  It is well-known that the non-extremal anti-de-Sitter (AdS) black holes show
various van der Waals type criticalities, such as the $P-V$, $T-S$, $Y-X$,
$Q^2-\Psi$ etc. In these cases, the phase space diagram of the relevant
quantities are similar in behaviour to the isotherms on $P-V$ diagram for the
usual van der Waals gas system. The existing thermogeometric descriptions of
these criticalities for the black holes deal with the whole thermogeometric
manifold and its curvature. However, while observing the criticality, one only
notices the behaviour of the relevant phase space variables and keep all other
thermodynamic quantities as fixed. Therefore it is quite natural to investigate
the geometrical behaviour of the induced metric, defined by these constant
macroscopic variables, corresponding to the whole manifold of the
thermogeometry. Using geometrothermodynamics (GTD), we precisely address this
issue. It is observed that the extrinsic curvature of this induced metric also
diverges at the critical point. This provides an alternative but important
aspect of the thermogeometric description of phase transition of black holes.
The entire formalism in this paper is very general as it is valid for any
arbitrary black hole which shows the van der Waals type phase transition.

General Relativity and Quantum Cosmology;Statistical Mechanics;High Energy Physics - Theory