Classification des actions hamiltoniennes complètement intégrables de rang deux. (Classification of completely integrable Hamiltonian actions of rang 2.)
type: Item
instance of: scholarly article
title: Classification des actions hamiltoniennes complètement intégrables de rang deux. (Classification of completely integrable Hamiltonian actions of rang 2.)
zbMATH Open document ID: 0711.58017
DOI: 10.1007/BF00055020
author: Thomas Delzant
published in: Annals of Global Analysis and Geometry
publication date: 1990-01-01T00:00:00Z
review text: According to the definition of \textit{A. S. \textit{Mishchenko}} and \textit{A. T. Fomenko} [Funkts. Anal. Prilozh. 12, No.2, 46-56 (1978; Zbl 0396.58003)] the action of a Lie group G is completely integrable if it is locally free in a point, at least, and if the dimension of the manifold is the sum of the dimension and the rank of G. This highly abstract paper presents the proof of the following theorem. Let G be a compact connected Lie group of rank two, and \(M_ 1\) and \(M_ 2\) two manifolds on which the action of G is completely integrable; assume that these manifolds have the same image by the moment map, and that their principal isotropy groups are the same, then there exists a symplectic G-equivariant isomorphism of one onto the other.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 22E20
zbMATH DE Number: 4169709
zbMATH Keywords: completely integrable Hamilton actions
zbMATH Keywords: symplectic manifold
zbMATH Keywords: moment map
reviewed by: M. Farkas
MaRDI profile type: Publication
cites work: Polar Coordinates Induced by Actions of Compact Lie Groups
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cites work: Fixed points and torsion on Kähler manifolds
cites work: Q3999811
cites work: Convexity properties of the moment mapping
cites work: Q3346437
cites work: Convexity properties of the moment mapping. III
cites work: Q5565588
cites work: Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups
cites work: Generalized Liouville method of integration of Hamiltonian systems
cites work: The local structure of Poisson manifolds
cites work: Neighborhood classification of isotropic embeddings
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description: scientific article; zbMATH DE number 4169709