hyperbolic Gauss±Kronecker curvature and the other is the hyperbolic mean curvature. Here we consider only the geometric meaning of the hyperbolic Gauss± Kronecker curvature. Compared with the ordinary Gauss±Kronecker curvature, the hyperbolic Gauss±Kronecker curvature is not an intrinsic invariant. It depends on the embedding of the hypersurface into hyperbolic space. One of our conclusions asserts that the hyperbolic Gauss±Kronecker curvature is a local invariant which describes the contact between hypersurfaces and hyperhorospheres. This means that we establish the`horospherical geometry' of hypersurfaces in hyperbolic space. We do not know whether or not these hyperbolic invariances are essentially new. This is the authors' future problem.In § § 4, 5 and 6, we apply mainly the theory of Legendrian singularities for the study of hyperbolic Gauss indicatrices. Basic notions and results of the theory of Legendrian singularities are given in the last part of the paper as an appendix. Almost all the results in the appendix are already known at least implicitly. However, the topological theory of Legendrian singularities has not been written down in any context except in [11], so we summarise it here.All maps considered here are of class C 1 unless otherwise stated.

Abstract. We de®ne the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these objects and geometric invariants of curves under the action of the Lorentz group.

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Some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic n-space are studied as an application of the theory of Legendrian singularities.

The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying curves and give a method for constructing framed rectifying curves. In addition, we reveal the relationships between framed rectifying curves and some special curves.

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

We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal¯eld. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that beinģ -umbilic with respect to a parallel light-like normal¯eld implies conformal°atness for submanifolds of dimension n ¡ 2 > 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal¯eld is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone. 375

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.