Some remarks on G_2-structures

By Robert L. Bryant. 2003

410 citations recorded by the source corpus.

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of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler.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

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f3ff74c272df12bfdc954f4e3d29d8c5852d113

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness (2015, methodology): “As in [5] (up to a constant factor), we define an operator iφ : S 2T M → Λ3T M locally by iφ(h) = 1 2 hiφljkdx i ∧ dx ∧ dx = 1 6 (hiφljk − h l jφlik − h l kφlji)dx i ∧ dx ∧ dx (2.”
Short-time behaviour of a modified Laplacian coflow of G2-structures (2012, positive): “This technique was first used by Hamilton for the Ricci flow [15, 14] and Bryant and Xu for the Laplacian flow of ϕ [4].”
Laplacian Flow of Closed G$$_2$$2-Structures Inducing Nilsolitons (2013, methodology): “For any closed G2-structure on a manifold M , in [3] Bryant introduced a natural flow, the so-called Laplacian flow, given by { d dt φ(t) = tφ(t), φ(0) = φ0,”
Notes on the octonions (2010, result): “This as well as the analogous result for Spin(7) can also be found in Bryant [Bry87, Section 2].”
A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces (2017, methodology): “The convention we choose to use in the present paper is the one used by Bryant [3] and the first author [20], but differs from the convention used by Bryant–Salamon [4] or Harvey–Lawson [17].”
Some Notes on G_2 and Spin(7) Geometry (2006, background): “Some references for G2 and Spin(7) structures are [1, 2, 3, 5, 6, 7, 8].”