Finsler spaces with infinite dimensional holonomy group

TL;DR

This paper investigates the holonomy groups of Finsler surfaces using infinite dimensional Lie theory, revealing that certain classes have infinite-dimensional holonomy algebras and the holonomy group closure relates to the diffeomorphism group of the circle.

Contribution

It introduces the infinitesimal holonomy algebra for Finsler surfaces and characterizes the holonomy group structure for specific classes like Randers and Funk metrics.

Findings

Infinitesimal holonomy algebra coincides with curvature algebra for some Randers surfaces.

Holonomy algebra is infinite-dimensional for projectively flat Randers surfaces with non-zero constant flag curvature.

Holonomy group closure for Funk metric is the full orientation-preserving diffeomorphism group of the circle.

Abstract

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. We obtain that the topological closure of the holonomy group contains the exponential image of any tangent Lie algebra of the holonomy group. A class of Randers surfaces is determined, for which the infinitesimal holonomy algebra coincides with the curvature algebra. We prove that for all projectively flat Randers surfaces of non-zero constant flag curvature the infinitesimal holonomy algebra has infinite dimension and hence the holonomy group cannot be a Lie group of finite dimension. Finally, in the case of the Funk metric we prove that the infinitesimal holonomy algebra is a dense subalgebra of the Lie algebra of the full diffeomorphism group and hence the topological closure of the holonomy group is the orientation preserving diffeomorphism group of the circle.