Ms'nifolds, Tensor Analysis, and Applications
RAt.PH ABRAHAM
Department of Mathematics Univei'sity of California, Santa Cruz
Santa Cruz, California JERROLD E. �ARSDEN
Department of Mathematics University of California, Berkeley Berkeley, California
TUDOR RATIU Df!partment of Mathematics University of Arizona Tucson, Arizona
1983 ADDISON-WESLEY PUBLISHING COMPANY, INC.
Advanced Book ProgramjWorld SCience Division Reading, Massachusetts London· Amsterdam. Don Mills, Ontario· Syd n ey. Tokyo
Some figures and text reproduced from "Foundations of Mechanics," 2nd Edition, by Ralph Abraham and Jerrold E. Marsden, (') 1978 by Addison-Wesley Publishing Company.
Library of Congress Cataloging In Publication Data Abraham, Ralph. Manifolds, tensor analysis, and applications. (Global analysis, pure and applied) Bibliography: p. Includes index. 1. Global analysis (Mathematics) 2. Manifolds (Mathematics) 3. Calculus of tensors. I. Marsden, Jerrold E. II. Ratiu, Tudor S. III. Title. IV. Series. QA614.A28 1983 514'.3 82-13737 ISBN 0-201-10168-8
American Mathematical Society (MOS) Subject Classification (1980): 34. 58, 70,76.93. Copyright· f 1983 by Addison-Wesley Publishing Company. Inc. Published simultaneously in Canada.
Ait rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means. electronic, mechanical, photocopying, recording, or otherwise, with'HJt thr' prtor written permission of the publisher, Addison-Wesley Publishing Company, Inc., Advanced Book ProgramjWorid Science Division, Reading, Massachusetts 01867, U.S.A. ABCDEFGHIJ-HA-S9S7654l
Manufactured in the United States of America
Contents
SarI. Editors' Foreword . ........................... xl Preface . ........................................ xv Background Notation . . . . . . . . . . .................... xix
Topology............................. .. 1 1.1. Topological Spaces . . . . . . . . . . . . . . . . . '. . ... . . . . I 1.2 Metric Spaces. . . . . . . . .. : . . . . . . . . . . . . . . . . . .9 1.3 Continuity ..................... " ........ 14 1.4 Subspaces, Products, and Quotients ............ 18 ) .5 Compactness ............................ 2A 1.6 Connectedness ......... ~ ................. 31
Ch8f»ter 1
Chapter 2
Banach Spaces and Differential Calculus ..... . 37
2.1 Banach Spaces ........................... 37 2.2 Linear and Multilinear Mappings .............. 52 2.3 The Derivative .......................... ; 67 2.4 Properties of the Derivative .................. 75 2.5 The Inverse and Implicit Function Theorems ..... 102 vii
v;;;
CONTENTS
Chapter 3
Manifolds and Vector Bundles ............ . 122
3.1 3.2 3.3 3.4 3.5
Manifolds..................... ......... 122 Submanifolds. Products, and Mappings. '........ 129 Vector Bundles .......................... 133 The Tangent Bundle ...................... 150 Submersions. Immersions and Transversality ..... 161
Chapter 4
Vector Fields and Dynamical Systems ....... 184
4.1 Vector Fields and Flows. . . . . . . . . . . . . . . . . ... 184 4.2 Vector Fields as Differential Operators ......... 207 4.3 An Introduction to Dynamical Systems ......... 240 4.4 Frobenius' Theorem and Foliations ............ 260 Chapter 5 Tensors .................. . . . . . . . . . . . .271
5.1 5.2 5.3 5.4 5.5
Tensors in Linear Spaces ................... 271 Tensor Bundles and Tensor Fields ............. 283 The Lie Derivative: Algebraic Approach ........ 293 The Lie Derivative: Dynamic Approach ......... 304 Partitions of Unity ....................... 309
Chapter 6 Differential Forms ...................... . 323
6.1 6.2
Exterior Algebra ......................... 323 Determinants. Volumes. and the Hodge Star Operator .............................. 335 6.3 Differential Forms ....................... 351 6.4 The Exterior Derivative. Interior Product, and Lie Derivative ........................ 356 6.5 Orientation. Volume Elements. and the Codifferential ........................... 381
Chapter 7 Integration on Manifolds . ................ . 395
7.1 The Definition of the Integral ................ 395 7.2 Stokes'Theorem ......................... 403 7.3 The Classical Theorems of Green. Gauss. and Stokes ................................ 424 7.4 Induced Flows on Function Spaces and Ergodicity .433 7.5 Introduction to Hodge-deRham Theory and Topological Applications of Differential Forms.... 440
CONTENTS
Chapter 8
ix
Applications ........................... 461
8.1 Hamiltonian Mechanics . . . . . . . . ............ 462 8.2 Fluid Mechanics . . . . . . . . . . . . . . . .......... 481 8.3 Maxwell's Equations ...................... 498 8.4 Thermodynamics, Constraints. and Control ...... 511 Appendix A.
Zorn's Lemma and Balre Spaces ......... 522
Appendix B.
The Three Pillars of Linear Analysis ....... 526
Appendix C.
Unbounded and Self-adjoint Operators ..... 530
Appendix D.
Stone's Th~rem ..................... 541
Appendix E.
The Sard and Smale Theorems ........... 551
References ..................................... 562 Index ............... ~ ......................... 570
Foreword
AIM AND SCOPE OF THE SERIES
What II Global Analyall? From ancient times till Newton. mathematics meant geometry and, algebra. Then analysis (now called classical) was born. along with the foundations of physics, engineering, and modem science. Arnong the outstanding events of modem mathematics are the syntheses of these fields along common frontiers. The synthesis of classical analysis and geometry is DOW called global analysis. The Hietory of Global AM.,... .... Ita AppIIcatIOni Importint pioneers in the synthesis of global analysis were Henri Poincar6 (l11Os), Oeorp Birlmofl (19208). Marston Morse '(1930&). and Hassler Whitney (19405). The technical tools of differential topology (19505) made the rmal synthesis possible (19605). Through the efforts of Solomon Lefshetz (1950s), the work of the Russian school (Liapounov. Andronov. Pontriagin) on dynamics became widely known in the west and included in this synthesis. A veritable explosion of new results and applications followed in the 1910s. From the earliest work of Poincare and Liapounov onward. the applications of g~metry and analysis to astronomy. physics. and engin~ring provided the explicit motivation for much of this work. The current form of the theory reflects this pervasive influence in its direct applicability to these
xi
xii
FOREWORD
fields. It has already created new and powerful methods of applied mathematics. which complement existin& tools such as perturbation methods. asymptotics. and numerical techniques. Far from being the exclusive preserve of pure mathematicians. globill analysis has its roots in physical problems and can he redirected to these problems once again. often with startling results. ' Target of the SerI..: The Accessibility of Global Analysis
There is a great contrast between the potential importance of global analysis and the great dirriculty of learning about it. A growing number of scientists of all disciplines have discovered that the techniques of global analysis have important applications in their own fields: they are looking seriously for keys to these techniques. This series will attempt to provide the keys. Needed are books that introduce the basic concepts and their applications. texts that develop the prerequisites for more serious study accessibly and compactly. and advanced monographs which make the research frontier available to a wide audience of scientists and engineers who have acquired these prerequisites. To these ends. this series will deal with such suhjects as Theory Linear algebra and representation theory Calculus on manifolds and bundles Differential geometry and Lie theory Manifolds of mappings and sections Transversal approximations Calculus of variations in the large Dynamical systems theory and nonlinear oscillations Nonlinear actions of Lie groups Applications Classical mechanics and field theory Geometric quantization Hydrodynamics Elastomechanics Econometrics Social theory Morphogenesis Network theory and other topics of pure and applied glohal analysis.
FOREWORD
xiii
AUDIENCES
Sub-Series A. The advanced texts will provide reports on theory or applications from the research frontier in expository style for /ipecialists. or for nonspecialists who have the prerequisite mathematical background. For example. graduate students of science or engineering as well as mathematics will find them manageable. , Sub-Series B. The basic texts will provide a complete curriculum of essential prerequisites. starting with advanced linear algebra and calculu,s. for the advanced texts of Sub-Series A. These texts will be suitable for advanced undergraduate courses in pure and applied mathematics. or as reference works for research in engineering. the ~cicnl:l~~. or mathematics. UNIQUE FEATURES
Through the basic texts (Sub-Series B) covering all the prerequisites in a uniform style and the advanced texts (Sub-Series A) building on this foundation, it will be possible for readers to study the deta,i1ed applications of global analysis to their own fields (as they appear in the series). to form independent evaluations of the new methods, and to master the techniques for their own use if it is justi(ied. Sub-Series B starts from the post-calculus level. in textbook format with worked examples, exercises and adequate illustrations. The series will give a complete library of prerequisites. together with new contributions to global analysis and some outstanding examples of its applications. illustrating the n~w methods in applied mathematics. All the texts will be in English and conform as far as possible to a common notational scheme. ' RALPH ABRAHAM PHILIP JERROLD
E.
J.
HOI.MES
MARSDEN
PREFACE
The purpose of this book is to provide cOre background material in global analysis for mathematicians sensitive to applications and to physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors. and differential forms. Some applications to Hamiltonian mechanics, fluid mechanics. electromagnetism. and control theory are given in Chapter 8. using both invariant and index notation. Detailed treatments of these and additional applications are planned for other volumes in the series. The book does not deal with Riemannian geometry in detail or with Lie groups. or Morse theory. These too are planned for a subsequent volume. Throughout the text special or supplementary topics occur in ,boxes. This device enables the reader to skip various topics without disturbing the main flow of the text. Additional background material in the appen4ices is given for completeness. to minimize the necessity of consulting too many outside refetences. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications. using, say. manifolds of mappings. the study of infinite-dimensional manifolds is hard to motivate. except for its intrinsic interest. Chapter 8 gives a hint of these applications. In fact, some readers may wish to skip the infinitedimensional case altogether. To aid in this we have separated into boxes many of the technical points peculiar to the infinite-dimensional case.
xv
xv;
PREFACE
Our own research interests lean heavily toward physical applications, and the choice of topics is partly molded by what has been and remains useful for this kind of research. Some interesting technical side issues not consistent with these goals have been omitted or relegated to boxes. We have tried to be as sympathetic to our readers as possible, by providing ample interesting examples, exercises, and applications. When a computation in coordinates is easiest, we give it and don't try to hide things behind complicated invariant notation. On the other hand, index-free notation can often provide valuable geometric, and sometimes computational, insight so we have tried to simultaneously convey this navor. The only prerequisites required are a solid undergraduate course in linear algebra and a classical course in advanced calculus. At isolated points in the text some contacts are made with other subjects. For students. this provides a good way to link this material with other courses. These links do not require extra background material. but it is more meaningful when the bond is made. For example. Chapter I (and Appendix A) link with point-set topology, the boxes in Chapter 2 and Appendices B, C, D, are connected with functional analysis, Section 4.3 relates to ordinary differential equations. Chapter 3, Section 7.5, and Appendix E are linked to differential topology and algebraic topology. and finally Chapter 8 on applications is connected with applied mathematics, physics, and engineering. ' This book is intended to be used in courses as well as for reference. The sections are, as far as possible, lesson sized, if the boxed material is omitted. For some sections, like 2.5, 4.2, or 7.5 two lecture hours are required. A standard course for mathematics graduate students, for example, could omit Chapter I and the boxes entirely and do Chapters 2 through 7 in one semester with the possible exception of Section 7.4. The instructor could then assign certain boxes or appendices for reading and choose among the applications of Chapter 8 according to taste. A shorter course, or a course for advanced undergraduates probably should omit all boxes, spend about two lectures on Chapter I for reviewing background point set topology, and cover Chapters 2 through 7 with the exception of Sections 4.4, 7.4, 7.5 and all the material relevant to volume elements induced by metrics, the Hodge star, and codifferential operators in Sections 6.2, 6.4, 6.5, and 7.2. A more applications oriented course could omit Chapter I. review without proofs the material of Chapter 2 and cover Chapters 3 to 8 omitting the boxed materials and Sections 7.4 and 7.5. For such a course the instructor should keep in mind that while Sections 8.1 and 8.2 use only elementary material, Section 8.3 relies heavily on the Hodge star and codifferential operators, while Section 8.4 consists primarily of applications of Frobenius' theorem dealt with in Section 4.4. The appendices, included for completeness, contain technical proofs of facts used in isolated places in the text.
PREFACE
xvii
The notation in the book is as standard as conflicting usages in the literature allow. We have had to compromise among utility. clarity. clumsiness, and absolute precision. Some possible notations would have required too much interpretation on the part of the novice while others. while precise. would have been so dressed up in symbolic decorations that even an expert in the field would not recognize them. We have used hold face symhols to help the reader distinguish objects; for the most part. linear spaces. linear operators and abstract tensor fields are in boldface italics. while manifolds, points. point mappings. and tensor components are in lightface itaIlcs. Strict compliance is not always possible. This usage is only to help the reader distinguish symbols to which no further significance should be attributed. In a subject as developed and extensive as this one. an accurate history and crediting of theorems is a monumental task. espcl:ially when so many results are folklore and reside in private notes. We have indicated some of the important credits where we know of them. but we did not undertake this task systematically. We hope our readers will inform us of these and other shortcomings of the book so that. if necessary. corrected printings wm be possible. The reference list at the back of the book is confined to works actually cited in the text. These works are cited by author and year like this: deRham (1955). During the preparation of the book. much valuable advice was pro- ' vided hy Alan Weinstein. Our other teachers and collaborators from whom we learned the material and who inspired. directly and indirectly. various portions of the text are too numerous to mention individually. We hereby thank them all collectively. Finally. we thank Connie C'alica for her careful typing of the manuscript. RALPH ABRAHAM JERROLDE.
~ARSDEN
TUDOR RATIU
BACKGROUND NOTATION
The reader is assumed to be familiar with usual notations of set theory such as e. u. n and with the concept of a mapping. If A and B are sets and I:. A -- B is a mapping. we often write a ...... I( a) for the e,ffect of the mapping on the element a e A; "iff' stands for "if and only if' (="if' in definitions). Other notations we shall ,use without explanation include the following: R.C Z. Q
real. complex numbers integers. rational numbers AX B Cartesian product R". C " Euclidean n-space. complex n-space (xl •...• x")eR" pointinR" , A\B set theoretic difference I or Id identity map rl(B) inverse image of B under I rf = «x. I(x)~x e domain of f} graph of I infinimum (greatest lower hound) of A c R infA supremum (least upper hound) of A c R sup A basis of an n-dimensional vector space:: el.···.e" kernel and range of a linear transformation T ker T. range T end of an example end of a proof proof of a lemma is done. but the proof of the theorem goes on.
••
These modifications of the Halmos symbol • are notations of Alan Weinstein.
xix
Manifolds, Tensor Analysis, and Applications
CHAPTER
1
Topology
The purpose of this chapter is to introduce just the right amount of topology for later requirements. It is assumed that the reader has had a course in advanced calculus and so is acquainted with open, closed, compact, and connected sets in Euclidean space (see for ex~ple Marsden (197_] and Rudin (1976». If this background is weak. the reader may find the pace of this chapter too fast If the background is under control. the cbapter sbould serve to collect, review, and solidify concepts in a more· general context. • A key concept in manifold tbeory is tbat of a dirlerentiable map between manifolds. However, manifolds are also topological spaces and . differentiable maps are continuous. Topology is the study of continuity in a general context; it is therefore appropriate to begin with it. Topology often involves interesting excursions into pathological spaces and exotic.theorems. Sucb excursions are deliberately minimized here. The examples will be ones most relevant to later developments, and tbe main thrust will be to obtain a working knowledge of continuity, connectedness. and compactness.
1.1 TOPOLOGICAL SPACES Abstracting our ideas about open sets in R", we shall first consider the notion of a topological space. 1
2
1.1.1
TOPOLOGY
Definition. A top%giclll space is a set S together with a colle(·tion 0
0/ subsets called 0fIDt .Is such that
.
(Tl) IZJE fl and S E fl: (TI) 1/ VI' V z E fl. then VI n V2 E (T3) The union 01 any collection 01 open sets is open.
e:
A basic example is the real line. We choose S ... R. with fl consisting of all sets that are unions of open intervals. Thus, as exceptional cases. the empty set IZJ E and R itself belong to fl. Thus (TI) holds. For (TI). let VI and V2 E fl: to show VI n Vz E fl. we can suppose VI n V2 * "". If x E VI n Vz then x lies in an open interval )a l • hll C VI and x E )(12' hzl C Vl . Let lal.bl[nlaz.bz[=la.b[ (so a=max(a l .a 2 ) and h=min(h l .b2 »). Thus x E la. b[ C VI n Vz. Hence VI n Vz is the union of such intervals. so is open. Finally. (TJ) is clear by definition. Similarly. R" may be topologized by declaring a set to be open if it is a union of open rectangles. An argument similar to the one just given shows that this is a topology. called the standard topology on R". Open intervals in . R and open rectangles in Rn are each examples of a basis for a topology; i.e.• every open set is a union of sets in '!1\. Any set S can be topologized in an obvious manner in two ways. The trivial topology on S consists of fl = { IZJ • S}. The discrete topology on S is defined by fl = {A IA C S}; i.e.• r consists of all subsets of S. Topological spaces are specified by a pair (S. ~): we shall. however. just write S if there is no danger of confusion.
e
'i'
1.1.2 DeflnlUon. Let S be a topological space. A set A c S is called cto.d if its complement S \
A is open.
The collection 01 closed sets will be denoted e.
For example. the closed interval [0. I] c R is closed as it is the complement of the open set ]-oo.O[U]I.oo[.
1.1.3 PropollUon. (el)
(e2) (e3)
Proof.
The closed sets in a topological space satisfv:
IZJE e and SE e; . If AI' A2 E t' then AI U A2 E t'; the intersection 0/ any collection of closed sets ;s dosed.
(C1) follows from (TI) since IZJ = S\S. S = S\ 0. The relations S\(A I U A z ) = (S\A I )U(S\A z )
and
S\C~\ B,) = ,~/S\B,)
TOPOLOGICAL SPACES
3
e'
for {Bj }, a family of closed sets show that (C2).«(,3) are equivalent to (T2).(TJ), respectively. • Closed rectangles in R n are closed sets. as are clo~cd halls. one-point sets. and spheres. Not every set is either open or dosed. For example. the interval [0, II is neither an open nor a closed set. In the discrete topology on S any set A c S is both open and closed. whereas in the trivial topology any A . 0 or S is neither. Closed sets can be used to introduce a topology just as well as open ones. Thus, if is a collection satisfying 0, and R itself; only the last two are open neighborhoods. The set (x, x + 2( contains the point x but is not one of its neighborhoods. In the trivial topology on a set S, there is only one neighborhood of any point, namely S itself. In the discrete topology any subset contaiping p is a neighborhood of the point pES, since {p} is an open set. 1.1.5 Definition. A topological space is called lint COfIIIttlbk if for each u E S there is a sequence {V•• V2 • ••• } - {V,.} of neighborhoods of U SUM that for any neiglJborhood V of u, there is an n such that Vn c V. The topology is called ,econd COfIIIttlbk if it has a countable basi,.
Topological spaces of interest to us will largely be second countable. For example R n is second countable since it has the countable basis formed by rectangles with rational side length. centered at points all of whose coordinates are rational. Clearly every second-countahle space is also first countable, but the converse is false. For example if S is an infinite noncountable set, the discrete topology is not second countable. but S'is first countable, since { p} is a neighborhood of pES. The trivial topology on S is second countable (see Exercises 1.11. I.IJ for more interesting counterexamples). . A basic fact about second-countable spaces is the following statement due to LindelOf..
.-
TOPOLOGY
1.1.8 PropoelUon. E~ry covering of a set A in a .,econd-countable space S by a family of open sets Vft (that is U "V" ::> A) ("Ontains a countable subcollection {Va,. Va, • ... } also covering A. Proof. Let 'i'l == {Bn} be a countable basis for the topology of S. For each pEA there are indices n and a such that p E Bn C Va' let ~~' = {B"I there
exists an II such that Bn C Va}' Now let Va be one of the Vft that includes the element B" of'i'l'. Since ~' is a covering of A. the countable collection {Vft } covers A. • • The terminology of closure and interior is also very useful.
1.1.7 DefinlUon. ut S be a topological space and A C S. Then the elos",. of A, denoted cl( A) is the intersection of all closed sets containing A. The interior of A. denoted int(A) is the union of all open sets contained in A. The boMtJmy of A. denoted bd(A) is defined by bd( A ) = cl( A) ncl( S \ A) By (e3). cl(A) is closed and by (n). int( A) is open. Note that as bd(A) is the intersection of closed sets. bd(A) is closed. and bd(A) - bd(S\A). Note that A is open iff A = int( A) and closed iff A == c1( A). For example. on R. c1([0.1[) = [O.J).
int([O. I[) = (0.1)
and
bd([O.I[)-{O.I}.
The reader is assumed to be familiar with examples of this type from advanced calculus. Some notions building on these are as follows.
1.1.8 DellnlUoM. A subset A of S is called dense in S if d( A) - S and is called IIOwllere detue if S\cl(A);s dense in S. S ;s called septII'fIbk if it has a countable dense .,uhsel. A point in S is called an tlCcum_,ion poin' of the set A if eoc-h of its neighborhoods contains points of A other than itself. The sel of uuunw/Illion /lfJillts of A is called the derived se' of A and is denoted hy der( A). . A point of A is said to be UoIll'ed if it has a neighborhood in S COIttuining no other point of A than itself. The set A == [0, I[U {2} in R has 2 as its only isolated point. iot( A) - JO, 1(. cl(A) - (0, I]U{2) and def( A) == [0.1). In the discrete topolou OIl • let S. int{p}-c1{p} == {p}. for any peS.
TOPOLOGICAL SPACES
5
Since the set Q of rational numbers is dense in R and is countable. R is separable. Similarly R" is separable. A set S with the trivial toPology is separable since cl{p} = S for any pes. But S = R with the discrete topology is not separable, since c1( A) = A for any A e S. It is not hard to see that any second-countable space is separable. The converse is false: see Exercises 1.11, 1.11. A basic characterization of the operations of closure, interior. and boundary is as follows.
1.1.' Propolltlon. Let S he a topological spare and A e S. Then (i) u E c1( A) iff for every neighborhood U of u, unA. 121; (ii) u E int( A) iff there is a neighborhood U of u .vuch that U e A; (iii)' u E bd(A) iff for every neighborhood U of u, UnA.IZI and un
(S\A).IZI.
(i) u ~ cl( A) iff there exists a closed set C :l A such that u ~ C. But this is equivalent to the existence of a neighborhood of u not intersecting A. namely S\C. (ii) and (iii) are proved in a similar way.. .
Proof.
The basic properties of the operations c1. into bd. and der are given in the following.
1.1.10 PrVpoaltlon. Let A, B and A,. i E I be subsets of S. (i) If AeB, then int(A)eint(B). c1(A)ecl(B). der(A)eder(B) and bd(A) e bd(B); (ii) S\cl(A) "" int(S\A), S\int(A) = cI(S\A). and'cl(A) = A U der(A). (iii) cl(IZI) ... int(IZI)'" 121. cl(S) - int(S) - S. cl(c1(A»'" cl(A) and int(int( A» - int( A); (iv) c1(A U B)=-cI(A)Ucl(B). der(A U B)=der(A)Udcr(B). int(A U B) ::> int(A)Uint( B); (v) cl(A n B)ecI(A)ncl(B). der(A n B)e der(A)ndcr( B). int(A n B) = int(A)nint(B); (vi) CI(U;E,~;)::>U;E,cl(A;), cl(n;E,A,)en'E,c1(A;). int(U;E,A;) ::> U IE ,int(A i ), int( n;E ,Ai) e :';E ,int(A i ). Proof. (i), (ii). and (iii) are immediate consequences of the definition and of Proposition 1.1.9. Since for each i E I. Ai e U IE ,A;. by (i) cl(A,)c c1(U iE ,A i ) and hence u;E,cI(A,)ecl(UiE,A;). Similarly. since niElA, e Ai e cI( A;) for each i E I. it follows that n iE,A; is a subset of the ~Iosed set n;E,cl(A,); thus by (i) cl(n;ElA;)ecl(n'E,cI(A,»=-n;e,cl(A;).
6
TOPOLOGY
The other formulas of (vi) follow from these and (ii). This also proves all the other formulas in (iv) and (v) except the ones with equalitiC5. Since c1(A)Ucl( B) is closed by (e2) and AU Be c1( A)ucI( B). it follows hy (i) that c1(AUB)ccl(A)Ucl(B) and hence equality by (vi). The formula int( A n B) = int( A)nint( B) is a corollary of the previous formula via (ii).
•
The inclusions in the above proposition can be strict. For example. if 14=]0.1[. B"'[I.2[. then c1(A)= der(A) = [0.1). cl(B)-der(B)-[1.2). int( A) -= ]0.1[. int( B) = )1.2[. A U B = )0.2[. and A n B - (lJ so that int( A)U int(B) -)0. I(U)I.2[" JO.2( - int( A U B). cl( A n B) - 0 .. (I) - cl(A)n c1(B). Let A
"
=]-.!. .!.[
n" n '
n= 1.2 •... :
then oc
n A" = {O}, ,,-I and int(
int( A,,) = A" for all n
nA,,) -0-
,,-I
{O} -
n
,,-I
int(A,,).
Dualizing this via (ii) gives
Uc1(R\A,,)=R\{O}*R=c1( ,,-I U(R\A,,»).
,,-I
Limits of sequences can be defined in topological spaces as follows. 1.1.11 Definition. Let S be a topological space and {un} a seqUl'lIc(, of points in S. The sequence is said to etHIW~ if there i.v a point u E S .fuch (hat for every neighborhood V of II. there is an N such that n ~ N implies urI E V. We say that u" CtIIIfJIrIft t" II, or II is Q limit po;"1 of {II,,}.
For example. the sequence {I / n} in R converges to O. I t is obvious that limit points of sequences II" of distinct points are accumulation points of the set {II,,}. In a first-countable topological space any accumulation point of a set A is a limit of a sequence of elements of A. Indeed. if {V,,} denotes the countable collection of neighborhoods of a E der( A) given by definition U.S, then choosing for each n an element a" E V" n A. a,," Q, we see that {all} converges to a. We have proved the following.
TOPOLOGICAL SPACES
7
1.1.12 PropoalUon. Let S be a first-countable space and A c S. Then u E c1( A) iff there is a sequence of points of A that converge to u (in the topology of S). It should be noted that a sequence can be divergent and still have accumulation points. For example-{2.0.3/2. -1/2.4/3. - 2/3 •... } does not converge but has both I and - I as accumulation points. Also. in arbitrary topological spaces. limit points of sequences are in general not unique. For example. in the trivial topology of S any sequence converges to aU points of S. In order to avoid such situations several separation axioms have been introduced. or which the three most impt)rtant ones will he mentioned.
1.1.13 DeftnlUon. A topological space S is called Hau.fdotff if each two distinct points have disjoint neighborhoods (that is. with empty intersection). S is called regu/Iu if it is Hausdorff and if each closed set and point not in this set have disjoint neighborhoods. Similarly. S is called IIOrmal if it is Hausdorff and if each two disjoint closed sets have disjoint neighborhoods.
AU standard spaces in analysis are normal. The discrete topology on any set is normal. but th:: trivial topology is not even Hausdorfr. It turns out that "Hausdorff' is the necessary and sufficient condition for uniqueness of limit points of sequences in first-countable spaces (see Exercise J.JE). Since in Hausdorff spaces single points are closed (Exercise 1.1 F). we haye the implications, normal =- regular =- Hausdorff. Counterexamples for each of the converse' of these implications are given in Exercises 1.11 and I.IJ.
1.1.14 Propoaltlon. A regular second-countable space is normal. Proof. Let A. B be two disjoint closed sets in S. By regularity. for every point pEA there are disjoint open neighborhoods Up of p and UB of B. Hence d( Up)n B ""' 0. Since {Up Ip E A} is an open covering of A. by the Lindel~r theorem 1.1.6. there exists a countable collection {Uklk = 1.2•... } covering A. Thus 00
l::.I
Uk:J A
and
d( Uk ) n B
= 0 .
4-1
Similarly. find a family {Vk } such that 00
U Then the sets
Vk:J B
and
cl(Vk)n If ==0.
k-O II
GII + 1-Un + 1\ U cl(V.). .-0
II
HII-v,,\ U cl(U.), Go -lIo, 4-0
8
TOPOLOGY
are open and
00
G- U G~:::lA,
.-0
H- U H.:::lB
,,-0
are also open and disjoint. •
ConventIon In the remainder of this book Euclidean n-space R· will be understood to have the standard topology (unless explicitly stated to the contrary).
Exerct... l.lA 1.1 B l.1C 1.1 D 1.1 E 1.1 F
1.10
l.lH
Let A - {(x, y, z) e R310 ""' x < I and y2 + Z2 ""' I}. Find int(A). Show that any finite set in R" is closed. Find the closure of {Ijnln = 1,2, ... } in R. Let A cR. Show that sup( A) e cI( A) where sup( A) is the supremum (l.u.b.) of A. Show that a first-countable space is Hausdorff iff all sequences have at most one limit point. (i) Prove that in a Hausdorff space, single points are closed. (ii) Prove that a topological space is Hausdorff iff the intersection of all closed neighborhoods of a point equals the point itself. Show that in a Hausdorff space S the following are equivalent: (i) S is regular; (ii) for every point peS and any of its neighborhoods U, there exists a closed neighborhood V of p such that V c U; (iii) for any closed set A the intersection of all closed neighborhoods of A equals A. (i) Show that if "Ir( p) denotes the set of all neighborhoods of peS, then the following are satisfied: (VI) If A:::l U and U E C'f{( p), then A E 'V(p); (V2) Every finite intersection of elements in 'V( p) is an element of
CV(p); (V3) P belongs to all elements of 'V( p); (V4) If V e 'V( p) then there is a set U
E 'V( p)
such that for all
qEU, Ue'V(q). (ii) If for each peS there is a family 'V( p) of suh:,clS of S satisfying (VI)-(V4), prove there is a unique topology eonS such that for each peS, 'V( p) is the set of neighborhoods of p
e. e
in the topology (Hint: Prove first uniqueness and then' define elements of as being subsets A c S satisfying: for each peA, we have A e'V(p).)
METRIC SPACES
1.11
9
Let S={p=(x.y)ER2IY~O} and let D,(p)={qlllq-plI~E} denote the usual e-disk about p in the plane R2. Define .
( {D,(P)()S. if p=(x,y) with y>O B. p)= {(x,y)ED,(p)IY>O}U{p). if p-(x,Q). Pr~ve the following: (i) 'Y( p) = {U C Sithere exists H,( p) C U} satisfies (VI)-(V4) of Exercise 1.1 H. Thus S becomes a topological space. (ii) S is first countable. (iii) S is Hausdorff. (iv) S is separable. (Hint: {(x. y)E Six. r"E Q. r > O} is dense in S.)
\.\ J
(v) S is not second countable. (Hint: Assume the contrary and get· a contradiction by looking at the points (\.0) of S.) (vi)· S is not regular. (Hint: Try to separate (x(l.!» from {( x.O) Ix E R}\{( x o, On.) With the same notations as in the preceding exercise. except changing B,(p) to if P"" (x, y), Y>O D.(x. e)U{p}. if p = (x.O).
D.(P)()S. B (P) = { . ,
.\ K
Show that (i)-(v) of 1.11 remain valid and that (vi) S is regular. (Hint: Use 1.10.) (vii) S is not normal. (Hint: Try to separate cI( B,( p» for p = (xo.O) from {(x.O)lx E R}\{( xo.O)}.) Prove the following properties of the boundary operation and show by example that each inclusion cannot be replaced by equality (i) bd(A)=bd(S\A); (ii) bd(bd( A» c bd( A); (iii) bd( A U B) c bd( A)ubd(B) c bd(A U B)U A U B; (iv) bd(bd(bd( A))) = bd(bd( A». (These properties may be used to characteri7.e the topology.) 1.2 METRIC SPACES
One of the common ways to form a topological space is through the use of a distance function. also called a metric. For example. on R" the standard distance d(x.y)= (x,1/2
(t,-I
y,)2)
10
TOPOLOGY
between x == (XI'" .,xn ) and y = (YI" .. ,Yn) can be used to construct the open disks and from them the topology. The abstraction of this proceeds as follows.
1.2.1
Definition. Let M he a set. A metric on M is a function d: M X M ..... R such that
d(ml' m2) == 0 iff m l = m 2; (definiteness); (symmetry); and (M3) d(ml' m) .. d( mi' m;)+ d(m 2 • m J ); (triangle inequality). (MI)
(M2) d(m l ,m2)==d(m 2,m l );
A metric spIICt! is the pair ( M, d); if there is no danger of confusion just write M for (M,d).
Taking m l == m) in (M3) shows that d is necessarily a non-negative function. It is proved in advanced calculus courses (and is geometrically clear) that the standard distance on R n satisfies (M I )-(M~). The topology determined by a metric is ddined as follows.
1.2.2 Definition. For r> O. and m EM. the t-disk about m is defined by D.{m) = {m'E Mld(m'. m) < t}. The collection of subsets of M that are unions of such disks is the metric topology of the metric space (M. d). Two metrics on a set are called equivalent if the.' induce the same metric topology.
1.2.3 Propolltlon.
(i) The de.~cripti(/n of open Jets Killen in the precedinK definition is a topology. (ii) A set U C M i.f open if and only if for each m E U there is an r > 0 such that D.( m) C U.
(i) (TI) and (TJ) are clearly satisfied. To prove (T2). it suffices to show that the intersection of two disks is a union of disks, which in tum is implied by the fact that any point in the intersection of two disks sits in a smaller disk included in this intersection. To verify this. let p E D.(m)n D,(n) and O O. Find the distance hetween the graph of I and (0.0).
12C Show that every separable metric space is sC'l'Ond countahle. 1.2D Show that every metric space has an equivalent metric in which the diameter of the space is I. (Hint: Consider d l (",. fl) = d( m. n )/(1 + demo n».) 1.2E In a metric space M. let '\.( m) = {V c MI there exists E > 0 such that D,(m)CU}. Show that 'V(m) satisfies (VI)-(V4) of Exercise I.IH. This shows how the metric topology can be defined in an alternative way starting from neighborhoods. 1.2F
In a metric space show that d( A) = {u E Mld( u. A) = O}.
Exercises 1.2G-I use the notion of continuity from calculus (see Section 1.3). 1.2G
Let M denote the set of continuous functions f: [0. I interval [0. I). Show that d(f. g) =
r
-+ ~
on the
11/(x) - g(.\')1 d:c 1
II
1.2H
is a metric. Let M denote the set of continuous functionsJ: [0.1] -+ R. Set
d (f. g ) = sup {If ( x ) - g ( x)1 I 0 .;:; x .;:; I).
1.21
(i) Show that d is a metric on M. (ii) Show that /" -+ I in M iff In converges un;rorm~v to f. . (Iii) By consulting theorems on uniform convergence from your advanced calculus text. show that M is a complete metric space. Let M be as in the previous exercise and define T: M -+ M by
'.
T(f)(x)=a+ fo'K(x.)')/(Y)d.Y. where a is a constant and K is a continuous function of two variables. Let k
=
and suppose k < I.
sup{
10'1 K ( x. Y)I dy I0 .;:; x ~ I}
14
TOPOLOGY
(i) (ii)
Show that T is a contraction. Deduce the eltistence of a unique solution of the integral, equation f(x)=a+l'K(x.)')f(y)dy. (I
(iii) Taking a special case of (ii). prove the "existence of e' ".
1.3 CONTINUITY Continuity is one of the basic ingredients used in manifold theory. We begin with the definition in the context of topological spaces.
1.3.1
Definition. Let Sand T be topological spaces and cp: S -+ T be a . mapping. Then cp is COIftilluoru tit u E S if for every neighborhood V of cp(u) there is a neighborhood U of u such that cp(U) C v. If. for every open set Vof T. cp-I(V)={uESlcp(U)E V} is open in S. cp is COIftilluoru. (Thus. cp is continuous iff cp is continuous at each u E S.) If cp: S -+ T is a bijfftiott (that is. one-to-one and onto). cp. and cp 1 are continuous. then cp is a Itomeomorpllism and Sand Tare Itomeomorpltic. For example. notice that any map from a discrete topological space to any topological space is continuous. Similarly. any map from an arbitrary topological space to the trivial topological space is continuous. Hence the identity map from the set S topologized with the discrete topology to S with the trivial topology is bijective and continuous. but its inverse is not continuous. hence it is not a homeomorphism. From 1.3.1 it follows by taking complements and using the identity S\!p-I(A) =11'- I(T\A). that 11': S -+ T is continuous iff the inverse image of every closed set is closed.
1.3.2 Propottltlon. Let S, T be topological spaces and 11': S -+ T. The following are equivalent: (i) II' is continuous; (ii) cp(cl(A»C cl(cp(A» for every A C S; (iii) cp - I(int( 8» C int( cp - 1( 8» for every 8 C T.
Proof. If II' is continuous, then II' Icl(!p( A» is closed. But A c!p Icl(!p( A» and hence cl( A) C II' - Icl(!p( A». or 11'(cl( A) C cl(!p( A». Conversely. let B C T be closed and A = !p-I(B). Then cl(A)C !p-I(B)= A. so A is closed. A similar argument shows that (ii) and (iii) are equivalent. • This proposition combined with l.I.l2 (or a direct argument) gives the following.
CONTINUITY
15
1.3.3 Corollary. Let Sand T he topological spaces with S lirst cOimtable and 'P: S -+ T. Then 'P i.f continuous iff lor every sequence {u m } converging to u. ('P( u,,)} converges to 'P( u), lor all u E S.
1.3.4 Propoeftlon. ous map.
The composition
01 two continuous maps is a
continu-
Proof. If 'PI: SI -+ S2 and 'P2: S2 -+ S) are continuous maps and if V is open in S3' then ('P2 0 'PI)- I(V) = 'PI I( 'Pi I( is open in SI since 'P2 I(U) is open in s,. by continuity of 'P2 and hence its inverse image by 'PI is open in SI by continuity of 'PI' •
V»
1.3.5 CorOlla". The set 01 all homeomorphisms of a topological space to itself lorms a group under composition.
Proof. Composition of maps is associative and has for an identity element the identity mapping. Since the,inverse of a homeomorphism is a homeomorphism by definition, and since for any two homeomorphisms 'PI' 'P2 of S. 'PI 0 'P2 and ('PI 0 'P2)- I = 'Pi 10 'PI I are continuous by 1.3.4. the corollary follows. •
1.3.8 Pr0P,08lUon. The space 01 continuous mUfI.f algebra under pointwise addition and multiplication.
I: S -+ R
lorm.I' un
Proof. We' have to show that if I and g are continuous. then so are I + g and Ig. Let So E S be fixed and e> O. By continuity of I and g at ,fo. there exists an open set V in S such that lI(s)- l(so)1 < e/2, Ig( s)- g(.fu)1 < E/2 for all s E V. Then· 1(f+g)(s)-(f+g)(so)IEOI/(s)-/(so)I+I.,,(.f)
g(so)l<e
Similarly. for e> O. choose a neighborhood V o(.so such that I/(s)- l(so)1 < 6.lg(s)- g(so)1 < 6 for all s E V ,where 6 is any positive number satisfying (6 + I/(so)l)6 + Ig(so)16 < e. Then l(fg )(s)- Ug )(so)1 EO lI(s )Ug( s) - g(so)1
+ if( s)- l(so)llg(so)1
< (6 + I/(so)1)6 + 6Ig(so)1 < e.
Therefore, I + g and Ig are continuous at so' •
16
TOPOLOGY
Continuity is defined by requiring that inverse images of open (closed) sets are open (closed). In many situations it is important to know whether the image of an open (closed) set is open (closed).
1.3.7 Deftnltlon. A map 'P: S -+ T. where Sand T are topological spaces. is called opm (cloud) if the image of every open (closed) set in S is open ( closed) in T. Thus a homeomorphism is a bijective continuous open (closed) map. It should be noted however that not every open (closed) map is closed (open). Examples and applications of this notion will be given in the next section. For the moment note that the identity map of S topologized with the trivial and discrete topologies on the domain and range. respectively. is not continuous but is both open and closed. Let us now turn our attention to continuous maps between metric spaces. For these spaces. continuity may be expressed in terms of f'S and 6's familiar from calculus.
1.3.8 Propo8lt1on. Let (MI' d l ) and (M2 • d 2 ) be metric spaces. and 'P: MI -+ M2 a given mapping. Then 'P: MI -+ M2 is continuous at u l E MI iff for every f> 0 there is a 6> 0 such that dl( u l ' u;) < 6 implies d 2('P( u l ). 'P( U;» <E.
Proof.
Let 'P be continuous at
UI
and consider D,2( 'P( UI»' the f-disk at
«p( u l ) in M 2 • Then there exists a 6-disk D~( u l ) at u l in MI such that
»
'P(Dl(ul»CD/('P(u l (by 1.3.1); i.e.. d(u l,u;) 0 such that 'P( D~( u l c D,2( (fi( III » by the fl. . regoing argument. Thus'P is continuous at u l . •
»
In a metric space we also have the notions of uniform continuity and uniform convergence.
1.3.9 Deftnltlon. (i) Let (MI.d l ) and (M 2 .d2 ) be metric spaces and 'P: MI -+ M 2· We say 'P is lIIIi/ormly COlttinuous if for every f> 0 there is a 6 > 0 mch that dl( u. v) < 6 implies d 2 ( 'P( u). 'P( v» < E. (ii) Let S be a set. M a metric space and 'PII: S -+ M. II = 1.2.... and 'P~ S -+ M be given mappings. We say 'Pn -+ 'P lIIIi/orml), if for every 10 > 0 there is an N such that d('P,,(u).'P(u» < f for all n ~ N and all uE S. For ellample. a map satisfying d( 'P( u). 'P( v» ~ Kd( u. v) for a constant K is uniformly continuous. Uniform continuity and uniform convergence
CONTlNUI.TY
17
ideas come up in the construction of a metric of the space of continuous maps. This is considered next. .
1.3.10 ~ltIon. Let M be a topological space and (N.d) a complete metric space. Then the collection C( M. N) 01 all bounded continuous maps cp: M - N lorms a complete metric space with the metric
Proof. It is readily verified that dO is a metric. Convergence of a sequence In E C( M. N) to IE C( M. N) in the metric dO is the same as unilorm convergence. as is readily checked. (See Exercise 1.211" Now if In is a Cauchy sequence in C( M. N). then I .. (x) is Cauchy for each x E M since.
Thus In converges pointwise. defining a function II x). We must show that I" - I uniformly and that I is continuous. First of all. given e> O. choose N such that dOC In. 1m) < e/2 if n. m ~ N. Then for any x E M. pick N" ~ N so that d(fm(x)./(x» < f'/2 if m., N", Thus with n ~ Nand m ~ N".
dU,,(x). I(x») ~ dUne x). Im(x »)+ dUm(x). I{x») < e/2+ e/2 = e.
so I" - I uniformly, The reader can similarly verify that f is continuous (see Exercise l.3F; look in any advanced calculus text such as Marsden'[1974a] for the case in R" if you get stuck). _ . Exercl... 1.3A Show that a map cp: S - T between the topological spaces Sand Tis continuous iff for every set BeT. cI(cp-I(B»ccp-l(cl(B». Show that continuity of cp does not imply any inclusion relations between cp(int A) and int( cp( A». 1.38 Show that a map cp: S - T is continuous and dosed if for every subset U C S. cp( cl( U» = c1( cp( U » 1.3C Show that compositions of open (closed) mappings are also open (closed) mappings. I.3D Show that cp: ]0. oc[ - ]0. oc[ defined by cp( x) = 1/x is continuous but not uniformly continuous. 1.3E Show that if d is a pseudometric on M, then the map d( '. A): M - R for A C M a fixed subset is continuous.
18
TOPOLOGY
l.3F If S is a topological space. T a metric space and CPn: S -+ T a sequence of continuous functions uniformly convergent to a mapping cP: S -. T. then cP is continuous. 1.4 SUBSPACES, PRODUCTS, AND QUOTIENTS
This section concerns the construction of new topological spaces from old ones. The first basic operation of this type we consider is the formation of subset topologies. 1.4.1 Definition. If A is a suhset of a topological space S. the relative topology on A is defined by
From the identities l. 0nA=0.SnA=A. 2. (V. n A)n(U2 n A) = (V. n U2 )n A. and 3. ua(VanA)=(UPa)nA. we see that ~A is indeed a topology. Thus. for example. the topology on sn I = ( X E R" Iti( x. 0) = I} is the relative topology induced from Rn; i.e .. a neighhorhood of /I point x F S" I is a subset of sn -. containing D,< x)n sn I for some E> O. Note that an open (closed) set in the relative topology of A is in general not open be a covering of S x T by open sets, Each Aa is the union of sets of the form V x V with V and V open in Sand T, respectively. Let {Vp x Vp} be a covering of S x T by open rectangles. If we show that there exists a finite subcollection of Vp x Vp covering S x T. then clearly also a finite subcollection of {A,,} will cover S x T. A finite subcollection of {Vp x Vp } is found in the following way. Since {s}X T is .compact there exists a finite collection
covering it. If I,
then V, is open. contains sand U. x Vp" ... ; V, x Vp, covers {s}X T. Let W, = V, x T; {w,} is an open covering of S x T and if 'we show that only a finite number of these W, cover S x T, then since I.
W,
=
U (V, x Vp ).
j-I
I
it follows that a finite number of Vp x VII will cover S x T. Now look at S x {t}, for t E T fixed, Since this is compact. a finite suhcolle~tion W" ..... W" covers it. But then k
U W, =SxT. • I-I
"
As we shall see shortly in 1.5.9, [-1,1) is compact. Thus TI is compact.' It follows from 1.5.3 that the torus T2. and inductively Tn, are compact. Thus if 71': R2 -+ T2 is the canonical projection we see that T2 is com-
26
TOPOLOGY
pact without R2 being compact; i.e.• the converse of 1.5.2 (ii) is false. Nevertheless it sometimes occurs that one does have a converse: this leads to the following.
1.5.4 Definition. A mapping
!p: S ..... T is proper if it is continuous and if the inverse image of every compact set is compact.
Note that the composition of proper mappings is again a proper mapping. Without the explicit assumption of continuity. such mappin'gs may be discontinuous. For example. the identity mapping of a finite set with the trivial and discrete topologies is not continuous. but the inverse images of compact sets are compact since any finite set with the trivial topology is compact. Note that any continuous map defined on a compact space is proper in view of 1.5.2(i). Compactness is also characterized in terms of convergence of sequences.
1.5.5 Boluno-Welerstras. Theorem. If S is a compact Hausdorff space. then every sequence has a convergent subsequence. The converse is also true in metric and second-countable Hausdorff spaces. Proof. Suppose S is compact and {un} contains no convergent subsequences. Then we may assume that the points of the sequence are distinct. Then each un has a neighborhood v" that contains no other Urn. for otherwise un would be a limit of a subsequence. Since the single-point sets {un} are closed (Exercise I.IF) the sets S\{u,,} form an open covering of S with no finite subcollection covering S. a contradiction. Let S now be second countable. Hausdorff. and such that every sequence has a convergent subsequence. If {Ua} is an open covering of S. by the Lindelof theorem there exists a countable collection {V"I n = 1.2.... } also covering S. Thus we have to show that {V"I n = I. 2.... } contains a finite collection covering S. If this is not the case. the family (S\ U ~-IV,) consists of closed nonempty sets and S\ U 7-1V, ::> S\ u 7: IV, for m ~ n. Let p" E S \ u 7- IV,. If {p"ln ... 1,2•... } is infinite. by hypothesis it contains a convergent subsequence; let its limit point be denoted by p. But then pES \ U 7- IV, for all 'I, contradicting the fact that {U"ln = I. 2•... } covers S. Thus {p"ln = 1.2•... } must be a finite set; i.e .• for all n ~ no, Pn = PIOn' But then again p" .. E S \ U :_ IV, for all n, contradicting the fact that {V"I n = 1,2.... } covers S. Hence S is compact. Let S be a metric space such that every sequence has a convergent subsequence. If we show that S is separable. then since S is a metric space it is second countable (Exercise 1.2(,). and by the preceding paragraph. it will be compact. Separability of S is proved in two steps.
COMPACTNESS
27
First we show that for any l > 0 there exists a fi"ife set { PI" .. • p,,} such that S - U ;'. ,D,( p,). If this were false. there would exist an l > 0 such that no finite number of e-disks cover S. Let p, E S he arhilrary. Since D,( PI)" S.there exists P2 E S\D,< p,). Since D,( p,)U D,( P~ j""" S. there exists P.l E S\(D.(p,)UD,(P2»' etc. The sequence {p"ln=I.2.... } is infinite and d( Pi' Pj) ~ e. But this sequence has a convergent subsequence by hypothesis. so this subsequence must be Cauchy. co~tradicting d( Pi' Pi) ~ e for all i. j. Second. we show that the existence for every l > 0 of a finite set {p, .....p"",} such that S = U 7~',)D,( p;) implies S is separable. Let -1" denote this finite set for l = 1/ n and A = U :'_, A". Thus A is countable and it is easily verified that d( A) ... S. • A property that came up in the preceding proof turnll out to be important.
1.5.8 DeflnlUon. Let S be a metric space. A suhset A c S IIoruIMd if for any
E>
i.~ called totally 0 there exists a finite set {Pl .... •p,,} in S .~uch that
..
Ac
U
D,(p;).
;-1
1.5.7 Corollary. A metric space is compact iff it is complete and total!.l· bounded: A subset of a complete metric space is relatil'ely compact iff it is tota/~v bounded. Proof. The previous proof shows that compactness implies total boundedness. As for compactness implying completeness. it is enough to remark that in this context. a Cauchy sequence having a convergent subsequence is itself convergent. Conversely. if S is complete and totally bounded. let {p"ln = I. 2.... } bj: a sequence in S. By total bounded ness. this scquenl'e contains a Cauchy subsequence. which by completeness. converges. I'hu .. S is compact hy the 8olzano-:-Weierstress theorem. The second statement 'hlW readily follows .•
1.1.1 PropoalUon. In a metric space £'Ompact S('t.V arC' c/o.fed and hounded.
Proof. The first assertion is a particular case of the previous corollary but can be easily proved directly in Hausdorff spaces. If A is compact. it can be finitely covered bye-disks: A"" U 7_ID.< Pi)' Thus
.
diam(A).,;
E diam(D.(p;»=2ne. •
From 1.5.2 and 1.5.8. if S is compact and 'P: S -0 R is continuous. then 'P(S}, is closed and bounded. Thus 'P attains its sup and inC ,<see Exercise I.ID).
28
TOPOLOGY
1.5.8 Helne-8oreI Th.."......
In R" a closed and bounded set is compact.
Proof. By 1.5.2(i) it is enough to show that closed bounded rectangles are compact in R". which in tum is implied via 1.5.3 by the fact that closed bounded intervals are compact in R. To show that [- a, a], a > 0 is compact, it suffices to prove (by 1.5.7) that for any given e> 0, [- a, a] can be finitely covered by intervals of the form (p - e. p + e). since we are accepting completeness of R. Let n be a positive integer such that a < ne. Let t E [- a, a] and k be the largest positive or negative integer satisfying kt: E; t. Then -n4iOk4iOn and ke4iOt«k+l)e. Thus any tEI-a.a] ~ils in an interval of the form (ke-f!.ke+e). for k--n ..... O•.••• n and hence {eke - f. ke + e)lk ... o. ± I ..... ± n} is a finite covering of (- a. a]. • This theorem is also proved in virtually every textbook on advanced calculus. As is known from calculus. continuity of a function on an interval [a. b] implies uniform continuity. The generalization to metric spaces is the following.
1.5.10 ProposlUon. A continuous mapping cp: MI ..... M 1 • where MI and M2 are metric spaces and MI is compact. is uniformly continuous. Proof. The metrics in MI and M z are denoted d l and d z• Fix e> O. Then for each p E MI' by continuity of cp there exists 8, > 0 such that if d I ( p, q) oj ( '" ( 11/, ). '" ( m » < f:
so ':f is totally bounded.
•
The.Heine-Borel theorem then implies the following. 1.5.13 Corollary. If M is a compact metric space. a set ~'t C C( M.R") is relatively compact iff it is equicontinuous and uniform(r hounded (i.e .• llcp(m~1 :E;; constant for all cr E ~. m EM).
30
TOPOLOGY
The following example shows how to use the Arzela-Ascoli theorem. 1.5.14 Example.
Let /,,: [O.IJ -- R be continuous and be such that
1I.(x)1 ~ 100 and the derivatives /,: exist and are uniformly bounded on
10.1[. Prove I. has a uniformly convergent subsequence. We verify that the set Un> is equicontinuous and bounded. The hypothesis is that 1f~(x)1 ~ M for a constant M. Thus by the mean-value theorem. I/',(x)- I.(Y)I~ Mlx-
Yi.
so given t we can choose 8 = tiM. independent of x, y. and n. Thus {/,,} is equicontinuous.1t is bounded because II In II = sUPo .. , .. dln(x>l" 100. • Exercl...
l.SA Show that a topological space S is compact iff every family of dosed subsets of S whose intersection is empty contains a finite subfamily whose intersection is empty. !.SB Show that every compact metric space is separable. (Hint: Use total boundedness.) I.SC Show that the space of Exercise 1.11 is not locally compact. (Hint: look at the sequence I/n).O).) !.SD Show that every closed subset of a locally compact space is locally compact. I.SE Show that S x T is locally compact if h(lth Sand T arc locally compact. 1.5F Let S be a compact topological space and - an equivalence relation on S. so that SI - is compact. Prove that the following conditions are equivalent (d. 1.4.10): (i) The graph P of - is closed in S x S; (ii) - is a closed equivalence relation; (iii) SI - is Hausdorff. I.5G Let S be a Hausdorff space that is locally homeomorphic to a locally compact Hausdorff space (that is, for each u E S. there is a neighborhood of S homeomorphic. in the subspace topology. to an open subset of a locally compact Hausdorff space). Then show that S is locally compact. In particular, Hausdorff spaces locally homeomorphic to R· are locally compact. Is the conclusion true without the Hausdorff assumption?
«
CONNECTEDNESS
31
1.5H Let MJ be the set of all 3 X 3 matrices with the topology obtained by , regarding M3 as R 9 , Let S0(3) = {A e M~IA is orthogonal and det A = -I}. (i) Show that S0(3) is compact. (ii) Let P={QeS0(3)IQ is symmetric} and let cp: Rp2->S0(3) be given by cp(l) = rotation by 17 about the line Ie R3. Show that cp maps Rp2 homeomorphically onto P. 1.51
Let fn: [a, b) -> R be uniformly bounded continuous functions. Set
F" (x ) = [fn ( t ) dt.
(/ ~ x '" h.
u
Prove that
F" has a uniformly convergent subs~qllence,
1.8 CONNECTEDNESS
There are three basic varieties of connectedness that topological spaces may have. These are referred to as arcwise connectedness. connectedness. and simple connectedness. 1.8.1 Definition. Let S be a topological space and I = [0. I] c R, A n are cp in S is a continuous mapping cp: I ~ S. If cp(O) = u. cp( I) = v. we ,WlI' cp joins u and v; S .is called are woe connected if evet:v two points /1/ S can he joined by an arc in S. A space is called locaHy arewise connected if each poiqt has an arcwise connected neighborhood (in the relative topology),
For example. R n is arcwise and locally arcwise connected: 'any two points of R n can be joined by the straight line segment connecting them. A set A c R n is called convex if this property holds for any two of its points. Thus convex sets in Rn are arcwise and locally arcwise connected, A set with the trivial topology is arcwise and locally arcwise connected. but in the discrete topology it is neither (unless it has only one point). 1.8.2 Definition. A topological space S is corrnected if (lj and S are the only subsets of S that are both open and closed . .if subset of Sis ctJM«IMI if it is connected in the relative topology. A COIrIptIMnt A of S is a nonempty connected subset of S such that the only connected .rubset of S containing A ;s A; S is called locaHy corrnected if each point has a connected neighborhood. ' The comptIIfeIIts of a subset T c S are the components of the points of T in the relative topology of T in S.
For example, R n and any convex subset of R n are connected and locally connected. The union of two disjoint open convex sets is discon-
32
TOPOLOGY
nected but is locally connected: its components are the two convex sets. The trivial topology is connected and locally connected. whereas the discrete topology is neither: its components are all the one-point sets. Connected spaces are characterized by the following. 1.8.3 Proposition. A space S is not connected iff either of the following holds. There is a nonempty proper subset of S that is both open and closed. (ii) S is the disjoint union of two nonempty open sets. (iii) S is the disjoint union of two nonempty closed sets. (i)
The sets in (ii) or (iii) are said to dUCOIIIIect S. Proof. If there is a nonempty proper set A that is both open and closed. then S=AU(S\A) with A.S\A open and nonempty. Conversely. if S == AU B with A. B open and nonempty. then A is also closed. and thus A
is a proper nonempty set of S that is both open and closed. The equivalence of (ii) and (iii) is a direct consequence of the definition of closed sets as complements of open sets. • The behavior of these two notions of connectedness under mappings is as follows. 1.8.4 ProposlUon. Iff: S -+ T is a continuous map of topological spaces and S.is connected (resp. arcwise connected) then so is f(S). Proof. Let S be arcwise connected and consider f(s,), f(s2) E f(S) cT. If c: 1-+ S. c(O) == Sl' c(I)'" s2 is an arc connecting Sl to S2' then clearly f 0 c: 1-+ T is an arc connectingf(s,) to f(s2); i.e.• f(S) is arcwise connected. Let S be connected and assume f(S) c U U V. where U and V are open and un V=0. Thenrl(U) andr'(V) are open by continuity off.r I(U)U r I(V)= rl(U U V);:)r 1(f(S»= S. and rl(U)nr I(V)= rl(U n V) ... r I( 0)'" 0. thus contradicting connectedness of S by 1.6.3. Hence f(S) is connected. •
We now prove that arcwise connected spaces are connected. To do so, however, requires the following. 1.6.5 Lemma. The on~v connected sets of R are the intervals (finite, infinite, open. closed, or half-open).
