  We have investigated electronic structures of antiperovskite GaCMn$_3$ and
related Mn compounds SnCMn$_3$, ZnCMn$_3$, and ZnNMn$_3$. In the paramagnetic
state of GaCMn$_3$, the Fermi surface nesting feature along the $\Gamma{\rm R}$
direction is observed, which induces the antiferromagnetic (AFM) spin ordering
with the nesting vector {\bf Q} $\sim \Gamma{\rm R}$. Calculated
susceptibilities confirm the nesting scenario for GaCMn$_3$ and also explain
various magnetic structures of other antiperovskite compounds. Through the band
folding effect, the AFM phase of GaCMn$_3$ is stabilized. Nearly equal
densities of states at the Fermi level in the ferromagnetic and AFM phases of
GaCMn$_3$ indicate that two phases are competing in the ground state.
