球面中具有平行平均曲率的子流形

Submanifolds With Parallel Mean Curvature on a Sphere

通过研究m+p维的单位球面中m维的具有平行平均曲率向量的Riemann流形,试图得到流形全脐的充分条件.为此构造了函数f(x),并用f(x)来刻画球面子流形"全脐"的这个性质.通过对△f的估计得到:当2≤m≤5时,若流形的第二基本形式长度的平方S满足S≤2m/3,则该流形是全脐的.

By studying the Riemannian m-submanifolds in Riemannian m ＋ p -manifolds with parallel mean curvature, we tried to get the property of totally umbilical submanifolds. Thus, a functionf（x） was constructed to depict the property. By estimating A f, it was proved that the submanifolds are totally umbilical, when Riemannian m-submanifolds （2 ≤ m ≤ 5 ） with S, square of the length of the second fundamental form satisfying with S ≤ 2m/3.