CONNECTEDNESS
33
Let us prove that [a. b[ is connected; all other possibilities have identical proofs. If not. [a. b[ = V U V with U. V nonempty closed sets in [a, b[. Assume that a E V. If x = suP(U). tben x E V since V is closed in [a. b[ and x < b since V. f2J. But then Ix. b[ c V and since V is closed. X E V. Hence x E V n V. a contradiction. , Conversely. let A be a connected set in R. We claim tbat [x. yJc A whenever x. yEA, which implies tbat A is an interval. If not. there exists zE[x,y] witb zEA. But then ]-oo.z[nA and )z,oo[nA are open nonempty sets disconnecting A. • Proof.
1.8.8 Proposition. If Sis arcwi.fe connected then it is connected. Proof. If not. there are two nonvoid. disjoint open sets Vo and VI whose union is S. Let Xo E Vo and XI e VI and let fP be an arc joining' Xo to XI' Then Vo = fP-I(VO) and VI = fP -I(VI ) disconnect [0,1). •
A sf,andard example of a space that is connected but is not arcwise connected nor locally connected, is {(x,y)eR2Ix>0 and y=sin(l/x))U{(O.y)I-I O. there exist /II,. M. > 0 such that Df" (0) c Dk (0) C The first inclusion says that if IlIxlll lit IIxll.s;; RIM,. Thus. if 1'''' O. then
D~t: (0)
M,. then IIxll lit R. i.e .• ir IlIxlll lit I. then
II.
I' 111'11 R 1111'111 = lIIelll.s;; MI' "
that is. 111'11 lit (R I M\) 1111'111 for all I' E E. Similarly. the second inclusion is equivalent to (MdR) 1111'111 lit 111'11 for all I' E E. Thus the two topologies are
44
BANACH SPACES AND DIFFERENTIAL CALCULUS
the same if there exist constants NI > O. N2 > 0 such that
for all , E E. Taking M = max( N2 .1/ N I ) gives the stalemenl or Ihe proposition. •
If E and F are normed vector spaces. the map 11'11: E x F -+ A defined by 11(.,.')11-11.11+11.11 is a norm on Ex F inducing the product topology. Equivalent norms on Ex Fare (.,.') .... max(I'",1I.11> and ( ••••) .... (11.11 2 +11.11 2 )1/2. The normed vector space E X F is usually denoted by eeF and called the dir.ct sum of E and F. Note that EeF is a Banach space iff both ' E and F are. These statements are readily checked. In finite dimensions some special things occur as follows. 2.1.10 Propo8lt1on. ut E be a finite-dimensional real or complex vector
space. Then (i)
there is a norm on E.
(ii) all norms on E are equivalent; (iii) all norms on B are complete.
Proof. Let ., .... ,." denote a basis of E. where n is the dimension of E. (i) A norm
on E is given for example by 11-11-
"
E loll.
n
E 0/./.
where - -
i-I
Note that this norm coincides with the A" or e"-norm
" loll. 111(0' ..... 0")111= E i-I
(ii)
Let 11'11' be any other norm on E. If
.= E "
the inequality
a l• i
and
I-I
1=
n
L bi.,. i-I
111.11'-11/11'1< II. -/1r..;; L" la i - billle;II' i-I
..;; max 01 •. :~:( a l , .... 0")- (b l , ... ,b")\\\. 1..,/< "
BANACH SPACES
45
shows that the map (xl •...• x")ec" ....
II;~IX;'.r ero.co[
is continuous with respect ~o the III·III-norm on C". Since the set S = .{ x e C "\lIIxlll"" I} is closed and bounded. it is compact. The restriction of this map to S is a continuous. strictly positive function. so it attains its minimum MI and maximum M2 on S; that is.
for all (xl •...• x")eC" such that IIKx' •...• x"~1I = I. Thus
or
for
Taking M - max(M2 .I/M1). Proposition 2.1.9 shows that 11·11 and equivalent. (iii) It is enough to observe that
n·U' are
" (XI ••.•• X") e C" .... Ex'" e I:: i-I
is a nonn-preserving map (i.e.• an isometry) between (e".III·III> and (E.II·II). •
•
The unit spheres for the three common norms on R 2 are shown in Fig.
2.1.1 .. The foregoing proof shows that compactness of tbe unit sphere in a finite-dimensional space is· crucial. This fact is exploited in the following box.
If.
IANACH '/lAC'S AND DIFFERENTIAl. CAI.CUI.US ..
Y
Y
x
muCx.
Y
x
x
y)
figure 2.1.1
BOX 2.fA A CHARACTERIZATION OF FINITE-DIAfENSlONAL SPACES
2.1.11 PropoaIIIon. .A normed vector space is finite dimensional iff ;t ;s locally compact. (Local comptlCtneu is equivalent to the fact that the closed un;t disk is compact.)
Proof. If E is finite dimensional. the previous proof of
(iii) shows that E is locally compact. Conversely, assume the closed unit disk D,(O) c ~ is compact and let {D I / 2 (x,)I; -I, .. .• n} be a finite cover with open disks of radii 1/2. Let F - span (Xl.' .. ,x,,). Then F is finite dimensional. hence homeomorphic to CA (or AA) for some k < n. and thus complete. Being a complete subspace of the metric space (E.II'II). it is closed. We shall prove F - E, Suppose the contrary. that is. there exists "E E, V. ~ F. Since F- cl(F). the number d - inf{II"- ell Ie E F} > O. Let r > 0 be such that D,(,,)nF.flJ. The set D,(,,)nF is closed and bounded in the fmite-dimensional space F. and thus is compact. Since inf{II" - ell I e E F}'" inf{lI" - '111' E D,( ,,) ~ F} and the continuous function. E D,( ,,)n F .... 11" - '11 E )0. co[ attains its minimum. there exists a point '0 E D,( ,,)n F such that d'"" II" - '011. But then there exists Xi such that
,,- eo II 1 II 11"eoll- Xi < 2' so that
'
BANACH SPACES
47 .
Since eo + 110 - eollx; E F. it follows that 110 - "" - ( II v - eo II )x;1I ~ d. which is a contradiction. •
2.1.12 Exampl... A. Let X be a set andF a nonned vector space. Let B(X. F) = {f: X-+FlsuPxExll/(x)lI H. as in 2.1.12. Show that the inclusion map EI -+ Eo is wmpal·t; i.e.. the unit ball in EI has compact closure in Eo. (Hint: lise the Arzela-Ascoli theorem.) 2.1 D Let (E. (-. be an inner product space amI A. B subseis of E. Let A + B = {a + bla E A. b E B}. Show that '(i) Ac B implies B.I. c A.I.; (ii) A.I. is a closed subspace of E; (iii) A.I. = (cl(span(A))).I.. (A.I. ).1. = cl(span( A»; (iv) (A+B).I.=A.I.()B.I.; (v) (cl(span(A»()cl(span(B»))l =Al +B J (not neces~arily a direct sum). 2.1 E A sequence {en} c E. where E is an inner product space. is said to be weakly convergent to I' E E iff all the numerical sequences (v. en) converge to (v.e) for all vEE. Let f2(C)={{an>ianEC and E:'_lla n I2 < co} and put ({an}.{bn }) = E:'_la"f;n. Show that (i) in any inner product space convergence implies weak convergence; (ii) f 2(C) is an inner product space; (iii) the sequence (1.0.0.... ).(0.1.0•... ).(0.0. I •... ). . .. is not convergent but is weakly convergent to 0 in f 2 (C). Note.f 2 (e) is in fact complete. so it is a Hilbert space.'The ambitious reader can attempt a direct proof or consult 1\ book on real analysis such as Royden [t 968). 2. t F Show that a normed vector space is a Banach space iff every ~bsolutely convergent series is convergent. (A series E:'_IXn is call~ absofutely convergent if E:'_ I IIxnll converges.) 2.1 G Let E be a Banach space and FI c F2 C E be dosed subspaces such . that F2 splits in E. Show that FI splits in E iff FI splits in F2. 2.1 H Let F be closed in E of finite codimension. Show that if G is a subspace of E containing F. then G is closed. 2.11 Let E be a Hilbert space. A set {e j } " I is called orthonormal if (e j • ei ) = 8, ,. the Kronecker delta. An orthonormal set {e,}, e I ,is (i) (ii)
-»
52
BANACH SPACES AND DIFFERENTIAL CALCULUS
called a Hilbert basis. if c1(span{ej }j e I) = E. (i) Let {_/},., be an orthonormal set and (,'o ..... ,,) be any finite subset. Show that
for any I' E E. (Hint: 1" - I' - Ej_,( e. e j )e j is orthogonal to all (ej[j=I ..... n}.) , I , (ii) Deduce from (i) that for any positive integer n. the set {i E 111(e.ej)l> lin} has at most n 2 11ell 2 elements. Hence at most countable many i E 1 satisfy <e, 1',) '* 0, for any e E E. (iii) Show that any Hilbert space has a Hilbert hasis. (Hint: Use Zorn's lemma (Appendix A) and 2.1.17.) (iv) If {ej}je 1 is a Hilbert basis in E,e E E. and {e,} is the 0 such that IIA41I1 ~ MI-+ Te is in 1.( L( E. F).
n.)
If IE c1(:;([a. hl.H) and vE
then
(/(I)vdl
F. =
((/(1) dt )v.
(Hinl: I ..... multiplication by I in Fis in L(R. L(F. 1'»; apply (i).) (iii) Let X be a topological space and I: [a. b)X X .... E be continuous. Then the mapping g: X -> E. g(x) = 1:1(1. x) dl is continuous. (Hinl: For IE R. Xo E X and E> 0 given. II/(s. x)-, 1(1. xo)1I < E if (s. x) E U, X Ux •• ,; use compactness of [a. hI to
find Uxo as a finite intersection and such that 1I/(I.x)1(1. xo)1I < E for all t E [a. b). x E Ux ".) 2.2G Show that the Banach isomorphism theorem is false for normed incomplete vector spaces in the following way. Let E be the space of all polynomials over H normed by lIa o + alx + ... + anx"lImax(laol ... ·.Ianl}· (i) Show that E is not complete. (ii) Define A: E-+EbyA
a ( Lna;x') =a,,+ Ln ...J.. x' ;-0
(iii) 2.2H
2.21
and show that A E L( E. E). Prove that A Show that A - I is not continuous.
,-I
I:
I
E -> E exists.
Let E and I' be Banach spaces and A E L(E. F). If A(E) has finite codimension. show that it is closed. (Hinl: If 1'0 is an algehraic complement to A (E) in F. show there exists a continuous linear isomorphism ElkerA "" FIFo; compose its inverse with Elker A .... A(E).) (Symmetrization operator). Define Sym": L k(E. F) .... L k(E. F). by . k
Sym A
=
I kI
~
~ aA •
. .. es,
THE DERIVA T1VE
67
where (aA)(e., ... ,eli:)'" A(e,,(I)' .,e,,(k»' Show that (i) Syrrr'«Lk(E,F»=L~(E,F). (ii) (Syrrr' I follows by induction. Similarly a/(u + e) = a/(II}+ aDI(u}'e' + ao(e)'" al(u)+ aDI(u)'e + o(e}. • 2.3.5 Proposition. Let /;: U c E ..... F;.1
2.3.1 Propoeltion. Suppose U C R" is an open set and I: U - R'" is di/l"entiable. Then the partial derivalives ap/ax; exist. and the matrix 01 the linear map Ol(x) wilh respect 10 the slandard bases in R" and R m is given by
!£ ax'
~ ax 2
~) ax"
!£ !£ ax'
ax 2
a/ 2 ax"
al'" ax'
aIm ax 2
al no ax"
THE DERIVA TlVE
'71
where each partial derivative is evaluated at x == (x I •...• X"). This matrix is called the Jacobillll matrix 01 f. Proof. By the usual definit~on of the matrix of a linear mapping from linear algebra. the jith matrix element 0 1; of DI(x) is given by the jth component of the vector DI(x)e;. where e l... .• e" is the standard basis of A". Lettingy -= x + he;. we see that
II/( y)- I(x)- DI(x)( y - x)1I lIy-xll
h =
Xll/(xl ..... x; + h ..... x")- l(xl ......l"")_vDI( x)e,1I
approaches zero as h -+ O. so the jth component of the numerator does as well; i.e., '
1-
1 I/J{ x I ..... x ; + h , .... x ") - li{ x I ..... x " ) - ha,j - 0 • Ii m -h
h .... O
which means that ail = ap/ax;. • In doing practical computations one can usually compute the Jacobian matrix easily. and this proposition then gives Df. In some books, D/is called' the differential or the total derivative of f. 2.3.7 Example. Let I: A2 -+ Al, I(x, y) = (x 2, xly, X 4y 2). Then D/(x, y) is the linear map whose matrix in the standard basis is
ax
all
all ay
al 2 ax
ap ay
all ax
all ay
=
( 2.2
3x y 4xly2
o 1
Xl
2x4y
where/l(x, y) ... x 2,f2(x. y) = xly,fl(X, y) = X·y2. • One should take special note when m -1, in which case we have a real-valued function of n variables. Then DI has the matrix
... YL) (YL axl ax"
72
BANACH SPACES AND DIFFERENTIAL CALCULUS
and the derivative applied to a vector t' = (a l ••••• a n ) is Df(x)·e=
af i • L" -.a i-I
ax'
It should be emphasized that DI is a linear mapping at each x E U and the definition of D/(x) is independent of the basis used. If we change the basis from the standard basis to another one. the matrix elements will of course change. If one examines the definition of the matrix of a linear transformation. it can be seen that the columns of the matrix relative to the new basis win be the derivative DI(x) applied to the new basis in R n with this image vector expressed in the new basis in R"'. or course. the linear map DI(x) itself does not change from basis to basis. In the case m = I. Df( x) is. in the standard basis. a I x n matrix. The vector whose components are the same as those of Df(x) is called the gradient of f. and is denoted grad for vj. Thus for f: U eRn -+ R.
gradf =
( : : . •...•
:!n ).
(Sometimes it is said that grad f is just Df with commas inserted!). The formation of gradients makes sense in a general inner product space as follows. 2.3.8 Definition. (i) ut E be a normed space and I: U c E --+ R be differentiable so thaI D/(II) E L( E.R) = E*. In this case we sometimes write d/(lI) for Df(lI) and call df the differential off. Thus d/: U -+ E*.
(ii) II E is a Hilbert space. the Ilnulia, 01 I is
(Jej"r~'J
by
grad/-v/: U-+E (v/(II).e) - dl(II)·e. where d/(II)'e means the linear map d/(lI) applied to the vector e.
Note that the existence of v/(lI) requires the Riel%. re.Y1II5entation theorem (se. ~,2·5) The notation 81/"" instead of (grad J )(U) II) is also in widt' II,"~ ("l)CCially in the case in which E is a space of functions.
=,-/(
(.»
2.3.8 Example. Let (E. be a real inner product space and / ( II) = 11_11 2, Since 1111112=11110112+2(110.11-110)+1111-110112. we obtain df(lIo)·e
THE DERIVATIVE
73
== 2(uo'~> and thus vf(u) = 2u. Hence f is of class C l . But since Of(u) = 2(u, . >E E· is a continuous linear map in u E E, it follows that 02f(u) ... Of E L(E, E·) and thus OAf = 0 for k ;;d. Thus f is of class CfX>. The mapping f considered here is a special case of a polynomial mapping (see 2.2.10) .. We close this section with the Fundamental Theorem or Calculus in real Banach spaces. First a bit or notation. If cp: U c R -- F is differentia· ble, then Dcp(t)E L(R, F). Now L(R, F) is isomorphic to F by A ..... A(I). IE R; note thatIlAII"'IIA(I~I. We denote cp'(/)'"
~~ =
Ocp(/)·I,
IE R.
Clearly
'( ) . I. cp(t + h)-cp(t) cp t = 1m h 11-0
and cp is differentiable iff cp' exists.
2.3.10 Fundamental Theorem of Calculua. (i) If g: [a, b] ~ F;s continuous, where F is a real Banach space, the map f: ]a. b[ -- F;f(t),. f~g(s)ds is differe!'tiable and we have f' = g. (0) Iff: [a, b] ..... F is continuous, is differentiable on ]a, b[ and/, extends to a continuous map on (a, b], then f(b)....,f(a)
= tf'(S)df. a
Proof.
(i) Let to E ]0, b[. Since the integral is linear and
continuous~
IIf(t o + h)- f(to)-hg(to)II .... IIJ,~o+II(g(s)- g(/o» dr~
4iOlhi sup{ IIg(s)-g(t o )lllto ..;s..;to +h} .. The sup on the right-hand side -- 0 as Ihl-- 0 by continuity of g at ' 0 • (ii) Let h(t)- (ff'(S)ds)- f(t).
By (i) h'(t) == 0 on la, b[ and h is continuous on [a, b]. If for some.t E [a, b] h( t) =I> h( a), then by the Hahn-Banach theorem there exists tJ E such that (tJ 0 h Xt) =I> ( tJ 0 h Xa). Moreover 11 0 h is differentiable on ]a, .b( and its
r
74
BANACH SPACES AND DIFFERENTIAL CALCULUS
derivative is zero (Exercise 2.3D). Thus by elementary calculus. 11 0 h is constant on [a.b]. a contradiction. Hence h(t)=h(a) for all tE[a.b]. In particular heal - h(b). • Exercl... 2.3A
Let B: E X F -. G be a continuous bilinear map of normed spaces. Show that B is COO and that
DB(" • .,)(e, f)=B(", f)+B(e,.,). 2.3B
Show that the derivative of the map is unaltered if the spaces are renormed with equivalent norms.
2.3C
If IE Sk(E. F), show that
and 2.3D
Let I: VeE -. F be a differentiable (respectively C) map and A E L( F. G). Show that A 0 I: VeE --+ G is differentiahle (respectively C) and D'( A 0/)(") = A 0 D'/( u). (Hint: Use induction.)
2.3E
Let I: VeE -. F be r times differentiable and A E L(G. E). Show that
2.3F
exists for all i ... r, where.,E A -I(V). and IJI •••.• IJ, E G. (i) Show that a complex linear map A E L(C.C) is necessarily of the form A(z) = >.z, for some>. E C. (ii) Show that the matrix of A E L(C.C) when A is regarded as a real linear map in L (R 2. R 2) is of the form
(~ -a h). (Hint: >. = a
(iii)
+ ih.)
Show that a map I: V c C --C.I = g + ih. g. h: V CR2--+R is complex differentiable iff the Cauchy- Riemann equations
ag = ah ! . La ax ay'ay
_ ah
are satisfied. (Hint: Use (ii) and 2.3.6.)
ax
PROPERTIES OF THE DERIVATIVE
75
Let (E.(.» be a complex inner product space. Show that the map . 1(11)= 1111112 is not differentiable. Contrast this with 2.3.9. (Hint: DI(II). if it exists. should equal 2 Re«II,' ».) 2.3H Show that the matrix of D 2/(x)eL 2 (R".R) for I: VeR"-R. is given hy 2.3G
~
~
ax'ax 2
a2n ax'ax"
~
a21 ax"ax2
a21 ax"a.J("
ax'ax'
ax"ax'
(Hint: Apply 2.3.6. Recall from linear algehra that the matrix of a bilinear mapping BeL(R".Rm;R) is given by the entries B(e,. ") (first index = row index. second index = column index). where {e, •...• e,,} and {/, •...•lm} are ordered bases of R" and R m respectively.) 2.4
PROPERTIES OF THE DERIVATIVE
In this section some of the fundamental properties of the derivative are developed. These properties are analogues of rules familiar from elementary calculus. Let us begin by strengthening the fact that differentiability implies continuity. 2.4.1 Propoaltlon. Suppose VeE is open and I: V ~ F;s differentiable on V. Then I is continuous. In lact. lor each "0 e V there i.~ a constant M> 0 and a 80 > 0 such that 1111 - 11011 < 80 implies 11/(11)- I(IIo~1 ~ MIlII - 110 11. (This is called the Lipsmit:. property.)
Proof.
1111(11)- 1(lIo)II-IIDI(uoHu -11 0 )111 =lIo(u-u()II~lIu-uoll
for
~
11/(11)- I(u o )- DI(uo)(u -11 0 )11
IIII-u olI O. The mIJfJ
r
0,: CO(M,U)-CO(M,F) de/inedby O,(/)=go/ tlbis is tmninolo&Y of Abraham (1963). Various results of this type can be traced badt to earlier works of SoboIev 11939) and Eells (1958).
PROPERTIES OF THE DERIVATIVE
97
is also 01 class C'. The derivative of D. is DO.(f)·h = [(Dg)o f]·h [DO.(f)·h ](x)'" Dg(f(x »·h( x).
Le ••
The formula for DO. is quite plausible. Indeed. we have d . DO.(n·h(x)"" deO.(f+eh)(x)I,-o
d - de/{ (f(x)+eh(\»I._u
By the chain rule this is Dg(f(x»·h(x). This shows that if 0,11 is differentiable. then DO. must be as stated in the propllsition.
Proof. Let! E
CO(M.U). By continuity of g and compactness of M.
1I0.(f)-O,(/')II- sup IIg(f(m»- g(f'(m)~·1 meM
is small as soon as point I .
III - f11 is small; i.e.• 0. is continuous at each
. Let
A,: CO{M. L~(E: F» .... L~(CO(M.E); CO(M. F»
be given by A;(H)(h ...... h;)(m) ... H(m)(h.(m) ..... h;(m») for H e CO(M. L'(E; F». h ...... h, eCo(M. B) and '" eM. The maps A; are dearly linear and are continuous with IIA,II]11 +11/,,(11
+"
)-1,,(11)- DI,,(II)'''II
+IIDI,,(II)'" - gl(II)·/t1l
<ell"l1· For general r use the converse to Taylor's theorem.) 2.4K (ex lemma). In the context of 2.4.18, let
Show that ex, is continuous linear and hence is C". 2.4L Consider the map ~: CI([O. CO([O. given by
I» -.
1»
~(f)(x) - exp[/'(x)]
Show that
~
is Coo and compute
D~.
PROPERTIES OF THE DERIVATIVE
2.4M (H. Whitney [I 943aJ). Letf: U c E expansion
-+
101
Fbe of class Ck+p with Taylor
f(b) .. f(a)+Df(a}·(b-a}+ .. · +
~!Dkf(a)(b-a)·
(i) Show that the remainder Rk(a, b) is Ck+p for b ... a and C P for all a, bEE. If E = F = R, Rk(a, a) = 0, and lim (lb-al;D;+PRk(a,b»)=O,I~i~k.
h-u
.
(For generalizations to Banach spaces, see Toan and Ang [ 1979).) (ii) Show that the conclusion in (i) cannot be improved by consid- ' eringf(x)- Ixlk+p+1/2. 2.4N (H. Whitney (l943b)). Let f: R -+ R be an even (odd) function; i.e. f(x) - f( - x)(f(x) - - f( - x». (i) Show thatf(x) - g(x 2)(f(x) - xg(x 2 ) for some R. (ii) Show that if f is C lk (C 2 k+ I) then g is C k (Hint. Use the converse to Taylor's theorem) (iii) Show that (ii) is still true if k = 00 ,(iv) Let f(x) = Ix12k+ 1+112 to show that the conclusion in (ii) cannot be sharpened. 2.40 (M. Buchner, J. Marsden, and S. Schecter (1983)). Let E = L 4«(0, I)) and let cp: R -+ R be a Coo function such that ep'('-) = I, - I ~ '- ~ I and ep'(A) ... 0 if IAI ~ 2. We assume ep is monotone inc~easing with , ep'" - M for '- ~ - 2 and ep = M for A ~ 2. Let h: E -+ R be given by
h(u) = ~ fo'ep([u(x W)dx (a) ,Show that his C 1 using the converse to Taylor's theorem. (b) Show that if q is a positive integer and E = U«(O, I)), then h is c q - I but is not c q• (c) The formal L2 gradient of h is given by
Vh(u)
=
1
"31/1'(u)
where 1/1('-) = ep(>t). Show that vh: E -+ E is CO but is not CI.
102
BANACH SPACES AND DIFFERENTIAL CALCULUS
(d)
Let f(u)""
111
2
0
lu(x)1 2 dx+h(u). Show that on L 4 ,
f has a
formally non-degenerate critical point at O. yet this critical point is not isolated. 2.5 THE INVERSE AND IMPLICIT FUNCTION THEOREMS
The inverse and implicit function theorems are pillars of nonlinear analysis and geometry. so we give them special attention in this section. Throughout. E. F... . ,are assumed to be Banach spaces. In the finitedimensional case these theorems have a long and complex history; the infinite-dimensional version is due to Hildebrandt and Graves (1927). We first consider the inverse function theorem. This states that if the linearization of the equationf(x) = y is uniquely invertible then locally so is f; i.e. we can uniquely solvef(x)'" y for x as a function of y. To formulate the theorem, the following terminology is useful. 2.5.1 DefInition. A map f: U c E -+ V c F (U, V open) is a C'diffeo".""",_ iff is of class C, is a bijection (that is, one-to-one and onto V), and I is also of class c.
r
2.5.2 1m..... Mapping Theorem. Let f: U c E -+ F be of class C, r ~ I. Xo E U. and suppose Df(x o ) is a linear isomorphism. Then f is a C' diffeomorphism of some neighborhood of Xo onto some neighborhood of f(x o ) and Dr '( y) = [Df( y »)- , for y in this neighborhood of /( X o )'
r- '(
Although our immediate interest is the finite-dimensional case, for Banach spaces keep in mind the Banach isomorphism theorem: if T: E -+ F is linear, bijective, and continuous, then T- I is continuous. (See Box 2.21'.) c. Proof of17teomn 1.5.1. We begin by assembling a few standard lemmas. First recall the contraction mapping principle from Section 1.2. 2.5.3 Lemme. Let M be a complete metric space with distance function d: M X M -+ R. Let F: M -+ M and assume there is a constant >., 0 ~ A < I such that for all x, y E M, d( F(x), F(y».,. Ad(x, y). Then F has a unique fixed point Xo EM; that ;s F(xo) = xo'
This result is the basis of many important existence theorems in analysis. The other fundamental fixed point theorem in analysis is the Schauder fixed point theorem. which states that a continuous map of a
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
103
compact convex set (in a Banach space, say) to itself, has a fixed point-not necessarily unique however. 2.5.4 Lemma. Lei GU E. F) denole Ihe sel of linear isomorphisms from E onlo F. Then GUE. F) c L(E. F) is open. Proof. ' Let
lIall = sup lIa( e )11 ' I'E 1;.
111'11-1
be the norm on L(E. F) relative to given norms on E and F. We can assume E = F. Indeed. if CPo E GUE. F). the map"' ...... CPo 1 0 ", from L(E, F) to L(E, E) is continuous and GL(E. F) is the inverse image of GL({lE). For cp E GUE, E), we shall prove that any'" sufficiently near cp is also invertible, which will give the result. More precisely. II'" - cpll < IIcp III I implies", E GUE, E). The key is that 11'11 is an algehra norm. That is. liP 0 all ~ IIPliliall for aE L(E, E) and PE L(E, E) (see Section 2.2). Since '" = cp 0 (I -' cp - I 0 (cp _. '" cp is invertible. and our norm assumption shows that IIcp - I (cp - '" ~I < I. is sufficient to show that I - ~ is invertible whenever II~II .-1(+ _ .>.-1. Again,
usin& II' 0 alllIii lIallII'li for caE L(E, F), PE
L(F, G),
11+-1(+ _ .>.-I( + _ .>.-111" 11+-11111+ _ .11 211.- 111 2. With this inequality. the limit is clearly zero.
...
To prove the theorem it is useful to note that it is enough to prove it under the simplifying assumptions that Xo = 0, /(x o) = 0, E = F, and Df(O) is the identity. (Indeed, replace/by hex) = Df(xo)-I o(f(x + xo)- /(xo»)') Now let g(x) -"" x - lex) so Dg(O) = O. Choose, > 0 such that II xII'" implies IIDg(xllllii i, which i, ,.~ible by continuity of Dg. Thus by the mean value inequality, IIxll ~ r Implies IIg(x~11Iii ,/2. Let 0.(0) - {x E Flllxll IIii e}. For y E D,./2(0), let g,v(x) - y + x - f(x). By the mean value inequality, if then
and Thus by Lemma 2.5.3, g" has a unique fixed point x in 8,(0). This point x is the unique solution of /( x ) - y. Thus / has an inverse
From (ii) we have (iii) IIr I(YI)_
ous.
r
I(Y2~1" 211 VI
-
Y211. so
r
I is continu-
From Lemma 2.5.4 we can choose, small enough so that Df(x) -I will exist for x E D,(G). Moreover, by continuity, IID/(x)-Ill" M for some M
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
and all x E D,(O) can be assumed as well. If YI' Y2 and X2 = '(Y2)' then
r
IIr I(y,)_ r
l
E
D,/2(O),
XI =
105
r I(y,),
(Y2)- D/(X2)-'(Y' - Y2)1I =
IIx, - X2 - D/(x 2) -1(j(X,)_ /( x2)]11
-IlD/(x2)-'{D/(x 2Hx, - x 2)- /(x,)+ /(x 2)}11 .. MII/(x,)- /(xz)- DI(xz)(x, - x2)1I.
r'
This, together with (iii). shoM! that is differentiahle with derivative ' DI(x)-' at I(x); i.e., D(r ') = ~ 0 DI 0 r' on Vo = D,/l(O). This formula. is C· -, then the chain rule. and Lemma 2.5.5 show inductively that if I C· for I ~ k ~ r. •
r'
r is
This argument also proves the following: If /: U --. V is a C',homeomorphism where U c E, and V c F are open sets, and D/(U)E GL(E. F) for U E U, ,then / is a C' diffeomorphism.
BOX 2.M THE SIZE OF THE NEIGHBORHOODS IN THE INVERSE MAPPING THEOREM
An analysis of the preceding proof also gives explicit estimates on the size of the ball on which I(x) = Y is solvable. t Such estimates are sometimes useful in applications. The easiest one to use in examples involves estimates on the second derivative. '
2.5.1 Corollary. Suppose I: U c E a,.d D/(x o ) is an isomorphism. Let
-+
F is 01 class C', r ;;, 2, Xo E U
L-IiD/(xo)1I and M-II DI(xo)-'U. Assume IID2/(x)U~K for IIx-xoll~R and DR(XO)CU. Let
R, - rnin{ 2~M' R} , R 2 ...
rnin{ ~I
'
2M(L
~ KR I )}
and R) ...
tWe thank M. Buchner for providing this formulation.
:t·
106
BANACH SPACES AND DIFFERENTIAL CALCULUS
Then f maps the ballllx - xoil..;; R2 dil/eomorphically onto an open set containing the balllIy - f(xo~l..;; R]. For YI' Y2 E DR,(f(xo», we have II/I(YI)- II(Y2~1";; 2LIIYI - Y211. . Proof.
We can assume Xo = 0 and I(xo) = O. From
DI(x) = DI(O) + foID2/(tX)Xdt we get IIDI(x~1 ~ L + Kllxll for x
E
DR(O). From the identity
DI(x) - DI(O){ 1+ [DI(O)] " Ifo'D2/(tX)Xdt} and the fact that
11(1+ A)-'II ... I +IIAII+IIAII2 + ."
=
I-:IAII
for IIAII < I (see the proof of 2.5.5) we get
IIDI(x)-'1I..;;2L if IIxll ... R and
IIXII"'2~K'
i.e., if IIxll ..;; R I'
As in the proof of the inverse function theorem. let gy(x) = [DI(On- I(y + DI(O)x - I(x». Now '
IIg,,(x)lI..;; M(IIYII+ IlfoID2/(tX)Xdtll) ... M(IIYII+ Kllxll). Hence for lIylI";; RI/2M. S" maps DR,(O) to DR(O). We similarly get from the mean value inequality and the estimate
IIg,.(xl)-g,.(X2~1"'!lIxl-X211
if IIxll..;; R I • Thus. as in the previous proof. II: DR,/2M(O) -+ DR, is defined. Note that by the mean value inequality,
II/( x )11 ... (L + Kllxll)lIxll so if IIxll..;; R 2 , then II/(x~I'" R I /2M. The rest now follows as in the proof of the inverse mapping theorem. •
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
107
In the study of manifolds and submanifolds. the argument used in the following is of central importance.
2.5.7 Implicit Function Theorem. Let VeE, V c F be open and I: V X V - G be C', r ~ I. For some Xo E V, Yo E V assume Dzi(x o' Yo): F- G is an isomorphism. Then there are neighborhoods l.{) of XII and Wo of fCl(). Yo) and a unique C' map g: V() X!Vo - V such that for all (x. w) E Vo X Wo' I(x,g(x,w»=-w.
Proof. Consider the map 41: VXV-EXG,(x.y)'+(x'/(x.y». Then D4I(x o• Yo) is given by
~4I(xo. Yo)-(x
i•
YI) = (DI/(x/(). "0) ),1
D2 f( XII' 0 Yo
»)( YIXI')
which is easily seen to be an isomorphism of E X F with E X G. Thus 41 has a unique C' local inverse. say 41 I: l.{) X J.t() - V x V.( \. w) .... (x. g(x. w». The g so defined is the desired map. _ Applying the chain rule to the relation explicitly compute the derivatives of g: '
I( x, g( x, w» =
w.
one can
Dlg(x. w) = - [Dzf(x. g(x. w»r 10 Dd(x. g(x, w»
2.5.8 Corollary. Let VeE be open and I: V - F be C. r ~ l. Suppo.se DI(xo) is surjective and kerDI(x o ) is complemented. Then I(V) contains a neighborhood oll(x o )' .
Proof.
Let EI = ker Df(x o ) and E = E. X E2 • Then Dzf(xo}: E"l -+ F is' an isomorphism. Thus the hypotheses of Theorem 2.5.7 arc satisfied and SO I(V) con~ains Wo provided by that theorem. _ . Since in finite-dimensional spaces every subspace splits. the fotesoina corollary implies that if I: VcR II - R m, n ~ m and the Jacobian of I at every point of V has rank m, then I is an open mapping. 'Ibis statement generalizes directly to Banach spaces, but it is not· a consequence of the implicit function theorem any more. since not every subspace is split. This
108
BANACH SPACES AND DIFFERENTIAL CALCULUS
,0e5
result back to Graves (1950). The proof given in Box 20SB follows Luenberaer (1969).
2.5.9 Propoeltlon (L«Ill S"rjectivity).
Iff: VeE -+ F is C l and Df(uo) is onto for some U o E V, then f is loca/~v onto: i.e., there exist open neighborhoods VI of U o and VI of f( u) such that fl VI: VI -+ VI is onto. In particular, if Df(u) is onto for u E V, then f is an open mapping.
BOX 2.5B PROOF OF THE LOCAL SURJECTIVITY THEOREM Proof. The key is to recall from Section 2.1 that E/ker Df(u o ) = Eo is a Banach space with norm lI[xll = inf{lIx + uiliu E ker Df(u o )}, . where [x) is the equivalence class of x. To solvef(x) .. y we set up an iteration scheme in Eo and E simultaneously. Now Df(uo) induces an isomorphism T: Eo -+ F, so r- I E L(F, Eo) exists by the Banach isomorphism theorem. Write
f(x) = y
as f(uo+h)-y,
i.e.,
T-I(y-f(uo+h»-O,
where x .. Uo + h. To solve this equation, define a sequence L" E Ejker Df(u o ) (so L" is a coset of kerDf(uo» and h" E L" c E inductively by Loker Df(u o ), ho e Lo small, and
(I) and selectin, h" E L" such that
(2) The latter is possible since ilL" - L,,-.I1 = infOlh - h,,_ .Illh E Ln}' Since h"_1 E L"_I' L"_I = r-1(Df(uo)h,,_I)' so
Subtractin, this from the same expression for L" _ I gives L" - L,,_I - - r-1(f(UO + h,,_I)- f(uo - Df(uo)(h,,_I- h,,-2»'
+ h,,_2) (3)
109
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
For e> 0 given. there is a neighborhood U of U o such that II D/{ u)II e U, since/is ct. Assume inductively that 110 + e U arid "0 + h,,-2 e U. Then from the mean value inequality, '
h._,
D/{lIo~1 < e for
(4) By (2),
Thus if e is small.
Starting with ho small and inductively in U since
IIh, - holl < !lIholl, Uo + h"
remains
IIh"lI..; IIholl+llh,- holl+llh 2 - h.ll+ ... +lIh" - h,,_.l1 ..; ( 1+ ! + ... + 2..t-, )lIholi ..; 21lholl· It also follows that h" is a Cauchy sequence, so it converges to some point. say h. Correspondingly. L" converges to L aild h E L. Thus, from (I),
and so
This proves that for y near Yo = f(uo).!(x) = y has a solution. If there is a solution g(y)= x which is C', then Df(xo)o Dg(yo) = I and so rangeDg(yo) is an algebraic complement to ker Df(xo)' It follows that if range Dg(yo) is closed, then ker Df(x o } is split. In many applications to nonlinear partial differential equations. methods of functional analysis and elliptic operators can be used to show that ker D/(xo) does split. even in Banach spaces. Such a splitting theorem is called the Fredholm alternatwf', For illustrations of this idea in geometry and relativity, see hscher and Marsden [1915),[1919), and in elasticity, see Marsden and Hughes [1982. ch. 6). For such applications, 2.5.8 suffices.
The locally injective counterpart of this theorem is the following.
110
BANACH SPACES AND DIFFERENTIAL CALCULUS
2.5.10 PropoIltIon (LOCtIIllfjen;.,;ty). Let I: U c E - F be a C· map. DI(uoXE) be closed in F and DI(uo)E G1..(E. DI(uoXE». Then there exists a neighborhood V 01 u o• V c U on which I is injective. The inverse I( V) - U is Lipschitz continuous.
r-.:
Since (DI(uoW·
Proof.
E
L(D/(uo)(E). E). there exists M > 0 such that
II DI( Uo )ell ~ Mllell for all e E E. By continuity of DI. there exists r > 0 such that IID/(u)- D/(uo~1 < M/2 whenever lIu - uoll < 3r. By the mean value inequality, for e •• e2
E
D,(u o )
II/(e.)- l(e2)- D/(uo}(e. - e 2 )11 Eo;
sup IIDI{e.
+ f(e2
- e.»- DI(uo)II·lIe. -
e.~1
is called the linearization of I about xo' (Phrases like "first variation," "first-order deformation." and so forth are also used.) 2.5.11
Example.
Let E = alI C· functions I: [0. I] -+ R with
11111. =
sup I/(x)1 + sup , E
(0 •• (
,
E
Id/(X) I ~
(0 •• 1
and F= all CO functions with 11/110 = sup, E' (Il.II I/(x)!. These are Banach spaces (see Exercise 2.1 G). Let F: E -+ F. F(/ ) = dfI dx + t-'. It is easy to check. that F is C«l and DF(O) = dldx: E -+ F. Clearly DF(O) is surjective (fundamental theorem of calculus). Also. ker DF(O) consists of E." all
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
111
constant functions. This is complemented because it is finite dimensIonal; explicitly, a complement consists of functio~s with zero integral. Thus Corollary 2.5.8 yields the following. There is an t> 0 such that if g: [0.1) -+ R. is a continuous lunction with Ig(x)1 < t. then there is a C' lunction I: [0.1) -+ R such that
The following series of consequences of the inverse function theorem are important technical tools in the study of manifolds, The first two results give. roughly speaking. sufficient conditions to "straighten out" the domain (respectively. the range) of I in a neighborhood of a point. thus \llaking I look like an inclusion (respectively. a projection), 2.5.12 Locallmm....lon Theorem. Let I: U c E -+ F be 01 ciass C. r ;. I. Uo E U and suppose that D/(uo) is one to one and has a closed split image' F, with closed complement F2, (II E = R m and F = R n. assume only that DI(u o ) has trivial kernel.) Then there are open sets U' c F. and V c EeF2 where I( uo) E U' and a C' diffeormorphism cp: U' - 0 V such that (cp 0 1)( e) (e.O) lor all e E V neE X{O}) c E. The intuition for E = £, = R2. £2 = R (i,e.• m = 2.,. = 3) is given in Fig. 2.5.1. The function cp nattens out the image of f. Notice that this is intuitively correct; we expect the range of I to he an m-dimensional' "surface" so it should be possible to natten it to a piel:e of R m , Note that the range' of a linear map of rank m is a linear suhspace of dimension eltactly m. so this result eltpresses. in a sense. 'a generalization of the linear case. Proof. Define g: UXF2 cExF2 -+F=F,eF2 by g(u.v)=/(u)+(O.o) and note that g(u.O)=/(u). Now Dg(uo,O)=(DI(uo),i,)eGL(Ee F2 • F) by the Banach isomorphism theorem (IFl denotes the identity mapping of F2 and (A. B) E L(EeE, FeF') for A e L(E. F). and Be L(E. F) is defined by (A. B)(e. e') = (Ae.&').) By the inverse function, theorem there eltist open sets U' and V such that (uo.O) E V c EeF2• and g(uo'O)I( U o ),E U' c £ and a C' diffeomorphism cp: U' -+ V such that cp - • - gl V. Hence for (e.O) E V. (cp 0 I)(e) = (cp 0 g)(e.O) ... (e.O),. .
2.5.13 Local Subm....lon Theorem. Let I: U c E -+ F be 01 closs C. r ~ I. U o e U and suppose D/(u o ) is surjective and has split kernel E2 with closed complement E •. (II E = R'" and F= R". assume only that rank (DI(u o n.) Then there are open sets U' and V such that eU'cU C B
»-
"0
112
BANACH SPACES AND DIFFERENTIAL CALCULUS
Figure 2.5.1
and V C FeE2and a C' diffeomorphism 1/1: V --+ U' such that (f 0 1/1)( u, v) = u for all (u, v) E V.
The intuition for EI ... E2 = F = R is given in Fig. 2.5.2, which should be compared to Fig. 2.5.1. Proof. By the fundamental isomorphism theorem (Section 2.2); Df(u o) E 0L(E1, F). Defineg: U C E 1eE2 --+ FeE2 by g(u l • u2)= (/(u l , U2)' u2)and note that
so that Dg(u o ) E OL(E, FeE2 ). By the inverse function theorem there exist open sets U' and V such that U o E U' cUe E, V c FeE2 and a C' diffeomorphism 1/1: V --+ U' such that 1/1 1 = gl U'. Hence if (u, v) E V. (u, v) - (g 0 1/1)( II. t') = (/( 1/1 ( u, t' )).1/I2( II. I'» where 1/1 = 1/11 X 1/12: i.e.. 1/12( II. ,,) = v . and (/0 I/I)(u. v)= u. • .
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
/0 '" -
/ =
constant
113
constant
.~ .J-------E,
F
F
F1gu... 2.5.2
2.5.14 Corollary (Loclll Representation Theorem). Let f: U c E -+ F be of class C. r ~ I. U o E U and suppose Df( u o) has closed split image F, with closed complement F2 and split kernel E2 with closed complement Pl' (II E - R"', F- R If , assume that rank (DI(u o k, k" n, k" m, so that F2 R If - k, F," Rk. E, - Rk, E2 - R",-k.) Then there are open sets U' and V such that Uo E U~ cUe E, V c F,eE2 and a C' difleomorphism "': V -+ (j' such that (f 0 '" Xu, v) = (u. 1J( u. v». where 1J: V -+ F2 is a C r map satisfying D1J(u o ) = o.
»"
Proof. Write f = f, X f2' where /;: U --+ F;. i -1,2. Then f, satisfies the conditions of 2.5.13, and thus there exists a C' diffeomorphism!/-: V c Fie E2 -+ U'c E such that U, !/-)(u. v) = u. Let 1J = 12 !/-. • 0
0
To use 2.5.12 (or 2.5.13) in finite dimensions, we must have the rank of Df equal to the dimension of its image space (or the domain space). However, we can also use the inverse function theorem to tell us that if Df(x) has constant rank k in a neighborhood of x(). then we can straighten out the domain of f with some invertible function", such that f 0 ' " depends only on k vanables. Then we can apply proposition 2.5.12. This is the essence of the following theorem. Roughly speaking, the theorem says that if DI has rank k on R "', then m - k variables are redundant and can be eliminated. As a trivial example, if I: R2 --+ R' is defined by I(x, Y)- x - Y. DI has rank I, and so we can express I using just one variable. namely, let !/-(x. Y) = (x + Y, y) so that (f o!/-)( x. y) ... x. which depends only on x. Let f: U c E -+ F be of class C', r ~ I, U o E U and suppose Df( u o) has closed split image F, with closed complement F2 and split kernel E2 with closed complement E,. In addition. Q.fsume thatlor'all u in a neighborhood 01 U o E U. Df(u)(E) is a closed subspace of F and DI(u)IE,: E,-+Df(u)(E) is a Banach spare isomorphism. (If E=R":' and F=RIf,
2.5.15 Rank Theorem.
114
BANACH SPACES AND DIFFERENTIAL CALCULUS
assume only that rank (D/(u» == k lor u in a neighborhood of uo.) Then there exist open sets VI C Fl eE2,v2 C E, VI C F. V2 C F and C' dilleomorphisms cP: VI - V2 and 1/1: VI - V2 such that (cp ° I ° 1/1 )(x,e) = (x.O).
The intuition is given by Fig. 2.5.3 for E = H2. F = H2 and k == 1.
£,
~~~~' f =
constant
---_. £2
range of /
.
~
v,
£2
F,
Figure 2.5.3
Rmlilric. It is clear that the theorem implies Eleker(O/(,,» = E and Ol( u)( E)eF2 - F for u in a neighborhood of U o in V. because cp 0 I ° 1/1 has these properties. These seemingly stronger conditions can in fact be shown directly to be equivalent to the hypotheses in the theorem by the use of the openness of 01..(E. E) in L(E. E).
By the local representation theorem there exists a C' diffeomorphism 1/1: V.cF.eE 2 -V2 cE such that j(x.y)=(/ol/l)(x.y)= (x.1J(x.y». Let P,:I F-F. be the projection. Since OJ(x.y)·(w.e)= _ ( w. 01J( x. y ).( w. e». it follows th~t (P" ° OJ( x. y»( w. e) == ( w.O). for we F•• ~E E2 • In particular P" ° DJ(x. y)IF. X{O}-l". which shows ~hat Proof.
· THE INVERSE AND IMPLICIT FUNCTION THEOREMS
115
Dj(x. y)lFj X {O}: FI X{O} -+ Dj(x, y)(F I EBF2 ) is injective. In finite dimensions this implies that it is an isomorphism, since dim FI = dim(Dj(x, y) (FI ED E 2 In infinite dimensions this is our hypothesis. Thus D/(x, y)o PF ID/(x, y)(FI eE2 ) = identity. Let (w, D.,,(x, y)(w, E Dj(x, y)(FI EB'E2 ). Since
».
e»
(Di(x. y) 0 PF ,)( w, D.,,(x, y). (w,
e»
= Di(x. y ).( w,O) = (w. D.,,(x. y ).( w.O» = (w, D I
,< x. y)·w),
we must have D21/(x. y)·e = 0 for all e E E 2 : i.e .. D21/(x. y) = O. But D 2 j(x.y)·e=(O.D21/(x,y)·e). says that D 2 i(x.y)=0; i.e.• i does not depend on y E E 2 • Define i(x) = j(x. y) = U 0 x. y),' so j: PF,(V) C FI -+ F where Pi,: FIEBE2 -+ FI is projection. Now f satisfies the conditions of 2.5.12 at Pi ('" I( Un» and hence there exists a C' diffeomorphism' ep: , VI -+ V2 , where VI' V2 c F. such that (ep 0 /)(z) = (z,O): i.e.• (ep 0 f 0 ' " )(x, y)
+)(
-
= (x.O).
•
2.5.16 Example (Functional Dependence). Let C' c R" he an open set and let the functions fl .... ./,,: U -+ R be smooth. The functions fl' ..../" are said to be functionally dependent at Xu E U if there i~ a neighhorhood. Vof the point UI(xo).... ./,,(xo))ER" and a smooth function F: V-+R such that DF'* 0 on a neighborhood of (fl( x o )•... •/,,( xu». an,d
F( fl ( X ) •... ./" ( x»
=
0
for all X in some neighborhood of xo' (i) Show that if fl .... ,f" are functionally dependent at determinant of Df, denoted
(ii)
XO'
then the
H.
on a neighborhood of Xo. then show that f ..... ./" are functionally dependent, and further, that
for some G.
116
BANACH SPACES AND DIFFERENTIAL CALCULUS
s.t.tio& Let 1- (f, •... •1,,)· (i) We have Fo/-O. so DF(f(x» ° DI(x) = O. Now ifJ/(xo). o. DI(x) would be invertible in a neighborhood of Xo. implying DF( I( x» - o. By the inverse function theorem. this implies DF(y)'" 0 on a whole neighborhood of I(x o)' (0) The conditions of (ii) imply that DI has rank n - I. Hence by the rank theorem. there are mappings fP and '" such that fP 0/ 0",(x, , .. . ,x,,) = (XI'" .• X,,_I'O).
Let F be the last component of fP. Then
F(/I •... .!,,) = 0 Since fP is invertible. DF. O. It .follows from the implicit function theorem that we can locally solve
F(f,.···.I,,)=O for I" -G(f, •...• I,,-I). provided we can show,;\ = saw before.
aF/ay" .0. Now, as we
DF(f(x» ° DI(x) ... 0, or, in components withy -/(x).
(;F ,... ,;F) YI
y"
= (0,0 •...• 0).
If aF/ay" == 0, we would have
- (0,0, ... ,0)
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
117
or
aF ( a"'" 'I
since the
~uare
aF ) -a- (0,0, ... ,0) ' .. -I
matrix is invertible by the assumption that
a(/I,···,/.. _I) a(xl" ... ,X.. _I )
.. 0
This implies D/ - 0, which is not truC!. Hence aF/ aY.. .. 0, and we have the desired result. The reader should note the analogy between linear dependence and functional dependence, where rank or determinant conditions are replaced , by the analogous conditions on the Jacobian matrix. • BOX 2.5C THE INVERSION !MAP
,
Let E and F be isomorphic Banach spaces and consider the inversion map 9: GL(E, F) -+ GL(F, E); 9(.)- .-1. We have shown that 9 is C"" and
+
for" e GL(E, F) and E L(E, F). We shall give below the formula for D"9. The proof is straightforward and done by a simple induction argument that will be left to the reader. Define alt+l:L(F,E) X ••• XL(F,E)-+'LIr(L(E.F);L(F,E»
(~ + 1) tim~s by
where
XI
e L(F, E), ; -I, ... ,k + I and +, E L( E. F). ; -I ..... k. Let
9x· .. x9:GL(E,F)-+GL(F,E)X ... xGL(F,E).
O} .
."i,.±( X I•••• ,x "+1) -_ (I x , ... ,x ;-1 ,.x ;+1 , ... ,x "+1)., 1{1; projects the hemisphere containing the pole (0 ..... ± 1..... 0) onto the open unit ball in the tangent space to the sphere at that pole. It can be easily verified that this atlas and the previous one in 3.1.2 with two charts are equivalent. The overlap maps of this atlas are
(1{Il D1{I;±fl(yl ..... y") _ (y' ..... yi-I ±b-(y,)2- ... _(yi_I)2_(yl+I)2_ ... _(y")2.yi+I ..... y ").
Next we show how any manifold is a topological space.
3.1.5 Deftnltlon. Let M be a differentiable manifold. A subset A C M is ""." ffor efJch a E A there is an admissible local chart (U. Ip) such that a E U andOcA. . 3.1.8 propolltlon. The open sets in M define a topology. Proof. Take as basis of the topology the family of finite intersections of chart domains. • tThis defines a Coo manifold. A C' manifold is defined similarly. We have not burdened the text by C"s since the C" case is tbe one th;tf occurs most commonly.
126
MANIFOLDS AND VECTOR BUNDLES
3.1.7 DefInition. A differentiable manifold M is an ,,-manifold when every chart has values in an n-dimensional vector space. Thus for every point a E M there exists an admissible local chart (U, cp) with a E U and cp( U) c R". We write n - dim M. An n-man;fold will mean a Hausdorff, differentiOble n-manifold in this book. A differentiable manifold is a !inite-dimnuional lIIIIIIifolti if its connected components are all n-manifolds (n can vary with the component). A differentiable manifold is called a Hilbert manifoltl if the model space is a Hilbert space. t No assumption on the connectedness of a manifold has been made. In fact. in many applications the manifolds are disconnected (see Exercise 3.1 C). But since manifolds are locally arewise connected. their components are both open and closed.
3.1.8 Exampl..
.
A. given by B. given by
Every discrete topological space S is a O-manifold. the charts being ({ s}. cp,). CP,: $'"-+ O. S E S. Every Banach space is a manifold its differentiable structure being the atlas with the single identity chart. C. S" with a maximal atlas generated by the atlas with two charts described previously makes S" into a n-manifold. The resulting topology is the same as that induced on S" as a subset of R,,+I. D. A set can have more than one differentiable structure. For example. R regarded as a set has the following incompatible charts
They are not compatible since CP2 0 CPII is not differentiable at the origin. Nevertheless. these two resulting structures tum out to be diffeomorphic (Exercise 3.2H). but two structures can be essentially different on more complicated sets (e.g.• S7). E. Essentially the only one-dimensional connected manifolds are R and SI. This means that all others are diffeomorphic to R or SI (diffeomor-
*
t One can form a manifold modeled on any linear space in which one has a theory of differential calculus. For example mathematicians often speak of a "Frechet manifold." a "LCTVS manifold,".etc. We have chosen to stick with Banach manifolds here to avail ourselves of the inverse function theorem. (See Exercise 2.5G). *nat S1 has two nondifCeomorphic differentiable structures is a famous result of J. Milnor (1956). Recently similar phenomena have been found on A 4.
MANIFOLDS
127
.Flgure 3.1.3
s'2 + "handles"
Figure 3.1.4
phic will be precisely defined later). For example. the circle with a knot is diffeomorphic to st. (See Fig. 3.1.3). See Milnor (1965) for proofs. F. A general two-dimensional connected manifold is the sphere with "handles" (see Fig. 3.1.4).t This includes. for example. the torus, whose precise definition will be given in the next section. G. (Grassmann Manifolds). Let E be a Banach space and Consider the set G(E) of all split subspaces of E. For FE G(E). let G denote o~e of its complements. i.e.• E = FeG, and define UG = {H E G(E)IE = HeG}. and 'PF.G: U(;...co L(F. G) by 'PF.dH) = 'If(H.G)o 'If(H. F)
where 'ifF: E
...co
-I.
F. 'lfG: E...co G denote the canonical projections and 'If(H. G)
tTbe classification of two-manifolds is described in Mas. and '" to open subsets of U and V. then I" ..~ .. is also C. Finally. note that if I is C on open submanifolds of M. then it is C on their union. That I is C' now follows from the fact that any chart of M can be obtained from the given collection by change of diffeomorphism. restrictions. and/or unions of domains. all three operations preserving the C' character of I. The converse is a direct consequence of the definition. •
cr.
cr.
Any C' map from (open subsets of) E to F in the usual sense is C in the sense of Definition 3.2.5. Other examples of coo maps are the antipodal map x .... - x of Sit and the translation map on T" by (8 1......8,,) given by (e l "
..... e l '.) ....
(e l ("
... ,,) •• ••• e ll
'.+'.».
3.2.7 Propoeltlon. II I: M -- Nand g: N -- Pare C maps. then so is
go f. Proof. This follows directly from the previous proposition and the composite mapping theorem. •
3.2.8 DefInition. A map I: M -- N. where M and N are manilolds. is called a C dif/eomorpIIum ill is 01 class C. is a bijection. and I: N -- M is 01 class C'. II a diffeomorphism exists between two manilolds, they are called dif/eomorpIIk.
r
I t follows from 3.2.7 that the set Diff'( M) of C diffeomorphisms of M to M forms a group under composition. This very large and complex group will be encountered again several times in the book.
VECTOR BUNDLES
133
Ex...... 3.2A Sbow that (i) if (U. If) is a chart of M and I/!: 'J'( U) .... Y c F is a diffeomorphism. then (U. I/! 091) is an admissible chart of M and (ii) admissible local charts are diffeomorphisms. 3.2B A C' diffeomorphism that is also a C' map is a C diffeomorphism. (Hillt: Use the comments after the proof of 2.5.2.) 3.2C Show that if N; c M, are submanifolds. i = 1•...• 11 then NI X •.• x N" is a submanifold of MI X ••• x M". 3.20 Show tbat every submaoifold N is locally closed in M; i.e.• every point n e N has a neighborhood U in M such that N () U is closed in U.
3.2E
Show that /;: Mi .... N;. i == I •...• n are all C' iff II X ... X fit: MI X .... X Mit .... NI X ... X Nit is C'.
Let M be a set and {Mi},EI a covering of M. each Mi being a manifold. Assume that for every pair of indices (i. i). MI () M j is an submanifold in both M j and ~. Show that there exists a unique manifold structure on M for which the M; are open submanifolds. The differentiable structure on M is said to he obtained by the collation of the differentia~le structures of MI' 3.20 Show that the map F ..... FO - {u E r l ulF= O} of G(E) into G( E*) is a COO map. If E = E·· (i.e.• E is reflexive) it restricts to a Coo diffeomorphism of G"(E) onto G,,(E) for all n == 1.2'" . 3.2H Show that the two differentiable structures of R defined in 3.1.80 are diffeomorphic. (Hint: Consider the map .lC ... XII'). 3.2F
open
3.3 VECTOR BUNDLES Roughly speaking, a vector bundle is a manifold with a vector space attached to each point. During the f!irmal definitions we may keep in mind the example of the n-spheres S" c R' . I. The collection of tangent planes to S" (regarded as vector spaces) form a vector bundle. Similarly. the collection of normal lines to Sit form a vector bundle. The definitions will follow the pattern of those for a manifold. Namely. we obtain a vector bundle by smoothly patching together local'vector bundles. The following terminology for vector space products will be usefut
3.3.1 DefInition. Let E and F be vector spaces with U an opell subset of E. We call the Cartesian product U X Fa IoeIII f'«tIIr""". We call U the ".. ¥'Gu. which can be identified with U X{O}. the zem ,ect;"'. For u E U. {u}X F is called the liber over u. which we can endow with the vector space
134
MANIFOLDS AND VECTOR BUNDLES
structUl'e of F. The map .: U X F - U given by .( u, /) = u is called the 0/ U X F, [Thus, for u E U, the fiber over u is .-'(u). Also note that U X F is an open subset of Ex F and so is a local manifold.)
projectiolr
Next, we introduce the idea of a local vector bundle map. The main idea is that such a map must map a fiber linearly to a fiber.
