Graduate Texts in Mathematics Editorial Board F W Gehring C C Moore 94 P R Halmos (Managing Editor) Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTI!ZARING Introduction to Axiomatic Set Theory 2nd ed OxTOBY Measure and Category 2nd ed ScHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI!ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules GoLUBITSKYIGUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HoLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI!SAMUEL Commutative Algebra Vol I ZARISKIISAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra 1: Basic Concepts JACOBSON Lectures in Abstract Algebra II: Linear Algebra JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEYINAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Aigebras KEMENY/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/1ERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LotvE Probability Theory I 4th ed LotvE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and continued after Index Frank W Warner FOUNDATIONS of DIFFERENTIABLE MANIFOLDS and LIE GROUPS With 57 Illustrations I Springer-Verlag Berlin Heidelberg GmbH Frank W Warner Vniversity of Pennsylvania Department of Mathematics El Philadelphia, PA 19104 V.S.A Editorial Board P R Halmos M anaging Editor Indiana Vniversity Department of Mathematics Bloomington, IN 47405 V.S.A F W Gehring Vniversity of Michigan Department of Mathematics Ann Arbor, MI 48109 V.S.A c C Moore Vniversity of California at Berkeley Department of Mathematics Berkeley, CA 94720 V.S.A AMS Subject Classification: 58-01 Library of Congress Cataloging in Publication Data Warner, Frank W (Frank Wilson11938Foundations of differentiable manifolds and Lie groups (Graduate texts in mathematics; 94) Reprint Originally published: Glenview, Ill.: Scott, Foresman, 1971 Bibliography: p Inc1udes index Differentiable manifolds Lie groups Title II Series QA614.3.w37 1983 512'.55 83-12395 Originally published © 1971 by Scott, Foresman and Co © 1983 by Frank W Warner Softcover reprint of the hardcover 1st edition 1983 AlI rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag Berlin Heidelberg GmbH, 765 432 ISBN 978-1-4419-2820-7 ISBN 978-1-4757-1799-0 (eBook) DOI 10.1007/978-1-4757-1799-0 This book provides the necessary foundation for students interested in any of the diverse areas of mathematics which require the notion of a differentiable manifold It is designed as a beginning graduate-level textbook and presumes a good undergraduate training in algebra and analysis plus some knowledge of point set topology, covering spaces, and the fundamental group It is also intended for use as a reference book since it includes a number of items which are difficult to ferret out of the literature, in particular, the complete and self-contained proofs of the fundamental theorems of Hodge and de Rham The core material is contained in Chapters I, 2, and This includes differentiable manifolds, tangent vectors, submanifolds, implicit function theorems, vector fields, distributions and the Frobenius theorem, differential forms, integration, Stokes' theorem, and de Rham cohomology Chapter treats the foundations of Lie group theory, including the relationship between Lie groups and their Lie algebras, the exponential map, the adjoint representation, and the closed subgroup theorem Many examples are given, and many properties of the classical groups are derived The chapter concludes with a discussion of homogeneous manifolds The standard reference for Lie group theory for over two decades has been Chevalley's Theory of Lie Groups, to which I am greatly indebted For the de Rham theorem, which is the main goal of Chapter 5, axiomatic sheaf cohomology theory is developed In addition to a proof of the strong form of the de Rham theorem-the de Rham homomorphism given by integration is a ring isomorphism from the de Rham cohomology ring to the differentiable singular cohomology ring-it is proved that there are canonical isomorphisms of all the classical cohomology theories on manifolds The pertinent parts of all these theories are developed in the text The approach which I have followed for axiomatic sheaf cohomology is due to H Cartan, who gave an exposition in his Seminaire 1950/1951 For the Hodge theorem, a complete treatment of the local theory of elliptic operators is presented in Chapter 6, using Fourier series as the basic tool Only a slight acquaintance with Hilbert