Some remarks on G2-structures

Abstract

This article consists of loosely related remarks about the geometry of G2-structures on 7-manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its representation theory, a set of techniques is introduced for calculating the differential invariants of G2-structures and the rest of the article is applications of these results. Some of the results that may be of interest are as follows: First, a formula is derived for the scalar curvature and Ricci curvature of a G2-structure in terms of its torsion and covariant derivatives with respect to the ‘natural connection’ (as opposed to the Levi-Civita connection) associated to a G2-structure. When the fundamental 3-form of the G2-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. These formulae are also used to generalize a recent result of Cleyton and Ivanov [3] about the nonexistence of closed Einstein G2-structures (other than the Ricci-flat ones) on compact 7-manifolds to a nonexistence result for closed G2-structures whose Ricci tensor is too tightly pinched. Second, some discussion is given of the geometry of the first and second order invariants of G2-structures in terms of the representation theory of G2. Third, 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 of this work was subsumed in the work of Hitchin [12] and Joyce [14]. 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.