3.3.2 Definition. Let U X F and U' X F' be local vector bundles. A map 'P: U X F - U' X F' is called a C' Ioclll veclor btuuIk map if it has the form 'P(U, f)- ('P.(u).IP2(u)·f) where 'P.: U -U' and'P2: U -> L(F, F') are C. A loeal vector bundle map that ;s a bijection ;s called a local vector bll1lllk i.roInorpIWm. (See Figure 3.3)
A local vector bundle map 'P: U X F -> U' X F' maps the fiber {u} X F into the fiber {'P.(u)}X F' and so restricted is linear. By Banach's isomorphism theorem it follows that a local vector bundle map 'P is a local vector bundle isomorphism iff 'P2(u) is a Banach space isomorphism for every uEU.
BOX 3.3A SMOOTHNESS OF LOCAL VECTOR BUNDLE MAPS
In some examples, to check whether a map 'P is a local vector ' bundle map. one is faced with the rather unpleasant task of verifying that 'P2: U -> L(F, F') is coo. It would be nice to know that smoothness of 'P as a function of two variables suffices. This is the context of the next proposition. We state the result for Coo, but the proof contains a C' result (with an interesting derivative loss).
3.3.3 Propo.atlon. A map 'P: U X F -> U' X F' ;s a Coo local vector bundle map i/l 'P is Coo and is of the form 'P(U, f) = ('P.(u), 'P2(U)· f), where 'P.: U -> U' and 'P2: U -> L(F. F'). Proof. (M. Craioveanu and T. Ratiu [1976]) The evaluation map ev: L(F, F')X F - F'; ev(T, f) = T( f) is clearly bilinear and continuous. First assume 'P is a C' local vector bundle map. Then 'P2: U -> L ( F, F') is COO. Now write 'P2(U)·/- (ev o ('P2 X I)(u, f). By the composite mapping theorem. it follows that 'P2 is COO as a function of two
variables. Thus'P is Coo by 2.4.12 (iii). Conversely, assume ,,(u. f) ... (".(u), 'P2(U)· f) is Coo. Then again by 2.4.12 (iii). ",(u) and "2(U)'/ are Coo as functions of two variables.
VECTOR BUNDLES
.
135
We have to show that '1'2: U - L(E. F') is COO. For this it suffices to prove the following: if h: U X F - F' is C. r ~ I and such that h( u, . ) e L (F, F') for all u e U. then the map h: U - I. ( F. F'). ii( u) h(u,') is C,-I. This will be shown by induction on ·r. Ir r - I we prove continuity of h in a disk around Uo e U in the following way. By continuity of Dh. there exists f > 0 such that for all u e D.(u o} and .,e D.(O).IID,h(u. "~I ~ N for some N '> O. The mean value inequality yields IIh( u.,,)- h( u',")11 ~ Nllu - 11'11
for. all u. u'e D.(uo) and "e D.(O). Thus IIh(u)-h(u')II= sup 11"11 co I
IIh(u,")-h(u'.")II~ Nllu~u11. f
proving that h is continuous. Let r > I and inductively assume that the statement is true for r -I. Let S: L(F, L(E. F'» - L(E. L(F. F'» be the canonical ispmetry: S(T)(e)'/= T( f)·e. We shall prove that
(I)
Dh=So Dlh,
C:
where D,h(u)'v= Dlh(u.v). Thus. if his Dlh is C- I • by induction Dlh is C- 2. and hence by (I) Dh will be C' 2. This will show that his C'- I. In order that relation (I) make sense we need to first show that Dlh(u.·) e L(F. L(E. F'». Since Dlh(u.o)·",'"" lim [h(u + t"')- h(u)]'v/t- lim A"o. 1-0
"-"00
for all v e F. where
A,,'" n( h( u+
*"'
)-h(u») e L(F. F'),
it follows by the uniform boundedness principle (or rather its coroUary 2.2.21) that D.h(u, )o..,e L(F, F). Thus (0...) .... D1h(u• .,),,,, is Hnear continuous in each argument and hence is bilinear continuous (Exercise 2.21). and consequtatly ., .... Dlh(u,o) e L(E. F) is linear and continuous. Relation (1) is proved in the following way. Fix Uo e U and let t and N be positive constants such that 0
IID,h(u,.,)- Dlh(u',"~I" Nllu - u11
.(2)
136
MANIFOLDS AND VECTOR BUNDLES
for all II, II' E Dz.( uo) and v E D.(O). Fix for the moment u' E Dz,( uo), v E D,(O) and apply the mean value inequality to the C-' map g(u) - h(u, v)- D,h(u', v)·u to get IIh(u+ w,v)-h(u,v)-D,h(u',v)'wll -lIg(u + w)- g(u)1I ~ IIwll sup IIDg(u + Iw)1I IEIO.')
-lIwll sup IID.h(u + tw,v)- D.h(u', v)1I IEIO.')
for w E
D.< uo). Letting u' ..... u and taking into account (2) we get IIh(u + w, v)- h(u, v)- D.h(u, v)·wIl4it Nllwll2;
i.e.,
lIiI(u + w)·v- iI(u)·v- [(S 0 D,h )(II),w](v)1I4it Nllwllz for all v E D,(O), and hence
lIiI(1I + w)-il(II)-(S 0 D,h )(II)'wIl4it N llwll2 e
thus proving (I). • Any linear map A E L(E, F) defines a vector bundle map " .. : Ex E ..... E X F by ,,( II,.) - (u, A.). Another example of a vc;ctor bundle map was encountered in Section 2.4. If I: U c E ..... V c F (s C+', then Tf: U X E ..... Vx Fis C' and is given by TI(u,.) = (f(u), DI(u)·.). Using these local notions, we are now ready to define a vector bundle.
FIguN 3.3.1
VECTOR BUNDLES
137
3.3.4 DefInition. Let S be a set. A local buntIk mart of S is a pair ( w, 'P ) where We Sand 'P: W c S ..... U X F is a bijection onto a local bundle U X F.
(U, F depend on 'P.) A vector blllfllk tltltu on S is a family ~ - {(w,. cPl)} 0/ local bundle charts satisfying: (VBI) .. (MAl of 3.1.1) (~ covers S) and, (VB2) For any two local bundle, charts (w,. 'Pi) and (W,. 'Pj) in ~ with W, n JJj ., 12J, 'PI ( W, n JJ:;) is a local vector bundle. and the overlap map 1/Ij l = 'Pj 0 'PI- I restricted to 'PI ( W, n Hj) is a Coo local vector bundle isomorphism. If ~ I and ~2 are two vector bundle atlases on S. we say they are VB -.",iNlnlt i/ ~ I U '!S2 is a vector bundle atlas: A vector "-dk strwtllr. on S is an equivalence class of vectot bundle atlases. A vector""" E is a pair (S, CV), where S is a set and CV is a vector bundle structure on S. A chart in an atlas ofCV is an admissibk vector""" dtiIrt of E. As with manifolds. we often identify E with the underlying set S.
The intuition behind this definition is depicted in Figure 3.3.2.
FIgu... 3.3.2
138
MANIFOLDS AND VECTOR BUNDLES
As in the case of manifolds. if we make a choice of vector bundle atlas
'!6 on S then we obtain a maximal vector bundle atlas by including all charts whose overlap maps with those in '!6 are COO local vector bundle isomorphisms. Hence a particular vcctor bundle atlas suffices to specify a vcctor bundle structure on S. Vector bundles are special types of manifolds. Indeed (VBI) and (VB2) give (MA I) and (MA2) in particular. so '\I'induces a differentiable structure on S. 3.3.5
Definition.
For a vector hundle E
=
(S. '\I') we define the ~el'O
section ( or lHIse) hy
B = {e E Elthere exists (W. 'P) E '\I'with e = 'P I( u.O)}. Hence B is the union of all the zero sections of the local vector bundles (identifying W with a local vector bundle via 'P: W --+ U X F). If (U. fP) E ~ is a vector bundle chart. and bE U with 'P( b) = (u.O). let E,,;. denote the subset fP - I({ u} X F) of S together with the structure of a vector space induced by the bijection 'P.
The next few propositions derive basic properties of vector bundles that are sometimes included in the definition.
3.3.6 Proposition. (i) If b lies in the domains of two local bundle charts 'PI and 'P2' then E",., = E" .• ,. where the eQ!Jality means equality as topological spaces and as vector spaces.
'
For e E E. there is exactly one b E B such that e E E",." for some (and therefore 0/1 ) (U.'P,). (iii) B is a submanifold of E. (iv) The map 'IT. defined by 'IT: E --+ B. 'IT(e) = b [in (ii)l is surjective and COO.
(ii)
Proof.
(i) Suppose 'P,(b)=(ul.O) and 'P2(b)=(u 2 .O). We may assume that the domains of 'PI and 'P2 are identical. for Eh . q• is unchanged if we restrict'P to any local bundle chart containing h. Then /l = 'PI 0 'P! I is a local vector bundle isomorphism. But we have
Hence E",., = E" .• , as sets. and it is easily seen that addition and scalar multiplication in E",., and E".", are identical as are the topologies. For (ii) note that if eEE. fP,(e)=(ul.J.). fP2(e)=(u 2./2)' b;= 'P.'(u,.O). and b2 'Pi '(U2'0). then a(u2./2) os (u,.I,). so CI gives a linear OK
VECTOR BUNDLES
139
isomorphism {u 2}X F2 -+ {ul}X Fl' and therefore CPI(b2 ) = 0(U2'0) = (ul.O) = CPI(b l ). or b2 = bl' To prove (iii) we must verify that for b E B there is an admissible chart with the submanifold property. For such a manifold chart we choose an admissible vector bundle chart (W. cp). b E W. Then cp( W () B) = U x {O) = cp(U)()(E X {O})
Finally. for (iv). it is enough to check that." is C'" using local bundle charts. But this is clear. for such a representative is of the form.(u,. ( U I' 0). That." is onto is clear. •
n ...
The fihers of a vector hundle inherit an intrinsic vector space structure and a topology independent of the charts. but there is no norm that is chart independent. Putting particular norms on fibers is extra structure to he . considered later in the book. Sometimes the phrase Banachable space is used to indicate that the topology comes from a complete norm but we are not nailing down a particular one. The following summarizes the basic properties of a \ector hundle.
3.3.7 Thebrem.
Let E be a vector bundle. The '[.ero section (or btue) B of -> B called the projection that is of class COO. and is surjective (onto). Moreover,for each h E B• .,,-I(h). called the fiber over b. has a Banachahle vector space structure induced by any admissible vector bundle chart. with b the zero element . E. is a submanifold of E and there is a map .,,: E
.
Because of these properties we sometimes write" the vector hundle .,,: E -+ B" instead of "the vector bundle ( E. 'Y )." Fibers art' often denoted hy E" ,.. ." - I( b). If the base B and map." are understood. we just ~ay .. the vector bun~le E."
3.3.8 Example A. Any manifold M is a vector bundle with zero-dimensional fiber. namely M X {O}. B. The cylinder E = S' xR is a vector hundle with w: E -+ B = Sl just the projection to the first factor (J;ig. 3.3.3). This is a trivial vector bundle in the sense that it is a product. C. The MiJbiur band is a vector hundle .,,: E. -+ .\.1 with one-dimensional fiber obtained in the following way. On the product manifold R ,:,. 11/(' following veclor bundle atlas: (('II'-I(U)X'II"-I(U'). cpXtIt)I('II' 1(U).q'). UcB and ('II" -I(U 1)•. tIt). U'c B' are vector bUlldle charts of J-.' tlml H'. respectivel,,}.
It is straightforward to check that the prodtict atlas vcrifies conditions (VBI) and (VB2) of 3.3.4. This section closes with a general construction. special cases of which are used repeatedly in the rest of the book. It allows the tranSfer of vector space constructions into vector bundle constructions. The abstract procedure will become natural in the context of examples given below in 3.3.22 and later in the book. 3.3.20 Detlnnlon. Let I and J be finite sets and consider a pair of families t;. = (E'>A e IUJ' ~~' = (Ei.)A e IUJ of Banachable spaces. Let L(b~.t.;') =
n L{E,. En x n I.(E;. E,)
,E
I
IEJ
and let
i.e .• A/ E L(Ej • E;'). i E I. and A . E L( E;. E;). j E J. An a.v.vignment 0 taking any family t;; to a Banach space ~~~ and any sequence of Iilll'llr maps ( A A ) to a linear continuous map O(A,,) E L(O&. Ob;') Sllli.vb'illg O( 'f,) = 0« B,,) 0 (A,,» = O( B, ) 0 O( A" ) (composition is takell C(mll'tmenlwi.n·) and i.r such that the induced map 0: L(t;;.t;,') -+ L(Ob).O~~·) i.v C'. will he called a temoritll COIfSt",ctiofl 0/ type ( I. J).
'm.
3.3.21
&=
PropoelUon.
Let 0 be a tensorial constructioll of type (I. J) and Let
(E" h e I U J be a family of vector bundles with the same base B.
Then 0& has a unique vector bundle structure over B with SU;" = Ob;" and '11': 0& -+ B sending 0&" to h E B. whose atlas is given by the ('hllrts ('II' - I( U). tit )). where tit: 'II'-.I(U) -+ U'x O(FA) is defined as follow.f. l.el ('fT,,-I( U).,,"). "A: '11',- I( U)
-+
U' X
F'. cpA ( e") = (CPI( 'II'A (e" »).,,~(eA»)
148
MANIFOLDS AND VECTOR BUNDLES
be vector bundle charts on Ek inducing the same manifold chart on B. Define tf(x)- ('P,(".(x».O(tf ..(X»(x» by tf .. (X) == (tf~(.,» where tf~(X) = ('P~(X»-" for i Eland I/I~(x) == ('P;(X» for j E J.
Proof. We need to show that the overlap maps are local vector bundle isomorphisms. We have
( 1/1' 1/1- ')( u. e) = ( 'PI 'PI')( u) and 0 (( 'P~/ 0
0
0 (
'P~) - '( u) ). e ).
the first component of which is trivially Coo. The second component is also Coo since each ql is a vector bundle chart by the composite mapping theorem. and by the fact that 0 is smooth. •
3.3.22 Exampl.. A. Whimey sum. Choose for the tensorial construction the following: J = (2). 1- {l ..... n}. and 0& is the single 8anach space E, X •.. X E". Let O(A,) == A, X •.• X An' The resulting vector bundle is denoted hy E, EB •.• EBEn and is called the Whitney sum. The fiher over he B is just the sum of the component fibers. B. Vector bIIIuIIn 0/ bruulle maps. Let E,. £2 be two vector hundles. Choose for the tensorial construction the following: I. J are one-point sets I- (J},J =- {2}. O(E,. '£2)'" L(E2 • E,). O(A,. A 2 )·S = A, 0 S 0 A2 for S E L(E"E,). The resulting bundle is denoted by L(E2 .E,). The fiber over b E B consists of the linear maps of (E2 h to ( E, h. C. DIIIII brmdIe. This is a particular case of Example 8 for which E = E2 and E, == B XR. The resulting bundle is denoted E*; the fiber over b E B is the dual space E:. D. V«tor""" 0/ muhilinetlr IIIIIpS. Let Eo, E, .... ,E, be vector bundles over the same base. The space of I-multilinear maps (in each fiber) L(E, ..... E,; Eo) is a vector bundle over B by the choice of the following tensorial construction: 1 = {O}. J = {I •.. .• 1}. O( Eo •... ,E,) = L'(E, •...• E,;' Eo).O(Ao.A, ..... A,)·S=AooSo(A, x .. · XA,) for SEL'(E, ..... E,; Eo)·
.'
Exercises
3.3A 3.38
Find an explicit example of a fiber-preserving diffeomorphism between vector bundles that is not a vector bundle isomorphism. (i) Let".: E .... B be a vector bundle and f: B' .... B a smooth map. Define the pull-back bundle /*".: /* E .... B' by /*E = {( e. b')I"'( e) = f( b')}.
/*".( e, b') =
b'
and show that it is a vector bundle OW! R' Show that h: ' /*E .... E. h( e. b') == e is a vector bundle Ill.':
VECTOR BUNDLES
149
(ii) If g: B" ..... B' show that (f 0 g )0.,,: (f 0 g)O E ..... B" is ismorphic to B". (iii) If p: E' ..... B' is a vector bundle and g: E' -+ E is a vector bundle map inducing the map f: B' ..... B on the zero 'sections, then prove there exists a unique vector bundle map. go: E' ..... E inducing the identity on B' and is such that h 0 gO == g. ' (iv) Let a: F ..... B be a vector bundle and u: F -+ E be a v!'Ctor bundle map inducing the identity on B. Show that there exists Ii unique vector bundle map F ..... E inducing the identity on B' and making the diagram
gOr.,,: gOrE . . .
r
ru: r
rF J F
/"u
u
r
rE { ·E
commutative. (v) If .,,: E ..... B. '1/": E' -+ B are vector bundles and if .:1: B -+ B x B is the diagonal map b ...... (b. b). show that £6)/::' .. .:1*( Ex E'). (vi) Let '1/': E ..... B and .,,': E' -+ B' be vector bundles and denote by P.: B X B' ..... Band P2: B X B' ..... B' the projections. Show that Ex E'i!!! pT(E)$p1(E') and that the following sequences are split exact: . 0 ..... E.,... E$E' ..... E'-+O. 0-+ E'-+ E$E' -+ E -+ O. 0-+ pT(E) -+ E X E' ..... p1(E') ..... 0. 0 ..... p1(E') ..... E X E' ..... pT(E) -+ O. 3.3D
(9 (ii)
Show that G,,(R") is a submanifold of G" + I (R"+ I). Denote by ;: G,,(R")- GH I(R"+ I). ;(F)= F XR the canonical inclusion . ' map. If p: R X G,,(R") .... G,,(R") is the trivial bundle show that ;0("1,I{IW) = U'c E. be a ·Iocal vector
150
U~NIFJjLDS AND VECTOR BUNDLES
1iundle chart for E. where V is open in B. If ~: V -+ E is a local section whose local representative is u .... (u.l,,(u». we can ~",: V' - F tbe principal part of the local representativ~ of €. (i) Let €. 'IJ be local sections defined on V. The pairs (~. bl)' ('IJ. b2 ), where bl' b2 E V are said to have k 1b order contact. k if b l - b2 and Di€",(b l ) - Di'IJ",(bl) for 0 ~ i "k. Show that this defines an equivalence relation independent of the vector bundle chart. Denote by jlt~(b) the equivalence class of (€, b) and call it the k-jet 01 € at b. Let J"E be the set of all
"r.
j"(€)(b).
(ii) Define JIt'IT: JlcE - B by j"~(b) .... b. If fJi: 'IT -I(V) - V'x F is a vector bundle chart of E, show that the map J"fJi: (J"'IT)-I(V) - V' X ei"_oL~(E, F) given by /'~(b) .... (u, ~",(u). D~",(u), ... ,DIe~",(u». where u=fJil(b). is bijective. Show that the coUection of all such J"fJi, when fJi ranges over the vector bundle atlas of E. forms a cr- It vector bundle atlas of JltE, thereby proving that J"'IT: JleE - B is a C'-" vector bundle over B ca1Jed the k-jet bundle of E. (Hint: Use Box 3.3A). (iii) Show that if~: V - E is a local section of 'IT, its k-jet extension j"~: V - JIcE is a C,-It local section of J"'lTo (iv) For I" k define the restriction map rlc/: JA( E) -+ .I ' ( E) by j*(~)(b)""l~(b). Show that rH is a surjective bundle homomorphism whose kernel Kltl(E) splits in each fiber. Show that Kle/(E) is formed by k-jets of sections of E having contact of order "I with the zero section. 3.4 THE TANGENT BUNDLE Recall that for I: VeE - V c F of class cr+ I we define the tangent of I, TI: TV-TV by TV=VXE, TV=VxF. and TI(u,e)= (f(u), DI(u)·e). Hence, TI is a local vector bundle mapping of class cr. Also T( log) - TI 0 Tg. Moreover, for each open set V in some vector space E, let TU: TV - V be the projection (as usual. identify V with the zero section V X (O». Then the diagram TI TV TU
• TV
1
1
TV
V
•V
I is commutative. that is. I
0
TV -
"'.0 • Tf.
THE TANGENT BUNDLE
151
The operation T can now be extended from this local context to the context of differentiable manifolds and mappings. During the definitions it may be helpful to keep in mind the example of the ramily of tangent spaces of the sphere Sit c R If + I. A major advance in differential geometry occurred when it was realized how to define the tangent space to an abstract manirold independent of any embedding in R It. t Several alternative ways to do this can be used according to taste. (See below and Spivak [1979] for further information.)
Coordinate approach. Using transformation properties of vectors under coordinate changes, one defines a tangent vector at m E M to be an equivalence class of triples'(U,cp.e), where cp: U-+ E is a chart and e E E. with two triples identified if they are related by the tangent of the CQrresponding overlap map evaluated at the point corresponding to . 2
mEM. Derivation approach. This approach characterizes a vector by specifying
a map that gives the derivative of a general function in the direction of that vector. 3 The ideal approach. This is a variation of alternative 2. Here T,.,M is the dual of (~!O)/~!I). where ~.!!) is the ideal of functions on Mvanishing up to order j at m. 4 The curves approach. This is ~he method followed here. We abstract the idea that a tangent vector to a surface is the velocity vector of a curve in the surface.
Deflnltlon. Let M be a manifold and m E M. A CIIIW lit m is a C' map c: I .... M from an open interval I c R into' M with 0 E I and c(O) - m. Let c 1 and C2 be curves at m and (U, cp) an admissible chart willi Iff e U. Then we say C1 and C2 are tMgelft lit m willi rap«t 10 cp if and ortIy if. • c. and cp 0 Ci are tangent at 0 (in the sense of Section 2.3- we may ratrict 1M . domain of C1 such that cp 0 c; is defined - see Fig, 3.4.1). 3.4.1
Thus two curves are tangent with respect to cp if they have identical tangent vectors (same direction and speed) in tbe chart cp.
3.4.2 PropoeItIon. Let c, and C2 be two CII1W$ at Iff eM. SIIpJlOH (U" cp,) are admissible charts wilh Iff e~. II -1,2. Then c, __ C2 1ft tangent at Iff with respect to 'I, if and ortIy 1/ tlrq tI1'e tangat lit Iff mp«t to 'fl.
wi'"
tTbe history is DOt completely dear to us, but this idea seems to be primarily due to Riemann. Weyi. IDd Levi-Cmtl aDd was "well kaown" by 1920.
152
MANIFOLDS AND VECTOR BUNDLES
R
1
o
FIgu.. 3.4.1
Note that C I and Cz are tangent at m with respect to CPI iff c i )(0) - D( CPI 0 Cz )(0). By taking restrictions if necessary we may suppose VI = Vz. Hence we have cpz 0 C; = (CP2 0 cpll )o( CPI 0 c;). From the C l composite mapping theorem it follows that D( CP2 0 C I )(0) = D( CP2 0 C 2 )(0). Proof.
D( 'PI
0
•
This proposition guarantees that the tangency of curves at mE M is a' notion that is independent of the chart used. Thus we say CI' C 2 are tangent at m EM if c l ,C2 are tangent at m with respect to cpo for any local chart cP at m. It is evident that tangency at mE M is an equivalence relation among curves at m. An equivalence class of such curves is denoted [clm. where c is a representative of the class.
3.4.3 Definition. For a manifold M and m E M the 'ange,,' spue to M
til
m is the set of equivalence classes of curves at m
T", (M) ... {[ cJ 1ft I c is a curoe at m} For a subset Ac M. let TMIA= UlftEATm(M). We call TM-TMIM the IIIItgeIfI btmdle of M. The mapping TM : TM ..... M defined by TM([cl m ) = m. is the tangellt '-die proi«t- of M.
Now we will work toward a derivation that T M : TM ..... M is a vector bundle. First, however we need to show that in case VeE is open. TV as defined here can be identified with V x E. This will establish consistency with our use of T in Section 2.3.
THE TANGENT BUNDLE
·153
3.4.4 Lemma.
Let U be an open subset of E. and e a euftll' at u E U. Then there is a unique e E E such that the curve c u defined by Cu., ( t ) = u + te (on . some interval I such that c u.,( I) cUI is tangent to c at u. Proof. By definition. Dc(O) is the unique linear map iii L(R. E) such that the curve g: R - E given by g(t) = u + De(O)·t is tangent to c at t = O. If e = Dc(O)·I. then g = cu.,, •
Define a map i: UXE-T(U) by i(u.e)=\cu.,III; the preceding lemma says i is a bijection. Moreover. we can define a local vector bundle structure on' T( U) by means of i. For example. the fihcr over Il E U is ;«u}X E). Then i becomes a local vectur bundle i'somorphism, It will be convenient to define the tangent of a mapping before showing that T M : TM - M is a vector bundle. The idea is simply that the derivative of a map can be characterized by its effect on tangents to' curves; see . Section 2.4.
3.4.6 Lemma. Suppose CI and c2 are curves at mE M and are tangent at m. Let f: M - N be of class C I . Then focI and f 0 c2 are tangent at f(m)EN.
.
From the C I composite mapping theorem it follows that focI and are of class C I . For tangency. let (V.",) be a chart on N" with fern) E V. We must show that ('" ° f ° cl)'(O) = (1//0 f 0 ( 2 ),(0). But", 0 f ° ea =("'ofocp-l)o(cpoCa ). where (U.CP) is a chart on M with f(U)cV. Hence the result follows from the C I composite mapping theorem. • Proof.
f
O
('2
This justifies the following.
3.4.6 Definition.
Iff: M - N is of class C I • we define Tf: TM - TN by Tf([ c]m) = (f 0 e]'(m)
We call Tf the tagent of f·
TI is well defined. for if we choose any other representative from [elm. say C I' then C and C I are tangent at m and hence 1 0 C and f 0 clare tangent at I(rn). That is. If 0 cl,(m)" If 0 c,l'(m)' By construction. the following diagram commutes. Tf TM .... ·---·TN TM
1
1
TN
M
·N
I The basic properties of T are summarized in the following,
154
MANIFOLDS AND VECTOR BUNDLES
3.4.7 Compoalle Mapping Theorem. (i) SUppOSE' f: M -+ Nand g: N ..... K are C r maps of manifolds. Then go f: M ..... K is of class C' and T(Ko f) .. TgoTf· II h: M ..... M Is the Identity map. then Th: TM ..... TM is the identity map. If f: M -+ N is a diffeomorphism. then Tf: TM ..... TN is a bijection and
(ii)
(iii)
(Tf)-I =
Proof.
and
T(r I).
(i) Let (V.IP). (V. I/! ). (W. p) be charts of M. N. K. with Then we have. for the local representatives.
I( V) c
V
g( V) C W.
(g °
n
'f'P
=
P ° g ° f ° IP - 1
=pogo1/l-'o1/lofolP- ' = gtj.p
° f.,tj..
By the composite mapping theorem in Banach spaces this. and hence go f. is of class cr. Moreover. T( g ° f)[ c) m = [g ° f ° c 1g o/(m)
and
(Tg 0 Tf)[c)m
=
Tg((fo C1/(m)
= [go fo C1go/(m).
Hence T( g f) ". Tg Tf. Part (ii) is an immediate consequence of the definition of T. For (iii). f and r 1 are diffeomorphisms with for I the identity on N. while flo I is the identity on M. But then using (i) and (ii). Tf ° Tf I is the identity on TN while Tj loTI is the identity on TM. Thus (iii) follows. • 0
0
Next let us show that in the case of local manifolds. Tf as defined in Section 2.4, which we temporarily denote /'. coincides with Tf as defined here. 3.4.8 Lemma. Let VeE and V c F be local manifolds (open subsets) and f: V ..... V be of closs C 1• Let i: V X E -+ TV be the map defined following 3.4.4. Then the diagram
/' VxE-VxF
illi
T ( V ) - T(V) Tf
commutes. that is. Tf 0 i = i
0 /'.
THE TANGENT BUNDLE
. Proof.
155
For (u. e) E U x E. we have (Tf 0 i)( u.e) "" Tf· [c II ."] ,,
Also. (i 0 f')(u. e) - i(f(u). Df(u)·e) = [C/(u,.D/(II""]/(II" These will be equal provided the curves I .... feu + Ie) 'and I .... f(U)+/(Df(u)'e) are tangent at I = O. But this is clear from the definition of the derivative and the comPosite mapping theorem. • This lemma states that if we identify U x E and T( U) by means of i then we can identify f' and Tf. Thus we will just write Tf and will suppress the identification. One more fact will be useful:
3.4.8 Lemma. If f: U c E - V c F is a diffeomorphism. Ihen Tf: U
X
E
- V X F is a local veclor bundle isomorphism.
Proof. Since Tf(u.e) = (f(u). Df(u)'e), Tf is a local vector bundle mapping. But 'as f is a diffeomorphism. (Tf)-I .". T(F I) is also a local vector ' bundle mapping. and hence Tf is a vector bundle isomorphism. •
For a chart (U.IJI) on a manifold M. we can construct TIJI: TUThen TIJI is a bijection. since IJI is a dirfeomorphism. Hence. on TM we can regard (TU. TIJI) as a local vector bundle chilrt. In the target of TIJI note that we have a special local vector bundle. where the fibers have the same dimension as the base. T( IJI(
U».
3.4.10 Theorem. LeI M be a manifold and fi an Cltlas 4 admissible charls. Then 1Cl.';" {(TU, TIJI)I(U,IJI)E fi) is a veclor bundle alICl,~ of TM called Ihe .
IIIItlllfll Gt/tu.
Proof. Since the union of chart domains of fi is M. the union of the corresponding TU is TM. Thus we must verify VB2 HenCe, suppose we have T~ n T~ .21. Then ~ n ~ • 21 and the overlap map 91 can be formed by restriction of Cff' to IJIj(~n~). '!'1l- we must verify that TIJII o(TlJlj)-1 - T( IJII 01Jl)-1) is a local vector bundle isomorphism. But this is , guaranteed by 3.4.9. •
·.i'
lienee TM has a natural vector bundle structure induced by the differentiable structure of M. If M is n-dimensional, Hausdorff: and second countable, TM will be 2n-dimensional. HausdQrff, and second countable.
156
MANIFOLDS AND VECTOR BUNDLES
We shall now reconcile the bundle projection TM with that for an arbitrary vector bundle. 3.4.11 Propoaltlon. II mE M, then TAi l(m) = TmM is a liber 01 TM and ils base B is difleomorphic 10 M by Ihe map TMIB: B -. M.
Proof. Let (V,,,) ,,(m) -
II.
be a local chart at m E M. with ,,: V -. ,,( V) C E and Then T,,: TMI V -. ,,(V)X E is a natural chart of TM,
by definition of T". and this is exactly T,"M. For the second assertion. TMIB is obviou!ily a bijection. and its local representative with respect 10 T" and" is the natural identification q>( U) x {O} -. q>( U). a local diffeomorphism. .' Thus M is identified with the zero section of TM and TM with the bundle projection onto the zero section. It is also worth noting that the local representative of TM is (" 0 TM 0 T" - 1)( II, e) = II, i.e. just the projection of cp(V)X E to cp(V). Since tangent bundles are vector bundles, all constructions of the previous sections can be applied to them. In particular, if N is a submanifold of M, the restriction (TM)N of TM to N is customarily denoted by TNM.
Let us next develop some of the simplest properties of tangent maps.. First of all, let us check that tangent maps are vector bundle maps. 3.4.12 PropoaItIon. LeI M and N be manifolds, and leI I: M -. N be 01 class cr+ I. 71ren Tf: TM -. TN is a oector bundle mapping 01 class cr. Proof. It is enough to check that TI is a local vector bundle map using the natural atlas. For m E M choose charts (U• .p) and (V, "') on M and N so m F. "./(m)E V and I• .,. - '" 0 I 0 cp- lis of class cr+ I. Then using (TV, Tcp) fOl I M and (TV, T",) for TN. we must verify that (Tf)T•. T." is a local vector bundle map of class cr. But we have (Tf)T•. T." - T",. TI 0 Tcp-l_ T(/• .,,), and TI• .,,(II,e) = (/• .,,(11), DI• .,.(II)·e), which is a local vector bundle map of class cr.• Now that TM has a manifold structure we can form higher tangents. For mappings I: M -. N of class C. we define T'/: T'M.-. T'N inductively to be the tangent of Tr-l/: T'-IM -. T" IN. Induction readily yields the following: Suppose /: M -. Nand g: N -. K are C mappings of manifolds. Then g • / is of class C' and T'( g • /) = T'g 0 T'I. Let us now apply the tangent construction to the manifold TM, forptting for the moment it' 'cctor bundle structure We get the tangent
THE TANGENT BUNDLE
157
bundle of TM. namely Tn/: T(TM)-'TM. In co(lnlinales. if (U,ep) is a chart in M, then (TU, Tep) is a chart of TM and-eTC TV). T(Tep» is a chart of T(TM) and-thus the local representative of Tnl is (Tep 0 Tnl 0 T(Tep--I»: (u,., "I' "1) .... (u,,,). On the other hand, taking the tangent of the map TM : TM ... M. we get TTM: T(TM) -0 TM. thus getting another vector bundle structure on T(TM) in addition to the usual one discussed before. The local representative of TTM is
Applying the commutative diagram for Tf following 3.4.6 with f = TM• we get what is commonly known as the duo/tangent rhomhi(·.
Let us now tum to a discussion of tangent bundles of product manifolds. Here and in what follows. tangent vectors will he denoted by hold race letters such as I)E T",M.
3.4.13 PrOpoeauon. Let MI and M1 be manifold, and P,: MI X M2 --+ Mi' i -1.2 the two canonical projections. The map (Tpl' Tp2): T(MI X M2)-o TMIXTMz defined by (TPI' TP1)(1) = (Tp,(I),TP2(1)) is a vector bundle isomorPhism of the tangent bundle T(M I X M z ) and the product bundle TM,XTMz'
Proof. The local representative of this map is (u l ' u 2 • "I' e 2 ) E
VI X U2 X EI X Ez .... «u •• ".). (UZ'''2»E(UI X EI)X(U1XE1). which clearly is a local
vector bundle isomorphism "sition follO\llfS •
'rio Tp2) covers the identity. the propo-
Since ~he tangent is just a global version of the derivative. statements concerning partial derivatives might be expected to have analogues on manifolds. To effect these analogies. we globalize the definition of partial derivatives. Let MI' M z and N be manifplds and f: MI X M2 -0 N a C'map. For (p.q)E M, X MI' define ip"' M1 ... MI X M2 and i q : MI -. MI X M2 by
i,(y) = (P. y). iq(x) = (x.q).
158
MANIFOLDS AND VECTOR BUNDLES
With these notations the following proposition giving the behavior of T under products is a straightforward verification using the definition and local differential calculus.
3.4.14 Propoeltlon. LeI MI' M 2• N. P be manifolds and 8/: P - Mi' i 1.2. f: MI x M2 - N be C' maps. r ~ I. Identify T( M, x M 2 ) with TMI X TM2. Then T(gl X 82)= T81 X Tg 2 Tf(lIp. Up) = T.t(lI p )+ Td(vq)./or lip E TpMI and Vq E TqM 2 (iii) (lmplic;t F""ct;on Theorem) If Td( p. q) is an isomorphism then there exist open neighborhoods V of p in MI' W of f( p. q) in N and a unique C r muP g: V X W - M2 such that for all (x. w) E V X W (i)
(ii)
f(x. g(x, w» = w. In addition. Tlg(x,w) = -(Td(x.g(x.w») T2g(X.W)'" (Td(x.g(x,w»)
I
o(T,f(x. g(x.w»
I
Throughout this book the tangent bundle and vector bundle constructions on M play a central role and we shall learn a variety of ways of representing them. For the moment we content ourselves with the direct analysis of tangent bundles for some simple manifolds.
3.4.15 Exampl... A. The langent bundle TS I of the circle. Consider the atlas with the four charts {(~ ± .1/1/ )1; = 1.2} of Sl = {(x. y) E R21x 2 + ).2 = I}. (See 3.1.4.) Let us construct the natural atlas for TS I = «(x. y). (U.lJ»E R2 XR21x 2 + y2 = I. «x. ,v). (u. v» - O}. Since
1/-;: VI' ={(x.y)ES ' lx>O}---+l-I.I(.
~;(x.y)=y.
by definition of the tangent we have
J; •. r)+~ (II. v) - (y. v).
T1/I~: TVt -
]-I.l[ xR.
THE TANGENT BUNDLE
159
Proceed in the same way with the other three charts. Thus, for example, 1(x.YI"'i" (u, v) =- (x, u) and hence for x E ]-1,0[,
This gives a complete description of the tangent bundle. But in this case, considerably more can be said. Namely, thinking of Sl as the multiplicative group of complex numbers with modulus I, we shall show that the group ~ 1 operations are Coo: The inversion $ .... $- has local repr~sentatives ("'I 0 ~ I)(X) = - x and the composition
o("'r ).
has local representatives
(here ± can be taken in any order). Thus for each $ E Sl the map Ls(s')., ss', L,: Sl ..... Sl is a diffeomorphism. This enables us to define a map A: TS 1 ..... Sl xR by A(vs )= (s, T1L.-1(v,» (I is the identity element of Sl), which is easily seen to be a vector bundle isomorphism over the identity. Thus TS 1 is a vector bundle isomorphic to SiX R. See Fig. 3.4.2. B. The tangent bundle TT" to the II-tOrus. Again one could start writing charts, but since T" = Sl X ... X Sl (n times) and TS 1 ~ Sl xR. it follows that TT" a; T" XR". C. The tangent bundle TS 2 to the sphere. The previous examples yielded trivial vector bundles. In general this is not the case. The simplest such example is the tangent bundle to the two-sphere which we now
~c-£ s"
Trivial langenl bundle'
Non trivial lanpnt bundle
•
160
MANIFOLDS AND VECTOR BUNDLES
:describe. Choose again the atlas with six charts {(U/. 1/1/ )1; = 1.2,3) of S2 that were given in 3.1.4. Since == {( X l • X 2 • x) >E
1/1:: ut
.
S 21 Xl> o} -+ D 1(0) =
{( x.
y) E R 21 x 2 + Y 2 < I}.
I/I~(Xl,x2,X3) = (X 2 .X 3 ),
we have
.r, + (1 2 3) _ (2 1 2 3) T.(x'.x'.xl)"1 v.v.v - x ,x ,v.v ,
where XlVI + X 2v 2 + X 3V 3 = O. Similarly construct the other five charts. For example one of the twelve overlap maps is
(TI/I3
o
(TI/I:>-I)(X,)'.U,ll)
- (b - x 2 -
y2 • X. -
ux/';':::~:'-y2 - vY/.fl=~2-':"7 ,u). for
x 2 + y2 < I. Y < O.
One way to see that TS 2 is not trivial is to use the topological fact that any vector field on S2 must vanish somewhere. We shall prove this later; see Theorem 7.6.10 in Chapter 7. •
Exercl... 3.4A 3.48
Let N c M be a submanifold. Show that TN is a subbundle of TNM and thus is a submanifold of TM. Let M. N be manifolds and I: M -+ N. (i) Show that (a> lis Coo iff graph (f) = {(m,f(m»
E
M
X
Him E M}
is a Coo submanifold of M x Nand (b)
7;",./''''11( M
x N) a 7;",./(m))(graph (f»e!J7'(m,N for all
mE
M.
(ii) If I is Coo show that the canonical projection of graph ( .r) onto M is a diffeomorphism. (iii) Show that 7;"'.f,,,,))(graph (f» a graph(T",f)
'"' «"... Tlft/(.,..,» I ~...
E
T... M} C T",M
x T"""N.
SUBMERSIONS, IMMERSIONS, AND TRANSVERSALITY
161
Let p: R X S" -. S" and 0: R"+' X S" -+ S" he trivial vector bundles. Show that TS"e(R x S") ~ (R"+' X S").
3.4C
(Hint: Realize p as the vector bundle whose one-dimensional fiber is
3.4D
the normal to the sphere.) (i) Show that there is a map s",: T(TM) -. T(TM) such that s", 0 s'" = identity and .f",
T(TM).
T(TM)
T1"i"\ s'" ft( TAl) TM
commutes. (Hint:
In local natural coordinate charts.'
s"'(u ••••••• 2 )-(u••••••• 2 )..) One calls s'" the canonical involution on M and says that T( TM) is a symmetric rhombic. (ii) Verify that for f: M -+ N of class C 2• T2/ 0 .~\1 = SN 0 T2f. (iii) If X is a vector field on M. that is. a section of T",: TM -+ M. show TX is a section of TT",: T2M -+ TM and X, = SM 0 TX is a section of TT"': T2M -. TM.
Show that T(M. x M2)~ p1(TM.)ep!(TM2 ) wherep;: M. x M 2 -+ M;. i = 1.2 are the canonical projections and p~( TM;) denotes the . pull-back bundle defined in Exercise 3.38. 3.4F (Requires some basic algebraic topology). In algebraic topology. the simply connected universal covering space Sf of a topological space M is constructed together with a projection 71': Sf -+ M which is a local homeomorphism. Show that if M is a manifold. so is Sf and 71' is . a smooth map which is a local diffeomorphism.
3.4E
3.5 SUBMERSIONS, IMMERSIONS, AND TRANSVERSALITY This set:tion brings the powerful implicit function theorem into play. The notions of submersion. immersion. and transversality are geometric ways of stating various hypotheses needed for the inverse function theorem. These ideas are central to large portio"s of calculus on manifolds (lne immediate benefit is easy proofs that various subsets of manifolds are actually submanifolds. Let us begin with a simple consequence of the inverse function theorem.
3.5.1 Theorem. Let M and N be 11Ulnifolds. f: M -. N be of class C. r ;;, I and m E M. Suppose Tf restricted to the fiber over m E M is an
162
MANIFOLDS AND VECTOR BUNDLES
isomorphism. Then f is a C diffeomorphism from some neighh'!rhood of m onto some neighborhood of f(m). Proof. In local charts. the hypothesis reads: (Df",oJ,Xu) is an isomorphism. where ,,( m) "" u. Then the inverse function theorem guarantees that f",oJ, restricted to a small neighborhOod of u is a C diffeomorphism. Composing with chart maps gives the result. •
The implicit function theorem also yields the "local onto" theorem. Let M and N he maniloldr and f: M ~ N he 01 clau C; I. Suppose TI restricted to the liber TnrM is mrjectil'e to T,'m,N. 711£'''
3.5.2 Theorem. ~
r
f is loral/\' 0"'0 lit 111: i.e. there exist Ilei",hhorhoods U of 1/1 and Vof f( m) .ruch that II V: U ~ V is onto: in particular. if T/ i.l' .I'urjectit'e on each tangent space. the" f i.l' an open mapping: (ii) if in addition the kernel ker( Tmf> is split in Tm M there are charts (V. ,,) and ev.~) with me V.j(V)C V.,,: U"" V'x V'.,,(m) = (O.O).~: V .... V' and f",oJ,: U· x V' .... V· is the projection onto the second factor. (i)
Proof. It suffices to prove the results locally. But these follow directly from 2.5.9 and 2.5.13. •
The notions of submersion and immersion correspond to the local surjectivity and injectivity theorems from Section 2.5. Let us first examine submersions. building on the preceding theorem.
3.5.3 Definition. Suppo.re M and N are manifolds with f: M ~ N of c/a.r.s C. r ~ I. A point n E N is cal/ed a reglllllr vallie of I if for etlch m E f I ({ n}). Tmf is surjective with split kernel. Let R f denote the .ret of regular tlalues 01 f: M .... N; note N \f( M) c R feN. If. for each m in a .ret S. Tmf is .fUl'jective with split kernel. we say f is a slIbmersion on S. (Thus n E R / iff I is a submersion on f- I({n». If T",f = O. me M is called a critical point and n = f( m) E N a critical vallie of I.
Note that statement (ii) in 3.S.2 at every point 111 E M is equivalent to f being a submersion. The next result is important when wnsidering level surfaces of maps.
3.5.4 Theorem. Suppose.f: M
r
-+
N is 01 class COY 11111/ "E R f' Then
l(n)=(mlmE M.j(m)=n} is a .mhmanilold of M and T",f- I (I1)=
kerT",/.
r
Proof. If I( n) = 0 the theorem is satisfied. Otherwise. for m E find charts (V. ,,). (V. ~) as described in 3.5.2.
f
I( n)
we
SUBMERSIONS, IMMERSIONS. AND TRANSVERSAL/TY
163
Then it must be shown that the chart (V,,,) has the submanifold = I;~(O) = V' X {O}. This is exactly the subproperty. But ,,(V () I( manifold property. (See Fig. 3.5.1.) Since I •• : V' X V' ..... V' is the projection onto the second factor, where V' C E and V' C F, we have
r n»
T,,(t;~(O») ... TP'- E .... ker(T"/•• ) for u E V', which is the local version of the second statement. • If N. is finite dimensional and E R /' observe that codim (f' I( dim N, as can he easily seen from the second statement of 3.5.4. This makes sense even if M is infinite dimensional. Sard:s theorem. discussed in Appendix E. implies that R r is dense in N.
n
n» ...
3.5.5 Exampl... A. We shall use the preceding theorem to show that S" eRn + I is a submaniCold. Indeed. let I: R"+I ..... R be defined by 1(.1')-11.1'11 2 • Thus S" ... I( I). To show that S" is a submaniCold. it suffices to show that I is a
r
FlguNa.I.,
164
MANIFOLDS AND VECTOR BUNDLES
regular value of f. Suppose f( x) = I. Then identifying TR" + I = R" + I X R" + I. and the fiber over x with elements of the second factor, (T... f)(v) - Df(x)·v-= 2(x.v).
Since x • O. this linear map is not zero, so as the range is one-dimensional. it is surjective. The same argument shows that the unit sphere in Hilbert . space is a submanifold. B. Let Stem. n; k) = {A
E
L(R'" .R")lrank A = k}.
where k..; mine m. n).
We shall prove using the preceding theorem that St(m,n; k}is a submanifold of L(Rm.R") of codimension (m - k)(n - k); this manifold is called the Stiefel manifold and plays an important role in the study of principal fiber bundles. In order to show that St( m. n; k) is a submanifold. we will prove that every point A E St(m. n; k) has an open neighborhood U in L(R"'.R") such that St(m. n; k)nu is a submanifold in L(R"'.R") of the right codimension; since the differentiable structures on intersections given by two such U obviously coincide (being induced from the manifold structure of L(R"',R"». the submanifold structure of St(m, n; k) is obtained by collation (Exercise 3.2F). Let A E St(m, n; k) and choose bases of Rm.R" such that A = [:
with
II
:]
an invertible k X k matrix. The set
U = { [:
~]
Ix is an invertible k
X k
matrix in the foregoing bases}
is clearly open in L(R"'.R"). A matrix [: iff., - tx- Iy = O. Indeed [ -
~]
with x invertible has rank k
t~-I ~]
is invertible and
so rank[!..
Y.,]=rank[Xo
Y].
v-zx-1y ,
SUBMERSIONS,IMMERSIONS, AND TRANSVERSALITY
165
r
The preceding remark shows that 1(0) = St(m, n; k)n U and thus if f is a' submersion, 1(0) is a submanifold of L(Rm,R") of codimension equal to
r
To see that f is a submersion it is enough to remark that for x, y, z fixed, the map v~ v - zx-1y is a diffeomorphism of L(Rm-Ic,R,,-A). • Next we tum to immersions. 3.5.6 Deflriltlon. A C' map f: M ..... N, r ~ I, is called an immers;of'I at m if Tmf is injective with closed split image in ~(m)N. Iff is an immersion at each m, we just say f is an ;mmers;OII. 3.5.7 Theorem. For a C' map f: M ..... N, where equivalent:
r ~
I. the following are
f is an immersion at m. There are charts (V, (JI) and (V, 1f) with mEV, f( U) c V, (JI: U ..... U',~: V ..... V' X V' and (JI (m ) = 0 such that f",oi-: V' ..... V' X V' is the inclusion u-(u,O); (iii) There is a neighborhood V of m such that f( V) is a submanifold in N and f restricted to V is a diffeomorphism of V onto f( V). (i) (ii)
Proof. ' The equivalence of (i) and (ti) is guaranteed by the local immersion Theorem 2.5.12. Assuming (ii), choose V and V given by that theorem to conclude that f(U) is a submanifold in V. But V is open in N and hence , f(U) is a submanifoJd in N proving (iii). The converse is a direct application , of the definition of a submanifold. • It should be noted that the theorem does not imply that f(M) is a submanifold in N. For example f: SI ..... R 2, given in polar coordinates by r = 00528, is easily seen to be an immersion (by computing Tf using the' curve c(9)=OO528) on SI but f(SI) is not a submanifold of R2; any neighborhood of 0 in Rl intersectsf(SI) in a set with "comers" which is not diffeomorphic to an open interval. In such cases we say f is an immersion with self-intersections. See Fig. 3.5.2.
166
MANIFOLDS AND VECTOR BUNDLES
5' --~
FIgu,,'.1.2
In the preceding ellample I is not injective. But even ir I is an injectiVe immersion. I( M) need not be a submanirold or N. as the rollowing ellample shows. Let I: ]0.2.".[ .... R2 be given in polar coordinates by r = sinO (Fig. 3.5.3) Again the problem is at the origin: any neighborhood or zero does not have the relative topology given by N. If I: M .... N is an injective immersion. I(M) is called an immersed submanilold or N.
y
I
~ (
)
o
2".
x
,.....3.5.'
SUBMERSIONS. IMMERSIONS. AND TRANSVERSALITY
167
3.5.8 Definition. An immersion I: M --+ N that is a homeomorphism onto I( M) with the relative topology induced Irom N is called.an embedding. Thus. if I: M --+ N is an embedding. then I( M) is a submanifold of N. The following is an important situation in which an immersion is guaranteed to be an embedding; the proof is a straightforward application of the definition of relative topology.
3.5.9 Proposition. An injective immersion which onto its image is an embedding.
IS
an open or clmed map
The condition "I: M --+ N is closed" may be replaced by the following. Assume that lor each sequence Xn EM such that I(x n ) converges in N. there is a convergent subsequence in M, Indeed, if this hypothesis holds. and A is a closed subset of M. then I(A) is shown to be closed in N in the following way. Let xn EA. and suppose I(x n ) = Yn converges to yEN. Then there eltists a subsequence {xn,} of {x n }. such that x n , ... x. Since A - cl(A). x e A and by continuity of l.y=/(x)E/(A); i.e.• I(A) is closed. In finite dimensiQns, this hypothesis is assured by the condition "I is proper"; i.e .• the inverse image of every compact set is compact. This is clear since in the preceding hypothesis one can choose a compact neighborhood Vof the limit of I(x n ) in N so that for n large enough all xn belong to the compact neighborhood I I( V) in M. The reader should note that wliile both hypotheses in the proposition are necessary. properness of I is only sufficient. In fact there are injective non proper immersions whose image is a submanifold. For example
I: JO,oo[ --+R2,/(t)= (tcos.!.,tsin!) t I . is an injective nonproper immersion which is an open map onto its image so 3.5.9 applies; the submanifold I(JO, ooD is a spiral around the origin. Third, we examine transversality.
3.5.10· Definition. A C' map I: M --+ N, r ~ I, is said to be trrIIISVn'StII to the submanilold P 01 N (denoted Il"I P) il either I( P) = 0, or il lor every
r
m Erl(p),
I. (Tm/)(TmM)+ 1f(m)p = 1f(m)N and 2. the inverse image (Tm/)- 1(1f(nt)P) 01 1f(",)p sp/iu in T",M.
The first condition is purely algebraic; no splitting assumptions arc made on (TmJ)(T",M), nor need the sum be direct. Also note that if j( ~a Hilbert manifold, in particular finite dimensional. then the splitting CClII'lCJi. tion 2. in the; definition is automatically satisfied.
.
168
MANIFOLDS AND VECTOR BUNDLES
3.5.11 Exampl... A. If each point of P is a regular value of f. then Ifrl P since (T",f)(T",M) = 7i(",)N in this case. B. Assume that M and N are finite-dimensional manifolds with dim(P)+dim(M) < dim(N). Then/frl P implies/( M)n P = 0. This is seen by a dimension count: If there would exist a point mEl '( P)n M. then dim(N) = dim«T",f)(T",M) + 7i(m)P) '" dime M)+dim(P) < dim(N) which is absurd. C. Let M = R2. N = R\ P = the (x. y) plane in R\ (l E R and define fa: M -+ N, by fa(x. y) = (x, y. x 2 + y2 + a). Then Ifrl P iff a ... ·0; see Fig. 3.5.4. This example also shows intuitively that if a map is not transversal to a submanifold it can be perturbed very slightly to a transversal map; for an informal discussion of this phenomenon we refer to Box 3.58. .. image of
f.
..
.. a>O
x
a 1 and MI n Mz is the union of two circles if 0 codim(kerTmf> = constant and f is thus a subimmersion by 3.5.16(ii). We have already encountered subimmersions in the study of vector bundles. Namely. the condition in 3.3.16(i) (which ensured that for a vector bundle map f over the identity ker f and 1m fare subhundles) is nothing else but f being a subimmersion, The fibration theorem has important applications in the study of Lie group actions; it can be used to prove that orbits are immersed submanifolds. We cloSe this section with a study of quotient manifolds. 0=
3.5.19 Definition. An equivalence relation R on a manifold M is called reguJar if the quotient space MIR carries a manifold structure such that the canonical projection 71': M -+ MI/!. is a submersion. If R is a regular equivalence relation. then M IRis called the quotient lIUlIIi!old of M by R.
Since submersions are open mappings, 71' and hence the regular equivalence relations R are open. Quotient manifolds are characterized by their effect on mapping~.
3.5.20 Proposition. Let R be a regular equivalence relation on M. (i) A map f: MIR-+ N is C, r ~ I iff f 0 71': M -+ N is C. (ii) Any C' map g: M -+ N compatible with R. i.e. xRy implies g(x) = g(y), defines a unique C map g: MIR -+ N such that go 71' = g. (iii) The manifold structure 0/ MIR is unique if we demand that 71' is a submersion. Proof. (i) If f is C, then so is / 071' by the composite mapping theorem. Conversely, assume f 0" is C'. Since 71' is a su~mersion it can be locally expressed as a projection and thus there exist charts (V, fJI) at m E M and (V, 1/1) at 7T(m)E MIR such that fJI(U) = VI X V2 C E l fBE 2 , 1/I(V) = V2 C E 2 • and 7T",I/-(X, y) = y. Hence if (W. X) is a chart at (f 7T)(m) in N with (f 0 7T)(U) C W. then fXI/- = (f 0 'II')"'I/-I{O} x V2 and thus fXI/- is C. (ti) The mapping g is uniquely determined by go 71' = g. It is C by (i). (iii) Let (MIR)I and (MIRh be two manifold structures on MIR having 71' as a submersion. Apply (ii) for (MIR)I with N=(MIRh and g = 71' to get a unique Coo map hi: ( M I R) I -+ ( M I R h such that h 0 71' = 71'. Since 71' is suIjective, h = identity fhanging the roles of the indices I 0
174
MANIFOLDS AND VECTOR BUNDLES
and 2, shows that the identity mapping induces a COO map of (M/Rh to (M/R) •. Thus, the identity induces a diffeomorphism. • 3.5.21 Corolla". Let M and N be manilolds, Rand Q regular equivalence relations on M and N, respectively, and I: M -- N a C' map. r ~ I, compatible with Rand Q; i.e., il xRy thenl(x)QJ(y). Then I induces a unique C map): M/R -+ N/Q and the diagram
commutes.
Proof. The map) is uniquely determined by '!TN 0I = )0 '!TM' Since '!TN 0I is c.l is C' by 3.5.20. The diagram is obtained by applying the chain rule to "N
0I - )0 '!TM'
•
The manifold M / R might not be Hausdorff. By 1.4. \0 it is Hausdorff iff the graph of R is closed in M X M (R is open since it is regular). For an example of a non-Hausdorrr'tlUQtient manifold see Exercise 3.5H. There is, in fact, a bijective ~ndence between surjective submersions and quotient manifolds. More precisely. we have the following. . 3.5.22 Proposition. Let f: M -+ N be a submersion and let R he the equivalence relation defined hr f; i.e .. xRy ilf I(x) = ltv). Then R i,~ re1{ular. M/R is dilfeomorphic to I( M). and f( M) is open in N.
Proof. As/is a submersion. it is an open mapping. so/<M) is open in N. Moreover, since I is open, so is the equivalence R and thus I induces a homeomorphism of M/R onto I(M) (see the comments following 1.4.9). Put the differentiable structure on M/R that makes the homeomorphism into a diffeomorphism. Then M/ R is a manifold and the projection is clearly a submersion, since I is. • This construction provides a number of examples of quotient manifolds. 3.5.23 EumpIea. A. The base space of any vector bundle is a quotient manifold. (Taite the submersion to be the vector bundle projection.)
SUBMERSIONS, IMMERSIONS, AND TRANSVERSALITY
175
B. The circle S' is a quotient manifold of R defined by the submersion IJ 0-+ e i'; we can then write S' ... R/2"Z. including the differentiable structure. C. Th~ Grassmannian G,,(E) is a quotient manifold in the following way. Let D'" {(x, •...• .1'" )1.1', E E. x, •...• .1'" are linearly independent). D is open in Ex··· X E(k times). Thus one can define the map ,,: D ~G,,(E) by ,,(x, •...• xh)=span (.1', •.•.• .1',,). Using the charts described in Example 3.1.8(G). one finds that " is a submersion. In particular. the projective spaces RP" and CP" are quotient manifolds. D. The Mobius band as explained in 3.3.8C is a quotient manifold. E. Quotient bundles are quotient manifolds (see 3.3.16). ... We close with an important characterization (If regular equivalence relations. 3.5.23 Theorem.
