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An Introduction to Riemann-Finsler Geometry

  • Textbook
  • © 2000

Overview

Authors:
  1. D. Bao

    1. Department of Mathematics, University of Houston, University Park, Houston, USA

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  2. S.-S. Chern

    1. Department of Mathematics, University of California at Berkeley, Berkeley, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Z. Shen

    1. Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 200)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xx
    Download chapter PDF

  2. Finsler Manifolds and Their Curvature

    1. Finsler Manifolds and the Fundamentals of Minkowski Norms

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 1-26

    2. The Chern Connection

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 27-48

    3. Curvature and Schur’s Lemma

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 49-80

    4. Finsler Surfaces and a Generalized Gauss-Bonnet Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 81-110

  3. Calculus of Variations and Comparison Theorems

    1. Variations of Arc Length, Jacobi Fields, the Effect of Curvature

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 111-138

    2. The Gauss Lemma and the Hopf-Rinow Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 139-172

    3. The Index Form and the Bonnet—Myers Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 173-198

    4. The Cut and Conjugate Loci, and Synge’s Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 199-224

    5. The Cartan—Hadamard Theorem and Rauch’s First Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 225-256

  4. Special Finsler Spaces over the Reals

    1. Berwald Spaces and Szabó’s Theorem for Berwald Surfaces

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 257-280

    2. Randers Spaces and an Elegant Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 281-310

    3. Constant Flag Curvature Spaces and Akbar-Zadeh’s Theorem

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 311-350

    4. Riemannian Manifolds and Two of Hopf’s Theorems

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 351-382

    5. Minkowski Spaces, the Theorems of Deicke and Brickell

      • D. Bao, S.-S. Chern, Z. Shen
      Pages 383-418

  5. Back Matter

    Pages 419-435
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Keywords

About this book

In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe?
It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.

Reviews

"This book offers the most modern treatment of the topic and will attract both graduate students and a broad community of mathematicians from various related fields."
EMS Newsletter, Issue 41, September 2001

Authors and Affiliations

  • Department of Mathematics, University of Houston, University Park, Houston, USA

    D. Bao

  • Department of Mathematics, University of California at Berkeley, Berkeley, USA

    S.-S. Chern

  • Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, USA

    Z. Shen

Bibliographic Information

  • Book Title: An Introduction to Riemann-Finsler Geometry

  • Authors: D. Bao, S.-S. Chern, Z. Shen

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-1268-3

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2000

  • Hardcover ISBN: 978-0-387-98948-8Published: 17 March 2000

  • Softcover ISBN: 978-1-4612-7070-6Published: 03 October 2012

  • eBook ISBN: 978-1-4612-1268-3Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XX, 435

  • Topics: Geometry

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