G2 manifold

Summary

A $G_2$ manifold is a Riemannian manifold of dimension $7$ whose holonomy group lies in the exceptional Lie group $G_2$. The structure is encoded in a single $3$-form that is parallel, and the metric is determined by this form. $G_2$ manifolds are Ricci-flat and form one of the two exceptional cases of Berger's list of holonomy groups. Compact examples are hard to build and are studied in geometry and in M-theory.