Far East Journal of Applied Mathematics

The Far East Journal of Applied Mathematics publishes original research papers and survey articles in applied mathematics, covering topics such as nonlinear dynamics, approximation theory, and mathematical modeling. It encourages papers focusing on algorithm development.

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FINSLER METRICS WITH BUSEMANN CURVATURE BOUNDS

Authors

  • Chang-Wan Kim

Keywords:

Finsler metric, Busemann curvature, Berwald metric

DOI:

https://doi.org/10.17654/0972096023014

Abstract

We prove that a Finsler metric has Busemann curvature bounded above (below, respectively) by $\kappa$ if and only if it is the Berwald metric with flag curvature bounded above (below, respectively) by $\kappa$. Combining this with Szabó’s Berwald metrization theorem, we can obtain that such a Finsler metric is affinely equivalent to a Riemannian metric with sectional curvature bounded above (below, respectively) by $\kappa$.

Received: July 13, 2023
Revised: August 25, 2023
Accepted: September 1, 2023

References

D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics 200, Springer-Verlag, 2000.

N. Boonnam, R. Hama and S. V. Sabau, Berwald spaces of bounded curvature are Riemannian, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 33 (2018), 339-347.

Y. Burago, M. Gromov and G. Perelman, A. D. Alexandrov spaces with curvature bounded below, Russ. Math. Surv. 47 (1992), 3-51.

H. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948), 259-310.

S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific, 2005.

S. Ivanov and A. Lytchak, Rigidity of Busemann convex Finsler metrics, Comment. Math. Helv. 94 (2019), 855-868.

E. Kann, Bonnet’s theorem in two-dimensional G-spaces, Comm. Pure Appl. Math. 14 (1961), 765-784.

M. Kell, Sectional curvature-type conditions on metric spaces, J. Geom. Anal. 29 (2019), 616-655.

P. Kelly and E. Straus, Curvature in Hilbert geometries, Pacific J. Math. 8 (1958), 119-125.

V. S. Matveev and M. Troyanov, The Binet-Legendre metric in Finsler geometry, Geom. Topol. 16 (2012), 2135-2170.

Z. I. Szabó, Positive definite Berwald spaces, Tensor (N.S.) 35 (1981), 25-39.

Z. I. Szabó, Berwald metrics constructed by Chevalley’s polynomials, 2006. Preprint, Available at: arXiv:math/0601522.

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Published

2023-10-12

Issue

Vol. 116 No. 3 (2023)

Section

Articles

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How to Cite

FINSLER METRICS WITH BUSEMANN CURVATURE BOUNDS. (2023). Far East Journal of Applied Mathematics, 116(3), 249-262. https://doi.org/10.17654/0972096023014