On Compact Riemannian Manifolds with Harmonic Weyl Curvature

Abstract

We give some rigidity theorems for an n-dimensional ( $$n\ge 4$$ ) 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 $$\mathbb {S}^4$$ .