An equivalence relation R on a mani/old M is regular iff
graph(R) is a submanifold of M X M. and (ii) PI: graph(R) .... M. p,(x. y) = x is a submersion. (i)
Proof. First assume that R is regular. Since ,,: M .... M/R is a submersion. so is "X ,,:M X M ~ (M/R)X(M/R) so that graph(R) - (" X '1r) - '(I1 M/ R ). where I1M/R = {(Ix J.lx))l[xJ E M/ R) is the diagonal of M/ R X M/R. is a submanifold of M X M (3.5.12). Let (x. y)E graph(R) and vx E ~!tf. Since '1r is a submersion and ,,(x) = ,,(y), there exists Vv E T,.M such that Tvw( v,.) = T,"( v,); i.e .• (v,. vv) E 1(,. I,,(graph( R» by 3.5.12. But then 1(, .•"p,(v;. v.. ) = v" showing that" is a sU,bmersion. . Conversely, assume (i) and (ii) are satisfied. We construct a chart at [X)E M/R in the following way. First let us show that" is an open mapping. Indeed. if V is open in M. then (V X V )ngraph( R) is open in the submanifold graph( R) and since p,: graph( R) .... M is a submersion it is open (3.5.2) so that V X V)ngraph( R» == ,,-'( w( V» is open; i.e., " is open. Let (x.X)EI1 M • Since graph(R) is a submanifold. there is a chart cp: V X V .... V'x V'c E X E such that cp«V X V)ngraph( R)) = (V'X V')n( F X{O}) .... V". where Ex E = F$G. cp(V X V)= V;' V', V" c F, and V'c G. Since p,: graph( R) .... M is a submersion, its local representative in the chart cp is also a submersion. We still denote it by PI: V" -4 E. Now use the local submersion theorem to change the chart cp inside V" so thal p, becomes projection onto the first factor; i.e.• find an open set of the form V, X VI C V", where cp(x, x) E VI X VI' and a diffeomorphism of an open set in V" onto V, X V, such that in this new chart PI is projection onto VI c E. Here V, cHand E$H'"= F.
p,«
176
MANIFOLDS AND VECTOR BUNDLES
Now for a sufficiently small neighborhood VI
X V'
c HeG;; E (since
Ex E., EeGeH) we have
I. w( '" - I( VI X V') is open in M / R (since w is open); 2. w( '" - I( VI X (O}» = w( '" - I( VI X V'» (by construction); 3. there is a homeomorphism induced between
4.
(again by construction) define x: w( '" - I(VI X (O})) --+ VI by
X( W",-I( v.O»
=
v.
so X is bijective. It is a homeomorphism by 3. The maps X are the charts for M/R. We only need to show that the overlap maps are Coo. But X is essentially detennined from the implicit function theorem and we know frc;>m 2.5.7 that the implicit function is a Coo function of any parameters in the problem. Using this idea, the reader can easily complete the proof. • This theorem essentially occurs in classical works concerning quotients of Lie groups and of manifolds by group actions on them. (See for example Chevalley [1946] and also see Abraham and Marsden [1978. ch. 4] for some applications to mechanics).
BOX 3.5A
LAGRANGE MULTIPLIERS
Let M be a smooth manifold and i: N'-+ M a submanifold of M. i denoting the inclusion mapping. If I: M --+ R, we want to determine necessary and sufficient conditions for n E N to be a critical point of /IN. the restriction of I to N. Since liN = I 0 i. we have T"UIN) = T,.! 0 Tlti; i.e., n E N is a critical point of I iff T,,/l T"N = O. This condition takes a simple form if N happens to be the inverse image of a point under a submersion.
3.5.24 Propo8lt1on. Let g: M --+ P be a smooth submersion. N = g - I( Po) and let I: M --+ R be cr. r ~ I. A point n E N is a critical point 01 liN iff there exists>. E T;.P. called a Lagrange multiplier. such that 1',,1 = >. 0 T"g.
SUBMERSIONS. IMMERSIONS. AND TRANSVERSAL/TY
Proof.
177
First assume such a ). exists. Since
i.e., 0 = (). 0 T"g)1 T"N = T"fl Tn N. Conversely, assume Tnfl TnN = O. By the local form for submersions, there exists a chart (V,cp) at n,cp: V-V, x V,cEX F such that cp( V () N) = {O} X V, and q>( n) = (0.0). and a chart ( V. I/- ) at p()' 1/-: V-VIc.E where g(V)cV.IHpo)=O. and such that g'l't(x. y .>= (l/-ogocp-l)(x.y)=xforall(x.y)EVI X VI' If!'I'=!oq>- : VIXV, -R, we ,have for alii E F, Dd'l'(O, 0)· I = 0 since Tn!1 TnN = O. Thus, letting 11 == Dlf'l'(O, O) E E*,e E E and IE F, we get
Df'l'(O,O)' (e. f) = Il( e) = (11 0 Dg'l'''')(O.O)· (e. f); i.e.,
Df'l'(O.O) = (11 0 Dg'l'''')(O,O). To pull this local calculation back to M and P"let ). = 11 0 1',.1/- E r;op, so composing the foregoing relation with T"cp on the right we get Tnf =). 0 T"g. •
3.5.25 Corollary. LeI g: M .... P be tran.wersal to the submanifold W of P, N = g-I(W), and let f: M .... R be C, r ~ I. LeI E,(n) be a closed complement to T,ln)W in TIf(n)P so T,(n)P = T,(n)WeEIf(n) and let 'IT: T,(n)P .... EIf(n) be the projection. A point n EN is a critical point of fl N iff there exists ). E E~ n) called a Lagnurge mllitiplkr such that Tnf =). 0 'IT 0 T"g. By 3.5.12, there exists a chart (V, cp) at n, with q>( V) = VI X U2 eEl X E 2 , q>(V () N) = (O}X V2' and q>(n) = (0,0), and a chart (V,I/!) at g(n) with I/!(V) = VI X VI C FI X F, I/-(V () W) = (O}X VI,I/!(g(rr» =(0,0), and g(V)CV, such that g'l'",(x,Y)=(l/-ogocp-I)(X,y)= (x, 1/(x, y» for all (x, y) E VI X VI" Let p: EI X F -+ E. be the canonical projection. By the previous proposition applied to the map po g"",: VI X V2 .... VI.(O,O) E V, x V2 is a critical point of fl{O} X V2 iff there exists" E Er such that Df,,(O,O) 0 p 0 Dg'l''''(O.O). Composing this relation on the right with T"cp and letting ). = T,(n)l/!. 'IT = (T,(n)¥- )-IIEIf(n) 0 p 0 TIf(n)l/-: 1~\,,) /' • ElfIn)' we get Tnf =). 0 'IT 0 T"g . Proof.
="
,,0
•
If P is a Banach space F, then 3.5.24 can be formulated in the following way.
178
MANIFOLDS AND VECTOR BUNDLES
3.5.21 Corollary. Let F be a Banach space. g: M -+ F a smooth submersion. N - g-I(O). and I: M -+ R be cr. r Oil> I. The point n E N is a critical point 01 II N iff there exists lI. E F*. called a Lagrange mIIItipilr. such that n is a critical point 011 - lI. 0 g. Notes
lI. depends not just on II N but also on how I is extended off N. This form of the Lagrange multiplier theorem is extensively used in the calculus of variations to study critical points of functions with constraints: cr. Caratheodory (1965) 3. We leave it to the reader to generalize 3.5.25 in the same spirit. I. 2.
The name lAgrange multiplier was first used in conjunction with the previous corollary in Euclidean spaces. Let V be an open set in R".F ... R'.g_(gl ..... g'): V-+RP a submersion and I: V-+R smooth. Then x eN - g-I(O) is a critical point of fiN. iff there exists p
lI.=
L AieiE (RP)*. i-I
where e l •... • e P is the standard dual basis in R P such that n is a critical point of p
I -lI.
0
g == I
-
L ";gl.
I-I
In classical calculus the real numbers A; are referred to as Lagrange multipliers. Thus. to find a critical point x-(xl •...• x'")eNcR'" of
liN
one solves the system of m + p equations
al
-(x)-
ax'
ag' LP ",-(x) = o. i_I ax}
j=l ....• m
i .. I ..... p
g'(x)=O.
Ar
for the m + p unknowns XI •••.• X'". "1 ..... For example. let N - S2 C Rl and I: R -+R;/(x. y. z)- z. Then II S 2 is the height function on the sphere and we would expect (0.0. ± I) to be the only critical points of II S2; note that I itself has no
SUBMERSIONS. IMMERSIONS. AND TRANSVERSAL/TY
179
critical points. The method of Lagrange multipliers. with g(x. y, z)= x 2 + y2 + Z2 -I, gives 0-2xA-O;
0-2yA - 0; 1-2zA =0;
The only solutions are A = ± ~. x = O. Y = O. z ~ ± I. and indeed these correspond to the maximum and minimum points for / on S2. See an elementary text such as Marsden and Tromba 119R I J for additional examples. For more advanced applications. see Luenberger (1969). The reader will recall from advanced calculus that maximum and minimum tests for a critical point can be given in terms of the Hessian. i.e.• matrix of second derivatives. For constrained problems there is a similar test involving bordered Hessian.v. See Marsden and Tromba [1981. pp. 224-30) for an elementary treatment. We invite the reader to formulate such results intrinsically on manifolds.
BOX 3.5B
TRANSVERSALITY THEOREMS
This box addresses two questions. H /: M -+ N is a C'-map and PeN is a submanifold. what can one say about thc points m at which / is transversal to P? Consider'the set 1"1' ( M. N; ,.) = {l: M -+ N 1/ is of class (" and /1"1 P). What can one say about this set in the set of all (" maps from M to N? The first question is relatively casy.
3.5.27 Proposition. Let f: M
-+
N be C' and!'
The set (m E Mil is transversal to P at m) is ope"
II 11/
.l"IIhlllani/o/d of N. M.
Proof. Assume I is transversal to P at m E M. Choose a submanifold chart (V.q'I) at I(m) E P. q'I: V -+ F, X F2 • q'I( V () 1') = ,.', X{O}. Hence if 'IT: F, x F2 -+ F2 is the canonical projection. V () " = If '( F, x {O}) = ('IT 0 q'I) - '(O). Clearly. 'IT 0 q'I: V () P -+ F2 is a submersion so that by 3.5.4. ker1j(m)('lT o q'l)=Tf (m)P, Thus/is transversal to P at/(m) iff Tf(p)N = ker T'(m)('lToq'l)+ Tf(p,P and (T,.,l) '(Tr(m)!') = ker Tm( 'IT 0 q'I 0 I) is split in Tm M. Since q> 0 'IT is a submersion this is equivalent to 'IT 0q'I 0/ being submersive at mE M (see also Exercise . 2.2E). But as in Proposition E.6 in Appendi){ E. the set where 'IT 0 q'I 0 / is submersive is open in U. hence in M. where U is a chart domain such that/(U)c V. •
180
MANIFOLDS AND VECTOR BUNDLES
The answer to the second question is one of the fundamental theorems in differential topology. Since this-and the techniques involved-would take us too far afield. we shall limit ourselves here to the description of the so-called classical Thom Iransversality Theorem. The reader is referred to Abraham and Rohhin [19671. Hirsch (IQ76J. and Golubitsky and Guillemin (1974) for the proofs and many applications. Guillemin and Pollack (1974) is an excellent introduction to the subject with as little technical machinery as possible. The first task is to give C( M. N) = {f: M -+ Nil of class C} a reasonable topology. The idea is that two C'-maps should be "close" if their graphs and the graphs of the first r derivatives are close and "at infinity" they approximate each other arbitrarily well. This is formalized in the following way. assuming the manifolds in question are finite dimensional and second countable. The Whitney C'-topology is defined for 0" r < 00 in the following manner. Let f1 == {U•• CP.)}.. e A be a locally finite atlas of M; i.e.• every point of M has a neighborhood intersecting only finitely many V,.. Let ~ = {K,.},.e A' K,. c V,. be a family of compact sets in M and ~ = Y,1I)}lIe B' an atlas of N. For every family e = (e,.}ae A of positive numbers. define
«Vp.
"V,.s.x .• (f)'" (g E C'(M. N)lg(K.) C J-I,(." sup
IIDJ( Y,II(., 0
I cP,.)( x)- DJ( Y,1I(ClI 0
0
g 0 cp,.)(x)1I < e. for all
Of
.te".(K.I. j-O ..... ,
The Whitney C'-topology is by definition generated by these as a basis of open sets. If r = 00. (resp. '" (analytic». the COD (resp. C W ) Whitney topology is the initial topology induced by the inclusions COD( M. N )(resp. CW( M. N» .... C'( M. N) for all r. This topology is extremely fine and has rather unpleasant properties. It has uncountably many connected components and is not first countable. thus not me&rizable. It has however a useful property that makes it suitahle to discuss approximations in C'( M. N). namely. C( M. N) with the Whitney C r -topology is a Baire space. Moreover. interesting special classes of mappings have useful properties in C r ( M. N). Thus C' diffeomorphisms. embeddings. dosed embeddings. immersions. submersiOns. and proper maps are dense in the corresponding C' classes 01 maps 01 C'(M. N)/or I < r. r == 1..... 00. Col. Moreover. the same classes 01 C maps are all open in C'(M. N). (r ~ I for diffeomorphisms; homeomorphisms are not open in Co( M. N ).)
E
A} .
SUBM,ERSIONS.IMMERSIONS. AND TRANSVERSALITY
181
This implies in particular that two C' manifolds C' diffeomorphic. I < r. are also C' diffeomorphic. As far as density is concerned. the situation is considerably more complicated. Thus, if dime N ) ;;.>. 2 dim( M), immersions are dense in C(M, N). But embeddings are dense in the C'-proper maps only for dim( N) ;;.>. 2dime M) + l. Since proper maps always exist (see Exercise 5.50), this theorem implies the Whitney embedding theorem: every C' n-manifold is diffeomorphic to a closed submanifold of R 2" + I. Let us return to the most important theorem of this kind. Let 7ft'(M,N;P)-{feC(M,N)lr;;.>.l,f is transveral to P at every point of M}. 3.5.28 Theorem (R. 11Jom). 7ft' (M. N; P) is residual, and hence dense, in C(M. N). If P is closed this set is also open. One can even be more precise regarding the approximation. Let VI' V2 be open in M such that cl( VI) C V2 • Then given f e C( M. N) there exist functions g e C( M, N) arbitrarily close to f. coinciding withfon VI and gi"l P off V2 • This classical transversality theorem has .. parameteric" versions (see Abraham and Robbin (1967» and generalizations to jet bundles (the "jet transversality theorem"), which are very useful in differential topology (Hirsch [1976J, Oolubitsky and Ouillemin (1974».
Exercl... 3.5A Show that the set of elements of L(Rn,R") with determinant I is a submanifold. 3.58 Show that the set O(n) of elements of L(R",R") that are orthogonal (i.e.• QQT = I) is a submanifold. (QT denotes the transpose of Q.) 3.5C (i) Let P c 0(3) be defined by P= {QeO(3)ldetQ= + I.Q=Q T }
Show that P is a two-dimensional compact submanirold of 0(3).
(ii) Define f: RP 2 - 0(3), f(l) = the rotation through 'IT about the line I. Show that f is a diffeomorphism of RP 2 onto P. 3.5D If N is a submanifold of dimension n in an m-manifold M, show that for each x e N there is an open neighborhood V c M with x e V and a submersionf: VcM-R m -", such that NnV=rl(O). (Hint: Choose a submanifold chart and compose it with an appropriate projection.)
182
MANIFOLDS AND VECTOR BUNDLES
3.5E
Show that Rpl is a submanifold of Rp2, which is not the level set of any submersion of Rp2 into Rpl; in fact, there are no such submersions. (Hint: Rpl is one-sided in Rp2, that is, there is no continu()us choice of normal direction to Rpl in RP2.) 3.5F (i) Show that if f: M --+ N is a subimmersion, g: N ...... P an immersion and h: Z --+ M a submersion, then go I 0 h is a subirnmersion. (ii) Show that if /;: M; --+ N;, i = 1,2 are immersions (submersions, subimmersions), then so is II X 12: MI X M2 ...... NI X N2. (iii) Show that the composition of two immersions (submersions) is· again an immersion (submersion). Show that this fails for subimmersions. (iv) Let f: M --+ N and g: N --+ P be C, r ~ I. If g 0 I is an immersion, show that I is an immersion. If g 0 I is a submersion and if I is onto, show that g is a submersion. .
3.SG
Let M be a manifold. R a regular equivalence relation and S another equivalence relation implied by R; i.e.• graph R c graph S. Denote by SjR the equivalence relation induced on MjR. Show that S is regular iff SjR is and in this case establish a diffeomorphism (MjR)j(SjR) ...... MjS. (ii) Let M;. i-I, 2 be manifolds and R, be regular equivalences on Mi' Denote by R the equivalence on MI X M2 defined by RI X R 2 • Show that MjR is diffeomorphic to (MljRI)X (MzlR 2 )· 3.SH (The line with two origins). Let M be the quotient topological space of (R X{O»U(R X{I» by identification of (/,0) with (t,I) for '" O. Show that this is a one-dimensional non-Hausdorff manifold. Find an immersion R --+ M. 3.51
(i)
Let f: M --+ N be C'" and denote by h: TM --+ r(TN) the vector bundle map over the identity uniquely defined by the pull-back.' Prove the following: (i) (ii) (iii)
h
I is an immersion iff 0 ...... TM --+ r(TN) is exact; h I is a submersion iff TM --+ r( TN) --+ 0 is exact; I is a subimmersion iff ker h and range h are subbundles.
3.SJ
Let A be a real nonsingular symmetric n X n matrix and c a nonzero real number. Show that the quadric surface {x E R"I(Ax, x) = c) is an (,. - 1)-submanifold of R". 30SK (Steinefs Roman Surface). Let f: S1 ...... R4 be defined by 1(.1', y, z) = (yz, xz, xy • .t 2 + 2y2 +3z 2 )
SUBMERSIONS,IMMERSIONS, AND TRANSVERSAL/TY
3.5L
3.5M 3.5N 3.50 3.5P 3.5Q 3.5R
3.5S
·183
(i) Show that f( p) = f(q) if and only if p = ± q. (ii) Show that f induces an immersion j: RP 2 -+ R 4. (iii) Let g: RP2 -+ R3 be the first three components of f Show that g is a .. topological" immersion and try to draw the surface g(Rp 2 ) (see Spivak (1979) for the solution). (Covering Maps). Let f: M -+ N be smooth and M compact. dime M) ... dime N) < 00. If n is a regular value of f. ~how (hat I( n) is a finite set {ml ..... m k } and that there exists an open neighborhood V of n in N and disjoint open neighborhoods Vi ..... Uk of ml ..... m A such thatrl(V)=U,U'" UVk • and flU,: L~-+I·.i=I. .... k are' all diffeomorphisms. Show k is constant if M is connected. Let f: M -+ Nand g: N -+ P be smooth maps. such (hat gi"\ V where Vis a submanifold of P. Show thatf/l'\g-I(V) iffg 0 f/l'\ V. Show that an injective irl)mersion f: M -+ N is an embedding iff f( M) is a closed submanifold of an open submanifold of N. Show that the map p: St(n. n; k) -+ Gk(R") defined hy p( A) = range A is a surjective submersion. Show that f: RP" xRP'" -+RP",ntmh given by (.\".y)>-> [xoYo; XOJ'I" .. x,Y,.· ... x"Ym) is an embedding. Show that {(x. Y) ERP" xRpmln ~ m.I:7_ox,Y, = O} is a (m + n -I) manifold. It is usually called a Milnor manifold. (Fiber product of manifolds). Let f: M -+ P and g: N -+ P be ('00 mappings such that 10' we speak of a semi-flow. In physics, it is usually not F,. to that is given. but rather the laws of molion. In other words, some differential equations are given that we must solve in order to find the now. These equations of motion have the form ds dl
-
=
X(s)
'
s(O) = So
where X is a (possibly time-dependent) vector field on S. ".1.3 Example. The motion of a particle of mass m under the innuence of the gravitational force field in Example 4.1.2 is determined by Newton's second law:
i.e.• by the ordinary differential equations
Letting q = (x, y. z) denote the position and p these equations become
=
m (dr/dt) the momentum.
~ .. F(,). The state space here is the manifold (R 1 \{0))XR 3, i.e .• the tangent manifold
VECTOR FIELDS AND FLOWS
187
of R 3 \{O}. The right-hand side of the preceding equations define a vector field on this six-dimensional manifold by X(q.p)= «q. p).( p/m. F(q))). In elementary courses in mechanics or differential equations. it is shown how to integrate these equations explicitly. producing trajel:tories. which are planar conic sections. These trajectories comprise the n(.w of the vector field. • Let us now tum to the elaboration of these ideas when a v.ector field X is given on a manifold M. Ir M = V is an open subset of a Banach space E. then a vector field on Visa map X: V .... V X Eofthe form X(x) = (x. V(x». We call V the principal parI of X,. However. having a separate notation for the principal part turns out to be an unnecessary hurden. Thus. in linear spaces we shall write. by abuse of notation. a vector field simply as a map X: V .... E and shall mean the vector field x'" (x. X ( When it is necessary to be careful with the distinction. we shall he. Ir M is a manifold and tp: V eM .... VeE is a chari for M. then a vector field' X on M induces a vector field X on E called the' local represenlalive of X by defining X( x) = Ttp· X( tp - I(.\: n. If E = R" we can identify the principal part of the vector field X with an n-component vector function (XI(x) •...• X"(x». Thus we sometimes just say "the vector field X whose local representative is (XI) = (XI ..... X")." Recall that a curve c at a point m of a manifold M is a C l map from an open interval I of R into M such that 0 E I and c(O) = m. For such a curve we may assign a tangent vector at each point ('(/), I F I. hy c'(/) = TC(/.I).
,l».
iIIt.'"
4.1.4 Definition. LeI M be a manifold and X E <x ( M), An euiw at mE M is a curve c al m such Ihal C'(/) - X(c(t» for each I E I.
0/ X
In case M ... U c E. a curve c(t) is an integral curve of X: V .... E when
t(/)" X(C(/». where t = dc/dl. Ir X is a vector field on a manifold M and X denotes the principal part of its local representative in a chart tp. a curve c on M is an ' integral curve of X when
: (t)-X(l(/».
.
where C" cp. c is the local represenlative of c. If M is an II-manifold and &he local representatives of X and c are (XI ..... X .. ) and «('I ..... C.. ). respectively. then c is an integral curve of, X when the following system of ordinary
188
VECTOR FIELDS AND DYNAMICAL SYSTEMS
differential equations is satisfied: cl(t)= XI(CI(t) •...• c"(t»)
c"(t) "" X"(cl(t) •...• c"(t)).
The reader should chase through the definitions to verify this assertion. These equations are autonomous, corresponding to the fact that X is time independent. If X were time dependent. a t would appear explicitly on the right-hand side of the foregoing equations. As we saw in Example 4.1.3. the preceding system of equations includes equations of higher order (by their usual reduction to first-order systems) and the Hamilton equations of motion as special cases. For a system of ordinary differential equations there are well-known existence and uniqueness theorems. The form that we shall need is the following. 4.1.5 Theorem. Let E be a Banach space and X: VeE ..... E be of class COO. For each Xo E V, there is a curve c: I ..... Vat Xo such that c'(t) = X( c(l» for all tEl. Any two such curves are equal on the intersection of their domains. Moreover, there is a nei~hborhood Vo of Xo E V. a real number a> 0, and a COO mapping F: Vo X I ..... E. where J = ( - a. a) such that the curve c,,(t): I ..... E, defined by c .. {t) = F(u, t) is a curve at u E E satisfying the differential equations c~( t) - X( cII( t» for all tEl. The proof proceeds in several steps. We begin with existence and uniqueness. 4.1.8 Lemma. Let E be a. Banach space. U c E an open set, and X: U c E ..... E a Lipschitz map; that is IIX(x)- X(y~1 CO Kllx - yll for all x. y E V and a constant K. Let Xo E V and suppose the closed ball of radius b.Bb(xo)-(XEElllx-xoll~b} lies in V, and IIX(x~I~M for all xE Bb(x o )' Let toER and leI o:=min(l/k,b/M). Then there is a unique C l curve x(t), t E [to - 0:, to + 0:] such that x(t) E Bb(x o) and {
x'(t} = X(x(t» . x(t o } = Xo
Proof. The conditions x'(t)=X(x(t».x{to}=x o are equivalent to the integral equation x(t}=x o + j'X(x(s»ds.
'0
VECTOR FIELDS AND FLOWS
189
Put xo(t) = Xo and define inductively X"+I(I)=X O + j'X(x,,(s»ds. 10
Clearly x"(t) E B,,(x o ) for all n and IE [10 - a, 10 + a] hy definition of a. We also find by induction that
Thus x,,(t) converges uniformly to a continuous curve x(l). Clearly x(t) satisfies the integral equation and thus is the solution we sought. For uniqueness, let y(t) be another solution. By indul,tion we find that IIx,,(t)- y(tlll ~ MK"II-lol"+ I; thus. letting n - 00 give~ x(1) =~,(t). • Let us observe that exactly the same result holds if X depends explicitly on I and/or on a parameter p, is jointly continuous in (t. p, x), and is Lipschitz in x uniformly in I and p. The next result is a basic estimate used to study the dependence on initial conditions. It is Gronwall's inequality.
4.1.7 Lemma. LeI f. g: [a, b[ - R be conlinuous and nonnegative. Suppose
f(I)~A+ f'f(s)g(s)ds./oraconslantA~O. (J
Then f(t)" Aexp(fg(s)ds)
forall
tE[a,b[.
Proof. First suppOse A> O. Let h(t) = A + J:f(s)g(s) dv; thus h(t) > O. Then h'(t) = f(t)g(t) " h(t)g(t). Thus h'(t)/h(t) ~ g(t). Integration gives h(t)" if exp O. If if = 0, then we get the result by replacing if by E> 0 for every E> 0; thus h and hence f is zero. •
4.1.. Lemma. Let X be as in Lemma 4.1.6. Let F, (x o ) denote the solution (-integral curve) of X'(I)=X(x(t»,x(O)=x o' Then there is a neighborhood V of Xo and a number Il > 0 such Ihal for every y E V there is a unique integral curve x(t) = F,(y) salisfying x'(t) = X(x(t) for IE [- E, II and
190
VECTOR FIELDS AND DYNAMICAL SYSTEMS
x(O) -= y. Moreover. 1IF,(x)- F,( Y)II" eK1'Ilix -
ylI·
Proof. The first part is clear from Lemma 4.1.6. For the second. let /(1) -1IF,(x)- F,(Y~I. aearly
/(1) =- Ilfo'X(~(X»)- X(~(y» dr + x -
so the result foUows from Lemma 4.1.7.
yll" IIx - yll+ Kfo'/{s}
dr.
•
This result shows that F,(x) depends in a continuous. indeed Lipschitz. manner on the initial condition x and is jointly continuous in (t. x). Again. the same result holds if X depends explicitly on 1 and on a parameter p; the evolution operator F!.,o(x) is the unique integral curve x( t) satisfying x'(t) - X(X(I). I, p) and X(lo) = x. Then F,~,,,(x) is jointly continuous in (to. I. P. x). and is Lipschitz in x. uniformly in (to. I. p). The next result shows that F, is C· i~ X is. and completes the proof of 4.1.5.
4.1.8 Lemma. LeI X in Lemma 4.1.6 be 0/ class C k • I ~ k ~ 00. and leI F,(x) be defined as be/ore. Then locally in (t. x). F,(x) is 0/ class C* in x and is Ck+ I in the I-variable. Proof. We deCine "'(I. x) E L(E. E). the continuous linear maps of E to E. to be the solution of the "linearized" or "first variation" equations:
d
dl"'(I.x)-DX(F,(x»)o",(t.x).
'" (0. x) = identity where DX( y): E .... E is the derivative of X taken at the point y. Since the vector field"' ..... DX(F,(x»o '" on L(E, E) (depending explicitly on I and on the parameter x) is Lipschitz in "'. uniformly in (I. x) in a neighborhood of every ('0' x o ). by 4.1.8 it foUows that "'(I. x) is continuous in (I. x) [using. the norm topology on L(E. E»).
VECTOR FIELDS AND FLOWS
We claim that DF,(x) == 1/1(/, x). To show this, lIet 8(t. h) = F,(x F,(x) and write
191
+ h)-
'(I, II )-1/1(/, x )·11 - fo'{X( F,(x + II» - x(f~( x»} ds - fo'[DX(F,(x» 0 1/I(s, x»)·hds· - fo'DX( F,(x»· ['(s, 1I)-1/I(.f. x )·11] tis
+ !'<X(f,(x + h»- X( F,(x»- DX(F,(x»'[F,(x + h)- F,(x)]} tis. Since X is of class C l , given e> 0, there is a I) > 0 such that IIhll < I) implies the second term is dominated in norm by fJellF,(x + h)- F,(x~1 dv, which is, in tum, 'smaller than Aellhll for a positive constant A. By GroDwall's inequality we obtain 118(1,11)-1/1(1, x)'hll" (constant)ellhll. It follows that DF,(x)·1I = 1/I(t,x)·h. Thus both partial derivatives of F,(x) exist and are continuous; therefore, F,(x) is of ~Iass CI. We prove F,(x) is c" by induction on k. Now
d dt F,(x) =
X( F,(x»
so d d . dl dlF,(x) - DX(F,(x»'X(F,(x»
and d dl DF,(x)'" DX( F,(x »·DF,(x).
Since the right-hand sides are Cit -~, so are the solutions by induction.· Thus F itself is Cit. • Again there is an analogous result for the evolution opera~(X) for a time-dependent vector field X( x, I, p), wbK:h may also .. extra parameters p in a Banach space P. II X is ~", then:~tX) is C~ -. •. . variables ~ is C" + I in , and '0' The variable p can be • dealt with .., _pending X to a new vector rleld obtained by appendiDa." ..... . differential equation p. 0; this dermes a vector rleld on BX r ... 44$" may be applied to iL The now on'E X Pisjuat F,(x.p)-(I1{X~;lJ;;'
192
VECTOR FIELDS AND DYNAMICAL SYSTEMS
For another more "modem" proof of 4.1.5 see Box 4.1C. This proof has a technical advantage: it works easily for other types of differentiahility on X on F,. such as Holder or Sobolev differentiability; see Ehin and Marsden (1970] for details. The mapping F gives a locally unique integral curve e. for each u E ~). and for each 1 E I. F, = FI(uo X (t}) maps ~) to some other set. It is convenient to think of each point u being allowed to .. flow for time t" along the integral curve c.. (see Fig. 4.1.2 and our opening motivation). This is a picture of a Uo "nowmg." and the system (Uo• a. F) is a local flow of X. or flow box. The analogous situation on a manifold is given by the following.
Figure 4.1.2
4.1.10 DeflnlUon. Lei M be a manifold and X a r
~
(i)
(ii) (iii)
I. A
cr tleelor field on
M
f10w box of X al me M is a triple (Uo• a. F). where
Uo eM is open. me Uo. and a E R. where a> 0 or a = + 00; F: Uo X I" -+ M is of class cr. where la = (- a. a); for each u E Uo. eu : I" curve of X al u;
-+
M defined by eu(t) = F(u./) is an integral
(iv) if F,: Uo -+ M is defined by F,(u) = F(u.n. Ihen for open. and F, is a C' diffeomorphism onto its image.
1E
la. F,(Uo) is
Before proving the existence of a flow box. it is convenient first to establish the following. which concerns uniqueness.
4.1.11
Prop08lUon. Suppose c. and c 2 are two integral curves of X at
me M. Then c. = c 2 on the intersection of their domains.
VECTOR FIELDS AND FLOWS
193
Proof. This does not follow at once from 4.1.5 for c, and C2 may lie in different charts. (Indeed. if the manifold is not Hausdorff. Exercise 4. t L shows that this proposition is false.) Suppose c,: I, - AI and c;: 12 . - M. Let I = I, n 12, and let K = (III E I and c,(t) = C,2(1»; K is closed since M is Hausdorff. We will now show that K is open. From 4.1.5. K contains some neighborhood of O. For IE K consider cl and c~. where c'(s) = c( I + of). Then c: and c~ are integral curves at c.(t) = c2(1). Again by 4.1.5 they agree on some neighborhood of O. Thus some neighborhood of I lies in K. and so K is open. Since I is connected, K"" I. • The next proposition gives elementary properties of now boxes.
4.1.12 Prop08It1on. Suppose (Uo• a, F) is a Iriple .,alis/i'ing (i), (ii), lind (iii) 0/2.1.3. Then lor I, s, and I + s E la' we have and
Fo is Ihe identUv map.
Moreover, i/ U, - F,(Uo) and U, n Uo *0, Ihen F,I U., n Uo: U _/ n Uo Uo n U, is a diffeomorphism and ils inverse is F _,I UO n u,.
Proof.
F,+.(u)" COl(1 + s), where cu is the integral curve defined by Fat u. But d(t) = F,(F,(u» = F,(cu(s» is the integral curve through cues) and /(/) -'C,,(I + s) is also an integral curve at cu(s). Hence by 4.1.11 we have F,( F,(u» - C.,( 1+ s) .. F,+.( u). For F, •• = F. 0 F,. merely note that F, • • = F,+,-F,oF,. Since C.,(/) is a curve at u, c,,(O)=u, so f;, is the identity. Finally, the Jast statement is a consequence of F, 0 F / = F .. , 0 F, = identity. Note, however, that F,(Uo)n Uo .. 0 can occur. •
Now we are ready to state and prove the existence and uniqueness of flow boxes. (E%i8~ _ U.~ of Flo .., Bo.'(es). C' veclor field on a ""''Ii/old M. For each m E M there is a flow m. Suppose (Uo• a. F), (Uo• a', F) are two flow boxes at m E M. F are equal on (UO n Uo)x(/.. n I ... ).
4.1.13 Theorem
Lei X be a box 0/ X at Then F and .
Proof (Uniqueness ). Again we emphasize that this does not follow at once
from 4.1.5, for Uo;Uo need not be chart domains. However, for each u E uon Uo we have FI{u}X I .. F'l{u}X I, where I -I.. n I .... This follows from 4.1.11 and 4.1.10 (iii). Hence F= F' on (UonUo)x I. (Exislence). Let (U, cp) be a chart in M with m E U. It is enough to establish the result in cp( U) by means of the local representation. That is, let (Uci. a, F') be a flow box of i. the local representative of X, at cp(m) as.
194
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Biven by 4. J.5. with
and let
F: UoX!" ..... M; Since F is continuous, there is abE (0, a) c R and ~, c Uo open. with mE Vo. such that F(Vo x I h ) c Uo. We contend that h. F) is a now box at m (where F is understood as the restriction of F to x/,.). Parts (i) and (ii) of 4.1.10 follow by construction and (iii) follows from the n:marks following 4.1.4 on the local representation. To prove (iv). note Iha. for t E Ih' F, has a C' inverse, namely. F _I as V, () Uo = v,. It follows thai /';1 is open. And. since F, and F I are both of class C. F, is a C' diffeomorphism. •
(v.,. v.,
v.,)
As usual. there is an analogous result for time- (or parameter-) dependent vector fields. The following result. called the "straightening out theorem." shows that near a point m that is not a critical point. that is. X( m) "" O. the now can be modified by a change of variables so that the integral curves oecome straight lines.
••1.14 Theorem.
Let X he a vector field on a manifold M and suppose at
me M, X( m) ... O. Then there is a local chart (U, cp) with m E U such that
(i) cp(U) = V x IcE X R, VeE. open, and 1 = (- a, a) c R. a > 0; (ii) cp-ll{v}X I: I ..... M is an integral curve of X at cp-I(V,O),for aI/v E V; (iii) the local representative K has the form K(y, t) = (y,t; 0,1).
Proof. Since the result is local. by taking any initial coordinate chart. it suffices to prove the result in E. We can arrange things so that we are working near 0 E E and X(O) = (0.1) E E = E1eR where EI is a complement to the span of X(O). Letting (~,. h, F) be a now box for X at 0 where Uo = VO X( - e. e) and Vo is open in E •. define
But Dfo{O.O) = Identity since
aF,(O,o)
at
I,-0 -
X(O)
=
(0,1)
and
Fo = Identity.
VECTOR FIELDS AND FLOWS
195
By the inverse mapping th~rem there are open neighhorhoods V X 10 c Vo X 1b and U- f(V X la) of (0,0) such thatfX I,,: V X 10 -+ U is a difrcomorphism. Then, I: U -+ V X la can serve as chart for (i). Notice that c ... /I{y}X I: 1-+ U is the integral curve of X through (y.O) for all y E V. thus p'roving (ii). Finally. the expression of the vector field X in this is D,I(y./)·X(f(y.I)I=D/ l(c(t)I·c'(I)= local chart given by (f- 1 0 c)'( I) = (0. I). since (f- 1 0 c)( I) = (y. I). thus proving (iii). •
",IV
,I
Now we turn our attention from local nows to global considerations. These ideas center on considering the now of a vector rield as a whole. extended as far as possible in the I-variable.
4.1.15 Definition. Given a manifold M and a tle('tor field X on M. let "i)x c M X R be the .ret of (m. I) E M X R such Ihat there is an integral ('urtJe
c: 1-+ M of X at m with (E I. The vector field X is complete if6fJx = M xR. Also. a poinl mE M is called (J complfle. where (J = +, -. or ±. if ('ilx n({m}XR) contains all (m. t) for I > 0,1 < 0,' or IE R. respeclively. Lei T': (resp. T;) denole Ihe sup (resp. in/) of Ihe limes of exiSlence of Ihe inlegral curves through m: T': (resp. T';;) is called Ihe positiw (".,me) life,u.ofm.
Thus. X is complete iff each integral curve can be extended so that its domain becomes ( - 00.(0): i.e.• T': - 00 and T; - - 00 for all m E M.
4.1.11 Exempt... A. For M - R 2. let X be the constant vector field. whose principal part is (0,1). Then X is complete since the integral curve of X through (x, y) is' .... (x.y+t). B. On M = R 2 \{O}. the same vector field is not complete since the integral curve of X through (0. - I) cannot be extended beyond' = I; in fact as I -+ I thiS. integral curve tends to (0.0). Thus ~; _I) = I. while ~o: -1)-00.
e.
On R consider the vector field X(x) = I + x 2 • This is not complete since the integral curve c with c(O) - 0 is c( (I) - tan (I and thus it cannot be continuously extended beyond - .,,/2 and .,,/2; i.e.• To t = ± .,,/2. •
4.1.17
P~ltIon.
Let M be a manifold and X
E
'X'( M), r ~ l. Then
(i) 6j)x::> M x {O}; (ii) 6fJx is open in M X R; (iii) Ihere is a unique C' mapping Fx: oDx -+ M such ,hal Ihe mapping I .... Fx(m,/) is an inlegral curve al m. for all mE M. (iv) for(m.t)E 6j)x.lhe pair (Fx(m. I).S)E oTJx'iff(m. (+ s)E "i)x: in this CtUe Fx (";, t + s) - Fx(Fx(m, I). s).
196
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Parts (i) and (ti) follow at once from the flow box existence theorem. and (iii) by the uniqueness of integral curves: "Dx is the union of all now box domains Uo x I" and Fx is the unique extension of the now box maps to UDx . Thus (iv) is a reformulation of the first part of 4.1.12. • Proof.
Thus. if X is complete, (M. 00. Fx) is a flow box. 4.1.18 DeflnlUon. Let M be a manifold and X E <X( M), r ~ 1. Then the mapping Fx is called the ;"tegl'Gl of X. and the curoe t ..... Fx (m, t) is called the mtIXimtII ;"tegrwl CfIrW of X at m. In case X is complete. Fx is called the /10M' ofX.
Thus. if X is complete with now F. then the set {F,I t E R} is a group of diffeomorphisms on M. sometimes called a one - parameter group of diffeomorphisms.t For incomplete flows. (iv) says that F, 0 F. = F,+ .• wherever it is defined. Note that F,(m) is defined for t E ]T'; • T,;! (. The reader should write out similar definitions for the time-dependent case and note that the lifetimes depend on the starting time to' Next we describe some basic criteria that ensures completeness of a vector field. 4.1.19 ProposlUon. Let X be C. where r ~ 1. Let c(t) be a maximal integral curoe of X such that for every finite open interoal ]a. b( in the domain 14(0)' T.·;O)( of c. c(]a. b() lies in a compact subset of M. Then C' is defined for alitER. Proof. It suffices to show that a E I. bEl. where I is the interval of definition of c. Let ttl E ]a, b(. ttl .... b. By compactness we can assume some subsequence c(l,,) converges. say. to a point x in M. Since the domain of the flow is open. it contains a neighborhood of (x,O). So there are e> 0 and T > 0 such that integral curves starting at points (such as c(l",) for large k I closer than f! to x persist for a time longer than T. This serves to extend c to a time greater than b, so bel since c is maximal. Similarly, a E I. •
The support of a vector field X on a manifold M is the closure of the set (1ft E MIX(m). O}.
4.1.20 Corollary. A C vector field with compact support on a manifold M is complete. In particular. a C' vector field on a compact manifold is complete. Since F" - (F,)" (the ntb power), the notation F' is sometimes convenient and is used where we use F,.
t
VECTORFIELOSANOFLOWS
197
Completeness corresponds to well-defined dynamics persisting eternally. In some circumstances (shock waves in fluids and solids. singularities in general relativity. etc.) one has to live with incompleteness or overcome it in some other way. Because of its importance we give two additional criteria. In the first result we use the notation X[ fl == df· X for the derivative of f in the direction X. Here 1: E - Rand df stands for the derivative map. In standard coordinates on R".
af af ) df(x) ... ( - , •...• ax" ax
and
X(f] ==
n / af LX -,' ax
/ _,
4.1.21 Proposition. Suppose X is a c k vector field on E. k ~ I. and f: E - R is a C' proper map (that is. the inverse images of compact sefs are compact). Suppose there are constants K. L IX(f](m)1 ~ Klf(m)1
~
0 such that
+ L forall mE E.
Then the flow of X is complele. Proof. that
From the chain rule we have (a/at)f(F,(m»== X[fJ(F,(m», so
f{F,(m»)- f(m) ;'l'X(f](~(m» dT. o Applying the hypothesis and Gronwall's inequality we see that 1f(F,(m»1 is bounded on any finite I-interval. so as f is proper, F,(m) lies in a compact set. Hence 4.1.19 applies. • Note that the same result holds if we replace "properness" by "inverse images of compact sets are bounded" and assume X has a uniform existence time on each bounded set. This version is useful in many infinite dimensional examples. The next theorem gives a completeness criterion in terms of the norm' on a Banach space.
4.1.22 ProposlUon. Let E be a Banach space and X a C' vector field on E. r ~ 1. Let" be any integral curve of X. Assume IIX( ,,(t)~1 is bounded on , finite t-intervals. Then the flow of X is complete. Proof. have
Suppose IIX(,,(t)~1 < A for tEla, b[ and let
"''(In)-''(I,,,)I~j'''lfJ'(t)ldl= I"
In
~ h. For
tn
< I", we
j'm Il X(,,(t»IIc/I< AI/",-I.I. 'n
198
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Hence G( t n) is a Cauchy sequence and therefore. converges. Now argue as in 4.1.19. • The following is a typical application. 4.1.23 IXllmple.
Let E be a Banach space. Suppose X(x) "" .... ·x + B(x).
where .... is a bounded linear operator of E to E and B is sublinear; i.e.• B: E .... E is C' with r;;. I and satisfies IIB(x~11Iit Kllxll+ L for constants K and. L. We shall show that X is complete by using 4.1.22 (In RIO. 4.1.21 can also be used withf(x) -lIxIl 2 ). Let x(t) be an integral curve of X. Then x(t) == x(O)+
fo'( .... ·x(s)+ B(x(s») dx.
Hence
IIx(t)lIlIitllx(O)II+ 1'(11 .... 11+ K)lIx(s)lItb + Lt. D
By Gronwall's inequality.
IIx(t)II'" (LT+llx(O)U)eUIAII+KI'. Hence x(t) and so X(x(t» remain bounded on bounded t-intervals.
•
BOX "-1A PRODUCT FORMULAS
A result of some importance in both theoretical and numerical work concerns writing a flow in terms of iterates of a known mapping. Let X E ~(M) with flow F, (maximally extended). Let K,(x) he a given map defined in some open set of [0. co[ X M containing {O} X M and taking values in M. and assume that (i) 'Ko(x)-x
(ii) K.(x) is C' in the "algorithm."
I
with derivative continuous in (e, .t). We call K '
Let X be a C' vector field. r> I. Assume that the algorithm K,(x) is ,....,., with X in the sense that X(x)-
a
a,K,(x)J.-o
VECTOR FIELDS AND FLOWS
199
Then, if (I, x) is in Ihe domain of F,(x), K,i,,(x) is defined for n sufficienlly large and converges 10 F, (x) as n --> 00 . Conversely.. if K,i,,(x) is defined and converges for 0 Et lEtT, then (T. x) is in Ihe domain of F and the limit is F,(x).
In the following proof the standard notation O( x ). x E R is used for any continuous function in a neighborhood of the origin such thaI O( x)/ x is bounded. Recall from Section 2.1 that o( x) denotes a. continuous function in a neighborhood of the origin satisfying limx_oo(x)/x - O. Proof. First, we prove that convergence holds locally. We begin by showing that for any x o, the iterates K,i,,(x o ) are defined if I is sufficiently small. Indeed, on a neighborhood of x o' K,(x) == x + O(e), so if K!/j(x) is defined for x in a neighborhood of x(). for j = 1,00" n -I, then
+ ... +(K,/,,(x)-x) ==O(t/n}+ .. · +O(I/n) =
OCt}
This is small, independent of n for I sufficiently small; so. inductively, K,i,,(x) is defined and remains in a neighborhood of -"0 for x near xo' Let P be a local Lipschitz constant for X so that 1IF,(x)- F,(yXI ~ e~ltlllx - yll. Now write
200
VECTOR FIELDS AND DYNAMICAL SYSTEMS
wherey.. " K,~,,(x). Thus 1IF,(x)-K,i,,(x)lI.;;
E" elll,l(n-k)lnllF,ln(Yn~kl)-K,/nCVn
~ 1)11
.. -I
..;nelll'lo(l/n)-O
as n ..... oo
since F.(y)- K,(y) - o(e) by the consistency hypothesis. Now suppose F,(x) is defined for 0.;; I .;; T. We shall show K,i,,(x) converges to F,(x). By the foregoing proof and compactness. if N is large enough, F,/N -lim n _ oa K,inN uniformly on a neighborhood of the curve t>-+ F,(x). Thus, for 0.;; I';; T,
By uniformity in t. Fr(x)= limKL(x). I
~oo
Conversely. let K,in{x) converge to a curve c(t).O~/~T. Let S = (t IF,( x) is defined and c( I) = F,( x )}. From the local result. S is a nonempty open set. Let E S.l k - t. Thus F,,(x) converges to c(t). so by local existence theory. F,(x) is defined, and by continuity, F,(x)= C(/). Hence S= [0. T) and the proof is complete. •
'4
4.1.25 Corollary. LeI X. Y E 'X,(M) wilhflows F, and G,. Let S, be + Y. Then for x E M.
the flow of X
The left-hand side is defined iff the right-hand side is. This follows from 4.1.24 by setting K,(x) - (F. 0 G,)(x). for example. for n X n matrices A and B. 4.1.25 yields the classical formula e(A+8)= lim (e A1 "e 8/,,)". n-oo
To see this. define for any n X n matrix C a vector field Xc E 'X,(R") by XC< x) - Cx. Since Xc is linear in C and has now F,(.r) - e'c.r. the formula follows from 4.1.25 by letting I = I. The topic of this box will continue in Box 4.20. The foregoing proofs were inspired by Nelson (1969) and Chorin et al. (1978).
VECTOR FIELDS AND FLOWS
201
BOX 4.18 INVARIANT SETS If X is a smooth vector field on a manifold M and N c M is a submanifold. the now of X will leave N invariant (as a set) ifr X is tangent to N. If N is not a submanifold (e.g.• N is an open subset together with a non-smooth boundary) the situation is not so simple; however. ·for this there is a nice criterion going hack to Nagumo [1942]. Our proof follows Brezis (1970).
4.1.21 Theorem. Let X be a locally Upschitz vector field on an open set U c E. where E is a Bana-:h space. Let G cUbe relatively closed and set d(x. G) = infOlx - Yilly E G}. The following are equivalent: Iimlr~o(d(x+hX(x).G)/h)=O locally uniformly in xEG (or just pointwise if E == R n); and . (ii) if x(t) is an integral curve of X starting in G. then xU) E G for all t'~ 0 in the domain of x(·).
(i)
Note that x(t) need not lie in G for t.s; 0; so G is only + invariant. (We remark that if X is only continuous the the~rem fails.) We give the proof assuming E = R n for simplicity. Proof. Assume (ii) holds. Setting x(t) = F,(x), where F, is the now of X. we get
d(x+hX(x).G).s;lIx(h)-x-hX(x)II=lhl II x(h) h - x -X(x) II
from which (i) foIlows. Now assume (i). It suffices to show x(t)E G for small t. Near x'" 'x(O). say on a ball of radius r. we have
and
We can ~ssume 1IF,(x)-xll 0. The local flow is F(x.t)=x+ H,(x)(tlr), as in step I. By' 2.4.17 (differentiability of the evaluation map), F is C·. +( E. x, y) ..
204
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Next we prove the result for k ~ 2. Considet the Banach space F-Ck-'(cl(U),E) and the map "'x:F .... F; 1J ..... Xo1J. This map is C' by the O-lemma (remarks following 2.4.18). Regarding "'x as a vector field on F, it has a unique C' integral curve 1J, with 1Jo ... identity. by Step I. This integral curve is the local flow of X and is C k - . since it lies in F. Since k ~ 2, 11, is at least C' and so one sees that D1J, =", satisfies .,Idt ... DX( 11;)''',. so by Step I again, II, lies in C k - '. Hence 11, is Cit. • The following is a useful alternative argument for proving the result for k - I from that for k ~ 2. For k -I. let X" .... X in C', where X" are C 2 • By the above, the flows of X" are C 2 and by Step I. converge uniformly i.e. in Co, to the now of X. From the equations for DlI,", we likewise see that DTI," converges uniformly to the solution of dM,ldt,. DX(TI,)·",. "0" identity. It foUows by elementary analysis (see Exercise 2.41 or Marsden [1974a, p. 109]) that TI, is C· and DTI, = II,. • This proof works with minor modifications on manifolds with vector fields and flows of Sobolev class H S or Holder class C k ~"; see Ebin and Marsden (1970) and Bourguignon and Brezis [1974]. In fact the foregoing proof works in any function spaces for which the O-Iemma can be proved. Abstract axioms guaranteeing this are given in Palais [1968].
Exercl... Find an explicit formula for the flow F,: R 2 .... R2 of the harmonic oscillator equation .f+ ",2X "" O. '" ERa constant. 4.lB Show that if (UO. a. F) is a flow box for X. then (Uo• a. F _) is a flow box for - X. where F _Cu. t) == F(u. - t) and (- X)(m) = -(X(m». 4.1 C Show that the integral curves of a C' vector field X on an n-manifold can be defined locally in the neighborhood of a point where X is nonzero by n equations ,,;(m, t) = cj == constant. i .. l •... ,n. Such a system of equations is called a local complete system 0/ integrals. (Hint: Use the straightening-out theorem.) 4.1 D Prove the foUowing generalization of Gronwall's inequality. Suppose vet) ~ 0 satisfies v(t)" c + fJ Ip(s)lv(s) tis, c ~ O. Then 4.1A
v(t) < cexP({IP(s)1 tis ). 1 I",.
this to generalize 4.1.23
,- -iilX.
In
allow A to be a time-dependent
VECTOR FIELDS AND FLOWS
4.1H
205 '
now
(VariatiON of COIUltmlllomtIIIa). Let F, - e'x be the of a linear vector field X on E. Show that the solution of the equation
.t- X(x)+ I(x) with initial condition Xo satisfies tbe integral equation
4.1 F
Let F( m. I) be a efta mal>ping of R x M to M such that F, •• - F, 0 F, and Fo - identity (wbere F,(m) - F(m, Show that there is a unique efta vector field X wbose now is F.
I».
4.10 Let a(l) be an integral curve of a vector field X and let g: M .... R. Let T(I) satisfy T'(I)= g(a(T(I))). Then show t .... CJ(T(t» is an inte&ral curve of gX. Show by example that even if X is complete. gX need not be. 4.1H
(i) (Gradienl Flows) letf: R" -+R be e l and let X= (iJf/ax l ••••• ilf/ ax") be the gradient of f. Let F be the now of X. Show that f(F,(x» ~ f(~(x» if I ~ s. (ii) Use (i) to find a vector field X on R" such that X(O) = o. X'(O) = 0, yet 0 is globally attracting; i.e.. every integral curve converges to 0 as I -+ 00.
4.11
let X= y 2 a/ax and Y=x 2 il/ay. Show that X and Yare complete on R 2 but X + Y is not. (ii) Prove the following theorem: Let H be a Hilbert space and let X and Y be locally Lipschitz vector fields tbat satisfy the following: (a) X and Yare bounded and Lipschitz on bounded sets; (b) there is a constant fJ ~ 0 sucb tbat (i)
(Y(x). x) < fJllxlf' (c)
for all
x EX
there is a locally Lipschitz monotone increasing function C(I) > O. I ~ 0 such that JOOdx/c(x)'" + 00 arid if X(I) is an integral curve of X.
!lIx(t)1I < c(lIx(t)lI) Then X. Y and X + Yare positively complete. NOle: One may assume IIX(xo~I'" c(lIxolI) in (c) instead of (d/dl~I%(t~1 < cOI%(I~I>. (Hinl. Find a differential inequality
206
VECTOR FIELDS AND DYNAMICAL SYSTEMS
for t 1. Let
so
'2 is a Coo function that is 1 if s < 1 and 0 if s > 3. Finally, let
The nonn on a real Hilbert space is Coo away from the origin. The order of differentiability of the norms of some concrete Banach spaces is also known; see Bonic and Frampton (I966J and Yamamuro (1974).
4.2.14 II",
Carol'' ' '
Let M be a manilold modeled on a C' Banach space. II
e r:,M, then there is an I e ~'( M) such that d/(m) =
11m'
.
Proof. If M .. E, SO TmE g; E, let I(x) = II",(X), a linear function on E. Then dl is constant and equals II",. The general case can be reduced to E using a local chart and a bump function as foUows. Let cp: U ..... U' c E be a local chart at m with cp( m) = 0 and such that U' contains the ball of radius 3. Let a", be the local representative of II", and let h be a bump function 1 on the ball of radius 1 and zero outside the ball of radius 2. Let i( x) = am (x) and let
1_ {(hJ)oCP, 0,
on U on M\U
It is easily verified that I is C' and d/( m) -
II",.
•
4.2.15 Propoeltlon. (i) Let M be a manifold modeled on a C' Banach space. The collection 01 operators Lx lor X e CX'(M), defined on C'+ '(M, F) and laking values in C'(M, F) lorms a real vector space and IJ(M) module with (/Lx )(g) -/(LxS), and is isomorphic to CX(M) as a real vector space and as an ~(M) module. In particular, Lx =0 iff X=O; and LfX = /Lx. (ii) Let M be any manilold.11 LxI = olor aliI e C'(U, F),lorall open subsets U 01 M, then X = O.
I'rrIof. (i) Consider the map a: X .... Lx. It is obviously R and
IJ(M)
linear; i.e.
Lx,+/x,'" Lx, + /Lx, To show that it is one-to-one, we must show that Lx = 0 implies X ... O. But if Lx/(m)'" 0, then 4j(m)X(m) == 0 for all f. Hence, 1I",(X(m»'" 0 for all II", e r:,( M) by 4.2.14. Thus X( m) .. 0 by the Hahn- Banach theorem.
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
217
(il) This has an identical proof with the only exception that one works in a local chart, so it is not necessary to extend a linear functional to the entire manifold M as in 4.2.14. Thus the condition on the differentiability of the norm of the model space of M can be dropped. •
Now we are ready to give the main result on derivations.
4.2.18 Theorem. (i) If M is finite dimensional, the collection of all derivations on iJ(M) is a real vector space isomorphic to ~x'(M). In particular, for each derivation 8 there is a unique X E ~ ( M) such that 8 = Lx. (ii) Let M be a manifold modeled on a Coo Banach space E i.e. E has a Coo norm away from the origin. The collection of all (R-linear) derivations defined on Coo(M, F) (for all Banach spaces F), furms a real vector .fpace isomorphic to ~(M).
We prove (ii) first. Let 8 be a derivation. We wish to construct X such that 8, = Lx. First of all, we note that 8 is a loca/ operator; that is, if he Coo(M, F) vanishes on a neighborhood V of m, then 8(h)(m)= o. Indeed, let g be a bump function equal to one on a neighborhood of m and zero outside V. Thus h = (1- g)h and so Proof.
8(h)(m)
=
8(1- g)(m)·h(m)+8(h)(m)(l- gem»~ = O.
(I)
If U is an open set in M, and f E Coo(U, F) define (8/uXf)(m) = 8( gf)( m), where g is a bump function equal to one on a neighborhood of m and zero outside U. By the prevlOi.1~ remark, (81 UXf)(m) is independent of g, so 81 U is well defined. For convenience we write 8 = 81 U. Let (U, 'P) be a chart on M, m E U, and f E C oo ( M, F) where 'P: U -+ U' c E; we can write, for x E U' and a '"" 'P( m), ('P.f)(x)= ('P.f)(a)+1 1 aa ('P.f)[a+t(x-a)]dt 0 t
,
= ('P.f)(a)+ foID('P.f)[a+t(x-a)].(x-a)dt.
This formula holds in some neighborhood 'P(V) of a. Hence for u E V we have (2) feu) = f(m)+ g(uH'P(u)-a), whereg E Coo(V, L(E, Applying 8 to (2) gives
F» is given by g(u) = fJD('P.f){a + t('P(u)~ a)]dt.
, 8f(m) = g(m)· (8!p )(m) = D( 'P.f)(a)· (8'P)(m)
(3)
218
VECTOR FIELDS AND DYNAMICAL SYSTEMS
since 8 was given globally, (3) is independent of the chart. Now define X on U by its local representative
X.(x) - (x,e(4p)(u». where x - cp( u) E U'. It follows that XI U is independent of the chart cp and hence X E 'X(M). Then, for IE C«J(M, F), the local representative of LxI is
Hence Lx = 8. Finally, uniqueness follows from 4.2.14. The vector derivative property was used only in establishing (I) and (3). Thus, if M is finite dimensional and e is a derivation on ~ ( M), we have as before
I(u) - I(m)+ g(u)'(4p(u)-a) -/(m)+
"
L (c,I(u)-a')8,(u). ,-,
where g, E Cj(V) and
0= (0', ... ,0"). Hence (3) becomes
8/(m) =
L" gi(m)8(4pi)(m)
i-'
and this is again independent of the chart. Now define X on U by its local representative
(x, 8( 4p')( u), ... ,e( 4p")( u» and proceed as before. • There is a difficulty with this proof for derivations mapping C .. , to C. Indeed in (2), 8 is only C if I is C+', so e need not be defined on g.
