The holonomy of spherically symmetric projective Finsler metrics of constant curvature

TL;DR

This paper characterizes the holonomy group of spherically symmetric projective Finsler metrics with constant curvature, showing it is maximal and isomorphic to the diffeomorphism group of the sphere, including for notable metrics like Funk and Bryant-Shen.

Contribution

It provides the first explicit description of the holonomy group for non-Berwaldian n-dimensional Finsler manifolds with constant curvature.

Findings

Holonomy group is maximal for spherically symmetric cases.

Holonomy group is isomorphic to the identity component of diffeomorphisms of the sphere.

Results include the standard Funk and Bryant-Shen metrics.

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.