$$G_2$$-structures on flat solvmanifolds

Abstract

In this article we study the relation between flat solvmanifolds and \(G_2\)-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of \(\mathsf{GL}(n,\mathbb {Z})\) for \(n=5\) and \(n=6\). Then, we look for closed, coclosed and divergence-free \(G_2\)-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free \(G_2\)-structure whose finite holonomy is cyclic and contained in \(G_2\), and examples of compact flat manifolds admitting a divergence-free \(G_2\)-structure.