  From the Aharonov-Bohm effect to general relativity, geometry plays a central
role in modern physics. In quantum mechanics many physical processes depend on
the Berry curvature. However, recent advances in quantum information theory
have highlighted the role of its symmetric counterpart, the quantum metric
tensor. In this paper, we perform a detailed analysis of the ground state
Riemannian geometry induced by the metric tensor, using the quantum XY chain in
a transverse field as our primary example. We focus on a particular geometric
invariant -- the Gaussian curvature -- and show how both integrals of the
curvature within a given phase and singularities of the curvature near phase
transitions are protected by critical scaling theory. For cases where the
curvature is integrable, we show that the integrated curvature provides a new
geometric invariant, which like the Chern number characterizes individual
phases of matter. For cases where the curvature is singular, we classify three
types -- integrable, conical, and curvature singularities -- and detail
situations where each type of singularity should arise. Finally, to connect
this abstract geometry to experiment, we discuss three different methods for
measuring the metric tensor, namely via integrating a properly weighted noise
spectral function and by using leading order responses of the work distribution
to ramps and quenches in quantum many-body systems.
