Universitext Editorial Board (North America). S.Axler K.A.Ribet
Universitext Editorial Board (North America): S. Axler K.A. Ribet
Loring W.Tu An Introduction to Manifolds Springer
Loring W. Tu An Introduction to Manifolds
Loring W.Tu Medford,MA 02155 loring.tu@tufts.edu N) K.A.Ribet Praisc C Berkeley.CA 94720-3840 at Berkeley USA USA axler@sfsu.cdu ribet@math.berkeley.edu ISBN-13:978-0-387-48098-5 e-SBN-13:978-0-387-48101-2 Mathematics Classification Code(2000):58-01.58Axx.58A05.58A10,58A12 Library of Congress Control Number:2007932203 008 Springer Science+Business Media.LLC. NY 10013.US subjectoproprietary rights Printed onacid-free paper. 987654321 www.springer.com (JLS/MP)
Loring W. Tu Department of Mathematics Tufts University Medford, MA 02155 loring.tu@tufts.edu Editorial Board (North America): S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu ISBN-13: 978-0-387-48098-5 e-ISBN-13: 978-0-387-48101-2 Mathematics Classification Code (2000): 58-01, 58Axx, 58A05, 58A10, 58A12 Library of Congress Control Number: 2007932203 © 2008 Springer Science + Business Media, LLC. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.springer.com (JLS/MP) NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, All rights reserved. This work may not be translated or copied in whole or in part without the written
Dedicated to the memory of Raoul Bott
Dedicated to the memory of Raoul Bott
Preface It has been more than two decades since Raoul Bott and Ipublished Differential Forms in Algebraic Topology.While this book has enjoyed a certain success,it does assume some familiarity with manifolds and so is not so readily accessible to the average first-year graduate student in mathematics.It has been my goal for quite some time to bridge this gap by writing an elementary introduction to manifolds assuming only one semester of abstract algebra and a ve ear of real analysis.Moreover,given the emen hetw een t ot only bu mathematicians and advanced undergraduates,but also physicists who want a solid foundation in geometry and topology. With so many excellent books on manifolds on the market,any author who un- dertakes to write another owes to the public,if not to himself,a good rationale.First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed,to the point where for example he or she can compute de rham cohomolooy of simple s A secondc nside in the prerequ Most books labo a as a su a Euclidean space.This has the disadvantage of making quotie manifolds,of which a projective space is a prime example,difficult to understand. My solution is to make the first four chapters of the book independent of point-set topology and to place the necessary point-set topology in an appendix.While reading the first four chapters,the student should at the same time study Appendix A to acquire the point-set topology that will be assumed starting in Chapter 5. The book is meant to be read and studied by a novice.It is not meant to be encvclopedic.Therefore.I discuss only the irreducible minimum of manifold theor which I think eve ery math allows the centra deas to emerge more clearly.In several years of teaching.I hav generally been able to cover the entire book in one semester. In order not to interrupt the flow of the exposition,certain proofs of a more routine or computational nature are left as exercises.Other exercises are scattered throughout the exposition,in their natural context.In addition to the exercises embedded in the
Preface It has been more than two decades since Raoul Bott and I published Differential Forms in Algebraic Topology. While this book has enjoyed a certain success, it does assume some familiarity with manifolds and so is not so readily accessible to the average first-year graduate student in mathematics. It has been my goal for quite some time to bridge this gap by writing an elementary introduction to manifolds assuming only one semester of abstract algebra and a year of real analysis. Moreover, given the tremendous interaction in the last twenty years between geometry and topology on the one hand and physics on the other, my intended audience includes not only budding mathematicians and advanced undergraduates, but also physicists who want a solid foundation in geometry and topology. With so many excellent books on manifolds on the market, any author who undertakes to write another owes to the public, if not to himself, a good rationale. First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute de Rham cohomology of simple spaces. A second consideration stems from the self-imposed absence of point-set topology in the prerequisites. Most books laboring under the same constraint define a manifold as a subset of a Euclidean space. This has the disadvantage of making quotient manifolds, of which a projective space is a prime example, difficult to understand. My solution is to make the first four chapters of the book independent of point-set topology and to place the necessary point-set topology in an appendix. While reading the first four chapters, the student should at the same time studyAppendixAto acquire the point-set topology that will be assumed starting in Chapter 5. The book is meant to be read and studied by a novice. It is not meant to be encyclopedic. Therefore, I discuss only the irreducible minimum of manifold theory which I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly. In several years of teaching, I have generally been able to cover the entire book in one semester. In order not to interrupt the flow of the exposition, certain proofs of a more routine or computational nature are left as exercises. Other exercises are scattered throughout the exposition, in their natural context. In addition to the exercises embedded in the
