# Singularities of Null Hypersurfaces of Pseudonull Curves

Authors: Xiupeng Cui; Donghe Pei
Journal: Journal of Function Spaces
 (2015)

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## 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.


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## Body

## 1. 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 [1–7]. Pseudonull helices, pseudonull Mannheim curves, pseudonull rectifying curves, and pseudonull osculating curves are considered, respectively, in [1], [2], [3], and [4]. And the relations are gotten in [5] between pseudonormal and pseudorectifying curves in Minkowski space-time. Moreover, the involute-evolute of the pseudonull curve is studied in [6] and they prove that there is no involute of pseudonull curves in Minkowski 3-space.However, there are few articles about pseudonull curves from the view point of singularities theory. In this paper, we focus on the singularities of null hypersurfaces of pseudonull curves. Here, we investigate the pseudonull curves withk1(s)≠0. We get the relationship between the singularities of null hypersurfaces of pseudonull curves and the functions ρ and η. When k1(s)=0, the third normal vector equals γ(3)/γ(3). This situation is considered in [7]. The authors, Petrović-Torgašev et al., make a contribution to the research of pseudonull curves. For example, they obtain the Frenet equations of pseudonull curves with only two curvatures and classify all such curves with constant curvatures. Besides curves in semi-Euclidean space, regarding singularity, have been studied extensively by Pei et al. and they make some achievements [8–11].Here we discuss null hypersurfaces and nullcone Gaussian surfaces of pseudonull curves inR24. There are some differences between Minkowski space-time (i.e., R14) and R24, for instance, the appearance of nullcone along a timelike pseudonull curve. Thus, we consider a one-parameter family of null Gauss indicatrices. We call it nullcone Gaussian surface. It can locally be a spacelike surface or a Lorentzian surface or a 1-lightlike surface depending on the characteristic of tangent plane of the surface. Moreover, we point that the nullcone Gaussian surface is also a regular surface. The null hypersurface is a surface bundle along a pseudonull curve whose fibres are the nullcone Gaussian surfaces. To allow a useful study of these singularities, we consider Lorentz distance-squared functions (which are denoted by G:I×R24→R, G(s,v)=〈γ(s)-v,γ(s)-v〉). These functions are the unfolding processes of these singularities in the local neighborhood of (s0,v0), and these functions only depend on the germ that they are unfolding. In this paper, we create these functions by varying a fixed point v in the Lorentz distance-squared function G(s,v)=〈γ(s)-v,γ(s)-v〉, to obtain a family of functions. We show that these singularities are versally unfolded by the family of Lorentz distance-squared functions. If the singularity of gv is Ak-type and if the corresponding four-parameter unfolding is versal, then, by applying Bruces theory (cf. Bruce and Giblin [12]), we know that the discriminant set of the four-parameter unfolding processes is locally diffeomorphic to a cusp or a swallowtail or a butterfly under certain conditions. Thus, we have completed the classification of the singularities of a null hypersurface of a pseudonull curve.The main result is Theorem2 which is about the geometric meanings of the singularities of null hypersurfaces of pseudonull curves. Our work must be conducive to the research of pseudonull curves and be an exploration to null submanifolds in this space. And we will consider the situation that γ′′ is not a null vector in our other paper [13]. We assume throughout the paper that all manifolds and maps are C∞ unless explicitly stated otherwise.


