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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1

# Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

Liang Chen
Published Online: 2016-11-14 | DOI: https://doi.org/10.1515/math-2016-0072

## Abstract

In this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.

MSC 2010: 53A35; 58k05

## 1 Introduction

This paper is written as a part of our research project on the study of Lorentz pairs in semi-Euclidean space with index 2 from the viewpoint of Lagrangian/Legendrian singularity theory. Our aim is to investigate the geometric properties of different Lorentzian pairs by constructing a unified way. A Lorentzian pair consists of a Lorentzian hypersurface W in semi-Euclidean space with index two and a timelike hypersurface M in W. AdS4/AdS5, for example, is a Lorentzian pair which is one of the space-time models in physics. As the first step of this research, we consider the simplest Lorentzian pair, i.e., a timelike curve on a Lorentzian surface in semi-Euclidean 3-space with index 2. However, a Lorentz-Minkowski 3-space is diffeomorphic to a semi-Euclidean 3-space with index 2, although the causalities of these two spaces are different. For the geometric properties, we can investigate a spacelike curve on a timelike surface in Lorentz-Minkowski 3-space instead of a timelike curve on a Lorentzian surface in semi-Euclidean 3-space with index 2.

On the other hand, singularity theory tools are useful in the investigation of geometric properties of submanifolds immersed in different ambient spaces, from both the local and global viewpoint [1-16]. The natural connection between geometry and singularities relies on the basic fact that the contacts of a submanifold with the models (invariant under the action of a suitable transformation group) of the ambient space can be described by means of the analysis of the singularities of appropriate families of contact functions, or equivalently, of their associated Lagrangian and/or Legendrian maps. This is our main motivation for the investigation of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space from the viewpoint of singularity theory.

The organization of the paper is as follows. We construct the framework of local differential geometry of spacelike curves on a timelike surface in Section 2. We give the Frenet-Serret type formula corresponding to the spacelike curves. Moreover, we define the lightcone Gauss image and the normalized Gauss map. We also define new invariants ${K}_{L}^{±}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\stackrel{~}{K}}_{L}^{±}$ and call them lightcone Gauss-Kronecker curvature and normalized lightcone Gauss-Kronecker curvature, respectively. We investigate their relations. We can prove that these two Gauss-Kronecker curvature functions have the same zero sets. In Section 3 we introduce the notion of height functions on spacelike curves on a timelike surface which are useful to show that the normalized lightcone Gauss map has a singular point if and only if the lightcone Gauss-Kronecker curvature vanishes at such point. According to the general results on the singularity theory for families of function germs (cf. [17]), we study the relationship of these height functions (cf. Theorem 3.5). In the last section, we define a curve in the lightcone, named the lightcone pedal, as a tool to study the geometric properties of singularities of the normalized lightcone Gauss map from the contact viewpoint.

## 2 The local differential geometry of spacelike curves on a timelike surface

In this section, we investigate the basic ideas on semi-Euclidean (n+1)-space with index two and the local differential geometry of Lorentzian pairs in semi-Euclidean (n+1)-space. For details about semi-Euclidean geometry, see [18]. Let ℝ3 ={(x1, x2, x3)|xi ∈ ℝ, i = 1, 2, 3} be a 3-dimensional vector space. For any vectors x = (x1, x2, x3 and y = (y1, y2, y3 in ℝ3, the pseudo scalarproduct of x and y is defined to be 〈x,y〉 = -x1y1 + x2y2 + x3y3. We call (ℝ3, 〈,〉) the Minkowski 3-space and write ℝ13 instead of (ℝ3, 〈,〉).

We say that a non-zero vector x in ℝ13 is spacelike, lightlike or timelike if 〈x,x〉 > 0, 〈x,x〉 = 0 or 〈x,x〉 > 0 respectively. The norm of the vector x ∈ ℝ13 is defined by $\parallel x\parallel =\sqrt{|\left\{{x}_{;}x\right\}|}.$

For any x = (x1, x2, x3), y = y1, y2, y3 ∈ ℝ13, we define a vector xy by $X∧Y=−e1e2e3x1x2x3y1y2y3,$

where {e1, e2, e3} is the canonical basis of ℝ13. For any ω ∈ ℝ13, we can easily check that ${w,x∧y}=det(w,x,y),$

so that xy is pseudo-orthogonal to both x and y. Moreover, by a straightforward calculation, we have the following simple lemma.

