Singularities of Flat Dual Surfaces of Cuspidal Edges in the Three-Sphere from Duality Viewpoint

We focus on investigating the differential geometric properties of cuspidal edge in the three-sphere from a viewpoint of duality. Using Legendrian duality, we study a special kind of flat surface along cuspidal edge in three-dimensional sphere space. This kind of surface is dual to the singular set of the cuspidal edge surface. Thus, we call it the flat \(\Delta \)-dual surface. Flatness of a surface can be defined by the degeneracy of the dual surface. It is similar to the case for the Gauss map of a flat surface in Euclidean space. Moreover, classifications of singularities of the flat \(\Delta \)-dual surface are shown. We also investigate the dual relationships of singularities between flat \(\Delta \)-dual surface and flat approximations of the original cuspidal edge surface. At last, we consider a global geometry of the singular set of a cuspidal edge surface using the flat \(\Delta \)-dual surface.