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Some remarks on G_2-structures

Robert L. Bryant

TL;DR

This work develops a detailed algebraic and geometric framework for ${ m G}_2$-structures on 7-manifolds, relating curvature, torsion, and differential identities. It provides explicit formulas for scalar and Ricci curvature in terms of the intrinsic torsion components $( au_0, au_1, au_2, au_3)$ and analyzes the Laplacian flow for closed G2-structures, including Hitchin's gradient-flow interpretation of volume. Key contributions include refined decompositions of exterior forms under ${ m G}_2$ and the structure of intrinsic torsion, as well as exact evolution equations for the torsion and metric along the Laplacian flow. The results illuminate rigidity and degeneration phenomena, pinching estimates, and potential pathways toward constructing holonomy ${ m G}_2$ metrics via flow, with notable insights into homogeneous examples and obstructions for compact Einstein closed G2-structures.

Abstract

This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has since been subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. A formula is derived for the scalar curvature and Ricci curvature of a G_2-structure in terms of its torsion. When the fundamental 3-form of the G_2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. This version contains some new results on the pinching of Ricci curvature for metrics associated to closed G_2-structures. Some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data.

Some remarks on G_2-structures

TL;DR

This work develops a detailed algebraic and geometric framework for -structures on 7-manifolds, relating curvature, torsion, and differential identities. It provides explicit formulas for scalar and Ricci curvature in terms of the intrinsic torsion components and analyzes the Laplacian flow for closed G2-structures, including Hitchin's gradient-flow interpretation of volume. Key contributions include refined decompositions of exterior forms under and the structure of intrinsic torsion, as well as exact evolution equations for the torsion and metric along the Laplacian flow. The results illuminate rigidity and degeneration phenomena, pinching estimates, and potential pathways toward constructing holonomy metrics via flow, with notable insights into homogeneous examples and obstructions for compact Einstein closed G2-structures.

Abstract

This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has since been subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. A formula is derived for the scalar curvature and Ricci curvature of a G_2-structure in terms of its torsion. When the fundamental 3-form of the G_2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. This version contains some new results on the pinching of Ricci curvature for metrics associated to closed G_2-structures. Some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data.

Paper Structure

This paper contains 41 sections, 8 theorems, 142 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Algebra
  3. The group ${\rm G}_2$
  4. Associated structures
  5. Some ${\rm G}_2$ actions
  6. The $\varepsilon_{}$-notation
  7. Matrix and vector representations
  8. The ${\rm G}_2$-type decomposition of exterior forms
  9. More ${\rm G}_2$ representation theory
  10. The standard representation
  11. The adjoint representation
  12. Other representations
  13. An example of ${\rm G}_2{$ G_2$}$-type decomposition
  14. Definite forms
  15. On general $7$-dimensional vector spaces
  16. ...and 26 more sections

Key Result

Proposition 1

For any ${\rm G}_2$-structure $\sigma\in\Omega^3_+(M)$, there exist unique differential forms $\tau_0\in\Omega^0(M)$, $\tau_1\in\Omega^1(M)$, $\tau_2\in\Omega^2_{14}(M,\sigma)$, and $\tau_3\in\Omega^3_{27}(M,\sigma)$ so that the following equations hold:

Theorems & Definitions (36)

  • Definition 1: The group ${\rm G}_2$
  • Remark 1: The vector cross product definition of ${\rm G}_2$
  • Definition 2: Definite $3$-forms on manifolds
  • Remark 2: Alternative terminologies
  • Definition 3: Associated metric, orientation, and vector cross product
  • Remark 3: Existence of ${\rm G}_2$-structures
  • Remark 4: ${\rm G}_2$-structures with the same associated metric and orientation
  • Proposition 1: The torsion forms
  • proof
  • Definition 4: The torsion forms
  • ...and 26 more