Dual framed surfaces of traveling trajectories of geometrical particles

By confining the traveling trajectories of geometrical particles to the null de Sitter space, we describe the trajectory as a spacelike framed surface in the semi-Euclidean space, whereby the Legendrian duality theory and contact manifolds theory, their geometric properties of these spacelike framed surfaces and their singular dual surfaces are investigated in detail. It is of great significance to explore the classifications of singularities of spacelike framed surfaces. We further demonstrated that these spacelike framed surfaces can be interpreted as one-parameter family of framed curves and established the relations between them.