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  • © 1997

Riemannian Manifolds

An Introduction to Curvature

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

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  • ISBN: 978-0-387-22726-9
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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xv
  2. What is Curvature?

    • John M. Lee
    Pages 1-10
  3. Connections

    • John M. Lee
    Pages 47-64
  4. Riemannian Geodesics

    • John M. Lee
    Pages 65-89
  5. Geodesics and Distance

    • John M. Lee
    Pages 91-113
  6. Curvature

    • John M. Lee
    Pages 115-129
  7. Riemannian Submanifolds

    • John M. Lee
    Pages 131-153
  8. The Gauss-Bonnet Theorem

    • John M. Lee
    Pages 155-172
  9. Jacobi Fields

    • John M. Lee
    Pages 173-191
  10. Curvature and Topology

    • John M. Lee
    Pages 193-208
  11. Back Matter

    Pages 209-226

About this book

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

Reviews

"This book is very well writen, pleasant to read, with many good illustrations. It deals with the core of the subject, nothing more and nothing less. Simply a recommendation for anyone who wants to teach or learn about the Riemannian geometry."
Nieuw Archief voor Wiskunde, September 2000

Authors and Affiliations

  • Department of Mathematics, University of Washington, Seattle, USA

    John M. Lee

Bibliographic Information

  • Book Title: Riemannian Manifolds

  • Book Subtitle: An Introduction to Curvature

  • Authors: John M. Lee

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/b98852

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1997

  • Hardcover ISBN: 978-0-387-98271-7

  • eBook ISBN: 978-0-387-22726-9

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XV, 226

  • Topics: Differential Geometry

Buying options

eBook USD 64.99
Price excludes VAT (USA)
  • ISBN: 978-0-387-22726-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book USD 84.95
Price excludes VAT (USA)