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Geometry of nondegenerate \({\mathbb R}^n\)-actions on \(n\)-manifolds. (English) Zbl 1339.37043

This long paper is devoted to a systematic study of the geometry of nondegenerated \(\mathbb{R}^n\)-actions on \(n\)-dimensional manifolds. The main motivations for such a study come from both integrable dynamical systems and geometry.
From a dynamical point of view this paper concerns a subclass of integrable systems, namely the systems of type \((n, 0)\) formed by \(n\) commuting vector fields (and \(0\) function) on a manifold of dimension \(n\). This class is of particular importance in the geometry of “minimal” invariant manifolds where the number of commuting vector fields is exactly equal to the dimension of the invariant (sub)manifold. From a geometrical point of view the case of nondegenerate singularities of these actions is fully studied and allows to reobtain the toric and quasi-toric manifolds.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58K50 Normal forms on manifolds
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
58K45 Singularities of vector fields, topological aspects
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
58K10 Monodromy on manifolds
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