## 2. Preliminaries

In this section we introduce some basic notions for semi-Euclidean 4-space with index 2 and pseudonull curves. LetR4 be a 4-dimensional vector space. For x=(x0,x1,x2,x3) and y=(y0,y1,y2,y3) in R4, thepseudoscalar product of x and y is defined as 〈x,y〉=-x0y0-x1y1+x2y2+x3y3. We call (R4, ,) asemi-Euclidean4-spacewithindex2. We write R24 instead of (R4, 〈,〉). We say that a nonzero vector x∈R24 isspacelike, null, or timelike if 〈x,x〉>0, 〈x,x〉=0, or 〈x,x〉<0, respectively. Thenorm of the vector x∈R24 is defined as x=x,x. The signature of a vector x∈R24∖{0} is defined as(1)sign(x)=1xisspacelike0xisnull-1xistimelike.We callNCα={x=(x0,x1,x2,x3)∈R24∣〈x-α,x-α〉=0} anullcone withvertexα and denote NC∗=NC0∖{0}.For anyx1,x2,x3∈R24, we define the vector x1∧x2∧x3 as(2)x1∧x2∧x3=-e0-e1e2e3x10x11x12x13x20x21x22x23x30x31x32x33,where {e0,e1,e2,e3} is the canonical basis of R24 and xi=(xi0,xi1,xi2,xi3). We can easily show that (3)x,x1∧x2∧x3=det⁡(x,x1,x2,x3),so that x1∧x2∧x3 is pseudoorthogonal to any xi(i=1,2,3).Letγ:I→R24 be a smooth regular curve (i.e., γ′(t)≠0), where I is an open interval. For any t∈I, the curve γ is calledspacelike, null (lightlike),ortimelike if the velocity of the curve is 〈γ˙(t),γ˙(t)〉>0, 〈γ˙(t),γ˙(t)〉=0, or 〈γ˙(t),γ˙(t)〉<0, respectively.Letγ:I→R24 be a unit speed timelike curve, parameterized by the arc length parameter s; that is, 〈γ′(s),γ′(s)〉=-1. For any s∈I, the curve γ is called apseudonull curve if γ′′(s) is a null vector (see [7]).One can define the null frameE={t,n1,n2,n3} of R24 which is positively oriented 4-tuple of vectors satisfying(4)n1,n1=n2,n2=0,n1,n2=12,t,t=-1,n3,n3=1,t,n1=t,n2=t,n3=n1,n3=n2,n3=0.For the sake of the simple forms and the convenience of our research objects, we take 〈n1,n2〉=1/2. The Frenet formula of γ with respect to the frame E is as follows:(5)t′(s)=n1(s)n1′(s)=k1(s)n1(s)+k2(s)n3(s)n2′(s)=12t(s)-k1(s)n2(s)+k3(s)n3(s)n3′(s)=-2k3(s)n1(s)-2k2(s)n2(s).When k1(s)=0, n3(s)=γ(3)(s)/γ(3)(s) which is the situation that the authors consider in [7]. We focus on k1(s)≠0. And we construct the null frame of γ in a different way. In [7], they also investigate W-curves, that is, the pseudonull curves with constant curvatures, and obtain the classifications of W-curves. Here, we do not discuss W-curves. We focus on the fact that k2(s) is not a constant. In particular k2(s)≠0; otherwise γ′′,γ3,γ4,…, are all null vectors with the same direction. It means γ′′ is a lightlike straight line.Without loss of generality, we suppose thatγ(3) and γ(4) are not null vectors. And we have 〈γ(3),γ(4)〉≠0 under the condition that k2(s) is not a constant. Thus, we take (6)t(s)=dγds,n1(s)=γ′′(s),n2(s)=-γ4,γ44γ3,γ32(γ′′+2γ3,γ3γ4,γ4γ(4))(s),n3(s)=γ3,γ4γ3,γ33/2(γ′′+γ3,γ3γ3,γ4γ(3))(s),which are appropriate for conditions (4). Thus, we have(7)k1(s)=-γ4,γ42γ3,γ32·γ3,γ′′+2γ3,γ3γ4,γ4γ4(s),k2(s)=γ3,γ4γ3,γ33/2γ3,γ′′+γ3,γ3γ3,γ4γ3(s),k3(s)=γ3,γ4γ3,γ33/2n2′,γ′′+γ3,γ3γ3,γ4γ3(s).We define thefamily of null Gauss indicatrices of γ(8)ngγ:U⟶NC∗by ngγ(s,θ)=tanθn1(s)-cotθn2(s)+n3(s), where