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

TL;DR

This paper derives higher-order Bochner formulas for the Weyl tensor on four-dimensional Einstein manifolds, extending classical results and providing new integral identities involving the Weyl tensor and its derivatives.

Contribution

It introduces higher-order Bochner formulas for the Weyl tensor on 4D Einstein manifolds, extending classical formulas and deriving new integral identities.

Findings

Derived second Bochner type formula for the Weyl tensor.

Established integral identities involving the Weyl tensor and its derivatives.

Extended classical Bochner formulas to covariant derivatives level.

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 some integral identities involving the Weyl tensor and its derivatives on a compact four dimensional Einstein manifold.