Frölicher-Nijenhuis bracket and geometry of \(G_2\)-and \({\rm Spin}(7)\)-manifolds

We extend the characterization of the integrability of an almost complex structure \(J\) on differentiable manifolds via the vanishing of the Fr\"olicher-Nijenhuis bracket \([J, J] ^{FN}\) to an analogous characterization of torsion-free \(G_2\)-structures and torsion-free Spin(7)-structures. W...

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Bibliographic Details
Published in:arXiv.org 2017-08
Main Authors: Kawai, Kotaro, Lê, Hông Vân, Lorenz Schwachhöfer
Format: Article
Language:English
Subjects:
Classification
Torsion
Online Access:Get full text
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Summary:We extend the characterization of the integrability of an almost complex structure \(J\) on differentiable manifolds via the vanishing of the Fr\"olicher-Nijenhuis bracket \([J, J] ^{FN}\) to an analogous characterization of torsion-free \(G_2\)-structures and torsion-free Spin(7)-structures. We also explain the Fernández-Gray classification of \(G_2\)-structures and the Fernández classification of Spin(7)-structures in terms of the Fr\"olicher-Nijenhuis bracket.
ISSN:2331-8422