Geodesic
Liftings of vector fields to $1$-forms on the $r$-jet prolongation of the cotangent bundle
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 565-573.

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For natural numbers $r$ and $n\geq 2$ all natural operators $T_{\vert \Cal M f_n}\rightsquigarrow T^* (J^rT^{*})$ transforming vector fields from $n$-manifolds $M$ into $1$-forms on $J^r T^{*}M=\{j^r_x (\omega)\mid \omega \in \Omega^1(M), x\in M\}$ are classified. A similar problem with fibered manifolds instead of manifolds is discussed.
Classification : 58A20, 58A32
Keywords: natural bundle; natural operator 
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     title = {Liftings of vector fields to $1$-forms on the $r$-jet prolongation of the cotangent bundle},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {565--573},
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Mikulski, W. M. Liftings of vector fields to $1$-forms on the $r$-jet prolongation of the cotangent bundle. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 565-573. http://geodesic.mathdoc.fr/item/CMUC_2002__43_3_a16/