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Contact and equivalence of submanifolds of homogeneous spaces. (English) Zbl 1137.53008

Kubarski, Jan (ed.) et al., Geometry and topology of manifolds. The mathematical legacy of Charles Ehresmann. On the occasion of the hundredth anniversary of his birthday. Proceedings of the conference, Bȩdlewo, Poland, May 8–15, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 76, 201-206 (2007).
The bundle \(C^{k,p}(M)\) of \(p\)-dimensional contact elements of order \(k\) of an \(n\)-dimensional manifold \(M\) is the factor space of the space \(J_0^k({\mathbb R}^m,M)\) of regular \(k\)-jets with source \(0\in{\mathbb R}^p\) and arbitrary target \(m\in M\) by the action of the group \(\text{Gl}^k(p,{\mathbb R})\) of \(k\)-jets at \(0\in {\mathbb R}^p\) of local diffeomorphisms of \({\mathbb R}^p\) leaving the origin fixed. It is a \(k\)-th order analogue of the Grassmann bundle \(G_m(M)\). Any immersion \(f:Y^p\to M\) of an \(p\)-dimensional manifold induces a map \(f^{(k)}:Y^m\to C^{k,p}(M)\) by considering the \(k\)-jet prolongation of \(f\).
The author considers a homogeneous space \(M\) with group \(G\) and defines differential systems of order \(k\) as submanifolds \(\Omega^k\subset C^{k,p}(M)\) such that the restriction of the projection \(\pi^k:C^{k,p}(M)\to M\) to \(\Omega^k\) is a submersion. The union of the \(G\)-orbits of the contact elements of a submanifold of \(M\) is such a differential system. A solution of \(\Omega^k\) is a submanifold whose contact elements belong to \(\Omega^k\). He enunciates theorems on existence and uniqueness of solutions. Proofs are postponed to forthcoming publications.
We mention that in the case of parametrized submanifolds where jets of immersions are considered rather than contact elements, results on existence and uniqueness of solutions follow from the Frobenius theorem [H. Gollek, Beitr. Algebra Geom. 9, 103–130 (1980; Zbl 0471.58002)].
For the entire collection see [Zbl 1112.57300].

MSC:

53B25 Local submanifolds
58A20 Jets in global analysis
53A55 Differential invariants (local theory), geometric objects
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 0471.58002
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