Manifolds with parallel differential forms and Kahler identities for G2-manifolds

Let M be a compact Riemannian manifold equipped with a parallel differential form to. We prove a version of the Kahler 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 the pseudocohomology of M. When M is compact and Kahler, and omega is its Kahler form, (H-c* (M), d) is isomorphic to the cohomology algebra of M. This gives another proof of homotopy formality for Kahler manifolds, originally shown by Deligne, Griffiths, Morgan and Sullivan. We compute H-c*(M) for a compact G(2)-manifold, showing that H-c(i)(M) congruent to H-i(M) 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) -> Lambda(3)(M). (C) 2011 Elsevier B.V. All rights reserved.