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
219
The result is. however. still true in this case. hut rcquires a different argument (see Box 4.2C). For finite-dimensional manifolds. the preceding theorem provides a local basis for vector fields. If (U. "'). "': U - VcR" is a chart on M defining the· coordinate functions Xi: U --+ R. define n derivations a/ax' on ~(U) by
These derivations are linearly independent with coefficients in
(Lli~)(XJ)=f'=O
Eli-;=O. then ax
ax
i_I
~'f(U).
for if
forall j=I ..... n.
since (a/axi)x i = 8/. By 4.2.16, (a/ax i ) can be identified with vector fields on U. Moreover, if X E 'X( M) has components Xl •...• X" in the chart ",. then
Lxl=X(fJ=
E
Xi
i-I
ali ax
=
(E, _ X'~)/. ax I
i.c.
X=
EX'~. ax'
i_I
Thus the vector fields ( a/ ax I). i - I •...• n form· a local basis for the vector fields on M. It should be mentioned however that a global basis of 'X( M). i.e.• n vector fields. XI ..... X n E 'X(M) that are linearly independent over ~(M) and span 'X(M). does not exist in general. Manifolds that do admit such a global basis for ~ (M) are called parallelizable. It is straightforward to sho~ that a finite-dimensional manifold is parallelizable iff its tangent bundle is trivial. For example. it is shown in differential topology that S3 is parallelizable but S2 is not (see Hirsch (1976». This completes the discussion of the Lie derivative of functions. Turning to the Lie derivative of vector fields. let us begin with the following.
4.2.17 Propoaltlon.
II X
and Yare C' vector fields on M. then [~x. Ly) H I( M. F) to C,-I(M. F).
... Lx Ly-:- Ly Lx is a derivation mapping C 0
0
Proof. More generally, let 81 and 6z be two derivations mapping C'· I to C and C to C- I • Clearly [8 1, 6z1 = 81 06z - 6z 08 1 is linear and maps C+ I to C- I . Also. if IE C'+ I (M. F), g E C+ I (M,G). and BE L(F,G; H),
220
VECTOR FIELDS AND DYNAMICAL SYSTEMS
then
(1 •• 8z](B(f. g» ... (I. o8z)(B(f. g»-(8z o8.)(B(f. g» - 1.{B(8z(f), g)+ BU. 8z(g))}- 8z{B(8.(f). g)+ BU.I. (g))} ==
B(I.(8z(f». g)+ B(8z(f), 8.(g»+ B(I.(f). 8z( g» + BU. 1.(8z( g»)- B( 8z(8.(f». g) - B(8.(f). 8z(g» - B(8z(f),I.(g»- BU.~(8.(g)))
=
B([8 •. ~](f). g)+
BU. [8 •. ~]( g».
•
Because of 4.2.16 the following definition can he given. 4.2.18 Definition. Let M be a manifold modeled on a COO Banach space M). Then (X, Y I is the unique vector field such that L [X. Y J = and X, Y E ( Lx, L y). This vector field is also denoted Lx Yond is called the Lk tkrivatit¥ 0/ Y willi ~$pect to X, or the Lk brtJcht 0/ X tuUI Y.
ex ""(
Even though this definition is useful for Hilbert manifolds (in particular for finite-dimensional manifolds). it excludes consideration of C' vector fields on Banach manifolds modeled on nonsmooth Banach spaces, such as L' function spaces for p not even. We shall, however, establish an equivalent definition, which makes sense on any Banach manifold and works for C' vector fields. This alternative definition is based on the following result. 4.2.18 Theorem. Let M be as in 4.2.18 and X, Y have (local) flow F,. Then
E
'X,(M) and let X
d dt (F,"'Y) = F,"'(LxY) (at those points where
Proof.
F, is defined).
If t = 0 this formula becomes
ddt
I,-0
F,"'Y= Lx Y
(4)
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
221
Assuming (4).
d (P,'Y)- .1dF:+.Y=P,' i d.1_ dI ~ ~
.-0
I.-0J;Y=p,"LxY'
Thus Jhe formula in the theorem is equivalent to (4). which is proved in the following way. Both sides of (4) are clearly vector derivations. In view of 4.2.16, it suffices then to prove that both sides are equal when acting on an arbitrary function f e COO( M. F}. Now
: (p'"Y)[f](m)l._o = =
:
I.-o{ df(m).(TF,(m,F._,
0
yo F,)(m)}
:1._op'"(Y[F!,f])(m).
Using 4.2.10 and Leibniz' rule, this becomes
X[Y[f)](m)- Y[X(fJ1(m) ... [X, Y][f](m) . • Note that in the preceding proof we derived the following fact of general utility: if.,: M ..... N is a' diffeomorphism and Y E ~. (N). then for
f e6J(M),
(5)
Since the formula for LxY in Eq. (4) does not use the fact that the norm of E is Coo away from the origin. we can state the following defmition of the Lie derivative on any Banach manifold M .
••2.20 Allermdlve DefInition of Lie bnIcket. If X,Ye~'(M), r;;.1 and X has flow F" the C- I vector field LxY - IX, Y] on M defined by [X,Y)- ddl I
,-0(p,'Y)
is called the Lk deriwltiw of Y with respect to X, or the Lk brrlebt of X andY. . From the point of view of this more general definition, 4.2.19 can be rephrased as follows .
••2.21 ~. Let X, Y e 6X(M), 140; r. Then IX, Y]- LxY is the uniqUe C- I vector field on M satisfying [X. Y][f] == X[Y[fJ] - Y[X(f]] for all f e C+ I(U, F), where U is open in M.
222
VECTOR FIELDS AND DYNAMICAL SYSTEMS
The derivation approach suggests that if X, Y E 6.X.'(M) then IX, Y] might only be C,-l, since [X, Y] maps C+ 1 functions to C- I functions, and differentiates them twice. However 4.2.20 (and the coordinate expression below) show that [X, Y] is in fact c- I. We can now derive the basic properties of the Lie bracket.
4.2.22 Propoeltlon. The bracket IX, Y] on 'X(M), together with the real v«tor space structure of 'X ( M), form a Lk lIIgebra. That is, (i) (,] is R bilinear: (ii) [X, X) - 0 for all X E 'X( M): (iii) [X,[Y, Z)]+(Y,[Z, X))+[Z.[X, Y)) = 0 for all X. Y, Z ( JlICObi ilkntily).
E ~X(M)
The proof is straightforward, applying the brackets in question to an arbitrary function. Unlike 'X(M), the space 'X'(M) is not a Lie algebra since [X. YJE 'X,-I(M) for X. Y E 'X'(M). Note that (i) and (ii) imply that [X. Y] = - [Y. X). for
[X + Y. X+ YJ -0 "" [X. X)+[X. Y)+ [yo X)+ I Y. Y). Also, (iii) may be written in the following suggestive way: Lx[Y. Z) = [LxY, Z) + [y, LxZ):
i.e., Lx is a Lie bracket derivation. Strictly speaking we should not use the same symbol Lx for both definitions of Lxf and LxY. However. the meaning is generally clear from the context. The analog of 4.2.7 on the vector field level is the following.
4.2.23 PropoalUon. (i)
Let ep: M
--+
N be a diffeomorphism and X
E
'X ( M). Then Lx: 'X ( M) --+ ~ ( M) is natural wit" IYspen to p_-1ol'WtU'd by ep. That is. L".xep.Y" ep.LxY, or [ep.X, ep.Y) = ep.[X. Y), or the following diagram commutes: 'J'.
CX(M)--• CX(N)
I
LxI
L".x
CX(M)-- ~(N)
••
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
223
(ii) Lx is """"'" witII rnp«t 10 rntrictiolu. ThaI is. for U c M open, we have [XI U, fIUJ- IX, fHU; or Ihe following diagram commule.f: IV ~(M)----' ~(U)
Lxl ~(M)
Proof.
any Z
we get
!L xlu ~(U)
IV
For (i),letf E "(V), vbe open in N. and tp(m) = n E V. By (5). for E ~x'(M)
fro~
«IP.Z }[f])(n)
= Z [f
° IP)( m).
4.2.21
(IP.[X, f])[f]{n) - [X, f](f°IP]{m)
... X[f(f ° IP ]](m)- f[X(f ° IP ]](m) = X [( IP.Y Hf] ° IP ](m)- f [( IP. X
)(f] ° IP](m)
... (IP.X)[( IP. f)[ f]]( n) - (IP. Y)[ (IP.X)[ f]]( n) ... [IP.X.IP.Y)[f)(n).
Thus IP.[X, fJ- [IP.X,IP.fJ by 4.2.15(ii). (ii}"rollows from the fact that d(1I U) ... d/I u. • let us now compute the local expression for [X. fJ. Let IP: U ... VeE be a chart on M and let the local representatives of X and f be X and Y respectively, so X, Y: V ... E. By 4.2.22, the local representative of IX, fJ is
[X, fJ. Thus,
[X, t][JJ(x) - X[y[JJ](x)- Y[X [JJ](x) ... D( y[j])(x )·X(x)- D( X[i])( x )·Y(x) .. Now f[lJ<x)- D](x)'Y(x) and its derivative may be computed by the product rule. The terms involving the second derivative of ] cancel by symmetry of D 2J(x} and so we are left with
D](x)· (DY{x).·X(x)- DX(x)· Y(x )}.
224
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Thus the local representative of (X. Y} is
DY·X-DX·Y. If M is n-dimensional and the chart '" gives local coordinates (xl •...• x") then this calculation gives the components of IX. Yl as
(6) Part (i) of 4.2.22 has an important generalization to ",-related vector fields. For this. however. we need first the following preparatory proposition.
4.2.24 Propoeltlon. Let "': M -+ N be a C' map of C' manifolds. X E CX,-I(M). and X' E CX,-I(N). Then X - X' iff (X1f)) ° '" - Xlf ° "'lfor all
f
"
E C§'I(V). where V is open in N.
Proof. By dermition, «X11l)o ",)(m) -
d!(",(m»·X'(",(m». By the chain
rule. X(f ° ",)(m) ... d(fo ",)(m)·X(m) .. df( ",(m»·T",,,,(X(m».
If X! X'. then T", ° X - X' ° '" and we have the desired equality. Conversely. if X[f ° "'] - (X'lll)° '" for all f E '!f1(V), and all V open in N, then choosing V to be a chart domain and f the puD-back to V of linear functionals on the model space of N. we conclude that II,,· (X' ° '" )( m) - II,,· (T", ° X)(m), where n = ",(m). (or all II" E Using the Hahn-Banach theorem. we conclude that (X'o ",)(m) = (T", 0 X)(m). for all mE M. • , It is to be noted that under dirferentiability assumptions on the norm of N (as in 4.2.16), the condition "for a1lf E '!f1(V) and all V eN" can be replaced by "for all f E 6J1(N)" by using bump function', This holds in particular for Hilbert (and hence for finite-dimensional) n.illifolds.
r:N.
4.2.25 PropoeHIon. Let "': M -+ N be a C'map of manifolds. X. Y E CX·-I(M). and X',Y'E~:-I(N). If X!.X, and Y!Y'. then
IX. Y] !·[X'. Y1.
·VECTOR FIELDS AS DIFFERENTIAL OPERATORS
225
Proof.' By 4.1.14 it suffices to show that ([X'. Yll1) ° cp = lX. YII cp] for aU I e ~I(Y). where V is open in N. We have 0
([ X'. Y'][/» ° cp'" X'[Y'[/]]· cp - Y'[X'[ I]] cp 0
- X[(Y'[/). cp)- Y[( X'[/). cp) ==
X[Yl/ 0cp)) - Y[xl/ 0.,,)]
==
[X. Y][/·.,,] . •
The analog of 4.2.9 is the following.
4.2.21 Propoeltlon. For every X e ~ (M). Lx ;s a derivation on (Cft( M). ~(M». That is. Lx is R linear on each. and Lx(/ Y) - (I.x/)Y + I(LxY).
' For g E coo(U, E), where U is open in M, we have
Proof.
[X,/y][g] - Lx{Lfyg)- LfyLxB - LxULyg)- fLyLxB - (Lxf}Lyg + fLxLyg- fLyLxB. so [X,/y) - (Lxf}Y + /[X, Y] by 4.2.IS(ii). • Commutation of vector fields is characterized by their nows in the followin& way.
4.2.27 PropoeIUon. Let X, Ye ~r(M), r .. I, and let flows. The loIlowlng are equioalent.
F"G, denote their
(i) lX, y) ... 0; (ii) FiY-,Y;
G:
(iii) X -' X; (iv) F, • G. - G. ° F,. (In (ii)-(ro), equoIity is undentood, as usual, where the expressions are
defined.)
F, • G• .. G, 0 F, iff G. - F, • G. • r, I. which by 4.2.4 is equivalent to y ... FiY; i.e., (iv) is equivalent to (ii). Similarly (iv) is equivalent to (iii). If FiY-Y, then
Proof.
[X,Y)=
dtdl ,-0F,*Y""O.
226
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Conversely. if IX. YJ - LxY "" O. then·
so that p,'Y is constant in t. For t = O. however. its value is Y. so that p,'Y - Y and we have thus showed that (i) and (ii) are equivalent. Similarly . (i) and (iii) are equivalent. • Just as in 4.2.10, the formula for the Lie derivative involving the flow can be used to solve special types of first-order linear n x n systems of partial differential equations. Consider the first-order system: .
f. (XJ(K)aY'(K.t)_YJ(K.t)aXi(K.t») ax' ax}
aaYi(K.t)", (P,,) {
t
J_ I
yl(K.O) "" gi(K) for given smooth functions Xi( K). gi( K) and scalar unknowns yi( K. t ). i = 1•..•• n. where K -= (Xl •. .•• x"). 4.2.28 Proposition. Suppose X= (XI •.... X") hav a complete /1011' F,. Then letting Y == (yl •...• Y") and G = (gl ..... g,,).
y= p,'G is a solution of the foregoing problem (P,,). (See Exercise 4.2C for uniqueness).
ay
at = =
d
dt p,'G = P,'[ x. G) = [F,* X. p,'G)
[X. Y)
since P,' X = X and Y = p,'G. The expression in the problem (P,,) is the ith component of IX. Y). • 4.2.21 Example. ayl
-
at
ay2
Solve the system of partial differential equations: =
ayl
ayl
ay2
ay2
(x+ y ) - -(x+ y) _ _ yl_y2 ax ay
-=(x+y)--(x+ .-)_+yl+y2 at ax . ay yl(X. y.O)
=
x
y2(X. y.O) = y2.
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
227
The vector field X(x, y) = (x + y, - x - y) has the complete flow F,(x, y) "" «x + y)1 + x, -(x + y)t + y), so that the solution is given by Y(x. y, t) ... F:(x, y2); i.e., { .
yl( x, y, t) ...
«x + y)t + x)( 1- t) - I [y - (x + y)t]2
y2(x, y, I) = t«x
+ Y)I + x)+(t + I)[y - (.t + y)tY. •
In later chapters we will need a flow type formula for the Lie derivative of a time-dependenl vector field. In Section 4.1 we discussed the existence and uniqueness of solutions of a time-dependenl Vl'l'lor field. Let us formalize and recall the basic facts.
4.2.30 DaflnlUon. A C tiIM-depent/Dft vector field i.f a C map X: RxM ..... TM such that X(t,m)ETmM for all (t,m)E:RXM; i.e:, X,E ~.'(M),
lor
where X,(m) = X(t, m). The time-de",_nt flo ... or ft:JOhIlw"opelYl-
F, .• of!l is defined by the requirement thai I .... F, ..,(m) be Ihe integral
curoe of X slarling at m altime t ""' s; i.e.,
d dl F, .• (m) = X(/, ~ .• (m»
and
F. .• (m) = m.
By uniqueness of integral curves we have F, .• F,,, = F,., (replacing the flow property F,+. = F, F.), and F,., = identity. It is customary to ~rite F, - F,.o· If X is time independent, F, .• " F,- •. In general F:X," X,. However, the basic Lie derivative formulae still hold, 0
0
4.2.31 Theorem. Lei X, E ~'(M), r., I for each t and suppose X(/, m) is continuous in (I, m). Then F,.• is of class C' and for f E C+ '(M, F), and Y E 'X'(M). we have
(i) d dlFi..Y= Fi..([X,. Y)) = Fi..(Lx,Y).
(ii)
Proof. The rarst part was proved in Section 4.1. For functions, the proof is a repeat of 4.2.10:
~(Fi..f)(m)- ~(fo F, .• )(m)=df(F, .• (m»· dF,'d!m) - d/(F,.,(m»·X,( F, .• (m» == (Lx,/)( F,.Am»= Fi..(Lx./)(m).
228
VECTOR FIELDS Afl/D DYNAMICAL SYSTEMS
For vector fields, note that from (5), (~.Y)[f1 = ~.(Y[F;,/])
(7)
since F.., = F,~". The result (ii) will be shown to follow from (i), (7). and the next lemma.
4.2.32 Lemma. d
dtF;,/= - X,[~,/]. Proof. Differentiating ferential equation:
F.., F, .• = 0
identity in t. we get the backward dif-
d
dt F.., =
-
TF.., 0 X,.
Thus
d
dt F:"f(m) - - dfU:.,(m »·TF.,,( X,(m»
... - d(f 0 F..,)·X,(m) = - X,(f 0 F..,](m).
9
Thus from (7) and (i). d
dt (~.Y)[f] -= ~.( X, [Y[F:,,/]])- ~.(Y[X,[~,fJ]). By 4.2.21 and (7). this equals (~.[X" Y)[f). • If f and Yare time dependent, then (i) and (ii) read
(a
d -P. dt , ...1 =P. ,.. -atf+ L x/ )
(8)
and
d
dt~'Y= ~s
(ay at + [X,. Y,] ).
(9)
Unlike time-independent vector fields. we generally have
Time-dependent vector fields on M can be made into time-independent ones on a bigger manifold. Let , E ~ (R x M) denote the vector field given
VECTOR FIELDS AND DIFFERENTIAL OPERATORS
229
Figure 4.2.4
by '(s, m) -
(~,
1),Om) e 1(•. m)(R X ~);; T,R X TmM. Define the .suspen-
sion of X by Xe 'X,(R X M) where XU, m) - «t, 1), X(t, m» and observe that X - , + X. Since b: I .... M is an integral curve of X at miff b'( I) ... X(t, b(t» and b(O) ... m. a curve c: J .... R X M is an integral curve of X at (0, m) iff c(t) - (t. b(I». Indeed, if c(1) == (a(t), b(/» then C(/) is an integral curve of X iff C'(I)" (a'(t), b'(/» = X(C(I»; that is a'(t) == I and b'(/) - X(a(/), b(/». Since a(O) - 0. we get aCt) - t. These observatians are
summarized in the following (see Fig. 4.2.4.)
,
4.2.33 P~ltIon. Lei X be a C-time-dependent veclor field on M wilh evolution oPerator F"., The flow F, of the suspension Xe CX'(R X M) is given by F,(s. m) = (t + s. F,+.,.(m». • Proof. In the preceding notations. b(l) = F"o(m). c(t) = F,(O, m) = (I. F"o( m», and so the statement is proved for s = O. In general. note that ~(s. m) = F,+s(O. Fo,.(m» since 1 - F, .. • (0. i;,.•(m» is the integral curve of X. which at 1=0 passes through F,(O.m)=(s.U:.nol·;,.,)(m»=(s.m). Thus F,(s. m) = F,+$(O. Fn. ,(m» = (I + s. (F,+"II ° 1;,. • )(m» = (t + .f. F,+s .• (m». •
BOX 4.2A
AUTOMORPHISMS OF FUNCTION ALGEBRAS
The property of flows corresponding to the derivation property of vector fields is that they are algebra preserving:
P,'(fg) ... (P,'f)( P,'g).
230
VECTOR FIELDS AND DYNAMICAL SYSTEMS
In fact it is obvious that every diffeomorphism induces an algebra automorphism of ~ ( M). The following theorem shows the converse. t 4.2.34 Theorem. Let M be a paracompact second-countable finitedimensional manifold. Let a: ~(M) - ~(M) be an invertible linear mapping that satisfies a(fg)=a(f)a(g) lor all I. g E ~(M). Then there is a unique Coo diffeomorphism fJ!: M - M such that a(f)=fOfJ!.
Remark. A. There is a similar theorem for paracompact second-countable Banach manifolds; here we assume that there are invertible linear maps a.T: COO( M. F) - C oo ( M. F) for each Banach space F such that for any bilinear continuous map B: F x G - H we have
for I ECoo(M. F) and gECOO(M.G). The conclusion is the same: there is a Coo diffeomorphism fJ!: M - M such that
for all F and all I E C oo ( M. F). Alternative to assuming this for all F. one can take F = R and assume that M is modeled on a Banach space that has a norm that is Coo away from the origin. We shall make some additional remarks on the infinite-dimensional case in the course of the proof. B. Some of the ideas about partitions of unity are needed in the proof. Although the present proof is self-contained. the reader may wish to consult Section 5.6 simultaneously. C. In Chapter 5 we shall see that finite-dimensional paracompact manifolds are metrizable. so by \.6.14 they are automatically second countable. Proof. (Uniqueness). We shall first construct a Coo function x: M - R which takes on the values I and 0 at two given points mi' m2 E M. m, tWe tIumk Alan Weinstein for suggesting this theorem. We think this result must be known in the literature. but we could not locate a specific reference except for an analogous result of Mackey (1962) for measurable functions and measurable auto-
morphisms.
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
23.1
"" m 2 • Choose a chart (U. cp) and mi' such that m 2 1i:. U and such that cp( U) is a ball of radius r, about the origin in E. cp( m, ) = O. Let V C U be the inverse image by cp of the ball of radius r2 < r, and let 8: E -+ R be a Coo-bump function as in Lemma 4.2.13. Then the function x: M -+ R given by .
80 CP.
x= { 0,
U M\U
on on
is clearly Coo and x(m,) = I. x(m 2 ) = O. Now let us assume that cp./ == If·/ for two different diffeomorphisms "'.If of M. for all/ E tj(M). Then there exists a point me M such that cp(m)-If(m) and thus we can find XE~'t<M) such that (X 0 cp)(m) = I,(X 0 If)(m) = 0 contradicting cp·X = If·X. Hence cp == If. The proof of existence is based on the following key lemma. 4.2.35 Lemma. Let M be a (/inite-dimensiona/) parucompact mani" /old and p: 'fJ( M) -+ R be a nonzero algebra homomorphism. Then there is a unique point m E M such that PU) /(m'>. =0
Proof. (Following suggestions of H. Bercovici). Uniqueness is clear. as before. since for m, "" m 2 there exists a bump function / E tj(M) satisfying/(m,) == O. /(m 2 ) = I. To show existence. note first that P(l) = 1. Indeed P(I) = P(12) = P( I )P( 1) so that either P( I) = 0 or P( I) = 1. But P< I) = 0 would imply P is identically zero since PU) = P(l·/) = P(I)·PU). contrary to our hypothesis. Therefore we must have P( I) = I and thus P( c) = c for cER. For mEM, let Ann(m) = {/E tj(M)I/(m) = O}. SecOnd, we claim that it is enough to show that there is an mE M such that kerp={/E~T(M>lPU)=O}==Ann(m). Clearly. if ~U)-/(m) for some m, then kerp = Ann(m). Conversely, if this holds for ~ mE M and / It:. kerp, let c,. PU) and note that / - cE kerPAnn(m), 'SO/(m) == c and thus P O. Let V", be an open neighborhood of the closure. c1(U",). Since M is paracompact. we can assume that {V",lm eM} is locally finite. Since M is second countable. M can be covered by {V",V eN}. Let ~ = /m and let Xj be bump functions equal to 1 on cl(U",) and v~ishing in M\Vm • If an < IlIn 2sup(x,,(m)/,,2(m)lm eM}). the function I = l::"_la nXnln2 is C"" (since the sum is finite in a neighborhood of every point). I> 0 on M and the series defining I is uniformly convergent. being majorized by E:'_ln-2. If we can show that II can be taken inside the sum. then lIe I) = o. This construction then produces I e ker11./ > 0 and hence 1'"' (I/!)/ e kerll; i.e.• kerll = IJ(M). contradicting the hypothesis fl .. O. To show that II can be taken inside the series sign. it suffices to prove the following "g-estimate:" for any g e IJ(M), III(g)1 '" sup(lg(m~lm eM}. Once this is done, then
~ supl r, a"x .. /"2 " -I
II-
0
as n -+ 00 by uniform convergence and boundedness of all functions involved. Thus 11(/) = E':_lfl(a"x"I,,2). To prove the g-estimate. let " > sup(lg(m~lm e M} so that" ± g "" 0 on M; i.e.• "± g are both invertible functions on M. Since II is an algebra homomorphism. 0-11(" ± g) -" ± lI(g). Thus. ± fl(g) """ for all sup(lg(m~lm eM}. Hence III(g)1 '" sup(lg(m~lm eM}. •
,,>
Aemn. For infinite-dimensional manifolds. the proof of the lemma is almost identical with the following changes: we work with fl: COIl ( M. F) -+ F. absolute values are replaced by norms. second countability is in the hypothesis and the neighborhoods V", are chosen in such a way that 1",1 Vm is a bounded function (which is possible by continuity of 11ft)'
VECTOR FIELDS AND DIFFERENTIAL OPERATOR8
233
Proof of em'... iII·f 71. For each mE M, define the algebra • R by /J",(f) - a(f Xm). Since a is inhomomorphism /J",: . vertible. a(l) .. 0 and !oil i~'; a(1) - a(12) - a( 1)a(1). we have a(1 ).- I. Thus /J",'." 0 for all m E M. By 4.2.35 there exists a unique point, which we call .,,(m)EM such that /J",(f)=/(.,,(m»-(.,,*/Xm). This dermes a map.,,: M - M such that a(f)- rp*/for alii E 'J(M). Since /I is an automorphism, " is bijective and since a(f) - .,,*1 E ~(M),a-I(f) - .".1 E 'J(M) for all I E 'J(M), both rp, rp- I are C'" (take for I any coordinate function multiplied by a bump function to show that in every chart the local representatives of cpo rp - 1 are C"') .
•
The proof of existence in the infinite-dimensional case proceeds in a similar way.
BOX 4.28 THE METHOD OF CHARACTERISTICS
The method used to solve problem (P) also enables one to solve first-order quasi-linear partial differential equations in R". Unlike 4.2.1 J. !he solution will be implicit, not explicit. The equation under consideration in R" is
.;. " I) al. -I... Xi( x 1 ,. ..• x. i-I
I
ax"
y( x 1•...• x. " I) ,
(Q)
where/=/(xl, ... ,x") is the unknown function and Xi,Y.i-J .... ,n are C' real-valued functions on R II + I, r ~ I. As initial condition one takes an (n -I)-dimensional submanifold r in R"+ 1 that is nowhere tangent to the vector field
Thus, if r is given parametrically by X ' ... X i (t l , ... ,t"_I),i==1 ..... n
and
1-=1(11 ..... 1"-1).
234
VECTOR FIELDS AND DYNAMICAL SYSTEMS
this requirement means that the matrix
x· ax·
at;
X" ax" a/.
al art
ax· al,,_.
ax" al"_.
al al"_.
y
has rank n. It is customary to require that the determinant obtained by deleting the last column be '* O. for then, as we shall see. the implicit function theorem gives the solution. The function / is found in the following way. Consider F,. the flow of the vector field r a/ax; + ya / 1 in R" + • and let S be the manifold obtained by sweeping out r by F,. That is. S'" U{F,(f)11 E R}. The condition that xia / axi + ya/al never be tangent to r ensures us that the manifold r is "dragged along" by the flow F, to produce a manifold of dimension n. If S is described by / .. /(x· •...• x") then 1 is the solution of the partial differential equation. Indeed. the tangent space to S contains the vector E7_.X 1a/8xl + Y8/a/; i.e.• this vector is perpendicular to the normal (81/ax· ..... al/ax".-I) to the surface/=/(x· ..... x") and thus (Q) is satisfied. We work parametrically and write the components of F, as Xi=XiU ...... I"_ •• /). i=I ..... n and /=1(1 ...... 1,,-1./). Assuming that
r:.7_. r:.7_.
a
x· ax·
o.
at; ax· al"_.
X" ax" a/. ax" al"_ •
ax· al ax· a/. ax· ar" _.
ax" al ax" a/. ax" ar"_.
one can locally invert to give 1= r( x· .... ,x"), I, = r,( x·, .... x"), i = I ..... n. Substitution into / yields / = /(x· ..... x"). The fundamental assumption in this construction is that the vector field E':_lxla/aXi + ya/al is never tangent to the (n - 1)manifold r. The method breaks down if one uses manifolds r. which are tangent to this vector field at some point. The reason is that at such a point. no complete information about the derivative of 1 in a
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
235
complementary (n - 1)-dimensional subspace to the characteristic is known. 4.2.38
Examples.
with initial condition one-manifold
Consider the equation in R 2 given by
r=
{(x. y. f)lx ... s. y'" !S2 - S. f = s}. On· this
f
s s- 1
=-1*0
so that the vector field a/ax + fa/ay +3a/afis never tangent to r. Its flow is F,(x. y. f) = (t + x.(3/2)t 2 + fl + y.3t + f) so that the manifold swept out by r along F, is given by x(t. s) = t + s. y(t. s) = (3/2)/ 2 + sl + (I/2)S2 - s. f(/. s) = 3t + s. Eliminating I. s we get
The solution is defined only for I - 2X2 + 4x + 4.1' ~ O.
•
. Another interesting phenomenon occurs when S can no longer be described in terms of the graph of f: e.g.• S "folds over." This corresponds to the formation of shock waves. Further information can be found in Chorin and Marsden (1979), Lax (1973). Guillemin and Sternberg (1977), John (1975), and Smoller (1982).
4.2C, DERIVATIONS ON
C" -1
FUNCTIONS
4.2.37 Theorem (A. Blasst). Let M be a C A , 2 finite-dimensional manifold: where k ~ O. The colleclion of all R linear derivations 8 from ':t h + I ( M ) to 'ff" ( M) is isomorphic to 'X" ( M) as a real t'ector space. The proof of 4.2.16 sho~s that there exists a unique C" vector field X such that 81~k+2(M)= Lxl~"+2( M). So all we have to do is show that 8 and Lx agree on C"+ I functions. Replacing 8 with e- Lx. Proof.
t
Private communication.
236
VECTOR FIELDS AND DYNAMICAL SYSTEMS
we can assume that 8 annihilates al1 C k + 2 functions and we want to show that it also annihilates all C k + 1 functions. As in the proof of 4.2.16, it suffices to work in a chart, so we can assume without loss of generality that M-R" . .For a Cl I are the projection maps. then the composite mapping
has a tangent that is an isomorphism at each point, and so by the inverse mapping theorem. it is a diffeomorphism onto an open submanifold. Let WI be its image. and 8 the composite mapping. We now show that 8: Wo ..... WI is a Poincare map. Obviously (PM I) is satisfied. For (PM 2). we identify V and V x I by means of !p to simplify notations. Then 'IT: W2 ..... WI is a diffeomorphism. and its inverse ('lT1 W1 ) - I: WI -I> W2 C WI xR is a section corresponding to a smooth function 0: WI ..... R. In fact. 0 is defined implicitly by
=
('lTF.(wo),PF.(wo)-I)(lIh»
=
('lTF.( wo),O)
=
8( wo).
Finally, (PM 3) i~ nhvious as (V.!P) is a flow box. (ii) The prOl . ,Irdensome because of the notational complexity in the definition of local conjugacv. so we will be satisfied to prove this uniqueness under additional simplIl)'ing hypOtheses that lead to global conjugacy (identified by italics). The general case will be left to the reader. We consider first the special case m "" m'. Then we choose a flow box chart (V.!P) at m. and assume SUS' c V. and that Sand S' intersect each . orbit arc in Vat most once, and that they intersect exactly the same sets of orbits. (These three conditions may always be obtained by shrinking Sand S'.) Then let W2 = S, W:i = S'. and H: W2 -I> W; the bijection given by the orbits in V. As in (i). this is easily seen to be a diffeomorphism. and Ho8=8'oH. Finally, suppose m .- m'. Then Fu( m) = m' for some a e )0. T(. and as 6D x is open there is a neighborhood V of m such that V X {a} C "Dx ' Then
254
VECTOR FIELDS AND DYNAMICAL SYSTEMS
F,,(U () S) -. s" is a local transversal section of X at m' E y. and H = Fa effects a conjugacy between a and a" - Fa 0 a 0 Fa- , on S". By the preceding paragraph. a" and a' are locally co~ugate. but conjugacy is an equivalence relation. This completes the argument. • If 'Y is a closed orbit of X E 'X (M) and mE y. the behavior of nearby orbits is given by a Poincare map a on a local transversal section Sal m. Clearly Tma E L(T",S. TmS) is a linear approximation 10 8 al m. By uniqueness of a up to local conjugacy. Tm·8' is similar to Tm8. for any other Poincare map 8' on a local transversal section at m' E y. Therefore. ' the eigenvalues of T",8 are independent of m E 'Y and the particular section S atm.
".3.10 DeftnlUon. If'Y is a closed orbit of X E 'X(M). the characteristic y are the eigenvalues of T",8. for any Poincare map e at
llUlltiplie" of X at any mE y.
Another linear approximation to the flow near y is given by Tm FT E L(T",M. T",M) if mE y and 'I' is the period of y. Note that fi(X(m» = X(m). so TlftF. always has an eigenvalue I corresponding to the eigenvector X( m). The (n - I) remaining eigenvalues (if dim( M) = n) are in fact the . characteristic multipliers of X at y.
Propoeltlon. If y is a closed orbit of X E 'X ( M) of period 'I' and cy is the set of characteristic multipliers of X at y. then cy U{l} is the set of eigenvalues of TmF.. for any m E y.
".3.11
Proof. We can work in a chart modeled on R n and assume m = O. Let V be the span of X(m) so R n = TmSe V. Write the flow F,(x. y)'" (F,'(x. y). f;2(X. y». By definition. we have
and
Thus the matrix of T", F. is of the form
where A .. D,F.2(m). From this it follows that the spectrum of TmF. is' {I)Ucy'
•
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
255
If the characteristic exponents of an equilibrium point lie (strictly) in the left half-plane. we know from Liapunov's theorem that the equilibrium is stable. For closed orbits we introduce stability by means of the following definition. 4.3.12 DeflnlUon. Let X be a vector field on a manifold M and y a closed orbit of X. An orbit F,(m o ) is said to. w_ tOwtlrd y if mo is + complete and for an}' localtran.rver,ral S to X at me y there is a to su('h that 'F,Il( mo) E S and successive applications of the Poincare map yield a sequence of points that converges to m. See Fig. 4.3,10. '
Figure 4.3.10
4.3.13 Theorem. Let y be a closed orbit of X E ',,\ ( M) and let the characteri~tic multipliers of y lie strictly inside the unit circle. Then there is a neighborhood U of y such that for any mo E U, the orbit through ""0 winds toward y. The proof of 4.3.13 relies on the following lemma.
4.3.14 Lemma. Let f; S ..... S be a smooth map on a manifold S with f( so) = So for some so' Let the spectrum of lie inside the unit circle. Then there is a neighborhood U of So such that if s e U, fer) E U and I"(s)"'" So as n ..... IX>, where I" - f 0 f 0 • • • 0 fen times).
T,J
The lemma is proved in the same way as 4.3.4. The Iinear'transformation T,.freplaces e ....(11 in the argument. Compitctness of S· then aUows the argument to be extended from the Poin M be a vector bundle and mn EM an element of the zero section. Show that TmoE is isomorphic to TmnM E9 Em" in a natural, chart independent way. (ii) If E: M -> E is a section of E, and E(m o) = 0, define E'(mo): TmoM -> Emo to be the projection of TE< m n ) to Em". Write out E'( mo) relative to coordinates. (iii)
Show that if X is a vector field. then X'( mo) defined this way coincides with Definition 4.3.1.
4.3B
Prove that the equation 1J+2klJ- qsin(J = O(q > 0, k > 0) has a saddle'point at (J = 0, IJ= O.
4.3C
Consider the differential equations ;= ar' - br. 6= I using polar coordinates in the plane. (i) Determine those a, b for which this system has an attractive periodic orbit. (ii) Calculate the eigenvalues of this system at the origin for various a, b.
4.3D Floquet's theorem in differential equations deals with the fundamental matrix solution of a linear homogeneous system x= A(t)x, where A is a T-periodic matrix. The fundamental solution G(t) satisfies G= AG and G(O) = I. Floquet's theorem states that G(t) has the form P(t)e·' where P is I-periodic and B is constant. Look up the proof in Hale (1969) or Hartman (1973). (ii) Relate this result to the stability analysis of periodic orbits, as outlined in the text. (i)
4.3E
Let X E 'X (M), cp: M -> N be a diffeomorphism and Y = cp. X. Show that (i) mE M is a critical point of X iff cp(m) is a critical point of Y and the characteristic exponents are the same for each. (ii) y c M is a closed orbit of X iff cp( y) is a closed orbit of Yand their characteristic multipliers are the same.
260
4.3F
VECTOR FIELDS AND DYNAMICAL SYSTEMS
The energy for a symmetric heavy top is
pJ
+ J + MglcosO where I, J> 0, b, Po/!' and Mgl> 0 are constants. The dynamics of the top is described by the differential equations 0= aHlflO. p, = - aHlaO. (i) Show that 0 = O. p, = 0 is a saddle point if 0 < Po/! < 2{Mgli (a slow top) (ii) Verify that cosO = 1- ysech2({1Fi 12). where y = 2- b 21P and P= 2MgljI, describes both the outset and inset of this saddle point. (This is called a homoclinic orbit.) (iii) Is8 ... 0, p, =- 0 stable if Po/! > 2..jMgll? (Hint: Use the fact that H is constant along the trajectories.) 4.4 FAOBENIUS' THEOREM AND FOLIATIONS
The three main pillars supporting differential topology and calculus on manifolds are the implicit function theorem, the existence theorem for ordinary differential equations, and Frobenius' theorem. which we discuss brieny here. First some definitions:
*
4.4.1 Definition. Let M be a mani/old and let E c TM be a subbundle 0/ its langent bundle. (So E is a vector bundle over M and is a submani/old 0/ TM by means 0/ a vector bundle chart 0/ E.) We call E a dutributiorl (or a p/IIM /kill) on M.
We say E is invoIutiw if/or any two vector /ields X and Y defined on \open sets 0/ M and which take values in E,[X. YJ takes values in E as well. (ii) We say E is iII,egrabk if for any mo E M there is a (local) submanifold N eM. called a (local) iIItegrvl numi/oill of E at mo containing mo whose tangent bundle is exactly E restricted to N. (i)
The situation is shown in Fig. 4.4.1.
*According to Lawson (1977). the theorem or Frobenius is due to A. Clebsch and F. Deahna.
FROBENIUS' THEOREM AND FOLIATIONS
281
E"'II
Figure 4.4.1
4.4.~
Examples A. Any subbundle E of TM with one-dimensional fibers is involutive; E is also integrable. and this is seen in the following way. Using local bundle charts for TM at mo e M with the lIubbundle property for E, we can find in an open neighborhood of mo. a vector field that never vanishes and has values in E. Its local integral curves through mo have as their tangent bundles, E restricted to these curves. The situation is nicer if the'vector field can be found globally and has no zeros. Then through any point of the manifold there is exactly one maximal integral curve of the vector field. and this integral curve never reduces to a point. We shall return to this crxample later. B. LCt I: M ~ N be a submersion and consider the bundle ker TI c TM. This bundle is involutive since for any X. Y e <X( M). where X, Y take values in ker TI, TIl X, Y) == 0 by' 4.2.25. The bundle is also integrable since for any mo e M, the submanifoldr l(f(mo» has the restriction of ker TI to itself as its tangent bundle (see Section 3.5). C. Let T" be the n-dimensional torus, n ~ 2. Let I ~ k ~ nand consider E-«vl .... ,v,,)eTT"lvHI == .. • = v" = O}. This distribution is involutive and integrable; the integral manifold through (t I ' " .• In) is T k x (/HI'· ... /,,). D. E =
TM is involutive and integrable; the integral submanifold through any point is M itself. E. An example of a noninvolutive distribution is as follows. Let M == S0(3), the rotation group (see Exercise 3.5S). The tangent space at
262
VECTOR FIELDS AND DYNAMICAL SYSTEMS
I - identity consists of the 3 X 3 skew symmetric matrices. Let
E,- {AeT,SO(3)A-
[~
o o -p
a two-dimensional subspace. For Q E S0(3), let
Then
is a distribution but is not involutive. In fact. one computes that the two vector fields with p "'" I, q = 0 and p = O. q = 1 have a bracket that does not lie in E. (Some additional background in the theory of Lie groups makes this example natural). • Frobenius' theorem asserts that the two conditions in 4.4. J are equivalent.
4.4.3 Frobenlu.' Theorem. only if it is integrable.
A subbundle E of TM is involutive if and
Proof. The" ir' part is obvious since the bracket of two vector fields on a. manifold N is again a vector field on N. If E has one-dimensional fibers, Frobenius' tneorem amounts to the existence theorem for differential equations. The proof of the "only ir' part actually proceeds along these lines. By choosing a vector bundle chart, one is reduced to this local situation: E is a model space for the fibers of E, F is a complementary space, and U x VeE x F is an open neighborhood of (0,0), so U x V is a local model for M. We have a map
f: UXV-L(E,F)
such that the fiber of E over (x, y) is E(J (cp*t)( ~*(lI, ...• CP*(I'. cP* Xl •...• cp* X,) = cp*( t{ (II •.... a'. Xl •... • X,)) for cP a diffeomorphism. Hence 9x is a differential operator. It remains to
306
TENSORS
show that ex coincides with Lx on "OW.. is closed. so by normality there exists an open set v..o such that Co C c1( Va.) C Wa.. If Vy is defined for all y < a. put Ca - S\( V y < aVy V V y > eWy») and by normality find Va such that C.. C c1(V.. )C W... The collection {V.. }.. eA' is the desired locally finite refinement of (W.. }.. e A' provided we can show it covers S. Let s E S. By local finiteness of {W,,}. of he\ongs only to a finite collection W.. ,..... W.... If fJ,., max(al ..... a n ) then clearly s E Wy for all y > fJ. so that if in addition sEVy for all y < fJ. then s E Cp C Vp, i.e. s E Vp. •
5.5.15 Lemma (A. H. Stone). compact.
Every pseudometric space is para-
Proof. Let {V.. }.. EA be an open covering of the pseudometric space S with distance function d. Put Vn."-={xEV,,ld(x.S\Vn)~1/2n}. By the triangle inequality we have d(U,•. ". S\U. I I.,,)? 1/2 n -1/2" I I = 1/2 n t I. Well-order the indexing set A and let u,~ n = U•. ,,\ V II < II V. f l . p . If y,6 E A. we have Vn~y C S\V. t I,3' if Y < 6, or u,~" c S\ Vn+I.y. if 8 liN. then the neighborhood{u E VII.,(e) > 1/ N) has empty intersections with all Vm for m > N. Third we show that the open set V is a C"-carrier. By the second step. V - U "V". with Vn a locally finite open covering of V by C" carriers. Then I = Enl" is C". i(e) ~ 0 for all e E E and carr 1= V. • The separability assumption was used only in showing that (iii) implies (i). There is no general theorem known to us for nonseparable Banach spaces. Also. it is not known in general whether Banach spaces admit bounded C" carriers. for Ie ~ I. However. we have the following. 5.~.1' Propo8ltlon. II the Banach space E has a norm C" away Irom its origin. k ~ I. then E has bounded C"-carriers.
Proof. By 4.2.13 there exists cp: R ..... R.C"" with compact support and equal to one in a neighborhood of the origin. If 11'11: E\{O} ..... R is C". k ;a. I, then cp °11'11: E\{O} ..... R is a nonzero map which is C". has bounded support 11'11- 1 (suppcp), and can be extended in a C~ manner toE. • Theorem 5.5.12 now follows from 5.5.19,5.5.18. and 5.5.16. The situation with regard to Banach subspaces and submaru.foJds is clarified in the following proposition, whose proof is an immediate consequence of 5.5.18 and 5.5.16.
320
TENSORS
5.5.20 ProposlUon. (i) If E i.f a Banach space admitting C A partitions of unity then so docs any closed suhspace. (ii) If a manifold admits C k partitions of unity subordinate to any open covering, then so does any submanifold. We shall not develop this discussion of partitions of unity on Banach manifolds any further, but we shall end by quoting a few theorems that show how intimately connected partitions of unity are with the topology of the model space. By 5.5.18 and 5.5.19. for separable Banach spaces one is interested whether the norm is C k away from the origin. Restrepo [1964] has shown that a separable' Banach space has a C I norm away from the origin if and only if the dual is separable. Bonie and Reis [1966] and Sundaresan [1967] have shown that if the norms on E and E· are differentiable on E\{O} and E·\{O}. respectively, then E is reflexive. for E a real Banach space (not necessarily separable). Moreover. E is a Hilbert space if and only if the norms on E and E· are twice differentiable away from the origin. This result has been strengthened by Leonard and Sundaresan [1973]. who show that a real Banach space is isometric to a Hilbert space if and only if the norm is C 2 away from the origin and the second derivative of e ..... UeU 2/2 is bounded by I on the unit sphere; see Rao [1972] for a related result. Palais [1965b] has shown that any paracompact Banach manifold admits Lipschitz partitions of unity. Because of the importance of the differentiability class of the norm in Banach spaces there has been considerahle work in the direction of determining the exact differentiabilit\ . l:J~s of concrete function spaces. Thus Bonic and Frampton [1966] haH: shown that the canonical norms on the spaces LP(R).IP(R). p ~ I. p < 00 are COO away from the origin if p is even. C p - I with DOl-liP-I) Lipschitz, if p is odd. and C[pi with Dlp)(II'UP) HOld"r continuous of order p -[pl. if p is not an integer. The space Co of sequences of real numbers convergent to zero has an equivalent norm that is Coo away from the' origin. a result due to Kuiper. Using this result, Frampton and Tromba [1972] show that the A-spaces (closures of Coo in the HOlder spaces) admit a Coo norm away from the origin. The standard norm on the Banach space of continuous real valued functions on [0. I] is nowhere differentiable. Moreover. since CO([O.I).R) is separable with nonseparable dual. it is impossible to find an equi\ .• lent norm that is differentiable away from the origin. To our knowledge it is still an open problem whether CO([O.I).R) admits C k partitions of unity for k;;d. Finally. the only results known to us for nonseparable Hilbert spaces are those of Wells [1971]. [1973]. who has proved that nonsep-
PARTITIONS OF UNITY
321
arable Hilbert space admits ("2 partitions cif unity. The techniques used in the proof. however. do not seem to indicate a general way to approach this problem.
Exercl... 5.5A
5.5B
Show that any closed set Fin R" is the inverse image of 0 hy II C' real-valued positive function on Rn. Generalize this to any n-manifold. (Hint: proof of 5.5.11 for R"\F.) In a paracompact topological space. an open suhset need not he paracompact. Show that (i) If every open subset of a paracompact space is paracompact. then any subspace is paracompact. (ii) Every open submanifold of a paracompact manifold is parac:ompICt. (HiIU: Uk chart domains to coru:lude metIUabilily.)
Let ..: £ - M be a vector bundle. £' c £ a 5ubbundle and assume III . admits C· partitions of unity subontiDate to any open coverin&. Show that £' splits in E, i.e.. there exists • subbundle £11 such that E - E'eE". (H;III: The result is trivial for local bundles. Construc:t for Cvay element of a locally fmite covering (ll;) a vector 'bundIe map /, wbose kernel is tt, .. complement of E', u,. for u" II» a C· partitioa of unity, put; 0 such that Y -AX ii ~ S.SE
0 (resp. < 0). Note that the last statement is independent of the representative of the orientation [w). for if w' e [w]. then w' = cw for some c> 0, and thus w'(el •...• e,,) and w(el, ... ,e,,) have the same sign. A vector space E has always exactly two orientations: one given by selecting an arbitrary dual basis el, ... ,e" and taking [el " ... "e"); the other is its reverse orientation. The definition of orientation, given previously, is related to the concept of orientation from calculus as follows. In R), a right-handed coordinate system like the one in Fig. 6.2.1 is by convention positively oriented, as are all other right-handed systems. On the other hand. any left-handed coordinate system, obtained for example from the one in Fig. 6.2.1 by interchanging XI and Xl' is by convention negatively oriented. Thus one would call a positive orientation in R' the set of all right-handed coordinate systems. The key to the abstraction of this construction for any vector space lies in the 9bservation that the determinant of the change of ordered bases of two right-handed systems in R3 is always strictly positive. ThUs, if E is an /I-dimensional vector space. define an equivalence relation on the set of ordered bases in the following way: 1- {el'" .• e,,} and , - {e, •.•. ,e~} are equivalent iff del. > O. where. e OL(E) is given by fP(e/) - e;. i ... 1••••• n. We can relate n forms to the bases by associating to a basis el, ... ,e" and its dual basis el •...• e" the n-form w - e l " ... "e". The following proposition shows that this association gives an identification of the corresponding equivalence classes. 8.2.7 Propoeltlon. An orientation i/l a vector space;s uniquely determined by an equivalence class 01 ordered bases. Proof. If [wI is an orientation of E there exists a basis {el' ...• e,,} such tlt(lt w(e l... .• e,,)'" 0 since w'" 0 in A"(E). Changing the sign of e l if necessary. we can find a basis that is positively oriented. Let {e;, ... ,e~} be an equivalent basis and.e OL(E), defined by cp(ej)=e:, ; = 1, ... ,/1, be the change of basis isomorphism. Then
w( e" ...• e~) ... w( .(el ) •...• cp(e,,» ... (cp·w)(el.···,e,,) .. (detcp)w(el, ...• e,,) > 0 That is, [w) uniquely determines the equivalence class of {el, ... ,e,,}.
DETERMINANTS. VOLUMES. AND THE STAR OPERATOR
341
Conversely. let {el ••.. •e,,} be a basis of E and let Co) - e l " ..• " e". wbere {el ••••• e"} is the dual basis. As before. Co)'(e; ••• .• e;) > 0 for any Co)' E [Co)] and {e; ••..• e~} equivalent to {e l •.•. ,e,,}; tbus. the equivalence class of tbe ordered basis {e l ••••• ell } uniquely determines the orientation (Co)). • Next we shall discuss volume elements in inner product spaces. An important point is that to get a particular volume element on E requires additional structure. although tbe determinant does not. The idea is based on the fact that in R) the volume of tbe parellelipiped P == P(x i • X2' x) spanned by, three positively oriented vectors XI' X2' and X) can be expressed independent of any basis as
wbere (x/.xj ») denotes the symmetric 3x3 matrix whose entries are (x/. Xj). If XI' x 2 • and X) are negatively oriented. det(x,. x)J < 0 and so the formula has to be modified to Vol(P) == ( Idet(x,. x)1 )
1/2
.
( I)
This indicates that besides the volumes. there are quantilles involving absolute values of volume elements that are also important. This leads to the notion of densities.
'.2.8 Deftnltlon. Let a be a real number. A continuow mopping II: Ex· .. X E ..... R (n factors of E for E an n-dimensional vector space) is called an a· . . .ity if 11(.,( 1)•...• .,( q,» -Idel .,1-11(01 •... •q,).for all "I' ...• q, E E and all., E L(E. E). LeI IAI-(E) denote the a·densities on E. With a -=1. l-densities on E are simply called dsuitin and IAII(E) is'denoted by IAI(E) ..
°
The determinant of ., in this definition is, taken with respect 'to any volume element of E. As we saw in 6.2.2. this is independent of the choice of the volume element. Note that IAlfl(E) is one-dimensional. Indeed. if d l and 112 E IAla(E). III" O. and e l ••••• ell is a basis of E. then 112 (e l ••• •• ell ) == adl(e l ••..• ell ). for' some constant a E R. If "I •...• "" E E. le~ "/ = .,(e,). defining., E L(E. E). Then
i.e.• 112 = ad l •
342
DIFFERENTIAL FORMS
Alpha-densities can be constructed from volume elements as follows. If ,.,eA"(E). define l"'I"eIAI"(E) by 1,.,1 ..(el ..... en)=I,.,(el ..... enW where e l •... • e~ e E. This association defines an isomorphism of N( E) with IAI"(E). Thus one often uses the notation 1,.,1" for a-densities. We shall construct canonical volume elements (and hence a-densities) for vector spaces carrying a bilinear symmetric nondegenerate covariant two-tensor. and in particular for inner product spaces. First we need to recall a fact from linear algebra.
6.2.9 Propoaltlon. Let E be an n-dimensional vector space and g = (.) e T2°(E) by symmetric of rank r; i.e., the map e e E ...... g(e.·) e E* has r-dimensional range. Then there exists an ordered basis {e l .... • e n } of E with dual basis el ..... e" such that
,
g
L
=
c;e;®e;.
;-1
where c, = ± I and r
s
n, or equivalently. the matrix of g is
o c,
o
o
o
This basis {el ..... e"} is called a g-orthonormal basu. Moreover, the number of basis vectors for which g(e,.e;)=1 (resp. g(e;.e;)= -I) is unique and equals the maximal dimension of any subspace on which g is po.vitive (resp. negative) definite. The number s = the number of + Is minus the number of -Is is called the s;gtUltun of g. The number of - Is is called the index of g and is denoted Ind(g). If g is an inner product (i.e .. is positive definite) this proposition (and the proof that follows) is the Gram-Schmidt argument proving the existence of orthonormal bases.
Proof.
Since g is symmetric. the following polarization identity holds:
Thus if, .... O. there is an e. e E such that '(e •• el) .... O. Rescaling. we can
DETERMINANTS, VOLUMES, AND THE STAR OPERATOR
343
',e,..•• )-
assume (' •• ± 1. Let E. be the span of '1 and E 2 " {, e EI,(e •• ,)'" O}. Clearly E. n E2 = {O}. Also, if % e E. Ihen ;: - c.g(%. e. lei e E2 so that E'"," E. + E2 and thus E = E.(f)E2. Now if g '* 0 on E2• thert~ is an e 2 e E2 such that ,(e2• e2) = ('2 = ± 1. Continue indl'ctively 10 complete the proof. For the second part. let E. = span{',lg(e,. e,) = I}. E2 = span{e,lg(e,. e,)'" - I} and kerg =' (elg(e. e') = 0 for all " e E}. Nole Ihat kerg = span{eilg( ei • e,) = O} and thus E = E.(f)E2(f)kerg
Let F be any subspace of E on which g is positive definite. Then clearly F nkerg = {O}. We also have E2 n F= {O} since any ve E2 n F mu~t simultaneously satisfy ,(v, v) > 0 and g(v. v) < O. Thus F n( F2 (f)kerg)'" {O} and consequently dim F s dim E •. A similar argument shows that dim E2 is the maximal dimension of any subspace of E on which g is negative definite. • Note that the number of ones in the diagonal representation of , is (r
+ s)/2 and the number of minus-ones is Ind(,) ... (r - ,f)/2. Nondegen-
eracy of , means thaI r = n. In this case any e e E may be written' e = 1:7_.[g(e. ei)/c;]e;. where c, = gee,. e,) = ± I and {e,} is a ,",orthonormal basis. For g a positive definite inner product. r'" nand Ind(,) '"' 0; for, a . Lorentz inner product r = nand Ind(,) ... 1.
8.2.10 Proposition. Let E be an n-dimensional vector space and T20( E) be nondegenerale and symmetric. (i)
,,=[w) If
is an orientation of E there exists a unique volume element ,,(g) E [wI. called the ,-~. such that ,,( ..........,,) - I lor all positively oriented g-orthonormal bases {" .... ",,} 01 E. In lacl. if {, •• , .. •e"} is the dual basis. then" ....' 1\ ••• 1\ .". MOI'e gennrJlly. il {f....../,,} is a positively orienled basis wilh dual basis (It·..... It .. ). then " ... Idet(,( /;.
(ii)
,E
·/2
ti )) 1 It' 1\ ••. 1\ It"·
There exists a unique a-density 1"1". called the g-a-Malty. such that I lor all g-orthonormal btue.f (., ........) 0/ E. II {.' ...... "}. is th. dual basis. then hal"· 1.'/\ ... II: ."f". MOI'e If!IIUo ally. il "._ .... "" E E. then 1111"( v, •.... ",,) -Idet,( ",. ,~./2.
1"1·('.......,,) -
Proof. First a relation must be established between the determiDaDtI of the foilowiDa three matrices: j)-cfiaa(CI..... C") (see 6.2.9). -l'Pbism. 6.2C A map • e L(E, F), where (E.l,,» and (F.lw» are oriented vector . spaces is called orientation preserving if.·" E I w]. If dim E - dim F. and • is orientation preserving, show that. is an isomorphism. Give an example for F - E - R 3 of an orientation-preserving but not volume-preserving map. • 6.20 Let E and F be n-dimensional real vector spaces with nondegenerate . symmetric two-tensors" E T2o(E) and" E T2o( F). Then.., E L(E. F) is called an ;som,try if ,,(.(,)• ..,(~'»=,( •• e'). for all ,.e'E E. (i) . Show that an isometry is an isomorphism. and ,,(II). (ii) Consider on E and F the,. and "-volumes Show that if .., is an orientation-preserving isometry. then cp-
lie,)
350
DIFFERENTIAL FORMS
commutes with the Hodge star operator; i.e.• the diagram Ak(F)----· A,,-k(F)
~·l
1~.
Ak(E)
6.2E
• A,,-k(E)
commutes. If ~ is orientation reversing. show that • (~·a) == - ~.(. a) for aE Ak(F). Let g be an inner product on R3. Use 6.1.I1C and 6.2.14C to show that if {el.e2.e3} is a positively oriented basis then ei X ej = (detg)'/2g k1el
where ( 6.2F
.1·~·k3) E S3 has sign + I.
I. j.
Let E be an n-dimensional oriented vector space and g E T2o(E) be symmetric nondegenerate of signature s. Using the g volume. define the Hodge star operator .: Ak(E. F) .... A,,-k(E. F). where F is another finite-dimensional vector space by •a
= (•
a' ) /, .
where a i E A*(E). (fl'" ·./m} is a basis of F and a= aJ,. Show the . following: (i) The definition is independent of the basis of F. (ii) •• - (_1)(,,-·)/2+k(,,-*) on N(E; F)
If" E Tzo(F) and if it( /.a)= (. ai)h(f. /,). then • it( /. a) = h(/•• a). (iv) If 1\ is the wedge product in A(E; F) with respect to a given bilinear form on F. then for Cl, II E Ak(E. F).
(iii)
( • a) 1\ II =
(•
II) 1\ a
and a
1\ ( •
II) = II 1\ ( • a).
(v) Show how g and h induce a symmetric nondegenerate covariant two-tensor on A*(E. F) and find formulas analogous to (7)-(10). 6.2G Prove the following identities in R 1 using the Hodge star operator:'
II" X vll2 = 1I,,1I 211v1l 2- (". V)2 . and
• X(vX.,) - (.,.,)O-(M· v).,.