For any non-zero vectors x,y ∈ ℝ13, we assume thatx,xand xy z. Then we have the following assertions:

1. If x is a timelike vector y is a spacelike vector then zx = y yz = -x.

2. If x is a spacelike vector and y is a timelike vector then zx = -y, yz = x.

3. If both x and y are spacelike vectors, then zx = -y, yz = -x.

For a vector ν ∈ ℝ13 and a real number c, we define the plane with the pseudo-normal ν by $P(v,c)={x∈R13|{x,v}=c}.$

We call P(ν,c) a timelike plane, spacelike plane or lightlike plane if v is spacelike, timelike or lightlike, respectively. We define the hyperbolic 2-space by $H2(−1)={x∈R13|{x,x}=−1},$

the de Sitter 2-space by $S12={x∈R13|{x,x}=1},$

the (open) lightcone at the origin by $LC∗={x∈R13∖{0}|{x,x}=0}.$

We call $L{C}_{+}^{\ast }=\left\{x\in L{C}^{\ast }|{x}_{1}>0\right\}$ the future lightcone. We also define the spacelike lightcone circle by $S+1={x=(x1,x2,x3)∈LC∗|x1=1}.$

For any lightlike vector x = (x1, x2, x3) ∈ LC*, we have $x~=(1,x2x1,x3x1)=1x1x∈S+1.$

We study the local differential geometry of spacelike curves on a timelike surface as follows. Firstly, let Y : V → ℝ13 be a regular surface (i.e., an embedding), where V ⊂ ℝ2 is an open subset. We denote W = Y(V) and identify W with V via the embedding Y. The embedding Y is said to be timelike if the induced metric I of W is Lorentzian. Throughout the remainder of this paper we assume that W is a timelike surface in ℝ13. We define a vector N(v) by $N(v)=Yv1(v)∧Yv2(v)∥Yv1(v)∧Yv2(v)∥.$

