Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$

TL;DR

This paper investigates the geodesic flows of non-reversible Finsler metrics with constant positive flag curvature on S^2, revealing their conjugacy to Katok's examples and establishing invariants and integrability properties.

Contribution

It demonstrates that all such Finsler metrics have geodesic flows conjugate to Katok's examples, introduces the shortest closed geodesic length as an invariant, and shows geodesic flow integrability in any dimension.

Findings

Geodesic flow conjugate to Katok's examples

Shortest closed geodesic length as a flow invariant

Complete integrability of geodesic flow in any dimension

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