Geometric thermodynamics: black holes and the meaning of the scalar   curvature
  In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures. 
Miguel \'A. Garc\'ia-Ariza, Merced Montesinos, Gerardo F. Torres del   Castillo
gr-qc hep-th math-ph math.MP physics.chem-ph