An Introduction to Riemann-Finsler Geometry

✍ Scribed by D. Bao, S.-S. Chern, Z. Shen (auth.)

• Publisher: Springer-Verlag New York
• Year: 2000
• Tongue: English
• Leaves: 454
• Series: Graduate Texts in Mathematics 200
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xx
Finsler Manifolds and the Fundamentals of Minkowski Norms....Pages 1-26
The Chern Connection....Pages 27-48
Curvature and Schur’s Lemma....Pages 49-80
Finsler Surfaces and a Generalized Gauss-Bonnet Theorem....Pages 81-110
Variations of Arc Length, Jacobi Fields, the Effect of Curvature....Pages 111-138
The Gauss Lemma and the Hopf-Rinow Theorem....Pages 139-172
The Index Form and the Bonnet—Myers Theorem....Pages 173-198
The Cut and Conjugate Loci, and Synge’s Theorem....Pages 199-224
The Cartan—Hadamard Theorem and Rauch’s First Theorem....Pages 225-256
Berwald Spaces and Szabó’s Theorem for Berwald Surfaces....Pages 257-280
Randers Spaces and an Elegant Theorem....Pages 281-310
Constant Flag Curvature Spaces and Akbar-Zadeh’s Theorem....Pages 311-350
Riemannian Manifolds and Two of Hopf’s Theorems....Pages 351-382
Minkowski Spaces, the Theorems of Deicke and Brickell....Pages 383-418
Back Matter....Pages 419-435

✦ Subjects

Geometry