J. Korean Math. Soc. 2021; 58(2): 401-418
Online first article July 28, 2020 Printed March 1, 2021
https://doi.org/10.4134/JKMS.j200086
Copyright © The Korean Mathematical Society.
Guangyue Huang, Qianyu Zeng
Henan Normal University; Henan Normal University
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)
2007; 44(1): 211-235
2007; 44(5): 1093-1102
2009; 46(5): 895-905
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