Contact polarizations and associated metrics in geometric thermodynamics

Abstract In this work we show that a Legendre transformation is nothing but mere change of contact polarization from the point view geometry. Then, construct set Riemannian and pseudo-Riemannian metrics on manifold by introducing almost para-contact structures analyze their isometries. We it not possible to find class metric tensors which fulfills two properties: one hand, be independent i.e. transformations are corresponding isometries and, other, induces Hessian into submanifolds. This second property motivated well known geometric description thermodynamics based space equilibrium states whose properties related fluctuations system. define structure with such necessary abandon idea an associated or structure. even extending thermodynamic phase desiderata cannot fulfilled.