Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes

Abstract

We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces $M$ in $H_{+}^3(-1)$. We study the generic singularities of the projections of $M$ to horospheres and planes. We give geometric characterizations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koenderink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.