---
tag: P1TFW9QWAD
kind: concept
name: G2 manifold
msc: [53C29, 53C10, 53C38]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/G2_manifold
same_as:
  wikidata: Q5512692
  wikipedia: 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.

## Explanations

### L1 human

The group $G_2$ is the symmetry group of the octonions, an eight-dimensional number system.
Its imaginary part is seven-dimensional, and $G_2$ preserves a special way of multiplying vectors there, a seven-dimensional cross product.
A $G_2$ manifold is a seven-dimensional curved space in which this cross product exists on every tangent space and does not change under parallel transport.
The flat space $\R^7$ and the flat torus $T^7$ are the simplest examples.