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  "body": "\n## Summary\n\nA $G_2$ manifold is a Riemannian manifold of dimension $7$ whose holonomy group lies in the exceptional Lie group $G_2$.\nThe structure is encoded in a single $3$-form that is parallel, and the metric is determined by this form.\n$G_2$ manifolds are Ricci-flat and form one of the two exceptional cases of Berger's list of holonomy groups.\nCompact examples are hard to build and are studied in geometry and in M-theory.\n\n## Explanations\n\n### L1 human\n\nThe group $G_2$ is the symmetry group of the octonions, an eight-dimensional number system.\nIts imaginary part is seven-dimensional, and $G_2$ preserves a special way of multiplying vectors there, a seven-dimensional cross product.\nA $G_2$ manifold is a seven-dimensional curved space in which this cross product exists on every tangent space and does not change under parallel transport.\nThe flat space $\\R^7$ and the flat torus $T^7$ are the simplest examples.\n",
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