Manifolds With 2-Nonnegative Ricci Operator

In this paper we study compact manifolds with 2-nonnegative Ricci operator, assuming that their Weyl operator satisfies certain conditions which generalize conformal flatness. As a consequence, we obtain that such manifolds are either locally symmetric or their Betti numbers between 2 and n − 2 vanish. We also study the topology of compact hypersurfaces with 2-nonnegative Ricci operator.