On Compact Riemannian Manifolds with Harmonic Weyl Curvature

Abstract

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