  In the study of fluid phases, the Ruppeiner geometry provides novel ways for
constructing the phase boundaries (known as the $R$-crossing method) and the
Widom line, which is considered by many to be the continuation of the
coexistence curve beyond the critical point. In this paper, we revisit these
geometry-based constructions with the aim of understanding their limitations
and generality. We introduce a new equation-of-state expansion for fluids near
a critical point, assuming analyticity with respect to the number density, and
use this to prove a number of key results, including the equivalence between
the $R$-crossing method and the standard construction of phase boundaries near
the critical point. The same conclusion is not seen to hold for the Widom line
of fluids in general. However, for the ideal van der Waals fluid a slight tweak
in the usual formulation of the Ruppeiner metric, which we call the
Ruppeiner-$N$ metric, makes the Ruppeiner geometry prediction of the Widom line
exact. This is in contrast to the results of May and Mausbach where the
prediction is good only up to the slope of the Widom line at the critical
point. We also apply the Ruppeiner-$N$ metric to improve the proposed
classification scheme of Di\'osi et al. that partitions the van der Waals state
space into its different phases using Ruppeiner geodesics. Whereas the original
Di\'osi boundaries do not correspond to any established thermodynamic lines
above (or even below) the critical point, our construction remarkably detects
the Widom line. These results suggest that the Ruppeiner-$N$ metric may play a
more important role in thermodynamic geometry than is presently appreciated.
