  In the present work, we present a detailed discussion of a Riemannian metric
structure originally introduced in [Gori et al., \textit{J. Stat. Mech.},
\textbf{9} 093204 (2018)] on the configuration space and on phase space
allowing us to interpret the derivatives of the configurational microcanonical
entropy and of the canonical entropy in terms of integrals of extrinsic
geometrical quantities associated with the equipotential level sets.
