Projections of surfaces in hyperbolic spaces and families of curve congruences

Professor S. Izumiya has recently developed a theory for studying the extrinsic geometry of manifolds in hyperbolic spaces. In particular, he has defined the Gauss map on a hypersurface in a hyperbolic space and looked at the singularities of height functions on these hypersurfaces.One of my research interest is in the qualitative theory of implicit differential equations. These model for instance the following geometric pairs of foliations on surfaces in the Euclidean 3-space: lines of curvature, asymptotic curves, characteristic curves. In a recent work with J.W. Bruce, we looked at families of curve congruences that link any two of the above pairs of foliations.Professor Izumiya invited me in November 2005 to Hokkaido University. During my visit we had fruitful conversations about joining our expertise to investigate the following problems. The first is to give a good definition of a projection of a submanifold in a hyperbolic space to a flat subspace and study its properties. The second is to study in detail the case of 2-dimensional surfaces anddefine the analogues of the above pairs of foliations for surfaces in hyperbolic spaces aswell as the corresponding families of curve congruence. Observe that for surfaces in the Euclidean 3-space, the asymptotic curves are linked to the singularities of orthogonal projections to planes. One of our aims is also to find a similar link in thehyperbolic spaces