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

TL;DR

This paper extends the use of the Fr"olicher-Nijenhuis bracket to characterize torsion-free $G_2$- and Spin(7)-structures, linking geometric integrability conditions with algebraic bracket properties.

Contribution

It generalizes the characterization of integrability from almost complex structures to special holonomy structures like $G_2$ and Spin(7), and relates their classifications to the Fr"olicher-Nijenhuis bracket.

Findings

Characterization of torsion-free $G_2$-structures via the Fr"olicher-Nijenhuis bracket

Characterization of torsion-free Spin(7)-structures via the Fr"olicher-Nijenhuis bracket

Connection of Fernandez-Gray classifications with the Fr"olicher-Nijenhuis bracket

Abstract

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 Fern\'andez-Gray classification of $G_2$-structures and the Fern\'andez classification of Spin(7)-structures in terms of the Fr\"olicher-Nijenhuis bracket.