Ruppeiner geometry and 2D dilaton gravity in the thermodynamics of black holes

We resolve the controversial issue of the geometric approach to the black hole thermodynamics. The geometric description of the equilibrium thermodynamics comes from Ruppeiner geometry based on a metric on the thermodynamic state space. For this purpose, we consider the Reissner-Nordstr\"om-AdS (RN-AdS) black hole which provides two different ensembles: canonical ensemble for fixed-charge case and grand canonical ensemble for fixed-potential case. Two cases are independent and cannot be mixed into each other. Hence, we calculate different Ruppeiner curvatures for two ensembles. However, we could not find the consistent behaviors of Ruppeiner curvature corresponding to those of heat capacity. Alternatively, instead of the Ruppeiner curvature, we newly propose the curvature scalar in the 2D dilaton gravity approach which shows the features of extremal, Davies and minimum temperature points of RN-AdS black hole, clearly.