Singularities of Dual Hypersurfaces and Hyperbolic Focal Surfaces Along Spacelike Curves in Light Cone in Minkowski 5-Space

Abstract

In this paper, we consider spacelike curves in the light-cone 3-space that is canonically embedded in the light-cone 4-space and the de Sitter 4-space in Minkowski 5-space. To study the differential geometry of spacelike curves in the light cone, we propose a new type of frame, called the 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 hypersurface and the sphere-cone dual hypersurface are revealed. We respectively define the light-cone focal surface and the sphere-cone focal surface as the critical value sets of both the light-cone dual hypersurface and the sphere-cone dual hypersurface. It is also revealed that the projections of the critical value sets of both the light-cone focal surface and the sphere-cone focal surface along a spacelike curve are the hyperbolic evolute of the spacelike curve in the light cone. Using the classical unfolding theory, the singularities of these two hypersurfaces, the singularities of the hyperbolic evolute of the original curve, and the singularities of these two focal surfaces are differentiated using several equivalent conditions.