An example of classification of germs of Lagrangian foliations in the neighbourhood of a compact leaf
type: Item
instance of: scholarly article
title: An example of classification of germs of Lagrangian foliations in the neighbourhood of a compact leaf
zbMATH Open document ID: 0939.53019
DOI: 10.1016/S0019-3577(98)80018-9
author: Pierre Molino
author: Carlos Currás-Bosch
published in: Indagationes Mathematicae. New Series
publication date: 2000-07-02T00:00:00Z
review text: The authors study some cases of the classification of the germs of Lagrangian foliations with linearizable holonomy in the neighbourhood of a compact leaf. If this leaf is the 2-torus equipped with the flat complete affine structure with integer holonomy in the sense of \textit{N. H. Kuiper} [Colloq. Int. CNRS 52, 79-87 (1953; Zbl 0053.13003)], the above mentioned type of germs of foliations are classified (up to a symplectomorphism) by the germs at the origin of smooth functions of one variable \(f(w)\) (obtained by an integral formula) such that \(f(0)=1\). In higher dimensions, the authors present a general method to reduce this study to the case of an ambient Poisson manifold. The main ingredient of this method is the extension theorem proved by the authors here: if the torus \(T^{k}\) acts (freely) by means of translations on a compact leaf with linearizable holonomy, this action may be uniquely extended to a Hamiltonian action in a neighbourhood of the leaf which is tangent to the foliation. An interesting interpretation of the invariant \(f(w)\) in terms of affine geometry is also presented.
reviewed by: Eugen Pascu
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R32
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D17
zbMATH DE Number: 1251262
zbMATH Keywords: Lagrangian foliation
zbMATH Keywords: compact leaf
zbMATH Keywords: classification of germs of foliations
zbMATH Keywords: Poisson manifold
zbMATH Keywords: symplectic geometry
MaRDI profile type: Publication
cites work: Q3141818
cites work: Neighbourhood of a compact leaf in a Lagrangian foliation: The problem of symplectic linearization
cites work: Q4297432
cites work: Sur la géométrie des sous-fibrés et des feuilletages lagrangiens
cites work: Q4046802
cites work: Q5821651
cites work: The affine structures on the real two-torus. I
cites work: Symplectic manifolds and their Lagrangian submanifolds
full work available at URL: s0019-3577(98)80018-9
OpenAlex ID: W2054819592
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description: scientific article; zbMATH DE number 1251262