U=I×(0,π/2). It is also called thenullcone Gaussian surface of γ. By a straightforward calculation, we have (9)∂ngγ∂θ=(1+tan2θ)n1+(1+cot2θ)n2,∂ngγ∂s=-12cotθt+(k1tanθ-2k3)n1+(k1cotθ-2k2)n2+(k2tanθ-k3cotθ)n3.The above two vectors are definitely linearly independent, because cotθ≠0. Therefore, the nullcone Gaussian surface ngγ is a regular surface. Obviously, ∂ngγ/∂θ is a spacelike vector. Let (10)ι=(2+tan2θ+cot2θ)(2k2∂ngγ∂θ+(1+cot2θ)∂ngγ∂s)-2k2∂ngγ∂θ+1+cot2θ∂ngγ∂s,∂ngγ∂θ∂ngγ∂θ.We have 〈ι,∂ngγ/∂θ〉=0 and the vector ι might be spacelike, timelike, or null. Therefore, the nullcone Gaussian surface ngγ might locally be a spacelike surface or a Lorentzian surface or a 1-lightlike surface depending on the characteristic of tangent plane of the surface. In [14], the authors introduce the notion of r-lightlike surface. As a lightlike submanifold, its null space is not empty and its nullity degree is just r. In other words, there will always be some vectors simultaneously pseudoorthogonal to the tangent space as well as the normal space. And we will give some results associated with 1-lightlike nullcone Gaussian surface in our other paper [13]. On the other hand, when the nullcone Gaussian surface degenerates onto n1(s) or n2(s), it can be constructed by the sum or minus of the spacelike vector n1(s)+n2(s) and the timelike vector n1(s)-n2(s). We can use similar method in [13], which we do not consider here.Remark 1.
If we takeU=I×(-π/2,0), we can have the same discussion. For the sake of the continuity of the surface, we only consider one situation.We define the null hypersurface alongγ(11)nhγ:U×R⟶R24by nhγ(s,θ,t)=γ(s)+tngγ(s,θ). If we fix θ0, the null hypersurface turns into a null ruled surface along γ.We also define new invariants of the pseudonull curve inR24 by (12)η(s)=(2k22k3-k1′k2k1-k1k22+k2′)(s).LetF:R24→R be a submersion and γ:I→R24 a timelike curve. We say that γ and F-1(0) have k-point contact for t=t0 if the function h(t)=F∘γ(t) satisfies h(t0)=h′(t0)=⋯=h(k-1)(t0)=0, h(k)(t0)≠0. We also say that γ and F-1(0) haveat leastk-point contact for t=t0 if the function h(t)=F∘γ(t) satisfies h(t0)=h′(t0)=⋯=h(k-1)(t0)=0. For any fixed v0∈R24, we have a model surface NCv0. It is a light cone with vertex v0. We now consider the following conditions.(A 1)
The number of pointsp of γ(I) where the model surface at p having five-point contact with the curve γ is finite.(A 2)
There is no pointp of γ(I) where the model surface at p has greater than or equal to six-point contact with the curve γ.Here, we present the main results in this paper.Theorem 2.
Letγ:I→R24 be a timelike pseudonull curve. Let v0=nhγ(s0,θ0,t0); one has the following.
(1)NCv0 and γ have at least 2-point contact at s0.
(2)NCv0 and γ have 3-point contact at s0 if and only if there exists θ0∈(0,π/2) such that (13)ρ(s0,θ0)≠0,γ(s0)-v0=-2cotθ0ngγ(s0,θ0),where ρ(s,θ)=-1/2k1(s)cotθ+k2(s). Under this condition, the null hypersurface nhγ at v0 is locally diffeomorphic to C(2,3)×R2 and the null focal set nfγ isnonsingular.
