Abstract
Abstract 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.
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