EffectiveL p pinching for the concircular curvature

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

“…The complete manifolds with harmonic curvature have been studied in literature (e.g., [5,6,9,10,11,12,13,14,17,20,22,24,27,29,30,32]). Some isolation theorems of Weyl curvature tensor of positive Einstein manifolds are given in [7,14,17,20,29], when its L p -norm is small. Some scholars classify conformally flat manifolds satisfying some curvature L p -pinching conditions [6,12,13,17,27,32].…”

“…Some isolation theorems of Weyl curvature tensor of positive Einstein manifolds are given in [7,14,17,20,29], when its L p -norm is small. 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].…”

Some rigidity characterizations on critical metrics for quadratic curvature functionals

“…Proof of Theorem 1.2. We recall that if E is a Codazzi tensor, then it satisfies the following sharp inequality (for the proof, for instance, see [10]. This inequality was first observed by Bourguignon [4]):…”

Complete noncompact manifolds with harmonic curvature

“…In 1996, Hebey and Vaugon [7] proved the L n/2 -type rigidity results for compact manifolds with harmonic curvature and positive scalar curvature. Recently, Kim [9] studied the noncompact manifolds with non-positive scalar curvature and proved the following result.…”