  We calculate the Schwinger pair production rates in $\mathbb{R}^{3,1}$ as
well as in the positively curved space $S^2 \times \mathbb{R}^{1,1}$ for both
spin-$0$ and spin-$\frac{1}{2}$ particles under the influence of an external
$SU(2) \times U(1)$ gauge field producing an additional uniform non-abelian
magnetic field besides the usual uniform abelian electric field. To this end,
we determine and subsequently make use of the spectrum of the gauged Laplace
and Dirac operators on both the flat and the curved geometries. We find that
there are regimes in which the purely non-abelian and the abelian parts of the
gauge field strength have either a counterplaying or reinforcing role, whose
overall effect may be to enhance or suppress the pair production rates.
Positive curvature tends to enhance the latter for spin-$0$ and suppress it for
spin-$\frac{1}{2}$ fields, while the details of the couplings to the purely
abelian and the non-abelian parts of the magnetic field, which are extracted
from the spectrum of the Laplace and Dirac operators on $S^2$, determine the
cumulative effect on the pair production rates. These features are elaborated
in detail.
