DUAL SURFACES ALONG SPACELIKE CURVES IN LIGHT CONE AND THEIR SINGULARITY

In this paper, we consider spacelike curves in the light-cone 2-space that is canonically embedded in the light-cone 3-space and the de Sitter 3-space in Minkowski space-time. To study the differential geometry of spacelike curves in the light cone, we propose a new type of frame called a light-cone frame, moving along a spacelike curve. Concerning the framework of the theory of the Legendrian dualities between pseudo-spheres, the dual relationships between these spacelike curves and the light-cone dual surface, the de Sitter dual surface, and the sphere-cone dual surface are revealed. Using the classical unfolding theory, the singularities of the hyperbolic evolute of the original curve and a classification of the singularities of these surfaces is found using several equivalent conditions. It is also revealed that the projections of the critical value sets of both the light-cone dual surface and the sphere-cone dual surface along a spacelike curve are the hyperbolic evolute of the spacelike curve. Finally, some relevant counterexamples are indicated.