Degenerate Hessian structures on radiant manifolds

We present a rigorous mathematical treatment of Ruppeiner geometry, by considering degenerate Hessian metrics defined on radiant manifolds. A manifold $M$ is said to be radiant if it is endowed with a symmetric, flat connection $\bar\nabla$ and a global vector field $\rho$ whose covariant derivative is the identity mapping. A degenerate Hessian metric on $M$ is a degenerate metric tensor $g$ that can locally be written as the covariant Hessian of a function, called potential. A function on $M$ is said to be extensive if its Lie derivative with respect to $\rho$ is the function itself. We show that the Hessian metrics appearing in equilibrium thermodynamics are necessarily degenerate, owing to the fact that their potentials are extensive (up to an additive constant). Manifolds having degenerate Hessian metrics always contain embedded Hessian submanifolds, which generalize the manifolds defined by constant volume in which Ruppeiner geometry is usually studied. By means of examples, we illustrate that linking scalar curvature to microscopic interactions within a thermodynamic system is inaccurate under this approach. In contrast, thermodynamic critical points seem to arise as geometric singularities.