(3)NCv0 and γ have 4-point contact at s0 if and only if there exists θ0=θ(s0)∈(0,π/2) such that (14)ρs0,θs0=0,ηs0≠0,γ(s0)-v0=-2cotθ(s0)ngγ(s0,θ(s0)),where ρ(s,θ)=-1/2k1(s)cotθ+k2(s). Under this condition, the null hypersurface nhγ at v0 is locally diffeomorphic to SW×R, the null focal set nfγ is locally diffeomorphic to C(2,3,4)×R, and the singular value set of nfγ is a regular curve.
(4)NCv0 and γ have 5-point contact at s0 if and only if there exists θ(s0)∈(0,π/2) such that (15)ρs0,θs0=ηs0=0,η′s0≠0,γ(s0)-v0=-2cotθ(s0)ngγ(s0,θ(s0)),where ρ(s,θ)=-1/2k1(s)cotθ+k2(s). Under this condition, the null hypersurface nhγ at v0 is locally diffeomorphic to BF, the null focal set nfγ is locally diffeomorphic to C(BF)×R, and the singular value set of nfγ is locally diffeomorphic to the C(2,3,4,5)-cusp.One callsC(2,3)={(x1,x2)∣x1=u2,x2=u3} a (2,3)-cusp,C(2,3,4)={(x1,x2,x3)∣x1=u2,x2=u3,x3=u4} a (2,3,4)-cusp (Figure 1), C(2,3,4,5)={(x1,x2,x3,x4)∣x1=u2,x2=u3,x3=u4,x4=u5} a (2,3,4,5)-cusp (Figure 2), SW={(x1,x2,x3)∣x1=3u4+u2v,x2=4u3+2uv,x3=v} aswallowtail,BF={(x1,x2,x3,x4)∣x1=5u4+3vu2+2wu,x2=4u5+2vu3+wu2,x3=u,x4=v} abutterfly (Figure 3), and C(BF)={(x1,x2,x3,x4)∣x1=6u5+u3v,x2=25u4+9u2v,x3=10u3+3uv,x4=v} a c-butterfly (i.e., the singular value set of the butterflies).Figure 1
(2,3)-cusp and swallowtail.Figure 2
Projection of a (2,3,4,5)-cusp, respectively, onx1x2x3-space, x1x2x4-space, x1x3x4-space, and x2x3x4-space.
(a)
(b)
(c)
(d)Figure 3
Whenw=0, projection of a butterfly, respectively, on x1x2x3-space, x1x2x4-space, x1x3x4-space, and x2x3x4-space.We will give the proof of Theorem2 in Section 4.


## 3. A Family of Lorentz Distance-Squared Functions of Pseudonull Curveγ(s)

In this section we introduce one very useful family of functions on a pseudonull curve. For a pseudonull curveγ, we define the function: (16)G:I×R24⟶R,G(s,v)=γs-v,γs-v.This function is called theLorentz distance-squared function of the pseudonull curve γ. We use the notation gv(s)=G(s,v) for any fixed vector v in R24. They describe the contact between γ(s) and a nullcone. As we study this family of functions, it will become clear how singularities and the corresponding catastrophes arise.By using (5) and by making tedious calculations, we can state Proposition 3.Proposition 3.
