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.

中文翻译：

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$$G_2$$ - 平面求解流形上的结构

在本文中，我们研究了平面解流形与\(G_2\)几何之间的关系。首先，我们使用\(\mathsf{GL}(n,\mathbb {Z})\)对于\(n=5\)和\(n =6\)。然后，我们寻找与它们的平面度量兼容的封闭、共封闭和无散\(G_2\)结构。特别是，我们提供了具有无扭转\(G_2\)结构的紧凑平面流形的显式示例，其有限完整是循环的并包含在\(G_2\)中，以及允许无散\( G_2\) -结构。