Flows of G_2 Structures, I

Abstract

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.

50 pages, no figures. Minor rewrites. References added and corrected. For Version 3: Minor revisions and clarifications. This final version is a longer, more detailed version of the shorter paper to appear in the Quarterly Journal of Mathematics

References in corpus (1)

• Some Notes on G_2 and Spin(7) Geometry

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• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
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• On Aspects of Holographic Thermal QCD at Finite Coupling
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• Deformation theory of G_2 conifolds
• Laplacian Solitons and Symmetry in G_2-geometry
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• Some Notes on G_2 and Spin(7) Geometry
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• Flows of ${\mathrm{G}}_2$-structures on contact Calabi--Yau $7$-manifolds
• Modified Laplacian coflow of $G_2$-structures on manifolds with symmetry
• A class of eternal solutions to the G${}_2$-Laplacian flow
• Geometric Partial Differential Equations from Unified String Theories
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• Deformation theory of nearly ${\mathrm{G}}_2$ manifolds
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• Singular foliations for M-theory compactification
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• Laplacian solitons: questions and homogeneous examples
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• The Space of Closed $G_2$-Structures. I. Connections
• Real analyticity of the modified Laplacian coflow
• Cohomogeneity one solitons for the isometric flow of $G_2$-structures
• Isolated singularities in $G_2$-structures with torsion
• Parallel spinors for ${\mathrm{G}}_2^{\ast }$ and isotropic structures