By the definition of wedge product, we have 〈N(ν), Yνi (u) = 0 (for i = 1,2). This means that N(ν) ∈ NqW, where ν = (ν1, ν2) ∈ V, q = Y (ν) ∈ W and NqW denotes the normal space of W in ℝ13 at q. Since the embedding Y is timelike, N is unit spacelike (i.e., N(ν) ( S12)$. Moreover, we define a regular curve on W by γ : IW, where I ⊂ ℝ is an open interval. We call this curve the spacelike curve, if t(t) = (dγ/dt)(t) is spacelike at any point t Ȉ I, and denote γ(I) = M. Since γ is a regular spacelike curve on W, it may admit the arc length parametrization s = s(t) . Therefore, we can assume that γ is a unit speed spacelike curve, namely, $t\left(s\right)={\gamma }^{\prime }\left(s\right)=\left(d\gamma /ds\right)\left(s\right)\in {S}_{1}^{2}.$ Throughout the remainder in this paper we assume that M is a spacelike curve on the timelike surface W. We define a smooth mapping $\overline{\gamma }:I\to V\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{\gamma }\left(s\right)=v.$ For any s Ȉ I, we have $\gamma \left(s\right)=\mathrm{Y}\left(\overline{\gamma }\left(s\right)\right).$ It follows that the unit spacelike normal vector field of W along γ can be defined by $n\left(s\right)=N\left(\overline{\gamma }\left(s\right)\right),$ for any s Ȉ I. We can also define another unit normal vector field e by e(s) = n(s)t(s). Since t and n are spacelike, e is timelike. Then we have a pseudo-orthonormal frame {t(s), n(s), e(s)} of ℝ13 along γ(s). By a straightforward calculation, we arrive at the following Frenet-Serret type formula: $t′(s)=κn(s)n(s)−κg(s)e(s),n′(s)=−κn(s)t(s)+τg(s)e(s),e′(s)=−κg(s)t(s)+τg(s)n(s),$ where ${\kappa }_{n}\left(s\right)=\left\{{t}^{\prime }\left(s\right),n\left(s\right)\right\},{\kappa }_{g}\left(s\right)=\left\{{t}^{\prime }\left(s\right),e\left(s\right)\right\},{\tau }_{g}\left(s\right)=\left\{{e}^{\prime }\left(s\right),n\left(s\right)\right\}.$ We call them the normal curvature, geodesic curvature and geodesic torsion of γ at point p = γ(s) , respectively. We remark that γ is a spacelike geodesic curve on W if and only if κg = 0; γ is a spacelike asymptotic curve on W if and only if κn = 0; γ is a spacelike principal curve on W if and only if τg = 0. Let n(s) = (n1(s), n2(s), n3(s)) and e(s) = (e1(s), e2(s), e3(s)). Since n(s) ± e(sLC*, we can define a mapping ${\mathrm{G}}_{L}^{±}:I\to L{C}^{\ast }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{G}}_{L}^{±}\left(s\right)=n\left(s\right)±e\left(s\right).$ We call it the lightcone Gauss image. Moreover, we define another mapping ${\mathrm{G}}_{L}^{±}:I\to {S}_{+}^{1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{G}}_{L}^{±}\left(s\right)=n\left(s\right)±e\left(s\right)=\frac{1}{{\xi }^{±}\left(s\right)}{\mathbb{G}}_{L}^{±}\left(s\right),$ where ${\xi }^{±}\left(s\right)={n}_{1}\left(s\right)±{e}_{1}\left(s\right).$ We call it the normalized lightcone Gauss map. By a straightforward calculation, we have $GL±′(s)=n′(s)±e′(s)=−(κn(s)±κg(s))t(s)±τg(s)GL±(s).$ Let ${\pi }^{T}:{T}_{p}M\oplus {N}_{p}M\to {T}_{p}M.$ It follows that $-{\pi }^{T}\circ {\mathrm{G}}_{L}^{{±}^{\prime }}\left(s\right)=\left({\kappa }_{n}\left(s\right)±{\kappa }_{g}\left(s\right)\right)t\left(s\right).$ Moreover, we also have $-{\pi }^{T}\circ {\mathrm{G}}_{L}^{{±}^{\prime }}\left(s\right)=\frac{1}{{\xi }^{±}\left(s\right)}\left({\kappa }_{n}\left(s\right)±{\kappa }_{g}\left(s\right)\right)t\left(s\right).$ According to the above calculations, we can define new invariants ${K}_{L}^{±}$ and ${\stackrel{~}{K}}_{L}^{±}$ by ${K}_{L}^{±}\left(s\right)={\kappa }_{n}\left(s\right)±{\kappa }_{g}\left(s\right)$ and ${\overline{K}}_{L}^{±}\left(s\right)=\frac{1}{{\xi }^{±}\left(s\right)}\left({\kappa }_{n}\left(s\right)±{\kappa }_{g}\left(s\right)\right),$ respectively. We call them the lightcone Gauss-Kronecker curvature (or, lightcone G-K curvature) of M and normalized lightcone Gauss-Kronecker curvature (or, normalized lightcone G-K curvature) of M, respectively. By the definitions, we have ${K}_{L}^{±}\left(s\right)=0$ if and only if ${\stackrel{~}{K}}_{L}^{±}\left(s\right)=0,$ for sȈ I. For $v\in {S}_{+}^{1},c\in \mathbb{R}$ we denote ${S}_{L}\left(v,c\right)=P\left(v,c\right)\cap W$ and call it the lightlike slice. Then we have the following proposition. Using the above notations, the following conditions are equivalent 1. ${K}_{L}^{±}\equiv 0.$ 2. ${\stackrel{~}{\mathbb{G}}}_{L}^{±}$ 3. There exists a constant lightlike vector ν ∈ S±1 such that}Im γ ⊂ SL(ν,c). We first assume that ${K}_{L}^{±}\equiv 0$ This condition is equivalent to ${\stackrel{~}{\mathbb{K}}}_{L}^{±}\equiv 0$ It follows that$G~L±′(s)=−ξ±′(s)ξ±2(s)G~L±(s)±τg(s)ξ±(s)G~L±(s)=(−ξ±′(s)ξ±(s)±τg(s))G~L±(s).$ Since the first component of ${\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)$ is 1, the first component of ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s\right)$ is 0. Therefore ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s\right)=0$ for any sI. It follows that ${\stackrel{~}{\mathbb{G}}}_{L}^{±}$ is constant. Moreover, if${\stackrel{~}{\mathbb{G}}}_{L}^{±}$ is constant, then ${K}_{L}^{±}\equiv 0$Thus the conditions (1) and (2) are equivalent. On the other hand, suppose that ${\stackrel{~}{\mathbb{G}}}_{L}^{±}$ is constant. We assume that $v={\mathrm{G}}_{L}^{±}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{Then}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left\{\gamma \left(s{\right)}_{;}v{\right\}}^{\prime }=0$ This means that 〈γ(s),ν=c, where c ∈ R is a constant real number. Therefore, γ is a part of P(νc) ∩ W. Moreover, we assume that there exists a constant lightlike vector$v\in {S}_{+}^{1}$ ν ∈ S+1 such that$\gamma \subset P\left(v,c\right)\cap W.$ It follows that 〈 γ(s),ν〉 = c. Therefore, 〈t(s),ν〉 = 0. This means that $v\in {N}_{p}M\cap {S}_{+}^{1}.$Thus, $v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)$ for any sI. It follows that the conditions (2) and (3) are equivalent. This completes the proof. As an application of the above proposition, we have the following corollary. Using the same notations, with the above proposition},$s_{0}${\it is a singular point of the normalized lightcone Gauss map ${\stackrel{~}{\mathbb{G}}}_{L}^{±},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}only\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{L}^{±}\left(s\mathrm{o}\right)=0.$ mathrm{G}_{L}^{\pm}$ {\it if and only if} \$K_{L}^{\pm}(s\mathrm{o}) =0.

## 3 Height functions on spacelike curves

In this section we define three families of functions on the spacelike curve γ on W which are helpful for investigating the geometric properties of the spacelike curve.

Let γ : IW be a spacelike curve on W. We first define a function$H:I×{S}_{+}^{1}\to \mathbb{R}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{b}y\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H\left({s}_{,}v\right)=〈\gamma \left(s\right),v〉.$ We call it the lightcone circle height function on γ. For any fixed ν ∈ S+1, we denote hv(s)= H(s,v). Then we have the following proposition.

Suppose that γ : I → W is a unit-speed spacelike curve on W. Then we have the following assertions}:

1. hγ′(s0)=0 if and only if ${\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right).$

2. hγ′(s0) = hγ″(s0)=0 if and only if$v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{L}^{±}\left({s}_{0}\right)=0.$

3. ${h}_{v}^{\prime }\left({s}_{0}\right)={h}_{v}^{″}\left({s}_{0}\right)={h}_{v}^{‴}\left({s}_{0}\right)=0$ if and only if $v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right),{K}_{L}^{±}\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0.$

Since {t(s),n(s),e(s)} is a pseudo orthonormal frame of 13 along γ(s) and ν ∈S+1 there exist η α λ ∈ ℝ with η2 + α2 - λ2 = 0 such that $v=\eta t\left(s\right)+\alpha n\left(s\right)+\lambda e\left(s\right)$ By the definition of hv(s), we can show that hν′(s0)=0 if and only if η = 0 Therefore, we have ν α(n(s0) ± ဉ(s0)) since ν ∈ S+1, we have $v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)$

Since hν″(s0)=0 〈 t′(s0),ν〉 = 0 By the assertion (1) and the Frenet-Serret type formula, we have hν′(s0) = hν″(s0) = 0, if and only if $v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\mathrm{o}{\right)}_{\mathrm{D}}{\kappa }_{n}\left({s}_{0}\right)±{\kappa }_{g}\left({s}_{0}\right)=0$ Therefore, ${K}_{L}^{±}\left({s}_{0}\right)=0.$

Since hν‴(s0) = 0 we have t′(s0),ν〉 = 0. By the assertions (1), (2) and the Frenet-Serret type formula, we have we have${h}_{v}^{\prime }\left({s}_{0}\right)={h}_{v}^{″}\left({s}_{0}\right)={h}_{v}^{‴}\left({s}_{0}\right)=0$ if and only if, we have$v={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}{\right)}_{;}{K}_{L}^{±}\left({s}_{0}\right)={0}_{;}{K}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0.$

Moreover, we define another function we have$I×{S}_{+}^{1}×{\mathbb{R}}^{\ast }\to \mathbb{R}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{H}\left({s}_{,}{v}_{,}r\right)=\left\{\gamma \left(s\right),v\right\}-r$. We call it the extended lightcone circle height function. For any fixed we have$v\in {S}_{+}^{{1}_{;}},\phantom{\rule{thinmathspace}{0ex}}r\in {\mathbb{R}}^{\ast }$ we have ${\overline{h}}_{\left(}{v}_{;}r\right)\left(s\right)=\overline{H}\left({s}_{,}{v}_{,}r\right)$. By almost the same arguments as in above proposition, we have the following proposition.

Suppose that γ : I → W is a unit speed spacelike curve on W. Then we have the following assertions:

1. $h¯(v,r)(s0)=h¯(′v,r)(s0)=0ifandonlyifv=G~L±(s0),r=〈γ(s0),G~L±(s0)〉.$

2. $h¯(v,r)(s0)=h¯(v,r)′(s0)=h¯(v,r)″(s0)=0ifandonlyifv=G~L±(s0),r=〈γ(s0),G~L±(s0)〉andKL±(s0)=0.$

3. $h¯(v,r)(s0)=⋯=h¯(v,r)‴(s0)=0$if and only if $v={\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left({\mathit{s}}_{\mathit{0}}\mathit{\right)}\mathit{,}\mathit{r}\mathit{=}〈\gamma \mathit{\left(}{\mathit{s}}_{\mathit{0}}\mathit{\right)}\mathit{,}{\stackrel{\mathit{~}}{\mathbb{G}}}_{\mathit{L}}^{±}\mathit{\left(}{\mathit{s}}_{\mathit{0}}\mathit{\right)}〉\mathit{,}{\mathit{K}}_{\mathit{L}}^{±}\mathit{\left(}{\mathit{s}}_{\mathit{0}}\mathit{\right)}\mathit{=}\mathit{0}$and ${K}_{L}^{{±}^{\prime }}\left({\mathit{s}}_{\mathit{0}}\mathit{\right)}\mathit{=}\mathit{0.}$

Furthermore, we define the third function $I×L{C}^{\ast }\to \mathbb{R}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\mathit{~}}{\mathit{H}}\mathit{\left(}{\mathit{s}}_{\mathit{,}}\mathit{v}\mathit{\right)}\mathit{=}〈\gamma \mathit{\left(}\mathit{s}\mathit{\right)}\mathit{,}\stackrel{\mathit{~}}{\mathit{v}}〉\mathit{-}{\mathit{v}}_{\mathit{1}}$, where v=(v1,v2,v3) ∈ LC*. We call it the lightcone height function on γ. For any fixed vLC*, we have ${\stackrel{~}{h}}_{v}\left(s\right)=\stackrel{~}{H}\left({s}_{,}v\right)$. By a straightforward calculation, we have the following proposition.

