An Introduction to Riemann-Finsler Geometry

1. Front Matter

   Pages i-xx

   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

   PDF

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.

• Calc
• DEX
• Jacobi
• Lemma
• Natural
• Riemann-Finsler Geometry
• Riemannian Geometry
• Volume
• calculus
• calculus of variations
• constant
• curvature
• form
• theorem
• tool

"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

• 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

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