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

Jan Kurek; Włodzimierz Mikulski

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014), / Harvested from The Polish Digital Mathematics Library

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Résumé

If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ 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 $M$ .

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:289739

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     title = {The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds},
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     volume = {68},
     year = {2014},
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Jan Kurek; Włodzimierz Mikulski. The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_59/

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