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Debreu's 3-Webs and Affinely Flat Bi-Lagrangian Manifolds Links with Transverse Symplectic Foliation of Souriau's Dissipative Lie Groups Thermodynamics
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DOI：10.1007/978-3-032-03921-7_32.png)
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Abstract
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We shall elucidate the foliation structures, namely, the 3-web and the bi-Lagrangian (Kunneth) structure, that were jointly employed by the physicist and mathematician Jean-Marie Souriau in his Lie Groups Thermodynamics, extended to include dissipation models, and by Gerard Debreu, the Nobel Laureate in Economics, within the context of preferences and utility theory. Debreu examined the conditions under which a web is considered trivial, that is, whether there exists a change of coordinates rendering the web equivalent to a standard orthogonal grid, integrable into a potential function. He employed the Frobenius theorem and Pfaffian forms to investigate whether certain distributions, in the sense of foliations, satisfy the necessary conditions for integrability. Souriau developed a symplectic model of thermodynamics based on symplectic foliation, wherein dissipation dynamics are defined on a transverse Riemannian foliation, thereby inducing a web structure linked to a bi-Lagrangian manifold. A bi-Lagrangian manifold may be endowed with a metric 3-web structure via a Riemannian metric. For every 3-web, one can associate a canonical Chern connection, whose flatness guarantees additive separability. This connection, originally introduced by Hess for Lagrangian 2-webs, also known as bipolarised symplectic manifolds, is particularly employed in the study of bi-Lagrangian manifolds.
Keywords:
3-web
Bi-Lagrangian
Symplectic Foliations
Lie Groups Thermodynamics
Integrable systems
pfaffian forms
Frobenius theorem
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