrn-k
is a nonsingular matrix, hence T h e o r e m 2.4.1 implies t h a t G is a diffeomorphism of a small enough n e i g h b o r h o o d Z of 0 in IRm onto a neighborhood Z = G ( Z ) of 0. Taking W smaller, if necessary, we can assume t h a t ~ ( W ) c_ Z. F r o m the formulas, it is clear t h a t ao()oF-l(wl,...,wn
on W.
) =
(w 1.... ,w]~,0,...,0)
[]
E x e r c i s e 2.4.7. D e d u c e T h e o r e m 2.4.1 from T h e o r e m 2.4.6. Since we deduced T h e o r e m 2.4.6 from T h e o r e m 2.4.1, the two theorems are equivalent. T h e r e are two particularly i m p o r t a n t cases of T h e o r e m 2.4.6, the immersion t h e o r e m and t h e submersion theorem. D e f i n i t i o n 2 . 4 . 8 . Let U C IRn and V C_ IRm be open subsets. A s m o o t h m a p : U -~ V is a submersion if it has constant rank m on U. It is an immersion if it has c o n s t a n t rank n on U. R e m a r k t h a t , if r is a submersion, t h e n n _> m. If it is an immersion, then m _> n. If it is b o t h a submersion and an immersion, t h e n n = m and ~ is locally a diffeomorphism by T h e o r e m 2.4.1. T h e next two corollaries are i m m e d i a t e applications of T h e o r e m 2.4.6.
58
2. LOCAL THEORY
C o r o l l a r y 2.4.9 (Submersion theorem). Let 9 : U ~ V be a submersion and let p C U. Then there are open neighborhoods W o f p in U and Z D_ (~(W) of ~(p) in V, together with smooth coordinate changes F:W-*
W,
G:Z~Z, such that the new formula ~ for 9 on W is
~ ( w l ...,?/)n) • (wl . . . , W i n ) . C o r o l l a r y 2.4.10 (Immersion theorem). Let q~ : U --~ V be an immersion and let p E V. Then there are open neighborhoods W o f p in V and Z D ~ ( W ) of (P(p) in V, together with smooth coordinate changes F:W~W, G:Z-~2, such that the new formula ~ for ~ on W is
~ ( w l , . . . , w ~) = ( w l , . . . , w ~ , o , . . . , o ) . Thus, submersions look locally like projections onto the first m coordinates and immersions look locally like the canonical imbeddings ~
=R ~ • {(0,...,0)}
N TM.
~
C o r o l l a r y 2.4.11 (Implicit function theorem). Let U C_ l~~ be open and let p C U. Let f : U --~ ]R be smooth with f(p) = a, If
of
ox k (p) r o, then, on some open neighborhood W of p in U, the set of solutions to the equation f ( x ) = a is the graph of a smooth function X k ---
g(xl,...,X
k-l,xk+l,...,xn).
E x e r c i s e 2.4.12. Use the proof of Theorem 2.4.6 to prove the implicit function theorem. 2.4.13. Use Corollary 2.4.11 to prove t h a t the unit sphere S n is a topological submanifold of IR~+1 of dimension n. Exercise
2.5. S m o o t h
Submanifolds
of Euclidean
Space
We already know what is meant by a topological submanifold of R n (Definition 1.5.3). We extend this notion to the smooth category. The model will be the s t a n d a r d imbedding 1Rr ~ N n, r _< n, given by
(xl,... ,x T) H (xl,... , s , 0 , . . .
,0).
n--T Whenever we view ~
C_ ]R~, we understand R ~ to be the image of this imbedding.
2.5. SMOOTH SUBMANIFOLDS
59
F i g u r e 2.5.1. A submanifold is locally flat D e f i n i t i o n 2.5.1. Let U C_ I~n be open. A topological subspace N C_ U is said to be a s m o o t h submanifold of U of dimension r < n if, for each z E N, 3 Ux _C U, an open neighborhood of z, a n d a diffeomorphism f : Ux --~ Q onto an open subset Q c_ R n such t h a t f ( N n U s ) = Q N N r. The empty set 0 C U is a s m o o t h submanifold of every dimension r _< n. T h a t is, if we view f as a local change of coordinates, we see t h a t N looks locally like the flat i m b e d d i n g of II~r in I~n. This is illustrated in Figure 2.5.1. L e r n m a 2.5.2. If N C U is a smooth submanifold of dimension r, then N is also a topological submanifold of dimension r of U. Proof. Let z E N and use the n o t a t i o n of Definition 2.5.1. T h e n N n Us is an open neighborhood of z in the relative topology of N in U. Since f carries N N U~ homeomorphically onto the open subset Q n N" of R", it follows that N, with the relative topology, is locally Euclidean of dimension r, the inclusion map i : N ~ U being a topological imbedding. As a topological subspace of Euclidean space, N is Hausdorff a n d second countable. []
T h e o r e m 2.5.3. Let U C N ~ and V C_ R m be open and let ,5 : U ~ V be a smooth map of constant rank k. Let q E V . Then ~ - l ( q ) is a smooth submanifold of U of dimension n - k. P r o @ If (I)-l(q) = 0, the assertion is true by convention. Assume t h a t this set is n o n e m p t y a n d let z be one of its points. Choose Us to be the neighborhood W as in T h e o r e m 2,4.6. W i t h o u t loss of generality, we can replace W with W and ~ I W with G o q5 o F -1 on W, all as in t h a t theorem. T h a t is, on U~ we assume t h a t
(I)(~1 ...,~rt)~_ (~1 ... ,yk 0,. . . , 0 ) . Thus
q-~ ( a l , . . . , a k , 0 , . . . , 0 )
60
2. L O C A L T H E O R Y
a n d U~ n ~ - l ( q ) is the set of all points in U~ of the form
(ai,...,ak,yk+l,...,yn). T h e desired diffeomorphism f will be
f(yl...,y~)
=
(yk+l,...,ynyl
_
al...,yk _ak). []
E x a m p l e 2.5.4. Theorem 2.5.3, applied to the function you used to carry out Exercise 2.4.13, implies t h a t S " C IR~+l is a smooth submanifold of dimension n. E x a m p l e 2.5.5. Recall t h a t Gl(n) denotes the group of nonsingular matrices over IR (the general linear group). The special linear group Sl(n) is defined to be Sl(n) = {A C Gl(n) I det(A) = 1}. This is clearly a subgroup. We have noted in Example 2.3.13 t h a t Gl(n) is an open subset of 9:~(n) = ~ n2 a n d we now show t h a t Sl(n) is a smooth s u b m a n i f o l d of
el(n) T h e d e t e r m i n a n t function det : DI(n) -+ ]R is a polynomial, hence is smooth. We claim t h a t det : Gl(n) --+ IR has constant rank 1. To prove this, we need to show t h a t , for arbitrary A E Gl(n), the linear m a p d(det)A : TA(Gl(n)) ~ Tdet(A)(~) has r a n k 1. For this, we only need to find an infinitesimal curve (S)A such t h a t det.A (S)A = (det OS)det(A) % 0. Define s(t) to be the m a t r i x obtained by multiplying the first row of A by 1 + t. T h e n s(0) = A and the fact t h a t Gl(n) is open in IRn2 implies t h a t s(t) G Gl(n) for Itl small enough. Since det(s(t)) = (1 + t) det(A) a n d det(A) :r 0, it is clear t h a t (detos)det(A) ~ 0. By the above remarks a n d T h e o r e m 2.5.3, St(n) = d e t - l ( 1 ) is a smooth submanifold of Gl(n) of dimension n 2 - 1.
D e f i n i t i o n 2.5.6. If N C U is an r-dimensional, smooth submanifold of the open set U c_ R ~, a n d if x E N, a vector v E Tx(U) is t a n g e n t to N at x if, as an infinitesimal curve, v = (S)x has a representative s : ( - e , e) --~ U such t h a t s(t) C N, - e < t < c. The subset T~(N) C_ T~(U), consisting of all vectors t a n g e n t to N at x, is called the t a n g e n t space to N at x. 2.5.7. If N C_ U is an r-dimensional, smooth submanifold of the open set U C ]~n, and if x E N, the tangent space Tx(N) is an r-dimensional vector subspace of T~ (U).
Lemma
Proof. For the "model" case N = ~ r c ll~n, the assertion is evident. Let all n o t a t i o n be as in Definition 2.5.1. If the smooth p a t h s : ( - c , c) ~ U has image in N a n d s(0) = x, t h e n the diffeomorphism f : U~ ~ Q sends s to a smooth p a t h f o s in ~ r t h r o u g h f(x). Thus, the linear isomorphism dA : T~(U~) -~ Tf(~)(Q) carries T~(N) into the vector space T/(~)(Q (~ ~ " ) = ]R~. But Q n II~~ is m a p p e d onto U~ ~ N by f - 1 and the same a r g u m e n t shows t h a t the inverse isomorphism d(f-1)/(~) = (dfx) -1 carries T/(~)(Q N R ~) into Tx(N). The assertion follows. []
2.5. S M O O T H S U B M A N I F O L D S
61
E x a m p l e 2.5.8. The subspace TI(SI(n)) C 9)I(n) is the space of matrices of trace 0. To prove this, we first show t h a t this t a n g e n t space is the kernel of det.~ a n d t h e n t h a t d e t . i = tr : 9J~(n) -~ R. Both TI(SI(n)) a n d k e r ( d e t . i ) have dimension n 2 - 1; so equality will follow if we show t h a t the first is a subspace of the second. If v C r l ( S l ( n ) ) is t h o u g h t of as a n infinitesimal curve, t h e n v = {s)i where s : ( - e , e ) --~ Sl(n), s(0) = [. Thus, det os - 1 is a c o n s t a n t curve, a n d so d e t . r ( v ) = (det os) l = 0. The linear functionals d e t . / and tr on 9)I(n) will be equal if they agree on a basis. T h e matrices Eij, 1 <_ i, j <_ n, having i in the (i, j ) position and 0s elsewhere, form a basis. As an infinitesimal curve at I, E i j = (sij) I where s~j(t) = I + tE~j. But l+t, i=j,
det(sij(t)) =
1
i r j,
from which it follows t h a t (det osij)l =
l=tr(Eij), 0 = tr(Eij),
i=j, i r j.
A n o t h e r interesting example is the subgroup O(n) C Gl(n) of orthogonal matrices. These are the matrices having as columns an o r t h o n o r m a l basis of ]Rn. Equivalently, A T = A -1, for each A c O(n). Evidently, this is a subgroup, b u t less evident is the fact t h a t it is a compact submanifold. The following exercise leads you through a proof of this fact a n d the d e t e r m i n a t i o n of T l ( O ( n ) ) C 9J[(n). E x e r c i s e 2.5.9. T h e m a p 9 : Gl(n) ~ Gl(n), defined by ~ ( Y ) = y T y , is s m o o t h since its coordinate functions are quadratic polynomials. Prove the following: (1) Relative to the s t a n d a r d identification T I ( G I ( n ) ) = 9Jr(n), the differential dePi : T I ( G I ( n ) ) ~ TI(GI(n)) has the formula
dq, i ( A ) = A T +A. (2) T h e m a p dp has constant rank n ( n + 1)/2. (3) Using the above, conclude t h a t the orthogonal group O(n) C Gl(n) is a smooth, compact submanifold of dimension n(n - 1)/2. (4) Show t h a t the vector subspace T l ( O ( n ) ) C fOl(n) is the space of skew symmetric matrices. Let U be an open subset of IRn and let N C_ U be a smooth r-dimensional submanifold. Exactly as we did for S n in Section 1.2, we can define the topological subspace T ( N ) _C IR~ x R ~ = R 2~ to be the set of pairs (x, v), where x E N a n d v E T~(N). We define p: r(X)
-~ N
by p ( x , v ) = z. T h e fiber p - l ( x ) = T~(N) is a vector space, Vx C N.
D e f i n i t i o n 2 . 5 . 1 0 . T h e structure p : T ( N ) --~ N is called the t a n g e n t b u n d l e of the submanifold N C_ U. T h e total space of-the b u n d l e is T ( N ) , the base space is N, a n d p is called the b u n d l e projection.
62
2. L O C A L T H E O R Y
In the case of the model submanifold Ii~" C_ ] ~ n , T ( R ~) = R ~ x IR~ C_ IR" x R n. We will say t h a t this b u n d l e is trivial (cf. Definition 2.5.11). If Y C N is an open subset, t h e n Y is also a smooth submanifold a n d T ( Y ) = p - l ( y ) is the total space of the t a n g e n t b u n d l e p : T ( Y ) ~ Y . In the following definition, the term "diffeomorphism" refers to a bijection between subsets of Euclidean spaces that, together with its inverse, is s m o o t h in the sense of Definition 2.3.17. D e f i n i t i o n 2 . 5 . 1 1 . The t a n g e n t b u n d l e p : T ( N ) c o m m u t a t i v e diagram T(N)
~
~
N is trivial if there is a
, N x ]R~
Pl
~pN
N
}
N
id
where PN is projection onto the factor N and ~ is a diffeomorphism with the p r o p e r t y that, g y E N , p I T ~ ( N ) --, {y} x N r is a linear isomorphism.
D e f i n i t i o n 2 . 5 . 1 2 . If the t a n g e n t b u n d l e p : T ( N ) ---* N is trivial, the s u b m a n i f o l d N is said to be parallelizable. D e f i n i t i o n 2.5.13. If N C_ R n is a s m o o t h submanifold as above, a s m o o t h vector field on N is a m a p X : N ---* T ( N ) C_ R 2n t h a t is smooth as a m a p from the subset N C N n into R 2~ and satisfies p o X = idN. E x e r c i s e 2 . 5 . 1 4 . Show t h a t the r-dimensional submanifold N is parallelizable if and only if there are r smooth vector fields on N that, at each point of N , are linearly independent. Using this, show that the submanifolds SI(n) C ~ 2 and O(n) C ]Rn2 are parallelizable. E x e r c i s e 2 . 5 . 1 5 . Let N C ]Rn be a s m o o t h submanifold of dimension r a n d let x E N . Prove the following: (1) There is an open neighborhood Y C N of z such t h a t p :T(Y) ~ Y
is trivial. (We say t h a t the t a n g e n t b u n d l e of N is locally trivial.) (2) T h e subspace T ( N ) C II~2n is a smooth submanifold of dim 2r. Remark. W h e n we get to the global theory, smooth manifolds will be defined without need of a n a m b i e n t Euclidean space and their t a n g e n t bundles will be defined in an intrinsic way. A key property of these bundles will be local triviality.
2.6. C o n s t r u c t i o n s of S m o o t h F u n c t i o n s T h e m a i n goal in this section is the proof of the following special case of the C ~ U r y s o h n Lemma.
Theorem
2.6.1. Let K C_ ]Rn be compact and let U C IRn be an open neighborhood of K . Then, there is a smooth map f : ]Rn ~ [0,1] such that f I K -~ 1 and s u p p ( f ) C U.
2.6. C O N S T R U C T I O N S O F S M O O T H F U N C T I O N S
63
Define h : IR ~ [0, 1) by
~e -1/t2, t r O, h(t) = ( 0 ,
t = O.
E x e r c i s e 2.6.2. Prove t h a t the function h is Coo, even at t = 0, where the derivatives are
dnh dtn (o) = o,
for all integers n > 1. (One says that h is C ~ - f l a t at t = 0.) The graph of h is depicted in Figure 2.6.1.
F i g u r e 2.6.1. The graph of h Tile functions h:L : IR ~ [0, 1) are defined by
h+(t)
=
e -1/t2, O,
t > O, t <_ O,
e -1/t2,
t < O,
h_(t)
=
o,
t >_ o.
F i g u r e 2.6.2. The graph of h+
These are s m o o t h a n d C~176 at t = 0 by exactly the same reasons t h a t h is. T h e graphs are depicted in Figures 2.6.2 and 2.6.3 respectively. C o m b i n i n g these functions produces a C ~ function k : R + [0, 1),
k(t) = h _ ( t where a < b. Figure 2.6.4).
This function is C a .
b)h+(t-
a),
It is positive exactly for a < t < b (see
64
2. LOCAL THEORY
I Figure 2.6.3. The graph of h_
F i g u r e 2.6.4. The graph of k 2.6.3. L e t A = ( a l , b l ) • . . . • (an,bn) C ~ n be an open, bounded, n d i m e n s i o n a l interval. T h e n there is a s m o o t h f u n c t i o n g : ]~n __. [0, 1) such that g > O on A and gl(II~n \ A) =- O. Lemma
Proof. T h e definition of k gives functions ki, by taking a = ai and b = bi in t h a t definition, 1 < i < n. T h e n g(Xl,:~2,...
,Z n) ~- ]~l(xl)k2(x2)... ]~n(x n)
is as desired.
[]
Next we define a s m o o t h function g : R --~ [0, 1] by
e(t) - f'a k(X) dx
dx' where a a n d b are the n u m b e r s in the definition of k. This function is weakly m o n o t o n i c increasing, t~(t) -- 0 for t < a a n d g(t) =- 1 for t >_ b. The graph is depicted in Figure 2.6.5.
F i g u r e 2.6.5. The graph of
2.7. S M O O T H
VECTOR
FIELDS
65
Proof of Theorem 2.6.1. Let K and U be as in the s t a t e m e n t of the theorem. For each x E K , let A~ be an open, bounded, n - d i m e n s i o n a l interval, centered at x a n d having A~ C U. Apply L e m m a 2.6.3 to o b t a i n a smooth function 9~ : IRn --* [0, 1), strictly positive on A~ a n d vanishing identically outside of Ax. Since K is compact, it is covered by finitely m a n y A x l , . . . , Axq. The function G = g X l ~- " ' " ~ - g X q i s C o o on IRn, strictly positive on K , and has supp(G) = A~, U . . . U A~, C U. Since K is compact, we can also find m i n ( G I K ) = ~ > 0. In the definition of g : N - - * [0,1], take a = 0 and b = ~. T h e n f = C o G : R ~ -~ [0,1] is smooth, s u p p ( f ) C U, a n d f I K ~ 1. [] E x e r c i s e 2.6.4. Let U c_ R ~ be open, f : U ~ IR smooth, a n d p E U. Prove t h a t there is a Coo function f : R n -~ R such t h a t [f]p = [f]p in Op. Conclude that, for a r b i t r a r y p E/R ~, @p can be identified canonically with the set of germs at p of globally defined, smooth, real valued functions on R n. E x e r c i s e 2.6.5. Let C C tR~ be a closed subset, U C_ R ~ an open neighborhood of C. Show t h a t there is a smooth, nonnegative function f : IR~ ~ IR such t h a t flC > 0 a n d s u p p ( f ) C U'. 2.7. S m o o t h V e c t o r F i e l d s Let U _C Nn be an open subset. In particular, this is a s m o o t h submanifold and we consider the set of smooth vector fields on U (Definition 2.5.13). This set is c o m m o n l y denoted by ~ ( U ) . It is a vector space over R under the pointwise operations. If X E X(U), its value at a point x E U is c o m m o n l y denoted by X~. T h r o u g h o u t this section, the preferred way to think of Xx E Tx(U) will be as a derivative of the algebra ~!ix of germs (Definition 2.2.15). For intuition, however, it is helpful to identify Tx(U) = ]R~, viewing Xx as a column vector. Thus,
[
fl(~) 1
Lf i )j 9
i=1
where Di,~ = O/Ox~I~ is the i t h partial derivative at x. As x varies, each i f ( x ) varies smoothly, so
11] f2
X =
n
= ~
n
ffDi,
i=1
where Di = O/Ox ~ a n d fi E C~176 1 < i < n. Using the column vector interpretation, we can picture X as a smoothly varying field of directed line segments (arrows), issuing from points of U and possibly degenerating to zero length segments somewhere (Figure 2.7.1). While this makes sense only in Euclidean space, the second i n t e r p r e t a t i o n makes X a first order differential operator on the space Coo (U) a n d will continue to make sense on manifolds.
66
2. LOCAL
THEORY
F i g u r e 2.7.1. T h e vector field X can be pictured as a s m o o t h field of arrows on U. I n t e r p r e t i n g X E 3r
as an operator, we write
X(g)
= " - . ' f bi -Og 7'
Vg~Coo(u).
i=1
We will give an abstract, purely algebraic definition of the t e r m "first order differential o p e r a t o r " and t h e n show t h a t such operators are exactly the elements of
~:(u).
We view Coo(U) as an algebra over R ( c o m m u t a t i v e and associative, with unity the constant function 1). While ~ ( U ) is a vector space over IR, it has more algebraic s t r u c t u r e t h a n that. For example, it is a module over the algebra Coo(U) under pointwise scalar multiplication:
C~176 • X(U) -~ X(U), (f, X ) ~-+ f X , where ( f X ) x = f ( x ) X ~ , for each x C U. T h e formal definition of a m o d u l e over an algebra is as follows. D e f i n i t i o n 2.7.1. Let F be an algebra over ]R, ~ a vector space over IR. Suppose t h a t there is an 1R-bilinear m a p F x ~ - - ~ JV[,
such t h a t
Vp,,~CF, V~C:~.
(po-).~=p.(o-.~),
T h e n 3V[ is said to be a module over F. If the algebra F has a unity ~ E F , it is further required t h a t ~.#=>, V # C :IV[. T h e m o d u l e is free if there is a subset {#~}~e~ C ~V[, called a basis of :M over F , such t h a t each # E :M has a unique representation r
tt = ~_~Pi " ftc~i, i=1
2.7. S M O O T H V E C T O R F I E L D S
67
with coefficients Pi E F, 1 < i < r. E x a m p l e 2.7.2. It should be clear that ~(U) is a module over C~176 Because U is an open subset of R n, this module is free with canonical basis {D1, D 2 , . . . , D , } . While 3~(U) will continue to be a module over Coo(U) when U is an open subset of a manifold, it will not be true, generally, that this module is free. Thus, while a module over an algebra is analogous to a vector space over a field, one must not press this analogy too far. We are going to give a deeper algebraic interpretation of ~(U). For this, we need some definitions. D e f i n i t i o n 2.7.3. Let F be an (associative) R-algebra with unity. A linear map A : F --~ F such that A ( f 9 ) = A ( f ) 9 + fA(9), V f, 9 E F, is called a derivation of F. Derivations of Coo(U) are also called first order differential operators. The set of derivations of F will be denoted by 9 L e m m a 2.7.4. The set of der'ivations 9
is a vector space over R under the
linear operations (aA1 + bA2)(f) = a A l ( f ) + bA2(f), Va, bE R, VA1, A2 E fD(F), V f E F. The proof of this is completely elementary and is left to the reader. If the algebra F is commutative as well as associative, define an operation of "scalar" multiplication by
(fA)(g)=f(A(g)),
V f, 9 E F ,
VAE 9
E x e r c i s e 2.7.5. If F is commutative, prove that f A is an element of ~ ( F ) , V f E F , VA E 9 This makes 9 a module over the algebra F. L e m m a 2.7.6. The space )~(U) is a Coo(U)-submodule of 9
Pro@ Indeed, given X E ~(U), write it as X = ~ - - ~ f i D i, i=1
where fi E C~(U), 1 < i < n. The Leibnitz rule for each partial derivative Di implies that
X(hg) = X(h)g + hX(9),
Vh, g ~ Coo(U).
[]
The main goal in this section is to prove that, all derivations of C~(U) are vector fields. T h e o r e m 2.7.7. The inclusion map )C(U) ~ [D(Coo(V)) is surjective. Before comiRencing the proof, we consider another algebraic structure on the module of derivations of an algebra.
68
2. LOCAL
THEORY
D e f i n i t i o n 2.7.8. If A1, As E ~D(F), then the Lie bracket [A1, A2] : F -~ F
is the operator defined by [A1, A2](f) = A I ( A 2 ( f ) ) - A 2 ( A I ( f ) ) , V f E F . This is also called the commutator of A1 and As. E x e r c i s e 2.7.9. Prove that the Lie bracket satisfies the following properties: 1. [A,, A2] 9 9 V / ~ I , A 2 9 ~)(F); 2. the operation [., . ] : 9 x 9 --* 9 is R- bilinear; 3. [A1, A2] : --[A2, A1] , V A i , As 9 D ( F ) (anticommutativity);
4. [AI,[A2, A3]] : [[A1,A2],A3] ~-[A2,[A1,A3]], VA1,A2, A3 9 ~)(F) (the Jacobi identity). Thus, we can think of the operation [.,.]: 9
x 9
-~ 9
as a bilinear multiplication making 9 into a kind of R-algebra. This algebra is nonassociative, however, with the Jacobi identity replacing the associative law. The algebra is also anticommutative and does not have a unity.
Remark. One way to remember the Jacobi identity is to notice that, by this identity, the operator [A, 9] : 9 -~ 9 is a derivation of the (nonassociative) algebra 9 VA 9 9 D e f i n i t i o n 2.7.10. A nonassociative algebra having the properties in Exercise 2.7.9 is called a Lie algebra. E x e r c i s e 2.7.11. It will follow from Theorem 2.7.7 that :~(U) is a Lie algebra. Here, you are to prove directly that iX, Z] 9 •(U),
VX, Y 9 )C(U).
For this, note that the composed operators X o Y and Y o X are second order differential operators, but show by direct calculation that the commutator X o Y Y o X is first order. L e m m a 2.7.12. If t E F is the unity, then A(ct) = 0, VA E 9
and Vc E R.
The proof is exactly like that of Lemma 2.2.19. In order to prove Theorem 2.7.7, we must establish the reverse inclusion
~D(C~(U)) c :~(u). For this, we need to show that a derivation of C ~ (U) can be localized to a derivative of the germ algebra ~5~, at each x E U. This is by no means evident. L e m m a 2.7.13 (Key Lemma). Let A E 9
f 9 C a ( U ) , and suppose
that V c U is an o;en set such that f t V -- O. Then A ( f ) I V -~ O.
Proof. Let x E V. By Theorem 2.6.1, we find ~ E C a ( U ) such that
~(~) =
0,
~l(U \ v) -
1.
2.7. SMOOTH
VECTOR
FIELDS
69
Indeed, since {z} is compact, w e find ~ E C a ( U ) such that g)(x) = 1 and supp(~b) C V. Then ~ = 1 - ~ is as desired. Since f l V =- O, we see that ~ f = f. Thus, A ( f ) = A((pf) = A ( ~ ) f + ~ A ( f ) . Hence A ( f ) ( x ) = A(~p)(x)f(x) + ~ ( x ) A ( f ) ( z ) . But f ( x ) = 0 = ~(x), and so A ( f ) ( z ) = O. Since x E V is arbitrary, A ( f ) I V = O. [] C o r o l l a r y 2.7.14. Let A C 9
f E C a ( U ) , and z E U. Then A ( f ) ( z ) depends only on A and the germ [fix E @z.
Proof. Let f , g E C~176 have the same germ If], = [g],. Choose an open neighborhood W C U o f x such that fl W = gl W. By the Key Lemma 2.7.13, ( A ( f ) - A(g))IW = A ( I - g)l W -- O, and so A ( f ) ( x )
=
A(g)(x).
[]
Given aJ E @~, x E U, there exists f E C a ( U ) such that aJ = [f]x. This is by Exercise 2.6.4. This allows us to make the following definition. D e f i n i t i o n 2.7.15. Given A E 9176
and x E U, Ax : @~ -~ IR is given by
Ax(aj ) = A ( f ) ( x ) ,
V~ = [f]x E |
where f E C a ( U ) . By tim above discussion, it is clear that Ax is well-defined, Vx E U, VA E P r o p o s i t i o n 2.7.16. I r A E 9
and x E U, then A , E T,(U).
Proof. Let [f]~, [g]x E @x, where f , 9 E C a ( U ) . Then, A~([f]~[g]x ) = A~[fg]~
= A(fg)(x) = (A(f)g + fA(g))(x) = A(f)(x)g(x) + f(x)A(g)(x) = Ax[f]~ex[g]~ + e~[f]~Ax[g]z. It is clear that A~ :
@ ~ -+
IR is linear; so Ax E T~(U).
[]
Given A C 9 define 2x : U ---, T(U) by A(x) = A~ E T~(U). This function satisfies p o A = idu. (Maps with this property are called sections of the tangent bundle.) If this section is smooth, then A E :E(U). Write
7, =
]+D+, i=1
relnarking that the smoothness of A is equivalent to ]+ E C ~ ( U ) ,
l
The following proposition completes the proof of Theorem 2.7.7.
then A E )C(U) and, as a derivation of C a ( U ) , the vector field 2x is identical with A.
P r o p o s i t i o n 2.7.17. /f A E 9
70
2. L O C A L T H E O R Y
Proof. Consider the coordinate functions x i E U, 1 < i < n. Let f i = A ( x i) e C~176 Then
fi=
Dj
(x i ) = fi,
l
[]
From now on, we think of vector fields either as sections of T(U) or as derivations of C ~ (U), but we denote the Lie algebra and Coo (U)-module of all such fields only by :~(U). E x a m p l e 2.7.18. One is often interested in certain Lie subalgebras of X(U). We give here an example on the group manifold Gl(n) to which we will be returning later. Write T(GI(n)) = Gl(n) x DI(n). A vector field X E X(GI(n)) can be written XQ = (Q, .~Q), where 3~: Gl(n) -+ DI(n) is smooth. Such a field is left-invariant if, for each P C Gl(n),
L p . ( X ) = X. This means that Lp.Q(XQ) = XpQ, for all Q E Gl(n), or equivalently, that P X Q = XpQ. In particular, such a field is completely determined by its value A -- )(I at I. We have
X p = P X I = P A = RA(P),
V P E Gl(n).
That is, a left-invariant vector field is identified with RA: Cl(n) -~ ~ ( n ) , where A c 9~(n). As a vector space, the set 9[(n) C :~(Gl(n)) of left-invariant vector fields is identified with ff)~(n) via R A e-~ A. In 9~(n), define the bracket to be the usual commutator of matrices
[A, B] = A B - BA,
V A, B c 928(n).
E x e r c i s e 2.7.19. Prove that the commutator operation makes 9Jr(n) into a Lie algebra and that the canonical identification 93I(n) = 9[(n) turns the commutator into the Lie bracket of vector fields. More precisely,
IRA, RB] = R[A,B], VA, B C 9J/(n); so g[(n) is a Lie subalgebra of X(Gl(n)) isomorphic to the Lie algebra 93I(n) of n • n matrices under the commutator bracket. As noted earlier, a diffeomorphism z : U --+ V onto an open subset V c_ ]~n can be thought of as a change of coordinates. Recall that such a change of coordinates translates formulas in differential calculus to new formulas relative to the new coordinates. In the present context, vector fields X E 3~(U), viewed as first order differential operators, are carried to vector fields z. (X) E 3~(V). One can think of this as a change of formula for the operator X in terms of the new coordinate system {V, z l , . . . , zn}.
2.8. LOCAL FLOWS
71
We analyze this "push forward" of vector fields more carefully. pullback o p e r a t i o n
Recall t h e
z* : c o o ( v ) -~ c o o ( u ) , z * ( f ) = f o z. T h e following is completely straightforward to check. L e m m a 2.7.20. I f z : U --* V is a diffeomorphism between open subsets of ]Rn, then z* : C ~ 1 7 6 --* Coo(U) is an isomorphism of algebras. D e f i n i t i o n 2 . 7 . 2 1 . If z : U -~ V is a diffeomorphism between open subsets of Nn, t h e n z. : ~ ( U ) ~ ~ ( V ) is defined by setting z,(X)(f)
= ( z - 1 ) * ( X ( z * ( f ) ) ) = X ( f o z) o z - 1 ,
for all X E :E(U) and all f E Coo(V). T h e fact t h a t z , ( X ) E X(V) is elementary, as is the following. These are left as exercises for the reader. L e m m a 2.7.22. I f z : U --~ V is a diffeomorphism between open subsets of N n, then z. : )~(U) --~ ~ ( V ) is an isomorphism of Lie algebras. E x e r c i s e 2 . 7 . 2 3 . Show that, if X E X(U) is viewed as a s m o o t h section of the t a n g e n t bundle p : T ( U ) --~ U, then the section Z = z. (X) of the t a n g e n t bundle 7r : T ( V ) --, V is given by Zr = dzz-~(r162 , V ( E V.
2.8. L o c a l F l o w s At this point we have two ways of viewing a vector field X E ~ ( U ) . It is a s m o o t h section of the t a n g e n t bundle and it is a derivation of the algebra C ~ (U). In this section, we view X as a field of infinitesimal curves and show that, in this guise, X is equivalent to a system of ordinary differential equations (O.D.E.) on U. Definition 2.8.1. to E (a,b) is
Let s : (a,b) ~
~(t0) = S.~o
U be smooth.
(d) d/to
T h e velocity vector of s at
c T~(~o)(U)
Remark. I f 7 = t + t 0 , a - t o < t < b - t o , t h e n c~(t) = s(7) has velocity d~(t) = i(~-) = i ( t + to). In particular, i(t0) = /~(0) = (cr)s(to). We will write this infinitesimal curve as (s)s(to). D e f i n i t i o n 2 . 8 . 2 . T h e m a p i : (a,b) --* T ( U ) is called the velocity field of t h e s m o o t h curve s : (a, b) --~ U. R e m a r k t h a t p o i = s, where p : T ( U ) --~ U is the bundle projection. remark t h a t ~ is smooth.
Also
D e f i n i t i o n 2.8.3. Let X E ~ ( U ) and z0 E U. An integral curve to X through z0 is a s m o o t h curve s : ( - 5 , c) -* U, defined for suitable 6, e > 0, such t h a t s(0) = x0 and i ( t ) = Xs(t), - 6 < t < e.
72
2. LOCAL THEORY
Suppose t h a t s is an integral curve to X E X(U) through x0 E U. W r i t e n
X = E
ffDi
i=1
and
8(t) ~-
(X l ( t ) , . . . , x n ( t ) ) .
At each t E ( - 5 , c), the Jacobian m a t r i x of s is
r dxl (t)l
The vector -~lt E Tt(-5, e) = R ~ is the canonical basis element 1 E R 1. So
[~xlrt)l dt.~ I
E
-~ dx ~ 9 -~(t)D{,s(t) E Ts(t)(U).
=
i=1 But
n Xs(t) = E
fi(xl(t)'''''xn(t))Di,s(t)"
i=l Thus, s is an integral curve to X if and only if
dx i dt (t) = f f ( x l ( t ) , . . . , x n ( t ) ) , - 5 < t < c, 1 < i < n. This is an (autonomous) system of O.D.E. with solution s s u b j e c t to the initial condition s(0) = x0. The existence and uniqueness of integral curves is g u a r a n t e e d by the following theorem. Theorem
2.8.4. Let V C_ ]W and U C_ ~n be open, let c > O, let
fiEC~((-c,c)
xYxU),
l
and consider the system of O.D.E. with parameters b = (b1. . . . , b~) E V
(*)
dxi
d-T .
fi(t,b, xl(t,b),
.
.
.
,xn(t,b)),
l < i < n.
If a = ( a l , . . . , a n) E U, there are smooth functions x i(t,b),
l
defined on some nondegenerate interval [-5, e] about O, that satisfy the system (*) and the initial condition xi(0, b) = a i,
1
Furthermore, if the functions 2i(t,b), 1 < i < n, on [-5,~] and satisfying the same initial condition, [-5, e] N [ - 5 , ~]. Finally, if we write these solutions emphasize the dependence on the initial condition),
give another solution, defined then these solutions agree on as x i = xi(t, b, a) (in order to there is a neighborhood W of
2.8. L O C A L F L O W S
73
a in U, a neighborhood B of b in V , and a choice of e > 0 such that the solutions x i ( t , z , x ) are defined and smooth on the open set ( - e , ( ) x B x W C_ ]R~+'+1.
This is the well-known theorem giving the existence, uniqueness, a n d s m o o t h dependence on initial conditions a n d parameters of solutions of systems of ordinary differential equations. The proof is given in A p p e n d i x C (where the s m o o t h dep e n d e n c e on initial conditions will require some elementary facts from calculus in B a n a c h spaces). D e f i n i t i o n 2.8.5. T h e system (.) is a u t o n o m o u s if the functions f i do n o t d e p e n d ont, l
• W-~
U
(written dp(t,a) = ~ t ( z ) ) , where W is a suitable open neighborhood of m0 in U, such that 1. q)0 : W --~ U is the inclusion W ~-+ U; 2. ~tl+t2(x) = ~st, (~t2(z)) whenever both sides of this equation are defined. If z E U, the flow line through z is tile curve or(t) = g)t(z), - e < t < e. T h e o r e m 2.8.8. Let X C )~(U) and xo E U. Then there is a local flow around xo such that the flow lines are integral curves to X . Two such local flows agree on their c o m m o n domain. P r o @ By T h e o r e m 2.8.4, we find a n open neighborhood W of x0 in U and a n u m b e r e > 0 such t h a t the integral curve Sz(t) through z is defined for - e < t < e a n d for all z E W. By the s m o o t h dependence on initial conditions, we define a smooth map q~ : ( - e , e ) x W--+ U
by 9 (t, ~) = s~(t). T h e o r e m 2.8.4 also assures us that, if ~ is another local flow a r o u n d z0 with flow lines integral to X , t h e n 9 a n d (~ agree on their c o m m o n domain. We must show t h a t dp is a local flow. Since q)0(z) = s~(0) = z, it is clear t h a t dp0 is the inclusion m a p W ~ U. F i x to E ( - e , e) and z c W and define the curves s(t) = s~(t + to) = e~+~o(~) and These are defined for small values of t and are both integral to X. T h e y satisfy s(O) = s~(to) and or(0) s~(to)(0) = s~(t0); so s(t) = a(t) whenever b o t h sides are defined. T h a t is, ~bt+to(z) = d&(d&o(Z)) whenever b o t h sides are defined. []
74
2. LOCAL THEORY
F i g u r e 2.8.1. Flow lines for X = xlD1 d- x21)2 D e f i n i t i o n 2.8.9. The local flow 9 associated to X E X(U) as in Theorem 2.8.8 is said to be generated by X. Also, the vector field X is called the infinitesimal generator of the local flow qb. E x a m p l e 2.8.10. Let U = ~2 and X = xlD1 + x2D2. The integral curves s(t) = (x 1(t), x2(t)) will satisfy dx I
dt dx 2 dt
-
-
-
-
xl
X 2"
The solution curve with initial condition s(0) = (a 1, a 2) is
s(t) = (ale t,a2e t) (see Figure 2.8.1). All of these solutions are defined for - o o < t < cx~. Remark that ~tl +t2 ( al, a2) = ( aletl+t2, a2et~+t2) = (Ptl (ePt2 ( al, a2)) 9 This curve is stationary for ( a l , a 2) = (0,0) and in all other cases follows a radial trajectory out of the origin, but is not parametrized linearly. The "speed" of the trajectory increases proportionally with the distance from the origin. E x a m p l e 2.8.11. Let U = ~2 and X = xlD1 - x2D2. The integral curves s(t) = (x l(t), x2(t)) will satisfy
dx 1 z xl dt dx 2 X 2" dt -
-
2.8. L O C A L F L O W S
75
F i g u r e 2.8.2. Flow lines for X = xlD1 - x2D2 T h e solution curve with initial condition s(0) = (a 1, a 2) is
s(t) = (ale~,a2e -~) (see Figure 2.8.2). All of these solutions are defined for - o c < t < oo. Again
~tiq_t2(al,a 2) = (algtiq-t2 a2g -ti-t2) = (~)tl((~Jt2(al,a2)), This curve is s t a t i o n a r y for (a*,a =) = (0,0). Trajectories of points on the x2-axis (other t h a n the origin) stay on the x2-axis and head toward the origin. Trajectories of points on the xl-axis (other t h a n the origin) stay on t h a t axis a n d head away from the origin. T h e remaining trajectories follow hyperbolic paths a s y m p t o t i c to the coordinate axes. E x e r c i s e 2 . 8 . 1 2 . As in Examples 2.8.10 and 2.8.11, discuss the vector field X = x2Di - miD2 on R 2. In particular, sketch the flow lines and show that they are defined for - o o < t < oo. d Let a E IR and E x e r c i s e 2 . 8 . 1 3 . On U = IR, consider the vector field X = e ~ aT" c o m p u t e the integral curve Sa to X through a. Be sure to find the largest open intervM (rl, w) on which sa(t) is defined. E x e r c i s e 2 . 8 . 1 4 . Let X E 5E(U) generate a local flow
@:(-e,e) x W-+U. Prove that, for each q E W, (r
= x<(q),
-e < t <
~.
So far, we have used vector fields to differentiate functions. Now we will show how a vector field can be used to differentiate another vector field. Let X E Y(U),
76
2. L O C A L T H E O R Y
q C U, and let r : ( - e , e) • W --~ U be a local flow about q generated by X. Let Y E ~(U). Generally, the vector
(r
Tq(U),
e
defined for - ( < t < c, differs from Yq, although they are equal if Y = X (Exercise 2.8.14). The difference quotient
z~(t):
(r
- yq c Z ( u ) t can be thought of as the average rate of change of Y near q along the integral curve to X through q. This lives in the vector space Tq(U) = N n, and so limt--o Zq(t) makes sense. If this limit exists Vq E U, we obtain a (conceivably not smooth) vector field L x ( Y ) : lira r - Y t~0
D e f i n i t i o n 2.8.15. If s of the field Y by the field X.
t
exists on W, it is called the Lie derivative (on W)
Remark. One can also define the Lie derivative of a function by the formula L x ( f ) = lim ~5~(f) - f t~0
t
The use of %t instead of gP-t is due to the fact that ~" pulls functions back, while e t . pushes vector fields for~ward. The reader should have no trouble seeing that
Lx(f) = X(f). T h e o r e m 2.8.16. If X, Y C )~(U), the Lie derivative of Y by X is defined and
smooth throughout U and L x ( Y ) : [X,Y]. The proof requires a couple of lemmas. We will fix q 6 U and a neighborhood W of q so that 9 : ( - c , e) x W --* U is defined. Write
Zq(t)
=~
(i(t)Di,q
i=1
and remark that limt~o Zq(t) exists if and only if limt~o Ci(t) exists, for 1 < i < n. L e m m a 2.8.17. The limit Aq = limt~o Zq(t) exists if and only if, for each f E C~(U), limt-~0 Zq(t)(f) exists, in which case this limit is Aq(f).
Proof. Assume that the limit Aq = ~i~=1 aiDi,q exists. That is,
1 < i < n. Then, for arbitrary f E Cc~(U), it is clear that
n
lira Zq(t)(f) = t~u~hn!~-~. r
t~O
,f ~__, i O f , ' Ox i (q) = 2._ a -~-~xiiq) = Aq(f)
i=1
i+1
exists. For the converse, suppose that, for each f E C~176 lim Zq(t)(f) = }i~ E
t~O
i=l
~i(t
)~@(q)
2,8. L O C A L F L O W S
77
exists. Choose f = x j and define a 3 = lira Z q ( t ) ( x 3) = }~1~ cJ(t), t~O
1 < j < n. Thus, l i m t ~ o Zq(~) = Aq exists and, as noted above, it follows t h a t
lira Z q ( t ) ( f ) = A q ( f ) ,
t--+O
for a r b i t r a r y f E C ~ ( U ) . Lemma
2.8.18.
[]
Given f C C~
there is a function
9 e c~176
e) x w )
such that 1. f ( ( P _ t ( x ) ) = f ( x ) - t g ( t , x ) , V x c W , - r < t < e; 2. X ~ ( f ) = g(O,x), V x c W . Proof. Define h ( t , x ) = f(d2_t(x)) - f ( x ) ; so h E C ~ ( ( - e , e ) x W ) and h(0, x) = 0. To simplify notation, we denote the partial of h with respect to t by Jz(t,x). Define g C C ~ 1 6 2 e) x W) by
/o' Then
du
= [ ' i~(v, x) dv = h(t, ~)
-
h(O, x)
= f(q~-t(x)) - f(x), giving the first assertion. For the second, consider
g(0, ~) = ~i2~ g(t, x) = lira f ( ( P - t ( x ) ) - f ( x ) t~O
--t
= lim f ( q & ( x ) ) - f ( x ) t ~O
[;
= X~(f). [] Proof of Theorem 2.8.16. Let f E C ~ ( U ) and let g be as in the preceding lemma. Let gt : W --+ U be given by gt(x) = g ( t , x ) . Thus, X ( f ) = go = l i m t ~ 0 g t , and so ~i~ YG(q)(gt) = Y q ( X ( f ) ) .
78
2. L O C A L T H E O R Y
We now c o m p u t e
lira Zq ( t ) ( f ) =
t~O
lim Ye,(q)(f o 62_t) - Yq(f)
t~O
t
= lira Y ~ ( q ) ( f - tgt) - Yq(f)
t~O
t
= lim Yc't(q)(f) - Yq(f) _ ~imY~(q)(gt)
t~O
t
= lim Y ( f ) ( ~ t ( q ) ) - Y ( f ) ( q ) _ lim Yq,t(q)(gt) t--+O
t
t--+O
= Xq(Y(I)) - Yq(X(f)) = [X, Y]q(f). Since f is arbitrary, L e m m a 2.8.17 gives the desired conclusion.
[]
Let X , Y E X(U) and let (b, ~ denote the respective local flows generated by these fields a b o u t some point q E U. We can choose 5q > 0 so t h a t #Ptk~s(q) a n d q2sg2t(q) are defined, -($q < s,t < 5q. As q E U varies, these b o u n d s 5q will also vary. T h e local flows vary too, b u t they agree on overlaps by T h e o r e m 2.8.4. D e f i n i t i o n 2.8.19. The local flows of X a n d Y c o m m u t e on U if
62tq2s(q) = qJsOt(q),
-5q < s,t < ~Sq, Vq E U.
T h e vector fields themselves c o m m u t e on U if [X, Y] ~ 0 on U. 2 . 8 . 2 0 . The vector fields X and Y commute on U if and only if their local flows commute on U.
Theorem
Proof. If the local flows c o m m u t e on U, then, for - a x < t < 5,, ~ t carries any flow line { ~ s ( x ) [ - ~ < s < 6~} of 9 onto another flow line of ~. By t a k i n g the infinitesimal curve point of view, we see immediately t h a t (~t).:~(Yx) = Y~,(:~), - 5 ~ < t < ($x, V x E U. T h a t is,
tArv,~1~ = lira 4)-t. (Y) - Y _ lira Y - - Y t--0 t t~0 t
I
0
t h r o u g h o u t U. For the converse, we assume t h a t [X, Y] = 0 on U and deduce t h a t the local flows commute. Let q E U, fix s E (-5q,($q), and let q' = ~s(q). Define v : (-5q, 5q) --+ Tq, (U) by the formula v(t) = 4)_t.(Ye~(q,)). (Here a n d elsewhere, in an a t t e m p t to streamline notation, we drop the subscript { on differentials f.~.) T h e n
2.8. L O C A L F L O W S
79
v(t) is a differentiable curve in the vector space Tq, (U) = R ~, and dv = lira (~-t-h),(Y,L+,(q,))- q~-t,(Y~dq,))
dt
h~o
h
= lim q)-t. h~0
~-h.(Y~+,xq'))
= ~ - t * lira h~0
-
Y~(q')
h
d~-h*(Y~h(~t(q'))) -- Y~L(q') h
C r,i,t(q,)(g) = ~ - t * [X, Y]e,(q') = 0,
- 6 q < t < 6q. It follows t h a t v(t) is constant o n (-6q, 6q); so
9 _t.(Yeodq,)) : Yq,,
--(~q < t < (~q.
B u t q' ranges over a(s) = 9~(q), -6q < s < 6q, an integral curve to Y. Thus, ~(s) = Y~(~) a n d 4)t.(~(s)) = Y~d~(4) as s and t range i n d e p e n d e n t l y over (-6q, 6q). Therefore, ~ t o a is also an integral curve to Y with initial condition (I)t(a(0)) : 'I~t(q). B u t ~ t g ~ ( q ) = 9 ~ t ( q ) , - 6 q < s , t < 6q, by the uniqueness part of Theorem 2.8.4. [] E x e r c i s e 2 . 8 . 2 1 . Given A c glI(n), view the right translation operation R A : Gl(n) -~ ~ ( n ) as a vector field RA E Z ( G I ( n ) ) . (1) For
show t h a t the local flow generated by RA has the formula
Vt C N, VQ C Gl(2). In particular, we o b t a i n a 9lobal flow 9 : ~{ • Gl(2) -~ Gl(2). (2) Note t h a t the formal definition
e~ A = [ + t A +
t2A2 tnAn 2! + " " + n ! +""
yields
C o m p u t e e tB for the m a t r i x
a n d make a n educated guess of a flow on Gl(2) generated by RB. Prove t h a t your guess is correct.
80
2. LOCAL
THEORY
2.9. C r i t i c a l P o i n t s a n d C r i t i c a l V a l u e s Let U C_ ll~n a n d V C ]l~m be open and let 9 : U -~ V be smooth.
D e f i n i t i o n 2.9.1. A point x E U is a regular point of (I) if 9 .~
: T x ( U ) -~ T~(~)(W)
is surjective. Otherwise, x is a critical point. Thus, s m o o t h maps from lower dimensions to higher dimensions have only critical points. At the other extreme, if 9 has only regular points, it is a submersion.
D e f i n i t i o n 2.9.2. A point y E V is a critical value of 4) if (I)-l(y) contains at least one critical point of 4). Otherwise, y is a regular value of ~. You must take care with these terms. If ~ - l ( y ) = 0, t h e n this set contains no critical points. T h a t is, a point y E V t h a t is not a value of 9 at all is a regular value of (I)! 2 . 9 . A . S a r d i s t h e o r e m . In this subsection, we prove the following fundam e n t a l result of Sard. It has i m p o r t a n t applications in topology, some of which will be t r e a t e d in the next chapter.
T h e o r e m 2.9.3. I f ~ : U --~ V is a smooth map, then the set of critical values has Lebesgue measure zero. We will u n d e r s t a n d the term "almost every" to m e a n "Lebesgue almost every". Thus, almost every point of V is a regular value. Our proof will follow closely t h a t given by J. Milnor [30]. First, however, we consider some examples a n d corollaries. E x a m p l e 2.9.4. It is well known t h a t one can construct a continuous surjection s : ~ --~ I~2 (a "space filling curve"). However, if s is smooth, every true value of s is a critical value; so s(R) C R 2 has measure zero. Smooth curves c a n n o t be space filling. More generally, smooth maps from lower to higher dimensions always have images of measure zero.
C o r o l l a r y 2.9.5. Let { ~ : Ui --~ V}i=l N , 1 <_ N < oc, be an at most countable family of smooth maps, each Ui C_ l~n~ being open. Then almost every y E V is simultaneously a regular value of Oi, 1 ~_ i < N + 1. Proof. Let Ci denote the set of critical values of Oi, 1 ~ i < N + 1. T h e n C = [_iN=1 Ci is a set of Lebesgue measure zero and the complement of C is the set of s i m u l t a n e o u s regular values. [] T h e fact t h a t almost every point is a regular value says t h a t the s i t u a t i o n in the following theorem is somehow "generic".
T h e o r e m 2.9.6. If (~ : U -~ V is a smooth map and if y E V is a regular value, then 9 - l ( y ) is a smooth submanifold of U of dimension n - m. Proof. If (I)-l(y) = 0, this is a submanifold of U of any desired dimension a n d we are done. We consider the interesting case, therefore, in which the regular value y is actually a value of 4). Let x0 E ~ - l ( y ) . Since JO(xo) has m a x i m u m rank m, there is a neighborhood Wxo of x0 in U such that J ~ ( z ) has rank m, Vz C W~ o. Let W = U Wx,
2.9. C R I T I C A L
POINTS
81
an open neighborhood of ~ - i (y) in U on which 9 has rank m. The assertion follows from Theorem 2.5.3. [] We turn to the proof of Theorem 2.9.3. Accordingly, let r : U ~ V be a smooth map, where U C IRn and V C_ IRm are open. The critical set C c_ U consists of those points x for which rank Jq5~ < m and we must prove that O(C) c_ V has Lebesgue measure zero. As usual, write q5 = (r . . . , qsm). D e f i n i t i o n 2.9.7. For each integer k _> 1, Ck C_ C is the set of points x E U such that all mixed partials of dPi of order < k vanish at x, 1 < i < m. It is clear that we obtain a nest C _D C1 _DC2 _D ... _D Ck _D ... of closed subsets of U. The basic estimates behind Sard's theorem are contained in the following proof. P r o p o s i t i o n 2.9.8. For ]~ >_ 1 sufficiently large, the set ~(Ck) has Lebesgue measure zero.
Pro@ Let Q C U be a compact cube. Since Ck n U is covered by countably many such cubes, it will be enough to find a value of k, depending only on m and n (not on Q) such that ~(Ck a Q) has Lebesgue measure zero. Let ~ denote the edge length of Q. Let p range over Ck a Q. The kth order Taylor series, expanded about p, takes the form 9 (p + v) - ~(p) = n(p, v), where the remainder term satisfies a uniform estimate
NJ~(P,V)II ~ CIIV]Ik+l, for all p C Ck n Q and all v E R n such that p + v E Q. Subdivide Q into r n subcubes of edge length 5 / r and let Q~ be one of these subcubes containing a point p E Ck. Every point of Q~ has the form p + v, where Ilvll <
- -
r
By the above estimate on the remainder term, we see that q~(Q~) lies in a cube, centered at O(p) and having edge length e/r k+l, with e = 2c(~v~) k+l. Thus, ~(Ck n Q) is covered by a union of at most r n cubes with total measure
V(r) <_ C > n-(k+l)m. For large enough k, the exponent of r is negative and limr~o~ V(r) = O.
[]
Sard's theorem is trivial when n = 0, so we make the inductive assumption that the theorem has been proven for the case U C_ R n - l , some n > 1, and deduce the case U C_ R n. L e m m a 2.9.9. Let p E C \ C1. Then there are coordinate neighborhoods of p in U and of ~2(p) in V relative to which the formula for ~2 becomes
~(yi ..., yn) = (yl ~ 2 ( r
yn),..., ~ (y~,..., yn)).
82
2. LOCAL THEORY
Pro@ By the definition of C1, p • C1 implies that some first order partial of some coordinate function of 4p fails to vanish at y. Suitably permuting the coordinates in ]Rn and ]R"~, we lose no generality in assuming that 0qr o z 1 (p) r o. Thus, by an application of the inverse function theorem, the map
( x l , x 2 , . . . ,x n) ~_+ ( q b l ( x l , x 2 , . . . , x n ) , x 2 , . . . ,x n) = ( y l , y 2 , . . . ,yn) is a diffeomorphism of some open neighborhood W o f p onto an open subset O(W) C IRn. This defines a coordinate system with the required property. [] P r o p o s i t i o n 2.9.10. The set ~ ( C \ C1) has Lebesgue measure zero.
Pro@ It will be enough to show that, for each p E C \ C1, there is a neighborhood Wp o f p in U such that ~ ( C n Wv) has measure zero. We assume that m _> 2 since, when m = 1, C = C1 and the assertion is vacuously true. By Lemma 2.9.9, we can assume that there is a neighborhood Wp in which carries each point (t, x 2 , . . . , x n) into the hyperplane {t} x IRm-1. For fixed t, the restriction of r to Wv Cq ({t} x II~~-1) can be viewed as a map ~t of that set into the hyperplane {t} x R m-1. Since the coordinate t is preserved, the critical set of r is C n Wp N ({t} x N ~ - I ) . Hence, by the inductive hypothesis, this is carried by ~t onto a set of (m - 1)-dimensional measure zero. Integrate with respect to the t coordinate, concluding by Fubini's theorem that ~ ( C n Wp) has m-dimensional measure zero. [] P r o p o s i t i o n 2.9.11. For each integer k >_ 1, the set q~(Ck \ Ck+l) has Lebesgue
measure zero. Proof. Again, it will be enough to show that, for each p E Ck \ Ck+l, there is a neighborhood Wp of p in U such that ~(Ck n Wp) has measure zero. Let ~ : U --* R be a kth order partial of a coordinate function ~ such t h a t some first order partial of ~ fails to vanish at p. Without loss of generality, we assume that O~ OX 1 (p) ys O.
Of course, ~ vanishes identically on Ck. The inverse function theorem again gives a change of coordinates
( x l , x 2 , . . . , X n) b--+ ( ~ ( x l , . . . , x n ) , x 2 , . . . , X
n) ~_ ( y l , y 2 , . . . , y n ) ,
defined on some neighborhood Wp of p, thereby coordinatizing Ck n Wp as a subset of the hyperplane {0} x ]Rn-1. Every point in this set is a critical point of the restriction q~0 of q5 to U n ({0} x R~-~), so the inductive hypothesis implies t h a t ~(Ck n Wp) = 02o(Ck n Wp) has Lebesgue measure zero. [] C o r o l l a r y 2.9.12. For each integer k >_ 1, the set ~(C'..Ck) has Lebesgue measure zero.
Proof. Indeed, C \ Ck = (C \ C1) U (C1 \ C2) U . . . U (Ck-1 \ Ck).
[]
By this corollary and Proposition 2.9.8, the proof of the inductive step is complete.
2.9. C R I T I C A L P O I N T S
83
2.9.B. N o n d e g e n e r a t e c r i t i c a l p o i n t s * . Of special note are the critical points of a smooth, real-valued function. Suppose that f : U --+ R is such a map, where U C_ R n is open. We present some facts that are the beginnings of "Morse theory", a remarkable application of critical point theory to topology due to M. Morse. This introduction to Morse theory will be continued in Sections 3.10 and 4.2. E x e r c i s e 2.9.13. Let p E U be a critical point of f.
If X , Y C X(U), then
X p ( Y ( f ) ) = Y p ( X ( f ) ) , and this number depends only on the tangent vectors Xp, Yp, not on their extensions to fields on U. Each element Yp C Tp(U) can be extended to a vector field Y E ~(U). Indeed, if Yp = ~ n = l aiDi,p, we can define Y = ~ i ~ 1 aiDi" This remark, together with the exercise, insures that the following definition makes sense. D e f i n i t i o n 2.9.14. If p C U is a critical point of f, the Hessian of f at p is the symmetric, bilinear form Hp(f) : Tp(U) • Tp(U) ~ ]~ defined by
Hp(f)(Xp, Yp) = X p ( Y ( f ) ) . D e f i n i t i o n 2.9.15. The critical point p E U of f is nondegenerate if the symmetric matrix representing the Hessian Hp(f), relative to some choice of basis, is nonsingular. The (Morse) index A of the critical point is the number of negative eigenvalues of this matrix. E x e r c i s e 2.9.16. Show that, relative to the standard basis of Tp(U), the matrix representing the Hessian is the matrix c~2f of 2nd partials of f at p. It is straightforward to check that a matrix representing the Hessian with respect to some coordinate system is nondegenerate of index )~ if and only if this is true relative to every coordinate system. E x a m p l e 2.9.17. Let U = 1Rn and let f : R n --~ Ii~ have the formula (*)
f ( z l , . . . , z n) = f(0) - ~-](zi) 2 + i=l
(zi) 2, i=A+I
relative to suitable coordinates. Then 0 is the only critical point and the Hessian at 0 is represented by the matrix In-),
"
It is evident that this matrix is nondegenerate of index A. The following, due to M. Morse, asserts that this is essentially the only example. T h e o r e m 2.9.18 (The Morse Lemma). Let p E U be a nondegenerate criticaI point
of index )~ of the smooth function f : U --~ R. Then there is an open neighborhood Up of p in U and a smooth change of coordinates (i.e., diffeomorphism) z : Up --+ W onto an open neighborhood W of 0 in ]~n such that, relative to the new coordinates z=(zl,...,zn),
84
2. L O C A L T H E O R Y
(1) p = 0;
(2) f has the formula (,). We need a preliminary lemma. By a translation, we assume that the critical point p = 0. L e m m a 2.9.19. In a suitable neighborhood V of O in U, there arc defined smooth,
real-valued functions his = hsi, 1 < i , j < n, such that f ( x l , . . . ,x ~) = f ( O ) + ~ , xixJhij(xl,... ,X n)
i,j=l and such that the matrix 2[hij(0)] represents the Hessian Ho(f). Proof. Choose V to be the open e-ball centered at 0, where e > 0 is small enough t h a t V C_ U. Write f ( x l , . . . ,x n) - f(O) = =
(tx~,... , t x n ) d t
~fa liOf'l
x -f~xiitx , . . . , t x ~ ) d t .
z=l
0
Thus, setting
gi(xl,... ,X n) ~--
~ x i ( t x , ' " ,txn)dt,
we write n f(xl,...,X
n)
- -
f(O) : ~-~X i gi( X 1 ,...,xn) 9 i=1
By differentiating the formula for gi under the integral sign, we compute
OV(o ) ~o~soz* "
~(o)Note t h a t
g i ( 0 ) = ~-~(0) = 0 ,
l
since 0 is a critical point of f. Thus, we can apply the same construction to each gi, obtaining smooth functions gis such that
f ( x l , . . . , x n) - f(O) = ~
xixSgis(xl,...,x~).
i,j=l Here,
giS(Xl'''"xn) =
~
1
OxJ ' txl "" . , t x n) dr,
SO
g~s(O) - 20xJOx ~" "" If we set his = (gij + gsi)/2, all assertions follow.
[]
2.9. CRITICAL POINTS
85
Proof of Theorem 2.9.18. Suppose inductively t h a t there are coordinates u = ( u l , . . . , u n) in a neighborhood V C_ U of 0 such t h a t (**)
~tiuJHij(u)
f ( u ) - f(0) = =:l=(?-tl)2 q - . . . 4- (ur--1) 2 -~ ~
i,j>_r on V, where the m a t r i x [Hij] is symmetric and nonsingular. L e m m a 2.9.19 gives the case r = 1, while the case r = n implies T h e o r e m 2.9.18 by a p e r m u t a t i o n of coordinates. It remains t h a t we prove the inductive step. As in the s t a n d a r d proof t h a t symmetric matrices can be diagonalized, a suitable linear change in the last n - r + 1 coordinates allows us to assume that H , , ( 0 ) r 0, hence t h a t IH,-,(u)l > 0 and is smooth t h r o u g h o u t a neighborhood V' c_ V of 0. We set h(u) = IHrr(u)l 1/2 on V' a n d introduce new coordinates v = v(u) by setting
vi =
u ~,
i ~ r,
[h(u)(u
r + E~>~ig~r(u)/g~(u)),
i = r.
T h e n v(0) = 0 a n d det J(v)(O) = h(0) > 0, so the inverse function theorem guarantees t h a t v = ( v l , . . . , v n) is a coordinate system on some neighborhood V" C_ V' of 0. We leave to the reader the rather tedious exercise of s u b s t i t u t i n g u i = v i, i ~ r, and
~,. _
_ _?3r
---~iH~4u(~))
h(~(~))
~.
H,.,.(~(~))
into the e q u a t i o n (**) and collecting terms. This exercise yields
f ( v ) - f(O) = •
2 :t=... + ( v ' ) 2 + ~
vivfHij(v),
i,j>r for suitable s m o o t h functions H i j . Symmetrizing these coefficients by
Hij(v)
= Hij(v) + Hji(v) 2
completes the inductive step.
[]
C o r o l l a r y 2 . 9 . 2 0 . Nondegenerate critical points are isolated. By contrast, degenerate critical points m a y easily fail to be isolated. For example, the function f ( x , y) = x 2 has the entire y-axis as its set of critical points, while f ( x , y) = x~y ~ , n, m > 1, has the union of b o t h axes as its critical set. Examples of isolated, b u t degenerate, critical points include f ( x ) = x 3, having 0 as its sole critical point, and the "monkey saddle" f ( x , y) = x a - 3xy 2. This latter has only the origin as critical point a n d the Hessian there is the zero matrix.
CHAPTER 3
The Global Theory of Smooth
Functions
Our present goal is to extend the theory of smooth functions, developed on open subsets of IRn in Chapter 2, to arbitrary differentiable manifolds. Geometric topology becomes an essential feature.
3.1. S m o o t h Manifolds and Mappings Let M be a topological manifold of dimension n. Tile locally Euclidean property allows us to choose local coordinates in any small region of M. D e f i n i t i o n 3.1.1. A coordinate chart on M is a pair (U, g)), where U is an open subset of M and qo : U --+ R n is a homeomorphism onto an open subset of IRn. One often writes ~o(p) = (zl(p),... ,zn(p)), viewing this as the coordinate ntuple of the point p E U. Relative to such a coordinatization, one can do calculus in the region U of M. The problem is that the point p will generally belong to infinitely many different coordinate charts and calculus in one of these coordinatizations about p might not agree with calculus in another. One needs tile coordinate systems to be smoothly compatible in tile following sense. D e f i n i t i o n 3.1.2. Two coordinate charts, (U, V)) and (V, ~b) on M are said to be C ~ - r e l a t e d if either U N V = (~ or o~-1 :W(UnV)~(UNV) is a diffeomorphism (between open subsets of IRn). This is illustrated in Figure 3.1.1. We think of ~ o ~ - l as a smooth change of coordinates on U N V. Thus, on U n V, functions are smooth relative to one coordinate system if and only if they are smooth relative to the other. Indeed, differential calculus carried out in UNV via the coordinates of ~(UNV) is equivalent to the calculus carried out via the coordinates of ~b(U n V). The explicit formulas will, of course, change from the one coordinate system to tile other. Furthermore, piecing together these local calculi produces a global calculus on M. The concept that allows us to make these remarks precise is that of a smooth atlas. D e f i n i t i o n a . l . a , a C ~ atlas on M is a collection r = {(U~, p~)}~e~ of coordinate charts such that 1. ( U ~ , ~ ) is C~176
s. M = U ~
to (U~,F~), Va,/3 E 9.1;
u~.
D e f i n i t i o n 3.1.4. Two C ~176 atlases J[ and A t on M are equivalent if JtUCV is also a C ~ atlas on M. It will be seen that global calculus carried out relative to A will be identical to global calculus carried out relative to the equivalent atlas A'.
88
3. G L O B A L T H E O R Y
F i g u r e 3.1.1. T h e s m o o t h coordinate change ~ o ~b-1 E x e r c i s e 3.1.5. Equivalence of Coo atlases is an equivalence relation. Each 6"~176 atlas on M is equivalent to a unique m a x i m a l 6 ,0o atlas on M . D e f i n i t i o n 3.1.6. A m a x i m a l C ~176 atlas .4 on M is called a s m o o t h s t r u c t u r e on M (also called a differentiable s t r u c t u r e or a Coo structure). T h e pair (M,.4.) is called a s m o o t h (or differentiable or Coo) n-manifold. By a typical abuse of notation, we usually write M for t h e s m o o t h manifold, the presence of t h e differentiable s t r u c t u r e A being understood. By Exercise 3.1.5, any 6 ,0o atlas (not necessarily maximal) on M completely determines the differentiable structure. N o t e t h a t the dimension n of a s m o o t h n-manifold is well-defined by P r o p o s i t i o n 2.3.12. E x a m p l e 3.1.7. T h e manifold IRn has a canonical s m o o t h structure, n a m e l y the set Ytn of all pairs (U, ~) where U C_ ]Rn is open and ~ : U --* IRn is a diffeomorphism onto an open set ~o(U) _C 1R~. E x a m p l e 3.1.8. If M and N are s m o o t h manifolds, d i m M = m and d i m N = n, w i t h respective s m o o t h structures A = { ( U ~ , ~ a ) } ~ and $ = {(V~,r t h e n M x N is canonically a s m o o t h (ra + n)-manifold. Indeed,
is a O0o atlas, d e t e r m i n i n g uniquely a m a x i m a l one, called tile Cartesian p r o d u c t of the two s m o o t h structures. E x a m p l e 3.1.9. If W C_ M is an open subset of a s m o o t h n-manifold, then W is a s m o o t h n-manifold in a natural way. Details are left as an easy exercise.
3.1. S M O O T H M A N I F O L D S
89
Remark. By substituting C k for C ~176 in the above discussion, one obtains the notion of a C k manifold, 1 < k < co. Similarly, one defines the notion of a real analytic (C ~) manifold. The reader should have no trouble in adapting the following discussion to these cases. Let M be a smooth n-manifold with a smooth atlas A = {(Ua, ~)}aE~*. We do not require this atlas to be maximal. Set
gaf~ = (~a o ~ 1 : ~ ( V a ~ V/3) ~ ~a(Va n Vf~). These local diffeomorphisms in Rn satisfy the coeycle conditions (1) gaf~ o gz'Y = g~'~ on ~g.y(Ua n U, n Uv) , (2) gaa = i d ~ ( u ~ ) , (3)
=
It should be noted that properties (2) and (3) follow from property (1). D e f i n i t i o n 3.1.10. The system {9aZ}a,Ze~ is called a structure cocycle for the smooth manifold M. The term "cocycle" is borrowed from algebraic topology due to certain formal similarities to cocycles in Cech cohomology.
Remark. It will be useful to see how to "reassemble" M out of the d a t a {Ua = Pa(Ua),ga~}a,ZE~. On the disjoint union aEP2
define the relation x~yc:~3a,~E91suchthatxEU~,
yCUz andy=gz~(x).
By properties (1), (2), and (3), this is an equivalence relation, so we form the topological quotient space M / ~ . We will show that this space is homeomorphic to M and exhibit a natural smooth structure on it. Let [z] E M / ~ denote the equivalence class of z E M. Define
~:M---* M / ~ by setting ~(x) = [ ~ ( x ) ] if x C Ua. If x C UZ also, then =
so ~ is well defined. It is also continuous. The map from M to M that takes z E U~ to ~ l ( z ) respects the equivalence relation, hence passes to a continuous map
r It is easy to see that ~ and ~p are mutually inverse, so M and M / ~ are canonically homeomorphic. Each U~ imbeds canonically in ff~'~/~ as an open subset and ida : Ua --* U~ c_ ]R~ defines a coordinate chart (Ua, ida) on M / ~ . These charts are C~176 via the cocycle {gaz}a,ZE~, SO M / ~ is canonically identified with M as a smooth manifold via the mutually inverse diffeomorphisms ~ and r (see Definition 3.1.18). E x e r c i s e 3.1.11. Prove that the topological n-manifold P~ (see Exercise 1.3.26) is a smooth n-manifold.
90
3. GLOBAL
Exercise 3.1.12. Show with just two charts. We
THEORY
that the manifold
turn to the smooth
maps
S n can be assembled
from a C a atlas
defined on a manifold.
Definition 3.1.13. A function f : M --* ~ is said to be smooth there is a chart (U, ~) E .4 such that x E U and
if, for each x E M,
is smooth.
will be denoted
The
set of all smooth,
real valued functions on M
by
C~176 The definition only requires us to be able to find some such chart about each point x E M, but the following assures us t h a t all charts will then work. L e m m a 3.1.14. The function f : M ---* N is smooth if and only if
f o ~21 : ~ ( U ~ ) ~ R is smooth, V ( U ~ , ~ a ) E A. Proof. Clearly this condition implies that f is smooth. For the converse, suppose that f is smooth and let z E U~ where ( U ~ , ~ ) E A. By Definition 3.1.13, choose (U~, ~o~) E A such t h a t x E U~ and f o (fl~l : ~ ( U ~ )
---+ ]t~
is smooth. Then, f o W21 : ~ ( U ~ N UZ) ~ IR is given by the composition ~(U,
n U,) ~
~ z ( U , n U,) s ~
1 a.
As a composition of smooth maps, this is smooth. T h a t is, f o ~ - 1 : ~a(gc~ ) _.+ is smooth on some neighborhood of the point qo~(x). But x E U~ is arbitrary, so f o ~ 1 is smooth on all of ~ ( U ~ ) . [] We think of f o ~21 as a formula for flU~ relative to the coordinate system ~a = ( x ~ , . . . , x~). We generally write (Ua, x ~ , . . . , x n) or (Ua, xa) for (Ua, ~a). D e f i n i t i o n 3.1.15. Let M and N be C a manifolds with respective smooth structures .4 and ~B. A map f : M ~ N is said to be smooth if, for each x E M, there are ( U ~ , ~ a ) E A and (Vz,r E N such that x E U~, f(U~) C_ V~, and r
o f o ~21 : ~ ( g ~ )
~ ~b~(V~3)
is smooth. L e r n m a 3.1.16. The map f : M ~ N is smooth if and only if, for all choices of
(u., ~ ) E ~ and (V~, r
E ~3 such that f(U.) C_ V,, the map ~3 o f o ~ 1 : qo~(U~) --~ ~b~(V~)
is smooth.
3.1. SMOOTH MANIFOLDS
91
The proof is similar to the previous one and is left to the reader. Again, we think of CZ o f o g)~l as a local coordinate formula for f. These two lemmas give an important part of the content of our remark that differential calculus in one coordinate chart is equivalent, in overlaps, to differential calculus in Coo-related neighboring charts. L e m m a 3.1.17. I f f : M ~ N and g : N ---, P are smooth maps between manifolds, then g o f : M --+ P is smooth. This is also elementary and is left to the reader. D e f i n i t i o n 3.1.18. A map f : M + N between smooth manifolds is a diffeomorphism if it is smooth and there is a smooth map g : N --+ M such that f o g = idN and 9 o f = idM. E x a m p l e 3.1.19. The maps : M/~-.
M,
~o:M+M/~ are mutually inverse diffeomorphisms (see the remark following Definition 3.1.0). L e m m a 3.1.20. I f (M,A) is a smooth n-manifold, U C_ M an open subset, and : U ~ R n a diffeornorphism of U onto an open subset ~o(U) of R n, then (U, qo) C A. Proof. By the definition of diffeomorphism, (U, qo) is C~176 A, so (U, ~) C A by the maximality of this atlas.
to every (Us, qoa) C []
If we write (U, ~o) as (U, x 1, x 2 , . . . , xn), the above discussion allows us to write f ( x l , . . . , x n) for f l U , whenever f : 5,I --+ N is smooth. This is logically a bit sloppy, but it is psychologically helpful. E x e r c i s e 3.1.21. Suppose that 54 is a smooth n-manifold and that r~:MI~
M
is a covering space. Prove that M ' has a unique smooth structure relative to which the projection ~r is locally a diffeomorphism. If p E M, it makes good sense to talk about germs at p of real valued C ~ functions defined on open neighborhoods of p. As before, these form an associative algebra ~Sp over R. The evaluation map ep : ~Sp --~ R is defined exactly as before. D e f i n i t i o n 3.1.22. A derivative of q3p is an R-linear map D : ~Sp--, R such that D(~r
= D(r162
+ ep(~)D(r
V~, ~ C I~ip. This operator D is also called a tangent vector to M at p and the vector space T p ( M ) of all derivatives of Op is called the tangent space to M at p. Definition 3.1.23. If f : M --~ N is a smooth map between manifolds and if p E M, the differential
f.p = dfp : T p ( M ) --~ Tf(p)(N)
92
3. G L O B A L T H E O R Y
is the linear map
defined by
(f,p(D))[g]/(p) = D[g o f]~, for all D E T p ( M ) and all [g]f(p) 6 ~)f(p). L e m m a 3.1.24 (Global chain rule). If f : M ~ N and g : N ~ maps between manifolds and x E M , then d(g o f ) x = dg/(z) o dfx,
P are smooth
Proof. Consider ((g o f),p(D))[h]g(f(p)) = D[h o g o f]p = (f,p(D)[h o glfo)) = 0,S(~)(Lp(D)))[h]~(S(,)
)
Since [h]g(f(p)) ~ Og(f(p)) and D E TB(M) are arbitrary, the assertion follows.
[]
It is clear that id,p = id : T p ( M ) ~ Tp(M), so the chain rule has the following consequence. C o r o l l a r y 3.1.25. I f f : M ~ N is a diffeomorphism, then f,p : T p ( M ) ~ T f o ) ( N ) is an isomorphism of real vector spaces, V p 6 M . C o r o l l a r y 3.1.26. I f M is a smooth manifold of dimension n, then T z ( M ) is a real vector space of dimension n, V x 6 M . Proof. Let (U,~) be a coordinate patch on M with x E V. Then, T~(U) = T ~ ( M ) and, by the previous corollary, ~ , x : Tx(U) ---* T~(~)(~(U)) is an ~-linear isomorphism. Since p(U) C_ R ~ is open, we know t h a t
T~(x)(~o(U) ) : ~n. [] The same kind of argument gives the following. C o r o l l a r y 3.1.27. I f f : M ~ N is a diffeomorphism, then d i m M = d i m N . Remark. We do not have a canonical basis for T ~ ( M ) since there is no preferred choice of local coordinates about x. Thus, we cannot write T x ( M ) = •n. The notion of infinitesimal curve (S)p makes sense in our context, as does the derivative D(s)p 6 T p ( M ) . Viewing tangent vectors as infinitesimal curves is preferable from an intuitive point of view, making the differential of a map "visible" and making the chain rule evident. If one is developing a theory of C k manifolds, defining tangent vectors to be infinitesimal curves rather than first order differential operators is essential [33]. The following is evident via local coordinates and the corresponding facts in ~n. L e m m a 3.1.28. The correspondence (S)p ~ DO) ~ between the set of infinitesimal curves at p C M f : M -~ N is a smooth mapping between manifolds, T / ( p ) ( N ) is given, in terms of infinitesimal curves,
is a one-to-one correspondence and T p ( M ) . Furthermore, if the differential f,p : T p ( M ) by
f,p( (S}p) = ( f o s) / o ) .
3.2. DIFFEOMORPHIC
93
STRUCTURES
Finally, for s m o o t h m a p s f : M --+ N, we define the notions of regular point, critical point, critical value, and regular value exactly as before.
3.2. Diffeomorphie Structures T h i s section is really an extended remark on some very deep theorems, the point of which can now be easily appreciated. Let M be a differentiable manifold with s m o o t h structure .q = { ( u ~ , ~ ) } ~ .
Let 9 : M --~ M be any homeomorphism. Set
aL[~ = {((!D-1 (Ucr), qPa o (]))}c~EN. P r o p o s i t i o n 3.2.1. The set Am is a Coo structure on M having the same structure cocycle as A. Proof. Indeed, g~
_- ~,~ o ~ F ~ _- ( ~
o ,~) o ( ~
o ,~)<,
and this m a p carries tile set
(~D/30 qT))((1D-I(Uc~) A (I)--l(v/3)) ~- ~9/3,(Uc, f"l Vj3 ) onto the set (~
o ~)(~-I(u~)
n ~-l(ue) ) = ~(U~
n Ue).
Finally, the m a x i m a l i t y of the Coo atlas Jtm follows from t h a t of Jt and the fact t h a t Atom 1 = A. [] 3.2.2. The smooth manifold M / H , whether defined from the C ~ structure .4 or Am, is the same. Consequently, ( M , A ) and (M, Am) are canonically diffeomorphic.
Corollary
We will let M e denote the s m o o t h manifold (M, Am). D e f i n i t i o n a . = . a . T w o C ~ structures Jt and ~, defined on the same topological manifold M, are said to be diffeomorphic structures if S = Am, for some homeom o r p h i s m q5 : M -~ M . It is clear t h a t diffeomorphism is an equivalence relation on the set of s m o o t h structures on a topological manifold M. It is natural to ask how m a n y diffeomorphism classes of s m o o t h structures a given topological manifold can support. T h e following e x a m p l e s are the deep facts, referred to at the beginning of this section, t h a t we cannot prove here. T h e m e t h o d s of proof are quite advanced. Example [42].
3.2.4. If n ~ 4, any two s m o o t h structures on R n are diffeomorphic
E x a m p l e 3.2.5. T h e case of IR4 was cracked by the combined work of a topologist, M. F r e e d m a n , and a global analyst, S. Donaldson, showing t h a t ]R4 admits a differentiable s t r u c t u r e not diffeomorphic to the usual one. In this structure, it is possible to find a c o m p a c t set t h a t cannot be surrounded by any s m o o t h l y i m b e d d e d S3! (For a discussion of this, see [11, Section 1].) Subsequently, various researchers found more "exotic" differentiable structures on R 4 (the first of these was R. Gompf, who found two new structures [12]). It is now known t h a t t h e n u m b e r of distinct diffeomorphism classes of differentiable structures on IR4 is
94
3. G L O B A L T H E O R Y
uncountably infinite. There is even a "universal" smooth (]R4, Jtu) such that every other smooth (]R4,A) smoothly imbeds as an open subset of (]R4, A~). E x a m p l e 3.2.6. Let a(n) denote the number of diffeomorphism classes of differentiable structures on S ~, up to oriented diffeomorphism (see Example 3.4.13). It was long known t h a t a(n) = 1 for n = 1, 2, 3. The value of a(4) remains a mystery. The following table was computed by M. Kervaire and J. Milnor [21].
a ( n n)
51
61 : 8
82 9 8
160 9112 12
133 124 16,256 15 1 11:1 7 1 1 8 16 16
T a b l e 1. Oriented differentiable structures on spheres
E x a m p l e 3.2.7. A topological manifold is said to be smoothable if it can be given a differentiable structure. For n = 1, 2, 3, it is known that all topological n-manifolds are smoothable. The first dimension in which there exist nonsmoothable manifolds is n = 4 [11, p. 23]. E x e r c i s e 3.2.8. Let 9 : ]R -+ ]R be the homeomorphism ~ ( x ) = x a. Show t h a t the identity map, viewed as id : ]Re --+ ]R, is not a diffeomorphism (although it is clearly a homeomorphism). On the other hand, show that, for any homeomorphism 4) : M --+ M of a differentiable manifold M, q5 : M~ --+ M is a diffeomorphism. 3.3. T h e T a n g e n t B u n d l e Let M be a C ~ n-manifold with smooth structure {(Uc~,~oa)}~e~. Consider the set T= Tx(M),
U
xEM
a disjoint union with, as yet, no topological structure. For each U~, ~ E 9.1, define
T(U~) = U
Tx(M) C_ T.
xEU~
Then the individual linear maps d~oax, x E Uc~, unite to define a set map
dqoo~ : T(U~) ~ T(~p~(U~) ) = ~o,~(Uc,) x ]R~ C_ ~2~. More precisely, if Vx denotes a tangent vector to M at x 6 U~,
d~,~(,,x)
=
(~.(x), d~.x(~,~)),
and this defines a bijection of T(Ua) onto an open subset of ]R2% Whenever U~ N UZ ~ ~, consider
d~. o dv~ 1 : T ( v z ( U . n U,)) --, T ( ~ ( U . n U~)). By the chain rule, this is
d g ~ : d~ofl(T(Vc~) N T(Uz) ) ---+d~o~(T(U~) N T(Uz)), a C ~ diffeomorphism between open subsets of ]R2~. We topologize the set T. If
W C_ d~,(T(Uc,)) : T(qo~(U~)) C_ ]R2n is an open set, then dqo~l(w) is to be an open subset of T.
3.3. T A N G E N T B U N D L E
95
E x e r c i s e 3.3.1. Prove that the above sets form the base of a topology on T and that, in this topology, T is a topological manifold of dimension 2n. Furthermore, show that the system {(T(Us), d ~ s ) } s e ~ is a (not maximal) C ~ atlas on T determining a maximal such atlas A. Finally, if T ( M ) denotes the differentiable manifold (T, A), show that the map 7r : T ( M ) + M,
~(~) =
x ** ~ r T ~ ( M ) ,
is smooth. D e f i n i t i o n 3.3.2. The system 7r : T ( M ) ~ M is called the tangent bundle of M. The total space is T ( M ) , the base space is M, and ~r is called the bundle projection.
Remark. It is often convenient to replace ~ ( U ~ ) x Nn with Uo x ]Rn, identifying d~s with the map v~ H (x,d~sz(v~)). This minor abuse of notation will be a major convenience in what follows. For each c~ C 92, we get a commutative diagram T(gs)
d~,o
U s x R '~
q us
us
id
where pl denotes projection onto the first factor and d~s is a diffeomorphism that restricts to be a linear isomorphism Tz(M) ---, {x} x N n, Vx E Us. Thus, T ( M ) is "locally" a Cartesian product of M and IRn, the projection 7r being "locally" the projection of the Cartesian product onto the first factor, and the fiber 7r-l(x) = T~(M) has a canonical vector space structure, Vx E M. D e f i n i t i o n 3.3.3. A vector field on M is a smooth map X : M --, T ( M ) (p H Xp) such that 7r o X = idM. The set of all vector fields on M is denoted by ~ ( M ) . Remark. Let X be a vector field on M, (U, x l , . . . ,x n) a coordinate chart on M. By this point in the book, the reader should be able to justify writing n
XIU = E
i 0
f ~x'
i=1
where ff : U - ~ R is smooth, 1 < i < n. Tangent bundles are examples of vector bundles. Vector bundles play a very important role in manifold theory, so we close this section with a brief discussion of the general theory. D e f i n i t i o n 3.3.4. Let M be a smooth m-manifold, E a smooth manifold of dimension (m + n), and 7r : E -~ M a smooth map. This will be called an n-plane bundle over M (or a vector bundle over M of fiber dimension n) if the following properties hold. (1) For each x C M, E~ = ~r ~(x) has the structure of a real, n-dimensional vector space.
96
3. GLOBAL THEORY
(2) There is an open cover { W j } j e j
~-~(w~)
wj
of M, together with commutative diagrams
~
id
,wj•
,
~
wj
such that ~j is a diffeomorphism, Vj E J. (3) For each j E J and z 6 Wj, ~j~ = ~jlE~ maps the vector space Ex isomorphically onto the vector space {x} x R n. As with tangent bundles, we call E the total space, M the base space, and 7r the bundle projection. We also call each Wj a trivializing neighborhood for the bundle and { W j } j e j a locally trivializing cover (of M) for E. An obvious example of an n-plane bundle is given by pl : M x IR~ - , M. Here, M itself is a trivializing neighborhood and the bundle is said to be trivial. D e f i n i t i o n 3.3.5. Let 71"1 : E 1 ~ M and 7r2 : E2 --~ M be n-plane bundles over M. A bundle isomorphism is a commutative diagram
El
~
E2
M
M id
such that ~ is bijective, smooth, and carries EI~ isomorphically (as a vector space) onto E2~, Vz E M. If E2 = M x R ~ with 1r2 the canonical projection, the isomorphism ~ is called a trivialization of El. Note that we did not explicitly require that ~-1 be smooth. One needs this to be true, however, in order that bundle isomorphism be an equivalence relation. The following lemma comes to the rescue. Remark that it is useful not to be required to check in each instance that ~ is a diffeomorphism. L e m m a 3.3.6. The m a p ~ in Definition 3.3.5 is necessarily a d i f f e o m o r p h i s m . Proof. Find a locally trivializing cover %[ of M for both E1 and E2 simultaneously. If U E 11, we use the local trivializations
1 r ~ - I ( u ) ~ U x I R ~, i = 1 , 2 , and the bundle isomorphism
~l~-~(u) : ~i-~(u) -~ ~ ( u ) to induce a commutative diagram UxlR~
~
U
~ UxR
,
n
U
id
which is also a bundle isomorphism. It will be enough to prove that ~ - 1 is smooth. But
~(x,v) = ( x , ~ ( x ) . v ) ,
3.3. TANGENT
BUNDLE
97
where O~ : U --+ Gl(n). We claim t h a t -y is smooth. Indeed, let ek C ]i{n denote the column vector w i t h 1 in t h e kth position, Os elsewhere. T h e n the m a p qok : U --+ U x IRn, defined by
~k(x) = ~(x, ek) = ( x , ~ ( * ) . ~k), is smooth. In particular, the kth column of O~(x) is 7(x).ek, hence depends s m o o t h l y on x. Since k is arbitrary, O' is smooth. T h e o p e r a t i o n of taking the inverse of a m a t r i x defines a m a p L: Gl(n) --, Gl(n) the coordinate functions of which are rational functions of the coordinates. m a p is smooth, so
~-~(x,~) =
This
~)
(x,~ o ~/(x).
is also smooth.
[]
D e f i n i t i o n 3.3.7. An n-plane bundle is trivial if it is isomorphic to the p r o d u c t bundle Pl : M x R n --+ M . D e f i n i t i o n 3.3.8. A section of the n-plane bundle ~r : E -~ M is a s m o o t h m a p s : M --+ E such t h a t rro s = idM. The set of all such sections is denoted by F ( E ) . Thus, ~ ( M ) = F ( T ( M ) ) . T h e following is elementary and is left to the reader. Lemma
3.3.9.
The set F ( E ) is a C~176
(flsl+f2s2)(x)
where f i E C~~ Definition
under the pointwise operations
: fl(x)sl(x)+f2(x)s2(x)
6 Ex,
V x E M,
and si C F ( E ) , i = 1, 2.
3.3.10. T h e manifold M is parallelizable if there are fields
xl, x2,..., xn c ~(M) such t h a t {XI~, X 2 ~ , . . . , X ~ , } is a basis of T~(M), g x C M . Proposition
3.3.11.
The manifold M is paralleIizable if and only if T ( M ) is a
trivial bundle. E x e r c i s e 3 . 3 . 1 2 . P r o v e t h a t the n-plane bundle ~r : E --+ M is trivial if and only if there exist s l , . . . , Sn C F ( E ) such t h a t {81 (x), . . . , Sn(X)} is a basis of E~, V x e M . In particular, this proves Proposition 3.3.11. So far, the only real examples of vector bundles t h a t we have seen are t a n g e n t bundles and trivial bundles. T h e following is the least complicated example of a nontrivial vector bundle. E x a m p l e 3 . 3 . 1 3 . We give an example of a 1-plane bundle (a "line" bundle) over the circle, known as the MSbius bundle. On 1I{ x R, define the equivalence relation (s,t) ~ (s + n, ( - 1 ) ~ t ) , n E Z. R e m a r k t h a t t ~ ( - 1 ) ~ t is a linear a u t o m o r p h i s m of IR. T h e projection (s,t) ~ s passes to a well-defined m a p rr : (IR x IR)/(~) --+ R / Z = S 1. It should be clear, intuitively, t h a t this is a vector bundle over S 1 of fiber dimension 1, but a rigorous proof of this involves checking m a n y details. E x e r c i s e 3 . 3 . 1 4 . Give a careful proof t h a t the MSbius bundle ( E x a m p l e 3.3.13) is truly a line bundle over S 1. Show t h a t this bundle is not trivial.
98
3. G L O B A L T H E O R Y
3.4. C o c y c l e s a n d G e o m e t r i c S t r u c t u r e s
This section is somewhat philosophical. The main point is to give the reader some insight into geometric structures on a manifold determined by subgroups G C Gl(n). A more elegant formulation of these ideas, using the notion of a principal bundle, will be taken up in Chapter 11. Let 7r : E --~ M be an n-plane bundle and let (Wj}jEj be a locally trivializing open cover for E, the trivializations being Cj : ~ - I ( W j ) -~ Wj • R ~. If Wi A Wj ~ O, consider @71
(wi n w~) • s n :_~ . - l ( w i n w~) -% (wi n wj) • R ~. This composition must have the form a1
a1
where 3'ji(x) E Gl(n), Vx E Wi N Wj. The following is proven by exactly the argument employed in the proof of Lemma 3.3.6. L e m m a 3.4.1. The map 7ji : Wi A Wj --* Gl(n) is smooth. These smooth maps have the "cocycle" property
(3.1)
7kj(~). ~ji(x) = 7ki(x),
Vx E Wi M Wj A Wk, Vi, j , k E J. As usual, this property implies also, for all appropriate choices of x and indices i, j E J, (3.2)
~/ii(x) = In,
(3.3)
Vii(x) = (Vji(x)) -1.
Again, this "cocycle" terminology comes from cohomology theory (see the remark following Exercise 3.3.14). D e f i n i t i o n 3.4.2. A Gl(n)-cocycle on M is a family O~= {Wj, %ji}ideJ such t h a t ( W j } j E J is an open cover of M and "~ji : Wi N W j --~ Gl(n) is a smooth map, V i , j E J, all subject to the cocycle condition (3.1). If the cocycle 7 arises as above from an n-plane bundle E, it is said to be a structure cocycle for E.
Just as a structure cocyele g = (U~, g~z}~,Ze~ for M gave all the d a t a necessary for reassembling the smooth manifold, up to diffeomorphism, so a structure coeycle 0/ for E gives all the d a t a necessary for reassembling the bundle, up to bundle isomorphism. Indeed, given any Gl(n)-cocycle 9/ = (Wj,'Yji}i,jEJ, one assembles an n-plane bundle for which it is a structure cocycle. Here is a quick sketch of the procedure. Set
E~ = U Wj x R~ jEJ
and define on/~7 an equivalence relation by setting (x, v) ~ (y, w) whenever (x, v) E Wj • ~ n , (y,w) E Wi • ~n, x = y E Wj A W i and w = ~ i j ( x ) . v . The fact that this is an equivalence relation is an obvious consequence of equations (3.1), (3.2), and
3.4, COCYCLES
99
(3.3). The standard projections Wj x N n ~ Wj then fit together as maps into M to define a m a p ~ : / ~ --~ M that respects the equivalence relation. We obtain : E~ = E ~ / ~ - - * M, and the reader can check t h a t this is a smooth n-plane bundle. In case the cocycle came from local trivializations ~pj : *r-l(Wj) ~ Wj x R n of a bundle ~r : E --* M, the diffeomorphisms r fit together to define ~ : /~v -~ E, again respecting the equivalence relation, and this defines a bundle isomorphism
r E~-~ E, as the reader again can check. D e f i n i t i o n 3.4.3. Two Gl(n)-cocycles
3` = {Wj, 3`ij}i,jEJ and 0
=
{Va,Oab}a,bEA
on the same manifold M are equivalent if they are contained in a common Gl(n)cocycle on M. The equivalence class of ~/will be denoted by [3`]. In order for [~/] to make sense, one nmst prove that equivalence of cocycles is an equivalence relation. The following exercise will be useful for this. E x e r c i s e 3.4.4. If two Gl(n)-cocycles on the same manifold contain a common Gl(n)-cocycle, show that they are contained in some common Gl(n)-cocycle. L e m m a 3.4.5. Equivalence of Gl(n)-cocycles is an equivalence relation.
Pro@ The only problem is transitivity. If ~/ ~ 0 and 0 ~ 6, let r be a cocycle containing both 3' and 0, ~ a cocycle containing both 0 and & Then ~ and ~ both contain 0, so Exercise 3.4.4 guarantees that they are contained in a common cocycle p. T h e n ~ / G p a n d 6 C _ p , s o 3 ` ~ 6 . [] Let Vectn(M) denote the set of isomorphism classes [E] of n-plane bundles E on M and let H ~(M; Gl(n)) denote the set of equivalence classes of Gl(n)-cocycles (notation borrowed from algebraic topology, again because of formal analogies with Cech cohomology). E x e r c i s e 3.4.6. (Bundle classification) If 0/ is a Gl(n)-cocycle on M, prove t h a t the isomorphism class [E,] E Vectn(M) depends only on the equivalence class [3'] E H i ( M ; Gl(n)). This defines a canonical bijective correspondence Vectn(M) ~ H ~(M; Cl(~)). By this exercise we identify V e c t , ( M ) with Hi(M; Gl(n)). In the case of the tangent bundle T(M), any smooth atlas { ( U ~ , ~ ) } ~ e a , with associated structure cocycle {g~}~,~c~ for M, provides a structure cocycle {Ua, Jg~z}~,Ze~. There are, of course, structure cocycles for T(M) t h a t are not obtained in this way, but these special cocycles tie together the bundle structure of T(M) and the smooth structure of M in an important way. D e f i n i t i o n 3.4.7. A structure cocycle {Ua, ggaz}a,Ze~ for T(M), associated to a smooth atlas on M, will be called a Jacobian cocycle.
100
3. G L O B A L T H E O R Y
E x a m p l e 3.4.8. If T(M) admits a cocycle {Us, 7 a Z } a , Z ~ such that 7aZ - I , for all a, ~ E P2, it turns out t h a t M is parallelizable (Exercise 3.4.9). A much stronger condition would be that T(M) admits a Jacobian cocycle {Us, Jg~}~,~e~ such t h a t Jg~z - I, for all a, p E P2. In this case, M is said to be integrably parallelizable. We will see in the next chapter t h a t this forces M to be diffeomorphic to T k x Nn-k, for some nonnegative integer k < n. E x e r c i s e 3.4.9. Prove that the following are equivalent for an n-plane bundle
7~: E--+ M. (1) The bundle ~ : E --~ M is trivial. (2) There is a Gl(n)-cocycle {Wj, 7ji}j,iEJ for the bundle such that 7ji(x) = In,
Vx E W j N W i , V i , j E J. (3) There is a smooth function f : E ~ I~n such that
fx= flEx:E~ ~R n is a linear isomorphism, Vx E M. In particular, setting E = T(M) and appealing to Proposition 3.3.11, we see that these conditions are equivalent to parallelizability of M. D e f i n i t i o n 3.4.10. Let G C Gl(n) be a subgroup and let ~ : E --+ M be an nplane bundle. We say that the structure group of E can be reduced to G if there is a Gl(n)-cocycle {Wj, 7j~}~,jeJ representing the isomorphism class of E such t h a t im(~'yi) C_ G, V j, i E J. Such a cocycle will be called a G-cocycle for E. Equivalence of G-cocycles can be defined exactly as for the case that G = Gl(n), giving a "cohomology set" H 1(M; G). Distinct elements of this set may correspond to distinct bundles or to the same bundle with inequivalent G-reductions. D e f i n i t i o n 3.4.11. If E admits a G-cocycle 7, then [7] E Hi(M; G) is called a G-reduction of E. D e f i n i t i o n 3.4.12. For a subgroup G C Gl(n), an infinitesimal G-structure on the n-manifold M is a G-reduction [7] of T(M). If [7] contains a Jacobian cocycle, then [7] is said to be integrable and will be called simply a G-structure or a geometric structure on M. In the case of the trivial subgroup I = {In} C Gl(n), M is parallelizable if there is an i n f i n i t e s i m a l / - s t r u c t u r e [7] on M. The manifold is integrably parallelizable if there is an integrable [7] E H 1(M; I). E x a m p l e 3.4.13. Let Gl+(n) C Gl(n) be the subgroup of matrices with positive determinant. Two ordered bases (vl,... , vn) and (wl,... , Wn) of an n-dimensional vector space V are said to have the same orientation if the unique matrix A E Gl(n) such t h a t (vl,... ,vn)A = ( w l , . . . ,wn) belongs to Gl+(n). This is an equivalence relation having exactly two equivalence classes, called orientations of V. A linear isomorphism L : V -~ W between n-dimensional vector spaces carries each orientation # of V to an orientation L(#) of W. Indeed, if ( v ~ , . . . , v n ) represents #, we take for L(#) the orientation represented by ( L ( v ~ ) , . . . , L(vn)). By linearity, the basis ( v ~ , . . . , v,OA will be taken to the basis (L(v~),..., L(v,~))A, det(A) > 0, so L(#) is well defined.
3.4. C O C Y C L E S
101
The standard orientation #n of Nn is the orientation class of the standard ordered basis ( e l , . . . ,en). The other orientation of R n will be denoted by the symbol - # ~ . A linear automorphism L of R ~ is orientation-preserving if L(#n) = #n and is orientation-reversing if L(#n) = - # n . Clearly, L is orientation-preserving if and only if det(L) > 0. Let 7r : E --* M be an n-plane bundle and let #x be an orientation of Ex, Vx E M. We will say that #x depends continuously on x if, for each x E M, there is a trivialization ~ : E{U --* U x R n, x E U, such that ~.y(#y) = #n, Vy E U. If t~ = {t~x}~EM depends continuously on x, we say that # is an orientation of E and that E is orientable. Given #, there is always the opposite orientation - # . An orientation # of T(M) is also called an orientation of M. If such exists, M is orientable and the pair (M, #) is an oriented manifold. If (M, #) and (N, u) are oriented n-manifolds, a diffeomorphism f : M --~ N is orientation-preserving (or, simply, oriented) if f.~(#~) = u/(~), Vx E M. An orientation of E determines a Gl+(n)-reduction of E. Indeed, cover M with local trivializations (Ui, ~i) as above and remark that the associated cocycle 3'ij(x) carries the orientation #~ of {x} x IR~ to itself, Vx E Ui N Uj. That is, 7~j(x) E Gl+(n). Conversely, given a Gl+(n)-reduction, the reader should be able to produce an associated orientation of E. Thus, an infinitesimal GI+ (n)-structure on M is an orientation of M. Every infinitesimal GI+ (n)-structure is integrable. Indeed, fix an orientation # of T(M) and let {Ua, Jgc~ }a,~c~ be any Jacobian cocycle such that each coordinate chart (U~,x~) is connected. The n-tuple of fields
defines a continuous orientation #~ of T(U~). Since U~ is connected, either #~ = #IU~ or #~ = - # l U g . In the latter case, replace the coordinate x~1 with - x ~1. This is an orientation reversing change of coordinates and the new coordinates give the correct orientation #~ to T~(M), Vx E U~. Carrying this out for each a C 9.1, we produce an atlas on M with associated Jacobian cocycle GI+ (n)-valued. E x e r c i s e 3.4.14. Let M be connected and set 0 ( M ) = {~#~ I x E M}. (1) P u t a topology and differentiable structure on the set 9 such that the projection 7r : 0 ( M ) --~ M sending 4-#~ ~ x, for each x E M, is a smooth covering map. (2) Prove that 0 ( M ) is connected if and only if M is nonorientable. In the orieatable case, 0 ( M ) falls into two components, each carried diffeomorphically onto M. (3) Prove that the manifold (_9(M) is orientable. (4) If M is simply connected, prove that M is orientable. E x a m p l e 3.4.15. Let G = O(n). An infinitesimal O(n)-structure is called a Riemannian structure and an n-manifold M with a Riemannian structure is called a Riemannian manifold. This is the starting point of Riemannian geometry, a subject that we will treat in Chapter 10. A Riemannian structure enables one to define lengths of tangent vectors and of curves, angles between vectors tangent at the same point and between curves through a point, curves that locally minimize length (geodesics), curvature of the manifold, etc.
102
3. G L O B A L T H E O R Y
E x e r c i s e 3.4.16. Prove that the manifold M admits an infinitesimal O(n)-structure if and only if there is a positive definite inner product (., '}x on Tx(M), V x E M , that varies smoothly with x in the following sense: given any local coordinate chart (U, x l , . . . , x n) in the C a structure of M, 0 is smooth on U, 1 _ i , j <_ n. This smoothly varying inner product on the fibers of T ( M ) is called a Riemannian metric on M. Using this metric, one defines the norm
H" I[ : T ( M ) -~ ]R+ by Ilvll -- Ilvll~ = ~ , whenever v e T~(M). Similarly, angles between elements of T~(M) are defined in the usual way, using the fiberwise inner product. Lengths of smooth curves s : [a, b] --~ M are defined by f
b
L(s) : ]o II~(t)ll d~. ARer it has been proven that open covers admit smooth, subordinate partitions of unity (Theorem 3.5.4), you will be invited to show that M1 manifolds admit Riemannian metrics (Exercise 3.5.9). For this, the fact that manifolds are 2nd countable is essential. On the contrary, very few manifolds admit integrable Riemannian structures ("flat" Riemannian manifolds). The obstruction to integrability is the Riemann curvature tensor. This will be treated in Chapter 10, where we will show (Theorem 10.6.7) that curvature 0 is equivalent to the geometry being locally Euclidean. D e f i n i t i o n 3.4.17. Let M and M ' be Riemannian manifolds with associated Riem a n n i a n metrics (., .) and (.,.)'. A diffeomorphism f : M ' -~ M is said to be an isometry if, for each x E M ' and all v , w E T~(M'), (f.x(v), f.z(w)}/(~) = (v, w}:. A local diffeomorphism with this property is called a local isometry. E x e r c i s e 3.4.18. If M is a Riemannian manifold and 7r : M ~ --+ M is a covering space, prove that there is a unique Riemannian metric on M ~ relative to which is a local isometry. E x a m p l e 3.4.19. One can take G = O(k, n - k) C Gl(n) the group of matrices that leave invariant the quadratic form
Qk(~l,...
, x D = (xl) 2 + . . .
+ (xk) 2 - (xk+l) ~ . . . . .
( ~ D ~.
