Nullcone Fronts of Spacelike Framed Curves in Minkowski 3-Space

Pengcheng Li,Donghe Pei

doi:10.3390/math9222939

Abstract

The investigation of objects in Minkowski space is of great significance, especially for those objects with mathematical and physical backgrounds. In this paper, we study nullcone fronts, which are formed by the light rays emitted from points on a spacelike curve. However, if the spacelike curve is singular, then we cannot use the usual tools and methods to study related issues. To solve these problems, we show the definition of spacelike framed curves in Minkowski 3-space, whose original curves may contain singularities. Then, the singularities of the nullcone fronts are characterized by using framed curvatures of spacelike framed curves. Finally, we exhibit some examples to illustrate our results.

Highlights

The pseudo-Riemannian space has a vital position in the research of differential geometry and physics, among which the Minkowski space is an important space model in relativity [1]
The nullcone front in this paper can be regarded as a surface formed by the trajectory of light rays emitted from each point on a spacelike curve
The nullcone fronts NFγ± we defined above can be regarded as the surfaces formed by the trajectories of light rays emitted from each point on a spacelike curve γ(t)

Summary

Introduction

The pseudo-Riemannian space has a vital position in the research of differential geometry and physics, among which the Minkowski space is an important space model in relativity [1]. Studied special curves formed by reflected light rays in different spaces and analyzed the singular properties of related submanifolds [7,8]. As we know, for singular curves, the Frenet frame does not exist at the singularities, such as cusps, inflection points, etc., which makes it impossible to utilize the Frenet frame to study the curve and its related submanifolds near the singularities For this reason, some authors constructed or adopted different tools to study the objects in different spaces [11,12,13]. To solve this kind of problem, in present paper, we extend the notion of framed curve to the Minkowski 3-space. We show several (singular) curves as examples in Section 5, whose nullcone fronts may have cuspidal edges or swallowtails

Preliminaries

Nullcone Fronts of a Spacelike Framed Curve

Examples

Conclusions