The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{\ast }M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{\ast }M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{(r)}M\widetilde{=}{T}^{r\ast }M$ between the $r$-th order vector tangent $T^{(r)}M=(J^r(M,R{)}_0{)}^{\ast }$ and the $r$-th order cotangent $T^{r\ast }M=J^r(M,R{)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M(g):T^{(r)}M\to T^{r\ast }M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.