Letγ:I→R24 be a timelike pseudonull curve. Supposing that γ(s0)≠v0, then one has the following:(1)
gv0(s0)=gv0′(s0)=0 if and only if there exist θ0∈(0,π/2) and μ∈R∖{0} such that γs0-v0=μngγs0,θ0;(2)
gv0(s0)=gv0′(s0)=gv0′′(s0)=0 if and only if there exists θ0∈(0,π/2) such that γs0-v0=-2/cotθ0ngγs0,θ0;(3)
gv0(s0)=gv0′(s0)=gv0′′(s0)=gv0′′′(s0)=0 if and only if there exists θ0∈(0,π/2) such that γ(s0)-v0=-2/cotθ0ngγ(s0,θ0) and -1/2k1(s0)cotθ0+k2(s0)=0, so one can write θ0=θ(s0);(4)
gv0(s0)=gv0′(s0)=gv0′′(s0)=gv0′′′(s0)=gv0(4)(s0)=0 if and only if there exists θ(s0)∈(0,π/2) such that γ(s0)-v0=-2/cotθ(s0)ngγ(s0,θ(s0)),   -1/2k1(s0)cotθ(s0)+k2(s0)=0, and ηs0=0;(5)
gv0(s0)=gv0′(s0)=gv0′′(s0)=gv0′′′(s0)=gv0(4)(s0)=gv0(5)(s0)=0 if and only if there exists θ(s0)∈(0,π/2) such that γ(s0)-v0=-1/cotθ(s0)ngγ(s0,θ(s0)) and -1/2k1(s0)cotθ(s0)+k2(s0)=η(s0)=η′(s0)=0.The above proposition also states that the discriminant set of the Lorentz distance-squared functionG is given by (17)DG=nhγ(U×R)={v=γ(s)+μngγ(s,θ)∣(s,θ)∈U,μ∈R},which is the image of the null hypersurface along γ. Therefore, a singular point of the null hypersurface is the point v0=γ(s0)+μ0ngγ(s0,θ0), where μ0=-(2/cotθ0).We definenfγ:U→R24 as (18)nfγs,θ=γs+μ0ngγs,θ;we call it thenull focal set ofγ. By definition, the null focal set is the singular value set of the null hypersurface nhγ.


## 4. Classifications of Singularities

In this section we classify singularities of the null hypersurface alongγ as an application of the unfolding theory of functions. Detailed descriptions could be found in [12]. Let F:(R×Rr,(s0,x0))→R be a function germ. We call F anr-parameter unfolding of f, if f(s)=Fx0(s,x0). We say f has Ak-singularity at s0, if f(p)(s0)=0 for all 1≤p≤k and f(k+1)(s0)≠0. Let F be an r-parameter unfolding of f and f has Ak-singularity (k≥1) at s0. We denote the (k-1)-jet of the partial derivative ∂F/∂xi at s0 by(19)jk-1∂F∂xis,x0s0=∑j=1k-1αjis-s0j,i=1,…,r.If the rank of k×r matrix (α0i,αji) is k(k≤r), then F is called aversal unfolding of f, where α0i=∂F/∂xi(s0,x0).Inspired by the proposition in the previous section, we have(20)DFl=x∈Rr∣∃s∈R,∂lF∂slhhhhhhhhhhFs,x=∂F∂ss,x=⋯=∂lF∂sl(s,x)=0,which is called adiscriminant set with orderl. Therefore, we have the following proposition.Proposition 4.
For a pseudonull curveγ, one has (21)DG=DG1=nhγU×R,DG2=nfγU,DG3isthesingularvaluesetofnfγ.Then we have the following classification theorem as a Corollary 6.6 in [12].Theorem 5.
LetF:(R×Rr,(s0,x0))→R be an r-parameter unfolding of f with Ak-singularity at s0. Suppose F is a versal unfolding of f; then one has the following assertions.(a)
Ifk=1, then DF is locally diffeomorphic to {0}×Rr-1 and DF2=∅.(b)
Ifk=2, then DF is locally diffeomorphic to C(2,3)×Rr-2 and DF2 is locally diffeomorphic to {0}×Rr-2 and DF3=∅.(c)
Ifk=3, then DF is locally diffeomorphic to SW×Rr-3,   DF2 is locally diffeomorphic to C(2,3,4)×Rr-3 and   DF3 is locally diffeomorphic to {0}×Rr-3 and DF4=∅.(d)
Ifk=4, then DF is locally diffeomorphic to BF×Rr-4, DF2 is locally diffeomorphic to C(BF)×Rr-4, DF3 is locally diffeomorphic to C(2,3,4,5)×Rr-4, and DF4 is locally diffeomorphic to {0}×Rr-4 and DF5=∅.For the proof of Theorem2 we have the following fundamental proposition in this paper.Proposition 6.
