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UniversitextEditorial Board(North America):S. AxlerK.A. RibetFor other titles in this series, go towww.springer.com/series/223

Loring W. TuAn Introduction to ManifoldsSecond Edition

Loring W. TuDepartment of MathematicsTufts UniversityMedford, MA [email protected] board:Sheldon Axler, San Francisco State UniversityVincenzo Capasso, Università degli Studi di MilanoCarles Casacuberta, Universitat de BarcelonaAngus MacIntyre, Queen Mary, University of LondonKenneth Ribet, University of California, BerkeleyClaude Sabbah, CNRS, École PolytechniqueEndre Süli, University of OxfordWojbor Woyczyński, Case Western Reserve UniversityISBN 978-1-4419-7399-3e-ISBN 978-1-4419-7400-6DOI 10.1007/978-1-4419-7400-6Springer New York Dordrecht Heidelberg LondonLibrary of Congress Control Number: 2010936466Mathematics Subject Classification (2010): 58-01, 58Axx, 58A05, 58A10, 58A12 c Springer Science Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connectionwith any form of information storage and retrieval, electronic adaptation, computer software, or by similaror 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 arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.Printed on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)

Dedicated to the memory of Raoul Bott

Preface to the Second EditionThis is a completely revised edition, with more than fifty pages of new materialscattered throughout. In keeping with the conventional meaning of chapters andsections, I have reorganized the book into twenty-nine sections in seven chapters.The main additions are Section 20 on the Lie derivative and interior multiplication,two intrinsic operations on a manifold too important to leave out, new criteria inSection 21 for the boundary orientation, and a new appendix on quaternions and thesymplectic group.Apart from correcting errors and misprints, I have thought through every proofagain, clarified many passages, and added new examples, exercises, hints, and solutions. In the process, every section has been rewritten, sometimes quite drastically.The revisions are so extensive that it is not possible to enumerate them all here. Eachchapter now comes with an introductory essay giving an overview of what is to come.To provide a timeline for the development of ideas, I have indicated whenever possible the historical origin of the concepts, and have augmented the bibliography withhistorical references.Every author needs an audience. In preparing the second edition, I was particularly fortunate to have a loyal and devoted audience of two, George F. Leger andJeffrey D. Carlson, who accompanied me every step of the way. Section by section,they combed through the revision and gave me detailed comments, corrections, andsuggestions. In fact, the two hundred pages of feedback that Jeff wrote was in itself amasterpiece of criticism. Whatever clarity this book finally achieves results in a largemeasure from their effort. To both George and Jeff, I extend my sincere gratitude. Ihave also benefited from the comments and feedback of many other readers, including those of the copyeditor, David Kramer. Finally, it is a pleasure to thank PhilippeCourrège, Mauricio Gutierrez, and Pierre Vogel for helpful discussions, and the Institut de Mathématiques de Jussieu and the Université Paris Diderot for hosting meduring the revision. As always, I welcome readers’ feedback.Paris, FranceJune 2010Loring W. Tu

Preface to the First EditionIt has been more than two decades since Raoul Bott and I published DifferentialForms in Algebraic Topology. While this book has enjoyed a certain success, it doesassume 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 sometime to bridge this gap by writing an elementary introduction to manifolds assumingonly one semester of abstract algebra and a year of real analysis. Moreover, giventhe tremendous interaction in the last twenty years between geometry and topologyon the one hand and physics on the other, my intended audience includes not onlybudding mathematicians and advanced undergraduates, but also physicists who wanta 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. Firstand foremost is my desire to write a readable but rigorous introduction that gets thereader quickly up to speed, to the point where for example he or she can computede 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 amanifold as a subset of a Euclidean space. This has the disadvantage of makingquotient manifolds such as projective spaces difficult to understand. My solutionis to make the first four sections of the book independent of point-set topology andto place the necessary point-set topology in an appendix. While reading the firstfour sections, the student should at the same time study Appendix A to acquire thepoint-set topology that will be assumed starting in Section 5.The book is meant to be read and studied by a novice. It is not meant to beencyclopedic. Therefore, I discuss only the irreducible minimum of manifold theorythat I think every mathematician should know. I hope that the modesty of the scopeallows the central ideas to emerge more clearly.In order not to interrupt the flow of the exposition, certain proofs of a moreroutine or computational nature are left as exercises. Other exercises are scatteredthroughout the exposition, in their natural context. In addition to the exercises embedded in the text, there are problems at the end of each section. Hints and solutions

xPrefaceto selected exercises and problems are gathered at the end of the book. I have starredthe problems for which complete solutions are provided.This book has been conceived as the first volume of a tetralogy on geometryand topology. The second volume is Differential Forms in Algebraic Topology citedabove. I hope that Volume 3, Differential Geometry: Connections, Curvature, andCharacteristic Classes, will soon see the light of day. Volume 4, Elements of Equivariant Cohomology, a long-running joint project with Raoul Bott before his passingaway in 2005, is still under revision.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 dela Recherche for a senior fellowship (bourse de haut niveau), the Institut HenriPoincaré, the Institut de Mathématiques de Jussieu, and the Departments of Mathematics at the École Normale Supérieure (rue d’Ulm), the Université Paris 7, and theUniversité de Lille, for stays of various length. All of them have contributed in someessential 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-Martinez, Christopher Watson, and especially Aaron W. Brown and Jeffrey D. Carlson for their detailed errata and suggestions for improvement, to Ann Kostant of Springer and herteam 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 ofNotations and the TEX files for many of the solutions. Special thanks go to GeorgeLeger for his devotion to all of my book projects and for his careful reading of manyversions of the manuscripts. His encouragement, feedback, and suggestions havebeen invaluable to me in this book as well as in several others. Finally, I want tomention Raoul Bott, whose courses on geometry and topology helped to shape mymathematical thinking and whose exemplary life is an inspiration to us all.Medford, MassachusettsJune 2007Loring W. Tu

ContentsPreface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiPreface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixA Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Chapter 1 Euclidean Spaces§1Smooth Functions on a Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . .1.1C Versus Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2Taylor’s Theorem with Remainder . . . . . . . . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3458§2Tangent Vectors in Rn as Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1The Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2Germs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3Derivations at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5Vector Fields as Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10101113141617§3The Exterior Algebra of Multicovectors . . . . . . . . . . . . . . . . . . . . . . . . . .3.1Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3Multilinear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4The Permutation Action on Multilinear Functions . . . . . . . . . . . . .3.5The Symmetrizing and Alternating Operators . . . . . . . . . . . . . . . . .3.6The Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7The Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8Anticommutativity of the Wedge Product . . . . . . . . . . . . . . . . . . . .3.9Associativity of the Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . .3.10 A Basis for k-Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1819202223242526272831

xiiContentsProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .§4nDifferential Forms on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1Differential 1-Forms and the Differential of a Function . . . . . . . . .4.2Differential k-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3Differential Forms as Multilinear Functionson Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.4The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5Closed Forms and Exact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.6Applications to Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7Convention on Subscripts and Superscripts . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32343436373840414444Chapter 2 Manifolds§5Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2Compatible Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4Examples of Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .484849525357§6Smooth Maps on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1Smooth Functions on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2Smooth Maps Between Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .6.3Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4Smoothness in Terms of Components . . . . . . . . . . . . . . . . . . . . . . . .6.5Examples of Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.6Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.7The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595961636365676870§7Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1The Quotient Topology . . . . . . . . . . . . . . . . . . . . . . .