A Characterization of Einstein Manifolds

Let $(M,g)$ be any Riemannian manifold. Our goal is to show that if $g$ and Ricci tensor $r_{g}$ are no locally constant, if, locally, their product is non-negative (respectively, non-positive), and if its scalar curvature $s_{g}$ is non-negative (respectively, non-positive), then $(M,g)$ is an Einstein manifolds. This result is a generalization of the characterization for compacts Einstein manifolds given by Hilbert.