Math 2.0

Differential geometry Definition

G2 manifold

Builds on Riemannian manifoldHolonomyLevi-Civita connectionDifferential formG-structureHodge star operator

Definition

On ℝ7\R^7 with coordinates x1,…,x7x_1, \ldots, x_7, write dxijk=dxi∧dxj∧dxkdx_{ijk} = dx_i \wedge dx_j \wedge dx_k and let φ0=dx123+dx145+dx167+dx246−dx257−dx347−dx356.\varphi_0 = dx_{123} + dx_{145} + dx_{167} + dx_{246} - dx_{257} - dx_{347} - dx_{356}. The group G2G_2 is the group of linear maps A∈GL(7,ℝ)A \in GL(7, \R) with A∗φ0=φ0A^* \varphi_0 = \varphi_0. It is a compact Lie group of dimension 1414 contained in SO(7)SO(7).

Let (M,g)(M, g) be a connected Riemannian manifold of dimension 77. Identify the tangent space at a point with ℝ7\R^7 by a suitable linear isomorphism. The manifold MM is a G2G_2 manifold if its holonomy group is then contained in G2G_2.

Remarks

Equivalently, MM carries a 33-form φ\varphi that agrees with φ0\varphi_0 at each point in suitable linear coordinates, that determines gg, and that is parallel for the Levi-Civita connection. Such a φ\varphi is a G2G_2-structure, and it is parallel exactly when dφ=0d\varphi = 0 and d∗φ=0d{*\varphi} = 0, where ∗* is the Hodge star of gg.

Example: ℝ7\R^7 with the Euclidean metric and φ=φ0\varphi = \varphi_0; the form has constant coefficients, so it is parallel and the holonomy is trivial. The flat torus ℝ7/ℤ7\R^7 / \Z^7 is a compact example with the same form.

Non-example: the round sphere S7S^7 is not a G2G_2 manifold. A metric with holonomy in G2G_2 is Ricci-flat, while the round metric has positive Ricci curvature.

Compact manifolds with holonomy exactly G2G_2 were first constructed by Joyce, by resolving singularities of quotients of T7T^7 by finite groups.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Differential form, concept, no evidence levelDifferential formconceptG-structure, concept, no evidence levelG-structureconceptHodge star operator, concept, no evidence levelHodge star operatorconceptHolonomy, concept, no evidence levelHolonomyconceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptG2 manifold, definition, no evidence levelG2 manifolddefinition
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  1. Differential form concept
  2. G-structure concept
  3. Hodge star operator concept
  4. Holonomy concept
  5. Levi-Civita connection concept
  6. Riemannian manifold concept
  7. G2 manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/G2_manifold
About this page
TagQG3BB9VZ3N
URIm2://QG3BB9VZ3N
Kinddefinition
WritingW0 stub
MSC53C29, 53C10, 53C38
DefinesG2 manifold
UsesRiemannian manifold, Holonomy, Levi-Civita connection, Differential form, G-structure, Hodge star operator
Changed2 November 2026

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