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 $π:M\rightarrow M_1$ 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.

In the second version of this paper, we corrected some mistakes and made some changes on the main theorem and the structure of the paper