α-Bach平坦流形的刚性刻画  

Rigidity Characterizations of α-Bach-flat Manifolds

作　　者：黄广月[1] 曾倩玉 HUANG Guangyue;ZENG Qianyu(College of Mathematics and Information Science,Henan Normal University,Xinciang,Henan,453007,P.R.China)

机构地区：[1]河南师范大学数学与信息科学学院,新乡河南453007

出　　处：《数学进展》2023年第2期358-370,共13页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.11971153)。

摘　　要：本文研究α-Bach平坦流形的刚性结果,其中α-Bach张量定义为B_(ij)^(α)=1/n-3W_(ikjl,lk)+α/n-2W_(ikjl)R_(kl,)这里的α是实常数.特别地,B_(ij)^(1)恰是Bach张量;B_(ij)^(0)=1/n-2C_(kij,k)是Cotton张量的散度形式.对于具有正数量曲率的闭流形,得到了一些刚性结果,这些结果涉及积分条件,或者涉及Weyl曲率和迹为零的Ricci曲率的逐点不等式.而且,对于一些L^(n/2)-积分不等式,也得到了一些类似的刚性结果.We study rigidity results of α-Bach-fat Riemannian manifolds,where the Q α-Bach tensor is defined by B^(α)_(ij)=W_(ikjl,lk)+W_(ikjl)R_(kl),with α a real onstant.In partiular,B_(ij)^(1),is exactly the Bach tensor;B^(0)_(ij)-=1/n-2 C_(kij.k) is the divergence of Cotton tensor.For closed manifolds with positive scalar curvature,we prove some rigidity results under integral condition,or pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature.Moreover,under some L^(n/2)-integral inequalities,we also provide some similar rigidity results.

关 键 词：Bach平坦 刚性 Yamabe不变量 EINSTEIN流形 

分 类 号：O186.12[理学—数学]