Abstract
We explain how a generalized completely integrable hamiltonian system on a symplectic manifold (MΩ) can be viewed as a generalized Duistermaat fibration: i.e. a fibration π:M 2n—>W nof a symplectic 2n-dimensional manifoldMover annmanifold W with isotropic tori of various dimensions as fibers. This definition, which contains as particular cases, completely integrable hamiltonian systems, hamiltonian actions, and Duistermaat (lagrangian) fibrations, is extended to the contact category, and the following famous results: Arnold-Liouville theorem, Atiyah-Guillemin-Sternberg convexity of the moment map theorem, Delzant realization theorem, Duistermaat theory have been shown to admit a generalization to the contact category in the paper [6] to which this expository paper may serve as an introduction. In short, this work traces the paths along the search of the “correct” definition of the complete integrability in Contact Geometry.
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Banyaga, A. (1999). The Geometry Surrounding the Arnold-Liouville Theorem. In: Brylinski, JL., Brylinski, R., Nistor, V., Tsygan, B., Xu, P. (eds) Advances in Geometry. Progress in Mathematics, vol 172. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1770-1_3
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DOI: https://doi.org/10.1007/978-1-4612-1770-1_3
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