# Article

Full entry | PDF   (0.2 MB)
Keywords:
natural operator; Weil bundle
Summary:
The paper contains a classification of linear liftings of skew symmetric tensor fields of type $(1,2)$ on $n$-dimensional manifolds to tensor fields of type $(1,2)$ on Weil bundles under the condition that $n\ge 3.$ It complements author's paper Linear liftings of symmetric tensor fields of type $(1,2)$ to Weil bundles'' (Ann. Polon. Math. {\it 92}, 2007, pp. 13--27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author's paper Affine liftings of torsion-free connections to Weil bundles'' (Colloq. Math. {\it 114}, 2009, pp. 1--8) and get a classification of affine liftings of all linear connections to Weil bundles.
References:
[1] Dębecki, J.: Linear liftings of skew-symmetric tensor fields to Weil bundles. Czech. Math. J. 55 (130) (2005), 809-816. DOI 10.1007/s10587-005-0067-0 | MR 2153104
[2] Dębecki, J.: Linear liftings of $p$-forms to $q$-forms on Weil bundles. Monatsh. Math. 148 (2006), 101-117. DOI 10.1007/s00605-005-0348-6 | MR 2235358
[3] Dębecki, J.: Linear liftings of symmetric tensor fields of type $(1,2)$ to Weil bundles. Ann. Polon. Math. 92 (2007), 13-27. DOI 10.4064/ap92-1-2 | MR 2318507
[4] Dębecki, J.: Affine liftings of torsion-free connections to Weil bundles. Colloq. Math. 114 (2009), 1-8. DOI 10.4064/cm114-1-1 | MR 2457274
[5] Doupovec, M., Mikulski, W. M.: On the existence of prolongation of connections. Czech. Math. J. 56 (131) (2006), 1323-1334. DOI 10.1007/s10587-006-0096-3 | MR 2280811 | Zbl 1164.58300
[6] Eck, D. J.: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133-140. DOI 10.1016/0022-4049(86)90076-9 | MR 0857563 | Zbl 0615.57019
[7] Gancarzewicz, J., Mikulski, W., Pogoda, Z.: Lifts of some tensor fields and connections to product preserving functors. Nagoya Math. J. 135 (1994), 1-41. MR 1295815 | Zbl 0813.53010
[8] Kainz, G., Michor, P.: Natural transformations in differential geometry. Czech. Math. J. 37 (112) (1987), 584-607. MR 0913992 | Zbl 0654.58001
[9] Kolář, I.: On natural operators on vector fields. Ann. Global Anal. Geom. 6 (1988), 109-117. DOI 10.1007/BF00133034 | MR 0982760
[10] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer-Verlag Berlin (1993). MR 1202431
[11] Kurek, J., Mikulski, W. M.: Canonical symplectic structures on the $r$th order tangent bundle of a symplectic manifold. Extr. Math. 21 (2006), 159-166. MR 2292745 | Zbl 1141.58002
[12] Luciano, O.: Categories of multiplicative functors and Weil's infinitely near points. Nagoya Math. J. 109 (1988), 69-89. MR 0931952 | Zbl 0661.58007
[13] Mikulski, W. M.: The geometrical constructions lifting tensor fields of type $(0,2)$ on manifolds to the bundles of $A$-velocities. Nagoya Math. J. 140 (1995), 117-137. MR 1369482 | Zbl 0854.53018
[14] Morimoto, A.: Prolongations of connections to bundles of infinitely near points. J. Diff. Geom. 11 (1976), 479-498. MR 0445422
[15] Weil, A.: Théorie des points proches sur les variétés différentielles. Colloques Internat. Centre Nat. Rech. Sci. 52 (1953), 111-117 French. MR 0061455

Partner of