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The Global Study of Riemannian-Finsler Geometry

  • Katsuhiro Shiohama
  • Bankteshwar Tiwari
Chapter

Abstract

The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W. Klingenberg. The key point in the study of Finsler manifolds is the non-symmetric property of its distance functions. We discuss fundamental results on the cut and conjugate loci of Finsler manifolds and note the differences between Riemannian and Finsler manifolds in these respects. The topological and differential structures on Riemannian manifolds, in the presence of convex functions, has been an active field of research in the second half of twentieth century. We discuss some results on Riemannian manifolds with convex functions and their recently proved analogues in the field of Finsler manifolds.

Keywords

Injectivity radius Cut locus Rauch conjecture Berger–Omori lemma Whitehead convexity Busemann function Klingenberg lemma Busemann-type geometry 

AMS Classification:

53C60 53C22 53C70 51H25 

Notes

Acknowledgement

The author’s “Katsuhiro Shiohama” work was supported by JSPS KAKENHI Grant Number15K04864.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Katsuhiro Shiohama
    • 1
  • Bankteshwar Tiwari
    • 2
  1. 1.Institute of Information ScienceFukuoka Institute of TechnologyFukuokaJapan
  2. 2.Centre for Interdisciplinary Mathematical Sciences, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

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