Ifg(s) has Ak-singularity (k=1,2,3,4) at s0, then G is a versal unfolding of g.Proof.
By definition, we have(22)Gs,v=-x0s-v02-x1s-v12+x2s-v22+x3s-v32,where γ(s)=(x0(s),x1(s),x2(s),x3(s)) and v=(v0,v1,v2,v3).
Thus, we get(23)∂G∂vis,v=2xis-vii=0,1-2xis-vii=2,3,∂2G∂s∂vi(s,v)=2xi′(s)i=0,1-2xi′(s)i=2,3.
For a fixedv0=(v00,v01,v02,v03), the 3-jet of ∂G/∂vi(s,v0) at s0 is(24)j(3)∂G∂vi(s,v0)(s0)=2xi′(s0)(s-s0)+xi′′(s0)s-s02+13xi′′′(s0)s-s03i=0,1-2xi′(s0)(s-s0)-xi′′(s0)s-s02-13xi′′′(s0)s-s03i=2,3.
The condition for versatility can be checked as follows.
(1) By Proposition3, g has A1-singularity at s0 if and only if there exist θ0∈(0,π/2) and μ∈R∖{0} such that γ(s0)-v0=μngγ(s0,θ0) and μ≠-(2/cotθ0). When g has A1-singularity at s0, we require the 1×4 matrix A1=(2(x0(s0)-v0),2(x1(s0)-v1),-2(x2(s0)-v2),-2(x3(s0)-v3)) to have rank 1, which it always does since v0≠γ(s0).
(2) It also follows from Proposition3 that g has A2-singularity at s0 if and only if there exists θ0∈(0,π/2) such that γ(s0)-v0=-2/cotθ0ngγ(s0,θ0) and ρ(s0,θ(s0))≠0. We require 2×4 matrix(25)A2=2(x0(s0)-v0)2(x1(s0)-v1)-2(x2(s0)-v2)-2(x3(s0)-v3)2x0′(s0)2x1′(s0)-2x2′(s0)-2x3′(s0)to have rank 2. Otherwise, if rankA2=1, it means that γ(s0)-v0 and t(s0) are linearly dependent. This contradicts with the fact that {t(s),n1(s),n2(s),n3(s)} is the pseudoorthogonal frame of γ.
(3) By Proposition3, g has A3-singularity at s0 if and only if there exists θ(s0)∈(0,π/2) such that γ(s0)-v0=-2/cotθ(s0)ngγ(s0,θ(s0)), ρ(s0,θ(s0))=0, and η(s0)≠0. We require 3×4 matrix(26)A3=2(x0(s0)-v0)2(x1(s0)-v1)-2(x2(s0)-v2)-2(x3(s0)-v3)2x0′(s0)2x1′(s0)-2x2′(s0)-2x3′(s0)x0′′(s0)x1′′(s0)-x2′′(s0)-x3′′(s0)to have rank 3. Otherwise, if rankA3=2, it means that γ′′(s0) can be generated by γ(s0)-v0 and t(s0). Through a straightforward calculation, we can easily show that it is a contradiction.
(4) By Proposition3, g has A4-singularity at s0 if and only if there exists θ(s0)∈(0,π/2) such that γ(s0)-v0=-2/cotθ(s0)ngγ(s0,θ(s0)), ρ(s0,θ(s0))=η(s0)=0, and η′(s0)≠0. We require 4×4 matrix(27)A4=2(x0(s0)-v0)2(x1(s0)-v1)-2(x2(s0)-v2)-2(x3(s0)-v3)2x0′(s0)2x1′(s0)-2x2′(s0)-2x3′(s0)x0′′(s0)x1′′(s0)-x2′′(s0)-x3′′(s0)13x0′′′(s0)13x1′′′(s0)-13x2′′′(s0)-13x3′′′(s0)to have rank 4.
