AnIntroductiontoManifolds,2ndeditionbyLoringW.Tu.

An Introduction to Manifolds, by Loring W. Tu.LoringW. TuAn introduction to manifoldsSecond edition②$prringerLoring w. TuDepartment of mathematicsTufts UniversityMedford. MA 02155loring. tu @tufts eduEditorial boardSheldon Axler, San Francisco State UniversityVincenzo capasso, Universita degli studi di milanoCarles casacuberta. universitat de barcelonaAngus MacIntyre, Queen Mary, University of LondonKenneth ribet, University of California, BerkeleyClaude sabbah, CNRS, Ecole PolytechniqueEndre suli, University of OxfordWojbor Woyczynski, Case Western Reserve UniversityISBN978-1-4419-7399-3e-ISBN978-1-4419-7400-6DOI10.1007/978-1-4419-7400-6Springer new york dordrecht heidelberg londonLibrary of congress control Number: 2010936466Mathematics Subject Classification(2010): 58-01, 58AXX, 58A05, 58A10, 58A12SPringer Science+Business Media, LLC 2011All 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 forbiddenThe 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 rightsPrinted on acid-free paperSpringerispartofSpringerScience+businessMedia(www.springer.com)Dedicated to the memory of raoul bottPreface 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 chaptersThe main additions are Section 20 on the lie derivative and interior multiplicationtwo 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 groupApart from correcting errors and misprints, I have thought through every proofagain, clarified many passages, and added new examples, exercises, hints, and solu-tions. In the process, every section has been rewritten, sometimes quite drasticallyThe 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 comeTo provide a timeline for the development of ideas, I have indicated whenever possi-ble the historical origin of the concepts, and have augmented the bibliography withhistorical referencesEvery 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 sectionthey 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, includg those of the copyeditor, David Kramer. Finally, it is a pleasure to thank philiCourrege, Mauricio Gutierrez, and Pierre Vogel for helpful discussions, and the Institut de mathematiques de Jussieu and the Universite paris diderot for hosting meduring the revision. As always, I welcome readers'feedbackPaF1aris. franceLoringW. TiJune 2010Preface 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 av-erage 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 topologyWith 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 spacesA 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 5The 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 clearlyIn 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