Characterization of projectively flat Finsler manifolds of constant curvature with finite dimensional holonomy group

TL;DR

This paper characterizes when the holonomy group of certain Finsler manifolds is finite dimensional, showing it occurs only in flat or Riemannian cases, thus linking geometric curvature properties with holonomy group structure.

Contribution

It provides a complete characterization of the holonomy group structure for simply connected locally projectively flat Finsler manifolds of constant curvature.

Findings

Holonomy group is finite dimensional iff the manifold is flat or Riemannian.

Holonomy groups of non-flat, non-Riemannian manifolds are infinite dimensional.

The result links curvature conditions with holonomy group properties.

Abstract

In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian.