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

Abstract

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