  We study the indefinite metric $G$ in the contact phase space $(P,\theta)$ 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 $(P,\theta
,G)$ with the Heisenberg Lie group $H_{n}$ endowed with the right invariant
contact structure and the right invariant indefinite metric. The lift $\tG$ of
the metric $G$ to the symplectization $\tP$ of contact space $(P,\theta)$, its
curvature properties, and its Killing vector fields are studied. Finally we
introduce the "hyperbolic projectivization" of the space $(\tP,{\tilde \theta},
\tG)$ that can be considered as the natural {\bf compactification} of the
thermodynamical phase space $(P,\theta, G).$
