拟常曲率空间中具常平均曲率的闭超曲面

设(Nn+1,g)是n+1维单连通完备黎曼流形,其黎曼曲率张量取如下形式:KABCD=a(gACgBD-gADgBC)+b(gACλBλD-gADλBλC+gBDλAλC-gBCλAλD),则称Nn+1为拟常曲率空间。又设M是Nn+1中具常平均曲率的连通闭超曲面,S为M的第二基本形式模长的平方。若Nn+1的生成元切于M,则(1)当S2(n-1)~(1/2)(a+b-b)时,M是全脐超曲面;(2)当S=2(n-1)~(1/2)(a+b-b)时,M是全脐超曲面或球面Sn+1(a)中的H(r)-环面S1(r)×Sn-1(t)。若Nn+1的生成元法于M,则(1)当S=2(n-1)~(1/2)a时,M是全脐超曲面;(2)当S=2(n-1)~(1/2)a时,M是全脐超曲面或Nn+1中的H(r)-环面S1(r)×Sn-1(t)。

Closed Hypersurfaces in Quasi-constant Curvature Space with Constant Mean Curvature

Assume that （ N^n＋1, g） be a n ＋ 1-dimensional complete and simple connected Riemannian manifold and its Riemannian curvature tensors KABCD=a（gAcgBD-gADgBC）＋b（gACλBλD-gADλBλC＋gBDλAλc-gBCλAλD）, then Nn＋1 is said to a quasi-constant curvature space. Let M be a connected and closed hypersurface in a quasi-constant curvature Nn＋1 with constant mean curvature, S be the square of the length of second fundamental form of M. If the generating elements of Nn＋1 are tangent to M, then（ 1 ）when S〈2√n-1（a＋b-｜b｜）,M is a umbilical hypersurface; （2）whenS=2√（a＋b-｜b｜）M is a umbilical hypersurface or a H（r）-torus S1（r）×Sn-1（t） of s^n＋1（a）.If the generating elements of Nn＋ 1 are normal to M, then （ 1 ） when S〈2√n-1a ,, M is a umbilical hypeurface;（2） when S〈2√n-1a, M is a umbilical hypersufface or a H（ r）-torus s1（r）×N^n-1（t） of N^n＋1.