Fr\"olicher-Nijenhuis cohomology on $G_2$- and ${\rm Spin}(7)$-manifolds

TL;DR

This paper explores a new cohomology theory derived from parallel differential forms on special manifolds, specifically G2 and Spin(7), generalizing known structures like the Dolbeault differential.

Contribution

It introduces a natural differential operator on forms and vector-valued forms on G2 and Spin(7) manifolds, extending the Fr"olicher-Nijenhuis cohomology framework.

Findings

Computed the cohomology groups of differential forms on these manifolds.

Provided a partial description of the cohomology of vector-valued forms.

Connected the construction to known structures like K"ahler geometry.

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 Fr\"olicher-Nijenhuis bracket. For instance, on a K\"ahler 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 ${\rm Spin}(7)$-manifold, respectively. We calculate the cohomology groups of $\Omega^\ast(M)$ and give a partial description of the cohomology of $\Omega^\ast(M, TM)$.