$G_2$-structures on flat solvmanifolds

TL;DR

This paper classifies 7-dimensional flat solvmanifolds and explores the existence of special $G_2$-structures on them, providing explicit examples of manifolds with torsion-free and divergence-free $G_2$-structures.

Contribution

It offers a classification of flat splittable solvmanifolds and constructs explicit examples of $G_2$-structures with specific properties on these manifolds.

Findings

Classification of 7-dimensional flat splittable solvmanifolds.

Existence of torsion-free $G_2$-structures with cyclic holonomy.

Examples of flat manifolds with divergence-free $G_2$-structures.

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.