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Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry

2023; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-031-38299-4_17

1611-3349

Frédéric Barbaresco,

Advanced Differential Geometry Research

We introduce a symplectic bifoliation model of Information Geometry and Heat Theory based on Jean-Marie Souriau's Lie Groups Thermodynamics to describe transverse Poisson structure of metriplectic flow for dissipative phenomena. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, we associate a symplectic bifoliation structure to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Mademoiselle Paulette Libermann's foliations, clarified as dual to Poisson Γ-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of “Séminaire Sud-Rhodanien de Geometrie”. The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. Paulette Libermann proved that a Legendre foliation on a contact manifold is complete if and only if the pseudo-orthogonal distribution is completely integrable, and that the contact form is locally equivalent to the Poincaré-Cartan integral invariant. Paulette Libermann proved a classical theorem relating to co-isotropic foliations, which notably gives a proof of Darboux's theorem. Finally, we explore Edmond Fédida work on the theory of foliation structures in the language of fully integrable Pfaff systems associated with the Cartan’s moving frame.