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arXiv · 1605.01508

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

Kotaro Kawai·Hông Vân Lê·Lorenz Schwachhöfer
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Abstract

We extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Frölicher-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 Frölicher-Nijenhuis bracket.

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Kotaro Kawai, Hông Vân Lê, Lorenz Schwachhöfer. 2017-08-02. Frölicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm Spin}(7)$-manifolds. https://arxiv.org/abs/1605.01508

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