Geodesic
The natural operators lifting connections to higher order cotangent bundles
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 509-518 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We prove that the problem of finding all ${\mathcal {M} f_m}$-natural operators ${C\colon Q\rightsquigarrow QT^{r*}}$ lifting classical linear connections $\nabla $ on $m$-manifolds $M$ into classical linear connections $C_M(\nabla )$ on the $r$-th order cotangent bundle $T^{r*}M=J^r(M,\mathbb R )_0$ of $M$ can be reduced to the well known one of describing all $\mathcal {M} f_m$-natural operators $D\colon Q\rightsquigarrow \bigotimes ^pT\otimes \bigotimes ^qT^*$ sending classical linear connections $\nabla $ on $m$-manifolds $M$ into tensor fields $D_M(\nabla )$ of type $(p,q)$ on $M$.
We prove that the problem of finding all ${\mathcal {M} f_m}$-natural operators ${C\colon Q\rightsquigarrow QT^{r*}}$ lifting classical linear connections $\nabla $ on $m$-manifolds $M$ into classical linear connections $C_M(\nabla )$ on the $r$-th order cotangent bundle $T^{r*}M=J^r(M,\mathbb R )_0$ of $M$ can be reduced to the well known one of describing all $\mathcal {M} f_m$-natural operators $D\colon Q\rightsquigarrow \bigotimes ^pT\otimes \bigotimes ^qT^*$ sending classical linear connections $\nabla $ on $m$-manifolds $M$ into tensor fields $D_M(\nabla )$ of type $(p,q)$ on $M$.
DOI : 10.1007/s10587-014-0116-7
Classification : 53C05, 58A20, 58A32
Keywords: classical linear connection; natural operator 
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Mikulski, Włodzimierz M. The natural operators lifting connections to higher order cotangent bundles. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 509-518. doi: 10.1007/s10587-014-0116-7

[1] Dębecki, J.: Affine liftings of torsion-free connections to Weil bundles. Colloq. Math. 114 (2009), 1-8. | DOI | MR

[2] Gancarzewicz, J.: Horizontal lift of connections to a natural vector bundle. Differential Geometry Proc. 5th Int. Colloq., Santiago de Compostela, Spain, 1984, Res. Notes Math. 131 Pitman, Boston (1985), 318-341 L. A. Cordero. | MR | Zbl

[3] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. I. Interscience Publishers, New York (1963). | MR | Zbl

[4] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer Berlin (1993). | MR

[5] Kurek, J., Mikulski, W. M.: The natural operators lifting connections to tensor powers of the cotangent bundle. Miskolc Math. Notes 14 (2013), 517-524. | DOI | MR

[6] Kureš, M.: Natural lifts of classical linear connections to the cotangent bundle. J. Slovák Proc. of the 15th Winter School on geometry and physics, Srní, 1995, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 43 (1996), 181-187. | MR | Zbl

[7] Mikulski, W. M.: The natural bundles admitting natural lifting of linear connections. Demonstr. Math. 39 (2006), 223-232. | DOI | MR | Zbl

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