Geodesic Vector fields of invariant $(\alpha,\beta)$-metrics on Homogeneous spaces

TL;DR

This paper characterizes geodesic vectors of invariant $(eta,eta)$-metrics on homogeneous spaces and provides explicit formulas for flag curvature on Lie groups, advancing understanding of Finsler geometry in symmetric spaces.

Contribution

It establishes conditions under which geodesic vectors of invariant $(eta,eta)$-metrics coincide with those of the underlying Riemannian metric and derives explicit flag curvature formulas.

Findings

A characterization of geodesic vectors for invariant $(eta,eta)$-metrics.

Conditions ensuring geodesic vectors of the Finsler metric match those of the Riemannian metric.

Explicit formula for flag curvature of bi-invariant $(eta,eta)$-metrics on Lie groups.

Abstract

In this paper we show that for an invariant $(\alpha,\beta)-$metric $F$ on a homogeneous Finsler manifold $\frac{G}{H}$, induced by an invariant Riemannian metric $\tilde{a}$ and an invariant vector field $\tilde{X}$, the vector $X=\tilde{X}(H)$ is a geodesic vector of $F$ if and only if it is a geodesic vector of $\tilde{a}$. Then we give some conditions such that under them, an arbitrary vector is a geodesic vector of $F$ if and only if it is a geodesic vector of $\tilde{a}$. Finally we give an explicit formula for the flag curvature of bi-invariant $(\alpha,\beta)-$metrics on connected Lie groups.