Preface text,there are problems at the end of each chapter.Hints and solutions to selected and are gathered at the end of the book.Ihave starred the problems This has been conceived as the first volume of a tetralogy on geometry and topology.The second volume is Differential Forms in Algebraic Topology cited above.I hope that Volume 3.Differential Geometry:Connections,Curvature,and Characteristic Classes,will soon see the light of day.Volume 4,Elements of Equiv- ariant Cohomology,a long-running joint project with Raoul Bott before his passing away in 2005.should appear in a year. This project has bee ten vearsin estation.during this time ihave benefited fron o the s tand hospitality of many i nstitutions in additic my ov the F ch Minis re deE et de la pe spe ior fellowship(bourse de ha aut niveau),the Instit t Henri Poincar he m ieu,and the Departments of Mathemati cs at the Ecole male Superieure (rue d'Ulm),the Universite Paris VII,and the Universite de Lille, for stays of various length.All of them have contributed in some essential way to the finished product. I owe a debt of gratitude to my colleagues Fulton Gonzalez.Zbigniew Nitecki ind Montserrat Teixidor-i-Bigas.who tested the manuscript and p mentsand students Cristian Go ally Aaron n for their detailed aands fo Kostant of Sp nnger and h er tea n John Spiegel oew for editing advice,typesetting,and manufacturing,respectively and to Steve Schnably and Paul Gerardin for years of unwavering moral support.I thank Aaron W.Brown also for preparing the List of Symbols and the TX files for many of the solutions.Special thanks go to George Leger for his devotion to all of my book projects and for his careful reading of many versions of the manuscripts.His encouragement,feedback,and suggestions have been invaluable to me in this book as well as in several others.Finally.I want to mention Raoul Bott whose courses on geor opol helped to shape my ma ath natical thinking and whose exemplary life is an inspiration to us all. Medford,Massachusetts Loring W.Tu June 2007
viii Preface text, there are problems at the end of each chapter. Hints and solutions to selected exercises and problems are gathered at the end of the book. I have starred the problems for which complete solutions are provided. This book has been conceived as the first volume of a tetralogy on geometry and topology. The second volume is Differential Forms in Algebraic Topology cited above. I hope that Volume 3, Differential Geometry: Connections, Curvature, and Characteristic Classes, will soon see the light of day. Volume 4, Elements of Equivariant Cohomology, a long-running joint project with Raoul Bott before his passing away in 2005, should appear in a year. This project has been ten years in gestation. During this time I have benefited from the support and hospitality of many institutions in addition to my own; more specifically, I thank the French Ministère de l’Enseignement Supérieur et de la Recherche for a senior fellowship (bourse de haut niveau), the Institut Henri Poincaré, the Institut de Mathématiques de Jussieu, and the Departments of Mathematics at the École Normale Supérieure (rue d’Ulm), the Université Paris VII, and the Université de Lille, for stays of various length. All of them have contributed in some essential way to the finished product. I owe a debt of gratitude to my colleagues Fulton Gonzalez, Zbigniew Nitecki, and Montserrat Teixidor-i-Bigas, who tested the manuscript and provided many useful comments and corrections, to my students Cristian Gonzalez, ChristopherWatson, and especiallyAaronW. Brown and Jeffrey D. Carlson for their detailed errata and suggestions for improvement, to Ann Kostant of Springer and her team John Spiegelman and Elizabeth Loew for editing advice, typesetting, and manufacturing, respectively, and to Steve Schnably and Paul Gérardin for years of unwavering moral support. I thank Aaron W. Brown also for preparing the List of Symbols and the TEX files for many of the solutions. Special thanks go to George Leger for his devotion to all of my book projects and for his careful reading of many versions of the manuscripts. His encouragement, feedback, and suggestions have been invaluable to me in this book as well as in several others. Finally, I want to mention Raoul Bott whose courses on geometry and topology helped to shape my mathematical thinking and whose exemplary life is an inspiration to us all. Medford, Massachusetts Loring W. Tu June 2007
Contents Preface.Vii 0 A Brief Introduction . Part I Euclidean Spaces 1 Smooth Functions on a Euclidean Space. Versus Analytic Functions +。,。 1.2 Taylor's Theorem with Remainder. 557 Problems. 9 Tangent Vectors in R"as Derivations 11 The Directional Derivative. 2.2 Germs of Functions. 13 2.3 Derivations at a Point. 14 24 Vector Fields 15 2.5 Vector Fields as Derivations 17 Problems. 18 3 Alternating k-Linear Functions. 。 19 3.1 Dual Space. 19 32 Perm ations 20 3.3 Multilinear Functions. 3.4 Permutation Action on k-Linear Functions. 2223 3.5 The Symmetrizing and Alternating Operators. 24 3.6 The Tensor Product. 25 3.7 The Wedge Product 25 38 Antico mutativity of the Wedge Product. 39 Associativity of the Wedge Product. 3.10 A Basis for k-Covectors . 30 problems. 31
