He claimed that thermodynamic systems can be represented by Riemannian geometry, and that statistical properties can be derived from the model.
This can be recognized if one considers the metric tensor gij in the distance formula (line element) between the two equilibrium states where the matrix of coefficients gij is the symmetric metric tensor which is called a Ruppeiner metric, defined as a negative Hessian of the entropy function where U is the internal energy (mass) of the system and Na refers to the extensive parameters of the system.
Proof of the conformal relation can be easily done when one writes down the first law of thermodynamics (dU = TdS + ...) in differential form with a few manipulations.
It is defined as a Hessian of the internal energy with respect to entropy and other extensive parameters.
It has long been observed that the Ruppeiner metric is flat for systems with noninteracting underlying statistical mechanics such as the ideal gas.
In addition, it has been applied to a number of statistical systems including Van der Waals gas.
The most physically significant case is for the Kerr black hole in higher dimensions, where the curvature singularity signals thermodynamic instability, as found earlier by conventional methods.