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# Horospherical flat surfaces in Hyperbolic 3-space

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 Title: Horospherical flat surfaces in Hyperbolic 3-space Authors: Izumiya, Shyuichi Browse this author Saji, Kentaro Browse this author Takahashi, Masatomo Browse this author Keywords: Hyperbolic 3-space Horosphere Horospherical geometry Horo-flat surfaces Singularities Issue Date: 20-Mar-2007 Publisher: Department of Mathematics, Hokkaido University Journal Title: Hokkaido University Preprint Series in Mathematics Volume: 838 Start Page: 1 End Page: 45 Abstract: Recently we discovered a new geometry on submanifolds in hyperbolic $n$-space which is called {\it 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 {\it 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. Type: bulletin (article) URI: http://hdl.handle.net/2115/69647 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics