The Global Study of Riemannian-Finsler Geometry

Katsuhiro Shiohama () and Bankteshwar Tiwari ()
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Katsuhiro Shiohama: Fukuoka Institute of Technology, Institute of Information Science
Bankteshwar Tiwari: Banaras Hindu University, Centre for Interdisciplinary Mathematical Sciences, Institute of Science

Chapter Chapter 16 in Geometry in History, 2019, pp 581-621 from Springer

Abstract: 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; 53C60; 53C22; 53C70; 51H25 (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-13609-3_16

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DOI: 10.1007/978-3-030-13609-3_16

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