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Differentiable spaces that are subcartesian. (English) Zbl 1473.58001

The paper under review deals with differentiable and differential spaces, introduced in [J. W. Smith, Tohoku Math. J. (2) 18, 115–137 (1966; Zbl 0146.19402)] and [R. Sikorski, Colloq. Math. 18, 251–272 (1967; Zbl 0162.25101)], respectively, and with subcartesian spaces, defined in [N. Aronszajn, “Sub-Cartesian and sub-Riemannian spaces”, Not. Amer. Math. Soc. 14, No. 111 (1967)] as Hausdorff spaces locally diffeomorphic to arbitrary subsets of \(\mathbb{R}^n\), where an atlas of singular charts gives the differential structure.
Based on the notion of a reflexive differential structure from A. Batubenge et al. [“Diffeological, Frölicher, and differential spaces”, Preprint, arXiv:1712.0457], in the reviewed paper the authors define the notion of a weakly reflexive differential structure. They consider a subcartesian space \(S\) and prove that if the differential structure \(C^\infty(S)\) is weakly reflexive, then it is a differentiable structure on \(S\).
In particular, the authors take a connected smooth manifold \(M\) and a connected Lie group \(G\) which has a proper action \(\phi\) on \(M\) and show that the differential structure of the space of \(G\)-orbits \(M/G = \{G\cdot m|m \in M\}\) in \(M\) is weakly reflexive, hence \(M/G\) is a differentiable space. Thus, it follows that the orbit space \(M/G\) has an exterior algebra of differential forms that satisfies Smith’s version of de Rham’s theorem. The authors prove that the differential space \((M/G,C^\infty(M/G))\) is a locally closed subcartesian space and they deal with vector fields on \(M/G\) and their flows.

MSC:

58A05 Differentiable manifolds, foundations
58A35 Stratified sets
58A40 Differential spaces
58A12 de Rham theory in global analysis
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References:

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