Suppose that γ: IW is a unit speed spacelike curve on W. Then we have the following assertions:

1. ${\stackrel{~}{h}}_{v}\left({s}_{0}\right)={\stackrel{~}{h}}_{v}^{\prime }\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}only\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v=〈\gamma \left({s}_{0}{\right)}_{,}{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left(s\mathrm{o}\right)〉{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right).$

2. ${\stackrel{~}{h}}_{v}\left({s}_{0}\right)={\stackrel{~}{h}}_{v}^{\prime }\left({s}_{0}\right)={\stackrel{~}{h}}_{v}^{″}\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}only\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v=〈\gamma \left({s}_{0}\right),{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left(s\mathrm{o}\right)〉{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{L}^{±}\left({s}_{0}\right)=0.$

3. ${\overline{h}}_{v}\left({s}_{0}\right)=\cdots ={\overline{h}}_{v}^{‴}\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}only\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v=〈\gamma \left(s\mathrm{o}{\right)}_{,}{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left(s\mathrm{o}\right)〉{\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right),{K}_{L}^{±}\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{L}^{{±}^{\prime }}\left({s}_{o}\right)=0.$

On the other hand, we will introduce some general results on the singularity theory for families of function germs as follows. Let F : ℝ × ℝr,(s0,x0))→ ℝ be a function germ. We call F an r-parameter unolding of f, where f(s)=F(s,x0). We say that f has an Ak-singularity at s0 if fp(s0)=0 for all 1 ≤ pk, and fk+1(s0)≠ 0. We also say that f has an Ak-singularity at s0 if fp(s0)=0 for all 1 ≤ pk. Let F be an unfolding of f and f(s) have an Ak-singularity (k ≥ 1) at s0. We denote the (k-1)-jet of the partial derivative $\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_{i}}$ at s0 by ${j}^{k-1}\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_{i}}\left({s}_{,}{x}_{0}\right)\right)\left({s}_{0}\right)=\sum _{j=1}^{k-1}{\alpha }_{ji}{s}^{j}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i={1}_{,}\cdots ,r.$ Then F is called an R+-versal unolding if and only if the (k-1)× r matrix of coefficients (αji) has rank k-1(k-1 ≤ r) . Moreover, we call F an R-versal unfolding if and only if the k × r matrix of coefficients (α0iji) has rank k(kr) , where ${\alpha }_{0i}=\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_{i}}\left(s{0}_{,}x\mathrm{o}\right).$

We now introduce important sets concerning the unfoldings relative to the above notions. The discriminant set of F is the set DF={x ∈Rr| there exists s with $F=\frac{\mathrm{\partial }F}{\mathrm{\partial }s}=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}at\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({s}_{,}x\right)$}.

The catastrophe set of F is the set $CF={(s,x)|∂F∂s(s,x)=0}.$

The bifurcation set of F is the set BF={x ∈ Rr| there exists s with $\frac{\mathrm{\partial }F}{\mathrm{\partial }s}=\frac{{\mathrm{\partial }}^{2}F}{\mathrm{\partial }{s}^{2}}=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}at\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({s}_{;}x\right)\right\}.$ Then we have the following well-known result (cf. [17]).

Let F: (R × R2,(s0,x0))→ R be a 2-parameter unolding of f(s) which has an A2-singularity at s0.Suppose that F is a R-versal unolding. Then DF is locally diffeomorphic to C. Here, $C=\left\{\left({x}_{1,}{x}_{2}\right)|{x}_{1}^{2}-{x}_{2}^{3}=0\right\}$is the ordinary cusp.

Now we can apply the above arguments to our case. Let γ : IW be a unit speed spacelike curve on W, H be the lightcone circle height function on $\gamma ,\overline{H}$be the extended lightcone circle height function on γ and $\overline{H}$be the lightcone height function on γ. According to Proposition 3.1, we have ${C}_{H}=\left\{\left(s,{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right)|s\in I\right\}$. By Propositions 3.2 and 3.3, we have ${\mathcal{D}}_{\overline{H}}=\left\{\left({\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s{\right)}_{,}〈\gamma \left(s{\right)}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right)|s\in I\right\}$and ${\mathcal{D}}_{\overline{H}}=\left\{〈\gamma \left(s\right),{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)〉{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)|s\in I\right\}$. Then we have the following propositon.

Let γ : IW be a regular spacelike curve on W, H be the lightcone circle height function on γ and $\overline{H}$ be the extended lightcone circle heightfunction on γ. Suppose that ${v}_{0}={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{h}_{{v}_{0}}\left(s\right)=H\left({s}_{,}{v}_{0}\right)$.Then the following assertions are equivalent.

1. hv0 has A2}-singularity at s0.

2. H is a R+-versal unolding of hv0.

3. $\overline{H}$is a R-versal unolding of hv0.

