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 {D}i\!f -3pt fₒ ({S}^n-1) D i f f o (S n - 1), the connected component of the identity of the group of smooth diffeomorphism on the n-1 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 {D}i\!f -3pt fₒ ({S}^n-1) D i f f o (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.