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.
Categories: math.DG