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AuthorBao, D. author
TitleAn Introduction to Riemann-Finsler Geometry [electronic resource] / by D. Bao, S.-S. Chern, Z. Shen
ImprintNew York, NY : Springer New York : Imprint: Springer, 2000
Connect tohttp://dx.doi.org/10.1007/978-1-4612-1268-3
Descript XX, 435 p. online resource

SUMMARY

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


CONTENT

One Finsler Manifolds and Their Curvature -- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms -- 2 The Chern Connection -- 3 Curvature and Schurโ{128}{153}s Lemma -- 4 Finsler Surfaces and a Generalized Gaussโ{128}{148}Bonnet Theorem -- Two Calculus of Variations and Comparison Theorems -- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature -- 6 The Gauss Lemma and the Hopf-Rinow Theorem -- 7 The Index Form and the Bonnet-Myers Theorem -- 8 The Cut and Conjugate Loci, and Syngeโ{128}{153}s Theorem -- 9 The Cartan-Hadamard Theorem and Rauchโ{128}{153}s First Theorem -- Three Special Finsler Spaces over the Reals -- 10 Berwald Spaces and Szabรณโ{128}{153}s Theorem for Berwald Surfaces -- 11 Randers Spaces and an Elegant Theorem -- 12 Constant Flag Curvature Spaces and Akbar-Zadehโ{128}{153}s Theorem -- 13 Riemannian Manifolds and Two of Hopfโ{128}{153}s Theorems -- 14 Minkowski Spaces, the Theorems of Deicke and Brickell


Mathematics Geometry Mathematics Geometry



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