Key points are not available for this paper at this time.
The second derivative matrices of internal energy or of entropy may be used to define a metric structure on the set of equilibrium states of a thermodynamic system. When expressed relative to the same coordinates, the metric matrices are proportional to each other with constant of proportionality given by the negative of the absolute temperature. This establishes the conformal equivalence of the two metric structures.
Salamon et al. (Sun,) studied this question.