Riemannian Manifolds

An Introduction to Curvature

  • John M. Lee

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

Table of contents

  1. Front Matter
    Pages i-xv
  2. John M. Lee
    Pages 1-10
  3. John M. Lee
    Pages 47-64
  4. John M. Lee
    Pages 65-89
  5. John M. Lee
    Pages 91-113
  6. John M. Lee
    Pages 115-129
  7. John M. Lee
    Pages 131-153
  8. John M. Lee
    Pages 155-172
  9. John M. Lee
    Pages 173-191
  10. John M. Lee
    Pages 193-208
  11. Back Matter
    Pages 209-226

About this book

Introduction

This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints.

Keywords

Riemannian geometry Tensor Volume curvature manifold

Authors and affiliations

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

Bibliographic information

  • DOI https://doi.org/10.1007/b98852
  • Copyright Information Springer Science+Business Media New York 1997
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-98322-6
  • Online ISBN 978-0-387-22726-9
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • About this book