Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics
Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics
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双曲空间中的平坦表面作为测地线全等的法线表面
DOI：
10.2748/tmj/1176734745
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发表时间：
2007
影响因子：
0.5
通讯作者：
Pedro Roitman
中科院分区：
文献类型：
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作者：
Pedro Roitman
我们提出了一些结果，关于平面在双曲3-空间，包括：·几何推导的局部Weierstrass表示的平面，作为一个出发点，旧的结果由于路易吉比安奇。·设M H是一个平坦的紧致连通光滑曲面，满足M M 6= M，横截于H的一个由horospheres构成的叶理。如果M与叶理的叶子成一个恒定的角度，那么M是到与叶理正交的测地线的等距曲面的一部分。·我们还考虑了与平行平面族相关联的焦散面，并证明了这样一个族的焦散面也是一个平面（可能具有奇点）。·对双曲空间中平坦曲面的完整例子进行了分类（我们只有horospheres和与测地线等距的曲面）。因此，研究具有奇点的曲面是很自然的。我们研究通过Weierstrass表示与奇点构造的平面，并证明了刚性的结果。粗略地说，它说：如果两个平面有相同的空间曲线作为一个奇异曲线，那么这两个平面重合。
We present some results concerning flat surfaces in hyperbolic 3-space, including: • A geometrical derivation for a local Weierstrass representation for flat surfaces, using as a starting point an old result due to Luigi Bianchi. • Let M ⊂ H be a flat compact connected smooth surface with ∂M 6= ∅, transversal to a foliation of H by horospheres. If ∂M makes a constant angle with the leaves of the foliation, then M is part of an equidistant surface to a geodesic orthogonal to the foliation. • We also consider the caustic surface associated to a family of parallel flat surfaces and prove that the caustic of such a family is also a flat surface (possibly with singularities). • The complete examples of flat surfaces in hyperbolic space are classified ( we only have horospheres and surfaces equidistants to a geodesic). Therefore it is natural to study surfaces with singularities. We study flat surfaces constructed via the Weierstrass representation with singularities, and prove a rigidity result. Roughly stated, it says the following: if two flat surfaces have the same space curve as a singular curve, then the two flat surfaces coincide.
DOI：
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发表时间：
2002-09
期刊：
arXiv: Differential Geometry
影响因子：
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作者：
M. Kokubu;M. Umehara;Kotaro Yamada
通讯作者：
M. Kokubu;M. Umehara;Kotaro Yamada