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

Jan Kurek, and Włodzimierz Mikulski. "The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 68.2 (2014): null. <http://eudml.org/doc/289739>.

@article{JanKurek2014,
abstract = {If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\mathrel \{\tilde\{=\}\}T^*M$ given by $v\rightarrow g(v,-)$ between the tangent $TM$ and the cotangent $T^*M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^\{(r)\}M\mathrel \{\tilde\{=\}\} T^\{r*\}M$ 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 $C_M(g):T^\{(r)\}M\rightarrow T^\{r*\}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.},
author = {Jan Kurek, Włodzimierz Mikulski},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {},
language = {eng},
number = {2},
pages = {null},
title = {The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds},
url = {http://eudml.org/doc/289739},
volume = {68},
year = {2014},
}

TY - JOUR
AU - Jan Kurek
AU - Włodzimierz Mikulski
TI - The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2014
VL - 68
IS - 2
SP - null
AB - If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\mathrel {\tilde{=}}T^*M$ given by $v\rightarrow g(v,-)$ between the tangent $TM$ and the cotangent $T^*M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{(r)}M\mathrel {\tilde{=}} T^{r*}M$ 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 $C_M(g):T^{(r)}M\rightarrow T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.
LA - eng
KW -
UR - http://eudml.org/doc/289739
ER -