GEODESIC BEHAVIOR FOR FINSLER METRICS OF CONSTANT POSITIVE FLAG CURVATURE ON S 2 - Archive ouverte HAL Access content directly
Journal Articles Journal of Differential Geometry Year : 2019

GEODESIC BEHAVIOR FOR FINSLER METRICS OF CONSTANT POSITIVE FLAG CURVATURE ON S 2

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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Dates and versions

hal-02361817 , version 1 (13-11-2019)
hal-02361817 , version 2 (22-01-2021)

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Cite

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. Journal of Differential Geometry, In press, 117 (1), pp.1-22. ⟨10.4310/jdg/1609902015⟩. ⟨hal-02361817v2⟩
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