Classification of integrable systems on two-dimensional manifolds and invariants in Fomenko’s models. (Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko.) (French) Zbl 0808.58025

Dufour, Jean-Paul; Molino, Pierre; Toulet, Anne

C. R. Acad. Sci., Paris, Sér. I 318, No. 10, 949-952 (1994).

The classification problem which the authors study here is related to triples of the type \((M, \omega, {\mathcal F})\) where \(M\) denotes an oriented compact connected 2-dimensional manifold, \(\omega\) is a symplectic form on \(M\) and \(\mathcal F\) is a Morse foliation on \(M\), that is a foliation with singularities having the property that for every leaf \(F\) of \(\mathcal F\) there exists a neighbourhood \(\mathcal U\), saturated by the foliation and a proper Morse function \(f_{\mathcal U}\) on \(\mathcal U\) such that the restriction \({\mathcal F}\mid_{\mathcal U}\) is defined by the level lines of \(f_{\mathcal U}\); the equivalences are the symplectomorphisms which preserve the foliations.
So, some invariants are associated to the triple \((M, \omega, {\mathcal F})\), among these being the Taylor coefficients corresponding to bifurcation points. Such invariants can be found in Fomenko’s models for Liouville’s torus bifurcations.

Reviewer: L.Maxim-Răileanu (Iaşi)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57R30	Foliations in differential topology; geometric theory

Keywords:

classification; Morse foliation; symplectomorphisms; invariants