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.

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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.

We study generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves. (2000): 58C27, 53A25, 53A05. Mathematics Subject Classification