Dualities and evolutes of fronts in hyperbolic and de Sitter space

We consider the differential geometry of evolutes of singular curves in hyperbolic 2-space and de Sitter 2-space. Firstly, as an application of the basic Legendrian duality theorems, we give the definitions of frontals in hyperbolic 2-space or de Sitter 2-space, respectively. We also give the notions of moving frames along the frontals. By using the moving frames, we define the evolutes of spacelike fronts and timelike fronts, and investigate the geometric properties of these evolutes. As a result, these evolutes can be viewed as wavefronts from the viewpoint of Legendrian singularity theory. At last, we study the relationships among these evolutes.