A discussion similar to the one above shows that such an infinitesimal structure corresponds to a nondegenerate, indefinite inner product or metric (also called a pseudo-Riemannian metric or structure) in each fiber Tx(M), varying smoothly with x. Manifolds equipped with such metrics are called pseudo-Riemannian manifolds. If the infinitesimal O ( k , n - k ) - s t r u c t u r e is integrable, the pseudo-Riemannian manifold and metric are said to be flat. One can again define the notions of "isometry" and "local isometry" between pseudo-Riemannian manifolds and prove that covering spaces of pseudo-Riemannian
3.4. C O C Y C L E S
103
manifolds are uniquely p s e u d o - R i e m a n n i a n in such a way t h a t the covering projections are local isometrics. T h e Lorentzian manifolds in relativity theory are 4-manifolds (space-time) w i t h an infinitesimal 0 ( 3 , 1)-structure. It is no longer true t h a t all manifolds a d m i t such structures. Integrability places an even more severe restriction on the manifold. T h e o b s t r u c t i o n to integrability is again a (Lorentz) curvature tensor, the physical i n t e r p r e t a t i o n of c u r v a t u r e being gravity. F l a t Lorentzian manifolds are those in which the c u r v a t u r e is everywhere 0 (no gravity), these space-times being locally equivalent to special relativity. E x a m p l e 3 . 4 . 2 0 . Consider the subgroup G l ( k , n - k) C Gl(n) consisting of all m a t r i c e s of the form
["1 0
C
'
where A E Gl(k), C C G l ( n - k), and B is an arbitrary k x ( n - k) matrix. Since the linear action Gl(k, n - k) x IRn ~ R n carries the subspace ]Rk C IRn isomorphically onto itself, a G l ( k , n - k)-reduction 7 of the n-plane bundle 7r : E ~ M selects a k-dimensional subspace F~ C E~, Vx E M . Indeed, the equivalence relation ~ on / ~ m a t c h e s the fiber {x} x R k c Wj x N ~ isomorphically to {x} x R k C Wi x ]~n, w h e n e v e r x E Wj N Wi, so the subspace {x} x 1Rk passes to a well-defined subspace F~ C (E~)x in the quotient.
D e f i n i t i o n 3 . 4 . 2 1 . A k-plane subbundle F of an n-plane bundle E over M is a k-plane bundle, t o g e t h e r with a c o m m u t a t i v e diagram F
i
M
~ E
,M id
such t h a t iz : Fx ~ Ez is a linear m o n o m o r p h i s m , Vx C M . A k-plane subbundle of T ( M ) is also called a k-plane distribution on M . We can view i as an inclusion map F ~-~ E. T h e choices {F~}xeM, defined by a Gl(k, n - k)-reduction, are the fibers of a k-plane subbundle. E x e r c i s e 3 . 4 . 2 2 . P r o v e t h a t a Gl(k, n - k)-reduction of the bundle E defines a k-plane s u b b u n d l e F of E with fibers Fx as above. Show that, given a k-plane s u b b u n d l e F ~-* E , there is a G l ( k , n - k)-reduction of E t h a t gives back this subbundle. Thus, an infinitesimal Gl(k, n - k)-structure on M is a k-plane distribution on M . If this s t r u c t u r e is integrable, it will be called a foliation of M of dimension k. These geometric structures will be studied in the next chapter, together w i t h the integrability condition, the Frobenius theorem. E x a m p l e 3 . 4 . 2 3 . T h e complex general linear group Gl(n, C) is the group of nonsingular, n x n matrices with complex entries. If we write tile elements of this group as A + x/-J-1B, where A and B are real matrices, then one can check t h a t
["1 -B
A
e Gl(2n).
104
3. G L O B A L T H E O R Y
In fact, this realizes the complex general linear group as a subgroup Gl(n, C) C Gl(2n). A 2n-manifold M , together with an infinitesimal Gl(n, C)-structure, is known as an almost complex manifold. In this case, one can define a bundle isomorphism J : T ( M ) --~ T ( M ) Such t h a t j 2 = _ idT(M) (the reader who has been successful with Exercises 3.4.16 and 3.4.22 will be able to check this). One extends the fiberwise scalar multiplication R • T(M) ~ T(M) to a complex scalar multiplication C • T ( M ) --~ T ( M ) by the formula (a + VZE-lb)v = av + bJ(v), Va, b E N, V v c T~(M), V x C M . Effectively, the tangent bundle becomes a vector bundle over C rather than N, of complex fiber dimension n, called the complex tangent bundle. If the almost complex structure is integrable, it can be shown that the manifold admits an atlas ( U ~ , ~ ) where qo~ : Us -o C n and the local coordinate changes g ~ are complex analytic. That is, M has the structure of a complex analytic manifold and the integrable almost complex structure will be called a complex analytic structure. One defines a "holomorphic tangent bundle" and shows t h a t it is isomorphic, as a complex vector bundle, to the complex tangent bundle associated as above to the ahnost complex structure. The details of all this, as well as the integrability condition, are advanced topics that will not be treated in this book. E x e r c i s e 3.4.24. If 7r : M ~ --, M is a covering space and if M admits an infinitesimal G-structure, show t h a t this structure "lifts" canonically to an infinitesimal G-structure on M I. Prove that the lifted structure is integrable if the one on M is integrable. 3.5. Global C o n s t r u c t i o n s of S m o o t h F u n c t i o n s The proof of Theorem 2.6.1 adapts easily to the global case. P r o p o s i t i o n 3.5.1. Let M be an n-manifold, let U C_ M be an open subset and K C U a compact subset. Then there is a smooth function f : M ---* R such that f [ K - 1 and s u p p ( f ) C U. Proof. For each p E K , choose a coordinate neighborhood (Up, zp) about p, Up C_ U, and an open, n-dimensional interval Ap with A p c Up, centered at p. Since K is compact, finitely many of the Ap cover K . The proof of Theorem 2.6.1 now goes through unchanged. [] This is not quite the global C ~ Urysohn Lemma (Corollary 3.5.5) in which the set K is assumed only to be closed, not compact. Actually, Proposition 3.5.1 suffices for most purposes and we will use one of its standard consequences, the existence of smooth partitions of unity, in order to prove the general version. One useful application of Proposition 3.5.1 is the following.
3.5. G L O B A L C O N S T R U C T I O N S
105
L e m m a 3.5.2. Let M be a smooth manifold and let x E M . Then the natural map Coo(M) ~ 03~ that carries f ~-* [fix is surjeetive. Proof. Indeed, given [g]~ E ~ ,
find p E Coo(M) with supp(~) C dom(g)
and g) - 1 on some compact neighborhood of x in dom(g ). Then qo9 extends by 0 to a smooth function f on M and [f]x = [g]~. [] Another application is to identify X(M) with the Lie algebra 9 of derivations of the function algebra Coo(M). Indeed, each X E 3~(M) is a derivation X : C ~ 1 7 6 -+ Coo(M) in the obvious way. In the local case, M = U _c R n, the proof of the reverse inclusion used Theorem 2.6.1 to show how to localize an arbitrary operator D E 9 to be a derivative Dx E T:~(U), and one uses Proposition 3.5.1 in the same way for D E 9 P r o p o s i t i o n a . 5 . a . The set Coo(M).
x(M) is
the Lie algebra of derivations of the algebra
One advantage to this point of view is that the Lie bracket is defined intrinsically on 2E(M). The alternative is to use the local definitions of bracket in coordinate charts and show that the formulas for the bracket transform correctly under coordinate changes so t h a t the local definitions fit together to give IX, Y] E 2E(M) globally. T h e o r e m 3.5.4. I f l l = {Us}se~ is an open cover of M , there is a C ~ partition of unity {)~s}sE~ subordinate to Ii. Proof. First remark that, if W = {Wz}oem is a locally finite refinement of II, a smooth partition of unity subordinate to W induces a smooth partition of unity subordinate to lI. Indeed, let i : ~ + 91 be a map such that W z C_ Ui(z), V/3 E ~3. If {#Z}Zem is a partition of unity subordinate to W, define As = ~ Z e i - ~ ( s ) p Z , Vc~ E 91. If i - l ( c t ) = ~, we understand that As - 0. Since W is locally finite, [-Jzei *(s)supptt~ is closed (Exercise 1.4.2), hence is the support of ,ks. It should be clear, then, t h a t {As}sE~ is a partition of unity subordinate to ~[. By the above remark and the fact that manifolds are locally compact, we lose no generality in assuming that each Us has compact closure in M. Thus, the precise refinement V = {Vs}se~ found in Lemma 1.4.8 has the property that Vs C Us is a compact subset, Vc~ E 91. In the proof of the existence of continuous partitions of unity (Theorem 1.4.11), appeal was made to the general form of Urysohn's lemma. Since each Vs is compact, Proposition 3.5.1 can be used in the same way to complete the proof of existence of smooth partitions of unity. [] At this point, we can remove the compactness hypothesis in Proposition 3.5.1. C o r o l l a r y 3.5.5 (The global smooth Urysohn lemma). Let U C_ M be an open subset of a smooth manifold and K c_ U a subset that is closed in M . Then there is a smooth function f : M ---+1R such that f l K - 1 and s u p p ( f ) C U.
106
3. GLOBAL
THEORY
Pro@ Let W = M \ K . Then {U,W} is an open cover of M, and we take a subordinate smooth partition of unity {AN, Aw}. Since A w I K -- O, Au + Aw - 1, the function f = Au is as required.
[]
Recall that, if X C IR~ is an arbitrary subset, a function f : X -* IRk is said to be smooth if it extends to a smooth function f : U --* IRk, where U is some open neighborhood of X in R n. This definition continues to work well when X C M, b u t it is not very convenient when the target space is a smooth manifold that is not an open subset of Euclidean space. The following result, another application of smooth partitions of unity, suggests a slightly different, backwardly compatible definition of smoothness that is more useful. P r o p o s i t i o n 3.5.6. If M is a smooth manifold and X C_ M , then
f : X - - . IRk is smooth if and only if, V x E X , 3 an open neighborhood Uz C_ M of x, and a smooth map f z : Uz --* IRk such that f~l(Ux N X ) = fl(U~ n X ) . Pro@ This property clearly follows from our definition of smoothness. We must recover our definition from this property. Let U = U z e x Uz. Then there is a smooth partition of unity {A~}zex on U, subordinate to the open cover { U z } z e x of the manifold U. Since each f~ is Rk-valued, A~fz makes sense and can be interpreted as a smooth map of U into IRk. Then define
f= EAxS , xEX
a smooth map of U into IRk. Evidently,
~xEX
Vy E X, so f is the required smooth extension of f to the neighborhood U of X. [] D e f i n i t i o n 3.5.7. A function f : X -~ Y from a subset X c_ M of a smooth manifold M into a subset Y C_ N of a smooth manifold N is said to be smooth if, for each x E X, there is an open neighborhood U~ C_ M of x and a smooth map fx : Uz ~ N such that fzl(Uz MX) = fl(U~ MX). Such a map is a diffeomorphism of X onto Y if it is bijective and both f and f - 1 are smooth. As an application, we prove and globalize a smooth version of Theorem 1.1.9. T h e o r e m 3.5.8 (Smooth invariance of domain). Let M and N be C ~ manifolds of the same dimension n. If U C M is open, if X C_ N , and if ~ : U --* X is a diffeomorphism, then X is open in N .
Proof. Let x0 E U and ~O(Xo) E X. Since ~-1 : X --* U is smooth, we can produce an open neighborhood V of ~(x0) in N and a smooth extension r : V --* M o f ~ o - l l ( V N X ) . Since ~ : U --* X is continuous, V = ~ - I ( V N X ) is an open neighborhood of x0 in U and
r o ~ l v = ~-1 o ~lV = ida.
3.6. M A N I F O L D S W I T H B O U N D A R Y
107
Since ~ : U -+ N is s m o o t h in the usual sense, the chain rule gives &P~(xo) o d~z o = idr~o(M), SO dp~ o : T ~ o ( M ) ~ T~(~0)(N ) is a linear isomorphism.
By the inverse function
theorem, there is an open neighborhood W C V C_ U of x0 t h a t is carried by diffeomorphically onto an open subset p ( W ) c_ N . B u t ~(x0) is an a r b i t r a r y point of X and p(x0) E ~o(W) C_ X , so X is an open subset of N . [] We close this section with some exercises t h a t can be solved using the tools we have developed. E x e r c i s e 3.5.9. Use the existence of partitions of unity to prove t h a t every s m o o t h manifold M a d m i t s a R i e m a n n i a n structure. Explain wily it is impossible to generalize this a r g u m e n t to prove the existence of an infinitesimal O(k, n - k)-structure. E x e r c i s e 3 . 5 . 1 0 . P r o v e t h a t a c o m p a c t n-manifold M cannot be diffeomorphic to any subset of IRa. E x e r c i s e 3.5.11. Let M be a s m o o t h manifold, K C M a closed subset, U D K an open neighborhood of K , v E 3Z(U). Prove t h a t vlK extends to a s m o o t h vector field on all of M . E x e r c i s e 3 . 5 . 1 2 . Let II = {U~}~e~ be an open cover of the s m o o t h manifold M . For each a E P,l, let p~ : M -~ R have a constant value ca > 0 on Us and be identically 0 on the c o m p l e m e n t M \ U~. Set
e~E~
and c o n s t r u c t a s m o o t h function f : M + R such t h a t 0 < f < ~o everywhere on M.
3.6. S m o o t h Manifolds w i t h B o u n d a r y Manifolds w i t h b o u n d a r y were introduced in Section 1.6 from the purely topological point of view. Here we introduce the s m o o t h version. Since Euclidean half-space ]HIn is a subset of the s m o o t h manifold R n, Definition 3.5.7 allows us to talk a b o u t s m o o t h m a p s and diffeomorphisms between open subsets of ]HIn. Thus, if M is a topological manifold with boundary, we can define the notion of C ~ - r e l a t e d n e s s of H a - c h a r t s on M in the obvious way. So we can define a differentiable s t r u c t u r e on M to be a m a x i m a l H a - a t l a s A of C ~ - r e l a t e d charts. As in the topological case, we define OM={xCMI3(U~,~)CA, i n t ( M ) = {x E M I 3 ( U ~ , ~ )
xcU~,
~(x)
~ A, z 6 u~, ~ ( u ~ )
E c~Ha}, _c int(H~)}.
T h e pair ( M , A ) is a (smooth) n-manifold with b o u n d a r y OM. Of course, all s m o o t h n-manifolds w i t h o u t b o u n d a r y are special cases, as is N a itself. The notion of s m o o t h m a p s between manifolds with b o u n d a r y is defined exactly as in the boundaryless case. T h e following is an i m m e d i a t e corollary of T h e o r e m 3.5.8.
P r o p o s i t i o n 3.6.1. If U C N n \ ON '~ is open, then U is not diffeomorphie to an open subset V C N n such that V N OH '~ r O.
108
3. G L O B A L T H E O R Y
C o r o l l a r y 3.6.2. If M is a manifold with boundary, then
M ", OM = int(M). Proof. Indeed, the proposition shows that OM = {x c M I f l ( U ~ , ~ )
9 A, x 9 Us, ~a(U~) c_ int(]HI~)}. []
C o r o l l a r y 3.6.3. If qz : M ---* N is a diffeomorphism, then ~(int(M)) = int(N),
~(OM) = ON. C o r o l l a r y 3.6.4. A smooth n-manifold M with boundary is also a smooth nmanifold ** M -- int(M) r OM -= ~. C o r o l l a r y 3.6.5. I f M is a smooth n-manifold with boundary, then OM is a smooth (n - 1)-manifold and int(M) is a smooth n-manifold. E x e r c i s e 3.6.6. Let M and N be smooth manifolds of dimension rn and n respectively. If 0 M ~ 0 = cON, show how to use the differentiable structures on these two manifolds to give M • N the structure of a smooth (m + n)-manifold with nonempty boundary. If both M and N have nonempty boundary, show that M • N is a topological manifold with boundary. Discuss the problem you encounter in trying to give M • N a natural smooth structure of manifold with boundary. This is, in fact, an example of what is called a smooth manifold with corners. For x E int(M), the n-dimensional tangent space T z ( M ) is defined as before. If x c OM, T~(OM) is only (n - 1)-dimensional, but we are going to define an n-dimensional tangent space T~(M) having Tx (0M) as a subspace. The more intuitively appealing approach is that of infinitesimal curves. Indeed, the notion of smooth curve s : [0, ~) - ~ M ,
8(0) = x,
makes good sense and we can define the corresponding infinitesimal curve (s}x. This can be thought of as a tangent vector to M at x. In this way, one obtains all the vectors tangent to the boundary and all the vectors that point "into" M. Unfortunately, the negatives of the inward vectors are not so obtained and the resulting set of tangent vectors is not a vector space. Alternatively, the definition of tangent vectors as derivatives of the algebra of germs of C ~ functions at x, while less intuitive, yields a vector space exactly as before. However, in order to show that this vector space is n-dimensional, we will find it convenient to t u r n to infinitesimal curves. If x E M, whether or not x E OM, the set C ~ ( M , x) of smooth, real valued functions defined in neighborhoods of x is defined and there is no problem defining germinal equivalence on this set. Thus, we obtain the St-algebra ~5~ of germs. We define T ~ ( M ) to be the vector space of derivatives D : ~x ~ St. If x E int(M), this agrees with our usual definition and is n-dimensional. If f : M --~ N is a smooth map between manifolds with boundary, the differentials
dfx = f . z : T~(M) ~ Tf(~)(N)
3.6. M A N I F O L D S
WITH
BOUNDARY
109
are defined by the usual formula
df~(D)[9]f(z) = D[g o f i x and are linear. The proof of the global chain rule (Lemma 3.1.24) goes through equally well in our present context.
L e m m a 3.6.7. If f : M --~ N and g : N --* P are smooth maps between manifolds with boundary, then 9 o f is smooth and, f o r each x C M , d(g o f ) x = dgf(z) o df~.
Corollary 3.6.8. If f : M --~ N is a diffeomorphism between manifolds with boundary, then dfz : T z ( M ) --~ T I ( z ) ( N ) is a linear isomorphism, V x C M . We have reached the point where a little work has to be done. We must show that T~(N n) is n-dimensional, even when x 6 0]HIn. The above considerations will then extend this property to arbitrary manifolds with boundary. When x E OHn C R ~, we will use the notation q ~ ( ] ~ ) for the algebra of germs of C ~ ( N ' ~ , x ) and q~x(H n) for tile germ algebra of C ~ ( H n , x ) .
L e m m a 3.6.9. Let x e ON '~ and let p : @x(R ~) --+ r
n) be defined by p[f]~ =
[fi(H n n dom(f)]x. Then p is a surjection. Proof. Let U C_ N n be an open neighborhood of x. If g : U --~ IR is smooth, there is a neighborhood V o f x in N ~ and a smooth extension ~ : V ~ I~ o f g l ( V F ) U A N ~ ) . Then [g]x e ~bx(IRn) and p[9]x = [g]x. [] For z 9 OlHI~, define
p* : Tx(l~ n) ~ Tx(I~ n) by setting
p*(D)[f]~ = D(p[f]~). It is elementary that this is linear.
L e m m a 3.6.10. p* is bijective. Proof. We prove that p* is one to one. If p*(D1) = p*(D2), then Dl(p[f]~) = D2(p[f]z), V[f]z 6 ~bx(lRn). Since p is surjeetive, it follows that Dl[g]z = D2[gJx, v[g]~ 9 ~ ( M ~ ) , so D~ = D~. We prove that p* is onto. Let v 9 Tz(]~ n) = ]~n. As an infinitesimal curve, this vector is represented by s(t) = x + tv. As an operator on germs, v = D(s L . Either v points into ]HI~ (we intend this to include the case that v is tangent to OH n) or v points out of ]HIn, in which case - v points into N ~. If v points into H ~, then s(t) 9 H ~, V t >_ O. Define D : r ~) --~ IR by D[g]x = lira g(s(t)) - g(x) t~O+
t
It is elementary that D 9 T~(IHI~) and that p*(D) = D<~L = v. I f v points out o f N ~, then s(t) 9 N ~, Vt _< 0. Define D : r
(IHIn) --, I~ by
D[9]z = lim g(s(t)) - 9(z) t--~O-
t
Again, D 9 T~(H ~) and p*(D) = v.
Corollary 3.6.11. The vector space T z ( H n) is n-dimensional, for all x 9 O]E~.
[]
110
3. G L O B A L T H E O R Y
C o r o l l a r y 3.6.12. Let M be a smooth n-manifold with boundary and let x E OM. Then the vector space T~(M) is n-dimensional. Proof. Let (U, ~) be an IE~Ucoordinate chart about x. Then ~.~:
T~(U) -~ T~(~)(~(U))
is an isomorphism. But T~(U) = Tz (M) and T~(~)(~(U)) = T~(~)(Hn). This latter is n-dimensional. [] At this point, one can define the tangent bundle 7r : T ( M ) ---, M E x e r c i s e 3.6.13. For an n-manifold M with boundary, mimic the construction of the tangent bundle 7r : T ( M ) --* M, showing that one obtains a smooth 2n-manifold with boundary, the projection 7r being identified locally with the canonical projection Pl :IN n x N n--~SIn . Vector fields on a manifold M with boundary are smooth sections o f T ( M ) . The smooth Urysohn lemma and its consequences extend to this context. In particular, vector fields are derivations of the function algebra C ~ ( M ) and open covers always admit smooth, subordinate partitions of unity. E x e r c i s e 3.6.14. Let M be a manifold with nonempty boundary. Show that there is a smooth function f : M --+ [0, oc) such that OM = f - l ( 0 ) .
3.7. S m o o t h S u b m a n i f o l d s We give a definition of "submanifold" that applies to manifolds with boundary. D e f i n i t i o n 3.7.1. Let M be an m-manifold, possibly with boundary. A subset X c M is a properly imbedded submanifold of dimension n if X is closed in M and, for each p E X, there is an Nm-coordinate chart (U, ~) about p in M in which ~(U N X) = ~(U) c~ INn, where INn c N "~ is the (image of the) standard inclusion. Remark that, in the above definition, (U A X,~pl(U N X ) ) can be viewed as an Nn-coordinate chart on X and that the collection of all such charts makes X a smooth n-manifold with boundary OX = X A OM. Thus, if OM = ~, then OX = 0 also. Note also that X cannot be tangent to OM at any point of OX. If OM = ~ = OX and we drop the requirement that X be a closed subset of M, but keep the requirement on local charts, X will be called simply a submanifold of M. E x a m p l e 3.7.2. The image of the standard inclusion II-lIn ~ imbedded submanifold.
N m is a properly
3.7. SMOOTH SUBMANIFOLDS
111
3.7.A. R e g u l a r v a l u e s a n d s u b m a n i f o l d s . The following is a globalization of Theorem 2.9.6. T h e o r e m 3.7.3. Let f : M ---* N be a smooth map between manifolds of respective dimensions m and n, and assume that ON = 0. If y C N is a regular value simultaneously for" f and for Of = f l O M , then f - l ( y ) is a properly imbedded submanifold of dimension m - n. Proof9 Since f is continuous, f - l ( y ) is a closed subset of M. Let p E f - l ( y ) , and find a suitable coordinate chart about p in M. There are two cases. Case 1. Suppose p E int(M). Choose a coordinate neighborhood (U,x) about p such that U c int(M). Then y is also a regular value of f l U , so Theorem 2.9.6 implies that f - l ( y ) N U is a smooth submanifold of U of dimension m - n. Case 2. Suppose p C O M and let (U, x 1, x 2 , . . . , x m) be an Nm-chart about p in M. Assume that f ( U ) C W , where (W, y l , . . . , yn) is a coordinate chart about y in which y = 0. Let Ofl(U • OM) be denoted by ~ ( x 2 , . . . ,x m) with component functions ~ 1 , . . . , ~n relative to the coordinates of W. Since p is a regular point for ~, U can be chosen so small that the matrix Ox."~
L
...
:,
has constant rank n on U A OM. By a permutation of the coordinates x 2 , . . . , x m, it can be assumed that the last n x n block
...
6q; n
is nonsingular on U A O M . Choosing U even smaller, if necessary, the corresponding n x n block in the matrix
is also nonsingu]ar. We then resort to the trick of recoordinatizing U near p by z i : X i, 1 < i < m -- n, and Z m - n + j ~-- f J , 1 <_ j < n. The inverse function theorem shows, by the above remarks, that this will define an NI~-chart on a small enough neighborhood (again called U) of p. But, relative to these coordinates,
setting
f(zl,z2,...,z
m)
=
(zm-n+l...,zm).
Then f-l(y) nU=
{(zl,...
,zm--n,o,...
T h a t is, f - l ( y ) N U = H m - n A U.
,0)}.
112
3. GLOBAL THEORY
E x a m p l e 3.7.4. Let f : ]HI~+1 ~ ]R be given by
f(xl,...,
n+l Xn + l ) = E ( x i )
2.
i=1
Then 1 E R is a regular value both for f and for Of. The hemisphere f - l ( 1 ) is the intersection S" MH n+ 1 and is an n-manifold wit h boundary f - 1(1) A 0IN~+ 1 = S . - 5 The following lemma shows that there are plenty of regular values as in Theorem 3.7.3. L e m m a 3.7.5. I f OM = 0 and f : N -~ M is smooth, then the set of points in M that are simultaneous regular values for f and Of is dense in M .
Proof. Clearly, if p E ON is a regular point for Of, it is also a regular point for f. Thus, y E M is a regular value both of f and 0 f precisely when it is a regular value both of f[ int(N) and of Of. Use countable coordinate coverings { U i } i e / o f int(N), { ~ }jeJ of ON, and {Wk}kEK of M. For each k E K, consider the countable family of smooth maps fik : U~ cl f - l ( w k )
----+ Wk,
Ofjk : Vj N O f - l ( W k ) - ~
Wk
obtained by restrictions. By Corollary 2.9.5, almost every y E Wk is a common regular value of all these maps. Doing this for each k E K, we complete the proof. [] E x e r c i s e 3.7.6. Suppose that U and V are open subsets of]HIn and that f : U --+ V is a smooth map such that f(OU) C OV. I f x E 0U and f.~ : T~(U) ~ Tf(~)(V) is an isomorphism, prove that f restricts to a diffeomorphism of some neighborhood U ~ of x onto some neighborhood V p of f ( x ) . This extends the inverse function theorem to open subsets of INn , hence to manifolds with boundary. 3.7.B. M a p s of c o n s t a n t r a n k a n d s u b m a n i f o l d s . For simplicity, the discussion in this subsection will be carried out only for the case of manifolds with empty boundary. D e f i n i t i o n 3.7.7. A smooth map f : N --* M of an n-manifold into an m-manifold has constant rank r if, for each p E N, the rank of the linear map f,p is r. The map is an immersion if f has constant rank n and it is a submersion if it has constant rank m. Exactly as Theorem 2.9.6 globalizes to Theorem 3.7.3, so does the constant rank theorem (Theorem 2.4.6). The statement follows and details are left to the reader. T h e o r e m 3.7.8 (Global constant rank theorem). If f : N --~ M has constant rank r and if p E f ( N ) , then f - l ( p ) C_ N is a smooth, properly imbedded submanifold of dimension n - r. This result is not very striking for immersions and, for submersions, it is just Theorem 3.7.3 for maps with no critical values. Since we are assuming empty boundaries, the term "proper imbedding" refers to a smooth imbedding with closed inlage. Note that this agrees with the usual topological notion of a "proper map", this being a map that pulls back compact sets to compact sets.
3.7. S M O O T H
SUBMANIFOLDS
113
3.7.9. If i : N --+ M is a one-to-one immersion, i ( N ) is called an i m m e r s e d submanifold of M . Definition
T h e reader should be warned t h a t m a n y authors call an immersed submanifold
i ( N ) C M simply a submanifold. This is misleading because tile relative topology inherited from M may not agree with the manifold topology of N. E x e r c i s e 3 . 7 . 1 0 . Let i : N --+ M be a one-to-one immersion, let X be a manifold, and let f : X ~ M be a s m o o t h m a p with f ( X ) C i ( N ) . (1) Show by an example t h a t i -1 o f : X --+ N m a y fail to be continuous. (2) If i -~ o f is continuous, prove t h a t it is smooth. 3 . 7 . C . I m b e d d i n g s in E u c l i d e a n s p a c e * . First, we note t h a t the existence of s m o o t h partitions of unity allows us to a d a p t the proof of T h e o r e m 1.5.7 w i t h o u t serious change to prove the following. 3.7.11. If M is a compact, differentiable n-manifold without boundary, then there is a smooth imbedding i : M ---+IRk, for some integer k > n.
Theorem
In fact, the following much more general theorem, due to H. W h i t n e y [50], is known. We will prove it for c o m p a c t manifolds. For a proof of the general case, see [2, C h a p t e r 6]. 3 . 7 . 1 2 ( W h i t n e y imbedding theorem). If M is an arbitrary differentiable n-manifold, then there is a smooth, proper imbedding of M into H 2n+l.
Theorem
Proof for M compact and OM = 0. Since OM = ~, we imbed in IR2'~+1. By Theorem 3.7.11, we choose a s m o o t h imbedding M C R k for a suitably large value of k >_ 2 n + 1 . I f k = 2 n + l , we are done. We assume t h a t k > 2 n + l a n d s h o w t h a t M imbeds s m o o t h l y in IRk-1. Finite repetition of this a r g u m e n t t h e n yields t h e t h e o r e m for the c o m p a c t case. View IRk-1 C IRk as the subspace x k = 0 and let p : R k + ]R be projection onto t h e kth component. For each unit vector v E S k-1 \ ]Rk - I , define pv : Rk ---+]Rt~-1 by
v(w)~ p~(w) = w - ~ T h a t is, p~ is the linear projection of IRk onto 1Rk-1 along v. T h e idea will be to choose v so t h a t p~]M actually imbeds M in IRk-1. To begin with, we choose v so t h a t p~]M is injective. Consider the diagonal
A
=
{(x,x) tx e M}
C
M x M
and the m a p
f:MxM..,A~S
a-1
defined by
x-y f ( x , y ) - iix - YlI' where Itwl[ denotes the usual Euclidean n o r m of w r IRk . Since k - 1 > 2n, Sard's t h e o r e m guarantees t h a t f is not surjective (cf.Example 2.9.4), so we choose v C S k-1 not in the image of f . T h a t is, v is not a scalar multiple of x - y, for any two distinct points x , y C M. Thus, pv(X - y) ~ O, whenever x and y are distinct points of M , proving t h a t pv[M is injective.
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If we can prove t h a t Pvl M is an immersion, then compactness of M and injectivity of p r i M implies t h a t this m a p is an imbedding. Equivalently, we must find v as above such that, for every nonzero tangent vector w E T ( M ) , v r w/[Iwll. L e t
S ( M ) = {w E Z ( M ) l liwll = 1}, the so-called unit tangent bundle of M . It is convenient to view this as a subset of M x S k-1 and, in Exercise 3.7.13, you will show t h a t it is a smooth, c o m p a c t submanifold of dimension 2n - 1. T h e canonical projection of M x S k-1 onto S k-1 restricts to a s m o o t h m a p
g : S ( M ) --* S k-1 t h a t can be viewed as parallel translation in iRk x iRk of unit t a n g e n t vectors to M to vectors issuing from the origin. Again, by the dimension hypothesis and Sard's t h e o r e m , there is v E i m g U i m f . Since Pv is linear, pv. = Pv at each point of M , so p , is b o t h one-to-one and an immersion. [] E x e r c i s e 3 . 7 . 1 3 . Prove t h a t S ( M ) is a s m o o t h submanifold of
T ( M ) C M x iRk. (Hint. F i n d a suitable m a p v : T ( M ) ~ iR having 1 as a regular value.) E x e r c i s e 3.7.14. If M is a compact n-manifold w i t h o u t boundary, show t h a t it a d m i t s a s m o o t h immersion into iR2n.
Remark. T h e existence of proper imbeddings M ~-~ ]HI2n+1 of c o m p a c t n-manifolds w i t h b o u n d a r y is proven by modifying carefully the above p r o o f (cf. [16, Theorem 4.3 on page 31], where proper imbeddings are called "neat imbeddings"). T h e n o n c o m p a c t case is proven by suitable modifications of the t r e a t m e n t in [2, C h a p ter 6]. E x e r c i s e 3 . 7 . 1 5 . Let M C ]HIk be a properly imbedded n-manifold w i t h b o u n d a r y and prove t h a t there is a R i e m a n n i a n metric on iRk agreeing with the s t a n d a r d E u c l i d e a n metric outside of a neighborhood of ON k and such that, at each point x of OM, the orthogonal complement of T~(M) in T~(IR k) lies in T~(OEIk). We say t h a t , relative to this metric, M meets 0]HIk orthogonally along its boundary. E x a m p l e 3 . 7 . 1 6 . If M C ]HIn is the hemisphere S n-1 N ]HIn, t h e n M is properly i m b e d d e d in this half-space and meets OIEn orthogonally along its boundary. Here it is not necessary to change the s t a n d a r d Euclidean metric on R ~ near 0IN~. If i : M -~ ]HIk is a smooth, proper imbedding of an n-manifold, we routinely identify M w i t h i(M), as above, realizing T ( M ) c M x R k as a s u b b u n d l e in the usual way. Via the inclusion II{k C irk, we view M C R k wherever convenient. M o d i f y i n g t h e Euclidean metric in N k as in Exercise 3.7.15, we define ~ ( M ) = {(x,v) E M x iRk I v • T~(M)}. If 0 M = ~, we view M C iRk and define v ( M ) via the s t a n d a r d Euclidean metric. E x e r c i s e 3 . 7 . 1 7 . Prove t h a t the p r o d u c t projection M x iRk ~ M restricts to a m a p 7c : ~ ( M ) --, M t h a t is the projection m a p of a vector bundle of fiber dimension k - n. T h i s is called the normal bundle of M in ]HIk.
3.7. S M O O T H S U B M A N I F O L D S
115
Remark that the normal bundle u(M) is a manifold of dimension k, generally with boundary u(M)IOM, and that this boundary is exactly the normal bundle of OM in 0/I-IIk = IRk-1. Define a smooth map : v ( M ) - , H k, ~ ( x , v) = z + v,
using the additive structure of IRk. Note that we can identify the (image of) the zero section {(x,0)lx E M} C u(M) with M and that, under this identification, v I M = idM. P r o p o s i t i o n 3.7.18. If M C N k is a smooth, properly imbedded submanifold of
dimension n, there is an open neighborhood U of M in ~(M) that is carried diffeomorphically by ~ onto an open neighborhood ~(U) of M in N k. If M is compact, this neighborhood cart be taken to be of the form U(e) = {(x,v) c ~(M) I Ilvll < e}, for suitably small e > O. Pro@ We give the proof for the case that M is compact, leaving the general case as Exercise 3.7.19. Let (x, 0) E M C u(M) and remark that there is a natural identification T(x,O)(~'(M)) = Tx(M) | ~,x(M). Relative to this identification, we can write ~.(~,0) = idT~(M) | id,x (M), so the inverse function theorem (if x C OM, use Exercise 3.7.6) guarantees that there is a neighborhood U~ of (x, 0) in ~(M) carried diffeomorphically by qz onto a neighborhood of x in H k. This neighborhood can be taken to be of the form
u~(~4 = {(y,v) e . ( M ) l y e W~, Ilvll < ~x}, where Wx is an open neighborhood of x in M and e~ > 0 is small enough. Cover M with finitely many sets of the form U~(e~), let e be the smallest e, that occurs, and consider the union of the corresponding neighborhoods U~(e). This is an open neighborhood of M in u(M) of the form
g ( 6 = {(x,v) ~ ~(M) I IIvll < d . Although ~ is locally a diffeon~orphism on U(e), it might fail to be globally oneto-one. We claim that, by choosing e > 0 smaller, if necessary, we can make sure that ~ is one-to-one on U(e). If not, we could choose sequences (x~, v~) r (Yn, wn) in U(e) such that
~(xn, ~n) = ~(y~, ~ ) while IIv~H < 1In and IIw~ll <_ 1In. Passing to a subsequence, if necessary, we assume that xn ~ x and yn --~ y in M, hence that (Xn, v~) --~ (x, 0) and (Yn, Wn) ---* (y, 0) as n --~ oc. By continuity, F(x, O) = ~(y, 0) and, since p is one-to-one on M, we conclude that x = y. Thus, all (x,~, Vn) and (y~, w~) ultimately belong to a neighborhood U~(e~) as in the first paragraph, contradicting the fact that ~ is one-to-one on this neighborhood. [] E x e r c i s e 3.7.19. Extend the above proof to properly imbedded noncompact manifolds. For this, use local compactness and 2nd countability to express M as a countable increasing union of open, relatively compact submanifolds Wi. The neighborhood U will be a union of neighborhoods U~(e~) of W~ in u(M) with e~ $ 0 as i --+ oo.
116
3. G L O B A L T H E O R Y
Remark that the bundle projection of u(M) onto M induces a map
7r : U---, M such t h a t 7rim = idM. Such a map is called a retraction and M is said to be a retract of U. The triple (U, Tc, M) is called a normal neighborhood of M in R k, although one usually calls U itself the normal neighborhood. 3.8. S m o o t h H o m o t o p y a n d S m o o t h A p p r o x i m a t i o n s We will study maps f : M --* N between manifolds. The set of all such smooth maps will be denoted by C ~ ( M , N), the set of continuous ones by C~ N). We will show that continuous maps admit arbitrarily small perturbations (homotopies) to smooth ones and that, in this way, the continuous homotopy classes of continuous maps correspond one-to-one to the smooth homotopy classes of smooth maps. The proof of the smoothing theorem and the equivalence of smooth and continuous homotopy can be omitted in a first reading without seriously disrupting later topics. The following subsection, however, contains basic definitions and results t h a t should not be omitted. 3 . 8 . A . S m o o t h h o m o t o p i e s . To begin with, we assume that OM = 0. This avoids manifolds with corners in the following definition. An alternative and equivalent definition of smooth homotopy will then be given that accomodates manifolds with boundary. D e f i n i t i o n 3.8.1. Elements fo, fl C Coo(M, N) are said to be smoothly homotopic if there is a smooth map H : M x [0, 1] ~ N such that (1) fo(x) = H(x,O), g z E M; (2) f l ( x ) = H ( x , 1), Vz E M. We write f0 ~ f l . The map H is called a (smooth) homotopy between f0 and f l . We frequently drop the words "smoothly" and "smooth", when this qualification is clear from the context. One should think of a homotopy as a deformation of one smooth map to another through smooth maps. It can be thought of as a "smooth curve" in COO(M,N) connecting f0 to f l . We write ft(x) = H(x,t), 0 < t < 1. Similarly, if Diff(M) denotes the set of all diffeomorphisms of M to itself, "smooth curves" in Diff(M) will be smooth deformations, called isotopies, of one diffeomorphism to another through diffeomorphisms. D e f i n i t i o n 3.8.2. If f0, f l E Diff(M), a (smooth) homotopy ft between f0 and f l will be called a (smooth) isotopy of f0 to f l if ft E Diff(M), 0 < t < 1. If such an isotopy exists, we say that f0 is isotopic to f l and we write f0 ~ f l . E x e r c i s e 3.8.3. If f0 ~ f l (respectively, f0 ~ f l ) , prove that there exists a homotopy (respectively, an isotopy) H : M x [0, 1] --. N such that ft = fo, 0 < t < e, and ft = f l , 1 - 5 < t < 1, for suitably small e > 0 and 5 > 0. Use this to prove t h a t homotopy (respectively, isotopy) is an equivalence relation on C~176 N) (respectively, on Diff(M)). The equivalence classes for these relations will be called, respectively, homotopy classes and isotopy classes. D e f i n i t i o n 3.8.4. A diffeomorphism f E Diff(M) is compactly supported if there is a compact subset K C_ M such that f l ( M \ K ) = idM-.K. The set of all compactly supported diffeomorphisms is denoted Diff,(M). A compactly supported isotopy
3.8. H O M O T O P Y
117"
between fo, fl E Diffc(M) is an isotopy such that there is a compact subset C C_ M with f t i ( M \ C) = i d M \ c , 0 < t < 1. E x e r c i s e 3.8.5. Prove that the set Diffc(M) is a group under composition. E x e r c i s e 3.8.6. Prove that compactly supported isotopy is an equivalence relation on the group Diffc(M).
Remark. For a compactly supported isotopy, ft belongs to Diffc(/1,J), 0 < t < 1. T h e o r e m 3.8.7 (Homogeneity lemma). If N is connected, boundaryless, and x, y E N , then there is f E Diffr and a compactly supported isotopy ft such that f ( x ) = y and f0 = idN, fl = f. The proof of this theorem uses flows and will be deferred until the next chapter. C o r o l l a r y 3.8.8. If g E C ~ ( M , N), y E N, and N is connected and boundaryless,
then g ~ ~ such that y is a regular value of'g. Proof. By Lemma 3.7.5, we choose a regular value x E N of g. Let f and H be as in Theorem 3.8.7. Since f ( x ) = y and f is a diffeomorphism, it follows that y is a regular value of ~ = f o g. But fi o g is a homotopy of g = f0 o g with ~ = fl o g. [] The definition of smooth homotopy and isotopy that we have given does not adapt nicely to manifolds with boundary. The problem is that [0, 1] is itself a manifold with boundary, hence M x [0, 1] will be a manifold with corners when OM ~ ~. This minor difficulty can be overcome by slightly modifying our definitions. D e f i n i t i o n 3.8.9. If f0, fl E Coo(M, N), these maps are (smoothly) homotopic if there is a smooth map H : M x R ---* N such that (1) H(x,O) = fo(x), g x E M; (2) H(x, 1) = f l ( x ) , V x E M. As usual, we set
f~(x) = H ( x , t ) and say that f0, fl E Diff(M) are isotopic if there is a homotopy between them such that ft E Diff(M), Vt E ]R. E x e r c i s e 3.8.10. Prove that, under the second definition of homotopy and isotopy, these continue to be equivalence relations. If 0 M = 0, prove that the second definition of homotopy and isotopy is equivalent to the first. D e f i n i t i o n 3.8.11. A map f c C~176 N) is a (smooth) homotopy equivalence if there is g E C ~ 1 7 6 such that f o g ~ idN and g o f ,.~ idM. In this case, we say that M and N are homotopy equivalent manifolds and that f and g are homotopy inverses of one another. E x a m p l e 3.8.12. Let M c H ~ be a compact, proper submanifold with normal bundle rr : ~(M) --~ M, and let U(e) be the open neighborhood of the zero section M C ~(M) as in Proposition 3.7.18. Then ~lU(e) e C ~ ( U ( e ) , M) is a homotopy equivalence with the inclusion i : M ~ U(e) as a homotopy inverse. Indeed, 7rlU(e ) o i = idM, so we investigate i o 7rlU(e ) : U(e) + U(e). But g :
u ( 6 x i - . u ( 6 , defined by
H((z,~),t)
=
(~,t~)
=
~(~,~),
118
3. G L O B A L T H E O R Y
is clearly a homotopy ioTrlU(e ) = 7r0 ~ ~rl = idu(6. Notice that each stage 7rt of this homotopy fixes M pointwise. In this situation, M is called a deformation retract of U(e). This is a very special type of homotopy equivalence. Since the normal neighborhood U of M in IHIk is just the image ~(U(e)) under the diffeomorphism of Proposition 3.7.18, we have shown that M is a deformation retract of its normal neighborhood. Intuitively, we can "shrink" the normal neighborhood U down to M while fixing IV/itself pointwise.
3.8.B. Smooth approximations*. Topologists usually formulate homotopy theory purely in the topological category. Differential topologists, on the other hand, like to take advantage of the differentiable structure of manifolds in using homotopy theory. The fact that smooth homotopy theory is equivalent (on manifolds) to the purely topological version is due to the approximation theory t h a t we now develop. The key to this is the following classical result. 3.8.13 (Stone-Weierstrass Theorem). Let X be a locally compact topological space, C ( X ) the algebra of real-valued, continuous functions on X . Let A C_ C ( X ) be a subalgebra containing the constant functions and separating points. That is, for arbitrary x , y E X such that x ~ y, there is f E A such that f ( x ) 7~ f ( y ) . Then, for each f E C ( X ) , each compact subset K C_ X , and each e > O, there exists g E A such that If(x) - g(x)] < e, for all x E K . Theorem
For a proof, see [8]. C o r o l l a r y 3.8.14. If W and V are open, relatively compact subsets of a manifold N such t h a t W C V, i r e > 0 and i f f E C~ Hk), there is "fC C~ Hk),
uniformly e-close to f , smooth on W and equal to f on the complement of V. Proof. Consider first the case k = 1. By the Stone-Weierstrass theorem, we can find f ' E C ~ ( M ) t h a t is e-close to f on V. Let O = N \ W and let {Av, Ao} be a smooth partition of unity subordinate to the open cover {V, O} of N. Then ? =
has the desired properties. For the general case, apply this argument to each of the coordinate functions of f = ( f l , . . . , fk), replacing e by e/x/~. [] We consider maps f E C~ M), appeal to Theorem 3.7.12, we imbed this does not use the full force of the enough dimension k will do. Define a Euclidean metric of INk. That is,
where N and M are both manifolds. By an M as a proper submanifold of INk. In fact, Whitney imbedding theorem since any large topological metric p on M by restricting the
p(x, y) = IIx - yll. By an e-small perturbation of f , we will mean a C O homotopy ft such t h a t f0 = f and p ( f ( x ) , ft(x)) < e, uniformly for x E N and 0 < t < 1. When we say "there is an arbitrarily small perturbation such that ...", this should be read: "for each e > 0, there is an e-small perturbation such t h a t ...". We also fix a choice of normal neighborhood 7r : U ~ M for the imbedded submanifold M. 3.8.15. If W and V are open, relatively compact subsets of N such that W C V and if f E C ~ there is an arbitrarily small perturbation ft
Proposition
3.9. D E G R E E T H E O R Y M O D U L O 2*
119
of f such that fl i8 smooth on W and ft agrees with f on the complement of V, 0
Proof. By Corollary 3.8.14, produce a map ]`E C ~
k) that is smooth on W, agrees with f on tile complement of V, and is uniformly so close to f on V that the image of ] ' l i e s entirely in the normal neighborhood U. Set ~ = t ] ` + (1 - t ) f and note t h a t IJf-ftll-< IIf-~l, 0
0. In any case, the homotopy ft takes its values entirely in U, so we can define ft = 7c o ft. Since 7r is smooth, f l will be smooth on W and the perturbation can be made as small as desired by choosing f uniformly sufficiently close to f . Since f agrees with f outside of V, so does ft, 0 < t < 1. [] T h e o r e m 3.8.16. Given f E C~
~1), there is an arbitrarily small perturbation o f f to a map ]`E C~176 M) and, if f , 9 E C~ N) are homotopic, then ]` and will be smoothly homotopic. Finally, if f is already smooth on some neighborhood of a closed subset X C N, then f can be chosen to agree with f on a smaller neighborhood of X .
Indeed, if M is compact, the first assertion is an immediate corollary of Proposition 3.8.15. The second assertion follows from the first by smoothly approximating the homotopy H C C ~ x R, M). Details of the full proof are left to the following exercise. E x e r c i s e 3.8.17. Prove the general case of Theorem 3.8.16 by an infinite sequence of applications of Proposition 3.8.15.
The set of homotopy classes in C~ M) and the set of smooth homotopy classes in C ~ ( N , M) are canonically the same. C o r o l l a r y 3.8.18.
Proof. If f E C ~ ( N , M), its smooth homotopy class [f]~ is a subset of its continuous homotopy class [f]0. This defines a map ~ : [f]o~ ~ [f]0. By Theorem 3.8.16, if f E C~ M), there is a smooth map f E [f]0, so ~ is surjective. If f , g E C ~ ( N , M) are continuously homotopic, we can choose the homotopy to be constant in t near the values 0 and 1, hence smooth on an open neighborhood of the closed subset X = N x {0, 1} of N x N. By Theorem 3.8.16, we approximate this homotopy by a smooth one that is unchanged on a neighborhood of X. T h a t is, f and g are smoothly homotopic, so the map t. is injective. [] D e f i n i t i o n 3.8.19. The set of homotopy classes of maps f : N ~ M, in either the topological or smooth category, will be denoted by 7r[N, M]. By the above corollary, this definition introduces no ambiguity. 3.9. D e g r e e T h e o r y M o d u l o 2*
Throughout this section, dim M = d i m N > 0 and OM = ~ = ON. The manifold M will be compact and N will be connected. If f E C ~ ( M , N), choose a regular value y E N of f. By Theorem 3.7.3, f - 1 (y) is a 0-dimensional submanifold, hence a set of isolated points. Being a closed subset
120
3. G L O B A L THEORY
of a compact space, it must therefore be a finite set. Let k = I f - l ( y ) [ denote the cardinality of this set, an integer _> 0. T h e o r e m 3.9.1 (Stack of records theorem). If, under the above hypotheses, k > 0, then there exists an open connected neighborhood U of y in N that is evenly
covered by f. Indeed, f - l ( U ) falls into k connected components, each carried by f diffeomorphicaIly onto U. Proof. By the inverse function theorem, choose an open neighborhood Wi of Pi in M t h a t is carried by f diffeornorphically onto an open neighborhood f(Wi) of y in N , 1 < i < k. Since M is Hausdorff, the neighborhoods W~ can be assumed to be pairwise disjoint. Since M is compact, so is X = f ( M \ U/k=1 Wi), and this set does not contain y. Then f(W1) N . . . N f(Wk) "-. X is an open neighborhood of y in N and we let U be the component containing y. Let Ui = f - 1 (U) Cq Wi, 1 < i < k. It is obvious that f carries each Ui diffeomorphically onto U and that k (.Ji=l Ui C_ f - l ( U ) . But, if x E f - Z ( u ) \ [.J/k=l Ui, then f(x) E X , contradicting the fact t h a t X A U = ~. [] C o r o l l a r y 3.9.2. The set R of regular values of f is an open, dense subset of N. The function ;~/ : R --~ Z +, defined by ;~I(Y) = [ f - l ( y ) [ , is constant on each connected component of R. Indeed, we already know that R is dense and Theorem 3.9.1 shows t h a t it is open. The theorem also shows that A/ is locally constant, hence constant on each component. D e f i n i t i o n 3.9.3. I f y E N is a regular value o f f , then d e g 2 ( f , y ) E Z2 is the mod 2 residue class of A/(y). L e m m a 3.9.4 (Homotopy lemma). If f , g C C ~ ( M , N) are smoothly homotopic and if y E N is a regular value for both f and 9, then deg2(f, y) = deg2(g , y).
Pro@ Let H be a smooth homotopy of f to g. We consider two cases. Case 1. The point y is also a regular value of H : M x [0,1] --* N. By Theorem 3.7.3, H - l ( y ) is a properly imbedded l-manifold in M x [0,1]. This submanifold is compact (as a closed subset of M x [0, 1]) and
OH-l(y) = H - l ( y ) A ( M x {0} U M x {1}) = f - l ( y ) x {0} O g-l(y) x {1}. It follows, by Corollary 1.6.15, that h / ( y ) + Ag(y) is an even integer, hence t h a t deg2 (f, Y) = deg2 (g, Y). Case 2. The point y is not a regular value of H. It is, however, a regular value for b o t h f and g, so Corollary 3.9.2 implies that there is an open neighborhood W of y in N on which both A/ and Ag are defined and constant. The set of regular values of H is dense, so we choose such a regular value z E W. By Case 1, we get deg2(f, z) = deg2(g,z), but .~f(y) =/~f(z) and Ag(y) = .~g(Z), so d e g 2 ( f , y ) = deg2 (g, Y). [] Let z E N. By Corollary 3.8.8, choose f ~ f for which z is a regular value. Then deg2(f, z) is independent of this choice, so we set deg2(f, z) = deg2(f, z) unambiguously, obtaining a function deg 2 (f) : N --+ Z2.
3.9. D E G R E E T H E O R Y M O D U L O 2*
By Theorem 3.9.1, this function is locally constant. deg2(f) is constant.
121
By the connectivity of N,
D e f i n i t i o n 3.9.5. The element deg2(f) E Z2 is called the degree (rood 2) of f E
C~176 C o r o l l a r y 3.9.6. If f , 9 C C~
N) are homotopic, then deg2 (f) = deg2 (9).
L e m m a 3.9.7. [f f : M ~ N is not surjeetive, then degz(f) = 0.
Proof. Any z c M that is not a value of f is a regular value, so degz(f) is the residue class rood 2 of A/(z) = ]~3]= 0. [] C o r o l l a r y 3.9.8. [ f N is not compact, degz(f) = 0. D e f i n i t i o n 3.9.9. A map f E C ~ ( M , N ) is essential if it is not homotopic to a constant map. A manifold M is contractible if idM is not essential. C o r o l l a r y 3.9.10. [f deg2(f) # 0, then f is essential. C o r o l l a r y 3.9.11. If M is compact and connected with empty boundary, then M
is not contractible. Pro@ Indeed, deg2(idM ) = 1, so idM is essential.
[]
T h e o r e m 3.9.12 (Boundary theorem). Suppose that M = OW for a compact man-
ifold W and let g E C ~ ( M , N ) . If g eztends to a smooth map G:W--*N, then deg2(g) = 0. Pro@ Let y C N be regular, both for G and 9 = OG. Then G -1 (y) is a compact, one-dimensional manifold with OG-l(y) = G-l(y) N O W = g -~(y). As usual, this set has an even number of elements, so deg2(9) = 0. [] E x a m p l e 3.9.13. Let f : C --+ C be smooth and let W C C be a compact region bounded by smooth, closed curves. A basic question is whether or not f has a zero in W, assuming that f has no zeros on OW. Define 9 : OW
----+ S 1
by
g(z)-
f(z)l,
If(z)
Vz C OW,
a smooth flmction between compact 1-manifolds, S 1 being connected. If f ( z ) r O, Vz C W, then g extends smoothly to G : W --* S 1 by
G(z) - If(z)f(z)[' Vz ~ W, hence deg 2 (g) = 0. This will be enough to prove "half" of the fundamental theorem of algebra. T h e o r e m 3.9.14 (Fundamental theorem of algebra). If f : C ~ C is a polyno-
mial of odd degree m, then f has a zero in C.
122
3. G L O B A L T H E O R Y
Proof. No generality is lost in assuming that f has leading coefficient 1. Write f ( z ) = z m + a l z m - 1 J- . . . q- am, and define a homotopy by
H ( z , t) = f t ( z ) = t f ( z ) + (1 - t)z m = z m J- t ( a l z m-1 + . . . + am). Then, fo(z) = z m and f l ( z ) = f ( z ) . Suppose that Wr C C is a closed disk, centered at 0 and of radius r > O. We claim that, for r sufficiently large, ft has no zeros on OW~, 0 < t < 1. Indeed,
ft(Z)z "~ - l + t
(~
a2 +7+...+77
am)
,
and the term in the parentheses converges to 0 as z --+ oo. Thus, for r > 0 sufficiently large, we define
G : OW~ x [0, 1] + S ~ by
H(z,t)
a(z,t)- qH(z,t)l This is a homotopy between
f(z)
Cl(z)-
If(z)l
and
Co(re ~~ = e~'~o, so degz(G0 ) = deg2(G1 ). But G o l ( y ) contains exactly m points, Vy E S 1, so deg2(G1 ) = 1 since m is odd. It follows, by Example 3.9.13, that f has a zero in Wr. [] There is an integer valued degree for maps f C C~ N ) when M and N are both orientable. This can be used to give a proof of the full fundamental theorem of algebra. We will take this up in Chapter 8 when we study differential forms and de Rham cohomology. D e f i n i t i o n 3.9.15. Let W be a compact manifold, possibly with boundary, and let X c_ W. A retraction of W to X is a s m o o t h map f : W - - , X such that f] x = idx. T h e o r e m 3.9.16. I f W is a compact manifold with O W # 0 connected, then there is no retraction f : W --* OW.
Proof. Indeed, d e g 2 ( f l O W ) = 1, since f l O W = idow, and this contradicts the existence of the extension f : W --+ O W by Theorem 3.9.12. [] C o r o l l a r y 3.9.17 (Brouwer fixed point theorem). If f : D~ ~ D~
is smooth, there is a point x G D n such that f ( x ) = x. Proof. Suppose f has no fixed point. Define g : D n --+ OD n = S n-1 as follows. For each x E D ~, construct the ray Rx starting at f ( x ) and passing through x ~ f ( x ) . Let g(x) be the unique point Rx N S ~-1. If x C O D n, it is clear that g(x) = x, so if g is smooth, we have contradicted Theorem 3.9.16. The smoothness of g is left for the reader to check. []
3.9. DEGREE
THEORY
MODULO
2*
123
E x e r c i s e 3.9.18. Prove that Corollary 3.9.17 holds when f is assumed only to be continuous. Another famous theorem that can be proven using mod 2 degree is the J o r d a n Brouwer separation theorem (smooth version). We introduce the key idea, that of "winding number", and then, in a series of exercises, lead you through a proof of the separation theorem in the plane (the smooth version of the Jordan curve theorem). D e f i n i t i o n 3.9.19. Let f : S 1 --~ IR2 be a smooth map and let p C IR2 \ f(S1). Define
fp : S 1 -+ S ~ by the formula
fp(Z) -
f(z)-
i]f(z)
p
P[t'
where [l" I[ denotes the usual Euclidean norm. Then the (mod 2) winding number of the closed curve f around p is
w2(f,p) = deg2(L). Remark that the winding number is defined for an arbitrary smooth closed curve f. It is not required that f be an imbedding or even an immersion. In the case that f is a diffeomorphic imbedding (a smooth Jordan curve), you will show in the exercises that the open set ]Re \ f ( S 1) has exactly two components, distinguished from one another by the fact that wz(f,p) = O, for every point p in one component, and w2(f,p) = 1, for every point p in the other. The component in which w2(f,p) = 0 is unbounded (and called the "outside" of f(S1)), while the other component is bounded (the "inside"). E x e r c i s e 3.9.20. Let f be a smooth Jordan curve, let U be a connected component of JR2 \ f(S~), and let p,q 9 U. Prove that w2(f,p) = w2(f,q). (Hint: The mod 2 degree is a homotopy invariant.) D e f i n i t i o n 3.9.21. If p 9 IR2, the ray in ]R2 out of p and having direction given by the unit vector v 9 S 1 will be denoted by Rp(V). E x e r c i s e 3 . 9 . 2 2 . If f : S 1 --+ ]R2 is a smooth Jordan curve and p 9 ]R2 \ f(S1), prove that v 9 S 1 is a critical value of fp : S 1 ---+S 1 if and only if the ray Rp(v) is somewhere tangent to the Jordan curve f.
E x e r c i s e 3.9.23. Prove the smooth Jordan curve theorem: I f f is a smooth Jordan
curve, then ]R2 ".. f ( S 1) has exactly two components, one of which (called the inside of f) is bounded (i.e., has compact closure) and the other of which (the outside of f) is unbounded. For every point p in the outside o f f , w2(f,p) = O, and for every p on the inside, w2(f,p) = 1. Finally, f ( S ~) is the set-theoretic boundary of each of these components. Proceed as follows. (1) If p 9 IR2 \ f(S1), prove that w2(f,p) is the number of points rood 2 in Rp(V)Af(S1), for v 9 S a any regular value of fp. (Hint: Use Exercise 3.9.22.) (2) Use (1) to prove that there are points p, q 9 IR2 ",, f ( S 1) such that w2(f,p) w2(f,q). By Exercise 3.9.20, conclude that IR2 \ f ( S 1) has at least two components. Also remark that the winding number is 0 about points in at least one of these components and is 1 about points in at least one of the other components.
124
3. G L O B A L T H E O R Y
(3) Using the fact that f is a smooth imbedding, choose a coordinate chart (U, u, w) about any point f(z) in which g = ((~,w)l-
2 < ~ < 2,-2
< w < 2}
and f ( S 1) A U is just a horizontal line segment w = 0. Show t h a t every point of ~2 \ f(S1) can be connected by a continuous path in R 2 \ f ( S 1) either to the point (0, 1) C U or (0, - 1 ) E U. This proves that R 2 \ f ( S 1) has at most two connected components. (4) Prove t h a t one of these components is bounded, that the other is not, and t h a t the winding number of f is 0 about the points in the unbounded component. (5) Show that each point of f ( S 1) lies on an arbitrarily short arc that meets both components. Conclude that f ( S 1) is the common set-theoretic boundary of these components. 3.10. M o r s e F u n c t i o n s * In Subsection 2.9.B, we defined the notion of a nondegenerate critical point of a function f c C ~ ( U ) , where U is an open subset of Euclidean space. Via local coordinates, this notion carries over to functions f E C ~ ( M ) on a r b i t r a r y manifolds. Since the definition of the Hessian (Definition 2.9.14) is coordinate-free, the actual choice of coordinates about the critical point is immaterial. D e f i n i t i o n 3.10.1. If f E C ~ ( M ) and a E N,
M~ = {, e M f f ( x ) <_a}. D e f i n i t i o n 3.10.2. A function f C C~~ is a Morse function if all of its critical points are nondegenerate and, for each a C IR, M ] is compact. E x a m p l e 3.10.3. We give a simple intuitive example of a Morse function. View the 4-holed torus M as imbedded in IRa as in Figure 3.10.1. The function f : M --+ 1R defined by orthogonal projection to the vertical axis has critical values as indicated. The ten corresponding critical points are all Morse singularities. The eight intermediate critical points have index 1 (saddle points), the minimum has index 0 and the maximum has index 2. Since M is compact, the requirement on M ] in the definition is automatic. It should be noted that, as a increases from its minimum value of 0 to its maximum value of 1, M~ undergoes a change in topology exactly when a critical level is passed. This sort of behavior, typical for Morse functions, will be examined further in Section 4.2 and illustrates the fundamental importance of such functions in differential topology. By the Morse lemma (Theorem 2.9.18), the critical points of a Morse function are isolated. In particular, Morse functions on compact manifolds have only finitely many critical points, In this section, we will demonstrate the following result. Theorem
3.10.4. If M is a manifold without boundary, it admits a Morse func-
tion. In fact, this will be improved considerably in Exercise 3.10.18, where you will show t h a t every smooth function on M can be approximated arbitrarily well on compact sets, together with its derivatives, by a Morse function.
3.10. M O R S E F U N C T I O N S *
125
F i g u r e 3.10.1. The height function for the 4-holed torus M The importance of Theorem 3.10.4 will become apparent in Section 4.2, where we will sketch the proof that a Morse function defines a decomposition of a compact manifold into a finite union of "handles", joined together in a very regular way. This so-called "handle-body decomposition" has deep topological applications, including S. Smale's proof of the Poinca% conjecture in dimensions > 5 [38]. We consider M C/R k via a proper imbedding. This does not use the full force of Theorem 3.7.12 since the exact value of k is not important. The normal bundle ~(M)is a properly imbedded submanifold of dimension k. Recall that ~(M) = {(z,v) Ix E M, and v Z Tx(M)}. We have earlier used the map : v(M) ~ IRk, ~o(x, v) = x + v to imbed a neighborhood of the zero section into IRk, but now we will be interested in this map on the entire manifold t~(M). Intuitively, we see v as a line segment issuing from x and perpendicular to M and take ~(x, v) to be the terminal point of this segment. D e f i n i t i o n 3.10.5. A point p E ]Rk is a focal point of M if it is a critical value of ~. If p = ~(q, v) is such a focal point, where the nullity of the linear map g~.(q,v) is # > 0, we say that p is a focal point of (M, q) having multiplicity #. E x a m p l e 3.10.6. Let M = S 1 • R C R 3 be the right circular cylinder with axis the z-axis and radius 1. Then this axis is exactly the set of focal points of M. If p = (0, O, z) is one of these focal points, then ~ - l ( p ) is the set of inwardly pointing unit normal vectors (q, v) at the points q C S 1 x {z}. Since T(q,v)(t~(M)) decomposes naturally into direct summands tangent to M and perpendicular to M, respectively, the reader should have little trouble seeing that the nullity of ~*(q,4 is 1.
126
3. G L O B A L T H E O R Y
E x a m p l e 3.10.7. The unit sphere S n - 1 C ]1~n has just one focal point, the origin. The preimage ~-1(0) is the set of inwardly pointing unit normal vectors (q,v), q C S n - l , and the multiplicity of the focal point at each q E S n-1 is n - 1.
Remark. These examples suggest that a focal point of M should be a point where arbitrarily nearby normal lines to M intersect. This is not exactly right, but it is correct in some "infinitesimal" sense. P r o p o s i t i o n 3.10.8. The set of points in ]Rk that are not focal points of M is an
open, dense subset. Proof. By Sard's theorem, the set is dense. Since dim u(M) = k, the inverse function theorem implies that this set is open. [] Given p E R k, we define fv : M --+ R by
fp(x) = ILx - pll 2, where H" II denotes the usual Euclidean norm. The following gives Theorem 3.10.4. T h e o r e m 3.10.9. I f p E ]Rk is not a focal point of M, then fp is a Morse function. In light of Proposition 3.10.8, it will follow that there are uncountably many Morse functions on M. This will be made sharper in Exercise 3.10.18. The proof of Theorem 3.10.9 will also show how to compute the index of a critical point of fp (Exercise 3.10.17). The following is evident. L e m m a 3.10.10. For fixed p E ]Rk and arbitrary a E JR, M~, is compact. It will be helpful to write the proper imbedding explicitly as { : M ~-~ R k. By the usual abuse, we write M for {(M). Relative to coordinate neighborhoods (U, u l , . . . , u n) on M, the vector-valued first derivatives O{/Ou i, 1 < i < n, form a basis of T{(~)(M). The formula for fp in the coordinates u = ( u l , . . . , u n) is
f ( u ) = fp({(u)) = ~(u). ~(u) - 2{(u) . p - p . p . Correspondingly, we have derivative formulas
(,)
o--~ =
2
(~ - p )
L e m m a 3.10.11. For fixed p C ]Rk, a point q E M is a critical point of fp if and only if there is a vector vo • Tq(M) in Tq(lRk), IIv01l = 1, and a real number A > 0 such that p = q + )~vo (= ~(q, Av0)).
Proof. Let (U, u l , . . . , u ~) be a coordinate neighborhood centered at q. That is, u~(q) = 0, 1 < i < n. By the local formula (.), q is a critical point of fp if and only if
o~
Oui(O) L ~ ( O ) - p = q - p , Equivalently, q - p s Tq (M).
l
E x e r c i s e 3.10.12. Fix a choice of unit vector vo 3_ Tq(M). If the coordinate neighborhood U of q in M is small enough, we can extend v0 to a smooth unit normal field v on U. That is, a(u) = (u,v(u)) is a section of the normal bundle such that ]lv(u)]l = 1. Here, in terms of the local coordinates, q = 0 and v(0) = v0.
3.10. MORSE FUNCTIONS*
127
If w c Tq(M), the derivative Dwv E Tq(R k) is taken by applying the directional derivative D~ to the coordinate functions of v(u). For Wl, w2 E T q ( M ) , prove that s
= (DwlV) .w 2
defines a symmetric bilinear form on Tq(M) that depends only on vo, not on the choice of extension v. (Hint. Extend wl and w2 locally to tangent fields and use the Leibnitz rule for differentiating the dot product.) D e f i n i t i o n 3.10.13. The symmetric bilinear form s mental form of M at q in the normal direction v0.
is called the second funda-
As the language second fundamental form suggests, there is also a first one. hi the terminology of classical differential geometry, the first fundamental form is just the Euclidean dot product restricted to Tq(M). This defines the intrinsic geometry of M. The second fundamental form detects the way in which M relates to the surrounding Euclidean space, the so-called "extrinsic geometry" (cf. Definition 10.2.14 ft.). The symmetric n x n matrix of "metric coefficients"
0~
or
gij(u) = Ou i . Ou i, evaluated at u = O, gives the matrix for the first fundamental form at q relative to the local coordinates. The following exercise gives the matrix of the second fundamental form relative to these coordinates. E x e r c i s e 3.10.14. Let v extend v0 = v(0) to a unit normal field on the coordinate neighborhood U as in Exercise 3.10.12. Set
02{ ~ij (u) = OuiOu----U 9v. Prove that the numbers fij(O) are the entries of the matrix for the second fundamental form Lvo of M at q. We fix the critical point q c M and the chart (U, u l , . . . ,u ~) as above. By a linear change of coordinates, we assume that
o~
0r
o ~ (0) . . . . ' a ~
(0)
is an orthonormal basis of Tq(AJ). That is, the metric coefficients satisfy gi5(O) = 6i/. L e m m a 3.10.15. I f p = q + Avo is as above, then p is a focal point of ( M , q ) of multiplicity # if and only if 1/A is an eigenvalue of the matrix [g~j(0)] of multiplicity #. (In classical terwzinology , A > 0 is a radius of curvature at q relative to the normal direction vo.)
Proof. The coordinate neighborhood U can be chosen to trivialize the normal bundle, so we choose linearly independent sections ac~ : U - ~ p ( M ) ,
1
Written in terms of the coordinates, these have the ~brm
~(~)
= (~, v~(~)).
128
3. G L O B A L T H E O R Y
By the G r a m - S c h m i d t process, we arrange that { v ~ , . . . , vk-~} be everywhere orthonormal. We can also arrange t h a t Vl(O) = v0. Since Ou I ( 0 ) , . . . ,
(0), vl ( 0 ) , . . . , vk-~(O)
form an orthonormal basis of IRk, we can rotate so that they are the s t a n d a r d coordinate basis. We then coordinatize rc-~(U) C ~(M) by k-rz
u
~=1
t
Relative to these coordinates, we write k-
=
+ v~=l
and 0~_ Ou i
O~ kc~ ~Vc~ 0 7 + ~ t -~--~ui,
1 < i < n,
a=l
O~
=v~,
l <_fl<_k-n.
Consider the matrix function A(u,t) =
[fo 1 >11 au, o~, o_z_ o_s
At the point ....
,o)
the matrix A(u, t) is simply the Jacobian matrix of ~. Explicitly, J~o(q,)wo) =
ou" 0
Ik-n
This matrix is singular if and only if q + Av0 is a focal point of (M,q), the nullity # of the matrix being the multiplicity of the focal point. This is the same as the nullity of the upper left-hand corner, which we claim is just the multiplicity of 1/A as an eigenvalue of the matrix [gij(O)]. Indeed, 0 -
~-~(vl 9 ~uJ) = OVlO~ Ou i " u-7 + gij, =o
and so we want the nullity of the matrix
[~ -
~t~j (0)].
But this is exactly the multiplicity of 1/A as an eigenvalue of the matrix [gij (0)].
[]
Let q E M , v0 2_ Tq(M), Hv0H = 1 and p = q +)w0. By Lemma 3.10.11, q is a critical point of fp. L e m m a 3.10.16. The critical point q of fp is nondegenerate if and only if p is not a focal point of (M, q).
3.10. M O R S E F U N C T I O N S *
129
Proof. We write f(u) = fp({(u)) and differentiate equation (*) to obtain the matrix equation
[0u~&JJ = 2 ~
&--7+ o~&---T ( ~ - p)
At the critical point q = {(0), the left-hand side of this equation becomes the Hessian matrix for fp (Exercise 2.9.16) and the right-hand side becomes
2[5,5 - ~ei~(0)]. The nullity of this matrix is exactly the multiplicity of 1/1 as an eigenvalue of [•ij(0)]. By Lemma 3.10.15, this is zero if and only if p is not a focal point of (M, q). []
Proof of Theorem 3.10.9. I f p C IRk is not a focal point of M, then Lemma 3.10.11 and Lemma 3.10.16 imply that every critical point of fp is nondegenerate. Together with Lemma 3.10.10, this proves that fp is a Morse function. [] E x e r c i s e 3.10.17. Using tile formula (**) for the Hessian, prove that the number of negative eigenvalues of this matrix (the Morse index of the critical point q) is equal to the number of focal points of (M, q), counted with multiplicity, on the line joining q to p. E x e r c i s e 3.10.18. Let g C C~176 let K C_ M be compact, let e > 0 and let rn > 0 be an integer. Prove that there is a Morse function h such that h and its derivatives of order _< rn are uniformly close, respectively, to g and its derivatives of order _< rn on the compact set K. Proceed as follows. (1) Given an imbedding ( : M ~ IRk-l, define = (9,
r
: M
---+ IR x I R k - 1 = IRk.
Prove that this is also a smooth imbedding. (2) Choose p = ( q - c, e 2 , . . . , ek) c IRk so that p is not a focal point of M. Note that c can be chosen as large as desired and the eis can be chosen as small as desired. (3) Show that h = (fp - c2)/2c is the desired approximating Morse function, for suitable choices of c and el, 1 < i < k. In particular, if M is compact, any smooth function can be approximated uniformly well by a Morse function and the approximation can be made uniformly close in as many derivatives as desired.
CHAPTER 4
Flows and Foliations In this chapter, we investigate the global theory of ordinary differential equations (flows), referred to as O.D.E., and the Frobenius integrability condition for k-plane distributions (foliation@ Although this latter topic concerns global partial differential equations, our approach will be largely qualitative, with very few explicit partial differential equations in evidence. Unless otherwise indicated, all
manifolds will have empty boundary. 4.1. C o m p l e t e Vector Fields The space X(M) of smooth sections of the tangent bundle is a module over the algebra C ~ ( M ) and a vector space over R. Viewed as the space of derivatives of C ~ ( M ) , ~ ( M ) is a Lie algebra over IR. By the local theory of O.D.E., for each q E M, a vector field X E X(M) generates a local flow (I) : ( - e , e) • V --~ U, where (U, x l , . . . ,x n) is a local coordinate chart about q, V is an open neighborhood of q with compact closure in U, and e > 0 is sufficiently small. Any two local flows generated by X agree wherever both are defined. The notion of a local flow about a point makes sense even when V and U are not coordinate neighborhoods. A system of suitably coherent local flows covering a manifold M will be called a local flow on M. Here is the precise definition. D e f i n i t i o n 4.1.1. A local flow ~5 on M is a family of smooth maps
written ~ s ( t , x) = ~ ( x ) , such d m t (1) Vs C Us C_ M are open sets and {V~}se~ covers M; (2) ~5~ : Vs --~ Uo is the inclusion map, V c~ C 91; (3) ~ stl+t2 = ~Ztt o ~ st2, wherever both sides are defined, Vc~,/3 C 91. D e f i n i t i o n 4.1.2. If 9 is a local flow on M and q C M, a curve of the form Sq(t) = q~(q), - e s < t < e~, where q E Vs, is called a flow line of q~ through q or the orbit of the flow through q. E x e r c i s e 4.1.3. If Sq and Sq ~ are flow lines through q, show that they agree on their common domain ( - e s , es) Cq(-eZ, eZ). Thus, the velocity vector Xq =
~q(O) 9 Tq(M)
is well defined, Vq 9 M. Prove that this defines a smooth field X 9 E(M) and that
D e f i n i t i o n 4.1.4. The vector field X obtained from the local flow (I) as above is called the infinitesimal generator of ~.
132
4. F L O W S A N D F O L I A T I O N S
E x e r c i s e 4.1.5. Show t h a t every vector field X C Y-(M) is the infinitesimal generator of a local flow r on M. If two local flows 02 and q~ have the same infinitesimal generator X, prove t h a t 02 0 9 is a local flow with the same infinitesimal generator X. Consequently, by partially ordering local flows by inclusion we see that, given a local flow 02 on M , there is one and only one maximal local flow on M containing 02.
There is a one-to-one correspondence between maximal local flows 02 on M and vector fields X E Y.(M) given by letting X be the infinitesimal generator of g2. C o r o l l a r y 4.1.6.
Working with local flows can be somewhat uncomfortable. Happily, there are n a t u r a l situations in which the m a x i m a l local flow of a vector field contains an honest (i.e., global) flow. D e f i n i t i o n 4.1.7. A (global) flow on M is a smooth map
02:R• w r i t t e n 02t(x) = 02(t, x), such t h a t (1) 020 = idM; (2) 02tl+t2 = 02tl ~ 02t2, V t l , t 2 E ~. If the m a x i m a l local flow of a vector field X E Y-(M) contains a global flow, we say t h a t X is a complete vector field. In this case, X is also called the infinitesimal generator of the global flow. E x a m p l e 4 . 1 . 8 . We define a g l o b a l flow on T 2 = S 1 • S 1. F i x p C ~ and, for each z = (e 2€ e 2r~ib) E T 2 and t C IR, define 02f(z) = (e 2'ri(a+t), e2'ri(b+pt)). It is clear t h a t this defines a global flow 02P on T 2. Of some interest is the way in which the qualitative behavior of this flow depends on the value of the c o n s t a n t p. If p is rational, there is a least positive integer k such t h a t pk is also a n integer. Thus, 2p
( 2~ria e2rrib) = 02P(e27ria e2rrib~ t+k\ e , t \ , 1"
One concludes rather easily t h a t each flow line 02P(]R • {z}) is an imbedded circle, the flow being periodic of period k (cf. Exercise 4.1.9). By contrast, if p is irrational, each flow line 02P(IR x {z}) is a one-to-one immersed copy of IR t h a t is everywhere dense in T 2. This is the two-dimensional version of a theorem of Kronecker t h a t we will prove in Chapter 5 (Example 5.3.9). The two-dimensional version will also follow from a result to be proven in Section 4.4 ( cf. Corollary 4.4.10 a n d Exercise 4.4.12). R e m a r k t h a t the infinitesimal generator X of the flow 02P "lifts" to a welldefined vector field on ]R2 relative to the canonical projection p : IR2 --. ]R2/Z 2 = T 2,
p(x, ,a) = (e =~x, e ~ ) . More precisely, the constant vector field )( E Y-(]R2), defined by X(~,y) - vp = (1,p)
4.1. C O M P L E T E
VECTOR FIELDS
133
satisfies p.(x,~))((:~,y) = Xp(x,y), for all (x, y) C I1{2. T h e reader can check this easily. It is n o t e w o r t h y t h a t the vector field )( on N 2 is also complete, generating the translation flow
~ f (w) = w + tv~. This lifted flow is quite tame, regardless of whether p is rational or irrational. E x e r c i s e 4.1.9. Let (I) be a flow on M and let x 6 M be a point not fixed by the flow, but such t h a t (I)c(x) = x for some c > 0. Prove t h a t there is a n u m b e r co > 0 and a s m o o t h i m b e d d i n g such t h a t In this case, the flow line Rx is the imbedded circle ~ ( S 1) and we say t h a t x is a periodic point of the flow of period c0. Not every vector field is complete. For example, Exercise 2.8.13 showed t h a t
e t ~ C ~ ( N ) is not complete. Lemma
4.1.10.
If the maximal local flow of X E ~ ( M ) contains an element of
the form 4~ : ( - e , e ) x M - + M
with e > O, then X is complete. Proof. Let t 9 ~. T h e n one can find k 9 Z and r 9 ( - e / 2 , e/2) such t h a t t = r + k . e/2. Given x 9 M , define
[~r(x),
k = 0.
If 02t(x) is well defined by this formula, - o c < t < oc, t h e n it will be an integral curve to X . To see this, remark t h a t ~T(x), - ~ < r < ( / 2 , is integral to X and use t h e fact t h a t (~• = X (Exercise 2.8.14). We show t h a t ~ t ( x ) is well defined. For simplicity, let t > 0. Obvious modifications of the a r g u m e n t give the general case. Suppose t h a t r + k . e/2 = t = .s + q . ~/2, where s, r 9 e - e / 2 , e/2) and k, q 9 Z. It follows t h a t r - s 9 ( - ~ , e), hence t h a t q - k = 0, 1, or - 1. If q - k = 0, then r - s = 0 and we are done. Assume, therefore, that q - k = • W i t h o u t loss of generality, take q - k = 1. Then, r - s = e/2, so
,~/~
o...o
~/,~ o~,.(~) = ,~/~
o...o
,~/~o,~+~/~(~)
= ,'F~/2 o ' " o 'F~/~o~(x).
k•
134
4. F L O W S A N D F O L I A T I O N S
Thus, (Pt(x) is well defined, - o o < t < oo, for each x E M , and is an integral curve to X . We must show t h a t d) : R x M ~ M is smooth. Let (to, x0) E ]R x M . For small enough z / > 0, we fix k0 E Z such t h a t t = r + k0 9 c/2, for each t E (to - r/, to + 7) and suitable r E ( - c / 2 , c/2). Then, (to - r], to + r]) x M is an open n e i g h b o r h o o d of (to, x0) in IR x M on which
ko
=
o .2.0
k0 < 0,
I ~0
[~r(X),
k0 = 0.
This is a s m o o t h function of ( r , x ) = (t - k0e/2, x), hence a s m o o t h function of [] Theorem
4.1.11. If X E :E(M) has compact support, then X is complete.
Pro@ Since s u p p ( X ) is compact, cover it with finitely many open subsets U 1 , . . . , U~ of M such t h a t the local flow of X contains elements o~ : ( - c i , ~i) x Ui ~ M, 1 < i < r. Let U0 = M \ s u p p ( X ) , an open set with XIUo ==-O. Define ~5~ : II~ x U0 --+ M r by qst~ ) ---- x, Vx E Uo, Vt E R. Since { U~i}i=o covers M and the 4)i agree on overlaps, we have a local flow on M generated by X. Let e = minl
~5:(-c,e) xM~M g e n e r a t e d by X. By L e m m a 4.1.10, X is complete. Corollary
[]
4.1.12. If M is compact, every vector field X E :E(M) is complete.
E x e r c i s e 4.1.13. Let r [0, 1] ~ [0, ~] be a s m o o t h map such t h a t ~(x) - 0 on 2 3 [0, ~] U [4, 1] and ~(x) -= g~r on [g, g] (smooth Urysohn lemma). E x t e n d this to a s m o o t h m a p ~ : ]R ~ [0, ~] by requiring periodicity: ~ ( z + 1) = ~(x). This extension is clearly smooth. In X(IR) define
X
=
x 2cos 2~(x) d ,
Y
=
x 2sin 2 ~ ( x ) d .
Prove t h a t X and Y are complete, but t h a t X + Y is not. As another application of Theorem 4.1.11, we return to the homogeneity l e m m a ( T h e o r e m 3.8.7). Let Diff~ C_ Diffc(M) denote the c o m p a c t l y s u p p o r t e d isotopy class of idM. We prove a local version of T h e o r e m 3.8.7. L e m m a 4 . 1 . 1 4 (Local homogeneity lemma). Let U = i n t ( D n) and let x0, Y0 E U. Then there is ~p E Diff~ such that ~(xo) = Yo.
4.1. C O M P L E T E
VECTOR FIELDS
135
Proof. Use the ordinary Euclidean norm from IR'~ and choose points e,71 E (0, 1) such t h a t o <
m*x{ll~oll, Itvoll} <,
Let f : U -~ IR be s m o o t h such t h a t
f(x) =-
{*, O,
<
~.
0 < II41 -< ~, e _< IIzN < 1.
Let y0-
z 0 = v = (c I , c 2 , . . .
, c ~) ~ IRn
a n d define
X = ~-~ ci f 2
c X(U ).
i=l
Since s u p p ( X ) c_ s u p p ( f ) is closed in U and b o u n d e d away from 0 D n, it is compact. By T h e o r e m 4.1.11, X is a complete vector field, and we let r be the flow it generates. This flow is stationary outside the compact set s u p p ( X ) . Let s(t) = Xo + tv = xo + (tc 1,tc2,.,. ,tcn), 0 < t < 1. T h e n IIs(t)ll = IltY0 + (1 - t)x0]l _< max{llz01l, UY011}< r/, so f(s(t)) - 1, 0 < t < 1. Thus,
X4t) = ~ c~ 0-~ 40 = ~(t), i=1
0 _< t _< 1. Therefore, s(t) is integral to X, 0 _< t _< 1, and it follows t h a t eel(z0) = Y0. T h e n aPt, 0 _< t _< 1, defines a compactly supported isotopy of <)o = idM to aPl = r where g, E Diff~ a n d qO(xo) = Yo. [] The proof of Theorem 3.8.7 is a fairly easy consequence of the local homogeneity l e m m a (Exercise 4.1.16). The idea is to show t h a t any two points in the same connected component, of M are isotopic in the following sense. D e f i n i t i o n 4 . 1 . 1 5 . Let xo,Yo E M. We say that x0 is isotopic to Y0, and write x0 ~ I Y0, if there is ~ C Diff~ such t h a t ~(xo) = Yo. E x e r c i s e 4 . 1 . 1 6 . Prove the homogeneity lemma, Theorem 3.8.7, proceeding as follows. (1) Show t h a t Diff~ is a (normal) subgroup of Diff,(M). (2) Show t h a t ~ I is an equivalence relation on M. (3) Prove t h a t the ~"I equivalence classes are exactly the connected c o m p o n e n t s of M . We close this section with a few more exercises. E x e r c i s e 4 . 1 . 1 7 . Let 9 : R x M --+ M be a flow. A subset C C_ M is said to be #P-invariant if apt(x) C C, Vx E C, Vt CIR. If x E M and Rx = {~Pt(X)}tE~ is the flow line through x, prove that the closure Rx is a ~ - i n v a r i a n t set.
136
4. F L O W S A N D F O L I A T I O N S
E x e r c i s e 4.1.18. Let q5 be a flow on M. A subset C C M is said to be a minimal set of q~ if C is a nonempty, closed, ~-invariant set containing no proper subset with these same properties. For example, if z is a periodic point (Exercise 4.1.9), the flow line R~ is a minimal set. Prove that, if M is compact, every closed, nonempty, q~-invariant subset of M contains at least one minimal set. (In particular, by Exercise 4.1.17, every flow line approaches at least one minimal set.) Show by an example that M itself may be a minimal set. E x e r c i s e 4.1.19. Let dp be a flow on 11I. One defines the a-limit co-limit set of a flow line R~ as follows: a(x)={yEMI3tk~-oosuchthat
set and the
lira ~stk(z ) = y } , k~oo
aa(x) = {y r M l q t k T oo such that
lira cbtk(x) = y}.
k~oo
If M is compact, prove that each of these limit sets is a compact, nonempty, (/,invariant set. Show by examples that c~(z) and w(z) may or may not be equal and may or may not be minimal. (Remark: The a- and w-limit set terminology is s t a n d a r d and seems to have its origin in a biblical quotation (Revelations 1:8).) 4.2. T h e G r a d i e n t F l o w a n d M o r s e F u n c t i o n s *
This section is really an extended example, showing how a certain flow associated to a Morse function on M leads to a detailed topological analysis of M. Together with Subsection 2.9.B and Section 3.10, this will complete a brief introduction to Morse theory. In nmttivariable calculus, the gradient V f of a function f on IRn is defined as the vector field Of Of ) Vf= O~,...,~-z~ . It is characterized as the field perpendicular to the level hypersurfaces f - 1 (a) and such that its dot product with an arbitrary vector v E Tp(R ~) is the derivative D v f . In order to generalize this notion to manifolds M, we must fix some choice of Riemannian metric {-, .) on M. This can always be done by gluing together local choices of metric with a smooth partition of unity (Exercise 3.5.9). L e m m a 4.2.1. Let f r C~176
Then, relative to a choice of Riemannian metric, there is a unique field V f r X ( M ) such that (Vf, X} = X ( f ) , for every field X r %(M). The field V f is called the gradient of f relative to the metric. Indeed, Xp ~ X p ( f ) defines a linear functional L on Tp(M). By nonsingularity of the Riemannian metric, there is a unique vector V f p C Tp(M) such that L = (.,Vfp}. Smoothness of the assignment p ~ Vfp is elementary and left to the reader. When the metric is fixed throughout a discussion, we will refer to V f simply as the gradient of f. The following is another elementary observation. L e m m a 4.2.2. A point p G M is a critical point o f f if and only if V fp = O. Fur-
therrnore, if a E ]R is a regular" value o f f and p E f - l ( a ) , then V fp • T p ( f - l ( a ) ) . The gradient will be used to prove the following theorem. Recall the definition of the set M2 (Definition 3.10.1). If a is a regular value of f , this is a manifold with OM] = f - l ( a ) . In any event, we fix the assumption that all of the sets M~
4.2. T H E G R A D I E N T F L O W A N D M O R S E F U N C T I O N S *
137
are compact. Recall (Definition 3.10.2) that this is part of the requirement that f be a Morse function. T h e o r e m 4.2.3. Suppose that a < b and that the level sets f - l ( a ) and f - l ( b ) are
nonempty. If f-l[a,b] contains no critical points of f, then there is a diffeomorphism q# : M~ --~ M~ that is the identity on M ] -C, where e > 0 is as small as desired. Proof. If we choose e > 0 small enough, f will have no critical points in f-~[a e, b + e] and, by Lemma 4.2.2, V f is nowhere zero on this compact set. Thus, we can choose a smooth, nonnegative function p E C~176 identically 0 on M} -~ and on M \ M} +~ and such that 1
plf-l[a, b] - (V f, V f) The vector field X = - p V f has compact support, hence generates a flow ~t on M. Since X vanishes identically on M ] -~, this flow fixes every point of that set. Also, for each q E M ,
d f(il)t(q)) = X~,dq)(f) = ( X , Vf)~,t(q)
= -P(ePt(q)) (V f, Vf)~,dv ) . In particular,
d f(~t(q)) <_ O,
Vt c IR,
d f(o&(q)) = - 1 ,
if ~t(q) E f-l[a,b].
Thus, f(g;t(q)) is nonincreasing as a function of t and, if q E f - l ( b ) , then
f(~t(q)) = b - t ,
O
It follows that the diffeomorphism (~b-a : M --4. M
carries M~ onto M~ and fixes every point in M~ -e.
[]
Now suppose t h a t f is a Morse filnction (Definition 3.10.2). In particular, M2 is always compact and the critical points of f are isolated. Note that, by 2nd countability, there can be at most a countable infinity of critical points and only finitely many of these can lie in any particular level set f - l ( a ) . E x e r c i s e 4.2.4. Using arbitrarily small "bump functions" in neighborhoods of critical points, show t h a t the Morse function f can be slightly perturbed to a Morse function y for which distinct critical points p~ lie in distinct levels f - ~ (a~). In light of this exercise, we assume that distinct critical points of f lie at distinct levels. Fix a critical point p and, replacing f with f - f ( p ) , assume that p E f - 1 (0). For e > 0 sufficiently small, p will be the only critical point of f in f - l [ - e , e]. We intend to analyze the change in the topology of M ] as a varies from - e to e. For this, we need tile notion of a handle.
138
4. FLOWS AND FOLIATIONS
D e f i n i t i o n 4.2.5. Let 0 < ,~ < n. If B ~ and B n-'~ are spaces homeomorphic to the closed unit balls in ]Ra and ]R~-~, respectively, then the Cartesian product B )' x ]~n-,X will be called a A-handle of dimension n. If N is a topological n-manifold and
~ : (OB ~) x B '*-;~ ~ ON is a homeomorphism onto a closed subset of ON, one forms the quotient space of N U B ~ x B ~-a that identifies points ~(y) and y. The resulting space, denoted by NtO~B ~ x B n-a, is said to be the result of attaching a k-handle to N. It is again an n-manifold. Note that, for a 0-handle, (OB ~ x B ~ is empty and we agree that such handles are "attached" via disjoint union. In the case of a 1-handle, (0B 1) x B n-1 has two components, the "ends" of the handle, but in all other cases, handles are attached along connected subsets of their boundary. Return now to the consideration of the sole critical point p C f - l [ _ e , el, f ( p ) = 0 and let ~ be the index of this critical point (Definition 2.9.15). T h e o r e m 4.2.6. The manifold M } is homeomorphic to the space obtained by attaching a )~-handle of dimension n to MT~.
Remark. If the attaching map is a smooth imbedding, one can put a differentiable structure on the resulting manifold. The differentiable structure generally depends on the attaching map. Thus, for instance, the result of attaching an n-handle to a 0-handle (both of dimension n) by a diffeomorphism of boundaries is a space homeomorphic to S ~. Milnor's constructions of exotic differentiable structures on spheres (Example 3.2.6) proceeded by suitably choosing these attaching maps. We will give a detailed sketch of the proof of Theorem 4.2.6. The first step is to use the Morse lemma (Theorem 2.9.18) to define a coordinate chart (U, z l , . . . , z n) about p such that f l U has the formula
f ( z l , . . . , z n) = - z + y where .X X = E(zi)
2,
i=l
v = fi
(zO 2
j=,k+l
In these coordinates, p is the origin. We consider three cases: 1. A = 0 ; 2. A = n ; 3. 0 < . X < n. Case 1. In this case, x = 0 and it is clear that p is a local minimum. Then, if e > 0 is sufficiently small, we obtain an n-ball B n = U N M} which is a connected component of M}. Also, g A M)-~ = 0. By Theorem 4.2.3, the manifold M} \ B n is diffeomorphie to M~-~, and so
M MFU n,
4.2. T H E G R A D I E N T F L O W A N D M O R S E F U N C T I O N S *
139
F i g u r e 4.2.1. The neighborhood U of p
a disjoint union. That is, M} is obtained by attaching a 0-handle of dimension n to M f e. Case 2. Here, y = 0 and p is a local maximum. Evidently, M~ n U = U,
M~ -~ N U = the complement in U of an open ball, where the second equality requires that c > 0 be sufficiently small. Clearly, M} is obtained from M f ~ by attaching an n-handle of dimension n. Case 3. This is the interesting case. To begin with, choose e > 0 small enough t h a t the coordinate chart (U, z l , . . . , z n) contains the closed bail B defined by n
2 < i=1
In Figure 4.2.1, we give a 2-dimensional schematic drawing of the neighborhood U of p, representing B as the disk bounded by the circle and representing the hypersurfaces f - 1 (• as hyperbolas. The horizontal axis represents the subspace on which the last n - A coordinates vanish, the vertical axis representing the space on which the first A coordinates vanish. The darker shading represents M -~ and the lighter shading represents f - 1 [-e, el. The flow of the gradient field is also indicated and it should be intuitively clear that, by adjusting the time parameter suitably, this flow can be used to deform M} to a subspace consisting of M ) -~ with a A-handle attached as in Figure 4.2.2. As indicated, the critical point p will be in the interior of this handle. The idea (following [28, w for a more rigorous approach, is to replace f with a function F t h a t agrees with f outside a small ellipsoidal neighborhood of p in U
140
4. F L O W S A N D F O L I A T I O N S
F i g u r e 4.2.2. Attaching a handle to M f *
(see Figure 4.2.2), has p as sole critical point in U, and satisfies
M~ = M}, M~ c = Mf ~ u~ B ~ x B n-x,
where p lies in the interior of the handle. Since there are no critical points in F-l[-e, eJ, an application of Theorem 4.2.3 proves that
M} ~ M~ c, giving Theorem 4.2.6. The flow in the proof of that theorem is indicated in Figure 4.2.2. More details follow. We are free to choose the Riemannian metric in any convenient way, patching together local choices by a partition of unity. Accordingly, we choose the metric in U so that, with respect to the coordinates z i, it is just the Euclidean metric. Thus, the gradient of any smooth function on U can be computed exactly as in standard multivariable calculus. Construct a smooth function # : R ~ [0, oc) with the following properties: 9 ~(0) > ~, 9 #(t)=0,2~_
Define F to coincide with f on M -, U and, in U, let F = - x + y - ~ ( x + 2y),
where x and y are the functions defined earlier. Since p ( x + 2y) vanishes for x + 2y _> 2q it vanishes outside the ellipsoidal ball E C B defined by x + 2y < 2e and the local definitions o f / 7 fit together smoothly. The solid E is indicated in Figure 4.2.2 by a dashed ellipse, in which the handle is shown to be inscribed. C l a i m 1. M} = M~.
4.2. T H E G R A D I E N T F L O W A N D M O R S E F U N C T I O N S *
141
Pro@ Everywhere, F _< f , and so M ] C_ M g, for every value of a. Since F and f are equal outside the ellipsoidal ball E, we have (,)
M~ \ E = M~, -../~.
Inside E, F < f _< (x + 2y)/2 _< ~, and so E g M~ n M~. Thus,
M r ( ~ .. E) u E_c (Mr -. E)uMr _c ML Mb c_ (Mb --. E) u E c (Mb \ E) u M~ c M~. These inclusions must all be equalities and, by (*), M} = M b.
[]
C l a i m 2. The function F has exactly the same critical points as f.
Pro@ Since F and f coincide outside of U, it is enough to show that FlU has p as its sole critical point. Write F = - z + y - #(x + 2y) in U and compute v r = ~ 9F v x' + N OFv~y' this being obvious since, in U, the formula for V F is the classical one. Purtherinore, the partials
OF Oz OF
--
0y
are never 0, so origin p.
1-#'(z+2y)
<-1+1=0,
= 1-2>'(z+2y)_>
1+0=
-
1,
Vf[U vanishes only where both V z and Vy vanish, namely, at the []
One notes that F ( 0 ) = - p ( 0 ) < - e , hence that there are no critical points of F in F-l[-e,e]. By Theorem 4.2.3, applied to the function F , we obtain C l a i m 3. M } ~ M ~ ~. Evidently, we can write
MF ~ = MT~ u H, where H is a compact set inscribed in E as in Figure 4.2.2. Thus, the final step in the demonstration of Theorem 4.2.6 would be to prove the following intuitively plausible fact (cf. [38]). We omit this step.
C l a i m 4. The space H is a A-handle of dimension n.
Remark. For compact manifolds, Morse theory gives a description of M as a space obtained by gluing together finitely many handles of dimension n. Each handle is attached only along tile part (ON~) x B ~-~ of its boundary, hence it is possible to "flatten" it appropriately, proving that M} has the homotopy type of the space M~-~UwB"x, obtained by attaching the A-cell B "x by an imbedding ~ : OB~ --~ OMj7~. This leads to a description of M, up to homotopy, as a finite "cell complex". For many applications of algebraic topology, this is more useful than the precise decomposition of M into handles. For more details, see [28].
142
4. FLOWS AND FOLIATIONS
4.3. T h e Lie B r a c k e t
As we have already remarked, 2r is a Lie algebra over IR under the Lie bracket. Vector fields D, when viewed as derivations of the function algebra C~176 localize to open subsets U C M. This localization is equivalent to the restriction DIU of D as a smooth section of the tangent bundle, so it is elementary to check the following. L e m m a 4.3.1. ff X, Y E X(M) and ifU C_M is open, then
[xlu,
YlU] = [x, Y]lU.
In particular, properties of the bracket that were proven with coordinates can be extended to global properties on M. We apply this remark to Lie derivatives. Since every vector field X E ~ ( M ) generates a local flow q) on M, the definition of the Lie derivative
L x ( Y ) = lira eP-t*(Y) - Y t--+O
t
makes sense pointwise on M and defines a new field L x ( Y ) E :~(M). The proof that L x ( Y ) = IX, Y] (Theorem 2.8.16) that was given in R n can be carried out in local coordinate charts hence, by Lemma 4.3.1, globalizes. T h e o r e m 4.3.2. If X , Y G X(M), the Lie derivative of Y by X is defined and
smooth throughout M and L x (Y) = [X, r ] . Similarly, Theorem 2.8.20 globalizes. Here, commutativity of local flows r = {~a}~e~ and g~ = {~e}eem means that q*t5 o r = ~5~ o q~t~ wherever both sides are defined. Commutativity of vector fields X, Y E X(M) means that [X, Y] - 0 on M. T h e o r e m 4.3.3. Vector fields X, Y on M commute if and only if the local flows
they generate on M commute. C o r o l l a r y 4.3.4. Complete vector fields X, Y e 2E(M) commute if and only if the flows that they generate commute. We are going to be interested in Lie subalgebras of X(M). Of course, ;~ C_ X(M) is a Lie subalgebra if it is closed under the vector space operations and the bracket. If F C_ T ( M ) is a k-plane subbundle (which we also refer to as a k-plane distribution on M), P(V) c ~(M) is a Coo(M)-submodule and a real vector subspace. It is not generally a Lie subalgebra but, when it is, there are important geometric consequences.
D e f i n i t i o n 4.3.5. The k-plane distribution F C_T(M) is a Frobenius distribution (or an involutive distribution) if P(F) is a Lie subalgebra of X(M).
Remark. If f,g r Coo(M) and X, Y E 2~(M), it is easy to verify the identity [fX, gY] = f X ( g ) Y - g Y ( f ) X + fg[X, Y]. Consequently, if F C T(M) is a k-plane distribution and if the fields X1, X 2 , . . . , X,. e p(F) span r ( f ) over Coo(M), then F will be a Frobenius distribution if and only if [Xi, Xj] C F(F), 1 _< i,j < r.
4.3. LIE BRACKET
143
E x a m p l e 4.3.6. On the group manifold Gl(n), we will define a couple of interesting Frobenius distributions. First recall (Example 2.7.18) that the space of left-invariant vector fields g[(n) C 3r is a finite dimensional Lie subalgebra, canonically identified with tile Lie algebra 9Jr(n) of n x n real matrices under tile commutator bracket. Let sl(n) C g[(n) be tile set of matrices of trace 0 and let o(n) C g[(n) be the subset of skew symmetric matrices. These are clearly vector subspaces and the reader should have little diMcnlty in computing dims[(n) = n 2 - 1, ~ ( n - 1) dim o(n) -
2
Since t r ( A B ) = t r ( B A ) , we see that tr[A, B] = 0, hence si(n) is a Lie subalgebra of l~[(n). If A, t3 c o(n), then [A,B] T = B T A T - - A T B
T = [B,A] = - [ A , B ] ,
so o(n) C l~[(n) is also a Lie subalgebra. If { S ~ , . . . , S ~ _ ~ } is a basis of s[(n), extend it to a basis of g[(n) by adjoining a n2 suitable vector Sn2 and view {S~}~=~ as a set of left-invariant vector fields on Gl(n). It is clear that, at each point P E Gl(n), these fields give a basis { P S I , . . . , PSn2} of TF(GI(n)). Let Sp C Tp(Gl(n)) be the (n ') - 1)-dimensional subspace spanned by ;l p .~qJ .i =pl ~ - I and remark that this subspace does not depend on the choice of basis of s[(n). Let
S=
[_J
Sp.
PEGI(n) n2
This is an (n ~ - 1)-plane distribution on GI(n). Indeed, the vector fields {S~}~=1 define an explicit triviMization of T(GI(n)) ~ Gl(n) x g[(n) relative to which S becomes the trivial subbundle Gl(n) x s[(n). (Warning: We are not using the standard trivialization of T(GI(n)) = Gl(n) x ffJt(n). This would give an imbedding GI(n) x M(n) C T(GI(n)) (tiff)rent from the one we have defined and not very interesting.) Similarly, extending a basis of 0(n) to one of g[(n) and viewing these as left-invariant vector fields on Gl(n), we obtain a distribution O C T(GI(n)) of fiber dimension n ( n - 1)/2 and independent of the choices. By tile remark preceding this example, the fact thai; s[(n) and 0(n) are closed under the bracket implies that S and O are Frobenius distributions on Gl(n). Also, by our construction, if P C Gl(n), then L p , (Sl) = Sp,
Lp,(Oi) = Op. Recall t h a t the special linear group is the subgroup Sl(n) C Gl(n) consisting of the matrices of determinant 1 (Example 2.5.5). In Example 2.5.8, we showed that Ti(Sl(n)) is the subspace of TI(GI(n)) = gJl(n) consisting of the matrices of trace 0. For each Q E Sl(n), LQ carries Sl(n) onto itself', so it follows from our construction that TQ(SI(n)) = S 0. Also, for arbitrary P C Gl(n),
TpQ( P . Sl(n)) = Lp,:ib(Sl(n) ) = Lp,( SQ) = Spq. We say t h a t tile left cosets P . Sl(n) are integral submanifoIds to tile distribution S. In exactly the same way, using Exercise 2.5.9, we see that the left cosets P . O(n) of tile orthogonal group are integral submanifolds to tile distribution O C T(GI(n)).
144
4. F L O W S A N D F O L I A T I O N S
These are the first examples of foliations in this book. The connected components of the left cosets of these groups are the leaves of the foliation integral to the respective distributions. A distribution F on M with (one-to-one immersed) integral submanifolds through each point of M is said to be integrable. We will see that the Frobenius property is precisely the integrability condition (ef. Example 3.4.20). This is the theorem of Clebsch, Deahna, and Frobenius, commonly called the Frobenius theorem. Generally speaking, a smooth map f : M --~ N does not push a vector field X C X(M) forward to a vector field f . ( X ) E X ( N ) . There are two problems. I f f is not surjective, there would be points of N where f . (X) would not even be defined. If f is not injective, there could be points of N where f . ( X ) would be multiply defined. Nevertheless, there are situations in which f fails to be bijective, b u t the following concept makes sense. D e f i n i t i o n 4 . 3 . 7 . If f : M --~ N is a smooth map between manifolds (possibly with boundary), vector fields X E ~ ( M ) and Y E ~ ( N ) are said to be f-related if, for each q C M , f . q ( X q ) = Yf(q)"
E x a m p l e 4.3.8. It is possible that X E 3~(M) is not f-related to any Y E 3r For example, let f : IR ~ S 1 be the map f ( t ) = e 27tit. Then
f,t
t-~
= tf, t
= 27rite 2~it e T / ( t ) ( S 1) C Ty(t)(C ) = C.
There is clearly no vector field on S 1 satisfying this. E x a m p l e 4.3.9. It is possible that X E X(M) may be f-related to many vector fields in ~ ( N ) . For example, let f : S 1 --~ C be the inclusion map. Let U C C be an open neighborhood of S 1 such that U ~ C, Let ~ : C --* [0, 1] be smooth such that supp(~) C U and ~IS 1 -= 1. Define the vector field Y~ = iz on C. Since z Z iz, we obtain X E X(S 1) by setting Xz = iz, V z C S 1. Clearly, X is f-related to both Y and ,kY and these are distinct fields on C. P r o p o s i t i o n 4.3.10. Let f : M --~ N be smooth and let X , Y C ~ ( M ) be f-related to X , Y C 3~(N), respectively. Then [X,Y] is f-related to [ X , Y ] . Equivalently, L x ( Y ) is f-related to L ~ ( Y ) .
P r o @ For each h E C ~ (N) and for each q C M, Y ( h o f ) ( q ) = Yq(h o f ) = f,q(Yq)(h) = Y/(q)(h) = ( Y ( h ) o f ) ( q ) . T h a t is,
Y(hof)=Y(h)of,
VhEC~176
There is a similar relation between X and X. Thus,
[2, Yls(
)(h) =
-
= f , q ( X q ) ( Y ( h ) ) - f,q(Yq)(_X(h)) = X q ( Y ( h ) o f ) - Yq(X,(h) o f ) = Xq(Y(h o f)) - Yq(X(h o f)) = [X,Y]q(ho f) = f.q([X,Y]q)(h),
4.4. COMMUTING FLOWS Vh E C~176
145
T h a t is, [X, Y]/(q) = f.q([X, Y]q),
Vq 9 M.
This is the assertion of the proposition.
[]
E x e r c i s e 4 . 3 . 1 1 . Let f : M --, N be a one-to-one immersion and let Y 9 X(N). Prove t h a t there exists a field X 9 ~ ( M ) t h a t is f-related to Y if a n d only if Yf(q) 9 f.q(Tq(M)), Vq 9 M. In this case, prove t h a t X is mfique (we call X the restriction of Y to the immersed submanifold). E x e r c i s e 4 . 3 . 1 2 . Let f : S 2~-1 ~-~ R 2n be the inclusion (an imbedding, hence a one-to-one immersion). F i n d Y 9 fl~(IR2~) t h a t restricts, as above, to a nowhere zero vector field X 9 X(S2n-1). (By Theorem 1.2.16, this is false for the inclusion f : S 2~ ~-~/R2~+1. This fact is more elementary t h a n Theorem 1.2.16, however, a n d will be proven later (Theorem 8.7.5).)
4.4. C o m m u t i n g Flows A n IRa-chart (U, x i, ... , x n ) about q C M determines n c o m m u t i n g vector fields
(0
Oxl,...
0}
,Ox n
c X.(U).
The corresponding local flows about q are of the form
(~(xl,...
,xi,...
,X n) =
(xl,...
, X i ~- t , . . . , x n ) .
Conversely, the following implies t h a t commuting, linearly i n d e p e n d e n t vector fields correspond to a coordinate chart. 4.4.1. Let M be an n-manifold without boundary, let q E M , let U be an open neighborhood of q, and let X 1 , . . . , X k E :E(U). If these vector fields commute and if { Xlq,... , Xkq } is linearly independent, then there is a local coordinate chart (W, ~) about q such that
Theorem
0 qo,(Xi]W) = 0x---7, 1 < i < k. Proof. Making U smMler, if necessary, we assume t h a t it is a coordinate chart, hence view it as an open subset of [R~. We can do this so t h a t q becomes the origin 0 a n d so t h a t the vectors
xl'
.
.
' x~. .--Ox~k+l 0' . .
0
~ Oxn
0
form a basis of To(U). Let
qsi
: ( - e , e) x W --* U
be a local flow a b o u t q = 0 generated by X i, 1 < i < k. We can choose W to be of the form ( - e , e) n with e > 0 so small that the formula o(xl,...,
x n) =
~
~
(o,...,
o, x
+z , . . .
is defined, V ( x l , . . . , x ~) C W. This defines a smooth m a p
O:W~U.
,
146
4. FLOWS AND FOLIATIONS
Since [Xi, X j] - 0 on U, 1 <_ i , j <_ k, we have O(X 1 ,
9 .
.
, X n)
=
dia(1) ~ x ~ ( 1
)
O
d)a(k) (0, ~xr~(k)
9 9 9 O
.
,. 0 ,. X. k +. l ,
.
,
X n)
for each p e r m u t a t i o n a of {1, 2 , . . . , k}. Also,
0 ( 0 , . . . ,0) = ( 0 , . . . ,0), k+l
d
_
(,~1 o . . . o %
d
dx i
( ~X~x(~1r I
o... o
, ~k ( 0 , . . . , 0, r k + ~ , . . . , rn))
O ' ' ' ~ r~7i ".'O~rk k
(0,... ,0, r k + l , . . .
, r~)))
ri
i
= Xo(r). A
Here, the n o t a t i o n r is a c o m m o n device for indicating omission of the term ~ . I n particular, 0.0 : IRn __, IRn is nonsingular, so we can assume (choosing e > 0 smaller, if necessary) that 0 : W -~ U carries W diffeomorphically onto an open neighborhood W of 0. By the above,
O. ( J ~ )
= Xi]W,
l
The desired coordinate chart (W, ~) is obtained by setting ~ = 0 -1.
[]
In particular, if X is a nonzero vector field defined in a neighborhood of q E M , t h e n there is a coordinate chart (U, x l , . . . , x ~) a b o u t q such t h a t 0 X[U
:
OX 1 .
D e f i n i t i o n 4 . 4 . 2 . A k-flow on M is a smooth m a p 9 : IRk x M -~ M , w r i t t e n q~(v,x) = q~v(X), such that 1. ~o = idM,
2. g~v+~ = ~ , o ~P~, Vv, w E IRk. It follows t h a t ~ _ ~ = ~ v 1, so ~
E Diff(M), Vv E IRk. The m a p
IRk __. Diff(M), defined by v ~-~ ~ v , is a homomorphism of the additive group 1Rk into the group Diff(M). We think of gr as a smooth action of the group IRk on M. Given q E M define ~i/q : IRk --4 M
by setting
~(~) = % ( q ) . In particular, qq (0) = q. D e f i n i t i o n 4.4.3. Given a k-flow ~, the ~ - o r b i t of q E M is qJq(Nk). choice of 9 is fixed, the ~ - o r b i t is also called the IRk-orbit of q.
When a
4.4. C O M M U T I N G F L O W S
147
Remark. Points p, q C M are said to be equivalent under the k-flow if and only if there exists v C ]Rk such that q)v(p) = q. The fact that this is an equivalence relation is a trivial consequence of Definition 4.4.2. The IRk-orbits are the equivalence classes. D e f i n i t i o n 4.4.4. The k-flow 9 is nonsingular if
9 ~,v : Tv(IRk) -~ T~q(~)(M) is one-to-one, Vv C ]Rk, Vq E M. L e m m a 4.4.5. The k-flow gJ is nonsingular if and only if qfl.o: To(Rk) --* Tq(M)
is one-to-one, V q E M. Proof. For fixed v E IRk, let Tv : IRk --* IRk denote translation by v. Then, qJq = ~v o ~q o r_~, Vv E IRk, Vq E M. By the chain rule, '~L = (~v).q o ~,0 ~ (r-~)*-. But (r_~).~ = idek and (q%).q is bijective, so ~qv is one-to-one if and only if koq0 is one-to-one. [] P r o p o s i t i o n 4.4.6. The set of k-tuples ( X 1 , . . . , X k) of complete, commuting vec-
tor fields on M is in natural, one-to-one correspondence with the set of k-flows qJ on M. The fields X 1 , . . . , X k are pointwise linearly independent if and only if g~ is nonsingular. Proof. Given the k-tuple ( X 1 , . . . , X k) of complete, commuting vector fields, let (Ir
, (I)k be the corresponding flows. Since these flows commute, the formula
~ ( ~ ...... 'o) = ~ 1 o . . . o ~ defines a k-flow on M. Conversely, given the k-flow ~, let { e l , . . . ,ek} be the standard basis of IRk and set
dp~ = ~tei, 1 < i < k. This defines k commuting flows and their corresponding infinitesimal generators, X 1, ... , X k are complete, commuting vector fields on M. Finally, these fields are linearly independent at q C M if and only if ~.q0 is one-to-one. Thus, the previous lemma gives the final assertion. [] T h e o r e m 4.4.7. Let M be a connected n-manifold. If there exists a nonsingular n-flow on M , then M is diffeomorphic to T k x R " - k for some integer k = O, 1 , . . . , n. Modulo one technical point, the proof of Theorem 4.4.7 is quite straightforward. Nonsingularity of the n-flow implies that, for each q E M, the smooth map ~q : IR~ ~ M has constant rank n. Consequently, each R~-orbit is an open subset of M. Being equivalence classes, distinct orbits are disjoint, so the connectivity of M implies that there is only one IR~-orbit and ~q is a surjection.
148
4. F L O W S A N D F O L I A T I O N S
F i x q C M . If v , w E ~ n , then eq(v) = eq(w) r
%@
= %@
ff2v-w(q) = q 9 q(v - w) = q V -- W E (~I/q)-l(q). It is clear t h a t G = ( ~ q ) - l ( q ) is an additive subgroup of II~n, so ~q passes to a well-defined h o m e o m o r p h i s m
r : ]~nlG --* M. If G were the subgroup Z k • {0} C Rk • R n - k , we would have a natural s t r u c t u r e of s m o o t h n-manifold on N ~ / G = T k • R n - k and, ~q being locally a diffeomorphism, the homeomorphism r
kxR ~-k-~M
would be a diffeomorphism. Therefore, it will be enough to find a linear autom o r p h i s m A : IR~ --. R ~ such t h a t A(G) = Z k • {0}. This is the technical point m e n t i o n e d above. D e f i n i t i o n 4.4.8. A n additive subgroup G c IR~ is a k-dimensional lattice if it is g e n e r a t e d by a linearly independent subset { v l , . . . ,vk} C R n. If G is a kdimensional lattice for some k = 0, 1 , . . . , n, t h e n G is called a lattice subgroup. For example, Z n C I1~= is an n-dimensional lattice. More generally, i f 0 < k < n, t h e n Z k x {0} C IRk x N ~-k is a k-dimensional lattice. In fact, if G is as in Definition 4.4.8, a linear a u t o m o r p h i s m of IR~ taking vi to the s t a n d a r d basis vector el, 1 < i < k, carries G to Z k x {0}. T h e o r e m 4.4.9. A nontrivial, additive subgroup G C ]R~ is a lattice subgroup if and only if there is a neighborhood U C N n of the origin such that G N U = {0}.
By t h e above remarks, this theorem will complete the p r o o f of T h e o r e m 4.4.7. Indeed, the fact t h a t ~Itq is locally a diffeomorphism implies i m m e d i a t e l y t h a t there is a n e i g h b o r h o o d of 0 c ]I~n meeting G = ( ~ q ) - l ( q ) only in the point 0. Before proving T h e o r e m 4.4.9, we discuss some other consequences. C o r o l l a r y 4 . 4 . 1 0 . I f G C ]~ is an additive subgroup that is isomorphic to Z k, some k > 2, then G is dense in R.
Proof. For dimension reasons, G is not a lattice subgroup. Given ( > 0, one can find a C G such t h a t lal < c ( T h e o r e m 4.4.9). Thus, { r a } ~ z C G partitions N into intervals of length < r so every t E N lies within e of a point of G and G is dense in ~ . [] C o r o l l a r y 4 . 4 . 1 1 . I f G C S 1 is a subgroup that is isomorphic to Z k, some k > 1, then G is dense in S 1.
Proof. Indeed, the s t a n d a r d projection p : N --. S 1 is a group surjection and p - 1 (G) is an a d d i t i v e subgroup of R t h a t is isomorphic to Z k+l. By Corollary 4.4.10, p - 1 (G) is dense in ~ and the assertion follows. []
4.4. COblMUTING
FLOWS
149
For the p r o o f of T h e o r e m 4.4.9, we need two lemmas. 4 . 4 . 1 3 . I f there exists a neighborhood U as in T h e o r e m 4.4.9, then every bounded subset of G is finite.
Lemma
Proof. If B _C G is bounded, let {g~}~%1 _C B. Since B is bounded, we can assume, w i t h o u t loss of generality, t h a t this sequence is Cauchy. Let e > 0 be so small t h a t the e-neighborhood of 0 in IR~ is contained in U and choose r > 0 such t h a t Iigi - gj[[ < e, V i , j >_ r. T h e n i , j >_ r ~ gi - gj E G N U = {0}, so the sequence must have only finitely m a n y distinct terms. [] Lemma
4.4.14.
If U is as in T h e o r e m 4.4.9 and n = 1, then G is infinite cyclic.
Proof. Let g E G f'/ (0, oc) be the element closest to 0. If there were no such element, we could produce an infinite, strictly decreasing sequence in G N (0, oo), c o n t r a d i c t i n g L e m m a 4.4.13. Let A = {rag [ m E Z}, an infinite cyclic subgroup of G. We claim t h a t A = G. Otherwise, find f E G \ A and m E g s u c h t h a t rng < f < ( r n + l ) g . Then 0 < f-rng < g and f - r a g E G, contradicting the choice of g. [] Proof of theorem ~.~.9. If G C R n is a lattice subgroup generated by the linearly i n d e p e n d e n t vectors v l , . . . ,vk, there is a nonsingular linear a u t o m o r p h i s m L : R '~ + R ~ such t h a t L(vi) = ei, the zth s t a n d a r d basis vector, 1 < i < k. Since L is also a h o m e o m o r p h i s m , we lose no generality in assuming vi = ei, 1 < i < k. In this case, e l e m e n t a r y g e o m e t r y shows t h a t [Igll -> 1, V 9 c G \ {0}. Thus, U = {v E IRn ] llvl] < 1} has the p r o p e r t y t h a t UTI G = {0}. For the converse, suppose U exists as desired and proceed by induction on n. T h e case n = 1 is true by L e m m a 4.4.14. For the inductive step, assume the t r u t h of the t h e o r e m for some n _> 1 and suppose t h a t G C R n+l and U C R n+l satisfy the hypotheses of the gheorenl. Let 0 ~ 9 E G and let V C IR~+1 be the one-dimensional vector subspace spanned by 9- T h e m a j o r step in our p r o o f will be to show, via the inductive hypothesis, t h a t G / ( G N V) is a lattice subgroup of
R~+I/V ~ R ~. By L e m m a 4.4.14, G n V is infinite cyclic, generated by some 90 E G (~ V. Let
{ f i } i o=o l be a sequence in G / ( G N V ) C IRn+I/V converging to 0 in t h a t vector space. Write f i = fi + (G N V), fi E G C R ~+1. T h e n the distance of fi from the line V -
-
approaches 0 as i --~ oo. Thus, for some constant c > 0, one can find rni e Z such t h a t ]]fi - m~90]l < c, Vi _> 1. By L e m m a 4.4.13, {fi - m~go}i~ C G contains only finitely m a n y distinct elements. T h a t is, f i = fi + (G n V) = (f~ - re,go) + (G N V) assumes only finitely m a n y distinct values as i --+ oo. Since limi~o~ f~ = 0, we conclude t h a t 7, = 0 for large enough values of i. Therefore, there is a neighborhood U' C I R n + I / v of 0 such t h a t U' N ( G / ( G N V)) = {0}. By the inductive hypothesis, G / ( G N V) is a lattice g e n e r a t e d by a linearly independent set {g~ + (G a V)}~= 1 c
R~+I/V. Given g E G, write
g + (any) = Z
+ ( c n v),
riEZ.
150
4. F L O W S A N D F O L I A T I O N S
T h a t is, there is ro C Z such that e
g - ~_~ rigi = rogo. i=1
Thus, {go,g1,... ,g/} generates G and this set is clearly linearly independent in R n+l ,
[]
E x e r c i s e 4.4.15. Let M be an n-manifold, O M = O. Prove that M is integrably parallelizable (Example 3.4.8) if and only if there exist pointwise linearly independent, commuting vector fields X 1 , . . . , X n (not necessarily complete). Use this to prove that a compact n-manifold is integrably parallelizable if and only if each component is diffeomorphic to T% Remark. Following J. Milnor [29], one defines the rank of an n-manifold M to be the maximum number of everywhere linearly independent, commuting vector fields that the manifold admits. By Proposition 4.4.6, the rank of a compact n-manifold M is the largest integer r < n for which there exists a nonsingular r-flow on M. It is a celebrated theorem of E. Lima [25] t h a t the rank of S 3 is 1. Since S 3 is parallelizable, it admits a nowhere 0 vector field, hence a nonsingular 1-flow. On the other hand, Theorem 4.4.7 implies that the only compact 3-manifold of rank 3 is T 3. The hard part of Lima's theorem is to show that S 3 does not have rank 2.
4.5. F o l i a t i o n s
Let F C_ T ( M ) be a k-plane distribution on M. For simplicity, we consider only the case in which O M = ~. D e f i n i t i o n 4.5.1. An integral manifold of F through q E M is a one-to-one immersion i : N --~ M of a k-manifold such that q E i ( N ) and i . x ( T x ( N ) ) = Fi(x),
V x E N.
We generally identify N and i ( N ) . This is similar to the customary identification of a curve s : [a, b] -~ M with its image. The correct topology on the subset i ( N ) of M is the manifold topology of N, not the relative topology. D e f i n i t i o n 4.5.2. A k-plane distribution F on M is said to be integrable if, through each q E M, there passes an integral manifold of F . E x e r c i s e 4.5.3. Let M = II(3 \ {0}and define Fv = {w E T v ( M ) I w • v}, Vv C M. Prove that F = [JveM F~ is an integrable distribution and describe the integral manifolds. E x a m p l e 4.5.4. Take M = R 3 and let F be the 2-plane distribution spanned by the pointwise linearly independent vector fields 0 ,
.
4.5. FOLIATIONS
151
Let 7r : ]Ra --+ R 2 be the projection ~r(x, y, z) = (x, y). Since 0 7r,q ( X q ) ~ -~x ":r(q) '
0 ~(q) ,
an integral manifold of F through q will be carried by lr locally diffeomorphically onto an open subset of IR2. That is, an integral manifold of F is locally a graph z = f(x, y) and
T h a t is, f(x, y) solves the system
of
o-; = 9(x, v),
Of o~ = h(x, y). This overdetermined system of P.D.E. implies that
Og = 02f _ 02f _ Oh Oy Oy Ox Ox Oy Ox T h a t is, a necessary condition for F to be integrable is that
Og Oy
Oh Ox
It turns out, as we will see, that this is also a sufficient condition for integrability. This integrability condition can also be written in terms of brackets. Indeed,
[X,y]= (Oh
09) 0
so the integrability condition becomes [X, Y] = 0. By our theory of commuting vector fields, this condition implies t h a t there is a local coordinate chart (U, u, v, w) about q in which 0 x = b-~u, ] / ' ~ _ _ _0
and, in this coordinate system, the integral manifolds are given by the equations w = const. Finally, in this coordinate neighborhood, arbitrary fields Z1, Z2 r F(F}U) are linear function combinations of X, Y, hence integrability implies that [Z1, Z2] E F ( F I U ). This is true in suitable coordinate charts about each point of 1~3, hence F ( F ) C Y(K 3) is a Lie subalgebra and F is a Frobenius 2-plane distribution on
152
4. F L O W S A N D F O L I A T I O N S
]I~3. This exemplifies the following theorem, due to Clebsch, Deahna, and Frobenius, but generally credited only to the last of this trio. T h e o r e m 4.5.5 (The Frobenius theorem). If F C_ T ( M ) is a k-plane distribution on M , the following are equivalent. (1) F is integrabIe. (2) F is a Frobenius distribution (Definition 4.3.5). (3) About each q E M there is a coordinate chart (W, x l , . . . ,x n) such that
0
Oz~ C r(FIW)'
l
E x e r c i s e 4.5.6. Define
Ev
=
{(a,b,c) e T~(M) l (b,e,a ) •
Vv E M. Prove t h a t E = UveM Ev is a 2-plane distribution and decide whether or not it is integrable. The proof of the Frobenius theorem is the primary goal of this section. Before giving a proof, however, we discuss some of the consequences. A coordinate chart as in (3) of Theorem 4.5.5 will be called a Frobenius chart. Theorem 4.5.5 allows us to find a C ~ atlas {(U~,~o~)}~e~ on M such t h a t the associated family of local trivializations {d~
: T(U~) ~ Us x
~}~
is contained in the Gl(k, n - k)-reduction of T ( M ) corresponding to F . This means that the associated Jacobian cocycle satisfies
d g ~ : U~ A U~ ---* Gl(k, n - k), V a , ~ 6 92. T h a t is, the infinitesimal G l ( k , n - k)-structure determined by F is integrable in the sense of Definition 3.4.12. Fix a coordinate cover {V~,x~,... ,x~}~en satisfying property (3) in Theorem 4.5.5 and such that 1. the index set 12 is at most countably infinite and {V~}~en is a locally finite cover of M; 2. x}~ ranges over the open interval ( - 2 , 2 ) , VA E 12, 1 < i < n; 3. if W~ C V~ is defined by the inequalities - 1 < x}~ < 1, 1 < i < n, then {Wx}~ec is an open cover of M. This is possible since M is 2nd countable and paracompact. We are primarily interested in the coordinate neighborhoods W~, with the V),'s playing an auxiliary role. D e f i n i t i o n 4.5.7. A coordinate cover { W a , x [ , . . . , X n~}xee with all of the above properties is called a regular cover for the integrable distribution F . If a = (ak+l,... ,a n ) where a i E ( - 1 , 1), k + 1 < i < n, the equations
x*~ = a ~,
k + l < i < n,
define an integral manifold P~,a to F[Wa, called a plaque of F in W~. Similarly, there are plaques/Sx,a of F in Va with P~,~ C/Sa,~. Remark that, by the definition of regular cover, the closure Wx is a compact subset of Vx and Pa,~ is a compact subset of Px,~. In fact, W x = [ - 1 , 1]~ in the coordinates of Vx.
4.5. F O L I A T I O N S
153
L e m m a 4.5.8. Suppose that F is an integrable distribution on M and that (W,x 1,
. . . . x n) is a Frobenius chart Wa or Vx as above. Then every connected integral manifold to F I W lies in some plaque of W. Consequently, if N1,N2 C M are arbitrary integral manifolds of F, then N1 A N.2 is an integral manifold of F. Proof. Let p C W and let P~ be the plaque through p. If N C W is also a connected integral manifold through p and if q E N, there is a smooth curve s : [0, 1] --~ N with s(0) = p and s(1) = q. Then i(t) C T~(t)(N) = F~(t), 0 < t < 1, so k
i(t) = E l i ( t )
O ~(t)
.
i=1
Thus, i f k + l _ ~ j < _ n a n d O < t <
d
1,
k Oxj xJ(s(t)) = s(t)(xJ) = E f i ( t ) ~ x i = O. i=1
That is,
x~ (sit) ) = xJ (p) = aJ, a constant, k + 1 < j < n, 0 < t < 1. This proves that q E Pa, Vq E N. Let N1 and N2 be arbitrary integral manifolds to F. If NI V) N2 = 0, there is nothing to prove. Let p C N1 N N2 and choose W about p as above. Then, the component of p in N1 N W lies in a plaque Pa. Since dim N1 = k, this component is an open subset of the plaque P~. Similarly, the component of p in N2 A W is an open subset of P~. Therefore, the component of p in N1 A N2 N W is an open subset of Pa, hence is an integral manifold to F. Since p E N1 n N2 is arbitrary, N1 N N2 is a (possibly disconnected) integral manifold to F. [] L e m m a 4.5.9. A plaque P;~,, of F in W;~ meets at most countably many plaques
of the regular cover. Proof. Since the cover {Wa}aee is locally finite, each point x C Pa,a has a connected neighborhood Ur~ in PX,a that meets only finitely many W ~ , 1 < i < r. The intersections of Ux with distinct plaques of Wx, are disjoint open (by Lemma 4.5.8) subsets of Ux, hence second countability of Ux implies that it meets at most countably many of the plaques of I/V~, 1 < i < r. Since P~,a is compact, it can be covered by finitely many of these neighborhoods U~ and the assertion follows. [] Remark. With a little more care, one can guarantee that each plaque meets only finitely many other plaques, but we do not need this. D e f i n i t i o n 4.5.10. If F is an integrable distribution on M, then points x, y C M are said to be F-equivalent, and we write x ~ F Y, if there exist connected integral manifolds N 1 , N 2 , . . . ,Nr of F such that x E N1, y C N~, and Ni N Ni+l r O,
l
relation ~ F is an equivalence relation on M and the equivalence classes are the maximal connected integral manifolds to F. Proof. The fact that ~ g is an equivalence relation is practically immediate. We show that each ~ F equivalence class L C_ M is (the image of) a one-to-one immersed submanifold integral to F.
154
4. F L O W S AND
FOLIATIONS
F i x p c L. Given q E L, choose connected integral manifolds N1, N2 . . . . . N~ such t h a t p C N1, q C Nr, and N i A N i + I # ~, 1 < i < r - 1. By the definition of ~ F , Ni C L, 1 < i < r. In particular, L is a union of integral manifolds to F and we can topologize L by letting the open subsets N C L be exactly the unions of integral manifolds to F that lie entirely in L. Then L itself is open, the empty subset is open by default, arbitrary unions of open sets are open and Lemma 4.5.8 guarantees t h a t finite intersections of open sets are open. This is a locally Euclidean topology of dimension k. Indeed, the topology in each of the integral manifolds in L is the usual manifold topology (open subsets of integral manifolds are integral manifolds). Since connected integral manifolds are path-connected, the definition of ~ F implies that L is path-connected in this topology. The fact t h a t the topology is Hausdorff is easy. We prove that L is 2nd countable. Let {(W~, x ~1 , . . . , x~)}~e~, n be a regular cover of M. Each plaque Pa C Wa is connected, hence each plaque that meets L lies entirely in L. Thus, L is a union of plaques and the chains N 1 , . . . , Nr in the definition of F-equivalence can be taken so t h a t each Ni is a plaque. It will be enough to prove that the set of plaques in L, defined by the given regular cover, is at most countably infinite. F i x a plaque P0 with p E P0 C_ L. Recall (Lemma 4.5.9) that each plaque meets at most countably many other plaques. Thus, for a fixed integer r > 0, there can be only countably many plaque chains P0, P 1 , . . . , P~ with P i - 1 A P i r ~, 0 < i <_ r. Thus, as r ranges over the positive integers, the number of such plaque chains, starting at the fixed Do, is at most countable. Since every plaque in L is reached by a finite plaque chain from P0, there are at most countably many distinct such plaques. We have shown that L is a connected topological k-manifold. But property (3) in Theorem 4.5.5 provides a smooth atlas on L and shows that this manifold is smoothly immersed in M. Being locally integral to F , L is itself a connected integral manifold, obviously maximal with this property by the definition of ~ F [] D e f i n i t i o n 4.5.12. The decomposition of M into F-equivalence classes is called a foliation 9- of M. Each F-equivalence class L is called a leaf of the foliation Y. If dim M = n and the leaves of 9" are k-dimensional, the dimension of the foliation is dim 9: = k and codim 9" = n - k is called the codimension of the foliation.
Example 4.5.13. If qJ : Nk x M --, M is a nonsingular k-flow, the IRk-orbits are the leaves of a k-dimensional foliation 9~ of M. Indeed, by Proposition 4.4.6, the k-flow is generated by a family { X 1 , . . . , X k} of everywhere linearly independent, commuting vector fields and these fields span a k-plane distribution F C__ T ( M ) . Since the fields commute, F is a Frobenius distribution, hence integrable, and there is a corresponding k-dimensional foliation 9" of M. For each q E M, 9 q : Rk ~ M is an immersion and G = (~Pq)-~(q) is a lattice subgroup of IRk. Thus, N k / G is a smooth k-manifold and ~q passes to a one-to-one immersion
r : R k / G --* M.
4.5. F O L I A T I O N S
155
This immersed submanifold is evidently a connected integral manifold to F , hence is an open neighborhood of q in the leaf Lq of 9" through q. The image of ~b is, in fact, the IRk-orbit of q. If it were not all of Lq, then this leaf would be the disjoint union of two or more such orbits, each open in Lq, contradicting the fact that Lq is connected. As we have seen, the foliation ~ can be quite complicated. As in Example 4.1.8, there is such a one-dimensional foliation of T 2 with each leaf everywhere dense in T 2. In fact, we will see that this is true for T '~, Vn > 2. (Example 5.3.9). In the same way, higher dimensional nonsingular k-flows on T '~ are readily produced having everywhere dense leaves (cf. Exercise 4.5.14 below). E x e r c i s e 4.5.14. Construct a nonsingular 2-flow on T 3 having each IR2-orbit diffeomorphic to the cylinder S 1 x R and everywhere dense in T 3. E x e r c i s e 4.5.15. Let M be an n-manifold without boundary. Let the map f : M --* N have constant rank k and prove that, as y ranges over f ( M ) , the connected components of the level sets f - l ( y ) range over the leaves of a toliation 9" of M of dimension n - k. E x e r c i s e 4.5.16. Let f : 1Ra --* IR be the submersion
f ( x , y , z ) = ( 1 - x 2 - ye)e~. By Exercise 4.5.15, there is an associated two-dimensional foliation 5 of ira (see Figure 4.5.1). Show the following. (1) The cylinder x 2 + y2 = 1 is a leaf L0 of 9". (2) The leaves interior to this cylinder L0 are diffeomorphic to ]R2. (3) The leaves exterior to L0 are diffeomorphic to cylinders. (But they are not geometric cylinders.) (4) The foliation :T is invariant under translations in the z-coordinate. E x e r c i s e 4.5.17". Let 9- be a foliation of M and let i : L ~-+ M be the one-to-one immersion of a leaf. Let X be a manifold and f : X --~ 5 I a smooth map such that f ( X ) C_ i(L). Then i -1 o f : X --+ L is smooth. (Hint: cf. Exercise 3.7.10.) We turn to the proof of Theorem 4.5.5. P r o p o s i t i o n 4.5.18. If F is art integrable distribution, then F is Frobenius.
Proof. Let X , Y E F ( F ) , q C M, and let i : N --+ M be an integral manifold of F through q. Let X, Y C ~ ( N ) be the unique restrictions of X, Y to N (Exereise 4.3.11). Then [X, Y] is/-related to IX, Y]. In particular, if q = i(p),
Ix, Y]q = i.p[x, Y]p 9 i.~(T~(N)) = rq [] This proves that (1) ~ (2) in Theorem 4.5.5. The following gives the implication (3) ~ (1). P r o p o s i t i o n 4.5.19. Let F be a k-plane distribution on M. If each point q 9 M has a coordinate neighborhood (U, x l , . . . , x n) such that
0
~0x 9 F(F[U),
then F is integrable.
1 < i < k,
156
4. F L O W S AND F O L I A T I O N S
F i g u r e 4.5.1. Foliation by the level sets (1 - x 2 - y2)eZ = c
P r o @ Indeed, if a i = xi(q), k + 1 < i < n, the level set
{ ( x l , . . . , x ~) l x i = a i , k + l < i < n} is a k-dimensional integral manifold of F through q.
[]
It remains t h a t we prove (2) => (3). This is the hard part. L e m m a 4.5.20. Let U C IRn be an open subset and let F be a k-plane distribution on U. Let 7r : IRn -+ IRk be the projection ( x l , . . . , x '~) ~-, ( x l , . . . , x k ) . Let p C U and suppose that rr.p : Fp --+ T~(p)(IR k) is bijective. Then there is an open neighborhood W of p in U such that rr.x : Fx -+ T~(~)(IR a) is bijective, V x E W . Proof. Let V be an open neighborhood o f p in U such that there are fields X 1 , . . . , X k C F ( F I U ) which give a basis of F~, g x E V. Write
l
Xi=~fjo@, j=l
Then
=
k j=l
i fj(x)
o
7r(z)
,
VxEV,
l
4.5.
FOLIATIONS
[,
Consider the k x k m a t r i x
157
1]
"fll(x)
.-.
f~(x)
f lk X)
...
k" f~(x)J
d(x) =
"
.
By assumption, det A(p) ~ O, so, for a small enough open neighborhood W of p in V, d e t A ( x ) 7~ 0, Vx E W. T h a t is, 7r. x : Fx ~ T,~(x)(IRk) is bijective, Vx E W. [] L e m m a 4 . 5 . 2 1 . Let F be a k-plane distribution on M and let q E M . Then there is a coordinate chart (W, x l , . . , x n) about q such that the map
7r : W---~ ]Rk, 9iven by 7r(xl,... , x n) = ( x l , .
,xk), has 7r.x : F~ --~ T.(~)(Nk)
bijective, V x E W . Proof. By L e m m a 4.5.20, we must choose the coordinates so that
~,q : ~q
-~
T~(~)(~ ~)
is bijective. T h e n we restrict to a smaller neighborhood, if necessary. Let (U, x l , . . . , x n) be a coordinate chart a b o u t q and let X 1 , . . . , X k E F ( F t U ) give a basis of Fq. P e r m u t i n g the coordinates suitably, we can assume t h a t
{Xq ,
0
,...
,
0}
0xn
is a basis of Tq(M). T h e n the surjection 7r,q : Tq(M) ~ T~(q)(I~ k) annihilates the last n - k of these vectors, hence it carries {Xql,... , Xqk} to a basis of T,~(q)(Rk). [] Lemma
4 . 5 . 2 2 . Let F, q E M , and 7r : W --. IRk
all be as in L e m m a 4.5.21. Given x E W and l < i < k, let Z~ E Fz be the unique vector such that 0 Then Z 1 , . . . , Z k E F ( F I W ). Proof. T h e only problem is to prove that Z i is smooth at each x E W, 1 < i < k. We can assume t h a t there is a trivialization T ( W ) ~ W x R ~ relative to which F I W ~ W x ]Rk. T h e s t a n d a r d trivialization of T(IR k) is given by the coordinate fields 0 0 Ox I , ' " , Ox k" We express the linear m a p 7r,~ : T~ ( W ) --~ T~(x)(IR k) relative to these trivializations by a m a t r i x [A(x),B(x)] where A(x) is k x k and B ( x ) is k x ( n - k ) . Since 7r,z carries F~ = {x} x R k bijectively onto {Tr(x)} x ]Rk, we see t h a t A(x) E Gl(k) and depends smoothly on x. Thus, A(x) -1 is also smooth in x a n d
:{~(~)}•
-~{x}•
158
4. FLOWS
AND
FOLIATIONS
has image Fz. The ~th column of this matrix is Z~, 1 _< i _< k, so this vector depends smoothly on x. [] We can now complete the proof of Theorem 4.5.5. P r o p o s i t i o n 4.5.23. If F is a Frobenius distribution on M , then about each q C M there is a coordinate chart (U, y l , . . . , y~) such that 0 - - C r(Flu), Oy i
1 < i < k.
Pro@ Let (W, x l , . . . , x n) be a coordinate neighborhood of q and Z 1 , . . . , Z k E all as in Lemma 4.5.22. Since Z i is 7r-related to the ith coordinate field, 1 < i < k, it follows that ~r.[Zi, Z j] -~ 0, 1 < i , j < k. But the Frobenius condition implies t h a t [Z i, Z j] C F ( F I W ) and, on F I W , 7r, is one-to-one, so [Z i, Z j] ~ O, 1 < i , j < k. Since these fields are pointwise linearly independent, Theorem 4.4.1 furnishes a coordinate chart (U, y l , . . . , yn) around q C W such that
F(FIW),
ZilU =
0 Oyi '
1 < i < k. []
E x e r c i s e 4.5.24. Although we have considered foliations only on manifolds M without boundary, one often relaxes this assumption by requiring special behavior for foliations near cOM. In the case of foliations of codimension one, it is natural to require t h a t each component of cOM be a leaf. Use Exercise 4.5.16 to produce a foliation of D 2 x S 1 having the boundary torus S 1 x S 1 as a leaf and having all other leaves diffeomorphic to IR2. (This is the famous "Reeb foliation" of the solid torus, pictured in Figure 4.5.2 and discovered by G. Reeb [37].) Nonsingular flows become foliations of dimension one if we forget the parametrization of the leaves. That is, the nonsingular vector field X that generates the flow is replaced by the one-dimensional, integrable distribution spanned by this field. The study of flows belongs to the branch of mathematics called "dynamical systems". Notions such as "minimal set" (see Exercise 4.1.18) and "limit set" (Exercise 4.1.19) are fundamental to the theory of dynamical systems and they carry over nicely to foliation theory, where they are likewise fundamental. The following exercises introduce these important ideas. E x e r c i s e 4.5.25. Let 9- be a foliation of M. A subset C C_ M is said to be 9-saturated if, for each x E C, the entire leaf Lx through x lies in C. Prove t h a t the closure C of an 9--saturated set is an 9--saturated set. E x e r c i s e 4.5.26. Let 9- be a foliation of M. minimal set of 9- if
A subset C C M is said to be a
(a) C r 0; (b) C is closed in M; (c) C is 9--saturated; (d) C contains no proper subset with all of these properties. For example, a closed leaf is a minimal set. Prove that, if M is compact, every closed, nonempty, 9"-saturated subset of M contains at least one minimal set. (In particular, by Exercise 4.5.25, every leaf of 9" closes on at least one minimal set.) Show by an example that M itself may be a minimal set.
4.5. F O L I A T I O N S
159
F i g u r e 4.5.2. The Reeb foliation of D 2 x S 1 E x e r c i s e 4.5.27. Let L C M be a leaf of a foliation 9". The limit set of L is defined by lim(L)= N (L\K), KEg(
where X is the family of compact subsets of the leaf L and the overline denotes closure in M. Prove the following. (1) If M is compact, then lim(L) is a compact, ~-saturated set. (2) If M is compact, lira(L) = ~} if and only if n is compact. (3) If L is dense in M (but not equal to M ) lira(L) = M. (4) If the leaf L is an imbedded submanifold of M, then L A lim(L) = ~.
CHAPTER 5
Lie Groups and Lie Algebras Lie groups and their Lie algebras play a central role in geometry, topology, and analysis. Here we can only give a brief introduction to this fascinating topic. 5.1. B a s i c D e f i n i t i o n s a n d Facts A topological group is a topological space together with a group structure on that space such that the group operations are continuous. A Lie group is a differentiable manifold together with a group structure on that manifold such that the group operations are smooth. Lie groups are also topological groups, but not vice versa. Here are the precise definitions. D e f i n i t i o n 5.1.1. A topological group G is a topological space that is also a group such that the operations # : G • G -~ G, t. : G ~
G,
#(x, y) = xy, ~(z)
= z -1
are continuous maps. If G is a smooth manifold without boundary and these operations are smooth, G is called a Lie group. Remark. In most of the literature, Lie groups are defined to be real analytic. That is, G is a manifold with a C ~ (real analytic) atlas and the group operations are real analytic. In fact, no generality is lost by this more restrictive definition. Smoott~ Lie groups always support an analytic group structure, and something even stronger is true. Hilbert's fifth problem was to show that if G is only assumed to be a topological manifold with continuous group operations, then it is, in fact, a real analytic Lie group. This was finally proven by the combined work of A. G1eason, D. Montgomery, and L. Zippin. The details are too deep to be discussed tlere. Evidently, every finite dimensional vector space over R or C is a Lie group under vector addition. Here are some more interesting examples. E x a m p l e 5.1.2. The group Gl(n) is evidently a Lie group, tile operations being given by rational flmctions of the coordinates. The subgroups Sl(n) and O(n) are smoothly imbedded submanifolds of Gl(n), hence the smoothness of the group operations on Gl(n) implies the smoothness of their restrictions to Sl(n) and O(n). It can be shown that Gl(n) and O(n) each have two connected components, distinguished by the sign of the determinant. The component of the identity I in each of these groups is itself a Lie group (Exercise 5.1.3), denoted by Gl+(n) and SO(n) respectively. The group SO(n) is called the special orthogonal 9roup. Recall that the orthogonal group O(n), hence also the special orthogonal group, is compact (Exercise 2.5.9).
162
5. L I E G R O U P S
E x e r c i s e 5.1.3. Prove t h a t the c o m p o n e n t of the identity in a Lie group is itself a Lie group. Example
5.1.4. T h e group G l ( k , n -
k) consists of matrices
[0 where A c GI(k), C E G l ( n - k), and B is an arbitrary k x (n - k) matrix. Thus, Gl(k, n - k) is a manifold diffeomorphic to Gl(k) x G l ( n - k) x II{k(n-k). This is an open subset of II~~2+(n-k)2+k(~-k) and the group operations are rational functions in t h e coordinates, so this is a Lie group. E x a m p l e 5.1.5. T h e group Gl(n, C) of nonsingular, n x n matrices over the complex field C is a Lie group, called the complex general linear group. R e m a r k t h a t GI(1, C) = C* is the multiplicative group of nonzero complex numbers. T h e unit circle S 1 C C* is a subgroup and a s m o o t h l y i m b e d d e d submanifold, hence is also a Lie group. E x a m p l e 5.1.6. If G and H are Lie groups, t h e n G x H is a Lie group under the usual C a r t e s i a n group operations and the s m o o t h product structure. In particular, T '~ = S 1 x . . . x S 1 i s a Lie group. E x a m p l e 5.1.7. Define V) : G I ( n , C ) ~ G I ( n , C ) by ~(A) = A T A . Here the overline indicates complex conjugation in each entry of the matrix. T h i s m a p has c o n s t a n t rank n 2 and ~ ( I ) = I. We define U(n) = T - I ( I ) , a s m o o t h l y imbedded, n o n e m p t y submanifold of G I ( n , C ) t h a t has dimension 2n 2 - n 2 = n 2. It is easy to check t h a t U(n) is also a subgroup, hence it is a Lie group, called the u n i t a r y group. For the same reasons t h a t O(n) is compact, the Lie group U(n) is compact. Since A E U ( n ) if and only if A -1 = ~ T , it follows t h a t I det(A)l = 1. Indeed, det : U ( n ) --* S 1 is a group h o m o m o r p h i s m and a submersion. One defines t h e s p e c i a l u n i t a r y g r o u p SU(n) = d e t - l ( 1 ) C U(n), a c o m p a c t Lie group of dimension n 2 - 1. E x e r c i s e 5.1.8. Check the various assertions in E x a m p l e 5.1.7, showing t h a t U(n) and SU(n) are c o m p a c t Lie groups of respective dimensions n 2 and n 2 - 1. E x a m p l e 5.1.9. Let IE denote the division algebra of quaternions. T h e nonzero q u a t e r n i o n s H* form a multiplieative group and a manifold diffeomorphic to ]R4 \ {0}. It is clear t h a t the group operations are smooth, so H* is a Lie group. T h e 3-sphere S 3 C H* consists of the unit length quaternions, hence it is closed under multiplication and passing to inverses. This gives a Lie group s t r u c t u r e on S 3" Usually, the identity element of a topological group or Lie group will be d e n o t e d by e. For m a t r i x groups, however, the c u s t o m a r y symbol for the identity is I. Because t h e Lie groups S 1 and S 3 are subgroups of the division algebras C and ]HI respectively, the identity elements in these groups are denoted by 1, t h e unity of the respective algebras. D e f i n i t i o n 5 . 1 . 1 0 . Let G be a topological group (respectively, a Lie group), a C G. Left t r a n s l a t i o n by a is the continuous (respectively, smooth) m a p L a : G ~ G defined by L ~ ( x ) = a x , V x c G.
5.1. B A S I C S
163
Remark that La-1 = L -la , so L~ E Homeo(G). Similarly, if G is a Lie group, L~ E Diff(G). Also, the inversion map L : G -+ G is continuous and equal to its own inverse, so t c Homeo(G) and, when G is a Lie group, c C Diff(G). The following discussion illustrates some of the striking ways in which algebra and topology interact in these structures. L e m m a 5.1.11. If G is a topological group and U C G is an open neighborhood of e E G, then there is an open neighborhood V C_ U of e with the property that v E V ~ v -1 E V. Such a neighborhood V ore is said to be symmetric. Proof. Indeed, t(U) is also an open neighborhood of e E G, so the neighborhood V = U n t(U) is as desired. [] L e m m a 5.1.12. If Z, W C O and if W is open in G, then the set Z W = { zw I z C Z and w C W } is open in G. Proof. Indeed, Z W = Uz~z L z ( W ) is a union of open sets.
[]
P r o p o s i t i o n 5.1.13. Let g be a connected topological group and let U C_ G be an arbitrary open neighborhood of the identity e C G. Then U generates the group G. Pro@ By Lemma 5.1.11, we lose no generality in assuming that U is a symmetric neighborhood of e. Using Lemma 5.1.12, we define open sets U n = UU n-1
by induction on n. Since e C U, these form an increasing nest U ~ U 2 c ... g U ~ g ... of open neighborhoods of the identity. By the symmetry of U and the formula for the inverse of a product, each U n is symmetric. Thus, we obtain an open, symmetric neighborhood oo
U~176 = U Un rt=l
of the identity. Clearly, U ~176 is closed under group nmltiplication; hence, being a symmetric neighborhood of the identity, U ~ is a subgroup of G. But the left cosets {aU~}aeO form a cover of G by disjoint open sets, hence the connectivity of G implies that there is only one coset. That is, U ~~ = G. [] We focus our attention on Lie groups G and their associated Lie algebras. It will be seen that the introduction of Lie algebras produces further remarkable interactions of algebra, topology and calculus. D e f i n i t i o n 5.1.14. A vector field X C X(G) is left-invariant if, for each a E G, L ~ . ( X ) = X . The set of left-invariant vector fields on G is denoted by L(G). The following is quite easy, but very important. P r o p o s i t i o n 5.1.15. The subset L(G) C 2~(G) is a Lie subalgebra. Proof. Indeed, the bracket of La-related fields is L~-related to the bracket of these fields. It follows immediately that the bracket of left-invariant fields is a leftinvariant field. []
164
5. L I E G R O U P S
D e f i n i t i o n 5.1.16. If G is a Lie group, its Lie algebra is LEG). E x a m p l e 5.1.17. We saw in Example 2.7.18 that L(Gl(n)) = glen) consists of the fields RA, A C 9)/(n), and t h a t this defines a canonical isomorphism of Lie algebras L(GI(n)) = g)/(n). Similarly, L(SI(n)) = s[(n) is the Lie algebra of n x n matrices of trace 0 and LeO(n)) = o(n) is the Lie algebra of skew symmetric matrices (cf. Example 4.3.6). E x e r c i s e 5.1.18. Identify L(U(n)) = u(n) and L(SU(n)) = su(n) as Lie algebras of complex matrices. P r o p o s i t i o n 5.1.19. The evaluation map e : LeG) ---* Tr Xe, is an isomorphism of vector spaces.
defined by e ( X ) =
Proof. This is clearly a linear map. The fact that it is injective follows immediately from Xa :La,(Xe), Vaff:G. We prove that e is surjective. Let v E Te(G) and define X a
:
L~.(v), Va C G.
This defines X : G ~ T ( G ) carrying each a E G to X~ C T~(G) and X~ = v. We must prove t h a t X is smooth, hence a vector field, and that it is left-invariant. For f E C ~ ( G ) , form the function X ( f ) : G ~ ~ by setting X(f)(x)
= X~(f),
V x e G.
If X ( f ) is smooth, V f c C ~ ( G ) , it will follow that X is smooth. Indeed, in local coordinates, smoothness of X ( x i) = fi implies smoothness (locally) of X. Note that X(I)(x) = X~(f) = (L~).r162 = X e ( I o Lz). The function g : G x G --* R, defined by 9(x,y) = f(Lx(y)) = f(xy), is smooth. Let (U, x l , . . . , x ~) be an arbitrary coordinate neighborhood in G. A b o u t e E G, choose coordinates (V, y l , . . . ,yn) relative to which e = ( 0 , . . . ,0). Then (x, e) C U x V has coordinates ( x l , . . . , x n, 0 , . . . , 0). Write '~ 8 i (0.... ,o) Xe = ~--~'~Ci ~Y i=1
Then X ~ ( f o L~) = " ~ c i
Og
,o)
(x 1,
is smooth in the arbitrary coordinate neighborhood (U, x l , . . . , x~), proving t h a t X e X(G). Finally, we prove that X is left-invariant. Indeed, if a, b E G, ( n a . ( X ) ) b = (La).a-lb(Xa-15) = (La).,-Ib((na
lb)*r
= (La o n a - l b ) . e ( X e ) = (Lb).e(Xe) = Xb.
5.1. BASICS
165
Since b is arbitrary, L a . ( X ) = X . Since a is arbitrary, X E L(G).
[]
C o r o l l a r y 5.1.20. If G is a Lie group, then dimL(G) = dimG. C o r o l l a r y 5.1.21. If G is a Lie group, there is a canonical trivialization of the
tangent bundle 7r : T(G) ~ G. In particular, every Lie group is parallelizable. Proof. Define ~ : G x L(G) ~ T(G) by ~ ( a , X ) = Xa. ~a = Pl({a} x L(G)) is an isomorphism
Then the restriction
qOa : {a} x L ( C ) ~ T a ( C )
of vector spaces, Va E G. If we show that ~ is smooth, it will be a bundle isomorphism. Fix a basis X 1 , . . . , Xn of L(G). Coordinatize L(G) via this basis, thereby defining an isomorphism G x L(G) ~ G x ]I{n. Relative to this coordinatization, has the formula
~(a, (bl,... ,bD) = ~-]biX~o. i=1
Since the fields Xi are smooth, so is ~, hence ~.
[]
In particular, S a is parallelizable (@ Theorem 1.2.13). This sphere is not, however, integrably parallelizable (cf. Exercise 4.4.15). D e f i n i t i o n 5.1.22. A 1-parameter subgroup of a Lie group G is a C ~ map s : R ~ C such that s(0) = e and
S(tl + t 2 ) = S(tl)s(t2),
V h , t 2 E R.
P r o p o s i t i o n 5.1.23. If G is a Lie group and X E L(G), there is a unique 1parameter subgroup s x : IR --~ G such that ix(O) = Xe. Furthermore, X is a complete vector field, the flow that it generates being given by
Proof. We first prove that X is complete. Indeed, if r : (-~,6 x U~
W
is a local flow about e E G generated by X I W and if a E G, then the formula
(p~(b) = La(4h(L~-~(b))), Vb 9 L~(U) defines a local flow (I~a : (--6_,(_) • L a ( U ) --+ L a ( W )
about a having infinitesinlal generator L a . ( X I W ) = X I L a ( W ). These fit together to give r : (-~,6
x
G--,G
and the field X is complete by Lemma 4.1.10. Let ~ x designate the global flow generated by X. Remark that we have also established the identity (5.1)
L~~
~
-~ =~xt , V a E G .
If a is any 1-parameter subgroup with initial velocity d~(O) = X~, then the identity ~(t + ~) : ~(t)~(~), for fixed but arbitrary t, ~- 9 R, implies that d(t + T) = no(t).(~(T)).
166
5. LIE GROUPS
In particular, take 7 = 0 and conclude that
~(t) = Lo(t), (~(0)) = Lo(t), (X~) = Xo(t). That is, a is the unique integral curve to X through e. By the previous paragraph, we must define
s x : IR -~ G by s x ( t ) = d~X(e) and prove that this is a 1-parameter subgroup of G. Indeed,
sx(O) = e and s x ( t l + t:) = 4)tlx ~-t2 ( e ) ---- t2 = ~X
t2(Sx(ti))
= Sx(t 1)8 X (t 1 ) - 1 r = sx(tl)
(8X(t 1)e)
(e)
= sx(tl)sx(t2), where the second-to-last equality is by (5.1). Evidently, i x ( 0 ) = X~. Finally, another application of the identity (5.1) yields
q}x (a) = La o d2X o La-1 (a) = a s x (t), for arbitrary values of t C R and a E G.
[]
E x a m p l e 5.1.24. Let A C 9Jr(n) = L(GI(n)). The series
exp(tA)=I+-~.
tA
t2A 2 +T+'"+
t~A ~ ~ +""
converges absolutely. Set s(t) = exp(tA). Clearly, s(0) = I and, by basic properties of the exponential series, s(tl +t2) = s(tl)s(t2). In particular, e x p ( t A ) e x p ( - t A ) = I, so the matrix exp(tA) is invertible, Vt C R. By these remarks, s(t) is a 1parameter subgroup of Gl(n). Finally, i(0) = A and, by Proposition 5.1.23, exp(tA) is the unique 1-parameter subgroup of Gl(n) with initial velocity vector A. This example provides the motivation for the following terminology. D e f i n i t i o n 5.1.25. If G is a Lie group, the exponential map exp : L(G) --* G is defined by exp(X) = sx(1), VX E L(G). In turn, we obtain a perfect generalization of Example 5.1.24. L e m m a 5.1.26. If G is a Lie group and X E L(G), then
s x ( t ) = exp(tX),
- e c < t < oa.
Proof. Fix r E R and set a(t) = s x ( r t ) . Then &(t) = rXo(t) and a(0) = e. By Proposition 5.1.23, cr = SrX. In particular, s x ( r ) = a(1) = SrX(1) = exp(rX), for each r E IR.
[]
5,1. B A S I C S
167
Hereafter, the s t a n d a r d n o t a t i o n for the 1-parameter subgroup associated to X 9 L(G) will be exp(tX). Via the identification L(G) = Te(G), we can view the exponential m a p as e x p : Te(G) ~ G, a m a p between s m o o t h manifolds of the same dimension. P r o p o s i t i o n 5.1.27. The map exp : Te(G) --+ G is smooth and carries some neighborhood of O in Te(G) diffeomorphieally onto a neighborhood ore in G.
Proof. Consider G • Te(G) = G • L(G) as a manifold a n d define the vector field Y 9 3~(G x L(G)) by
h~,x) = (x~, o),
v (g, x ) 9 a • L(a).
Clearly, the curve ~(t) = (g. exp(tX), x ) ,
- o o < t < o0,
is integral to Y with or(0) = (9, X). Thus, Y is complete and its flow is
9 : ~ • (G • L(G)) + G • L(G), ~t(g, X) = (9. exp(tX), X). In particular, define a smooth m a p ~ : L(G) --~ G x L(G) by = ~1({1} x {e} • L(G)). T h e n p ( X ) = (exp(X), X) and the smoothness of exp follows. Under the canonical identity To(Te(G)) = Te(G), we claim that exp.~ : we(c,) -+ w d c . )
is the identity map. Indeed, represent the t a n g e n t vector X to T~(G) at e as the infinitesimal curve represented by s(t) = t X . T h e n e x p o s is the curve e x p ( t X ) which, as an infinitesimal curve, represents X as well as e x p . e ( X ). The inverse function theorem t h e n gives the final assertion. [] E x e r c i s e 5 . 1 . 2 8 . If G is a Lie group a n d X , Y 9 Te(G), show t h a t the curve a(t) = e x p ( t X ) exp(tY) has initial velocity vector d(0) = X + Y. D e f i n i t i o n 5.1.29. If G a n d H are Lie groups, a Lie group h o m o m o r p h i s m ~ : G --+ H is a s m o o t h m a p t h a t is also a group homomorphism. If, in addition, qo is a diffeomorphism, it is called a Lie group isomorphism and G and H are said to be isomorphic Lie groups. R e m a r k t h a t a 1-parameter subgroup s : IR --+ G is a Lie group homomorphism, where R is a Lie group under addition. E x e r c i s e 5 . 1 . a 0 . A Lie algebra a is said to be abelian if [A,B] = O, V A , B 9 a. Prove the following. (1) A connected Lie group G is abelian if and only if its Lie algebra L(G) is abelian. (2) If the Lie group G is connected and abelian, the m a p exp : Te(G) --+ G is a surjective Lie group homomorphism, where the vector space Te(G) is viewed as a Lie group under vector addition.
168
5. LIE GROUPS
(3) Every connected, n-dimensional, abelian Lie group is Lie isomorphic to T k x N ~-k for some k = 0, 1 , . . . , n.
Proposition
5.1.31. If ~ : G ~ H is a Lie group homomorphism, then there is a unique linear map ~ , : L(G) ~ L ( H ) such that X is Q-related to p , ( X ) , V X E L(G). Thus, ~ , is a Lie algebra homomorphism.
Pro@ The R-linear map ~, will be the composition L(G) A, T~(G) ~*% T~(H) ~-~, L(H). If X e L(G), Y = ~ . ( X ) , and a 9 G, then ~.~(xo)
= ~,~((L~).~(X~)) = (r o L a ) , e ( X e )
= (n~,(a) =
o
(p),e(Xe)
(i~o(a))*e(~*e(Xe))
: (L~,(~)),~(Z~) = Y~o(a).
Thus, X is ~-related to Y. The left-invariant field Y is uniquely determined by Ye which, itself, is uniquely determined by the requirement that X be ~-related to Y. []
Remark. Using the canonical identifications L(G) = T~(G) and L ( H ) = T~(H), one obtains a Lie algebra structure on T~(G) and Te(H). Then the Lie algebra homomorphism ~ , becomes ~,~ : T~(G) ~ T~(H). D e f i n i t i o n 5.1.32. Let G be a topological group, F C_ G a subgroup. If there is a neighborhood U C_ G of the identity e E G such that U n F = {e}, then F is called a discrete subgroup of G. By Theorem 4.4.9, a discrete subgroup of the additive Lie group N n is exactly the same thing as a lattice F in N n. In this ease, the projection map p:N n~N~/F=T
k x N ~-k
is readily seen to be the universal covering. We consider the generalization of this to arbitrary Lie groups. E x e r c i s e 5.1.33. If G is a Lie group and F C G is a discrete subgroup, prove that the space a/r has a canonical smooth manifold structure relative to which the quotient projection p : G ~ G/F is a regular covering map, the covering transformations being the right translations by elements of F. If F is also a normal subgroup, show that G / F is a Lie group and that p : G ~ a/r is a Lie group homomorphism. Finally, note that p, : L(G) --* L(G/F) is an isomorphism of Lie algebras. P r o p o s i t i o n 5.1.34. Let G and H be Lie groups with H connected. Let ~ : G ~ H be a Lie group homomorphism such that ~,~ : T~(G) ~ T~(H) is an isomorphism of vector spaces. Then there is a discrete normal subgroup F C_ G such that H is isomorphic as a Lie group to a/C.
5.1. B A S I C S
169
P r o @ By tile inverse function theorem, qo carries some open neighborhood U C_ (7, of e C G diffeomorphically onto an open neighborhood ~(U) C H of e E H. Since qo respects left translations in these groups, it follows that ~(G) is an open neighborhood of the identity in H. Since ~(G) is a subgroup of H, Proposition 5.1.13 impIies that g)(G) = H. Finally, ~ being one-to-one on U C G, the normal subgroup P = ker(qo) is a discrete subgroup of G and g) induces a Lie group isomorphism : G/P ~ H. []
T h e o r e m 5.1.35. Let G be a connected Lie group with identity e and let 7r : (G, ~ ---+ (G, e) be the universal covering space. Then G is canonically a Lie group in such a way that g is the identity element and rr is a Lie 9roup homomorphism. E x e r c i s e 5.1.36. Prove Theorem 5.1.35, proceeding as follows. First construct a commutative diagram
GxG
"
, G P where p ( x , y ) = rr(x)rr(y) -1 and ~ is the unique lift such that ~ ( g , ~ = g. The existence and uniqueness of this lift are guaranteed by Corollary 1.7.40 and the fact that G, hence G x G, is simply connected. Since r~ is a local diffeomorphism, it is elementary that ~ is smooth. Given x , y C C,,, define y-1 to be ~(g,y) and x y to be ~ ( x , y - 1 ) . Next, consider a commutative diagram
G"
, G 7r
such that c~(e-") = g. Since c~ = id works, uniqueness of lifts shows that every (~ satisfying the condition is equal to the identity map. Using this, prove successively the identities (X-l) -1 = x and gx = x = x~. Use similar argmnents to prove that x-ix
= ~ = x x -1 and (xy)z = x(yz).
Thus, G is a group with smooth operations, hence a Lie group. From the definitions, it is obvious that 7r is a group homomorphism. Thus, combining Theorem 5.1.35 and Proposition 5.1.34, we see that every connected Lie group has the form G / F , where G is a simply connected Lie group and P is a discrete normal subgroup. All Lie groups sharing this same mfiversal covering group G have canonically the same Lie algebra. Conversely, connected Lie groups with isomorphic Lie algebras have isomorphic universal covering groups, but we will not prove this.
170
5. L I E G R O U P S
E x e r c i s e 5.1.37. Let 7r[M, N] denote the set of homotopy classes of maps M --* N. You may use the standard, but nontrivial fact that this set is canonically the same whether defined with continuous maps and continuous homotopies or smooth maps and smooth homotopies (cf. [16]). (1) If G is a Lie group, show that 7riM, G] has a natural group structure. (2) There is a natural map of sets ~ : 7rl(G,e) --* 7r[S1,G]. Define this map and show that it is actually an isomorphism of groups. Show simultaneously that this group is abelian. (3) Using the above, we easily conclude that a discrete normal subgroup of a simply connected Lie group is abelian. In fact, give a direct proof that a discrete normal subgroup of any path-connected topological group G is a subgroup of the center of G. E x e r c i s e 5.1.38. For each z E S a C N = R 4, define Az : IR4 --* IR4 by A z ( w ) = z w z -1 (quaternion operations). Prove the following, using standard facts about the skew field H and the norm ]z] = v / ~ on IHI. (1) A z is a nonsingular, norm-preserving linear transformation. That is, as a matrix, Az e 0(4). In fact, show that Az 6 SO(3) under canonical inclusions SO(3) C 0(3) C 0(4) of Lie subgroups. (2) The map A : S 3 -~ SO(3), defined by A ( z ) = A~, is a surjective homomorphism of Lie groups. (3) The kernel of the homomorphism A is the normal subgroup Z 2 = {4-1} C S 3. Thus, SO(3) is diffeomorphic to the projective space p3 and S 3 is the universal covering group. (4) Let Sl and s2 be i-parameter subgroups of S a such that the initial velocity vector ~i(0) E TI(S 3) C ]l~4 has Euclidean norm 1, i = 1,2. Using the previous step, show that there is an element z C S a such that ZSl(t)z -1 = s2(t), V t E R. (5) Using the above, prove that, up to parametrization, the 1-parameter subgroups of S 3 are exactly the great circles through 1 E S 3.
5.2. Lie Subgroups and Subalgebras We fix a choice of the Lie group G and discuss its Lie subgroups. D e f i n i t i o n 5.2.1. A subset H C_ G is a Lie subgroup i f H has a Lie group structure relative to which the inclusion map i : H ~ G is a one-to-one immersion and a group homomorphism. In particular, the inclusion i of a Lie subgroup is a Lie group homomorphism. We emphasize that the topology of H as a Lie group may not coincide with its relative topology in G.
Example 5.2.2. A nontrivial 1-parameter subgroup s : N --~ G is an immersion, generally not one-to-one. We will see that, if n > 2, uncountably many of the 1-parameter subgroups s : ~ --* T n are one-to-one immersions, each having image dense in T n (Example 5.3.9). By our definition, s(N) with its manifold topology and additive group structure is a Lie subgroup with i = s, but the relative topology of this subgroup in T n is wildly different from its manifold topology.
5.2. L I E S U B G R O U P S
171
D e f i n i t i o n 5.2.3. If it is a Lie algebra, a vector subspace I) C_ tJ is a Lie subalgebra if [? is closed u n d e r the bracket. 5.2.4. Let i : H r G be a Lie subgroup. Then i, : L ( H ) --~ L(G) imbeds L ( H ) as a Lie subalgebra of L(G). Lemma
Indeed, i,~ : T~(H) -* T~(G) is one-to-one, so the l e m m a follows from Proposition 5.1.3i. 5.2.5. The correspondence between Lie subgroups i : H ~-* G and their Lie subalgebras i, : L ( H ) ~ L(G) induces a one-to-one correspondence between the set of connected Lie subgroups of G and the set of Lie subalgebras of L(G).
Theorem
T h e proof of T h e o r e m 5.2.5 is the m a i n goal of this section. In light of the preceding lemma, what we have to prove is that, given a Lie subalgebra ~ C_ L(G), there is a u n i q u e connected Lie subgroup i : H r G such t h a t ~? = i , ( L ( H ) ) . T h e principal tool for this will be the Frobenius theorem. T h e evaluation m a p c : L(G) ---, T~(G) carries 0 one-to-one onto a vector subspace E~ C_ T~ (G). For each a C G, define
Ea : (La).e(Ee) C Ta(G). T h e n E = [-)aeg Ea is a k-plane distribution on G, where k = dim I?. Indeed, Lemma
5.2.6. The subset E C_ T(G) is an integrable k-plane distribution on G.
Proof. Let X 1 , . . 9 , Xk be a basis of the vector space b. This is a set of everywhere linearly independent, left-invariant fields on G, proving t h a t E is a k-plane distribution. R e m a r k t h a t b C_ F ( E ) spans F ( E ) as a C ~ 1 7 6 a n d [Xj,Xe] c D, so E is an integrable distribution by Theorem 4.5.5. [] Let H be the leaf through e of tile corresponding foliation 9{. T h e n H is a connected k-manifold together with a one-to-one immersion i : H ~-~ G with e E i(H) a n d i,b(Tb(H)) = Eb, Vb E H. Our first goal will be to show that this leaf is a Lie subgroup of G with i , ( L ( H ) ) = 0. Secondly, we will show t h a t this is the only connected Lie subgroup with this property. Wherever explicit reference to the immersion i is not needed, we generally denote i(H) by H . Similarly, if a C G, a H will denote the immersed submanifold
Laoi:H~--~G. Lemma
5.2.7. For each a C G, aH is the leaf of 9s through a.
Proof. We must show t h a t a H is the maximal integral manifold to E through a. To see t h a t it is an integral manifold to E , note t h a t b E H if and only if
ab C a H a n d ( La o i),b(Tb( H) ) = ( L~),b(i,b(Tb( H) ) ) = (La),b(Eb) = (L~),b(Lb),~(E~) = (n~ o Lb),~.(Er :
(Lab),e(Ee)
:
Nab.
We deduce the m a x i m a l i t y of a H from t h a t of H. Let j : K ~-* G be a connected integral manifold to E through a = j ( a ) . Then, L a - i o j : K ~ G is
172
5. L I E G R O U P S
a one-to-one immersion, integral to E and containing La-1 o j ( a ) = La-I (a) = e. By the maximality of H, La-, o j : K --+ G carries K into H. Thus, the image of j = La o L~-I o j is contained in all. [] Thus, 5C is the foliation of G by the "left cosets" all. C o r o l l a r y 5.2.8. If a E H, then a -1 C H. Pro@ Consider the leaf a - l H through a -1. Since a E H, e E a - l H , so this leaf coincides with H. [] C o r o l l a r y 5.2.9. If a, b c H, then ab E H. Pro@ As above, a -1 E H, so e = aa -1 C all, hence a H = H. This implies that ab E a H = H. [] Thus, H is an abstract subgroup and a one-to-one immersed submanifold of G. It remains to be shown that the group operations are smooth in H. If H were an imbedded submanifold, this would be immediate, but we must allow i to be only an immersion. L e m m a 5.2.10. The immersion i : H ~ G defines a connected Lie subgroup of G with i , ( L ( H ) ) = O. Proof. The multiplication map PH : H x H -+ G is given by the composition HxH
iXi~GxG
U-~G.
Since I.tu(H x H) C_ i(H), the map i-l o#H : H x H - + H is smooth by Exercise 4.5.17. This is the group multiplication in H. Similarly, the group inversion ~ : H ---, H is smooth. Finally, i,e(Te(H)) = Ee, so i , ( L ( H ) ) = 0. [] The following lemma completes the proof of Theorem 5.2.5. L e m m a 5.2.11. The Lie subgroup i : H ~ property that i . ( L ( H ) ) = ~.
G is the only connected one with the
Pro@ The Lie subgroup H is a leaf of the foliation 9~ determined by the Lie subalgebra ~1. Suppose that i' : H ' ~-~ G is also a connected Lie subgroup such that i'.(L(H')) = b. Then H ' must be a connected integral manifold through e to the distribution E determined by b. The maximal such integral manifold is H, and therefore H ' C_ H. Indeed, H ' is an open Lie subgroup of the connected Lie group H, so H ' = H by Proposition 5.1.13. [] E x e r c i s e 5.2.12. Let i : H ~-~ G be a Lie subgroup of G. Let exp H : Te(H) "-+ H, exp G :Te (G) --+ G be the respective exponential maps. Prove that the diagram Te(H)
H
i,~ , T~(G)
--~ i
is commutative.
G
5.3. C L O S E D
SUBGROUPS*
173
E x e r c i s e 5.2.13. Using the above exercise and the fact that exp: 9Yt(n, F) ~ Gl(n, F) is ordinary matrix exponentiation, F = ll~ or C, give a new proof that L(O(n)) = o(n) is the algebra of skew symmetric matrices over IR and determine the Lie subalgebra L(U(n)) = u(n) C 9Jr(n, C). E x e r c i s e 5.2.14. Let ~ : H --* G be a homomorphism of Lie groups and prove that p ( H ) is a Lie subgroup of G. 5.3. C l o s e d S u b g r o u p s * While a Lie subgroup i : H ~-~ G is generally only an immersed submanifold, we have seen a number of examples, such as Sl(n) C Gl(n), O(n) C Gl(n), and U(n) c GI(n,C), in which H is a properly imbedded submanifold. In this case, the Lie subgroup has the relative topology from G, making it easier to work with. Generally, imbedded submanifolds are not closed subsets, but this is true for Lie subgroups. P r o p o s i t i o n 5.3.1. If the Lie subgroup H C G is an imbedded submanifold, then
H is closed as a subset of G. (That is, H is properly imbedded.) Proof. We use the foliation ~s from the previous section. The components of H are leaves of 9s Find a neighborhood U of e in which the foliation 9 1 such that hnlhn+l E U,
V n >_ r.
That is, hnlhn+l E U V? H = Po. Also, since e E P0, it follows that both h~ C Lh,, (Po) and h~+l C Lh,L (Po). For r sufficiently large, Lh,~ (P0) C /5, where/5 is a plaque of 9qW. That is, when n > r, h~ and hn+lN lie in a common plaque P of W. Similarly, h~+l and h~+2 lie in a common plaque Pq Since hn+l E P N P ~, it follows that P = P ' . Proceeding in this way, we see t h a t / 5 contains hm, V m >_ r. [] This proposition has a surprisingly strong converse. T h e o r e m 5.3.2 (Closed subgroup theorem). If G is a Lie group and H C G is a
closed subset that is also an abstract subgroup, then H is a properly imbedded Lie subgroup. E x e r c i s e 5.3.3. Let G be a topological group whose underlying space is a topological manifold. Use Theorem 5.3.2 to prove that there is at most one differentiable structure on the topological manifold G making G into a Lie group. (The positive solution to Hilbert's fifth problem guarantees that a topological group-manifold does have a smooth (in fact, analytic) structure making it into a Lie group.)
174
5. L I E G R O U P S
Our proof will be modeled, in certain important ways, on the proof given in [14, pp. 105-106]. The problem with that proof is that it is based on a lemma [14, Lemma 1.8, p. 96] that assumes that Lie groups are real analytic groups. The fact t h a t C ~~ groups are, in fact, real analytic, will not be used in our proof. For a somewhat different presentation, also carried out in the smooth category, the reader can consult [49, pp. 110 112]. F i x the assumptions in Theorem 5.3.2. Define
b = { X e L(G) l e x p ( t X ) 9 H , - o o < t < oo}. This contains 0 9 L(G) and is closed under scalar multiplication. It is not evident t h a t I? is a vector subspace, let alone a Lie subalgebra. It is also unclear t h a t l) ~ 0 if H is not discrete. In fact, [1 will turn out to be the Lie algebra of an open Lie subgroup of H. Let V C L(G) be the vector space spanned by O. In the following proof, it will be convenient to use the notation a X = L a . ( X ) and X a = R a . ( X ) (where Ra denotes right translation by a), a 9 G, X 9 2E(G). Thinking of vectors as infinitesimal curves makes this notation particularly natural. L e m m a 5.3.4. The vector space V is a Lie subalgebra of L(G).
Proof. Since the Lie bracket is bilinear and V is spanned by I1, it will be enough to prove t h a t IX, Y] 9 V, VX, Y 9 I1. By Theorem 4.3.2 and Proposition 5.1.23, [X, r ] = lim Y e x p ( - t X ) - r t~0
t
Since Y is a left-invariant field, Y e x p ( - t X ) = exp(tX) Y e x p ( - t X ) , and, for a fixed t, this is a left-invariant field whose corresponding 1-parameter group is c~(r) = exp(tX) e x p ( r Y ) e x p ( - t X ) . Since X, Y 9 ~, a(r) is a product of elements of the subgroup H, hence a ( r ) 9 H, - o o < r < co. It follows that, for each value of t, Y e x p ( - t X ) 9 0 _c V. Thus, IX, Y] 9 V, as desired. [] Let H0 C G be the connected Lie subgroup with L(Ho) = V. L e m m a 5.3.5. There is an open neighborhood U ore in Ho (in the manifold topol-
ogy of rio) that is contained in H. Proof. Let {Y1,... ,Yq} C I) be a basis of V. The map qo : V --+ H0, defined by qa
tiYi
= exp(tlY1) exp(t2Y2)...exp(tqYq),
\i=1
satisfies ~o,0(Yi) = Yi, 1 < i < q, so the inverse function theorem implies t h a t ~o carries some neighborhood U0 g V of 0 diffeomorphically onto a neighborhood U c_ H0 of e. But exp(tiY/) 9 H, 1 _< i < q, and H is a subgroup, so U C_ H. [] C o r o l l a r y 5.3.6. The Lie group Ho is a subgroup of H.
Proof. By Proposition 5.1.13, H0 is generated by U C H, and so H0 C H.
[]
5.3. C L O S E D S U B G R O U P S *
175
Remark that, at this point, we know that I) = V, hence I? is the Lie algebra of Ho.
L e m m a 5.3.7. The subgroup Ho C_ H, with its manifold topology, is open in the
relative topology of H. Proof. (Compare [14, p. 106].) It will be enough to prove that some open neighborhood U of e, in the manifold topology of H0, is a neighborhood of e in the relative topology of H. Tile problem is that, for each such U, there might be a sequence o~ {X k}k=l C H \ U such that xk --~ e in the topology of G. Assuming that this is so, we deduce a contradiction. Find a direct sum decomposition L(G) = b O W and remark that, by the inverse function theorem, the map p : 0 | W --+ G, defined by ~(v, w) = exp(v) exp(w), carries some neighborhood N of 0 in L(G) diffeomorphically onto a neighborhood of e in G. Choose U = exp(% N N). Thus, for k sumciently large, we can write xk = exp(vk)exp(wk) e ~o(N), where vk C 0/q N and wk E I/V N N. Since xk ~ U, it is clear that wk ~ 0, for all large values of k. Select a bounded neighborhood TWoC W of 0 and positive integers nk such that, for k sufficiently large, n,:wk E Wo, but (nk + 1)wk g W0. Since W0 is bounded, we can assume that nkwk ---+w C W. Since wk --+ 0 and (nk + 1)wk ~ W0, we must have w ~ 0. For arbitrary t C IR, we will show that exp(tw) C H. That is, w G 0, hence 0 5r w C b ~ W, the desired contradiction. Write t n k = sk + tk, where sk E Z and Itkl < 1. Thus, tkwk --~ 0 and exp(tw)
=
lim exp(tnkwk) /c~oo
:
lim exp(skwk) exp(tkw/~) k~oo
= =
lira exp(skwk)
k~oo
lira exp(wk) ~k k~oo
=
lira ( e x p ( - v ~ ) . ~ ) s~ /c~oo
Since H is closed in G, it follows that exp(tw) E H.
[]
Proof of theorem 5.3.2. Let i : Ho ~-+ H be the inclusion map. We have proven that i carries Ho, with its manifold topology, homeomorphically onto an open subset of H in the relative topology. In particular, the manifold topology of H0 coincides with its relative topology, so H0 is an imbedded Lie subgroup of G. By Proposition 5.3.1, H0 is closed in G, so H0 = i(Ho) is a connected, open-closed subset of H. Thus, H0 coincides with the component of the identity in H. The other components La(Ho), a E H, of H are also properly imbedded submanifolds of G, so H is a Lie subgroup. Since H has the relative topology, each of its components is relatively open in H, so H is a properly imbedded Lie subgroup. [] C o r o l l a r y 5.3.8. Let F C G be an abstract subgroup Then the closure F in G is
a properly imbedded, Lie subgroup of G.
176
5. LIE GROUPS
E x a m p l e 5.3.9. Let v = ( a l , . . . , a n) C ]Rn be a point such that, when N is viewed as a vector space over the rational number field Q, the subset { a l , . . . ,a n} c ]R is linearly independent. Let p : 1Rn ~ T n be the standard projection and let g C ]Rn be the line through v and 0. A classical theorem of Kronecker asserts that this line projects one-to-one to a 1-parameter subgroup p(g) C T n that is everywhere dense in T n (for the case n = 2, cf. Example 4.1.8 and Exercise 4.4.12). It is now fairly easy to prove Kronecker's theorem. Indeed, one proves (Exercise 5.3.10) that, if v l , . . . ,vk E Z n and k V = ~2
xivi
i
i=1
for suitable coefficients x i E N, then k = n. By Corollary 5.3.8, the closure ~ C T n is a compact, connected, abelian Lie subgroup, hence a toroidal subgroup of dimension r < n (Exercise 5.1.30). It follows that v E g C V where V C N n is a subspace and = p ( V ) . In particular, V is spanned by V (3 Z n, so Exercise 5.3.10 implies that d i m V = n and ~ = T n. E x e r c i s e 5.3.10. Prove the assertion in Example 5.3.9 that the vector v E ]Rn, with rationally independent coefficients, cannot be expressed as a real linear combination of fewer than n elements of the integer lattice 7/.n. E x e r c i s e 5.3.11. Let v = ( a l , . . . , a n) E R n be a point such that the set of coefficients {1, a l , . . . , a n} is linearly independent over Q. Prove that the subgroup A C T n, generated by a = p ( v ) , is everywhere dense. (Hint: Every point in the coset qv + Z n has rationally independent coefficients, Vq C Z. Prove this and use it to show that every 1-parameter subgroup of T n meeting a nontrivial element of A is dense in Tn.) Following Helgason [14, pp. 107-108], we deduce the following classical result as another corollary of Theorem 5.3.2. For a proof that does not depend on that theorem, see [49, p. 109]. T h e o r e m 5.3.12. I f ~ : G --~ H is a c o n t i n u o u s group h o m o m o r p h i s m
between
L i e groups, t h e n qp is s m o o t h . Proof. The product G x H is a Lie group and the projections 7ra
GxH
~
G,
~rH
G x H
--*
H
are smooth group homomorphisms. Let F C G x H be the graph of ~. T h a t is, P = {(x, ~(x)) ] x C G}, clearly a closed subgroup of G x H. Thus, F is a properly imbedded Lie subgroup. Also, 7ral F = ~ : F --* G is a smooth group homomorphism and is bijective. If it can be shown that r : G --* F is smooth, then ~ = ~rHor -1 will be smooth and the assertion will be proven. By the inverse function theorem, it will be enough to show that r is bijective, Vy C F. Since P is a Lie group and ~ is a smooth homomorphism, it is enough to prove this at y = (e, e). Remark that the exponential maps for the groups G x H, G, and H are related
by eXpaxH = expa x exPH.
5.3. C L O S E D S U B G R O U P S *
177
This, together with the proof of Theorem 5.3.2, implies that L(F) = { ( X , Y ) r L(G) x L(H) [ (expGtX, e x P H t Y ) 9 F, Vt 9 ~}. Since r162
Y) = X, we must show that, for each X 9 L(G), there is a unique
Y 9 L ( H ) such that (X, Y) 9 L(F). We first show uniqueness of Y. If (X, Y) 9 L(F) and (X, Z) 9 L ( r ) , then the difference is (0, Y - Z) 9 L (F), implying that (e, eXpg t ( Y - Z)) 9 F, V t 9 N. Thus, e x P H t ( Y - - Z) = ~(e) = e, Vt 9 R, and Y - Z = 0. Choose open neighborhoods Uo C_ L(G) and Vo C L(H) of the origin and Ue C G and Ve G H of the identity such that 1. exp C : U0 --+ U~ is a diffeomorphism onto; 2. expH : Vo --~ Ve is a diffeornorphism onto;
3. ~(u~) c v~; 4. e x p a •
carries (U0 x V0)n
L(r)
diffeomorphically onto (U~ • V~)N F.
Let X
9 L(G) and choose an integer r > 0 such that ( 1 / r ) X 9 Uo. Thus, ~ ( e x p a ( 1 / r ) X ) 9 Ve and there is a unique Yr 9 V0 such that e x p g Yr = ~ ( e x p a ( l / r ) X ). There is also a unique Z,. 9 (Go x Vo) N L(F) such t h a t exPaxH Z~ = ( e x p a ( 1 / r ) X , exPH Y~). Since e x p c •
is one-to-one on Uo x Vo, this implies that
((1/r)X, Y~) = Z,. 9 L(P). Take Y = rY.~, obtaining (X, Y) = rZ,. r L(F).
[]
Corollary 5.3.13.
L e t ~ : C, + H be a continuous homomorphism of Lie groups and let K = ker(~). Then K is a properly imbedded, normal Lie subgroup of G, G / K is canonically a Lie group, and the induced map -~ : G / I ( ---+H is a one-to-one immersion of this Lie group as a Lie subgroup of H.
Proof. Indeed, K is a normal subgroup by standard group theory and K = ~ - 1 ( c ) is a closed subset of G, so Theorem 5.3.2 guarantees that K is a properly imbedded Lie subgroup of G. By Theorem 5.3.12, ~ is a smooth homomorphism, so Exercise 5.2.14 guarantees that ~(G) is a Lie subgroup of H. Obviously, ~ is an isomorphism of the group G / K onto ~(G), so ~ (:an be used to transfer the Lie structure of qo(G) back to G / K . [] E x e r c i s e 5.3.14. Maximal abelian subalgebras of Lie algebras play an important role in Lie theory, as do the maximal abelian subgroups of Lie groups. Prove the following. (1) Show t h a t every finite dimensional Lie algebra contains a nontrivial abelian subalgebra that is not itself contained properly in another such subalgebra. Similarly, show that every compact Lie group G contains a maximal subgroup that is Lie isomorphic to T k, some k >_ 1. This is called a maximal torus of G. (2) W h e n G is compact, prove that the correspondence between Lie subalgebras and connected Lie subgroups sets up a one-to-one correspondence between the maximal abelian subalgebras of L(G) and the maximal tori in G. (3) Let G be compact and connected, T C_ G a maximal torus. Prove that T is a maximal abelian subgroup. (4) There are maximal abelian subgroups of a connected Lie group G t h a t are not maximal tori. Find a finite subgroup of SO(3) that is maximal abelian.
178
5. LIE G R O U P S
E x e r c i s e 5.3.15. Let G be an n-dimensional Lie group and g = L(G) its Lie algebra. Let Gl(g) denote the group of nonsingular linear transformations of the vector space g and let Ant(g) C Gl(g) be the subgroup of Lie algebra automorphisms of g. Prove the following. (1) Gl(g) has a canonical Lie group structure under which it is (non-canonically) isomorphic to Gl(n). Also, for use in Exercise 5.3.16, show that L(GI(g)) is canonically the space End(g) of linear endomorphisms of the vector space g, the bracket in End(g) being the commutator product of endomorphisms. (2) Ant(g) is a closed subgroup of Gl(g), hence a properly imbedded Lie subgroup. (3) Assume that G is connected and let C C_ G denote the center of G, clearly a closed subgroup. Each element a 9 G determines an inner automorphism of G, denoted by Ad(a) and defined by Ad(a)(g) = aga -1,
Vg 9 G.
Prove t h a t {Ad(a)}aec is canonically a Lie subgroup of Ant(g), isomorphic as a group to G/C. This subgroup is denoted by Ad(G) and called the adjoint group of G. E x e r c i s e 5.3.16. Let G be a Lie group and again denote its Lie algebra by g. A derivation D : g --+ g is a linear transformation such that
D[X, Y] = [DX, Y] + [X, DY], VX, Y 9 g. Let 9
be the space of derivations of g.
(1) Prove that, under the commutator product, 9 is naturally identified as a Lie subalgebra of L(Gl(g)) (cf. Exercise 5.3.15, part (a)). (2) For each X 9 g, define a d ( X ) : g --+ g by ad(X)Y=[X,Y],
VY 9
and prove that ad(X) 9 9 (3) Prove t h a t ad : g --+ L(GI(g)) is a homomorphism of Lie algebras. Thus, ad(g) C_ L(GI(9)) is a Lie subalgebra. (4) Assume that G is connected and prove t h a t the connected Lie subgroup of Gl(g) corresponding to the Lie subalgebra ad(g) is exactly the adjoint group Ad(G). (Hint: Prove that a d ( e x p ( t X ) ) = exp(t ad(X)), V X 9 g.)
5.4. Homogeneous Spaces* Lie groups arise in many natural ways as transformation groups of differentiable manifolds. When the group action is transitive, the manifold is called a homogeneous space and one has considerable control over its structure. D e f i n i t i o n 5.4.1. Let M be a smooth manifold and G a Lie group. A smooth map
#:G•
M,
written p(g, x) = gx, is said to be an action of G (from the left) on M, and G is called a Lie transformation group on M, if (1) gl(g2x) = (9192)x, Vgl,g2 E G and Vx E M; (2) e x = x , V x 9
5.4. HOMOGENEOUS SPACES*
179
Remark. One can also define a right action #:MxG--~M by making the obvious changes in the above definition. D e f i n i t i o n 5.4.2. An orbit of the action
GxM~M is a set of points of the form {gxo I g C G}, where x0 E M. The action is transitive if M itself is an orbit, in which case M is said to be a homogeneous space of G.
Remark. It is elementary that the orbits of a group action are equivalence classes, two points x, y E M being equivalent under the action if 3 g C G such that g x = y. E x a m p l e 5.4.3. The orthogonat group O(n) acts on R n in the usual way, leaving invariant the unit sphere S n - : . Note that, if el C S n is the column vector with first entry 1 and remaining entries 0, then Ael is the first column of A C O(n). Every unit vector appears as the first column of suitable orthogonal matrices, so the action O ( n ) X S n - 1 ----> S n - 1
is transitive and S ~-1 is a homogeneous space of O(n). In a completely similar way, there is a transitive action U ( n ) x S 2 n - 1 --+ S 2 n - l ,
where S 2n-: C C n is the unit sphere in the standard Hermitian metric. D e f i n i t i o n 5.4.4. Let M be a homogeneous space of G and let x0 C M. isotropy group of z0 is the set Gxo = {g C G I gxo = z0}.
The
L e m m a 5.4.5. The isotropy 9roup Gzo as above is a properly imbedded Lie sub9roup of G.
Pro@ It is obvious that Gxo is an abstract subgroup of G. If {gn}n~=: is a sequence in Gxo converging to g C G, then, by the continuity of the group action, gxo=
lim g ~ x o =
n~oo
lim x 0 = x 0 .
n~oo
Thus, G~o is a closed subset of G. By Theorem 5.3.2, Gxo is a properly imbedded Lie subgroup. [] E x a m p l e 5.4.6. If el G S ~ - : is as in Example 5.4.3, the isotropy group O(n)~: is the set of matrices
where A r O(n - 1). Similarly, for e: E S 2n-1, U(n)~: is the set
where A C U(n - 1). We are going to show how to put a smooth structure on the quotient space G/G~o and prove that this manifold is diffeomorphic to the homogeneous space M. Under the identification M = G/Gxo, the G-action on M becomes the action
G x G/G~o --~ G/G~o , g(ha~o ) = (gh)G~o.
180
5. L I E G R O U P S
In what follows, we consider an arbitrary properly imbedded Lie subgroup H C_ G, put the quotient topology on G / H , and construct a natural smooth structure on this space. Throughout this discussion, we set t} = L ( H ) . Decompose L(G) = moil, where m is any fixed choice of complementary subspace. Let
r :m|247 be the map r
B) = exp(A) exp(B)
and choose a neighborhood V of 0 in il and a neighborhood W of 0 in m such that r sends W x V diffeomorphically onto a neighborhood U of e in G. Choose a compact neighborhood C C W of 0 with the property that - C = C and exp(C) exp(C) C U. We can assmne that coordinates x l , . . . ,x k in il define V by the inequalities - 1 < x i < 1, 1 < i < k. Similarly, coordinates y l , . . . ,yq for m define C by - 1 <_ yJ < 1, 1 <_ j <_ q. Coordinatize Q = r x V) by r : Q __~ C x V. The foliation of G by the components of the left cosets of H is given in Q by plaques r x V) that are level sets ( y l , . . . , yq) = c. Since H is properly imbedded in G, we can arrange that H n U = r x Y). L e m m a 5.4.7. Each coset a H meets Q in at most one plaque. Proof. Let cl,c2 E C be such that exp(cl)exp(V) and exp(c2)exp(V) lie in a common coset exp(cl)H = exp(c2)H. Then, e x p ( - c l ) exp(c2) e g n V = r
x V).
This implies that r
v) = exp(cl) exp(v) = exp(c2) = r
for some v E V. Since r is one-to-one, we conclude that v = 0 and c 1 = C2.
[]
E x e r c i s e 5.4.8. Let Co = int(C) and Q0 = int(Q). Prove that the map (p : Co x H--* QoH, given by
~(c, h) = exp(c)h, is a diffeomorphism. By this exercise, if ~c : C • H -~ C denotes projection onto the first factor, we obtain a submersion y = Trc o ~ - I : Q o H --, m = IRq. This smooth submersion assigns a coordinate q-tuple (yl ( a l l ) , . . . , y q ( a H ) ) to each coset a H c QoH, distinct cosets getting distinct coordinates. Cover G by open sets of the form aQoH, a C G. Such a set is also a union of cosets b H and we assign coordinates to each coset via the submersion Ya = Yo La-1. On overlaps a Q o H N bQoH, the coordinates Ya and Yb are related by Yb = Ya o Lab-1. T h a t is, the change of coordinates on overlaps is smooth. Let 7r : G -~ G / H be tile quotient map. This carries a Q o H onto an open set Ua C G / H and Ya induces Ya : Ua --* Nq. L e m m a 5.4.9. The map [la : Ua -~ ]Rq is a homeomorphism onto an open subset
ofR q.
5.4. HOMOGENEOUS SPACES*
181
Proof. Indeed, we have coordinatized m so t h a t the image of Ya is the open set Co. It is clear t h a t 9a is one-to-one a n d continuous. We must prove t h a t it is an open map. If Z C_ Ua is open, t h e n r e - l ( Z ) is open and y:, being a submersion, carries this open set onto an open set. But ~a(Z) = ya(rc-l(Z)). [] We view { ( U : , g a ) } a e a as a coordinate atlas on G / H . By the above remarks, this is a C ~ atlas on the locally Euclidean space G / H . The following exercise completes our analysis of G / H . E x e r c i s e 5.4.10. W i t h the above Coo atlas, prove t h a t G / H is a smooth manifold, t h a t the projection re : G --+ G / H is a submersion, a n d that the action # : G x G / H --+ G / H , defined by p(a, bH) = abH, is smooth. C o r o l l a r y 5.4.11. If g is a Lie group and H C_ G is a closed, normal subgroup, then the group G / H has a smooth structure in which it is a Lie group. We r e t u r n to the s m o o t h transitive action #:GxM-+M. Let x0 C M a n d let H = Gx o be the isotropy group. Define the m a p 0 : G / H --+ M by O(aH) = axo. This is induced by the smooth m a p 0 : G --+ M , 0(a) = axo, so 0 is continuous. Since a H = bH if and only if a - l b E H, we see t h a t x0 = a-lbxo, hence aa:o = bxo, if and only if a H = bH. T h a t is, 0 is well defined, one-to-one, a n d continuous. Since the action of G is transitive, 0 is a surjection. T h e following diagram is commutative: G x G/H
"
idx01 Gx M
, C/H
10 . /z
~ /14-
Thus, if we prove t h a t 0 is a diffeomorphism, 0 will be a canonical identification of G / H with M as a homogeneous space of G. Proposition
5.4.12.
The map 0 : G / H --~ M is a diffeomorphism.
Pro@ Let La denote left t r a n s l a t i o n by a E G on both G / H a n d M. T h a t is, La(bH) = abH, La(x) = ax,
VbH E G/H, V x C M.
Then 8=LaoOoLa-1,
VaCG.
Since La : M --+ M a n d La-1 : G / H --+ G / H are diffeomorphisms, it follows t h a t 0 will be s m o o t h at a H if a n d only if it is smooth at ell. Furthermore, if smoothness has been established, t h e n
O,aH = (La).xo oO.~i~ o ( L ~ - l ) . a , will be an isomorphism of Tall(G/H) onto T , , o ( M ) if a n d only if o,~. : Totda/H)
--+ T x o ( M )
182
5. LIE GROUPS
is an isomorphism. We show smoothness at ell. Consider the commutative diagram
M The m a p 41 exp(C0) : exp(C0) --+ M is smooth, and the map rr I exp(C0) : exp(C0) --+ G / H is a diffeomorphism onto the coordinate neighborhood Ue, so 0 is smooth in a neighborhood of ell. We show that {~,eH : T~H(G/H) --* Tzo(M) is an isomorphism. Again, this translates, via the commutative triangle, to showing that 0,r is an isomorphism of Te(exp(Co)) onto T , o ( M ). Let v E T~(exp(C0)) -- m and consider the curve s(t) = exp(tv)xo. Since ~,~(v) = i(0), we only need prove that ~(0) = 0 implies that v = 0. If a = exp(tov), then La(s(t)) = s(t + to), so L.a(8(0)) = 8(t0) and i(0) = 0 implies that i(t) = 0, Vt E IR. T h a t is, exp(tv)xo = x0, g t E IR, implying thatvEbNm={0}. [] E x a m p l e 5.4.13. Thus, as a homogeneous space of O(n),
S n-1 = O ( n ) / O ( n - 1), where O ( n - 1) is properly imbedded as a Lie subgroup of O(n) as in Example 5.4.6. Similarly, S 2n-1 = U ( n ) / U ( n - 1). E x e r c i s e 5.4.14. Let (Tn,k denote the set of k-dimensional vector subspaces of IRn. (1) Show how to make Gn,k into a compact manifold that is a homogeneous space of O(n). This is called the (real) Grassmann manifold of k-planes in n-space. (2) If xo e Gn,k is the standard ]Rk _C ]Rn, identify the isotropy group O ( n ) . o. (3) Show t h a t projective space p ~ - I is the Grassmann manifold Gn,1 and identify the standard two-to-one map S ~-1 -+ pn-1 as a map O ( n ) / O ( n - 1) --+ O ( n ) / O ( n ) ~ o. (Remark: Using C n instead of 1R~, one defines in a similar way the complex Grassmann manifolds Gn,k(C) as homogeneous spaces of U(n). Complex projective space is defined to be Pn (C) = Gn,1 (C). The real and complex Grassmann manifolds play an important role in differential geometry and topology.) E x e r c i s e 5.4.15. A k-tuple ( v l , . . . , vk) of orthonormal vectors in IR~ will be called an orthonormal k-frame in IRn. Let V~,k denote the set of all orthonormal kframes in IRn, identify this as a homogeneous space of O(n) (called the Stiefel manifold of k-frames in n-space). Remark t h a t V,~,I = S '~-1 and t h a t this case gives back Example 5.4.3. Using C ~ and the standard positive definite Hermitian inner product on C n, one obtains the complex Stiefel manifolds V~,k(C)VnkC@ and gn,l(C ) = S 2n-1.
CHAPTER 6
C o v e c t o r s and 1 - F o r m s An important analytic tool in our study of manifolds M has been the Lie algebra ~ ( M ) of smooth vector fields. In this chapter, we begin the study of the dual object, the space A I(M) of differential 1-forms on M. One would expect this space of "covector fields" to be neither more nor less useful than 3r but for many purposes it is much more powerful. One reason for this is the "functoriMity" of A I(M), as will be explained presently. Another is exterior" derivative and exterior multiplication, operations that produce higher order objects, called q-forms. These q-forms can be integrated over suitable q-dimensional domains and differentiated. A version of the fundamental theorem of calculus, called Stokes' theorem, relates these operations and, in the global setting, leads to a remarkable tool (de Rham cohomology) for analyzing the topology of M. This, in fact, is the beginning of a major mathematical discipline called algebraic topology. Finally, theorems stated in terms of differential forms sometimes provide interesting and useful alternatives to equivalent vector field versions. An example of this will be a differential forms version of the Frobenius theorem. These topics will require the next several chapters to do them justice. Here we deal only with 1-forms. 6.1. D u a l B u n d l e s
Let rr : E --+ M be a k-plane bundle. In parl~icult~r, E can be viewed as a parametrized family of /c-dimensional vector spaces Ex, where ttle parameter x ranges over M. There is a general philosophy that linear algebra constructions which do not involve a choice of basis, being canonically defined on every Ex, can be extended smoothly to the entire bundle. We will see many examples of this, beginning here with the construction of the dual bundle 7r : E* -* M. Recall that a vector space V has a dual space V*, this being the vector space V* = HomlR(V, IR) of all linear functionals on V. Similarly, one constructs the dual bundle E*, essentially by taking the vector space duals E~ of the fibers Ex Vx C M, and assembling them into a bundle by means of a suitable Gl(n)-cocycle as in Section 3.4. The dual space V* of V is abstractly, but generally not canonically, isomorphic to V. (As we see below, there is a canonicM choice of this isomorphism when V = IEk.) If ~o : V1 ~ V2 is a linear map between vector spaces, the adjoint ~o* : V ; ~
V~
is the linear map defined by 9~*(f) = f o ~,
VfeV~.
184
6. COVECTORS
AND
1-FORMS
E x a m p l e 6.1.1. We represent elements v E IRk by k x 1 matrices and elements f G R k* by 1 x k matrices. Then f ( v ) = f 9v is just matrix multiplication. Thus, the transpose operation v ~-* v W defines a canonical isomorphism between IRk and IRk.. If qo : IRk __. IRm is a linear map, let A be the m x k matrix representing ~. Then, relative to the canonical identifications IRk = IRk, and IRm = Nra*, the adjoint ~a* is represented by the k x m matrix A w. In the language of category theory, the associations V ~ V* and ~ ~ ~* define a contravariant functor on the category of real, finite dimensional vector spaces and linear maps. That is, morphism arrows are reversed under qa ~-+ ~o* and, consequently, (~or162 This contravariance is a slight problem when we try to find a Gl(k)-cocycle for the construction of the dual bundle E*. The following saves the day. 6 . 1 . 2 . If ~ : 1/1 ~ V2 is an isomorphism of vector spaces, then ~/ : VI* ~ 1/2" is the isomorphism ~ / = (~o.)-i.
Definition
Thus, if V1 = V2 = R k and A is the k x k matrix representing ~, then (AT) -1 = ( A - l ) w represents ~ / One clearly has covariant functoriality
(~ o r
=,
o r
corresponding to the matrix identity
((AB) T)-I = (AT)-l(B T)-I. Let 3' = { W ~ , ' ~ } ~ , ~ E ~ trivializations
r
be a Gl(k)-cocycle arising from a family of local
: 71--1(Wo~) " ~ Wo~ x IRk
of a k-plane bundle 7r : E --~ M. We can recover that bundle from the disjoint union
I Iwo• ~Eg.I
U ~Eg~
by quotienting out an equivalence relation. (The standard device of reducing a disjoint union to an ordinary union by adding the index as a factor will be notationally useful.) The equivalence relation identifies an element (x, v, ~) E WZ x IRk x {/3} with ( y , w , a ) E W~ x R k x {a} whenever x = y E W~ A W~ and w = 7 ~ ( x ) 9v. Indeed, it is straightforward to check that the quotient space E~ has a canonical vector bundle structure with projection
7r:E~ ~ M, ~([~, ~, ~]) = x. One then checks that the local trivializations ~b~ fit together to define a canonical bundle isomorphism r : E --~ E~. The reader who did not carefully think this through in Section 3.4 really should do so now. By the above remarks, the Gl(k)-cocycle ~/ = {W, 7~}~,~ee~ gives rise to a Gl(k)-cocycle 3,I,
and the bundle E 7, constructed from this cocycle will be called the dual bundle E*.
6.2. S P A C E O F 1 - F O R M S
185
P r o p o s i t i o n 6.1.3. For each x E M , the fiber (E~,)~ = E* is canonically isomorphic to the dual of the fiber (E~)z = E~.
Proof. Let the equivalence classes in (E~)x of elements (x, v, c~) of W~ x ]Rk • {c~} be denoted by Ix, v, ~] and those in (E~,)x = E x by [x, v, c~]*. We attempt to evaluate Ix, w, c~]* on [x, v, c~] by the formula
[x,w,~]*. [x,~,~] = ~ T
~.
We show that this is well defined. Indeed, if x C W~ N W~,
[x,~,~]* = [ ~ , % ~ ( x ) ~ , z ] * , and
(%~(x). ~)T .~,~(x). v = ((~,~(x) -I) T .~) T .~,~(x). v = ~vT .~,~(x)-1. ~(x). z
It) T
v
.V.
This action of [x, w, ~]* on Ex is clearly linear, and defines E~ as a vector subspace of the dual space (Ex)*. Since these spaces have the same dimension, E~ = (Ex)* canonically. [] E x e r c i s e 6.1.4. Let a : M -~ E* be a section, not necessarily smooth or continuous. Show that a is continuous (respectively, smooth) if the map M~R, x ~, ax(r~)
is continuous (respectively, smooth) for every continuous (respectively, for every smooth) section 7- of E.
Remark. The double dual of an n-plane bundle 7r : E --~ M is canonically isomorphic to the original bundle. That is, (E*)* = E. Indeed, it is immediate that
6.2. T h e s p a c e o f 1 - f o r m s
We apply the construction of the previous section to the tangent bundle T ( M ) . D e f i n i t i o n 6.2.1. Let M be a differentiable manifold, x E M. The dual space (Tx(M))* is called the cotangent space of M at the point x and will be denoted by T~ (M). Each element a E Tx (M) is called a cotangent vector to M at x. The dual bundle T* (M) to the tangent bundle is called the cotangent bundle of M. E x e r c i s e 6.2.2. Show that a choice of Riemannian metric on M induces a bundle isomorphism T ( M ) ~- T * ( M ) . Thus, in the case that M is an open subset of Euclidean space, the standard Euclidean inner product defines a canonical choice of isomorphism. (In classical physics and advanced calculus courses, it is quite common not to distinguish tangent vectors and cotangent vectors. It is precisely because these treatments are carried out in Euclidean domains that this identification is legitimate. In general, we do not identify T ( M ) and T* (M).)
186
6. COVECTORS
AND
I-FORMS
A typical cotangent vector is the differential of a map. Let U C M be open, x E U, and let f C C ~ ( U ) . Since Tf(x)(N) = R canonically, we obtain a linear functional df~: : T~(M) + R , so df~ c T ~ ( M ) . It is evident that dfx depends only on the germ [f]~ E qS~, so we obtain an IR-linear map d : q3x --+ T* (M). L e m m a 6.2.3. For each Z z e T z ( M ) , df~(Xx) = X x ( f ) .
Proof. Let (U, x l , . . . , x n) be a coordinate chart about x. Then df~ = Jf~ =
x),...,
x
.