DIFFERENTIAL FORMS
6.2H
351
(i) Prove the following identity for the Hodge star operator:
(.ex.P)'" (ex A P.,,). where exE A"(E) and PE A"-"C E). (ii) Prove the basic properties of • using (i) as the definition. Let E be an oriented vector space and S c T20(E) be the set of nondegenerate symmetric two-tensors of a fixed signature s. (i) Show that S is open. (ii) Show that the map Vol: g ..... /-I(g) assigning to each g E Sits g-volume element is differentiable and has derivative. at g given by" ..... ! - 1 is of class Coo, as is c.>1/- = (T1/I)* 0 c.> 0 1/1- I. The local representative of Pc.> is
which is of class Coo by the composite mapping theorem. For (ii), we merely note that it holds for the local representatives; (iii) follows at once from the definition; (iv) follows in the usual way from (ii) and (iii); and (v) follows from the corresponding pointwise result. • We close this section with a few optional remarks about vector-bundlevalued forms. As before. the idea is to globalize vector-valued exterior forms. 6.3.11
Definition.
Let 77: E
-+
B. p: F
-+
B be vector bundles Oller the
same base. Define
the vector bundle with base B of vector bundle homomorphisms over the identity from N( E) to F. If E == TB. Ak(TB: F) is denoted by N( B; F) and
356
DIFFERENTIAL FORMS
is called the vector bundle 0/ F-va/wd k-forms 011 M. 1/ F = B x F, we denote it by A"(B, F) and call its elements J¥ctol'-f)a/uedk-fomu ill M. The spaces 0/ sections 0/ these bundles are denoted respectively by O"(E; F). O"(B; F) and OIr(B; F). Finally,O(E; F) (resp. O(B; F), O(B, F» denotes the direct sum o/O"(E; F), k -1,2", ',n, ... together with its structure as an infinitedimensional real vector space and <J ( B)-module.
Thus, IIEO"(E; F) is a smooth assignment to the points h of B of skew symmetric k-linear maps IIh: Eh X ... X Eh .... Fh • In particular. if all manifolds and bundles are finite dimensional, then liE DIr( M,R r) may be uniquely written in the form 11= Ef_llljej• where ul.···.UPEOA(M). and (e l•·· ·,ep } is the standard basis of Rp. Thus uEOIr(E.RP) is written in local coordinates as l . dx j , ( a ',".'A-
"
•••
"dXi • ... aP
'
"1
0
. dx" " ... " dx i.) ",.
for i l < ... 1(U) (k = O.I.2..... n. and U is open in M). which we merely denote lIy d. called thl' exterior derit""iw on M. such that (i) d is a A-IIIItitkriwltiott. That is. dis R linear and for caE Ok(U) and II E O'(U).
d(ca A II) - dca A II + (-I)kca A df) (product rule) (ii) II f E iff(U). then dl is as defined in 4.2.5; (iii) d 2 = dod = 0 (that is. d H I(U)o d"(U) = 0); (iv) dis IIIIhllYII wi", rnpect to rntrictioru; that is. if U eVe M are open and ca E 0' ( V). then d( cal U) = (dca) IU. or the lollowing diagf.'am commutes:
IU O"(V)
d
• OIc(U)
1d
J
OHI(V)_ O"+'(U)
IU .' As usual. condition (io) means that d is a I«IIl """,'or.
Proof. We first establish uniqueness. Let (U.CP). where cp(u)= (x' ..... x .. )•.
, ,
i".
be a chart and ca- IIi ... 1 dx l , A ••• A.. dx i , E O"(U). i, < . " < If k = O. the local formula dll- (alii axl) dx' applied to the coordinate functions Xl. i - ....... n shows that the differential of Xl is the one-form dx'. From (iii). d( dx l ) - O. which corroborated with (i) shows that d( dx i , 1\ .•• A dx i ,) == Thus. again by (i).
o.
(I) and so d is uniquely determined on U by properties (i)-(iii), and by (iv) on any open subSet of M. "
358
DIFFERENTIAL FORMS
For existence, define on every 'chart (U,.,,) the operator d by Eq. (I). Then (ii) is trivially verified as is R-linearity. If p.,., Pio" ..},dx il /I. •.• /I. dx h E g'(U). then
..
.... p." ".
d(a /I. A) = d(a." I -
(101""10 ( _ _ _ II
(Ia,···i '_I dx' (Ix i
= __
+( - I)
.
~io""l'
(Ixi
It.
a·
/I.
dXi,
...
+0
1""/0
dx i, /I.
/I. '"
/I.
dxio
(I~""h).. - - dx' /I. dx" (Ix I
••• /I.
dx i,
.
/I.
.
p.
" ..."
/I.
dx io
/I. ••• /I.
/I. ••• /I.
dx il
dx h )
.. dx'o /I. dXl'
/I. ••• /I.
/I. ••• /I.
. dx"
dx h
apj, ..." . .
.
. dx" /I. ••• /I. dx" /I. - - d x ' /I. dx 1' /I. ••• /I. dx"
ax'
',""0
=-da/l. P +( -I)"a/l. tIP. and (i) is verified. For (iii), symmetry of the second partial derivatives shows that
Thus. in every chart (U• .,,). Eq. (I) defines an operator d satisfying (i)-(iii). It remains to be shown that these local ds define an operator d on any open set and (iv) holds. To do so, it suffices to show that this definition is chart independent. Let d' be the operator given by Eq. (I) on a chart (U'• .,,'), where U' () U .. ". Since d' also satisfies (i)-(iii), and local uniqueness has already been proved, d'a = da. on U () U'. The theorem thus follows. •
6.4.2 Corollary. Let.., E g"(U), where U c E is open. Then d..,( u)(~ •...• v.) ...
Ie
E (-I); D..,( u )·v,( ll(, •••• • e
,-n
i .....
v,,)
where v, denotes that v, ;.~ deleted. Also. we den()fe element.v (u. v) 0/ TU merely by v lor brevity. [Note that D..,(u)·VE L!(E.R) since ..,: U ..... L!(E.R).)
"-f.
It suffICeS to show that d dermed this way coincides with Eq. (I). For functions and one-forms this veriracation is trivial. The rest follows by induc:tion on the degree of..,. •
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, 'LIE DERIVA TlVE
359
8.4.3 Exampl..
A.' On R2.let a= f(x. y)dx + g(x. y)dy. Then da= df A dx
+ fd(dx)+dg
A
dy + gd(dy)
, by linearity and the product rule. Since d 2 = O. da-df
Adx +dg Ady -
ag + ( axdx
(¥XdX
+ if;dy)A dx
ag ) v I'dy. + ayd.
Since dx A dx = 0 and dy A d.y = O. this becomes da= af dyAdx+ a8dXAdV=(ag - af)dxAdy.
ay
ax'
ax
ai'
B. On R 3; letf(x. y. z) he given. Then af af af df = -dx + -dr + -dz, ax ay' ilz
so the components of df are those of grad f. That is, (grad.f )' = df, where t> is the index lowering operator defined by the standard metric of R3 (see Section 5.1). C. On R3. let
Computing as in Example A yields • (aF aF,- aF, ) dF"= -2- aF, - ) dxAdv- ( ax ay 'az ax' dXAdz aF3 - aF 2) +(dVAdz. ay az .
Thus associated to each vector field G = Gli + Gd + G3 i on R J is the one-form Gb and to this the two-form .(Gb) by
360
DIFFERENTIAL FORMS
where. is the Hodle operator (see Section 6.2); it is clear that 4Fb_(curIF)b. D. The divergence is obtained from 4 by 4- Fb - (divF) dx 1\ dy 1\ dz;
i.e.,
_4_,'0- divF. Thus, by associating to a vector field F on R J the one-form F" and the two-form - F,4 gives rise to the operators curl F and div F. From Db - • (curl F)b it is apparent that
tl4F b - 0 - 4-(curl F)'o - (div curl F) dx 1\ dy 1\ dz·. That is, 4 2 == 0 gives the well-known vector identity div curl F .... O. Likewise, MI-O becomes 4(gradf)b-O; i.e., -(curl gradf)"=O. So here 4 2 -0
becomes the identity curl grad 1- O.
•
The effect of mappings on 4 can now be considered. Recall that O(M) is the direct sum of all the O*(M). Let F: M - N be a C' map. As F*: O*(N)-D*(M) is R-Iinear, it induces a mapping on the direct sums, F*: O(N)-O(M). -+ N be of class C'. Then F*: D(N) -O(M) is a homomorphism of differential algebras; that is,
8.4.4 Theorem. Let F: M
(i) F*(+ 1\ Col) ... F*+ 1\ F*Col, and (ii) 4 is IItIIIIrrII willi rap«I 10 mtIpp;"p; that is, F*(4Col) - 4(F*Col), or the following diagram commutes:
Proof. Part (i) was already established in 6.3.11. For (ii) we shall show in fact that if m E M, then Ihl're is a neighborhood U of m EM such that 4(F*ColI U) "" (F*4Col)1 is sufficient, as F* and 4 are both natural with respect to restriction. Let (V, cp) be a local chart at F( m) and U a neighborhood of m E M with F( U) c V. Then for Col E O*(y), we can write
u. " '.
EXTERIOR DE'RIVATIVE, INTERIOR PRODUCT, LIE DERIVATIVE
where
and so
361
a a =-iJx;. '0
and by (i)
If ~ E gO(N), then d(F*~) = F*d~ by the composite mapping theorem, so
d(F*IiJIU)-F*(dliJ
~y
".
'... . )/\F*dx " /\ ... /\
F*(h;"
(i). •
•.4.5 Corollary. The operator d is natural with respect to push-forward by diffeomorphisms. That is, if F: M ..... N is a diffeomorphism, then F .dliJ = dF ,j,IiJ, or the following diagram commutes:
F.
gA(M)--_ glc(N)
14 .
1
d gAil(M)
• gk+ I(N)
F.
Proof. Since F. = ( r I)., the result follows from 6.4.4(ii). • The next few propositions give some important relations between the Lie derivative and the exterior derivative.
1.4.1 Propo8ItIon. Let X E CX(M). Then d is natural with respect 10 Lx. TIuJt is, for CIt E; OIc(M) we heme LxliJ E Olc( M) and
or the following diagram commutes: Lx Ok(M) ---gk(M)
d1
14
gk+l(M)
glc+l(M) Lx
362
DIFFERENTIAL FORMS
Proof. Let F, be the (local) now of X. Then we know that Lx..,(m) = ad (F,*..,)(m)1 I
1-0
.
Since F,*.., e nk( M). it follows that LxG) e nk( M). Now we have F,* d.., = d(F,*"'), Then. since d is R linear. it commutes with dldt and so dL x '" = Lx d..,. • In Chapter 5 contractions of general tensor fields were studied. For differential forms. contractions playa special role. 1.4.7 DetlnHlon. Let M be a manifold. X e ~(M). and.., e g"+ I(M). Then define i x '" e 6j"o(M) by
If.., e gO(M). we put i x '" = O. We call i x '" the intnior product or CfIfItNd_ of X and..,. (Sometimes X J.., is written for i x "") 1.4.8 TheoNm. We have ix: gk(M)_gk-I(M).k=I ..... n. and if II e M).II e M). and f e go( M). then
n"(
(i)
n'(
ix is a lI-IIIItitkrivtltiOll. That is. ix;s R-linear and i X (lIl1fl)= (ixG) II II + (- I)kll II (ixll);
(ii) ilxll= fixll; (iii) ixdf - Lxf; (iv) LxG- ixdG+ dixll; (v) LIXIl" /LxII + df II ixll.
Proof. That ixlle nk-I( M) follows at once from the definitions. For (i). R-linearity is clear. For the second pan of (i). write ix( II II 11)( X2 • X] •...• Xk+,) - (II II II)(X. X 2 .···.Xk+,)
and
+(
_I)d k + I-I)! A(II®i.\fl). k!(/-I)!
Noor." _"rite out the dd"mitioD 0( A in terms 0( permutations (rom defmition
'EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVA TlVE
363
6.1.1. The sum over all permutations in the last term can be replaced by tlie sum over aao. where ao is the permutation (2,3 •..•• k + 1.I.k +2•...• k + I) ..... (1.2.3 •...• k + I) whose sign is (_1)k. Hence (i) follows. For (ii). we note that CI is ~(M)-muhilinear. and (iii) is just the definition of L"I. Fo~ (iv) we proceed by induction on k. First note that for k = o. (iv) reduces to (iii). Now assume that (iv) holds for k. Then a k + I form may be written as Dlh " "'i' where ..,/ is a k form. in some neighborhood of m E M. _But Lx(tll"..,) = Lxtll " .., + til " Lx'" since Lx is a tensor derivation and commutes with A. (or from the definition of Lx in terms of nows). and so
ixtl(tll" ..,)+ tlix (til" ..,) =
-
= -
ix (til" tI..,)+ tI(ixtil " .., - til" i x "') ixtll " tI.., + til" ixtl.., + tlixtll "
Ii)
+ ixtll " tI.., + til " tlix'" =
til" Lx'" + tlLxl " ..,
by the inductive assumption and (iii). Since tiL "I =: 1."tll. the result fol-lows. Finally for (v) we have
L,x'" = i,xtl.., + di,x'" = lixtl.., + tI( li x "') =
lixtl.., + til "ix'" + Itlix'"
=
/Lx'" + til" i x""
•
Note that (i) and (ii) are valid without change in Banach manifolds. Formula (iv)
(a "magic" formula of Cartan) is particularly useful. It can be used in the foUowing way. 6.4.9 Exampl.. A. rr CI is a k-form such that tlCI = 0 and X is a vector field such that tli XCi = O. then P,CI = CI. where F, is the now of X. ~ndeed. :
P,CI
=
p,LxCl= p'(ixtlCl + dixCl) = O.
so FiCi is constant in t. Since Fo - identity. FiCl- Cl for aU t.
364
DIFFERENTIAL FORMS
B. Let M .. R], suppose div X = 0 and let u == dx /\ dy /\ dz. Thus tlu-O. Also
So tlixu-tl. Xb "" (divX)u== O. Thus by Example A, P'(dx /\ dy /\ dz) == dx /\ dy /\ dz. As we shall see in the next section in a more general context, this means that the flow of X is volume preseroing. Of course this can be proved directly as well (see for example Chorln and Marsden [1979]). For related applications to fluid mechanics, see Section 8.2. • The behavior of contractions under mappings is given by the following proposition. (The statement and proof also hold for Banach manifolds.) 6.4.10 Proposition. Let M and N be manifolds and f: M ~ N a C' mapping. If Co) E nk(N), X E 'X (N). Y E 'X (M). and Y is J-related to X. then
In particulttr, iff is a diffeomorphism, then
That is, interior products are natural with respect to diffeomorphisms and the following diagram commutes: Ok(N)
r
nk(M)
1irx
ix 1 nk-I(N)
Similarly for Y
E ~(M)
r
' nk-I(M)
we have the following commutative diagram:
nk (M) ----'f'-._ nk ( N)
iyj nk - I (M)
li,.y f.
nk
I(
N)
EXTERIOR DERIVA TlVE, INTERIOR PRODUCT, LIE DERIVATIVE
Proof. Let .,' ..... .,k-' e Tm(M) and
PI =
365
f(m). Then
iyr..,(mH .,' .... '.,k-I) = r..,(m)· (Y( m),.", ... ,v~ I) = ..,(n)·«TI 0 Y)(m). TI( v, ) ..... Tf( Vk-I»
... ..,(n)«X 0 f)(m), TI( v, ) ....• Tf( v"
_,»
.,,,_,»
=
ix..,(n )·(Tf( Vi) ... ·• Tf(
=
rix..,(m)·(." , .. ·,1'''_1) •
The next proposition expresses d in terms of the Lie derivative (cf. . Palais [1954]). 8.4.11 PropoalUon. we have
Let Xi e 'X(M).;,. O..... k. and.., e D"(M).Then
(i) (Lx •..,)(XI.· .. ,X,.) = Ld..,(XI.···,X,,»-
L" ..,(X" ...• LX.X, ... ~.Xk)
1-'
and Ie
(ii) d..,(Xo• XI •...• XIc )
...
L (-I)iLx,(..,(Xo ..... X, ..... XIc)) 1-0
E
+
(-I)i+j..,(Lx.(Xj ). X{) ..... Xi .... ~XJ .... 'Xk)
O,,;<j""
where
X, denotes that
X, is deleted.
Proof. Part (i) is exactly condition (001) in 5.3.1. For (ii) we proceed by induction. For k "'" O. it is merely d..,(Xo) = Lx."" Assume the formula for k - I. Then if .., e DIe ( M). we have, by 6.4.8(iv)
d..,(Xo• XI'· ..• X Ic )
""
(ix.d..,)(XI ... ·.XIc )
=
(Lx.'" )(X, ..... XIc ) - (d(ix."'))( X, .... ,XIc )
=
LX.(..,(XI.···.Xh »
- E" ..,(XI, .. ·,Lx.X, .... 'Xh ) I-I
-(dixo"')(XI .... ,Xh )
(by (;».
But ixo..,eD"-I(M) and we may apply the induction assumption. This
366
DIFFERENTIAL FORMS
gives. after a simple permutation k
(d(;XoCo)))(X, •...• X k ) =
L
'-I
(-I)'-'Lx/(Co)(Xo • X, •...• X, ..... X A »)
Substituting this into the foregoing yields the result.
•
Note that (i) and the next corollary hold as well for infinite-dimensional manifolds.
6.4.12
Corollary.
LetX.YE~-X(M).
[Lx.iy]=i[x.y[ and
Then [Lx.Ly]=L[x.y[
In particular. ix 0 Lx = Lx 0 i x ' Proof. It is sufficient to check the first formula on any k-form Co) E nk(U) and any X, •...• X k _, E ~ (U) for any open set U of M. We have by 6.4.II(i)
(iyLxCo)( X, ..... X k
_,)
=
(LxCo)( Y. X, ..... Xk _,)
=Lx(Co)(Y.X, •...• X k
_,»-
k - I
L
Co)(Y.X, ..... [X.X,] ..... X k _,)
I-I
k - ,
=
Lx «iyCo)( X, ..... Xk
,»- L (iyCo)(X, ..... (X. X,] ..... X '-I
A _ ,)
- (i,x.y,)( XI" ",Xk _,) =
(LXiyCo)( X, ..... X k-,)- (i,x.YlCo) }(X, ... "Xk _,)
one similarly proves (Lx. Ly] = L[x.y)' •
6.4A THE EXTERIOR DERIVATIVE ON INFINITE-DIMENSIONAL MANIFOLDS Now we are ready to discuss the eltterior derivative on infinite-dimanifolds. Tht\~rem 6.4.1 i~ rather awkward. primarily . ~'au!'(' we cann(\t. with,")ut a I(\t l,f techni~·alitie~. palOS fr(lm. for
men5-i\~nal
EXTERIOR QERIVA TlVE. INTERIOR PRODUCT. LIE DERIVA TIVE
367
example. one-forms to two-forms by linear comhinations of decomposable two-forms. i.e.• two-forms of the type I I " p. However. there is a simpler alternative available. , Adopt formula 6.4.ll(ii) as the definition of d nn any open subset of M. Note that at first it is defined as a multilinear function on vector fields and note that Lx is already defined. 2. In charts. 6.4.II(ii) reduCCI to the local formula 6.4.2. This or a direct computation shows tbat d: D"(M)-D"+'(M) is well defined. depending only on the point values of the vector fields. (cr. 5.2.19). 3. One checks the basic properties of d. This can be done in two ways: directly. using the local formula. or using the definition and the following lemma. easily deducible from the Hahn- Banach theorem: II a k-Iorm II) is zero on any set X, •...• X" E IX(U) lor all open sets U in M. then II) == O. This second method is slightly faster if one first proves the key formula I.
(2)
Proof 0/ Form,,1II (2). Let
II be a k-form and X, •...• X" a set ot k vector fields defined on some open subset of M. Writing Xo = X.
k
- I: (-I)'Lx,(II(Xo •...• X, ..... X,,))
'-0
~ L..
+
(
. . ) -I )' + j II ( Lx,X,. XO ..... X,., ..• X, ....• X4
o 0, F,(u) -Iu. Thus F, is a diffeomorphism and, starting at t -I, is generated by the time-dependent vector field X,(u) = ult;
that is. FI(u)- u and dF,(u)/dt = X,(F,(u». Therefore. since w is closed,
=
F*(di , x, w)
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVA TlVE
For 0 < to
oe;;
371
I, we get CAl- P,'CAl =
"
Letting to -- 0, we get
CAl =
d
.. f 'P,"xCAldt. 10
tIP. where
Explicitly,
(Note that
thi~
formula for
Pagrees with that in the previous proof.) •
The proofs of (ii) show in fact the following: if CAl E nA( U) is closed and U is contractible. then CAl is exact. See Exercise 6.41 for a relative Poincare lemma. It is not true that closed forms are always exact (fpr example. on a sphere). In fact, the quotient groups of closed forms by exact forms (called the deRham cohomology [{roups of M) are important algebraic objects attached to a manifold. A discussion may be found in Section 7.5. Table 6.4.1 summarizes some of the important algehraic identities involving vector fields and differential forms that have been Ilbtained. Table 6.4.1 I.
Vector fields on M with the bracket IX. VI form a Lie a1,ebra; that is. IX. VI is n:a1 bilinear. skew symmetric. and Jacobi'~ identity holds: [[X. Y).ZI +[[Z.X). Yl+ [(V.z).Xl- O.
Locally. [X.Yl- DY·X - DX·Y
2. 3. 4.
and on runctions. IX. YHJI = XI YI/II- YIXI/II· For dirreomorphisms ",.,y. ",.1 x. Y 1- I '" ~X. CPo VI and (cp 0,y). X = ",.1Jt. x. The forms on a manifold are a real associative algebra with 1\ as ~ultiplicalion. Furlhermore. u 1\ p - ( - I)"P 1\ U for k and '·forms u and p. respectively. For maps CP.,y. cpO( U 1\ p) - cp'u 1\ ",·P.(", 0,y )·u - .,....",.111.
372
DIFFERENTIAL FORMS
Table 6.4.1 Continued S.
tI ia a real linear map on forma and tI tI. - O. tI(. 1\
IS) -
tlo 1\ IS + ( - I)' a
1\
dII
for II a Ie·form
6 . . For II a Ie·form and Xo." .• X. vector fields: k
dll(Xo."·.X.)-
E (-I)'X/(II(Xo..... X.. " .. X.)) /-0
+
E (-I)" 'II([X,.X,).Xo." .• X, ..... X, .. ".X.) ;<j
Locally. k
tI", ( x )( 1\) ..... "k )
7. II.
9.
-
E (- I) / Dw ( x ). II, ( 1\) ..... ti,. " .• Ok)'
.-0
,,0
For a map ". da - d"oll. (Poincare lemma) U dll - O. then a is locally exact; that is. there is a neighborhood U about each point on which a - dIS. iXIl is real bilinear in X. II and ror h: M -- R. iua- hixll- ixha. Also ixixa = O. a~d
··60. For a diffeomorphism ". ,,°;1/(11- ;._1/(,,°11. II. Lxa-dix«+;xtl•. 12. Lx. is real bilinear in X. II and LX(II II IS> - LXII II II + II II LxlS. 13. For a diffeomorphism "."oLxa- L.-X"oll. 14. (LXIl)(X, •...• X,) - X(II(X, •... •X. »-t~_,I1( X, .... •IX. X/I.... •X.). Locally. k
(LXIl).·( 0, ..... Ok) -
Da.· X( II(). (""" ""k)+ Ea.· (""'''' DX.· 0/ ..... 0.) .. i-I
I S.
The following identities hold:
~XTERIOR
DERIVATIVE. INTERIOR PRODUCT. LIE DERIVATIVE
373
For vec:tor-valuJ k-forms one proceeds in a similar manner, adopting Palais' formula, 6.4.1 I(ii), as the definition on an open subset of M. Note again that this definition uses the fact that Lx is defined for vector-valued tenson, and apin one hal to prove that the local formula 6.4.2 holds. Then all properties in Table 6.4.1 are verified in the same manner as previously. For vector-valued forms we have however an additional formula on Ok(M; F)
·for any A E L(F, F). If F is finite dimensional, the definition and properties of d become quite obvious; one notices that if n
"''''' I: "'j"EOk(M;F),
where "'jEOk(M).
j-I
then, d", is given by n
d", =
I: d"'j"
j-I
and this formula can be taken as the definition of d in this case.
• OX I... DIFFERENTIAL IDEALS AND PFAFF/AN SYSmlttS
This box discusses a reformulation of the Frobenius theorem in terms of differential ideals in the spirit of E. Cartan. Recall that a subbundle E c TM is called involutive if for all pairs (X, Y) of local sections of E defined on some open subset of M, the brackef[X, Y] is also a local section of E. The subbundle E is called integrable if at every point m E M there is a local submanifold N of M such that T",N ... E",. Frobenius' theorem states that E is integrable iff it is involutive (see Section 4.4). J)efore starting the general theory, which is fairly heavy on algebraic formalism. let us illustrate by a simple example how forms and involutive subbundles are interconnected. Let (oj E 02( M) and assume that E.. = {vE TMli,,"'''' O} is a subbundle of TM. H X and, Y tlbe method described does not work for vector-bundle-valued forms. Additional '>cable to lift Lx. structure on the bundle is require'!
374
DIFFERENTIAL FORMS
are twO'sections of E... then
Thus, E.. is involutive iff ixiydll) = O. In particular, if II) is closed, then E.. is involutive. In this box we shall want to express conditions such as ixiydll) - 0 in terms of one forms for subbundles E of TM that are not necessarily defined by exterior forms. For any subbundle E, the k-annihilator of E defined by
is a subbundle of the bundle N( M) of k-forms. Denote by f( V. E) the cae sections of E over the open set V of M and notice that . oc
g(E)~
E9 f(M.
EO(k»)
k-O
is an ideal of OeM); i.e., if 11)1.11)2 Eg(E) and pEO(M), then + 11)2 E geE) and p 1\ WI E geE).
WI
6.4.15 Propoaltlon. The subbundle E of TM is involutive iff for all
n
n
open subsets V of M and 01111) E V, EO( I», we have dw E V, EO(2». If E is involutive, then II) E f(V. EO(k» implies dw E feU, EO(k + I). Proof.
For any
0 E
nU. E°(l»
do( X. Y) = X( o( Y
and X. Y E f( u. E). 6.4.11(ii) yields
»- Y( o( X»- 0(1 x. Y)
Thus E is involutive iff do( X. Y) = 0; i.e .• do E
= -
0(1 x. Y).
n U. E()(2».
•
The Frobenius theorem in terms of differential forms takes the following form.
6.4.16 Corollary. The subbundle E c TM is integrable iff for all ope" subsets U of M.w E
nu. E°(l» implies dWE feU. EO(2».
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVATIVE
375
The following considerations are strictly finite dimensional. They can be generalized to infinite-dimensional manifolds under suitable splitting assumptions. We restrict ourselves to the finite-dimensional situation due to their importance in applications and for simplicity of presentation.
6.4.17 DeflnlUon. Let M be an n-manifold and '\ C 0.( M) be an ideal. We say that 9 is locoIIy generuted by II - k ilukpelldellt one-forms. if every point of M has a neighborhood V and n - k pointwise llnearly independent one-form.r I&)I •...• I&),,_~ E gl(V) such that: n-k
(i)
E
If I&) e ~. then 1&)1 V ,
8/1\ 1&), lor
some 8, e 0.( M);
;-1
(ii) If I&) e 0.( M) and M is covered hy open sets V,fuch that for each V in this cover. n-k I&)IV=
L
,-1
8;1\1&);
for some 8; Eg(M).
then I&) E 'l. The ideal
9C
0.( M) is called a differelltial ideal if d'i c ,\.
Finitely generated ideals of 0.( M) are charal'lcrizcd hy heing of the form 1( E). More precisely. we have the following. Let!t be an ideal ofg( M) and let n = dime M). The ideal 9 is locally generated by n - k linearly independent one-forms iff there exists a subbundle E c TM with k-dimensional fiber such that g - geE). Moreover. the bundle E is uniquely determined by g.
6.4.18 Proposition.
If E has k-dimensional fiber. let X,,-H I ..... X" be a local basis of f( V. E). Complete this to a basis of ex (V) and let 1&); E 0 1(U) be the, dual basis. Then clearly I&) I'" .• 1&)" _ k are linearly independent and locally generate 1( E). Conversely. let 1&)1 ..... 1&)". k generate ;) to write Co) = ECJ>;Co) and let dOl; = CJ>;Co), 01 = EOl;.) (iii) Generalize this to forms on a closed submanifold of a manifold admitting C·-partitions of unity. (i) Let M be a manifold and let i,: M .... [0.1 JX M he the mapping i,(m)-(t,m). DefineH: (}k+I([O.I]XM) .... UA(M) by (ii)
6.4L
ex
(HOl)(XI.· .. 'Xk ) =
(ii)
fol(i~;i1la'CJ)( XI'···'Xk) dl.
Show that H is wen-defined and that doH + Hod = it - i~. (Hint: Use formula 6 in Table 6.4.1.) Two smooth mappings /. g: M .... N are called C-homolopic, r ~ 0, if there exists'a C mapping F: I X M .... N such that F(O, .) "" / and F(1,') == g. Let OlE (}k(N) he closed. Show that if / and g are C'-homotopic. then g·Ol- f·Ol E Ok( M) is exact. (Hint: Show that G - HoP with H as defined in (i) satisfies doG -G od- g.-
r.)
(iii)
Oeduce from (ii) that if M is contractible (see 1.6.13)· then
every closed form is exact. (Hint: use (ii) and the fact that the identity mapping is homotopic to the map sending all of M to a point.) , (iv) Use (iii) to give another proof of the Poincare lemma. the ideal generated by Co)I = x 2 dx I + Xl dx 4 , Co)2 = 6.4M Show that on x 3 dx 2 + x 2 dx l is a dirferential ideal. Find its integral manifolds.
R".
1.5 ORIENTATION, VOLUME ELEMENTS, AND THE CODIFFERENTIAL This section globatizes the setting of Section 6.2 from linear spaces to manifolds. All manifolds in this section are finite dimensional. t t For infUlite-dimensional analogues of orientability, see for instance, E1worthy and Tromba [197Obj.
382
DIFFERENTIAL FORMS
1.5.1 DefInition. A voIIuIw form on an n-manifold M ;s an n-form 1& E 0"( M) such that 1&( m) .. 0 for all m E M; M is called if there exists some volume form on M. The pair (M.I&) is called a vohune lIIIIIIi/o/iJ.
tJIV",,,,,
Thus. 1& assigns an orientation. as defined in 6.2.5. to each fiber of TM. For example. R' has the standard volume form 1& =- dx " dy" dz. 1.5.2 (i)
(ii)
P~ltIon.
Let M be a connected n-manifold. Then
Mis orientabie iff there is an element 1& E O"(M) such that every other 1&' E 0"( M) may be written 1&' == fl&for f E ~(M). M is orientable iff M has an atlas {( If;. 'Pi)}' where 'Pi: If; ..... If;' C R". such that the Jacobian determinant of the overlap maps is positive (the Jacobian determinant being the determinant of the derivative. a linear map from R" into R").
Proof. For (i) assume first that M is orientable. with a volume form 1&. Let 1&' be any other element of O"(M). Now each fiber of A"(M) is one-dimen-
sional. so we may define a map f: M ..... R by
We must show thatf E '!f(M). In local representation. we can write
and
I&(m)"'"(m)dx i ,,,
. . . ,,
dx i •
But ,,(m) .. 0 for all me M. Hence f(m)'" ,,'(m)/,,(m) is of dass Coco Conversely. if 0"( M) is generated by 1&. then 1'( m) .. 0 for all m E M since each fiber is one-dimensional. To prove (ii), let {( If;. 'P,)} be an atlas with 'Pi ( If;) -If;' C R". Also. we may assume that all If;' are connected by taking restrictions if necessary.. Now 'Pi. 1& = /; dx I " ••• " dx" = /;1&0' where 1'0 is the standard volume form on RIO. By means of a reflection if necessary. we may assume that /;( u') > 0 (/; .. 0 since I' is a volume form). However. a continuous real-val- ' ued function on a connected space that is not zero is always > 0 or always < O. Hence. for overlap maps we have
/;.
----".!...,-~I dx
h
0
Ifj 0 Ifj-
I
" ... " dx".
ORIENTA TlON, VOLUME ELEMENTS, AND CODIFFERENTIAL
383
But. 1/J*(u)(ul
"
...
"u") = D1/J(u)*'u l
"
D1/J(u)*'u 2 "
•••
"D1/J(u)*·u" •
. where D1/J(u)··u'(e)-u l (D1/J(u)·e). Hence. by definition of determinant we have
We leave as /in exercise for the reader the fact that the canonical isomorphism L(E; E) '"' L(E*; E*). used before does not affect determinants. For the converse of (ii). let {( V".1/J a)} be an atlas with the given property. and {(lI,. (JJj' gj)} a subordinate partition of unity. Let
and let _(
lA, m
)={g;(m)IA,(m) 0
Since supp g/' c lI,. we have
if m E l~ if m!ll~
ii; E 0"( M). Then let
. Since this sum is finite in some neighborhood of each point. it is clear from local representatives that Il E 0"( M). Finally. as the overlap maps have positive Jacobian determinant. then on U, () ~.Ilj * 0 and so
Since Ljg/ = 1. it follows that lA(m)* 0 for each mE M . • Thus. if (M.IA) is a volume manifold we get a map from 0"( M) to namely. for each 1" E 0"( M). there is a unique IE '!f(M) such that
~t( M);
lA' ,.. Ill·
.
6.5.3 DeftnlUon. Let M be an orientable manilold. Two tlo/ume lorms IAI and 1'2 on M are called equioalent il there is an I e ':t ( M) with I( m) > 0 lor
3tU
DIFFERENTIAL FORMS
1111 me M such that ". - f"2' (Thu is clearly an equivalence relation.) An __tIItitM of M is an equivalence class [Ii] of volume forms on M. An oriatal _i/oltl. (M.["D. is an orientable manifold M together with an orientation [,,] on M. If [,,] is an orientation of M. then [- ,,]. (which is clearly another orientation) is clllied the rewru oriallltitM.
The next proposition tells us when [,,] and [- ,,) are the only two orientations.
1.5.4 PropoalUon. Let M be an orientable manifold. Then M is connected iff M has exactly two orientations.
Pro9/. Suppose M is connected. and
Ii. Ii' are two volume forms with ,,' ... f". Since M is connected. and f(m) ... 0 for all m E M'/(m) > 0 for all m or elsef(m) < 0 for all m. Thus Ii' is equivalent to" or - Ii. Conversely, if M is not connected. let V ... '" or M be a subset that is both open and closed. If" is a volume form on M. define Ii' by
'(m) ... { "
,,(m) - ,,(m)
mEV mEV.
Obviously. Ii' is a volume form on M, and Ii' E [,,)U[ - ,,]. • A simple example of a nonorientable manifold is the MObius band (see Fig. 6.5.1 and Example 3.3.8C.
Flgu....5.1
ORIENTATION, VOLUME ELEMENTS, AND CODIFFERENTIAL
385
'.5.5 PropoeItIon. The equivalence relation in 6.5.3 is natural with respect to mappings and dif!eomorphisms. That is, ill: M -0 N is 01 class Coo. ,. Nand "N equivalent volume lorms on N. and IV ) is a volume form on ' M. thenr("N) is an equivalent volume form. III is a dil/eomorphism and "'M and,.:., are equivalent volume lorms on M. then 1.(",,,,) and 1.( ... :.,) are equivalent volume lorms on N.
r(,.
are.
Proof. This follows from the fac, that r(gw)== (gof)rw. which implies
1.(gw)=(gorl)/.w
when I is a diffeomorphism. •
8.5.' D......ltIon. Let M be an orientable manilold with orientation ItA). A chart (U.",) with ",(U) - U' e R" is called positiwly orimled if ",.( ... 1U) i.f equivalent to the standard volume lorm
From 6.5.5 we see that this defmition does not depend on the choice' of the representative from [tA). . If M is orientable. we can find an atlas in which every chart has positive orientation by choosing an atlas of conneCted charts and. if a chart has negative orientation. by composing it with a reDection. Thus. in 6.S.2(ii), the atlas consists of positively oriented charts. If M is not orientable. there is an orientable manifold M and a 2-to-l COD suljective local diffeomorphism w: M- M. The manifold if is called the orientable double covering and is useful for reducing certain facts to the orientable case. The double covering is constructed as follows. Let M = . {(m.I,.",)lm e M,l,.",] an orientation of T",M). Define a chart a~ (m,l,.",» in the following way. If,,: U-U'eR" is a chart of M at m, then 1".(,.",)] is an orientation of R". so that setting
D- {(u,[,..))lueU,[",.(,..)] -
c;: D.... u'.
[",.(,.",)]}.
C;(u.[,..]) = ",(u).
we get a chart (D. c;) of M. It is straightforward to check that the family
{( D, C;)}
constructed in this way forms an atlas. thus' making if into a differentiable n-manifold. Define w: if .... M by w( m, I ...... ]) ... m. I n local
386
DIFFERENTIAL FORMS
charts fT is the identity mapping. so that fT is a surjective local diffeomorphism. Moreover fT-'(m)={(m.["",J),(m. -["",))). so that fT is a twofold covering of M. Finally. M is orientable. a natural orientation on M is given on the tangent space 1( .... (.. ~J)M by I(T...fT)-"",].
'.5.7 Propoamon. Let M be a connected n-manifold. Then M i.' connected iff M is nonorientable. In fact. M is orientable iff M consists of two disjoint copies of M. one with the given orientalion. the other with the reverse orientation. Proof. The if part of the second statement is a reformulation of 6.5.4 and it also proves that if M is connected, then Mis nonorientable. Conversely if M is a connected manifold and if M is disconnected, let e be a connected component of M. Then since fT is a local diffeomorphism, fT(e) is open in M. We shall prove that fT(C) is closed. Indeed, if mE c1(fT(C».let "',,"'2 E M be such that fT(!",) = fT("'2) = "!: If there exist neighborhooc;!s V" VI' of "'I' "'2 such that V, () C - 0 and V2 () C = 0. then shrinking VI and V 2 if necessary. the open neighborhoods fT( V,) and fT( V2 ) of m have empty intersection with fT( C). contradicting the fact that mE c1( fT( Thus at least one of m ,. m 2 is in c1( C) = e; i.e.• m E fT( C) and hence fT( C) is closed. Since M is connected. fT(C) = M. But fT is a double covering of M so that M can have at most two components. each of them being diffeomorphic to M. Hence M is orientable. the orientation being induced from one of the connected components via ft. •
e».
Another criterion for orientability is the following.
'.5.'
Propoamon. Suppose M is an orientable n-manifold and V is a suhmanifold of codimension k with trivial normal bundle. ThaI is, there are Coo maps N;: V-TM, i-I •...• k such that N;(v)eTv(M), and N;(v) span a subspace w" such that Tv M = T., V $ Wv for all v e V. Then V is orientable. Let" be a volume form on M. Consider the restrict,ion "I V: . III V is a smooth mapping of manifolds. This follows by using charts with the submanifold property. where the local representative is a restriction to a subspace. Now define 110: V -+ A" A( V) as follows. For Proof.
V -+ h"( M). Let us first note that
0
X"oo.,X,,_1t e 'X(V), put
ORIENTATION, VOLUME ELEMENTS, AND CODIFFERENTIAL
387
It is clear that ILo( v) "" 0 for all v. It remains only to show that ILo is smooth, but this follows from the fact that ILl V is smooth. •
, If • is a Riemannian metric. then ,b: TM - T- M denotes the index,lowering operator and we write (,b)- I. For I E ~f( M), grad I - r(tl/) is called the gradient of I. Thus, grad I E ~X (M). In local coordinates, if gij=,(a/ax i; a/axil and gil is the inverse matrix, then
r-
( gradl ) i =
..
al
g'J_..
ax J
(I) ,
6.5.9 Coronary. Suppose M is an orientable manilold. H E ~(M) and c E R is a regular value of H. Then V = H- I( c) is an orientable submanifold of M of codimension one, if it is nonempty. Proof. Suppose c is a regular value of H and H - 1( c) = V .. 0. Then V is a submanifold of codimension one. Let g be a Riemannian metric of M and N = grad( H)IV. Then N( v) E T"V for v E V. because T..V is the kernel of dH( v), and dH( v)[ N( v») = ,( N. N)( lJ) > 0 as dH( I') .. 0 by hypothesis. Then 6.5.8 applies, and so V is orientable. •
Thus if we interpret V as the "energy surface:' we ~ce that it is an oriented submanifold for "almost all" energy values (hy SaHl's theorem; see Appendix E). Let us now examine the effect of volume forms under maps more closely. •
6.5.10 Definition. Let M and N be two orientable n-manifolds with volume , forms I'M and ILN' respectively. Then we call a Coo map I: M - N vobune prese"'i1Ig (with respect to IL M and I' N ) il f*IL N = IL M' and we call f orimtatiOll pNse",in, if f*( I' N ) E [I'M)' and orieNtation reversin, if f*( IL N ) E [ - I' At). From 6.5.5, (f*ILN) depends only on IILN)' Thus the first Pllrt of the definition explicitly depends on ILM and ILN' while the last two parts dClpend only on the orientations IILM J and IILN). Furthermore. we see from 6.5.5 that if f is volume preserving with respect to I'M and ILN' then I is volume preserving with respect to hIL", and IDAN iU h = g 0 I. It is also clear that if f is volume preserving with respect to IL", and ILN' then f is orientation preserving with respect to (I'M) and (IL N).
6.5.11
Proposition. Let M and N be n-manilolds with !'olume lorms IL M and ILN' respectively. Suppose f: M -+ N is 01 class Co f( V) is a diffeomorphism. (ii) If Mis connected. then f is a local diffeomorphism iff f is orientation preserving or orientation reversing.
r("
Proof. If f is a local diffeomorphism. then clearly N)( m) ., 0, by 6.4.4(ii). Conversely, if r("N) is a volume form, then the determinant of the derivative of the local representative is not zero, and hence the derivative is an isomorphism. The result then follows by the inverse function theorem. (ii) follows at once from (i) and 6.5.4. • Next we consider the global analog of the determinant. 8.5.12 DeIInlUon. Suppose M and N are n-manifolds with volume forms "M and "N' respectively. Iff: M -> N is of class COO. the unique Coo function J(,. ... ,.,."f e 'S(M) such that r"N'" (J(,. ... ,.,.,'/)"M is called the JlICObilm line",.. . ., of f (with respect to and "N)' If f: M -> M, we write J,..J'" )f. .
"M
'
Let (M. ,,) be a volume manifold.
M and g: M -> M are of class COO, then J,.(/og) = [(J,.f)og][J"g).
(ii) If h: M -> M is the identity. then J"h = I. Iff: M -> M is a diffeomorphism. then
(iii)
J,,(/-I) Proof.
= \/[ (J"f)
0
f
1].
For (i). J,,(/og),,= (/og)*"=g*r,, = g*(/)&fhl. = ((J"f) 0 g )g*" "" ((J,./) 0
g)( I)&g)"
ORIENTA TlON. VOLUME ELEMENTS, AND CODIFFERENTIAL
389
Part (ii) follows since h· is the identity. For (ill) we have
8.5.15 PropoIIUon. Let (M.[I'M)) and (N,[I'N)) he oriented manifolds and f: M -+ N be of class COO. Then I is orientation presen.Jing ill J(,..,.,.",,!( m) > 0 for all me M, and orientation reversing iff -,. Let (M. ,,) be a volume manifold. Prove the identity div,,[X. Y] = X [div"Y] - Y [div"X].
6.5G Let I: M -+ N be a diffeomorphism of connected oriented manifolds with boundary. If Tml: TmM -+ ~(m)N is orientation preserving for some m E Int( M). show that J( f) > 0 on M; i.e.• 1 is orientation preserving.
CHAPTER
7
Integration on Manifolds
The integral of an n-form on an n-manifold i~ ddll1l'd in terms (If integrals over sets in An by means of a partition of unitv ~l1hordinale to an atlas. The change-of-variables theorem guarantees thai the integral is well defined. independent of the choice of atlas and partition of unity. The basic theorems of integral calculus-namely. the change-of-variables theorem and . Stokes' theorem-are discussed in detail. along with some applications. 7.1
THE DEFINITION OF THE INTEGRAL
The aim of this section is to define the integral of an n-form on an oriented n-manifold M. We begin with a summary of the basic results in An. Suppose f: An ..... A is continuous and has compact support. Then If d:(1 ... dx" is defined to be the Riemann intt:gral over any rectangle containing the support of f. 7.1.1 Definition. Let U c An be open and support. If. relative to the standard basis of An.
WE
O"(U) have compact
I (). . I w(x)= n!"';, ... ;" x dx"/\ ···/\dx'·="'I .. ,,(x)dx /\ ... /\d.'(". where the components of w are defined by
"';, .. ;.<x) =
"'( x)( e;,."
.. eJ. 395 .
396
INTEGRA TION ON MANIFOLDS
then we define jll)= j"'I .. ,,(x)dx l
••
·dx".
The change-of-variables rule takes the following form.
7.1.2 Propoeltlon. Let U and V be open subsets of H" and suppose f: U .... V is an orientation-preseroing diffeomorphism. Then if II) E (1"( V) has compact support, /*11) E O"(U) has compact support as well and jf*1I) = jll), that is, the following diagram commutes: (1"(U)
r
(1"(V)
i\'/f H
Proof. If II) = "'I ... " dx l " ••• "dx", then f*1I) = ("'I." f)(1I1./)(1(;. where (10 = dx l " ••• "dx" is the standard volume form on Hn. As was discussed in Section 6.5, }gJ > 0 is the Jacobian determinant of f. Since f is a diffeomorphism, the support of f*1I) is I(SUpp 11). which is compact. Then 0
r
by the usual change-or-variables formula, namely,
we get jf*1I) =
III). •
Suppose that (U, cp) is a chart on a manifold M. and II) E (1"( M). Then if supp II) C U, we may form II) I U, which has the same ·support. Then CP.( II) I U) has compact support. and we may state the following.
7.1.3 Definition. Let M be ann-manifold with orientation [(1). Suppose (1"(M) has compact support C C U. where (U.CP) chart. Then we define 1(.)'" = !cp*( 11)1 U).
'" E
7.1.4 Propoeltlon. Suppose
I.~
a positively oriented
II) E 0"( M) has compact support C C U () V. where (U. cp). (V, '" ) are two posilit)e(v oriented charts on the oriented manifold
THE DEFINITION OF THE INTEGRAL
397
M. Then
f w-f (,,)
w
(oS-)
By 7..1.2. f'l'.(wl V} = f( ¥- 0 'I'~ I }.'I'.( wi V). Hence f'l'.(wl V) = [Recall that for diffeomorphlsms f. =
Let
f be any Coo function with compact support on
M. Then by
.7.1.1. /. P{fJloN) = /. {foF)(~P.M.P.N)F)flM /.Nfdp.N=/.fflN= N M M
400
INTEGRA TION ON MANIFOLDS
As in the proof of 7.1.9. this relation holds for f the characteristic function of F( A). That is,
7.1.11 Propoeltlon. Let M be an orientable manifold with volume form I' and corresponding measure ". Let X be a (possibly time-dependent) C l vector field on M with flow F,. The following are equivalent (if the flow of X is not complete. the statements involving it are understood 10 be where defined): (i) (ii) (iii) (iv) (v)
div" X - O. J,.F, -I for all 1 E R. F,." -" for all t E R. r,I' ... I' for aliI E R. Ifd" = l(f 0 F,) d" for all f E LI(M. 1') and all t E R.
Proof. Statement (i) is equivalent to (ii) by 6.5.19. Statement (ii) is equivalent to (iii) by 7.1.10 and to (iv) by definition. We shall prove that (ii) is equivalent to (v). If J,.F, = 1 for all 1 E R and f is continuous with compact support, then
l(f F,) d" for all integrable
Hence. by uniqueness in 7.1.9. we have If d" =
0
f. and so (ii) implies (v). Conversely, if
then taking f to be continuous wi th compact support. we see that
Thus. for every integrablef. f(f 0 F,) dp. and so (v) implies (ii). •
=
f( f F,)( J"F,) dp.. 0
HenceJ"F, = I.
The following result is central to continuum mechanics (see below and Section 8.2 for applications).
THE DEFINITION OF THE INTEGRAL
401
'7.1.12 Trarwpott Theorem. Let (M.~) be a volume IIltinifold and X a vt!ctor field on M with flow F,. For f E ~'i' ( M x R) and lettin/{ /, ( m) = f( m. t). we h.'lvt! '
d / -d 1
/,dp. - /
F,(UI
F,(UI
(!!1at + dlV... ( /,X).) dp.
for an open set U in M. Proof. · have
By the flow characterization of Lie derivatives and 6.5.17 (ii). we
=
a f -. F,*( a,~)+ F,*[( Lx/')~ + /,(dlv~X)~]
=
F,* [(
~~ + div~ ( /, X) ) ~ ] .
· Thus by the change-of-variables formula.
.
-/F,*[( aaft +diV~(/,X»)~] =/ I'
1-,1 U I
(i/}/I hli\~(/,X»)~ . • I
7.1.13 Example. Let p(x. t) be the density of an ideal fluid moving in a compact region with smooth boundary of R 3. One of the basic assumptions of fluid dynamics is conservation of mass: the mass of the fluid in the open · set U remains unchanged during the motion described by a flow F,. This , means that
:t f
F,(UI
p(x,t)dx=O
for all open sets U. By the transport theorem, this is equivalent t O} the 01 U and au - U nCR ,,- I X (O}) the botmdary 01 u.
_rior
au.
au
We clearly have U = Int U u InW is open in U, closed in U n Int U = Ilj. The situation is shown in Fig. 7.2.1. Note is not the topological boundary of U in R", but it is the topological that boundary of U intersected with that of R~. This inconsistent use of the is temporary. notation A manifold with boundary will be obtained by piecing together sets of the type shown in the figure. To carry this out, we need a notion of local smoothness to be used for overlap maps of charts.
(not in R "). and
au
au
au
7.2.2 Definition. Let U, V be open sets in R ~ and I: U -. V; We shall say that I is SIrItItIIII illor t!tICh point x E U there exist open neighborhoods U I 01 x and VI oll(x) in R" and a smooth map II: U I -+ VI such that II U n U I III U nUl' We then define DI(x)" DII(x). We must prove that this dermition is independent of the choice of II' that is. we have to show that if cp: W -+ R" is a smooth map with Wopen in
Xo
RO, inlU
au FIgu... 7.2.1
STOKES' THEOREM
405
R" such that'l'l W nR~ = 0, then Dcp(x) = 0 for all x E W nR.~. If x E Int(WnR'!.). this fact is obvious. If xEa(WnR~). choose a sequence x"elnt(WnR'!.) such that x" .... x; hut then 0= Dcp(x.. ) .... Dcp(x) and. hence Dcp(x) = O. which proves our claim. 7.2.3 Lemina. Let V C R~ be open. '1': V .... R~ be a smooth map. and assume that for some Xo e IntV.cp(xo)e aR~. Then Dcp(xn)(R")c aR~.
Let p.. : R ...... R be the canonical projection on to the nth (last) factor and notice that the relation
Proof.
'1'( Xo + IX) = '1'( Xo ) + Dcp( x n )· Ix + fI(tX) , where lim,~oo(/x)11 = O. together with the hypothesis (1'./0 'I')(y) ~ 0 for all y e V. implies 0 :IIi; (p" 0 'I' )(xo + Ix) = () + ( P.. 0 Ocr')( Xu )'Ix + p"C o(tx)). whence for I > 0 O:lli; (P.. 0 Dcp)(xn)'x
+ p,,( ~(tl ).
Letting t .... O. we get (P.. 0 Dcp)(xo)'x ~ 0 for all x e R". Similarly. for 1< 0 and letting I .... O. we get (P.. Dcp)(xo)'x:lli; 0 for all x e R", The conclusion is 0
Intuitively. this says that if cp preserves the condition XII ~ 0 and maps an interior point to the boundary. then the derivative must be zero.in the vertical direction. The reader may also wish to prove 7.2.3 from the implicit mapping theorem. Now we carry this idea one step further. 7.2.4 Lemma. Let V. V be open sets in R~ and f: U .... Va diffeomorphism. Then f restricts 10 diffeomorphism" Intf: IntV .... Int V and
av .... av.
.
at:
Proof. Assume first that av = 121. that is. that V n(R" I X (O}) = 121. We . shall show that av == 121 and hence we take Int f - f. If av. 121. there exists x e V such that f( x) E av and hence by definition of smoothness in R~. there are open neighborhoods VI C V and VI C R". such that x E VI and f( x) e VI' and smooth maps fl: .VI .... VI' .It I: Vi .... VI such that II VI = fl' gd V n VI = II V n VI' Let XII e VI' x" .... X.,," E VI\r1V. and,),. = fCx,,). We have
r
406
INTEGRA T/ON ON MANIFOLDS
and similarly Dgl(f(x»o DI(x)
=
Id R "
so that DI(x)-1 exists and equals Dgl(f(x». But by 7.2.3. DI(x)(R")C R,,-I X{O}. which is impossible. DI(x) being an isomorphism.
r
Assume that au .. 121. If we assume av = 121. then. working with I instead of I. the above argument leads to a contradiction. Hence av ... 121 • Let x E Int U so that x has a neighborhood UI C U. UI n au = 121. and hence au, == fZl. Thus. by the preceding argumen t. aI (UI ) = fZl • and I (UI ) is open in V\ av. This shows that I(lnt U) C Int V. Similarly. working. with I. we conclude that I(lnt U) :::> Int V and hence I: Int U - Int V is a diffeomorphism. But then I( au) = av and II au: au - iJV is a diffeomorphism as well. •
r
Now we are ready to define a manifold with boundary. 7.2.5 DeflnlUon. An n-lIIQIIiJoid with boundDry is a set M together with an lit"" 0/ chlufl with IJoIuuI4ry on M: ebllrtl "';th bolllUltl", are pairs (U. q» where U C M and q>( U) C and an atills on M is a lamily 01 charts with
R:
x·'
R" ,
R" ,
Figure 7.2.2
STOKES' THEOREM
407
boundary satisfying (MAl) and (MA2) of 3.1.1, with smoothness of overlap maps rpj; understood in the sense of7.2.2. See Fig. 7.2.2. Define Int M= U urp-I(lnt(rp(U))) and aM = U 1' O} where ., E E*, /- O. We leave the detailed formulation to the reader. Recall that before we can integrate a differential n-form over an n-manifold M, M must be oriented. If Int M is oriented, we want to choose an orientation on iJM compatible with it. In fact the reader will recall that in the classical Stokes' theorem for surfaces, it is crucial that the boundary curve be oriented in the correct sense, as in Fig. 7.2.3.
Figure 7.2.3
408
INTEGRATION ON MANIFOLDS
The rirst thing to do is to note that the earlier developments. including the tangent bundle. tensor rields. and differential forms. carry over directly to manifolds with boundary. In particular we can define T, M even if x E aM. Note that TxM is n-dimensional. even at points.v E aM. (See Fig. 7.2.4.) Thus. it makes sense to talk about volume forms and orientations on M up to and including the boundary points. .:.:: .............:......... . .'
iJM
..
",
.::,::.:. . . :::.:. . ,>::: . = I'M( x >( X/ex )1'; -
(ixl'M}(X)( v, ..... 1'" _
X"(x>
= r(X)I'aM(X)(V' ..... v"
and X"
= (X. "aM)'
a!" .1', ......",- ~) ,>
the corollary follows by Gauss' theorem. _
7.2.11 Corollary. If X is divergence-free on a compact boundarylen manifold with a volume eleme"t 1'. then X as an operator is skew-symmetric: that is. for f and g E t!i( M).
f. X (f )g dp. .. - f. f X [ g] dp.. M
101
412
Proof. Thus
INTEGRA TlON ON MANIFOLDS
Since X is divergence-free, Lx(h")=(Lxh),, for any hE~'f(M).
Integration and the use of Stokes' theorem gives the result. •
7.2.12 Corollary. If M is compact without boundary, X E ~X ( M). «E O"(M) and p E O"-"(M), then
Proof. Since « " PE 0"( M). the formula follows by the use of the Stokes' theorem integrating both sides of the relation di x ( « " P) = Lx ( « " P) = Lx«" P + LxP. •
«"
7.2.13 Corollary. If M is a compact orientable, boundaryless n-dimensional pseudo-Riemannian manifold with a metric g of index Ind( g), then d and 3 are adjoints, i.e.,
Proof.
Recall from 6.5.21 that
=
liP =
( - ) )n' k + 2)
I
I + Ind(,r) •
d •• so that
d( I. Let /,,{I) = f( nt) for all positive integers R and note that
If;(1)1
=
nlf'( nt)1 ~ Cn.
Define the Coo function
where the product is taken over integers z such that I;: I < 2 Na + I. Note that if Itl'< a + Ij4N and z E Z is chosen such that Ix - z/2NI < 1/2N. then fN(/- z/2N) =- 0 and Izi ~ 2NI/I +! < 2Na+ 1. so that cp(t)=O. Similarly if Itl>a+2IN and IZI liN so that cp(/)-1. Finally. let Ito - al < 21N and let Zo E Z be such that Ito - zo/2NI < liN. All factorsfN(/ o - z/2N) are one in a neighborhood of ' 0 , unless Ito - z/2NI ~ liN. But then Ik - kol"; Iz -2Ntol + 12Nlo - zol ~ 3. Hence at most seven factors in the product are not identically I in a neighborhood of to. Hence Icp'(t,,)1
~
TCN = A/f.
.,
420
INTEGRA TION ON MANIFOLDS
a,.
7.2.23 L....m..
Let K be a compact subset of the singular set of For every e> 0 there exists a neighborhood U, of K in V X R and a C'" function ",,: V X R ..... [0. I]. which vanishes on a neighborhood of K in V" is one on the complement of V,. and is such that
f.
(i)
vol(V.>
sup 1a"" (x) I] ~ E. [ zeA" ax
i = I ..... n
_r_;_
and
(ii) vol( lI,> < I. A( V, (i p,) < I. where '" i.~ the mea.Jure on PI auocialed with the oolume form). E O,,-I(P/)' and vol(Vr ) is the Lebesgue measure 0/ lI, in R".
