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

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.

References in corpus (2)

• Homogeneous geodesics in homogeneous Finsler spaces
• Finsleroid space with angle and scalar product