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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.

58A20 Jets in global analysis
53C05 Connections (general theory)
Full Text: DOI
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