If
Xz = ~.ai
o@ x E T~:(M),
i=1
then
dfx(X~) = J f z "
=
a'
x) = X x ( f ) . []
C o r o l l a r y 6.2.4. Relative to local coordinates x l , . . . , x n about x E M , the covectots dxlx,... , dx~ form a basis of T~ (M).
Proof. Since dim T* (M) = n, it will be enough to show that this set of covectors is linearly independent. By the lemma, 9
0
5ij.
Thus
bidxiz = 0 ~ 0 = ~ i=1
( O bidxiz
)~ l * < , J
_
_< n ,
i=1
r
l <_j < n . []
C o r o l l a r y 6.2.5. The linear map d : ~bx --+ T ~ ( M ) is surjective. T h a t is, every covector is the differential of a function. Let U C_ M be an open subset, f C C~176 and consider the assignment
xHdfxeTx(U),
VxeU.
In local coordinates,
gfx
~:-,Of r ,, i = 2_., ~ Vx~ axx. /=1
Since dxl(O/OxJlx) the map
= 5~y, it is clear that, for each smooth vector field X C ~(U), x~
dfx(X~)
6.2. S P A C E O F 1 - F O R M S
187
defines a smooth function on U. By Exercise 6.1.4, it follows that x ~ dfx defines a smooth section df of T*(U). More generally, smooth sections co of T*(U) have local coordinate formulas n
cox = ~
fidx i,
i=1
where fi E Coo(U), 1 < i < n. D e f i n i t i o n 6.2.6. The Coo (M)-module F (T* (M)) of smooth sections of the cotangent bundle is denoted by A l ( M ) = F(T*(M)). The elements of A I(M) are called covcctor fields or (more commonly) 1-forms on M. If co E AI(M), then its value at x E M is denoted by co~ E T*(M). the Coo(M)-module of all maps Denote by Homc~(M)(~(M),C~176 X(M) ---, Coo(M) that are Coo(M)-linear. If w E A l ( M ) and X E ~ ( M ) , we obtain co(X) E C~176 by setting
co(x)(x)
:
cox(Xx),
VxEM.
It is clear that
cv(fX) = f w ( X ) ,
V f E Coo(M),
so we can view this 1-form as an element
co E Homc~(M)()~(M), Coo(M)). This defines an injective homomorphism
A I ( M ) ~ Homc~(M)(X(M), Coo(M)) of C ~ ( M ) - m o d u l e s . We will now show that this is also a surjection, proving that these Coo (M)-modules are canonically isomorphic. Let c~ E Homc~(M)(~(M), Coo(M)) and let U _C M be an open subset. L e m m a 6.2.7. If X E :~(M) and XIU = O, then c~(X)IU = O.
Proof. Let x E U and choose f E Coo(M), vanishing at x and identically equal to 1 onM\U. Then f X = X a n d c~(X) = c~(fX) = fc~(X). This shows that
a(X)(x) = f ( x ) a ( X ) ( x ) = O. Since this is true for arbitrary x E U, it follows that c~(X)IU - 0.
[]
L e m m a 6.2.8. There is a canonical
E Homc~(u)(X(U), Coo(U)) such that
~(XlU) = ~(X)lU,
v x E X(M).
Proof. If Y E :~(U), define ~(Y) E Coo(U) as follows. For arbitrary y E U, choose f E Coo(M) such that f = 1 on some open neighborhood V C U of y and f l ( M \ U) - O. Then we can interpret f Y as a field defined on all of M ( - 0 outside of U) and f Y [ V = YIV. Define ~(Y)(y) = c~(fY)(y).
188
6. C O V E C T O R S
AND 1-FORMS
If f and V are different choices, Lemma 6.2.7 implies that the two definitions of ~(Y) agree at y (and, indeed, on the neighborhood V N V of y in U). It is clear that this defines 5 E Homc~(v)()~(U), C~(U)) and that 5(XIU) = c~(X)IU,
v x c X(M).
[]
By this lemma, we can define (~lU = 5, calling this the restriction of c~ to U. C o r o l l a r y 6.2.9. If (~ E Homc~(M)(i~(M),C~(M)) then a ( Z ) ( x ) depends on X s but not otherwise on X , Vx E M, V X E X(M).
Proof. Let x E M. Choose a neighborhood U of x in M over which T ( M ) is trivial and let y 1 , . . . , y n E :~(U) give a basis of the tangent space at each point of U. Then arbitrary X E :~(M) can be written on U as n
XIU : Z
fiYi
i=l
and
n
a(X)(x) = (cdU)(X4U)(x) = ~ fdx)(,~lu)(Yi)(x). i=1
On the right-hand side of this equation, the only dependence on X is in the values
fi(x), 1 < i < n.
[]
The property of (~ in the above corollary is called the tensor property. L e m m a 6.2.10. If ~ : :~(M) ~ C ~ ( M ) is an R-linear map, then ~1 has the tensor property if and only if ~ E Homc~(M)(i~(M), C ~ ( M ) ) .
Proof. We have proven the "if" part. For the converse, assume that T] has the tensor property and let f E C ~ ( M ) , X E :~(M). For each x E M, ~ ( f X ) ( x ) depends only on f ( x ) X s , hence ~ ( f X ) ( x ) -- ~ ( f ( x ) X ) ( x ) = f(x)~?(X)(x) by ~-linearity. Since x E M is arbitrary, ~ ( f X ) = f~(X).
[]
This equivalence between C~r and the tensor property will recur in the broader context of C~ later in this book. For x E M and a E H o m c ~ ( M ) ( E ( M ) , C ~ ( M ) ) , we define a s E T*(M) as follows. Given v E Ts(M), let X E X(M) be any vector field such that Xs = v. Define as(v) = a(X)(x). By the above, this depends only on v, not on the choice of extension X, so we get a s E T*(M), Vx E M. An application of Exercise 6.1.4 proves that the map x H az defines a smooth section of T*(M). This identifies a as an element of AI(M), completing the proof of the following. P r o p o s i t i o n 6.2.11. There is a canonical isomorphism,
A I ( M ) = Homc~(M)(X(M), C ~ ( M ) ) of C ~ (M)-modules. Remark. Similarly, :~(M) = Homc~( M) (A I ( M), C ~ (M) ). Let ~ : M ~ N be smooth. I f w E AI(N), define ~*(w) : M ~ T*(M) by
~*(~)s = ~;(~(s)),
Vx e M.
If f E C ~ ( N ) , define ~*(f) = f o ~ E C ~ ( M ) . lemmas will be left to the reader.
The proof of the following two
6.2. S P A C E O F l - F O R M S
189
L e m m a 6.2.12. [f ~o : M ~ N is a smooth map of manifolds and if ~ E A I ( N ) , then 9)*(w) E A I ( M ) and this defines a linear map
~* : A ~(N)
-~ A ~(M)
of vector spaces over R. Furthermore, if f e Coo(M), qo*(fw) = w*(f)y)*(w). L e m m a 6.2.13.
If M&N&P
= V*or
are smooth maps of manifolds, then ( r
on both Coo(P) a~d A I ( p )
Thus, A 1 is a eontravariant functor (an anti-homomorphism of categories) from the category of differentiable manifolds and smooth maps to the category of real vector spaces and linear maps. D e f i n i t i o n 6.2.14. If f C Coo(M), then df is called the exterior derivative of f. The R-linear map d : C ~ (M) --+ A 1(M) is called exterior differentiation. L e m m a 6.2.15. /f f, g E C~176
then d(fg) = f dg + g df .
Proof. Indeed, if X E 2E(M), then d(fg)(X) = X(fg) = X(f)g + fX(9) = 9 df(X) + f dg(X ) = (fdg + gdf)(X). Since X E ~ ( M ) is arbitrary, the assertion follows.
[]
This lemma is a Leibnitz rule for exterior differentiation. Exercise
6.2.16. If ~ : M ~ N is smooth, prove that the diagram
Coo(N)
~" , C ~
dI AI(N)
~d , AI(M)
is commutative. That is, d(~*(f)) = F*(df), V f E Coo(N). The property of d in this exercise is called the naturality of the exterior derivative. E x e r c i s e 6.2.17. Let X E X(M) and let (I) denote the local flow generated by X. One defines the Lie derivative
Lx : AI(M) ~ AI(M)
by (*)
L x ( w ) = lira qs•(co) - aa t~O
t
Voa E A I ( M ) , '
taken pointwise on M. Recall from Section 2.8 the analogous definitions of L x (f) and L x ( Y ) for f E Coo(M) and Y E ~ ( M ) . Prove that (*) is defined and satisfies the following identities for arbitrary f E Coo(M), a~ E A I ( M ) , and Y E X(M).
190
6. C O V E C T O R S AND 1-FORMS
(1) L x ( d f ) = d L x ( f ) . (2) L x ( f w ) = L x ( f ) w + f L x ( w ) . (3) L x ( w ( Y ) ) = L x ( w ) ( Y ) + w ( L x ( Y ) ) .
6.3. Line Integrals IfaJ 9 A I ( M ) and s : [a,b] --+ M is a smooth curve, then s*(w) 9 Al([a,b]). We can write s*(w) = f dt. D e f i n i t i o n 6.3.1. The line integral of w 9 A 1(M) along a smooth curve s : [a, b] --+ M is
is
aJ =
/abs* (w) = /j f ( t ) dt.
Line integrals are insensitive to orientation preserving changes of parameter and experience a sign change only under an orientation reversing reparametrization. It is not even necessary to require that the change of parameter be nonsingular or monotonic. L e m m a 6.3.2. Let s : [a, b] --+ M and u : [c, d] --+ [a, b] be smooth. Set a = s o u. Then~
(1) i f u ( c ) = a a n d u ( d ) = b, f s w = l o w , Vco 9 A I ( M ) ; (2) if u(c) = b and u(d) = a, - f co = f< co, V w 9 A I ( M ) . P r o @ Let t denote the coordinate of [a, b] and T the coordinate of [c, d]. Then
= ff u*(s*(~)) = ~au*(fdt) =
~
d(f
du
o u) T
dr.
In case (1), the rule for change of variable in integrals gives
In case (2), the same rule gives
[] L e m m a 6.3.3. Let Sl : [a,b] ~ M and s2 : [c,d] --+ M be smooth paths with the same initial point and the same terminal point. That % sl(a) = s2(c) = x and sl(b) = s2(d) = y. If f 9 C~176
then f~l df = f~2 df = f ( y ) - f ( x ) .
6.3. L I N E I N T E G R A L S
191
Pro@ A p p e a l i n g to L e m m a 6.3.2, we assume, w i t h o u t loss of generality, t h a t [a, b] = [c, d]. Then,
~sl df = ~abs~(df) = ~abd(fosl) = ffab d f ( s l ( t ) ) d t = f(Sl(b)) - f ( s l ( a ) ) = f(s2(b)) - f(s2(a))
=~df. 2
[] T h i s l e m m a is a 1-dimensional version of Stokes' Theorem. As the proof makes clear, it is just the f u n d a m e n t a l theorem of calculus. D e f i n i t i o n 6.3.4. A form w C A I ( M )
is said to be exact if w = df, for some
f E C~(M). E x e r c i s e 6.3.5. Show t h a t every 1-form on ]E is exact, but exhibit a l - f o r m on R 2 t h a t is not exact. L e m m a 6.3.3 says t h a t the line integral fs co of an exact l - f o r m co depends only on the endpoints of the p a t h s, not otherwise on s. In physics, the law of conservation of energy is a special case of this result. L e m m a 6.3.3 is a part of T h e o r e m 6.3.10, which will be stated and proven shortly. The notion of a line integral can be e x t e n d e d to allow integration of 1-forms along p a t h s s : [a, b] --~ M t h a t are only piecewise smooth. T h a t is, s is continuous and there exists a p a r t i t i o n a = to < t l ~ ' ' " ~ tq = b such t h a t si = sl[ti-l,ti] is smooth, 1 _< i _< q. We write s = sl + s2 + ... + Sq and define
i=1
i
Since it is not assumed t h a t the partition contains only points at which s is not smooth, it is necessary to observe t h a t this definition is independent of the choice of allowable partition. This is elementary and is left to the reader. T h e proof of the following consequence of L e m m a 6.3.3 is also left to the reader. 6.3.6. Let s~ : [a, b] ---* M and s2 : [c, d] ~ M be piecewise smooth paths with the same initial point x and the same terminal point y. Then, if f C C ~ ( M ) ,
Corollary
df = / 1
df = f ( y ) -
f(x).
2
6.3.7. If w C A 1 ( M ) and if, for every pieeewise smooth path s, the integral fs w = O, then aJ = O.
Lemma
192
6. COVECTORS
AND
1-FORMS
Proof. Otherwise, there is a point z C M and a vector v E Tz(M) such that Wz(V) > 0. Let s : [-e,e] --* M be smooth such t h a t s(0) = z a n d ~(0) = v. Choosing e > 0 smaller, if necessary, we can assume t h a t ~ s ( , ) ( ~ ( t ) ) > 0,
-~ < t <
~
A n elementary c o m p u t a t i o n shows t h a t s* (w)t = ws(t)(i(t)) dt, so =
/
s*(aJ) =
/=
aJs(t)(i(t))dt > 0,
contradicting the hypothesis.
[]
D e f i n i t i o n 6.3.8. We say t h a t w E A I ( M ) has p a t h - i n d e p e n d e n t line integrals if, for every piecewise smooth path s : [a, b] -~ M , L ~ depends only on s(a) a n d s(b) a n d n o t otherwise on s. D e f i n i t i o n 6.3.9. A piecewise s m o o t h p a t h s : [a, b] --* M is a loop if s(a) = s(b). Theorem
6 . 3 . 1 0 . For aJ E A I ( M ) ,
the following are equivalent.
(1) aJ is an exact f o r m . (2) fs w = 0, f o r all piecewise smooth loops s. (3) w has path-independent line integrals. Proof. We prove t h a t (1) ~ (2). If a = df is exact and s : [a, b] --* M is a piecewise s m o o t h loop, s(a) = q = s(b), then Corollary 6.3.6 implies t h a t
where q denotes the constant path q(t) = q, a < t < b We prove t h a t (2) =~ (3). Let sl a n d s2 be piecewise smooth curves s t a r t i n g at the same point x and ending at the same point y. W i t h o u t loss of generality, assume t h a t Sl is parametrized on [-1, 0] and s2 on [0, 1]. Let u : [0, 11 ~ [0, 1] be defined by u(t) = 1 - t. T h e n s2 o u starts at y a n d ends at x and Sl + s 2 o u
= s : [-1,1]-~ M
is a piecewise smooth loop. By our assumption,
1
where
we have used part (2) of Lemma
2
6.3.2 to write
2 ou
2
We prove t h a t (3) =~ (1) by using (3) to construct f C C ~ ( M ) such t h a t = dr. W i t h o u t loss of generality, we assume t h a t M is connected (otherwise, carry out the construction of f on each c o m p o n e n t individually). Fix a basepoint x0 c M. Given any point x C M , use connectivity to find a piecewise s m o o t h p a t h s : [a, b] --~ M such t h a t s(a) = xo and s(b) = x. (In fact, the homogeneity lemma, T h e o r e m 3.8.7, implies the existence of a s m o o t h path, b u t the present claim is more elementary and is left to the reader.) Set
f(~) = f ~ .
6.3. L I N E I N T E G R A L S
193
By the a s s u m p t i o n of path-independence, this is independent of the choice of piecewise s m o o t h p a t h s from x0 to x. R e m a r k t h a t f(xo) = O. We prove first t h a t f : M --+ IR is smooth. Let q C M be arbitrary and choose a c o o r d i n a t e chart (U, x l , . . . , x n) a b o u t q in which q is the origin and U = i n t D n, where D n is the unit ball in R ~. In these coordinates, we can write
colU = ~
gi dxi.
i=l
For each x C U, let Sx : [0, 1] --+ U be defined by sx (t) = tx, 0 < t < 1, and express flU by the formula
f(x) = f(q) + /
CO.
Js x
Thus, on U,
I ( x 1, .
,x. '~) .
f(O) . .+
.
. gi(tx 1,
z=l
,tx'~)~d (txi)d t
0
~ i f01 9i(txl,...
= f(O)+Ex
,tx~)dt.
i=1
T h i s is clearly smooth. Since q E M is arbitrary, f E C ~ ( M ) . Next, we prove t h a t co = df. Let s : [a, b] ~ M be an arbitrary piecewise s m o o t h path. Let c < a and let so : [e,a] --+ M be piecewise s m o o t h such t h a t so(c) = xo and so(a) = s(a). T h e n
O+S
o
T h a t is
It follows t h a t tile form c~ = w - df satisfies
f
c5 = 0,
for all piecewise s m o o t h paths s. By L e m m a 6.3.7, c~ = 0, so co = df.
[]
D e f i n i t i o n 6 . 3 . 1 1 . A 1-form co E A I ( M ) is locally exact if, for each x C M , there is an open n e i g h b o r h o o d U of x such t h a t colU E AI(U) is exact. Example
6 . 3 . 1 2 . On the manifold M = IR2 \ {(0, 0)}, define the 1-form
- Y dx + x - x 2 + y2 ~
dy.
We claim t h a t ~ is locally exact. Indeed, if q C M is not on the y-axis, a branch of 0 = arctan(y/x) is defined and s m o o t h on a neighborhood of q. A direct comp u t a t i o n gives dO = ~]. Similarly, if q C M is not on the x-axis, select a branch of 0 = - arctan(x/y) and check t h a t dO = r/. Since no point of M is on b o t h axes, this proves t h a t r/ is iocally exact. We claim, however, t h a t r] is not exact. Indeed, consider the s m o o t h loop s : [ 0 , 1] --+ M defined by s(t) = (cos2r4, sin 2rrt). Clearly,
rl4t) = - sin 2rrt dx4t ) + cos 2rrt dys(t),
194
6. C O V E C T O R S
AND
1-FORMS
and
s* (dx) = -27r sin 2~rt dr, s* (dy) = 27r cos 27rt dt, SO
s* (rj) = 27r(sin 2 2~rt + cos 2 2~t) dt = 27r dt. Thus
= 27r
/0
dt = 27r r O.
R e m a r k t h a t the form r/ in the above example cannot be extended to a 1-form on N 2. We are going to see shortly (Corollary 6.3.15) that, on N 2, every locally exact 1-form is, in fact, exact. The above example reflects a topological feature of R 2 -, {(0, 0)}, the missing point, t h a t distinguishes t h a t space from R 2. T h e n o t i o n of smooth homotopy extends to a notion of piecewise smooth hom o t o p y in a fairly obvious way. Here is the formal definition. D e f i n i t i o n 6.3.13. Let s0, sl : [a,b] -~ M be piecewise smooth loops. We say t h a t so is (piecewise smoothly) homotopic to sl, and write so ~ sl, if there is a continuous m a p H : [a, b] x [0, 1] ~ M a n d a p a r t i t i o n a = to < tl < . . . < tr = b such t h a t (1) Hl([ti_l,t~ ] x [0, 1]) is smooth, 1 < i < r; (2) H ( t , O ) = so(t) a n d H(t, 1) = sl(t), a < t < b; (3) H ( a , T ) = H(b, 7-), 0 < T < 1. As usual, (piecewise smooth) homotopy is an equivalence relation. E x e r c i s e 6 . 3 . 1 4 . If w c= A I ( M ) is locally exact and if the loops Sl a n d s2 are piecewise s m o o t h and homotopic, show t h a t
1
2
Proceed as follows. (1) Let U C R 2 be open, let R = [a,b] x [c,d] C U, a n d let w E A I ( U ) be locally exact. For e > 0, let R~ = (a - e, b + c) x (c - e, d + c). Show t h a t there is e > 0 such t h a t R~ C_ U and co[R~ is exact. (2) Let ~ : M --* N be a smooth m a p between manifolds a n d let cv E A 1 (N) be locally exact. Prove t h a t ~* (w) E A 1 (M) is locally exact. (3) Use these two results to prove the proposition. C o r o l l a r y 6 . 3 . 1 5 . Every locally exact 1-form on R n is exact.
P r o @ Let s : [a, b] ~ R n be a piecewise smooth loop and define H : [a, b] • [0, 11 --~ ]~n by H ( t , T) = Ts(t). T h e n H is a homotopy of the constant loop 0 to the loop s. If cv is locally exact, Exercise 6.3.14 implies t h a t
Since the loop s is arbitrary, Theorem 6.3.10 implies t h a t w is a n exact form.
[]
6.4. FIRST COHOMOLOGY
195
Corollary 6.3.15 is a special case of one version of the Poincar~ Lemma, to be treated later.
6.4. The First Cohomology Space In Example 6.3.12, we saw that a locally exact form on a manifold can fail to be exact and that this seems to be related to the topology of the manifold. This insight is formalized and exploited by the de Rham cohomology H 1(M), a vector space associated to the manifold M which measures, in some sense, how much the notions of "locally exact" and "exact" differ on M. D e f i n i t i o n 6.4.1. The space of (de Rham) l-cocycles on M is
Z 1(M) = {co E A 1(M) lco is locally exact}. The space of (de Rham) 1-coboundaries is B I ( M ) = {co E AI(M) lw is exact}.
Remark that, if we regard A 1(M) as a vector space over JR, then Z I(M) and It is also clear that B I(M) C_ Z I(M).
B I ( M ) are vector subspaces. They are not C~176
D e f i n i t i o n 6.4.2. The vector space
HI(M) = ZI(M)/BI(M) is called the first (de Rham) cohomology space of the manifold M. If co is a 1eoeycle, its cohomology class is [co] = co + B 1(M) C H I(M). Although, whenever d i m M > 0, the vector spaces Z I ( M ) and B I ( M ) are infinite dimensional, it frequently happens that H I ( M ) is finite dimensional. We will see, for instance, that this is the case whenever M is compact. Cohomology is a contravariant functor from the category of differentiable manifolds (smooth maps are the morphisms) to the category of real vector spaces (and linear maps). Indeed, by Exercise 6.2.16 an arbitrary smooth map ~ : M --+ N induces a linear map g)* : Z I ( N ) --~ Z I ( M ) and g)*(BI(N)) C B I ( M ) , so g)* passes to a well-defined linear map (of the same name) ~* : H 1(N) --+ H 1(M). It is trivial to check that (~0 o r
= ~b* o ~* and (idM)* = idH*(M).
Proposition
6 . 4 . 3 . Let co, ga C 2 l ( a ) . Then [co] z [~] C H I ( a ) /f and only if f~ co = f~ ~ as s varies over all pieeewise smooth loops in M. These numbers are called the periods of co and of the cohomology class [co].
Proof. The locally exact forms co, ~ E Z 1(M) have the same periods f~ co = fs ~, for every piecewise smooth loop s, if and only if fs (co - ~) = 0 for all such loops. By Theorem 6.3.10, this holds precisely when co - ~ belongs to B I(M). Equivalently, [co] = p]. [] E x a m p l e 6.4.4. Consider the sphere S n, n >_ 2. By stereographic projection, we know that the complement of a point in S n is diffeomorphic to IRn, so every piecewise smooth loop in S n that misses a point is homotopic to a constant loop. By Sard's theorem, no piecewise smooth curve in S ~ can be space-filling if n > 2 (see
196
6. COVECTORS AND 1-FORMS
E x a m p l e 2.9.4), so all piecewise smooth loops a on this sphere are homotopically trivial. By Exercise 6.3.14, jfa aJ = 0 , for all locally exact 1-forms ~o. By Proposition 6.4.3, we conclude t h a t
H I ( S n) = 0, Proposition
whenever n _> 2.
6.4.5. If fo, f l : M ~ N are smooth and homotopic, then
f~ = f~ : H I ( N ) --+ H I ( M ) . Proof. Let [aJ] 9 H I ( N ) . If s : [a,b] --> M is a piecewise s m o o t h loop, t h e n si = f i o s : [a,b] ~ N is also a piecewise s m o o t h loop, i = 0, 1. Let H : M x N --+ M be a h o m o t o p y of f0 to fa. Then, the composition
[a, b] x [0, 1]
sxid> M x If{ ~
N
is a h o m o t o p y of so to sl, so
~ f~(o:)= f bs*f3(o~) b f
(fo o S)*(w)
'1W (Exercise 6.3.14)
Since s is an arbitrary piecewise smooth loop, Proposition 6.4.3 implies t h a t
f;[w] = [f;(w)] = [f~(w)] = f~[w]. Since [co] C H 1 (M) is arbitrary, f~ = f~ at the cohomology level, as desired.
[]
D e f i n i t i o n 6.4.6. A smooth m a p f : M + N is a homotopy equivalence if there exists a s m o o t h m a p g : N --+ M such t h a t f o g ~ i d g and g o f ~ idM. C o r o l l a r y 6.4.7. A homotopy equivalence f : M --~ N induces a linear isomorphism f * : g 1 (N) --+ H 1 (M).
Proof. Since f o g ~ i d g , it follows by the (contravariant) functoriality of cohomology a n d Proposition 6.4.5 t h a t g* o f* = ( f o g ) * = id~v = i d H l ( y ) . Similarly, f* o g* = idHl(M), so f* and g* are m u t u a l l y inverse isomorphisms on cohomology. [] E x a m p l e 6 . 4 . 8 . Let f : {0} ~ D ~ be the inclusion. Let g : D ~ + {0} be the only map. These maps are smooth a n d g o f = id{0}. Consider the m a p f o g : D n ~ D n having image {0}. We claim t h a t this is homotopic to idD,,.
6.4. FIRST
197
COHOMOLOGY
Indeed, let ~) : IR --+ [0,1] be s m o o t h such t h a t ~(0) = 0 and qo(1) = 1. Define H : D ~ x IR + D ~ by
H(x, t) = ~(t)x. Then
H(x,O)=O=f(g(x)),
gxeD
~,
H(x, 1 ) = x = i d D , ~ ( x ) ,
V x E D n.
and T h i s establishes the desired h o m o t o p y and completes the proof t h a t f is a h o m o t o p y equivalence, so f* : H I ( D n) -+ H i ( { 0 } ) = 0 is an isomorphism. T h a t is, H 1(D '~) = 0 or, equivalently, every locally exact 1-form on D n is exact. A similar proof shows t h a t IRn is homotopically equivalent to a point, and we recover Corollary 6.3.15. Example
6.4.9. Let i : S n - 1 c__+ R n \
{0}
be t h e inclusion. Let g :Rn
\
{ 0 } --+ S n - 1
be the m a p defined by ?2
These m a p s are smooth, and g o i = idsn-1 . We claim t h a t i o g ~ id~.,..{o}, hence t h a t i is a h o m o t o p y equivalence. Indeed, define H : (R n \ {0}) • [0, 1] -~ ~ n \ {0) by the formula
H(v, t) -
V
t + (1 - t ) l l ~ l [
This is s m o o t h since II~ll > 0 implies t h a t t + (1 - t)>ll > 0, 0 < t < 1. Then H ( ~ , 1) = v,
W e ~
\ {0},
and
It follows t h a t H I ( R ~ \ {0}) = Hl(Sr~-l). In particular, t o g e t h e r With E x a m p l e 6.4.4, this proves t h a t H 1(N ~ \ {0}) is trivial, whenever n _> 3. Proposition
6.4.10.
There is a canonical isomorphism H I ( S 1)
11{.
198
6. COVECTORS AND 1-FORMS
We will prove this via three lemmas. Recall the universal covering m a p
p : ]~ --* S 1, p(t) = (cos 2~t, sin 27a). This is the m a p t h a t induces the s t a n d a r d diffeomorphism R / Z = S 1. Define O~
: Z l ( S 1 ) ----+]1~
by
a(~)
=
a linear map. Lemma
6.4.11.
So
p*(~),
The linear map a passes to a well-defined linear map a : H l ( S 1) --+ R.
Proof. Indeed, a = p[[O, 1] is a smooth loop and w C B I ( S 1) implies t h a t
a(~) = / ~ = 0, by T h e o r e m 6.3.10. Lemma
6.4.12.
[]
The linear map a : H I ( S 1) --* 1~ is injective.
Proof. Let co E ZI(S 1) be such that a(w) = 0. We must prove t h a t co E BI(S1). For n E Z, let rn : IR --~ R be the t r a n s l a t i o n rn(t) = t + n . Then, porn = p, so r* op* = p * : AI(S 1) ---+AI(]R). This a n d the change of variable formula for the integral gives
f t + n p * ( ~ ) = fot ~ ; ( p * ( ~ ) ) = ~ot p*(~),
VncZ,
o
dn
In particular, since a(w) = O, we o b t a i n fn+l
an Define fw E C~176
p* (~) = o = f n n v* W). +1
by fw (cos 27rt, sin 27rt) =
p* (w).
If this is well defined, it will be smooth. It will be well defined precisely if
f t-bn p*(~)= ~0 t p*(~),
VneZ,
vteR.
J0
If n = 0, this is obvious. If n > 0,
t+n JO
Jn
f+n =
n-1
i+1
,,'<+El ,,'< i=0 Ji ( ) p* W
an
= f~ p*(~). Jo
199
6.4. FIRST COHOMOLOGY
A similar c o m p u t a t i o n for the case n < 0 is left to the reader. For the lift f~ = we get
p*(f~o) C C~176
et
L(t) = / o v*(co), so the f u n d a m e n t a l theorem of calculus and Exercise 6.2.16 give
p*(w) = d L = d(p*(f~)) = p*(df~o). But p : [R --+ S 1 is a local diffeomorphism, so co and df~, are equal locally, hence globally. T h a t is, co E B I ( S 1) as desired. [] Recall the locally exact form
--y
X
rl - x 2 + y2 dx + ~
dy,
of E x a m p l e 6.3.12 and let
f i : i*(~) e z~(sl), where i is the inclusion m a p of S 1 into ]R2 \ {(0,0)}. s = i o a is as in E x a m p l e 6.3.12, and we showed t h a t
For the loop (7 = pl[0, 1],
~r*(~) = cr*(i*(r])) = (i o a)*(,?) = s*(r/) = 2rrdt. Lemma
6.4.13.
The linear map c~ : H I ( S 1) --, IR is surjective.
Proof. It is enough to show t h a t a is nontrivial. But [~ E H ~ ( S 1) and
~[~ = f01 p*(~) = f01 ~*(~ = 2~-. [] T h e p r o o f of Proposition 6.4.10 is complete. Corollary
6.4.14.
HI(IR 2 \ { 0 } ) = IR.
E x a m p l e 6 . 4 . 1 5 . Recall t h a t the Brouwer fixed point t h e o r e m for a r b i t r a r y s m o o t h (in fact, continuous) m a p s f : D 2 --~ D 2 follows from the nonexistence of a s m o o t h r e t r a c t i o n p : D 2 -~ 0 D 2 = S 1. The proof we gave using the f u n d a m e n t a l group can be mimicked using cohomology instead. Recall t h a t , for p to be a retraction, it is required t h a t the diagram S1
L
D2 P S1
c o m m u t e , where ~ is the inclusion of the b o u n d a r y circle. By the functoriality of cohomology, this produces a c o m m u t a t i v e diagram
HI(S 1) .
b*
H~(D 2)
H I ( S 1)
200
6. COVECTORS
AND
1-FORMS
T h a t is, R,
0
R
commutes, which is absurd. Our c o m p u t a t i o n of H l ( S 1) generalizes to the higher dimensional tori. E x e r c i s e 6 . 4 . 1 6 . Let exp : ]Rn --* T n be the homomorphism of abelian Lie groups defined by
oxp/x If a : [a, b] --~ R ~ is piecewise smooth such t h a t g~ = a(b) - a(a) C 7/~n, t h e n exp oa is a piecewise s m o o t h loop on T ~. Using this observation, prove t h a t H I ( T ~) = R ~, proceeding as follows. (1) Prove t h a t every piecewise smooth loop on T ~ is of the form exp oa as above a n d t h a t the homotopy class [exp oa] is completely d e t e r m i n e d by G (2) I f w E ZI(T~), define ~ : Z n --* ~ as follows. Given g E Z ~, choose a piecewise s m o o t h p a t h a : [a, b] --~ ~ such that g = G and set
~(~)
= fd e x p o a
~.
Show t h a t this is well defined and t h a t ~ linear functional of the same n a m e
(3) Prove t h a t the assignment w H ~
: Z ~ + ll~ extends uniquely to a
passes to a well-defined linear injection
~ : H I ( T n) ~ (Rn)* = ]~n. (4) Show t h a t there are forms 0 z , . . . ,0 n e ZI(T n) such t h a t exp*(0 i ) = d x i, 1 < i < n. Use this to show t h a t
: H I ( T ~) ~ ]~n is also surjective, hence is the canonical isomorphism we seek. E x a m p l e 6 . 4 . 1 7 . If n >_ 2, H 1 (S n) = 0 a n d H 1(T n) = ]R~, proving t h a t S ~ a n d T n are not homotopically equivalent. Of course, S 1 = T 1. E x e r c i s e 6 . 4 . 1 8 . We will say t h a t a locally exact form w E Z I ( M ) is integral if all of its periods are integers. For example, ~ / 2 ~ C Z I ( S 1) is integral. By P r o p o s i t i o n 6.4.3, ~ is integral if a n d only if every a / E [w] is integral, in which case we say t h a t [~] is an integral cohomology class. We denote by H i ( M ; Z) C H~(M) the subset of integral cohomology classes. (1) Prove t h a t H~(M; Z) is a subgroup of the additive group of the vector space H i ( M ) . We call H I ( M ; Z ) the integral cohomology of M.
6.4. F I R S T C O H O M O L O G Y
201
F i g u r e 6.4.1. The pair of pants P
(2) If f : M --~ N is smooth and w E Z I ( N ) is integral, prove that f*(w) C Z I ( M ) is also integral. Using this, show that integral cohomology is a contravariant functor from the category ~ of smooth manifolds and smooth maps to the category 9 of abelian groups and group homomorphisms. (3) Referring to Exercise 6.4.16, prove that H 1(Tn; E) is canonically the integer lattice ~ n C ]~n : H I ( T n ) .
(4) To each smooth map f : T ~ ~ T ~, show how to assign canonically an n • n matrix A / o f integers, depending only on the homotopy class of f , such that Afog = A g A / (matrix multiplication). If f is a diffeomorphism of T n onto itself, prove that A / is unimodular (i.e., has determinant • (5) Prove t h a t every n • n unimodular matrix of integers occurs as the matrix A f assigned to some diffeomorphism f : T n ~ T n. E x e r c i s e 6.4.19. If the vector space H i ( M ) has finite dimension k, prove that there is a set of piecewise smooth loops { a l , . . . , ak} on M such that the map
H i ( M ) ~ ll~k defined by
is an isomorphism of vector spaces. In light of Exercise 6.4.19, one might expect to generalize part (3) of Exercise 6.4.18 to all manifolds with finite dimensional first cohomology. T h a t is, one asks whether the loops in Exercise 6.4.19 can be chosen so as to carry H i ( M ; Z) isomorphically onto Z k. In fact, this can be done if M is compact, but we sketch an example t h a t shows what can go wrong in general.
202
6. C O V E C T O R S
AND
I-FORMS
E x a m p l e 6.4.20. Let P denote the 2-manifold with boundary obtained by removing two small, disjoint, open disks from the interior of D 2. The boundaries cl and c~ of these disks should be disjoint, each from the other and from co = OD2. The resulting surface, called by topologists a "pair of pants", is pictured in Figure 6.4.1, together with a dotted loop a that is homotopic to the outer boundary circle. In Figure 6.4.2, we cross this manifold with a closed interval and identify opposite ends with a twist through ~ radians. The result is a solid torus with a "wormhole" drilled out that winds around twice longitudinally. Denote this 3-manifold by V0. Note that this manifold is a kind of bundle with fibers diffeomorphic to P. A meridian on the outer boundary corresponds to the boundary curve co of P in Figure 6.4.1. By Exercise 6.3.14, the integral around Co of any locally exact form w is equal to the integral of w around the loop a in Figure 6.4.1 and this, in turn, is equal to the sum of the integrals around cl and c~. In V0, the loops cl and c~ are homotopic along the boundary of the wormhole, so we obtain
o
1
We now glue another copy of V0 (longitudinally) into the wormhole, obtaining a manifold V1 containing a loop c2 such that ff~ c o = 2 ~ 0
co=4~ i
co. 2
Inductively, a manifold V~ is obtained by gluing a copy of V0 longitudinally into the wormhole of Vn-1 and V~ contains a new loop cn such that /coco=2nfc
co"
Proceeding ad infinitum, we obtain a limit manifold Vcr This is the complement in the solid torus of a very complicated compact subspace E called the solenoid. If one first imbeds the solid torus in S 3 in the standard unknotted fashion and then removes the solenoid, the noncompact manifold M = S 3 \ E that results can be shown to have first de Rham cohomology H 1(M) = N and the set of loops chosen in Exercise 6.4.19 can be taken to be the singleton {co}. In fact, one can show that all periods of any locally exact form co are sums of periods of co corresponding to loops ci, i >_ 0. If co is a locally exact l-form that is integral, we obtain integers
~ co=hi,
i>O,
i
and no = 2nl . . . . . 2ini . . . . . This can only happen if no = 0, in which case every ni = 0 and co has all periods 0. That is, the isomorphism in Exercise 6.4.19 identifies Hi(M; Z) = 0. 6.5. D e g r e e T h e o r y o n S 1. Recall from Example 1.7.33 that ~1(S 1, 1) = Z. This is a canonical isomorphism, produced by lifting a loop a based at 1 in S 1 to a path ~ in the universal cover ~ starting at 0. The endpoint of this path is the integer corresponding to [a]. By Exercise 5.1.37 and the fact that S 1 is a Lie group, the group 71-[S1, S 1] is canonically isomorphic to 7rl (S 1, 1). This isomorphism, denoted by deg : 7r[S 1, S 1] --+ Z,
6.5. D E G R E E T H E O R Y ON S 1.
203
F i g u r e 6.4.2. Forming the manifold Vo
is called the degree map and deg([f]) is also called the degree of any f E [f] and denoted by deg(f). We are going to give two equivalent definitions of this degree, one using cohomology and one in terms of regular values. The first remark is that, since H 1(S 1) = R, f induces a linear map f* :IR ~ I R depending only on the homotopy class of f . Thus, f* is just multiplication by a certain constant a / E 1R and a / depends only on the homotopy class of f . By part (4) of Exercise 6.4.18, a I is an integer, being the sole entry in the 1 x 1 integer matrix A f . We are going to give another way to see that a / E Z. By an application of Theorem 1.7.39, if f : S 1 --~ S 1 is smooth, we can lift the m a p f o p : R --+ S 1 to a smooth map 37: N --~ N. T h a t is, the diagram
R
?,R
S1
, S1 I
is commutative. Also, since the group of covering transformations consists of translations by integers, f" : IR --* 1R will be a lift of f o p if and only if ]" = y + k, for some integer k.
Proposition 6.5.1. If f : S 1 ~
S 1 is smooth, then
a / = 37(1) - 37(0) = deg(f) e •, where 37 is any lift of f o p.
204
6. COVECTORSAND 1-FORMS
Proof. R e m a r k t h a t p ( f ( 1 ) ) = f(p(1)) = f(p(O)) = p ( f ( 0 ) ) ,
SO ?(1)
- ?(0)
= m 9 Z
Thus, let ~ 9 Z I ( S 1) be as in the c o m m e n t a r y following the proof of L e m m a 6.4.12 and c o m p u t e 2~ra/= ai~[~
=
~1 ( f o p ) * ( ~
~01( p o f ) * ( ~
=
f0
1 f* (2~ dt)
: 2~(f(1)
fo ep * ( f * ( ~ )
= c~(a/[~) = (~(f*[~) =
- f(0))
27r
f01
:
2~m.
f* (dt)
=
~01 f * ( p * ( ~ )
27T
j l?(
t) dt
T h a t is, a I = m E Z. The fact t h a t this integer is d e g ( f ) as defined above is e l e m e n t a r y and left to the reader. []
Remark. In Section 3.9, we defined deg2(f) e Z2 for s m o o t h m a p s between manifolds (without b o u n d a r y ) of the same dimension. We will see t h a t , for s m o o t h m a p s of the circle to itself, deg2(f) is just the residue class modulo 2 of d e g ( f ) (Corollary 6.5.4). 6.5.2. If f : S 1 --* S 1 is smooth, if f is a lift of f o p, and if t E ]R is arbitrary, then d e g ( f ) = f ( t + 1) - ]'(t).
Corollary
Proof. View p : ~ --* S 1 C C as a group homomorphism. Then
p(f(t +
1) - f(t))
-
p(y(t
+ 1))
p(i(t))
f(p(t + 1)) f(p(t)) so f ( t + 1) - f ( t ) E Z, Vt E ~. This function of t, being continuous and integervalued, is constant on N, hence equal to f ( 1 ) - f(0) = d e g ( f ) . [] We turn to the description of d e g ( f ) in terms of regular values. Let z0 E S 1 be a regular value of f : S 1 --* S 1. Then f - l ( z o ) = { Z l , . . . ,Zr}. Here, if f - l ( z o ) = ~), we take r = 0. Recall t h a t deg2(f) = r (rood 2). Choose ~/ c N such t h a t P(ii) = zi, 1 < i < r. The s m o o t h m a p f : S 1 ~ S 1 preserves orientation at zi if f~(~i) > 0 and reverses orientation at zi if ~ ( ~ / ) < 0. Let ei = ? ( ~ ) / I ] ) ( ~ i ) l e { - 1 , 1} and r e m a r k t h a t this depends only on f and zi. Proposition
6.5.3.
r
With the above conventions, d e g ( f ) = ~ = 1 ~i.
Proof. Choose a 9 R \ { p - l { z l , . . . ,z,.}}. T h e n p-l(z~) C~(a,a + 1) is a singleton and we choose this point as our z-i, 1 _< i _< r. We will also use the fact t h a t f ( a + 1) = f(a) + d e g ( f ) . Let p-l(zo) = {b + k}kez and consider the graph of s • f ( t ) over the open interval (a, a + 1), together with the horizontal lines s = b + k, k 9 Z. Each time
6.5. DEGREE
Figure
THEORY
ON
S 1.
205
6 . 5 . 1 . G r a p h of f
t h e g r a p h crosses a line s = b + k, t h e p a r a m e t e r t is equal to one of t h e Ei a n d ei records w h e t h e r t h e g r a p h crosses this line while increasing (ei = 1) or d e c r e a s i n g (ei = - 1 ) . F i g u r e 6.5.1 i l l u s t r a t e s a case in w h i c h r = 7, q = e2 = ea = e4 = e7 = 1, a n d e5 = e6 = - 1 . T h e s u m of t h e Qs p e r t a i n i n g to a single line s = b + k is 1, - 1 , or 0, t h e n e t n u m b e r of d i r e c t e d crossings. Clearly, t h e s u m of all t h e s e n e t n u m b e r s is r
E e i = "f(a + 1)
- f(a) =
deg(f).
i=1
(In F i g u r e 6.5.1, t h e degree is 3.) Corollary Example
6.5.4.
deg2(f)
[]
= d e g ( f ) ( m o d 2).
6 . 5 . 5 . For each n C Z, define f n : S 1 ~ S 1, A(z) = ~n
Here, of course, we view S 1 C C.
5(t)
W e c a n choose t h e liR f ~ : IR -~ R to b e
= he, so
deg(A) =/.(1) - into) : If z 9 S 1 is a regular value of fn, then
f~l(z)
= { P l , . . . ,PinS},
n.
206
6. C O V E C T O R S A N D 1 - F O R M S
where P l , . . . ,Plnl are the distinct n t h roots of z. Of course, if n = 0, t h e n f0 is constant and f o l ( Z ) = 0. I f n > 0, all r = +1 and, i f n < 0, all r = - 1 . Thus, M
n=E(i i=1
in all cases. T h e o r e m 6.5.6. A smooth map f : S 1 --~ S 1 extends to a smooth map F : D 2 --+ S 1 if and only if d e g ( f ) = 0. Proof. First suppose t h a t the s m o o t h extension F exists. T h a t is, f = F o i where i : S 1 ~ D 2 is the inclusion. T h e n f* = i* o F* and F * : H I ( s 1) -+ H I ( D 2) = 0, implying t h a t f* = 0. Therefore, d e g ( f ) = 0. For the converse, suppose t h a t d e g ( f ) = 0. Since the degree is a complete invariant for homotopy, it follows t h a t f ~ f0 - 1. By the C ~ U r y s o h n trick, choose t h e h o m o t o p y H:S 1 x[0,1]~S 1 so t h a t H(z, 1)= f(z), H ( z , t ) =- 1,
V z C S 1, 0
1/2.
T h e n H induces a s m o o t h m a p of the disk to the circle as follows. Define a s m o o t h surjection ( ; : S 1 x[O, 1]---*D2 C C
by w(z, t) = tz. T h e n W carries S 1 • (0, 1] diffeomorphically onto D 2 \ {0}. Define F :D2 ~ S1
by F ( ~ ( z , t)) = H(z, t). This is well defined. It is s m o o t h o n D 2 \ { 0 } and, on {w C D 2 I Iwl - 89 it is constant, so F is smooth. Evidently, F ( z ) = H ( z , 1) = f ( z ) , V z C S 1, so F extends
f.
[]
Recall t h a t the rood 2 degree allowed us to prove the f u n d a m e n t a l t h e o r e m of algebra for polynomials of odd degree ( T h e o r e m 3.9.14). T h e integer degree makes it possible to carry out essentially the same a r g u m e n t for all positive degrees. T h e o r e m 6 . 5 . 7 (Fundamental T h e o r e m of Algebra). Let f : C --~ C be a polynomial of degree n >_ 1. Then there is zo E C such that f ( z o ) = O. Proof. We can assume t h a t the leading coefficient is 1 and write f ( z ) = z n + a l z n-1 + ... + a n - l z + an. Suppose this has no root. For each positive real n u m b e r r, define a s m o o t h function F~ : D e --~ S 1
6.5. D E G R E E T H E O R Y ON S 1.
207"
by the formula f(rz) F~.(z) = i f ( r z ) I .
This is where we use the hypothesis t h a t f has no roots. Let g~ = F~I $1. T h e n set HT(Z,t) = (rz) ~ + t ( a l ( r z ) ~ - l
+ " '' + a~)
and note that, if r is large enough, this vanishes nowhere on S 1 x [0, 1]. Indeed, ~(rz)
-
l + t
~
+ ...+
approaches 1 as r --+ oc, uniformly on S 1 x [0, 1]. Thus, fix a large enough value of r and define H : S 1 x [0, 1] --~ S 1 by the formula
~(z,t) H(z,t)-
i~rr(z,t)l.
T h e n H ( z , 1) = gT(z) and H ( z , O) = z n, V z C S 1. Thus, gr ~ f ~ and deg(gT) = n > 0. But g,. extends smoothly to F~ : D 2 --~ S 1, a contradiction to Theorem 6.5.6. [] L e m m a 6.5.8. If f, g : S 1 --~ :{1 are smooth, then
deg(f
o
9) = d e g ( f ) deg(g ).
Indeed, functoriality of cohomology implies t h a t aio 9 = afag , so the lemma is immediate. Corollary
6.5.9. I f f , 9 : S 1 --* S 1 are smooth, then f o 9 and g o f are homotopic.
By Exercise 5.1.37, the Lie group structure on S 1 makes ~r[M, S 1] into an abelian group and one obtains the following. Theorem
6.5.10.
The m a p x: ~[M,S
1] ~
Hi(M; Z),
defined by x[f] = f*[g/27r], is an i s o m o r p h i s m of groups.
Here, the integral cohomology H i ( M ; Z) is defined as in Exercise 6.4.18. The fact t h a t [~/27r] is an integral class implies t h a t f*[~/27r] is also integral by t h a t same exercise. E x e r c i s e 6 . 5 . 1 1 . Prove Theorem 6.5.10. Proceed as follows. (1) Show t h a t X is a group homomorphism. (2) Let aJ C Z I ( M ) be an integral form. You are going to define a s m o o t h m a p f~ : M --* S 1 such t h a t f~(~/2~r) = aJ. For this, no generality will be lost in assuming t h a t M is connected (why?), so make t h a t assumption and fix a basepoint x0 C M . For each x E M , choose any piecewise s m o o t h p a t h s : [a, b] ~ M such t h a t s(a) = xo and s(b) = x, and show t h a t f~o(x)= ( / w
(modE))
elR/Z=S
1
208
6. C O V E C T O R S
AND 1-FORMS
depends only on x (and xo), not on the choice of p a t h s. (3) Prove t h a t fw : M --+ S 1 is s m o o t h and t h a t its h o m o t o p y class is independent of the choice of basepoint x0. (4) Prove t h a t f~(~/2rr) = co. In particular, conclude t h a t ;g is surjective. (5) Let f : M --+ S 1 be such t h a t co = f* (~/2rr) is an exact form. You are to prove t h a t f ~ 1, so note that, again, no generality is lost in assuming t h a t M is connected. In this case, show t h a t fo0, as defined in step (b), is actually well defined as a m a p f~ : M --+ IR and t h a t there is a constant e such t h a t f = p o (f~ + c). Conclude t h a t f ~ 1, hence t h a t X is one-to-one. E x e r c i s e 6.5.12. Use degree theory to show t h a t the group Diff(S 1) has exactly two isotopy classes. (Hint. An easy application of degree theory will show t h a t there are at least two isotopy classes. The hard step is to show t h a t , if f E Diff(S 1) and d e g ( f ) = 1, then f is isotopic to fl. It then follows fairly easily t h a t there are at m o s t two isotopy classes.)
CHAPTER 7
Multilinear Algebra and Tensors Smooth functions, vector fields and 1-forms are tensors of fairly simple types. In order to handle higher order tensors, we will need some rather sophisticated multilinear algebra. The reader who is well grounded in the multilinear algebra of R-modules can skip ahead to Section 7.4, referring to the first three sections only as needed. 7.1. T e n s o r A l g e b r a We will be working in the category ?V[(R) of R-modules and R-linear maps, where R is a fixed commutative ring with unity 1. In order to study R-multilinear maps, we build a universal model of multilinear objects called the tensor algebra over R. In the typical applications in this book, R will be either the real field N or the ring C a (M). D e f i n i t i o n 7.1.1. An R-module V is free if there is a subset B C V such that every nonzero element v E V can be written mfiquely as a finite R-linear combination of elements of B (terms with coefficient 0 being suppressed). The set B will be called a (free) basis of V. If R is a field, every R module is free. Another example is the integer lattice Z k, a free Z-module. At the other extreme, the abelian group Z2, when viewed as a Z-module, is not free. A basis would }lave to contain 1 C Z2, but 0 C Z2 would then have infinitely many representations a. 1, a E 2Z. The following example will be very important. E x a m p l e 7.1.2. Let 7r : E --- M be an n-plane bundle. Then F(E) is a free Coo(M)-module on a basis of n elements if E is trivial. Indeed, if E ~ M • N n, let { e l , . . . , en} be the standard basis of ]Rn, and define si C F(E) by the formula si(x) = (x, ei), 1 < i < n. An arbitrary section s(x) = (x, f l ( x ) , . . . , f n ( x ) ) has the n unique expression s = ~ i = 1 fisi. E x e r c i s e 7.1.3. Suppose that rr : E --+ M is an n-plane bundle and that F(E) is a free C ~ (M)-module with basis B. One easily checks that B must contain at least n elements. Using local triviality and the Coo Urysohn lemma, show that B has exactly n elements 81,... , 8n and that 81(X),..., 8n(X) form a basis of E , , for each x E M. Thus, E must be trivial.
Remarks. There are strong but limited analogies between vector spaces over a field and free R-modules. Here are some of the facts. (1) If V is free on the basis B, then R-linear maps g) : V --~ W into arbitrary R-modules W correspond one-to-one to maps ~ : B + W of sets, the correspondence being ~ = ~IB.
210
7. M U L T I L I N E A R A L G E B R A
(2) If V is a free R-module, it can be shown that any two bases of V have the same cardinality, called direr V. For example, dimz Z k = k. (3) On the other hand, there are important dissimilarities. A submodule W C V of a free R-module can fail to be free and, even when the submodule W is free, it may have no basis that extends to a basis of V. We give two examples illustrating Remark (3). E x a m p l e 7.1.4. If M C IRn is a nonparallelizable submanifold, then we have the canonical inclusion T ( M ) ~-~ M x ~ n of the tangent bundle as a subbundle of the trivial bundle. Thus, •(M) C F ( M x ~ n ) is a C ~ ( M ) - s u b m o d u l e . By Exercise 7.1.3, X(M) is not free, but by Example 7.1.2, F ( M x ~ n ) is free. E x a m p l e 7.1.5. The submodule 2Z C Z is a free Z-submodule of a free Z-module. But there are two bases {2} and { - 2 } of 2Z, neither of which extends to a basis of Z. Modules will not be assumed free unless that is explicitly stated. D e f i n i t i o n 7.1.6. If V1,1/2, 1/3 are objects in ?d(R), a map ~ : Vi x V2 -~ V3
is R-bilinear if
~(-, v2): 1/1 -~ v3 ~(Vl, .): v2 -~ V3 are R-linear, Yvi E V/, i = 1,2.
Remark. For fixed choices of V1,1/2, 1/3 E :SI(R), the set of R-bilinear maps q| : 1/1 • 1/2 ~ 1/3 is itself an R-module under the pointwise operations. D e f i n i t i o n 7.1.7. If V1 and 1/2 are R-modules, their tensor product is an R - m o d ule V1 | V2, together with an R-bilinear map | : Vix
V2 -.-> V1 |
2
with the following "universal property": given any R-module 1/3 and any R-bilinear map p : V1 x 1/2 --* 1/3, there is a unique R-linear map ~ such that the diagram | VIXV 2 , Yl| 2
Va commutes. We write |
w) = v | w.
Thus, the R-module of R-bilinear maps V1 x V2 ~ Va is canonically isomorphic to the R-module HomR(V1 | V2, 1/3) of R-linear maps Vi | V2 --~ V3,
T h e o r e m 7.1.8. Given V1,V2 a$ above, a tensor product Yl | V2 exists and is
unique up to a unique isomorphism. That is, if |
iV1 x V2 .~ Y l |
6 : Vl x V2 -~ Vl 6 V2
7.1. T E N S O R A L G E B R A
211
are two such tensor products, there is a unique isomorphism O : Vl O 72 ~ 7 1 ~ 72 of R-modules such that the diagram
|
VIXV2
,
71072
Ul g W2
commutes.
Pro@ First we prove uniqueness. If 0 and O are two tensor products, the universal property gives unique R-linear maps 01 and 02 making the following diagrams commute: | 71x72 v10v2 ,
VI @V2 71x 7~
, 71~ 72
710V2
Then the diagram 0
Vl x g2
. Vl O V2
~
[02o01 V1072
also commutes, as does | V~ x 72
tid
. 71o72
71072 By the universal property, we conclude that 02 o01 = id and, similarly, that 01002 = id, so 01 and 02 are mutually inverse R-linear isomorphisms. Since 01 is unique, we are done. The existence proof, though elementary, is a bit more long winded. Let W be the free R-module spanned by the set V1 x 72. The module W is just the set of all formal linear combinations k
E ai(vi, wi) i=1
212
7. M U L T I L I N E A R A L G E B R A
where ai E R and (vi,wi) E Vi x 8 9 This is an R-module under the obvious operations and each element 0 r w C W is uniquely expressed as an R-linear combination of finitely many members of the basis 171 x 1/2. Any linear combination with all coefficients 0 is equal to the 0 E W. Let [R C_ W be the submodule spanned by all elements of the form
(av + bu, w) - a(v, w) - b(u, w), (v, aw + bu) - a(v, w) - b(v, u) where a, b E R and u, v, w are in 171 or 89 appropriately. We think of :R as the submodule of bilinear relations and set
Vl | 89 = w/:~. The cosets of the elements of the basis 1/1 x 89 will be denoted by
(v,w)+ J~=vQw, and we define |
: 1/1 x 89 --, V1 | 1 8 9
by
|
= v | w.
Bilinearity follows immediately from tile definition of [R. For example,
(av + bu) | w = (av + bu, w) + R = a(v, w) + b(u, w) + :R = a(v | ~ ) + b(,, | ~). Note that, as a special case of bilinearity, (av) @ w = a(v | w) = v | (aw) and, in particular, v @ 0 = 0 = 0 @ v. We establish the universal property. Let ~o : V1 x 89 -+ V3 be an R-bilinear map. Since V1 x 89 is a free basis of W, there is a unique R-linear map
7:W~V3 such that -~(v,w) = ~ ( v , w ) , V(v,w) E 171 x 172. Since qDis bilinear, it follows that vanishes on the generators of :R, hence that ~IIR = 0. Consequently, ~ passes to a well-defined R-linear map
? : W/:R= VI | 89 --, 89 such that the diagram |
V~ x V2
, w~|189
va commutes. Since 1/1 | 89 is spanned by elements of the form v @ w, ~ is unique.
[]
7.1. TENSOR ALGEBRA
213
In a completely parallel way, one can consider R-trilinear m a p s and prove t h e existence and uniqueness of a universal R-trilinear m a p
Vlxv2xvs2s174174 sending (Vl, ~32, V3) H Vl @ V2 @ V3.
It is a trivial exercise to check t h a t tile composition
(Vl X 72) X V3 |
(V1 @ V2) x V.3 ~
(V1 @ V2) @V3
also has the universal property, as does V1 x (V,2 • V3) idvl •174 V1 • (V2 @V3) ~
V1 @ (V2 @ V3).
7.1.9. If Vi is an R-module, i = 1, 2, 3, there are unique R-linear iso(V1 @V2)@V3----VI@V2@V,3 identifyingvl|174 (V 1 | l)2) | V3 = V1 | V2 | V3, V V i E Vi, i = 1, 2, 3.
Corollary
morphisms V1 | 1 7 4
More generally, for each integer k >_ 2, there is a unique universal, k-linear m a p (over R)
vlXv2x...vk ~--~v~ov20...evk and canonical identifications "Vl e ( V2 e . . . e Vk ) = ( E l |
. . . e Vk_ l ) |
v k = v1 |
v2 |
...|
tk .
A n obvious induction shows t h a t all groupings by parentheses are equivalent, so parentheses can be d r o p p e d or used selectively as desired. D e f i n i t i o n 7.1.10. An element v c !/1 | ... | Vk is decomposable if it can be w r i t t e n as a m o n o m i a l v = vl | 999 | vk, for suitable elements vi E V~, 1 < i < k. Otherwise, v is indecomposable. By the construction of the tensor p r o d u c t in the proof of T h e o r e m 7.1.8, the decomposable elements span. 7.1.11. If V and W are flee R-modules with respective bases A and B, then V | W is free with basis C = { a | b l a E A, b c B }.
Lemma
Proof. An a r b i t r a r y element v E A | B can be written as a linear c o m b i n a t i o n of decomposables. A decomposable element v | w can be expanded, via the multilinearity of tensor product, to a linear combination of elements of C, proving t h a t C spans V | W. It remains for us to show that, if P,q
E
P,q
cijai | by = E
i,j=l
dijai | bj,
i,j=l
where ai C A and bj E B, 1 < i < p, 1 <_j < q t h e n all cij = dij. S u b t r a c t i n g one expression fronl tile other, we only need to prove t h a t P,q
(*)
E
cijai
@
bj
=
0
i,j=l
implies t h a t all cij = O. T h e bilinear functionals p : V x W --~ R correspond one-to-one to arbitrary flmctions f : A x B -~ R. T h e correspondence is p ~ ~I(A x B). Thus, the linear functionals ~ : V | W ~ R also correspond one-to-one to these functions
214
7. MULTILINEAR
ALGEBRA
f : A x B -~ R. If (a,b) C A x B, let fa,b : A x B -+ R be the function taking the value 1 on (a, b) and the value 0 on every other dement of A x B. The corresponding linear functional will be denoted by ~a,b. Applying ~a~,bj to equation (,), we see t h a t all cij = 0 as desired. []
By an obvious induction on the number of factors, this lemma generalizes to the following. C o r o l l a r y 7.1.12. I f V 1 , . . . , Irk are free R-modules having respective bases B 1 , . . . , Bk, then V1 | | Vk is a free R-module with basis B={Vl@...@vklviEBi,
l
P r o p o s i t i o n 7.1.13. If )~i : V~ --, Wi is an R-linear map, 1 < i < k, there is a unique R-linear map A~ | " " | ;~k : VI | " " | Vk --* WI |
. | Wk
that, on decomposable elements, has the formula (~1 |
|
~k)(vl
|
|
vk) = ~(vl)
|
9|
~k(vk).
Proof. Since the decomposables span, uniqueness is immediate. For existence, define the multilinear map ~ : V l X . . . x V k --, W l |
. . . e Wk
by ~(vl,... ,vk) = ~(v~) |
|
Then ~ | 9.. | ~k is defined to be the unique associated linear map.
[]
D e f i n i t i o n 7.1.14. The dual V* of an R-module V is HomR(V, R), the module of R-linear functionals. L e m m a 7.1.15. If V has a finite free basis { v l , . . . , v~}, then V* has a finite free basis { v ~ , . . . , v ~ } , called the dual basis and defined by v*(vj)=5}, Proposition
l<_i,j
7.1.16. There is a unique R-linear map ~:V?|
-* (V~ |
9| Vk)*
that, on decomposable elements, has the formula
~(~1 |
| ak)(Vl |
| vk) = v , ( v l ) ~ ( v ~ ) . . ,
ak(v~).
If the R-modules Vi are all free on finite bases, then ~ is a canonical isomorphism. Proof. Uniqueness is immediate by the fact that decomposables span. For existence, define the multilinear functional O : V~ x . . . x V~ x Vx x . . . x Vk ~
R
by 0(~1, 9 9 9 ~k, v ~ , . . . , vk) = ~1(Vl)~2 ( v ~ ) . . . ~k (vk).
By the universal property, this gives the associated linear functional ~ : V? e . . . e V~ |
V~ e . . . |
Vk ~
and we define
~: v,* | 1 7 4
--,(v~e...evk)*
R,
7.1.
TENSOR
ALGEBRA
215
by @ ) ( v ) = ~(~ e v). If {vi,1t 9 9 .,vi,m~} is a free basis of V/, 1 < i < k, let {V *i,1,'. . , "U*i,m,} be the dual basis. Let B and B* be the respective bases of V1 | 9 | Vk and VI* | 99@ Vk* given by Corollary 7.1.12. The formula 4vii,
|
| v~,j~ ) @1,il |
| vk,i~ ) = 54',1 9 9 9d ~ = ~jI~1 ~~
shows that t carries the basis B* one-to-one onto the basis dual to B, so t is an isomorphism. [] Let V be an R-module and view R as a module over itself. L e m m a 7.1.17. Scalar multiplication RxV~V, VxR~V induces canonical i s o m o r p h i s m s R @ V -- V | R = V relative to which 1 @ v =
v| Indeed, scalar multiplication is R-bilinear, so there are canonical R-linear maps R@V,--* V V@R~V.
These are inverted by the R-linear maps v~-~ l | v~-~v|
respectively. D e f i n i t i o n 7.1.18. Let V be an R-module. For each integer r >_ 0, the r t h tensor power of V is
7~-(v) =
v,
|174
r = 1,
~->2.
R e m a r k . By Lemma 7.1.17, ~Y~ | 7~(V) = Y~(V) = 7 n ( v ) @ 70(V). When n and m are both positive, the identity ~ ( V ) @ ~Ym(V) = 9-~+'~(V) is given by the associativity of the tensor product.
Set 9"(V) = {T'(V)}~= o and note that | defines an R-bilinear map
9 ~(v) x ~ ( v ) This makes IT(V) into a
~ ~(v) |
= ~+~(v).
graded algebra over R in the following sense.
D e f i n i t i o n 7.1.19. A graded (associative) algebra A over R is a sequence {Ar}~=0 of R-modules, together with R-bilinear maps (multiplication) An x A ~ Z ~ A ~ + ' ~ , V n , m Z O , (written (a, b) ~-+ a . b or, sometimes, (a, b) ~-+ ab) that is associative in the sense that the compositions ( A ~ x A m ) x A k .xid An+,~ x A k -& A ~+'~+k
216
7. MULTILINEAR ALGEBRA A n x (Am x A k) id x . A n x A 'n+k ~ A n+'~+k
are equal, Vn, m , k _> 0. D e f i n i t i o n 7.1.20. The graded algebra A is connected if A ~ = R and A ~ x A m - 4 A m .2- A m x A ~ are equal to scalar multiplication, V m _> 0. R e m a r k that a connected graded algebra has unity 1 r R = A ~ D e f i n i t i o n 7.1.21. If V is an R-module, then ~T(V), with multiplication •, called the tensor algebra of V.
is
It is clear t h a t the tensor algebra ~Y(V) is connected. D e f i n i t i o n 7.1.22. A homomorphism ~ : A --* B of graded R - a l g e b r a s is a collection of R-linear maps 9~n : A n ~ B n, V n >_ 0, such t h a t the diagrams A n x A "~
~ A n+m
,~x~m I B n X B m
~,~+" ~ B n+m
commute, V n , m > O. The homomorphism ~ is an isomorphism if 9 n is bijective, Vn>0. T h e o r e m 7.1.23. [f )~ : V ---* W is an R-linear map, then there is a unique induced h o m o m o r p h i s m ~Y()~) : ~Y(V) --* ~Y(W) of graded R-algebras such that T~ = idR and 9q ( ~ ) = ;~. This homomorphism satisfies ~ ( ~ ) ( ~ , | ~2 |
| ~n) = ~(vl) | ~ ( ~ ) |
| ~(~n),
V n >_ 2, Vvi E V , 1 < i < n. Finally, this induced homomorphism makes 9~ a covariant functor from the category of R-modules and R-linear maps to the category of graded algebras over R and graded algebra homomorphisms. Proof. T h e formula on decomposable tensors is imposed by the requirement t h a t 9"(,k) be a homomorphism of graded algebras, together with the stipulation t h a t 9"1 (,k) = ,k. Existence and uniqueness of the linear maps 7n(,~) are given by Proposition 7.1.13. T h e fact that 9"(~) preserves | multiplication is immediate. T h e final assertion a m o u n t s to the obvious identities
~'(~, o #) = ~(~) o g'(~), 9 (idv) = id~(v) 9 [] T h e following is an elementary consequence of Corollary 7.1.12 a n d Proposition 7.1.16. Theorem
7.1.24. If V is a free R-module with basis { e l , . . . , era} then {eil @ ' ' ' @ e i k } l < _ i 1 ..... i k < r n
is a free basis ofiTk(V) and ITk(V *) = 9"k(V) *. Remark. In particular, if V is a free R-module with d i r e r V = m, then dimR ~Tk(V) = d i r e r iTk(V *) = m k.
7.2. E X T E R I O R A L G E B R A
217
Terminology. The established terminology a b o u t covariance a n d contravariance of tensors in geometry is inconsistent with the usage of "covariant" a n d "contravariant" in category theory. For later reference, here are the geometer's definitions. Let V be a finite dimensional vector space over the field F.
D e f i n i t i o n 7 . 1 . 2 5 . For each integer r > 0, 7r(V*), viewed as the space of r-linear m a p s V ~ - " IF, is called the space of covariant tensors on V of degree r and is denoted by T~(V). D e f i n i t i o n 7.1.26. For each integer s >_ 0, 7~(V), viewed as the space of s-linear m a p s (V*) ~ - " IF, is called the space of contravariant tensors on V of degree s and is denoted by T~s(V ) D e f i n i t i o n 7.1.27. T h e space of tensors on V of type (r, s) is the tensor p r o d u c t
%(V) = %'(V)0~(V). A tensor c~ E '3-,"(V) is said to have covariant degree r and contravariant degree s. Obviously, 7~(V) is the space of (r + s)-linear maps V • ... x VxV*
x ... x V*-.
IF.
Y 7"
s
E x e r c i s e 7.1.28. Let V be a n R-module, V* its dual. (1) E x h i b i t a canonical R-linear m a p a : V* | V - . R. (2) If V is free, prove t h a t c~ is a surjection. If, in addition, V has a basis with one element, prove t h a t a is a bijection. (3) If V a n d W are R-modules (not necessarily free), exhibit a canonical R-linear m a p fl : V* @ W - . HomR(V, W). (4) If R is a field, prove t h a t / 3 is injective. Do not assume t h a t V a n d W are finite dimensional. (5) If R is a field, prove t h a t fl is surjective if a n d only if either d i r e r V < oo or dimR W < oc. E x e r c i s e 7 . 1 . 2 9 . Let A be a connected, graded R-algebra. (1) Show t h a t there is a unique homomorphism ~ : 7 ( A 1) - . A of graded algebras such that ~0 = idR and ~i = idAa. (2) Define a suitable notion of graded 2-sided ideal I C A so t h a t A / I = {An/[~}~=o is again a graded R-algebra. (3) If A is generated, as a graded algebra, by A 1, show t h a t there is a canonical ideal I C 'y(A1), with I ~ = {0} = 11, and a canonical isomorphism ,~ : r ( A ~ ) / I
~ A
such t h a t 7 o = idR and 71 = idm*.
7.2. Exterior A l g e b r a Let R be any c o m m u t a t i v e ring with unity 1 such that ~1 E R. T h a t is, if 2 = 1 + 1 E R, t h e n ~i E R has the property t h a t 1 . 2 = 1 . I n t t l e c a s e t h a t R = F is a field, this means t h a t the characteristic of F is not 2. Lemma
7.2.1. L e t V be a n R - m o d u l e ,
v E V.
Then
v = -v
~=> v = O.
218
7. M U L T I L I N E A R A L G E B R A
Proof. Evidently, v = 0 ~ v = - v . For the converse,
v=-v~2v=0~v=-l(2v)=l(0)= 2
0.
2
[] Let Ek be the group of permutations of {1, 2 , . . . , k}, a group of order k!. D e f i n i t i o n 7.2.2. The sign of a E Ek is
(-1)~ =
1, --1,
a an even permutation, cr an odd permutation.
D e f i n i t i o n 7.2.3. Let V and W be R-modules. An antisymmetric k-linear map o2 : V k -+ W is a k-linear map such that ~ ( v o o ) , . . . , va(k)) = ( - 1 ) ~ o ( v l , . 99 , vk), V v l , . . . ,vk E V, Va E Ek. Remark that this definition will be useful only because 1 ~ - 1 in R. As in the definition of tensor product, for each k > 2, define a universal antisymmetric k-linear map V x . . . x V 2+ Ak(V), written A ( v l , . . . ,vk) = vl A . . . A v k . Here, the subspace ~ of relations is generated by the k-linear relations and all elements ( v l , . . . ,vk) - ( - 0 ~ , ~ ( k ) ) , ~ ~ ~k. Existence and uniqueness, up to unique isomorphism, are established exactly as for tensor product. One sets A~ = R and AI(V) = V. D e f i n i t i o n 7.2.4. The R-module Ak(V) is called the kth exterior power of V. We set A ( 7 ) = {Aa(V)}~~ . D e f i n i t i o n 7.2.5. An element w E Ak(V) that can be expressed in the form vl A vs A 9.. Avk, where vi E V , 1 < i < k, is said to be decomposable. Otherwise, w is indecomposable. It is clear that Ak(V) is spanned by decomposable elements, but generally there are plenty of indecomposable elements as well. As before, R-linear maps ~ : V --~ W will induce canonical R-linear maps Ak(~) : Ak(V) -~ Ak(W) such that A~
= idm AI(~) = ~, and
Ak(A)(vl A . . . Avk) = /~(Vl) A ' "
A
~(Vk).
We are going to define a bilinear, associative multiplication AP(V) • Aq(v) ~ A P + q ( V ) that will satisfy (Vl A " " A Vp) A (Vp+l A ' ' " A Vp+q) = Vl A " " A Vp+q.
7.2. E X T E R I O R A L G E B R A
219
By the above remarks, this will make A a covariant functor from the category of R-modules and R-linear maps to the category of graded algebras over R and graded algebra homomorphisms. To define the algebra structure on A(V), we must relate A(V) more directly to
= o,
~xl ( v ) = o,
vl,v2 E V } ,
9.12(V) = span{v1 |
+ v2 |
~ ( v ) = span U
~(V) | ~(V) | r~(V),
p+q=k--2
Let ~o : V k --* W be an antisymmetric k-linear map. As a k-linear map, ~ can be interpreted as a linear map which, for clarity, we denote by ff : 7k(V) -~ W. L e m m a 7.2.7. I f T) : V k --~ W is antisymmetric, then ff(gJk(V)) = {0}.
Proof. It will be enough to show that ff vanishes on a set spanning ~lk(V). Thus, if w E 7P(V), u C ~q(v), p -t- q = k - - 2, and Vl, v2 E V, we will show that
?(w | (vl | v2 + v2 | vl) | u) = o. But the antisymmetry of @ implies that ~ ( w | Vl | v2 | u) = - f f ( w | v2 | vl | u), and the assertion follows from linearity.
[]
C o r o l l a r y 7.2.8. There is a canonical isomorphism
ak(v)
=
~rk(V)/~k(V)
of R-modules, V k > O. Thus, A(V) is a graded algebra over R (the exterior algebra of V) and A is a covariant functor from R-modules to graded R-algebras9 Pro@ For k = 0,1, it is clear that Ak(V) = 7k(V)/P2k(V). consider the k-linear map
For each k >_ 2,
v k ~ ~k(v) & ~k(V)/~k(V), where rr is the quotient projection 9 The reader will check easily that this is antisymmetric. Given an arbitrary antisymmetric k-linear map ~ : V k ~ W , we obtain the commutative triangle |
Vk
, 7k(v)
W
220
7. MULTILINEAR ALGEBRA
and ~lP.lk(V) = 0. Thus, c~ induces ~ : O ' k ( V ) / P l k ( V ) --+ W making the following diagram
commutative: V k
W
|
)
o-k(v)
)
W
id
~
, ~k(V)/~k(V)
,
id
W
T h a t is, the triangle ~oO v ~
~k(V)/~k(V)
.
W
commutes. Since g'k(V)/P.l;v(V) is spanned by
D(vl |
o vk) I vl,... ,~k c v}
and commutativity of the diagrams forces ~(Tr(Vl ~ ' ' ' O V k ) )
=
~O(Vl,... ,Vk),
we see t h a t ~ is the only linear map making the triangle commute. T h a t is, ~ o | : V k --~ g ' k ( V ) / P A k ( V ) has the universal property for antisymmetric, k-linear maps, hence is uniquely identified with A : V k --~ Ak(V). Finally, this identification makes vl A 9 .. A vk : vl | " " | vk + P2k(V), and all assertions follow. [] D e f i n i t i o n 7.2.9. A graded algebra A is anticommutative if a E A k and/3 E A r => a/3 = (-1)kr/3a.
C o r o l l a r y 7.2.10. T h e graded algebra A(V) is a n t i e o m m u t a t i v e . Proof. It is enough to verify Definition 7.2.9 for decomposable elements of Ak(V) and At(V). But that ease is an elementary consequence of the case k = r = 1, and this latter case is given by
v n w = vOw
+ ~12(V)
= -w | v + ~2(V) =
Vv,w
--W
A V,
E V.
[]
C o r o l l a r y 7.2.11. I f w 6 A2r+I(V), then w A w = O. Proof. Indeed, w A w :
By Lemma
7.2.1, w A w = 0.
Corollary
7.2.12.
For the remainder
(-I)(2"+I)(2~+I)w
If w 6 Ak(V)
A w =
-w
A w.
[] /s decomposable,
then w Aw
= O.
of this section, we specialize to the case in which V is a free
R-module on a basis { e l , . . . ,era}.
7.2. EXTERIOR ALGEBRA
Lemma
7.2.13.
221
I f V is as above, then { e i 1 A el2 A . . . A e i k } l < < i l < i 2 < , . . < i k < m
is a free basis of A k ( V ) , k > 2. In particular, dimRAk(V) =
and, i f k
> .~, A k ( V ) =
- kT(m - k)!
(0}.
P r o @ T h e basis {%|174
..... j~<_m
of 7 k ( V ) projects to a spanning set {ejl A . 9 9A ej~ }I_
= (-1)aeji
A...Aejk.
B = {ei, A'"
A eik}l<_il
Thus, tim set n
spans Ak(V). We must prove t h a t B is free. For 1 < il < i2 < ... < ik _< m, define an a n t i s y m m e t r i c k-linear m a p ~gili2...ik : V~' --9 R by stipulating t h a t ~9ili2...ik(ejl,ej2,
"
9
"
I
e j k ) = ~JlJ2...Jk ili2""ik
whenever 1 < j l < j2 < "'" < jk _< m. By a n t i s y m m e t r y and k-multilinearity, this uniquely determines the desired map. As a linear map, ~ i ~ . . . i ~ : h k ( V ) -+ R has the p r o p e r t y t h a t )gili2...ik(ejl
A ej2 A . . . A e j k ) = ~JlJ2...jk ili2...ik 9
This proves t h a t the spanning set B is free and provides, simultaneously, the dual basis B* = {~ili~...i~ }1 _q,
[] 7 . 2 . 1 4 . d i m R A k ( V ) = d i m R A m - k ( V ) , 0 < k < m.
We emphasize t h a t dimR Am(V) = dimR A ~
= 1
and, if { e l , . . . , e,~} is a basis of V, then el A . . . A em constitutes a basis of Am(V). E x e r c i s e 7 . 2 . 1 5 . If R is a field and dimR V = n, prove t h a t every element of A ~ - I ( v ) is decomposable. (Hint: Given 0 r a C A ~ - I ( v ) , consider the linear m a p c~A : V --4 A n ( v ) . )
222
7. M U L T I L I N E A R A L G E B R A
R e m a r k . If A : V --* V is linear, then det(A) C R is well defined. Indeed, if A, B are m x m matrices over R t h a t represent A relative to two choices of basis, t h e n there is an invertible m a t r i x P over R such t h a t A = P B P - 1 , so det(A) = d e t ( B ) .
L e m m a 7.2.16. I f A : V --~ V is linear, then A'~(A) : A m ( v ) --~ A m ( v ) is m u l t i plication by det(A). Proof. Relative to a basis { e l , . . . , e,~} of V, write m
A(ei) = E a f e j ,
1 < i < m.
j=l
Then, am(/~)(el A . . . A em) = /\(el) A . . . A ~(gm) =
=
ae
A...A
a
e
E a { ' . . . a ~ ~ ej, A . . . A l_<jl,... ,jm_<m
ey,.
A n y term w i t h a repeated j index vanishes. I f J = ( j l , j 2 . . . . ,j,~) contains no repetitions, there is a unique p e r m u t a t i o n o j E ~ m such t h a t Joa(~)=r,
l
Thus,
m ) elA...
--1) aa~(1 ) 1 ""aa(m)
Am(A)(el A " ' A e m ) =
Aem
\O'E~m
= d e t ( A ) < A . . . A era. [] L e m m a 7.2.17. I f R is a f i e l d , a set o f vectors w l , . . . i n d e p e n d e n t i f and only i f w l A . .. A wk 7~ O.
, w k C V , k > 2, is linearly
Proof. If the set is dependent, the existence of inverses in R allows us to assume, w i t h o u t loss of generality, t h a t k
W l : ~_~ a i w i . i=2
Then, k Wl A . " A wk = E
aiwi A w2 A " " A wk = 0 .
i=2
Conversely, if the set is linearly independent, extend it to a basis by suitable choices of wk+l, 9 9 ,wm E V. Then, wl A . . . AWk A . . . Awm is a basis of the one-dimensional space Am(V), hence is not 0. [] In an obvious sense, one can view A(V) and 7 ( V ) as graded R - m o d u l e s by ignoring multiplication. It will be convenient to construct a (graded) R-linear m a p A : A ( V ) -+ T ( V ) . T h a t is, A = { A k : Ak(V) + 7k(V)}~=0 .
7.2. E X T E R I O R
ALGEBRA
223
We e m p h a s i z e t h a t this will not be a h o m o m o r p h i s m of graded algebras. Define an a n t i s y m m e t r i c , k-linear map A k : V k -~ 9"k(V) by
Ak(vl"'" 'vk) : Z
(--])aVc~(1)|176
aEEk
By t h e universal p r o p e r t y of the exterior powers, we view this as a linear m a p
A k : A k ( V ) -~ 7~(V). The sequence A = {A/~}0
Lemma
Proof. Let { e l , . . . , era} C V be a basis and consider the basis {eil A ' "
A eik}l<_i,<...
of A k ( V ) . Let {e~,... ,e~} C V* be the dual basis. (Theorem 7.1.24), we obtain a subset
Since ~Tk(V *) = ~Tk(V) *
t h a t is part of a free basis. Then, since j l < " " < Jk and il < . . ' < ik,
(e;, | 1 7 4
e j ~ ) ( A k ( % A - - . A eik))
= (e;, | ' ' " | e~k) ( ~-~. ( - 1 ) ~ e M , ) | 99" | e,.(k)) \crE~ k
= (e), |
| e}~)(% |
|
A---Jk
= (~il ""ik
and t h e assertion follows.
[]
7.2.19. If 1/k! E R, then A k is one-to-one for every R-module V. In particular, if the rational field is imbedded as a subring Q c_ R containing the unity, then A : A ( V ) ~-~ 9 i V ) Lemma
is a canonical graded linear imbedding. Proof. First, consider A k as a multilinear map on V k and define a = ( 1 / k ! ) A k. D e n o t e by A k ( V ) the s u b m o d u l e of 'Yk(V) spanned by the image of a. Of course, this s u b m o d u l e is also spanned by the image of A k. We will show t h a t the ant i s y m m e t r i c , k-linear map a : V k --+ A k ( V ) has the universal property, hence is canonically the same as A : V k --~ Ak(V). By universality, the corresponding linear m a p a : Ak(V) --+ A k ( V ) = Ak(V) is the identity. In particular, as a linear map, A k = k!a is injective. We verify the universal property. Let qa : V k --+ W be an arbitrary a n t i s y m m e t ric, k-linear map. Since it is k-linear, ~ induces a unique linear m a p ~ : 9~k(V) ---, W and we restrict this linear m a p to the s u b m o d u i e A k ( V ) . T h e diagram
224
7. M U L T I L I N E A R
ALGEBRA
C~
vk
. 4k(v)
W commutes. Indeed,
_(1
? ( a ' ( v l , . . . , v k ) ) = 9s ~. E
)
(--1)ava(1)|174
aEEk
=
1
kq ~
(-1)~(v'(1)'""v~(k))
aEEk
=
1
kq E
(-1)a(-1)a~(Vl"'"Vk)
aEEk
= ~(vl, ... ,vk), where the second equality is by the universal property of ~k (V) a n d the third by the a n t i s y m m e t r y of ~. This c o m m u t a t i v i t y forces the definition of ~ on the s p a n n i n g set ira(a), so ~ is the unique linear m a p m a k i n g the diagram commute. [] Since A is canonical, we will generally suppress it from the notation, writing A(V) c ~'(V) Vl A ' " A V k - =
E (--1)aVa(1)| aEEk
'
provided either t h a t V is free a n d finite dimensional or Q c_ R. T h e o r e m 7.2.20. I f V is a free R - m o d u l e on a finite basis, there is a canonical i s o m o r p h i s m A a (V*) = A k ( V ) * . Proof. By L e m m a 7.2.18 a n d Theorem 7.1.24,
Ak(V *) c ~ ( v * )
= ~(v)*,
so Ak(V *) can be viewed as a space of k-linear maps V k ---* R. We prove first t h a t each 02 E Ak(V*), interpreted as o2 : V k --* R, is a n t i s y m metric. Indeed, let {e~ e * } c V* be a basis a n d suppose that o2 = e* A...Ae* k 1
~ Vk)
m E
(~ - - lY~Ce * |174 / " ia(1)
'
..
'
vk)
aEEk
= ~(
x--
We* / ia(1) (~1),,,CZa(k) (v~),
aEEk
T h a t is, o2(vl . . . . . vk) = det[e~'j (ve)].
Thus, if T C Ek, a2(v~(1), . . . , v~(k)) = (-- 1)~o2(v1, 99 9 , Vk),
proving t h a t o2 = e~l A ... A e*~k is antisymmetric, 1 _< il < ' " < ik _< m . These m o n o m i a l s range over a basis of hk(V*), so every element o2 of this space is antis y m m e t r i c as a k-linear m a p w : V k --~ R.
7.2. E X T E R I O R A L G E B R A
225
Consequently, we can view w C Ak(V *) as defining a linear map ~ : Ak(V) --, R. That is, ~ E Ak(V) * and, if
l<_il<...
.
.
Ae~k(e AA . . . .
Aejk )
i~...,:k
~ ~ 1 ""Jk"
It follows that e*
A . .. A e*
=
(ell
A ...
A eik)*
hence that the map co H ~ carries Ak(V *) isomorphically onto Ak(V) *.
[]
E x e r c i s e 7.2.21. If R is a field, dimR V = n, and 0 r a E A2(V), let r _> 1 be the integer such that the r-fold exterior power a A 999A ~ r 0, but the (r + 1)-fold power = 0. This integer is called the rank of c~. Show that there exists a basis { v l , . . . , v , } of V so that (2 = 1;1 A V 2 "~ V 3 A V 4 @ " " + V 2 r _ l A V 2 r .
(Hint: Proceed by induction on r. For the inductive step, show that there is always a basis { w l , . . . , wn} such that c~ = wl A w2 + c~' where a ' is a linear combination of t e r m s w i A w j w i t h 3 < i < j < n . ) E x e r c i s e 7.2.22. Recall the Grassmann manifold Gn,k of k-planes in R" (Exercise 5.4.14). If V is a vector space, one defines similarly the Grassmann manifold Gk(V) of k-dimensional subspaces of V. We use this notation in what follows. (1) Using exterior algebra, define a canonical imbedding (of sets)
.m c~(Rm) ~ CI(Ak(R~)). (Hint: Consider the decomposable elements of Ak(N'~)). (2) Exhibit a natural linear (hence, smooth) group action
Cl(~) • Ak(R ~) -~ A~(R~). (3) Using the above, exhibit a smooth action
GI(~) • C~(Ak(R~)) -~ al(Ak(Xm)). (4) Let x0 = span{el A e2 A . . . A ek} E GI(Ak(]Rm)) and show that, relative to the above action, the isotropy group of x0 is Cl(m)xo = Cl(k, m - k). (5) Show that i~(ak(Rm)) = Gl(m) 9 z0, hence Gk(IRm) is expressed as the homogeneous space G l ( m ) / G I ( k , m - k). (6) Finally, restrict the above action to the orthogonal group O(m) x GI(Ak(IRm)) --+ Gl(Ak(Rm)) and prove that this has exactly the same orbits as Gl(m). This gives Gk (IR"~) as a homogeneous space of O(m).
226
7. MULTILINEAR ALGEBRA
E x e r c i s e 7.2.23. If V is a free R-module on a finite basis and v C V, define the
interior product i~ : Ak(V *) --* A k - I ( v *) as follows. Viewing w E Ak(V *) as an antisymmetric k-linear map W : V k ~ R, let i,(w) be the antisymmetric (k - 1)- linear map defined by the formula
i~(~)(v~,..., ~ _ ~ ) = ~(~, ~ , . . . , ~k-~). If w C AP(V *) and 7/E Aq(v*), prove that
i~(~ A ,) = r
A , + ( - 1 F ~ A i~(~).
7.3. S y m m e t r i c A l g e b r a This will be an abbreviated treatment, not because the subject is unimportant, but because the ideas and proofs are so analogous to those for exterior algebra. There are, however, notable differences. Again, our initial hypothesis is that V is a module over a commutative ring R with unity. D e f i n i t i o n 7.3.1. A k-linear map ~ : V k ~ W is symmetric if, for each a c Ek,
~ ( ~ ( 1 ) , . . . ,v~(k)) = ~ ( ~ , . . .
,~k),
V V l , . . . ,vk E V. In the usual way, we build a universal, symmetric, k-linear map
vk :~ Sk(V), usually written with the dots suppressed:
(vl,v2,... ,vk) ~ v x v 2 " "vk. D e f i n i t i o n 7.3.2. The space Sk(V) is called the kth symmetric power of V, where, as usual, S~
:
R and ~l(Y)
= V.
We define the graded, 2-sided ideal G(V) C 9"(V), generated by all Vl ~ V 2 -- V2 ~ V l
E ~]'2(V)
and obtain
Proposition 7.3.3. There is a canonical isomorphism
sk(v) = ~k(V)/Gk(V) of graded R-modules. The connected, graded algebra S(V) multiplication ". ", is called the symmetric algebra of V.
=
Remark that, if a E $;(V) and/3 E s q ( v ) , then
a/3 =/3a ~ $v+q(v). If A : V --* W is R-linear, there is induced a homomorphism S(A): 8(V) -~ 8(W) of graded algebras such t h a t
sk(~)(VlV2 ' ' vk) = :~(~1)~(~2)-" ~(vk).
~SkrV k \ ~ )Jk=O~ with
7.4. MULTILINEAR BUNDLE THEORY
227
O n c e again, S is a covariant functor. Specializing to the case in which V is a free R - m o d u l e on a finite basis, we obtain Lemma
7.3.4. I f {el . . . . . e,~} is a basis of V, then { % % " " eit: }l
is a basis of Sk(V), k > 2. Remark. If the space V is nontrivial, so is Sk(V), Vk _> 0, in strong contradistinction to the fact t h a t Ak(V) = {0}, Yk > m. As for exterior algebras, we define a canonical m a p
s k : sk(v) -~ 7k(y), this being the linear m a p defined by the symmetric, k-linear m a p (Vl'''''Vk)
~ E
Va(1) @ ' ' " (~)vat:"
aEEt:
E x e r c i s e 7.3.5. If V is free on a finite basis or if Q c R is i m b e d d e d as a subring containing the unity, show t h a t S k : S(V) ~-~ T(V) is an inclusion of graded Rmodules. In the first case, use this to prove t h a t S~(V * ) = S k ( v ) *,
Vk_>2.
Show that, if {el, , era} is a basis of V, this identifies the m o n o m i a l e* e* ... e~ w i t h (ei, ei2 " " eit:)*. D e f i n i t i o n 7.3.6. Let V be a finite dimensional vector space over a field F of characteristic zero. A function f : V + F is a homogeneous polynomial of degree k on V if, relative to some (hence, every) basis { e l , . . . , era} of V,
I
}--~,e~
= P(~l . . . . . ~m)
i=1
is a homogeneous polynomial of degree k in the variables x l , . 99 , x ~ . The vector space of all homogeneous polynomials of degree k on V will be denoted by Tk(V). Exercise
7.3.7. For all k _> 0, establish a canonical isomorphism
0: sk(v *) -~ ~k(v) of vector spaces. For the case k = 2, construct 0 -5 explicitly. (~P2(V) is called the space of quadratic forms on V and the process 0 -1 of recovering the s y m m e t r i c bilinear form from its associated q u a d r a t i c form is called polarization.)
7.4. Multilinear B u n d l e T h e o r y J u s t as the linear construction of dualizing a vector space passes to the construction of dualizing a vector bundle, so the multilinear constructions of the previous three sections pass to corresponding constructions on vector bundles. Let 7ri : E.i --~ M be a ki-plane bundle, 1 < i < m. We want to define a bundle
7r : E1 |
"" Q Em --+ M
228
7. M U L T I L I N E A R A L G E B R A
with fiber over x C M canonically equal to E 1 x | 9 ' ' | E m x. The fiber dimension will be klk2 .." km. Let {Us, r be the maximal family of simultaneous trivializations Edg~
r
, Us x R k,
Us
,
U~.
id
Note that, if Aj C Gl(kj) is viewed as a nonsingular linear transformation of IRkj , 1 _ j _< m, then A1 | A2 | 9 9| Am is a nonsingular linear transformation of R kl | N k2 | 999| N k" , hence A1 | A2 |
| Am C G l ( k l k 2 . . . k m ) .
Here, we identify N kl | 9 | 1Rk,,` = ]Rklk2'"k,,~ by lexicographic order on the basis l<j<m
where {e~,... , e~; } is the standard basis of IRk;, 1 _< j _< m. The system {Us, r i gives rise to a Gl(k~)-cocycle { 7i~ } ~ e ~ , for each i = 1 , . . . , k , with the aid of which the bundle Ei can be assembled from the products Us x R k~, a E 92. We try to define a cocycle for assembling the tensor product of these bundles by setting % ~ : Us n U~ -~ G l ( k l k 2 . . . kin),
L e m m a 7.4.1. The map ~/aZ is smooth and ~/= {3'~}~,Ze~ is a cocycle. Proof. The cocycle property is rather obvious. For smoothness, it is enough to show that the vector-valued function x ~ ~.Z(x)(e~ |
|
is smooth on Us n Uz. But kj
s~(x)(%) ~=1
and the real-valued functions a~ (x) are smooth. Expanding ~o~/3(x)(e~l |
" . | eim) "
1 = %1A X )(%)|
"|
kl
=
m km
~ (=)e -
m
|174
ae (x)e= \t'=l
, "
we obtain a linear combination of the elements of the basis B with smooth functions of x as coefficients. []
7.4. M U L T I L I N E A R
BUNDLE THEORY
229
E x e r c i s e 7.4.2. Let rr : E --* M be the bundle determined by the cocycle { U ~ , % ~ } ~ and produce a canonical isomorphism
Ex = E l x | 1 7 4 1 7 4 for each x E M. In particular, given an n-plane bundle rr : E --+ M, we can form the tensor powers 7r :Irk(E) ~ M, the fiber over x E M being, canonically, Irk(E~). If E has an associated cocycle {%Z}~,~<~, then Irk(E) has {Irk(%z)}~,~e~ as an associated cocycle. L e m m a 7.4.3. The Oth tensor power T~
is canonically isomorphic to the trivial
bundle M x R. Proof. Indeed, for each x E M, TO(E,) = ]R and, i f x E U~NUz, then TO(%5(x)) = idm
[] The following is proven in the same way.
L e m m a 7.4.4. The 1st tensor power 9"I(E) is canonically isomorphic to E. We denote by 7r : ir(E) ~ M oo the collection {rr : Irk(E) ____~ M }k=0 and interpret this system as a "bundle" of graded N-algebras over M. In complete analogy with these constructions, we form the exterior powers
7r : Ak(E) --+ M of the bundle E over M and the "bundle" rr : A ( E ) + M of exterior R-algebras. Again, A~ = M x IR and AI(E) = E. Finally, one forms the symmetric powers gk(E) and the bundle g(E) of symmetric algebras, noting the identities g~ = M x ]R and g l ( E ) = E. E x e r c i s e 7.4.5. Let d i m M = n _> 2. Prove that M is orientable if and only if An(T *(M)) admits a nowhere 0 section. By identities proven in the previous sections, we obtain the following. L e m m a 7.4.6. If E is a vector bundle over llJ, then
Ir~(E*) : Irk(E)*, a k ( E *) = a k ( E ) *, gk(E*) = gk(E)*,
canonically for each integer k >_ O. Of particular note are the tensor bundles ~(E)
=
T'(E*) 0 Ir~(E),
of covariant degree r and contravariant degree s. Similarly, one can define bundles of linear homomorphisms. If E and F are vector bundles over M, we construct a vector bundle Horn(E, F ) , the fiber of which
230
7. MULTILINEAR ALGEBRA
over each x C M is the vector space HomR(Ex,F~). Since this vector space is canonically equal to E~ | Fx (Exercise 7.1.28), we define Horn(E, F ) = E* @ F. Likewise, the bundle of k-linear maps (Ex) k --* Fx, V x E M , can be defined as
L k ( E , F ) = O'k(E *) | F, and its antisymmetric and symmetric cousins are
ALk(E, F ) = Ak(E *) N F, S L k ( E , F ) = Sk(E *) N F, respectively. E x e r c i s e 7.4.7. Recall the operation of direct sum 1/1 | V2 of vector spaces. If E1 and E2 are vector bundles over M, show how to define a bundle E1 (9 E2 over M with fibers (El @ E2)x = EI~ | E2~, Vx C M. This is called the "Whitney sum" of the bundles. E x e r c i s e 7.4.8. If 1/1, 1/2, and W are finite dimensional vector spaces, construct a natural bilinear map
0:(v1 ey~) x w - ~ (yl
|174
hence a natural linear map
O: (VI e V~) O W -~ (Vl | W ) e (V2 O W). Prove that 0 is a linear isomorphism. Use this to prove that there is a canonical bundle isomorphism
(El 9 E2) | F = (El | F) 9 (E2 | F).
Remark. Recall the general philosophical principal mentioned when we constructed the dual bundle. If one views vector spaces as vector bundles over a point, then all "natural" linear and multilinear constructions for combining vector spaces to get new ones extend to analogous operations, fiber by fiber, on vector bundles. Here "natural" means that the constructions can be carried out without reference to choices of bases. Such constructions are also "canonical". Similarly, natural relations between vector spaces, such as the relation Hom(V, W) = V* | W, extend to analogous relations between vector bundles.
7.5. T h e M o d u l e o f S e c t i o n s We are going to view the set of all vector bundles over a fixed manifold M as the objects of a category ~ M . For this, we need to define the morphisms of the category. Let
~r : E - .
M,
p:F-~
M
be vector bundles (of possibly differing fiber dimensions).
7.5. M O D U L E O F S E C T I O N S
231
D e f i n i t i o n 7.5.1. A homomorphism of the n-plane bundle E to the m-plane bundle F is a commutative diagram E
M
~
id
~ F
, M
where ~o is smooth and, for each z E M, ~ox = ~lEx : E~. ~ F~ is linear. We denote by HOM(E, F ) the set of all bundle homomorphisms from E to F. E x a m p l e 7.5.2. The canonical fiberwise inclusions
A k : a k ( E ~ ) ~ irk(E~), S k : Sk(E~) ~ irk(E~) (Sections 7.2 and 7.3) assemble to give bundle monomorphisms
A k : Ak(E) ~ Irk(E), s k : Sk(E) ~ irk(E), the smoothness of these maps being easily checked via local trivializations. It is clear that the composition of bundle homomorphisms, whenever defined, is a bundle homomorphism and that idE : E --+ E is a bundle homomorphism, so VM is a category with bundle homomorphisms as its morphisms. To each object E C VM we associate a Coo(M)-module r(Z), tile space of smooth sections of E. If ~ : E -+ F is a bundle homomorphism, there is induced a C ~ (M)-linear map g). : F(E) ---, F(F), defined by
~ . ( ~ ) ( ~ ) : ~(s(~)),
w
~ M, v s c r ( z ) .
It is obvious that (idE). = idF(E) and that (~b o qa). = ~/J. o g)., so F is a covariant functor F : VM -~ ~ ( C o o ( M ) ) . (The squiggly arrow "~-~" is commonly used for functors.) E x e r c i s e 7.5.3. If E, F E 37M, exhibit a canonical identification HOM(E, F ) = F(Hom(E, F)) = F(E* | F). In particular, the set HOM(E, F) of homomorphisms of the bundle E to the bundle F is naturally a C ~ (M)-module. Tile main purpose of this section is to show that the funetor F transforms the IR-multilinear bundle constructions of Section 7.4 into the corresponding Coo(M)multilinear module constructions of Sections 7.1, 7.2 and 7.3. A fairly easy case in point is tile following. P r o p o s i t i o n 7.5.4. If E is a vector bundle over M , then there is a canonical isomorphism r(z*) = r(Z)* of C ~ ( M)-modules.
232
7. MULTILINEAR A L G E B R A
Indeed, Proposition 6.2.11 was a particular case of this proposition and the proof for the general case remains the same. Consider the tensor product F ( E ) | F ( F ) of Coo(M)-modules. Since these are also vector spaces over R, this notation can be ambiguous. In order to avoid such ambiguities the tensor product of R-modules A and B is often written A | B. For the time being, we will denote this C ~ ( M ) - m o d u l e by F ( E ) | F ( F ) and its decomposable elements by s | or. The vector space tensor product will be F ( E ) | F ( F ) and its decomposables will be s @a or. Since one seldom thinks of the set of sections as a real vector space, we will ultimately drop the subscript C~~ b u t retain | for the vector space case. Given s 9 F ( E ) and cr 9 F ( F ) , one produces ~(s, ~) 9 F ( E | F ) by setting
~(s, ~)(x) = s(x) | ~(x) 9 E~ | F~ = (E | F)~, V~ 9 M. We will write c~(s, a) = s | a, the pointwise tensor product of sections. It is not hard to check t h a t this defines a smooth section of E | F and that a : F ( E ) x F ( F ) -* F ( E | is C ~ ( M ) - b i l i n e a r . Denote also by ~ the associated C ~ ( M ) - l i n e a r map
0~: F(E) |
r(F) --+P(m | F).
We emphasize t h a t s | cr and s | a = c~(s | 0") are conceptually distinct. The following theorem asserts that this conceptual distinction can safely be disregarded. T h e o r e m 7.5.5. The C ~ ( M ) - l i n e a r Coo ( M )-modules
r(E) |
map a
F(F)
is a canonical isomorphism
= r(E
of
| F).
C o r o l l a r y 7.5.6. There are canonical isomorphisms r(~rk(E)) = ~rk(r(E)), r(hk(E)) = Ak(r(E)),
r(Sk(E)) = Sk(r(E)), of Coo ( M ) - modules. Pro@ Indeed, the first of these identities is an immediate consequence of Theorem 7.5.5. There are canonical inclusions
Ak: Ak(r(E)), ~ ~k(r(E)), Ak: r(hk(E)) ~ r(~k(E)). The first of these is by Lemma 7.2.19 and the observation that Q c C o o ( M ) as a subring of constant functions. The second comes from the bundle inclusion of Example 7.5.2. The images of these inclusions correspond perfectly under the identification ~rk(r(E)) = r(9"k(E)), proving the second identity. The third has exactly the same proof as the second. [] Combining this corollary with Proposition 7.5.4 gives C o r o l l a r y 7.5.7. There is a canonical isomorphism ~(P(E)) = F(~;(E)) of C ~ ( M ) - modules.
7.5. MODULE OF SECTIONS
233
E x a m p l e 7.5.8. There are many other natural identifications now available. For example, (~k(r(E)))*
= (r(~k(E))) * =
by Corollary 7.5.6 by Proposition 7.5.4
r(o-k(E) *)
= r(~k(E*))
by Lemma 7.4.6
: ~k(r(E*))
by Corollary 7.5.6
: ~k(r(E)*)
by Proposition 7.5.4,
and there are similar identities for A k and S k. E x a m p l e 7.5.9. A Riemannian metric (., .) oll a manifold M is a smooth section of g2(T* (M)) that is positive definite. In local coordinates, the metric has a formula
9i~(x) dx~ dxj l
{c~i |
rJ}i,j:l
is a free basis of F ( E ) @Ca(M) F ( F ) (Corollary 7.1.12). The set n,m {(7i @ Tj}i,j= 1 of pointwise tensor products of sections trivializes the bundle E | F , hence is a free basis of F ( E | F ) . Since a(~i |
Tj) = ai | T~,
for all relevant indices, we see that c~ is an isomorphism of C ~ ( M ) - m o d u l e s .
[]
In light of this lemma, we will reduce the general case of Theorem 7.5.5 to the case in which both bundles are trivial. The key to this is the following. Theorem
7.5.11. Given a vector bundle E over M , there exists a vector bundle E • over M such that the bundle E @ E • is trivial. For the moment, we accept this. Let E1 and E2 be vector bundles over M and define bundle homomorphisms b: ~1 ----+E1 @ E2
t~(v) = (v, 0),
p : E1 9 E2 ---, E1
p(v, w) = v.
It is clear that p o ~ = idEt, so the functoriality of F implies the following.
234 Lemma
7. MULTILINEARALGEBRA 7.5.12.
The composition p, o 5, is equal to idr(E~). In particular, 5, : F ( E , ) ---, F(E1 | E2)
is injective and
p.: r(E~ 9 E~) ~ r(E~) is surjective. Proof of theorem 7.5.5. By Exercise 7.4.8, (E | E • | (F | F • s u m m a n d E | F . Consider the commutative diagram F((E|
•174177
,
F(E | E • |
"l
splits off a direct
F(F | F •
It*@;'*
r(E | F)
F(E) |
r(F)
By Lemma 7.5.10, the top arrow is an isomorphism of C~176 lemma guarantees that the leftmost vertical arrow is injective. Since
The above
(p, | p,) o (5, | 5,) = (p, o 5,) | (p, o 5,) = idr(E) | idr(F), the rightmost vertical arrow is also injective. It follows that
~: r(E) |
F(F) ~ F(E | F)
is injective. Similarly, the diagram F ( E | E • @C~(M) F ( F 9 F •
F ( ( E 9 S ~) | ( r 9 F a ) )
P*I
~P.|
r ( z | F)
,--
V(E) O C t ( M ) r ( S )
commutes and the vertical arrows are surjective, implying that a: F(E)
|
F(F) ~ F(E|
is surjective.
