Finsler manifolds with non-Riemannian holonomy

TL;DR

This paper demonstrates that certain Finsler manifolds have holonomy groups fundamentally different from Riemannian cases, including non-compact and non-Lie group holonomies, answering a longstanding open problem.

Contribution

It proves that non-Riemannian Finsler manifolds can have holonomy groups not realizable in Riemannian geometry, including examples on the Heisenberg group.

Findings

Holonomy group of certain Finsler manifolds is not a compact Lie group.

Existence of Finsler manifolds with holonomy groups not corresponding to any Riemannian manifold.

Provided explicit example on the Heisenberg group with non-Lie holonomy group.

Abstract

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension >2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. In addition, we provide an example of left invariant Finsler metric on the Heisenberg group, so that its holonomy group is not a (finite dimensional) Lie group. These results give a positive answer to the following problem formulated by S. S. Chern and Z. Shen: "Is there a Finsler manifold whose holonomy group is not the holonomy group of any Riemannian manifold?"