Singularities of Null Hypersurfaces of Pseudonull Curves

Xiupeng Cui,Donghe Pei

doi:10.1155/2015/502789

Abstract

The main goal of this paper is to study the singularities of null hypersurfaces of pseudonull curves. To do this we construct a null frame and a Lorentz distance-squared function of the pseudonull curve. The relations are shown between singularities of the null hypersurfaces and those, of the Lorentz distance-squared function. And we reveal the geometric meanings of the singularities of such hypersurfaces. In addition, we also introduce some properties of the nullcone Gaussian surface of the pseudonull curve.

Highlights

A pseudonull curve is not a null curve, but its tangent curve is a null curve
To allow a useful study of these singularities, we consider Lorentz distance-squared functions (which are denoted by G : I × R42 → R, G(s, k) = ⟨γ(s) − k, γ(s) − k⟩)
These functions are the unfolding processes of these singularities in the local neighborhood of (s0, k0), and these functions only depend on the germ that they are unfolding. We create these functions by varying a fixed point k in the Lorentz distancesquared function G(s, k) = ⟨γ(s) − k, γ(s) − k⟩, to obtain a family of functions. We show that these singularities are versally unfolded by the family of Lorentz distance-squared functions

Summary

Introduction

A pseudonull curve is not a null curve, but its tangent curve is a null curve. Mathematicians have got many good results on the aspect of differential geometry of pseudonull curves [1,2,3,4,5,6,7]. We consider a one-parameter family of null Gauss indicatrices To allow a useful study of these singularities, we consider Lorentz distance-squared functions (which are denoted by G : I × R42 → R, G(s, k) = ⟨γ(s) − k, γ(s) − k⟩). These functions are the unfolding processes of these singularities in the local neighborhood of (s0, k0), and these functions only depend on the germ that they are unfolding. We assume throughout the paper that all manifolds and maps are C∞ unless explicitly stated otherwise

Preliminaries

Classifications of Singularities

Generic Properties