Umbilicity of space-like submanifolds of Minkowski space
Umbilicity of space-like submanifolds of Minkowski space
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DOI：
10.1017/s0308210500003267
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发表时间：
2002-07
期刊：
影响因子：
--
通讯作者：
S. Izumiya;S. Pei;M. Fuster
中科院分区：
文献类型：
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作者：
S. Izumiya;S. Pei;M. Fuster
我们研究了 Minkowski n 空间中类空间子流形的一些性质，其点都相对于某个法线场是脐带状的。根据这些结果以及 Asperti 和 Dajczer 论文中包含的一些结果，我们得出，相对于平行光法线场而言，ν 脐带意味着维度 n − 2 ≥ 3 的子流形具有共形平坦性。在曲面的情况下，我们将脐带条件与全半脐带条件（每个点的曲率椭圆的简并性）联系起来。此外，如果所考虑的法向场是平行的，我们证明当且仅当表面分别包含在双曲 3 空间、德西特 3 空间或三维光锥中时，它处处是类时间、类空间或类光。我们还给出了双曲 4 空间、德西特 4 空间和四维光锥中包含的表面的总半脐带特征。
We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.