单位球面低维子流形的Pinching定理

A Pinching Theorem for lower Dimensional Submanifolds of Unit Sphere

设M是n+p维单位球面S^(n+p)的n维紧致子流形,n=2,3,4;M具有平行平均曲率向量,若M的第二基本形式长度的平方S≤(2/3)n处处成立,则M是全脐点的或Veronese曲面。

Let M be an n-dimensional compact submanifold with parallel mean curvature tensor in unit sphere s^(n+p) with dimension n-p.S be the square of the length of second fundamental form of M; if n =2,3,4. and S satisfies S≤(2/3)n everywhere on M, then M must be totally umbilical or Veronese surface.