The minimal number of homogeneous geodesics depending on the signature of the Killing form

TL;DR

This paper investigates the minimum number of homogeneous geodesics in homogeneous Finsler manifolds, showing that indefinite Killing forms guarantee at least four such geodesics, with specific examples illustrating these results.

Contribution

It establishes a lower bound of four homogeneous geodesics for manifolds with indefinite Killing forms and provides explicit examples for both definite and indefinite cases.

Findings

Indefinite Killing form implies at least four homogeneous geodesics.

Examples of invariant Randers metrics with two or four geodesics based on Killing form signature.

Previous results on at least two geodesics are extended to a lower bound of four for indefinite cases.

Abstract

The existence of at least two homogeneous geodesics in any homogeneous Finsler manifold was proved in a previous paper by the author. The examples of solvable Lie groups with invariant Finsler metric which admit just two homogeneous geodesics were presented in another paper. In the present work, it is shown that a homogeneous Finsler manifold with indefinite Killing form admits at least four homogeneous geodesics. Examples of invariant Randers metrics on Lie groups with definite Killing form admitting just two homogeneous geodesics and examples with indefinite Killing form admitting just four homogeneous geodesics are presented.