Closed $G_2$-structures on compact quotients of Lie groups

Abstract

$G_2$-structures defined by a closed non-degenerate 3-form constitute the starting point in various known and potentially effective methods to obtain holonomy $G_2$-metrics on compact 7-manifolds. Albeit linear, the closed condition is quite restrictive, and no general results on the existence of closed $G_2$-structures on compact 7-manifolds are currently known. In this paper, we review some results regarding compact locally homogeneous spaces admitting invariant closed $G_2$-structures. In particular, we consider the case of compact quotients of simply connected Lie groups by discrete subgroups.