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.
@article{bwmeta1.element.ojs-doi-10_17951_a_2015_69_1_91,
author = {Mariusz Plaszczyk},
title = {The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {69},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2015_69_1_91}
}
Mariusz Plaszczyk. The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 69 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2015_69_1_91/
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