Rigidity of complete Riemannian manifolds with vanishing Bach tensor

摘要

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving Ln2-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.