[]
Everything now hinges on Theorem 7.5.11. We prove the case in which M is compact and then quote a theorem from dimension theory t h a t extends this proof to the noncompact case. Suppose t h a t E is an n-plane bundle over a compact manifold M. Compactness of M will be used only to find a finite open cover {Ui}i~=l such that E[Ui is trivial, 1 < i _< r. Let {Ai}i~l be a subordinate partition of unity and, for each i = 1 , 2 , . . . , r , let s ~ , . . . , sn E F(EIUi ) be everywhere linearly independent (hence a basis at e a c h x C Ui). Let a Ji C F ( E ) be the extension by 0 of ALSO, 1 < i < r, 1 _< j _< n. Then, for each x 6 M, {a~ (x)}i,j= r,n 1 spans E~. View F ( E ) as a vector space over ]R and consider the finite dimensional subspace r j~r,n V = span~tcr i ~i,j=l" Then Pl : M x V ~ M is a trivial vector bundle. We define a surjective homomorphism
p:MxV--,E of vector bundles by setting p(x, ~) = ~(x).
7.5. M O D U L E O F S E C T I O N S
235
The smoothness of p is elementary, as is the fact that it carries {x} x V linearly onto the fiber E~, Vx E M. Let E ~ C {x} x V be the kernel of this linear surjection and set
El= xEM The fact that E • is a vector subbundle of M x V is a case of the following result. L e m m a 7.5.13. If p : F --* E is a surjectivc homomorphism of vector bundles
over M, then
E•
U ker(p~) xEM
is a subbundle of F. Proof. It is enough to produce local trivializations. Let x E M and choose vectors Vl,. 99 vT- E Fx that are carried by pz one-to-one onto a basis of E~. Extend these to a basis {Vl,... ,v~,vr+l,... ,v~} of F~. By the local triviality of F , there is a neighborhood U of x in M and sections ai of FlU such that ai(x) = vi, 1 < i < n, and such that {ai (y)}i~=1 is a basis of Fy, V y E U. Consider the sections si = p o ai of EIU. Taking U smaller, if necessary, and appealing to the continuity of p, we arrange that {zi(y)}ir__l is a basis of Ey, Vy E U. Then there are unique expressions s,.+l(y)
=
s
fr
i=1
8n(y) = s f~(y)s@), i=l
where the coefficient functions f j are all smooth. Consider the smooth sections
TI(Y):~r§
-s i=1
r Tn_v(y ) • (Tn(y) -- E f i (y)o-i(y). i=1
It should be clear that these give a basis of ker(py), Vy C U, hence define a local trivialization of E • over U. [] Fix a positive definite inner product on V, viewing it as a Riemannian metric on the bundle M x V. For each x E M, let/~x C {z} x V be the subspace orthogonal to E ~ . We claim that the set xEM
is a subbundle of M x V. More generally, L e m r n a 7.5.14. If F C E is a vector subbundlc and if there is given a Riemannian
metric on E, then the subset F C_ E, fiberwise perpendicular to F, is a subbundle.
236
7. M U L T I L I N E A R A L G E B R A
P r o @ Again, local triviality is all t h a t needs to be proven. There are sections a l , . . . , a~, a ~ + l , . . . , an of EIU, trivializing t h a t bundle, where U is a neighborhood of a n a r b i t r a r y point of M. These can be chosen so t h a t a l , . . . , a~ are sections of F l U t h a t trivialize t h a t bundle. A n application of G r a m - S c h m i d t t u r n s these into fiberwise o r t h o n o r m a l sections S l , . . 9 s~, s ~ + l , . . 9 s~ with the same properties. It follows t h a t S ~ + l , . . . , s ~ are trivializing sections of F l U , proving t h a t F is a s u b b u n d l e of E as desired. [] The b u n d l e homomorphism pl/~ : /~ ~ E is an isomorphism, this being true fiber by fiber, so MxV=E•177174 This completes the proof of Theorem 7.5.11 in the case t h a t M is a compact m a n ifold. T h e compactness a s s u m p t i o n on M is removed by showing that, whether M is compact or not, there is always a finite trivializing cover { Ui}i=l ~ for E. This will follow from a theorem in dimension theory. T h e o r e m 7.5.15. Let M be a manifold of M admits a refinement W = { W a } a ~ 1 < i < r + 2, are all distinct, then
of dimension r. Then every open cover V such that, whenever the indices eq E 9~,
Wal N VF~2 A . . . A W ~ + 2 = ~. For the proof, see [20, Theorem V.8, page 67], together with E x a m p l e III.4 on page 25 of t h a t same reference. T h e o r e m 7.5.16. Let M be a manifold of dimension r and let E be a vector bundle over M . Then there is an open c o v e r {Uk}~+__11 such that EIUk is trivial, l
Proof. Let V be an open cover of M trivializing E. Let W = { W ~ } a e a be the refinement given by Theorem 7.5.15. In particular, E I W ~ is trivial, V a C 9.1. Let {A~}~e~ be a partition of unity s u b o r d i n a t e to W. If S C_ ~ is a finite subset, define Us = {x E M l m i ~ ) ~ ( x ) >
max ,kZ(x)}.
/3E~I-.S
Let ISI denote the cardinality of S. The following are elementary:
(i) {Us I S C_ 92 is finite} is an open cover of M; (ii) if $1 ~ $2 are finite subsets of P2 with ISll = ]$21, then Usl A Us2 = ~;
(iii) Isl > r + 1 ~ u s = O; (iv) UsC_W~,Va~S. For each integer k _> 1, set
Uk =
[ J Us. Isl=k ~o is an open cover of M . For each k > 1, E l g k is trivial ((ii) a n d By (i), { U k}k=l (iv)) and, by (iii), Uk = ~ for k > r + 1. [] T h e proof of Theorem 7.5.5 is now complete. D e f i n i t i o n 7.5.17. The space of covariant tensors of degree k on M is g-k (M) = F(g *k(T* (/1/I))) = 9-k (F(T* (M))) = g-k (A 1 (M)). T h e graded algebra It* (M) = {g"k (M)}k= oo 0 is called the covariant tensor algebra of M.
7.5. MODULE OF SECTIONS
237
R e m a r k . If T*(M) is viewed as F(T(T*(M))), the multiplication is by pointwise tensor product of sections. If it is viewed as 7 ( F ( T * ( M ) ) ) , the multiplication is just t h a t of the tensor algebra of the C~~ F(T*(M)). By the proof of Theorem 7.5.5, the two graded algebra structures agree. R e m a r k . The use of the asterisk to denote graded structures is standard, as is its use to denote duals. Which meaning is intended will usually be clear from the context.
D e f i n i t i o n 7.5.18. The space of contravariant tensors of degree k on M is Tk(M) = F(~'k(T(M))) = 9-k(F(T(M))) = 9"k(X(M)), oo is called the contravariant tensor algebra of M. and % ( M ) = {9"k(M )}k=0 D e f i n i t i o n 7.5.19. The space of (mixed) tensors of type (r, s) on M is 9 ;(M) = F(~(T*(M)) O = ~(r(r*(M)))
= W(A1 ( M ) ) |
|
~'(T(M))) rs(P(r(M)))
9-s(X(M)).
D e f i n i t i o n 7.5.20. The space of k-forms on M is Ak(M) = F(Ak(T*(M))) = A k ( C ( r * ( v ) ) ) : Ak(AI(/~/)). The exterior algebra A*(M) = F ( A ( T * ( M ) ) ) = A(F(T*(M))) = A ( A I ( M ) ) is called the Grassmann algebra of M. D e f i n i t i o n 7.5.21. The space of (covariant) symmetric tensors on M is sk(M)
= F(Sk(T*(M)))
= gk(F(T*(M)))
= gk(Al(~'/)).
The graded algebra g*(M) = F(S(T*(M))) = S(F(T*(M))) = S ( A I ( M ) ) is called the symmetric algebra of M. Remark that the graded algebras O'*(M), T . ( M ) , A * ( M ) , and g*(M) are all connected and t h a t T I ( M ) = X(M) and T I ( M ) = A I ( M ) = g l ( M ) . E x e r c i s e 7.5.22. Let M be an oriented n-manifold, let (U, x l , . . . , x n) and (V, yl, ..., y n ) be coordinate neighborhoods in M respecting the orientation, and let w C A n ( M ) be such that supp(co) is a compact subset of U n V. Let co = f d x 1 A .. . A d x n, co = h dy 1 A . . . A d y n
be the respective formulas for co in these coordinate systems. Finally, let g ( x l , . . . , x n) = ( y l , . . . , yn) be the formula for the change of coordinates. (1) Show that, on qo(U C/V), f = (h o g) det(J9). (2) Show how to define the integral fat co E IR and prove that your definition is independent of choices. (3) Denote the oppositely oriented M by - M and show that - hl co = -- / M CO"
CHAPTER 8
Integration of Forms and de Rham Cohomology In Chapter 6, we studied the first de Rham cohomology H 1 (M) of a manifold. This measures the difference between exactness and local exactness of 1-forms on M and was shown to have interesting topological applications. Here we generalize these ideas, using the full Grassmann algebra A*(M) to produce a graded algebra H * ( M ) , the de R h a m cohomology algebra. The proper generalization of "locally exact 1-form" is "closed p-form", defined as a p-form that is annihilated by "exterior differentiation". Exact forms are closed and HP(M) measures the extent to which closed p-forms may fail to be exact. By Stokes' theorem, the geometric boundary operator and exterior differentiation of forms are mutually adjoint operations in a certain precise sense. This is a generalization of the fundamental theorem of calculus and a powerful tool for computing cohomology. The reader who would like to pursue this theory further could hardly do better than to consult [5]. 8.1. T h e E x t e r i o r D e r i v a t i v e
Let U C_ N n be an open subset. Since A~ the exterior derivative
d: A~
= C~(U), we have already defined
~ AI(u)
(Definition 6.2.14). For p > 1, we can define the exterior derivative
d: AP(U) --~ AP+I(U) by the following formula: fil...i~ d x i l A . . 9 A d x i ~ ) l
=
E
d(fil""~p)Adxil A ' " A d x i ' "
l <_Q<...<_n
It is clear t h a t this operator is R-linear. It is not clear, but will be proven shortly, t h a t the definition is invariant under changes of coordinates. L e m m a 8.1.1. If V C_ U is open, then (da~)lV = d(aJIV). L e m m a 8.1.2. The composition
A~(U ) d A,+I(U) d Ap+2(U) is trivial (d 2 = 0). Proof. By the antisymmetry of exterior multiplication, the above formula for d gives the same answer whether or not the indices are in increasing order or are
240
8. I N T E G R A T I O N A N D C O H O M O L O G Y
distinct. Thus
Of. dx j dxil d ( d ( f dx il A . . . A d x i ' ) ) = d ( ~ - " \~=l OX' A
j=l k=l
.Adz ~) . .
•
and this vanishes by the equality of mixed partials and the antisymmetry of exterior multiplication. []
Remark. The equation d 2 = 0 is equivalent to the equality of mixed partials which, in turn, is equivalent to [c9/0x k, cg/OxJ] = 0, the commutativity of coordinate fields. The proof of the following is mechanical, hence is left to the reader. L e m m a 8.1.3. If ca E AP(U) and r~ E A q ( u ) , then
d(ca A r/) = (d~) /~ r~ + (-1)Pca A dr~. Remark. In particular, writing f r / f o r f A r/when f E A~
= C a ( U ) , we get
d(fr/) = df A r~ + fdr/. If r / = dfl A ... A dfp, where f~ e A~ two lemmas yields d r / = 0 and
1 <_ i < p, then repeated use of the above
d(fr/) = df A r/. C o r o l l a r y 8.1.4. If U C R n and V C_ R m are open subsets and if the map ~ : U ~ V is smooth, then
d o ~ * = ~* o d : AP(Y) ~ Ap+I(u), for all p > O. Proof. In the following computation, the third equality is by the above remark: d ( ~ * ( f dy il A ... A dyi')) = d(p*(f)~o*(dy ~ ) A . .. A ~*(dyi')) = d ( F * ( f ) d(~*(yi~)) A . . . A d ( ~ * ( y i ' ) ) ) = d ( ~ * ( f ) ) A d(~*(yi~)) A . . . A d(~o*(yi')) = 9)*(df) A ~*(dv i~) A . . . A ~*(dy/') = ~* (df A dy i~ A . . . A dy i') = ~o*(d(fdy i~ A . . . A dyi')). Since every r/C AP(V) is a sum of forms of the type used in the above computation, the claim follows. [] We want to extend the exterior derivative to an R-linear operator
d: A P ( M ) --* Ap+I(M) on all manifolds M and for all nonnegative integers p. We take an axiomatic approach, requiring that this operator satisfy the following: (1) (2) (3) (4)
f E A~ and X C )~(M) ~ d f ( X ) = X ( f ) . ca C A P ( M ) and r/C A q ( M ) :=> d(ca A r/) = dw A r~ + (-1)Pw A dr/. d 2 = 0. U C M open and ca 9 A P ( M ) =~ (dw)lU = d(calU).
8.1. EXTERIOR
DERIVATIVE
241
(5) ~ : M ~ N s m o o t h ~ * o d = d o ~ * .
Remark. If one considers only manifolds that are open subsets of Euclidean spaces, then all of the axioms hold for the exterior derivative as already defined. For d : A~ --~ A I ( M ) , which has been defined on all manifolds, we have seen the t r u t h of the first axiom (Lemma 6.2.3). D e f i n i t i o n 8.1.5. If an R-linear operator d : AP(M) ~ Ap+I(M), defined for all smooth manifolds M and all integers p _> 0, satisfies the above axioms, it is called an exterior derivative.
Theorem
8.1.6. There is a unique exterior derivative.
Pro@ First we prove uniqueness. By Axiom (4), it will be enough to show that, whenever U C_ M is a coordinate neighborhood and w E AP(M), the form d(colg ) c AP+I(U) is uniquely determined. Let ~ : U ~ R n be a diffeomorphism onto an open subset V and set ~i = ~*(x i) = x i o ~, 1 < i < n. Also, by functoriality, ( ~ - 1 ) . = ( ~ , ) - 1 , which is to say t h a t ~* : A*(V) ~ A*(U) is an isomorphism. Thus, since dp i = ~*(dx i) (Axiom (5)), the set { d ~ il A . . . A d~gip}l<_il<...
is a free basis of the C ~ ( U ) - m o d u l e AP(U). Thus,
wiN =
~
fi,...ip d99i~ A . . . A d~ i"
l<_it<...
and
d(~lU)=d(
<~< 1
=
_<:zl "
...
f/1...ipd~i' A 9 A d~ ip ) ' _ zp
E
dfil...i~ A d~ it A ... A d~9ip,
l~_il<..,
where the second equality uses Axioms (2) and (3). But dfi~...ip is uniquely specified by Axiom (1). Thus, d ( ~ l i ) is unique and, as remarked above, the uniqueness in general of the operator d follows. Remark that all five axioms have been used in this argument. We turn to the proof of existence. Let {u.,~}~a
be the maximal coordinate atlas for M. The diagrams
Ap+I(~a(Uo~VI U~))
,
Ap+I(~z(U~ n UZ))
make sense and commute, Va,/3 c ~1, since everything is written for open subsets of Euclidean space. For a; E AP(M), define (d~)tU~ =
~ (*d ( ~ -~* (~lU~)))
Vo~
e
~a.
242
8. I N T E G R A T I O N
AND
COHOMOLOGY
If Ua V/UZ • ~, we check that the two definitions agree on Us A UZ. Indeed, by the commutative diagram, * --1. ~.(d(~s (~lgs n Uz))) = ~ g* ~r , *3- - 1 o d 0 g *~ ( ~ s- - 1 . ( ~ l U . n uz)))
(gg-J o ~s)* o d o (~o~1 o g~)*(w[Us n UZ)
~o*~(d(~o~l*(~lUs n UZ) ) ). Thus, the local definitions of dw piece together to give dw E A p+I(M). Since the axioms are true for open subsets of Euclidean space, they are true locally on M, hence globally. [] D e f i n i t i o n 8.1.7. A form w E AV(M), p >_ O, such that doa = 0 is called a closed p-form on M. Closed p-forms are also called (de Rham) p-cocycles and the real vector space of all such forms is denoted by ZP(M). We set Z*(M) = {ZP(M)}~= 0. D e f i n i t i o n 8.1.8. A form w C AP(M), p >_ 1, is said to be exact if there is a form C A p-1 (M) such that du = ~. Exact p-forms are also called p-coboundaries and the real vector space of all such forms is denoted BP(M). For p = 0, we define B~ = {0} C A~ Finally, we set B*(M) = {BP(M)}~= o.
Remark. A form ~ C AP(M) is locally exact if every point x E M has an open neighborhood U such that wlU C BP(U). One version of the Poincar~ lemma (Section 8.3) asserts that the set of locally exact p-forms on M is precisely ZP(M). In particular, our earlier definition of ZI(M) (Definition 6.4.1) agrees with our present one. E x e r c i s e 8.1.9. This exercise is in anticipation of the Poincar~ lemma. Define a manifold M to be contractible if it is homotopy equivalent to a point. Prove that the following three versions of the Poincar~ lemma for 1-forms are equivalent, and verify (3) when n = 2. (Here, ZI(M) denotes the space of closed 1-forms, not the space of locally exact ones.) (1) The form w E AI(M) is locally exact if and only if it is closed. (2) If M is contractible, then ZI(M) = BI(M).
(3) Z I ( N n) = BI(]Rn). The formula d 2 = 0 is equivalent to the inclusion BP(A//) C_ ZP(M), p > 0. D e f i n i t i o n 8.1.10. For each integer p _> 0, the pth (de Rham) cohomology space of M is the real vector space HP(M) = ZP(M)/BP(M). If ~o : M ~ N is smooth, the formula ~* o d = d o ~o* implies that
~*(ZP(N)) C_ ZP(M), ~*(BP(N)) G BP(M), so ~o* induces an R-linear map
V~* : HP(N) ~ HP(M). As usual, we have functoriality. L e m m a 8.1.11. The pth cohomology H p is a contravariant functor from the cat-
egory of differentiable manifolds and smooth maps to the category of real vector spaces and linear maps.
8.i. EXTERIOR
DERIVATIVE
243
If w E ZP(M) and ~ C Zq(M), then d(~ A ~/) = d~ A ~ + (-1)Pw A d~/= 0. T h a t is, the exterior product of a closed p-form with a closed q-form is a closed (p+ q)-form. Thus, Z* (M) is a graded algebra over N under exterior multiplication. Both A*(M) and Z*(M) are anticommutative in the sense of Definition 7.2.9. L e m m a 8.1.12. The graded N-algebra Z*(M) is connected if and only if M is a
connected manifold. Pro@ The space Z~ consists of all f C C a ( M ) with exterior derivative df = O. T h a t is, Z~ is the space of locally constant, real-valued functions on M. Identifying R with the space of constant functions in C ~ ( M ) , we have R C_ Z~ The product in Z*(M) of a constant flmction and a form becomes naturally identified with scalar multiplication. But locally constant functions are all constant if and only if M is connected. [] C o r o l l a r y 8.1.13. The space H~
is one-dimensional if and only if M is conneeted. In this case, H~ = ~ canonically. Generally, H~ is a direct product of copies of ]~, one for each component of M.
Proof. Indeed, H~ = Z~176 functions, and all claims follow easily.
= Z~
the space of locally constant []
L e m m a 8.1.14. [f d i m M = n, then HP(M) = O, Vp > n.
Proof. Indeed, AP(M) = 0 for all integers p greater than tile dimension of M.
[]
L e m m a 8.1.15. The graded subspace B*(M) C_ Z*(M) is a 2-sided ideal, hence H*(M) = Z * ( M ) / B * ( M ) is a graded, anticommutative algebra over" the field R.
Pro@ If a~ E ZP(M) and r/ C Bq(M), q _> 1, then r~ = da for some a E A ~ - I ( M ) , hence wA~/ = w A d ( ~ = d~ A ( - 1 ) P ~ + ( - 1 F w A d((-1)P~) = d(~ f ( - 1 ) ' ~ )
Since r] A aJ = (-1)PqoJ A r/, it follows that B*(M) is a 2-sided ideal in Z*(M).
[]
Since smooth maps ~ : M --~ N preserve exterior multiplication and pass to well-defined maps in cohomology, we see that
y)* : H*(N) --~ H*(M) is a homomorphism of graded algebras. We have completed the proof of the following. T h e o r e m 8.1.16. The graded cohomology construction defines a contravariant fune
tot H* from the category of differentiable manifolds and smooth maps to the category of anticommutative graded algebras overN and graded algebra homomorphisms. The graded algebra H* (M) is connected if and only if M is connected. The graded algebra H* (M) is called the (de Rham) cohomology algebra of M. Whether or not it is connected, H* (M) has a unity, namely the constant function
1 e Z~
= H~
244
8. I N T E G R A T I O N A N D C O H O M O L O G Y
D e f i n i t i o n 8.1.17. The space of compactly supported p-forms on M is denoted by APc(M). It is clear that the exterior pactly supported. Indeed, it is Thus, each APc(M) is a module gebra A*(M) over C~176 It our purpose. Furthermore,
product of two compactly supported forms is comenough that one of them be compactly supported. over C~176 and these assemble into a graded alis also a graded algebra over IR, which is more to
d(A~(M)) C_ A~+I(M), so one can define the space
ZP(M) = {~ e AP(M) I &z = 0} and the vector subspace
B~(M) = {~ = da I a e A~-I(M)}. If we use the common convention that A -1 (M) = A~-1 (M) = 0, the above definition of BP(M) includes the case p = 0. As before, Z~(M) is a graded subalgebra of A*~(M) and B c ( M ) C Z*(M) is a 2-sided ideal. D e f i n i t i o n 8.1.18. The (de Rham) cohomology algebra with compact support is
H* (M) : Z c ( M ) / B c (M). Remark that H*(M) = H*(M) if M is compact. At the other extreme, if M has no compact component, the space Z~ of compactly supported, locally constant functions on M is trivial, so H~ = 0. In any event, each element of Z~ will vanish on all but finitely many components of M. These observations establish the following. L e m m a 8.1.19. The vector space H~
is isomorphic to a direct sum of copies of ~, one for each compact component of M.
Note the different conclusions in Lemma 8.1.19 and Corollary 8.1.13. Each element of a direct sum has terms from only finitely many summands, while elements of a direct product are allowed to have terms from infinitely many of the factors.
Remark. The graded algebra H i (M) generally does not have a unity unless M is compact. Recall that a smooth map F : M --~ N is said to be proper if, for each compact set C C_ N, the set T - I ( C ) is also compact. For example, id : M --+ M is always proper. If M is compact, ~ is always proper. In any event, the composition of proper maps is proper, so the class of differentiable manifolds and smooth, proper maps between them is a category. If qp : M --* N is proper and if co E AP(N), then F*(w) E AP(M). As usual, 9)* o d = d o ~*, so we get an induced homomorphism of graded algebras ~ * : H c(N) ~ H 2(M). T h e o r e m 8.1.20. Cohomology with compact supports is a contravariant functor
H~ from the category of differentiable manifolds and smooth, proper maps to the category of anticommutative graded algebras over ~ and graded algebra homomorphisms. E x e r c i s e 8.1.21. Prove that Hcl(~) ~- R. This is the one-dimensional case of the Poincar~ lemma for compactly supported cohomology.
8.2. STOKES' THEOREM
245
8.2. S t o k e s ' T h e o r e m a n d S i n g u l a r H o m o l o g y
In this section, we define integration of forms and give a detailed treatment of two versions of Stokes' theorem. As an application of the second (combinatorial) version, we define the singular homology of a manifold and relate it to de Rham cohomology, stating the celebrated de Rham theorem. A detailed sketch of the proof of this theorem will appear in Section 8.9. Throughout what follows, we assume that dim M = n and that M is oriented. We also allow 0 M ~ 0. T h e o r e m 8.2.1. For each oriented n-manifold M , there is a unique R-linear func-
tional ~
: A2(M ) ~
R,
called the integral and having the following property: if (U, ~) is an orientationrespecting coordinate chart, if co E A ~ ( M ) has supp(a~) C U, and if ~-l*(w) = g d x 1 A . . . A dx ~ e A2(p(U)), then /MCZ=~
(v)
g
(the Riemann integral).
Proof. First we prove uniqueness. Let {(Us, ~ ) } ~ e ~ be a smooth Hn-atlas on M respecting the orientation. Let {A~}~e~ be a smooth partition of unity subordinate to the atlas. If ~ E A~(M), then A ~ e A ~ ( M ) and A ~ r 0 for only a finite number of ~ C 92. This is because supp(w) is compact and the partition of unity is locally finite. Thus, = E AaaJ ~692
and this sum is actually finite. Then, if fM exists, linearity gives
and s u p p ( A ~ ) = supp(A~) N supp(~) is a compact subset of Us. By the local property of fM, each fM A ~ is uniquely given as
where g~ dx I /k. .. A dx, n : ~ l * ( ~ l V a ) . We give one way to define fM, establishing existence. This will depend on a choice of orientation-respecting coordinate atlas {U~, ~ } ~ e ~ and of subordinate partition of unity {A~}~e~. (One could remove some of this arbitrariness by requiring the atlas to be the maximal one, but the choice of partition of unity still could not be made canonical.) We appeal to the uniqueness already proven to show independence of the choices. If aJ C A~(M), only finitely many A ~ are not identically 0. Define =
~
~(u~)
)g~,
246
8. I N T E G R A T I O N A N D C O H O M O L O G Y
where g~ d x I A . . . A d x n
= ~jl*(o2lUc~).
Then
define
a finite sum. Defined in this way,
is an JR-linear map.
We must check that, if supp(w) C U, where (U, ~) is an arbitrary orientationrespecting coordinate chart (not necessarily in our atlas), and if cfl-i*(w) = g d x I A . . . A d z n,
then M w = L ( u ) g' First remark that
= E
~
U~m U)
c~e~ ~ (
(,,~ 0 (~a,1)go~,
since supp(w) C U. By Exercise 7.5.22, (Aa~ = L (Ac~o~-l)g, ~,(s~nu) (u~nu) for each a C 92. The fact that the charts are compatibly oriented is essential. Thus, aEPa
(U~nU)
=
(a~ c~C92
J1 =L
(u)
9. []
Remark. By Exercise 7.5.22 and the above proof, _MO2 = -- / M ~
The orientation of M induces an orientation of O M in the following way. Let {Us, x ~1 , . . . , x n~ } ~ be an IHI~-atlas on M respecting the orientation. By the deftnition of H ~, x~i < _ 0, wherever defined, and x~1 = 0 exactly on Us n OM, V a E 92. Let 9,1' = { a C 92 I Us N O M r 0} and consider the N~-l-atlas
{us n OM,xL...
8.2. S T O K E S ' T H E O R E M
247
of O/V/. Let g~z and gaz~ denote the respective changes of coordinates for these atlases. At x E U~ n UZ n O M , rOx~
0 gc~
] z
Here, since x~ decreases with x~, the upper left-hand entry in this matrix is strictly positive. Since d e t ( J g ~ ) x > 0, it follows that d e t ( J g ~ > 0. Thus, this IRn - I atlas on 0 M defines an orientation of 0 M . D e f i n i t i o n 8.2.2. The orientation of O M , produced as above, is said to be induced by the given orientation of M. We always assume that, when M is oriented, O M has the induced orientation. The following fundamental result asserts that exterior differentiation is the "adjoint" of passing to the boundary. For this reason, d is often referred to as the "coboundary operator". T h e o r e m 8.2.3 (Stokes' Theorem). L e t M be an oriented n - m a n i f o l d a n d let i : O M ~ M be the inclusion. T h e n , i f w E A ~ - I ( M ) ,
where, i f O M = ~, the r i g h t - h a n d side is interpreted as O. Proof. First we prove the local case. That is, M = H n and O M = IR~-1. Any a~ C A ~ - I ( H n) can be written as n
a) = E ( - 1 ) J - l f j d x
1 a'.'AdxJ
A".Adx
n,
j=l
where f j has compact support, 1 < j < n, and d x j indicates that this term is omitted. Then i*(co) = (fl
o
i) d x 2 A . . .
A
d x n E An-I(cO]H[n)
and dw =
_
OxJ J d x l A . . . A d x n C A~c~(tHIn).
By the fundamental theorem of calculus and the compactness of s u p p ( f j ) , for j = 2 , . . . , ?z,
/? 2o, oo""
oo ~ x J d x l "'" d x n =
~ '"
~ \J_~
OxJ d x j
d x l "" " d x j "" " d x n = O.
248
8. I N T E G R A T I O N
AND
COHOMOLOGY
Therefore,
L
dco = n
F
/-cx~; OO .
=
.
Co .
Ofl " 1
dx n
. Oo . ~ x l a.X
l i l?; "
l(O, x 2 , . . .
,z ~)dx 2"''dx ~
oo
9
fl oi
: L N ~ i*w.
Now we can prove the global case. Let i M : O M '---* M ,
iH~ : OH n ~ H n
denote the respective boundary inclusions and let
{u~,~ = (~L... , ~n ) } ~ , { U~, M O M , ~o~ = ( x 2 , . . .
,x2)}~,
be the orientation-respecting atlases on M and O M , respectively, as chosen above. If { A ~ } ~ is a smooth partition of unity subordinate to the atlas on M, then {ha o iM}aE~l' is a smooth partition of unity subordinate to the atlas on O M . Note that ~ 1 oiH~ = i M o (~0)--1, E For w E A n - I ( M~ ~ j, the fact that w = ~ A~w is a finite sum gives finite sums
VO~ ~[/
d~ = ~ d(a~), aE~
*
% M a) =
E ~;~(~).
aE~'
Therefore,
L
&o=
Y~ fuo d ( ~ )
c~E9.1
E L (r aE~l o(U~)
= ~ f~
aE~ ~ (U~)
=
EL,,
d(~21"(~))
~Hn~ _1,
c~6 9 / '
(d()~aca))
(~).
H~
The last equality is by the local version proven above. If O M = O, then supp(w) A O M = O,
8.2. STOKES' THEOREM
249
and this integral vanishes. Otherwise, we get dw =
r /o
=
a ~ j~ ,
(~)
~M(;~o)
( ~O ) -1..,~ M ( ~ ) ,,(U~nOM)
o~EgX'
~nOM
= F. /
(x~ o i,~,)iS(~)
o~Ega' U ~ n O M =
7,A/IOO. hi
[] Theorem
8.2.4. Let M be an oriented n-manifold with O M = ~). Then f
: H ~ ( M ) --,
is a well-defined, N-linear surjeetion. Proof. Since A ~ + I ( M ) = 0, we have Z ~ ( M ) = A ~ ( M ) . I f w = dr] E Bn~ ( M ) , then
Stokes' theorem and the fact that cgM = {3 imply that
Thus, the linear map
~ : Zg(M) -~ r
induces a well-defined linear map M : H n ( M ) --' N.
To prove surjectivity, we only need prove that this map is nontrivial. Let (U, x 1, ... , x n) be a compatibly oriented chart and let A G C ~ 1 7 6 have compact support contained in U, with A _> 0 everywhere and ,~ > 0 somewhere. Thus co = A dx 1 A ... A dx '~ can be interpreted as an element of Z 2 ( M ) and of A~(IR~), so /Me~ = s
A>0' []
A deeper fact, to be proven later (Theorem 8.6.4), is that, if M is both oriented and connected, then fM is a bijection from H ~ ( M ) to IR. In order to integrate p-forms, where p < dim M, it is necessary to define suitable p-dimensional domains of integration. For the case p = 1, we have already studied lille integrals, the domain of integration being a (piecewise) smooth curve in M. In general, it is convenient to use singular p-simplices (defined below) as domains for integrating p-forms. A singular 1-simplex is simply a smooth curve. Recall t h a t a subset A C ]Rp is convex if, for each pair of points v, w G A, the straight line segment joining v and w lies entirely in A. If C C IRv is an arbitrary
250
8. I N T E G R A T I O N A N D C O H O M O L O G Y
subset, the c o n v e x h u l l of C is defined to be the smallest convex set C containing C. Since an arbitrary intersection of convex sets is convex, and NP is itself convex, is just the intersection of all convex sets containing C. D e f i n i t i o n 8.2.5. The standard p-simplex /~p (~ ]I~p is the convex hull of the set { e 0 , e l , . . . ,ep}, where ei is the ith standard basis vector, 1 < i < p, and e0 = 0. Thus, A0 = {0}, a single point, and A1 = [0, 1]. The cases p = 2 and p = 3 are pictured in Figures 8.2.1 and 8.2.2, respectively.
F i g u r e 8.2.1. The standard 2-simplex
F i g u r e 8.2.2. The standard 3-simplex A more explicit definition of the standard p-simplex is Ap =
( x l , . . . , x P ) leach x i > 0 and ~ x
i _< 1 .
i=1
It is sometimes convenient to set A_I = 0. D e f i n i t i o n 8.2.6. A (smooth) singular p-simplex in a manifold M is a smooth map s : Ap --~ M. Thus, each point of M can be thought of as a singular O-simplex and smooth curves, up to parametrization, are singular 1-simplices. One could also define piecewise smooth singular p-simplices, but we will not do so.
8.2. S T O K E S '
THEOREM
251
D e f i n i t i o n 8.2.7. For 0 < i < p, the ith face of the standard p-simplex Ap is the singular ( p - 1)-simplex Fi : Ap-1 --* Ap defined by Fi(xl'""xP-1)
l , . . . ,xi--l,0, x i , . . . ,3::p - l ) = [(1--x 1 ..... xP-I,Xl,...,X p-l)
f(x
i f / > 0, if/=0.
The 0th face of A0 is considered to be defined but empty. If s : Ap ~ M is a singular p-simplex, the ith face of s is tile singular (p - 1)-simplex Ois = s o Fi. It is clear t h a t Fi : Ap_I --* Ap is a topological imbedding and t h a t the image of Fi is exactly the subset ordinarily thought of as the "face" of Ap opposite the vertex el. The ith face Ois of a singular p-simplex s is essentially the restriction of s to the ith face of Ap, but parametrized on the standard Ap_I. 8 . 2 . 8 . If s : Ap --+ M is a singular p-simplex and co E AP(M), then s*(co) has the form g d x I A . . . A dx p and we set
Definition
p
where the right-hand side is tile Rietnann integral. If s : {0} ~ M is a singular 0-simplex and co = f E A ~ the integral is interpreted to mean f f = f(s(O)). There is a combinatorial version of Stokes' theorem, according to which the integral of an exact p-form dr/over a singular p-simplex s is equal to the integral of r/over the "boundary" of s. T h e o r e m 8.2.9 (Combinatorial Stokes' Theorem). I f s : Ap --~ M is a singular p-simplex and r~ C A P - I ( M ) , then P i=0
is
Remarks. The signs in the combinatorial Stokes' theorem are dictated by comparing the s t a n d a r d orientation of Ap_ 1 with the induced orientation of Fi (A v_ 1 ) &q a part of the b o u n d a r y of Ap. We write the fornml combinatorial expression P
08 = y ~ ( - - 1 ) % S /=1
and express Stokes' formula as fdr/=
f0s r/.
This highlights the analogy with Theorem 8.2.3 and agrees with established usage in algebraic topology. For f E A ~ and a smooth curve s : [0, 1] --* M, Theorem 8.2.9 asserts that s d f = f(s(1)) - f(s(0)), which is just Lemma 6.3.3. T h a t lemma was a thinly disguised version of the fundamental theorem of calculus and Theorem 8.2.9 is a somewhat less thinly disguised version of the same fundamental theorem.
252
8. I N T E G R A T I O N
AND
COHOMOLOGY
Proof of theorem 8. 2.9. It is clearly sufficient to prove that
/ dv=~,(-1)is n;(V), P
P
i=O
p--1
where ~/is a (p - 1)-form defined on an open neighborhood of Ap in N p. We can write P
=
--~.s dx 1 A . . . A~XJ
A...
A dx v
j=l
and, by the linearity of the integral, prove the formula for each term of the sum. That is, without loss of generality, we assume that 1 _< j _< p and = f dx I A 999 A dxJ A 999A dx p. By the local formula for exterior differentiation,
of
@ = (-1) j-l~dx
1 A...Ad2,
and we are reduced to proving the formula (8.1)
fAp(-1)J-l~dxl
A...AdxP
= E(-1)
i
F * ( f d x 1 A . . . A dxJ
i=0
A...
A
dxP).
v- 1
The right-hand side of equation (8.1) can be simplified. For this, it will be helpful to let x i denote the coordinates in N p and z i the coordinates in N p-1. Remark t h a t fdz j-1 ~ - E P - : dz i
FG (dx~ )
i f j > 1,
if j = l ,
and, if i > 0, zj
if j < i, i f j = i,
( dz j - 1
ifj >i.
I~
F [ ( d x j) = One obtains the formula
F ~ ( f dx 1 A ... A dz"~ A ... A dx v) = ( - 1 ) J - l ( f o Fo)dz 1 A . . . A d z p-1 and, if i > 0, F*(fdxl
A A'"AdxJ
A'"AdxP)=
l0 ~,(foFj)dzl
A...Adz
p-1
if/~j, ifi=j.
Substituting these terms in the equation (8.1) and multiplying both sides by ( - 1 ) j - 1 reduces us to proving /A
O f dxl A . . . A dxP = / A ( f o Fo) dzl A . . . A dz p-I r, OxJ p1
-
f J~
( f o iaj) dz 1 A ' " p--1
Adz p - l ,
8.2. STOKES' THEOREM
253
which, rewritten in terms of the Riemann integral, becomes (8"2)
/ ~ p OxJ 0 f = //x ,,-1 f(1 -- Z 1 . . . . .
Zp - l , Z 1 , . . . , Zp - l ) -- f
f(zl, "'',zj-I,0,zj,''',zp-1)" p- 1
The linear change of coordinates in ]Rv-l, defined by f l-z 1 ..... w i = J z i-1 (z i
if i = 1, if2
z p-1
preserves volume ( i.e., the Jacobian deternfinant is -t-1) and carries Ap-1 diffeomorphically onto itself, as the reader will easily check. Also, Z j - 1 -~ 1 -- W 1 . . . . .
Wp - 1 .
While this coordinate change reverses orientation when j is odd, the Riemann integral is insensitive to orientation, so (8.2) becomes /A
Of =/A p OxJ
f(wl' v- 1
'wJ-l'l--wl . .
.
.
..... .
-- f
wP-I'wJ .
'WP-1)
f(zl, "'',zj-I,0,zj,''',zp-1)"
3 ZXp-1
To avoid notational confusion, replace the dummy variables w i with zi: (8.3)
/zx Of = /zx f(zl p c~xJ ~,_~ '9
'zJ-l'
] -- Z 1 . . . . .
zP-1 Z j . . . ~ z p - l )
f(zl'"" 'zJ-l'o'zJ"'" 'zP-1)"
-- / A p-1
Let nJ = [Pj(Ap_l) = {(xl,...
, x p) E A p I x j ~. 0},
the face of Ap opposite the vertex ej, 1 _< J -< P. Since j > O, it is evident that the linear diffeomorphism FS : Ap-1 -~ Z p - I ) ~- ( Z l , . . . ,
fj(zl,...,
A~,
Z j - 1 , O, Z J , . . . , Z p-1 )
preserves (p - 1)-dimensional volume, so the three integrals in (8.3) can be computed, respectively, by the multiple integrals
fA,, Of d x l . . , dx', OxJ ~
f
zl,...,X
j-l,1
-- E
xi'zj+l''"
, x p dx 1 "'" d z J . . . d x p,
ir
j,f ( x l , . . . ,
x j - i , O, x J + l , . . . , 2 ) dx 1 . .. dxJ 9 .. dx p.
With these substitutions, (8.3) is checked by standard manipulation of iterated integrals and an application of the fundamental theorem of calculus. The proof of Theorem 8.2.9 is complete. []
254
8. I N T E G R A T I O N A N D C O H O M O L O G Y
C o r o l l a r y 8.2.10. A f o r m co E A P ( M ) is closed if and only if los co = O, for every singular (p + 1)-simplex s in M . Proof. If co is closed, then
/0,
=0
For the converse, suppose that d~ = r/ r 0. Choose a point x E M such that r/x r 0. Choose vectors v ~ , . . . , v v + l E T x ( M ) such that r/~(vl A . . . Avp+x) > 0. These vectors must be linearly independent, so we can find a local coordinate chart (U, x l , . . . , x ~) about x in which vi is the value of the ith coordinate field ~i = O/Ox i at x, 1 < i _< p + 1. By making this chart sutficiently small, we can guarantee that r/(~l A ... A ~;+1) > 0 on all of U. Let s : Ap -~ U be any orientation-preserving, smooth imbedding into the coordinate (p + 1)-plane { ( x l , . . . , x n) E U [ x p+2 = . . . . x ~ = 0}. It follows that
[] Singular simplices are used to detect topological features of a manifold. For instance, a piecewise smooth, closed curve s = Sl + 999+ Sq in the punctured plane IR2 \ {(0, 0)} can detect the missing point, provided that s has nonzero "winding number" w ( s ) about the origin. The closed curve s is assembled from the singular 1-simplices s l , . . . , Sq which join together, end to end, to form a "i-cycle" and the winding number itself is defined by integrating the locally exact (hence, closed) 1-form r/ of Example 6.3.12 over s. Piecewise smooth closed loops s in R 3 \ {(0,0,0)} do not detect the missing point (Example 6.4.9). However, a map s : S 2 --~ IR3 \ {(0,0,0)} can snag the missing point. One effective way to use this observation is to triangulate S 2 (Section 1.3) and form singular 2-simplices Sl,. 9 Sq by restricting s to these triangles. It is necessary to assume only that each si is smooth, so we get a piecewise smooth map s: s 2 -~ R ~ \
{(0,0,0)}
and write s = Sl + "" + Sq by analogy with the case of loops. We call this a "singular 2-cycle". One should test whether or not the singular 2-cycle has snagged the missing point x by integrating a suitable closed 2-form co over this cycle: q
The possibilities for singular 2-cycles are richer than for 1-cycles. For instance, triangulations of T 2 and corresponding piecewise smooth maps s of T 2 into M define "toroidal" singular 2-cycles s = sl + "" + Sq in the manifold M and such a cycle might well detect a topological feature that would be missed by a "spherical" cycle. Again, a test of what this cycle detects is made by integrating closed 2-forms over the cycle. These remarks are extended and made precise by defining the singular homology of a manifold, a covariant functor H , from the category of smooth manifolds to the category of graded vector spaces over IR. The celebrated de Rham theorem asserts that this functor is dual to de Rham cohomology. We sketch the main
8.2. S T O K E S '
THEOREM
255
facts, illustrating the importance of the combinatorial Stokes' theorem for algebraic topology. D e f i n i t i o n 8.2.11. The set of all singular p-simplices in M, p > 0, is denoted by A v ( M ). The space C v ( M ) of singular p-chains on M is the free N-module (real vector space) generated by the set A v ( M ). By convention, if p < 0, A v ( M ) = and Cp(M) = O. Each p-form co E AP(M) can be viewed as a linear functional
co: Cp(M) --~ R as follows. An arbitrary p-chain c C Cp(M) can be written uniquely (up to terms with coefficient 0) as a linear combination m
C = Eajsj~ j=l
where sj E A p ( M ) , 1 <_ j _< m. One then defines the value of a~ on c to be
j=l
J
The face operators Ois = s o Fi have already been defined, Vs E A v ( M ) , and can be viewed as set maps 0i : ZX,(M) -~ C p _ , ( M ) . Since A p ( M ) is a basis of Cv(M), these set maps extend uniquely to linear maps
0~ : Cp(M) --+ Cp_I(M). D e f i n i t i o n 8.2.12. The boundary operator 0: Cp(M) --. C~_~(M), p _> 0, is the linear map P
o =
Z(-~)'o~. i=0
For co E A p - I ( M ) and c ~ @ ( M ) , the combinatorial Stokes' theorem asserts that cz(0c)=/acW=fcdW=da~(c)' T h a t is, the operators d and 0 are adjoint to one another. The boundary operator is an algebraic analogue of the geometric notion of a boundary. The following crucial property can be viewed as the algebraic analogue of the fact that the boundary of the boundary of a manifold is empty. E x e r c i s e 8.2.13. Prove that the composition
Cp(M) o Cp_~(M) a Cp_2(M) is trivial (0 2 = 0). D e f i n i t i o n 8.2.14. The subspace Zv(M ) C_ C v ( M ) of all (singular) >cycles is the kernel of the boundary operator cq: Cp(kJ) --~ Cp_I(M). The subspace Bp(M) C_ Cp(M) of all (singular) p-boundaries is the image of the boundary operator 0 :
Cp+I(M) ---+C,,(M). An immediate corollary of Lemma 8.2.13 is that Bp(M) C Z v ( M ).
256
8. I N T E G R A T I O N A N D C O H O M O L O G Y
D e f i n i t i o n 8.2.15. T h e pth singular homology of M is the vector space
Hp(M) = Z p ( M ) / B p ( M ) . If z e Zp(M), the homology class of z is the coset [z] E Hp(M) represented by the cycle z. E x a m p l e 8.2.16. Since C-1 (M) = 0, the b o u n d a r y operator vanishes identically on Co(M). T h a t is, Zo(M) = Co(M) is the real vector space with basis the set of points of M . Define c : Zo(M) ~ N by aixi \i=1
=
ai, i=1
where a l l a i E ]R and all xi E M. I f s E A I ( M ) , e(Os) = e ( s ( 1 ) - s ( 0 ) ) = 0, so B o ( M ) C_ ker(c) a n d ~ passes to a well-defined linear m a p g : Ho(M) --~ R. We assume t h a t M ~ !~, so there is a point x E M a n d ~([x]) = 1, proving t h a t ~ is a surjection a n d Ix] 7~ 0, Yx c M. If M is connected, then every two points x , y E M can be joined by a piecewise smooth p a t h s = sl + ... + sT a n d Os = y - x. This implies t h a t [x] = [y], hence that H0 (M) has basis consisting of a single element [x]. We have proven t h a t the 0th singular homology of a nonempty, connected manifold is canonically isomorphic to JR. E x a m p l e 8 . 2 . 1 7 . Let M be contractible (cf. Exercise 8.1.9) with contraction ~t : M --~ M. This is a homotopy of P0 = idM to a constant m a p ~1. Using this contraction, we are going to define linear maps
Lp : Cp(hl) --~ Cp+I(M), Vp > 0, with a remarkable property. T h e s t a n d a r d inclusion R T ~-~ Rp+I restricts to an inclusion A T ~-~ Ap+l which is j u s t the face m a p Fp+l. For each point v E Ap+l \ {eT+l}, there is a u n i q u e point v' E A T and a unique n u m b e r t C [0, 1] such t h a t
v = teT+l + (1 - t)v', a n d every point oflR p+I of such a form is a point in Ap+l. (For v = ep+t, v' is not unique, b u t t = 1, so this will cause no problem in what follows.) If s : Ap --~ M is smooth, define a s m o o t h m a p Lp(s) : Ap+l ~ M by the formula
Lp(s)(tep+l + (1 - t)v') = ~t(s(v') ). T h e fact t h a t this is well defined when t = 1 is due to ~1 being a c o n s t a n t map. We view s H Lp(s) as a set m a p
Lp : A p ( M ) ~ Cp+l(M) a n d take the linear m a p L T to be the unique linear extension of this set m a p to all of CT(M ). In Exercise 8.2.18, you are invited to check t h a t
(,)
0 o Lp = Lp-1 o 0 -k ( - 1 ) p+I idcp(M),
provided t h a t p _> 1. This is the remarkable property promised above. If z E Z T ( M ) a n d p > 1, it follows t h a t
O(LT(z)) = Lp_l(OZ ) + (--1)P+lz = (--1)T+lz, hence t h a t Z v ( M ) C Bp(M). The reverse inclusion also holds, so we have the result t h a t the singular p-cycles and the singular p-boundaries in a contractible space are exactly the same, Vp >_ 1. T h a t is, HT(M ) = 0 in all degrees p > 0.
8.2. S T O K E S ' T H E O R E M
257
E x e r c i s e 8.2.18. Prove the identity (*) in Example 8.2.17. The above two examples give T h e o r e m 8.2.19. I f M is a contractible n-manifold, then Hp(M)
= ~,
p = o,
[ 0,
p>0.
In particular, this is true for M = ]R~. P r o p o s i t i o n 8.2.20. I f co C ZP(M) and z ~ Zp(M), then the real number fzco depends only on the cohomology class [co] C H P ( M ) and the homology class [z] C
H~(M). Proof. Indeed, [co] is the set of all closed p-forms co + dTi, where ~ C AP-~(M). We have
by Stokes' theorem and the fact that z is a cycle, so
Similarly, [z] is the set of all p-cycles of the form z + Oc, where c E Cp+~ (M). Since
we obtain +0c
c
[] Thus, we can define an R-linear map D R : H P ( M ) -+ ~ , ( M ) * , by (DR[co])([z]) = [
[co]. )
T h e o r e m 8.2.21 (The de Rham Theorem). The linear map DR is a canonical isomorphism of vector spaces. This is a deep result. For the case in which M is compact, the proof will be discussed in some detail in Section 8.9. In that case, the vector spaces are finite dimensional (Theorem 8.5.8), so we also get H p ( M ) = H P ( M ) *. The following corollary generalizes Proposition 6.4.3. C o r o l l a r y 8.2.22. Let co,~ C ZP(M). Then [w] = [c5] if and only if fz w = fz ~ as z ranges over all singular p-cycles in M . These numbers are called the periods of w and of the cohornology class [co]. In particular, co is an exact form if and only if all of its periods are 0, which generalizes the equivalence of properties (1) and (2) in Theorem 6.3.10. E x e r c i s e 8.2.23. Let M be an n-manifold and let z E Z~+I(M). Assuming the de Rham theorem, prove that there is a chain c E C , + 2 ( M ) such that z = Oc. E x e r c i s e 8.2.24. Show that singular homology is a covariant functor, proceeding as follows.
258
8. I N T E G R A T I O N A N D C O H O M O L O G Y
(1) If f : M --~ N is a smooth map between manifolds, exhibit a canonical way to induce a linear map f # : Cp(M) ~ Cp(N), Vp >_O. (2) Prove that the diagram
Cp+l(M)
f#
~
Cp+l(N)
o~ Cp(M)
1o f#
,
Cp(N)
commutes, Vp _> 0. Conclude that the linear map f # passes to a linear map f , : H , (M) ~ / 4 , (N) of graded vector spaces. (3) Verify the properties ( f o g), = f , o g, and id, = id. (4) Under the de Rham isomorphism of g k ( M ) with Hk(M)*, show t h a t
f* : Hk(N) ~ Hk(M), f, : Hk(M) --+ Hk(N) are adjoint to each other. E x e r c i s e 8.2.25. W i t h o u t appealing to the de Rham theorem, extend the argument in Example 8.2.16 to show that Ho(M) is a direct sum of copies of N, one for each connected component of M. 8.3. T h e P o i n c a r 6 L e m m a
In the following discussion, R will stand for any nondegenerate, compact interval [a, b] or for JR. We consider an arbitrary n-manifold M, not necessarily orientable. If M has nonempty boundary, M x R will always denote M x ~, thereby avoiding manifolds with corners. Homotopies, therefore, will be understood in the sense of Definition 3.8.9 whenever convenient. We will agree to denote the s t a n d a r d projections by
lr : M x R--* M and
p:MxR--~R. The coordinate of R will be denoted by t and p* (dt) E A 1(M x R) will be denoted by dt (an abuse). A locally finite atlas {(W~,x~)}~e~ on M determines such an atlas {(W~ • R , ( x ~ , t ) ) } ~ e ~ on M x R. Here, i f 0 M = ~ and R = [a,b], we model ( n + l ) manifolds with boundary on Rn x [a, b] instead of on IHIn+l. A smooth partition of unity {A~}~e~ , subordinate to {W~}~e~, determines a smooth partition of unity
on M x R subordinate to {W~ x R}~e~. Here,
~ (x, t) = As (x). If w C A k ( M x R), let
~ = ~ l ( W ~ x R) and write
aJ~ = E f~ dxI A dt + E g~ dxJ' I J
8.3. POINCARE LEMMA
259
where we use the conventions I=
i1,i2,...
J=jl,j2,...
,/k-l,
l <_il < i 2 < . . . <
i k - i <_n,
,jk,
l <_ j l < j2 < "" < jk <_ n,
and dx~ = dx~~ A dx~2 A .. . A dx~k-~, dX Joe = d x jl A dxJot2 A 999 A dx~Jk .
Since o~E92
we write
=
xof
dxt A
+ J
For each a E 9/, choose Oa : M --+ [0, 1] with supp(O~) C W~ and O~]supp(A~) -= 1. Let O~ = 7r*(O~). Then, A~O~ = A~ and J
o~E ~I.
Each x E M has an open neighborhood U such that only finitely many indices a E 91 correspond to nonzero terms in the expression for wl~r-l(U). Also, for 7F(71),for some 77 E A q ( M ) . q = k o r k - 1 , 0- a d x ai~ A . . A . d x. k L e m m a 8.3.1. Each f o r m w E A k ( M x R) can be expressed as a locally finite sum of k-forms, each being one of the following two types: (i) f ( x , t ) dtATr*(77) , 77 E A k - I ( M ) , ( i i ) f(x,t)Tr*(~]), ~ E A k ( M ) . We construct an important operator which "integrates out" the dt component of forms on M • R. This operator is a special case of an operator in algebraic topology called "integration over the fiber". L e m m a 8.3.2. For each T E R and each integer k > O, there is a unique ]R-linear map ST : A k ( M • R) --~ A k - I ( M x R) which is additive over locally finite sums and satisfies t
,
(a) S~ ( f ( x , t) dt A w* (~l)) = (f~ f ( x , u)du)rc (~), ( b ) S , ( f ( x , t ) n * ( V ) ) = O. (Here we understand that A - ~ ( M x R) = O, so ST : A ~ x R) ~ 0 is trivial, consistent with the fact that all forms in A ~ x R) are of type (ii).) Proof. By the existence of a decomposition of w E A k ( M x R) into a locally finite sum of forms of the types (i) and (ii), the stipulated properties of ST force that operator to be unique, provided that it exists. But, if we fix the choice of locally finite atlas {Wa,x~}~c~, as well as the choice of subordinate partition of unity {A~}~E~ and of the functions {O~}~e~, we then have an algorithm for producing a locally finite decomposition of w E A k ( M x R) into the desired types of summands. We use (a) and (b) to define S~ on each of these summands and remark that the
260
8. INTEGRATION AND COHOMOLOGY
result is a locally finite system of (k - 1)-forms on M x R, hence that their sum is a well defined element ST(co) E A k - I ( M x R). It is clear that S~, defined in this way, is N-linear. The crucial fact that it is also additive on locally finite sums is left as Exercise 8.3.3. Uniqueness shows that the definition of Sv is really independent of the choices. [] E x e r c i s e s . a . a . Prove that the operator • on locally finite sums.
in Lemma 8.3.2
is, indeed, additive
For each r E R, let ir : M --+ M x R be given by iv(x) = (x,r). One version of the Poincard lemma is that, at the cohomology level, rr* and i ; are mutually inverse isomorphisms. Indeed, it is clear that i ; o 7r* = (Tr o iv)* is the identity at the level of forms, so it remains to show that rr* o i ; is the identity on cohomology. The main step is the following. E x e r c i s e 8.3.4. On A k ( M x R), prove that the operator Sv satifies the identity d o S r + Sv O d = id-~r* o i : ,
VrER,
Vk>O.
T h e o r e m 8.3.5 (Poincard Lemma, Version I). The map
~* : I 4 " ( v ) --, H * ( M • R)
is an isomorphism and its inverse is i~, g r E R. In particular, at the cohomology level, i; is independent of r. Proof. As remarked above, we only need to prove that, at the cohomology level, 7r* o i~ = id. I f w C ZP(M x R), we apply Exercise 8.3.4 to obtain co - ~*(i;(co)) = d ( & ( ~ ) ) + Sv(d(~)) = d ( & ( ~ ) ) . That is, co and rr*(i*(co)) differ by a coboundary, and we are done.
[]
T h e o r e m 8.3.6 (Poincard Lemma, Version II). If fo, fl : M --+ N are homotopic,
then f~ = f{ : H* (N) --+ H* (M). Pro@ Let F : M x N -+ N be the homotopy. Then f0 = F oi0 and f l = F o i l . By functoriality, fa = / ; o F * , f~ = i7 o F * . But i~3 = i~ by Theorem 8.3.5, so f~ = fi ~.
[]
Here are four more versions of the Poincar4 lemma. The first of these is iramediate by Theorem 8.3.6, and each implies the next. All of the implications are rather obvious. T h e o r e m 8.3.7 (Poincar4 Lemma, Version III). If f : M --~ N is a homotopy equivalence, then f* : H*(N) --~ H * ( M ) is an isomorphism of graded algebras. T h e o r e m 8.3.8 (Poincar6 Lemma, Version IV). If M is a contractible manifold, Hk(M)=
In particular, this holds for M = N n.
N,
k = 0,
0,
k>0.
8.3. POINCARF, LEMMA
Theorem
261
8.3.9 (Poincar~ Lemma, Version V). For k > O, every closed k-form on
a contractible manifold is exact. Since manifolds are locally contractible (each point has a neighborhood diffeomorphic to Rn), the next version follows. Theorem
8.3.10 (Poincar~ Lemma, Version VI). If k > O, a k-form on a mani-
fold M is closed if and only if it is locally exact. Thus, the definition of H ~(M) given in Chapter 6 agrees with our current definition. It can be shown t h a t Version VI implies Version I, so all versions are mutually equivalent. We will not prove this. D e f i n i t i o n 8.3.11. If OM = (~ and f0, f l : M --~ N are proper smooth maps, they are said to be properly homotopic if there is a proper smooth map
F:M
x [0,1] --~ N
such t h a t F(x,O) = fo(x) and F ( x , 1) = fl(x), g x E M. The map F is called a proper homotopy between f0 and f l . For compactly supported cohomology on manifolds without boundary, define the term "homotopy" using R = [0, 1] and remark that both 7c* and i ; are proper maps. Theorem 8.3.5 continues to hold in this situation and a suitably reworded version of Theorem 8.3.6 also holds (Exercise 8.3.19). One could call this the Poincar6 lemma, but what usually goes by that name for compact cohomology takes a rather different form. This is our next topic. First note that co E A ~ ( M x R) is a finite linear combination of forms of the types
(i) f ( x , t ) d t A 7r*(r}), where f ( x , t ) is compactly supported, but the form r; E A k-1 (M) may not be compactly supported.
(ii) f ( x , t)Tr*(r]), where f ( x , t) is compactly supported, but the form r/E A k ( M ) may not be compactly supported. One then defines an R-linear map
7r. : A ~ ( M • R) ~ A kc - I ( ~M ~ J , called "integration along R", by requiring that
(a) ~r,(f(x,t) dt A Tr*(~)) = ( f o 2 f ( x , t ) dt)r], N
~
(b) 7c.(f(x,t)Tr*(r])) = O. As before, there is a unique R-linear operator 7r, with these properties. Remark t h a t requiring the operator to be additive over locally finite sums is no longer necessary. It will also be convenient to define A - J ( M ) = 0, Vq > 0, to agree that d (= 0) is defined on this trivial module, hence to have H - q ( M ) defined and trivial. L e m m a 8.3.12.
With the above definitions, do 7r, = -Tr. o d: A ~ ( M x R) ~ Ak~(M),
V k C Z. That is, 7r. anticommutes with d. The proof is a straightforward computation on forms of types (i) and (ii). It is analogous to Exercise 8.3.4, only easier.
262
8. I N T E G R A T I O N A N D C O H O M O L O G Y
C o r o l l a r y 8.3.13. The linear map ~r. passes to a well-defined linear map ~ . : H k ( M x R) ~ H ~ - I ( M ) , VkEZ. We want to prove that this map is an isomorphism, so we need a candidate for its inverse. Choose a compactly supported function b : ~ ~ ~ such that f_~r b(t) dt = 1. Let/3 = b(t) dt E A~ (R). Finally, for each k E Z, define /3,: A k ( M ) -~ A ck+l ( M • ]~) by
/3,(~) = b(t) dt A 7r*(~). It is practically immediate that do~,
= -~, od,
and we draw the following conclusion. L e m m a 8.3.14. The linear map ~, passes to a well-defined linear map ~ . : Hkc(M) --+ H ~ + I ( M • N), VkEZ. Clearly,
and we will show that, at the level of compact cohomology, fl, o rr, is also the identity. Once again, we construct an operator
S : A ~ ( M • R) ~ A k - I ( M • R) such t h a t i d - ~ , o~r,.
doS+Sod= To define S, set
B(t) = and define S on forms of type (ii) by
s(f(x,
i
b(u) du oo
t>*(~))
= 0
and, on those of type (i), by
(C
S(f(x,t)dtATr*O?))=
f(x,u)du-B(t)
/?
f(x,u)du
cyo
)
~r*(~?).
Remark t h a t this form is, indeed, compactly supported. This is obvious in the x variable and, for t ~ - ~ , it is also clear. But, as t T ~ , the function in the parentheses ultimately becomes
/?
f ( x , u) du - 1.
F
f ( x , u) du = O,
so the support is bounded in all directions, hence compact. E x e r c i s e 8.3.15. Prove that the formula
doS+Sod holds on A~ ( M • IR), V k E Z.
= id-~.
o~.
8.3.
As in the proof of Theorem following. Theorem
POINCARI~
LEMMA
263
8.3.5, this is all that is needed to establish the
8.3.16 (Poincar6 lemma for compact supports). The map ~. : nk~(M x IR) ~ H ~ - I ( M )
is a canonical isomorphism with inverse ft., V k C Z. In particular, 13. does not depend (in compact cohomology) on the choice of the compactly supported function b(t) such that f~-~o b(t) dt = 1. Remark. The operators S~ and S, used to prove the Poincar6 lemmas for ordinary and compact de Rham theory, are examples of cochain homotopies in algebraic topology. One says that idA*(M• is cochain homotopic to rr* o %,'* writing 7c* oi* ~ idA.(MxR ) . Similarly, /9. o re. ~ idA~(Mx~) . Exactly as in the proof of Theorem 8.3.5, cochain homotopic maps induce the same map in cohomology. In the present situation, since one of the maps is the identity, they both induce the identity. In Example 8.2.17, we used a chain homotopy between the identity and a map that is 0 in positive degrees to show that the singular homology of a contractible manifold is trivial. C o r o l l a r y 8.3.17. For each integer n >_ O,
H~(}Rn) =
•, 0,
k = n, otherwise.
Proof. This is clearly true for n = 0. Inductively, suppose that it is true for a given value of n >_ 0 and appeal to Theorem 8.3.16 to get H ~ ( ~ n+l)
=
H ~ ( ~ '~ x R)
:
VkcZ.
H~-I(Rn), []
C o r o l l a r y 8.3.18. The linear map :Hg'(R n)-~R n
is an isomorphism. Indeed, since ]R'~ is orientable, we have seen that this is a surjection (Theorem 8.2.4). Since the cohomology space is one-dimensional, it is an isomorphism. E x e r c i s e 8.3.19. Show that Theorem 8.3.5 makes sense and holds for compactly supported cohomology, provided that 0 M = ~ and R = [a, b]. Using proper maps and proper homotopies, formulate and prove the analogue of Theorem 8.3.6.
264
8. I N T E G R A T I O N A N D C O H O M O L O G Y
8.4. E x a c t S e q u e n c e s A basic tool for computing cohomology will be the Mayer-Vietoris sequence (Section 8.5). In order to develop and apply this sequence, we will need some properties of exact sequences. This purely algebraic section may be a review for many readers. At any rate, the proofs are elementary and will be relegated to exercises. We fix a commutative ring R and consider modules A, B, C, etc., over R. All maps ~ : A --~ B will be R-linear. We also consider graded R-modules A*, B*, etc., over R, in which case p : A* --* B* will denote a homomorphism of graded R-modules. We will generally assume that the grading is indexed by Z rather than just Z +. No generality is lost since A* = {Ak}~~ can be replaced by {Ak}k=_~ by setting A -p = 0, Vp > 0. D e f i n i t i o n 8.4.1. A sequence
..--.A~B
~-~C . . . .
of module homomorphisms is said to be exact at B if im(c~) = ker(~). If a sequence of module homomorphisms is exact at each module (except the first and last), it is called an exact sequence. An exact sequence of the form
o---~AA, BmC---~o is called a short exact sequence. Similarly, the notions of exact sequence and of short exact sequence are defined for graded module homomorphisms. Remark that, in the short exact sequence, i is injective and j is surjective. E x e r c i s e 8.4.2. Let A
a
, B
A' a'
~
, C
-y
,I
-I
,I
I B'
I C'
) D'
fl'
6
~ D
~ E
'
~'
E'
6'
be a c o m m u t a t i v e diagram in which the two rows are exact. If I, #, p, a n d ~ are
isomorphisms, prove that y is an isomorphism. This is called the five lemma. D e f i n i t i o n 8.4.a. A cochain complex (A*, 6) is a graded R-module, together with a sequence
A p J . Ap+ 1 ~ Ap+ 2 such that 62 = 0. Similarly, a chain complex (C., 0) is a graded R-module and a sequence o Cp O Cp_ l o Cp_ 2 0 ___. . , .
----+
.
..
such that 0 2 = 0. As usual, one defines Z p : ker(6) A Ap (respectively, Zp = ker(O) N Cp) and B p = im(6) A A p (respectively, Bp = im(0) n @). In what follows, we explicitly consider cochain complexes, but everything goes through, with the obvious modifications, for chain complexes. In this book, we are mainly interested in the de Rham cochain complexes (A*(M),d) and (A*(M),d) and in the singular chain complex (C. (M), 0), although others will be mentioned on occasion.
8.4. E X A C T SEQUENCES
265
The condition that 52 = 0 implies that B* C_ Z* and the cohomology of the cochain complex is defined to be
H*(A*,5) = Z*/B*, a graded R-module. This can be viewed as a measure of the extent to which the sequence in Definition 8.4.3 fails to be exact. The corresponding construction for a chain complex (C.,0) is called the homology of the complex and denoted by H. (C., 0). D e f i n i t i o n 8.4.4. a homomorphism p : (A*, 5) --* (C*, 5) of (co)chain complexes is a homomorphism of the graded R-modules such that ~ o 5 = 5 o F. Evidently, a homomorphism p : (A*, 5) -~ (C*, 5) of cochain complexes induces a homomorphism ~* : H* (A*, 5) ~ H* (C*, 5) of graded R-modules. D e f i n i t i o n 8.4.5. A homomorphism A : H*(A*,d) ~ H*(C*,5) of degree p E 2~ is a sequence of R-linear maps
A : H~:(A*,5) --, Hk+P(C*,5), - o o < k < ec. This is sometimes written A : H*(A*,5) ~ H*+P(C*,5). For instance, in the Poincar5 lemma for compactly supported cohomology, we defined a homomorphism 7r,: H * ( M x R) --* H * - I ( M ) of degree - 1 . L e m m a 8.4.6. Let
0
> (C*, 6) 2. (D*, 6) j
(E*, 6) ~
0
be a short exact sequence of homomorphisms of cochain complexes. Then, there is canonically induced a homomorphism 5* : H*(E*,5) ---* H * + I ( c * , 5 )
of degree +1, called the connecting homomorphism. This homomorphism is "natural" in the following sense: if o
, (c*,5)
> (D*,5)
0
, (J*,6)
, (K*,5)
J
, (~*,5)
, o
(L*,6)
~0
,
J
is a commutative diagram with both rows exact, then H*(E*, 6)
6" > H . + I (C*, (5)
H*(L*,5)
, H*+I(J*,5) 5*
also commutes.
266
8. I N T E G R A T I O N
AND
COHOMOLOGY
In the case of a short exact sequence of chain complexes, the connecting homom o r p h i s m has degree - 1 . We show how to find 6*[e] E Hk+I(C*,5), where [e] E Hk(E*,5). Consider the c o m m u t a t i v e diagram 0
Ck
~
Dk
J
Ek
ck+l
i
)
Dk+l
j
)
Ek+l
) 0
ck+2
i
)
Dk+2
j
)
Ek+2
) 0
and choose e E E k representing [el. In particular, 6(e) = 0. Since j is surjective, choose e' E D k such t h a t j(e') = e. T h e n j(5(e')) = 6(j(e')) = 6(e) = 0 and exactness of the middle row implies t h a t there is a unique c E C k+l such t h a t i(c) = 6(e'). T h e n i(5(e)) = 5(i(e)) = 6(5(e')) = 0. Since i is one-to-one, it follows t h a t 6(c) = 0, so we define 5* [el = [c] E H k+l(C*, 5). More d i a g r a m chasing proves t h a t [c] is i n d e p e n d e n t of the choices of e E [e] and of e' E D k such t h a t j(e') = e. E x e r c i s e 8.4.7. P r o v e t h a t the connecting h o m o m o r p h i s m 5* is natural as defined in t h e s t a t e m e n t of L e m m a 8.4.6. E x e r c i s e 8.4.8. Prove t h a t a short exact sequence
0 --~ (c*, 5) -L (D*, 5) ~ (E*, 5) --~ 0 of cochain complexes induces a long exact sequence
... ~ Hk(C.,5 ) i~ Hk(D.,5) i_~ Hk(E.,6) ~
Hk+I(C.,5) ~ . . .
in cohomology. O n e s o m e t i m e s writes the long exact sequence more c o m p a c t l y as an exact triangle: H* (C*, 6)
i*
, H* (D*, 5)
H*(E*,5) E x e r c i s e 8.4.9. Let R be a field. If
A--LBJ-~C is an exact sequence of vector spaces over R, prove t h a t the dual sequence A* ~ i*
B* ~ J*
C*,
where i* and j* are the respective adjoints, is also exact. F i n d an e x a m p l e showing t h a t this m a y fail for modules over a c o m m u t a t i v e ring.
8.5. MAYER-VIETORIS SEQUENCES
267
8.5. M a y e r - V i e t o r i s Sequences Let U1 and U2 be open subsets of the n-manifold M and consider the inclusions
Jl
U1N U2 ~--+U1,
j:
U l n U: ~-- U~,
and
il : UI,~---~ UI U U2 i2 : U2 ~-~ UI U U2 . L e m m a 8.5.1. The above inclusions give rise to a short exact sequence
0 ~ (A*(U1 u U2), d) -~ (A*(U1) | A*(U2), d | d) ~ (A* (U1 vi U2), d) ---* 0 of cochain complexes, where i ( w ) = (i~(w),i~(w)),
V w e A*(Vl UU2),
and j(wi,w2) = Y ~ ( w l ) - i i ( w 2 ) ,
Vwe ~ A*(Ue), e = 1,2.
Proof. Indeed, a nontrivial form on U1 U U2 must be nontrivial on either U1 or U2, so i is one-to-one. Since j~" o i~ = j~ o i~, it is clear that ira(i) c_ ker(j). For the reverse inclusion, let (Wl,W2) E ker(j). Then Wl](U1 n U2) = w2](U1 N U2), so these forms fit together smoothly to define a form w on U~ U U2 and (~1,w2) = i(~). Finally we must prove that j is surjective. Let co be a form on U1 N U2. Let {A1, A2} be a partition of unity on U~ U U2 subordinate to {U1, U2} and set ~1 = A2~, ~2 = AlW. (Note that, since A2 is supported in /]2, A2~ extends smoothly by 0 to all of U1. Similarly, AlW is a form on/72.) Then, j(a~l, -w2) = a;l + w2 = oz. [] T h e o r e m 8.5.2. There is a long exact sequence
... d~ H q ( u ~ U U~) ~
Hq(u~) e H q ( U 2 ) AL ~ " ( U i n u : ) ~-L H , + I ( u 1 u U:) ~A~ ...
called the Mayer-Vietoris sequence. Indeed, the cohomology of the cochain complex (A*(U1)| A*(U2),d | d) is clearly H*(U1) 9 H*(U2), so we apply Proposition 8.4.8. We turn to the Mayer Vietoris sequence for compactly supported cohomology. Again, U1 and U2 are open subsets of some n-manifold M. Clearly, there are inclusions ae : A*~(U1 N U2) ~ A*~(Ue), g = 1,2, and ~e : Ac(Ue) ~ Ac(U1 U U2), g = 1, 2. It is evident that these inclusions commute with exterior differentiation, hence induce linear maps a~, t3~ in compact cohomology, g = 1, 2. L e m m a 8.5.3. The above inclusions induce a short exact sequence
0 ~ (A*~(U~ n U2),d) ~ (A~(U~)| A*~(U2), d | d) A (A~(U1 u U2),d) ~ 0 of cochain complexes, where ~(~) = ( ~ ( . ~ ) , - ~ 2 ( ~ ) ) ,
w
~ A;(UI n U~),
and 9(Wl,W2) = g x ( w i ) + Z 2 ( w 2 ) ,
Wgc
A*~(Ue), g = 1,2.
268
8. INTEGRATION AND COHOMOLOGY
Pro@ Everything is clear except, perhaps, the fact that fi is a surjection. If co is a compactly supported form on U1 U U2 and {A1,A2} is a partition of unity on U1 UU2 subordinate to {UI, U2}, then AlaJ has compact support in U1 and A2w has compact support in U2 (note the switch from the proof of Lemma 8.5.1). Then, /3(Alco, A2co) = w and/3 is surjective. [] Remark. We have chosen the signs differently than in Lemma 8.5.1. This is not necessary for our present needs, but will be useful in our treatment of Poincar~ duality. Theorem
8.5.4. There is a long ezact sequence
d* Hkc(U1 fl W2) ~
Mck(U1) @ Hck(W2)
---o/3"Hkc(U1U U2)
d ~ IILrk+l/rT c ktJl n U2) cz*' ""
called the M a y e ~ Vietoris sequence for compactly supported cohomology. D e f i n i t i o n 8.5.5. Let M be an n-manifold without boundary. An open cover {U~}~e~ of M is said to be simple if it is locally finite and every nonempty, finite intersection
U=U~onU~, n...nv~, is contractible and has H~ (U) = H~ (R~). By Theorem 8.2.19 and Theorem 8.3.8, simple covers have the following property. L e m m a 8.5.6. If ~[ is a simple cover of M and U is any nonempty, finite intersection of elements of ~[, then
H*(U) = H*(Nn), H , ( U ) = H,(Nn). T h e o r e m 8.5.7. If M is a manifold with OM = O, then every open cover of M admits a simple refinement. We will postpone the proof of this theorem to Section 10.5, since it requires methods from Riemannian geometry. The idea is to produce a locally finite refinement by geodesically convex open sets and to prove that a geodesically convex open set U has the property in Definition 8.5.5. Since finite, nonempty intersections of geodesically convex sets are geodesically convex, we obtain a simple refinement. Using Theorem 8.5.7 and Mayer-Vietoris sequences, we obtain the following interesting result. T h e o r e m 8.5.8. If M admits a finite simple cover, then H * ( M ) and H * ( M ) are finite dimensional. In particular, if M is a compact manifold without boundary, then H * ( M ) is finite dimensional.
Proof. Select a finite simple cover {Ui}i~=l of M. We proceed by induction on r. If r = 1, then M = U1 has the ordinary and compact cohomology of R ~. Thus, H* (U 1) H* (singleton), hence is finite dimensional. For H2 (U 1), the assertion is given by Corollary 8.3.17. Suppose, then, that it has been shown that H~ (N) and H* (N) are finite dimensional whenever N has a simple cover by r - 1 elements, r r--1 ui and remark some r > 2. Let M have the simple cover { U i}i=l, let U = Ui=l t h a t {U1 N (Jr, U2 n U~,... , U~-I n U~} is a simple cover of U N U~.. We consider the =
8.5. MAYER-VIETORIS SEQUENCES
269
compactly supported case. By the inductive hypothesis, Hc(U ) and H*(U n U,.) are b o t h finite dimensional as, of course, is H i (U~). Since M = U U U,-, the M a y e r Vietoris sequence gives an exact sequence
H*(U) | H2(U~ ) ~
H2(M ) d~+ H . + I ( u n [fir).
By s t a n d a r d linear algebra,
H2(M ) ~ ker(d*) | im(d*) = im(/3*) | ira(d*). Since/3* has finite dimensional domain and d* has finite dimensional range, the assertion for H i (M) follows. The proof for ordinary cohomology uses the appropriate Mayer-Vietoris sequence in the same way. []
Remark. Even if the compact manifold M has boundary, it is true that H* (M) is finite dimensional. One way to prove this is to show that int(M) has a finite simple cover and that M and int(M) are homotopically equivalent. E x e r c i s e 8.5.9. If the manifold M is connected, but not necessarily compact, prove t h a t the real vector spaces H$ (M) and H . (M) have dimension at most countably infinite. You may use the de Rham theorem. E x e r c i s e 8.5.10. Prove that
H~(S~) =
R, 0,
k = 0, n, otherwise,
for all n _> 1. There is also a Mayer-Vietoris sequence for singular homology. The proof is similar to those for cohomology except for one technical point, the proof of which is very tedious and would take us too far afield. Since we will need this sequence for the proof of the de R h a m theorem, we derive it here, referring the reader to standard references in algebraic topology for the bothersome technicality. The inclusions j,
Ul n U2 ~ U1,
j2
Ul n U2 ~ U2
il
Ul ~ U~ u U2,
and
i2 U2 ~---,Ul U U2 induce an exact sequence
(c.(ul) e (c.(u2), o e o) 2~ i(c. (Ul u u2), o), where j(c) = ( j l # ( c ) , - j 2 # ( c ) ) and i(cl,c2) = il#(Cl) + i2#(e2) (and the induced homomorphisms i1#, jl#, etc., are as in Exercise 8.2.24. The exactness is imme(S.4)
0 --+ ( C , ( U 1 I"l U2) , 0) 5_+
diate. If (*) were a short exact sequence, the Mayer-Vietoris sequence for singular homology would follow immediately, but it is generally false that i is a surjection. This brings us to the technical point. D e f i n i t i o n 8.5.11. Let Ii = {Uo}~c~ be an open cover of tile manifold M. A singular p-simplex s : a p ~ M is said to be ll-small if, for some a C 91, s(Ap) C_ U~. Tile set of ll-small singular p-simplices is denoted by A ~ ( M ) . The vector subspace of Cp(M) spanned by A ~ ( M ) is denoted by C~(M).
270
S. INTEGRATION AND COHOMOLOGY
By the definition of the singular boundary operator, it is immediate t h a t
O: CUp(M) --~ C~_I(M ). D e f i n i t i o n 8.5.12. The chain complex (C,~(M),O) is called the complex of ~ small chains. The homology of this complex is H~,(M), the ll-small homology of M. It is clear that the natural inclusion of the space of ~l-small chains into the space of all chains is a homomorphism of chain complexes
zu : (C~, (M),O) ~ (C,(M),O), so there is induced a canonical homomorphism in homology Z.u : H~,(M) __~H,(M).
We arrive at the technical result. P r o p o s i t i o n 8.5.13. The homomou)hism zu is a canonical isomorphism
Hr.~(M) = H, (M). Proofs of Proposition 8.5.13 will be found in the standard references in algebraic topology, such as [la, pp. 85-88] and [46, pp. 207-208]. The idea is to subdivide the singular simplices in each cycle z until all simplices in the subdivision are ll-small. W i t h appropriate choices of signs, there results a R-small cycle z' with [z'] = [z]. Thus, homology can be computed using ll-small chains, for any open cover II of M. In our situation, II = {U1, [72} is an open cover of the manifold U1 U Uu and we replace the sequence (*) with the short exact sequence
0 -e (C'. (U1 n U2), oq) & (C. (Ul) (~) (6'. (U2), O @ O) ~e (C.II (U1 u U2), O) ~ O. 8.5.14. There is a long exact sequence
Theorem
... o_:,HAUl n []2) At, H,(U1) r Hp(U2) ~ HAUl VU2) 2 V Hp-I(U1 nU~) 2 : + . called the Mayer Vietoris homology sequence. Using the result of Exercise 8.4.9, we obtain dual Mayer-Vietoris sequences. 8.5.15. The Mayer Vietoris sequences dualize to exact sequences
Theorem
i' "'" ---+ H q ( U l ) * ([]3Hq(U2) * ~
9 '
H q ( U l n g2)* d.~ H q + l ( u 1 U U2)* ~ "
Hg(U1)* eH:(U2)* ~A+gg(g, uGh)* ~ H ~+1~" riG2)*
"
c
j,
\~'1
j'
Hp(UI)* r Hp(U2)* ~+ Hp(Ux UU2)* a~+ Hp-I(U1 rid2)* --+"
where i' is the adjoint of i* (respectively, of i,), etc.
8.6. C O M P U T A T I O N S
OF COHOMOLOGY
271
8.6. Computations of Cohomology In this section, we compute the top dimensional cohomology of connected m a n ifolds. 8.6.1. If the open subsets U1, U:, and U1 A U2 of an n-manifold M all have the same compact cohomology as IRn and are coherently oriented, then
Lemma
s
H:(U1o U2) --+R 1UU2
is an isomorphism. Proof. Consider the diagram H~(U1 N U2)
oz
*
)
,fUlnU2I IR
Hn(u1) | H~(U2)
~* ~ Hn(u1 U U2)
ful ~,fu2 I A
~
IR |
IR
5
d*
)
0
,fu1uu2~
id l
IR
~ 0
>
where A(t) = ( t , - t ) and 5(s,t) = s + t, Vs, t C JR. C o m m u t a t i v i t y of the diagram is obvious (coherency of orientations is essential), exactness of the b o t t o m row is obvious, a n d the top row is exact by Theorem 8.5.4. The m a p fu~nu~ is a n isomorphism by the hypothesis t h a t H'~(U1 A U2) = R (as in Corollary 8.3.18). Similarly, fu~ Q fu2 is an isomorphism. Since the diagram can be extended harmlessly by a c o m m u t a t i v e square of 0s on the right, it follows from the Five L e m m a that feluu~ is a n isomorphism. [] 8.6.2. Let {U~}sc~ be a simple open cover of a connected, oriented nmanifold M without boundary, each Us being oriented coherently with the orientation of ll~[. Let w s , w z C A n ( M ) have respective supports in Us and Ufl, a, fl C 92. Then [w~] = [wfi] if and only if fM ~ = fM ~Z"
Lemma
Proof. If [cz~] : [czZ], t h e n we know that fM a& : IM aJZ. For the converse, assume equality of the integrals. Since M is connected, we can find a sequence of indices a = a 0 , a l , . . . , a ~ . = fl in 92 sucll that U~{_ 1 rhU~{ ~ ~, 1 < i < r. Choose a~, E A ~ ( M ) such t h a t supp(c~s{) C U~{, 1 < i < r, and such that aJ~o = a&, eva,. : :vfl, and
This is clearly possible. By L e m m a 8.6.1,
v
H n ( G ' - ~ U G~) ~ N
is an i s o m o r p h i s m , 1 < i < r, so
[(A). . . . ] = [(A)Si] ~ Hn(U C \ O~i--1 U Usi ) ,
T h a t is, there is a form t] E A nc - l f \U s i - 1 U Ua~) such that ws, = ws,_~ + &]. These forms all live in A~(M), so [w~,] = [w. . . . ] E H 2 ( M ) , 1 < i < r. In particular, [~sl = [czZ] as desired. []
272
8. I N T E G R A T I O N
AND
COHOMOLOGY
L e m m a 8.6.3. If {U~}~e~ is a simple cover of the connected, oriented n-manifold M with OM = O, then, for each a o r 92, the natural inclusion e : A'~(U~o) ~ A'~(M)
induces an isomorphism e. : H2(U~o ) ~ H : ( M ) . Proof. Indeed, since fUoo : Hn(u~o) --~ R is an isomorphism, [co] 9 H'~(U~o) is nontrivial if and only if fM co = fu,, o co # O. Since fM vanishes on B 2 ( M ) , it follows t h a t e.[co] 7~ 0, so e. is injective. We prove surjectivity. Let co 9 A'~(M) and use a partition of unity {/~}~e~, subordinate to the simple cover, to write
aEg.l
where )~,co 9
A2(U~,), 1 <
s Let
i=1
i < r. Choose coi 9 co~=L
A2(U~o)so that
A~,co.
r
i=1
and remark that, by the above lemma, [co/] = [A~co] 9 H n ( M ) , 1 < i < r. Thus, as classes in H2(M), =
[coi] i=1 ?-
i=1
=
[col.
T h a t is, viewing [c~] C H~(U~o) and [co] E H~'(M), we have proven that e.[c~] = [co], so e. is surjective. [] T h e o r e m 8.6.4. If M is a connected, oriented n-manifold, OM = O, then the linear map
M : H n ( M ) --~ ~ is an isomorphism. Proof. Fix a simple cover and let U be an element of that cover. Consider the commutative diagram H2(U)
/u , R
H2(M )
, R f~
8.6. C O M P U T A T I O N S
OF COHOMOLOGY
Since e. and fg are isomorphisms, so is f M Corollary
273
[]
8.6.5. [f M is a compact, connected, oriented n-manifold without bound-
ary, then M : Ha(M)
~
•
is an isomorphism. 8.6.6. If M is a connected, nonorientable n-manifold with empty boundary, then H g ( M ) = O. In particular, if M is also compact, H a ( M ) = O.
Theorem
Proof. Let ~ E A n ( M ) . We must show t h a t ~ = dO for suitable 0 ~ A ~ - I ( M ) . Choose a simple cover { U ~ } ~ . By a partition of unity argmnent, write aJ as a, finite sum of forms, each c o m p a c t l y s u p p o r t e d in one or another element of the cover. If each of these is the exterior derivative of a c o m p a c t l y s u p p o r t e d form, we are done. Thus, w i t h o u t loss of generality, we assume that supp(a~) C U~ 0. By nonorientability of M , there is a sequence U~0, U ~ , . . . , U ~ of elements of the simple cover and orientations #i of U~,, 0 < i < r, with the following properties: 1. U ~ _ ~ N U ~ r
2. ]/,i-1 and #i restrict to the same orientation of U~, ~ n U ~ , 1 < i < r; 3. U~, = Uao and #~. = - # 0 . Choose forms aJi E A 2 ( U ~ ) , 0 < i < r, such t h a t a~ = ca0 and
U~ ,tz~) l
U~i_ 1, ~ - 1)
Thus, 02i =COi--1
+dr]i-l, ?~i-1 ~ A2-1(U . . . . UUa~), 1 < i < r.
T h a t is, a~ = ~o + dv,
~ c A2-~(M).
On the o t h e r hand,
implying t h a t a~ = -a;o + d T ,
7 E A~-I(U,~o).
C o m b i n i n g these equations, we conclude t h a t a; = wo = dO for suitable 0 C
A2-1(M).
[]
If M is compact, any simple cover, being locally finite, is finite. If M is noncompact, it m a y or m a y not a d m i t finite simple covers, but it always admits infinite ones. A n easy variation on the proof of T h e o r e m 8.5.7 gives 8.6.7. If M is noncompact and connected, OM = O, then there is a countably infinite simple cover { U i } ~ 1 by relatively compact sets such that UiDUi+I ~ O,
Lemma
1
274
8. I N T E G R A T I O N A N D C O H O M O L O G Y
E x e r c i s e 8.6.9. If M is an n-manifold with OM compact and nonempty, prove
t h a t M and int(M) are homotopically equivalent. If M is connected, conclude t h a t H ~ ( M ) = 0. (Hint: There is a compactly supported vector field on M, pointing inward and nowhere 0 along OM. This generates a "half flow", parametrized on [0, oo) and stationary outside of a neighborhood of OM.) 8.7. D e g r e e T h e o r y * Let M and N be connected, oriented n-manifolds without boundary and f : M --~ N a proper map. Let y E N be a regular value of f . Then f - l ( y ) is compact and discrete, hence finite. Set f - l ( y ) = {Yl . . . . , yq} and let
ei =
{ +11 if f.y~ preserves orientation, _ if f.y~ reverses orientation,
l
oriented n-manifolds without boundary, and if y E N is a regular value, then f* : H ~ ( N ) --+ H n ( M ) is multiplication by degy(f). Proof. By Theorem 3.9.1, there is an open, connected neighborhood U C N of y such t h a t f - l ( U ) = U1 U . . . t2 Uq, a union of disjoint open sets such that Yi E Ui and f carries Ui diffeomorphicMly onto U, 1 < i _< q. If [a3] E Hg(N), we can choose a representative n-form w so that supp(a~) is a compact subset of U. Then, wi = f*(a~)lUi is compactly supported in Ui, 1 < i < q, and f : Ui --* U preserves or reverses orientation according as ~i = 1 or - 1. Thus, q
q
i=l ei
02
= degy(f) f g w. It follows that f* : H~(N) --+ Hcn(M) is multiplication by degy(f).
Remark. This has several obvious consequences: 1. f* : H~(N) --~ H ~ ( M ) is multiplication by an integer. 2. degy(f) = deg(f) (the degree of f ) is independent of y. 3. d e g ( f ) is a proper homotopy invariant of f.
[]
8.7. DEGREE THEORY*
275
4. If M is compact, d e g ( f ) is a h o m o t o p y invariant of f . 5. d e g ( f o g) = d e g ( f ) deg(g). T h e o r e m 8.7.3. Let W be an oriented (n + 1)-manifold with nonempty, connected boundary. Let N be a connected, oriented n-manifold without boundary and let f : OW -~ N be proper. I f f extends to a proper map F : W ---* N , then d e g ( f ) = 0.
Proof. Suppose t h a t f extends to a proper m a p F : W --~ N . If aJ C A n ( N ) , t h e n F*(w) E A 2 ( W ) and d(F*(w)) = F*(dco) = 0, since dw 9 A~*+I(N) = 0. Thus, by Stokes theorem,
L.
=
By P r o p o s i t i o n 8.7.2, it follows t h a t d e g ( f ) = 0.
L ~176
[]
T h i s t h e o r e m partially generalizes T h e o r e m 6.5.6. Let S ~ c N ~+1 be the unit sphere and let a ~ : S ~ -~ S ~ denote the a n t i p o d a l interchange m a p
O!n(Xl,... ,X n471) :
Proposition
(--xl,...,--xn+l).
8.7.4. deg(c~n) = ( - 1 ) n+1.
Proof. Since o~n is a diffeomorphism, every x 9 S ~ is a regular value and has pre-image a singleton. Thus, the question reduces to whether c ~ preserves or reverses orientation. T h e linear extension A : IR~+1 -~ R ~+1 of c ~ is represented by the m a t r i x - I ~ + 1 with d e t e r m i n a n t ( - 1 ) ~+1. This transformation, therefore, is orientation-preserving if and only if n is odd. T h e restriction of the t r a n s f o r m a t i o n to the unit ball D n+l is orientation-preserving if and only if n is odd. But S n = cgDn+l has orientation indnced by the orientation of D '~, hence AIS ~ = c~ : S ~ S n is orientation-preserving if and only if n is odd. [] T h e o r e m 8.7.5. if n is odd.
The sphere S n has a nowhere zero tangent vector fieId if and only
Pro@ For n odd, you constructed a nowhere vmfishing vector field in Exercise 4.3.12. Suppose, therefore, t h a t s : S ~ --+ ]I{n+l \ {0} is s m o o t h with s(v) • v, Vv 9 S ~. Equivalently, s is a nowhere zero section of T(Sn). Note t h a t v c o s 0 + s(v) s i n 0 r 0, V 0 9 IR, so we can define a s m o o t h m a p
F:S~x[O,1]---+S ~ by v cos tTr + s(v) sin tTr
F(v,
t) =
Ilv cos t~r
+ s(v)
sin tTrlt "
Then,
F(v,O)=v F(v, 1) -v
} V v C S n,
so ~n ~ id and 1 = deg(c~n) = ( - 1 ) n+l, implying t h a t n is odd. Corollary
[]
8.7.6. Every smooth flow on S 2~ has at least one stationary point.
For the sphere S 2, T h e o r e m 8.7.5 and its corollary are sometimes stated facetiously as the previously quoted "you c a n ' t comb the hair on a billiard ball". (People whose sensibilities are offended by hairy billiard balls s u b s t i t u t e "coconut".)
276
8. INTEGRATION AND COHOMOLOGY
T h e o r e m 8.7.7. If f : S n fixed point.
--*
S n i8 smooth and d e g ( f ) ~r ( - 1 ) n+l, then f has a
Proof. In fact, we will prove that, in the case t h a t f has no fixed point, f ~ c~, hence des(X) = ( - 1 ) n + l . We claim that t ( f ( v ) + v) r v, Vv E S ~, 0 < t < 1. Otherwise, t r 0 and f ( v ) = (1 - t ) v / t . Since IIf(v)ll = 1 = Ilvll, it follows that t = 1/2 and f ( v ) = v, contrary to assumption. Therefore, we can define F : S n x [0, 1] ~ S ~ by
F(v,t) =
t ( f ( v ) + v) - v IIt(/(v) + v) - v i i
This is a homotopy between f (t = 1) a n d c~,~ (t = 0).
[]
E x e r c i s e 8.7.8. If f : S ~ ~ S ~ has Jdeg(f)l • 1, prove that f has a fixed point and t h a t there is a point t h a t f carries to its antipode. E x e r c i s e 8.7.9. If f : p2~ ~ p2~ is smooth, prove that f has a fixed point. (Hint. Lift to the universal cover.) E x e r c i s e 8 . 7 . 1 0 . If M is a compact, connected, orientable, boundaryless manifold of d i m e n s i o n n a n d f : M -~ R ~+1 is a s m o o t h imbedding, use degree theory to prove t h a t R ~+l \ f ( M ) has exactly two connected components and f ( M ) is the set-theoretic b o u n d a r y of each. This is the J o r d a n - B r o u w e r separation theorem. (Proceed in analogy with Exercise 3.9.23. In fact, the rood 2 degree theory is a d e q u a t e for this.) E x e r c i s e 8.7.11. Let q , ~ 2 : S I --* R 3 be s m o o t h maps with disjoint images. Define the linking number L k ( o l , 02) to be tile degree of tile m a p f : S 1 x S 1 --, S 2 defined by f(x,y) =
al(x)-02(y) IIo-l(Z) - a2(y)l)"
Intuitively, it seems reasonable to define ol and o2 to be topologically unlinked if there is a compact, orientable 2-manifold N with ON = S 1 a n d Ol extends to a smooth map ~'1
:
N ----+R 3 \ O-2(S 1)
(or if the parallel condition holds, in which the roles of a l and o2 are interchanged). Prove that, if a l a n d o2 are topologically unlinked, t h e n L k ( o l , 02) = 0. Give an example showing, at least intuitively, t h a t the requirement t h a t N be orientable is necessary. (Hint: Consider the M5bius strip.)
8.8. P o i n c a r 6 D u a l i t y * Assume t h a t M is a connected, oriented n-manifold with e m p t y boundary. We s t u d y the pairing
~
: A~(M)
An-k(M) ~ N
I
defined by
w' @X ~/ ~--> ./M uJ A ~/-
8.8. POINCARI~ D U A L I T Y *
277
Tile fact t h a t one of the forms is compactly supported guarantees t h a t the integral is defined. One can view this pairing as an R-linear map P D : A~(M) ~ A n - k ( M ) *, where A n - k ( M ) * is the vector space dual of A n - k ( M ) and
PD(w) = f
JA /
co A {.}.
I f w = dr, some 3' C A ~ - I ( M ) a n d d r / = 0, it is clear t h a t
f ~ c~ A r/ = Jii d("/ A r/) = O by Stokes' theorem. Similarly, if co is closed and r~ is exact, the integral is 0. Thus, our pairing passes to
/ :HI(M)| Again, this can be interpreted as a linear map
PD: H I ( M ) --+ H n - k ( M ) *, called the Poincard duality operator. 8.8.1 (Poincar6 Duality Theorem). Suppose that M is a connected, oriented n-manifold, OM = ~, and that M admits a finite simple cover. Then the Poincard duality operator
Theorem
PD: H I ( M ) ~ H ~ - k ( M ) * is an isomorphism, Vk.
In particular', this holds for M compact and defines a
canonical isomorphism H k ( M ) = H n - k ( M ) * = H~_k(M).
Of course, the last equality depends on the de R h a m theorem. R e m a r k that, in the compact case, we can also say t h a t H k ( M ) ~ H n - k ( M ) , b u t the isomorphism is n o t canonical. T h e o r e m 8.8.1 will be proven by a series of lemmas. 8.8.2. If M has the same ordinary and compact cohomology as IR'~ , the Poincard duality operator
Lemma
P D : H~(M) ~ H n - k ( M ) *
is an isomorphism. Proof. Indeed, if k ~ n, b o t h H~(M) a n d H n - k ( M ) * are O, so the assertion is trivially true. If k = n, t h e n [co] E H ~ ( M ) = R is uniquely determined by fMa~, while c 6 H ~ = R is just the constant function e. Then, PD[co](C) = /M CCO= C/M co. T h a t is, PD[co] : IR --+ IR is just multiplication by fM co, proving t h a t P D is also an isomorphism in this case. []
278
8. I N T E G R A T I O N
AND COHOMOLOGY
The proof of Theorem 8.8.1 will use the Mayer-Vietoris sequences. Let U1, U2 C M be open subsets. Consider the diagram
Hkc(U1 N U2)
a*
H~-k(Uz n U2)*
,
Hck(U1) 9 Hck(U2)
, H~-k(V~) * r H~-k(V2)*
where ct* is as in the definition of the Mayer-Vietoris sequence for compact supports and j ' is as in the dual of the ordinary Mayer-Vietoris sequence. L e m m a 8.8.3. The above diagram is commutative.
Proof. Let [wI E Hkc(U1 N [72) and ([rh] ,[~2]) e H~-k(U1)| Hn-k(U2). Remark that
fa c~
fa ~e(w)Arle'
1NU2
e
g = 1, 2. Thus, j ' PD[w] (['r/~, [rJ2]) = =
PD[w](j*([rh] , [r12])) PD[w]([jt(rh)]- [j~(r12)])
=
t.OA ( 3 1 ( 7 / 1 ) - - 32 (~]2)) 19)U2
aNU2 =
171U2
/UIO~I(07) A~I--~U20~2(O-J) A~2.
But
(PD | PD)a* [~]([~I],[U2]) = =
(PD| ,[r/2l) PD(c~[co])[rh] - PD(c~[co])[rl2 ] 1
2
[]
giving the asserted commutativity. E x e r c i s e 8.8.4. Prove that the diagram /3"
)
H~(U1 U [72)
IPD
P D ~) P D i
Hn-k(U1) * 0 Hn-k(U2) *
)
Hn-k(U1 U U2)*
i'
is commutative. Remark that the different sign conventions for the Mayer-Vietoris sequences in ordinary and compactly supported cohomology are needed for this exercise and the previous lemma.
8.8. P O I N C A R I ~ D U A L I T Y *
27"9
Lemma 8.8.5. The diagram d*
Hkc(U1 U U2)
u k + l ~ r r NU2)
[,o gn-~(ga U g~)*
, H~-k-l(U~
ng~)*
commutes up to sign. More precisely, d t o P D = ( - 1 ) k+l P D od*. Proof.
We
fix [w] E H)(U1 U/]2) and [7] 9 g~-k-l(U1 n U2) and verify t h a t
(8.5)
PD(d*[~])([~]) = f dA1 A w A T , gU 1NU2
(8.6)
d'(PD[w])([7]) = ( - 1 ) k+l f d/~l A w A 7, Ju 1 N U2
where {A1, A2} is a partition of unity on U 1 U U 2 subordinate to {U1, U2}, In the d i a g r a m Ack(U1) G A~(U2)
~ , A)(U1
u
U2)
dGd~ A.k+l (U1 n U2) we
c~
,
k+l
(U1)|
A c
k+l
(/]2)
see t h a t
d @ d(/~lW, )~2w) = (d)~l A a;, -d)~l A w),
a(d~l A ~)
= (dA1/~ ~, -d~l A ~),
where tile second equation uses the fact t h a t dA1 +dA2 = d(/~l +/~2) = 0. Therefore, d*[w] = [d)~l A w] and equation (8.5) follows. In order to compute d'(PD[w])([7]) = PD([w])(d* [7]), we first compute d* [rl]. Here we set r = n - k and consider
A "-I (U1) | A"-I(U2)
J ~ Ar-l(U1
d~dl A~(U1 u v2)
,
A"(U1) ~ A'(U2)
and note t h a t
j(A2~],-A17) = 7, d ~ d(A27, - - / ~ 1 7 ) = (-d/~l A 7, -d/~l A 7), i ( - d A 1 A 7) = (-dA1 A 7, -dA1 A 7)-
n
U2)
280
8. I N T E G R A T I O N A N D C O H O M O L O G Y
It follows t h a t d* [r/] = [-dA1 A r/]. Then d' (PD [co])([r/]) = PD([ca])(d* Jr/l) = PD([ca])([-d~X1 A r/l)
= [
Jr/ 1 (~U2
ca A (-d/~l A r/)
= (--1) k+l f d/~l A ca A JU 1 AU2 which is equation (8.6). Here, the sign is due to permuting tile 1-form d~X1 past the k-form ca. []
Proof of theorem 8.8.1. Let { Ui}i=l " be a simple cover of M and proceed by induction on r. By Lemma 8.5.6 and the definition of simple covers, the case r = 1 is given by Lemma 8.8.2. If, for a given r > 2, the assertion has been proven whenever a manifold has a simple cover with r - 1 elements, then it holds for the manifold U = U1 U ... U Ur-1, for UT, and for U A Ur (which has the simple cover {UIK/U~-,... ,Ur_l AUr}). We must prove it for M = U U U r . By the M a y e r Vietoris compact cohomology sequence of Theorem 8.5.4 and the dual cohomology sequence of Theorem 8.5.15, together with Lemmas 8.8.3, 8.8.5, Exercise 8.8.4 and the Five Lemma (for which commutativity up to sign is fine), PD: H~(M) ~ H~-k(M) * is an isomorphism.
[]
E x e r c i s e 8.8.6. Let M be connected, oriented and n-dimensional with cgM = ~. Let N C M be a compact, oriented, k-dimensional submanifold with ON = O, 0 < k < n, and denote the inclusion map by i : N ~ M. If [ca] E H k ( M ) , we will write fN[ca] for fN i* [cal. (1) Show t h a t there is a unique compactly supported cohomology class [r/N] C H 2 - k ( M ) such that
V [ca] C H k ( M ) . For fairly obvious reasons, [r/N] is called the Poincar~ dual of N. (2) If U C M is any open neighborhood of N, prove that the representative, compactly supported form r/N E [~N] can be chosen so that supp(~N) C U. This is the localization principle for the Poinca% dual of N. (3) If i0, il : N --+ M are two smooth imbeddings, we say that they are isotopic if there is a homotopy it : N --* M between i0 and ii such t h a t it is a smooth imbedding, 0 < t < 1. In this case we also say that Ne = ie(N) are isotopic submanifolds of M, g = 0, 1. If No and N1 are isotopic submanifolds of M, prove that [r/N0] = [r/N~]. (4) Suppose that /)1 and P~ are compact, oriented, boundaryless submanifolds of M of respective dimensions ki and k2 such that n = kl + k2. Show that / p [r/P~] = ( - 1 ) k~k2 f 1
[r/P~]'
JP2
This is called the algebraic intersection number ~(P1, P2) of P1 with P2 and is an integer (but you are probably not prepared to prove that).
8.9. DE RHAM THEOREM*
281
(5) If P1 and P2 as above are isotopic to submanifolds P~ and P9', respectively, such that P~ N P~ = 9, prove that L(P1, P2) = 0. (6) If P is a compact, orientable n-manifold without boundary, let A p C P x P be the diagonal, A p = {(x, x) [ z r P}. If P has a nowhere vanishing vector field, prove t h a t c(Ap, A p ) = 0. E x e r c i s e 8.8.7. In part (6) of Exercise 8.8.6, you proved half of the Poincar6-Hopf theorem: There is a nowhere vanishing vector field on P if and only if L(Ap, A p ) = 0. Assuming this theorem, prove the following. (1) The diagonal A p C P x P can be isotoped completely off of itself if and only if its algebraic self intersection number ~(Ap, A p ) vanishes. (In particular, by Theorem 8.7.5, in S 2k x S 2k the diagonal cannot be isotoped completely off of itself.) (2) Every compact, orientable, odd dimensional manifold with empty boundary has a nowhere vanishing vector field.
