  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.
