欧氏空间中紧致子流形的拟高斯映照

On the quasi Gauss map for a compact sub-manifold in Euclidean space

摘要: 令 Mn 为 (n+p) 维欧氏空间 Rn+p 中 n 维定向的紧致无边子流形，而 σ 为 Mn 的拟高斯映照. 用 ξ 表示 Mn 的单位平均曲率向量场，而 Hi 表示 Mn 沿 ξ 方向的 i-平均曲率. 假设对某个整数 r(1≤r≤n-1) 而言有 Hi>0, i=1,2,…,r 而且 Hr 为常数.利用作者自己最近得到的一个积分公式，证明了：如果 σ(Mn) 落在一个开的 n 维半球面 Sn+ 中，则 Mn 必全拟脐.结果推广了有关欧氏空间中超曲面的一个相关定理.

Abstract: Let  σ be the quasi Gauss map of a compact and oriented  n-dimensional isometric immersion sub-manifold  Mn in the  (n+p)-dimensional Euclid space  Rn+p. Denote by ξ the unit mean curvature vector field to  Mn and denote by  Hi the  i-mean curvature along the direction  ξ.Assume that  Hi>0,  i=1,2,…,r for some integer  r  (1≤r≤n-1) and  Hr is a constant. By applying an integral formula recently given by themselves, it is proven that if the image σ(Mn) lies within an open  n-dimension unit semi sphere  Sn+ then  Mn must be totally quasi umbilical. This result generalizes a relevant theorem on hypersurfaces in Euclid space.