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Introduction to Riemannian Manifolds

  • John M. Lee

Part of the Graduate Texts in Mathematics book series (GTM, volume 176)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. John M. Lee
    Pages 1-8
  3. John M. Lee
    Pages 9-54
  4. John M. Lee
    Pages 55-83
  5. John M. Lee
    Pages 85-113
  6. John M. Lee
    Pages 115-150
  7. John M. Lee
    Pages 151-191
  8. John M. Lee
    Pages 193-224
  9. John M. Lee
    Pages 225-262
  10. John M. Lee
    Pages 263-282
  11. John M. Lee
    Pages 283-317
  12. John M. Lee
    Pages 319-344
  13. John M. Lee
    Pages 345-370
  14. Back Matter
    Pages 371-437

About this book

Introduction

This textbook is designed for a one or two semester graduate course on Riemannian geometry for students who are familiar with topological and differentiable manifolds. The second edition has been adapted, expanded, and aptly retitled from Lee’s earlier book, Riemannian Manifolds: An Introduction to Curvature. Numerous exercises and problem sets provide the student with opportunities to practice and develop skills; appendices contain a brief review of essential background material.

While demonstrating the uses of most of the main technical tools needed for a careful study of Riemannian manifolds, this text focuses on ensuring that the student develops an intimate acquaintance with the geometric meaning of curvature. The reasonably broad coverage begins with a treatment of indispensable tools for working with Riemannian metrics such as connections and geodesics. Several topics have been added, including an expanded treatment of pseudo-Riemannian metrics, a more detailed treatment of homogeneous spaces and invariant metrics, a completely revamped treatment of comparison theory based on Riccati equations, and a handful of new local-to-global theorems, to name just a few highlights.

Reviews of the first edition:

Arguments and proofs are written down precisely and clearly. The expertise of the author is reflected in many valuable comments and remarks on the recent developments of the subjects. Serious readers would have the challenges of solving the exercises and problems. The book is probably one of the most easily accessible introductions to Riemannian geometry. (M.C. Leung, MathReview)

The book’s aim is to develop tools and intuition for studying the central unifying theme in Riemannian geometry, which is the notion of curvature and its relation with topology. The main ideas of the subject, motivated as in the original papers, are introduced here in an intuitive and accessible way…The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research. (C.-L. Bejan, zBMATH)

Keywords

Riemannian geometry curvature manifold differential geometry textbook graduate mathematics textbook Riemannian geometry course textbook Riemannian metrics geodesics Levi-Cevita connection Riemannian submanifolds Gauss-Bonnet theorem Jacobi fields comparison theory curvature and topology tensor

Authors and affiliations

  • John M. Lee
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-91755-9
  • Copyright Information Springer International Publishing AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-91754-2
  • Online ISBN 978-3-319-91755-9
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site
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