Contents Preface . vii 0 A Brief Introduction . 1 Part I Euclidean Spaces 1 Smooth Functions on a Euclidean Space . 5 1.1 C∞ Versus Analytic Functions . 5 1.2 Taylor’s Theorem with Remainder . 7 Problems . 9 2 Tangent Vectors in Rn as Derivations . 11 2.1 The Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Germs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Derivations at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Vector Fields as Derivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Alternating k-Linear Functions. 19 3.1 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Multilinear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Permutation Action on k-Linear Functions. . . . . . . . . . . . . . . . . . . . . . 23 3.5 The Symmetrizing and Alternating Operators . . . . . . . . . . . . . . . . . . . 24 3.6 The Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7 The Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.8 Anticommutativity of the Wedge Product . . . . . . . . . . . . . . . . . . . . . . . 27 3.9 Associativity of the Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.10 A Basis for k-Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Contents 4 Differential Forms on 3 4.1 Differential 1-Forms and the Differential of a Function 33 ferential k-Forms 536 The Exterior Derivative. 36 4.5 Closed Forms and Exact Forms. 39 4.6 Applications to Vector Calculus. 39 4.7 Convention on Subscripts and Superscripts. 42 Problems. 42 PartⅡManifolds 5 Manifolds 47 5.1 Topological Manifolds 47 5.2 Compatible Charts. 48 5.3 Smooth Manifolds 50 54 Exa mples of Smooth Manifolds 51 problems. 53 6 Smooth Mans on a Manifold 57 nctions and Maps. 1 Partial Derivatives 6.3 The Inverse Function Theorem . 62 7 Qu otients 63 7.1 The Quotient Topology. 63 7.2 Continuity of a Map on a Ouotient. 64 7.3 Identification of a Subset to a Point. 65 74 A Necessary Condition for a Hausdorff Ouotient 65 75 Onen Eau valence relations 66 7.6 The Real Pre 68 77 The Sta nda Problems CAtlas on a Real Projective Space 73 Part II The Tangent Space 8 The Tangent Space 1 8.1 The Tangent Space at a Point. 8.2 The Differential of a Map.,. 78 83 The Chain Rule. 79 84 Bases for the Tangent Space at a Point 80 只5 Local Expression for the Differential 82 8.6C arves in a Manifold 83
x Contents 4 Differential Forms on Rn . 33 4.1 Differential 1-Forms and the Differential of a Function . . . . . . . . . . . 33 4.2 Differential k-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 Differential Forms as Multilinear Functions on Vector Fields . . . . . . 36 4.4 The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Closed Forms and Exact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6 Applications to Vector Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.7 Convention on Subscripts and Superscripts . . . . . . . . . . . . . . . . . . . . . 42 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Part II Manifolds 5 Manifolds . 47 5.1 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Compatible Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Smooth Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 Examples of Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6 Smooth Maps on a Manifold . 57 6.1 Smooth Functions and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 Quotients . 63 7.1 The Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Continuity of a Map on a Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.3 Identification of a Subset to a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.4 A Necessary Condition for a Hausdorff Quotient . . . . . . . . . . . . . . . . 65 7.5 Open Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.6 The Real Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.7 The Standard C∞ Atlas on a Real Projective Space . . . . . . . . . . . . . . 71 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Part III The Tangent Space 8 The Tangent Space . 77 8.1 The Tangent Space at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.2 The Differential of a Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.4 Bases for the Tangent Space at a Point . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.5 Local Expression for the Differential . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.6 Curves in a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Contents xi 8.7 Computing the Differential Using Curves . 85 8.8 Rank,Critical and Regular Points. 86 problems. Submanifolds 91 Submanifolds 9.2 The Zero Set of a Function. 94 9.3 The Regular Level Set Theorem. 95 9.4 Examples of Regular Submanifolds. 97 Problems 10 Categories and Functors 10 10.1 Categories. 10 l02 Functors.。.,。,.。,。.,.。.。 102 10.3 Dual Maps Problems. 104 11 The Rank of a Smooth Map 105 11.1 Constant Ran 106 0卡”中”+”卡中中”卡”卡”4”中4年中+卡卡”卡。” 1.2 Immersions and Submersions. 107 11.3 Images of Smooth Maps. 109 11.4 Smooth Maps into a Submanifold. 13 11.5 The Tangent Plane to a Surface in R3 .115 Problems. 116 2 The Tangent Bundle 119 12.1 The Topology of the Tangent Bundle. 119 12.2 The Manifold Structure on the Tangent Bundle. 12 12.3 Vector Bundles. 121 12 4 Smooth Sections 123 12.5 Smooth Frames 105 Problems. 126 13 Bump Functions and Partitions of Unity 127 13.I C Bump Functions.127 13.2 Partitions of Unity. 131 13.3 Existence of a Partition of Unity 132 Problems . 134 14 Vector Fields. .135 14 1 Smoothness of a Vector Field 135 14.2 Integral Curves. 136 14 3 Local Flow 138 14.4 The Lie Bracket 141 14.5 Related Vector Fields 14 14.6 The Push-Forward of a Vector Field.144 Problems . .144
