  In this paper, the new formalism of thermodynamic geometry proposed in [1] is
employed in investigating phase transition points and the critical behavior of
a Gauss Bonnet-AdS black hole in four dimensional spacetime. In this regard,
extrinsic and intrinsic curvatures of a certain kind of hypersurface immersed
in the thermodynamic manifold contain information about stability/instability
of heat capacities. We, therefore, calculate the intrinsic curvature of the
$Q$-zero hypersurface for a four-dimensional neutral Gauss Bonnet black hole
case in the extended phase space. Interestingly, intrinsic curvature can be
positive for small black holes at low temperature, which indicates a repulsive
interaction among black hole microstructures. This finding is in contrast with
the five-dimensional neutral Gauss Bonnet black hole with only dominant
attractive interaction between its microstructures.
