Riemannian geometric approach to critical points: General theory

The postulate that the thermodynamic Riemannian curvature scalar is inversely proportional to the free energy is generalized to cases with more than two independent thermodynamic variables. In the appropriate thermodynamic coordinates, the resulting partial differential equation has as a solution a free energy in the form of a generalized homogeneous function. In addition, linear transformations of the variables leave the functional form of the solution unchanged. These findings are consistent with expectations from scaling and universality. Analyzed in some detail are ``corrections to scaling,'' where one ``irrelevant'' variable is added to a ``relevant'' ordering field and the temperature. The ratio of the corrections to scaling amplitudes is computed for the heat capacity and the susceptibility along the critical isochore. Two solution branches result, in the form of exact equations in terms of the critical exponents. The first solution branch is in good agreement with other calculations of this universal ratio. With three variables, our scaled equation of state is not determined uniquely in terms of just a single set of critical exponents. How this relates to the modern theory of critical phenomena is discussed.