We study the singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space through their contact with hyperplanes.

We study some geometrical properties associated to the contacts of surfaces with hyperhorospheres in H4+(-1). We introduce the concepts of osculating hyperhorospheres, horobinormals, horoasymptotic directions and horospherical points and provide conditions ensuring their existence. We show that totally semiumbilical surfaces have orthogonal horoasymp- totic directions.

We classify singularities of lightlike hypersurfaces in Minkowski 4-space via the contact invariants for the corresponding spacelike surfaces and lightcones.

We study some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic n-space as an application of the theory of Legendrian singularities.

We study some properties of spacelike submanifolds in Minkowski n- space all whose points are umbilic with respect to some normal field. As a consequence of these and some results contained in [1] we obtain that being ~-umbilic with respect to a parallel lightlike normal field implies conformal flatness for submanifolds of dimension n-2>-3. In the case of surfaces we relate the umbilicity condition to that of total semiumbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel we show that it is everywhere timelike, spacelike or lightlike if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a 3-dimensional light cone respectively. We also give characterizations of total semiumbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and 4-dimensional light cone.

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

We consider the contact between curves and horospheres in Hyperbolic 3-space as an application of singularity theory of functions. We define the osculating horosphere of the curve. We also define the horospherical surface of the curve whose singular points correspond to the locus of polar vectors of osculating horospheres of the curve. One of the main results is to give a generic classification of singularities of horospherical surface of curves.

We study the geometry of the spacelike surfaces in Minkowski 4-space through their generic contact with lightlike hyperplanes.

In the first part ($2, $3), we give a survey of the recent results on application of singularity theory for curves and surfaces in Hyperbolic space. After that we define the hyperbolic canal surface of a hyperbolic space curve and apply the results of the first part to get some geometric relations between the hyperbolic canal surface and the center CllrVe.

In this paper we adopt the Hyperboloid in Minkowski space as the model of Hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in Hyperbolic space. The hyperbolic Gauss map has been introduced by Epstein[7]in the Poincar@ball model which is very useful for the study of constant mean curvature surfaces. However, it iS very hard to proceed the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study singularities of these. In the study of singularities of the hyperbolic Gauss map(indicatrix), we understand that the hyperbolic Gauss indicatrix is much easier to proceed the calculation. We introduce, the notion of hyperbolic Gauss-Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic GaUSS map(indicatrix). We also develop a 10cal differential geometry of hypersurfaces concerning on contact with hyperhorospheres.