Geodesic
The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 2.

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If (M,g) is a Riemannian manifold, we have 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, we generalize this isomorphism to the one T^(r)M=̃ 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→ T^r*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M. 
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Kurek, Jan; Mikulski, Włodzimierz. The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 68 (2014) no. 2. http://geodesic.mathdoc.fr/item/AUM_2014_68_2_a7/

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