摘要
设x:Mn→Sn+1是(n+1)维单位球面Sn+1中的无脐点的超曲面.Sn+1中超曲面x有两个基本的共形不变量,Mobius度量g和Mobius第二基本形式B.当超曲面维数大于3时,在相差一个Mobius变换下这两个不变量完全决定了超曲面.另外Mobius形式Φ也是一个重要的不变量,在一些分类定理中Φ=0条件的假定是必要的.本文考虑了Sn+1(n≥3)中具有消失Mobius形式Φ的超曲面:对具有调和曲率张量的超曲面进行分类,进而,在Mobius度量的意义下,对Einstein超曲面和具有常截面曲率的超曲面也进行了分类.
Let x ： M^n→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilics. Two basis invariants of x under the Mobius transformation group of S^n＋1 are the Mobius metric g and the Mobius second fundamental form B, which determine the hypersurface x up to a Mobius transformation if n ≥ 3. In addition, the Mobius form Ф is a important invariant. The assumption Ф = 0 is necessary in some classification theorems. In this paper, we consider the n-dimensional hypersurfaces （n 〉 3） with vanishing Mobius form Ф. We classify the hypersurfaces with harmonic Mobius curvature tensor. Moreover, we classify all Einstein hypersurfaces and all hypersurfaces of constant sectional curvature with respect to Mobius metric.
出处
《数学进展》
CSCD
北大核心
2008年第1期57-66,共10页
Advances in Mathematics（China）
基金
This work is supported by the Tianyuan Foundation of Mathematics．
