Sn+1中具有调和M(o)bius曲率张量的超曲面

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:M(o)bius度量g和M(o)bius第二基本形式B.当超曲面维数大于3时,在相差一个M(o)bius变换下这两个不变量完全决定了超曲面.另外M(o)bius形式Ф也是一个重要的不变量,在一些分类定理中Ф=0条件的假定是必要的.本文考虑了Sn 1(n≥3)中具有消失M(o)bius形式Ф的超曲面:对具有调和曲率张量的超曲面进行分类,进而,在M(o)bius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.

Hypersurfaces With Harmonic M(o)bius Curvature in Sn+1

Let x: Mn→Sn+1 be a hypersurface in the (n+1)-dimensional unit sphere Sn+1 without umbilics. Two basis invariants of x under the M(o)bins transformation group of Sn+1 are the M(o)bius metric g and the M(o)bius second fundamental form B，which determine the hypersurface x up to a M(o)bius transformation if n ≥ 3. In addition，the M(o)bius 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 M(o)bius form Ф. We classify the hypersurfaces with harmonic M(o)bius curvature tensor. Moreover, we classify all Einstein hypersurfaces and all hypersurfaces of constant sectional curvature with respect to M(o)bius metric.