  It is developed a Riemannian reformulation of classical statistical mechanics
for systems in thermodynamic equilibrium, which arises as a natural extension
of Ruppeiner geometry of thermodynamics. The present proposal leads to
interpret entropy $\mathcal{S}_{g}(I|\theta)$ and all its associated
thermo-statistical quantities as purely geometric notions derived from the
Riemannian structure on the manifold of macroscopic observables
$\mathcal{M}_{\theta}$ (existence of a distance
$ds^{2}=g_{ij}(I|\theta)dI^{i}dI^{j}$ between macroscopic configurations $I$
and $I+dI$). Moreover, the concept of statistical curvature scalar
$R(I|\theta)$ arises as an invariant measure to characterize the existence of
an \textit{irreducible statistical dependence} among the macroscopic
observables $I$ for a given value of control parameters $\theta$. This feature
evidences a certain analogy with Einstein General Relativity, where the
spacetime curvature $R(\mathbf{r},t)$ distinguishes the geometric nature of
gravitation and the reducible character inertial forces with an appropriate
selection of the reference frame.
