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

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.

v3: 25 p. final version with improved and shortened presentation, accepted for Annali di Matematica Pura ed Applicata. v2: 36 p. Completely revised and shortened version. The part on new cohomology invariants is removed and will be extended in a forthcoming paper. v 1: 55 p. Comments are welcome!