An introduction to Riemann-Finsler geometry

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.

From the contents: Finsler methods and the fundamentals of Minkowski norms.- The Chern connection.- Curvature and Schur's Lemma.- Finsler surfaces and a generalized Gauss-Bonnet theorem.- Variations of arc length, Jacobi fields, and the effect of curvature.- The Gauss lemma and the Hopf-Rinow theorem.- The index form and the Bonnet-Myers theorem.- The cut and conjugate loci, and Synge's theorem.- The Cartan-Hadamard theorem and Rauch's first theorem.- Berwald spaces and Szabo's theorem for Berwald spaces.- Randers spaces and a theorem from the Japanese school.- Constant flag curvature spaces, and ther Andar-Zadeh theorem.- Riemannian manifolds and two theorems of Hopf's.- Minkowski spaces, the theorems of Dickie and Brickell

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