Integrable systems and closed one forms

TL;DR

This paper explores the topological structure of manifolds with closed forms and provides new proofs for the Liouville theorem, showing that integrable systems' invariant sets are tori, using topological and geometric methods.

Contribution

It offers a new topological perspective on integrable systems and provides novel proofs of classical theorems in symplectic and Poisson geometry.

Findings

Manifolds with independent closed forms are fibrations over tori.

Invariant sets of integrable systems are tori.

New proof of Liouville theorem for symplectic and Poisson manifolds.

Abstract

In the first part of this paper we revisit a classical topological theorem by Tischler (1970) and deduce a topological result about compact manifolds admitting a set of independent closed forms proving that the manifold is a fibration over a torus. As an application we reprove the Liouville theorem for integrable systems asserting that the invariant sets or compact connected fibers of a regular integrable system are tori. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds.