Vanishing conditions on Weyl tensor for Einstein-type manifolds

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete manifold with fourth-order divergence-free Weyl tensor zero radial curvature is locally warped product $(n-1)$-dimensional Einstein fibers, provided the potential function proper. As consequence, result about nonexistence of multiple black holes in spacetimes.