# RIGIDITY CHARACTERIZATIONS OF COMPLETE RIEMANNIAN MANIFOLDS WITH α-BACH-FLAT

• Huang, Guangyue (College of Mathematics and Information Science Henan Normal University) ;
• Zeng, Qianyu (College of Mathematics and Information Science Henan Normal University)
For complete manifolds with α-Bach tensor (which is defined by (1.2)) 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.