Duality and singularities of world sheets and worldlines of geometric particles along framed base curves in world hyper-sheet

Abstract

In this paper, as applications of singularity theory, we consider the singularities of a world sheet and the traveling trajectory (i.e. a worldline) of a geometric particle when they are confined in a world hyper-sheet and are generated by a Sabban framed base curve in de Sitter 3-space. Using the approach of the unfolding theory in singularity theory, we establish the relationships between the world sheet, the worldline of the geometric particle and two invariants to characterize the local topological structures of singularities of the world sheet and the worldline of the geometric particle via these two invariants, respectively. Moreover, we also describe the dual relationships between the tangent curve of framed base curve and the worldline of the geometric particle. An important fact indicates that the worldline [Formula: see text] and the tangent curve [Formula: see text] of the original curve [Formula: see text] are [Formula: see text]-dual each other. Finally, two concrete examples are presented to interpret our theoretical results.