---
tag: 1GMSWCHKHE
kind: concept
name: Spin(7)-manifold
msc: [53C29, 53C27, 53C38]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Spin(7)-manifold
same_as:
  wikidata: Q7577346
  wikipedia: Spin(7)-manifold
---

## Summary

A Spin(7)-manifold is a Riemannian manifold of dimension $8$ whose holonomy group lies in the group $\operatorname{Spin}(7)$, viewed inside $SO(8)$.
The structure is encoded in a parallel $4$-form, the Cayley form.
Spin(7)-manifolds are Ricci-flat and form, with [[$G_2$ manifolds|P1TFW9QWAD]], the two exceptional cases of Berger's list.

## Explanations

### L1 human

The octonions form an eight-dimensional number system, and the group $\operatorname{Spin}(7)$ preserves a special $4$-form on them.
A Spin(7)-manifold is an eight-dimensional curved space in which such a form exists at every point and is unchanged by parallel transport.
The flat space $\R^8$ is the simplest example.
These spaces are cousins of the seven-dimensional $G_2$ manifolds: the product of a $G_2$ manifold with a circle is an example.