Journal of the
Korean Mathematical Society JKMS

For complete manifolds with $\alpha$-Bach tensor (which is defined by \eqref{1-Int-2}) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some $L^{\frac{n}{2}}$-integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.

Keywords: $\alpha$-Bach-flat, rigidity, Sobolev constant, Einstein

MSC numbers: Primary 53C24; Secondary 53C21

Supported by: The research of author is supported by NSFC (Nos. 11971153, 11671121)