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

Jan Kurek, and Włodzimierz M. Mikulski. "The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds." Annales UMCS, Mathematica 68.2 (2015): 59-64. <http://eudml.org/doc/270006>.

@article{JanKurek2015,
abstract = {If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) 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 ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.},
author = {Jan Kurek, Włodzimierz M. Mikulski},
journal = {Annales UMCS, Mathematica},
keywords = {Riemannian manifold; higher order vector tangent bundle; higher order cotangent bundle; natural tensor; natural operator},
language = {eng},
number = {2},
pages = {59-64},
title = {The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds},
url = {http://eudml.org/doc/270006},
volume = {68},
year = {2015},
}

TY - JOUR
AU - Jan Kurek
AU - Włodzimierz M. Mikulski
TI - The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds
JO - Annales UMCS, Mathematica
PY - 2015
VL - 68
IS - 2
SP - 59
EP - 64
AB - If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) 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 ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.
LA - eng
KW - Riemannian manifold; higher order vector tangent bundle; higher order cotangent bundle; natural tensor; natural operator
UR - http://eudml.org/doc/270006
ER -