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Orbits of families of vector fields on subcartesian spaces. (English) Zbl 1048.53060

A Hausdorff differential space \(S\) in the sense of Sikorski is called a subcartesian space if it is covered by open sets which are diffeomorphic to subsets of \(\mathbb{R}^ n\). Let \(\mathcal F\) be the family of vector fields on a subcartesian space \(S\), and let \(\varphi^ X_ t\) be the local one-parameter group of local diffeomorphisms of \(S\) generated by a vector field \(X\). The family \(\mathcal F\) is said to be locally complete if, for every \(X\), \(Y\in\mathcal F\), \(t\in\mathbb{R}\), and \(x\in S\), for which \(\varphi^ X_{t*}Y(x)\) is defined, there exist an open neighborhood \(U\) of \(x\) and \(Z\in\mathcal F\) such that \(\varphi^ X_{t*}Y_{| U}=Z_{| U}\).
In this interesting paper, the author generalizes the theorem of H. J. Sussmann [Trans. Am. Math. Soc. 180, 171–188 (1973; Zbl 0274.58002)] on orbits of families of vector fields on manifolds to smooth subcartesian spaces and investigates its applications. It is shown that each orbit \(M\) of a locally complete family \(\mathcal F\) of vector fields on a subcartesian space \(S\) is a smooth manifold and its inclusion into \(S\) is smooth.

MSC:

53D20 Momentum maps; symplectic reduction
58A40 Differential spaces
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
32C15 Complex spaces
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
58A35 Stratified sets

Citations:

Zbl 0274.58002
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References:

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