Flows of G_2 Structures, I

Details Flows of G_2 Structures, I Flows of G_2 Structures, I

2007-02-04

arxiv.org/abs/math/0702077v3

Quarterly Journal of Mathematics 60, 487-522 (2009)

Mathematics

Differential Geometry

50 pages, no figures. Minor rewrites. References added and corrected. For Version 3: Minor revisions and clarifications. This

Scientific paper

10.1093/qmath/han020

This is a foundational paper on flows of G_2 Structures. We use local coordinates to describe the four torsion forms of a G_2 Structure and derive the evolution equations for a general flow of a G_2 Structure on a 7-manifold. Specifically, we compute the evolution of the metric, the dual 4-form, and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernandez-Gray. As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G_2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G_2 geometry.

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