# zbMATH — the first resource for mathematics

On special types of nonholonomic contact elements. (English) Zbl 1229.58005
It is well known that the classical bundle of all contact $$(n,r)$$-elements on a smooth manifold $$M$$ can be expressed as $$K^r_nM=\text{reg}\,T^r_nM/G^r_n$$, where $$\text{reg}\,T^r_nM$$ denote regular $$n$$-dimensional velocities of order $$r$$, $$T^r_nM=J^r_0(\mathbb R^n,M)$$ and $$G^r_n$$ is the $$r$$-th order jet group. The author first discusses nonholonomic $$[r,s]$$-jets. In particular, the nonholonomic $$[r,s]$$-jet prolongation of a fibered manifold $$Y\to M$$ is defined as $$J^{r,s}Y=J^s(J^rY\to M)$$ and the elements of $$J^{r,s}(M,N):=J^{r,s}(M\times N\to M)$$ are called nonholonomic $$[r,s]$$-jets of $$M$$ to $$N$$. Using nonholonomic $$[r,s]$$-jets, he introduces the bundle $$K^{r,s}_nM$$ of iterated contact elements and proves $$K^{r,s}_nM=\text{reg}\,T^{r,s}_nM/G^{r,s}_n$$. Then, he introduces the concept of a regular subcategory $$F\subset\widetilde{J}^r$$ as a rule transforming every pair $$(M,N)$$ of manifolds into a fibered submanifold $$F(M,N)\subset\widetilde{J}^r(M,N)$$ satisfying certain properties. Finally, he introduces the bundle $$K^F_nM$$ of contact $$(n,F)$$-elements. He also clarifies that the incidence relation among the holonomic contact elements can be extended to each type of the nonholonomic ones.

##### MSC:
 58A20 Jets in global analysis 53C05 Connections (general theory)
Full Text:
##### References:
 [1] Čap, A.; Slovák, J., Parabolic geometries I, Mathematical surveys and monographs, vol. 154, (2009), AMS Providence, USA · Zbl 1183.53002 [2] Ehresmann, C., Oeuvres complètes et commentés, parties I-1 et I-2, Cahiers topol. Géom. diff., XXIV, Suppl. 1 et 2, (1983) · Zbl 0561.01027 [3] Kolář, I., On special types of nonholonomic jets, Cahiers topol. Géom. diff. catégoriques, XLVIII, 229-237, (2007) · Zbl 1134.58003 [4] Kolář, I., Weil bundles as generalized jet spaces, (), 625-665 · Zbl 1236.58010 [5] Kolář, I.; Vitolo, R., Absolute contact differentiation on submanifolds of Cartan spaces, Diff. geom. and its applications, 28, 19-32, (2010) · Zbl 1186.58004 [6] Kolář, I.; Michor, P.W.; Slovák, J., Natural operations in differential geometry, (1993), Springer Verlag · Zbl 0782.53013 [7] Libermann, P., Charles ehresmannʼs concepts in differential geometry, (), 35-50 · Zbl 1116.53001 [8] Sharpe, R.W., Differential geometry, Graduate texts in mathematics, vol. 166, (1996), Springer · Zbl 0876.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.