Bochner type formulas for the Weyl tensor on four dimensional Einstein\n manifolds

The very definition of an Einstein metric implies that all its geometry is\nencoded in the Weyl tensor. With this in mind, in this paper we derive\nhigher-order Bochner type formulas for the Weyl tensor on a four dimensional\nEinstein manifold. In particular, we prove a second Bochner type formula which,\nformally, extends to the covariant derivative level the classical one for the\nWeyl tensor obtained by Derdzinski in 1983. As a consequence, we deduce some\nintegral identities involving the Weyl tensor and its derivatives on a compact\nfour dimensional Einstein manifold.\n