  We investigate the intrinsic and extrinsic curvatures of certain
hypersurfaces in the thermodynamic geometry of a physical system and show that
they contain useful thermodynamic information. For an
anti-Reissner-Nordstr\"{o}m-(A)de Sitter black hole (Phantom), the extrinsic
curvature of a constant $Q$ hypersurface has the same sign as the heat capacity
around the phase transition points. For a Kerr-Newmann-AdS (KN-AdS) black hole,
the extrinsic curvature of $Q \to 0$ hypersurface (Kerr black hole) or $J \to
0$ hypersurface (RN black black hole) has the same sign as the heat capacity
around the phase transition points. The extrinsic curvature also diverges at
the phase transition points. The intrinsic curvature of the hypersurfaces
diverges at the critical points but has no information about the sign of the
heat capacity. Our study explains the consistent relationship holding between
the thermodynamic geometry of the KN-AdS black holes and those of the RN and
Kerr ones \cite{ref1}. This approach can be easily generalized to an arbitrary
thermodynamic system.
