Thermodynamic Geometry and Critical Behavior of Black Holes
  Based on the observations that there exists an analogy between the Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) black holes and the van der Waals-Maxwell liquid-gas system, in which a correspondence of variables is $(\phi, q) \leftrightarrow (V,P)$, we study the Ruppeiner geometry, defined as Hessian matrix of black hole entropy with respect to the internal energy (not the mass) of black hole and electric potential (angular velocity), for the RN, Kerr and RN-AdS black holes. It is found that the geometry is curved and the scalar curvature goes to negative infinity at the Davies' phase transition point for the RN and Kerr black holes.   Our result for the RN-AdS black holes is also in good agreement with the one about phase transition and its critical behavior in the literature. 
Jianyong Shen, Rong-Gen Cai, Bin Wang, Ru-Keng Su