On Compact Riemannian Manifolds with Harmonic Weyl Curvature

We give some rigidity theorems for an n-dimensional (n4) compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant sigma 2 curvature. Moreover, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant sigma 2 curvature is isometric to a quotient of the round S4.