Geodesic orbit Finsler space with $K\geq0$ and the (FP) condition

Abstract

In this paper, we study the interaction between the geodesic orbit (g.o.~in short) property and certain flag curvature conditions. A Finsler manifold is called g.o.~if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also concern the (FP) condition for the flag curvature, i.e., in any flag we can find a flag pole, such that the flag curvature is positive. The main theorem we will prove is the following. If a g.o.~Finsler space $(M,F)$ has non-negative flag curvature and satisfies the (FP) condition, then $M$ must be compact. Further more, if we present $M$ as $G/H$ where $G$ has a compact Lie algebra, then we have the rank inequality $\mathrm{rk}\mathfrak{g}\leq\mathrm{rk}\mathfrak{h}+1$. As an application of the main theorem, we prove that any even dimensional g.o.~Finsler space which has non-negative flag curvature and satisfies the (FP) condition must be a smooth coset space admitting positively curved homogeneous Riemannian or Finsler metrics.