Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner-type formulas for the Weyl tensor on a four-dimensional Einstein manifold. In particular, we prove a 2nd Bochner-type formula that, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdziński in 1983. As a consequence, we deduce new integral identities involving the Weyl tensor and its derivatives on a compact four-dimensional Einstein manifold and we derive a new rigidity result.