  Liquid water has anomalous liquid properties, such as its density maximum at
4\degree C. An attempt at theoretical explanation proposes a liquid-liquid
phase transition line in the supercooled liquid state, with coexisting
low-density (LDL) and high-density (HDL) liquid states. This line terminates at
a critical point. It is assumed that the LDL state possesses mesoscopic
tetrahedral structures that give it solid-like properties, while the HDL is a
regular random liquid. But the short-lived nature of these solid-like
structures make them difficult to detect directly. We take a thermodynamic
approach instead, and calculate the thermodynamic Ricci curvature scalar $R$ in
the metastable liquid regime. It is believed that solid-like structures signal
their presence thermodynamically by a positive sign for $R$, with a negative
sign typically present in less organized fluid states. Using thermodynamic data
from ST2 computer simulations fit to a mean field (MF) two state equation of
state, we find significant regimes of positive $R$ in the LDL state, supporting
the proposal of solid-like structures in liquid water. In addition, we review
the theory, compute critical exponents, demonstrate the large reach of the MF
critical regime, and calculate the Widom line using $R$.
