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Remarks on the behaviour of higher-order derivations on the gluing of differential spaces. (English) Zbl 1374.58004

Since a smooth manifold structure is a very strong assumption, not valid in many interesting cases (as in cosmology, when singularities appear), there have been several attempts to extend the mechanism and notions of the classical differential geometry of smooth manifolds to non-smooth spaces. Among these attempts, there exist three major directions: those of differential spaces, diffeologies (initiated by J. M. Souriau [Lect. Notes Math. 836, 91–128 (1980; Zbl 0501.58010)]) and differential triads (due to A. Mallios [Geometry of vector sheaves. An axiomatic approach to differential geometry. Vol. I, II. Mathematics and its Applications (Dordrecht) 439. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0904.18001, Zbl 0904.18002)]).
Differential spaces, founded by R. Sikorski [Colloq. Math. 18, 251–272 (1967; Zbl 0162.25101); Colloq. Math. 24, 45–79 (1971; Zbl 0226.53004)], were subsequently generalized by other authors to include more general cases [K. Spallek, Math. Ann. 180, 269–296 (1969; Zbl 0169.52901); M. A. Mostow, J. Differ. Geom. 14, 255–293 (1979; Zbl 0427.58005)].
The present paper, placed within the framework of Sikorski’s differential spaces, is dealing with the gluing of differential spaces. The process of gluing of differential manifolds is generalized for differential spaces in two ways: the global “generator gluing technique”, and the local one, which identifies subspaces via a diffeomorphism. The present paper offers a detailed exploitation of the latter technique for gluing of arbitrary order \(k\), and it discusses the \(k\)-th tangent spaces and vector fields of the resulting space. Finally two particular cases are examined: (i) the case where the identified subspaces are singletons and (ii) the case where products of differential spaces are considered.

MSC:

58A40 Differential spaces

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