Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold withharmonic curvature and positive scalar curvature. Assume that $M^n$ satisfiessome integral pinching conditions. We give some rigidity theorems on compactmanifolds with harmonic curvature and positive scalar curvature. In particular,Theorem 1.4, Corollary 1.6 and Theorem 1.9 are sharp for our conditions havethe additional properties of being sharp. By this we mean that we can preciselycharacterize the case of equality.
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机译：令$ M ^ n（n \ geq3）$是具有调和曲率和正标量曲率的$ n $维紧凑黎曼流形。假设$ M ^ n $满足某些积分收缩条件。我们给出了具有简谐曲率和标量曲率的紧流形上的一些刚性定理。特别是，定理1.4，推论1.6和定理1.9对于我们的条件而言是尖锐的，它具有尖锐的其他属性。这意味着我们可以精确地描述平等的情况。
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