Asymmetric free energy from Riemannian geometry

It is shown how to include free energies not symmetric in the ordering field in a recent Riemannian geometric theory of critical phenomena [G. Ruppeiner, Phys. Rev. A 44, 3583 (1991)]. A mixing of coordinates scheme, as is conventionally used to deal with asymmetry, is proved to be consistent with the geometric theory. Furthermore, with scaled forms of the free energy and the assumption of universality, this appears to be the only scheme for including asymmetry in this theory which does not introduce a singularity near the critical isochore.