水乃翔.共形平坦黎曼流形中具有平行第二基本形式的超曲面[J].数学研究及应用,1987,7(3):379~382 |
共形平坦黎曼流形中具有平行第二基本形式的超曲面 |
On hypersurfaces in a conformally flat Riemannian manifold with parallel second fundamental form |
投稿时间:1984-05-03 |
DOI:10.3770/j.issn:1000-341X.1987.03.004 |
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基金项目:中国科学院科学基金资助课题. |
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In this paper, we prove the followingTheorem. Let Cn+1 ( n >5 ) be a conformally flat Riemannian anifola of dimension n + 1 . If Mn is a hypersurface immersed isometrically in Cn+1 over which the second fundamental form is covariant constant, then there are three posible cases only:I . locally Mn= Sp×Sq×Sr, p+q+r=n;Ⅱ . locally Mn=Sp×Sq, p + q = n , where Sk is k- dimensional Riemannian space of constant curvature;III. Mn is umbilical and conformally flat. Moreover, if Mn is connected and complete, then the result holds globally. |
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