  An important phase transition in black hole thermodynamics is associated with
the divergence of the specific heat with fixed charge and angular momenta, yet
one can demonstrate that neither Ruppeiner's entropy metric nor Weinhold's
energy metric reveals this phase transition. In this paper, we introduce a new
thermodynamical metric based on the Hessian matrix of several free energy. We
demonstrate, by studying various charged and rotating black holes, that the
divergence of the specific heat corresponds to the curvature singularity of
this new metric. We further investigate metrics on all thermodynamical
potentials generated by Legendre transformations and study correspondences
between curvature singularities and phase transition signals. We show in
general that for a system with n-pairs of intensive/extensive variables, all
thermodynamical potential metrics can be embedded into a flat (n,n)-dimensional
space. We also generalize the Ruppeiner metrics and they are all conformal to
the metrics constructed from the relevant thermodynamical potentials.
