Finsler 2-manifolds whose holonomy group is the diffeomorphism group of the circle

TL;DR

This paper demonstrates that certain projectively flat Finsler 2-manifolds with constant curvature have holonomy groups whose topological closure is the entire diffeomorphism group of the circle, revealing fully infinite-dimensional Finslerian holonomy structures.

Contribution

It establishes the maximality of the holonomy group for a class of Finsler 2-manifolds, including Funk planes and Bryant-Shen-spheres, as the connected component of the circle's diffeomorphism group.

Findings

Holonomy group closure is isomorphic to the circle's diffeomorphism group.

Includes examples like Funk plane and Bryant-Shen-spheres.

First complete description of infinite-dimensional Finslerian holonomy.

Abstract

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.