摘要
本文研究伪欧氏空间中的伪球面子流形的特征,寻求伪欧氏空间中的类空子流形是伪球面子流形的充要条件。将子流形的位置向量ψ分解成水平分量ψ~T和垂直分量ψ~⊥,运用活动标架法进行研究。本文探讨类空的伪球面子流形,得到伪球面子流形的一个特征:子流形上存在一单位法向量场满足给定的三个条件。特别地,对于Chen子流形,若它具有非迷向的平行平均曲率向量场,且支撑函数有固定符号,则它是伪球面子流形。
对于法向量全是类时向量的情况本文特别地作了研究,得到紧致类空子流形是伪球面子流形的三个充要条件:(1)函数F=是常数,其中H是M的平均曲率向量场;(2)M的Ricci张量S和函数F满足:S(ψ~T,ψ~T)≥n~2(1+F)~2;(3)向量场ψ~T是调和的。同时,给出具有平行平均曲率向量场的子流形是伪球面子流形的一个充要条件。
Decomposing the position vectors of a submanifold into the tangential and normal components and by means of moving frame, the present paper studies the characterizations of pesudo-spherical submanifolds of a pseudo-Euclidean space. We firstly study the space-like pesudo-spherical submanifolds and obtain a necessary and sufficient condition: there exists a unit normal vector satisfying the given three conditions. Specially , we also study Chen submanifolds and obtain that they are pseudo-spherical submanifolds if the mean curvature vector is parallel and not isotropic and the support function doesn't chang its sign.We also study the compact submanifolds whose normal vectors are all time-like. It is observed that the tangential components of the position vectors being harmonic , the support function of the submanifold being contant andthe function F=and the Ricci curvature in the direction of Ψ~Tsatisfying the condition: S(Ψ~T,Ψ~T) ≥n~2(1 + F)~2 provides three necessaryand sufficient conditions. At the same time, we obtain another necessary and sufficient condition for the submanifold with parallel mean curvature vector.
引文
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