  As a measure of the quantum distance between Bloch states in the Hilbert
space, the quantum metric was introduced to solid-state physics through the
real part of the so-called geometric Fubini-Study tensor, the imaginary part of
which corresponds to the Berry curvature measuring the emergent gauge field in
momentum space. Here, we first derive the Ginzburg-Landau theory near the
critical superfluid transition temperature, and then identify and analyze the
geometric effects on the effective mass tensor of the Cooper pairs. By showing
that the quantum metric contribution accounts for a sizeable fraction of the
pair mass in a surprisingly large parameter regime throughout the BCS-BEC
crossover, we not only reveal the physical origin of its governing role in the
superfluid density tensor but also hint at its plausible roles in many other
observables as well.
