  In the present work, we discuss how the functional form of thermodynamic
observables can be deduced from the geometric properties of subsets of phase
space. The geometric quantities taken into account are mainly extrinsic
curvatures of the energy level sets of the Hamiltonian of a system under
investigation. In particular, it turns out that peculiar behaviours of
thermodynamic observables at a phase transition point are rooted in more
fundamental changes of the geometry of the energy level sets in phase space.
More specifically, we discuss how microcanonical and geometrical descriptions
of phase-transitions are shaped in the special case of $\phi^4$ models with
either nearest-neighbours and mean-field interactions.
