On compact Riemannian manifolds with harmonic weyl curvature

Abstract: We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $\sigma_2$ is isometric to a quotient of the round $\mathbb{S}^4$.