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

R. L Bryant (1) , P. Foulon (2) , S. V Ivanov (3) , V. S Matveev (4) , W. Ziller (5)

1 Duke University [Durham]
2 CIRM - Centre International de Rencontres Mathématiques
3 Odessa State Environmental University
4 Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany]
5 University of Penssylvania

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.

Domaines: Géométrie métrique [math.MG]; Systèmes dynamiques [math.DS]

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⟩