For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{(r),a}$ over $n$-manifolds such that $T^{(r),0}$ is the (classical) vector tangent bundle $T^{(r)}$ of order $r$. For integers $r\ge 1$ and $n\ge 3$ and a real number $a<0$ we classify all natural operators $T_{\vert M_n}\rightsquigarrow TT^{(r),a}$ lifting vector fields from $n$-manifolds to $T^{(r),a}$.
@article{107733,
author = {W\l odzimierz M. Mikulski},
title = {The natural operators lifting vector fields to generalized higher order tangent bundles},
journal = {Archivum Mathematicum},
volume = {036},
year = {2000},
pages = {207-212},
zbl = {1049.58010},
mrnumber = {1785038},
language = {en},
url = {http://dml.mathdoc.fr/item/107733}
}
Mikulski, Włodzimierz M. The natural operators lifting vector fields to generalized higher order tangent bundles. Archivum Mathematicum, Tome 036 (2000) pp. 207-212. http://gdmltest.u-ga.fr/item/107733/
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