spaces is presumed I wish to thank Jerry Kazdan, who spent a large portion of the summer of 1969 educating me to the whys and wherefores of inequalities and who provided considerable assistance with the preparation of this chapter I also benefited from notes on lectures by J J Kohn and Stephen Andrea, from several papers of Louis Nirenberg, and from Partial Differential v Preface vi Equations by Bers, John, and Schechter, which the reader might wish to consult for further references to the literature At the end of each chapter is a set of exercises These are an integral part of the text Often where a claim in a chapter has been left to the reader, there is a reminder in the Exercises that the reader should provide a proof of the claim Some exercises are routine and test general understanding of the chapter Many present significant extensions of the text In some cases the exercises contain major theorems Two notable examples are properties of the eigenfunctions of the Laplacian and the Peter-Weyl theorem, which are developed in the Exercises for Chapter Hints are provided for many of the difficult exercises There are a few notable omissions in the text I have not treated complex manifolds, although the sheaf theory developed in Chapter will provide the reader with one of the basic tools for the study of complex manifolds Neither have I treated infinite dimensional manifolds, for which I refer the reader to Lang's Introduction to Differentiable Manifolds, nor Sard's theorem and imbedding theorems, which the reader can find in Sternberg's Lectures on Differential Geometry Several possible courses can be based on this text Typical one-semester courses would cover the core material of Chapters 1, 2, and 4, and then either Chapter or or 6, depending on the interests of the class The entire text can be covered in a one-year course Students who wish to continue with further study in differential geometry should consult such advanced texts as Differential Geometry and Symmetric Spaces by Helgason, Geometry of Manifolds by Bishop and Crittenden, and Foundations of Differential Geometry (2 vols.) by Kobayashi and Nomizu I am happy to express my gratitude to Professor I M Singer, from whom I learned much of the material in this book and whose courses have always generated a great excitement and enthusiasm for the subject Many people generously devoted considerable time and effort to reading early versions of the manuscript and making many corrections and helpful suggestions I particularly wish to thank Manfredo Carmo, Jerry Kazdan, Stuart Newberger, Marc Rieffel, John Thorpe, Nolan Wallach, Hung-Hsi Wu, and the students in my classes at the University of California at Berkeley and at the University of Pennsylvania My special thanks to Jeanne Robinson, Marian Griffiths, and Mary Ann Hipple for their excellent job of typing, and to Nat Weintraub of Scott, Foresman and Company for his cooperation and excellent guidance and assistance in the final preparation of the manuscript Frank Warner This Springer edition is a reproduction of the original Scott, Foresman printing with the exception that the few mathematical and typographical errors of which I am aware have been corrected A few additional titles have been added to the bibliography I am especially grateful to all those colleagues who wrote concerning their experiences with the original edition I received many fine suggestions for improvements and extensions of the text and for some time debated the possibility of writing an entirely new second edition However, many of the extensions I contemplated are easily accessible in a number of excellent sources Also, quite a few colleagues urged that I leave the text as it is Thus it is reprinted here basically unchanged In particular, all of the numbering and page references remain the same for the benefit of those who have made specific references to this text in other publications In the past decade there have been remarkable advances in the applications of analysis-especially the theory of elliptic partial differential equations, to geometry-and in the application of geometry, especially the theory of connections on principle fiber bundles, to physics Some references to these exciting developments as well as several excellent treatments of topics in differential and Riemannian geometry, which students might wish to consult in conjunction with or subsequent to this text, have been