TEXTS AND READINGS IN MATHEMATICS 22. A Course in Differential Geometry and Lie Groups

Transcription

1 TEXTS AND READINGS IN MATHEMATICS 22 A Course in Differential Geometry and Lie Groups

2 Texts and Readings in Mathematics Advisory Editor C. S. Seshadri, Chennai Mathematical Institute, Chennai. Managing Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi. Editors V. S. Borkar, Tata Institute of Fundamental Research, Mumbai. R. L. Karandikar, Indian Statistical Institute, New Delhi. C. Musili, University of Hyderabad, Hyderabad. K. H. Paranjape, Institute of Mathematical Sciences, Chennai. T. R. Ramadas, Tata Institute of Fundamental Research, Mumbai. V. S. Sunder, Institute of Mathematical Sciences, Chennai. Already Published Volumes R. B. Bapat: Linear Algebra and Linear Models (Second Edition) R. Bhatia: Fourier Series C. Musili: Representations of Finite Groups H. Helson: Linear Algebra ( Second Edition) D. Sarason: Notes on Complex Function Theory M. G. Nadkarni: Basic Ergodic Theory (Second Edition) H. Helson: Harmonic Analysis ( Second Edition) K. Chandrasekharan: A Course on Integration Theory K. Chandrasekharan: A Course on Topological Groups R. Bhatia (ed.): Analysis, Geometry and Probability K. R. Davidson: C - Algebras by Example M. Bhattacharjee et af.: Notes on Infinite Permutation Groups V. S. Sunder: Functional Analysis - Spectral Theory V. S. Varadarajan: Algebra in Ancient and Modern Times M. G. Nadkarni: Spectral Theory of Dynamical Systems A. Borel: Semisimple Groups and Riemannian Symmetric Spaces M. Marcolli: Seiberg-Witten Gauge Theory A. Bottcher and S. M. Grudsky: Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis A. R. Rao and P. Bhimasankaram: Linear Algebra ( Second Edition) C. Musili: Algebraic Geometry for Beginners A. R. Rajwade: Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem

3 A Course in Differential Geometry and Lie Groups s. Kumaresan University of Mumbai ~HINDUSTAN U LQj UBOOKAGENCY

4 Published by Hindustan Book Agency (India) P 19 Green Park Extension, New Delhi Copyright 2002 by Hindustan Book Agency ( India) No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest exclusively with Hindustan Book Agency (India). Unauthorized export is a violation of Copyright Law and is subject to legal action. ISBN ISBN (ebook) DOI /

5 Dedicated to the memory of my mother S. Susila

6 Contents Preface 1 Differential Calculus 1.1 Definitions and examples Chain rule, mean value theorem and applications 1.3 Directional derivatives Inverse mapping theorem Local study of immersions and submersions 1.6 Fundamental theorem of calculus Higher derivatives and Taylor's theorem 1.8 Smooth functions with compact support 1.9 Existence of solutions of ODE Manifolds and Lie Groups 2.1 Differential manifolds Smooth maps and diffeomorphisms 2.3 Tangent spaces to a manifold 2.4 Derivatives of smooth maps 2.5 Immersions and submersions 2.6 Submanifolds Vector fields Flows and exponential map 2.9 Frobenius theorem Lie groups and Lie algebras 2.11 Homogeneous spaces 3 Tensor Analysis 3.1 Multilinear algebra 3.2 Exterior algebra 3.3 Tensor fields.... ix

7 viii 3.4 The exterior derivative, 3.5 Lie derivatives 4 Integration 4.1 Orient able manifolds 4.2 Integration on manifolds. 4.3 Stokes' theorem... 5 Riemannian Geometry 5.1 Covariant differentiation 5.2 Riemannian metrics The Levi-Civita connection 5.4 Gauss theory of surfaces in 1R Curvature and parallel transport 5.6 Cartan structural equations. 5.7 Spaces of constant curvature.. A Tangent Bundles and Vector Bundles B Partitions of Unity Bibliography List of Symbols Index CONTENTS

