On compact Riemannian manifolds with harmonic weyl curvature

Abstract

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

arXiv admin note: text overlap with arXiv:1512.00256