On a class of hypersurfaces in [FORMULA]n × ℝ and ℍn × ℝ
On a class of hypersurfaces in [FORMULA]n × ℝ and ℍn × ℝ
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发表时间：
2010
影响因子：
0.7
通讯作者：
R. Tojeiro
中科院分区：
文献类型：
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作者：
R. Tojeiro
本文给出了乘积空间[FORMULA] n × n和[FORMULA] n × n × n中具有平坦法丛的所有超曲面，当它们被看作是基础平坦空间[FORMULA] n+2 <$n × n和[FORMULA] n+2 <$n × n的余维为2的子流形时，它们的完全刻画.本文证明了[FORMULA] n × n（分别为n × n）中的任何这样的超曲面都可以由[FORMULA] n（分别为n × n）中的平行超曲面族和一元光滑函数构造。然后，我们证明了这类常平均曲率超曲面对应于基空间中的等参族和由等参族的平均曲率函数明确确定的光滑函数。作为一般结果的另一个推论，我们对[FORMULA] n × n和n × n × n的常角超曲面进行了分类，即具有单位法向量场与第二因子n的单位向量场成常角性质的超曲面。这推广了Dillen，Fastenakels，货车der Veken，Vrancken和Munteanu关于[FORMULA] n × n和[FORMULA] n × n × n中曲面的先前结果。我们的方法还得到了所有欧氏超曲面的分类，其性质是环境空间中常数向量场的切分量是主方向，特别是单位法向量场与固定方向成常角的欧氏超曲面。
We give a complete description of all hypersurfaces of the product spaces [FORMULA] n × ℝ and ℍ n × ℝ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces ℝ n+2 ⊃ [FORMULA] n × ℝ and [FORMULA] n+2 ⊃ ℍ n × ℝ. We prove that any such hypersurface in [FORMULA] n × ℝ (respectively, ℍ n × ℝ) can be constructed by means of a family of parallel hypersurfaces in [FORMULA] n (respectively, ℍ n ) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class correspond to an isoparametric family in the base space and a smooth function that is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of [FORMULA] n × ℝ and ℍ n × ℝ, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor ℝ. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in [FORMULA] n × ℝ and ℍ n × ℝ Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.