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 metrics 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 have constant flag curvature

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• Riemann-Finsler Geometry and Lorentz-Violating Scalar Fields
• Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds
• Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics
• The number of geometrically distinct reversible closed geodesics on a Finsler sphere with $K\equiv 1$
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
• Homogeneous Finsler sphere with constant flag curvature
• Homogeneous Finsler spaces with only one orbit of prime closed geodesics