Lifting vector fields to the rth order frame bundle

J. Kurek, and W. M. Mikulski. "Lifting vector fields to the rth order frame bundle." Colloquium Mathematicae 111.1 (2008): 51-58. <http://eudml.org/doc/284029>.

@article{J2008,
abstract = {We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^\{r\}M = inv J₀^\{r\}(ℝ^m, M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^\{r\}M$. In both cases we deduce that the spaces of all operators in question form free $(m(C^\{m+r\}_\{r\}-1) + 1)$-dimensional modules over algebras of all smooth maps $J₀^\{r-1\}T̃ℝ^m → ℝ$ and $J₀^\{r-1\}Tℝ^m → ℝ$ respectively, where $Cⁿ_k = n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we find that the vector space over ℝ of all natural linear operators lifting vector fields X on m-manifolds M to vector fields on $L^\{r\}M$ is $(m²C^\{m+r-1\}_\{r-1\}(C^\{m+r\}_r - 1) + 1)$-dimensional.},
author = {J. Kurek, W. M. Mikulski},
journal = {Colloquium Mathematicae},
keywords = {natural bundle; natural operator; jet},
language = {eng},
number = {1},
pages = {51-58},
title = {Lifting vector fields to the rth order frame bundle},
url = {http://eudml.org/doc/284029},
volume = {111},
year = {2008},
}

TY - JOUR
AU - J. Kurek
AU - W. M. Mikulski
TI - Lifting vector fields to the rth order frame bundle
JO - Colloquium Mathematicae
PY - 2008
VL - 111
IS - 1
SP - 51
EP - 58
AB - We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r}M = inv J₀^{r}(ℝ^m, M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$. In both cases we deduce that the spaces of all operators in question form free $(m(C^{m+r}_{r}-1) + 1)$-dimensional modules over algebras of all smooth maps $J₀^{r-1}T̃ℝ^m → ℝ$ and $J₀^{r-1}Tℝ^m → ℝ$ respectively, where $Cⁿ_k = n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we find that the vector space over ℝ of all natural linear operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$ is $(m²C^{m+r-1}_{r-1}(C^{m+r}_r - 1) + 1)$-dimensional.
LA - eng
KW - natural bundle; natural operator; jet
UR - http://eudml.org/doc/284029
ER -