We determine all natural operators transforming vector fields on a manifold $M$ to vector fields on $T^*T^2_1M$, $\operatorname{dim}M \ge 2$, and all natural operators transforming vector fields on $M$ to functions on $T^*TT^2_1M$, $\operatorname{dim}M \ge 3$. We describe some relations between these two kinds of natural operators.
@article{107543,
author = {Ji\v r\'\i\ M. Tom\'a\v s},
title = {Some natural operators on vector fields},
journal = {Archivum Mathematicum},
volume = {031},
year = {1995},
pages = {239-249},
zbl = {0844.58007},
mrnumber = {1368261},
language = {en},
url = {http://dml.mathdoc.fr/item/107543}
}
Tomáš, Jiří M. Some natural operators on vector fields. Archivum Mathematicum, Tome 031 (1995) pp. 239-249. http://gdmltest.u-ga.fr/item/107543/
The canonical isomorphism between $T^kT^*M$ and $T^*T^kM$, C. R. Acad. Sci. Paris 309 (1989), 1509-1514. (1989) | MR 1033091
Natural liftings of vector fields to tangent bundles of $1$-forms, Mathematica Bohemica 116 (1991), 319-326. (1991) | MR 1126453
Covariant Approach to Natural Transformations of Weil Bundles, Comment. Math. Univ. Carolinae (1986). (1986) | MR 0874666
On Cotagent Bundles of Some Natural Bundles, to appear in Rendiconti del Circolo Matematico di Palermo. | MR 1344006
On the Natural Operators on Vector Fields, Ann. Global Anal. Geometry 6 (1988), 109-117. (1988) | MR 0982760
Natural Operations in Differential Geometry, Springer – Verlag, 1993. (1993) | MR 1202431
Natural Transformations of Second Tangent and Cotangent Bundles, Czechoslovak Math. (1988), 274-279. (1988) | MR 0946296
Some results on second tangent and cotangent spaces, Quaderni dell’Università di Lecce (1978). (1978)