  We investigate the impacts of the quantum geometry of Bloch states,
specifically through the band-resolved quantum-metric tensor, on Cooper pairing
and flat-band superconductivity in a three-dimensional pyrochlore-Hubbard
model. First we analyze the low-lying two-body spectrum exactly, and show that
the pairing order parameter is uniform in this four-band lattice. This allowed
us to establish direct relations between the superfluid weight of a multiband
superconductor and ($i$) the effective mass of the lowest-lying two-body branch
at zero temperature, ($ii$) the kinetic coefficient of the Ginzburg-Landau
theory in proximity to the critical temperature, and ($iii$) the velocity of
the low-energy Goldstone modes at zero temperature. Furthermore, we perform a
comprehensive numerical analysis of the superfluid weight and Goldstone modes,
exploring both their conventional and geometric components at zero temperature.
