On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

Abstract

Abstract. Let M n (n ≥ 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 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.

On critical point equation of compact manifolds with zero radial Weyl curvature

“…To do so, we shall present the following estimate for an arbitrary n-dimensional Riemannian manifold. Such estimate appear, for instance, in [19,Lemma 2.5] and here, we present its prove for convenience of the readers. Lemma 6.…”

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

“…Some scholars classify conformally flat manifolds satisfying some curvature L p -pinching conditions [6,12,13,17,27,32]. Recently, Tran [31] obtain two rigidity results for a closed Riemannian manifold with harmonic Weyl curvature, which are a generalization of Tachibana's theorem for non-negative curvature operator [30] and integral gap result which extends Theorem 1.10 for manifolds with harmonic curvature in [11]. We are interested in some pinching problems for compact Riemannian manifold with harmonic Weyl curvature.…”

“…Remark 2.7. We follow these proofs of Proposition 2.1 in [7] and Lemma 4.7 in [3] to prove this lemma which was proved in [11]. For completeness, we also write it out.…”

Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature

“…Recently, two rigidity theorems of the Weyl curvature tensor of positive Ricci Einstein manifolds are given in [4,11,12], which improve results due to [14,16,20]. The second author and Xiao have studied compact manifolds with harmonic curvature to obtain some rigidity results in [8,9,10]. Here when a Riemannian manifold satisfies δRm = {∇ l R i jkl } = 0, we call it a manifold with harmonic curvature.…”