Some remarks on G_2-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consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler.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.

31 pages, no figures, corrections of various typos and a reorganization of some material to improve clarity. A purported compact example of an extremally Ricci-pinched closed G_2 structure was removed from this version, as it turns out that the argument was seriously flawed

References in corpus (2)

• Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
• On the geometry of closed G2-structure

Cited by in corpus (21)

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• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
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• Supersymmetric AdS_3 solutions in Heterotic Supergravity
• Existence, Convergence and Limit Map of the Laplacian Flow
• Eleven-Dimensional Supergravity in 4D, $N=1$ Superspace
• Einstein locally conformal calibrated $G_2$-structures
• Heterotic strings on G_2 orbifolds
• Almost complex structures on spheres
• A second look at gauged supergravities from fluxes in M-theory
• The Geometry of Almost Einstein (2,3,5) Distributions
• Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors
• Hermitian Curvature Flow
• Proper Orbifold Cohomology
• The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions
• Deformations of nearly parallel G_2-structures
• Supersymmetry Enhancement of Heterotic Horizons
• Geometry of minimal energy Yang-Mills connections
• Locally conformally cocalibrated G2-structures
• Spectral asymmetry for manifolds of special holonomy