An example of classification of germs of Lagrangian foliations in the neighbourhood of a compact leaf
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.
scientific article; zbMATH DE number 1251262
scientific article