The holonomy of spherically symmetric projective Finsler metrics of constant curvature

Abstract

In this paper, we investigate the holonomy group of $n$-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to $Diff_o({\mathbb S^{n-1}})$, the connected component of the identity of the group of smooth diffeomorphism on the $(n-1)$-dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant-Shen metrics are maximal and isomorphic to $Diff_o({\mathbb S^{n-1}})$. These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.