On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

Abstract

Let $$M^n(n\ge 3)$$ be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $$M^n$$ satisfies some integral pinching conditions. We give some rigidity theorems. In particular, Theorems 1.4 and 1.10 are sharp for our conditions have the additional properties of being sharp. By this, we mean that we can precisely characterize the case of equality.