Contents xi 8.7 Computing the Differential Using Curves . . . . . . . . . . . . . . . . . . . . . . 85 8.8 Rank, Critical and Regular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9 Submanifolds. 91 9.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9.2 The Zero Set of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.3 The Regular Level Set Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.4 Examples of Regular Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10 Categories and Functors . 101 10.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.3 Dual Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 11 The Rank of a Smooth Map . 105 11.1 Constant Rank Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 11.2 Immersions and Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.3 Images of Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 11.4 Smooth Maps into a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 11.5 The Tangent Plane to a Surface in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 12 The Tangent Bundle . 119 12.1 The Topology of the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 119 12.2 The Manifold Structure on the Tangent Bundle . . . . . . . . . . . . . . . . . . 121 12.3 Vector Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.4 Smooth Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12.5 Smooth Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 13 Bump Functions and Partitions of Unity . 127 13.1 C∞ Bump Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 13.2 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 13.3 Existence of a Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 14 Vector Fields . 135 14.1 Smoothness of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 14.2 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 14.3 Local Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 14.4 The Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 14.5 Related Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 14.6 The Push-Forward of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xii Contents Part IV Lie Groups and Lie Algebras 15 Lie Groups ,149 15.I Examples of Lie Groups. 149 15.2 Lie Subgroups. 152 15.3 The Matrix Exponential 15.4 The Trace of a Matrix. 155 15.5 The Differential of det at the Identity. 16 Lie Algebras. 161 16.1 Tangent Space at the Identity ofa Lie Group. 16 16.2 The Tangent Space to SL(n,R)at I . 161 16.3 The Tangent Space to O(n)at I 162 16.4 Left-Invariant Vector Fields on a Lie Group. 163 16.5 The Lie Algebra of a Lie Group. 165 16.6 The Lie Bracket on gl(n,R). 166 16.7 The Push-Forward of a Left-Invariant Vector Field. 167 16.8 The Differential as a Lie Algebra Homomorphism. .168 Problems 170 Part V Differential Forms 17 Differential 1-Forms. 175 17.1 The Differential of a Function. 17.2 Local Expression for a Differential 1-Form. 176 17.3 The Cotangent Bundle 177 17.4 Characterization of c 1-Forms. 17 17.5 Pullback of1-foms.179 Problems 179 18 Differential k-Forms. .18 18.1 Local Expression for a k-Form .182 18.2 The Bundle Point of View 183 18.3 C k-Forms 183 18.4 Pullback of k-Forms . 184 18.5 The Wedge Product. 18.6 Invariant Forms on a Lie Group. 186 Problems. 186
xii Contents Part IV Lie Groups and Lie Algebras 15 Lie Groups . 149 15.1 Examples of Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 15.2 Lie Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 15.3 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 15.4 The Trace of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 15.5 The Differential of det at the Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 16 Lie Algebras. 161 16.1 Tangent Space at the Identity of a Lie Group . . . . . . . . . . . . . . . . . . . . 161 16.2 The Tangent Space to SL(n, R) at I . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 16.3 The Tangent Space to O(n) at I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 16.4 Left-Invariant Vector Fields on a Lie Group . . . . . . . . . . . . . . . . . . . . 163 16.5 The Lie Algebra of a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 16.6 The Lie Bracket on gl(n, R). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 16.7 The Push-Forward of a Left-Invariant Vector Field . . . . . . . . . . . . . . . 167 16.8 The Differential as a Lie Algebra Homomorphism . . . . . . . . . . . . . . . 168 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Part V Differential Forms 17 Differential 1-Forms . 175 17.1 The Differential of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 17.2 Local Expression for a Differential 1-Form . . . . . . . . . . . . . . . . . . . . . 176 17.3 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 17.4 Characterization of C∞ 1-Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 17.5 Pullback of 1-forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 18 Differential k-Forms . 181 18.1 Local Expression for a k-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 18.2 The Bundle Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 18.3 C∞ k-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 18.4 Pullback of k-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 18.5 The Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 18.6 Invariant Forms on a Lie Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186