A characterization of holonomy invariant functions on tangent bundles

TL;DR

This paper characterizes holonomy invariant functions on tangent bundles, linking them to local parallelisms, and provides a new perspective on generalized Berwald manifolds, including an example that is not Berwald.

Contribution

It establishes a new characterization of holonomy invariant functions via local parallelisms and offers a novel example of a generalized Berwald manifold that is not Berwald.

Findings

Holonomy invariance is equivalent to the existence of compatible local parallelisms.

Provides a characterization of generalized Berwald manifolds.

Constructs an example of a non-Berwald generalized Berwald manifold.

Abstract

We show that the holonomy invariance of a function on the tangent bundle of a manifold, together with very mild regularity conditions on the function, is equivalent to the existence of local parallelisms compatible with the function in a natural way. Thus, in particular, we obtain a characterization of generalized Berwald manifolds. We also construct a simple example of a generalized Berwald manifold which is not Berwald.