Killing forms on G2- and Spin7-manifolds

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We prove that on a compact manifold with holonomy G(2) or Spin(7) any Killing form has to be parallel. The main tool is a universal Weitzenbock formula. We show, how such a formula can be obtained for any given holonomy group and any representation defining a vector bundle. (c) 2005 Published by Elsevier B.V.