Projections of surfaces in the hyperbolic space along horocycles
Preprint Series # 887
Izumiya, Shyuichi and Tari, Farid Projections of surfaces in the hyperbolic space along horocycles. (2008);
We study in this paper orthogonal projections of embedded surfaces $M$ in
$H^3_+(-1)$ along horocycles to planes. The singularities of the projections capture the extrinsic geometry of $M$ related to
the lightcone Gauss map. We give geometric characterisations of these singularities
and prove a Koenderink type theorem which 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.
|Uncontrolled Keywords:||Bifurcation sets, contours, Legendrian duality, projections, profiles, hyperbolic space, singularities, de Sitter space, lightcone|
|Subjects:||53-xx DIFFERENTIAL GEOMETRY|