The existence of homogeneous geodesics in special homogeneous Finsler spaces

Abstract

A well known result by O. Kowalski and J. Szenthe says that any homogeneous Riemannian manifoldadmits a homogeneous geodesic through any point.This was proved by the algebraic method using the reductive decomposition of the Lie algebra of the isometry group.In previous papers by the author, the existence of a homogeneous geodesic in any homogeneous pseudo-Riemannianmanifold and also in any homogeneous affine manifold was proved.In this setting, a new method based on affine Killing vector fields was developed.Using this method, it was further proved that any homogeneous Lorentzian manifoldof even dimension admits a light-like homogeneous geodesicand any homogeneous Finsler space of odd dimension admits a homogeneous geodesic.In the present paper, the affine method is further refined for Finsler spaces and it is proved that any homogeneousBerwald space or homogeneous reversible Finsler space admits a homogeneous geodesic through any point.