The affine approach to homogeneous geodesics in homogeneous Finsler spaces

TL;DR

This paper corrects a previous claim by proving that all homogeneous Berwald and reversible Finsler spaces admit homogeneous geodesics through any point, using an affine method instead of algebraic approaches.

Contribution

It introduces an affine method to study homogeneous geodesics in Finsler spaces and proves the existence of such geodesics in broader classes of spaces.

Findings

Homogeneous Berwald spaces admit homogeneous geodesics through any point.

Homogeneous reversible Finsler spaces admit homogeneous geodesics through any point.

The affine method effectively addresses gaps in previous algebraic proofs.

Abstract

In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present paper, homogeneous geodesics in Finsler homogeneous spaces are studied using the affine method, which was developed in earlier papers by the author. The mentioned statement is proved correctly and it is further proved that any homogeneous Berwald space or homogeneous reversible Finsler space admits a homogeneous geodesic through any point.