欧氏空间中全拟脐子流形

Totally Quasi-umbilical Sub-manifolds in Euclid Space

令R^(n+p)为n+p维欧氏空间,而M^(n)为R^(n+p)中n维定向的等距浸入紧致无边子流形.记ξ为M^(n)的单位平均曲率向量场,而H_(i)为M^(n)沿ξ方向的i-平均曲率.如果存在一个整数r 1≤r≤n-1使得H r和H r+1均为非零常数,则M^(n)必全拟脐.

Let R^(n+p)be the(n+p)-dimensional Euclid space and let M^(n)be a compact and oriented n-dimensional sub-manifold of isometric immersion in R^(n+p)without boundary.Denote byξthe unit mean curvature vector field to M^(n)and by H_(i)the i-mean curvature of M^(n)along the directionξ.In this paper we prove that if there exists an integer r 1≤r≤n-1 such that H_(r)and H_(r+1)are both non-zero constants,then M^(n)must be totally quasi-umbilical.