Geodesic
The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 1.

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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 JrTM → JrT*M between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.
Keywords: Riemannian manifold, higher order prolongation of a vector bundle, natural tensor, natural operator 
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Plaszczyk, Mariusz. The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 1. http://geodesic.mathdoc.fr/item/AUM_2015_69_1_a7/

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