Singularities of worldsheets in spherical space–times

In this paper, the singularities of the geometry for four classes of worldsheets, which are respectively, located in three-dimensional hyperbolic space and three-dimensional de Sitter space–time are considered. Under the theoretical frame of geometry of space–time and as applications of singularity theory, it is shown that these worldsheets have two classes of singularities, that is, in the local sense, these four classes of worldsheets are, respectively, diffeomorphic to the cuspidal edge and the swallowtail. The first hyperbolic worldsheet and the second hyperbolic worldsheet are Formula: see text-dual to the tangent curves of spacelike curves. Moreover, it is also revealed that there is a close relationship between the types of singularities of worldsheets and a geometric invariant Formula: see text, depending on whether Formula: see text or Formula: see text and Formula: see text, the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.