Bulletin of the
Korean Mathematical Society BKMS

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 $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.

Keywords: Sobolev constant, rigidity, vanishing Bach tensor

MSC numbers: Primary 53C24; Secondary 53C21

Supported by: The research of the rst author is supported by NSFC(Nos. 11371018, 11671121) and the second author is supported by NSFC(No. 11401179).