Graduate Texts in Mathematics 218

Graduate Texts in Mathematics

Series Editors:

Sheldon AxlerSan Francisco State University, San Francisco, CA, USA

Kenneth RibetUniversity of California, Berkeley, CA, USA

Advisory Board:

Colin Adams, Williams College, Williamstown, MA, USAAlejandro Adem, University of British Columbia, Vancouver, BC, CanadaRuth Charney, Brandeis University, Waltham, MA, USAIrene M. Gamba, The University of Texas at Austin, Austin, TX, USARoger E. Howe, Yale University, New Haven, CT, USADavid Jerison, Massachusetts Institute of Technology, Cambridge, MA, USAJeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USAJill Pipher, Brown University, Providence, RI, USAFadil Santosa, University of Minnesota, Minneapolis, MN, USAAmie Wilkinson, University of Chicago, Chicago, IL, USA

Graduate Texts in Mathematics bridge the gap between passive study and creativeunderstanding, offering graduate-level introductions to advanced topics in mathe-matics. The volumes are carefully written as teaching aids and highlight character-istic features of the theory. Although these books are frequently used as textbooksin graduate courses, they are also suitable for individual study.

For further volumes:www.springer.com/series/136

http://www.springer.com/series/136

John M. Lee

Introduction toSmooth Manifolds

Second Edition

John M. LeeDepartment of MathematicsUniversity of WashingtonSeattle, WA, USA

ISSN 0072-5285ISBN 978-1-4419-9981-8 ISBN 978-1-4419-9982-5 (eBook)DOI 10.1007/978-1-4419-9982-5Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012945172

Mathematics Subject Classification: 53-01, 58-01, 57-01

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Preface

Manifolds crop up everywhere in mathematics. These generalizations of curves andsurfaces to arbitrarily many dimensions provide the mathematical context for un-derstanding space in all of its manifestations. Today, the tools of manifold theoryare indispensable in most major subfields of pure mathematics, and are becomingincreasingly important in such diverse fields as genetics, robotics, econometrics,statistics, computer graphics, biomedical imaging, and, of course, the undisputedleader among consumers (and inspirers) of mathematicstheoretical physics. Nolonger the province of differential geometers alone, smooth manifold technology isnow a basic skill that all mathematics students should acquire as early as possible.

Over the past century or two, mathematicians have developed a wondrous collec-tion of conceptual machines that enable us to peer ever more deeply into the invisi-ble world of geometry in higher dimensions. Once their operation is mastered, thesepowerful machines enable us to think geometrically about the 6-dimensional solu-tion set of a polynomial equation in four complex variables, or the 10-dimensionalmanifold of 5 5 orthogonal matrices, as easily as we think about the familiar2-dimensional sphere in R3. The price we pay for this power, however, is that themachines are assembled from layer upon layer of abstract structure. Starting with thefamiliar raw materials of Euclidean spaces, linear algebra, multivariable calculus,and differential equations, one must progress through topological spaces, smooth at-lases, tangent bundles, immersed and embedded submanifolds, vector fields, flows,cotangent bundles, tensors, Riemannian metrics, differential forms, foliations, Liederivatives, Lie groups, Lie algebras, and morejust to get to the point where onecan even think about studying specialized applications of manifold theory such ascomparison theory, gauge theory, symplectic topology, or Ricci flow.

This book is designed as a first-year graduate text on manifold theory, for stu-dents who already have a solid acquaintance with undergraduate linear algebra, realanalysis, and topology. I have tried to focus on the portions of manifold theory thatwill be needed by most people who go on to use manifolds in mathematical or sci-entific research. I introduce and use all of the standard tools of the subject, andprove most of its fundamental theorems, while avoiding unnecessary generalization

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or specialization. I try to keep the approach as concrete as possible, with picturesand intuitive discussions of how one should think geometrically about the abstractconcepts, but without shying away from the powerful tools that modern mathemat-ics has to offer. To fit in all of the basics and still maintain a reasonably sane pace,I have had to omit or barely touch on a number of important topics, such as complexmanifolds, infinite-dimensional manifolds, connections, geodesics, curvature, fiberbundles, sheaves, characteristic classes, and Hodge theory. Think of them as dessert,to be savored after completing this book as the main course.

To convey the books compass, it is easiest to describe where it starts and whereit ends. The starting line is drawn just after topology: I assume that the reader hashad a rigorous introduction to general topology, including the fundamental groupand covering spaces. One convenient source for this material is my Introduction toTopological Manifolds [LeeTM], which I wrote partly with the aim of providing thetopological background needed for this book. There are other books that cover sim-ilar material well; I am especially fond of the second edition of Munkress Topology[Mun00]. The finish line is drawn just after a broad and solid background has beenestablished, but before getting into the more specialized aspects of any particularsubject. In particular, I introduce Riemannian metrics, but I do not go into connec-tions, geodesics, or curvature. There are many Riemannian geometry books for theinterested student to take up next, including one that I wrote [LeeRM] with the goalof moving expediently in a one-quarter course from basic smooth manifold theoryto nontrivial geometric theorems about curvature and topology. Similar material iscovered in the last two chapters of the recent book by Jeffrey Lee (no relation)[LeeJeff09], and do Carmo [dC92] covers a bit more. For more ambitious readers,I recommend the beautiful books by Petersen [Pet06], Sharpe [Sha97], and Chavel[Cha06].

This subject is often called differential geometry. I have deliberately avoidedusing that term to describe what this book is about, however, because the term ap-plies more properly to the study of smooth manifolds endowed with some extrastructuresuch as Lie groups, Riemannian manifolds, symplectic manifolds, vec-tor bundles, foliationsand of their properties that are invariant under structure-preserving maps. Although I do give all of these geometric structures their due (afterall, smooth manifold theory is pretty sterile without some geometric applications),I felt that it was more honest not to suggest that the book is primarily about one orall of these geometries. Instead, it is about developing the general tools for workingwith smooth manifolds, so that the reader can go on to work in whatever field ofdifferential geometry or its cousins he or she feels drawn to.

There is no canonical linear path through this material. I have chosen an order-ing of topics designed to establish a good technical foundation in the first half ofthe book, so that I can discuss interesting applications in the second half. Once thefirst twelve chapters have been completed, there is some flexibility in ordering theremaining chapters. For example, Chapter 13 (Riemannian Metrics) can be post-poned if desired, although some sections of Chapters 15 and 16 would have to bepostponed as well. On the other hand, Chapters 1921 (Distributions and Foliations,The Exponential Map, and Quotient Manifolds, respectively) could in principle be

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inserted any time after Chapter 14, and much of the material can be covered evenearlier if you are willing to skip ove