Proof. Partition R" - I by closed cubes
D of edge length 4//3. I ~ I. At most 2" such cubes can meet at a vertex. The set 'IT( K). where 'IT: V X R ..... V is the projection. can be covered by finitely many open cubes C of the edge length 21. each one of these cubes containing a cube D and having the same center as D. Since 'IT( K) and K have measure zero. choose I so small that for given 8 > 0,
(i) The (n - I )-dimensional volume of U :"'IC; is smaller than or equal to 8; and (ii) "('IT I(U:"'IC,)npf)~8.
Since f is locally Lipschitz and 'IT( K) is compact. there exists k > 0 such that If(x)- f( .r)1 ~ kllx -.rll for x. y E 'IT( K). We can assume k ~ I without loss of generality. In each of the sets 'IT I( C,) = C; x R. choose a box P; with base C; and height 2kl such that 'IT( K) is covered by parallelipipeds 1';' with the same center as I'; and edge lengths equal to two-thirds of the edge lengths of Pi' Let V ... U f-, Pi' Then 'IT( V) = U f_,C; and since at most 2" of the I'; intersect. vol(V) - 2k12" vol( 'IT(V»
< 2"+ Ikl8 < 2"+ 1k8
and By the previous lemma. for each I'; there exists a COD function "';: V x R ..... [0. I] that vanishes on p;,. is equal to I on the complement of PI' and sUPzER"lIa",Jax}1I < A/I. Let", == nf-,,,,,, Clearly "': V x R ..... [0. I) is C..,. vanishes in a neighborhood of K and equals one in the complement of V. But at most 2" of the P, can intersect. so that
Iax' I Ii-,_1 ax} ".,n I . -a", ....
~
-a",;
"'4.r
~"·1
.
1=
I •...• n
STOKES' THEOREM
Hence VOI(V)[ sup .. eA"
421
Iax'a'Pl] ~2"+lkI82"A/I+221'lk8A .
Now let Ii = min{E. E/2 2 ,,+ IkA}. 'P, = 'P, and V,
• .Proof of Eq. (I). Let w=E7_1""dx l l\ ... 1\ dx' 1\ ..• 1\ dx", dw - bdx l 1\ ••• 1\ dx", and i·w - a)... Then "',. band U ilre continuous and bounded on V X R and PI respectively; i.e., '''',(x)1 ~ M, Ib(x)1 ~ N for xEV X Rand la(y)l ~ N ror yE P" where M, N > 0 are constants. Let V, and 'P, be given hy the previous Iemmil applied to supp( w)U af . But then 'P,W vanishes in a neighborhood or af and lemma 7.2.21 is applicable; that is, =
V.
1r, d(~,w) =1r,'P,w
(2)
We have
1w-1r,'P,wl ~ If
1 ,r,
a( 1-
P,
'P,);\I ~ N;\( u, () PI) ~ NE
(3)
and
11r, dw - 1r, d( 'P,W)I ~ 11.I, (dw - 'P,dW)/ + 11." d'P. 1\ w/
I I
acp +E f. 1"'11 ~ ,_I r, ax II
dx l 1\ •.• 1\ d,"
II ~ N vol( V.) + ME sup [
,_1 "eA"
10!Jl(X)I] - ' - ,vol( V. ) ax
(4) From (2)-(4) we get
1r, dW-lwl~(2N+nM)E r,
1
for all E> O. which proves the equality. •
422
INTEGRA TlON ON MANIFOLDS
In analysis one also desires hypotheses on the smoothness of c.> that are as weak as possible as well as on the boundary. Our proofs show that Co) need only be C I • An effective strategy for sharper results is to approximate Co) by smooth forms Co)k so that both sides of Stokes' theorem converge as k ..... 00. A useful class of forms for which this works are those in Sobolev spaces. function spaces encountered in the study of partial differential equations. The Lipschitz nature of the . boundary of N in Stokes' theorem is exactly what is needed to make this approximation process work. The key ingredients are approximation properties in M (which are obtained from those in R") and the Calderon extension theorem to reduce approximations in N to those in R". (Proofs of these facts may be found in Stein [1970]. Marsden [1973]. and Adams [1975].) Exercl...
7.2A
Let M and N be oriented n-manifolds with boundary and f: M ..... N an orientation-preserving diffeomorphism. Show that the change-ofvariables formula and Stokes' theorem imply that rod = d r.
7.28
Let M be a manifold and X a smooth vector field on M. Let CI E Ok ( M). We call CI an invariant k-form of X iff Lx ex = O. Prove the following theorem of Poincare and Cartan: Let X be a smooth vector field on an-manifold M with flow F,. and let CI E Ok( M). Then ex is an invariant k-form of X iff for all oriented
0
compact k-manifolds with boundary (V. aV) and C<X:J mappinKs 'P: V ..... M. such that the domain of F, contains 'P( V). 0 ~ t ~ T. we have
f v( F,
0
'P )*CI =
f v'P*CI.
(Hint: For converse show that the equality between integrals holds
for any measurable set and conclude that ( FA cp )*CI "" cp*ex; then let V be a portion of various subspaces in local representation.) 0
7.2C Let X be a vector field on a manifold M and CI. II invariant forms of X. (See Exercise 7.28) Prove the following: (i) iXCl is an invariant form of X. (ii) dCl is an invariant form of X. (iii) LxY is closed iff dy is an invariant form. for any y E O*(M). (iv) CI /\ P is an invariant form of X. (v) Let f.£x denote the invariant forms of X. Then (:I'x is a /\ subalgebra of ll( M). which is closed under d and i x. 7.2D Let X be a vector field on a manifold M and ClE Ok(M). Then CI is called a relatively inf)ariant k· form of X if L XCI is closed. Prove the following theorem of Poincare and Cartan.
STOKES' THEOREM
423
Let X be a vector lield with flow F, on n-manilold M. and leI .. e O· - I ( M). Then .. is a relatively invariant (k - I )-Iorm 01 X ifflor
all oriented compact k-manilolds with boundary (V. ;W ) and cae map.f cp: V .... M such that the domain 01 F, contains cp( II ) lor 0 .e;; t .e;; T. we have
where i: 7.2E
av .... V is the inclusion map.
If X e ~X (M). let tf x be the set of all invariant forms of X, ~ x the set of all relatively invariant forms of X, the set of all closed forms in O( M). and f, the set of all exact forms in O( M). Show that
e
e
(i) (f x C ~'R x' t;, c c ~,~ x' ff x is a differential subalgebra of O( M), but ~ x is only a real vector subspace. . I· x
;
"
(ii)
the sequence 0 .... (fx"" O(M)"" O(M) .... O( M)jlm(L x ) ~ 0 is exact.
(iii)
0~
e ....; I3t x ....4 tf x ....." Cf. x If, n (f. x .... 0 IS. exact.
(iv) d«(fx)C tf. x and ix«(f.x)C (ix' Let M be a compact (n + l)-dimensional manifold with boundary,/: aM .... N a smooth map and", e N) where d", = O. Show that if I extends to M, then faM/*'" ""' O. 7.20 Let M be a boundaryless manifold and I: M .... R a COO mapping having a regular value a. Show that 1«( a, oo() is a manifold with boundary I(a). 7.2H Let I: M .... N be a COO mapping. M having boundar\' aM. and let PeN be a submanifold of N, where P and N are houndaryless. Assume I is transverse to P. Show that I-I(p) is a manifold with boundary arl(p)- rl(p)naM and dimM-diml I(p)= dimN -dimP. (Hint: At the boundary. work with a boundary chart with the technique of the proof of 3.S.II; then use the preceding exercise.) 7.21 Let M be a paracompact manifold with boundary. Show that there exists a positive smooth function I: M .... [0. oo[ with 0 a regular 1(0). (Hint: First do it locally and then value, such that aM patch the local functions together with a partition of unity.) 7.21 (Col/an). Let M be a manifold with boundary. A col/or for M is a diffc:omorphism of aM x [0, I[ onto an open neighborhood of aM in M that is the identity on aM. Show that a manifold with boundary and admitting partitions of unity has a collar:. (Hint: Via a partition 7.2F
one
r
r
r
424
INTEGRATION ON MANIFOLDS
of unity, construct a vector field on M that points inward when restricted to aM. Then look at the integral curves starting on aM to define the collar.) 7.2K (71re boundlJlyless double). Let M be a manifold with boundary. Show that the topological space oblained by identifying the points of aM in the disjoint union of M with Itself is a boundaryless manifold in which M embeds, called the boundar),less double of M. (Hint: Glue together the two boundary charts.) 7.2L (L. Loomis and S. Sternberg (1968). and R. Rasala). Let M be a compact Riemannian manifold with boundary. not necessarily oriented. Let X be a vector field on M. let p be the Rjema~nian density and let Px be a density on aM defined by Px( v,,",, vn - I) = p( v,,",, v,,-I' X) and let,x(x) = I if X points out of M. 0 if X is tangent to aM. and -I if X points into M. Show that
J. (divpX)p =1 M
'xPx
aM
and that this reduces to the usual divergence theorem if M is oriented. 7.2M Let M be a compa,·' orientable boundaryless n-manifold and oE 0" - I( M). Show that do vanishes at some point. 7.3 THE CLASSICAL THEOREMS OF GREEN, GAUSS, AND STOKES
This section shows explicitly how to obtain these three classical theorems as consequences of Stokes' theorem for differential forms. We begin with Green's theorem. Green's theorem relates a line integral along a closed piecewise smooth curve C in the plane R 2 to a double integral over the region D enclosed by C. (Piecewise smooth means that the curve C has only finitely many comers.) Recall from advanced calculus that the line integral of a one-form w - Pdx + Qdy along a curve C parametrized by y: la. b) -0 R2 is by definition equal to
{w = { R. But this latter integral equals
ffJ
~: + ~;) dxd.v = fIr}divX) dxd.v
by 7.3.1. • laking P(x. y) = x and Q(x. y) = y in Green's theorem, we get the following.
7.3.3 Corollary. Let D be a region in R2 hounded hy a closed piecewise smooth curve C. The area
1
0/ D is given by ! x dy - }' dx. C
The classical Stokes theorem relates the line integral of a vector field around a simple closed curve C in R3 to an integral over a surface S for which C = as. Recall from advanced calculus that the line intexral of a vector field X in R 3 over the curve a: [a, hI -> R) is defined hy
l X 'd., "
=
j"X(a(t))'a'(t)dt II
The surface integral of a compactly supported two-form (a) in R 3 is defined to be the integral'of the pull-back of (a) to the oriented surface. If S is an oriented surface. " is called the outward unit normal at xES if " is perpendicular to TxS and {II. el' e2} is a positively oriented hasis of R) whenever {el,e2} is a positively oriented basis of 1~S. Thus S is orientable iff the normal bundle to S -which has one-dimensional fiber-is trivial. In particular. the area element tIS of S is given by 6.5.8. That is.
THE THEOREMS OF GREEN, GAUSS, AND STOKES
427
"th.
We want to express J... '" in a form familiar from vector calculus. Let,., - Pdy" dz + Qdz" dx + R dx" dy so that,., - • Xb. where ~nd IIY - dx " dy
x-
a
P ax
a
a
+ Q ay + R az .
Recall that CI" P ... (CI, P) I' where I' is the volume form. Letting CI = ·"b. Xb and I' - IIY. we get lIb " • Xb - (X·,,) IIY. Applying both sides to (",VI'Oz) and ':Ising (2) gives .
P-
(3) (Jhe base point x is suppressed). The left side of (3) is • Xb( VI' Oz) since n b is one on n and zero on VI and Oz. Thus (3) becomes • Xb= (X'n)IIS.
(4)
Therefore. is identified with a surface integral familiar from vector calculus. A physical interpretation of fs( X·,,) dS may be useful. Think of X as the velocity field of a fluid. Then X is pointing in the direction in which the fluid is moving across the surface S and X· n measures the volume of fluid p,assing through a unit square of the tangent plane to S in unit time. flence the integral fs( X· n) dS is the net quantity of flUid flowing acrou the sruface per unit time, i.e .• the rate of fluid flow. Accordingly, this integral is also called the flux of X across the surface.
Theo.......
7.3.4 Classical Stok. .' Let S be an oriented compact sruface in R3 and X a C l vector field on S and its bountlmy. Then f.<curIX)'ndS=
s
1asx· •. ·
where n is the oulward unil normal to S (Fig. 1.3.2). Proof. First extend X via a bump function to all of R 3 so that the extended X still has compact support. By definition. fils X •tis = fasXb where b denotes the index lowering action defined by the standard metric in RJ. But flXlI - .(curl X)II (see 6.4.3 (C» so that by (2) and Stokes' theorem,
1asx·. -: 1 Xb_f.4Xb =f. •(curl X)b ... f.(curl X'n )dS. ilS
S
S
S
•
428
INTEGRATION ON MANIFOLDS
x
Figure 7.3.2
7.3.5 EX8mplft A. The historical origins of this formula are connected with Faraday's law, which is discussed in Chapter 8 and Example 8 below. In nuid dynamics, this theorem is related to Kelvin's circulation theorem, to be discussed in Section 8.2. Here we concentrate on a physical interpretation of the curl operator. Suppose X represents the velocity vector field of a nuid. Let us apply Stokes' theorem 10 a disk D, of radius T at a point P E R3 (Fig. 7.3.3). We get
/, X·lb= /, (curIX)'nds=(curIX'n)(Q)'lTT 2 , liD,
liD,
the last equality coming from the mean value theorem for integrals: here QED, is some point given by the mean value theorem and 'lTT2 is the area of D,. Thus «curIX)·n)(P)= lim
~1aD,X·lb.
,-0 'ITT
Figure 7.3.3
THE THEOREMS OF GREEN, GAUSS, AND STOKES
429
The number leX·. is called the circulation of X around the closed curve C. It represents the net amount of turning of the nuid in a counterclockwise direction around C. The preceding equality gives the following physical interpretation for curl X, namely: (curl Xl'" is the circulation 0/ X per unil area on a sur/ace perpendicular to ,.. The magnitude of (curl X)·,. is clearly maxi~ when ,. - (curl X)/Ucurl XU. Curl X is called the vorticity vector. B. A basic law of electromagnetic theory is that if I:.' (t. x. y. z) and H(t, x, y, z) represent the electric and magnetic fields at time t. then V X E = - aN/at, where V x E is computed by holding t fixed. and aH/at is cdmputed by holding x. y, and z constant. Let us use Stokes' theorem to determine what this means physically. Assume S is a surface to which Stokes' theorem applies. Then
{ E·.=l(VXE)'dS'""-l aH. dS ..l-!...l N . dS .
JliS
S
S
(The last equality may be justified if N is
f
ifS
E·tIs =
at
at
S
e l .) Thus we obtain
at iSN . dS . .
-.!
This equality is known as Faraday's law. The quantity lasE·tIs represents the" voltage" around as, and if as were a wire, a current would now in proportion to this voltage. Also IsH·dS is called the flux of N, or the magnetic nux. Thus, Faraday's law says that the voltage around a loop equals the negative 0/ the rate 0/ change 0/ magnetic flUX through the loop. C. Let X E 'X (R'),'Since R' is contractible. the proof of the Poincare lemma shows that curl X = 0 iff X - grad/ for some function / E ~(R3). This in turn is equivalent (by Stokes' theorem) to either of the following: (i) for any oriented simple closed curve e, leX' tis = 0, or (ii) for any oriented simple curves cl.ez with the same end points. Ic,X·. = fe,X· ... The function / can be found in the following way: /(x,y,z)=l x X I (t,0.0)dt+1 Y XZ(x,t,O)dt+ o 0
l'·x (x.y,t)dt. 3
0
.
Thus, for. example, if a a a X= Y + YCOS(YZ)-aZ . . Yax +(ZCOS(YZ)+X)-a
curl X = 0 and.so X = grad/, for some f. Using the foregoing formula, one
430
INTEGRATION ON MANIFOLDS
finds
f(x. y. z) ... xy +sinyz D. The same arguments apply in R 2 by the use of Green'S' theorem. Namely if
lhen.X'" gradf, for some f E ~1(R2) and conversely. E. The following statement is again a reformulation of the Poincare lemma: div X - 0 iff X - curl Y for some Y E ~x. (R 3).
7.3.8 CluelCliI Gau.. Theorem. Let n be a compact set with nonempty interior in R3 bounded by a surface S that is piecewise smooth. Then for X a c' vector field on 0 U S, fo(diVX) dV= L(x'") dS,
where dV denotes the standard l'olume elemenl in R) (Fig. 7.3.4). Proof.
Either use 7.2.10 or argue as in 7.3.4: By (4)
• .... ~.'.' -:':' ": '.: .::-:.: :: O. we conclude that ., - O. Thus (A - iAO)-1 makes sense and (A - iAr- 1 is its closure. It follows from Appendix C that A is the closure of Ao. •
7.4.2 Propoaltlon. Let X be a Coo diverKen("(!~free I'c'clor /it·/J on ( M. 1') with a complete flow F,. Then iX is an essentiu/~v selj-tl((;oilll operator on CcOO = the Coo functions with compact support in the' complex lIi1bert spa('e L2( M. II). Proof.
Let U,f -= f
0
F _I be the unitary one-parameter group induced from
F,. A straightforward convergence argument shows that U,f is continuous in t·in L2(M. II). In Lemma 7.4.1. choose Do = Coo functions with compact support. This is clearly invariant under If f E Do. then
u,.
so the generator of U, is an extension of - X O. Similarly, if the defect index of iX is zero in the lower half-plane. the flow is essentially complete for t < O. The converses of these more general results can be established along the lines of the proof of 7.4.1.
Proof. Suppose that there is a set E of finite positive measure such that if x
E
xE
E, F,(x) fails to be defined for t sufficiently large. Let Er be the set of E for which F,(x) is undefined for I ~ T. Since E = U7=-IET' some'Er
436
INTEGRATION ON MANIFOLDS
has Positive measure. Replacing E by ET • we may assume that all points of E "move to infinity" in a time ..,. T. If I is any function on M, we shall adopt the convention that I( F,(x» - 0 if F,(x) is undefined. For any x E M, if t < - T. F,(x) must be either in the complement of E or undefined; otherwise it would be a point of E that did not move to infinity in time T. Hence we must have XE( F,( x» = 0 for 1< - T, where XE is the characteristic function of E. We now define a
function on M by
g(x) =
fIX; e-Txd FT(x» dT. -00
Note that the integral converges because the integrand vanishes for I < - t. In fact. we have 0.;; K(X)';; f'XTe 'dT = e T. Moreover. K is in L2. Indeed. because F, is measure-preserving. where defined, we have IIx,: f~II!.;; IIx/JI2 (where II 112 denotes the L2 norm), so that 0
IIgll2 410 f~e 'lixE £.112 dT';; IIXf:lbe T • 0
The function g is nonzero because E has positive measure. Fix a point x EM. Then F,(x) is defined for I sufficiently small. It is easy to see that in this case £,(F,(x» and FT+/(x) are defined or undefined together, and in the former case they are equal. Hence we have XE( FT ( F,(x» - XE( F.+/(X» for I sufficiently small. Therefore, for t sufficiently small
= e'g(x).
Now if 'P is Coo with compact support. we have
· = I1m
Ig(F I(X»-K(.")-
= lim
e I-I I --g(x)q>(x)dl1
I ~ II
1-0
= -
I
I
I g(x)'P(x)dl1
q>
( x )dP-
INDUCED FLOWS ON FUNCTION SPACES AND ERGODICITY
437
(These equalities are justified because on the support of fJ' the flow F, exists for sufficiently smallt and is measure-preserving.) . Thus g is orthogonal to the range of X + I. and therefore the defect index of iX in the upper half-plane is nonzero. The case of completeness for t < 0 is similar. • Methods of functional analysis applied to L2(M. f&) can. as we have seen. be used to obtain theorems relevant to flows on M. Related to this is a measure-theoretic analogue of the fact that any automorphism of the algebra ~t( M) is induced by a diffeomorphism of M (see Box 4.2A.). This result. due. to Mackey (1962). states that if U, is a linear isometry on 1.2( M. f&). which is multiplicative (i.e .. U,(fg) = U,f·U,g. where defim;d). then U, is induced by some measure preserving flow F, on M. This may he used to give another proof of the Povzner-Nelson theorem. 7.4.3. A central notion in statistical mechanics is that of ergodicity; which is intended to capture the idea that a flow is random or chaotic. In dealing with the motion of molecules. the founders of statistical mechanics. particularly Boltzmann and Gibbs. made such hypotheses at the outset. One of the earliest precise definitions of randomness of a dynamical system was minimality: the orbit of almost every point is dCIl O. set
F. = { -q-tp q + tp ,(q.p)
if if
q>O. q>O.
q+tp>O q+tp 0 has a root.
hoD/. Assume without loss of generality that p(z)=z"+a"
IZ,,-I
+ ... + ao, where Q j E C, and regard p as a smooth map from R2 to R2. If
INTRODUCTION TO HODGE-DeRHAM THEORY
451
p has no root, then we can define the smooth map f(z)= p(z)/lp(z)1 whose restriction to Sl we denote by g: Sl -+ Sl. Let R > 0 and define for (E (0, I) and Z E SI, p
p,(z)-(Rz)"+/[a,,_I(Rz)"- 1+ ... + ao]. Since p,(z)/(Rz)"=I+/(a,,_I/(Rz)+ ... +an/(Rz)") and the coefficient of I converges to zero a! R -+ 00; we conclude that for sufficiently large R, none of the p, has zeros on S I. Thus
F: [0. I]XSI
p (z)
-+
SI defined by F(/. z} = ~( )
./ Ill, z I
is a homotopy of d,,(z) == zIt with g. On the other hand, G: [O.I)X SI -+ Sl defined by GU, z) == i(lz) is a homotopy of the constant mapping c: Sl -+ Sl, c(z) - /(0) with g. Thus d" is homotopic to a constant map. Hence if Ca) = d8 E (}1(SI) is the volume form of SI. then by 7.5.8.
f d:Ca) f c*Ca) =
.')'
=
0
:;-'
since c*Ca) = O. On the other hand if we parametrize Sl by ar~ length 8, 0", 8 '" 2'17. then d" maps the segment 0 '" (J '" 2'1T/n onto the segment 0", (J '" 2'17 since d" has the effect e;' >-+ e''''. Using this fact and the change of variables formula. we get
J..'), d:Ca) == nJ.s' Ca) = 2'1Tn . Thus for n ... 0 we get a contradiction. •
BOX 7.U ZERO AND n-DIMENSIONAL COHOMOLOGY AND THE DEGREE OF A MAP In this box we shall compute HO( M) and H"( M) for a connected n-manifold M. Recall that Hk(M)= kerdk/ranged k - I • where d k : (}k( M) -+ (}k+ I( M) is the exterior differential. Thus HO(M)-{fEIj(M)ldf=O}aR since any locally constant function on a connected space is constant. If M were not connected, then HO( M) = R e , where c is the numher of connected components· of M. By the Poincare lemma, if M is contractible. then H'( M) ... 0 for q .. O.
452
INTEGRATION ON MANIFOLDS
The rest of this box is devoted to the proof and applications of the following special case of deRham's theorem.
7.5.13 Theorem. ut M be a boundaryless connected. compact nmanifold. (i)
If Mis orientahle. then H"( M) ~ A. the isomorphism being given bv integration: [Co)) ..... fMCo). In particular Co)EO"(M) is e.'Wct iff ~Co)=Q
(ii)
•
If Mis nonorientahle. then H"( M) -
o.
Before starting the actual proof. let us discuss (i). The integration mapping 1M: 0"( M) - A is linear and onto. To see that it is onto. let Co) be an n-form with support in a chart in which the local expression is Co)'" fdx' 1\ ••• 1\ dx" withfa bump function. Then fMCo) = fR"f(x)dx > O. Since we can multiply Co) by any scalar. the integration map is onto. Any Co) with nonzero integral cannot be exact by Stokes' theorem. This last remark also shows that integration induces a mapping. which we shall still call integration. fM: H"(M) -A. which is linear and onto. Thus. in order to show that it is an isomorphism as (i) states. it is necessary and sufficient to prove it is injective. i.e.• to show that if IMCo) - 0 for Co) E 0"( M), then Co) is exact. The proof of this will be done in the following lemmas; the main technical ingredients are Exercise 6.4K. Stokes' theorem. and the Poincare lemma.
7.5.14 Lemme. The theorem holds for M
=
S I.
Proof. Letp: A -Sl be given by p(t)~eil and Co)E01(SI). Then p.Co) = fdt for f E ~(A) a 2.".-periodic function. Let F be an antiderivative of f. Since
0=
1s· == f ' Co)
+2"
f( s ) d~ - F( t
+ 27T ) -
F( t )
I
for all tEA. we conclude that F is also 2.".-periodic. so it induces a unique map G E ';'t(SI). determined by p·G = F. Hence p.Co) == dFp·dG implies Co) ... dG since p is a surjective submersion. ...
7.5.15 Lemme. The theorem holds for M
= S". n >
1.
Proof. This will be done by induction on n. the case n = I being the previous lemma. Write S" ... NUS. where N ... {or E S"lx"'" I .. O} is the closed northern hemisphere and S = {x E S"lx"+ 1 "O} the closed
INTRODUCTION TO HODGE-DeRHAM THEORY
453
southern hemisphere. Then N n S = S" - I is oriented in two different ways as the boundary of Nand S, respectively. Let ON = {x E S"lx"+ I> - t}, Os = {x E S"lx"+ I < t} be open contractible neigh~ borhoods of Nand S, respectively. Thus by the Poincare lemma, there exists an aN EO" I( ON)' as EO" I( P.. ) such that daN = w, on ON' da s '= W on Os. Hence by hypothesis and Stokes' theorem,
o=f w=fw+fw=fdaN+fdas = ( aN+f as
sn
-1.
.~"
N
aN I
S
1.
,fi"
as'" I
N
1.
JilN
S
.'i"
,IS
(aN - as); I
the minus sign appears on the second integral becaus~ th~ {lrientations of S,,-I and as are opposite. By induction, aN - a.\ E U" I(S" I) is exact. Let 0 = ON n Os and note that the map r: 0· ... S" I, sending each xES to r( x) E S" - I, the intersection of the meridian through x with ~e equator S"-I, is smooth. Let j: SrI I -+ S" he the inclusion of S" - I as the equator of S". Then r 0 j is the identity on S" I. Also, j 0 r is homotopic to the identity of 0, the homotopy being given by sliding x E 0 along the meridian to r( x). Since d( aN - as) = w - w = 0 on 0, by Exercise 6.4K we conclude that (aN - a s )- r*j·( aN - Us) is exact on O. But we just showed that j*( aN - as) E O,,-I( S" .. I) is exact, and hence· r*j*(u N - as) E 0,,-1(0) is also exact. Hence aN - as E 0,,-1(0) is exact. Thus, there exists p E 0" 2(0) such that aN - as = dP on O. Now use a bump function to extend p to a form yEO" 2( SrI) so that on 0, p = y and y - 0 on S"\V, where V is an open set such that cl U c V. Now put xEN xES
and note that by construction
~
is C'" and
d~ =
w.
•
7.5.16 Lemma. A compactly supported .n.-form wE O"(R") is the exterior derivative of a compactly supported (n - I )-form on R" iff f.ow = o.
Proof. Let 0: S" -+ R" be the stereographic projection from the north pole (O..... I)E S" on R". By the previous lemma, o·w = do., for some aE O,,-I(S") since 0 = f.ow = fsoo*w by the change-of-variables for-
454
INTEGRA TlON ON MANIFOLDS
mula. But o·w = dCl is zero in a contractible neighborhood. V of the north pole, so that by the Poincare lemma. CI = dP on V. where peO,,-2(v). Now extend P to an (n-2)-form yeO,,-2(S") such that P== y on V and y == 0 outside a neighborhood of cl( V). But then 0.( CI- dy) is compactly supported in R n and do.( CI- dy) = o.dCl == w. "
7.5.17 L....
m..
Let M he a compact connected If-manifold. Then H n ( M) is at most one-dimensional.
Proof.
Let
{( u,. cr, n. i =
I ..... p, where cr,: U,
--+
B = {x
E
R" I II xII
O. This agrees with the common experience that increasing the surrounding pressure on a volume of fluid causes a decrease in occupied volume and hence an increase in density. Many gases can often be viewed as satisfying our hypotheses. with p = ApT where A and y are constants and y ~ I. Cases A and B above are rather opposite. For instance. if p == Po is a constant for an incompressible fluid. then clearly p cannot be an invertible function of p. However. the case p'" constant may be regarded as a limiting case p'( p) -0 00. In case B. p is an explicit function of p. In case A. p'is implicitly determined by the condition div II == O. Finally, notice that in neither case A ~r B is the possibility of a loss of total energy due to friction taken into account. This leads to the subject of viscous fluids. not dealt with here. Given a fluid flow with velocity· field lIe x. t). a streamline at a fixed time is an integral curve of II; i.e.• if xes) is a streamline parametrized by s at the instant t. then xes) satisfies dx
ds =u(x(s),t).
t fixed.
On the other hand. a trajectory is the curve traced out by a particle as time
492
APPLICA TIONS
progresses, as explained at the beginning of this section: i.e., is a the differential equation dx dt =u(x(t),t)
~olution
of
with given initial conditions. If u is independent of t (Le., au/at = 0), then, streamlines and trajectories coincide. In this case, the flow is called stationary or steady. This condition means that the "shape" of the fluid flow is not changing. Even if each particle is moving under the flow, the global configuration of the fluid does not change. The following criteria for steady solutions for homogeneous incompressible now is a direct consequence of Euler's equations, written in the form au~ / at + P( L.u~) = 0, where P is the Hodge projection to the co-closed I-forms.
'.2.2 Proposition. Let u, be a solution to the Euler equations for homogeneous incompressible flow on a compact manifold M and cP, its flow. Then the following are equivalent: (i)
_, is a steady flow (i.e., (a_/at),. 0).
(ii) cp, is a one-parameter group: CPt+. == cP, 0 CPs' (iii) is an exact I-form. (iv) is an exact I-form.
L.o-t ;.o""t
It follows from (iv) that if U o is a harmonic vector field; i.e.•
Uo
satisfies
aut == 0 and ""t '"' O. then it yields a stationary flow. Also. it is known that
there are other steady flows. For example. on a closed two-disk, with polar coordinates (r.O).u==f(r)(a/aO) is the velocity field of a steady flow because u·Vu=-vp.
where p(r,O)= io'f 2 (s)sds.
Cleatiy such a _ need not be harmonic. We have seen that for compressible ideal isentropic flow. the total energy fM( ipl_1 2 + pw)tip is conserved. We can refine this a little for stationary flows as follows.
1.2.3 Bernoulli'. Theorem. For stationary compressible ideal isentropic flow. with p a Junction of P.
ilul 2 + f!!£. p
==
ilul 2 + w + I!..p
is constant along streamlines where f dp / f' = w + p / p denotes a potential for the one form dp/p. The same hold.f for statIOnary' homogeneous (p == constant
FLUID MECHANICS
·493
in space'" Po) incompressible flow with Jdp /p rep/aced by p /Po. If body forces deriving from a potential U are present i.e .. bb ... - dUo then the conserved . qutlntity is
Proof. Since (L."b)." =
d("b(,,».,,:
for stationary ideal compressible or incompressible homogeneous nows we have
so that
=
iJ"b I.s, _·,,(x(s»ds=O iJs S2
since x'(s)'" ,,(x(s». • The two-form II) = tbl b is called vorticity. (In R) we can identify II) with curl ".) Our assumptions so far have precluded any tangential forces and thus any mechanism for starting or stopping rotation .. Hem:e. intuitively. we might expect rotation to be conserved. Since rotation is intimately related to the vorticity. we can expect the vorticity to be involved. We shall now prove that this is so. . Let C be a simple closed contour in the nuid at t = 0 and let C, be the contour carried along the now. In other words.
where fIi, is the nuid now map. (See Fig. 8.2.3.) The circulation around C, is defined to be the integral
494
APPLICATIONS
c,
-,----...., Figure •• 2.3
8.2.4 Kelvin Circulation Theorem. Let M be a manifold and I c M a smooth closed loop. i.e., a compact one-manifold. Let II, solve the Euler equtltions on M for ideal isentropic compressible or homogeneous incompressible flow and I( t) be the image of I at time t when each particle moves under the flow 'P, of "'; i.e., 1(1) = 'P,(/). Then the circulation is constantin time; i.e .•
df
-
dt
IIb=O
1(/1'
•
/'roof. Let 'P, be the now of ",. Then I(I)-'P,(I). and so changing variables.
which becomes, on carrying out the differentiation,
However, L."b + a"b/ at is exact from the equations of motion and the integral of an exact form over a closed loop is zero. • In practical nuid mechanics. this is an important theorem. One can obtain a lot of qualilative information about specific nows by following a • closed loop in time and using the fact the circulation is constant. We now use Stokes' theorem. which will bring in the vorticity. If E is a surface (a two-dimensional submanifold of M) whose boundary is a closed
FLUID MECHANICS
495
"
Figure 8.2.4 contour C. then Stokes' theorem yields
See Fig. 8.2.4. Thus. as·.a corollary of the circulation theorem. we can conclude:
8.2.5 Helmholtz' Theorem.
Under the hypotheses of 8.2.4, the flux of vorticity across a surface moving with the fluid is constant in time. We shall now show how 10) and'll = Io)/p are Lie propagated by the flow.
8.2.8 Propoaltlon. For isentropic or homogeneou.r incompressible llow. we have (i)
iIlo)
-+L 10)=0 and
al
•
ii'll +L ... - ... div(u)=0
at
,,""
called the vorticity-stream equatiOll and
(ii) where 'II,(X) = 'II(x. t) and J( 'P,) is the Jacobian of 'P" . Proof. Applying d to Euler's equations for the two types of nuids we get , the vorticity equation: iIlo)
iii + L.Io)=O.
Thus
-a'll + L ilt
•
'II = -I ( -alo) + L 10) ) - -10) ( -iIp + dp' u ) p ilt • p2 ilt = - -'II (il -p
P iIt
='II divu by conservation of mass.
+ d P . u + Pd'IV.II) + 'II d'IV U
496
A PPLICA TlONS
From aw/at + L.w - 0 it follows that (a/at)(tp;ow,) - Wo. Since tp;op, - Po/J(tp,) we also get tp,."" = J(tp,)",o' •
0 and so tp,·w,
In three dimensions we can associate to '" the vector field equivalently) ie!' ==",. Thus t = (curl ,,)/p.
t=
*'" (or
'.2.7 Corolla". If dim M - 3. then t is transported as a wctorby tp,: i.e .•
t, = tp,.to or Proof. co
t, (tp,( x») =
Dtp, (x H, (0).
tp;o1l, = J( tp, )110 by 8.2.6, so tp;oi,,!' = J( tp, )i,,,!'. But tp;oi,,!' = ;"~,,tp;o!, Thus i.,~,,!, = i,,, .... which gives tp,·t, = to. •
i,,~e/( tp, )....
Notice that the vorticity as a two-form is Lie transported by the flow but as a vector field it is vorticity /P. which is Lie transported. Here is another instance where distinguishing between forms and vector fields makes an important difference. The flow tp, of a fluid plays the role of a configuration variable and the velocity field" plays the role of the corresponding velocity variable. In fact. to understand fluid mechanics as a Hamiltonian system in the sense of Section 8.1. a first step is to set up its phase space using the set of all diffeomorphisms tp: M --+ M (volume preserving for incompressible flow) as the configuration space. The references noted at the beginning of this section carry out this program (see also Exercise 8.21). Exercl...
"·V"
In classical texts on fluid mechanics, the identity = ! v(,,· ,,)+ x ")X,, is often used. To what does this correspond in this section? 8.2B A flow i!o ~alled potential flow if "b = dIP for a function tp. For (not necessarily stationary) homogeneous incompressible or isentropic flow prove Bernoulli's law in the form aIP/at+11"12+ Jdp/p= constant on a streamline. 8.2C Complex variables texts "show" that the gradient of tp(r. 8) = (r + l/r)cos8 describes stationary ideal incompressible flow around a cylinder in the plane. Verify this in the context of this section. 8.2D Translate 8.2.2 into vector analysis notation in R J and give a direct proof. 8.2E Let dim M = 3. and assume the vorticity w has a one-dimensional kernel. (i) Using Frobenius' theorem. show that this distribution is integrable. (ii) Identify the one-dimensional leaves with integral curves of t (see 8.2.7)-these are called l'Ortex line.~. (iii) Show that vortex lines are propagated by the flow. 8.2A
(V
FLUID MECHANICS
497
8.2F
Assume dim M - 3. A vortex tube T is a closed oriented two-manifold in· M that is a union of vortex lines. The .~trenRth of the vortex tube is the nux of vorticity across a surface }: inside T whose boundary lies on T and is transverse to the vortex lines. Show that vortex tubes are propagated by the now and have a strength that is constant in time. 8.2G Let f; R) -+ R be a linear function and g: S2 -+ R its restriction to the unit sphere. Show that dg gives a stationary solution of Euler's equations for now on the two-sphere. 8.2H (Stream Functions) (i) For incompressible now in R 2, show that there IS a function ~ such that u l = iNliJy and u 2 = - iJI/Jlih. One calls ~ the stream function (as in Batchelor (1967» (ii) Show that if we let • I/J = I/J dx 1\ dy be the associated two form. then u b = ,; • y,. (iii) Show that u is a Hamiltonian vector field (see Section 8.1) with energy I/J (i) directly in R 2 and then (ii) for arbitrary twodimensional Riemannian manifolds M. (iv) Do stream functions exist for arhitrary fluid flow on T 2? on S2?
8.21
(v) Show that the vorticity is '" = fl. I/J. (Clebsch Variables; Clebsch (1359]). Let '!tbe the space of functions on a compact manifold M with the dual space ,?r*. taken to be densities on M; the pairing between f E ~ and p E \~* is (f. p) =
/"dp·
(i) On the symplectic manifold ':t x \~* x \'f x \'f* with variables (a. J\. p.. p). show that Hamilton's equations for a given Hamiltonian Hare . 8H ~= - 8a . wher~
. 8H p= - 8p. .
6H 16~ is the functional derivative of H defined by 6H . . ( 8~ ,~) = DH(~)'~'
(ii)
In the ideal isentropic compressible nuid equations: set M
=
pub, the momentum density. If dx denotes the Riemannian volume form on M, identify the density o( x) dx E ~* with the
function O(X)E~~ and write M=-(pdp.+~da)dx. For momentum densities of this form show that Hamilton's equations for the variables (a, ~,p., p) imply the Euk'r equation and the equation of continuity.
498
APPLICA TIONS
8.3 ELECTROMAGNETISM Classical electromagnetism is governed by Maxwell's field equations. The form of these equations depends on the physical units chosen. and changing these units introduces factors like 4fT, c = the speed of light. EO = the dielectric constant and Po = the magnetic permeability. This discussion assumes that EO' Po are constant; the choice of units is such that the equations take the simplest form; thus c = EO = Po = I and factors of 4fT disappear. We also do not consider Maxwells equations in a material, where one has to distinguish E from D. and B from H. Let E. B, and J be time dependent CI-vector fields on R J and p: R} x R ..... R a scalar. These are said to satisfy Maxwell's equation.f with charge density p and current density J when the following hold: divE = p
(Gauss's law)
(I)
divB = 0
(no magnetic sources)
(2)
(Faraday's law of induction)
(3)
(Ampere's law)
(4)
aB at
curIE+-=O
BE
curIB--=J
at
E is called the electric field and B the magnetic field. The quantity Igp dV = Q is called the charge of the set Sl c R 3. By the classical Gauss theorem, (I) is equivalent to
{ E'ndS= IpdV=Q
Jill!
(5)
1/
for any (nice) open set Sl c R'; i.e .. the electric flux out of a dosed slIrface equals the total charge inside the .~Ilrface. This generalizes Gauss' law for a point charge discussed in Section 7.3. By the same reasoning. Eq. (2) is equivalent to
(6)
{ B·IIdS=O. Jill!
That is. the magnetic flux out of any closed surface is zero. In other words there are no magnetic sources inside any closed surface. By the classical Stokes theorem, (3) is equivalent to
/, E·ds = l(curIE)'"dS = as
s
~ at iB,"dS
(7)
ELECTROMAGNETISM
499
for any closed loop as bounding a surface S. The quantity !,1SE·tb is called the voltage around as. Thus, Faraday's law of induction (3) says that the voltage around a loop equals the negative of the rate of change of the magnetic flux through the loop. Finally, again by the classical Stokes theorem, (4) is equivalent to
1 il·tb= !.(curIB)'nds=: !.E.ndS+ !.J.ndS. as
t s
s
s
(8)
Since !sJ'ndS has the physical interpretation of currelll, this form states that if E is constant in time, then the magnetic potential difference !;lS B· d, around a loop equals the current through the loop. In general. if E varies in time. Ampere's law states that the magnetic potential difference around a loop equals the total current in the loop plus the rate of challge (If electric flux through the loop. We now show how to express Maxwell's equation!> in terms of differential forms. Let M = R4 -= {(x. y, z. t)} with the Lorentz metric If on R4 of diagonal form (1.1.1. -I) in standard coordinates (x. y. z,l).
8.3.1 PropolHlon. There is a unique two-form F on R 4, ('alled the Faraday two-form such that (9) Bb= - iii. F.
(10)
ii,
(Here the b iJ Euclideall ill R 3 and the. iJ Lorentzi~n in R4.) Proof
If
F= F,\.dx
1\
+ F"dx
dv 1\
+ ~,dz 1\ dx + F..,dy 1\ dz
dt + Fv,dy
1\
dt + ~,dz
1\
dt,
then (see Example 6.2.14E)• • F = F,\.dz " dt + F,..,dy " dt + Fv:dx
1\
dt
- F"dy" dz - F..,dz" dx - F,.,dx" d.l'
and so - if,F= F"dx
+ F,.,dy + F,.,dz
500
APPLICA TIONS
and
i~
-
• F= Ftldz +
ii,
~,d.v
+ F,.:dx
Thus, F is uniquely determined by (9) and (10), namely F,.., E' dx
1\
dt
+ E 2 d.v 1\ dt + E 3 dz
+ B) dx 1\ dy + B2 dz
1\
1\
dt
dx + B' dy
1\
dz.
•
We started with E and B and used them to construct F, but one can also take F as the primitive object and construct E and B from it using (9) and (10). Both points of view are useful. Similarly. out of p and J we can form the source one -form j = - pdt + J. dx + J 2 dy + J) dz; i.e.,j is uniquely determined by the equations - iil/il') .. p and iil/il,. j = • Jb; in the last relation. J is regarded as being defined on R4.
'.3.2 Propoeltlon.
Maxwell's equations (1)-(4) are equivalent to the
equations
dF= 0 and 3F= j on the manifold R4 endowed with the Lorentz metric.
A straightforward computation shows that
Proof.
dF ... (curl E +
~~), d.v 1\ dz 1\ dt + (curl E + ~~) ,. dz 1\ dx 1\ dt
+ (curl E +
a;:): dx
1\
dy
1\
dt
+ (div B) dx 1\ d.v 1\ dz.
Thus dF = 0 is equivalent to (2) and (3). Since the index of the Lorentz metric is I, we have Ii = • d •. Thus 3F= .d. F= .d( - E,dy 1\ dz - E 2dz
1\
dx - E)dx
+ B. dy 1\ dt + B2 d.r 1\ dt + B)dz = • [ -
(div E) dx
+ (curl B =
(curl B -
1\
dy
1\
dz + ( curl B -
~~
dy
dy
1\
dt)
1\
dz
1\
dt +
~~ ),. dz 1\ dx 1\ dt + (curl B - ~~): dx 1\ 4y 1\ dt]
aE) dx + (curl B - aE) dy at .. ilt ...
+ (curl B -
L
1\
~~): dz -
Thus 3F= j iff (I) and
(4)
(div E) dl
hold. •
ELECTROMAGNETISM
501
As a skew matrix, we can represent F as follows
F-
x
y
[- ~,
8
82 -£'
z 3
0
-8' -£2
- 82 8' 0
- £3
x
E'l £2
Y
£3
;:
0
Recall from Section 6.5 and Exercise 7.5G, the formula
Since Idetgl = I. Maxwell's equations can be written in terms of the Faraday two-form F in components as
(II) and F'·'A
(12)
= -}'.
where F".A ,., 8F,,/8x·, etc. Since a2 ... 0, we obtain
0= a21" = 6j = = • [(
•
d • j = • d( - p dx /\ dy /\ dz + ( • j~) /\ dt)
a;: +diVJ) dx /\ dy'/\ dz /\ dt] = a;: +divJ;
a,
i.e.• 8p/ +divJ = 0, which is the continuity equation (see Section 8.2). Its integral form is, by the classical Gauss theorem dQ dl
=~ipdV=f dt n
J·ndS
,III
for any bounded open set D. Thus the continuity equation says that the flux of the currenl density oul of a closed surface equals the ratc of change of the lotal charge inside the surface. Next we show that Maxwell's equations are Lorentz invariant, i.e., are special-relativistic. The Lorentz group ~ is by definition the orthogonal group with respect to the Lorentz metric If; i.e.,
502
APPLICA TIONS
Lorentz invariance means that F satisfies Maxwell's equations withj iff A*F satisfies them with A*j. for any A E e. But due to Proposition 8.3.1 this is clear since pull-back commutes with d and orthogonal transformations commute with the Hodge operator (see Exercise 6.2D) and thus they commute with 8. As a 4 x 4 matrix, the Lorentz transformation A acts on F by F ...... A* F "" AFAT. Let us see that the action of A E f mixes up E's and B's. (This is the sOurce of statements like: "A moving observer sees an electric field partly converted to a magnetic field.") Proposition 8.3.1 defines E and B intrinsically in terms of' F. Thus, if one performs Ii Lorentz transformation A on F. the new resulting electric and magnetic fields, E' and B' with respect to the Lorentz unit normal A *( a/ at) to the image A (R 3 x 0) in R 4 are given by
(E,)b - - i,. • .!..A*F, 1(B,)b - - i,. • .!.. *A*F ii,
ii,
For a Lorentz transformation of the form x'== {
x-l'I .• r'= l'.Z'=Z.
1- v 2
'
,
t,'= {1-ll . '2 t'\'
(the special-relativistic analogue of an observer moving uniformly along the' x-axis with velocity v) we get
E2 - vB 3 E3 + vB2 ) E'= ( E'
. ';1- v 2
'
';1- v 2
and
We leave the verification to the reader. By the way we have set things up, note that Maxwell's equations make' sense on any Lorentz manifold: i.e., a four-dimensional manifold with a pseudo Riemannian metric of signature ( + , + . + . - ). Maxwell's vacuum equations (i.e.• j = 0) will now be shown to be conJormally invariant on any Lorentz manifold ( M. ,). A diffeomorphism cp: ( M.,) .... (M.,) is said to be conformal if 'cp*, -f'" for a nowhere vanishing function f.
8.3.3 PropoaHlon.t Let FE 02( M). where (M.,) is a Lorentz manifold, salisfy dF = 0 and 8F = j. LeI cp be a conformal diffeomorphism. Then cp* F tSce Fulton. Rohrlich. and Witten (1962) for a review of conformal invariance in physics and the original literature references.
ELECTROMAGNETISM
503
satisfies drp*F= 0 and arp*F= f2rp*j. Hence Maxwell's vacuum equations (with j == 0) are conformally invariant; i.e. if F satisfies them, so does cp* F.
Proof. Since rp* commutes with d. dF = 0 implies d'P* F = O. The second equation implies rp*&F= rp*j. By Exercise 7.5H. we have 8q .• lI'rp*P = rp*l\P. Hence &F= j implies &"'ttrp*F= rp*j= 8"II'/fIt 2 from hoth sides. V
a (divA .
2 fI21(> I(> - - , = -
p- -
at"
at
al(>- ). + -.
at
It is apparent that (16) and (17) can be considerably simplified if one could choose. using the gauge freedom. the vector potential A and the function cP such that divA +
~~ =0.
So. assume one has Ao. CPo and one seeks a function f such that A = An + gradl and cp='Po-afla, satisfy divA + acp/al=o. This becomes. in terms of I. 0= div( At) + grad f) + :, ( CPo _ . acpt) 2 - divA" + -a + V 1I'
a 1.
2 -2 •
at
:~)
506·
APPLICATIONS
( 18)
i.e.,
This equation is the classical inhomogeneous wave equation. The homogeneous wave equation (right-hand side equals zero) has solutionsf(t. x. y, z) -1/I(x - t) for any function 1/1. This solution propagates the graph of 1/1 like a wave-hence the name wave equation. Now we can draw some conclusions regarding Maxwell's equations. In terms of the vector potential A and the function cpo (I) and (4) become
( 19)
which again are inhomogeneous wave equations. Conversely. if A and cp satisfy the foregoing equations and divA + acp / at = 0, then E = - grad cp aA / at and B - curl A satisfy Maxwell's equations. Thus in R 4, this procedure reduces the study of Maxwell's equations to the wave equation, and hence solutions of Maxwell's equations can be expected to be wavelike. We now repeat the foregoing constructions on R4 using differential forms. Since dF- 0, on R4 we can write F= dG for a one-form G. Note that F is unchanged if we replace G by G + df. This again is the gauge freedom. Substituting F .. dG into 8£ = j gives 8dG = j. Since 6 = d8 + 8d is the Laplace-DeRham operator in R 4 , we get 6G = j - d8G. (20) Suppose we try to choose G so that 8G = 0 (a gauge condition). To do this, given an initial Go' we can let G = Go + df and demand that
0= 8G = 8Go + 8df = 8Go + 6f so f must satisfy 6f = - 8Go. Thus, if the gauge condition
6f= -8Go
(21 )
holds, then Maxwell's equations become
6G=j.
G-
(22)
Equation (21) is equivalent to (18) and (22) to (19) by choosing Ab + "dl (where b is Euclidean in R 3 ).
EL ECTRt;JMAGNE TISM
507
BOX B.3A MAXWELL'S VACUUM EQUATIONS AS AN INFINITE-DIMENSIONAL HAMILTONIAN SYSTEM
We shall indicate here briefly how the dynamical pair of Maxwell's vacuum equations (3) and (4) with J = 0 are a Hamiltonian system (see Box 8.1 A). (A proper understanding of the "constraints" div E = p.divB = 0 requires a procedure called reduction; we refer the interested reader to Marsden and Weinstein (1982].) As' the configuration space for Maxwell's equations. we tak,e the space ~I of vector potentials. (In more general situations. one should replace ~I hy the set of connections on a principal hundle over configuration space.) The corresponding phase space is then the contangent bundle .,..~ with the canonical symplectic structure. Elements of .,..~ may be identified with pairs (A. Y) where Y is a vector field density on R). (We do not distinguish Yand Yd.t.) The pairing between A's and Y's is given by integration. so that the canonical symplectic structure Co) on .,..~ is given by
with associated Poisson bracket (2) where SF/SA is the vector field defined by D... F(A. Y)·A'=
f :~ ·A'dx
with the one-form SF/SY defined similarly. With the Hamiltonian (3) Hamilton's equations are easily Computed to be
ay =
-
a,
-curlcurlA
and
aA
-
at
... Y.
(4) .
If we write B for curl A and E for· - Y. the Hamiltonian becomes the
508
APPLfCA TIONS
usual field energy (5)
Equation (4) implies Maxwell's equations
aE = curl Band -aB = - curl E at at'
(6)
-
and the Poisson bracket of two functions F(A. E).G(A. E) is . (F,G}(A,E)=-
8F 6G 6G 8F) 1 (6A' 8E- 6A' 6E d:c
(7)
H'
We want to express this Poisson bracket in terms of E and B = curl A when F and G are functions of Band E. For this. we need to compute 8F/8A·8G/8E in terms of Band E. Let f be a function of Band E and let F(A.E)=Fc.B.E). where B=vxA. Let L he the linear map A ...... V x A. We have Df'( A). 6G =
6E
f 6A6F . 6A
6(; /. t.\
by definition of 6F/6A and. by the chain rule.
f BF 6B
8G = DF(A)· -BG = (DF{B)o(DL(A»)·6E 6E
-
. curl ( -BG
6E
) dx
since DL(A) = Land 8G/6E = M1/6E. Thus the Poisson bracket (7) becomes
--
{F.G}(B. E) = -
-
f( 8f 6G 6B ·curl 6E -
6G f( 6f 6E ·curl 6B -
6G 6B
6f) ·curl 6E dx
6G
6f ) 6E ·curl 6B dx
(8)
(Using integration by parts we see that f X· curl Y dx = Jy. curl X dx). This bracket was found by Born and Inreld [1935] by a different method. With respect to the Hamiltonian
f
II ( B. E) = ! (II BII2 + II EII2 ) dx.
ELECTROMAGNETISM
509
(3) and (4) are easily verified to be equivalent to the Poisson bracket equations
f- (f. if) (For one implication use the chain rule. for the other. usc the functions Fe B, E) "7 IBi dx and Fe B. E5 = 1£' dx).
Exercf... Assume that the Faraday two-form F depends only on t - x. (i) Show that dF= 0 is then equivalent to B) = £2. B 2 ... - £1, BI ... 0. (ii) Show that IF= 0 is then equivalent to Bl = £2. B2 = - £1, £1 =0. These solutions of Maxwell's equations are called plane electromagn~tic waves; they are determined only by £2. E) or B2. B). respectively. . , IUR Let" = ii/ ilt. Show that the Faraday two-form F E 02(R 4) is given in terms of E and B E ','X (R 4) by F ... lIb A £~ - • (u b A B~). 8.3(' Show that the Poynting vector satisfies
8.3A
where" =
a/at and E. B E ~·X (R4)
8.3D Let (M.,) be a Lorentzian four-manifold and" E ' \: ( M) a timelike unit vector field on M; i.e., ,(u. u) = -I. (i) Show that any CI E 02( M) can be written in the form
(ii) Show that if ;.CI = 0, where ClE 02( M)("CI is orthogonal to u"), then • CI is decomposable. i.e. • CI is the wedge product of two one-forms. Prove that CI is also locally decomposable. (Hint: Use the Darboux theorem.) 8.3E ,The field of a stationary point charge is given by
E=~
47Tr J '
B=O.
where r is the vector xi + yj + zk in Rl and r is its length. Use this
510
APPLICA TlONS
and a Lorentz transformation to show that the electromagnetic field produced by a charge e moving along the x-axis with velocity I' is
and. using spherical coordinates with the x-axis as the polar axis,
8.3F
(the magnetic field lines are thus circles centered on the polar axis and lying in planes perpendicular to it). (c. Misner. K. Thome. and J. Wheeler [I973J.) The following is the Faraday two-form for the field of an electric dipole of magnitude PI oscillating up and down parallel to the z-axis. E=Re
. . [2cos8 (I?-? iW) d,/\dt {Ple,wr-u.J/
. (1,J
2
iw - w ) ,d8/\dt +sm8 - - ,2 ,
(2)
iw +sin8 ( -?--;d,/\,d8
]} .
Verify that dE = 0 and 3E = O. except at the origin. 8.30 Let the Lagrangian for electromagnetic theory be
Check that ae; agi ) is the stress-energy-momentum tensor T" (see Hawking and Ellis (1973. sec. 3.3». 8.3H Show that the Bom-Infeld Poisson bracket{.} on functions of E and B defined by equation (8) in Box 8.3A defines a unique (weak) symplectic structure on the space {( E. B) I div E = 0 and div B = O}. Verify that the Hamiltonian vector field for the energy H gives the vacuum Maxwell equations for aE; ill and ilB; ill.
THERMODYNAMICS, CONSTRAINTS, AND CONTROL
. 511
8.4 THERMODYNAMICS, CONSTRAINTS, AND CONTROL The applications in this final section all involve the Frobenius theorem in some way. Each example is necessarily treated briefly. but hopefully in enough detail so the interested reader can pursue the subject further by utilizing the given references. We begin with a discussion of equilibrium thermodynamics. This subject has a rather bad reputation for being sloppily taught and full of mysterious jargon. This is partly because the basic physical laws involve notions of. irreversibility. whereas laws for particle motion are reversible. This apparent contradiction and the difficulty in properly explaining it contributes to the problem. On the matbematical side. the formalism can be made clean and precise. It is the latter aspect we wish to describe. For additional reading. three sources we find useful are Sommerfeld (1964). Truesdell (1969]. and Hermann (1973). To motivate the formalism we need a little physical background and intuition and so begin with it. A. thermodynamic system is usually thought of as a continuum or a set of particles. possessing internal energy and interacting with its surroundings by either delivering or absorbing heat and thereby doing work. For instance. the particles' could be molecules in an internal combustion engine. Let M represent the set of thermal equilibrium states of the given system. i.e.• the states at which the system does not exchange energy (and thus heat) w,ith its surroundings. In the mathematical formulation of thermodynamics the basic assumption is that M. called the phase space. is a given connected finitedimensional manifold. such as Rn. The first principle of thermodynamics was first formulated by Rudolf Clausius around 1850. building on earlier work of ('Iapeyron. Carnot. and Thompson (Lord Kelvin) as follows. "In all cases in which work is produced by the agency of heat: a quantity of heat is consumed that is proportional to the work done; and conversely by the expenditure of ~n equal quantity of work an equal quantity of heat is produced." Thus, the rust principle is just the principle of conservation of energy for a closed system (a totally isolated thermodynamic system). Let E denote the internal energy of the system. W the work done. and Q the heat delivered by the system. Regard W and Q as given one-forms on M and E a given function on M. Then they are said to satisfy the first principle when Q=W+dE.
(In many texts one sees this written as IJQ = IJW + 8E.) A thermodynamic system often utilizes three additional real-valued functions on M: the, volume V. the temperature T. and pressure P. The work is expressible in
512
APPLICATIONS
terms of the volume and pressure as W = P dV. An equation of state consists of functions relating P and E to Vand T: P = P( V. T). E = E( v. T). These equations of state are specific functions whose exact form depends on the nature of the particular substance under investigation. The second principle 0/ thermodynamics states that IjT is an integrating factor for Q. In other words, QIT is an exact one-form (at least locally). If Q IT - dS, the function S is called the entropy of the system. Thus the second principle states that locally I I dS- r Q - r(dE
+ PdV).
