# Article

 Title: The natural operators lifting connections to higher order cotangent bundles (English) Author: Mikulski, Włodzimierz M. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 64 Issue: 2 Year: 2014 Pages: 509-518 Summary lang: English . Category: math . Summary: 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 $M$. (English) Keyword: classical linear connection Keyword: natural operator MSC: 53C05 MSC: 58A20 MSC: 58A32 idZBL: Zbl 06391509 idMR: MR3277751 DOI: 10.1007/s10587-014-0116-7 . Date available: 2014-11-10T09:54:46Z Last updated: 2016-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/144013 . Reference: [1] Dębecki, J.: Affine liftings of torsion-free connections to Weil bundles.Colloq. Math. 114 (2009), 1-8. MR 2457274, 10.4064/cm114-1-1 Reference: [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. Zbl 0646.53028, MR 0864879 Reference: [3] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. I.Interscience Publishers, New York (1963). Zbl 0119.37502, MR 0152974 Reference: [4] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry.Springer Berlin (1993). MR 1202431 Reference: [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. MR 3144087 Reference: [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. Zbl 0905.53018, MR 1463520 Reference: [7] Mikulski, W. M.: The natural bundles admitting natural lifting of linear connections.Demonstr. Math. 39 (2006), 223-232. Zbl 1100.58001, MR 2223893 .

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