Calibrated submanifolds and reductions of $G_{2}$-manifolds

Abstract

(Co)associative submanifolds in a $G_{2}$-manifold with a free $S^{1}$ or $T^{2}$ action are characterized by submanifolds in the quotient space. Using our method, we construct various examples of (co)associative submanifolds and fibrations on $G_{2}$-manifolds with the $T^{2}$-symmetry such as the cone of the Iwasawa manifold.