Configurational microcanonical statistical mechanics from Riemannian geometry of equipotenital level sets

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.