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The Global Study of Riemannian-Finsler Geometry

In: Geometry in History

Author

Listed:
  • Katsuhiro Shiohama

    (Fukuoka Institute of Technology, Institute of Information Science)

  • Bankteshwar Tiwari

    (Banaras Hindu University, Centre for Interdisciplinary Mathematical Sciences, Institute of Science)

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.

Suggested Citation

  • Katsuhiro Shiohama & Bankteshwar Tiwari, 2019. "The Global Study of Riemannian-Finsler Geometry," Springer Books, in: S. G. Dani & Athanase Papadopoulos (ed.), Geometry in History, chapter 0, pages 581-621, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-13609-3_16
    DOI: 10.1007/978-3-030-13609-3_16
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