Thermodynamic geometric analysis of D-dimensional RN black hole

This paper studies the thermodynamics and Ruppeiner geometry of D-dimensional RN black holes. We analyze the thermodynamic curvature scalar $R$ in various thermodynamic ensembles. It is found that in an ensemble of fixed charge (canonical ensemble), the Ruppeiner curvature is curved and diverges at a critical point, indicating the existence of a phase transition for $D>4$. In contrast, when all extensive variables are allowed to fluctuate (for example, in a grand-canonical ensemble or with pressure fixed), the Ruppeiner geometry can appear flat. We also demonstrate that the thermodynamic geometric metric has a one-to-one correspondence with the periodicity of the Euclidean path integral method. In particular, the inverse temperature (the Euclidean time period) serves as a bridge connecting the thermodynamic geometry and the Euclidean action approach.