Geodesic orbit Finsler metrics on Euclidean spaces

Abstract

A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a symmetric Finsler space $M_1$ of non-compact type such that each fiber $M_2$ is a totally geodesic nilmanifold with a step-size at most 2, and the projection $\pi:M\rightarrow M_1$ is a Finslerian submersion. Furthermore, when $M_1$ has no Hermitian symmetric factors, the fiber bundle description for $M$ can be strengthened to $M=M_1\times M_2$ as coset spaces, such that each product factor is totally geodesic in $(M,F)$ and is a geodesic orbit Finsler space itself. Finally, we use the techniques in this paper to discuss the interaction between the geodesic orbit spaces and the negative (non-positive) curved conditions, and provide new proofs for some of our previous results.