Let $v=\left({1}_{,}\mathrm{cos}{\theta }_{,}\mathrm{sin}\theta \right)\in {S}_{+}^{1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\gamma \left(s\right)=\left({r}_{1}\left(s{\right)}_{,}{r}_{2}\left(s{\right)}_{,}{r}_{3}\left(s\right)\right)\in W$we have $H(s,v)=−r1(s)+r2(s)cos⁡θ+r3(s)sin⁡θ=H(s,θ).$

By a straightforward calculation, we have $∂H∂s(s;θ)=−r1′(s)+r2′(s)cos⁡θ+r3′(s)sin⁡θ;∂H∂θ(s;θ)=−r2(s)sin⁡θ+r3(s)cos⁡θ.$

It follows that $∂2H∂s∂θ(s;θ)=−r2′(s)sin⁡θ+r3′(s)cos⁡θ.$

For s=s0, we suppose that $-{r}_{2}^{\prime }\left({s}_{0}\right)\mathrm{sin}{\theta }_{0}+{r}_{3}^{\prime }\left(s\right)\mathrm{cos}{\theta }_{0}=0$ and $t\left({s}_{0}\right)=\left({r}_{1}^{\prime }\left({s}_{0}{\right)}_{,}{r}_{2}^{\prime }\left({s}_{0}\right){,}_{}{r}_{3}^{\prime }\left({s}_{0}\right)\right)$. Since $〈t\left({s}_{0}{\right)}_{,}{v}_{0}〉=-{r}_{1}^{\prime }\left({s}_{0}\right)+{r}_{2}^{\prime }\left({s}_{0}\right)\mathrm{cos}{\theta }_{0}+{r}_{3}^{\prime }\left({s}_{0}\right)\mathrm{sin}{\theta }_{0}=0,$, we have $-{r}_{1}^{\prime }\left({s}_{0}\right)\mathrm{sin}{\theta }_{0}+{r}_{3}^{\prime }\left({s}_{0}\right)=0$. Moreover, we also have $-{r}_{1}^{\prime }\left({s}_{0}\right)\mathrm{cos}{\theta }_{0}+{r}_{2}^{\prime }\left({s}_{0}\right)=0$. Therefore, $t\left({s}_{0}\right)={r}_{1}^{\prime }\left({s}_{0}\right)\left({1}_{;}\mathrm{cos}{\theta }_{0;}\mathrm{sin}{\theta }_{0}\right)$. It follows that t(s) is a unit spacelike vector for any sI, so we have a contradiction. Thus, $\left({\mathrm{\partial }}^{2}H/\mathrm{\partial }s\mathrm{\partial }\theta \right)\left({s}_{0},{\theta }_{0}\right)\ne 0$ Therefore, the rank of the matrix $\left(-{r}_{2}^{\prime }\left({s}_{0}\right)\mathrm{sin}{\theta }_{0}+{r}_{3}^{\prime }\left(s\right)\mathrm{cos}{\theta }_{0}\right)$is one. We have thus shown that the assertions (1) and (2) are equivalent

On the other hand, for extended lightcone circle height function $\overline{H}$, we have $H¯(s,v,r)=−r1(s)+r2(s)cos⁡θ+r3(s)sin⁡θ−r=H¯(s,θ,r).$

Thus, $∂H¯∂r(s,θ,r)=−1,∂H¯∂θ(s,θ,r)=−r2(s)sin⁡θ+r3(s)cos⁡θ.$

It follows that $∂2H¯∂s∂r(s,θ,r)=0,∂2H¯∂s∂θ(s,θ,r)=−r2′(s)sin⁡θ+r3′(s)cos⁡θ=∂2H∂s∂θ(s,θ).$

Suppose that $A= (∂H¯/∂θ)(s0,θ0,r0)(∂2H¯/∂s∂θ)(s0,θ0,r0)(∂H¯/∂r)(s0,θ0,r0)(∂2H¯/∂s∂r)(s0,θ0,r0)= −r2(s0)sin⁡θ0+r3(s0)cos⁡θ0 −r′2(s0)sin⁡θ0+r′3(s0)cos⁡θ0 −1 0$

Since $\left({\mathrm{\partial }}^{2}H/\mathrm{\partial }s\mathrm{\partial }\theta \right)\left({s}_{0,}{\theta }_{0}\right)\ne 0$, we have $\left({\mathrm{\partial }}^{2}\overline{H}/\mathrm{\partial }s\mathrm{\partial }\theta \right)\left({s}_{0,}{\theta }_{0,}{r}_{0}\right)\ne 0$. This means that rankA =2. Therefore, the assertions (1) and (3) are equivalent. This completes the proof.

## 4 Lightcone pedal curves

We now define a mapping ${P}_{L}^{±}:I\to L{C}^{\ast }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{P}_{{L}^{±}}\left(s\right)=〈\gamma \left(s\right),{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)〉{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)$.

We call it the lightcone pedal curve. We also define another mapping $C{P}_{L}^{±}:I\to {S}_{+}^{1}×{\mathbb{R}}^{\ast }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}C{P}_{L}^{±}\left(s\right)=\left({\stackrel{~}{\mathrm{G}}}_{L}^{±}\left(s{\right)}_{,}〈\gamma \left(s{\right)}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)〉\right)$, where ℝ * = ℝ \ {0}. We call CPL± the cylindrical lightcone pedal curve. By definitions, we have $\left\{{P}_{L}^{±}\left(s\right)|s\in I\right\}={\mathcal{D}}_{\stackrel{~}{H}}$ and $\left\{C{P}_{L}^{±}\left(s\right)|s\in I\right\}={\mathcal{D}}_{\overline{H}}$. In order to investigate the relationship between the lightcone pedal curve and the cylindrical lightcone pedal, we define a mapping $\varphi :{S}_{+}^{1}×{\mathbb{R}}^{\ast }\to L{C}^{\ast }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\varphi \left(v,r\right)=rv$. It is easy to check that Φ is a diffeomorphism and $\varphi \left(C{P}_{L}^{±}\left(s\right)\right)={P}_{L}^{±}\left(s\right)$, this means that CPL± and PL± are diffeomorphic. Therefore, the singular sets of ${\mathcal{D}}_{\stackrel{~}{H}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{D}}_{\overline{H}}$ are diffeomorphic.

