Abstract
In this review, we establish the mathematical framework of geometrothermodynamics (GTD) as a formalism capable of describing nonextensive, quasi-homogeneous, self-gravitating systems in a Legendre-invariant manner. We argue that the fundamental equations of black holes are quasi-homogeneous functions, a property that invalidates the standard Euler identity of laboratory thermodynamics. We derive the metrics for the equilibrium manifold and analyze their curvature singularities for the Reissner–Nordström, Kerr and Kerr–Newman black holes. Furthermore, we establish a direct correspondence between the curvature singularities of the equilibrium space and phase transitions, as determined by the divergences of the corresponding heat capacities.