Homogeneous Finsler spaces with only one orbit of prime closed geodesics

TL;DR

This paper investigates whether homogeneous Finsler spaces with a single orbit of prime closed geodesics are necessarily compact rank-one Riemannian symmetric spaces, providing positive results under certain conditions.

Contribution

It extends the understanding of prime closed geodesics in homogeneous Finsler spaces, proving a characterization in specific cases and suggesting methods for broader contexts.

Findings

Positive answer for even-dimensional cases

Positive answer for reversible metrics

Potential for extending techniques to non-homogeneous spaces

Abstract

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two prime closed geodesics to the equivariant situation, we purpose the question if a closed Finsler manifold has only one orbit of prime closed geodesic if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler geometry, and get a positive answer when the dimension is even or the metric is reversible. We guess the rank inequality and algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.