On the other hand, let F : W → ℝ be a submersion and γ : IW be a spacelike curve on W. We say that γ and F−1(0) have k-point contact for t=t0 provided the function g defined by g(t)=F(γ(t)) satisfies $g(t0)=g′(t0)=⋯=g(k−1)(t0)=0,g(k)(t0)≠0.$

We also say that the order of contact is k. Dropping the condition g(k) (t0) ≠ 0 we say that there is at least k-point contact (cf. [17]). Then we have the following result.

Let γ : IW be a spacelike curve on W. For ${v}_{0}^{±}={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}{\right)}_{,}{s}_{0}\in I$, we assume that ${h}_{{v}_{0}}±:I\to \mathbb{R}$ is the lightlike circle height function and $I\to L{C}^{\ast }$is the lightcone pedal curve. Then we have the following:

(A)The following conditions are equivalent.

1. ${K}_{L}^{±}\left({s}_{0}\right)\ne 0.$

2. ${\mathrm{G}}_{L}^{±}$is non-singular at s0.

3. hv0± hasA1 singular point.

4. γ and ${S}_{L}^{±}\left({\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}{\right)}_{,}〈\gamma \left({s}_{0}{\right)}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)〉\right)$have 2-point contact at} s0.

5. PL±is an immersion at s0.

(B) The following conditions are equivalent.

1. KL±(s0)=0 and KL±'}(s0)≠ 0.

2. ${\stackrel{~}{\mathbb{G}}}_{L}^{±}$has A1 singular point at s0.

3. hv0± has A2 singular point.

4. γ and ${S}_{L}^{±}\left({\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}{\right)}_{,}〈\gamma \left({s}_{0}{\right)}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)〉\right)$ have 3-point contact at s0.

5. PL±is is an ordinary cusp at s0.

6. $H¯$is a R-versal unfolding of hv0±.

7. H is a R+-versal unfolding of hv0±.

We first consider (A). According to Corollary 2.3, the assertions (1) and (2) are equivalent. By Proposition 3.1 (1), the ${K}_{L}^{±}\left({s}_{0}\right)\ne 0$if and only if $\left(\mathrm{\partial }{h}_{{v}_{0}}/\mathrm{\partial }s\right)\left({s}_{0}\right)=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left({\mathrm{\partial }}^{2}{h}_{{v}_{0}}/\mathrm{\partial }{s}^{2}\right)\left({s}_{0}\right)\ne 0$.This means that the assertions (1) and (3) are equivalent. Moreover, we define a mapping $\overline{\mathcal{H}}:W\to \mathbb{R}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}by\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathcal{H}\left(u\right)=〈{u}_{,}{v}_{0}〉-{r}_{0}$, where v0S+1, r0 ∈ ℝ. It follows that $\overline{\mathcal{H}}\left(\gamma \left(s\right)\right)={h}_{{v}_{0}}\left(s\right)-{r}_{0}$. Since ${\overline{\mathcal{H}}}^{-1}\left(0\right)={S}_{L}\left({v}_{0,}{r}_{0}\right)$and 0 is a regular value of $\overline{\mathcal{H}},{h}_{{v}_{0}}$has an Ak-singularity at s0 if and only if γ and SL(v0,r0) have (k+1)-point contact for s0. This means that the assertions (3) and (4) are equivalent. By the definition, we have $C{P}_{L}^{{±}^{\prime }}\left(s\right)=\left({\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s{\right)}_{,}〈\gamma \left(s{\right)}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s\right)〉\right)$. It follows that $C{P}_{L}^{{±}^{\prime }}\left(s\right)\ne 0$ if and only if ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s\right)\ne 0$. Since the singular sets of ${P}_{\mathrm{\Lambda }}^{±}$are diffeomorphic, we have $C{P}_{L}^{{±}^{\prime }}\left(s\right)\ne 0$if and only if ${P}_{L}^{{±}^{\prime }}\left(s\right)\ne 0$. Therefore, the assertions (2) and (5) are equivalent.

