Homogeneous geodesics in homogeneous Finsler spaces

TL;DR

This paper investigates the properties of homogeneous geodesics in Finsler spaces, providing criteria, existence results, and curvature analysis, with special focus on Lie groups and Randers spaces.

Contribution

It introduces a criterion for geodesic vectors, explores geodesics in Lie groups with bi-invariant metrics, and studies curvature properties in homogeneous Finsler spaces.

Findings

Homogeneous geodesics on Lie groups are cosets of one-parameter subgroups.

Infinitely many homogeneous geodesics exist on compact semi-simple Lie groups.

S-curvature vanishes along homogeneous geodesics.

Abstract

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.