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 \({\mathcal {D}}i\!f \hspace{-3pt} f_o({\mathbb {S}}^{n-1})\), 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 \({\mathcal {D}}i\!f \hspace{-3pt} f_o({\mathbb {S}}^{n-1})\). 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.