黎曼流形里满足:_LH_(ijp)=0的m维曲面

THE m-DIMENSIONAL SURFACES IN RIEMANNIAN MANIFOLDS SATISFYING _1H_(110) = 0.

本文应用n维黎曼空间M^a中沿m维子空间M^m(m维曲面)的广义共变导数,探讨了黎曼流形里具有平行曲率的m维曲面,得到广义的Ricci公式,Weingarten公式和广义的Codazzi方程,Gauss方程的一种新的特殊形式。并应用这些公式和方程推导了几个定理,[2]中的平行曲率超曲面是本文的特殊情形。

LetM^m be an m-Subspace (i, e. m-dimensional surface) in a Riemannian space M^n of n-dimen-sion. By applying to M^m with parallel curvature in M^n the generalized coariant derivative along M^m,we obtained in the present paper new special forms of generalized formulas of Ricci, Weingarten, andequations of Codazzi, Gauss, respectively. With these formulas and equations, several theorems werederived. It was also pointed out that hypersurface with paralle curvature in [2] is aspecial case in oursense.