The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one J r TM → J r T*M between the r-th order prolongation J r TM of tangent TM and the r-th order prolongation J r T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D M (g) : J r TM → J r T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.