A characterization of holonomy invariant functions on tangent bundles

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. M.S.C. 2010: 53B05, 53B40. Key words: holonomy invariance; parallel translation; parallelism; generalized Berwald manifold; one-form manifold.