  We investigate topological features of electronic structures which produce
large anomalous Hall effect in the non-collinear antiferromagnetic metallic
states of anti-perovskite manganese nitrides by first-principles calculations.
We first predict the stable magnetic structures of these compounds to be
non-collinear antiferromagnetic structures characterized by either $T_{1g}$ or
$T_{2g}$ irreducible representation by evaluating the total energy for all of
the magnetic structures classified according to the symmetry and multipole
moments. The topology analysis is next performed for the Wannier tight-binding
models obtained from the first-principles band structures. Our results reveal
the small Berry curvature induced through the coupling between occupied and
unoccupied states with the spin-orbit coupling, which is widely spread around
the Fermi surface in the Brillouin zone, dominantly contributes after the
$k$-space integration to the anomalous Hall conductivity, while the local
divergent Berry curvature around Weyl points has a rather small contribution to
the anomalous Hall conductivity.
