Some remarks on G_2-structures

TL;DR

This paper explores the geometry of G_2-structures on 7-manifolds, deriving formulas for curvature in terms of torsion, and discusses implications for scalar curvature, Ricci curvature, and the evolution under Laplacian flow.

Contribution

It provides new formulas relating curvature to torsion in G_2-structures and offers insights into the behavior of closed solutions under Laplacian flow.

Findings

Scalar curvature is nonpositive for closed G_2-structures.

Torsion-free G_2-structures have zero scalar curvature.

Derived evolution equations for torsion and metric under Laplacian flow.

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of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler.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