Integrable systems and closed one forms
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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 is a torus. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds.
CitationMiranda, E., Cardona, R. Integrable systems and closed one forms. "Journal of geometry and physics", vol.131, p. 204-209, 2018.