8.9. T h e de R h a m T h e o r e m *
We will prove the following case of Theorem 8.2.21. T h e o r e m 8.9.1 (de Rham Theorem). If M is a manifold without boundary which has a finite simple cover, then the de Rham map
DR: H*(M) ~ (H.(M))* is a canonical isomorphism of graded vector spaces. The proof follows exactly the pattern of proof of Theorem 8.8.1. L e m m a 8.9.2. I f the n-manifold M has the same singular homology and cohomology as R n, then the de Rham homomorphism
D R : H k ( M ) ---* H k ( M ) * is an isomorphism. Proof. I f n = 0, the result is immediate, so we assume n > 0. I f k r 0, both H k ( M ) and H k ( M ) * are 0, so the assertion is trivially true. Finally, H ~ = Z~ = IR is the space of constant functions on M and H o ( M ) = ]R has canonical basis the singleton {Ix]}, where x C M is fixed but arbitrary. If c r H ~ then DR(c)([x]) = c(x) = c, so DR is an isomorphism as claimed. [] We will use the appropriate Mayer Vietoris sequences. Let U1,U2 C M be open subsets. Consider the diagram
Hk(U1AU2 )
d* , Hk+I(U1UU2 )
DR+ Hk(U1 A U2)*
+DR o'
) Hk+I(U1 U U2)*
with d* as in Theorem 8.5.2 and 0' as in Theorem 8.5.15. L e m m a 8.9.3. The above diagram is commutative.
282
8. I N T E G R A T I O N A N D C O H O M O L O G Y
Proof. Let [co] E Hk(U1 n U2) a n d [z] C gk+l(U 1 U g2). Let lI denote the open cover {U1, U2} of U1 U U2. We can choose the representative cycle z C [z] to be l i - s m a l h T h a t is, z = Zl + zz, where zi E Ck+l(Ui), i = 1,2. Note t h a t zl a n d z2 m a y not, individually, be cycles. All t h a t is required is t h a t O(zl + z2) = 0, so 0(Zl)
---- --C9(Z2) E
Ck(gl A g 2 ) .
As a singular chain, z2 is a linear combination of 81,..., Sq, where si : Ak+l --~ U2,
l
We define the support of this chain to be q
Iz=l = [_J si(Ak+i), i=1
a compact subset of U2. It is easy to choose a smooth p a r t i t i o n of unity {AI,A2} on U1 U U2, s u b o r d i n a t e to It and having the property that A2 - 1 on a Iz21, hence /~1 ---- 0 on a ]z2]. R e m a r k that, for 1 < i < q,
s~(d~:)
= d(~;(a2))
= d(1)
-
0,
with a similar remark for dA1. Given these choices, we will show t h a t
DR(d*[co])([z]) = f
(8.7)
d)~2 A CO~
Jz 1
(8.8)
O'(DR[col)([z]) =
/ d~
A
Jz 1
Since [w] a n d [z] were arbitrary elements of the respective vector spaces, c o m m u t a tivity of the diagram will follow. In the diagram
Ak(U1) 9 Ak(U2)
J
, Ak(U1 A [/2)
dOdl Ak+I(u1) 9 Ak+l (U2)
Ak+I(U1 U [/2) i
we see t h a t
j(~2co,-~lco) = co, d | d(Azco, -Azco) = (d)~2 A co, dA2 A co),
i( d,~2 A co) = ( dA2 A co, d,k2 A co), where the second equation uses the fact t h a t d/~l = -d/~2. [dA2 A co] a n d
DR(d'Ecol)(Izl) : i Jz
since A2 ~- 1 on Iz21. This is equation (8.7).
A co : [ Jz 1
Aco
Therefore, d*[co] =
8.9. DE RHAM THEOREM*
283
In the diagram
i
Ck+l (U1) (~ Ckq-1 (g2)
Ck+I(U1U u2)
0~0~
G(u1 n u2)
,
G(u~) 9 G(u2)
we see that
i(Zl, z2) = z, 0 (~) O(Zl, Z2) : (OZl, --Oz1), j(OZl) ~-- (OZl,--OZ1), where the second equation uses the fact that Ozl = -Oz2. It follows that 0,[z] = [Ozl]. Then, O'(DR[co]) ([z])
= fo
,[z]
[~]
= fo~l ~1~ + fozl ~2~
= / dA2 A cd~ dz 1 since Ozl = -Oz2 and A1 vanishes identically on
Iz21. This
is equation (8.8).
[]
Commutativity of the remaining squares is easier since the connecting homomorphisms are not involved. E x e r c i s e 8.9.4. The diagram
Hk(U1)OHk(U2)
J* , Hk(U1NU2)
DRODRI Ha(U1)* ~ Hk(U2)*
IDR j,
, Hk(U1 N U2)*
is commutative. E x e r c i s e 8.9.5. The diagram
H k ( u 1 I J U2)
, Hk(U1) 9 Hk(U~)
~DRQDR Hk(U~ u U2)* is commutative.
' Hk(U1)* 0 Hk(U2)*
284
8. I N T E G R A T I O N A N D C O H O M O L O G Y
r Proof of theorem 8.9.1. Let { g i}/=1 be a simple cover of M and proceed by induction on r. By Lemma 8.5.6, the case r = 1 is given by Lemma 8.9.2. If, for a given r _> 2, the assertion has been proven whenever a manifold has a simple cover with r - 1 elements, then it holds for the manifold U = U1 o ... tO U,--l, for U,., and for U NUr (which has the simple cover {U1 A Ur,... , U,--1 Cl Ur}). We must prove it for M = U tO U,-. By the Mayer-Vietoris cohomology sequence of Theorem 8.5.2 and the dual homology sequence of Theorem 8.5.15, together with Lemma 8.9.3, Exercises 8.9.4 and 8.9.5 and the Five Lemma, DR : H k ( M ) ---, Hk(M)* is an isomorphism. []
There are many versions of the de Rham theorem. The version we have proven identifies de R h a m theory as a graded vector space with the dual of singular homology. Actually, this latter can be defined directly from a singular eochain complex (the dual of the singular chain complex) and is called singular eohomology. There is a natural graded algebra structure in singular cohomology (the multiplication is called "cup product" for some obscure reason) and a stronger version of the de R h a m theorem asserts that DR is an isomorphism of graded algebras. Also, the requirement that OM = 0 was convenient for our approach, but is quite inessential. In Appendix D, we will prove a version of the de Rham theorem for the Cech cohomology algebra _fi/*(M), a cohomology theory fashioned out of the family of open subsets of M. By a parallel argument, we will also show that the Cech cohomology algebra is isomorphic to the singular cohomology algebra defined using all of the continuous singular simplices instead of only the smooth ones. A very interesting consequence is the following. 8.9.6. The de Rham cohomology algebra H*(M) depends only on the underlying topological manifold M, not on the choice of differentiable structure.
Theorem
We discuss here the equality HP(M) = /5/P(M) for p = 0, 1. This will also motivate the use of the term "cocycle" in our earlier discussion of differentiable structures (Definition 3.1.10) and in vector bundle theory (Definition 3.4.2), as well as the cohomology notation H 1(M; Gl(n)) in Exercise 3.4.6. Let l[ = { U ~ } ~ be an open cover of the manifold M. A Ceeh 0-cochain on ~l is a function 0 which, to each U~ o E ~[, assigns a real number 0~ 0 = O(U~o). A 12ech l-cochain on ~1 is a function 9' which assigns a real number %o~1 = ~ ( G o , G 1 ) to every ordered pair (U~o, U ~ ) of elements U~o, U ~ c L[ such that U~ 0 A U ~ ~ 0. Similarly, a Cech 2-cochain ~ assigns a real number ~ o ~ 2 to each ordered triple (U~o,U~,U~2) of elements of ~ with U~o N U~I n U~2 ~ 0. The general p a t t e r n is clear, but we will stick with p-cochains for p = 0, 1, 2. The set of p-cochains is denoted by CP(ll). As real-valued functions on a set, p-cochains can be added and they can be multiplied by real scalars. This makes C'P(~) into a vector space over N. Define (~ech coboundary operators o
Z d~
Z 0~(u) Z 0~(u)
by (~0) . . . . (~7)~o~
=G~ = ~
if 0 C ~,o ('R),
- Go -7~o~
+ %o~
if V E 01
(~).
8.9. DE RHAM THEOREM*
285
It is a m o m e n t ' s work to check that 6 2 = 0, so we o b t a i n the space ZP(11) of Cech p-cocycles, the space /9p(11) of Cech p-coboundaries, and the pth Cech cohomology space/2/p(11), for p = 0, 1.
Remark. A cochain "y E d '1 (ll) is a cocycle precisely if it satisfes the cocycle condition ~/c~ = 2/c~ + 73~, whenever Us N U3 N U,7 r 0. The Gl(n)-cocycles in b u n d l e theory had a completely analogous definition, except for the multiplicative n o t a t i o n forced by the multiplicatire s t r u c t u r e of Gl(n). Indeed, i f ~ is a Gl+(1)-cocycle on II, t h e n logo~ is a Cech 1-cocycle on 11. T h e set of Gl(1)-cocycles forms an abelian group under operations inherited from Gl(1), bul,, for n > 1, the Gl(n)-cocycles do not form a group of any kind because of the n o n c o m m u t a t i v i t y of Gl(n). 8.9.7. If each Us E Ii is connected, the space I:I~ is canonically isomorphic to the space of locally constant, real valued functions on M. In particular, [t0(11) = HO( M). Lemma
Proof. R e m a r k that/2/0(11) = ~0(11) and t h a t this is the space of 0-cochains 0 such t h a t 0s = 0 3 whenever Us N U3 7~ 0. T h i n k i n g of 0s as a constant function on Us, g ~ E 91, we see t h a t these constant functions agree on overlaps of their domains, hence unite to form a coherent locally constant function 0 on M. Conversely, if 0 : M --+ I~ is a locally constant function, its restriction 0~ = OIUs is constant by the connectivity of Us, V ~ E ~. [] R e m a r k t h a t simple covers 11 satisfy L e m m a 8.9.7. 8.9.8. If 11 is a simple cover, there is a canonical linear isomorphism l~ 1(11) = H a(M).
Lemma
Remark. It is an i m m e d i a t e consequence of Lemmas 8.9.7 and 8.9.8 t h a t [tP(11) does not d e p e n d on the choice of simple cover, hence this vector space can be denoted b y / ~ / ~ ( M ) , p = 0, 1. For the c ~ e p = 0, the purely topological condition on 11 in L e m m a 8.9.7 proves that HO(M) and /:/~ are topological invariants. However, the definition of a simple cover requires a difl>rentiable structure, so we c a n n o t conclude from L e n m m 8.9.8 that H i ( M ) and t715(M) are topological invariants of M. T h e proper definition of Cech cohomology involves passing to an algebraic limit over the directed set of all open covers of M, thus o b t a i n i n g a true topological invariant. In Section 10.5, we will show that every open cover has a simple refinement and, in A p p e n d i x D, use this fact to prove Theorem 8.9.6. We sketch the construction of the isomorphism in L e m m a 8.9.8 a n d leave verification of several details to the exercises. Fix the choice of simple cover We define a linear m a p ~ : S 1 ( M ) ---+ Iv/1 (11).
Given [co] E H I ( M ) , select a representative w E [co]. By simplicity of the cover, H i ( u s ) = 0, so the restriction cos = colU~ of the closed 1-form co is exact, V(~ C 9.1. Thus, we can choose f s E A~ such that co~ = dfs, Vc~ E 91. On Us0 NU~x # (~, d(f~ o - f ~ ) = w - co - 0, so f~o - f ~ is locally constant on Uso A Us,. The cover
286
8. INTEGRATION AND COHOMOLOGY
being simple, this set is connected, so fao - f~l = cao~l E IR is a constant. This defines a Cecil 1-cochain c E ~,1(ll). But (5c)~o~,~
2 = c~m:
-
Oil UO:0 [~ Uoq N U0r we call set
c,~o~ 2 + c . . . .
SO C 9 21(r
=
(f~
-
f,~2) -
(f,~o -
f~2)
+ (f~o
-
f~l)
-- 0
If [c] 9 /;/1(1/) depends only on [co] 9 H I ( M ) ,
~([co]) = [ 4 E x e r c i s e 8.9.9. Prove t h a t the class [c] defined above is independent of the choice of representative co 9 [co] and of the choices of f~ 9 A~ such t h a t df~ = co<<. Consequently, ~o is a well-defined linear map. We define a linear m a p r
(U) --+ H 1 (M).
For this, we will need to fix the choice of a smooth partition of unity {k~}~Eet s u b o r d i n a t e to R. Given [c] 9 / / 1 ( l i ) , choose a representative cocycle c 9 [c]. For each a0 9 9.1, define f~o 9 A~ by f~o = E
C~o~k~.
aEg.
Then, on U~ o M U ~ :~ (~,
~e~
aCQI
I t follows t h a t 7S~ = 7S~o on U~ o N U ~ , so these exact forms assemble to give
a well-defined locally exact 1-form co 9 Z ~(M). If [co] 9 H 1(Mr) depends only on [C] 9 x~l(~-[), w e
can
set r
= [02].
E x e r c i s e 8.9.10. Prove t h a t the class [co] defined above is independent of the choice of representative c 9 [c]. Consequently, %b is a well-defined linear map. Lemma
8.9.11.
The homomorphisms qo and r are mutually inverse.
Proof. Given [c] 9 /:/I(U), tile definition of r = [co] produces functions f~ 9 A~ such t h a t colU~ = df~, g a 9 92. Using this choice in the definition of qo([co]) gives back the representative cocycle c. T h a t is, ~o o %b= id. For the reverse composition, the definition of ~,o([co]) = [c] selected the functions f~ 9 A I ( U ~ ) such t h a t all df~ = coIU~ and f~o - f~l = c . . . . . The definition of ~b produces different functions c~EQI
c~E92
where h = - ~
faAa E A ~
aE~
Tile closed form c~ obtained by piecing together the exact forms df~ is related to co by b-co=dh, so [9] = [w], proving t h a t 9 o ~ = id.
[]
8.9. D E R H A M T H E O R E M *
287
In particular, although "(a was defined relative to a choice of partition of unity, it inverts 9~ which did not depend on that choice, so ~b is, in fact, independent of the choice also. We close this section with some remarks about triangulations and cohomology. Let S = {e0, e l , . . . , e~} be the set of vertices of the standard n-simplex An. The convex hull of any subset ~ C_ S of cardinality p + 1 is a p-simplex. It lies in tile boundary of A~ and will be called a p-face of Am. The natural ordering of the indices of the points eie C ~ defines a canonical identification of this p-face with the s t a n d a r d p-simplex Ap. More precisely, this natural ordering defines a canonical linear imbedding A v ~ AN with image the given p-face. If s : An ~ M is a singular n-simplex, its p-faces are the singular p-simplices obtained by restricting s to the p-faces of A,~. Recall from Section 1.3 the fact that compact surfaces can be triangulated. A corresponding theorem for compact, differentiable n-manifolds also holds. T h a t is, the manifold can be divided up into a union of smoothly imbedded n-simplices A 1 , . . . A~, any two of which either do not meet at all or meet along exactly one common lower dimensional face. This theorem is intuitively plausible, but rather difficult to prove. If A stands for a choice of triangulation of M, we obtain a chain subcomplex (C.a(M), 0) C (C. (M), 0) by using only those singular simplices that are the inclusion maps of simplices of the triangulation. (By the simplices of the triangulation, we mean all of the p-faces of the n-simplices of A 0 <_ p _< rz.) This is called the simplicial chain complex associated to the triangulation A. Remark that C ~ ( M ) is a f n i t e dimensional vector subspace of Cp(M), 0 < p <_n, and vanishes if p > n. Let ia : (C,~(M),O)~, ( C , ( M ) , O ) be the inclusion map, a homomorphism of chain complexes, and let H .a (~i) be the homology of the simplicial chain complex. The following theorem is standard in algebraic topology (cf. [39, p. 191], wtlere it is proven more generally for simplicial complexes.) 8.9.12. The inclusion homomorphism i a induces a canonical isomorphis,~ H ? (M) = H. (M). Theorem
The beauty of this result is that the problem of finding tile homology of compact manifolds is reduced to a finite set of computations. Note that the theorem assures independence of the choice of triangulation, so one normally chooses A to have the fewest possible simplices. The triangulation of b'2 depicted in Figure 1.3.10 has the fewest simplices of any triangulation of S z. Triangulations can be used to give a proof, without appeal to Rienmnnian geometry, of tile existence of a simple cover of a compact manifold. We do not pursue this, but remark that it leads to a very simple proof that the Cech cohomology of this simple cover and the simplicial cohomology (Ha(M))* are canonically isomorphic. E x e r c i s e 8.9.13. Using the minimal triangulation of S 2 depicted in Figure 1.3.10, give a direct computation of the homology of S 2. E x e r c i s e 8.9.14. Fix a triangulation A of the compact n-manifold M and let cp denote the number of p-simplices of A, 0 _< p < n. Note that ev = dimCpa(M). Let hj, = d i m H ~ ( M ) = d i m H p ( M ) (called the pth Betti number of M). Define
288
8. I N T E G R A T I O N
AND COHOMOLOGY
the Euler characteristics of A and M by n
x(A) = E ( - 1 ) P c p , p= O n
%(M) = E ( - 1 ) P h p , p= O
respectively. (1) Prove that %(A) = x(M). Thus, this important topological invariant can be computed from a triangulation, but does not depend on the choice of triangulation. (2) Compute %(S2) and give an intuitive proof, not using part (1), that this number is independent of the choice of triangulation. (3) Prove that x(M) = 0 for compact, odd-dimensional manifolds M. (4) In fact, it can be proven that, if M is orientable,
)c(M) = 4AM, AM), the algebraic self intersection number of Exercise 8.8.6, part (4). The Poincar~-Hopf theorem (Exercise 8.8.7) then asserts: There is a nowhere
vanishing vector field on M if and only if the Euler characteristic of M vanishes. (In fact, the orientability condition, required in the earlier statement, can now be dropped.) Assuming this theorem, give a new proof that S n admits a nowhere vanishing vector field if and only if n is odd.
CHAPTER 9
Forms and Foliations In Section 4.5, we proved the vector field version of the Frobenius integrability theorem: a k-plane field E on a manifold M is integrable if and only if F ( E ) C iE(M) is a Lie subalgebra. In this chapter, we develop an equivalent version of this theorem, stated in terms of the G r a s s m a n n algebra A*(M) of differential forms. Useful consequences of this point of view will be treated. 9.1. T h e F r o b e n i u s T h e o r e m R e v i s i t e d Let M be a n n-manifold without b o u n d a r y and let E C_ T ( M ) be a s m o o t h k-plane distribution on M. D e f i n i t i o n 9.1.1. For each integer p >_ 0, the degree p annihilator of E is
IP(E) = {oa e AP(M) Lw(~) = 0, V~ C AP(F(E))}, where we u n d e r s t a n d that, for p = 0, I~
= 0. The annihilator of E is
I ( E ) = { I ' ( E ) } , , > 0 C A*(M). It is clear t h a t I ( E ) is a graded C ~ 1 7 6 true.
of A*(M), b u t more is
L e m m a 9.1.2. The annihilator I ( E ) is a 2-sided graded ideal in A*(M). Indeed, this follows by applying tile following lemma fiber-by-fiber. L e m m a 9.1.3. Let V be a finite dimensional vector space and let E C V be a
subspace. Then, the annihilator I ( Z ) = {co C A ' ( V * ) t ~(r ) = 0,V~ e A~'(E)}p_>0 is a 2-sided graded ideal in A(V*). Proof. Let d i m V = n, d i m e = k, and let { e l , . . . ,ek, f l , . . . ,f,~-k} be a basis of V with ei E E, 1 < i < k. Let {e~,... , e ~ , f { , . . . ,fn_k} be the dual basis of V*. T h e n , in particular, { f { ~ , . . . , f ~ - k } is a basis of 71 (E). Consider 71 = e*~, A ...
A ei* ' A
f;*~ A
...
A f!'a,._,,
9
A~"(V*),
where 1 < il < ... < ip < k, 1 < jl < "'" < Jr-p <_ n - k, a n d it is allowed t h a t either p = 0 or p = r. If p ~ r, it is clear that r/vanishes on a n y t h i n g of the form er~l A ... A e,~._, hence it vanishes on all ~ 9 A~'(E). I f p = r, 7! ranges over a basis of At(E*). It follows t h a t the set of all the forms 71, with p ~ r, is a basis of P ' ( E ) , r _> 1, clearly implying t h a t I ( E ) is a 2-sided graded ideal. [] D e f i n i t i o n 9 . 1 . 4 . A graded ideal g C_ A*(M) is a differential graded ideal if d(3) C_
290
9. F O R M S
AND
FOLIATIONS
T h e o r e m 9.1.5 (The Frobenius theorem). The following are equivalent for a hplane distribution E C_ T ( M ) :
(1) E is integrable; (2) [ ( E ) is a differential graded ideal; (3) d ( I I ( E ) ) c_ I2(E). For the proof, we need the following. L e m m a 9.1.6. L e t w E A I ( M ) and let X , Y E X ( M ) . Then
d w ( X A Y ) = X ( w ( Y ) ) - Y ( w ( X ) ) - w([X,Y]). Pro@ Define g : X ( M ) x X ( M ) ---* Coo(M)
by the formula ~(X, Y ) = X ( w ( Y ) ) - Y ( w ( X ) ) - w([X, Y]). This is clearly ]R-bilinear and antisymmetric. We claim, in fact, that ~ is Coo(M)bilinear. Indeed, let f E C ~ 1 7 6 and compute -~(IX, Y ) = f X ( w ( Y ) ) - Y ( f w ( X ) ) - w ( [ f X , Y]) = f X ( w ( Y ) ) - f Y ( w ( X ) ) - Y ( f ) w ( X ) - w ( f [ X , Y] - Y ( I ) X ) = f(x(~(z)) = f~(x,
- y(~(x))
-
~([x, Y]))
Y)
By antisymmetry, we also have ~(X, f Y ) = - ~ ( f Y , X ) = - f ~ ( Y , X ) = f ~ ( X , Y). Thus, ~ E T2(M). By antisymmetry, ~ C A:(M). In order to prove the equality of ~ and dw, it will be enough to show that these forms agree on any coordinate chart (U, x l , . . . , xn). (The previous paragraph was needed so that ~IU would make sense.) Write n
IU = y~ g~ dx ~, i=1 n
d~lU = ~ d(gi d.~), i=1
and remark that no generality is lost in assuming that w is of the form g dx i. By permuting the coordinates, assume that w = g dx 1, so &o= -
~Og ~ x i d X l A dx i. i=2
Thus, if k < j, -Og/OxJ
if k - - 1.
9.1. F R O B E N I U S
REVISITED
291
Since 1 _< k < j , we have d x l ( O / O x j) = 0, so
- 9 dx~
_
0]) (5))
o x k' cgxJ
0 (9
dxl
= fO -O9/cOxJ
[
(0
ifkT~ 1 if k = 1 O)
Since dcotU = FlU for an arbitrary coordinate neighborhood, dco = ~.
[]
Proof of theorem 9.1.5. We prove that (1) ~ (2). Thus, it is assumed that E is integrable and we must prove that, if co E I q (E), then dco C I q+l(E). By the integrability condition, it will be enough to prove this in a coordinate chart (U, x l , 999 , z n) such t h a t { O / O x l , . . . , O/Ox k } spans F(EIU). Then { d x k + ~ , . . . , dz n } spans I I ( E I U ) . In these coordinates, we write co
=
E
fil...iq dxil A ... A dx i" .
l<_ii<...
Then, if 1 _< il < .-- < iq _< k, fil"'iq =
A "" A
co
= O,
so every nonzero term in the expression for coIU contains at least one dx ij E I I ( E ] U ) . The same will then hold for dcoiU and, I ( E I U ) being an ideal, we see that dco[U E I q+~ ( E IU). Covering M with such charts, we conclude that dco C I q+l(E). Tile implication (2) ~ (3) is trivial. We prove that (3) ~ (1). Thus, we are given that d ( I I ( E ) ) C I 2 ( E ) . Let X, Y ff F ( E ) be given and choose an arbitrary element w C I I ( E ) . Then, since d~ C I 2 ( E ) , 0 = dco(X A Y ) =
x(co(r))
-
Y(co(x))
=-o =
-
co(IX,Y])
_=o
-co(Ix, Y]).
Since co E I S ( E ) is arbitrary, it follows that [X,Y] C F ( E ) . Since X , Y E F ( E ) are arbitrary, it follows that F ( E ) C_ X(M) is a Lie subalgebra. By the vector field version of the Frobenius theorem, E is integrable. [] E x e r c i s e 9.1.7. Let J" be a foliation of codimension q and integral to tile distribution E. Prove t h a t the exterior product of any q + 1 elements of I ( E ) vanishes identically. We note that Lemma 9.1.6, which played a key role in the proof of Theorem 9.1.5, is a special case of the following exercise.
292
9. FORMS
AND
FOLIATIONS
E x e r c i s e 9.1.8. I f w E Aq(M) and X 1 , ' " ,Xq+l E X(M), prove that
q+l dco(X 1 A . . . AXq+l) ~- E ( - 1 ) i W 1 x i ( c o ( X l A . . . A Xi A . " /k Xq+l) ) i=1 q- E ( - 1 ) i + J c o ( [ X i , X j ] i<j
A X1 A ... A s
A " " A X j A . . . A Xq+I).
Remark. If q = 0, the formula in Exercise 9.1.8 is understood to reduce to df(Xl)
= (-1)~+l Xl (f),
V f E A~
For q _> 1, the formula is noteworthy in that it gives a completely coordinate-free definition of the exterior derivative. It is useful in other ways, one of which is given in the following example. E x a m p l e 9.1.9. The formula in Exercise 9.1.8 is closely related to one t h a t occurs in a purely algebraic context. Let s be a finite dimensional Lie algebra over IR and let A(~*) be the exterior algebra of the dual vector space s Thus, Aq(13*) is the dual space of A q (2`) or, equivalently, the space of antisymmetric q-linear functionals on ,g. One defines a coboundary operator
5: Aq(2` *) --+ Aq+l(2` *) by the formula 5co(vl, . . . , vv+l)
SEi<j(--1)i+JCO([Vi,Vj],Vl,...,V"i,...,~j,...,Vq+I),
q ~ 1,
-L
[0,
q = 0,
where co E Aq(s is viewed as a multilinear functional and vi E ~ is arbitrary, 1 < i < q + l . The antisymmetry of~co is part of Exercise 9.1.10, so 5co E Aq+l (2`*). In Exercise 9.1.10, you are also asked to prove that ~: = 0, so (A(s 5) is a cochain complex. The cohomology of this complex is denoted by H* (~) and is called the cohomology of the Lie algebra s This is of considerable interest in algebra, b u t we want to remark on its use in studying the de Rham cohomology of Lie groups. If G is a Lie group and we apply the above construction to its Lie algebra 2, = L(G), we obtain a finite dimensional cochain complex (A(L(G)*),5). Remark t h a t the subspace of left-invariant q-forms in Aq (G) is canonically isomorphic to Aq(Te(G)*) = Aq(L(G)*). This can also be identified as the subspace of all co E Aq(G) such that co(X1,... ,Xq) is a constant function, for each choice of leftinvariant vector fields X 1 , . . . , X v E ~(G). The formula for dw in Exercise 9.1.8, when evaluated on L(G) C ~(G), reduces to 5co, Vco E Aq(L(G)*). It follows that there is an injective homomorphism (A(L(G)*),5) ~-~ (A*(C),d) of cochain complexes. There is induced a canonical homomorphism
L: H*(L(G)) ~ H*(G) of graded algebras and a surprising theorem, which we will not prove, asserts that, if G is both compact and connected, then ~ is an isomorphism (el. [47, Chapter IV]). Since, as a vector space, L(G) is completely determined by T~(G) and, as a Lie algebra, by the Lie derivatives at e of left-invariant vector fields, it follows that
9.2. T R A N S V E R S A L I T Y
293
the cohomology of the manifold G is entirely determined by infinitesimal data at e. This is a remarkable case of recovering global data from linear approximations at a single point. E x e r c i s e 9.1.10. For a; C Aq(~*), prove that tile coboundary operator 6, defined in Example 9.1.9, does produce an element &o E Aq+l(s Prove also that 62 = 0. E x e r c i s e 9.1.11. Using the theorem cited in the remark above, compute the de Rham cohomology algebra of the torus T n. E x e r c i s e 9.1.12. Prove that, if G is a Lie group, the set of connected Lie subgroups K C_ G corresponds one-to-one to the set of graded ideals g C A(L(G)*) such that 6(3) C_ 3. 9.2. T h e N o r m a l B u n d l e a n d T r a n s v e r s a l i t y
Let P C_ A I ( M ) be a C ~ ( M ) - s u b m o d u l e that is closed under locally finite sums. For example, if E C T ( M ) is a k-plane distribution on M, we might take I s = 1 I(E). For each z E M, set
Q~ = { ~ c T; (M) I~ c I ~} L e m m a 9.2.1. For each x C M , Qx is a vector subspace of T * ( M ) . This is rather obvious. Indeed, if we view P as a vector space over N, we see that co ~-~ a;~ defines an N-linear map 11 --* 77*(M) with image Q~.. Let q~ = dim Qx. D e f i n i t i o n 9.2.2. The C ~ ( M ) - s u b m o d u l e I s c_ A I ( M ) has constant rank q if
q~=q, VxE M. E x e r c i s e 9.2.3. Let M be an n-manifold, 11 C_ A t ( M ) a C~176 is closed under locally finite sums. Prove that the following are equivalent:
that
(1) 11 has constant rank q; (2) I I = F(Q), for some q-plane subbundle Q C_ T*(M); (3) I ~ = 1 I(E), for some (n - q)-plane subbundle E C_ T ( M ) . D e f i n i t i o n 9.2.4. If E is a p-plane distribution on M, set q = n - p, the codimension of E. Then the q-plane subbundle Q _C T* (M) such that I 1(E) = F(Q) is called the normal bundle of E. If E is integrable and 9" is the corresponding foliation, then Q is called the normal bundle of :t and q is called the codimension of 9: (codim 9").
This terminology comes from the following observation. 9.2.5. Let E be a p-plane distribution on M and fiz a choice of Riemannian metric on M . Let Z ~ : {v ~ T x ( M ) I v • Z~}, V x 6 M . Then Lemma
E~ = U E~ c T(M) x E A,I
is a vector subbundle isomorphic to the normal bundle Q. Proof. Denote the Riemannian metric on T~(M) by (.,.)~ and define an isomorphism ~ : T(M) -~ T*(M) of bundles by
~ ( v ) = (v,.L, Vx ~ M, Vv c ~ ( M ) , It is clear that ~ carries E • onto Q.
[]
294
9. FORMS AND FOLIATIONS
Our way of defining the n o r m a l b u n d l e as a s u b b u n d l e of the cotangent b u n d l e is intrinsic. Defining it in T ( M ) via a R i e m a n n i a n metric gives a s u b b u n d l e t h a t depends on the choice of the metric. D e f i n i t i o n 9.2.6. Let M be a manifold with a foliation 9- t h a t is integral to the d i s t r i b u t i o n E. A smooth m a p f : N ~ M is transverse to 9" if, Vx E N , f . x ( T x ( N ) ) U El(x) spans Tf(~)(M). In this case we write f ch 9". R e m a r k t h a t no assumption is made a b o u t the relative dimensions of N a n d M other t h a n what is implicit in the definition: dim N must be large enough t h a t its s u m with the dimension of the leaves of 9" is at least as large as dim M. R e m a r k also t h a t a submersion f : N --~ M is automatically transverse to every foliation of M. 9.2.7. Let 9: be a foliation of M with normal bundle Q. A smooth map f : N --~ M is transverse to 9: if and only if f~ : T](~)(M) ~ T * ( N ) is one-to-one on Qf(x), V x E N.
Lemma
Proof. Assume t h a t f ch 5. I r a E Qf(~) and f~(c~) = 0, then, for every v e Tx(N), o~(f.x(v)) = fx(O~)(v) = O. B u t a also vanishes on Ef(~), hence, by transversality, it vanishes on all of Tf(~)(M). T h a t is, a = 0. For the converse, suppose that f* : Qf(,) --~ T * ( N ) is one-to-one. Let v c Tf(~)(M). We are to prove t h a t v is the sum of an element of Ei(x) and an element of f.~(T~(N)). Choose a basis { a l , . . . ,aq} C Qf(~) and let ai = a i ( v ) , 1 5 i _< q. Since { f ~ ( a l ) , . . . , f ; ( a q ) } is linearly independent, there is a vector w e T~(N) such t h a t ai = f~(ai)(w), 1 <_ i <_ q. T h a t is,
ai(f.x(W) - v) = ai - a~ = O, l < i < q, from which it follows t h a t f . x ( w ) - v E El(z).
[]
9.2.8. Let 5: be a foliation of M, codimg- = q, and suppose that f : N --+ M is smooth and transverse to 9:. Then there is a canonically defined foliation f - l ( 9 ~ ) (called the pullback of 9: by f ) of N of eodimension q such that f carries each leaf o f f - i ( 9 ") into a leaf of g:. Theorem
Pro@ Let 9" be integral to the distribution E and have normal b u n d l e Q. T h e n f * ( I I ( E ) ) C_ A~(N) is a vector subspace a n d we let 11 C_ A I ( N ) be the C ~ ( N ) s u b m o d u l e , closed under locally finite sums, t h a t is generated by this subspace. A n a r b i t r a r y element r1 E 11 can be w r i t t e n locally as
7] = E
fil]i'
i=1
where~?i = f*(a~i) a n d w i E I I ( E ) = F(Q), 1 < i < f. B y L e m m a g . 2 . 7 , 11 will have c o n s t a n t r a n k q, so we can write ]1 :
Z 1 ( ~ ) = r((~)),
as in Exercise 9.2.3. It is clear t h a t f * ( I ( E ) ) C_ I(E). Furthermore, writing 7] E i I ( ~ ) locally as above, we see that, locally, g
dr] = E ( d f i A f*(wi) + fif*(da;i)) e i2(/~), i=1
9.2. T R A N S V E R S A L I T Y
295
since f*(doa~) E f * ( I ( E ) ) by the integrability of E (Theorem 9.1.5). B u t this implies t h a t E is integrable, again by Theorem 9.1.5. Define f-1(9-) to be the foliation integral to E. The normal bundle is s so codlin f - l ( 9 ~) = q. It remains to be shown t h a t each leaf of f-1(9-) is carried by f into a leaf of 9-. Since the leaves of 9- (respectively, of f - l ( ~ ) ) are m a x i m a l connected integral manifolds to E (respectively, to 2 ) , it will be enough to show t h a t f . ~ ( E ~ ) C_ El(x), Vz E N. But, if ct C Qf(x) and v ff Ex, then
e~(f.z(v)) = f~(c~)(v) = 0, since f ~ ( a ) E s
Since a C Qf(~) is arbitrary, it follows t h a t
f.~(v) e S f ( x ) ,
VV C Ex.
[] E x a m p l e 9.2.9. Let f : N --+ M be a submersion, dim M = q, dim N = n. T h e n , as y ranges over M , the connected components of tile submanifolds f - l ( y ) range over the leaves of a foliation of N of codimension q. Indeed, the u n i q u e 0-plane d i s t r i b u t i o n on M is trivially integrable, the leaves of the corresponding foliation being the points of M. This foliation is of codimension q, so there is a pullback foliation f-1(9-) on N of codimension q. Each leaf L of f - 1 ( 5 ) is a connected s u b m a n i f o l d of N of dimension n - q and is carried by f into a point y of M. By the c o n s t a n t r a n k theorem, f - l ( y ) is also a submanifold of dimension n - q, so L must be a connected c o m p o n e n t of f - 1 (y). E x a m p l e 9 . 2 . 1 0 . Let El, E2 C_ T ( M ) be integrable subbundles with corresponding foliations 9-i of codimension qi, i = 1, 2. Let Ei have fiber dimension Pi = rt - qi, i = 1, 2, where n = d i m M . If, for each leaf L of $1, the inclusion m a p L : L ~-+ M (a one-to-one immersion) is transverse to 9-2, we will say t h a t 9-1 is transverse to 52 a n d write 9-1 ch 9-~. This simply means that, for each z C M , Elm U E2x spans T~(M), so the relation is symmetric (9-1 gb 9-2 ~ ~2 ch 9-1). In this case, if z E M and ~ : L~ ~-+ M is the inclusion of the leaf through z of ~1, t h e n ~21(9-2) is a foliation of L~ of codimension q2- Thus, the leaves of L~-I(2"2) have t a n g e n t spaces contained in the restriction of E2 to L~(Lx). These t a n g e n t spaces also lie in E1 and their dimension is pl - q2 = Pl + p2 - n, the fiber dimension of the b u n d l e E1 n E2. Since each point of M lies in a leaf of E l , it follows that E1 n E2 is integrable and t h a t the leaf through z C M of the corresponding foliation is the the leaf t h r o u g h x of ~21(9~2). These leaves are just the connected components of the intersections (when n o n e m p t y ) of leaves of 9-1 with leaves of 9-2. We can denote this foliation of M by 9:1 A 9-2. It is of codimension n - (Pl + p2 - n) = ql + q2.
D e f i n i t i o n 9.2.11. A p-plane distribution E on M is transversely orientable if its n o r m a l b u n d l e Q is orientable. A foliation 9- of M of dimension p is transversely orientable if it is integral to a transversely orientable p-plane distribution.
Remark. If 9: is transversely orientable and M is orientable, t h e n each leaf of 9- is an orientable manifold. Indeed, T ( M ) ~ E | Q a n d it follows easily t h a t E is an orientable vector bundle. Thus, the t a n g e n t b u n d l e to a leaf L, being the restriction of E to L, is orientable.
P r o p o s i t i o n 9 . 2 . 1 2 . If ~ is a transversely orientable foliation of M and f : N M is transverse to 9-, then f-1(9:) is transversely orientable.
296
9. F O R M S AND FOLIATIONS
Proof. Let Q be the normal bundle of 9" and Q the normal bundle of f - l ( 9 . ) . These are q-plane subbundles of the cotangent bundles of M and N, respectively. Since Q is orientable, there is a nowhere vanishing section w of Aq(Q). By Lemma 9.2.7, f*(w) is a nowhere vanishing section of Aq(Q), proving that (~ is orientable. [] E x e r c i s e 9.2.13. Let E C_ T ( M ) be a p-plane distribution on the n-manifold M, and let Q c_ T*(M) be its normal bundle, a q-plane bundle where q = n - p. Assume that E is transversely orientable, hence that there is a nowhere zero q-form co C F(Aq(Q)). (1) For each x C M, prove that Ex is the set of all vectors v C Tx(M) such t h a t cox(v A vl A . . . A ~2q_l) ~-- 0, for all choices of v l , . . . , vq,1 E Tx(M). We call E~ the nullspace of co~ and we also say that E is defined by the partial differential equation (P.D.E.) co = 0. If E is integrable, the leaves of the foliation 9. integral to E are said to be the maximal solutions to the P.D.E. w = 0. In this case, w is said to be integrable. (2) Prove that a: is integrable if and only if there is a form ~7 E A 1(M) such that
dco = ~l A co. (3) Let the foliations 9.1 and 9"2 in Example 9.2.10 be transversely orientable and let the P.D.E. coi = 0 define the bundle Ei, i = 1,2. Show t h a t the P.D.E. COl A a~2 = 0 defines E1 A E2 and verify the integrability condition d(Wl A ~S) ---- ~ A w l A w'~
as a direct consequence
of the integrability of col and w2.
Exercise 9.2.14. Let 9. be a transversely orientable foliation of codimension q with normal bundle Q and tangent distribution E. Let w E F(Aq(Q)) be nowhere zero. Let r/E At(M) be such that dw = uAw. (1) Prove that d~7 E I2(E). (2) Prove that 7] A (d~) q E A2q+i(M) is a closed form. (3) Show that [r/A (dr/)q] C H 2q+l(M) does not depend on the allowable choices of co and of rl. (Hint: First hold co fixed and prove independence of the choice of rl. Then prove independence of the choice of w.) This class is denoted by gv(9") and called the Godbillon-Vey class of 9.. (4) If f : N ~ M is transverse to 9~, prove t h a t
f* (gv(9.)) ----g v ( f -1(9.)). This is called the naturality of the Godbillon-Vey class. The Godbillon-Vey class was discovered in the early 1970s, leading to a formidable b o d y of research into the algebraic topology of foliations.
9.3. Closed, Nonsingular 1-forms* The topology of foliations is a fascinating and subtle topic. In this book we can only scratch the surface, but, with the tools developed so far, there are some interesting questions about foliations of codimension one that are accessible. One of these starts with the question: which compact, connected, boundaryless n-manifolds M admit closed, nowhere zero 1-forms co? A nowhere zero form is also said to be
nonsingular. Throughout this section, we fix the hypothesis that the n-manifold M is compact and connected with OM = O.
9.3. C L O S E D , N O N S I N G U L A R 1-FORMS*
297
L e m m a 9 . 3 . 1 . In order that M , as above, admit a closed, nonsingular f o r m cz E A I ( M ) , it is necessary that H I ( M ) ~ O. Indeed, such a f o r m determines a nontrivial element [w] C H 1 ( M ) .
Proof. If w = df, some f C C ~ ( M ) , the compactness of M implies the existence of a critical point x C M of f . For instance, a point where the n m x i m u m is a t t a i n e d will do. B u t df~ = 0 at any critical point x, contradicting the assmnption t h a t aJ = df is nonsingular. [] It is known, however, t h a t this condition is not sufficient. T h e following observation relates the question to foliations. Lemma
9.3.2. Let ~ C A I ( M ) be closed and nowhere O. At each x c M , define
E~ = {v E T~(M) I ~x(~) = 0}. Then E = U~eM E~ is an integrable distribution on M . Pro@ Indeed, let 11 C A I ( M ) be the C ~ generated by oz. T h a t is, 11 is the set of all fa~, f E C ~ ( M ) , hence is of constant rank 1. T h e n E is clearly the (n - 1)-plane distribution such t h a t 11 = P ( E ) . Since ~ is closed, we have d ( f w ) = df A w C I 2 ( E ) , so E is integrable by T h e o r e m 9.1.5. [] We let ~ denote the foliation of M corresponding to the closed, nonsingular 1-form aJ. T h e codimension of 5~ is 1. E x a m p l e 9 . 3 . 3 . Since S 1 is parallelizable, it admits a nowhere 0 form 0 E A 1 ($1). By default, dO = 0. T h e corresponding foliation is the zero-dimensional foliation having each point of S 1 as a leaf. Admittedly, this is not an interesting example, but it leads to a class of interesting examples, namely the manifolds t h a t fiber over S 1. T h a t is, we consider a compact, connected, boundaryless n-manifold M , t o g e t h e r with a s m o o t h m a p 7r : M - ~ S 1 t h a t is locally trivial in a sense quite similar to the local triviality of vector bundles and of principal bundles: there is a c o m p a c t (possibly not connected) (n - 1)manifold F w i t h o u t b o u n d a r y such t h a t each point z E S 1 has a neighborhood U and a c o n m m t a t i v e d i a g r a m
7r-l(U)
~
U
, U x F
,
U
id
such t h a t F is a diffeomorphism. We say t h a t 7r : M + S 1 is a fibration (or a fiber bundle with fiber F ) . In particular, 7r : M -~ S 1 is a submersion and the ( n - 1)-manifolds 7c-1(z) are all diffeomorphic to F. Tile connected c o m p o n e n t s of these fibers are tile leaves of a (codimension one) foliation, as was observed in E x a m p l e 9.2.9. Also, w = 7r*(0) is a closed, nonsingular 1-form on M and $~ is exactly tile foliation by the components of the fibers. E x a m p l e 9.3.4. Let T 3 = S 1 x S 1 x S 1. T h e r e are three obvious fibrations 7ri : T 3 --~ S 1 given by 7ri(zl,z2,z3) = zi, 1 < i < 3. These are trivial cases of E x a m p l e 9.3.3, but there are more interesting examples of closed, nonsingular forms cz c A I ( T 3 ) . Indeed, let a3~ = 7r~(0), 1 < i < 3. These are pointwise linearly
298
9. F O R M S A N D F O L I A T I O N S 3
independent over II~, so every nontrivial linear combination w = Y'~i=l aiwi is closed and nonsingular. If, when we view N as a vector space over the rationals Q, the set {ax,a2,a3} is linearly independent, then the corresponding foliation 9:~ of T a has each leaf diffeomorphic to IR2 and dense in T a. Similarly, if we require t h a t two of these numbers, say {al,a2}, be linearly independent over Q, but not all three, each leaf of 9:~ is diffeomorphic to IR x S 1 and is dense in T a. Finally, if {al} is linearly independent over Q, but {ai,a2} and {al,a3} are not, each leaf of 9:~ is diffeomorphic to T 2 and these leaves are the fibers of a fibration of T a over S 1. Remark t h a t any triple {al, a2, a3} can be uniformly well approximated by triples of this last type. Thus, there is a sense in which the linear foliations of T a by dense planes or by dense cylinders can be uniformly well approximated by fibrations over S 1. These assertions are left as an exercise. E x e r c i s e 9.3.5. Fill in the details in Example 9.3.4 D e f i n i t i o n 9.3.6. Let 9" be a foliation of M of codimension one. Let the flow q) : II~x M --~ M be smooth, nonsingular (i.e., no stationary points), with flow lines everywhere transverse to 9:. If, for each leaf L of 9: and each t E R, ~t(L) is also a leaf of 9:, we say that ~5 is a transverse, invariant flow for 9:. L e m m a 9.3.7. The flow 9 : IR x M --~ M (not necessarily nonsingular and not
necessarily transverse to 9:) carries leaves of 9: to leaves of g: if and only if the infinitesimal generator X C Z ( M ) of 9 satisfies [X, F(E)] C F ( E ) ,
where E C T ( M ) is the integrable distribution of tangent spaces to 9:. Proof. If 9 carries leaves to leaves, then '5-t.(Ex) C_ Ee_dx),
Vx E M,
Vt CIl~.
Thus, if Y E F ( E ) ,
[X,Y]
= lira ' f - t * ( Y ) - Y e r ( ~ ) . t~0
For the converse, suppose that IX, r(E)] c r(E) and remark that it will be sufficient to show that, in any Frobenius chart (U, x l , . . . ,x n) for 9:, (I)t carries plaques to plaques for small enough values of t. Here, we assume that the plaques are the level sets x ~ = eonst. In these coordinates, we write
x=
fiTxS. i=l
The condition t h a t
[0 ] ~xJ X
EF(EIU),
l<j
implies that fn = f~(x n) is independent of x l , . . . , x n-1. Thus, the local system of O.D.E. for/i) is
dx i dt dx n dt
-
fi(xl,...,x~),
-
fn(xn).
1
9.3. C L O S E D ,
NONSINGULAR
1-FORMS*
299
Consequently, given the initial condition a = ( a S , . . . , a n) E U, the n t h coordinate depends only on a n a n d t. T h a t is, the plaque a n = a ~ is carried into the plaque x n = xn(a n, t). []
xn(a, t) of the flow line r
9.3.8. The foliation 5: admits a transverse, invariant flow if and only if g- = 9-~o, for some closed, nonsingular 1-form co E A I ( M ) .
Theorem
Proof. Assume t h a t 9 is a transverse, invariant flow for 9" and, using the infinitesimal generator X E / E ( M ) of the flow, define a nonsingular 1-form co by co(x) cole
-
1,
-=
0,
where E is the d i s t r i b u t i o n of t a n g e n t spaces to 9". Clearly, cox spans tile n o r m a l fiber Qx, Vx c M. We m u s t prove that d~ = 0. For this, let x E M a n d choose a basis { v l , . . . , v n - 1 } C Ex. We can extend vi to a field Yi C F ( E ) , 1 < i < n - 1. Since F ( E ) C X ( M ) is a Lie subalgebra, dco(~, ~ ) = E ( c o ( ~ ) ) - ~ ( c o ( ~ ) ) - co[~, ~1 = 0,
1 < i , j _< n -
1. Also, since co(X) - 1,
dco(X, ~ ) = X(co(~)) - ~(co(X)) - co[X, ~ ] = -co[X, ~ ] , 1 < i < n - 1. By the invariance property and L e m n m 9.3.7, [X, Y~] E F ( E ) , a n d it follows that & ( X , ~ ) = -co[X, ~ ] = O. In particular, (dco)x vanishes on all pairs from the basis { v , , . . . , V n - l , X x } of T x ( M ) , hence (dco)~ = O. Since x C M is arbitrary, dco = O. Conversely, suppose t h a t Y = Y~, where dco = O. We must produce the transverse, invariant flow. Let{Us, x ~ , . . . , x~}s~__l be a cover of M by Frobenius charts, the Y-plaques in Us being the level sets x~ = const. Then, cos = colUs = f s
d,~,
where f s ~ 0 on Us. Set 1 X s
0
-
f~ O x a ' a vector field transverse to the plaques and satisfying
cos(Xs) = 1. Let {~s}s'~l be a p a r t i t i o n of unity s u b o r d i n a t e to the Frobenius atlas a n d set nl
x = Z Asxs
X(M)
oL=I
This vector field satisfies co(X) K 1 and, in particular, the flow 9 t h a t it generates is everywhere transverse to 9". Furthermore, if Y C F ( E ) , 0 = c/co(X, Y) = X(co(Z)) - Z(co(X)) - co[X, Y] = -co[X, Y], implying t h a t [X, Y] E F ( E ) . Since Y C F ( E ) is arbitrary, L e m m a 9.3.7 completes the proof t h a t q) is a transverse, invariant flow for 9-~. [] Fix the hypothesis t h a t co, Y~, and ~5 : IR x M --+ M are all as above. Let L denote the 1-dimensional foliation of M by the flow lines of d).
300
9. F O R M S A N D F O L I A T I O N S
9 . 3 . 9 . For arbitrary leaves L and L' of :Y~o, there are values t E IR such that (Pt(L) A L' ~ ~, in which case {~t carries L diffeomorphically onto L'.
Lemma
Proof. It is clear t h a t ~ t ( L ) N L' # 0 if and only if ~ t carries L diffeomorphically onto L'. Thus, we o b t a i n an equivalence relation on the set of leaves by setting L ~ L' if and only if there is such a value of t. Since the flow is leagpreserving a n d transverse to if, an easy application of the inverse function theorem proves t h a t : ]R x L --- M is a local diffeomorphism, hence it has as image an open subset of M . This image is the union of the leaves equivalent to L, hence, by the connectivity of M, all leaves are equivalent. [] If L is a leaf of flu, denote by P(L, co) the set {t E IR I q)e(L) = L}. 9 . 3 . 1 0 . I l L and L' are leaves, then P(L, co) = P(L',co) and this set, call it P(co), is an additive subgroup of R.
Lemma
Proof. Let t E P ( L , co). Let r E ]R be such that (I)~(L) = L'. T h e n ~)t(L') = Ot+r(L) = r
= (Pr(L) = L'.
Thus, P(L, co) C_ P(L', co) a n d the reverse inclusion is proven in the same way. This set P(w) carries eve W leaf to itself. By the properties of a flow, it is clear t h a t P(w) is closed under addition and multiplication by - 1 , hence is an additive s u b g r o u p of IR. [] 9.3.11. Let {U~,x~,... 1 n m ,x~}~=~ be a Frobenius atlas for 9",~. Let a : [a, b] --~ M be a piecewise smooth loop. Prove that a is homotopic to a piecewise s m o o t h loop r with the following property: there is a p a r t i t i o n Exercise
a = to < tl < ... < t p = b such that, for 1 < i < p, rt[t~_l,t~ ] C U~, and this segment either lies in a plaque of 5 or in a plaque of L. C o r o l l a r y 9.3.12.
The group P(co) is exactly the set of periods of the closed l-form
CO.
Proof. If a C P(co), the segment sl(t) = (Pt(xo), where t ranges from 0 to a, has b o t h e n d p o i n t s in the leaf L of ff through x0. Let s2 be a piecewise s m o o t h p a t h in L from the e n d p o i n t of Sl to the initial point of sl. T h e n s = sl + s2 is a piecewise s m o o t h loop in M a n d it is clear t h a t f, co = a. T h a t is, a is a period of co. For the reverse inclusion, choose a piecewise s m o o t h loop a and deform it to r as in Exercise 9.3.11. Let rk = rl[pk,vk], 0 < k < r, (taken in increasing order) be the segments of r t h a t lie in L-plaques and let Tl[Vk,#k+l], 0 < k < r, lie in if-plaques, with r ( # r + l ) = r(p0). T h e n ~ c o = frco= ~-~, fr w = ~-~ak=a, k=l
k
k=l
a n d a is a period of co. All periods can be obtained in this way. Let Lk denote the leaf t h r o u g h r(#k), 0 < k < r + 1 (hence Lr+l = L0). T h e n O~k(Lk) = Lk+l, 0 < k < r, a n d q)~(L0) = L0, proving t h a t a r P(co). [] L e m m a 9.3.13.
dense in R.
The subgroup P(co) C I~ is either infinite cyclic or everywhere
9.3, C L O S E D , N O N S I N G U L A R 1-FORMS*
301
Proof. Indeed, if P(oa) r 0, this follows from Corollary 4.4.10. But, by Theorem 6.3.10, P(w) = 0 implies t h a t [co] = 0, contradicting L e m m a 9.3.1. [] 9 . 3 . 1 4 . If the period group P(co) of a closed, nonsingular f o r m w E A I ( M ) is infinite cyclic, then the leaves of g:~o are the fibers of a suitable fibration p : M ~ S 1 . If P(co) is not infinite cyclic, each leaf of :Y,, is dense in M .
Theorem
Proof. If P(co) is n o t infinite cyclic, then L e m m a 9.3.13 implies t h a t a dense set of real n u m b e r s t has the property t h a t Or(L) = L, for an arbitrary leaf L of 9~. Since ~5 : IR x L -+ M is onto, it follows easily that L is dense in M . Suppose t h a t P(aa) is infinite cyclic and let a E (0, oo) be the smallest positive period. Then, replacing co by co~a, we lose no generality in assuming t h a t P(co) = Z. Fix x0 E M and consider all piecewise smooth paths ~r : [a, b] -+ M with a(a) = x0. If a(b) = x, define t ) ( z ) = e 2 ~ i J; ~
In order to see t h a t this is well defined, let r be another such p a t h from x0 to x. Let -or denote the p a t h from x to x0 obtained by reversing the p a r a m e t r i z a t i o n of a n d consider the loop ~ = r + ( - ~ ) . Since the period f~ aa = k is an integer, it follows t h a t
e2~ri.I; w = e2~i(ta+.l; ~,) = e2rciJ" (w). Furthermore, since M is assumed to be connected, p(x) is defined for every x C M. The reader can verify t h a t p : M + S 1 is smooth. We claim t h a t p(x) is constant as x ranges over a given leaf L of 5~. Indeed, given a r b i t r a r y points x, y C L, let a be a p a t h from xo to y and let r be a path in L from y to x. B u t f~ co = O, since co vanishes on vectors in E, so p ( x ) = e 2rrif~+. w = e27ri C w =
p(y).
This shows t h a t plL is constant. By the definition of P(a~), it is clear that p : M S 1 sets up a one-to-one correspondence between the leaves of Yw a n d the points of S 1. Finally, we prove local triviality. Indeed, given z C S 1, let U C S 1 be the open arc {ze 2"it [ - 1 / 2 < t < 1/2}. Let Lz = p - l ( z ) . The m a p @ : g • Lz ~
p-l(u),
defined by r = (I)t(w), is a diffeomorphism and the inverse diffeomorphism = r makes the diagram
p_l(U)
vo ~ U x Lz
Pl S 1
commute.
Im ~
S 1
[]
Thus, the possibilities illustrated in Example 9.3.4 are the typical ones. T h e last s t a t e m e n t in t h a t example, that the linear foliations of T a with dense leaves can be arbitrarily well approximated by fibrations over S 1, is also typical. 9 . 3 . 1 5 (D. Tischler). If co is a closed, nonsingular 1-form on M , then there is a sequence {co~}~=1 of closed, nonsingular 1-forms with P(coi) ~ Z, Vi >_ 1 such that limi~oo wi = w uniformly on M .
Theorem
9
o o
302
9. F O R M S
AND
FOLIATIONS
C o r o l l a r y 9.3.16. There is a closed, nonsingular form a; E A I ( M ) if and only if
M fibers over S 1. This corollary is an answer to the opening question of this section. The proof of Theorem 9.3.15, which will be found in [45] and in [6, Section 9.4], is not difficult, but it uses some algebraic topology not developed in this book. Among other things, one needs the fact, mentioned (but not proven) after Exercise 6.4.19, t h a t the loops in that exercise define an isomorphism of H I(M; Z) onto Z k if M is compact. E x e r c i s e 9.3.17. Let 3" be a foliation of M of codimension 1 and let ~[ :
{Ua,xl,...
n 77~ ,Xo~}~= 1
be an atlas of Frobenius charts. Thus, on Us N UZ, the change of coordinates has the form
z~i = x ~i ( x ~1, . . . , z ~ ) , n
n
1
n
z~ = x~(x~).
(1) If there is a closed, nonsingular 1-form a~ such that 2~ = 9:~, prove t h a t the Probenius atlas can be chosen in such a way that the second equation above always takes tile form n
X a = X~-~Caj3
~
where caz E R is a constant, defined whenever Ua A Ufl ~ O. (2) Conversely, if the Frobenius atlas can be chosen as in part (a), prove that there is a closed, nowhere vanishing 1-form w such that :Y = :Y~. (3) In this situation, show that c~z = c(U~, UZ) defines a Cech cocyele on the open cover ~. (4) It is a fact that, in the above situation, the Frobenius cover ~ can be chosen to be simple. Assuming this, prove that the de Rham isomorphism H 1(M) = /:/1 (~/), as defined in Section 8.9, identifies [c] C/:/1 (~A) with [~] E H 1(M).
C H A P T E R 10
Riemannian Geometry Properly speaking, geometry is the study of manifolds that are equipped with some additional structure that permits meas~rerner~ts. For example, nowhere in the definition of a piecewise smooth curve is there anything that would enable us to measure tile length of the curve. Likewise, on a compact, oriented n-manifold, we can integrate n-forms, but which of these integrals should be interpreted as the volume of the manifold? And given intersecting curves, how could we measure the angle they make at an intersection point? The additional structure that is needed is a metric tensor, Riemannian metrics and, to a lesser extent, pseudo-Riemannian metrics, being the main examples. Such a tensor makes it easy to define the quantities mentioned above and provides much more. For instance, the metric tensor gives rise to the "Levi-Civita connection", which can be thought of as a way of parallel transporting vectors along curves. One is led to study special curves that are "straight" in the sense that the velocity field is parallel along them. These are "geodesics", the analogues in Riemannian geometry of straight lines in Euclidean geometry. These geodesics are locally length minimizing, but this may fail in the global sense. For instance, if two points on a sphere are not antipodal, then, in the standard metric on the sphere, the great circle through these points is a geodesic. It falls into two imbedded arcs joining the points, one of which is the shortest curve joining them, but the other clearly is not. Parallel transport along curves holds some surprises for Euclidean "flatlanders". For instance, consider the geodesic triangle cr on S 2 in Figure 10.0.1. Imagine that you are a two-dimensional native on S 2. Starting at point A, you walk down the first leg of the triangle, holding the initial tangent vector straight ahead so as to keep it (in your world) always parallel to its original position. Upon arriving at point B, you start moving sideways along the equatorial geodesic, determinedly keeping the vector pointing in a direction always parallel to its earlier positions. Finally, at C, you start up the last leg of your journey, walking backwards and again holding the vector in front of you in a constant parallel direction. Upon arriving at A, you find that, despite your best efforts not to change the direction of the vector, it ends up pointing in a different direction at A than it started with! Although, at the beginning of this experiment, you may have been convinced that your world was a Euclidean plane, you now have evidence of intrinsic "curvature", something you (as a two-dimensional creature) probably cannot imagir~e, but can nevertheless conceive. If you are Gauss, you may even be able to figure out how to compute the curvature of your world via experiments such as the above.
304
10. R I E M A N N I A N G E O M E T R Y
F i g u r e 10.0.1. A parallel field along a in S 2 10,1. C o n n e c t i o n s Let U C_ ~ Write
be open. Given
X,Y E
:~(U), define 0
X= Ef~-zi, i=1
j=l
075'
and define 0 j=l
Ogj 0 =
i,j=l
f~
axe
We can view D as an R-bilinear map D : X(U) • X(U) --~ :~(U). It has the following properties:
Dx(Y) C :~(U) as
follows.
10.1, C O N N E C T I O N S
305
(1) D I x ( Y ) = f D x ( Y ) , V f E C~176 VX, Y E 2C(U). (2) D x ( f Y ) = X ( f ) Y + f D x ( Y ) , V f E Coo(U), V X , Y E 2C(U). This is an example of a connection. D e f i n i t i o n 10.1.1. Let M be a smooth manifold. A connection on M is an IRbilinear map V : X(M) • X(M) -~ X(M), written V(X, Y) = V x Y or V x (Y), with the following properties: (1) V I x Y = f V x Y , V f E C~176 VX, Y E R~(M); (2) V x ( I Y ) = X ( f ) Y + f V x Y , V I E C~176 V X , Y E X(M). The connection D, defined above on open subsets U C ]Rn, is called the Euclidean connection.
Remark. By property (1), V is a tensor in the first argument. Thus, if v E Tz(M) and Y E X(M), V~,Y E T=(M) is defined. By property (2), however, V is not a tensor in the second argument. Property (2), together with the Coo Urysohn lemma, allows us to prove in standard fashion that V~Y E Tx(M) depends only on the values of Y in an arbitrarily small neighborhood of z. Thus, connections can be restricted to open subsets of M. We describe an important class of examples, the Levi-Civita connections for submanifolds of Euclidean space. Let M c R m be a smoothly imbedded n-manifold. If z ff M, then Tx(R m) = IRm canonically. Also,
Tx(~~) = Tx(M) 9 ~ ( M), where ~ , ( M ) = {v E T~(IR") I v • T~(M)}, perpendicularity being defined by the Euclidean metric in IRm. Then w(M) = U
~:~(M) C T(1Rm)iffI
zEM
is an (ra - n)-plane bundle over M and
T(Rm)l M = T ( M ) | u(M) is a canonical direct sum bundle decomposition. The summand ~(M) is the normal bundle of M in IRm introduced in Subsection 3.7.C. The canonical projection
p: T(R'~)I M = T ( M ) @ L,(M) ~ T ( M ) is a surjective homomorphism of bundles. Let X , Y E ~ ( M ) . Given x E M, there is a neighborhood U of x in N m and extensions X, Y of XI(U N M) and YI(U N M), respectively, to fields on U. Then, depends only on 0(~ = X~ and on Y. Represent X~ = <s>x as an infinitesimal curve, where s : ( - e , e) -+ M is smooth and s(O) = z. Then,
(D29)~
Dx,.Y
=
-~(P~(t) d )~=0
=
d(y,(,)) ~=0
and this depends only on Y[(U N M), not on the choice of extension !/. Therefore, D x Y is a well-defined element of F(T(Rm)IM).
306
10. R I E M A N N I A N
GEOMETRY
D e f i n i t i o n 10.1.2. If X, Y C 32(M), t h e n the operator V : X(M) • X(M) -~ X ( M ) , defined by
VxY = p(DxY) is called the Levi-Civita connection on M C IRm. T h e following is totally elementary, as the reader can check. Lemma
10.1.3.
The Levi-Civita connection is a connection on M.
If V is a connection on M and (U, x l , . . . ,x ~) is a coordinate chart, set ~i =
O/Ox i a n d write
v~j = ~ r~& k=l
D e f i n i t i o n 10.1.4. The functions F ~ E C~176 are called the Christoffel symbols of V in the given local coordinates. D e f i n i t i o n 10.1.5. Let V be any connection on a manifold M. T h e torsion of V is the IR-bilinear m a p T : ~ ( M ) x ~ ( M ) --+ ~ ( M ) defined by the formula
T ( X , Y) = V x Y - V z X - [X, Y]. If T _= 0, t h e n V is said to be torsion free or symmetric. E x e r c i s e 10.1.6. Prove t h a t the torsion T of a connection V on M is Coo(M)bilinear. T h a t is, T C 'J~I(M) and T ( v , w ) E Tx(M) is defined, Vv, w C T z ( M ) , Vx E M . Torsion is a tensor. E x e r c i s e 10.1.7. Prove that the connection V is torsion free if a n d only if, in every local coordinate chart (U, x l , . . . , xn), the Christoffel symbols have the s y m m e t r y k k Fij = Fji ,
1 < i , j , k < n.
This is the reason that torsion free connections are also said to be symmetric. E x e r c i s e 10.1.8. If M C IRm is a smoothly imbedded submanifold, prove t h a t the Levi-Civita connection is symmetric. We use connections to define a way of differentiating vector fields along a curve. Indeed, if X E 2E(M) a n d s : [a, b] --* M is a s m o o t h curve, then, at each point s(t), one can c o m p u t e
X's(t) = V~(t)X e Ts(t)(M). B u t we will also be interested in differentiating vector fields along s t h a t are only defined along s. In fact, it is often n a t u r a l to consider fields Xs(t) along s t h a t are also p a r a m e t r i z e d by the p a r a m e t e r t, allowing Xs(tl) ~ Xs(t2) even if s (t~) = s (t2), t l r t2. For instance, Xs(t) = i(t), the velocity field, may exhibit such behavior. In these cases, it is not immediately clear how to use a connection to produce the desired derivative. D e f i n i t i o n 10.1.9. Let s : [a,b] ~ M be smooth. s m o o t h m a p v : [a, b] -~ T ( M ) such t h a t the diagram
A vector field along s is a
10.I. C O N N E C T I O N S
307
T(M)
[a, b]
s
9M
commutes. The set of vector fields along s is denoted by Y(s). We have already seen two examples, namely, the restriction (VIs)(t) = Ys(t) to s of a vector field Y E X(M) and the velocity field i(t). Via pointwise operations, it is evident that Y(s) is a real vector space and, indeed, a C~[a, b]-module. D e f i n i t i o n 10.1.10. Let V be a connection on M. An associated covariant derivative is an operator V --
dt
: X(s)
-~ X(s),
defined for every smooth curve s on M, and having the following properties:
1. V/dt is N-linear; 2, (V/dt)(fv) = (df/dt)v + f V v / d t , V f E C~[a,b], Vv E X(s); 3. if Y E ~ ( M ) , then
V (yls)(t) = V~(t)(Y) E Ts(t)(M), a < t < b. Remark. By property (3), V/dt is associated only to the one connection V. T h e o r e m 10.1.11. To each connection V on AJ, there is associated a unique co-
variant derivative V/dt. Proof. We prove uniqueness first. For this, it is enough to work in an arbitrary coordinate chart (U, x l , . . . ,x~). Let F~j be the Christoffel symbols for V, set ~i = O/Ox i, consider a smooth curve s : [a, b] --~ U, and let v c ~(s). Write
v(t) = ~ v~(t)~ ~(,), i=1
~(t) = ~ ~J (t)~j ~(~). j=l
Then any associated covariant derivative must satisfy
Vv
--
=
dt
~
k,/dv~
~
+ v
i V~i J
i=1 n
k
dv = ~ -j~k + ~-~.v~V~ k=l
i=1
dv
=
k=t
-
-
viuJF~i~k
+
i,j,k=l
n
i,j=l
)
308
10. R I E M A N N I A N
GEOMETRY
evaluated along s(t). This is an explicit local formula in terms of the connection, proving uniqueness. We turn to existence. In any coordinate chart (U, x l , . . . ,zn), use the above formula to define V / d t for curves lying in the chart. The reader can easily check t h a t the three properties in the definition of covariant derivative are satisfied, where M = U. Thus, in U, the connection VI U has an associated covariant derivative and, by the preceding paragraph, this covariant derivative is unique. Consequently, on overlaps U N V of charts, the two sets of Christoffel symbols must define the same covariant derivative for VIU Cl V. (Classical geometers and physicists defined connections and covariant derivatives by Christoffel symbols and they checked this invariance via explicit change of coordinate formulas.) Thus, along any smooth curve s : [a, b] + M, these local definitions of V / d t can be pieced together to give a global definition. [] In particular, for the Euclidean connection D in ]R"~, all Fi~ - 0 and we get the expected formula: m
i=1
-
Dv dt -
m
= ~ - -dv~
i
i ,(t).
For M C N "~, the covariant derivative associated to the Levi-Civita connection U on M is dt
P
the orthogonal projection into T ( M ) of the usual Euclidean covariant derivative. Convention. From now on, we adopt tile "summation convention" of Einstein. According to this convention, the summation symbol is omitted and it is understood that any expression is summed over all repeating indices. For example, at +
=~=1 \ dt + ~
i,j=l
It is necessary that the indices repeat for terms in a product, not just in a sum, and it is customary t h a t the repeated index occur once as a superscript and once as a subscript (a custom we will usually honor). Thus, dvk + viuJP~i - dv k ~-, 9 dt - d----t+ ~ v*uaP~i' i,j=l
the index k being repeated only in terms separated by +. D e f i n i t i o n 10.1.12. Let M be a manifold with a connection V. Let v E 2E(s) for a smooth p a t h s : [a, b] --~ M. If V v / d t -~ 0 on s, then v is said to be parallel along s (relative to the given connection). T h e o r e m 10.1.13. Let V be a connection on M , s : [a,b] -* M a smooth path, c C [a,b], and vo E T~(~)(M). Then there is a unique parallelfield v C ~ ( s ) such that v(e) = vo. This field is called the parallel transport of vo along s.
10.1. C O N N E C T I O N S
309
Pro@ In local coordinates, write
~(t) = ~J(t)~ 5 s(~), ~(t) = ~ ( t ) ~ s ( ~ ) , VO = ai~is(c).
Here, as promised, we are using the summation convention. The condition that v be parallel along s becomes the equation
~-Z(t). + v~(t)uJ ( t ) r ~ ( s ( t )
o = / dv k
)~
or, equivalently, the linear O.D.E. system (10.1)
dv k dt --
viuJF~i, 1 < k < n,
with initial conditions
vk(c) = a k, l < k < n. By the existence and uniqueness of solutions of O.D.E., there is e > 0 such that the solutions vk(t) exist and are unique for c - e < t < c+e. In fact, these equations being linear in the vks, it is standard in O.D.E. theory (Appendix C, Theorem C.4.1) that there is no restriction on e, so the unique solutions vk(t) are defined on all of [a, b], l
D e f i n i t i o n 10.1.15. Consider a piecewise smooth loop
s : [a,b] --+ M based at x0 = s(a) = s(b). Then the holonomy of V around s is the map
defined by setting
h~(vo) = v(b), where v E :g(s) is the parallel transport of v0 E T~o(M). Here, ~(s) denotes the space of continuous, piecewise smooth fields along s.
310
10. R I E M A N N I A N G E O M E T R Y
Since the parallel transport v of vo along s is the solution of the linear system (10.1), it follows that, if w is also the parallel transport along s of a vector wo C T x o ( M ) , then v + w E Z(s) is the parallel transport of v0 + w0. Similarly, if a C R, av E ~ ( s ) is the parallel transport of avo. This proves the following. Lemma
10.1.16. The holonomy h~ : Tx 0 (M) --* T~ 0 (M)
of V around the piecewise smooth loop s is a linear transformation.
D e f i n i t i o n 10.1.17. If s : [a,b] --+ M is piecewise smooth, a weak reparametrization of s is a curve s o r, where r is a piecewise smooth map r : [c, d] --* [a, b] carrying {c, d} onto {a, b}. If r(c) = a and r(d) = b, the reparametrization is said to be orientation-preserving. If r(c) = b and r(d) = a, it is said to be orientationreversing. L e m m a 10.1.18. Let s : [a,b] --* M be a piecewise smooth loop at xo and let = s o r : [c, d] --~ M be a weak reparametrization. If the reparametrization is orientation-preserving, then h~ = h~ and, if it is orientation-reversing, h~ = h71. Proof. W i t h o u t loss of generality, assume that s and r are smooth. Set
~(T) = s(r(T)), ~(T) = ~(r(T)), hence
eJ(T) =-~(T)~J(r(T)), aT
d5 k
dr
dv k
d7 ( T ) = ~ ( T ) - - j T - ( r ( T ) ) , and the linear system (10.2)
d~ k dT-
viuJr~i
is obtained from the system (10.1) by multiplying through by dr/d~-. Since the system (10.1) is assumed to be satisfied, so is the system (10.2). Thus, if r(c) = a and r(d) = b, h~(vo) = 5(d) = v(b) = hs(vo).
If r(c) = b and r(d) = b, then we take the initial condition to be ~(c) = v(b) = hs(vo) and h~(h~(vo)) = h~(v(b)) = ~(d) = v(a) = vo. []
Let f~(M, x0) denote the set of all piecewise smooth loops in M based at x0, loops being identified if one is a weak reparametrization of the other. More precisely, on the set of piecewise smooth loops, we quotient out the smallest equivalence relation t h a t identifies each loop with all of its weak reparametrizations. If sl, s2 E ~ ( M , xo), then sl + s2 E ~ ( M , xo) and hsl+s2 = hs2 o hsl.
Also, h_s = h s 1, so hs is a nonsingular linear transformation of T x o ( M ) . These considerations give the following.
10.2. RIEMANNIAN MANIFOLDS
311
L e m m a 10.1.19. The set {h,}sEU(M,xo) is a subgroup Jgxo(Aq) _C Gl(Txo (.A//)),
called the holonomy group (at xo) of the connection V. Remark. If M is connected and x0, x1 E M, then the groups 5~xo(M) and 9(xl (M) are isomorphic, but generally not canonically isomorphic. Indeed, fix a piecewise smooth path ~ : [0,1] --+ M with or(0) = x0 and a(1) = Xl. Then, given any s E f~(M, Xl), note that + s +
c
x0)
and that the correspondence hs ~-+ h~+~+(_~) defines a group homomorphism from ~C~, (M) to ~ o (M). By replacing c~ with - a , we get the inverse group homomorphism ~ o (M) --+ ~ , (M), proving that these are group isomorphisms. Generally, the isomorphism depends on the choice of a, so it is not canonical. E x e r c i s e 10.1.20. The standard imbedding of S 1 as the unit circle in R 2 defines a standard imbedding of T ~ = S 1 x ... x S 1 in ]R2~. Prove that the associated Levi-Civita connection on T ~ has holonomy ~ ( T ~) = {id}, g x E T ~. (Hint: Find a coordinate atlas relative to which the ChristoffeI symbols vanish.) E x e r c i s e 10.1.21. We say that V is a 91obally flat connection if (as in Exercise 10.1.20) its holonomy group ~C~(M) is trivial, g x E M. (The reason for this terminology will become apparent when we study the relationship between curvature and holonomy.) Prove that a manifold M has a globally flat connection if and only if M is parallelizable. E x e r c i s e 10.1.22. By Exercise 10.1.21, the spheres S 3 and S 7 support globally flat connections. Prove that, for n _> 2, the Levi-Civita connection relative to the standard imbedding S n C IR~+1 is not globally fiat. Finally, we should point out that connections are ubiquitous. T h e o r e m 10.1.23. Every manifold M has a connection.
Pro@ Let {U~,x~,... ,x2}~E~ be an atlas on M and let D ~ be the Euclidean connection on U~ relative to the coordinates x~, .. . , x~. n Let {A~}~E~ be a s m o o t h partition of unity subordinate to the atlas. Given X, Y E 2E(M), write Xc~ = X[U~ and Y~ = YIU~. Then, define V = E
A~D~: ~ ( M ) x ~ ( M ) --* X(M),
o~Eg] where
vxz
=
x.uioz,
e Z(M)
c~E92
It is entirely straightforward to verify that V is a connection.
[]
10.2. R i e m a n n i a n M a n i f o l d s
A Riemannian manifold is a pair (M, (., .)) consisting of the manifold M and a choice of Riemannian metric on T(M). From now on, such a choice of metric is fixed and we will speak of "the Riemannian manifold M".
312
10. R I E M A N N I A N
GEOMETRY
D e f i n i t i o n 10.2.1. If v c T~(M), some x C M, then the length of v is the nonnegative n u m b e r
Jlvll =
~.
W h e n no ambiguity is likely, we will often dispense with the subscript x on (v, w L , where v, w 9 Tx (M). Definition 10.2.2. If v , w 9 Tx(M), some x 9 M , then the angle between v a n d w is the unique 0 9 [0, 7r] such t h a t
cos 0 = Definition
10.2.3.
(v, w)
If s : [a, b] ~ M is smooth, t h e n the length of s is F b
Isl = J~ II&(t)ll at. If s = sl + . . . + Sq is piecewise smooth, each si being smooth, t h e n the length is
Ist= I s l l + . . Lemma
+lsql
10.2.4. / f u : [c, d] --~ [a, b] is a weak reparametrization, then ]s[ = ]s o u].
T h e proof should be familiar from advanced calculus. Notice that it does not m a t t e r whether u preserves orientation or reverses it. If the R i e m a n n i a n manifold M is oriented, we also get a canonical volume form 9 A n ( M ) (where n = dim M). Consider a local trivialization of T ( M ) . T h a t is, we are given a n open set U C M and a smooth frame ( X 1 , . . 9 , X~) of vector fields defined on U t h a t determines the correct orientation at each point of U. Relative to this trivialization, we express the R i e m a n n i a n metric by
h = det[hij] > O. Let { w l , . . . , ~ n } C A I ( U ) be the dual basis: ~ i ( X j ) = ~}. 10.2.5. The R i e m a n n (or R i e m a n n i a n ) volume element on U, relative to the given local trivialization of T ( M ) , is Definition
x/h~
~ A . . . A ~n
9 A~(U)
In particular, if ( X 1 , . . . ,Xn) is an orthonormal frame, the volume element becomes w 1 A .. - A wn. This agrees with intuition. The following theorem shows t h a t the volume element is i n d e p e n d e n t of the choice of local trivializations. 10.2.6. If M is an oriented, Riemannian n-manifold, there is a globally defined form ~ 9 A n ( M ) that, relative to any orientation-respecting local trivialization o f T ( M ) , coincides with the Riemann volume element.
Theorem
10.2. R I E M A N N I A N M A N I F O L D S
313
Proof. Indeed, let (U, X 1 , . . . , Xn) and (V, Z 1 , . . . , Zn) be two such trivializations with U n V r ~. Let the respective dual bases of l-forms be {a~l,... ,a~n} and {71,... ,r]~}. Let 7(z) = [Tij(x)] be the Gl(n)-valued function on U A V such that
(zl,..., Zn)7 =
(xi ..., xn)
on U A V. Since the frames are coherently oriented, det 7 > 0. Let
hij : (xi, xj>, fij = (Z,, Zj>. Then, [h~j] = [Tki]T[fke][Tej] and it follows that = (act 7 ) ~ / 7 . Also, (det7')~ 1 A . . . At/n = a~1 A . . . AaJ n, where 7 ~ = (7 T ) - I . P u t t i n g this information together, we obtain v / h ~ 1 A . . . AcJ ~ = (det 7)X/-f(det 7') r/I A . . . Arl n = v / f r l I A . - . Ar] n. Thus, the locally defined volume forms fit together coherently to define f~ as desired. [] D e f i n i t i o n 10.2.7. Let (U, z 1, ... , z ~) be a coordinate chart with coordinate fields
~i = O/Oz i. Then the functions 9ij = ([i,~j>, 1 <_i,j<_n, are called the metric coefficients. Thus, in correctly oriented local coordinate charts (U,z 1, element is given in terms of the metric coefficients by ~[U
. ,zn), the volume
= x/~ dx 1 A . . . A dx n,
where g = det[gij]. D e f i n i t i o n 10.2.8. Let M be an oriented Riemannian n-manifold and let U C_ M be a relatively compact, open subset. Then the volume of U is
IuI = ~ . Remark. On the a-algebra of Borel sets of M, this Riemannian volume generates a measure, finite on the relatively compact Borel sets. Even if M is not orientable, f~ is defined locally up to sign and, for small, connected, open sets U,
Igl = S~ ~
f
This also leads to a Borel measure, finite on relatively compact Borel sets. D e f i n i t i o n 10.2.9. A connection V on the Riemannian manifold M is a RiemannJan connection if, for all X, Y, Z C ~ ( M ) ,
X
314
10. R I E M A N N I A N G E O M E T R Y
E x a m p l e 10.2.11. Let M C_ ]~m be a smoothly imbedded n-manifold. The standard inner product (., .) on ~ m viewed as a Riemannian metric on the tangent bundle T(]~m), restricts to a Riemannian metric (., '}M on T(M) C T(I~m)[M. By Exercise 10.1.8, the connection V on M, constructed in the previous section and called there the Levi-Civita connection, is symmetric. The Euclidean connection D on R m clearly satisfies
X (Y,Z) = (DxY, Z} + (Y, D x Z ) , VX, Y , Z C :~(I~m). If X, Y, Z E X(M), then D x Y = V x Y + W, where W E r ( . ( M ) ) , hence W _L Z everywhere on M. Consequently,
(DxY, Z) = ( V x Z , Z)M and, similarly,
(z, D x Z ) = (Z, V x Z ) M Thus, V is Riemannian, hence is a Levi-Civita connection in our new sense as well. E x e r c i s e 10.2.12. Prove that a Riemannian manifold M has a unique Levi-Civita connection, proceeding as follows. (1) Uniqueness. Show that a Levi-Civita connection V must satisfy the identity
2 ( V x Y , Z} = X (Y, Z} + Y (X, Z) - Z (X, Y} + (IX, Y], Z} + (IX, X], Y} + ([Z, Y], X ) ,
v x , Y, z e ~(M). (2) Existence. Use the identity in (1) to define V : X(M) • X(M) --~ X(M) and prove that V is a Levi-Civita connection. In any local coordinate chart, the matrix [gij] of metric coefficients is nonsingular, so we can define The coefficients gk~ are rational functions of the metric coefficients gij. By definition, they satisfy
gikg ke = 5'i . E x e r c i s e 10.2.13. Let Fi~ denote the Christoffel symbols of the Levi-Civita connection and find a formula for Fik that involves only the gk~'s and first derivatives of the gij's. D e f i n i t i o n 10.2.14. A property of the Riemannian manifold M is intrinsic if it depends only on the metric. Otherwise, the property is extrinsic. It is geometric if it does not depend on choices of local coordinates. For example, the functions gij and gke are intrinsic, but not geometric. By Exercise 10.2.13, the Christoffel symbols Fi~ for the Levi-Civita connection are also intrinsic, but not geometric. In particular, the Levi-Civita connection for M c_ I~m is intrinsic, even though our initial definition of it used the normal bundle v ( M ) , a structure that can be proven to be extrinsic. Our definition of "intrinsic" may seem a bit too informal by current standards. For a more formal definition, one needs the notion of an isometry.
10.3. G A U S S C U R V A T U R E
315
D e f i n i t i o n 10.2.15. An isometry ~ : M1 --* M2 between two Riemannian manifolds, with respective metrics (., "}i, i = 1, 2, is a diffeomorphism such that
for arbitrary x C M1 and v,w E T~(M1). Now we see that a property of Riemannian manifolds should be called intrinsic if and only if it is preserved by all isornetries. 1 0 . 2 . 1 6 . If ~ : M --+ N is a diffeomorphism and V is a connection on N, show" how to define the pullback connection ~*V on M. If p is an isometry between Riemannian manifolds and V is the Levi-Civita connection on N, prove that ~*V is the Levi-Civita connection on M. This is the formal proof that the Levi-Civita connection is an intrinsic property of a Riemannian manifold. Exercise
10.2.17. Let v, w C X(s). Show that the covariant derivative defined by the Levi-Civita connection (indeed, by any Riemannian connection) satisfies Exercise
dt (It follows from this exercise that fields parallel along a curve s, relative to a LeviCivita connection, make a constant angle with each other and have constant lengths along s.) 10.3.
Gauss
Curvature
Throughout this section, unless otherwise stated, we assunle that 0 M = 0 and dim M = 2, and that we are given a fixed imbedding M ~-~ R 3 with normal bundle p(M). As usual, we use the metric induced on M by the Euclidean metric of IR3 and the associated Levi-Civita connection V. The Euclidean connection will be denoted by D. Given x c M, we find a connected neighborhood U C M of x and a smooth section g C F(~(M)IU ) such that Ilgll - 1. Remark that r~ is determined up to sign. Remark also that one can interpret g as a smooth map
~:U~S
2.
D e f i n i t i o n 10.3.1. The map ~ : U --~ S 2 is called the Gauss map. Intuitively, the area of ~(U) C_ S 2 seems to have something to do with the
curvature of M in U. Thus, if U is an open subset of a 2-plane in IR3, rT(U) degenerates to a single point, hence has area 0 (Figure 10.3.1). We say that the plane has (Gauss) curvature 0. Similarly, if U lies in a right circular cylinder, g(U) will lie in a great circle in S 2 (Figure 10.3.2). Again the area of if(U) is 0 and we say that the cylinder has (Gauss) curvature 0. The point here is that the cylinder can be "unrolled" to a portion of a plane without any metric distortions. On the other hand, if U c S 2, then ~ = • id and the area of d(U) is the same as the area of U, this being a positive number. We say that the sphere has positive (Gauss) curvature. It is not possible to flatten out any portion of S 2 to be planar without distorting the metric properties. For similar reasons, every convex surface has nonnegative curvature everywhere (Figme 10.3.3). By introducing a notion of "signed area", one obtains cases of negative curvature, a saddle shaped surface
316
10. R I E M A N N I A N
GEOMETRY
F i g u r e 10.3.1. The flat plane
F i g u r e 10.3.2. The flat cylinder being the typical example (Figure 10.3.4 and Exercise 10.3.8). In order to put these ideas into precise form, we introduce the Weingarten map. If v C Ty(M), some y c U, then Dvg E IRa makes sense. The equation 0 = v <{, ~) =
L(v) = D ~ , is called the Weingarten map. This map is well defined up to sign. As soon as the sign of the Gauss map has been fixed, the sign of the Weingarten map is determined also. Remark t h a t Ty(M) = T~(y)(S 2) in 11{3, since these 2-planes have the common normal vector g(y). Thus, we are allowed to view the Weingarten map as
L : Ty(M) -+ Tn(v)(S2). L e m m a 10.3.3. The Weingarten map L : Ty(M) --~ T~(y)(S 2) is the differential at y of the Gauss map.
10.3. GAUSS
CURVATURE
317
F i g u r e 10.3.3. Nonnegative curvature
F i g u r e 10.3.4. Negative curvature
Proof. Represent a n a r b i t r a r y vector in Ty(M) as i(O) for a suitable arc s : ( - e , e) --+ M. T h e n r~,y(a(O)) = ~ { ( s ( t ) )
= D5(o)~ : L(.~(O)). t:O
Lemma
10.3.4.
The Weingarten map is self adjoint. That is,
. = <~, L(~)>~, Vv,w E Ty(M).
[]
318
10. R I E M A N N I A N G E O M E T R Y
Pro@ Let v , w C Ty(M) and, by making the open set U C_ M smaller, if necessary, extend these vectors to fields X, Y C :~(U). Then (L(X), Y)M = (Dxff, Y) = X (#~,Y) - (#,, D x Y ) = - (~, D x Z )
= - (~, [X, Z] + D r X )
= - {~, D y X }
=
{L(Y),X>M. []
Remark that the symmetric bilinear form {L(v), W}Mis just the second fundamental form in the normal direction r7 as defined in Section 3.10. By Lamina 10.3.4, the matrix of L is symmetric relative to any choice of orthonormal basis in Ty (M). By the diagonalization theorem for symmetric matrices, there is an orthonormal basis {el, e2} C Ty(M) relative to which the matrix for L is
[o1 l That is, ei is an eigenvector for L with eigenvalue ~i, i = 1, 2. We agree to number these so that ~l _< ~2. L e m m a 10.3.5. As v ranges over the unit circle in Ty(M), the quadratic form (L(v), V}M takes minimum value I.~1 and maximum value ~2. Indeed, if
is the matrix of a quadratic form Q on R 2, it is standard that the extreme values of Q(v) on S 1 are the eigenvalues of the matrix (the method of Lagrange multipliers). Remark that, for the eigenvectors ei, ~i = (Ve~g, ei) measures the rate at which the normal vector ~ is turning in the direction ei. This motivates the following definition. D e f i n i t i o n 10.3.6. The numbers ~1 and ~2 are called the principal curvatures of M at y. The product ~1~2 = detL is called the Gauss curvature of M at y and is denoted by n (or ~(y)).
Remark. There is another important kind of curvature, the mean curvature h of M, which we will not treat in any detail. It is defined by h = t r L = ~1 + ~2. Unlike the Gauss curvature, this quantity depends, up to sign, on the choice of the unit normal field v. It can be shown that surfaces of mean curvature h - 0 are exactly the ones that,, in a certain precise sense, locally minimize surface area. Such surfaces are called minimal surfaces. They arise, for instance, when one considers the possible shapes of soap films spanning wire loops of various configurations. Such a soap film will be modeled by a 2-manifold S with boundary and M = S \ 0 S will be a minimal surface. The work of D. Hoffman and W. H. Meeks ([19], [17],
10.3. G A U S S C U R V A T U R E
319
[18], et al.), inspired and illuminated by some spectacular conlputer graphics, has revealed an astounding array of complete, unbounded minimal surfaces. Let ft' denote the Riemann volume form on S 2 and let ft denote the volume form on M. Define tile "signed area" of tile Gauss map on U to be
A(~(U))=/, ~*(~') and, as usual, let the area of U be
A(U) = ~ ~. As U shrinks down oi1 {y}, one can try to form a kind of "derivative" of the Gauss map g with respect to area:
d g ( y ) _ lira A(g(V)) dl2 v~{y} A(U) In order to define this precisely, define a(v,y)
= sup IIx - yll, xEU
using the ordinary Euclidean norm in N a, and characterize this derivative, if it exists, as the unique number such that, for each e > 0, there is a > 0 for which U ~ y and ~(U, y) < ~ ~
d'ff(y) dr2
A(g(U)) A(U) < e.
E x e r c i s e 10.a.7. For each y E M, prove that this derivative exists and that d~(y) _ ~(y). df~ E x e r c i s e 10.3.8. Let M C ]Ra be the graph of the equation z = x 2 - y2. Prove that ~(x, y, z) < 0, g (x, y, z) E M. In the above discussion, the normal field g played a central role. The curvature of M was seen as a measure of how much this field "spreads" infinitesimally at a point. Thus, curvature appears to be an extrinsic property of the surface. But Gauss proved a remarkable theorem (he called it his "Theorema Egregium") that showed the Gauss curvature to be intrinsic. A two-dimensional inhabitant of the surface can take measurements leading to the computation of curvature. We turn to this theorem. Let X, 17, Z E 5~(U) and extend these to fields X, Y, Z E 2E(U), where U C IRa is an open set such that U n M = U. L e m m a 10.3.9. For fields chosen as above,
D ~ Y = V x Y - (L(X), Y} r~ along U. Proof. Along U, D 2 Y depends only on X and Y. As in the proof of Lemma 10.3.4, (DxZ, ~) = - ( n ( x ) , Y) , so the component of D x Y perpendicular to M is - (L(X), Y) g. By the definition of the Levi-Civita connection, V x Y is the component of D x Y tangent to M. []
320
10. R I E M A N N I A N G E O M E T R Y
The Euclidean connection D satisfies a simple commutator relation:
[D~, Dr] = DE~,~1 This is because D ~ and D~p operate on a vector field Z by applying .~ and respectively to the individual components of Z. It turns out that, on M, curvature is an obstruction to this commutator relation for V. D e f i n l t i o n 10.3.10. The curvature operator n ( x , v ) : ~(M) -~
X(M)
is defined, for arbitrary X, Y E ~(M), by R(X,Y)Z = VxVyZ - VyVxZ
- V[x,y]Z,
v z E X(M). The fact that this operator is related to curvature is far from obvious. It is the content of the Theorema Egregium. By Lemma 10.3.9, at every point of U we have D ~ ( D ~ 2 ) = D ~ ( V y Z - (L(Y), Z) ~) = D x ( V y Z ) - (L(Y), Z) L ( X ) - X (L(Y), Z) = Vx(VyZ)
- (L(Y), Z) L ( X ) - X (L(Y), Z)
- (L(X),VyZ)~.
Similarly, at every point of U, - D~(D~2) = -Vy(VxZ)
+ (L(X),Z) L ( Y ) + Y (L(X), Z) ~ + (L(Y), V x Z ) ~.
Finally, at every point of U, - D [ ~ , 9 1 2 : - V [ x , y ] Z +
R ( X , Y ) Z = (L(Y), Z} L ( X ) - (L(X), Z) L ( Y ) .
Summing the coefficients of~ also gives 0. Applying the fact that V is a Riemannian connection and using the fact that Z varies freely over all tangent fields, the reader can obtain the Codazzi-Mainardi equation (10.4)
L([X, Y]) = V y L ( X ) - V x L ( Y ) .
One immediate consequence of equation (10.3) is the following. L e m m a 10.3.11. The expression R ( X , Y ) Z has value at y E U depending only on the vectors Xy, Yy, Zy. Titus, this expression is a tensor, called the Riemann curvature tensor R. Indeed, the right-hand side of equation (10.3) is clearly a tensor in all three vector fields. That is, R E 9"13(M). Note also that, since V is an intrinsic and geometric property of the surface, so is the Riemann curvature tensor.
10.3. GAUSS CURVATURE
321
T h e o r e m 10.3.12 (Theorema Egregium). Let y E U and let {el,eg} be an orthonormal basis of T y ( M ) . Then
(R(el, ~)e~, ~) = '~(V). In particular, the Gauss curvature of a surface in ]R3 is intrinsic and geometric. Proof. By equation (10.3), (~(el, e2)e2, el> = (L(e2), r
(L(el), el} - (L(el), e2} (L(e2), el>.
Since the basis is orthonormal, it is true (and easily checked) that the corresponding matrix representation of L is the 2 x 2 matrix with ( i , j ) t h entry (L(ej), e~). Thus, ( / ~ ( e l , e2)e2, e l ) = det L - ~(y).
[] By Exercise 10.2.16, the Levi-Civita connection is preserved by isometrics, hence we obtain the more formal version of the statement that ~ is an intrinsic property.
Corollary 10.3.13. If f : M ~ N is an isometry between two surfaces in R 3 and if ~M and t~N are the respective Gauss curvatures of these surfaces, then t~N(f(x)) = ~;M(X),
Y x E M.
Theorem 10.3.12 suggests a definition of curvature for a general connection on an n-manifold M. D e f i n i t i o n 10.3.14. Let M be an n-manifold and let V be a connection on M. The curvature operator R of V is given, for each choice of the fields X, Y, Z E :~(M), by R(X, Y)Z = VxVyZ - ~Ty~xZV[x,y]Z. E x e r c i s e 10.3.15. Prove that the curvature operator R ( X , Y ) Z of a connection V is C~176 in the fields X,Y, Z. Consequently, for each x E M and u , v , w E T x ( M ) , R ( u , v ) w E T x ( M ) is well defined. If M is a Riemannian manifold and V is the Levi-Civita connection, then R is called the Riemann tensor. We will return to the study of this tensor later. It turns out that, in Riemannian geometry, the Riemann tensor is exactly the obstruction to the geometry being locally Euclidean. In the non-Riemannian geometry of spacetime, there is an analogue of the Levi-Civita connection and Einstein represents gravity by the curvature tensor of this connection. Special relativity ("flat" spacetime) is the case in which this curvature tensor vanishes identically. E x e r c i s e 10.3.16. Let M C R a be a compact 2-manifold. You are to prove that it is not possible that ~c _< 0 on all of M. Proceed as follows. (1) By compactness, choose a point v0 E M at which the function A : M--~ IR,
~(v)--IlvLI ~ assumes its maximum. Prove that 0 7r v0 l T, o(M). (2) If s : ( - e , e ) --~ M is smooth with s(0) = v0 and ~(0) r (~(0), g(v0)) is strictly negative.
0, prove that
322
10. R I E M A N N I A N G E O M E T R Y
(3) Using the above, prove that the principal curvatures nl and ~2 are both nonzero and have the same sign at v0, hence ~(v0) > 0. E x e r c i s e 10.3.17. One calls a point v E M at which ~1 = ~2 an umbilic point. Let U C_ M be the set of points that are not umbilic. (1) Prove that U is open in M and that ~1 and ~2 are smooth functions on U. (2) Prove that each v C U has a neighborhood V C_ U on which there is a smooth, orthonormal frame field (X1, X2) such that L ( X i ) = ~iXi, i = 1,2. (3) For V c_ V and X~,X2 E 2~(V) as in part (2), define f l , f 2 C C ~ ( V ) by
X2(~1)
f2--
--XI(E2) t~2 - - E1
Prove that
•x1X1 = - f i X 2 , V x 2 X 1 = f2X2,
fiX1, Vx2X2 = -f2X1,
YxIX2 =
and that
IX1, X2] = A X 1 - f2X2. (4) Using the formulas in part (3), show that, if v E U is a critical point for both ~1 and ~2, then, at v,
x~(~2) - x~(,~,) /~2 -- /~1
E x e r c i s e 10.3.18. Let M C R a be a compact, connected 2-manifold with constant curvature t~ ~ a. By Exercise 10.3.16, a > 0. You are to prove that M is a 2-sphere, centered at some point w0 E R 3 and of radius 1/x/-a. Proceed as follows. (1) Prove that v E M is a point at which ~2 is maximum if and only if it is a point at which ~1 is minimum. (2) Use part (4) of Exercise 10.3.17 to show that ~2 can be maximum only at an umbilic point. (3) Prove that every point of M is an umbilic and that ~1 -- v ~ -- ~2- (Hint: This is the maximum value of n2.) (4) Deduce the form of the Gauss map, drawing the desired conclusion. Be sure to make clear how you use the hypothesis that M is connected.
10.4. C o m p l e t e R i e m a n n i a n M a n i f o l d s This section presents the Hopf-Rinow theorems and related matters. The author first learned this material from J. Milnor's beautiful exposition [28, pp. 55-64], and its influence will be evident in what follows. The goal here is to use the Riem a n n i a n metric to obtain a topological metric on M and to relate the topological notion of "completeness" to the problem of extending geodesics indefinitely. In the process, one also discovers, without the use of variational calculus, that geodesics locally minimize arc length. In Euclidean geometry, straight lines play a central role. They can be characterized as the unique smooth curves whose tangent fields are parallel. Here, parallelism under the Euclidean connection is clearly identical with the absolute
10.4. C O M P L E T E R I E M A N N I A N M A N I F O L D S
323
parallelism in Euclidean space. Taking our cue from this observation, we define the n o t i o n of a geodesic for a general connection. D e f i n i t i o n 10.4.1. Let M be an n-manifold with a connection V. A smooth curve s : [a, b] --4 M is a geodesic for V if ~(t) is parallel along s(t), a < t < b. We are interested in the case in which V is the Levi-Civita connection of a R i e m a n n i a n manifold, so we make t h a t a s s u m p t i o n from here on. We emphasize t h a t we are considering general R i e m a n n i a n manifolds, not just surfaces in R 3. D e f i n i t i o n 10.4.2. A s m o o t h curve cr : [a, b] --~ M is said to be evenly parametrized if Ild(t)ll = c, a < t < b, for some constant c > 0. By Exercise 10.2.17, the following is immediate. L e m m a 10.4.3. A geodesic s on a R i e m a n n i a n manifold M is necessarily evenly parametrized. In local coordinates, the definition of a geodesic translates into a system of nonlinear, second order, ordinary differential equations. Indeed, write s(l) = ( x l ( t ) , . . . , x n ( t ) ) , a(t) = (a?l(t),... ,a?n(t)), = ~ ( t ) ~ i s(t), a n d write down the parallelism condition for ~(t): 0-
V(~(t)) dt 9i
V
= ~(t)~,.(o + x (t)-d7 (~(~)) = ~ ( t ) ~ ( 0 + x~(t)v~(0 (~) = ~k(t)~k ~(0 + zi(t)JcJ(t)Pkij(s(t))~k ~(t) = (~k + e~esr~j)~k. Equivalently, this is the second order system
(10.5)
~k + ~ S F ~
= 0, 1 < k < n.
By setting u i = .~i, 1 < i < n, we get an equivalent, nonlinear, first order system ~Zk q-uiuJ['~j ~-- O, S=u
e,
1 < k < n, 1
Given initial conditions 2 x e ( 0 ) = a ~,
l
u k ( 0 ) = b k,
l
there is a unique solution
s ( t ) = ( , l ( t ) , . . . ,xn(~)), defined on some interval - e < t < e. The initial condition can be w r i t t e n
s ( o ) = (a',... ,a~), ~(0)=(bl,...,bn).
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10. R I E M A N N I A N
GEOMETRY
The existence and uniqueness theorem for solutions of O.D.E., together with the smooth dependence of the solutions on initial conditions, gives the following. T h e o r e m 10.4.4. Let xo 9 M . Then, there is an open neighborhood U of xo in M and numbers q , e 2 > 0 such that, for every x 9 U and every vx 9 T x ( M ) with I[vxll < q , there is a unique geodesic Crvx : (-e2, e2) --* M with
~vx (o) = x,
evx (o) = Vx. Furthermore, in local coordinates, x = ( x l , . . .
~v~(t) = ~ ( x l ,
,xn),
,x n,vl,
v x ~-- v i ~ i x ,
and
,<,t)
defines a smooth function of 2n + 1 variables. The domain of the function ~ is W x (-e2, e2), where
w=
{ v 9 T ( g ) llbll < ~1},
an open neighborhood of 0~o. E x e r c i s e 10.4.5. Let M C IR3 be a smoothly imbedded surface with the relativized metric and Levi-Civita connection. Let a : [a, b] --+ M be evenly parametrized and suppose that im(a) C_ P N M , where P is a 2-plane in R 3 such that uo(t)(M ) C P, a < t < b. Prove that a is a geodesic. E x e r c i s e 10.4.6. If M is a surface of revolution (as defined in freshman calculus), identify a natural, infinite family of geodesics. Discuss whether or when the "circles of latitude" are geodesics. E x e r c i s e 10.4.7. Show that every geodesic on the unit sphere S 2 C ]R3 must lie along a great circle. Remark. It will be convenient to reparametrize geodesics so that their domain of definition always contains the closed interval [-1, 1]. The following trick will be found in [28, p. 57]. The system (10.5) is homogeneous of degree 2. This implies that, if a(t) is a geodesic, so is a~(t) = a(ct), for a fixed constant c, and ~r = cd(ct). Let 0 < e < ele2/2. If Ilvll < e a n d It[ < 2, then
112v/e21[ < q , 1r
< ~2,
so the curve
,~v(t) = ~2v/c2(e2t/2),
- 2 < t < 2,
defines a geodesic
"/v: ( - 2 , 2) --, such that
"/v (0) = x, "t,(0) = v. If
Bx(C) = {v~ c T~(M) I Ilvxll < d ,
10.4. C O M P L E T E
RIEMANNIAN
MANIFOLDS
325
t h e n % ( t ) varies s m o o t h l y with
(v,t) 9 U B~(~) • ( - 2 , 2 ) , xEU
an open n e i g h b o r h o o d of (0~o, 0) in T ( M ) x ]R. Definition
10.4.8. For x 9 U, v 9 T~(M), Itvll < e, all as above,
expA~) = % ( 1 )
By the homogeneity of the system (10.5), we obtain % ( T ) = Vrv(1)following is an i m m e d i a t e consequence. Lemma
10.4.9.
The
For x e U, v 9 T z ( M ) , Hv]l < e, all as above, 7v(t) = expx(tv),
- 1 < t < 1.
Conventions. In w h a t follows, we routinely view M C T ( M ) by the imbedding of M as the 0-section of T ( M ) . Thus, x 9 M is identified with 0x 9 Tx(M). Since the t a n g e n t space Tv(V) at any point v of a finite dimensional vector space V is identified canonically with V itself, we will also identify To~ (Tz(M)) with T~(M). Theorem
10.4.10.
If ex > 0 is sufficiently small, then exp~ : Bx(ex) --~ M,
called the exponential map at x, is a diffeomorphism onto an open neighborhood of x inM. Proof. Write ~ = exp~ : B~(e) ~ M. Clearly, V(0x) = x. We c o m p u t e ~.o~ : To~(B~(e)) : T~(M) ~ Tx(M). For 0 # v 9 To:r(B~(e)) = Tx(M), set s(t) = tv, -e/llvlt < t < e/llvll , a s m o o t h curve on B~(e) w i t h s(0) = 0~, i(0) = v. T h e n r / o s defines a curve on M , r/(s(O)) = r/(0z) = x and r/(s(t)) = expx(tv ) = vv(t). Thus,
~7,0~(~(0)) = , , 0 ~ ( v ) = %(0) = v T h a t is, r/.0x is the identity under the identification To~ (B~(e)) = T~(M). By the inverse function theorem, 7/ will be a diffeomorphism for e = e~ > 0 sufficiently small. [] R e m a r k t h a t , if the manifold is compact, the n u m b e r e = e~ > 0 can be chosen to be i n d e p e n d e n t of x. D e f i n i t i o n 1 0 . 4 . 1 1 . T h e R i e m a n n i a n manifold M is geodesically complete if exPx(V ) is defined for all x 9 M and for all v 9 Tx(M). Equivalently, every geodesic segment extends (uniquely) to a geodesic 7(t), - o o < t < oo. D e f i n i t i o n 1 0 . 4 . 1 2 . A p a t h a : [a, b] ~ M is regular if it is a s m o o t h immersion. If there is a partition a = to < tl < ... < tT = b such t h a t al[ti_l,ti I is regular, 1 < t < r, then a is piecewise regular. N o n d e g e n e r a t e geodesics cr are regular. Indeed, if d~(t0) = 0, for some to, then t h e fact t h a t a is evenly p a r a m e t r i z e d implies t h a t # - 0 and cr degenerates to a c o n s t a n t path.
326
10. RIEMANNIAN GEOMETRY
D e f i n i t i o n 10.4.13. Let M be a connected Riemannian manifold. Then the Riem a n n (or Riemannian) distance function p : M x M --+ [0, oc) is defined by
p(x,y) = inf M, where a ranges over all piecewise regular paths from x to y in M. Fixing the hypothesis that M is a connected, Riemannian n-manifold with OM = 0, we are going to prove the following results. P r o p o s i t i o n 10.4.14. The Riemann distance function is a topological metric on the Riemannian manifold M and the metric space topology is the same as the manifold topology. T h e o r e m 10.4.15 (Hopf Rinow I). I f the Riemannian manifold M is geodesicalIy complete, x , y C M , then there is a geodesic "7 from x to y such that [7[ = p(x, y). In particular, expx : Tx (M) ~ M is surjective. T h e o r e m 10.4.16 (Hopf-Rinow II). The Riemannian manifold M is a complete metric space in the metric p if and only it is geodesieally complete. C o r o l l a r y 10.4.17. If M is compact, then M is geodesically complete in any Riemann metric. Because of these results, it is standard to use the term "complete Riemannian manifold" when either geodesic completeness or metric completeness is intended. A number of preliminary considerations are necessary for the proofs of Proposition 10.4.14 and the Hopf-Rinow theorems. To begin with, remark that the distance function p has the following properties:
1. p ( x , x ) = O, V x c M; 2. p(x,~) = p(y,x), w , y 9 M; 3. p(x,y) < p(x,z) + p(z,y), Vx, y,~ 9 M. Thus, to prove that p is a topological metric, it is only necessary to prove that
p(x,y)
=
0 ~
x = y.
Let xo 9 M C T ( M ) , choose an open neighborhood U C_ M of x0, and let e > O. Then W = {v~ 9 T~(M) I x 9 U and IIv~JI < e} is an open neighborhood of x0 = 0~0 in T ( M ) . If U and ~ are chosen small enough,
c(v) = (~(~), exp~(v)(v)) is defined and smooth as a function O : W - ~ M x M. We can take U to be a coordinate neighborhood, with coordinates x l , . . . ,x n. We can also assume that there is a smooth, orthonormal frame field Z 1 , . . . , Zn defined on U. We obtain, thereby, coordinates
(xl,... ,xn,yl,..
,y~) ~-, y~Z~(~:~...... ,,)
on 7r-l(U) such that n
W:
{ ( x l , . . . , x n , y l , . . . , y n)
(xl,...
,X n) C U, E ( y i ) 2 < i~1
= U x B(Q.
e 2}
10.4. C O M P L E T E
RIEMANNIAN
MANIFOLDS
327
L e m m a 10.4.18. Given xo C M , the neighborhood W of xo in T ( M ) can be chosen, as above, so small that G : W --4 M x M carries W diffeomorphieally onto an open neighborhood of (xo,xo) in M x M .
Proof. Indeed, let (i represent the basic coordinate fields for x i and (j those for yJ, 1 < i , j <_ n, and observe that
G,0~o (~i0~o) = (~ xo, ~xo),
G,o,,,o (r
= (O~o, zs~o).
Thus, G,0xo is bijective and the assertion follows by the inverse function theorem. [] We fix the choice of W as in this lemma. Let V be an open neighborhood of x0 in M such that V x V C_ G ( W ) a n d V C _ U . Given ( z , y ) E V x V, let v,~ E W be the unique element such that ( x , y ) = G(v~) = (x,expx(v~)). That is, exPx(tV~ ), 0 < t < 1, is the unique geodesic of length < e going from x to y and parametrized on [0, 1]. The point exp:~(tvx) depends smoothly on (v~,t) E W x [0, 1] and G is a diffeomorphism on W, so exp~(tv,~) also depends smoothly on (x, y, t). Less formally, we say that the unique geodesic of length < joining x, y E V depends smoothly on its endpoints. Remark that, even though x , y E V, there is no reason why the geodesic expx(tV~ ) should stay in V for all vahles of t C [0, 1]. It is a fact, as we will see later, that V can be chosen so that the geodesics of length < e joining any points x , y C V do remain in V. This is J. H. C. Whitehead's theorem on the existence of geodesically convex neighborhoods which will give us the existence theorem for simple covers that we used in the discussion of de Rham cohomology. Finally, note that this discussion also shows that, for each x E V C U, exp~ maps the open e-ball B~(e) C T ; ( M ) diffeomorphically onto an open neighborhood of x in M that contains V. Let U C_ R 2 be open and consider smooth maps s : U --+ M. In complete analogy with the case of curves, we obtain the space ~(s) of all vector fields along s, these being all smooth maps v : U ~ T(I~J) such that the diagram
T(M)
8
commutes.
E x a m p l e 10.4.19. Let (r, t) be the coordinates of U. Then we define &s/Or and Os/Ot E :~(s) by
OS (r,t) Or
-
-
z
s.(,-t)
Os (r, t) = s,<,t) Ot
328
10. R I E M A N N I A N
GEOMETRY
J u s t as for curves, given v C 2E(s), we have partial covariant derivatives Vv Vv
Or Ot 6 ~(s), o b t a i n e d by taking covariant derivatives along the respective curve families t -= constant, r -
constant.
E x e r c i s e 1 0 . 4 . 2 0 . For s : U --+ M as above, prove t h a t
~
-~
=~
~
Let x E V and e > 0 be as before. For 0 < a < e, define the spherical shell a r o u n d x of radius a to be S~ = {expx(v) I v E T~(M) and
Ilvll
= a},
clearly a s m o o t h hypersurface (submanifold of codimension one). Lemma 0
10.4.21.
The 9eodesics out of z meet the spherical shell Sa orthogonally,
Proof. Let v(t) be a s m o o t h curve in Tz(M), p a r a m e t r i z e d on IR, with IIv(t)tl ~ 1. For Irl < e, define a s m o o t h function
f(r, t) = exp~r For fixed to E R, f(r, to) describes the geodesic out of m with initial velocity V(to). For fixed r0 E ( - e , e), f(ro, t) describes a s m o o t h curve on the shell S,-0. Since v(t) was arbitrary, it will be enough to prove t h a t the curves f(ro, t) and f(r, to) m e e t o r t h o g o n a l l y at the point f(ro, to). Since (ro, to) is to be arbitrary, w h a t we really need to show is t h a t
os os\ Or' Ot / =- O. Since the r-curves (t = constant) are geodesics, we have
V O f =o. Or Or Also, by Exercise 10.4.20,
v o~ Or Ot
v o~ Ot Or'
SO
_(af 2ot ~---0.
va/} '-oT
)
10.4. C O M P L E T E
RIEMANNIAN MANIFOLDS
329
T h e last equality is due to the fact t h a t []Of/Orll is the length of the velocity field along the geodesic r v-~ exp~(rv(t)) and this length is the c o n s t a n t IIv(t)ll = 1. It follows t h a t (Of/Or, Of lOt) is c o n s t a n t in r. But, at r = O, Of lOt - O, so
of o:\
7 , - g - i / -- o. [] Let x E V a n d set V~ = exp~(Bx(e)). T h e n V C_ Vx. Let
~: [a, b] -~ V~ \ {x} be a piecewise regular curve. Since exp~ : B~(e) --~ V~ is a diffeomorphism, there is a u n i q u e pieeewise regular curve 5 in Bz(c) "-. {0}, parametrized on [a, b] and such t h a t a = expz o~-. We write
~(t) _ r(t)v(t), ~(t) = II~(t)ll Ile(t)ll where
o < r(t) = Ile(t)ll < e, ilv(t)il- I. Thus, a(t) = exp~(r(t)v(t)) with r(t) and v(t) subject to these conditions. Lemma
1 0 . 4 . 2 2 . For a as above, 1
.b
I~1 = Jo II~(t)ll
at >_ [r(b) - r(a)l.
If equality holds, then r(t) is strictly monotone and piecewise regular and v(t) is constant. Consequently, any shortest path joining two concentric shells around x is a piecewise regular reparametrization of a radial geodesic segment. Proof. Let f ( r , t ) = exp~(rv(t)). Thus, a(t) = f ( r ( t ) , t ) , so d(t) -- dr Of Of dE Or + Ot ' Clearly, IlOf/Orll =- ~ and, by L e m m a 10.4.21, Of~Or • Of~Or. Thus, dr 2
Ila(t)ll2=
dt
Of
~
+ -0-[
dr ~
>- at
"
If equality holds, then Of/Ot - O, so ~)(t) -= 0 a n d v(t) is constant. Therefore,
I~1 =/b > >-
II~(t)ll dt dt i b dr dt
= It(b) - r(a)t. If equality holds, n o t only is v(t) = v constant, b u t dr~dE c a n n o t change sign and, since r(t)v = 5(t) is piecewise regular, dr/dt can never be O. T h a t is, r(t) is strictly m o n o t o n i c a n d piecewise regular. []
330
10, R I E M A N N I A N G E O M E T R Y
1 0 . 4 . 2 3 . For V and r > 0 as above, let V : [0, 1] ---+ M be the unique geodesic of length < e joining two points x, y E V. Let a : [0, 1] ---+M be an arbitrary piecewise regular path joining the same two points. Then IV] <- ]cr], where equality holds if and only if a is obtained from 7 by a piecewise regular reparametrization. Theorem
Pro@ Set y = e x p x ( r v ) , w h e r e 0 < r < e a n d IivH = 1. Then, i f 0 < 5 < r < e, a contains a segment joining the spherical shell $5 to Sr and lying between these shells, hence lying in Vx. By L e m m a 10.4.22, this segment has length at least r - 5, so lal _> r - 5 . L e t t i n g 5 ~ 0, we conclude t h a t lal _> r = 171. I f l a l = r, t h e n the segment from each $5 to S, must be a reparametrization of (the same) radial geodesic. Since a is piecewise regular, the reparametrization r(t) is continuous and § has only j m n p discontinuities, occurring only finitely often as 5 I 0, so a itself is a piecewise regular reparametrization of V. Conversely, if a is a piecewise regular r e p a r a m e t r i z a t i o n of V, then 1~7]= IV]. [] C o r o l l a r y 1 0 . 4 . 2 4 . Let a : [a,b] ---+ M be piecewise regular and have minimal
length for any piecewise regular path from a(a) to a(b). Then a is obtained from a geodesic by piecewise regular reparametrization. If a is regular, it is a regular reparametrization of a geodesic. If ]]~]] is constant, a is a geodesic. Proof. Consider any segment of a lying in a n open set V as above and having length < r By the above, this segment must be a piecewise regular r e p a r a m e t r i z a t i o n of a geodesic. Since every interior point of a lies in the interior of such a segment a n d a is m a d e up of finitely m a n y such segments, a must be a piecewise regular r e p a r a m e t r i z a t i o n v ( r ( t ) ) of a geodesic V. If cr is regular, then
dr
0 # ~(t) = 7{a/(r(t)) a n d this implies t h a t r(t) is a regular change of parameter. Since ]]~(r(t))l] is constant, dr/dr will be constant if IIa(t)II is. In this case, r(t) = ct + e for suitable constants c ~ 0 a n d e, so a(t) = v(ct + e) is a geodesic. [] T h e next two results complete the proof of Proposition 10.4.14. Proposition
1 0 . 4 . 2 5 . If x , y C M and p(x,y) = O, then x = y.
Proof. Suppose x # y. Choose V and e > 0 as usual, b u t such that x E V, y r V. For a suitable choice of r] E (0, e), expx(B,(r/)) C V. Then, every piecewise regular p a t h a from x to y must meet the spherical r/-shell centered at x, so ]or] k r/ and p(x, y) >_ rl > o.
[]
Thus, the R i e m a n n distance function is a topological metric on M. Proposition
1 0 . 4 . 2 6 . The topology induced on M by the metric p coincides with
the manifold topology. Proof. Let x C M a n d choose e > 0 so small t h a t exp~ : B~ (e) + M is a diffeomorphism onto an open neighborhood U~ (e) of x in the manifold topology. T h e set of all such U~ (e) is a base for the manifold topology of M. But
(10.6)
v~(~) = {v c M I P(~, v) < ~}.
Indeed, if y E Ux(e), then p(x,y) < e. Furthermore, if z C M \ U~(e), every piecewise regular a from x to z must meet every ~?-shell, 0 < ~] < e, centered at x, so Ia] > e and, consequently, p(x, z) >_ e. This proves the assertion (10.6), showing
10.4. C O M P L E T E R I E M A N N I A N M A N I F O L D S
331
F i g u r e 10.4.1. The Hopf-Rinow setup
t h a t {Ux(e) [ x E M, e > 0} is also a base for the topology induced by the metric p. [] We note also the following useful fact. 1 0 . 4 . 2 7 . If C C M is compact, there exists 5 > 0 such that any two points x, y E C with p(x, y) < 5 are joined by a unique geodesic (parametrized on [0, 1]) of length < 5. This geodesic is the shortest piecewise regular path from x to y and depends smoothly on its endpoints. In particular, if M is compact, 5 can be chosen uniformly for all of M . Proposition
Proof. Cover C by open sets V~ with corresponding e~ as in the above discussion. Select a finite subcover { 1/.~ } i r= 1 ' Let 5 > 0 be a Lebesgue n u m b e r for this cover (i.e., i f p ( x , y ) < 5 a n d x , y E C, t h e n x and y lie in a c o m m o n V~). We can also d e m a n d t h a t 5 < minl_
Proof of theorem 10.4.15. We assume that M is geodesically complete a n d choose a r b i t r a r y x, y E M , r = p(x, y). We must prove t h a t there is a geodesic 3' from x to y such t h a t 13'1= r. This is trivial if x = y, so we exclude t h a t possibility. Let V a n d e > 0 be chosen as usual, with x E V, and let Sa C V be a spherical shell a r o u n d x, 5 < r sufficiently small. Since Sa is compact, there is x0 E Sa such that p(xo, y) = rain p(z, y). zESa
Write x0 = expz(Sv0 ), Iiv01I = 1. The geodesic ~/(t) = exp~(tv0) is defined for all real values of t because M is geodesically complete (see Figure 10.4.1). We will prove that, contrary to the possibility allowed in Figure 10.4.1, y = 7(r) = expz(rv0) , thereby proving T h e o r e m 10.4.15. Our procedure will be to prove, for each t E [5, r], the proposition F ~ : p('~(t),y) = r - t. In particular, F~ asserts t h a t p(3'(r), y) = 0, giving y = 7(r).
332
10. R I E M A N N I A N
GEOMETRY
(a) We prove Fa. Every piecewise regular curve from x to y must cross Sa, hence
= p(x, u) = inf
zESe
(p(x,z) +p(z,y)) 5
= 5 + inf
zC=Sa
p(z,y)
= 5 + p(zo, y). Therefore, p(z(a),
u ) = p ( ~ o , u ) = ~ - 5.
(b) Assuming the truth of Fto, some to C [5, r), we will prove the existence of a maximal half-open interval [to, r]) such that ~ < r and Ft is true for all t E [to, r]). Indeed, for 5' > 0 sufficiently small (as usual), let S~, be the spherical shell of radius 5' around 7(to). Let x~ E S~, be a point with p(xto,y) minimum. Then, as before, ,- -
to = p ( ~ ( t o ) , ~ ) =
z~f, (,~(~(to), z)+~(~, y)) 5,
=
a'
+ p(x'o, y),
SO
p(4,
u ) = r - to - s ' = r -
(to + 5 ' ) .
We will show that x~) = ")'(to + fi'). Indeed, P ( X , 4 ) >- P(x,Y) - P ( > 4 ) =
r -
(r
-
to -
5')
= to + 5'. But the path consisting of the segment of -y from x to "y(t0), followed by a minimal geodesic from "y(to) to x~, has length to + 5', hence is a piecewise geodesic, parametrized by arc length, joining x to x~ and of minimal length. By Corollary 10.4.24, this path is a geodesic. It coincides with 3, on [0, to] and to > 0, hence it coincides with 7 on [0, to + 5']. That is, x~ = "y(to + 5'), as claimed. We have proven that p(7(to + 5'), y) = p(x~, y) = r - (to + ~'), which is the assertion Fto+~,. Since 5' > 0 was arbitrarily small, there is a half-open interval [to, ~'), r / < r, on which Ft holds. The union of all such intervals produces the maximal one [to, rl). (c) Since F~ holds, let [5, r]) be the maximal half-open interval on which Ft holds. But the truth of Ft on [5, rl) implies Fv, by continuity. Thus, if r / < r, we could apply (b) to obtain a contradiction to the maximality of [5, 7). Consequently, r / = r and Fr holds. []
Proof of theorem 10.3.16. We first assume that M is geodesically complete and we prove that M, as a metric space under p, is complete. For this, it will be enough
10.4. COMPLETE
RIEMANNIAN
MANIFOLDS
333
to prove that, whenever B c_ M is a p - b o u n d e d subset, the closure B is compact. Choose any x C B a n d consider the continuous map exp x : T x ( M ) + M, defined because M is geodesically complete. Since B is bounded, there is a n u m b e r r > s u p y e u p(x,y). If D C T x ( M ) is the closed ball of radius r, t h e n B C_ e x p , ( D ) (Theorem 10.4.15) a n d expx(D) is compact. Thus, B C_ expx(D ) is compact. Next, a s s u m i n g t h a t M is complete in the metric p, we prove t h a t M is geodesically complete. Let x C M and let v C T ~ ( M ) have u n i t norm. Let (a,b) denote the m a x i m a l open interval a b o u t 0 in R such that exp~(tv) is defined, Vt E (a, b). We must show t h a t a = - o o and b = oo. If b < oo, choose {ta:}~=x C (a, b) such t h a t tk T b strictly. This is a Cauchy sequence. Set xk = exp~(tkv) and remark t h a t p(xe,xk) <_ te - tk, whenever h < ~. Indeed, the segment of expz(t), tk <_ t <_ te, is a geodesic of length te - t k oo joining these two points. Therefore, {X k}k=l is Cauchy in the metric p and, by the completeness of this metric, x~: + y C M. Define 7 :[0, b] + M by 7(t)
fexp~(tv),
0 _< t < b,
[y,
t=b.
By the previous paragraph, 7 is continuous on [0, b]. It is s m o o t h on [0, b). If 7 is also s m o o t h at b, it is a geodesic and can be extended as a geodesic to [0, b + r/), some rl > 0. This would contradict the m a x i m a l i t y of (a, b), proving that b = oc. In order to prove t h a t ~/ is smooth at b, choose a neighborhood V C M of y a n d a n u m b e r e > 0 such t h a t expz(w ) is defined, V z C V, V w C T~(M) with Ilwll < e. For a Cauchy sequence {x~ = expz(t/~v)}~=l, chosen as above, :rL= E V a n d b - tk < e, for all sufficiently large values of k. Let k be large enough and set vk = "~(tk) C T ~ k ( M ). Since Ilvkll = 1, exp~k(tvk ) is defined for 0 < t < b - t k < ~. But, for 0 _< t < b - tk, this curve coincides with "y(tk + t). By continuity, 3'(b) = exp~ k ((b - tk)v~), completing the proof t h a t 7 is smooth at b. A completely parallel a r g u m e n t shows that a = - o c . [] E x e r c i s e 1 0 . 4 . 2 8 . Let M be a complete R i e m a n n i a n manifold, 9- a foliation of M , and let L be a leaf of 9". The R i e m a n n i a n metric (., .) oil M induces a R i e m a n n i a n metric {-, }L on L via the one-to-one immersion i : L ~ M. Let PL denote the corresponding topological metric on L. Generally, this is not the restriction of the oo metric p of M. Let {X k}k=l C L be pr-Cauchy. (1) Prove t h a t {Xk}k~__l is p-Cauchy, hence t h a t xk ~ x E M. (2) Let (U, y l , . . . ,yn) be a Frobenius neighborhood of x. Prove that all b u t finitely m a n y of the points xk lie on the same 9--plaque in U as x. (3) Using the above, conclude that, as a R i e m a n n i a n manifold in the induced metric {., ")r,, L is complete. In particular, this exercise implies that a leaf L in a compact, foliated manifold (M, 9-) is complete in any metric (-, ')c that; arises, as above, by restricting to L an a r b i t r a r y R i e m a n n i a n metric (., .) on M. The leaf L need not, itself, be compact. The geodesics on the R i e m a n n i a n manifold (L, (., .)L) are not, generally, geodesics in (M, (-, .)).
334
10. R I E M A N N I A N
GEOMETRY
10.5. G e o d e s i c C o n v e x i t y We will prove a theorem of J. H. C. Whitehead that, in particular, will guarantee the existence of simple refinements of open covers. This result was anticipated and used in our treatment of de Rham cohomology (Chapter 8). Throughout this section, M is a Riemannian n-manifold with empty boundary. D e f i n i t i o n 10.5.1. A subset X C_ M is star shaped with respect to a point x0 E X if each x c X can be joined to x0 by a unique shortest geodesic in M and if this geodesic always lies in X. D e f i n i t i o n 10.5.2. A subset X a_ M is geodesically convex if it is star shaped with respect to each of its points. Equivalently, X is geodesically convex if any two of its points are joined by a unique shortest geodesic in M and this geodesic lies in X. The following is immediate. L e m m a 10.5.3. An arbitrary intersection of 9eodesieaUy convex sets is 9eodesi-
cally convex. T h e o r e m 10.5.4 (Whitehead). Let W C_ M be open, x E W .
Then there is a
geodesically convex, open neighborhood U C W of x. Before proving this result, we show how it implies the existence of simple refinements. L e m m a 10.5.5. A set X a_ M, star shaped with respect to xo E X , is contractible.
Proof. Indeed, each x C X determines uniquely vx C T~0(M) and tx > 0 such that Ilvxll = 1 and x = exPxo(txvz ). Then, F : X x [0,1] --+ X, defined by
F ( z , r) = expx o (rtxvx), is the desired contraction.
[]
It seems that open, star shaped sets U C M are always diffeomorphic to IRn, but this is extremely difficult to prove. The problem is that the set theoretic boundary OU may be very badly behaved. For instance, the "radius function" r : S ~-1 zo
--,
[0, oc],
even if it takes only finite values, may not be continuous, let alone smooth. This function is defined on the sphere of unit vectors in Tx0 (M) and assigns to v C S xo n-1 the supremum r(v) of the numbers r > 0 such that expx o(tv) CU,
O
Keep this possible bad behavior of r in mind while attempting the following exercise. E x e r c i s e 10.5.6. Let U C_ M be open and star shaped with respect to x0 C U and let C C U be compact. Prove that there is an open set V C U, also star shaped with respect to x0, such that V is a compact subset of M and C C V C V C U. P r o p o s i t i o n 10.5.7. I f U C_ M is an open set, star shaped with respect to xo, then
H~ (U) = H; (~tn).
10.5. G E O D E S I C
CONVEXITY
335
Pro@ For a suitable value p0 > O, the open ball Bxo(PO) C Txo(M ), centered at 0 with radius P0, is carried by expx o diffeomorphically onto an open set Upo C U with compact closure in U. Since B~ o(P0) is diffeomorphic to R n, the same is true for Upo. In particular, H2(Upo ) = H2(Rn). The inclusion i : Upo ~-~ U induces homomorphisms i . : A*~(U~o) ~ A*~(U), i. : H:(U~o)
~
H:(U),
so it will be enough to prove that the second of these is an isomorphism. Let C C U be compact. By Exercise 10.5.6, find open sets V C W C U, star shaped with respect to x0 and such that V C W C W c U, where V and W are compact, and Uoo t2 C C V. Also, fix 0 < a < b < P0 and the corresponding open balls Ua C Ub C Upo. By the smooth Urysohn lemma, find f : U --* [0, 1] such that
fI(U \ W) =- O, fl(V \ Ub) > O, flUa=-O. Let Z E Y-(U \ {x0}) be nowhere 0, tangent to the radial geodesics out of x0, and everywhere pointing toward x0. Let Ft : U ~ U denote the flow, defined for all time t, generated by the compactly supported vector field f Z E Y.(U). Then, since C ".. Upo is contained in the interior of the support of f Z , as is 0U;o , there is a value 7 > 0 such that F~(C) C U;o. Set r = F_~. Since ~L~ is a compactly supported diffeomorphism of U onto itself, isotopic through such diffeomorphisms Ft to F0 = idu, it follows that r : H~(U) ~ H~(U) is the identity. Let aJ E Z~(U) and let C = supp(a@ By the previous paragraph, we obtain r U ~ U such that r E Z~(Upo ) and r = [a~] E H~(U). It follows that [aJ] E im(i.), hence that i. carries H;(Upo ) onto H~(U). Suppose that a~ E ZP(Upo) and that i.[~] = 0. Choose a > 0 as above such that supp(cJ) C U~. Viewing '~ = i.(w) in ZP(U), we find r~ E AP~-I(U) such that w = dr/. Let C = supp(r/) and obtain r : U -+ U, as above, so that
~*(~]) = r/o e Ap-I(Upo), but dr/o = d r
= ~*(dr/) = r
= ~.
That is, [w] = 0 in H*(Upo), proving that i. is one-to-one.
[]
C o r o l l a r y 10.5.8. Every open cover of M admits a simple refinement.
Proof. By Theorem 10.5.4, each open cover admits a refinement by open, geodesically convex sets. With a little care, one chooses this refinement to be locally finite (Exercise 10.5.9). By Lemma 10.5.3, any finite intersection of elements of this refinement is also an open, geodesically convex set, hence star shaped. By Lemma 10.5.5 and Proposition 10.5.7, this refinement is simple. [] E x e r c i s e 10.5.9. Check the assertion in the proof of Corollary 10.5.8 that the refinement by open, geodesicafly convex sets can be chosen to be locally finite.
336
10. R I E M A N N I A N
GEOMETRY
E x e r c i s e 1 0 . 5 . 1 0 . Let x c U C M where U is open in M and star shaped w i t h respect to x. Let r : S~ -1 -+ [0, oo] be tile radius function for U. T h a t is, $2 -1 C T x ( M ) is the unit sphere and U = {exp~(tv) I v 9 $2 -1 and 0 _< t < r(v)}. If r is finite-valued of class C ~ , prove t h a t U is diffeomorphic to ]Rn. E x e r c i s e 1 0 . 5 . 1 1 . Let x 9 U C_ M and r : S~ -~ -+ [0, oo] be as in the preceding exercise, but do not assume t h a t r is s m o o t h or even continuous. Prove t h a t r is lower semicontinuous. T h a t is, r - l ( a , oo] is open in S zn-1 , V a E R . E x e r c i s e 1 0 . 5 . 1 2 . Let x 9 U C_ M and r : S~ -1 --* [0, oo] be as in t h e preceding exercises. C o n s t r u c t an example in which r is finite-valued everywhere and discontinuous on a dense subset of S Sn - - 1 . We t u r n to the proof of T h e o r e m 10.5.4. Let x 9 W, as in the s t a t e m e n t of the theorem. As in Section 10.4, choose a neighborhood V of x in W and a n u m b e r e > 0 such t h a t any two points y, z 9 V can be joined by a unique geodesic ay,z in kcr of length < e. As usual, ay,z is p a r a m e t r i z e d on [0, 1] and depends s m o o t h l y on ( y , z ) 9 V • V. Choose 5 > 0 such t h a t the open ball Bx(a) c T x ( M ) of radius a is carried diffeomorphicaliy by expx onto a neighborhood Ux C_ V of x. Let ( v z , . . . ,v~) be an o r t h o n o r m a l frame for T s ( M ) and coordinatize this vector space by
(xl,... ,xD
U n d e r the diffeomorphism exp~-T : Us --+ B~(a), these become coordinates on U~ (called a normal coordinate system on U~). The corresponding coordinate fields are {i 9 :~(Us), 1 < i < n. If y 9 Us has coordinates (bz,... , b~), t h e n 7Z
y)" = Z
i=1
If 0 < 5. < 5, if $5. C U~ is the spherical shell of radius 5., centered at x, if y = ( b l , . . . ,b,~) 9 $5., and i f v = aJ~j 9 r y ( S 5 . ) , then
bia i = O. T h e key l e m m a for the proof of T h e o r e m 10.5.4 is the following. Lemma
10.5.13.
I f & 9 (0,5) is small enough, then every geodesic "y : ( - rl , rl) ---, M ,
such that 7(0) = y C $5. and "9(0) E T y ( S & ) , has the property that p(x, v(t)) > & , for all sufficiently small values of ]tlr o. Proof. As above, denote the normal coordinates of y by ( b l , . . . , bn). Let 5. be so n small that, for ~ i = 1 b~ _< 5., the s y m m e t r i c m a t r i x O = 215ke - biFkg(bl,... i ,bn)] is so close to [25ke] as to be positive definite. Let "y(t) =
(xl(t),...
,xn(t)),
--7"] < t < l],
be a geodesic in M , tangent to $5, at 7(0) = y = ( b l , . . . , bn), and let ~(0) = ai{i. Define n
F ( t ) = p(x,'~(t) ) 2 - 52. = ~-~ xi(t) 2 i=i
52,.
10.6. C A R T A N S T R U C T U R E E Q U A T I O N S
337
For small values of Ill, this is a smooth function and V(0) = 0,
r ' ( o ) = 2~(o>~(o) = 2bia i
~0, F"(t) = 2(2i(t)2i(t) + xi(t)~i(t)). Since 7(t) is a geodesic, it satisfies
~i
=
_~k:bgpi he, 1 < i < n,
giving r"(o)
= 2(x~(O) ~ - ~ q o > k ( o ) S ( o ) p ~ e ( x ~ ( O ) , . . .
,.'~(o)))
= 2((ai) 2 - a~ae(biFike(bl,... , b~))) _- [ a l
...
, a~]Q
,
the value of a positive definite quadratic form on the vector "~(0) r 0. T h a t is, F"(O) > 0. Plugging this d a t a into the 2nd order Taylor expansion of F(t) about t = 0, we see that
F(t) = ~ F ' ( O ) + O(t a) > 0, for small enough values of Itl r 0.
[]
Proof of theorem 10.5.3. Choose N , = exp~(Bx(5.)), where 5, is chosen by the above lemma. Let R C_ N : • N : be the subset of all (y,z) such that (r>z lies entirely in N : . By the smooth dependence of this geodesic on its endpoints, R is an open subset. It is also clear t h a t / g ~ 0. If we prove that R is also a closed subset, then, by the connectivity of N : x N~, R = N: • N: and N~ is geodesically convex. Let {(Yk,Zk)}~=l C R with limk--oo(yk,zk) = (Yo, Zo) in Nx x X~. If (yo, zo) r then Cr~o,zo meets ON= = Sa.. If ayo,~o is tangent to the spherical shell at some point of intersection, an application of the lemma shows that ayo,zo contains points in Ux ".. Nx. But smooth dependence on (Yo, zo) implies that this remains true for all values of (y, z) sufficiently near (Y0, z0), hence for (Yk, zk), k sufficiently large. This contradicts the fact that (Yk, zk) C R. But if an intersection point of ayo,zo with the shell is not a point of tangency, it is clear that a:o,yo contains points in U~ \ N , , leading to the same contradiction. Thus, (xo,Yo) C R, proving that R is closed in N : . [] 10.6. T h e C a r t a n S t r u c t u r e E q u a t i o n s We return to the torsion and curvature tensors that were introduced earlier for a connection V. The key to understanding the geometric significance of these tensors is a pair of equations, written in terms of differential forms, called the equations of structure. In tiffs section, we derive these structure equations in local coordinate charts. (In tile next ct~apter, where we treat principal bundles, we will be able to obtain global, coordinate-free versions of these equations by lifting them
338
10. R I E M A N N I A N G E O M E T R Y
to the frame bundle.) As an application, we will prove that the Riemann tensor is exactly the obstruction to the integrability of the Riemannian structure. T h a t is, the vanishing of curvature is equivalent to the existence of a coordinate atlas {U~,x~,l.. . ,x~}~e~ such that the coordinate fields O/Ox~, 1 < i < n, form an orthonormal frame field on Us, for each ~ C 92. Equivalently, the Riemannian manifold is locally isometric to Euclidean space. In what follows, V is a general connection on the n-manifold M, n > 2. To begin with, we will work in an open, trivializing neighborhood U for T ( M ) and fix the trivialization by a choice of a smooth frame ( X 1 , . . . ,X~) on U. Define 0 i r A I ( U ) by Oi(Xj) = 6~, 1 < i , j < n. Then, each X E X(U) can be written
X = Oi(X)Xi. Remark. Elie Cartan called the frame field a "moving frame". The discussion in this section concerns his "method of moving flames". In the next chapter, we will use principal bundles to globalize this method and give some geometric applications. Define forms wji E A I(U), 1_
=
vx
9
The fact that these are forms, i.e., that wji ( f X ) = fw}(X), V f 9 C~176 follows from the fact t h a t V x X j is a tensor in X. In order to express the torsion and curvature tensors of V in terms of the frame field, we introduce forms 7 , f ~ jic i A2(U) by the formulas
T ( X , Y) = ~-i(X, Y ) X i , R ( X , Y ) X j = f~}(X, Z ) X i . The fact t h a t these are antisymmetric tensors (i.e., 2-forms) follows from the same properties of T and R. Theorem
10.6.1 (Cartan structure equations). The above forms satisfy the iden-
tities (10.7)
9 dOi = - w ji A 0j + T ~,
(10.8)
d ~ = -w~9 A wjk +ft}.
Proof. The proof is a computation. We carry this out for equation (10.7). The computation of equation (10.8) is more of the same and will be left as an exercise. For arbitrary X, Y 9 ~(U), 7"i(x, Y ) X i = V x Y - V y X - [X, Y] = vx(oJ(Y)Xj)
- vv(oJ(x)xA
-
vl)xj
= (X(OJ(Y)) - Y ( o J ( x ) ) - oJ([x, Y]))Xj + oJ(Y)VxXj - OJ(X)VyXj = dOJ(X, Y ) X j + o J ( Y ) w j ( X ) X i - OJ(X)w}(Y)Xi = (dO
(X, Y ) +
-
But we claim t h a t
w~( X)OJ (Y) - w} (Y)OJ (X ) = a;} A OJ(X, Y).
10.6. C A R T A N S T R U C T U R E E Q U A T I O N S
339
Indeed, the standard inclusion A2(U) ~ ~oo(U) takes COj
A 0j
~-+ COji @
0j
--
0 j @ co}
(Lemma 7.2.18). Thus, the coefficients of X i on each side of the above being equal, 1 < i < n, we obtain Ti(X, Y) = (dO~ + co} A o J ) ( x , Y ) . Since X and Y are arbitrary, equation (10.7) follows.
[]
E x e r c i s e 10.6.2. Verify equation (10.8). Using matrix notation, we can write the equations of structure more compactly. Set
I~ 0
T
co = [co;l,
The n-tuples 0 and r can be thought of as Rn-valued forms. The matrices co and ft can be thought of as L(Gl(n))-valued forms. In Chapter 11, we wil] lift these local forms to the frame bundle of T ( M ) , where they will fit together coherently to define global forms. D e f i n i t i o n 10.6.3. Tile Rn-valued forms 0 and 5- are called the trivializing coframe field and the torsion form, respectively. The L(GI(n))-valued forms co and ft are called the connection form and tile curvature form, respectively. The structure equations are (10.9)
dO = -co A 0 + %
(10.10)
dco = -co A co + ft,
where we multiply matrices of forms by the usual rules of matrix multiplication, but use exterior multiplication for products of entries. L e m m a 10.6.4 (Key Lemma). Let U C_ M be an open, connected subset, together with a connection V and frame field ( X 1 , . . . , X n ) on U with associated curvature f o r m f~ -- O. Then, given q E U and B C ~ ( n ) , there is a connected neighborhood V of q and a unique smooth map A : V ~ JV{(n) such that A(q) = B and dA = A A co. Proof. By hypothesis, the second structure equation becomes
(*)
dco = -co A ua.
340
10. R I E M A N N I A N G E O M E T R Y
Let (V, x l , . . . , X n) be a coordinate chart about q in U, let P -- V x JV[(n) and let p : P ~ V be projection onto the first factor. Coordinatize P by the coordinates x i on the V factor and the standard coordinates z~ on the :lV[(n) factor. Then Z -- [z~] can be interpreted as a matrix-valued function on P, constant in the coordinates x i, and the matrix w of connection forms can be interpreted as a matrix of 1-forms on P, constant in the coordinates z~. Define
A = dZ - Z A w , a matrix of 1-forms on P. Then
dA = - d Z A ~ -
Z A dw = - ( d Z -
Z Aw) Aw = -AAw,
where we have used (*). This equation looks very much like a Frobenius integrability condition. Indeed, the n 2 entries of the matrix A are of the form 9
k
hence are linearly independent pointwise on P. Then Eu =
ker)~ju, N l
VuEP,
defines an n-plane distribution on P. The 1-forms A~ generate the annihilator ideal I ( E ) C A* (P) and the equations
show that d I l ( E ) C I 2 ( E ) , so E is integrable by Theorem 9.1.5. Let 9" denote the associated foliation. Notice that the restriction of A~ to any factor {x} • ?d(n) is just dz~, so a vector tangent to this factor and annihilated by every A~ must be 0. It follows that p.~ : E~ --* Tp(u)(V ) is an isomorphism, for each u C P, hence that p restricts to a local diffeomorphism on each leaf of 5. In particular, making V smaller, if necessary, we can assure that the leaf through (q, B) is the graph FA of a smooth function A : V --+ 3V[(n). That is, the leaf is the image of the section
a:V~P, a ( x ) = (z, m(x)). The necessary and sufficient condition that I~A be a leaf is that c~*(A) -= 0. Since a* (Z) = Z o c~ = A and a* (w) = w, we see that FA is a leaf if and only if
0 =- a * ( d Z -
ZAw)
= dA-
A A~.
The condition that A(q) = B is equivalent to the condition that the leaf FA pass through the point (q,B). This uniquely determines the leaf, hence the function A. [] The Frobenius theorem played a key role in the above proof, showing that curvature is the obstruction to integrability of a certain n-plane distribution. In the case of the Levi-Civita connection for a Riemannian metric, we are about to show that this integrability is equivalent to the integrability of the infinitesimal O(n) structure defined by the metric. Let V be the Levi-Civita connection of a Riemannian metric. In this case, the frame field ( X 1 , . . . , X ~ ) can and will be chosen to be orthonormal. Since this connection is torsion free, the torsion form ~ vanishes. The following exercises are straightforward computations.
10.6. CARTAN STRUCTURE EQUATIONS
E x e r c i s e 10.6.5. Let R be the Riemann tensor and let ( X 1 , . . . thonormal frame field. For arbitrary X, Y C 2~(U), prove that
341
,Xn)
be an or-
(R(X, Y)Xj, Xk} = - (Xj, R( X, Y)X~:) . Exercise 10.6.6. Prove that the connection form co and curvature form f~ of V, relative to an orthonormal frame field, take values in the Lie algebra L ( O ( n ) ) of antisymmetric matrices. For the curvature form, appeal to the previous exercise.
T h e o r e m 10.6.7. The R i e m a n n i a n manifold M is locally isometric to Euclidean space (also said to be fiat) if and only if the R i e m a n n tensor R - O. Since tile Riemann tensor for the Euclidean metric does vanish identically, the "only if" part of this result is evident. Thus, we assume that R - 0 and prove that every point q E M has a coordinate neighborhood (V, y l , . . . ,yn) such that the coordinate frame field (~1,..., ~n) is orthonormal on V. In the Key Lemma 10.6.4, choose the matrix B to be an element of O(n). Under this and our other current hypotheses, we have the following. L e m m a 10.6.8. The function A : V ---+:~(n) takes its image in the group O(n). Proof. Since A(q) = B and B B T = I, it will be enough to show ttmt A A T is constant oil V. But d ( A A T) = dA A A T + A A (dA) T = AAwAAT+AA
(AAw) T
= AAwAAT-AAwAA
T
=-0, since, by Exercise 10.6.6, co T = --co. By the connectivity of V, it follows that A A T is constant. [] Proof of Theorem 10.6.7. Write A = [a~], an orthogonal matrix of functions by Lemma 10.6.8. Define F=AA0, a column vector of l-forms p~ = a~Ok. Since A is nonsingular and 0 is the coframe to X 1 , . , . , Xn, we see that the l-forms ~i are linearly independent pointwise on V. Furthermore, dp=dAAO+AAdO = AAwAO+
AA(-wAO)
--0, where we use the fact that, the Levi-Civita connection being torsion-free, the first structure equation becomes dO = - w A O. We lose no generality in taking V to be contractible, so the closed forms ~ will be exact: @i = dyi for suitable yi C C ~ ( V ) , 1 < i < n. Since these forms are linearly independent at q, we can choose V smaller, if necessary, to guarantee that (yl . . . , y n ) : V ~ ]~n
342
10. RIEMANNIAN GEOMETRY
is a diffeomorphism of V onto an open subset of IRn. T h a t is, (V, y l , . . . ,yn) is a coordinate chart and we will show that the associated coordinate frame field ( ~ 1 , . . . , ~n) is everywhere orthonormal. R e m a r k that
~ ( X j ) = aikOk(Xj) = aj, "
i
i m p l y i n g t h a t X j = a}~i. T h a t is, ( ~ , . . . , (~)A = ( X 1 , . . . , Xn). Since A is O(n)-valued a n d ( X ~ , . . . , ( ~ 1 , . . . , ~n) is also orthonormal.
Xn) is
a n orthonormal frame, it follows t h a t []
Remark. This theorem is a special case of the fact that, in a certain precise sense, curvature determines the R i e m a n n i a n geometry up to local isometry. We will n o t prove this more general result, b u t the reader will find an extensive t r e a t m e n t of these m a t t e r s in [40, Chapter 7].
10.7. Riemannian Homogeneous Spaces* This will be a very quick, introductory look at a large topic. We assume t h a t (M, (.,.)) is a connected, R i e m a n n i a n manifold. The group of all isometrics : M --, M will be denoted by I(M). T h e action of I ( M ) on M preserves all intrinsic properties. In particular, it preserves the R i e m a n n distance function, hence is a group of metric space isometrics. We note, without proof, the following classical result [32].
Theorem 10.7.1 (Myers and Steenrod). If M is a Riemannian manifold, the group I ( M ) , with the compact-open topology, is isomorphic, as a topological group, to a Lie group such that the natural action, I ( M ) x M ---* M,
defned by (~, x) ~ ~(x), is smooth. D e f i n i t i o n 10.7.2. A R i e m a n n i a n manifold M is homogeneous if the action I ( M ) x M --~ M is transitive. T h e fact t h a t I ( M ) is a Lie group implies t h a t a homogeneous R i e m a n n i a n manifold is of the form M = I ( M ) / K , where K is a closed subgroup, hence a properly imbedded Lie subgroup of I(M). We will not use this fact. 10.7.3. If M is a homogeneous Riemannian manifold, then it is a complete Riemannian manifold.
Proposition
Proof. At any x C M , let B~(e) be the diffeomorphic image, under expx , of the open c-ball in T~(M). For every y E M, choose ~ E I ( M ) such t h a t ~xY(X) = y. T h e n Bv(e ) = ~Y(B~(e)) is the diffeomorphic image, under expy, of the open c-ball in Ty(M). T h e point is t h a t this ~ is uniform for all points of M. If s : (a,b) --* M is a geodesic with m a x i m a l parameter interval, we must show t h a t b = ec. The same proof will give a = - e c . We may assume t h a t s is p a r a m e t r i z e d by arclength. If b < oc, find c C (a, b) such t h a t b - c < c. T h e n B~(c) (e) is as above, so there is a geodesic segment a of length e out of s(c), having initial velocity i(c). T h e n a must agree with s on [c, b) and la I = c > b - c, so s extends to the interval (a, c + e) where c + e > b, contradicting maximality. []
10.7. R I E M A N N I A N H O M O G E N E O U S S P A C E S *
343
D e f i n i t i o n 10.7.4. A (Riemannian) symmetric space M is a R i e m a n n i a n manifold with the property that, for each x E M , there is r E I ( M ) such r = x and ( ~ ) , ~ = - idT,, (v). Remark that r
reverses every geodesic s through x. T h a t is, s(O) : 9 ~ r
: s(-t).
As obvious examples, IR~ with the Euclidean metric and S ~ with its usual metric are b o t h s y m m e t r i c spaces. There are m a n y other examples a n d the theory of s y m m e t r i c spaces is highly developed (cf. [14]). Proposition
10.7.5. If M is a symmetric space, then M is a complete Riemann-
Jan manifold. Proof. Let s : (a, b) --~ M be a maximal geodesic. If b < oo, choose c E (a, b) closer to b t h a n to a. By reparametrizing, if necessary, we can assume t h a t c = 0, hence t h a t b < - a . Then, for - b < t < - a , ~(t) : C s ( 0 ) ( s ( - t ) )
is a geodesic t h a t coincides with s on ( - b , b). These fit together to form a geodesic parametrized on ( a , - a ) , contradicting the maximality of b. Thus, b = oo and, similarly, a = - o o . [] C o r o l l a r y 10.7.6. If M is a connected symmetric space, then M is a homogeneous
Riemannian manifold. Proof. Indeed, if x , y E M, completeness and connectedness allow us to find a (shortest) geodesic s joining them (Theorem 10.4.15). Parametrize this geodesic on [ - 1 , 1], s ( - 1 ) = x and s(1) = y. T h e n r E I ( M ) reverses this geodesic, hence carries x to y. [] D e f i n i t i o n 10.7.7. Let (., .} be a R i e m a n n i a n metric on a Lie group G. This metric is left-invariant if every left translation in G is an isometry. It is right-invariant if all right t r a n s l a t i o n s are isometries. If the metric is both right- a n d leR-invariant it is said to be bi-invariant. 10.7.8. Relative to a left-invariant (respectively, right-invariant) Riemannian metric, a Lie group is a homogeneous Riemannian manifold.
Lemma
This is clear, as is the existence of such metrics. 10.7.9. If (., .) is a right-invariant metric on the Lie group G, if Y, Z E :~(G), and if X 6 L(G), then Lemma
([X, Y], Z) = - (Y, [X, Z]). Proof. Since X is leR-invariant, it generates the flow ~t(x) = x . e x p ( t X ) ,
344
10. R I E M A N N I A N
GEOMETRY
where exp(tX) is the one-parameter group of X. Thus,
= t--0 lim -/((Y" 1 e x p ( - t X ) , Z} - (Y, Z}) = t~o lira 71 ((Y' Z . exp(tX)} - (Y, Z}) = - (y, [ x , z ] ) .
[] C o r o l l a r y 10.7.10. If the Lie group G admits a bi-invariant metric and V is the corresponding Levi-Civita connection, then VxX--O,
V X G L(G).
Proof. Indeed, by Exercise 10.2.12, if X , Y E L(G), we get (V x X, Y) = ([Y, X], X} = - ( [ X , Y ] , X )
= (Y, [ X , X ] } - O. []
C o r o l l a r y 10.7.11. The geodesics on G, relative to a bi-invariant metric, are the cosets (left or right) of the one-parameter subgroups. Proof. Indeed, due to the bi-invariance, we only need to show that the geodesics out of e are exactly the one-parameter subgroups. But s(t) = exp(tX) is integral to the left-invariant field X and V x X = O, so the one-parameter groups are geodesics. But these groups are in one-to-one correspondence with their initial velocity vectors Xe, as are the geodesics, so every geodesic out of e must be of this form. [] C o r o l l a r y 10.7.12. If the connected Lie group G has a bi-invariant metric, then every element of G can be reached from e by a one-parameter subgroup. This follows from the geodesic completeness of G by Theorem 10.4.15. E x a m p l e 10.7.13. This last corollary gives a necessary condition for a Lie group to admit a bi-invariant metric. Consider the connected Lie group S1(2). If
then det(A) = 1 and ab + bd = btr(A), ac + dc = ctr(A), a 2 + bc = a 2 + a d - 1 = atr(A) - 1 d 2 + bc = d 2 + a d - 1, = dtr(A) - 1. These give A 2 = t r ( A ) A - I, hence t r ( a ~) = t r ( A ) 2 - 2,
V A e Sl(2).
10.7. RIEMANNIAN H O M O G E N E O U S SPACES*
345
Let
so tr(A) = - 3 . If there is E c L(SI(2)) such t h a t A = exp(E) = e E, set B = e E/2 and get A = B 2. This gives - 3 = t r ( B 2) = tr(B) 2 - 2 _> - 2 , a clear contradiction. Thus, this element A E S1(2) cannot be reached from I by a o n e - p a r a m e t e r Lie group. The group S1(2) cannot have a bi-invariant metric. Proposition
1 0 . 7 . 1 4 . If G has a bi-invariant metric, it is a symmetric space rel-
ative to that metric. Proof. By homogeneity, it will only be necessary to produce '~bx for x = e. For this, we take r = y - l , Vy E G. To see t h a t this is an isometry, let X, Y E L(G), r e m a r k t h a t r (X) a n d ~b~.(Y) are right-invariant fields, hence the constant (X, Y) a n d the c o n s t a n t ( r are both equal to (Xe,Ye) = ( - X e , - Y e ) . [] Theorem
1 0 . 7 . 1 5 . Every compact Lie group has a bi-invariant metric.
Proof. Choose a left invariant R i e m a n n i a n metric (., .) on G. The corresponding R i e m a n n volume form ~t is left invariant also, so the Borel measure # t h a t it defines is left invariant. The corresponding integral satisfies /Gfd#=/Gf~
Va~G.
By m u l t i p l y i n g the metric by a suitable positive constant, we normalize this integral:
l d # = 1. We define an inner product on L(G) by (X, Y)' = . / : (R~. (X), Rx. (Y)) dp(x). It is trivial t h a t this is a positive definite bilinear form. As an inner product on L(G), it determines a left invariant R i e m a n n i a n metric on G. But we claim t h a t this is also a right-invariant metric. Indeed, for all a C G and X, Y E L(G),
(F~a.(X),~I~a.(Y)) , = ./~, (F~x.17~a.(X),~U~x.J~y.(Y)) dl~(x)
=/~_ (R~.(X).R~.(Y)> d,(~) = ./~ (Rx.(X), R~.(Y)) d#(x) (left-invariance of #) = (X, Y)'. [] Thus, the compact Lie groups provide an interesting set of examples of symmetric spaces. E x e r c i s e 1 0 . 7 . 1 6 . Let G be an n-dimensional, connected Lie group with a given bi-invariant R i e m a n n i a n metric. Prove that G is fiat if and only if G is abelian. (By Exercise 5.1.30, G = T k • ]R~-k.)
346
10. RIEMANNIAN
GEOMETRY
E x e r c i s e 10.7.17. Let G be a compact, connected Lie group, a : G -~ G a Lie group automorphism such that a 2 = id, and let
K = {x e G la(x) = x}. (1) Prove that K is a compact Lie subgroup of G. (2) Find a bi-invariant Riemannian metric g on G relative to which a is an isometry. (3) Show that g passes to a Riemannian metric on the homogeneous space G / K , making that manifold a symmetric space. (4) Show that the map ~ : G --* G defined by ~o(x) = xa(x -1) passes to a smooth imbedding V : G / K r G. (5) Write L(G) = L(K) | 9X, where ~rt _l_ L(K) relative to the metric g. Prove that exp(g)l) = p ( G / K ) . (6) Prove that ~ carries each geodesic in G / K onto a geodesic in G, exactly doubling arclength and preserving angles. (7) Conclude that the properly imbedded submanifold M = exp(gJ[) of G, under the metric gIT(M), is itself a symmetric space and a totally geodesic submanifold of G (i.e., the geodesics of M are also geodesics in G).
C H A P T E R 11
Principal Bundles* A good command of the theory of principal bundles is essential for mastery of modern differential geometry (@ [22], [23]). In recent years, principal bundles have also become central to key advances in mathematical physics (Yang-Mills theory) which have in t u r n generated exciting new mathematics (e.9., S. K. Donaldson's work on differentiable 4-dimensional manifolds [7], [11]). This chapter will be a brief introduction to principal bundles. 11.1. T h e F r a m e B u n d l e Before defining the term "principal bundle", we discuss the central motivating example. Let V be a real vector space of dimension n. An n-frame in V is an ordered basis t~ = (Vl,... , v,~). If V = R n, each vi is a column vector and the frame t~ is a nonsingular matrix. That is, the set of all n-frames in R n is naturally identified with the manifold Gl(n). Generally, we denote the set of all n-frames in V by F(V) and topologize this set as a subset of V ~ = V x ... x V. Let ~ : V -~ R n be an isomorphism of vector spaces and define a diffeomorphism 7 : Vn --~ gII(n) by ~(Yl,.*.
, Y n ) ~-- ( ~ ( V l ) , . . .
,~(Vn)).
Clearly ~(F(V)) = Gl(n), hence F(V) C V n is an open subset, diffeomorphic via to Gl(n). D e f i n i t i o n 11.1.1. The smooth manifold F(V) is called the frame manifold of V. One might try to make F(V) into a Lie group via ~, but this is a bad idea since, if ~1 and ~2 are two isomorphisms of V to R n, it is not generally true that the diffeomorphism ~1 o (~2) -~ : Gl(n) -~ Gl(n) is a group isomorphism. That is, there is no canonical way to make F(V) into a Lie group. There is, however, a natural right action
F ( v ) • GI(~) -~ F(V) defined by formal matrix multiplication
(121 . . . .
,vn)'
[all9 "" aln. 1 = (~=1 ailvi,... ,Eainvi n )9 [an1 . . . annJ i=1
This is a smooth, transitive action with trivial isotropy group (such an action is said to be simply transitive). Remark that the right action of Gl(n) on itself, defined by the group multiplication, is also smooth and simply transitive and that -~: F(V) --~ Gl(n) respects these actions in the following sense.
348
11. PRINCIPAL BUNDLES*
L e m m a 11.1.2. If ~o : V --+ ]Rn is a linear isomorphism and if 13 C F ( V ) , A E G l ( n ) , then
~(13. A) = ~(13)A. One says t h a t the m a p ~ is Gl(n)-equivariant or, more simply, equivariant. There is a n a t u r a l map F ( V ) • R n ~ V defined by formal m a t r i x multiplication
(11.1)
(tJ,5) ~
13. a =
(Vl,... ,Vn).
: Eaivi. i=1
For each fixed 13 ff F ( V ) , this defines a linear isomorphism 13 : ]Rn --+ V, and every linear isomorphism is of this form. We summarize. L e m m a 11.1.3. The map defined by (11.1) sets up a canonical identification of F ( V ) with the set of all linear isomorphisms 13 : IR'~ --+ V. One can canonically reconstruct the vector space V from the frame manifold F ( V ) as follows. Define the left action of Gl(n) on the space F ( V ) x ]R~ to be the "diagonal action" (11.2)
A . (13,5) = (13. A - 1 , A S ) ,
where A ranges over Gl(n) and (13,5 ) ranges over F ( V ) x IRn. It is e l e m e n t a r y t h a t this is a smooth, left action of the group Gl(n), so there is an associated equivalence relation on F ( V ) x ]Rn with the Gl(n)-orbits as equivalence classes. We denote the quotient set by F ( V ) xGl(n ) IR~. The equivalence class of (13,/5) will be denoted by [13,5]. R e m a r k t h a t (11.3)
[v. A , S ] = [13,A51, VA e Gl(n).
W e p u t a vector space structure on the quotient set F ( V ) x Ol(n) ]R~ by fixing a frame 13 E F ( V ) and defining
[13,51] + [13,52] = [13,< + 521, c[13, ~1 = [13,cSl. To see t h a t these definitions do not depend on the choice of frame, let u be another choice, let A C Gl(n) be the unique m a t r i x such t h a t u = 13 9 A, and use the relation (11.3) to o b t a i n [u, 511 + [u, 52] = [13,A511 + [13,A52 ]
= [13,A(51 + 52 )] = [u,< +<], c[u, ,7] = c[13, AS]
= [13,AcS]
= [u, c5] The m a p defined by (11.1) passes to a well-defined linear m a p r : F ( V ) xol(n) IRn --* V. F i x i n g a frame 13 C F ( V ) , we define j : V --+ F ( V ) x al(,~) IRn
11.1. T H E F R A M E B U N D L E
349
by j ( ~ . if) = [~,~1. It is clear that Ib and j are mutual inverses, hence that j does not really depend on the choice of frame ~. We summarize these remarks in the following. L e m m a 11.1.4. The vector space operations on F ( V ) XCl(n) ~n are well defined,
independently of the choice of the frame ~, and the map r is a canonical isomorphism of vector spaces, the inverse j being well defined independently of the choice
o/frame. These remarks generalize to vector bundles fiberwise. Let 7r : E -~ M be an n-plane bundle over an m-manifold. Each fiber Ez = 7r-l(x) is an n-dimensional real vector space, so we form the associated frame manifold F ( E x ) and the disjoint union
F(E) = U F(Ex). xEM
We also define p : F(E) ---* M by p(F(Ex)) = x, Vx 9 M. If 71" - I ( U )
~
' U x I~ n
U
,
U
id
is a local trivialization of E, then 9)x : E~ --* It~~ is a linear isomorphism, Vx E U, inducing a diffeomorphism G:
F ( G ) -* Cl(n).
This determines a commutative diagram U x Gl(n)
q U
,
U
id
where ~ is bijective, hence defines a topology and smooth structure on p - l ( U ) . It is straightforward to check that, on overlaps p-1 (U) n p-1 (V), corresponding to two local trivializations of E, the two smooth structures and underlying topologies coincide and that this makes F(E) into a smooth manifold of dimension m + n 2. Furthermore, p : F(E) ~ M is a smooth submersion and the inclusion map, ix : F(E~) ~ F(E) smoothly imbeds the fiber p - l ( x ) as a proper submanifold,
VxEM. We have defined a new kind of bundle over M, called the frame bundle of E. The fibers F(Ex) are not vector spaces. Instead, they are manifolds diffeomorphic to Gl(n) that admit a canonical right action of Gl(n). This defines a right action
F(E) x GI(n) --* F(E) that is simply transitive on each fiber. By Lemma 11.1.2, the local trivializations t u r n this action into the action (U x Gl(n)) x Gl(n) ~ U x Gl(n)
350
11. PRINCIPAL BUNDLES*
defined by (x, B ) . A = (x, B A ) . This is clearly a smooth action, so F ( E ) • Cl(n) -~ F ( E ) is locally, hence globally, a smooth action. By Lemma 11.1.3, the fiber F ( E x ) over x C M of the frame bundle of E is exactly the set of linear isomorphisms of the "standard fiber" R n to the specific fiber Ex. The construction for recovering a vector space V from its frame manifold F ( V ) globalizes in a natural fashion to give a canonical way of recovering the vector bundle E from its associated frame bundle F ( E ) . The diagonal action of Gl(n) given by (11.2) extends by the same formula to a left action Gl(n) x ( F ( E ) x N ~) --~ F ( E ) x ~ and we let F ( E ) xCl(n ) N n denote the corresponding quotient space (with the quotient topology). The bundle projection p : F ( E ) --~ M passes to a well-defined map ~ : F ( E ) xGl(~) ]~n ~ M and the map F ( E ) • X n ~ E, again defined as in (11.1), passes to a well-defined map r : F ( E ) xc1(~) R n ~ E. These maps are smooth and the diagram F(E)
XGl(n ) ]~n
M
~
id
, E
,M
commutes. It has also been arranged, by the very definitions, t h a t r restricts to r on the vector space p - - l ( x ) = F(E~) Xgl(n ) Rn C F ( E ) xcl(n) R n,
carrying it isomorphically onto the vector space E~, Vx C M. The inverse j~ of r although defined by a choice of frame v G F(E~), is independent of t h a t choice (Lemma 11.1.4), g x G M, and these fit together to give a global inverse j : E -+ F ( E ) xGI(n ) R ~. In a local trivialization of E, one can choose n linearly independent smooth sections, using these to define j . It follows that j is locally, hence globally, smooth and r is a diffeomorphism. We summarize this discussion in a theorem. T h e o r e m 11.1.5. The s t ~ c t u r e ~ : F ( E ) xol(n ) ]~n ___+M is a vector bundle, canonically isomorphic to 7r : E ~ M . The association E ~ F ( E ) is a one-to-one correspondence between the set of vector bundles over M and the set of associated frame bundles.
11.2. P R I N C I P A L G-BUNDLES
351
Exercise 1 1 . 1 . 6 . Let p : Gl(n) x IR~ --* IR~ be the s m o o t h action defined by
t,(A, ~) = (A -1) T ~. Given an n-plane bundle E over M , define E* = F ( E ) xp R n by analogy w i t h t h e definition of F(E) xGl(n ) ]I;~n and prove t h a t E* is an n-plane bundle over M w i t h each fiber /?7* canonically isomorphic to the dual space of E~. Again, it will be helpful to consider first the case F(V) xp ]R~ = V*, t h e n the case of a p r o d u c t bundle, and finally the general case. This is exactly the dual bundle E* introduced in Section 6.1
11.2. Principal G-Bundles T h e frame bundle F(E) is an example of a principal G-bundle where G = Gl(n). We will give the general definition after the following.
Definition 11.2.1. Let M and N be manifolds, together with s m o o t h actions from the right
M xG--~ M, N •
N,
each w r i t t e n as (x,g) ~ x 9 g. A s m o o t h m a p f : M --. N is G-equivariant (or equivariant, if the G-actions are understood from the context) if
f(x.g)=f(x).9,
Vx 9
V9 9
D e f i n i t i o n 11.2.2. Let M and P be s m o o t h manifolds, G a Lie group, and p : P --~ M a s m o o t h map. Suppose t h a t there is all open cover {U~}~eL of M and, V c~ E 92, a c o m m u t a t i v e diagram p-l(Ua)
~'~
) Ua • G
Pl
I p*
u~
,
u~
id
where ~b~ is a diffeomorphism. Suppose also t h a t there is a s m o o t h right action P x G --~ P, simply transitive on each fiber p - l ( z ) . Finally, assume t h a t %ha is equivariant with respect to this action and the right G-action
(U~ x G) • G--+ U~ x G,
((x,v), h ) H (.,gh), g ~ E 92. Then, p : P --+ M , together with this G-action, is called a principal G - b u n d l e over M . The group G is called the structure 9roup of the bundle. E x a m p l e 11.2.3. Let p : P ~ M be a principal G-bundle, a : M --+ P a s m o o t h m a p such t h a t p o a = idM. As in the case of vector bundles, a is called a section of tile principal bundle. While a vector bundle always admits sections (e.g., the 0 section) this generally fails for principal bundles. Indeed, if a is a section, define
~:MxG--+P by
~(~, g) = ~(~). v,
352
11. P R I N C I P A L BUNDLES*
obtaining a diffeomorphism such that the diagram MxG
~
~ P
m~
IF
M
,M id
commutes. As in the case of vector bundles, ~ is called a trivialization of the principal bundle P. Thus, the existence of a section is equivalent to triviality of a principal G-bundle. E x a m p l e 11.2.4. Suppose that the n-plane bundle 7r : E --~ M is given an explicit O(n)-reduction. Equivalently, there is a positive definite inner product (.,.)~ on E~ that varies smoothly with x E M. That is, (sl(x), s2(x))~ is a smooth function of x, Vsl,s2 E F(E) (cf. Exercise 3.4.16 for the case E = T(M)). In fact, this smoothly varying inner product can be thought of as a smooth "field" of inner products, often called a Riemannian metric on the bundle E. One can then define O(E) c F(E) to be the subset of frames that are orthonormal with respect to this smooth field of inner products and restrict p to a projection p : O(E) -~ M. If U c_ M is an open, locally trivializing neighborhood for E, let (sl,... ,Sn) be a smooth frame field on U (i.e., a smooth section of F(EIU)). By an application of the Gram-Schmidt process to this frame field, we can assume that it is everywhere orthonormal, hence is a smooth section a of O(EIU ). All of O(EIU ) can be swept out by applying right actions of O(n) to a and one obtains a local trivialization
UxO(n)
0 ,o(Elu)
U
,
U
id
by setting O(x,A) = a(x). A, Vx E U, VA E O(n). Thus, O(E) C F(E) can be thought of as a subbundle that is invariant under the right action of O(n) C Gl(n). In fact, O(n) is simply transitive on the fibers and O(E) is a principal O(n)-bundle. The standard application of partitions of unity shows that there are infinitely many choices of Riemannian metrics on E and corresponding orthonormal frame bundles. E x a m p l e 11.2.5. Every n-plane bundle 7r : E --* M admits O(n)-reductions, but we know that it may or may not admit a Gl(k, n - k)-reduction. In fact, such a reduction is equivalently a k-plane subbundle Ek C_ E (Example 3.4.20). Given such a subbundle, let Fk(E) G F(E) consist of the frames of E whose first k entries form a frame of Ek. As in the previous example, this forms a locally trivial subbundle of F(E) that is invariant under the right action of Gl(k, n - k), a simply transitive action on each fiber. This is a principal Gl(k, n - k)-bundle. E x a m p l e 11.2.6. An O(n, n - k ) - r e d u c t i o n of E corresponds to an indefinite inner product (., .)z on E~ (cf. Example 3.4.19) that varies smoothly with x E M. Such a reduction may or may not exist but, if it does, one obtains a principal O(k, n - k)bundle of frames that are "orthonormal" with respect to this indefinite metric. The reader can supply the details. E x a m p l e 11.2.7. Let G be a Lie group, H C G a closed subgroup. Then the quotient projection p : G ~ G / H is a principal H-bundle over the homogeneous space
11.2. P R I N C I P A L G - B U N D L E S
353
G/H. Indeed, the local triviality is by Exercise 5.4.8 and the required p r o p e r t y of the right H - a c t i o n is obvious. E x a m p l e 11.2.8. T h e case of a principal bundle with discrete s t r u c t u r e group G is noteworthy. A discrete group is simply an abstract group with the discrete topology (each point is an open set). This is a topological group and, if G is at most c o u n t a b l y infinite, it can also be viewed as a Lie group of dimension 0. A principal G - b u n d l e p : P -+ M is exactly a regular coverin9 space (Definition 1.7.13) if G is a discrete group. Indeed, the local triviality of the bundle guarantees t h a t each p o i n t z E M has an evenly covered neighborhood U _C M . It can be seen t h a t the group of h o m e o m o r p h i s m s 9 : P --+ P such t h a t p o 9 = P is exactly the group G, acting (from the right) on the total space P of the bundle. This is the group of covering t r a n s f o r m a t i o n s and, by the definition of principal bundles, it p e r m u t e s the fiber simply transitively. We will want an a p p r o p r i a t e definition of isomorphism for principal G - b u n d l e s over M . As in the case of vector bundles, it is useful to give an apparently weaker definition t h a n one really wants and t h e n prove t h a t the stronger p r o p e r t y holds. 11.2.9. If p : P --~ M and p' : P ' --+ M are principle G-bundles, an isomorphism ~o : P --+ P ' is a smooth, G-equivariant m a p such t h a t the diagram Definition
p
~
M
~ p'
,M id
commutes. 11.2.10. If 9~ : P --+ P ' is an isomorphism of principal G - b u n d l e s , prove t h a t ~ is a diffeomorphism and t h a t qo-1 is also an isomorphism. Thus, isomorphism is an equivalence relation. Exercise
Associated to a principal G-bundle over M and any s m o o t h left action of G on a manifold F , there is a s m o o t h bundle over M with fiber F . This associated bundle is constructed using exactly the same technique whereby we recovered a vector bundle E from its associated frame bundle. D e f i n i t i o n 1 1 . 2 . 1 1 . A (locally trivial) bundle over M with fiber a manifold F is a s m o o t h m a p rr : E + M with the following property. For each z C M , there is an open n e i g h b o r h o o d U of z in M and a c o m m u t a t i v e diagram lr-l(U)
~
, axF
l U
,
U
id
in which ~o is a diffeomorphism and Pl is projection onto the first factor. E x e r c i s e 1 1 . 2 . 1 2 . Let p : P --+ M be a principal G-bundle and let F be a m a n ifold. If p : G x F --+ F is a s m o o t h action, define tile "diagonal" action of G on P x F in analogy w i t h the definition you gave in Exercise 11.1.6 and denote the quotient space by P x o F . Show t h a t the bundle projection p passes to a welldefined m a p re : P x 0 F -~ M , and t h a t this is a smooth, locally trivial bundle with
354
11. PRINCIPAL BUNDLES*
fiber F . As usual, it will be helpful to consider first the case in which M reduces to a single point (i.e., exhibit a canonical identification G xp F = F), then the case in which P = M x G is a trivial bundle, and finally the general case. D e f i n i t i o n 11.2.13. The locally trivial F - b u n d l e constructed in Exercise 11.2.12 is called the F - b u n d l e associated to the principal G-bundle by the group action p. Normally, when the action and the principal bundle are understood from context, one refers simply to the "associated F-bundle". E x e r c i s e 1 1 . 2 . 1 4 . If p : P -+ M and p' : P ' --~ M are isomorphic principal G - b u n d l e s over M and p : G x F --* F is a smooth action, prove that the associated bundles fit into a commutative diagram
PxpF
M
f
~ P' x p F
id
,
M
in which f is a diffeomorphism. One says that f is an isomorphism of these locally trivial F-bundles. E x e r c i s e 11.2.15. If E is the associated F - b u n d l e as above, show how each element g E P can be interpreted as an inclusion map g : F ~-~ E of the fiber over
P(9). E x e r c i s e 11.2.16. Given a vector bundle E over M, show how to obtain the various associated tensor bundles, constructed in Section 7.4, as locally trivial bundles associated to the frame bundle F(E). 11.3.
Cocycles
and Reductions
The notion of a G-cocycle ~/was defined in Section 3.4 (Definition 3.4.10). There we considered subgroups G C Gl(n), but the definition works equally well for arbitrary Lie groups G. The notion of equivalence of Gl(n)-cocyctes (Definition 3.4.3) extends without change to G-cocycles. As in the comment following Definition 3.4.10, the set of equivalence classes [~] of G-cocycles is denoted by H I(M; G), the cohomology notation being motivated by analogies with Cech cohomology. Given a principal G-bundle, a family of local trivializations p-I(u~)
r
' Uc~ • G
p~
~pl
u~
,
g~,
id
where {U~}aega covers M, gives rise to a G-cocycle "y = {'YaZ}a,Ze~ in a fairly obvious way. Indeed, over each nonempty intersection Ua C/Off, the trivializations give a transition function ~b~ o ~b/71 of the form
r If we set %Z(x) = r
or
= (x, r
g)).
e), then the G-equivariance of ~b~ and ~b5 implies that ~/)c~ O ~/)~-l (x, g) = (X,'yct/~(X)g).
11.3. C O C Y C L E S A N D R E D U C T I O N S
355
The cocycle property for 3' = {3`~}~,~e9~ is obvious, as is the fact that the principal G-bundle p : P --+ M determines 3` up to equivalence. E x e r c i s e 11.3.1. Mimic the procedure in Section 3.4 for constructing n-plane bundles from Gl(n) to show how to construct a principal G-bundle from a G-cocycle. Show that the set of isomorphism classes of such bundles over M is canonically identified with H I(M; G). By a common abuse of terminology, we often refer to H I(M; G) simply as the set of principal G-bundles over M. If H C G is a properly imbedded Lie subgroup, we can view H-cocycles on M as G-eocycles. If two such cocycles are equivalent as H-cocycles, they are also equivalent as G-cocycles, so we obtain a natural map of sets A: H I ( M ; H ) ~ H I ( M ; G ) . It is possible that two H-cocycles not be equivalent, but become equivalent when viewed as G-cocycles, and so A is not generally injective. It is not generally surjective either, since there is no reason a priori that a G-cocycle should be equivalent to some H-cocycle. D e f i n i t i o n 11.3.2. If [3`] C H I ( M ; G ) , [7] r H I ( M ; H ) and [3`] = A[~/], then the principal H-bundle [~/] is said to be an H-reduction of the principal G-bundle [3']. The following exercise lays bare the geometric meaning of this definition. E x e r c i s e 11.3.3. Let [3`] = A[r/] as in the above definition and let
q:Q~M p:P---~ M be, respectively, the principal H-bundle with cocycle ~7 and the principal G-bundle with coeycle 3`. Let p : H x G ~ G be the canonical action of H on G by left multiplication and build the associated G-bundle
7r : Q x p G -~ M, as in Exercise 11.2.12. Note that G has a natural right action on the total space Q xp G of this bundle and prove that, equipped with this action, the associated G-bundle is a principal G-bundle isomorphic to p : P --~ M. Use this to produce a commutative diagram Q i ~ p
M
,M id
in which i is a proper imbedding of Q as a submanifold of P that is invariant under the right action of H. Conversely, given such an imbedding of a principal H-bundle, show that the class [3'] has the form A[r/]. The infinitesimal G-structures of Section 3.4 (see Definition 3.4.12) are now seen to be G-reductions of the frame bundle F(T(M)). E x a m p l e 11.3.4. In Example 11.2.4, we saw that a Riemannian metric on an nplane bundle E led to an O(n)-reduction O(E) C F(E) exactly as in Exercise 11.3.3. Conversely, suppose that Q c F(E) is an O(n)-reduction as in that exercise. We
356
11. PRINCIPAL BUNDLES*
will recover canonically a Riemannian metric on E giving this reduction. Indeed, let x E M and v C Q~, write v = ( V l , . . . , v n ) as a frame, and define a positive definite inner product {., '}o on E~ by requiring that (Vi,Vj}t~
dij ,
:
1 < i,j <__n.
I If 0~ = ( v ~ , . . . , V n) E Qx is another choice, there is unique A c O(n) such that v' = v 9A and it is a routine computation to check that
( V' i , V j'} tJ =~ij,
l <- -i , y < n- - .
This shows independence of the choice of frame. In order to see that this inner product varies smoothly in a neighborhood of x, let U C M be such a neighborhood, small enough that there is a smooth section a : U --* Q. Then, for arbitrary v, w E F(EIU), the expression
(v(y),~(y)L(y) depends smoothly on y and the assertion follows. The converse of Example 11.2.6 can be verified analogously. E x e r c i s e 11.3.5. Check the converse of Example 11.2.5. That is, given a Gl(k, n k)-reduction, produce the k-plane subbundle of E giving rise to that reduction. E x e r c i s e 11.3.6. With the hypotheses and notations of Exercise 11.3.3, assume further that
#:GxF---~F is a smooth action on the manifold F. Let # denote also the restricted action
#:HxF-~F and exhibit an isomorphism
f :Qx, F ~ Px, F of associated bundles. E x a m p l e 11.3.7. If the n-plane bundle admits a (pseudo-) Riemannian metric, let O(E) C F(E) denote the orthonormal frame bundle. Let O stand for the respective groups O(n) or O(k, n - k ) according to whether the metric is Riemannian or pseudoRiemannian. Let p : OXl~n----+]I~n
be the standard left action. By Exercise 11.3.6, we then see that O(~J)
Xp ~n
~_
F(E) XGl(n) IRn = E,
canonically. This shows that E can be assembled using an O-cocycle. Generally, Exercise 11.3.6 shows that H-reductions of principal G-bundles allow us to assemble the associated bundles using an H-cocycle. This is the origin of the terminology H-reduction.
11.4. F R A M E
BUNDLES AND THE EQUATIONS
OF STI~UCTURE
357
11.4. F r a m e B u n d l e s a n d t h e E q u a t i o n s o f S t r u c t u r e
In Section 10.6, we showed that a connection V on an n-manifold M, together with a choice of frame field a = (X1,.. 9 Xn) on a trivializing neighborhood U C M for T ( M ) , gives rise to a pair of equations relating certain differential forms (equations (10.9) and (10.10)) called the Caftan structure equations. These equations depend on the choice of local section (7 of F ( M ) = F ( T ( M ) ) , but it turns out t h a t they can be lifted to the total space F ( M ) to be globally defined independently of choices. The pullback by (7 of these globally defined equations gives back the local equations. To emphasize the provisional nature of the local structure equations, we will write them with tildas over the forms as follows (11.4)
d O = - V A 0 + ~,
(11.5)
d~ = -~ A ~ +
fi,
reserving the forms without tildas to denote the canonical global forms to be produced on the manifold F ( M ) . If V is the Levi-Civita connection of a Riemannian metric, the structure equations can be lifted to the total space 0 ( 3 I ) = O ( T ( M ) ) of the reduced orthonormal frame bundle. In the case of a pseudo-Riernannian metric, there is also a unique Levi-Civita connection V (just mimic Definition 10.2.10 and Exercise 10.2.12) and the structure equations will again lift globally to the total space Ok(M) of the orthonormal frame bundle. We will be particularly interested in both of these cases and so, in what follows, p : P --* M will denote either the full frame bundle or the orthonormal frame bundle associated either to a Riemannian or pseudo-Riemannian metric. Similarly, G will denote any one of Gl(n), O(n) or O(k, n - k), as appropriate. Our main application of this theory will be to prove the following basic result characterizing flatness of Riemannian and pseudo-Riemannian manifolds. (cf. Theorem 10.6.7). T h e o r e m 11.4.1. Let M be a (pseudo-) Riemannian manifold, V the Levi-Civita
connection, and R the curvature tensor of V. The following are equivalent. (1) R ~ O . (2) There is a smooth atlas in which the metric coefficients are everywhere gij :t:hij, the negative sign occurring exactly for
k+l<_i=j<_n. (3) There is a smooth atlas in which the Christoffel symbols are everywhere Fi~ =_ O. (4) Each x E M has a neighborhood Ux such that the holonomy of V around each loop (7 E f~(U~, x) is the identity transforvnation. (5) The Riemannian (respectively, pseudo-Riemannian) metric, as an infinitesimal O(n)-strueture (respectively, O( k, n - k )-structure), is integrable.
A (pseudo-) Riemannian manifold in which one, hence all, of these holds is said to be fiat. E x e r c i s e 11.4.2. As a review of the structure equations and for later use, check that, when V is Levi-Civita for a Riemannian metric, then the L(Gl(n))-valued forms ~ and fi actually take values in the subalgebra L(O(n)). Similarly, if V is
358
11. PRINCIPAL BUNDLES*
Levi-Civita for a pseudo-Riemannian metric, show that these forms are L(O(s n h))-valued. We will lift equations (11.4) and (11.5), as promised, by finding It{n-valued forms 0 and r on P and L(G)-valued forms co and f~ on P satisfying (11.6)
dO = -co A 0 + r,
(11.7)
d~ = -co A co + ft.
Furthermore, the choice of frame ( X 1 , . . . , Xn) on U is a choice of smooth section ~r : U ~ PIU and we will prove that = o*(0), = ~*(~),
5 = ~*(a).
Let ( E P, x = p((), and set Px = P - I ( z ) . The vertical space at ( E P will be
v~ = T~(Px) c T d e ) D e f i n i t i o n 11.4.3. The vertical subbundle V C T ( P ) is V = U(EP Vr ments X E F(V) are called the vertical fields on P.
The ele-
Each E E L(G) can be viewed as a vertical field on P. Indeed, E is an n x n matrix and etE is the one-parameter subgroup of G generated by E. Using the right action P x G --~ P, we obtain, for each r E P, a curve r 9etE which is at r at time t = 0. This curve lies in the fiber of P through r so the corresponding infinitesimal curve is '
:
' e
>t=0
E
As r varies over P, this defines a vertical field E on P. The mapping E H E is a canonical linear injection L(G) ~ F(V). If U C M is an open, trivializing neighborhood for P, the trivializations PIU ~U x G are in one-to-one correspondence with the sections ~ E F(PIU ). Indeed, o ( z ) . B ~-~ (x,B). Fix the choice of ~. Since B E G C Gl(n) is a nonsingular n x n matrix and, as remarked above, each E E L(G) is an n x n matrix, the value of the vertical field E at ( = (z, B) E PIN is r E = (w, B E ) , where B E is the matrix product. This is because E, as a left-invariant vector field on G, has value at B E G given by B E . Thus, this way of viewing a matrix E E L(G) as a vertical field E on P is quite analogous to the way that E is viewed as a left-invariant field on G.
The right action P x G --~ P induces a linear action
x ( P ) • a -~ :~(P) via the differential. If X E 3~(P) and B E G, it will be natural to denote this action by X ~ X - B. Consider also the automorphism Ad(B) : G ~ G, called the adjoint action and defined by Ad(B)(A) = B - l A B (cf. Exercise 5.3.15). The differential A d ( B ) . : ~(G) -~ X(G) restricts to an automorphism of L(G) where it will also be called Ad(B) and written A d ( B ) ( E ) : B - X E B . The following is practically immediate.
11.4. F R A M E
BUNDLES AND THE EQUATIONS
OF STRUCTURE
359
L e m m a 11.4.4. Under the inclusion L ( G ) ~ F(V), A d ( B ) ( E ) H E . B , V B E G, V E E L(G). In particular, L(G) C F(V) is invariant under the right action of G. From now on, we denote E by E, identifying L(G) as a vector subspace of P(V). We will also write E B for the right t r a n s l a t e E . B of this vector field by
BEG. Remarh. A n y basis of L(G) c F(V) gives a trivialization of V. If desired, it is possible to specify a canonical choice of this basis. In the case t h a t G = GI(n), this will be {E)}l<_i,j<_,~, where E} is the n • n m a t r i x having ( i , j ) t h entry 1 and all r e m a i n i n g entries 0. T h e Lie algebra of O(h, n - h) consists of all matrices
where A is a skew symmetric, ]c x k matrix a n d C is (n - h) x (n - k) and skew symmetric, as is easily checked. For 1 _< i < j _< h and k + 1 _< i < j _< n, set A} -~ E ~ - E ~ . For all other i < j , set A} = E} + U j. Then, {Aj}l<_i<j<_ni is a canonical basis of L(O(k, n - k)). The case in which G = O(n) is j u s t the special case in which h = n, so the canonical basis is given by A} = / ~ } - E~, 1 < i < j < n. We are going to show t h a t tile connection V defines a direct s u m decomposition T ( P ) = V ~ H,
and a canonical choice of basis {Ei}l
P(H).
We call tile b u n d l e H the
Fix ~o E P , x0 = P(C0) E M , and let s : [0, e) --* M be a smooth curve with s(0) = x0. The frame @ = (Vl,... , v~) can be parallel t r a n s p o r t e d along s. T h a t is, each vi is parallel t r a n s p o r t e d and these remain linearly independent. In fact, in the case t h a t V is Levi-Civita for a (possibly indefinite) nletric, the Nct t h a t V respects tile metric implies t h a t the orthonormal frame parallel translates to o r t h o n o r m a l frames. Therefore, this parallel t r a n s p o r t can be interpreted as a lift s ~ = (X~s(t),... ,X~ns(t)), 0 _< t < e, of s to P starting at @. T h a t is, the diagram P
[0, e)
s
, M
commutes. We think of s ~ as a "horizontal" lift and say that tile initial velocity ~(0) C Tr ) is horizontal. If we can show t h a t the set of all vectors in Tr ) t h a t can be o b t a i n e d in this way is a vector subspace of dimension n, this will be our c a n d i d a t e for Hr Similarly, ~>(t) C H~(t), justifying the term "horizontal lift". A section a E r(FIu), where U is a suitably small neighborhood of x0, gives another way to lift s. If a = ( X 1 , . . . , X~) and a(x0) - C0, set
s~(t) = (x~(t),...
x~,~(~)),
Then,
s"(0) = C0 and the diagram
0 _< t < e.
360
11. P R I N C I P A L
BUNDLES*
P
[o, ~)
s
, M
commutes. Since sb(t) and s~(t) lie in the fiber P~(t), there is a unique element A(t) 9 G such that
J ( t ) = sb(t)A(t),
(11.8)
0 < t < e.
Clearly, A : [0, e) --+ G is smooth and A(0) = I, the identity matrix. Finally, we use the local frame cr = ( X 1 , . . . , X,~) to define the forms 0", ~, ~, and ~ on V satisfying equation (11.4) and (11.5). L e m m a 11.4.5 (Key Lemma). A(0) = D(i(0)) E L(G). Before proving Lemma 11.4.5, we deduce some important consequences. C o r o l l a r y 11.4.6. If a 9
r(PIU )
with a(xo) = @ E P~o, if s:[O, e) ~ U
is smooth with s(O) = xo, and if s b : [0, e) --~ P i g
is the horizontal lift with s~(O) = ~o, then ~(0) = -r
~(~(0)) + ~,~o(~(0)).
Proof. Indeed, differentiate s~(t) -- s~(t)A(t) at 0, obtaining i~(0) = ib(0). A(0) + sU(0) 9A(0)
= ~(o) + r
~(~(o)). []
But i~(0) = a.~o(i(0)). We define h xo ~o : T x o ( M ) -~
T~o(P)
by h~xo (~(0)) = ~(0), this being linear by Corollary 11.4.6. Furthermore, P*r o h~~ = idTxo(M) , since - - @ ' ~(i(0)), being vertical, is annihilated by P*r
and
p
H~o = im(h~x~) The horizontal subbundle is
H=U ,
11.4, F R A M E B U N D L E S A N D T H E E Q U A T I O N S O F S T R U C T U R E
a61
E x e r c i s e 11.4.8. Prove that the horizontal bundle H really is a vector bundle and that T(P) = V | H, each summand being an invariant subbundle under the right action of G. In the following proof and subsequently, we will use the summation convention without further comment.
Proof of the key lemma. We take U to be a coordinate neighborhood and write s(t) = ( x l ( t ) , . . . ,x~(t)). Write :
[a>],
where a{(0) = 6~. The smooth frame <7 = (X1,... ,Xn) can also be written as a matrix valued function. Writing Xk ~(t) = u~(t)({ s(t) as a column vector, we get
~(~) =
[~(t)].
Similarly, write
s ~(t) = ( x ~ ( ~ ) , . . . ,
< ~(,)) : [~](t)].
In this notation, equation (11.8) becomes
vji(t)aJ~(t)
(11.9)
=
{ ). uk(t
Using the covariant derivative to express the fact that s ~ is a parallel frame along s gives VX~
o:
d~ = (~] + ~ r ~ ) ~ ( ~ ,
SO
(11.10)
.i vj(t)
:
-v~(t)ScP(t)rza(s(t)).
Differentiate equation (11.9) and get (11.11)
i .j ) iJj(t)aJk(t) + va(t)ak(t
=
.i ). uk(t
In equation (11.11), set t = O, note that
[v}(O)] = s~(O) = s~(O) = b}(O)] , and get .j
ug(O) = S~(O)ag(O) + vj(O)a~(O) = ~(o) + ~}(o)ag(o). Use equation (11.10) at t = 0 to conclude that (11.12)
/~(0) + u~(0)a~a(0)pS~(Xo) = uj(0)ak(0) . i .y
By the definition of the local connection form ~, we obtain ~(~(O))u~(O)~k~o = ~(~(O))Xi~ o = V~(o)Xj c~
= V~(o)(U 5
~)
= (~)(0) + ~ ( 0 > e ( 0 ) r ~ ( . o ) )
~k=o.
362
11. P R I N C I P A L B U N D L E S *
Setting the coefficients of ~k xo equal and applying equation (11.12), we obtain u~ (0)~) (~(0))) = ~ (0) + u ; (0)~ ~ (0)Cw (xo) = ~ (0)~} (0). In terms of matrix products, this says
[~(o)] [~}(~(o))] = [~(o)] [~(o)]. Since the matrix [~(0)] is nonsingular, we conclude that A(O) = ~(~(0)) e L(G). [] D e f i n i t i o n 11.4.9. The tautological horizontal frame (E~,... ,E~), at r 9 P, is the unique n-frame of vectors in H i such that
r = (p<(E~),... ,p<(E~)). Equivalently, in terms of a local section a = ( X 1 , . . . ,Am) of P,
EJ(~) : h~(~)(Xj x), 1 _< j _< n. As ~ varies over P, the reader can check that E~ varies smoothly. That is, EJ 9 F(H) and the tautological frame field ( E a , . . . , E ~) is a canonical trivialization of H. This, together with the remark following Lemma 11.4.4, gives the following. L e m m a 11.4.10. Given the connection V, the manifold P is canonically paral-
Iclizable. Indeed, for the case P = F ( M ) , the canonical basis of :~(P) is i {Ei}n= 1 U {E~}l<_i,j<_n
and, for the case P = O(M) or Ok(M), it is
{E~}~=I u {g}h<~<j<,. D e f i n i t i o n 11.4.11. The canonical coframe field for V on P is the N~-valued 1form 0 such that OIV -
0;
O(aiE i) = Recall that, for each r 9 P, the vertical space is canonically Vr = L(G). This identification is understood in the following definition. D e f i n i t i o n 11.4.12. The connection form of V on P is the L(G)-valued 1-form w such that
wlH=O; wr r
V( 9
11.4. F R A M E BUNDLES AND T H E E Q U A T I O N S OF S T R U C T U R E
363
Remark. If we write 0 =
,o col d
02
co J
Lcor then, in the case t h a t G = Gl(n), the set {0i}~=1 U
i {a)j}l<_i,j<_n
of 1-forms is dual to the canonical basis
{E }L1 u
i
For G -- O(n) or O(k, n - k), the dual to the canonical framing of P is u
E x e r c i s e 1 1 . 4 . 1 3 . For B E G, let RB : P -~ P denote the right action of B and check t h a t R}~(oJ) = Ad(B) o co. T h a t is, if ( E P and v E T<(P), t h e n (R~(co))((v) = c o ( e ( v . B) = ad(B)(co((v)).
Remark. One can use this to generalize the notion of a connection to principal Gbundles p : P --* M , where G is a general Lie group. A connection on P will be an L ( G ) - v a l u e d 1-form co on P such t h a t (1) if ( E P and L ( G ) = V< C Tr Vr --~ L(G); (2) R~(co) = Ad(b) ~ Vb E G.
is tile vertical space at (, then co( = id :
By the first of these conditions, H = ker(co) is an n-plane subbundle of T ( P ) c o m p l e m e n t a r y to V. T h a t is, T ( P ) = V @ H , and one calls H the "horizontal distribution". By the second condition, tile horizontal distribution is invariant under the right action P x G -~ P. Piecewise s m o o t h curves s : [a, b] --* M have "horizontal lifts" s b : [a, b] --~ P , uniquely determined by the initial point sb(a) = ( E P~(a) and the requirement t h a t .~(t) E H ~ ( t ), a < t < b. This assertion is proven by the basic t h e o r e m of O.D.E. T h e horizontal lift is interpreted as "parallel t r a n s p o r t " of r along s. F u r d m r m o r e , if F is some manifold and G x F --+ F is a s m o o t h left action, there is an associated bundle 7c : P x c F -~ M , with fibers diffeomorphic to F , and the connection on P defines a notion of parallel t r a n s p o r t (or horizontal lifting) is, this bundle along curves s : [a, b] + M. Using the n o t a t i o n [(,x] E P x a F for the equivalence class of ( ( , x ) E P x F , we define the parallel t r a n s p o r t of [(,x] E r r - l ( s ( a ) ) to be g(t) = [s~(t),x], where sb(a) = (. We r e t u r n to the frame bundle P = F ( M ) , O ( M ) or O k ( M ) and the connection V. T h e following definitions are dictated by equations (11.6) and (11.7). Definition given by
11.4.14.
T h e torsion form of V on P is the R < v a l u e d 2-form r on P
T = dO+wAO.
364
11. P R I N C I P A L
BUNDLES*
The connection is said to be symmetric or torsion free precisely if T -- 0. D e f i n i t i o n 11.4.15. The curvature form of V on P is the L(G)-valued 2-form on P given by ~
=
d~v+wAw.
The connection is said to be fiat precisely if ~ =- 0. L e m m a 11.4.16. Let a = ( Z l , . . . , Xn) e F ( P I U ) and let O, ~, ~, 5 be the associated forms on U as in equations (11.4) and (11.5). Then,
~ = ~*(o), = ~*(~), = ~,(~),
fi = ~*(~). Proof. If we verify the first two equations, the remaining two follow from the equations of structure. We use the notation in the remark following Definition 11.4.12. For the first equation, we use OIV -~ O. Thus, for arbitrary x e U and 1 < j < n, 9
i
X
z
= O~(~)(h~(x)(Xj 5) + a ( x ) . b ( X j 5)) = O~(z)(h~(~)(Xj~))
T h a t is, a*(0 i ) = ~ i l < i < n , soa*(0)=0. For the second equation, we use w~lH - 0 and consider the case of a general connection (P = F ( M ) ) . Thus, for arbitrary x C U and 1 _< j _< n, 9
l
e
- ~ k~(x)(hx~
(X~)+
= ~(~)(~(~).
a(x)
.~(Xj~))
~(Xj~))
~T
= ~k ~(~)(% ( x j ~)E~r ~(~)) ~r
g
r
~r ~k = %Axh~)~,.~
= ~x(Zj~). For the case P = O(M) or Ok(M), replace E~o(~) in the above with A~a(~ ) and sum only over the indices 1 < r < q < n, again obtaining * ~(~k~(~))(xj~) = ~(xj
~).
Since x E U and 1 _~ j _~ n are arbitrary, this gives a* (w~) = wk,-~so a* (w) = ~. L e m m a 11.4.17. The following are equivalent for the connection V: (1) o- ~ O;
[]
11.4. F R A M E B U N D L E S A N D T H E E Q U A T I O N S O F S T R U C T U R E
365
(2) ~-t(H 9 H) =_ O; (3) V is symmetric.
Proof. (1) ~ (2) is immediate. We prove (2) e* (3). Let @ C P, set x0 = P(Co) E M , a n d let U c M be an open neighborhood of xo such t h a t PIU is trivial. We can choose the section cr E F ( P I U ) so t h a t a(xo) = Co and so t h a t G,xo(Txo(M)) = Hr For this choice, we have ~oI(H~o • H~o ) - 0 ** 0 = G;o(~o) = ~xo the torsion tensor Tx o = 0. Since xo E M is arbitrary, the assertion follows. In order to prove (3) ~ (1), let @, x0, and U be as above, a n d assunle t h a t V is symmetric. Given any direct s u m decomposition one can choose cr E F ( P I U ) such that ~,zo(T~o(M)) = ~rr a*X0 (~-(o) = ~ o = 0 implies that
RoI(H~o • H~o) Given v C VG a n d w E Heo, choose/tr ~r G. This is clearly possible. T h e n
Then, the fact t h a t
-- 0.
as above, and u E H G such t h a t v + u , w E
0 = ~o(V + ~,, w) = ~o(V,W) + Ro(U,W) = ~o(V,W). This proves t h a t ~ol(V~o • H~o ) - 0. By antisymmetry, we also have
RoI(H~o • V~o) -= 0. In order to prove t h a t 3-r = 0, it remains to prove that ~Col(V~o • V
0 = r~o (v + Ul, w + u2) = Ro (v, w). We have proven t h a t Tr = 0 for a r b i t r a r y (o E P .
[]
The following is proven by exactly the same argument. Lemma
(])
f~
11.4.18. -
The following are equivalent for the connection V:
o;
(2) f~l(H| -= 0; (3) The curvature tensor R - O. We can now deduce the following result. 1 1 . 4 . 1 9 . The following properties of V are equivalent: The n-plane distribution H on P is integrable; ~---0,' R =- O; for each x E ]F[, there is an open neighborhood U of x on which the holonomy group of V is Hx(U) = {id}.
Theorem (1) (2) (3) (4)
366
11.
PRINCIPAL
BUNDLES*
Furthermore, if V is symmetric, these properties are equivalent to (5) [E~,E j] =- O, 1 <_ i,j <_ n.
Proof. We prove (1) r E
(2) r
(3). The distribution H is exactly ker(~). Thus,
=
k,E9
=
+
A
E
-
-
:
Therefore, integrability of H is equivalent to gtl(H | H ) -= 0, which is equivalent to t2 _= 0 and to R = 0 by L e m m a 11.4.18. W e prove t h a t (1) ~ (4). Thus, assume H to be integrable and choose x c M , r C p - l ( x ) , and let L be the leaf through r Since p.< carries T<(L) : He isomorphically onto T~(M), there is an open neighborhood W c_ L of r carried diffeomorphically by p onto an open neighborhood U C_ M of x. Let s : [a, b] -* U be a piecewise s m o o t h loop at x. Let hs : Tz(M) ~ Tz(M) be the holonomy t r a n s f o r m a t i o n around s. The horizontal lift s ~ : [a, b] --~ P with initial point sb(a) : { : ( V l , . . . ,v~) has terminal point s~(b) : (hs(vl),... ,h~(vn)). The curve s ~ must lie in W, hence it must be be the loop in W carried by p back to s. T h a t is, s~(b) : s~(a) : r and h~(vi) : vi, 1 < i < n. Since the vi's form a basis of T~(M), hs : idT~(M). Since s E gt(M, x) was arbitrary, the holonomy group H~(U) of V in U is trivial. Conversely, supposing t h a t (4) holds, we deduce (1). Let (N, x l , . . . , x ~) be a coordinate chart in M and set W = p - l ( N ) . It will be enough to show t h a t H I W is integrable. Let Z{ C F ( H I W ) be the unique horizontal field t h a t is p-related to (i, 1 < i < n. These fields span H I W and their brackets [Zi, Zy] are p-related to [(i, (j] = 0. This implies t h a t [Zi, Zj] E F ( V I W ), 1 _< i,j <_ n, but we will show the stronger fact t h a t [Zi, Zy] z 0, 1 _< i,j <_ n, so H I W is integrable. Let ~{ be the local flow in N generated by ~i, 1 < i < n. These flows lift to local flows 9ib on p - l ( N ) with infinitesimal generator Z{ E E(H]W), 1 < i < n. Let x E N and let U _c N be an open neighborhood of x so small t h a t H~(U) : {id}. Let s~i be the flow line of 9{ in N from x to y : ~ it~(x). Similarly, s~ is the flow line of ~J in N from y to z = ~tJ (y), - s ~i is the flow line of ( ~ i ) - 1 in N from z to w = ~ i-t~, and - s ~ is similarly defined from w to x' : 9J_t2(w). For sufficiently small values of h and t2, the p.r. curve 8 =
i
S~ +8~y
"
i
"
-- 8 z -- 8 3
stays in U. Since these local flows commute, x' = x and s must be a loop at x. Since the holonomy group H~(U) is trivial, the horizontal lift s ~ to any r 6 p - ~ (x) is a loop at r and this implies t h a t
for all small values o f h and te. Since x 6 N and r 6 p-~(x) are arbitrary, it follows t h a t [Zi, Z3] -~ O, 1 <_ i, j <_n. W e t u r n to p r o p e r t y (5). I f [ E k , E el -- 0, 1 < k,~?< n, it is clear t h a t H i s integrable. To complete the proof of the theorem, we assume t h a t V is s y m m e t r i c
11.4. FRAME
BUNDLES
AND
THE
EQUATIONS
OF
STRUCTURE
367
and prove that the integrability of H implies that [E k, E e] - O. Indeed,
0 = r i ( E k, E e) = dOi(E k, E e) + co~ A OJ(E k, E e) = Ek(0i(E~))
- Ee(0~(Ek))
-
O~([Ek, E q )
: -o~([z k, Ee]) This implies that [E k, E e] E P(V), hence, when H is integrable, that this bracket vanishes identically, 1 _< k, g < n. [] E x e r c i s e 11.4.20. Deduce Theorem 11.4.1 From Theorem 11.4.19.
APPENDIX A
C o n s t r u c t i o n of the Universal Covering In this brief appendix, we give the construction of universal covering spaces, proving Theorem 1.7.29. The idea is thae the path-lifting property of covering spaces suggests a way to form a covering space that has the path-lifting property tautologically. This is somehow the most natural covering space and the universal property will itself be a tautology. F i x a path-connected, locally simply connected, pointed space (X, zo) and let {2(X, x0) denote the set of paths a : [0,1] -~ X with a(O) = x0. Let ~" = {P(X, xo)/ "5, the set of homotopy classes [a] rood the endpoints of paths cr E 9~(X, z0). Let z0 = [z0] and define
p: (2,~0) ~ (x,~0) by p([cr]) = or(l). We will put a topology on 3~ relative to which p becomes a covering map. Since X is locally simply connected, there is a base N of the topology of X consisting of simply connected open sets. For each z C X, each basic neighborhood U C ~B of z, and each [~] E p - l ( z ) , define
@1 = {[71 ~ 2 1 7 = ~ . ~ where im a c U and c~(0) = z}. Call such subsets of J( basic neighborhoods and let ~ be the family of all basic neighborhoods. L e m m a A . 1 . ~ is the base of a topoloq 9 on X .
Pro@ We only need to show that the intersection of two basic neighborhoods is the union of basic neighborhoods. Accordingly, let
and write
[7] : [~. ~],
[7]-
[~'. ~'],
where a(O) = a(1), a'(O) = a'(1), i m a C U and i m a ' C V. Let W C_ U V / V be a simply connected neighborhood of 7(1). It is then evident that
w-M c
n
Since [7] was an arbitrary point of U[~] N ~ , 1 , it follows that this intersection is a union of elements of ~ .
[]
As yet, we have not used the fact that ~ consists of simply connected neighborhoods. This is required for the following.
370
A. C O N S T R U C T I O N OF T H E U N I V E R S A L C O V E R I N G
L e m m a A . 2 . The function p : X ~ X is a covering map. Proof. If U E 113, it is clear that p - l ( U ) is the union of all basic neighborhoods of the form U[~], where a E [P(X, x0) ranges over all paths with a(1) E U. Thus, p is continuous. We claim that, for arbitrary 5[ol, the map p : U[o] --~ U is bijective. Indeed, surjectivity is trivial (X is path-connected), so we prove injectivity. Suppose c~ a n d / 3 are paths in U originating at a(1) such that p([o.
~])
= p([~/3]).
Thus, a and/3 are paths in U with the same initial points and the same terminal points, so the simple connectivity of U implies that a ~ a / 3 and [a. a] = [G./3]. We have proven that each element 5[~] of ~ is carried by p one-to-one onto an element U of N. So p is an open map and p[U[o] is a homeomorphism onto U. It only remains for us to show that
5[~] n 5[.] # ~ ~ 5i~ ] = 5 M Indeed, for arbitrary [a 97] E 5[~1, we will show that this element belongs also to 5[r The reverse inclusion follows by the same proof, proving equality of sets. By assumption, there are paths a and/3 in U such that
~(0) : a(1), /3(0) = T(1), [~. ~l : p./3]. The last equality implies that a(1) = /3(1), and so we can define a p a t h 7 ~ = (/3. a -1) "7. Then T" 71 ~D (T" /3)" (Of -1 " 7) ~ 0 (0'' OL)" (O! - 1 " 7) ~ a G "7. T h u s [G. 7] = [ r . 7'] E Uid.
