Geodesic
Non-holonomic $(r,s,q)$-jets
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1131-1145
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We generalize the concept of an $(r,s,q)$-jet to the concept of a non-holonomic $(r,s,q)$-jet. We define the composition of such objects and introduce a bundle functor ${\tilde{J}}^{r,s,q}\: \mathcal{F}\mathcal{M}_{k,l} \times \mathcal{F}\mathcal{M}$ defined on the product category of $(k,l)$-dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor $\tilde{J}^{r,s,q}_1\: 2\text{-}\mathcal{F}\mathcal{M}_{k,l} \rightarrow \mathcal{F}\mathcal{M}$ defined on the category of $2$-fibered manifolds with $\mathcal{F}\mathcal{M}_{k,l}$-underlying objects.
We generalize the concept of an $(r,s,q)$-jet to the concept of a non-holonomic $(r,s,q)$-jet. We define the composition of such objects and introduce a bundle functor ${\tilde{J}}^{r,s,q}\: \mathcal{F}\mathcal{M}_{k,l} \times \mathcal{F}\mathcal{M}$ defined on the product category of $(k,l)$-dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor $\tilde{J}^{r,s,q}_1\: 2\text{-}\mathcal{F}\mathcal{M}_{k,l} \rightarrow \mathcal{F}\mathcal{M}$ defined on the category of $2$-fibered manifolds with $\mathcal{F}\mathcal{M}_{k,l}$-underlying objects.
Classification : 58A05, 58A20, 58A32
Keywords: bundle functor; jet; non-holonomic jet; Weil bundle 
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     title = {Non-holonomic $(r,s,q)$-jets},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1131--1145},
     year = {2006},
     volume = {56},
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     mrnumber = {2280799},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a4/}
}
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Tomáš, Jiří M. Non-holonomic $(r,s,q)$-jets. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1131-1145. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a4/

[1] M. Doupovec and I. Kolář: On the jets of fibered manifold morphisms. Cah. Topologie et Géom. Différ. Catég. 40 (1999), 21–30. | MR

[2] C. Ehresmann: Extension du Calcul des jets aux jets non-holonomes. C. R. Acad. Sci. Paris 239 (1954), 1763–1764. | MR | Zbl

[3] I. Kolář: Bundle functors of the jet type. Diff. Geom. Appl., Proc. of the Satelite Conference of ICM in Berlin Diff. Geometry and its Applications, Brno (1998). | MR

[4] I. Kolář: Covariant approach to natural transformations of Weil functors. Comm. Math. Univ. Carolinae 27 (1986), 723–729. | MR

[5] I. Kolář, P. W. Michor and J. Slovák: Natural Operations in Differential Geometry. Springer Verlag, 1993. | MR

[6] I. Kolář and W. M. Mikulski: On the fiber product preserving bundle functors. Diff. Geom. and Appl. 11 (1999), 105–115. | DOI | MR

[7] M. Kureš: On the simplicial structure of some Weil bundles. Rend. Circ. Mat. Palermo Ser. II, Num. 54 (1997), 131–140. | MR

[8] W. M. Mikulski: Product preserving bundle functors on fibered manifolds. Arch. Math. 32-4 (1996), 307–316. | MR | Zbl

[9] W. M. Mikulski: On the product preserving bundle functors on $k$-fibered manifolds. Demonstratio Math. 34-3 (2001), 693–700. | MR | Zbl

[10] W. M. Mikulski and J. M. Tomáš: Liftings of $k$-projectable vector fields to product preserving bundle functors. Acta Univ. Jagellon. Cracow 37-3 (2004), 447–462. | MR

[11] W. M. Mikulski and J. Tomáš: Product preserving bundle functors on fibered fibered manifolds. Colloq. Math. 96-1 (2003), 17–26. | MR

[12] J. Pradines: Representation des jets non holonomes per des morfismes vectoriels doubles soudes. C. R. Acad. Sci. Paris 278 (1974), 1523–1526. | MR

[13] J. Tomáš: On quasijet bundles. Rend. Circ. Mat. Palermo Ser. II, Num. 63 (2000), 187–196. | MR

[14] J. Tomáš: Natural operators transforming projectable vector fields to product preserving bundles. Rend. Circ. Mat. Palermo Ser. II, Num. 59 (1999), 181–187. | MR