On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

TL;DR

This paper investigates the existence of homogeneous geodesics in singular homogeneous Finsler spaces, especially Kropina spaces, establishing conditions for their existence and providing examples involving Lie groups with Randers metrics.

Contribution

It extends the existence results of homogeneous geodesics to singular homogeneous $( ext{alpha},eta)$-spaces, particularly Kropina spaces, and characterizes geodesic vectors in Douglas type spaces.

Findings

Homogeneous Kropina spaces admit at least one homogeneous geodesic through any point.

Conditions are identified under which $( ext{alpha},eta)$-homogeneous spaces have homogeneous geodesics.

A necessary and sufficient condition for geodesic vectors in Douglas type spaces is provided.

Abstract

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(\alpha,\beta)$-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any $(\alpha,\beta)$-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.