Smooth maps on convex sets. (English) Zbl 1533.57068
Magnot, Jean-Pierre (ed.), Recent advances in diffeologies and their applications. AMS-EMS-SMF special session, Université de Grenoble-Alpes, Grenoble, France, July 18–20, 2022. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 794, 97-111 (2024).
There are several generalizations of smooth manifolds in the literature. Two of them are diffeological spaces [J. M. Souriau, Lect. Notes Math. None, 91–128 (1980; Zbl 0501.58010); in: Feuilletages et quantification géométrique, Journ. lyonnaises Soc. math. France 1983, Sémin. sud-rhodanien Géom. II, 73–119 (1984; Zbl 0541.58002)] and Chen spaces [K.-T. Chen, Ann. Math. (2) 97, 217–246 (1973; Zbl 0227.58003); Trans. Am. Math. Soc. 206, 83–98 (1975; Zbl 0301.58006); Bull. Am. Math. Soc. 83, 831–879 (1977; Zbl 0389.58001); Lect. Notes Math. 1174, 38–42 (1986; Zbl 0593.58004)]. Every Chen space has an underlying diffeology. It is shown (Theorem 6.5) that among the Chen structures with the same underlying diffeology there is a maximal one characterized by a property (called exhaustive), and the natural functor from exhaustive Chen spaces to diffeological spaces becomes an isomorphism of categories.
Another generalization is Sikorski spaces [R. Sikorski, Colloq. Math. 24, 45–79 (1971; Zbl 0226.53004)]. Both a diffeology and a Sikorski structure that determine each other are called reflexive. It is shown (Theorem 4.1) that closed convex sets are reflexive. It is also shown (Theorem 5.6) that any Sikorski space that is locally Sikorski diffeomorphic to closed convex subsets of cartesian spaces is reflexive.
For the entire collection see [Zbl 1531.53005].
Another generalization is Sikorski spaces [R. Sikorski, Colloq. Math. 24, 45–79 (1971; Zbl 0226.53004)]. Both a diffeology and a Sikorski structure that determine each other are called reflexive. It is shown (Theorem 4.1) that closed convex sets are reflexive. It is also shown (Theorem 5.6) that any Sikorski space that is locally Sikorski diffeomorphic to closed convex subsets of cartesian spaces is reflexive.
For the entire collection see [Zbl 1531.53005].
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 57R55 | Differentiable structures in differential topology |
| 58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |
Citations:
Zbl 0501.58010; Zbl 0541.58002; Zbl 0227.58003; Zbl 0301.58006; Zbl 0389.58001; Zbl 0593.58004; Zbl 0226.53004References:
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