8 Preface This book arose out of the courses offered by me at T.I.F.R. Centre, Bangalore, T.I.F.R., Bombay (twice), Indian Institute of Technology, Kanpur and Ramanujan Institute of Mathematics, University of Madras. I plunged into writing this book thanks to the encouragement and persuasions of the audience of my courses. The book covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and a short but motivated introduction to basic differential and Riemannian geometry. This book will be suitable for a course for students of Physics and Mathematics at the graduate level of western universities or at M.Phil. level of Indian universities. While the topics are traditional, the discerning reader will find the approach to the topics and the proofs at many places quite novel. Our main emphasis is on the geometric meaning of the concepts, so that the reader will feel confident and acquire a working knowledge. For this reason, motivations are given, many simple exercises are included and illuminating nontrivial examples are discussed in detail. Some of the salient features of the book are the following: 1. Geometric and conceptual treatment of Differential Calculus with a wealth of nontrivial examples. 2. A thorough discussion of the much-used result on the existence, uniqueness and smooth dependence of solutions of ODE. 3. Special care in introducing the concept of tangent space to a manifold. 4. An early and simultaneous treatment of Lie Groups and related concepts as we develop the basic topics in differential manifolds. 5. An early and elementary proof of the fact that all classical groups are. Lie groups. 6. A motivated and highly geometric proof of the Frobenius theorem. 7. A constant reconciliation with the classical (such as tensor calcu Ius) treatment or notation and the modern approach.

9 x Preface 8. Simple proofs of the Hairy-Ball theorem and Brouwer's fixed point theorem. 9. Construction of manifolds of constant curvature a la Chern. 10. Merits and comparisons of different view points, whenever possible. Major portion of this book was typeset when I was at the Tata Institute and I thank the Tata Institute for it. A substantial portion of the preliminary version was typed by G. Santhanam and was proof-read by C.S. Aravinda. Kirti Joshi and Kapil Paranjape introduced me to the world of 'lex and Jb.'lEX. V. Muruganandam went through the preliminary version ~ made suggestions for improvement. V. Nandagopal formatted the book for the TRIM series. Ajit Kumar drew the figures on computers and helped me a lot in the final version. It gives me great pleasure to record my sincere thanks to all these friends. The book was in hibernation for about ten years and the sustained efforts of many of my friends, especially S. Ilangovan, led to its seeing the light of day. I take this opportunity to record my sense of gratitude to one of my relatives Mr. G.Gnanasambandam, B.E., who honed my scientific thinking by getting me into serious discussions on scientific matters during my high-school days. I also record my deep sense of appreciation for the numerous hours of pleasant discussions on a variety of topics in Mathematics which I had with Adimurthi, Akhil Ranjan, K. Okamoto, M.S. Raghunathan and R. Parthasarathy during my days at the Tata Institute. My ideas, knowledge and appreciation of Mathematics owe a lot to these people. I thank many of my colleagues, especially juniors, at the Tata Institute who asked me to give many seminars and discussed their difficulties and problems, mathematical as well as personal. These helped me perr.eive the difficulties of a beginner and also made me a better human being. I also record my appreciation for Rajendra Bhatia, Managing Editor of the series and the referees. Their corrections and persuasive suggestions made me realize that my enthusiasm for dissemination of knowledge cannot be an excuse for being sloppy. I am sure that paying heed to their suggestions has enhanced the value of this book and eliminated some of the egregious errors and tactless remarks. I thank my wife Kalai and my children Sivaguru and Bharathi who bore with me while I was busy with the preparation of the book. Books and expositions at this level, as a rule, are written in a formal and precise way. I broke away from this and wrote the book in a con-

10 Preface xi versational tone. I hope that beginners will find it easy for self-study. I crave the indulgence of experts in case they find my style a bit too abrasive. I hope that this book will instill confidence in the reader, provide a working knowledge and create a desire for further studies. I shall be happy if readers appreciate my efforts in making their learning curve of this subject less steep. I welcome corrections and suggestions for improvement. Mumbai December 2001 S. K umaresan kumaresa@math.mu.ac.in