On the other hand, we consider (B). By Proposition 3.1 (2), the assertions (1) and (3) are equivalent. Moreover, by Proposition 3.1 (1), we have $\left(\mathrm{\partial }H/\mathrm{\partial }s\right)\left(s,{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right)=0$. If we take the derivative of the equation, then we have $\frac{d}{ds}\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }s}\right)\left({s}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right)=\frac{\mathrm{\partial }{}^{2}H}{\mathrm{\partial }{s}^{2}}\left({s}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right)+\frac{\mathrm{\partial }{}^{2}H}{\mathrm{\partial }s\mathrm{\partial }v}\left({s}_{,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left(s\right)\right){\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left(s\right)=0.$

For s=s0, we have ${h}_{{v}_{0}}^{″}\left({s}_{0}\right)+{\mathrm{\partial }}^{2}H/\mathrm{\partial }s\mathrm{\partial }v\right)\left({s}_{0,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\right){\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0$where ${v}_{0}={\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)$. By the proof of Proposition 3.5, we have $\left({\mathrm{\partial }}^{2}H/\mathrm{\partial }s\mathrm{\partial }v\right)\left({s}_{0,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\right)\ne 0$. It follows that ${h}_{{v}_{0}}^{″}\left({s}_{0}\right)=0$if and only if ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0.$By a similar calculation as above, we have ${h}_{{v}_{0}}^{‴}\left({s}_{0}\right)+\left({\mathrm{\partial }}^{2}H/\mathrm{\partial }s\mathrm{\partial }v\right)\left({s}_{0,}{\stackrel{~}{\mathbb{G}}}_{L}^{±}\left({s}_{0}\right)\right){\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{″}}\left({s}_{0}\right)=0$. This means that ${h}_{{v}_{0}}^{‴}\left({s}_{0}\right)=0$if and only if ${\stackrel{~}{\mathrm{G}}}_{L}^{{±}^{″}}\left({s}_{0}\right)=0$. Therefore, the assertions (2) and (3) are equivalent. Moreover, if we consider the mapping $\overline{\mathcal{H}}:W\to \mathbb{R}$ defined in the proof of (A), then we obtain that the assertions (3) and (4) are equivalent. Furthermore, we consider the cylindrical lightcone pedal curve $CPL±(s)=(G~L±(s),〈γ(s),G~L±(s)〉)=(G~L±(s),H(s,G~L±(s)))$

and denote $v(k)=G~L±(k)(s)(k=0,1,2,3)$ and $v0=G~L±(so)$.Then we have $CPL±′(s)=(v;′∂H∂v(s,v)v′),$ $CPL±″(s)=(v″,∂2H∂s2(s,v)+2∂2H∂v∂s(s,v)v′+∂2H∂v2(s,v)(v′)2+∂H∂v(s,v)v″),$ $CPL±‴(s)=(v‴,∂3H∂s3(s,v)+3∂3H∂v∂s2(s,v)v′+3∂3H∂v2∂s(s,v)(v′)2+∂3H∂v3(s,v)(v′)3+3∂2H∂v2(s,v)v′v″+3∂2H∂v∂s(s,v)v″+∂H∂v(s,v)v‴).$

The previous discussion shows that $C{P}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0$ if and only if ${\mathrm{G}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0.$ Then we obtain that CPL± (s0) is an ordinary cusp if and only if ${\stackrel{\mathbb{~}}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0$ and CPL±″(s0), CPL±‴(s0) are linearly independent, where we have used the well-known fact that CPL±(s0) is an ordinary cusp if and only if $C{P}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0$ and $C{P}_{L}^{{±}^{″}}\left({s}_{0}\right),C{P}_{L}^{{±}^{‴}}\left({s}_{0}\right)$ are linearly independent. Moreover, we denote $A=G~L±″(s0)∂2H∂s2(s0,v0)+∂H∂v(s0,v0)G~L±″(s0)G~L±‴(s0)∂3H∂s3(s0,v0)+3∂2H∂s∂v(s0,v0)G~L±″(s0)+∂H∂v(s0,v0)G~L±‴(s0).$ $B=G~L±″(s0)∂2H∂s2(s0,v0)G~L±‴(s0)∂3H∂s3(s0,v0)+3∂2H∂s∂v(s0,v0)G~L±″(s0).$

Then rank A = rank B. Since the assertions (2) and (3) are equivalent, condition (2) or (3) holds if and only if ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0,{\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{″}}\left({s}_{0}\right)\ne 0$ and ${h}_{{v}_{0}}^{\prime }\left({s}_{0}\right)={h}_{{v}_{0}}^{″}\left({s}_{0}\right)=0,{h}_{{v}_{0}}^{‴}\left({s}_{0}\right)\ne 0.$ Therefore, the assertion (2)is equivalent to the conditions ${\stackrel{~}{\mathbb{G}}}_{L}^{{±}^{\prime }}\left({s}_{0}\right)=0,$ rank B = 2. Thus, the assertions (2), (3) and (5) are equivalent. By Proposition 3.5, we have the assertions (3), (6) and (7) are equivalent. This completes the proof.

## Acknowledgements

This work is supported by NSF of China (Grant No. 11101072) and STDP of Jilin Province (Grant No. 20150520052JH).

The author would like to thank Professor Shyuichi Izumiya and Professor Masatomo Takahashi for their constructive suggestions.

The author would like to thank the referee for helpful comments to improve the original manuscript.

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Accepted: 2016-09-02

Published Online: 2016-11-14

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 889–896, ISSN (Online) 2391-5455,

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