  Let $M^n(n\geq3)$ 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 on compact
manifolds with harmonic curvature and positive scalar curvature. In particular,
Theorem 1.4, Corollary 1.6 and Theorem 1.9 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.
