We define two surfaces, the horospherical surface and the hyperbolic dual surface of a spacelike curve in the de Sitter 3-space, in the Lorentzian-Minkowski 4-space. These surfaces are, respectively, in the lightcone 3-space and in the hyperbolic 3-space (other pseudospheres). We use techniques from singularity theory to obtain the generic shape of these surfaces and of their singular point sets. Furthermore, we give a relation between these surfaces from the viewpoint of the theory of Legendrian dualities between pseudo-spheres.
MSC: 58K05 53D10 53B30
Keywords:Curves on a timelike surface Pseudo-spherical normal Darboux image Minkowski space Lightlike surface a b s t r a c tIn this paper we introduce the notion of pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz-Minkowski space. We investigate the singularities and geometric properties of these pseudo-spherical normal Darboux images. Furthermore, we investigate the relation of the de Sitter and of the hyperbolic normal Darboux images of a spacelike curve in S 2 1 with the lightlike surface along this spacelike curve.
We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal set is not defined. We use singularity theory techniques to carry out our study and investigate the focal set near lightlike points.
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