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Projections of surfaces in the hyperbolic space along horocycles

Published online by Cambridge University Press:  30 March 2010

Shyuichi Izumiya  and
Farid Tari
Shyuichi Izumiya
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan ([email protected])
Farid Tari
Affiliation:
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK ([email protected])
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Abstract

We study orthogonal projections of embedded surfaces M in H3+ (−1) along horocycles to planes. The singularities of the projections capture the extrinsic geometry of M related to the lightcone Gauss map. We give geometric characterizations of these singularities and prove a Koenderink-type theorem that relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 2 , April 2010 , pp. 399 - 418
Copyright
Copyright © Royal Society of Edinburgh 2010

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