Math 2.0

Differential geometry Definition

Spin(7)-manifold

Builds on Riemannian manifoldHolonomyLevi-Civita connectionDifferential formG2 manifold

Definition

Let φ0\varphi_0 be the standard 33-form of G2G_2 on ℝ7\R^7, and let ∗7*_7 be the Hodge star of ℝ7\R^7. The Cayley 44-form on ℝ8=ℝ⊕ℝ7\R^8 = \R \oplus \R^7 is Φ0=dx0∧φ0+∗7φ0\Phi_0 = dx_0 \wedge \varphi_0 + *_7 \varphi_0. The group Spin⁡(7)\operatorname{Spin}(7) is identified with the group of linear maps A∈GL(8,ℝ)A \in GL(8, \R) with A∗Φ0=Φ0A^* \Phi_0 = \Phi_0. It is a compact Lie group of dimension 2121 contained in SO(8)SO(8).

Let (M,g)(M, g) be a connected Riemannian manifold of dimension 88. Identify the tangent space at a point with ℝ8\R^8 by a suitable linear isomorphism. The manifold MM is a Spin(7)-manifold if its holonomy group is then contained in Spin⁡(7)\operatorname{Spin}(7).

Remarks

Equivalently, MM carries a 44-form Φ\Phi that agrees with Φ0\Phi_0 at each point in suitable linear coordinates and is parallel for the Levi-Civita connection; such a Φ\Phi determines gg.

Example: ℝ8\R^8 with Φ=Φ0\Phi = \Phi_0, whose constant coefficients make it parallel. If (N,φ)(N, \varphi) is a G2G_2 manifold, then N×S1N \times S^1 with Φ=dt∧φ+∗φ\Phi = dt \wedge \varphi + *\varphi is a Spin(7)-manifold, with holonomy contained in G2G_2.

Non-example: the round sphere S8S^8 is not a Spin(7)-manifold, since a metric with holonomy in Spin⁡(7)\operatorname{Spin}(7) is Ricci-flat and the round metric is not.

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

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Differential form, concept, no evidence levelDifferential formconceptG2 manifold, concept, no evidence levelG2 manifoldconceptHolonomy, concept, no evidence levelHolonomyconceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSpin(7)-manifold, definition, no evidence levelSpin(7)-manifolddefinition
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  1. Differential form concept
  2. G2 manifold concept
  3. Holonomy concept
  4. Levi-Civita connection concept
  5. Riemannian manifold concept
  6. Spin(7)-manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spin(7)-manifold
About this page
TagAJ7P4G261J
URIm2://AJ7P4G261J
Kinddefinition
WritingW0 stub
MSC53C29, 53C27, 53C38
DefinesSpin(7)-manifold
UsesRiemannian manifold, Holonomy, Levi-Civita connection, Differential form, G2 manifold
Changed2 November 2026

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