---
tag: AJ7P4G261J
kind: definition
name: Spin(7)-manifold
msc: [53C29, 53C27, 53C38]
defines: 1GMSWCHKHE
uses: [7CBT4SM83Y, WHDZ3MK56B, CAZYGE1T4E, 0E3G0GRPZ6, P1TFW9QWAD]
sources:
  - url: https://en.wikipedia.org/wiki/Spin(7)-manifold
---

## Definition

Let $\varphi_0$ be the standard $3$-form of $G_2$ on $\R^7$, and let $*_7$ be the Hodge star of $\R^7$.
The Cayley $4$-form on $\R^8 = \R \oplus \R^7$ is $\Phi_0 = dx_0 \wedge \varphi_0 + *_7 \varphi_0$.
The group $\operatorname{Spin}(7)$ is identified with the group of linear maps $A \in GL(8, \R)$ with $A^* \Phi_0 = \Phi_0$.
It is a compact Lie group of dimension $21$ contained in $SO(8)$.

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $8$.
Identify the tangent space at a point with $\R^8$ by a suitable linear isomorphism.
The manifold $M$ is a *Spin(7)-manifold* if its [[holonomy|WHDZ3MK56B]] group is then contained in $\operatorname{Spin}(7)$.

## Remarks

Equivalently, $M$ carries a [[$4$-form|0E3G0GRPZ6]] $\Phi$ that agrees with $\Phi_0$ at each point in suitable linear coordinates and is parallel for the [[Levi-Civita connection|CAZYGE1T4E]]; such a $\Phi$ determines $g$.

Example: $\R^8$ with $\Phi = \Phi_0$, whose constant coefficients make it parallel.
If $(N, \varphi)$ is a [[$G_2$ manifold|P1TFW9QWAD]], then $N \times S^1$ with $\Phi = dt \wedge \varphi + *\varphi$ is a Spin(7)-manifold, with holonomy contained in $G_2$.

Non-example: the round sphere $S^8$ is not a Spin(7)-manifold, since a metric with holonomy in $\operatorname{Spin}(7)$ is Ricci-flat and the round metric is not.

Compact manifolds with holonomy exactly $\operatorname{Spin}(7)$ were first constructed by Joyce.