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

created by catino on 02 Dec 2016
modified on 18 May 2018

[BibTeX]

Accepted Paper

Inserted: 2 dec 2016
Last Updated: 18 may 2018

Journal: Int. Math. Res. Not.
Year: 2018

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 second Bochner type formula which, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdzinski 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.