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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)
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