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