容有常平均曲率的全脐m维曲面的黎曼流形的一些性质

SOME PROPERTIES OF RIEMANNIAN MANIFOLD WHICH ADMITS m DIMENSIONAL SURFACES WITH CONSTANT MEAN CURVATURE AND TOTALLY UMBILICAL POINTS.

1 记号、概念和公式的说明1.1 {M^n,g}表示以g为黎曼度量的n维黎曼流形,为了简便用M^n表示。{M^m,g}表示M^n里黎曼度量为g的m维曲面。(1<m<n)常用M^m表示。1.2 指标α,β,γ,…λ,μ,ν,…取值范围为:1,2,…,n. a,b,c…,i,j,k,…取值范围为:1,2,…,m. A,B,C,…,P,Q,R,…取值范围为:m+1,m+2,…,n.1.3 Ricci主方向。即满足:(R_(αβ)-Tg_(αβ))V~β=0的方向(V~β)。一般有n个,且彼此正交。其中函数T称为对应于方向(V~β)的Ricci主曲率。

In recent years, LiZhonglin has researched Quasi-Einsteinian manifolds and obtained a system re- sults. In paper[1], he has found some properties of n(>3)dimensiomal- Riemannian manifold which ad- mits a family of totally umbilical hyperfaccs with constant mean cuvature. what properties of n(>3)dimensional Riemannian manifold admits a family umbilical m dimensional- surfaces with con- stant mean curvature have? In this paper, we have found some properties of n(>3)dimensional-Riemannian manifold in this respect and obtain severl theorems and propositions. It was also pointed out theorem 2. 1 in[1]]is a special case of this paper in our sense.