具有平行平均曲率向量场的三维子流形

3-dimensional Submanifolds with Parallel mean Curvature Vectors

本文给出了空间形式F^(3+p)(c)(P>1)中具有平行平均曲率向量场的三维紧致子流形M^3是全脐点的Ricci曲率的Pinching条件。

Let F3 +p(c) be a 3+p-dimensional space form with constant sectional curvature C, and let M3 be a 3-dimensional submanifold with parallel mean curvature vector. In this paper, we obtain the following : Theorem 1. Let M3 be a 3-dimensional submanifold in F3+p(c)(c≥0, P>1) with parallel mean curvature vector ξ. If the Ricci curvature of M3 is not less than 5/4-(c+ ||ξ||2), then M3 must be totally umbilical. Theorem 2. Let M3 be a 3-dimensional submanifold in F3+p(c)(p>1) with parallel mean curvature vector ξ. If the sectional curvature of M3 is positive and the Ricci curvature of M3 is not less then k(c+||ξ||2) with k = min then M3 is totally umbilical.