On compact manifolds with harmonic curvature and positive scalar curvature

TL;DR

This paper establishes rigidity theorems for compact n-dimensional manifolds with harmonic curvature and positive scalar curvature under integral pinching conditions, precisely characterizing equality cases.

Contribution

It provides new rigidity results for such manifolds, including sharp theorems that precisely identify equality cases under integral pinching conditions.

Findings

Rigidity theorems for harmonic curvature manifolds

Characterization of equality cases in pinching conditions

Sharpness of the main theorems

Abstract

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.