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July, 2010 Horospherical flat surfaces in Hyperbolic 3-space
Shyuichi IZUMIYA, Kentaro SAJI, Masatomo TAKAHASHI
J. Math. Soc. Japan 62(3): 789-849 (July, 2010). DOI: 10.2969/jmsj/06230789


Recently we discovered a new geometry on submanifolds in hyperbolic n-space which is called horospherical geometry. Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of SO(n)), but it has quite interesting features. For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants. In this paper, we investigate the horospherical flat surfaces (flat surfaces in the sense of horospherical geometry) in hyperbolic 3-space. Especially, we give a generic classification of singularities of such surfaces. As a consequence, we can say that such a class of surfaces has quite a rich geometric structure.


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Shyuichi IZUMIYA. Kentaro SAJI. Masatomo TAKAHASHI. "Horospherical flat surfaces in Hyperbolic 3-space." J. Math. Soc. Japan 62 (3) 789 - 849, July, 2010.


Published: July, 2010
First available in Project Euclid: 30 July 2010

zbMATH: 1205.53065
MathSciNet: MR2648063
Digital Object Identifier: 10.2969/jmsj/06230789

Primary: 53A35
Secondary: 57R45 , 58K40

Keywords: horo-flat surfaces , horosphere , horospherical geometry , hyperbolic 3-space , singularities

Rights: Copyright © 2010 Mathematical Society of Japan


Vol.62 • No. 3 • July, 2010
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