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

Configurational microcanonical statistical mechanics from Riemannian geometry of equipotenital level sets