Two-step homogeneous geodesics in some homogeneous Finsler manifolds

TL;DR

This paper extends the concept of two-step homogeneous geodesics from Riemannian to Finsler spaces, providing conditions for their existence and examples in specific Finsler manifolds.

Contribution

It introduces the notion of two-step homogeneous geodesics in homogeneous Finsler spaces and identifies conditions under which these spaces admit such geodesics.

Findings

Certain $( ext{α,β})$ spaces admit two-step Finsler geodesic structures.

Decomposable cubic spaces can possess invariant Finsler metrics with two-step geodesic properties.

Examples demonstrate the existence of these geodesic structures in specific homogeneous Finsler manifolds.

Abstract

A natural extension of a homogeneous geodesic in homogeneous Riemannian spaces $G/H$, known as a two-step homogeneous geodesic, can be expressed of the form $\gamma(t)=\pi(\exp(tx)\exp(ty))$, where $x$ and $y$ are elements of the Lie algebra of $G$. This paper aims to expand this concept to homogeneous Finsler spaces. We provide certain sufficient conditions for $(\alpha,\beta)$ spaces and decomposable cubic spaces to possess a one-parameter family of invariant Finsler metrics that can be classified as two-step Finsler geodesic orbit spaces. Additionally, we present some illustrative examples of these spaces.