Liang Chen

Cuspidal edges represent the generic singularities of wave fronts. We apply Legendrian dualities, originally introduced by Izumiya in the Minkowski space and subsequently extended to the semi-Euclidean space by Izumiya and the third author of the paper, to establish the relations of principal curvatures between cuspidal edges and their dual surfaces.Additionally, we determine the relationships among several parallel surfaces associated with cuspidal edges and their corresponding dual surfaces.Based on the aforementioned two types of relations, we explore the connections between the singularity types of parallel surfaces and the geometric characteristics of the initial cuspidal edges, as well as those of the dual surfaces of the initial cuspidal edges, respectively. Finally, we analyze the correspondence between the singularities and geometric properties of parallel surfaces that are Legendrian dual to each other.

A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In fact, we consider the focal surfaces and evolutes of hyperbolic framed curves by using Legendrian dualities which developed by Chen and Izumiya. The focal surfaces are the dual surfaces of tangent indicatrix of original curves. Moreover, classifications of singularities of the serval dual surfaces are shown. By this, we give the relationship among focal surfaces, evolutes and dual surfaces of evolutes. Finally, we study duality relations of singularities between focal surfaces and dual surfaces of evolutes.

For a nonlightlike nondegenerate regular curve, we have the arc-length parameter and the Frenet-Serret type formula by using a moving frame like a regular space curve in the Euclidean space. If a point of the curve moves between spacelike and timelike regions, then there is a lightlike point. In this paper, we consider mixed types of not only regular curves but also curves with singular points. In order to consider mixed type of curves with singular points, we introduce a frame, so-called the lightcone frame, and lightcone framed curves. We investigate differential geometric properties of lightcone framed curves.

For a non-lightlike non-degenerate regular curve, we have the arc-length parameter and the Frenet-Serret type formula by using a moving frame like as a regular space curve in the Euclidean space. If a point of the curve moves between spacelike and timelike regions, then there is a lightlike point. In this paper, we consider mixed types of not only regular curves but also curves with singular points. In order to consider mixed type of curves with singular points, we introduce a frame, so-called the lightcone frame, and lightcone framed curves. We investigate differential geometric properties of mixed type curves by using the lightcone frame.

There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we define slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame field along the lightlike locus defined by Honda etc. We obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus. Furthermore, we investigate the relationship between slant focal surfaces and slant evolutes. We also consider the relationship between slant evolutes and the lightlike locus on the lightcone.

We consider envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one‐parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one‐parameter families of frontals in hyperbolic and de Sitter 2‐space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.