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Graduate Texts in Mathematics 94 Editorial Board F. W. Gehring P. R. Halmos (Managing Editor) C. C. Moore

Graduate Texts in Mathematics TAKEUTI!ZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 OxTOBY. Measure and Category. 2nd ed. 3 ScHAEFFER. Topological Vector Spaces. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 5 MACLANE. Categories for the Working Mathematician. 6 HUGHEs/PIPER. Projective Planes. 7 SERRE. A Course in Arithmetic. 8 T AKEUTI!ZARING. Axiometic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. II CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FuLLER. Rings and Categories of Modules. 14 GoLUBITSKYIGUILLEMIN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 ROSENBLATT. Random Processes. 2nd ed. 18 HALMOS. Measure Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed., revised. 20 HUSEMOLLER. Fibre Bundles. 2nd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNEs/MACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HoLMES. Geometric Functional Analysis and its Applications. 25 HEWITT/STROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARISKI!SAMUEL. Commutative Algebra. Vol. I. 29 ZARISKIISAMUEL. Commutative Algebra. Vol. II. 30 31 32 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk. 2nd ed. 35 WERMER. Banach Algebras and Several Complex Variables. 2nd ed. 36 KELLEYINAMIOKA et al. Linear Topological Spaces. 37 MONK. Mathematical Logic. 38 GRAUERT/FRITZSCHE. Several Complex Variables. 39 ARVESON. An Invitation to C*-Aigebras. 40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 42 SERRE. Linear Representations of Finite Groups. 43 GILLMAN/1ERISON. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LotvE. Probability Theory I. 4th ed. 46 LotvE. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and 3. 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. 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 Wilson11938- Foundations of differentiable manifolds and Lie groups. (Graduate texts in mathematics; 94) Reprint. Originally published: Glenview, Ill.: Scott, Foresman, 1971. Bibliography: p. Inc1udes index. 1. Differentiable manifolds. 2. Lie groups. 1. 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 1 st 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, 9 8 765 432 1 ISBN 978-1-4419-2820-7 DOI 10.1007/978-1-4757-1799-0 ISBN 978-1-4757-1799-0 (eBook)

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 It is of point set topology, covering spaces, and the fundamental group. 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 4. 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 3 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 is presumed. tool. Only a slight acquaintance with Hilbert spaces 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 6. 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 5 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 3 or 5 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 do 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

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