[]
Examining this proof, the reader should see t h a t the requirement t h a t X be locally simply connected can be weakened to"semi-locally simply connected" (cf. the remark following Theorem 1.7.29). Observe that the path-lifting property, valid for all covering spaces, is transparent for this one. Indeed, if a E T(X, x0) and t E [0, 1], define Gt(s) = G(ts),
O < S < I.
Then ~(t) = [Gt], 0 < t < 1, defines a continuous path ~ in )(, issuing from ~0, and p(~(t)) =p([at]) = G ( t ) ,
O < t < 1.
This is what we meant by the introductory remark that p : 9~ ~ X "has the path-lifting property tautologically". Similarly, since this lift of cr terminates at the point ~(1) = [a], and since [a] is an arbitrary point of )(, we see t h a t X is path-connected. Finally, if a is a loop at x0, ~ will be a loop at x0 if and only if [a] = Ix0]. Thus, suppose 7 : [0, 1] --, X is a loop at ~0. Then p o 7 is a loop in X at x0 and 7 is, by definition, the lift of p o 7. By the preceding remarks, p o 7 ~ o xo and the homotopy lifting property then implies that 7 ~ a 20. These remarks prove the following. L e r n m a A . 3 . The space ~2 is simply connected.
A. C O N S T R U C T I O N
OF THE UNIVERSAL
COVERING
371
The following completes the proof of Theorem 1.7.29. T h e o r e m A.4. The function p : X -+ X is a universal covering map. Furthermore, a covering space ~ : X --+ X is isomorphic to the universal covering if and only if X is simply connected. Proof. Let ~ : X --~ X be a covering space with .~ connected. Choose a basepoint ~0 E ~-l(x0). We define (the unique) map f making the following diagram commutative:
2
X
~
,
X
If [G] E )(, we let ~ denote the unique lift of the path a to a path in J~ starting at x0. We then define f([cr]) = 8(1), a function such that ~'o f = p. As V ranges over ~B, the connected components o f ~ - l ( v ) range over a base ~ for the topology of X. Evidently, if V is a component of ~-1 (V), where V C 5, the components of f - 1 (p) are among the components of p - l ( V ) and it follows that f is continuous. Thus, p : )( --+ X is the universal cover. If ~ is also universal, the uniqueness theorem (Lemma 1.7.26), together with Lemma A.3, implies that )~ is simply connected. Conversely, suppose that X is simply connected. By Exercise 1.7.24, f is a covering map and, by Exercise 1.7.28, X will be evenly covered by f. Since X is pathconnected, it follows that f is a homeomorphism, proving that ~ : X + X is also the universal cover. []
APPENDIX
B
The Inverse Function Theorem T h e following simple l e m m a will be used in the proof of T h e o r e m 2.4.1 and in t h a t of T h e o r e m 2.8.4. B . 1 ( C o n t r a c t i o n m a p p i n g lemma). Let X be a complete metric space with metric p and let T : X --~ X be a mapping. If there is a constant c E (0, 1) such that p ( T ( x ) , T ( y ) ) <_ cp(x,y), Vx, y E X, then T has a unique fixed point xo E X . Furthermore, for each x E X ,
Lemma
lim Tn(x) = xo. n~oo
Proof. Let x E X and remark t h a t 1 + c + c ~ + . . . .
C < oc, so
k
fl(T n (X), T n+k (x) ) ~ e n E P(Ti-1 (x), T i (x) ) < cnVp(x, T ( x ) ) , i=1 implying t h a t {T~(x)}~=l is a Cauchy sequence. Our hypothesis also implies t h a t T is continuous. By completeness, set x0 = lim T~(x). n---*oc
Then,
T(xo) = T( lim Tn(x)) n~oo = lim T ( T n ( x ) ) (by continuity) n--*oo
= lira T n + l ( x ) X0.
Since T fixes x0 and strictly reduces distances, the fact t h a t x0 is the only fixed point is clear, as is the remaining assertion of the lemma. [] D e f i n i t i o n B . 2 . A m a p p i n g T : X --~ X , as above, is called a contraction mapping. A n o t h e r useful tool in the proof of the value t h e o r e m from multivariable calculus. ~5 : U --~ V a m a p of class C k for some 1 < t)p}o
~(q) - d)(p) =
(/01
inverse function t h e o r e m is the m e a n Let U and V be open subsets of N n, k < oo. If the line segment {tq + (1 value t h e o r e m asserts t h a t
JO2(tq + (1 - t)p) dt
)
9 (q - p).
Turning to the proof of T h e o r e m 2.4.1, we let q) : U --~ V be a C k m a p between open subsets of R n and assume t h a t J~(p) is nonsingular, for some p E U. We
374
B. I N V E R S E F U N C T I O N T H E O R E M
must find an open neighborhood Wp of p in U that 4) carries Ck-diffeomorphically onto an open neighborhood of 0(p). By appropriate changes of coordinates, we lose no generality in assuming that p = 0 = (I)(p) and that J~(O) = In. Thus, the associated mapping ko : U -* ~ n defined by 9 (x) = x - ~(x),
satisfies 9(0) = 0 and J ~ ( 0 ) -- 0. Finally, for each positive real number T/, let By denote the closed ball in R n of radius U and centered at 0. Since J ~ ( x ) is continuous in x, we can choose ~/ > 0 so small that J O ( x ) is nonsingular, Vx E Bv. We fix this condition and, in fact, the following. L e m m a B.3. There is a value of ~ > 0 so small that, if x l , x 2 C Bn, then ]l~(xz)
- ~(x2)ll
_< Ilxl - ~2]]/2,
IIo(~l)
- ~(~)ll
> IIxx - x~l]/2.
Proof. Since 9 is of class C 1 and Yk0(0) = 0, an application of equation (B.1) to the mapping ko implies that, for r / > 0 sufficiently small, I]~(zi)
- ~(z~){]
< I]~1 - ~1]/2.
Since ~(x) = x - kO(x), it follows that H~(x~)
- ~(~2)11
> IIz~ - ~21] - I]~(xl)
- ~(x~)l]
_> IlXl - x211/2.
[] C o r o l l a r y B.4. For each y E Bn/2, there is a unique x C Bv such that ~2(x) = y.
Proof. Define Ty on B~ by T~(z) = y + ~ ( z ) . By the first inequality in Lemma B.3, it is clear that Ty(B~) C_ Bv. By this same inequality, ]lTy(xi) - Ty(x2)[] = I 1 ~ ( ~ ) - O(x~)ll < I1~ - ~ 1 1 / 2 , so Ty is a contraction mapping on the complete metric space B n. Let x E Bn be the unique fixed point and remark that
= G(x) = y + ~(x) = y + x - ~(x) is satisfied if and only if ~(x) = y.
[]
Let Z = int(Bv/2) and W = (I)-I(Z). These are open neighborhoods of 0 in V and U, respectively. C o r o l l a r y B.5. The mapping (]) : W --~ Z is a homeomorphism.
Proof. We have shown that (I) maps W one-to-one onto Z, so it remains to be proven that (I)-1 is continuous. But the equations (I)(xl) = Yl and (I)(x2) = Y2 and the second inequality in Lemma B.3 imply that II~-lCyl)
- o-l(y2)ll
--
IJxi
-
x211 < 211yi - y211,
proving the assertion.
[]
L e m m a B.6. The map q~-lis differentiable at each point of Z and
J(e-~l = ( ~ ) - ~
o e-1.
B. I N V E R S E F U N C T I O N T H E O R E M
375
Proof. Let b = ~5(a) E Z, a E W. By differentiability at a, we can write d)(x) - c)(a) = Jep(a) . (x - a) + IIx - all~(x,a), where
lim ~'(x, a) = 0.
x~a
Since Jq,(a) is nonsingular, we can write x-
a = Jd2(a) - 1 . (,5(x) - 62(a)) - H x - al]J~(a) -1 9 ~'(x,a).
W r i t i n g x = q)-I (y), we o b t a i n ap-I(y) - o2-I(b) = Je2(a) - I . (y - b) - ] I x = J~(a)
-I
9 (y - b) -
allJ~(a) -1. g(x,a)
IlY - bll~(y,
b),
where 2(y, b) -
]lxlly- - ~
J4i'(a)-l " ~(x, a).
B u t y --+ b if a n d only if x --~ a and we have the inequality IIx - al_______~] < 2 lly-
bll -
by L e m m a B.3. T h a t is,
lim ~'(y, b) = 0
y~b
and ~5-1 is differentiable at b E Z with J 4 p - l ( b ) = JqS(a) -1 = Jq~(qs-l(b)) -1. Since b C Z is arbitrary, all assertions follow.
[]
C o r o l l a r y B . 7 . The m a p (p-1 is of class C k on Z. Proof. Since 9 is of class C k, the entries in the m a t r i x ( j ~ ) - i are functions of class C k-1 a n d Corollary B.5, together with L e m m a B.6, implies that J((I)-1) is continuous. T h a t is, ~ - 1 is of class C 1. If k = 1, we are done. Otherwise, feeding this new fact back into L e m m a B.6 implies t h a t ~ - 1 is of class C 2. C o n t i n u i n g in this way (forever, if k = oc), we complete the proof. [] We have proven the C k version of T h e o r e m 2.4.1, 1 < k < oo. R e m a r k . It is not hard to a d a p t the above proof to work for m a p p i n g s F:U-*F, where U _C E is open a n d E, F are Banach spaces over N. One says t h a t F is differentiable at p E U if there exists a b o u n d e d linear t r a n s f o r m a t i o n J F ( p ) : E --* F (the Jacobian of F at p) such that lira F(~)
- F(p)
- JF(p)
9 ( x - p ) = 0.
As usual, one shows t h a t such a linear t r a n s f o r m a t i o n is mfique a n d t h a t its existence implies the continuity of F at p. If this condition holds for all p E U, we obtain a map J F : U --* L ( E , F ) ,
376
B. INVERSE
FUNCTION
THEOREM
where L(E, F) denotes the Banach space of bounded linear transformations from E to F. If the map J F is continuous, we say that F is of class C 1 on U. As usual, one obtains the chain rule J ( F o G) = J F o J G for C 1 functions, as well as the fact that a bounded linear transformation is its own Jacobian. Inductively, one defines F to be of class C k on U, k > 1, if J F is defined and of class C k-1 on U. If F is of class C k on U, Vk > 1, then F is of class C ~~ on U. If F is a C k mapping of U, one-to-one onto an open subset V C_ F, k >_ 1, and if F -1 : V -~ U is also of class C k, then F is said to be a C k diffeomorphism of U onto V. In our proof of the inverse function theorem, we chose r] > 0 so small that JiP(x) is invertible, Vx E Bv. The usual determinant argument for this is unavailable in infinite dimensions, but it remains elementary that the subset of elements in L(E, F) with bounded inverses is open (cf. [24, pp. 71-72]). Finally, the mean value theorem (equation (B.1)) is completely elementary for general Banach spaces (cf. [24, p. 107]), so the proof that we have given for the finite dimensional case of the inverse function theorem goes through unchanged. T h e o r e m B.8. Let F and E be Banach spaces, U c_ E an open subset, and let F : U ---* F be of class C k on U, 1 < k < oo. I f p E U and J F ( p ) is an isomorphism of Banach spaces, then there is an open neighborhood W of p in U that is carried C k diffeomorphically by F onto an open neighborhood F ( W ) of F(p) in F. The Jacobian o f f -1 at F(x) is the inverse of J F ( x ) , V x E W . Here, of course, by an "isomorphism of Banach spaces" we mean a bounded linear transformation with bounded inverse. The final statement of Theorem B.8 is just the equation J ( F -1) = ( J F ) -~ o F -~ (Lemma B.6). There is a corresponding version of the implicit function theorem (Corollary 2.4.11) for Banach spaces. This will give a remarkably elegant way of proving the smooth dependence on initial conditions in the fundamental theorem of O.D.E. In order to state this implicit function theorem, we will need some notation. Let E, F, and H be Banach spaces, U C_ E and V C F open subsets, and let F : U x V --~ H be of class C k on U x V, 1 < k < oo. Denoting the variables in U and V by x and y, respectively, let (p, q) E U x V and form the "partial Jacobian" JvF(p, q) E L(F, H) (respectively, J~F(p, q) E L(E, H)) by holding x (respectively, y) fixed and treating F as a function of the remaining variable. T h e o r e m B.9. Let F : U x V --+ H be of class C k as above, let (p, q) E U x V, F(p, q) = c E H, and assume that JyF(p, q) : F ---* H is an isomorphism of Banach spaces. Then there exists an open neighborhood W of p in U and a unique C k map : W ---* V such that ~(p) = q and, on W , p ( x , ~(x)) - c.
Proof. Let G:UxV~E• be defined by the formula C ( x , y) = (x, r ( x , y)),
B. I N V E R S E F U N C T I O N T H E O R E M
a r
map
377
with Jacobian
This is an isomorphism of the Banach space E x F onto E x H, so the inverse function theorem provides an open neighborhood of (p, q) in U x V that is carried by G diffeomorphically onto an open neighborhood of (p,c) in E x H. Since the Banach space E x F has the Cartesian product topology, this neighborhood of (p, q) can be taken to be of the form W x W ~, where W is an open neighborhood of p and W ~ an open neighborhood of q. On G ( W x W~), the inverse transformation has a formula
a-l(x,z) = (x,H(x,z)) for a unique C k map H : G ( W • W ~) ~ Wq formula ~(x) = U(x, c). Indeed,
Thus, the desired map 9~ has the
(x, F(x, ~(x))) = G(x, (p(x) ) = C(x, H(x, c) ) = (x, c). Since G is a diffeomorphism, ~ is unique.
[]
APPENDIX C
Ordinary
Differential
Equations
We prove T h e o r e m 2.8.4. R e m a r k t h a t the general system (time d e p e n d e n t with parameters z = ( z l , . . . ,z r) E V C_ R ~)
dx i d-T = f f ( t ' z ' x l ( t ' z ) " ' " x ~ ( t ' z ) ) '
l < i < n,
- ~ < t < e,
on a n open subset U C ~ , can be viewed as an a u t o n o m o u s system without parameters on the open subset ( - ~ , r • V • U C R ~+~+1, by adjoining the equations
dt dt dz ] dt
O,
dz r DzO.
dt
Consequently, we formulate the proof for the a u t o n o m o u s case w i t h o u t parameters on an open subset U C_ ~ :
(*)
dXid-7= f f ( x l ( t ) " " . ,xn(t)),
1 < i < n.
We will assume t h a t 1 < k < oc and t h a t f i E Ck(U), 1 < i < n, a n d prove t h a t the solution defines a local flow of class C k. This will involve an i n d u c t i o n on k in which the r e m a r k in the previous paragraph becomes crucial. No generality will be lost by t a k i n g U = ]~n and assuming t h a t the vector field X = ( f l , . . . , f ~ ) is compactly supported. Indeed, we are proving a local theorem near x0 C U, so X can be d a m p e d off to 0 outside of a relatively compact region W C W C U, containing a given closed ball By(x0), t h e n extended by 0 to all of 1 ~ . I n this way, we will be considering a complete vector field X on ]I~n a n d will find a uniform p a r a m e t e r interval ( - c , c) on which the solution curves are defined for all choices of initial condition x E R ~. Restricting to x C B~(xo) and t a k i n g c > 0 smaller, if necessary, we see t h a t integral curves to X , s t a r t i n g in B~(xo) a n d p a r a m e t r i z e d on ( - c , c), must stay in the region W where X has not been altered.
C.1. Existence and uniqueness of solutions Since we will n o t be t h i n k i n g of the vector field X as a differential operator, we will write X ( x ) for Xx. A curve s : ( - 5 , c) -~ R n is integral to X if a n d only if
s(t) = s(0) +
f
x(s(u)) du, -~ < t <
~.
380
C. ORDINARY DIFFERENTIAL EQUATIONS
This formula suggests a mapping of a certain complete metric space into itself which will turn out to be a contraction mapping (Definition B.2). Let K be a Lipschitz constant for X. This exists because X is C 1 and compactly supported. Let 0 < c < 1/K. In what follows, E ( R ~) will denote the Banach space of all continuous paths
s: [ - c , c ] - ~ , with the sup norm. L e m m a C . l . 1 . For each a ~ R ~, the transformation
T~ : E ( R ~) -~ E(R~), T~(s)(t)=a+
X(s(u))du,
-c
is a contraction mapping. Pro@ If Sl, s2 C E(IRn), then ][T~(sl)(t) -
Ta(s2)(t)]]
=
.~at(x(81(u))
_< c sup --c
<_ c K
- X ( 8 2 ( u ) ) du
IIX(s~(~))-X(s2(~))ll
sup
IlSl(U)-S2(u)ll.
--c
T h a t is, I]Ta(Sl) - Ta(s2)[I ~ cKI]sl - 8~11 and 0 < c K < 1, so Ta is a contraction mapping.
[]
By the contraction mapping lemma, it follows t h a t there is a unique curve s C E(IR n) with T~(s) = s. As remarked above, this says, equivalently, that there is a unique solution s(t) = ( x l ( t ) , . . . , x n ( t ) ) to (*), parametrized on [-e,c] and having initial condition 8(0) = a. L e m m a C.1.2. The solution curve s : I - e , c] --* IRn is of class C k+l.
Proof. Indeed, the equation 8(t) = a +
/0
X ( s ( ~ ) ) du,
together with the fact that X is C k, implies that, if s is of class C j, some 0 _< j _< k, then s is actually of class C j+l. But s E E ( R n) is of class C ~ by definition, hence induction on j gives the assertion. []
Remark. If the vector field X is only required to be Lipschitz, the above argument goes through to provide a unique C 1 solution parametrized on I - c , c]. We have not quite proven the uniqueness of solutions as formulated in Theorem 2.8.4. Suppose that s~ and s2 are two solutions, parametrized on respective closed, nondegenerate intervals J1 and J2 about 0, and both satisfying the initial condition 81(0) = s2(0) = a. Let J C_ Yl n J2 be the largest closed, nondegencrate subinterval about 0 on which sl = s2. By what has just been proven, J is not empty. If J 7~ J1 N J2, then one endpoint ~ of J lies in both J1 and J2 and = 81(7") = 82(T ). Then cri(t) = si(w + t) is a solution of (*) with ~ri(0) = ~,
C.1. E X I S T E N C E
AND UNIQUENESS
OF SOLUTIONS
381
i = 1, 2, so it follows from what has just been proven that Sl and s2 agree on a larger subinterval than J after all. T h a t is, J = gl A J2 and we have proven the existence and uniqueness part of Theorem 2.8.4. Define : [-c, c] x ~
-~ X n
such that, for each x E ]Rn and - c < t < c, ~ ( t , x ) describes the unique integral curve to X with ~(0, x) = x. Denote this integral curve by sx and define 7) : ]~n .__. E(]Rn)
by ~(x) = s~. L e m m a C.1.3. The map ~ admits a Lipschitz constant B . That is,
II~(x) - ~(y)ll -<
BID
-
YII,
V x , y E ]Rn.
In particular, 7) is continuous. Proof. Let x, y E W and remark that
llsx -
=
Ty(sx)ll
IlZx(s~)
- Z~(sx)ll
=
IIx
- yll
Using notation established above, set c = c K c (0, 1) and write q
q
I I s ~ - zq(s~)ll _< ~-2~[[ry3-e(sx)- TJ(sz)l[ _< j=l
~-2~eJ-allz-yII. j=l
Since T~(s~) ~ sy in E(II~n) as q --~ oo and oo
--2ej - 1 = B < o% j=l
[]
the assertion is established. C o r o l l a r y C.1.4. The map 9 : (-c,c)
x Rn
+ll~n
is a local U ~ flow.
We emphasize that, because of our simplifying assumption that X is compactly supported, the parameter interval ( - c , c) is uniform for all initial values x E II~n. The proof of Lemma 4.1.10 is applicable, therefore, and gives C o r o l l a r y C.1.5. The compactly supported vector field X 9enerates a unique C o ]tow ~ : I R x l I ~ n ~ R n. The hardest part of the proof of Theorem 2.8.4 is to show that this flow is of class C k. It turns out that, if we can prove it to be C 1, a rather ingenious recursive argument yields an inductive proof that 9 is of class C k. In order to prove that 9 is C 1, we will verify the hypotheses of the implicit function theorem (Theorem B.9) for the map F : ~n • E(Rn) _~ E(R~), defined by F(x,
s) = z -
s +
/0
X
o s.
382
C. ORDINARY DIFFERENTIAL EQUATIONS
This notation is understood to define a continuous curve in IRn by the formula F(x, s)(t) = x - s(t) +
f
X(s(u))
du,
-c
< t < c.
One has F ( x , s) = 0 if and only if s = sx is the integral curve to X with initial value sx(O) = x. The implicit function theorem will guarantee that the map ~(x) = sx, satisfying F ( x , ~p(x)) = O, is of class C 1 (in fact, of class Ck). It is not obvious that this implies C 1 smoothness for the flow q~ itself, but this will be the case. Before giving the details, we make a small digression.
C.2. A digression
concerning
Banach
spaces
Let G : F --* H be a C k map of Banach spaces, 1 < k < oe, and let E ( F ) and E ( H ) denote the associated Banach spaces of continuous paths, parametrized on a fixed, bounded, nondegenerate interval I-c, c]. We define a map F : E ( F ) --~ E ( H ) by the formula F(s)
=
/0
a o s.
We are going to prove that F is also of class C k. First, remark that
f0
*JV~ 8 eE(L(F,H))
can be interpreted as an element of L ( E ( F ) , E ( H ) ) , E ( F ) , then riot J G ( s ( u ) ) 9a(u) du E H,
V s C E ( F ) . Indeed, if r E
-c < t
c,
so f o J G o s can be viewed as a transformation sending cr C E ( F ) to
fo
*(JV o 8)'(7 9 E(H).
This is evidently a linear transformation of E ( F ) into E ( H ) and it is bounded because J G ( s ( u ) ) is uniformly bounded, - c < u < c. L e m m a C.2.1. The map F is of class at least C 1 and, as s ranges over E ( F ) , JF(s) =
I
J G o s.
Proof. Let so 9 E ( F ) . Then lira
F ( s ) - F(so) - f o ( J G o so)" (s - so)
,~o
lis - soil s~solimv~* G o s - G o s0][s_70~[( JG o so) " (s - so) = 0
since the integrand converges to 0 uniformly on I-c, c]. Thus, f o J G o so satisfies the definition of J F ( s o ) , V s0 9 E ( F ) . To see that J F is continuous, write
IIJf(Sl)-Jf(s2)ll
=
f0
*(JGOSl-Jao
~2)
< cllJaOSl-
Jao
s21l
C.3. SMOOTH
DEPENDENCE
383
and a p p e a l to the continuity of JG.
[]
Suppose, now, t h a t it has been proven t h a t F is of class C r, some 1 _< r < k, and t h a t J r F ( s ) = f o J " G o s, Vs 9 E ( F ) . T h e Iemma gives the case r = 1 and it also provides the inductive step: J r F is of class C 1 and
Jr+iF(s) =
/0
Jr'+lGos,
Vs 9
Corollary C.2.2. If G : F --~ H is of class C k, 1 < k < oc, then F is also of class C k and JR(s) =
/o
J a o s,
Vs 9 E(F)
C.3. S m o o t h d e p e n d e n c e on initial c o n d i t i o n s We r e t u r n to the m a p F of ]I~n x E ( R n) into E ( ~ ~) given by the fornmla F(x,s)
= x-
s+
2 Xos
F r o m the previous section, this has the same smoothness class C k as the vector field X and
J s F ( x , s) = - idE(~,~) +
I*
J X o s.
Let NJXII = M . For the following lemma, we take c > 0 smaller, if necessary, so t h a t c < 1 / M . L e m m a C . 3 . 1 . For each (x, s) E IRn x E ( R n ) , the bounded linear operator J s F ( x , s) has a bounded inverse.
P r o @ Indeed, for a r b i t r a r y ~r E E(IRn),
fo tJx(s(u)).~(~)
du
< eMil
- e < t < c.
It follows t h a t the o p e r a t o r L = f o J X o s has n o r m [[Lt[< c M < 1, hence t h a t oo
R = E(-1)J+IL j=o
j
converges. T h a t is, R E L ( E ( R n ) , E(IRn)) and R o (L - idE(~,~)) = ( L -
idE(~,~)) o R = idE(x-)
as asserted.
[]
Thus, the hypothesis of T h e o r e m B.9 is verified.
Corollary C . 3 . 2 . The map g) : IRn --~ E(IRn), defined by qo(x) = s~, is smooth of class C k. In fact, we only need to know t h a t ~ is of class C I.
Corollary c . a . a .
The 91obal flow ~ on ]Rn is smooth of class at least C 1.
384
C. O R D I N A R Y D I F F E R E N T I A L
EQUATIONS
Proof. By Corollary C.3.2, ~ is smooth of class at least C 1, so ~(x + h) - p(x) = J~(x) . h + Ilhll~(h), where lim 5(h) = 0 in E(]R~).
h~0
T h a t is, lim (~(h)(t) = 0 uniformly on [-c, c].
h--~0
Thus, from
r
+ h) - r
= ~(x + h)(t) - v ( x ) ( t )
= ( J ~ ( x ) . h)(t) + ]lhllS(h)(t) we deduce t h a t the partial Jacobian
JxO(t,x) . h = ( J~o(x) . h)(t) exists and, for each h C ]Rn, is continuous in (t, x). By successively s u b s t i t u t i n g h = ei, 1 < i < n, we conclude t h a t all entries of the m a t r i x JxO2(t, x) are continuous functions of (t, x) E [ - c , c] • ]Rn. Since the flow lines are integral to X , we also see that 0r _ X(r
Ot is continuous in (t,x). Thus, all entries of Jd2(t,x) are continuous on [-c,c] x ]I~n a n d 9 is of class C 1 there. T h e p a r a m e t e r interval [-c, c] of this local C 1 flow being uniform for all initial values x C R n, we o b t a i n the unique global C 1 flow a5 : N i x IR~ ~ R ~ generated by X.
[]
Remark. In order to prove L e m m a C.3.1, we had to allow c to be chosen small enough. Our final conclusion, however, was t h a t the global flow ~5 is C 1. We are going to prove inductively that, for 1 <_ q _< k, 9 is of class C q (the case q = 1 is Corollary C.3.3). At each step of the induction, c may have to be chosen smaller. Nonetheless, at each step the conclusion is a b o u t the global flow, so the i n d u c t i o n works even for the case k = oo. Let 2 < j < k a n d assume that it has been shown t h a t the flow (p(t,x) is of class C j - 1 . We will show t h a t cg~(t, x)/Ot a n d J ~ ( t , x) are b o t h of class C j-1 in (t, x), concluding t h a t ~5(t, x) is of class C j. Lemma
C.3.4.
The expression Or
is o/ class C j-1 in (t,x).
Pro@ Indeed, or a n d X is of class C k, if) of class Lemma
C.3.5.
x)/Ot = x ( r C j-1 ,
x))
and j _< k.
[]
The expression JxO~(t, x) is of class C j-1 in (t, x).
Pro@ Since we know that (I) is of class at least C 1, we can differentiate r
= x +
/0
x(r
d~
C.4. T H E L I N E A R C A S E
385
under the integral sign with respect to the variables x, obtaining JzcP(t,x) = ide,~ +
JX(d2(u,x))Jzd2(u,x)du.
Then, S~ Jx(VP(t,x)) = J X ( V P ( t , x ) ) J z ( g ; ( t , x ) ) , and this can be interpreted as a time dependent system of O.D.E. with parameters x. The unknown functions are the entries of the matrix Jx(ff(t,x)), and so the system is linear. The coefficients are entries of the matrix J X ( g p ( t , x ) ) , hence are of class C J-1 in (t, x). Thus, this system is of class C j - 1 . As remarked at the beginning of this appendix, time dependent systems with parameters are equivalent to autonomous systems without parameters, so the inductive hypothesis guarantees t h a t the entries of Jx(ep(t, x)) are smooth of class C j - 1 . [] The proof of Theorem 2.8.4 is complete. Remark. It would have been possible to formulate and prove the O.D.E. theorem entirely in the context of Banach spaces (see, e.g., Lang [24, pp. 132-145], who credits the idea of using the implicit function theorem to Pugh and Robbin), but we have avoided significant technical details by resisting the temptation to do so. On the other hand, judicious use of the implicit function theorem in infinite dimensions does seem to simplify the proof of the finite dimensional theorem. C.4. T h e L i n e a r C a s e
In the proof of Theorem 10.1.13, we appealed to the theorem t h a t a linear system of O.D.E. has solutions that are defined on the largest parameter interval (b, c) on which the system itself is defined. Here is the formal theorem. T h e o r e m C.4.1. Let A : (b,c) --* 9Jr(n) be smooth of class C k. Then the system
~(t) = A(t) . s(t),
b < t < c,
with initial condition s(to) = a has solution s(t) defined for b < t < e. Proof. Let (b',c') C_ (b,c) be the maximal open interval about to on which the solution s(t) is defined. If c' < c, we deduce a contradiction as follows. Fix a basis { w l , . . . , Wn} of N n and, for 1 < i < n, let ai(t) be the unique solution of (Ti(t) = A(t) . ai(t), ~(e')
= ~,i,
defined on some interval (c' - e, c' + ~) c_ (b, c). Choose t, E (c' - c, c'), so close to c' t h a t { a l ( t . ) , . . . , ~n(t.)} is also a basis of N n. This is possible by the continuity of the solutions ai. Thus, there are constants c l , . . . , c~ E R such that
fi c~i(t,) = a. i=1
By the linearity of the system, it is clear that n
or(t) = E i=1
cicri, Ct - ~ < t < C; + ~,
386
C. O R D I N A R Y
DIFFERENTIAL
EQUATIONS
is also a solution and t h a t a ( t . ) = s ( t . ) . By uniqueness of solutions, ~ and s agree on (c' - c, c'), so a extends the solution s to the larger interval (b', c' + e). This contradicts the m a x i m a l i t y of (b', c'), proving t h a t c' = c. Similarly, b' = b. []
APPENDIX D
The de Rham Cohomology
Theorem
In this appendix, our goal is to prove the de Rham theorem for Cech and singular cohomology. In order to avoid any possible confusions between cohomology theories, we will denote the de Rham cohomology by H~)R(M ). Our proof will show that the graded algebra structures are isomorphic. In particular, this will show that the de Rham cohomology algebra H~)R(M) is a purely topological invariant of the manifold M (Theorem 8.9.6). D.1. (~ech
cohomology
This section consists largely of definitions and statements of basic properties. Few proofs will be given because they are mostly routine (and tedious) computations. The first step is to define the cohomology of an open cover ~/ = {U~}~e~ of the space X with coefficients in a commutative ring R with unity. D e f i n i t i o n D . I . 1 . If p > 0 is an integer, a (Cech) p-simplex of l[ is an ordered (p + 1)-tuple (U~o, U~I,... , U~p) of elements of U such that U~o A ... n U~p is nonempty. D e f i n i t i o n D.1.2. If p > 0 is an integer, an R-valued (Cech) p-cochain on ~/is a function ~ that, to each p-simplex (U~0, U~I,... , U~.), assigns an element w(U,o,U~l,...
, u ~ , ) = w , 0 , 1 .... . ~ R.
The set of all R-valued p-cochains on ~/is denoted by CP(I[; R). Evidently, the operations of "simplexwise addition" of cochains and "simplexwise scalar multiplication" make C'B(I[; R) into an R-module. More precisely,
(~+r
....p = ~ 0 ~
.... ~ + r
.... ~
and (a~) . . . . ...~p = a(~ . . . . ...ap), Ya E R and V ~ , r C CP(~[; R). D e f i n i t i o n D.1.3. The ((~eeh) coboundary operator
is the R-linear map defined by the formula p+l
i=0
388
D. DE
RHAM
THEOREM
Thus, we obtain a sequence r
~ dP+~(11;R) •
dP+2(11;R) ~ . . . .
The following is a routine computation. L e m m a D.1.4. The sequence of coboundary operators satisfies 52 = O. D e f i n i t i o n D.1.5. The module of ((~ech) p-cocycles is 2P(U; R) = ker(5) N C'P(11; R) and the module of (Cech) p-coboundaries is /)P(11; R) = im(5) A alP(U; R).
As usual,/)P(ll; R) C_2~(11; R). D e f i n i t i o n D.1.6. The pth (2ech cohomology of 11, with coefficients in the ring R, is /:/~(11; R) =/)P(U; R)/2P(U; R). We have defined /2/* (11; R) as a graded R-module. It is made into a graded algebra by the "cup product". D e f i n i t i o n D.1.7. If ~ E C'P(11;R) and r E Cq(ll;R), the cup product ~or E 6"P+q(11; R) of these cochains is defined by (~r
= ~ o .... p C ~ p . . . ~ + q .
This makes C*(ll; R) into a graded algebra. Another straightforward computation proves the following. L e m m a D.1.8. / f ~ E OP(~/.;R) and r E 0q(~/.;R), then ~(~r
= (6~)r + (-1)P~(5r
This is formally the same as the formula for the exterior derivative of the wedge product of a p-form and a q-form. As in that case, we get the following consequence. L e m m a D.1.9. The graded module Z*(~I; R) of Cech cocycles is a graded subalgebra of C*(~; R) and B*(~A;R) C_ 2* (]g; R) is a 2-sided ideal. Consequently, cup
product is well defined on/2/*(ll; R), making that graded R-module into a graded algebra over R. Finally, we will define the Cech cohomology algebra of the space X to be the "direct limit" /2/* (X; R) = li__.m/2/.(11; R) over finer and finer open covers. We make this precise. Let { V * } ~ be a family of graded R-algebras, indexed by a partially ordered set 92. That is, there is a partial ordering c~ ~ ~ on ~ such that, whenever a, ~ E 92, 3 3' E 91 with a -21_3' and ~ _~ ~/. Assume also that, whenever c~ _~/3, there is given a homomorphism ~ : V* --* V~ of graded algebras and that, whenever ~ -~ ~ ~ 3", then ~o~ o ~a = ~ ' * We say that {V2, ~}~,~ is a directed system of graded R-algebras.
D.1. C E C H C O H O M O L O G Y
389
On the disjoint union
v*= Hvd, c~e~
define the equivalence relation ~ generated by ~
where v E Vc~ and c~ ~ ft. Then V * / ~ has a natural graded R-algebra structure. Indeed, scalar multiplication a[v] = [av] is clearly well defined. As for addition, if [v], [w] E V * / ~ are represented by v c Vd and w E V~, find 7 c 92, c~ ~ V, fl -< 7, and set Iv] + [w] = [ G ( v ) + ~7~(w)]. It is trivial to check that this is well defined and that these operations make V * / ~ into a graded R-module. Similarly, the algebra multiplication passes to a well defined multiplication making V*/~ into a graded R-algebra. D e f i n i t i o n D . I . 1 0 . In the above situation, we set lim Vd = V*/~ and call this graded R-algebra the direct limit of the directed system of graded R-algebras. E x a m p l e D . I . l l . For a differentiable manifold M, let ~lx denote the set of open neighborhoods U of x E M. This is a directed system under the partial order U _~ V ev U _D V. The graded algebras {A*(U)}ueux form a directed system under the restriction homomorphisms
= lV, where a~ E A*(U) and U _D V. Then = lira A * ( U )
is just the graded algebra of germs at x C M of smooth forms. Let D ( X ) denote the set of open covers of X. This is partially ordered by: ~1 ~ V e$ V is a refinement of ~. Since any two open covers of X have a common refinement, this makes O ( X ) into a directed system. If U = {U~}~ev~, V = {Vfl}Ze~ , and ~1 _~ V, then there is a choice function i : ~3 --+ 92 such that VZ C_ Ui(~), Vfl C lB. This induces a homomorphism i # : O * ( ~ ; R ) --* C'*(V; R) of graded algebras, where i # (~)Zo&..-G = ~(Zo)~(;h)..4(~,J" The following is trivial. Lemma D.1.12. ~ o i #.
The homomorphism i # : 0"(~1; R) --~ C*(V; R) satisfies i # o (~ =
We cannot use i # as a homomorphism 9~v C'* (L[; R) C* (V; R) for a directed system of algebras. The problem is that i # depends on arbitrary choices, so we could never guarantee that ~ w o ~vu= ~wu. But the above lemma implies t h a t i # induces i* :/:/* (]~; R) --~/:/* (V; R) and it turns out that, at the level of cohomology, the arbitrariness disappears.
390
D. DE
Definition D.I.13. If11 ~ V in above, and if p E Z, define
RHAM
D(X),
THEOREM
ifi, j : ~ --* 9/are two choice functions, as
S : CP(U; R) --~ C p - I ( v ; R) by the formula p-1
~=0
As usual, if p - 1 < 0, we understand that CP-I(1L;R) = 0 and S = 0. The following is checked by a routine (if somewhat tedious) computation, left to the reader. L e m m a D.1.14. If i, j, and S are as above, then
S o 5 + 6 o S = j # - i #.
Consequently, i* = j* :/:/* (ll; R) --*/:/* (V; R). By this lemma, whenever ~A__ V in D(X), we define a homomorphism ~
= i * : / r * ( U ; R) -~ H*(V; R)
of graded algebras that is independent of the (allowed) choice of i : fl~ --* 9/. L e m m a D.1.15. IfU -< V-~ W in D(X), then ~oW oqo~
~w
Pro@ Indeed, set u = {u~}~, V = {v~}~, w = {w~},~,
and let i : f13 --* 92, j : ~ --. ~ be suitable choice functions. Then i o j : E --~ 9/is an allowed choice function relative to the refinement 11 ~ W. But qow = ( i o j ) * = j * o i *
= ~ W o q o uv. []
v,
V
Thus, we get a directed system {H (11;R), ~u}u,ve~(z) of graded algebras over R. D e f i n i t i o n D . l . 1 6 . The Cech cohomology of the space X with coefficients in R is the direct limit = hm H (11;R), taken over the directed system D(X). Let f : X --* Y be a continuous map between spaces. Given 11 E D(Y), define f - l ( ~ l ) E D(X) by the usual pullback construction. If we are given a psimplex ( f - i ( U ~ 0 ) , . . . , f - i ( U a , ) ) of f - l ( l l ) , then it is clear that (U~o,... ,Uap) is a p-simplex of ~1. Consequently, each Cech cochain 0 E CP(11; R) has a natural pullback f#(O) E ~ p ( f - i (ll); R). This defines a homomorphism f # : C*(ll;R) ~ C * ( / - I ( l l ) ; R ) of graded algebras.
D.2. DE RHAM-(~ECHCOMPLEX LemmaD.l.17. If f : X --+ Y, as above, then f # o($ = 5 o f # canonically defined an induced homomorphism
391 and there is
f* :/:/*(Y; R) -+/:/*(X; R) of graded algebras over R. This makes Ceeh eohomology into a contravariant functor on the category of topological spaces and continuous maps.
D e f i n i t i o n D . l . 1 8 . If P/ is a directed system, a cofinal subsystem ~ _C 92 is a directed subsystem with the property that, whenever a C 9/, ~7 E ~ such that a_~7. Finally, the proof of the following lemma is a straightforward application of definitions9 L e m m a D.1.19. Let {Vo:,F~}~,~e~ be a directed system of graded R-algebras. If C_ P2 is a cofinal subset, then there is a canonical isomorphism hm V~ = hm V~ of graded R-algebras, where the first limit is taken over all c~ C P2 and the second is taken over all 7 c ~.
By Corollary 10.5.8, the family of simple covers (Definition 8.59 tiable manifold is a cofinal subset of D ( M ) .
of a differen-
C o r o l l a r y D.1.20. The Cech cohomology H * ( M ; R ) of a differentiable manifold can be computed by taking the limit only over the directed system of simple covers. D.2. T h e de R h a m - ( ~ e c h c o m p l e x The proof we will give of the de Rham-Cech theorem is essentially that of Andr@ Weft [48]. The main step is to prove the following. T h e o r e m D.2.1. If11 is a simple cover of M, there is a canonical isomorphism e u : / : / * ( U ; JR) --+ H~)R(M ) of graded algebras and, if II ~_ V, where both are simple covers, then the diagram II
/7/*(11;]I~) / ~ 1 " ~
. ~ / / ) ~ 2 ' /7/*(~; R)
HSR(M) is commutative.
The equivalence of de Rham theory and Cech theory follows easily. Indeed, the Cech cohomology can be computed by passing to the limit over the simple covers only (Corollary D.1.20), so the isomorphisms e u induce a well-defined homomorphism ~b:/:/*(M;]R) --+ H~)R(M ). The fact that each e u is an isomorphism implies the same for ~b.
392
D. D E R H A M T H E O R E M
T h e o r e m D . 2 . 2 (de Rham). There is a canonical isomorphism H~)R(M ) = / : / * ( M ; R )
of graded ]~-algebras. For use in the following section, we record the following corollary, implicit in the above argument. C o r o l l a r y D . 2 . 3 . If]1 is a simple cover, the natural homomorphism of/:/*(]1;N)
into the limit/:/*(M;N) is an isomorphism of graded algebras. In order to prove Theorem D.2.1, we will build an enormous cochain complex of graded algebras that includes both (A* (M), d) and (C* (]1; R), 5) as subcomplexes. If ]1 is simple, we will prove that the inclusions of these subcomplexes induce isomorphisms in cohomology. F i x an open cover ]1 = {U~}~e~ of M. For the following definitions, it is not necessary that ]1 be simple. D e f i n i t i o n D . 2 . 4 . A Cech p-cochain on ]1 with values in A q is a function qo that, to each p-simplex (U~o, U ~ I , . . . , U~p) of ]1 assigns
~c*oal .... v C Aq(Uao N Uc~l N . . . N Uav ). The set of all such cochains will be denoted EP'q(]1) = CP(~/; Aq). Although the coefficient ring Aq(u~ 0 A U~I A . . . N U~,) changes with each simplex, one can still add cochains simplexwise and multiply them by real scalars. These operations make EP'q(]1) = CP(I/; A q) into a real vector space. There is also a bigraded multiplication
EP,q(]1) | E',S(]1) -~ EP+~,q+s(]1). In defining this and other operations, we make the notation less bewildering to the eye by abusing it (the notation, that is). Whenever respective forms have been defined on respective open sets with common, nonempty intersection, addition of such forms and exterior products of such forms are understood to be defined on their common domain. For instance, if (Uao,U~,Ua2) is a 2-simplex of ]1 and COalo~~ E Aq(Uai CI Uo~j), then
a;c~1o~2-- Wc~oa24- a;o~oo~l C Aq(U,~o N Uoq N Uo~2). Similarly, if ~oaoa~ C Aq(U~o N U~,~) and r
C As(Uc,~ M U,~2), then
W i t h these conventions understood, we define the bigraded multiplication as follows. If ~ C EP'q(]1) and r C E~'~(]1), then ~ r 9 EV+~,q+~(]1) is defined on a (p + r)-simplex (U~o,... , U ~ , , . . . , U~,+~) by (~r
.... ,+~ = ( - 1 ) q ~ , o .... p Ar
.... ,+~ 9 Aq+s(U~o n . . . n U ~ + ~ ) .
Often we suppress explicit reference to the simplex on which this formula is being evaluated and write
~r = ( - 1 ) q ~
A
r
We say t h a t E** (~) = {E p'q (~A)}~,~=0 is a bigraded algebra under this multiplication. Note how this operation combines the cup product from Cech theory with the exterior multiplication from de Rham theory. The strange sign ( - 1 ) q~ will be needed in the proof of Lemma D.2.8.
D.2. D E R H A M - ~ E C H
COMPLEX
D e f i n i t i o n D . 2 . 5 . The de Rham operator e : EP'q('~) setting
(c~) . . . .
= (-1)Pd(~aoal...a,) E Aq+l(Uao
...c~,,
393 ----*
n
EP'q§
Ual
is defined by
n . . . lq
for arbitrary ~ C CP(II; A q) and for every p-simplex (U~o, U ~ , . . .
Uc~r,),
, U~,) of II.
D e f i n i t i o n D . 2 . 6 . The Cech operator 0 : EP'q(ll) --* EP+~'q(~) is defined by setting p+l
(0~:')~o~
.... ,+,
= E(-1)i(Fo~o...a,
.... p+, C A q ( U ~ o
n U,~, n . . .
r~ U , ~ p + , ) ,
i=0
V~ C (~'P(~[; A q) and for every p-simplex (Uao, U ~ , , . . . , U ~ ) of II. The following are evident: 9 E2=0, 9
02 =
9
~o0=-0oe.
O,
Remark t h a t the sign ( - 1 ) p in the definition of e is responsible for the anticommutativity of e and 0.
/~0,2(~[)
5
, E1,2(~
E0,~(lt)
~ , Eu(lt)
Eo,o(~)
~
, E1,~
)
5
~
, E2,2(~ )
5
, E2,1(~)
5 ) 9
, E2,~
5
) --.
F i g u r e D . 2 . 1 . The de Rham-(~ech complex
D e f i n i t i o n D . 2 . 7 . For each integer rn >_ O, Em(~i) = (~p+q=m EP'q(~'i) and the total differential operator D : Em(~l) ----,Em+l(ll) is D = c + 5. It is a good idea to picture E**(~i) laid out as a first quadrant array in the (p,q)-plane, having EP'q(~I) at the point (p,q) of the integer lattice as in Figure D.2.1, with the de R h a m operators e as vertical arrows and the Cech operators 5 as horizontal arrows. This array is called the de R h a m - C e c h complex. The total degree of an element ~ C EP'q(~i) is p + q and Em(~l) is spanned by the elements of total degree m. One can view E'~(~2) in this diagram as lying along the diagonal p + q = rn. If ~ E EP'q(~I), where p + q = m, then
D(~) : ~(~) + 0(~) c E~'~+I(U) 9 E~+I,~(u) c Em+~(u).
394
D. DE
RHAM
THEOREM
L e m m a D.2.8. The pair (E*(1/),D) is a cochain complex in which E*(1/) is a graded algebra over ]R and D(~r
= D(~)r + (-1)m~D(r
where ~ E Era(t[). Proof. Indeed, E*(1/) = {Em(1/)}~m=o is a graded vector space and it is clear that the bigraded multiplication in E** (11) induces a graded algebra structure on E* (1/). Because of the anticommutativity of r and 5, D 2 =r162
=0.
Finally, it is only necessary to verify the Leibnitz formula for ~p E EP'q(1/) and r E E r's (1/), p + q = m. Suppress reference to the (p + r)-simplex (U~o,... , Ua,+~) and compute
r162
= (-1)P+~d((-1)'q~ = (-1)P+r+rq(d(~)
A
r
A r -}- (--1)q~P A d(~)))
~- (--1)P+r+rq+P+(q+l)r(E((D)r
-~ (--1)P+r+rq+q+r+rq(~g(~)))
= ~(~)r + (-1)P+q~s(r That is, (D.1)
~(~r
= E(~)r + (-1)m~zs(r
Similarly, suppress reference to the (p + r + 1)-simplex (U~o,... ,Uap+,.+l ) and compute 5(~r
= (-1)ra(5(9~) A r + ( - 1 ) P ~ A 8(r = (-1)'q(-1)rqs(~)r
+
(-1)P+'q(-1)q(r+l)~5(r
= 5(~)r + ( - 1 ) ' + q ~ 5 ( r That is, (D.2)
5(Vr
= 5(~)r + (-1)ra~5(r
By adding equation (D.1) and equation (D.2), we obtain the desired Leibnitz rule for D. [] L e m m a D.2.9. There are canonical inclusions
(A*(M),d)
i
(E*(1/),D),
(C* (ll; R), 8) ~ (E* (1/), D)
of subcomplexes, respecting the graded algebra structures. Proof. Indeed, ifw 6 Aq(M), i(w) E ~0(1/; A q) = E0,q(1/) assigns to each 0-simplex (U~o) the element i(~J)~ o = wlU~ o. Since (5(i(~))).o~,
= ~ l g ~ o n u s , - wlg~o n u ~ l = 0,
we see that
n(i(w) ) = E(i(w) ) = i(dw). Likewise, if ~ E CP (1/; R), j (~) E CP (1/; A ~ assigns to each p-simplex (U~o,... , Us,) the O-form on the open set UaoN...NUa, that is the constant function ~ o . . . a , E ~.
D.2. D E R H A M - C E C H
COMPLEX
395
Clearly, e(~O)so...s, = O, and so j also commutes with the coboundary operators of these complexes. It is clear from these definitions t h a t
i(~ A ~?) = i(aJ)i(~), whenever w 9 AV(M) and r / 9 A~(M), and that
j(go%b) = j(~o)j(r whenever g) 9 OP(11; 1R) and %b9 O~(ll; ]R). Corollary D.2.10.
[]
There are canonical homomorphisms i * : H~)R(M ) --~ H*(E*(~I),D)
and
/*:/~*(u; R) -~ H* (E* (U), D) of graded algebras. We augment the rows of the diagram in Figure D.2.1 by i. T h a t is, the new rows are
Aq(M) & E~
a+ EI,q(U) A .
Similarly, we augment the columns by j :
Op(u;~) & Ep,~ Lemma D.2.11.
& E~,I(U) &
The augmented diagram has exact rows.
Proof. If w e Aq(M), we have seen in the proof of Lemma D.2.9 that 5(i(w)) = 0. Conversely, if ~ C d'~ A q) and 5(~) = 0, then the forms ~ o C Aq(U~o) and ~sl E Aq(u~I) must agree on Uso n U~I, if this intersection is nonempty. Hence, the forms p~ E Aq(u~) piece together smoothly to give a form co C Aq(M) such t h a t i(cz) = ~. This proves exactness at E~ We prove exactness at EP'q(~l) = d'P(ll; A q) when p > 1. Let {As}~ea be a smooth partition of unity subordinate to ~i. Define
as follows. Given ~ E OP(~I; A q) = EP'q(~I), define A(~) E 0P-l("d; A q) to be the element, the value of which on the (p - 1)-simplex (U~o,... , Us,, 1) is
a(~)oo...~_l : ~ ~s~sso .... .-1, s
where each term of this locally finite sum is interpreted, in the obvious way, as a q-form on Uso N . . . N U~,_I. If 5(qD) = O, the reader can check that 9~ = a(A(w)), proving exactness at EP'q(~). [] L e m m a D . 2 . 1 2 . If the cover ~i is simple, the augmented diagram has exact columns.
Proof. If ( E CP(ll; R), we have seen in the proof of Lemma D.2.9 t h a t e ( j ( ( ) ) = 0. Conversely, it is clear that, if ~ C CP(U, A ~ and c(~) = 0, then, d ( ~ o.... ,,) = 0, for each p-simplex ( U s o , . . . , Us,). The fact that U~ o n ... N U~,, is connected implies t h a t ~o.--s,, e A~ N . . . n Uo,) is constant. Thus, we can define ( C CP(~I;R) by (~o...s~, = the constant ~so...~, and j ( ( ) = ~. This proves exactness at E p'~
396
D. DE RHAM T H E O R E M
If q _> 1, exactness at F,P'q(~[) follows from the Poincar~ l e m m a a n d the fact t h a t U~ o N 9-. N U ~ , if not empty, has the de R h a m cohomology of ~ n . [] Lemma
D . 2 . 1 3 . If the cover ~1 is simple, the homomorphisms i* and j* are sur-
jective. Proof. Let ~ E E m ( l l ) be a D-cocycle. We show t h a t the element [~] E H*(E*(II), D) is in the image b o t h of i* and j*. If m = 0, t h e n ~ = 0 = c~ and the assertions follow from L e m m a s D.2.11 and D.2.12, respectively. Assume, therefore, t h a t m > 1. T h e cocycle ~ lies along the diagonal p + q = m, so we write
r = }-~r p=0
where (~ E E P ' m - ~ ( l l ) , 0 _< p _< m. The equation D(~) = 0 implies t h a t e(@) = 0, so L e m m a D.2.12 allows us to find {0 E E~ with e({0) = @. T h e n - D ( { 0 ) has 0 as its c o m p o n e n t in E~ and is D-cohomologous to ~. Suppose, inductively, t h a t {k E E m - l ( ~ ) has been found so t h a t ~ - D({k) has 0 as its c o m p o n e n t in EP'm-P(~[), 0 <_p <_ k. Since this is still a D-cocycle, exactness of the c o l u m n p = k + 1 allows us to find 0 E E k + l ' m - k - 2 ( R ) such t h a t ~ - D({k) - D(O) has 0 as its c o m p o n e n t in EP"~-P(~d), 0 _< p_< k + l . We take {k+l = { k + 0 . By finite induction, there is {m E E m-l(~) such t h a t ~ - D({m) is concentrated in Em'~ Thus, w i t h o u t loss of generality, we assume t h a t ~ E E m ' ~ = C m ( l l ; A~ T h e fact t h a t this is a D-cocycle implies t h a t e(~) = 0 = 5(~). Thus, there is c E C ' m ( ~ ; R ) such t h a t j(c) = ~ and j(5(c)) = 5(r = 0. Since j is one-to-one, 5(c) = 0, so [c] E/:/rn(ll; R) and if[c] = [~]. A n entirely parallel argument, using L e m m a D.2.11 (exactness of the rows), shows t h a t we can assume that ~ is concentrated in E~ a n d t h a t there is co E A'~(M) such t h a t dw = 0 and/*[co] = [r [] L e m m a D . 2 . 1 4 . If the cover ~ is simple, the homomorphisms i* and j* are in-
jective. Pro@ Suppose t h a t w E A'~(M) has dw = 0 a n d i*[w] = 0. I f r n = 0, t h e n w is a locally c o n s t a n t function a n d i(co) vanishes on every 0-simplex (U~o). T h a t If m >_ 1, t h e n i(co) E E~ is of the form is, co = 0, so [co] = 0 E H ~ i(co) = D(~), ~ E E m - l ( l l ) . Write m-1
~= ~ , p=0
where ~p E EP"~-l-P(ll), 0 _< p _< m - 1. Since the component of i(co) in E m ' ~ is 0, 5(~,~-1) = 0 and L e m m a D.2.11 implies t h a t ~m-1 = 6(0), 0 E E'~-2'~ Thus, ~' = ~ - D(O) has component 0 in E ' ~ - I ' ~ a n d D(~') = i(co). Again using L e m m a D.2.11 a n d finite induction, we see t h a t no generality is lost in assuming t h a t ~ is concentrated in E~ a n d t h a t 5(~) = 0. By one more appeal to L e m m a D.2.11, we find a unique r/ E A m - I ( M ) such t h a t i(r/) = (. T h e n , i(dr/) = e(i(r/)) = i(w) and, i being injective, w = dr/. T h a t is, [col = 0 as desired. A completely parallel argument, using L e m m a D.2.12, proves t h a t j* is injective. []
D.3. SINGULAR COHOMOLOGY
397
If the cover 11 is simple, we define Cu = (i.)-1 o j* :/:/*(l~; N) --, H ~ a ( M ) , an isomorphism of graded R-algebras by Lemmas D.2.13 and D.2.14. The following completes the proof of Theorem D.2.1. L e m m a D.2.15. If11 ~ V are simple covers of M, then the diagram ~oIIv
H~)R(M) is commutative. Proof. Indeed, if ~ = {U~}~e~ and V = {V~}ze ~ recall that Fv is induced by a choice function g : ~3 -+ 91 such that V/~ C_ U~(~), V~ E ~ . The same choice function g defines a homomorphism ~
: H* (E* (U), D) --+ H* (E* (V), D)
and the diagram H*(M)
i* , H*(E*(ll),D) ~ j"
~i
idl H;R(M)
/:/*(~A;R)
~~
i* , H * ( E * ( V ) , D ) ,
y"
commutes.
/:/*(V;N) []
Remark. In fact, the de Rham-Cech isomorphism r
N) --+ H ~ a ( M )
is functoriaL That is, whenever f : M -~ N is a smooth map between manifolds, the diagram
H*(N;N)
s"
, ZP(M;N)
H;R(N )
> H;R(M ) S* is commutative. That is, on smooth manifolds, Cech theory and de Rham theory are equivalent as functors. Checking this functoriality is straightforward.
D.3. Singular Cohomology We will define the graded singular cohomology algebra H* (M; N) and prove the following. T h e o r e m D.3.1. There is a canonical isomorphism /:/* (M; N) = H * ( M ; R ) of graded R-algebras.
398
D. DE
RHAM
THEOREM
We note that with little extra effort, the proof of this theorem can be carried out with N replaced by an arbitrary commutative ring with unity. It can also be generalized to a larger class of topological spaces than manifolds. The proof of Theorem D.3.1 will be analogous to that of Theorem D.2.2. Coupled with Theorem D.2.2, this will prove the de Rham theorem for singular cohomology. T h e o r e m D.3.2 (de Rham). There is a canonical isomorphism
H~)R(M) = H*(M;N) of graded N-algebras. Again, these isomorphisms are easily checked to be functorial. The singular cohomology should be defined via duality at the chain level. Recall (Definition 8.2.11) that Cp(M;R) denotes the real vector space with basis the set Ap(M) of smooth (respectively, continuous) singular p-simplices in M. This is called the space of singular p-chains. Set
C'(M; N) = Home(Cp(M; R), N), the space of singular p-eochains. The boundary operator 0 : Cp+,(M; N) --+ C,(M; N) has adjoint 0* : CP(M;N) --* Cv+I(M;N), called the singular coboundary operator and the identity 02 = 0 dualizes to 0 *2 = 0. In the standard fashion, this gives rise to the vector spaces Z* (M; N) and B* (M; N), called the spaces of singular cocycles and singular coboundaries, respectively. The singular cohomology theory is then the quotient
Z*(M;N)/B*(M;N). The above construction is functorial. That is, smooth (respectively, continuous) H*(M;N)
=
maps f : N --~ M induce graded, N-linear maps f# : C, (N; N) ~ C, (M; N), f # : C*(M;N) --* C*(N;N). These commute, respectively, with 0 and 0", hence pass to graded, R-linear maps, f, and f* on homology and cohomology, respectively. The usual functorial properties are satisfied, making singular homology into a covariant functor, singular cohomology into a contravariant functor. For homology, this is the content of Exercise 8.2.24. Dualizing this gives the corresponding result for cohomology. It remains that we define the graded algebra structure on H* (M; N). Multiplication in this algebra is called the singular cup product and will be denoted by a dot "." to distinguish it from the Cech cup product. For this, let n = p+q, p, q >_O, and consider the maps O'p : A p --~ A n , 6rq : Aq ~
/~n~
defined by
o,(xi,...,xp)
= (x~,...,xp,o,...,o),
~q(xl,...,xq)
= (0,...,0,x~,...,xq).
D.3. S I N G U L A R C O H O M O L O G Y
399
One calls O-p the front p-face operator and a q the back q-face operator. Given c C P ( M ; N ) and r C Cq(M;N), the cup product ~ . r C C n ( M ; N ) will be completely determined by its values on the set A n ( M ) of singular n-simplices on M. This is because this set is a basis for the vector space Ca(M; R). If s E A n ( M ) , define ~. r = ~(s o ~ ) r o ~q). A little combinatorics gives the following expected relation. L e m m a D.3.3. [f ~ E CP(]ll;]I~) and r E Cq(M;N), then 0"(~o. r
= (0*p). r + ( - 1 ) v ~ 9 (0"r
As usual, we get the following consequence. L e m m a D . 3 . 4 . The graded vector space Z*(M;]R) of singular cocycles is a graded subalgebra o f t * (M; ]R) and B* (M; JR) C Z* (M; ~) is a 2-sided ideal. Consequently, cup product is well defined on H*(M;I~), making that graded vector space into a
graded algebra over JR. Finally, if f : N --~ M is smooth, the induced map f* : H * ( M ; ~ ) -* H * ( N ; N )
is a homomorphism of graded algebras. Remark. In the above discussion, we allowed Aq(M) to be either the set of smooth singular simplices in M or the set of continuous ones. This yields two possibly different singular cohomologies, the smooth and the continuous. The proof that we will give of Theorem D.3.1 works equally well in either case, hence both theories, being canonically isomorphic to Cech cohomology, are identical. Also, since continuous maps between manifolds are homotopic to smooth ones (Subsection 3.8.B), the homotopy invariance of singular theory (not proven here, but cf. [13]) implies t h a t these theories are canonically isomorphic as functors. C o r o l l a r y D . 3 . 5 . The singular cohomology algebra, computed by using smooth
singular simplices, is canonically and functorialIy equivalent to that computed by using continuous singular simplices. Let ~ = {Us}ae~ be an open cover. We are going to mimic the construction of the de Rham-(~ech complex to produce a singular-Cech complex E**(~). Ultimately, we will need ~1 to be simple. The proof of Theorem D.3.1 will then proceed almost exactly as that of Theorem D.2.1. D e f i n i t i o n D . 3 . 6 . A (~ech p-cochain on 11 with values in C q is a function ~ that, to each p-simplex (U~o , U s ~ , . . . , Uo~,) of II assigns ~sos~ .... , ~ c q ( U s o n us~ n . . .
n
Us,;~).
The set of all such cochains will be denoted F,P'q(]~) = CP(~; cq). Once again, this is naturally a real vector space. We have simply replaced
A q(Uso (3 Us~ N . . . N Uo,,) with C q(Uso n Us~ N . . . n Us,,; R) in the earlier definition. Similarly, we get a bigraded multiplication on the resulting double complex and, setting c = ( - 1 ) P 0 * : EP+q(~) --+ E p'q+l and defining 5 in complete analogy with Definition D.2.6, we complete the definition of the double complex. The total differential is D = e + 5. The first significant difference between the current and former construction is t h a t there is no natural inclusion of (C*(M;R),O*) into (E*(11),D). If ~ C
400
D. D E R H A M T H E O R E M
Cq(M;~), go~ E ~1,, and if i~ : U~ ~-* M is the inclusion, then we can define the restriction of ~o to U~ by ~IU~ = i~(~). This defines a homomorphism i : (Cq(M; R), 0") -~ (C~
cq), ~),
i(~)~ = wlu~ of cochain complexes. However, it is quite possible that ~ ~ 0, but that ~IU~ = 0, for every ~ E 91, and so i will not be injective. The solution to this is the delicate subdivision process for singular homology (see commentary following Proposition 8.5.13) and cohomology that allows computation of these theories using only "11small" singular simplices (Definition 8.5.11). The sketch for homology accompanying Definitions 8.5.11 and 8.5.12 dualizes to a similar procedure for cohomology and the references for details are the same. We note that the definition of cup product also works for this ~[-small theory and record the following. T h e o r e m D.3.7. The inclusions AUq(M) ~-~ Aq(M), q >_ O, induce canonical isomorphisras of graded vector spaces H.U(M;N) = H . ( M ; R ) , H ~ ( M ; R ) = H*(M;]R).
In the case of cohomology, this is an isomorphism of graded algebras. One
now
notes that
i: (C~(M;N),O*) --~ (d~
Cq),e)
is injective. Lemma D.3.8.
There are canonical inclusions (Cu(M,N),O*) ~ (E*(U),D), (d'*(li;]R), 5) ~ (E*(II),D)
of subcomplexes, respecting the graded algebra structures. Proof. Indeed, for each Cech 1-simplex (Uao , U~I ) and each cochain ~o E C~(M; JR), we see that (~i(~))~o~1 = ~]V~o n U ~ - ~lV~o n V~, = 0 for the above definition of i. Thus, D(i(~)) = ~(i(~)) = i(0*~). For the definition of j, note that, for each Cech p-simplex, (U~o,...,U~,), C~ ~ ... N Uc~p) is just the set of arbitrary JR-valued functions on the open set U~ o N ... Cl U~p (the singular 0-simplices in a space are just the points). A Cech 0-cochain r E ~,0(~; JR) assigns to each p-simplex (U~o,... , U ~ ) an element r .... ~, E IR. We define j ( r E d'P(U; C ~ by letting J(r .... ~ denote the constant function on U~o N ... N Ua~ with value r .... p. Evidently, j ( r = 0 if and only if = 0, and so j is injective, By the definition of the singular coboundary operator, the coboundary of a constant 0-cochain vanishes, so
D(j(r
= 5(j(r
-- j(5r
D.3. SINGULAR COHOMOLOGY
401
The fact that i and j respect the graded algebra structures can be checked by the reader. [] Again we augment the rows of E** (11) by i, obtaining the sequence i C q ( M ; IR) c__+ E~
~
5 E l ' q ( 1 1 ) ---+.
,
and we augment the columns by j, obtaining the sequence Ov(11;R) & EP'~
& EP'l(11) ~ . .
L e m r n a D.3.9. If the cover 11 is locally finite (in particular, if II is simple) the
augmented diagram has ezact rows. Pro@ We have seen that a o i = 0. Conversely, if ~ E 6'~ see that, for each 1-simplex (U~o, U~I) ,
C q) and a~ = 0, we
~olS~o n g~l = ~ , l g ~ o n U~,, and so the ~b,s patch together to define ~b' E C~(M) such that i(~b') = ~b. This proves exactness at E ~ (11). We prove exactness at E;'q(~) = CP(11; C q) when p _> 1. For this, we construct A : Ev,q(11) -~ SP-1'q(11), p > 1, such that ~
= 0 ~
~ = ~A(~).
Since a 2 = 0, this will prove the lemma. Let ~ : AqU(M) --, Z + be defined by ~(a) = cardinality of {Ua E 11[ o(~Xq) C Ua}. The hypothesis that 11 is locally finite guarantees that ~(a) is finite. If U~ E 1-{ and (Uao,..., U~,_,) is a Cech (p - 1)-simplex, define
ia o.... ~-~ : c ~ ( e ~ n U ~ o n . . . n u ~ , _ , ) ~ c ~ ( U ~ o e .
.nV~,_,)
by i~o .... v-*(~9)(~)= ~ ( a ) '
ifcr(Aq) C U a n U a ~
nUa, ....
otherwise.
to Finally, for each ~o E EP'q(~A), define 1(~) E Ep-1'q(11) by a(~)<,,o...<~,,_,
,.,1 E z a a ~ ~, ,(VV~ao...~,,_l)" aE~
A direct computation proves the desired identity.
[]
L e m m a D.3.10. If the cover II is simple, the augmented diagram has ezact columns.
Proof. We have seen that e o j = 0. Conversely, if ~b E CP(11; C ~ and e(~b) = 0, the definition of the singular coboundary operator implies that, for each Cech p-simplex (U~0,... , U ~ ) , ~b~o...~, is constant on each component of U~o n . . . n U ~ . Since the cover is simple, this open set is connected and ~b~0...~, is constant. That is, ~bE imj. Finally, exactness at Ev'q(lI), q > 1, is proven by dualizing the construction in Example 8.2.17, proving that the singular cohomology of a contractible space is the same as that of a point. Since the cover is simple, the open set U~o N 9 n U~, is contractible, for each Cech p-simplex (U~o,..., U~,), and the desired exactness follows. []
402
D. D E R H A M T H E O R E M
At this point, the arguments of the previous section can be reproduced, practically word for word, to prove the following. T h e o r e m D.3.11. If lI is a simple cover, there is a canonical isomorphism /:/*(ll;lt~) = H~(M;~) of graded algebras.
By Corollary D.2.3,/:/* (ll; ~) =/:/* (M; R) and, by Theorem D.3.7, H~ (M; ]I{) = H*(M; R), completing the proof of Theorem D.3.1, hence of Theorem D.3.2. Remark. The advantage of this proof of the de Rham theorem is that it demonstrates the equivalence, at the level of cohomology, of the singular cup product with exterior multiplication of forms. The disadvantage is that it completely disguises the fact that the equivalence of cohomology classes of forms with those of singular cocyeles is achieved by integration of forms on (smooth) singular cycles.
Bibliography [1] F. Adams, Vector fields on spheres, Ann. Math. 75 (1962), 603-632. [2] L. Auslander and R. E. MacKenzie, Introduction to Differentiable Manifolds, McGraw-Hill, New York, NY, 1963. [3] W. Boothby, An Introduction to Differentiable Manifolds and Differential Geometry, Academic Press, New York, NY, 1975. [4] R. Bott and J. Milnor, On the paraUelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89. [5] R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York, NY, 1982. [6] A. Candel and L. Conlon, Theory of foliations, i., American Mathematical Society, Providence, RI, 1999. [7] S. K. Donaldson, An application of gauge theory to the topology of 4-manifolds, J. Diff. Geo. 18 (1983), 269 316. [8] J. Dugundji, Topology, Allyn and Bacon, Boston, MA, 1966. [9] B. Eckmann, Gruppentheoretischer Beweis des Satzes yon Hurwitz-Radon iiber die Komposition quadratischer Formen, Comm. Math. Helv. 15 (1942), 358-366. [10] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, N J, 1952. [11] D. Freed and K. Uhlenbach, Instantons and Four-Manifolds, Springer-Verlag, New York, NY, 1984. [12] R. Gompf, Three exotic ~4 ~S and other anomalies, J. Diff. Geo. 18 (1983), 317-328. [131 M. Greenberg and H. Harper, Lectures on Algebraic Topology, The Benjamin Cummings Publishing Co., Reading, MA, 1981. [14] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, NY, 1962. [15] N. J. Hicks, Notes on Differential Geometry, D. Van Nostrand, New York, NY, 1965. [16] M. W. Hirsch, Differential Topology, Springer-Verlag, New York, NY, 1976. [17] D. Hoffman and W. H. Meeks III, A complete, embedded minimal surface with genus one, three ends and finite total curvature, J. Diff. Geo. 21 (1985), 109 127. [18] _ _ , Embedded minimal surfaces of finite topology, Ann. Math. 131 (1990), 1 34. [19] _ _ , Minimal surfaces based on the catenoid, Amer. Math Monthly 97 (1990), 702-730. [20] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, N J, 1948. [21] M. Kervaire and J. Milnor, Groups ofhomotopy spheres, I, Ann. Math. 77 (1963), 505-537. [22] S. Kobayashi and K. Nomizu, The Foundations of Differential Geometry, I, Wiley (Interscience), New York, NY, 1963. [23] , The Foundations of Differential Geometry, II, Wiley (Interscience), New York, NY, 1969. [24] S. Lang, Real Analysis, 2nd edition, Addison-Wesley, Reading, MA, 1983. [25] E. L. Lima, Commuting vector fields on S 3, Ann. Math. 81 (1965), 70-81. [26] W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace, and World, Inc., New York, NY, 1967. [27] J. Milnor, Some consequences of a theorem of Bott, Ann. Math. 68 (1958), 444-449. [28] _ _ , Morse Theory, Princeton University Press, Princeton, N J, 1963. [29] _ _ , Problem list, Seattle Topology Conference, 1963. [30] , Topology from a Differentiable Viewpoint, University of Virginia Press, VA, 1965. [31] E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, New York, NY, 1977.
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BIBLIOGRAPHY
[32] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. Math. 40 (1939), 400-416. [33] W. F. Newns and A. G. Walker, Tangent planes to a differentiable manifold, J. London Math. Soc. 31 (1956), 400-407. [34] B. O'Neill, Elementary Differential Geometry, Academic Press, New York, NY, 1966. [35] H. Poincar@, Analysis situs, Journal de l']~cole Polytechnique 1 (1895), 1-121. [36] T. Rad6, Ober den Begriff der Riemannsche Fldche, Acta Univ. Szeged 2 (1924-26), 101-121. [37] G. Reeb, Sur Certaines Propridtds Topologiques des Varidtds Feuilletdes, Hermann, Paris, 1952. [38] S. Smale, Generalized Poincard's conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391-406. [39] E. Spanier, Algebraic Topology, McGraw-Hill, Inc., New York, NY, 1966. [40] M. Spivak, Differential Geometry, Volume II, Publish or Perish Publishing Co., Boston, MA, 1970. [41] , Differential Geometry, Volume I, Second Edition, Publish or Perish, Inc., Houston, Texas, 1979. [42] J. Stallings, The piecewise linear structure of euclidean space, Proc. Camb. Phil. Soc. 58 (1962), 481-488. [43] N. Steenrod, The Topology of Fiber Bundles, Princeton University Press, Princeton, N J, 1951. [44] J. J. Stoker, Differential Geometry, Wiley (Interscience), New York, NY, 1969. [45] D. Tischler, On fibering certain foliated manifolds over S 1, Topology 9 (1970), 153-154. [46] J. W. Vick, Homology Theory, Springer-Verlag, New York, NY, 1982. [47] S. Halperin W. Greub and R. Vanstone, Connections, Curvature and Cohomology, Volume II, Academic Press, New York, NY, 1972. [48] G. Wallet, NuUitd de l'invariant de Godbillon-Vey d'un tore, C. R. Acad Sci. Paris, Sdrie A 283 (1976), 821-823. [49] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, NY, 1983. [50] H. Whitney, The self intersections of a smooth n-manifold in 2n-space, Ann. Math. 45 (1944), 220-246.
Index 1-form, see also form
A*(M), Ak(M), AI(M), A*(M), AP(M),
244 237 183, 187 237 244
action diagonal, 348 left, 178 right, 179 simply transitive, 347 smooth o f ~ k, 146 of a Lie group, 178 ad, 178 Ad(a), 178 Adams, F., 6 ad(9), 178 adjoint, 247, 255 group, see also group of a linear map, 183
ad(X), 178 algebra, 46 associative, 46 bigraded, 392 commutative, 46 exterior, 217-226, 237 definition of, 219 of a free R-module, 220 graded, 215 anticommutative, 220 connected, 216 with unity, 216 Grassmann, 237 Lie, 161-178 abelian, 167 of a Lie group, 164 of linear endomorphisms, 178 ~[(n), 164 9~(n), 164 o(n), 164 si(n), 164 5u(n), 164 u(n), 164 multilinear, 209-237 symmetric, 226-227, 237
definition of, 226 tensor, 209-217 contravariant, 237 covariant, 236 definition of, 216 algebraic intersection number, 280 algebraic topology, see also topology almost every, 80 analytic group structure, 174 angle between curves, 101 between tangent vectors, 101, 102, 312 annihilator, 289 annulus, 8 atlas C ~ , 87 differentiable or smooth, 87 H n-, 107 maximal C ~176 , 88 maximal smooth, 88 Aut(9), 178 automorphism of covering space, 27 autonomous system, see also differential equations
B*(M), BI(M), BP(M), Up(M),
244 195 242 255
t)p(u),
285 t)p(u; R), 388 Banach spaces, 375 base space, see also bundle basepoint, 33 basis, 209 cardinality of, 210 Betti number, 287 bigraded algebra, see also algebra multiplication, 392 bilinear map, see also map Borel measure, 313 Bott, R., 5 boundary of a 1-manifold, 26
406 of a smooth manifold, 107 of a topological manifold, 23 operator, see also operator point, 23 singular, 255 Boundary theorem, 121 cOis, 251 cOs, 251 bracket f-related, 144 Lie, 105, 142-145 Brouwer fixed point theorem, 39, 122, 199 Brouwer, L. E. J., 2 bundle associated to a principal bundle, 354 base space, 4, 61, 96 classification, 99 dual, 183-185 frame, 347-351 orthonormal, 352 isomorphism, 96 k-plane subbundle, 103 line, 97 locally trivial, 353 MSbius, 97 normal, 114, 305 of a foliation, 293 n-plane, 95 orientable, 101 principal, 98, 347-356 general definition, 351 reduction, 355 section of, 351 trivialization, 352 with discrete structure group, 353 projection, 4, 61, 96 section(s), 69, 97 module of, 230-237 tangent, 50, 94-97 base space, 95 complex, 104 holomorphic, 104 of S '~, 4 of a submanifold, 61 of manifold with boundary, 110 projection, 95 total space, 95 total space, 4, 61, 96 trivial, 62, 96, 97 unit tangent, 114 vector, 95 multilinear theory, 227-230 of exterior algebras, 229 of graded algebras, 229 of linear homomorphisms, 229 of symmetric algebras, 229 trivialization, 96 W h i t n e y sum, 230
INDEX
C,a(M), 287 (C.U(M), cO), 270 Cp(M), 255
0F(u), 284
CP(1/; Aq), 392
C~(M), 269 r R), 387 canonical, 15, 230 coframe field, 362 Cartan structure equations, 337-342 lifted to frame bundles, 357-367 Cartesian product local, 95 of smooth structures, 88 category, 34 morphisms of, 34 objects of, 34 of R-modules, 219 of graded algebras, 219 of groups, 34 of pointed spaces, 33 cell complex, 141 center of a Lie group, 178 chain complex, see also complex homotopy, see also homotopy singular, 255 chain rule, 50, 109 general, 52 global, 92 matrix form, 52 charts C~176 87 coordinate, 87 H n-, 107 C~ 107 smoothly compatible, 87 t~ech coboundary, see also coboundary coboundary operator, see also coboundary cochain, see also cochain cocycle, see also cocycle cohomology, see also cohomology operator, see also operator simplex, see also simplex Christoffel symbols, 306 Coo(M), 90 C ~ atlas, see also atlas C ~ ( V , p ) , 43 C C ~ ( M , N ) , 116 C~ N), 116 class differentiability, 41 equivalence, 15 smoothness, 41, 42 Clebsch, 144, 152 Closed subgroup theorem, 173
INDEX cobordant, 24 cobordism, 24 coboundary Cech, 285, 388 de R h a m , 195, 242 operator, 247 Cech, 284, 387 singular, 398 singular, 398 cochain Cech, 284, 387 with values in A q, 392 with values in C q, 399 complex, see also complex homotopy, see also homotopy cocycle, 98-104, 284 (~ech, 285, 302, 388 condition, 285 conditions, 89 de Rham, 195, 242 equivalence, 99, 100 G-, 100 Jacobian, 99, 100 property, 98 singular, 398 structure, 89, 98 Codazzi-Mainardi equation, 320 codimension of a foliation, 154 coframe field, s e e also field(s) cohomology Cech, 285, 387-391 algebra, 284 class, 195 integral, 200 de Rham, 122, 183, 239 288 as a contravariant functor, 243 definition of, 242 first, 195-202 first (definition), 195 functoriality of, 242 graded algebra structure, 243 with compact support, 244 integral, 200 of a contractible manifold, 260 of Lie algebras, 292-293 of spheres, 269 of tori, 293 simplicial, 287 singular, 397-402 l~-small, 400 top dimensional, 271-274 of manifolds with boundary, 274 of noncompact manifolds without boundary, 273 of nonorientable manifolds, 273 of oriented manifolds without boundary, 272 commutator product, 178
407 commuting flows, see also flow(s) vector fields, see also field(s) complex chain, 264 homology of, 265 singular, 264 cochain, 264 cohomology of, 265 de Rham, 264 de Rham-Cech, 391-397 definition of, 393 singular-(~ech, 399-402 component of the identity, 161 connected semi-locally simply, 35 simply, 31, 32, 38 connected sum, 9 connection, 304-311 definition of, 305 Euclidean, 305 existence, 311 flat, 364 form, 339, 362 globally flat, 31I Levi-Civita, 303, 306, 313 existence and uniqueness, 314 on a principal bundle, 363 pullback, 315 Riemannian, 313 symmetric, 306, 364 torsion free, 306, 364 conservation of energy, 191 constant rank, see also map constant rank of a submodule of A 1 (M), 293 Constant rank theorem, 56 global, 112 contractible manifold, see also manifold contraction mapping, 373 Contraction mapping lemma , 373
convex hull, 12, 250 set, 249 coordinate chart, 87 coordinates, 42 differentiable or smooth change, 87 normal, 336 cotangent space, see also space
vector, s e e also vector covector, 183-208 field, see also field(s) cover locally trivializing, 96 refined, 18 simple, 268, 287 simple refinement, 334
408
INDEX
covering group, 27 map, see also m a p regular, 30, 353 space, see also space transformation, 27 universal, 33 construction, 369-371 critical points, 80-85, 93 definition of, 80 nondegenerate, 83-85 definition of, 83 critical values, 80-85, 93 definition of, 80 cup product, 284, 388 curvature, 101,303 form, 339, 364 Gauss, 315-322 infinitesimal spread of normal field, 319 mean, 318 operator, 320 principal, 318 R i e m a n n tensor, 102, 320 tensor, 337 tensor (of a general connection), 321 curve evenly parametrized, 323 infinitesimal, 44, 92 as a derivative, 47 integral, 71 piecewise regular, 325 regular, 325 space filling, 80 cycle, 254 singular, 255 D(s)p, 46 ~D(C~(M)),
~p(M), 255 ~(M), 269
105
vdt 307 V 328 ~-7, de R h a m cohomology, see also cohomology isomorphism of graded algebras, 284 map, 281 operator, see also operator theorem, 257, 281-288, 387-402 for singular cohomology, 398 for Cech cohomology, 392 de R h a m - C e c h cohomology theorem, 391-397 complex, 391-397 definition of, 393 Deahna, 144, 152 deck transformation, see also transformation deformation retract, see also retract deg, 203, 274
deg2, 121, 204 degree local, 274 of a m a p of oriented manifolds, 274 degree theory modulo 2, 119-124 definition of deg 2, 121 on S 1, 202-208 definition of, 203 equivalent definitions of, 203 on oriented manifolds, 274-276 derivation of a Lie algebra, 178 derivations, 105 Lie algebra of, 105 derivative, 41, 46 covariant, 307 partial, 328 uniqueness and existence, 307 directional, 43 exterior, 183, 189, 239-244 axioms for, 240 coordinate-free formula, 292 existence and uniqueness of, 241 naturality of, 189 Lie, 76, 142 of a form, 189 of germ algebra, 91, 108 operator, 46 determinant, 53, 60 Diff(M), 116 Diffc(M), 116 diffeomorphic structures, see also structures diffeomorphism, 52, 62 between manifolds, 91 compactly supported, 116 on arbitrary subsets, 106 oriented, 101 differentiable manifold, see also manifold differential, 50, 109 as a cotangent vector, 186 form, 122 graded ideal, see also graded ideal of a linear map, 52 of a map of manifolds, 91 differential equations existence and uniqueness of solutions, 379-383 ordinary, 379-386 ordinary (O.D.E.), 71 partial (P.D.E.), 151 smooth dependence of solutions on initial conditions, 383-385 system autonomous, 73 existence and uniqueness of solutions, 72 linear, 309, 385-386 ordinary, 71
INDEX without parameters, 73 dimension local, 1 of a foliation, 154 of a locally Euclidean space, 3 of a smooth (differentiable) manifold, 88 of a topological manifold, 3 of an open subset of R n, 52 theory, 234, 236 direr (for free R-modules), 210 direct product, 244 sum, 244 directed system, 389 cofinal subsystem, 391 of algebras, 388 limit, 388, 389 directional derivative, see also derivative distribution Frobenius, 142 horizontal, 363 integrable, 144, 150 involutive, 142 k-plane, 103, 142 integrable, 103 Donaldson, S., 93, 347 double dual, 185 DR, 257, 281 dual R-module, 214 bundle, see also bundle free R-module, 214 vector space, 183 dynamical system, 158 limit set, 158 minimal set, 158 Einstein summation convention, 308 equations of structure, 337 342 lifted to frame bundles, 357-367 equivalence class, 15 F-, 153 germinal, 45 homotopy, 117, 196 infinitesimal, 44 of atlases, 87 of points under a k-flow, 147 relation, 15 equivariant map, see also map essential map, see also map Euclidean connection, see also connection Euler characteristic, 14, 288 evaluation map, see also map evenly covered, 26 exact sequence, 264 266 long, 266 short, 264 exactness of dual sequence, 266
409 exp, 166 exponential map, see also map expx, 325 exterior algebra, see also algebra derivative, see also derivative differentiation, 189 multiplication, 183 power, 218 decomposable, 218 indecomposable, 218 of a bundle, 229 extrinsic property, see also property face operator, see also operator field(s) coframe, 339, 362 commuting as coordinate fields, 145 covector, 183, 187 vector, 95 along a curve, 306 commuting, 78, 142 complete, 131-136 f-related, 144 left-invariant, 70, 163 on S n, 5 on a submanifold of ~n, 62 on manifolds with boundary, 110 parallel, 308 piecewise smooth, 309 restriction to submanifold, 145 smooth, 65 71 velocity, 71 vertical, 358 Five Lemma, 264 flow(s), 131 136 a-limit set, 136 commuting, 78, 145-150 global, 79, 132 commuting, 142 gradient, 136-141 infinitesimal generator, 132 invariant set of, 135 k-, 146 nonsingular, 147 line, 131 linear on T 2, 132 local, 71-79 commuting, 142 definition of, 73 infinitesimal generator, 74, 131 maximal, 132 on a manifold, 131 minimal set, 136 existence, 136 w-limit set, 136 period, 133 periodic point of, 133
410 transverse, invariant for a foliation, 298 focal point, 125 foliation(s), 103, 144, 150-159, 289-302 algebraic topology of, 296 definition of, 154 integral to a distribution, 144 leaf of, 144, 154 pullback, 294 Reeb, 158 form 1-, 183-208 closed and nonsingular, 296-302 exact, 191 integral, 200 locally exact, 193 restriction of, 188 closed, 242, 254 compactly supported, 244 connection, 339, 362 curvature, 339, 364 exact, 242 k-, 237 locally exact, 242 torsion, 339,363 frame bundle, see also bundle manifold, see also manifold n - , 347 orthonormal, 182 tautological horizontal, 362 Freedman, M., 93 Frobenius, 152 chart, 152 distribution, see also distribution integrability condition, 152 theorem, 103, 144, 152, 183 differential forms version, 290 :3"-saturated set, 158 function C r , 42 C ~ , 42 differentiable at a point, 41 differentiable or smooth, 42 on an arbitrary subset of Rn, 54 constructions, 104 107 global theory, 87-129 local theory, 41 85 on arbitrary subsets, 106 on manifolds, 90 infinitely smooth, 42 locally constant, 243 Morse, 124-129, 136-141 definition of, 124 functor, 34 contravariant, 34, 184, 189 covariant, 34 functoriality, 183 of de Rham-singular isomorphism, 398
INDEX of de R h a m - C e c h isomorphism, 397 of singular-Cech isomorphism, 398 fundamental form first, 127 second, 127 group, see also group Fundamental theorem of algebra, 121,206 Fundamental theorem of calculus, 251 F ( V ) , 347 F(V) XGi(n ) ~ n , 348 F(E), 97 r ( f ) , 142 Gauss curvature, see also curvature equation, 320 map, see also m a p Gauss-Bonnet theorem, 14 F(E), 231 general linear group, see also group genus, 12 geodesic, 101,303, 323 completeness, 325 convexity, 334-337 existence and uniqueness, 324 smooth dependence on endpoints, 327 geodesically convex, 334 geometric property, see also property structure, see also structure geometry extrinsic, 127 locally Euclidean, 102 Riemannian, 20, 101 germ, 45, 91 algebra, 91 algebraic operations, 46 germinal equivalence, see also equivalence Gleason, A., 161 GI(~), 178 Gl(k, n - k), 103 G k ( Y ) , 225 Gl(n), 60 g[(n), 70 G l ( n , C), 103 Gl+(n), 100 Gn,k, 182
G~,k(C), 182 Godbillon-Vey class, 296 naturality, 296 gv(9"), 296 Gompf, R., 93 @p, 45, 91 graded algebra, see also algebra ideal, 217, 289 differential, 289
INDEX gradient in ~ n , 136 in Riemannian manifolds, 136 Grassman algebra, see also algebra Grassmann manifold, 182 complex, 182 gravity, 103 G-reduction, 100 group abelian, 209 acting simply, 30 acting simply transitively, 30 adjoint, 178 complex general linear, 103, 162 covering, 27 fundamental, 36-40 general linear, 53 holonomy, 311 isometry, 342 isotropy, 179 Lie, 17, 161-178 abelian, 167 C*, 162 definition of, 161 Gl(k, n - k), 162 Gl(n), 161 Gl(n, C), 162 Gl+(n), 161 H*, 162 identity component of, 162 O(n), 161 S 1, 162 S 3, 162 Sl(n), 161
SO(n), 161 SU(n), 162
T n, 162 transformation, 178 U(n), 162 of Lie algebra automorphisms, 178 orthogonal, 61 permutation, 218 real analytic, 161, 174 special linear, 60 special orthogonal, 161 special unitary, 162 structure of a principal bundle, 351 reduction of, 100 topological definition of, 161 manifold, 173 transformation, 178 unitary, 162 H.a (M), 287
411
H*(M), 2S4 H}L(M), 270 /=/* (X; R), 390 Hi(M), 195 HI(M;G), 100 Hi(M; Gl(n)), 99 H;(M), 244 H~)R(M), 387 HP(M), 242 Hi(M; Z), 200
Hp(M), 256 fir(u), 2s5 //p(u; R), 388 half-space, 22, 107 handle, 125, 138 attaching, 138 handlebody decomposition, 125 A(notation for omission of an entry), 146 Hessian, 83 matrix, 129 Hilbert's fifth problem, 161, 174 Hoffman, D., 318 holonomy, 309 holonomy group, see also group Homc~(M) ( 2~(M), COO(M)), 187 HOM(E, F), 231 Hom(E, F), 229 Homogeneity lemma, 117 proof of, 135 homogeneous, 342 polynomial, 227 space, see also space homology class, 256 of simplicial chain complex, 287 singular, 254 258 as a covariant fimctor, 257 definition of, 256 of IOn, 257 of a contractible manifold, 256, 257 Z-small, 270, 400 homomorphism connecting, 265 definition of, 266 in cohomology degree of, 265 induced in (co)homology, 265 in de R h a m cohomology, 243 in homology, 258 natural, 265 of (co)chain complexes, 265 of covering spaces, 28 of graded algebras, 216 of Lie groups, 167 continuous, 176 of vector bundles, 231 homotopy, 30
412 chain, 263 class, 30, 116 cochain, 263 continuous smoothly approximating, 119 differentiable or smooth, 116-119 alternative definition, 117 definition of, 116 equivalence, see also equivalence lifting property, 31 m o d C, 30 m o d the endpoints, 31 proper, 261 relative, 30 Hopf-Rinow theorems, 322, 326 horizontal distribution, see also distribution frame (tautological), see also frame lift, s e e also lift space, see also space subbundle, see also subbundle 9 ( x ( M ) , 311 identity component of a Lie group, 161 I(M), 342 imbedding proper, II0, 112 topological, 20-22 definition of, 20 immersion, 57, 112 one-to-one, 113 topological, 20-22 definition of, 20 Immersiontheorem, 58 Implicit function theorem, 58 for Banach spaces, 376 induced map, s e e also m a p infinitesimal curve, see also curve equivalence, s e e also equivalence generator, s e e also flow(s) structure, see also structure inner product indefinite, 102 positive definite, 102 integrability, 144 integrability condition, 151 integrable distribution, s e e also distribution integral curve, s e e also curve manifold, see also manifold f M , 249 integration defined on H n ( M ) , 249 of n-forms global, 245 local, 237 of forms, 245-258 on singular simplices, 251
INDEX over the fiber, 259 Riemann, 20, 245 interior of a smooth manifold, 107 of a topological manifold, 23 point, 23 interior product, 226 intrinsic property, s e e also property Invariance of domain, 2 differentiable or smooth, 106 Inverse function theorem, 54, 373-377 isometry, 102, 315 local, 102 isomorphism of covering spaces, 28 of graded algebras, 216 of Lie groups, 167 of principal bundles, 353 isotopy, 116 between points, 135 between submanifolds, 280 class, 116 compactly supported, 116 differentiable or smooth, 116 alternative definition, 117 isotropy group, see also group Jacobian cocycle, see also cocycle for maps of Banach spaces, 375 matrix, 51 Jordan curve, 123 Jordan curve theorem, 123 Jordan-Brouwer separation theorem, 123, 276 ,% 318 Kervaire, M., 5, 94 Klein bottle, 6 Kronecker, 176 A(E), 229 Ak (E), 229
hk(V), 218 lattice, 17 integer, 17, 209 k-dimensional, 148 subgroup, 148 leaf, s e e also foliation(s) left translation, 162 left-invariant, see also field(s) Leibnitz rule, 47, 189 length of a curve, 101, 102, 312 of a tangent vector, 101,312 Levi-Civita connection, s e e also connection L ( G ) , 163 L(Gl(n))-valued form, 339 Lie algebra, see also algebra
INDEX bracket, see a l s o bracket derivative, s e e also derivative group, see a l s o group subalgebra, see also subalgebra subgroup, see also subgroup lift horizontal, 359, 363 to covering space, 28 Lima, E., 150 limit set a-, 136 of a dynamical system, 158 of a leaf, 159 w-, 136 line bundle, see also bundle integral, 190-195 definition of, 190 invariant under reparametrization, 190 path-independent, 192 long, 1 linear approximation, 41 linear endomorphism, 178 linking number, 276 local flows, see a l s o flow(s) Local homogeneity lemma, 134 locally Euclidean, see also space finite, 17 trivial, 62 locally finite sum, 259 long line, see a l s o line M~, 124 manifold 4-dimensional, 347 almost complex, 104 C ~ , 88 complex analytic, 104 contractible, 121, 242 differentiable or smooth, 87-93 with boundary, 107-110 frame, 347 integral, 150 maximal connected, 153 Lorentzian, 103 flat, 103 orientable, 13, 101, 229 oriented, 101 pseudo-Riemannian, 102 fiat, 102, 357 real analytic, 89 Riemannian, 101, 311-315 complete, 322-333, 342 flat, 102, 341, 357 geodesically complete, 325 topological, 3-6 definition of, 3
413 with boundary, 22-26 with boundary (definition), 23 with corners, 108 map basepoint-preserving, 33 continuous smoothly approximating, 118-119 covering, 27 differentiable or smooth, 50-54 between manifolds, 90 composition, 91 definition of, 50 equivariant, 348, 351 essential, 121 evaluation, 46, 91, 164 exponential associated to a Riemannian metric, 325 of a Lie algebra, 166 of matrices, 166 Gauss, 315 induced, 15 left multiple, 53 multilinear antisymmetric, 218 symmetric, 226 of constant rank, 54-58, 112 definition of, 56 proper, 112 R bilinear, 210 right multiple, 53 Weingarten, 316 maximal torus, s e e also torus Mayer-Vietoris sequences, 267-270 dual, 270 for compact supports, 268 for ordinary cohomology, 267 for singular homology, 270 mean curvature, s e e also curvature Mean value theorem, 373 for Banach spaces, 376 Meeks, W. H., 318 metric bi-invariant, 343 coefficients, 127, 313 indefinite, 102 left-invariant, 343 pseudo-Riemannian, 303 pseudo-Riemannian, 102 flat, 102 Riemannian, 102, 233, 303 existence, 107 flat, 102 on arbitrary vector bundles, 352 right-invariant, 343 Milnor, J., 5, 94, 150, 322 minimal set of a dynamical system, 158 of a flow, 136
414
INDEX
of a foliation, 158 minimal surface, 318 unbounded, 319
0n(n), 53 M6bius strip, 6, 8 module, 66 free, 66, 209, 211 of bilinear maps, 210 R-, 209 monkey saddle, 85 Montgomery, D., 161 morphism, see also category Morse function, see also function index, 83, 129 lemma, 83 theory, 83, 136 Morse, M., 83 ~ ( R ) , 209 multilinear, 209 algebra, see also algebra map, see also m a p Myers-Steenrod theorem, 342 natural, 230 non-Hausdorff, 15 norm, 102 normal bundle, see also bundle neighborhood, 116 space, see also space O.D.E., see also differential equations object, see also category O(k, n - k),
]02
f ~ ( M , x o ) , 310
O(n), 101 0(n), 143 operator boundary singular, 255 Cech (in the de R h a m - C e c h complex), 393 de R h a m (in the de R h a m - 0 e c h complex), 393 derivative, 46 differential first order, 65 second order, 68 face, 255 total differential, 393 orbit of a k-flow, 146 of a flow, 131 of a group action, 179 ordinary differential equations, see also differential equations orientability of a surface, 13 orientable, 13
orientation, 12, 100 clockwise, 13 continuous, 101 counterclockwise, 13 induced on the boundary, 247 of a manifold, 101 of a vector space, 100 standard, 13, 101 oriented coherently, 13 orthogonal group, see also group P.D.E., see also differential equations paracompact, 18 parallel transport, 303, 308, 363 existence and uniqueness, 308 parallelizable, 5, 62, 97, 311, 362 integrably, 100 partition of unity, 17-20 definition of, 19 smooth, 20, 105 for manifolds with boundary, 110 subordinate, 19 path-lifting property, see also property PD, 277 periods of a p-cycle, 257 of a 1-form, 195,300 of a cohomology class, 195, 257 permutation, 218 even, 218 odd, 218 sign of, 218 ~1 (notation for conjugate inverse), 184 piecewise smooth homotopy, 194 loops, 192 paths, 191 ~r[M,N], 119
9 k(v), 227 plaque, 152 plaque chain, 154 p n , 17 p n ( c ) , 182 Poincar6 dual localization principle, 280 duality, 276-281 operator, 277 theorem, 277 lemma, 195, 242, 258-263 for compact supports, 263 Version I, 260 Version II, 260 Version III, 260 Version IV, 260 Version V, 261 Version VI, 261 Poincar6-Hopf theorem, 281, 288
INDEX polarization, 227 principal bundle, see also bundle projective plane, 7 projective space, see also space property extrinsic, 314 geometric, 314 intrinsic, 314 path-lifting, 29, 369 universal, 210, 369 pseudo-Riemannian manifold, see also manifold metric, see also metric pullback of a foliation, 294 quadratic form, 227 quaternions, 162 quotient space, see also space topology, see also topology Rad6, T., 12 rank constant, 54-58 of a 2-form, 225 of a manifold, 150 Reeb, G., 158 refinement, 18 regular cover, 152 covering, see also covering points, 80, 93 space, see also space values, 80, 93, 204 simultaneous, 80, 112 relativity, 103 special, 103 retract, 116 deformation, 118 retraction, 116, 122, 199 Riemannian curvature, see also curvature distance function, 326 geometry, see also geometry homogeneous space, see also space manifold, see also manifold metric, see also metric structure, see also structure symmetric space, see also space volume element, 312 Riemannian geometry, 303-346 R ( X , Y ) Z , 320 Sard's theorem, 80 section, see also bundle self intersection number, 281, 288 signed area, 319 simple cover, see also cover, 395-397 simplex
415 Cech, 387 singular face of, 251 smooth, 250 l~-smaI1, 269 standard, 250 2-, 12 3-, 14 face of, 251 simplicial chain complex, 287 simply connected, see also connected singular 1-cycle, 254 2-cycle, 254 boundary, see also boundary chain, see also chain cochain, see also cochain cohomology, see also cohomology complex, see also complex cycle, see also cycle homology, see also homology simplex, see also simplex continuous, 284 Ek, 218 S(E), 229 gk(E), 229 Sk(M), 237 S(V), 226 r (V), 226 Sl(n), 60 s[(n), 143 smooth action, see also action atlas, see also atlas manifold, see also manifold smoothable manifold, 94 soap film, 318 solenoid, 202 space coset, 17 cotangent, 185 covering, 26-40 regular, 30, 353 Euclidean half-, 22 homogeneous, 178-183, 353 Riemannian, 342-346 horizontal, 360 locally Euclidean, 1-3 definition of, 1 non-Hausdorff, 1 normal, 19 of k-forms, 237 of contravariant tensors, 237 of covariant tensors, 236 of mixed tensors, 237 of singular chains, 255 of smooth sections, 231 of symmetric tensors, 237
416 paracompact, 18 pointed, 33 projective, 17, 182 complex, 182 quotient, 6-17 by a subgroup, 17 definition of, 15 regular, 2 Riemannian symmetric, 343 tangent, 44 basis of, 48 equivalent definitions, 49 to a manifold, 91 to a manifold with boundary, 108 to a submanifold, 60 vector of homogeneous polynomials, 227 of quadratic forms, 227 vertical, 358 space-time, 103 special linear group, see also group special orthogonal group, see also group special unitary group, see also group squiggly arrow for functors, 231 Stack of records t h e o r e m , 120 standard fiber, 350 standard simplex, see also simplex star shaped, 334 Steiner's surface, 21 stereographic projection, 3 Stiefel manifold, 182 complex, 182 Stokes' theorem, 183, 191, 245-258 combinatorial, 251 for manifolds with boundary, 247 structure group, see also group structure(s) C ~162 , 88 complex analytic, 104 diffeomorphic, 93-94 differentiable or smooth, 88 canonical on ~ n , 88 diffeomorphic, 93 exotic, 93 on R 4 93 on Nn, 93 on manifolds with boundary, 107 on spheres, 94 G-, 100 geometric, 98-104 Gl(k, n - k) integrable, 152 infinitesimal G-, 100 Cl+ (n)-, 101 Gl(k, n - k)-, 103 Cl(n, C)-, 104
INDEX O(k, n - k)-, 102 O(n)-, 101 integrable, 100 pseudo-Riemannian, 102 integrable, 102, 357 Riemannian, 101 integrable, 102, 338, 357 subalgebra Lie, 170-173 definition of, 171 of •(M), 142 subbundle, 103 horizontal, 359, 360 vertical, 358 subgroup closed of a Lie group, 173-178 discrete, 168 Lie, 170-173 1-parameter, 165, 170 definition of, 170 one-parameter, 344 properly imbedded, 173, 175 submanifold differentiable or smooth, 110-116 immersed, 113 integral, 143 isotopy of, 280 of manifold with boundary, 110 parallelizable, 62 Poincard dual of, 280 properly imbedded, 110 smooth, 58 topological, 21 immersed, 21 submersion, 57, 112 Submersion theorem, 58 s u m m a t i o n convention, 308 S ( U , p ) , 43 support, 19 of a singular chain, 282 symmetric algebra, see also algebra neighborhood of the identity, 163 power, 226 of a bundle, 229 of a free R-module, 227 tangent bundle, see also bundle space, see also space vector, see also vector tensor, 209-237 algebra, see also algebra contravariant, 217 space of, 217 covariant, 217 space of, 217 decomposable, 213, 218
INDEX degree contravariant, 217 covariant, 217 indecomposable, 213, 218 of type (r, s), 217 power, 215 0th, 229 1st, 229 of bundles, 229 product, 210 associativity, 213 existence and uniqueness, 210 of bundles, 228 of free R-modules, 213, 214 of R-linear maps, 214 pointwise, 232 property, 188 Tk(E), 229 O ' k ( M ) , 236 T k ( M ) , 237 ~*(M), 236 | and | 232 9"~s(M), 237 T~(V), 215 T~'~(V), 217 ~T(V), 215, 216 Theorema Egregium, 319, 321 Tischler, D., 301 topological group, see also group topology algebraic, 183 geometric, 87 of a Lie subgroup, 170 quotient, 6 torsion, 306 form, 339, 363 tensor, 306, 337 torus g-holed, 10 maximal, 177 T n, 4
two-holed, 10 total degree, 393 total differential operator, see also operator total space, see also bundle transformation covering, 27 deck, 27 group, see also group translation left, 53 right, 53 transversality, 293-296 definition of, 294 triangulation, 12, 287 nonorientable, 13 of compact manifolds, 287
417 orientable, 13 trivializing neighborhood, 96 umbilic point, 322 unimodular matrix, 201 unitary group, see also group unity, 46 universal R 4 94 antisymmetric multilinear map, 218 covering, see also covering model, 209 property, 33, 210, 369 Urysohn lemma, 19 smooth, 62 for manifold with boundary, 110 global, 104, 105 H-small singular complex, 270 Vectn (M), 99 vector bundle, see also bundle cotangent, 185 field, see also field(s) tangent, 42-50 at a boundary point, 108 to a manifold, 91 to a submanifold, 60 velocity, 71 vector field problem for spheres, 5 velocity field, see also field(s) vector, see also vector vertical field, see also field(s) space, see also space subbundle, see also subbundle VM, 230 Vn,k, 182 Y n , k ( C ) , 182 volume of an open set, 313 weak reparametrization, 310 Weingarten map, see also map Whitehead, J. H. C., 334 W h i t n e y stun, see also bundle winding number, 254 modulo 2, 123 X(M), 95 3~(s), 307 x ( u ) , 65 Yang-Mills theory, 347 Z~*(M), 244 Z I ( M ) , 195 Z P ( M ) , 242 Zp(M), 255 Z k , 209
418
INDEX
2P(U), 285 ZV(lI; R), 388 Zippin, L., 161
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