Homogeneous geodesics of left invariant Finsler metrics

TL;DR

This paper investigates the properties and criteria of homogeneous geodesics in left-invariant Finsler metrics on Lie groups, providing new conditions for geodesic vectors and characterizing bi-invariant Randers metrics.

Contribution

It introduces a simple criterion for geodesic vectors, characterizes when left-invariant Randers metrics are of Berwald type, and links homogeneous geodesics to critical points of restricted Finsler metrics.

Findings

A criterion for geodesic vectors in left-invariant Finsler metrics.

Necessary and sufficient conditions for bi-invariant Randers metrics to be Berwald.

Existence results for homogeneous geodesics on Lie groups.

Abstract

In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient condition that left-invariant Randers metrics are of Berwald type is given. Finally a correspondence of homogeneous geodesics to critical points of restricted Finsler metrics is given. Then results concerning the existence homogeneous geodesics are obtained.