included in the bibliography Finally, I want to thank Springer for encouraging me to republish this text in the Graduate Texts in Mathematics series I am delighted that it has now come to pass Philadelphia, Pennsylvania October, 1983 Frank Warner vii n s 11 22 30 34 41 so MANIFOLDS Preliminaries Differentiable Manifolds The Second Axiom of Countability Tangent Vectors and Differentials Submanifolds, Diffeomorphisms, and the Inverse Function Theorem Implicit Function Theorems Vector Fields Distributions and the Frobenius Theorem Exercises TENSORS AND DIFFERENTIAL FORMS 54 62 69 73 77 Tensor and Exterior Algebras Tensor Fields and Differential Forms The Lie Derivative Differential Ideals Exercises LIE GROUPS 82 89 92 98 101 102 109 110 112 117 120 132 viii Lie Groups and Their Lie Algebras Homomorphisms Lie Subgroups Coverings Simply Connected Lie Groups Exponential Map Continuous Homomorphisms Closed Subgroups The Adjoint Representation Automorphisms and Derivations of Bilinear Operations and Forms Homogeneous Manifolds Exercises Contents INTEGRATION ON MANIFOLDS 138 140 153 157 Orientation Integration on Manifolds de Rham Cohomology Exercises SHEAVES, COHOMOLOGY, AND THE DE RHAM THEOREM 163 173 176 186 189 191 200 205 207 214 216 Sheaves and Presheaves Cochain Complexes Axiomatic Sheaf Cohomology The Classical Cohomology Theories Alexander-Spanier Cohomology de Rham Cohomology Singular Cohomology Cech Cohomology The de Rham Theorem Multiplicative Structure Supports Exercises THE HODGE THEOREM 220 222 240 243 250 251 The Laplace-Beltrami Operator The Hodge Theorem Some Calculus Elliptic Operators Reduction to the Periodic Case Ellipticity of the Laplace-Beltrami Operator Exercises 260 BIBLIOGRAPHY 262 SUPPLEMENT TO THE BIBLIOGRAPHY 264 INDEX OF NOTATION 267 INDEX 227 ix [I] Bers, L., F John, and M Schechter Partial Differential Equations New York: John Wiley & Sons, Inc., 1964 [2] Bishop, R L., and R J Crittenden Geometry of Manifolds New York: Academic Press, 1964 [3] Bredon, G E Sheaf Theory New York: McGraw-Hill, 1967 [4] Cartan, H Seminaire 1950/1951 Paris: Ecole Normale Superieure, 1955 [5] Chevalley, C Theory of Lie Groups I Princeton, N.J.: Princeton University Press, 1946 [6] Fleming, W H Functions of Several Variables Reading, Mass.: Addison-Wesley, 1965 (2nd ed Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1977.) [7] Godement, R Topologie Algebrique et Theorie des Faisceaux Paris: Hermann, 1958 [8] Gunning, R C Lectures on Riemann Surfaces Princeton, N.J.: Princeton University Press, 1966 (9] Helgason, S Differential Geometry Lie Groups and Symmetric Spaces New York: Academic Press, 1978 [10] Hodge, W V D The Theory and Applications of Harmonic Integrals 2d ed Cambridge: Cambridge University Press, 1952 [11] Hurewicz, W Lectures on Ordinary Differential Equations New York and Cambridge, Mass.: John Wiley & Sons, Inc., and MIT Press, 1958 [12] Jacobson, N Lie Algebras New York: John Wiley & Sons, Inc., 1962 (Reprinted by Dover, 1979.) [13] Kelley, J L General Topology Princeton, N.J.: Van Nostrand Company, Inc., 1955 (Graduate Texts in Mathematics, vol 27, Springer-Verlag, New York, 1975.) [14] Kervaire, M A Manifold which does not admit any differentiable structure Comment Math Helv., 35(1961), 1-14 260 Bibliography 261 [15] Kobayashi, S., and K Nomizu Foundations of Differential Geometry, vols New York: John Wiley & Sons, Inc., 1963 and 1969 [16] Kohn, J J Introduccion a Ia teoria de integrales harmonicas Lecture notes issued by the Centro de Investigacion del lPN, Mexico, 1963 [17] Lang, S Introduction to Differentiable Manifolds John Wiley & Sons, Inc., 1962 [18] Loomis, L H., and S Sternberg Advanced Calculus Reading, Mass.: Addison-Wesley, 1968 [19] Milnor, J On manifolds homeomorphic to the 7-sphere Ann of Math., 64(1956), 399-405 New York: [20] Montgomery, D., and L Zippin Topological Transformation Groups New York: Interscience, 1955 (Reproduction by Krieger, Melbonrne Florida, 1974.) [21] Newns, N., and A Walker Tangent planes to a differentiable manifold J London Math Soc., 31(1956), 400-407 [22] Nirenberg, L On elliptic partial differential equations Ann Scuola Norm Sup Pisa, 13(1959}, 115-162 [23] Pontrjagin, L S Topological Groups Princeton, N.J.: Princeton University Press, 1939 [24] de Rham, G V arietes Differentiables (3rd Ed., 1973.) [25] Samelson, H Uber die Sphiiren die als Gruppenriiume auftreten Comment Math Helv., 13(1940), 144-155 [26] Simmons, G F Introduction to Topology and Modern Analysis New York: McGraw-Hill, 1963 [27] Singer, I M., and J Thorpe Lecture Notes on Elementary Topology and Geometry Glenview, III.: Scott, Foresman and Company, 1967 (Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.) [28] Spanier, E H Algebraic Topology 1966 [29] Spivak, M Calculus on Manifolds New York: W A Benjamin, Inc., 1965 [30] Sternberg, S Lectures on Differential Geometry Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1964 [31] Woll, J W., Jr Functions of Several Variables New York: Harcourt, Brace & World, Inc., 1966 Paris: Hermann, 1960 New York: McGraw-Hill No attempt at completeness was made for either the original bibliography or the following supplement which is being added for the Springer edition My goal is simply to provide the reader with a few basic references and sources for alternate treatments or additional readings Quite extensive bibliographies on differentiable manifolds and their role in the many aspects of modern analysis and geometry can be found in volume II of Kobayashi-Nomizu, listed above, and volume of Spivak, listed below For a comprehensive, but leisurely and very readable treatment of differential and Riemannian geometry, I recommend Spivak, M Differential Geometry, vols Boston, Mass.: Publish or Perish, Inc., 1970-1975 For a beautiful introduction to manifolds and a geometric treatment of many central topics of differential topology, see Guillemin, V., and A Pollack Differential Topology Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1974 A rapid introduction to manifolds and a very beautiful development of basic Riemannian geometry is given in Carmo, M Geometria Riemanniana Rio de Janeiro: IMPA, 1979 For a very recent treatment of smooth manifolds furnished with metric tensors of arbitrary signature (pseudo-Riemannian manifolds), with applications to the theory of relativity, see O'Neill, B Semi-Riemannian Geometry New York: Academic Press, 1983 The reader who wishes to study the theory of characteristic classes should consult the excellent exposition in Milnor, J., and J Stasheff Characteristic Classes Annals of Mathematics Studies, no 76 Princeton, N.J.: Princeton University Press, 1974 The following three texts are additional sources of relatively selfcontained treatments of basic elliptic theory The Griffiths/Harris and Wells texts develop Hodge theory for compact complex manifolds The 262 Supplement to the Bibliography 263 Lang text has an appendix on elliptic partial differential equations including the regularity theory on a torus and in Euclidean space Griffiths, P., and J Harris Principles of Algebraic Geometry New York: John Wiley & Sons, Inc., 1978 Lang, S SL (R) Reading, Mass.: Addison-Wesley, 1975 Wells, R.O., Jr Differential Geometry on Complex Manifolds Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1973 (Graduate Texts in Mathematics, vol 65, Springer,Verlag, New York, 1979.) For an application of elliptic operator theory to very remarkable and far-reaching connections between analysis and topology, see Palais, R S Seminar on the Atiyah-Singer Index Theorem Annals ofMathematics Studies, no 57 Princeton, N.J.: Princeton University Press, 1965 A systematic development of the general theory of second order quasilinear elliptic partial differential equations and the required linear elliptic theory is given in Gilbarg, D., and N Trudinger Elliptic Partial Differential Equations of Second Order 2nd ed Berlin Heidelberg New York Tokyo: Springer-Verlag, 1983, in prep For very recent and striking applications of analysis to problems in geometry, see Aubin, T Nonlinear Analysis on Manifolds Monge-Ampere Equations Grundlehren der mathematischen Wissenschaften, vol 252 New York Heidelberg Berlin: Springer-Verlag, 1982 Yau, S-T, ed Seminar on Differential Geometry Annals of Mathematics Studies, no 102 Princeton, N.J.: Princeton University Press, 1982 Lie groups are central to the theory of fibre bundles and connections For some recent and very significant applications of these theories to the study of gauge theories in physics, one should consult the following two references: Drechsler, W., and M E Mayer Fiber Bundle Techniques in Gauge Theories Lecture Notes in Physics, no 67 Berlin Heidelberg New York: Springer-Verlag, 1977 Bleecker, D Gauge Theory and Variational Principles Mass.: Addison-Wesley, 1981 Reading, 19 T(M), kMm, goj (x- x(m)), 20 id Jk