A differential complex for locally conformal calibrated $G_{2}$-manifolds

Abstract

We characterize $G_{2}$-manifolds that are locally conformally equivalent to a calibrated one as those $G_{2}$-manifolds $M$ for which the space of differential forms annihilated by the fundamental $3$-form of $M$ becomes a differential subcomplex of de Rham's complex. Special properties of the cohomology of this subcomplex are exhibited when the holonomy group of $M$ can be reduced to a subgroup of $G_{2}$. We also prove a theorem of Nomizu type for this cohomology which permits its computation for compact calibrated $G_{2}$-nilmanifolds.