On admissible groups of diffeomorphisms. (English) Zbl 0888.57030

Rybicki, Tomasz

Slovák, Jan (ed.), Proceedings of the 16th Winter School on geometry and physics, Srní, Czech Republic, January 13–20, 1996. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46, 139-146 (1997).

The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the following
Theorem. Let \((M_i, \alpha_i)\), \(i= 1,2\), be a geometric structure such that its group of automorphisms \(G(M_i, \alpha_i)\) satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3’, 4, 5, 6 and 7, and \(M_i\) is compact, or axioms 1, 2, 3’, 4, 5, 6, 7, 8 and 9. Then if there is a group isomorphism \(\Phi: G(M_1, \alpha_1) \to G(M_2, \alpha_2)\) then there is a unique \(C^\infty\)-diffeomorphism \(\varphi: M_1\to M_2\) preserving \(\alpha_i\) and such that \(\Phi(f) =\varphi f\varphi^{-1}\) for each \(f\in G(M_1, \alpha_1)\).
The axioms referred to in the theorem concern a finite open cover of \(\text{supp} (f)\), \(\text{Fix} (f)\), leaves of a generalization foliation \({\mathcal F}\), etc., and \(\alpha_i\) is a volume element or a symplectic form. Several examples are given.
For the entire collection see [Zbl 0866.00050].

Reviewer: I.Pop (Iaşi)

MSC:

57S05	Topological properties of groups of homeomorphisms or diffeomorphisms
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R50	Differential topological aspects of diffeomorphisms

Keywords:

automorphism group; geometric structure