Manifolds with parallel differential forms and Kaehler identities for G_2-manifolds

Abstract

Let M be a compact Riemannian manifold equipped with a parallel differential form \omega. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient $(H^*_c(M), d)$, called {\bf the pseudocohomology} of M. When M is compact and Kaehler and \omega is its Kaehler form, $(H^*_c(M), d)$ is isomorphic to the cohomology algebra of M. This gives another proof of homotopy formality for Kaehler manifolds, originally shown by Deligne, Griffiths, Morgan and Sullivan. We compute $H^i_c(M)$ for a compact G_2-manifold, showing that it is isomorphic to cohomology unless i=3,4. For i=3,4, we compute $H^*_c(M)$ explicitly in terms of the first order differential operator $*d: \Lambda^3(M)\arrow \Lambda^3(M)$.