Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space

Abstract

In this paper, we consider the spacelike curves in de Sitter space and we investigate the singularities of lightcone dual surfaces and hyperbolic dual surfaces of these spacelike curves in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space. We classify the singularities of these subjects and reveal the relationships between their singularities and geometric invariants of spacelike curves under the action of the Lorentz group. As application and illustration of the main results, an example is given.

Legendrian dualities between spherical indicatrixes of curves and surfaces according to Bishop frame

“…For any z = j l f (0) in J (1, 1), we have the orbit L l (z) given by the action of the Lie group of l-jet diffeomorphism germs. If f has an A k -singularity, then the codimension of the orbit is k. There is another characterization of R-versal unfoldings as follows [4,13]. Proposition 5.5.…”