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January 2021 Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$
R. L. Bryant, P. Foulon, S. V. Ivanov, V. S. Matveev, W. Ziller
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J. Differential Geom. 117(1): 1-22 (January 2021). DOI: 10.4310/jdg/1609902015

Abstract

We study non-reversible Finsler metrics with constant flag curvature $1$ on $S^2$ and show that the geodesic flow of every such metric is conjugate to that of one of Katok’s examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable.

Finally, we give an example of a Finsler metric on $S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.

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R. L. Bryant. P. Foulon. S. V. Ivanov. V. S. Matveev. W. Ziller. "Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$." J. Differential Geom. 117 (1) 1 - 22, January 2021. https://doi.org/10.4310/jdg/1609902015

Information

Received: 21 October 2017; Published: January 2021
First available in Project Euclid: 6 January 2021

MathSciNet: MR4195750
Digital Object Identifier: 10.4310/jdg/1609902015

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 1 • January 2021
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