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© 2012

Introduction to Smooth Manifolds

  • New edition extensively revised and clarified, and topics have been substantially rearranged

  • Introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier in the text

  • Added topics include Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures

Textbook
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Part of the Graduate Texts in Mathematics book series (GTM, volume 218)

Table of contents

  1. John M. Lee
    Pages 515-539
  2. John M. Lee
    Pages 540-563
  3. John M. Lee
    Pages 564-595
  4. Back Matter
    Pages 596-708

About this book

Introduction

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research—smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Keywords

Frobenius theorem Lie group Sard’s theorem Smooth structures Stokes's theorem Tangent vectors and covectors Whitney approximation theorem Whitney embedding theorem de Rham cohomology differential forms first-order partial differential equations foliations immersed and embedded submanifolds smooth manifolds tensors vector bundles vector fields and flows

Authors and affiliations

  1. 1., Department of MathematicsUniversity of WashingtonSeattleUSA

About the authors

John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of four previous Springer books: the first edition (2003) of Introduction to Smooth Manifolds, the first edition (2000) and second edition (2010) of Introduction to Topological Manifolds, and Riemannian Manifolds: An Introduction to Curvature (1997).

Bibliographic information

  • Book Title Introduction to Smooth Manifolds
  • Authors John Lee
  • Series Title Graduate Texts in Mathematics
  • DOI https://doi.org/10.1007/978-1-4419-9982-5
  • Copyright Information Springer Science+Business Media New York 2012
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4419-9981-8
  • Softcover ISBN 978-1-4899-9475-2
  • eBook ISBN 978-1-4419-9982-5
  • Series ISSN 0072-5285
  • Edition Number 2
  • Number of Pages XVI, 708
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Differential Geometry
  • Buy this book on publisher's site

Reviews

From the reviews of the second edition:

“It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. … the book under review is laden with excellent exercises that significantly further the reader’s understanding of the material, and Lee takes great pains to motivate everything well all the way through … . a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject.” (Michael Berg, MAA Reviews, October, 2012)