【摘要】 设M是n维完备黎曼流形,等距浸入(n+p)维单位球空间Sn+p,具有平行的单位平均曲率向量.则或者M局部地是Sn+p的一个(n+1)维全测地子流形Sn+1中的超曲面片;或者supSa≥n.其中supS是M的第二基本形式长度的平方的上确界.进一步,若n≤7,或者M整体地是Sn+p的一个(n+1)维全测地子流形Sn+1中的超曲面;或者supS(1+12sgn(p-2))>n.所得结果推广了具有平行的平均曲率向量的紧致子流形的结果.更多还原

【Abstract】 The complete submanifolds with parallel unit mean curvature vector in a sphere S^n+p（p>1） are studied, with some characteristics of these submanifolds obtained by using the generalized maximal principle. It is shown that if M is complete submanifolds with parallel mean curvature vector in S^n+p, then S=Sn+1, M is a hypersurface of a n+1 dimensional totally geodesic submanifolds Sⁿ⁺¹ of S^n+p, or sup Sa≥n. And it is further shown that if n≤7, then S=S（n+1）, M is a hypersurface of a n+1 dimensional totally geodesic submanifolds Sⁿ⁺¹ of S^n+p, or sup S（1+12 sgn （p-2））>n.更多还原