Geometrical particles and Legendrian dualities related to null curves

Abstract

In this paper, we investigate the special properties of geometrical particles with null paths in de Sitter 3-space–time, new Frenet equations and an important invariant associated with null paths are presented. By means of unfolding theory, the local topological structure of the lightlike dual surfaces is revealed. It is found that the lightlike dual surface has some singularities whose types can be determined by the invariant. Based on the theory of Legendrian dualities on pseudospheres and the theory of contact manifolds, it is shown that there exists the [Formula: see text]-dual relationship between the lightlike transversal trajectory of the particle and the lightlike dual surface. In addition, an interesting and important fact mentioned is that the contact of lightlike transversal trajectory with lightcone quadric and the contact of lightlike transversal trajectory with null hyperplane have the same order when they are related to the same type of singularities of the lightlike dual surface.