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.