The Indefinite Metric of R.mrugala and the Geometry of the Thermodynamical Phase Space

We study the indefinite metric G in the contact phase space (P, θ) of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of P constitutive surfaces of different homogeneous thermodynamical systems. We established an isomorphism of the space (P, θ,G) with the Heisenberg Lie group Hn endowed with the right invariant contact structure and the right invariant indefinite metric. The lift G̃ of the metric G to the symplectization P̃ of contact space (P, θ), its curvature properties, and its Killing vector fields are studied. Finally we introduce the ”hyperbolic projectivization” of the space (P̃ , θ̃, G̃) that can be considered as the natural compactification of the thermodynamical phase space (P, θ,G).