Frölicher-Nijenhuis cohomology on $G_2$- and $\mathrm{Spin}(7)$-manifolds

Abstract

In this paper we show that a parallel differential form $\Psi$ of even degree on a Riemannian manifold allows to define a natural differential both on ${\Omega }^{\ast }(M)$ and ${\Omega }^{\ast }(M,TM)$, defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel $4$-form on a $G_2$- and $\mathrm{Spin}(7)$-manifold, respectively. We calculate the cohomology groups of ${\Omega }^{\ast }(M)$ and give a partial description of the cohomology of ${\Omega }^{\ast }(M,TM)$.

Revised version published in Internat. J. Math. 29 (2018), no. 12