We shall now reconcile this formulation involving the Frobenius integrability condition on the Pfaffian equation Q = 0 to the classical formulation of Kelvin: "It is impossible. by means of an inanimate material agency, to derive a mechanical effect from any ponion of matter by cooling it below the temperature of the coldest of the surrounding objects." That is. it is impossible in a closed system to transfer heat from a hot to 'a cold body without making other changes as well. t This reconcilation will be done by means of the Caratheodory inaccessibility theorem. A path y: [0,1) -. M is called a quasi-static adiabatic path of the closed thermodynamic system if Q. dyldt,.. 0 (the one-form Q acting on the vector dyldt). This says that along y no heat is added. If m EM is a fixed thermal equilibrium state of the system and if all other points of M can be connected to m by a quasi-static adiabatic path-i.e .. any thermal equilibrium can be obtained from any other one without addin/{ heat. then the work done by a closed system would be entirely due to its intrinsic internal energy (such as by molecular movements. atomic vibrations. etc.). Thus one could get work done in a machine without any heat transfer: a perpetual motion machine. The following theorem of Caratheodory [1909] states that this is impossible, recovering Kelvin's formulation of the second principle. Here we see the fundamental difference between the first and second principle. The first principle allows all possibilities consistent with conserving energy, whereas the second principle gives serious restrictions to these possibilities.
1A.1 C8ralheodo..,'. lnecceulbillty Theorem. Let M be a smooth n-manifold without boundary and Q E Ol( M) a nowhere l'ani.vhing one-form.
t
Tbcrc is also a formulation of 1M second law for dynamic process that is related to
die popular statcllleDt that '"eatropy increases with time- or .. things tmel to become rmdom." The usual s..tement is called 1M Clausius- Duhem ineqvtllity, but it is IOl'IICWhat coatrovenial; lee Manden and Huahes (1983) for a discussion.
,THERMODYNAMICS, CONSTRAINTS, AND CONTROL
513
Then the following are equivalent: (i) (ii)
Q = 0 is an integrable Plalfian system; i.e .• local(v Q = TdS lor function.s Tand S. For each Xo e M there exists an open neighborhood V of Xo on M such that each neighborhood W of Xo. W c V. contains a point x e W that cannot be connected to Xo by a ( pieceWise smooth) quasi-static adiabatic path.
If Q - 0 is integrable. then the distribution defined hy its annihilator defines a foliation by the Frobenius theorem. (Sec Section 4.4 and Box 6.4A.) In particular. locally the submanifolds of the foliation are defined hy' S = constant. where S is the entropy of the system (recall that TdS = Q). But as ,Q is nowhere zero. these (n -I)-dimensional submanifolds do not intersect. so it is impossible to reach all points in the given neighborhood by curves lying only on these submanifolds. Thus statement (i) implies (ii). To prove the converse. let V be a neighborhood of xI) e M and let Proof.
E=
{"e T.MIQ(x)·,,= O. x e V}
be the annihilator of Q in TV. If we show that the subhundle E of TV is involutive. then by the Frobenius theorem. Q .. 0 is integrable in V and hence in M. Let E' be the smallest involutive subbundle of TV containing E and let A = (K e Vlx cannot be joined to Xo by a path y: [0. I) ..... V satisfying Q(x)·dy/dt .;. O}. By (ii). A .. V. so that E' is not equal to TV. In partiCular. the fiber dimension of E' is """ n - 1. But E':::> E and E has fiber dimension n - 1. so that E has fiber dimension n -I and hence E = E'. Consequently E is involutive. • • Thermodynamics has a connection with the symplectic geometry of Section 8.1 through the notion of a "Lagrangian submanifold." This treatment is beyond the scope of this book. The reader may consult Hermann (1973). Marsden and Hughes [1983]. Abraham and Marsden (1978) and Oster and PereISGn (1973) for inforniation and further references. We tum next to the subject of holonomic constraints in Hamiltonian systems. A Hamiltonian system as discussed in Section 8.1 can have a "~to available points in phase space. Such a condition imposed that lill' , condition is a constraint. hi! .:xample. a ball tethered to a string of unit length in R3 may be considered to be constrained only to move on the unit sphere S2 (or possibly interior to the sphere if the string is collllpsible). If the phase space is 1'*Q and the constraints are all derivable from cOnstraints imposed only on the configuration space (the q ·s). the constraints are ~lIed
514
APPLICA TlONS
holonomic. For example. if there is one constraintf(q) = 0 for f: M -+ R. the constraints on T* M can be simply obtained by differentiation: df = 0 on T* M. If the phase space is TM. then the constraints are holonomic iff the constraints on the velocities are saying that the velocities are tangent to some constraint manifold of the positions. A constraint then can be thought of in terms of velocities as a subset E C TQ. If it is a subbundle. this COIUlrainl is IhllS holonomic iff it is integrable in the sense of Frobenius' theorem. Constraints that are not holonomic. called nonholonomic constraints. are usually difficult to handle. Holonomic constraints can be dealt with in the .enle that one understands how to modify the equations of motion when the constraints are imposed. by adding forces of constraint. such as centrifugal force. See, for example Goldstein (1980. ch. I]. and Abraham and Marsden (1978. sec. 3.7]. We shall limit ourselves to the discussion of two examples of nonholonomic constraints. A classical example of a nonholonomic system is a disk rolling without slipping on a plane. The disk of radius a is constrained to move without slipping on the (x. y)-plane. Let us fix a point P on the disk and call fJ the angle between the radius at P and the contact point Q of the disk with the plane. as in Fig. 8.4.1. Let (x. y. a) denote the coordinates of the center of the disk. Finally. if" denotes the angle between the tangent line to the disk at Q and the x-axis. the position of the disk in spal:c is completely determined by (x. y, fJ. fJ». These variables form elemcnts of our configuration space M = R 2 X s' X st. The condition that there is no slipping at Q
..
Flgunt'.4.1
...
THERMODYNAMICS. CONSTRAINTS. AND CONTROL
515
means that the velocity at Q is zero; i.e.•
dx dB -- +a-coscp=O dl dl •
dB. -dy + asm
=
0 and
.v+ alJsinq> =
0
such that IX. Y] is not in E; i.e.• use Frobenius' theorem directly rather than using Pfaffian systems. 8.4E Justify the names wriggle and slide for the vector fields Wand S in the example of Fig. 8.4.2 using the product formula in Exercise 4.20. Use these formulas to explain the following statement of Nelson 11967. p. 35]: "the Lie product of Steer and Drive is equal to Slide and Rotate ( - a/ aq» on 8 - 0 and generates a now which is the simultaneous action of sliding and rotating. This motion is just what is needed to get out of a tight parking spot." 8.4F The word holonomy arises not only in mechanical constraints as explained in this section but also in the theory of connections
TH~RMODYNAMICS. CONSTRAINTS. AND CONTROL
521
(Kobayashi-Nomizu (1963. II. sec. 7.8]). What is the relation between' the two uses? 8.40 In linear control theory (I) is replaced by N
"~t)=X·"'(t)+
L
,-1
p;(t)Y,.
where X is a linear vector field on A" and Y, are mflStant vectors. By using the methods used to prove 8.4.2. rediscover for yourself the Kalman criterion for local controllability, namely. {X'Y,lk = 0.1, .... n - I, i = I, ... ,N} spans A".
APPENDIX
A
The Axiom of Choice and Balre Spaces
This appendix presents. for completeness. supplementary topics in topology that were used in the main text in a few technical proofs. For additional details. see Kelley (1975) and Choquet [1969). THE AXIOM OF CHOICE
One of the most widely used axioms in set theory is the axiom of choice. A.1 AxIom of ChoIce. Let S; be a collection of nonempty sets. Then there exists a function x: ~ -+ USE" S such that X(S) E S for every S E S;.
The function X chooses one element from each S E S; and is called a choice function. Even though this statement seems self-evident. it has been shown to be equivalent to a number of nontrivial statements using other axioms of set theory. To discuss them. we need a few definitions. An order on a set A is a binary relation. usually denoted by .. ~ .. satisfying a a
~
~
a~
522
a (renexivity); band b ~ a implies a = b (antisymmetry). and band b ~ c implies a ~ (" (transitivity)
APPENDix A: THE AXIOM OF CHOICE AND BAtRE SPACES
523
An ordered set A is called a chain if for every a, b EA. a '* h we have a ~ h or b EO a. The set A is said to be well ordered if it is a chain and if every nonempty subset B has a first elqnent; i.e., there exists an element bE B such that b ~ x for all x E B. An upper hound u E A of a chain C c A is an element for which c ~ u for all c E C. Finally a maximal element m of an ordered set A is an element for which there is no other /I E A such that m ~ a. a" m; in other words. x ..,; m for all x E A that are comparable to m. We state the (ollowing without proof.
A.2 Theorem.
Given other axioms of set theory. the lollowin?, statemellts
are equivalent: (i)
(ii) (iii) (iv)
The (/.lciom ol choke. If {A,}, E ' i.v a collection of nonemplY set.r then the product [1, E ,A, = {(x,)lx, E A,} is nonemplY (product axiom). Any set can be well ordered (Zermelo's theorem). If A is an ordered set for which every chain has all upper hound (i.e .• A is induct~/y ordend). then A has at least one maximal element ( Zorn's lemma).
BAIRE SPACES
The Baire condition on a topological space is fundamental to the idea of "genericity" in differential topology and dynamical systems.
· A.3 Definition. Let X be a topological space and A c X a subset. Then A · is called nsidluJl if A is the intersection of a ('Ountable famill' of open dense subsets of X. A space X is called a Bain space if every residual set is dense. A set B c.X is called a first category set if Be U jCn where (:, is c/o.red with int( Cn ) = 0 . A second category set is a sel not of the first category. A set Be X is called nowhere dense if int(cI(B» =0. so that X\A is residual iff A is the union of a countable collection of nowhere dense closed sets. i.e.• if X.\ A is of first category. Clearly. a countable intersection of residual sets is residual. In a Baire space X. if X = U ::".. IC" where the c" are closed sets. then · int( Cn ) ., 0 for some n. For if all int( Cn ) = 0 then 0" = X \ Cn are open. dense. and () jO" == X\ U ::"_ICn - 0 contradicting the definition of Baire space. In other words. Baire spaces are of second Cale1{orV. The following is often useful.
A.4 Propoaltlon. LeI X be a locally Baire space; that is. ('(Ich point x E X has a neighborhood U such Ihal d( U) is a Raire spal'/'. TII('II X is a Baire space.
524
APPENDIX A: THE AXIOM OF CHOICE AND BAIRE SPACES
Proof. Let A c X be residual: Yo
A= nO" II-I
where c1( 0,,) ""' X. Then
A n c1( U ) =
n'"
(0" n d( U )).
II-I
Now On ncl(U) is dense in c1(U) for if II e c1(U) and II e 0. where 0 is an open set. then on U • (lJ and nun 0" • (lJ. Hence cl( U ) c cl( A) and so c1(A)'"' X. •
°
The most important examples of Baire spaces are given by the following theorem. A.S 88lre Category Theorem. pact spaces are Boire spaces.
Complete pseudometric and local(l' com-
Proof. Let X be a complete pseudometric space. Let U c X I}e open and A ... n JOIl be residual. We must show UnA .. (lJ. Now as c1( 0,,) = X. U n 0" .. (lJ and so we can choose a disk of diameter less than one, say VI' such that c1( VI ) C U () 01. Proceed inductively to obtain cl( V,,) c U () 0" n V,,_ I_ where v" has diameter < lin. Let x" e c1(V,,). Clearly (.t ll ) is a Cauchy
sequence. and by completeness has a convergent subsequence with limit point x. Then X E
n I
cI( v"
) and so U n no,,'" (21 : "t I
i.e.. A is dense in X. If X is a locally compact space the same proof works with the following modifications: V" are chosen to be relatively compact open sets. and {XII} has a convergent subsequence since it lies in the compact set cl( VI ). • To get a feeling for this theorem. let us prove that the set of rationals Q cannot be written as a countable intersection of open sets. For suppose Q - () jOlt. Then necessarily each 0 ,~dense in R since Q is and sO CII = R\O" is closed and nowhere .1.' ,,', Since R = Q u U ::'-IC" is a com-
APPENDIX A: THE AXIOM OF CHOICE AND BAIRE SPACES
525
plete metric space (as well as a locally compact space). it is of second category. so some C" should have nonempty interior (hecause intQ = 0,. But this is a contradiction. The notion of category also imposes restrictions on a set. For e"ample in a nondiscrete Hausdorff space. any countable set is first category since, the one-point set is closed and nowhere dense. Hence in such a space every second calegory .fel i.t uncounlahie. In particular. nonfinite complete p~udo metric and loCally compact space... are uncountable.
APPENDIX
B
The Three Pillars of Linear Analysis
We give here the classical proofs of the three fundamental theorems of linear analysis in the setting of Banach spaces. See Chapter 2 for a discussion and applications. 8.1
Hahn-Benach Theorem. ut E he a real or complex vector space. -0 R a seminorm and F § E a subspac·e. If I E F* satisfies I/(")! ~ lIell for all e E F. then there exists a linear map i: E -0 R (or C) suc·h Ihal il F ... I and lj(e)1 ~ lie II for all e E E.
11·11: E
Proof. Real Case. First we show that I E F* can be extended with the given property to FCDspan (eo}. for a given eo E F. For e,.e2 E F we have
I( e. ) + I( e2) - I( e.
+ e2) " lie. + e211 " lie, + eo II + lI e2- eoll.
so that and hence
Let a E R be any number between the sup and inf in the preceding expression and define FEB span (e.} -0 R hy j( e + leo) = I(e)+ tc/. It is
i:
526
APPENDIX B: THE THREE PILLARS OF LINEAR ANAL YSIS
clear that i is linear and that il F = /. To show that note that by the definition of a,
1ft f' + lefl)j "
527
lie + leo II.
/( e2 )-lIe 2 - e"II" a" lIe , + e"l1- /(e , ). so that multiplying the second inequality by t ~ 0 and the first by t < 0 we get the desired result. Second, one verifies that the set :;={(G.g)IFcGcE,G is a subspace of E. If E G*, IfIF = /. and Ig(e)I" lIell for all e E (i) is inductively ordered with respect to the ordering (G"lf,)" (Gz.lfz)
iff G , cG2 .lf2IG,=If,.
Thus by Zorn's lemma (Appendix A) there exists a maximal element (f;,. I,,) of:;. ; Third. using the first step and the maximality of (F". /n)' one concludes that £., = E. Complex Case. Let / = Re / + ilm / and note that complex linearity' implies (1m 1)( e) = - (Re /)( ie) for all e E F. By the real case. Re / eXJends to a real linear continuous map (Ref) -: E ..... R. such that I(Ref) - (e)I" lIell for all e E E. Definei: E ..... C by fie) = (Ref) - (e)- i(Ref) - (ie) and note that / is complex linear and il.F = /. To show that Ifte)I" lIell for all e E E. write fie) = lfie)lexp(i(l). so complex linearity of implies ft e' exp( - i(l» E R and hence
i I.il e)1 = it. e·exp( -
i(l» =.(Re f) - (e·exp( - i8»
.;;lIe·exp( - i(l)1I = lIeli . • The foliowing is the form of the theorem that
was u)o~'d
in Section 2.2.
B.2 Corollary. Lei (E.II·IO be a normed space. F c E a subspace and / E F* (/he topological dual). Then there exists
i
E E* such lhat
ilF =
/ and
lIill""' II/II· We can assume
/-0. Then lIIelil-II/liliell is a norm on
E and the pr~ing theorem we 'get a linear map i: E ..... R (or C) such that II F""' / and lIe e) I .;; IIlelll for all e E E. This says that II ill.;;1I1I1 and since i extends /. it follows that 11111 " II II; i.e.• II ill = II/II and E E*. • Proof.
lI(e)I" 1I/1I'lIell-lIIelll for all e E F.
i
Appl~ing
i
B.3 Open Mapping Theorem of Banach and Schauder. Let E and F he Banach spaces and Suppo.fe A E L ( E, F) is onto. Then A is an open mapping.
528
APPENDIX B: THE THREE PILLARS OF LINEAR ANAL YSIS
Proof. To show that A is an open mapping. it suffices to prove that A(c1( D1(0))) contains a disk centered at zero in F. Let r> O. Since E = U:'_IDM(O). it follows that F= U'::'_I(A(D",(O» and hence U :'_Icl( A( D",(O») - F. Completeness of F implies that at least one of the c1( A( D",(O») has nonempty interior by the Baire category theorem (A.5). The mapping , E E ... n, E E being a homeomorphism. this says that cI(A(D,(O» contains some open set V c F. We shall prove that in fact the origin or F is in int(cl(A( D,(O»». Continuity or ('1"2) E E x E""I -'2 E E assures the existence of an open set U c E such that U - U = {'I '21'1.e2 E U} c D,(O). Thus eI( A( DI(O))):::> cI( A(U)- A(U»:::> cI(A(U»cI(A(U»:::> V-V. But V - V = U r'" ,( V - ,,) is open and clearly contains o E F. It follows that there exists a disk D, (0) c F such that D, (0) c eI( A ( D,(O)). Now let E" = 1/2'" I. II - 0.1.2 ..... so that 1= 1:;:'_of". By the forego. ing result there exists an 11" > 0 such that D~.. O. eo E E such that q>( e) ~ M. for all , E \."I( D,( en))' where M > 0 is some constant. . For each i E I. and lIell = I. we have IIA,(re + en~1 ~ q:>(re + eo)~ M. so that
~ (M+q>(en))/r.
i.e..
IIA,II~(M+q:>('n»)jr
for all i E I. •
APPENDIX
c
Unbounded and Self-adjoint Operatorst
In many applications involving differential equations. the operators one meets are not defined on the whole Banach space E and are not continuous. Thus we are led to consider a linear transformation A: D,. c E ~ I:.' where DA is a linear subspace of E (the domain of A). If D,. is dense in E. we say A is densely defined. We speak of A as an opera/or and this shall mean linear operator unless otherwise specified. Even though A is not usually continuous. it might have the important property of being closed. We say A is closed if its graph rA
rA -
{(x. Ax) E EX Elxe DA }
is a closed subset of E x E. This is equivalent to (x"EDA.x,,-XEE
and
implies (xEDA
Ax,,-YEE)
and
Ax=)').
An operator A (with domain D,. ) is called c/o.fable if d( fA I. the closure. of the graph of A. is the graph of an operator. say. A-: We call the closure of A. It is easy to see that A is closable irr (Ix. ED•. xn ~ 0 and Ax. -0)')
..r
tThis appendix was written in collaboration with P. Chernorr.
530
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
531
implies y == O}. Clearly A- is a closed operator that .is an extension of A; i.e.• D.i"~ D,4 and A-- A on D,4. One writes this as A-~ A. The closed graph theorem asserts that an everywhere defined closed operator is bounded. (See Section 2.2.) However. if an operator is only densely' defined. "closed" is weaker than" bounded." If A is a closed operator. the map x ..... (x. Ax) is an isomorphism between D,4 and the closed subspace r,4' Hence if we set
V,4 becomes a Banach space. We call the norm III ilion VA the graph norm. Let A be an operator on a real or complex Hilbert space H with dense domain D,4' The adjoint of A is the operator A* with domain D,4o derined as follows:
D,4o={yeHlthereisazeH
such that (Ax.J')=(x.z) for all x e I~~ }
and A*:D,40---H.
...
J......
From the fact that DA is dense we see thaI A* is indeed well defined (there is at most one such z for any J' e H). It is easy to ~ee that if A ::'l B then B*::> A*. If A is everywhere defined and bounded, it f(lll()w~ from the Riesl representation theorem (Box 2.2A) that A* is everywhere ddined: moreover it is not hard to see that, in this case, IIA*II = IIAII. An operator A is symmetri(' (Hermitian in the complex case) if A* ~ A: i.e., (Ax. y) = (x. Ay) for all x. y e D,4. If A* = A (this includes the , condition D,4o = D,4)' then A is -called self-adjoint. An everywhere defined symmetric operator is bounded (from the closed graph theorem) and so is self-adjoint. It is also easy to see that a self-adjoint operator is closed. . One must be aware that. for technical reasons, it is the notiQn of self-adjoint rather than symmetric. which is important in applications. Correspondingly. verifying self-adjointness is often difficult while verifying symmetry is usually trivial. ' . Sometimes it is useful to have another concept at hand. that of essential self-adjointness. First. it is easy to check that any symmetric operator A is dosahle. The closure A- is easily seen to be symmetric. One says that A is euentia[(r self-adjoint when its closure A-is self-adjoint. A related concept is this: Let A be a self-adjoint operator. A dense subspace C cHis said to be a core of A. if C c D... and the closure of A
632
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
restricted to C is again A. Thus if C is a core of A one can recover A just by knowing A on C. We shall now give a number of propositions concerning the foregoing concepts. which are useful in applications. Most of this is classical work of von Neumann. We begin with the fo))owing.
C.1 PropoeitIon. Let A be a closed symmetric operator 0/ a complex
Hilbert space H. 1/ A Is self-adjoint then A + "AI is surjective lor every complex number A with 1m A ... 0 (I is the identity operator). Conversely, i/ A is symmetric and A - il and A + il are both surjective then A is self-adjoint.
Proof. Let A be self-adjoint and A= a + iP, P '* O. For x E
DA we have
II(A + A)x1l 2= II(A + a)xll2 + ip(x, Ax) - iP(Ax. x) + P21ixll 2
-II(A + a)xll2 + P211xll2 ~ P211xll2, where A
+ A means A + AI. Thus we have the inequality II(A
+ A)xll ~ (ImA)lIxll.
(I)
Since A is closed, it fonows from (I) that the range of A + A is a closed set for 1m A .. O. Indeed, let y" - (A + A)x" ... y. By the inequality (I), IIx" - x ... 11 < II y" - Y... II/IIm All so x" converges to, say, x. Also Ax" converges to Y - Ax; thus x E DA and y - Ax = Ax as A is closed. Now suppose y is orthogonal to the range of A + AI. Thus (Ax+Ax,y)==O
fora))
xEDA ,
or (Ax, y) == - (x, Ay).
By definition, y E DAo and A· y ... - Xy. As A = A·, y E DA and Ay - - ~, or (A + XI)y - O. By inequality (I) it fonows that y = O. Thus the range of A + Al is all of H. Conversely, suppose A + i and A - i are onto. Let y E DA •• Thus for all xEDA •
«A + ;)x. y) = (x.(A· - ;)y) = (x,(A - i)z) for some
zE
DA since A - i is onto. Thus
«A + i)x. y)
=
«A + i)x. ~
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
and it follows that, = z. This proves that DA• C DA and so DA result follows. •
=
533
DA•. The
If ..4 is self-adjoint then for 1m>. .. O,AI -..4 is onto and from (I) is one-to-one. Thus (>. I - ..4) - I: H .... H exists, is bounded, and we have ,
II(AI- A)-III C; 1/IIm>'l.
(2)
This operator (>'1-..4)-1 is called the resolvent of A. Notice that even though..4 is an unbounded operator, the resolvent is bounded. The sam~ argument shows the following.
C.2 Propo8lUon. A symmetric operator A is essentially Jell-adjoint iff the ranges of..4 +'iI and A - il are dense. If ..4 is a (closed) symmetric operator then the ranges of ..4 + il and ..4 - il are (closed) subspaces. The dimensions of their orthogonal complements are called the deficiency indices of ..4. Thus C. t and C.2 can be . restated as: a closed symmetric operator (resp., a symmetric operator) is self-adjoint (resp., essentially self-adjoint) iff it has deficiency indices (0,0). If ..4 is a closed symmetric operator then from (I), A + il is one-to-one and we can consider the inverse (..4 + i/) - I, defined on the range of ..4 + il. One calls (..4 - iI )(..4 + il) - I the C;ayley transform of ..4. It is always isometric, as is easy to check. Thus..4 is se/f-a4joint iff its Cayley transform is unitary. Let us return to the graph of an operator..4 for a moment. The adjoint can be described entirely in terms of its graph and this is often convenient. Define an isometry J: HeN .... HeR by J(x,,) "" (- y, .1'); note that
J2 = -I. C.3 Propo8lUon. Let A be densely defined. Then (fA) 1.
r
= J( A.) and - fA'''' J(fA ).I.. In particular, A* is closed, and if..4 is c1osed,then
NeH
=0
fAeJ(rA .),
where NeN carries the usual inner product: , « .1'1 • .1'2 ),( 'I' JI2}) ,Proof, have
>=
(XI' 'I) + (.1'2.12)·
Let (z.y)eJ(fA.), so yeDA• and ::=-..4*y. Let xeDA. We «.1' • ..4.1').( - A*y.
and so J(fA .) c
fl.
,»
=
(.1'. - ..4*,) + (Ax. y) = O.
534
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
Conversely if (z. y) E
rl.
then
(x.z)+(Ax. y)-O for all xED... Thus by definition. y ED... and z - - A- y. This proves the opposite inclusion. • Thus if A is a closed operator, the statement HeH = fAeJ( fA') means that given '. I E H. the equations {
=,
X -
A-),
Ax
+Y= I
have exactly one solution (x. y). If A is densely defined and symmetric. then Ac A- since A- is closed. There are other important consequences of C.3 as well.
C.4 Corollary. For A densely defined and closable. we have
.o4=A--.
(i)
and
(ii)
.».1.
I'mDf. (i) Note that r A.... - J{(rA')~}'" -(J(rA isometry. But
- (J( rA
.»
.I.
= -
(ii) follows since rj -
(J 2 rl
since J is an
) .I. = ri .I. = rA =' r.i.
ri. •
Suppose A: DA c H - H is one-to-one. Then we get an operator A - I defined on the range of A. In terms of graphs:
rA
•
= K(fA ),
where K(x, y) - (y. x); note that K2 = I. K is an isometry and KJ = It follows for example that if A is self-a4ioint, so i.f A I. si nce r(A .• ) .... -
-
JK.
Jri. = - JKri = KJri = KrA • = rAe •.
Next we consider possible self-adjoint extensions of a symmetric operator.
c.s
PropoeItIon. Let A be a symmetric densely defined operator on. H. The following are equivalent: (i)
A is essentially self-adjOint.
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
535
(ii) A* is self-adjoint. (iii)
A**:::> A*.
(iv) A has exactly one self-adjoint extension. (v)
A-. A*.
Proof. By definition, (i) means (X)* = A-: But we know (X)* == A* and A-= AU by C.4. Thus (i), (ii), (v) are equivalent. These imply (iii). Also (iii) implies (ii) since A c A-c A* c A** = A- and so A* = AU. To prove (iv) is implied let Y be any self-adjoint extension of A. Since Y is closed, Y:::> A-: But A = A* so Yextends the self-adjoint operator A*; i.e. Y:::> A. Taking adjoints, A* = A:::> y* = Y so Y= A.
The proof that (iv) implies the others is more complicated. We shall in fact give a more general result in C.7 below. First we need some notation. Let D + = range( A + il) .1 C II and D _ == range( A - ;/ )
II
.1 C
called the ptAfi'it~ and 1tt'.fllJtlIV .k/n', ,~"''''c''~. l Ish'~ easy to check that D~ =
Ih,' .\I~'"lk·1\1 tIIl',l II I~
{x E DAoIA*x = ix}
and D -' x
C.I Lemma. sum
c
II". A- II
-
"
.
V,fing the graph norm on VAo, we have
'''1' orthowmal (IIrc'c't
D,.o = D,.-eD+eD_.
Since D +, D _ are closed in H they are closed In D~o, Also D,i"C D,.o is closed since A* is an extension of A and hence of A. It IS easy to see that the indicated spaces are orthogonal. For example let x E D,.- and y ED. Then using the inner product
Proof.
«x, y» = (x. y) + (A*x. A* y) «x, y» = (x, y) + (A*x. - iy) = (x. y) - i(A*x, y),
gives Since x
E
D,.-- D,.o by (v). we get «x, y»
= (x.
y) - i(x. A* y)
= (x, y) - (x, y) = O.
536
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
To see that D,.o "" D,.-$O +$0 _ it suffices to show that the orthogonal complement of D,.-$D +$D _ is zero. Let" e (D,.-$D+$O_).J., so
«", x» ... «", y»
=
«",:» ... 0
for all xeDA-.yeO+,:ED_. From «",x»""O we get (",x)+ (.4.", A·x) -0 or A·"e DAo and A·A·" = -". It follows that (/ - iA·)" eD •. But from «",y»=O we have «(I-iA·)".y)=O and so (1jA.)" - O. Hence" e D _. Taking: ... " gives" = O. •
C.7 Propoeltlon. The self-adjoint extensions of a .rymmetric densely defined operator A (if any) are obtained as follows. Let T: D, ..... D be an isometry mapping D + onto D _ and let r reD + $ D _ be it.f Kraph. Then the restriction of A· to DA-$ r T is a self-adjoint extension of A. Thus A has self-adjoint extensions iff its defect indices (dim D., dim D _) are equal and these extensions are in one-to-one correspondence with all isometries of D + onto D_. Assuming this result for a moment, we give the following. Completion of Proof of C.S. If there is only one self-adjoint extension it follows from C.7 that D .. = D _ = {O} so by (",2. A is essentially self-adjoint.
•
Proof of C. 7. Let B be a self-adjoint extension of A--: Then (A)* = A· :J B so B is the restriction of A* to some subspace containing DA-. We want to show that these subspaces are of the form DA-$ r T as stated. Suppose first that T: D + ..... D _ is an isometry onto and let A I be the restriction of A· to DA-$ r T' First of all, one proves that A I is symmetric: i.e.• for II. x e DA-and 0, y ED. that (Ax + A*Y + A·Ty,,, + v + Tv) = (x + Y + Ty, A" + A·v + A*Tv).
This is a straightforward computation using the definitions. To show that AI is self-adjoint, we show that DAr c DA,. If this does not hold there exists a nonzero: e DAr such that either Ar: ... i: or Ar: "" - i:. This follows from Lemma C.6 applied to the operator AI' (Observe that Al is a closed operator-this easily follows). Now A I :J A so A*:J Ar. Thus : e D. or: e D_. Suppose Z E D+. Then: + Tz E D,., ~\' as «DA = 0 « denotes the inner product relative to A I)'
,.:»
(.»
0= «: + Tz.z» = «z.z»+«Tz.z» = 2(z.z). since T: ED _. Hence: = O. In a similar way one sees that if : e D _ then z - O. Hence A I is self-adjoint.
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
537
We will leave the details of the converse to the reader (they are similar to the foregoIng). The idea is this: if A 1 is restriction of A* to a subspace D,.-$V for V c 0+$0_ and 01 is symmetric. then Vis the graph of a map T: WcO+-O_ and (Tu.Tv) ... {u.v). for a subspace WCO+. Then , self-adjoint ness of A 1 implies that in fact W = 0+ and T is onto. • A convenient test for establishing the equality of the deficiency indices is to show that T commutes with a conjugation l/; i.e .. an antilinear isometry U: H - H satisfying V 2 = I (antilinear means V( ax) = iiVx for complex scalars a and V(x + y) "" Vx + Vy for .1'. y E " ) . In fact it is easy to see that V is the isometry required from D. t(l n (use 0 + = range (A
+ il).I.). .
As a corollary, we obtain an important classical result· of von Neumann: Let H be· L2 of a measure space and let A be a (closed) symmetric operator that is real in the sense that it commutes with complex conjugation. Then A admits self-adjoint extensions. (Another sufficient condition of a 'different nature, due to Friedrichs, is given below.) This result applies to many quantum mechanical operators. However. one is also interested in essential self-adjointness, so that the self-adjoint extension will be unique. Methods for proving this for specific operators in quantum mechanics are given in Kato (1966) and Reed and Simon [1974). For corresponding questions in elasticity, see Marsden and Hughes (1983). We now give some additional results that illustrate methods 'for handling self-adjoint operators,
C.B Propos'Don_ LeI A he a self-adjoint and B a bounded self-adjoint operator. Then A + B (with domain D,.) is self-adjoint, If A is essentially self-adjOint on DA then so is A + B.
Proof. A + B is certainly symmetric on D,., Let y
E DCA
I
II,' so that for all
XED,.,
«A + B)x, y) -= (.I', (A + B)*y). The left side is
(Ax. y) + (Bx, y)
=
(Ax. y) + (.I'. By)
since B is everywhere defined, Thus
(Ax, y) = (.I'. (A
+ B)*y - By).
HenceyE DA • = D,. and Ay= A*y= (A DA •
+ B)*y - By.
Hence y E
DA
+.
=
I.e • •
_-I. -._.ISh.-.-w_ .
*'
I.- . - ..- - .
~l'Il.
FIr _
~,...
el);. " ' _ a MJ
.., ...... -b.. n.:. ••
• ..ai;::a. ).- ....-- a-
•
' .... ED. tad' lliI£ .... E~
-L
In genera!. Ihe sum of two self-adjoinl operators need not be selfadjoint. (See Nelson (1959) and Chernoff (1974) for this and related examples.)
C.t Propoeltlon. Let.AI be a symmetric operator. If the range of.AI is all of H then.AI is self-adjoint.
Proof. We first observe that A is one-ta-one. Indeed let Ax = O. Then for any yeD,f.O-(Ax.y) = (x. Ay). But A is onto and so x=O. Thus A admits an everywhere defined inverse A - I. which is therefore self-adjoint. Hence A is self-adjoint (we proved earlier that the inverse of a self-adjoint operator is seJr-adjoint). _ We shall use these results to prove a theorem that typifies the kind of techniques one uses.
C.10· PropoeHIon. Let A be a symmetric operator on H and suppose A < 0; that is (Ax. x) < 0 for x e D,f. Suppose 1- A ha.~ dense ran~e. Then A is essentially self-adjoint.
Proof. Note that
«1- A)u.u) .. (u.u) -(Au.u);;. nun
2
and so by the Schwarz inequality we have
n(l- A)un ~ nun· It follows that (I - A) = 1 - A-has closed range. which by hypothesis must be all of H. By C .9. 1 - A is self-adjoint and so by e.s. Xis self-adjoint. •
C.11 Corolla". If.Al;s .~elf-adjo;nt and A EO O. then for any A > O. A- A is onto, (A - .AI)-I exists and
II(A-A)-III.4' Upper semi-bounded is dermed similarly. If A is either upper or lower semi-bounded then A is called semi-bounded. Observe that if A is positive or negative then A is semi-bounded. As an example. let A ... - V 2 + V where V 2 is the Laplacian and let V , be a real valued continuous funclion and bounded below. say Vex) ~ cr. Let H - L 2(R".C) and DA the COO functions with compact support. Then - V 2 is positive so
(AI. J) =
(-
v 2/. J) + (VI. J) ~ cr( I, J),
and thus· A is semi-bounded. , We already know that this operator is real so has self-adjoint extensions by von Neumann's theorem. However. the self-adjoint extension constructed below (called the Friedrichs extension) is "natura,l." Thus the actual construction is as important as the statement:
C.12 Theorem. A semi-bounded symmetric (densely defined) operator admits a self-adjoint extension.
Proof. After multiplying by - I if necessary and replacing A by A + (I cr)1 we can suppose (Ax. x) ~ IIx1l2. Consider the inner product on DA given by y» - (AX, y). (Using symmetry of A and the preceding inequality one easily checks that "this is an inner product.)
«x,
540
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
Let H 1 be the completion of D;4 in this inner product. Since the H 1 norm is stronger than the H norm, we have Hie H (i.e.. the injection °D;4 c H elltends uniquely to the completion). Now let H- 1 be the dual of H I. We have an injection of H into H' 1 defined as follows: if y is fixed and x .... (x, y) is a linear functional on H, it is also continuous on HI since
I(x. y) lor;; II xliII yllor;; III xIII II )'11, where 111·111 is the norm on HI. Thus HI c H C H I . Now the inner product on H I defines an isomorphism B: HI I Let C be the operator with domain Dc = (x E H II B( x) E " ). and C .. = Bx for x E Dc. Thus C is an elltension of A. This will be the extension we sought. We shall prove that C is self-adjoint. By definition, C is surjective; in fact C: Dc - H is a linear isomorphism. Thus by C.9, it suffices to show that C is symmetric. Indeed for x. y E Dc we have. by definition.
--+"
(Cx, y) ... «x, y»
= « y. x» == (CY. x) = (x.Cy) . •
The self-adjoint extension C can be alternatively described as follows: Let H I be as before and let C be the restriction of A* to D;40 n H I. We leave the verification as an ellercise.
APPENDIX
D
Stone's Theoremt
Here we give a self-contained proof of Stone's theorem for unhounded self-adjoint operators on a complex Hilbert space H. This guarantees that the one-parameter group e"..4 of unitary opetators exists. In fact. there is a one-ta-one correspondence between self-adjoint operators and continuous one-parameter unitary groups. A 'continuous one-parameter unitary group is a homomorphism t ...... U, from R to the group of unitary operators on H. such that for each x E H the map t ...... u,x is continuous. The infinitesinwl generator A of U, is defined by
I
.
Uh(x)-x iAx = -d u.x = hm ....!!.":"""";'--dt ' ,-0 h ... O h its domain D consisting of those x for which the indicated limit exists. We insert the factor i for convenience; iA is often called the generator.
Theorem (Stone 119320. bl). Let U, be a conlinrlOu., one-parameter unitary group. Then the generator A of U, is self-ad.Joint. (In particular. hy Appendix C, it is closed and dense(v defined.) Conversely. let A be a gh'en self-a4joint operator. Then there exists a unique one-parameter unitary group U, whose generator is A. D.1
t
This appendix was written in collaboration with P. Chernoff. 541
542
APPENDIX D: STONE'S THEOREM
Before we begin the proof. let us note that if A is a bounded self-adjoint operator then one can form the series
U, = ei,A = I + itA +
;!
(itA)2 +
;!
(itA)' + ....
which converges in the operator norm. It is straightforward to verify that U, is a continuous one-parameter unitary group and that A is its generator. Because of this. one often writes e".4 for the unitary group whose generator is A even if A is unbounded. (In the context of the so-called "operational calculus" for self-adjoint operators. one can show that e i ,.4 really is the result of applying the function e ill " to A; however, we shall not go into these matters here.) Proof of S'OM's ,/womn ( lirst IuIIf). Let U, be a given continuous unitary group. In a series of lemmas, we shall show that the generator A of U, is self-adjoint. D.2 ......ma. The domain D of A is invariant under each A U,x - U,Ax for each xeD. PIvof.
u,. and moreover
Suppose xeD. Then
*(U"U,x - U,x) u,( *< U"x - x»). =
which converges to U,(iAx) = iU,Ax as h .... O. The lemma follows by the definition of A. •
D.3 Corollary. A is clo.fed. Proof.
If xeD then. by 0.2
~ U,x =
iA U,x == iU,Ax.
(I)
Hence
Now suppose that x" e D. X,,"" x. and Ax" .... y. Then we have, by (I).
U,x = lim U,x"
"-or:
= lim {x" II-')C
+ il'U.Ax"dT}. 0
APPENDIX D: STONE'S THEOREM
543
(2)
Thus
(Here we have taken the limit under the integral sign because the convergence is unirorm; indeed IIU.Ax" - U.YII = IIAx" - "II --.0 independent or T E [0. I).) Then. by (2). d dt
u,xl
=
iy.
1-0
Hence xED and y - Ax. Thus A is closed. •
D.4 Lemma. A is densely defined. Let x E H. and let (fI be a Coo runction with compact support on R. Derine x. -.J~oo(fl(t)U,xdl. We shall show that each x.,. is in D. and that x = lim" _ ""x•• ror a suitable sequence {(fin}' To take the latter point rirst. let (fI,,(I) be nonnegative. zero outside the interval [0.1/11). and such that f~ oo(fl,,( I) dl - I. By continuity. ir , > 0 is given one can rind N so large that lIU,x - xII < , ir III < 1/ N. Suppose that n> N. Then Proof.
II x •• - xII = 1I{0<Xi(fI,,( 1)( U,x - x) cllil
1o
1/"
=11
L
(fI,,(I)(U,X-X) dIll
illi
.,.; 0
(fI,,(I)IIU,x-xlidl
1
"';E 0
1/"
(fI,,(t)dl=,.
Finally. we show that x. E D; moreover. we shall sh()w that iAx. = - x.'. Indeed.
. . -fO
- GO
da'(fI'(a)·1'u.+fl xdT 0
544
APPENDIX D: STONE'S THEOREM
Integrating by parts and using the fact that cp has compact support. we get
-1'o U.x",d.,
=
JOO (UO+IX - U.,X )cp( 0) do -00
=
(U,_l)jOO U.,xcp(o) do. -00
That is.
-l'o U.x",d.,
=
U,x" - x".
rrom which the assertion follows. • Thus far we have made no significant use of the fact that U, is unitary. We shall now do so. 0.5 Lemme. A is symmetric. Proof.
Take x. y
E
(Ax.y)=
D, Then we have
~ddt(u,x.Y)1 = !~(x·u,*J')1 , ,_ 0 ,dt , _ ()
I -(x.U d = -:-
, dt
= -
,y)
~, (x. iAy) =
II_
0
= -
I -(x.u,y) d -:,
dl
I
1-0
(x. Ay). •
We can now complete the proof that A is self-adjoint. Suppose that y E D*. Pick any xED. Then by (I). (0.1) and (0.5). (U,y.x) = (y.U_lx) -(y.X)+( y.i!'U• AXd .,) =
(y. x) - i1o-/( y.U.Ax)d.,
=
(y. x)- il- /( y. AU.x)d., o
=
(y. x) + i 1o/(UTA* J'. x) d.,
= (
y + ;{UTA*J'dT. x).
APPENDIX D: STONE'S THEOREM
545
Because D is dense. it follows that
u,J'= y+ il'U A*ydT. T
(I
Hence. differentiating. we see that J' E D and A* y = AJ'. Thus A = 1'4*. Proof of SlOW's I_rem (uCOlld Iudf). We are now given a self-adjoint operator A. We shall construct a continuous unitary group U, whose senerator is A.
If>.. > O. then' + >..1'4 2 : DA , -+ H;s hijectil'e. (I + >"1'4 2 ).1: DA , is bounded by I. and DA ,. the domain of 1'4 2 • i.f detl.~('.
D•• Lemma. H
-+
Proof. If A is self-adjoint. so is .fA A. It is therefore enough to estahlish the lemma for>.. '- l. First we establish surjectivity. By C.3 and D.3. if : E H is given there exists a unique solution (x. ,,) to the equations
x- Ay=O Ax+y=:.
2"
From the first equation. x = Ay. The second equation then yields A + Y = ::. so, + A2 is surjective. For xEDA!. note that «(I+A2)x.x)~lIxIl2. so 1I(1+A2)xll~lIxli. Thus' + 1'4 2 is one-to-one and IK' + A2r 11I:s I. Now suppose that" is orthogonal to DA!' We can find a v such that II =< V + A 2v. Then 0= (II. v) = (v + A 2v. v) = IIvll2 + II A vII 2 • whence v'" 0 and therefore
II
=
O. Consequently DA , is dense in H. •
For>.. > O. define an operator A" by A" = A(I + >"A2)-I. Note that A" is defined on all of H because if xE.1! :,cn (1+>"Al)-IXEDA,CD. so A(I + >..A 2 )-IX makes sense. D.7 Lemma. A" is a bounded sel/-adjoint operator. Also. A" and A" commute for all >...fA > O. Proof.
Pick x E H. Then by D.6.
>"111'4 "xII 2 =
(>..A{I + >"A 2
= (>"1'4 2 ( ,
r
IX.
1'4(1 + >..1'4 2 ) -IX)
+ >"A2) -I X .(1 + >..1'4 2) -IX)
..A 2 )(1 + >..A 2) -lx.(I + >..1'4 2 ) -Ix)
. > 0 using power series or the results of Section 4.1. Since AA and A,. commute. it follows that A and u,,. commute for every sand t.
u.
0.8 L....m•• IlxEDthenlimA_oAAx=Ax. Proof.
If xED we have AAX - Ax= (I + >.A 2 ) 'Ax- Ax = -
>.A 2 ( 1+ >.A 2 ) "Ax.
It is therefore enough to show that for every y E H. >.A 2 (1 + >'A2)' I)' --+ O. From the inequality IKI + >'A2)YII2 ~ II>'A 2yII 2, valid for >. ~ O. we see that II>.A 2(1 + >'A 2 )-'II';; I. Thus it is even enough to show the preceding equality for all y in some dense subspace of H.
APPENDIX D: STONE'S THEOREM
547
Suppose Y E D.. J, which is dense by D.6. Then IIAA2(I + AA2) -\ yll = AII(J E;
+ AA2)
I A2YII
AIIA2YII.
which indeed goes to zero with A.••
D.I Lemma. For each x E H. limA _ ou,Ax exists. If we ('(II/ the limit u,x. then {u,} is a continuous one-pararneter unitary group. Proof.
We have
'U,AX-U"X= , ,
1,d 0
-{u.Au" dT T '-T }XdT
=ifo'lf.,Au,"-T{AAX- A"x)dT. whence IIu,AX
-
U,"xll
E;
(3)
Itl'IIAAX - A"xll·
Now suppose that xED. Thcn, by D.8, AAx - Ax, so that IIAAxA"xll- 0 as A, po - O. Because of (3) it follows that {U,Axh > 0 is uniformly Cauchy as A - 0 on every compact t-interval. It follows that limA _ p,Ax ... U,x exists and is a continuous function of t. Moreover. since D is dense and all the u,A have norm I. an easy approximation argument shows that the preceding conclusion holds even if x ff: D. It is obvious that each U, is a linear operator. Furthermore.
(U,x.u,y)
=
lim (u,AX.u,Ay) A-O
... lim (x, y) ... (x, y) A-O so U, is isometric. Trivially. Uo = I. Finally.
(V,u,x. y) .. lim (u,Au,X. y) A-O
=
' - ;A)-IX ED, (see Proposition C.I) and so by D.lt (>'-iA,,)(>.-iA)-IX-(>.-iA)(>.-iA)-'x=x
Because IK>' - iA,,)-11i ~ IRe>'I-1 it follows that
II( >. -
iA) - I X -
(>. -
iA"
r xll- O. I
•
as 1'-0.
550
APPENDI>' D: STONE'S THEOREM
In closing. we mention that many of the results proved have generalizations to continuous one-parameter groups or semi-groups of linear operators in Banach spaces (or on locally convex spaces). The central result. due to Hille and Yosida. characterizes generators of semi-groups. Our proof of Stone's theorem is based on methods that can be used in the more general context. Expositions or this more general context are found in. for example. Kato (1966) and Marsden and Hughes [1983. ch. 6).
APPENDIX
E
The Sard and Smale Theorems
This appendix is devoted to the classical Sard theorem and its infinitedimensional generalil.ation due to Smale. The exposition is inspired hy Ahraham and Rohhin (1967). Recall that a suhset A c R'" is said to have measure zero if, for every f> 0, there exists a countahle covering of A by closed cuhes K, '(with edges parallel to the coordinate axes) such that the sum of the volumes of K, is less than E.. Clearly a countable union of sets of measure zero has measure zero. Before we proceed to the local version of Sard's theorem it will be useful to have at hand two facts concerning sets of measure zero in R"'. E.1 Lemma. Let U c R m be open and A cUbe of mct/I'ure zero. If f: U -+R'" is a C l map, then f(A) has measure zero. Proof. First write A as a countable union of relatively compact sets Cn • If ' we show that A n CIt has measure zero. then A has measure zero since it will be a countable union of sets of measure zero. But Cn is relatively compact and thus there exists M> 0 such that IIDf(x~1 ~ M for all x E en' By the mean value theorem. the image of a cube of edge length d is contained in a cube of edge length dim M. •
E.2 Lemma (FItbi,,;). Let A be a countable union (If compact sets in R" and asslime that Ac = A n({c}XR,,-I) has measure zero for all c E R. Then A has measure zero. 551
552
APPENDIX E: THE SARD AND SMALE THEOREMS
Proof. It is enough to work with one element of the union. so we may assume A itself is compact and hence there exists an interval [a. b) su.:h that A c [a. b)XR,,-·I. Since A, is compact and has measure zero. for each c E [a. b) there exists a finite number of closed cubes Ke 1••••• Ke N in R,,-I the sum of whose volumes is less than E and such tilat (c}x' Ke., cover A,.• i-I ..... Ne • Find a closed interval I,. with c in its interior such that le.xKe.,cAcxR,,-I. Thus the family (Je.xK, .. ,li-I ..... N,.cE[a.b)} covers A n(ra. b)XR,,-I) - A. But since (int(/,.)lcE [a.b)} covers [a. b). we can choose a finite subcovering 1,., ..... 1,..,. Now find another covering J,., •... •./,.• such that each J e., is contained in some I" and such that the sum of the lengths of all J e is less than 2(b - a). Consequently {J,. x K, ,I j = 1•...• K. i = I ....• N, } cover A and the sum of their volumes' is les~ than 2(h - a)E.. '
We are now ready to prove the local version of Sard's theorem. It will be stated for CA-maps. k ~ I. but the proof we give is strictly for Coo-mappings and follows Abraham and Robbin (1967) and Milnor (1965). We shall indicate the only troublesome spot in the proof for the C k case and how one circumvents it; see Abraham and Robbin (1967) for the lengthy technical details. First we recall the following notations from chapter 3. Let M and N be C I manifolds and/: M -+ N. a C I map. A point x EM is a regular point of / iff T, / is surjective; otherwise. x is a critical point of /. If C c M is the set of critical points of /. then /( C) c N is the set of critical values of / and N \/( C) is the set of regular values of f. The set of regular values is denoted by (.'R f or (:~ (/). In addition. for A c M we define (:11 ,IA by (:11 ,IA = N \/( Ann. In particular. if U c M is open. (:11 ,I U = (:~ (/1 U).
E.3 Sard', Theorem In R". Let m ~ n. U c R'" be open. and /: U -+ R" be 0/ class CA. where k ~ 1 and k > max(O. m - n). Then the .fet 0/ critical values 0/ / has measure zero in R". Proof. . Denote by C = (x E Ulrank D/(x) < n} the set of critical points of If m = O. then.R'" is one
f. We shall show that/(C) has measure zero in R".
point and the theorem is trivially true. Suppose inductively the theorem holds for m - I. Let C, = (x E UIDJ/(x) = 0 for j = I ..... i}. Then C is the following union of disjoint sets C=(C\C1)U(C1\C2 )U'" U(CA_1\CdU{j. The proof that /(C) has measure zero is divided in thr< I.
2. 3.
/( C~ ) has measure zero. /(C\C 1 ) has measure zero. /( C.\C• • I) has measure zero for 1 ...
of ...
k - I.
1'"
APPENDIX E: THE SARD AND SMALE THEOREMS
553
Proof of Step 1. Since k ~ 1.( k -I)n ~ k -I; i.e.• kn ~ n + k -1. But we also have k ~ m - n + 1. so that n + k - I ~ m; i.e .• m $ kn. Let K c V be a closed cube with edges parallel tn the coordinate axes. We will show that/(C, () K) has measure zero. Since CAcan be covered by countably many such cubes. this will prove that I( CA) has measure zero. By Taylor's theorem, the compactness of K. and the definition of CA' we have
I( Y) = I(x)+ R(x. y)
where (I)
for x E CA() K and y E K. Here M is a constant depending only on DAf and K. Let e be the length of the edge of K. Choose an integer I. subdivide K into In! cubes with edge e / k. and choose any cube K' of this subdivision which intersects CA' For x E C, () K' and y E K'. we have' IIx - yll ~ .[m (e / /). By (I )./( K ') C L where L is the cube of edge Nit - I with center I(x); N = 2M«m)I/2/)A t I. The volume of L is N"I--II(" I). There are at' most I'" such cubes; hence. I( C, () K) is contained in a union of cubes ' whose total volume V satisfies.
, Since m ~ kn we have m - n(k + \) < O. so V - 0 as / ..... 00. and thus I( CA() K) has measure zero. Proolof Step 2. KII _I' where
Write C\C I = (XE VII Kq
=
~
rank D/(x) < fl} = KI U
... U
(xEVlrank D/(x) =q}
and it suffices to show that/( Kq) has measure zerQ for q = I. .... n - I. Since Kq is empty for q > m, we may assume q ~ m. As before it will suffice to show that each point of Kq has a neighborhood V such that I( V () Kq) has measure zero. Choose xE K q • By the Local Representation Theorem 2.5.14 we may assume that x has a neighborhood V"" VI X V2 where VI c R It, - q and V2 c R q are open balls such that for x E VI and t E V2 /(x.t)= (1J(x,t).t).
1J,(x)=1J(x.t)
for
XE,VI .
554
APPENDIX E: THE SARD AND SMALE THEOREMS
Then for I
E
V2
This is because, for (x, I) E
VI X V2' Df(x, t)
Df ( X,I ) = [
DlI,( x) 0
is given by the matrix
;J
Hence rank Df(x, I) = q iff DlI,(x) = o. Now 11, is C· and k ~ m - n = (m - q)-(n - q). Since q ~ I, by induction we find that the critical values of 11, and in particular 1I,({ x EVil DT/,( x) = O}) has measure zero for each IE V2 • By Fubini's lemma,J(K q () V) has measure zero. Since Kq is covered by countably many such V, this shows thatf(K q ) has measure zero.
Prool 01 Step J. To show f(C,\C•• I) has measure zero, it suffices to show that every x E C.\Cs+ I has a neighborhood V such that f( C, () V) has measure zero; then since C.\C.. I is covered by countably many such neighborhoods V, it follows that f( C,\C,. I) has measure zero. Choose Xo E C,\ C•• I' All the partial derivatives of f at Xo of order less than or equal to s are zero. but some partial derivative of order .f + I is not zero. Hence we may assume that Dlw( Xo )
'*' 0 and w( xu) = O.
where DI is the partial derivative with respect to .\" I lind w( x) = D" ... D,.f( x).
Define h: U _R m by
where x = (XI' X2'" •• x",) E U c R"'. Clearly h is C· - 'and Dh(xo) is nonsingular; hence there is an open neighborhood V of Xo and an open -set We R m such that
h: V-W is a C 4 • diffeomorphism. Let A = C. () V, A' = h( A) and g = h - I. We would like to consider the function fog and then arrange things such that we can apply the inductive hypothesis to it. If k = 00, there is no trouble.
APPENDIX E: THE SARD AND SMALE THEOREMS
555
But if k max(O. m - n). Then ~'~'f is residual and hence dense in N.
556
APPENDIX E: THE SARD AND SMALE THEOREMS
Proof. Denote by C the set of critical points of I. We will show that every x e M has a neighborhood Z such that ~II Z is open and dense. where Z- cl(Z). Then. since M is LindelOf we can find a countable cover {Zi} of X with ctlt/lZ, open and dense. Since ctlt / ... n i~/IZ,. it will follow that ~/is residual. Choose xe M. We want a neighborhood Z of x with ~/IZ open and dense. By taking local charts we may assume that M is an open subset of R m and N = R". Choose an open neighborhood Z of x such that Z is compact. ThenC""{xeMlrankDI(x) max(O.index(T,.f» for every x EM,. then ~It, n ~ ill is residual in N. Proof. By c;iefinition. Tx( af) == T,.(f)1 TxC aM) for x E aM. Thus. if."x is a regular point for then it is also regular for I. Hence yEN is a critical value of both I and a f iff it is a critical value of flint M or af. But lnt M and aM are both boundaryless. so Smale's theorem applies. giving '3I'f and Iilt al residual in N. Consequently '31, n ~ is residual in N. •
al.
a,
Sard's theorem deals with the genericity of the surjectivity of the derivative of a map. We conclude with a hrief consideration of the dual question of genericity of the injectivity of the derivative of a map.
E.12 Lemma.
The set IL(E. F) of linear continutJlIJ ,~plil injective maps is
open in L(E; F). Proof. Let A E lL(E. F). Then A(E) is closed and F = A( E)$G for G a closed subspace of F. The map A: EX G -+ F;Ale.g) == A(e)+ g is clearly linear, bijective. and continuous. so by Banach's isomorphism theorem AEGL(EXG.F). The map P: L(ExG.F) ..... L(E.F);P(B)==BIE is linear. continuous, and onto. so by the open mapping theorem it is also an open mapping. Moreover P(A) = A and P(GL(E X G. F)c IL(E. F) for if BE GL(E X G. F) then F== B(E)$B(G) where both B(E) and B(G) are
560
APPENDIX E: THE SARD AND SMALE THEOREMS
closed in F. Thus A has an open neighborhood P(GL(E x G. F» contained' in IL(E.F). • Note that even in finite dimensions IL(E. F) is empty if dim E > dim F and so E.12 is vacuously true in this case.
E.13 PropcHtlUon. Let f: M -.. N be a Cl-map of manifolds. The set p .. {x E Mlf is an immersion at x} is open in M. Proof. It suffices to prove the proposition locally. Thus if E and F are the models of M and N respectively and if f: U -.. E is immersive at x E U c E. then Df(x) E IL(E. F). By the lemma. (Df) -I(IL(E. F» is open in U since Df: U -.. L(E. F) is continuous. • This result can be improved if M is finite dimensional. This is done in the next theorem whose proof relies ultimately on the existence and uniqueness of integral curves of CI-vector fields.
E.13 Theorem. Let f: M -.. N be a C I injective map of manifolds. dim(M)= m. The set P = {x E Mlf is immersive at x} is open and dense in M. In particular. if dim{N) = n. then m ~ n. Proof. (D. Burghelea). It suffices to work in a local chart V. We shall use induction on k to show that ell!" .. " :::) V. where
U, .. ; = I
,
{XE vi TJ(~) ax'l ..... TJ(~) ax" linearly independent} .
The case k - n gives then the statement of the theorem. Note that by the preceding proposition. U,' ... " is open in V. The statement is obvious for I since if it fails would vanish on an open subset of V and thus f would be constant on V. contradicting the injectivity of f. Assume inductively that the statement for k holds: i.e.. U" ... " is open in V and cI U" ... ;':::) V. Define
k-
U'
'1. -I
=
{
X E
U
'1 "'"
TJ
I ( a) * T
xl ".'4.'
0
(1.\
.
and notice Ihat it is open in l~ , and thus in V. It is (by the case k = I) and hence i~ ~! hy induction. Let
u,~ .. '".
I
= { X E
l{ .ITxf( ~) ..... Txf( ~). I ax" iJx" . I
CU"".;,."
} 01'
,tense in U"."
linearly independent}
APPENDIX E: THE SARD AND SMALE THEOREMS
661
We prove that U:, ... i • • , is dense in 11;:. " which then shows that cll1;""i, .. ::::> V. If this were not the case, there exists an open set We Uf.., such that
'al(x)TJ(~)+ ... +ak(x)TJ(~)+TJ(_a_) ... o ax" ax', ax l ,., for some C l functions a l •. ..• a k nowhere zero on W. Let c: (- t, t) .... W be an integral curve of the vector field
a +--. a-E~XI( W). aI-a. + ... +a k --. ax" ax" ax" . , Then
(f 0 c )'(1) = T..("f(c'(I»'" O. so that foe is constant on (- e. e) contradicting injectivity of f.
•
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