单位球面中具有平行平均曲率向量的子流形(英文)

Submanifolds in a Unit Sphere with Parallel Mean Curvature Vector

设Mn是单位球面Sn+p中具有平行平均曲率向量的紧致可定向子流形,令|A|2为第二基本形式长度的平方.若|A |2< (2n(n-1)^(1/2))/(2θ(n-1)^(1/2)+n),则Mn是Sn+p中的标准球面;当|A|2=(2n(n-1)^(1/2))/(2θ(n-1)^(1/2)+n)时,还可以对子流形Mn进行分类.

Let M^n be a compact submanifold with parallel mean curvature vector immersed in the unit sphere S^n＊ p.Denote by ｜ A｜2 the square of the length of its second fundamental form. If ｜ A｜^2〈2n√（n-1）/[2θ√（n-1）＋n ]then M^n is a standard sphere in S^n＊p. We can also characterize M^n with ｜ A｜^2〈2n√（n-1）/[2θ√（n-1）＋n ]