In fact,(28)det⁡A4=43det⁡⁡γs0-v0,γ′s0,γ′′s0,γ′′′s0=83k2(s0)det⁡(n2,t,n1,n3)≠0.Therefore, rankA4=4.
In summary,G is a versal unfolding of g. This completes the proof.We now give the proof of Theorem2.Proof of Theorem2.
Letγ:I→R24 be a pseudonull curve. As v0=nhγ(s0,θ0,t0), we give a function G:R24→R, by G(u)=〈u-v0,u-v0〉; then we assume that gv0(s)=G(γ(s)). Because G-1(0)=NCv0 and 0 is a regular value of G,γ and NCv0 have (k+1)-point contact for s0 if and only if gv0(s) has Ak-singularity at s0. Thus γ and NCv0 have at least 2-point contact for s0 if and only if gv0(s0)=gv0′(s0)=0. By Proposition 4, we have (29)DG1=nhγU×R,DG2=nfγU,DG3isthesingularvaluesetofnfγ.By combining Proposition 3, Theorem 5, and Proposition 6, we get the results.


## 5. Generic Properties

In this section we consider the generic properties of pseudonull curves inR24. The main tool is transversality theorem. Assume that Embt(I,R24) is the space of timelike embedding γ:I→R24 equipped with Whitney C∞-topology. We also consider the function G:R24×R24→R defined by G(u,v)=〈u-v,u-v〉. We claim that Gv is a submersion for any v∈R24, where Gv(u)=〈u-v,u-v〉. For any γ∈Embt(I,R24), we have G=G∘(γ×idR24). We also have the l-jet extension(30)j1lG:I×R24⟶Jl(I,R)defined by j1lG(s,v)=jlgv(s); we consider the trivialization Jl(I,R)≡I×R×Jl(1,1). For any submanifold Q⊂Jl(1,1), we denote that Q~=I×{0}×Q. It is obvious that j1lG is a submersion and Q~ is an immersed submanifold of Jl(I,R). Then j1lG is transversal to Q~. We have the following proposition as a corollary of Lemma 6 in [15].Proposition 7.
Let Q be a submanifold ofJl(1,1). Then the set(31)TQ={γ∈Embt(I,R24)∣j1lGistransversaltoQ~}is a residual subset of Embt(I,R24). If Q is a closed subset, then TQ is open.Letf:(R,0)→(R,0) be a function germ which has Ak-singularity at 0. It is well known that there exists a diffeomorphism germ ϕ:(R,0)→(R,0) such that f∘ϕ(s)=±sk+1. This is the classification of Ak-singularities. For any z=jlf(0)∈Jl(1,1), we have the orbit Ll(z) given by the action of the Lie group of l-jet diffeomorphism germs. If f has Ak-singularity, then the codimension of the orbit is k. There is another characterization of versal unfolding as follows [16].Proposition 8.
Assume thatF:(R×Rr,0)→(R,0) is an r-parameter unfolding of f:(R,0)→(R,0) which has an Ak-singularity at 0. Then F is a versal unfolding if and only if j1lG is transversal to the orbit Ll(jlf(0))~ for l≥k+1. Here, j1lG:(R×Rr,0)→Jl(R,R) is the l-jet extension of G given by j1lG(s,x)=jlGx(s).The generic classification theorem is given as follows.Theorem 9.
There exists a dense and open subsetTLkl⊂Embt(I,R24) such that, for any γ∈TLkl, the lightlike hypersurface of γ is locally diffeomorphic to the cuspidal edge, the swallowtail, or the butterfly at any singular point.Proof.
Forl≥5, we discuss the decomposition of the jet space Jl(1,1) into Ll(1) orbits; we now define a semialgebraic set by(32)∑l={z=jlf(0)∈Jl(1,1)∣fhasanA≥5-singularity}.Then the codimension of ∑l is 5. Therefore, the codimension of ∑l~=I×{0}×∑l is 6. Then the orbit decomposition of Jl(1,1)-∑l is (33)Jl(1,1)-∑l=L0l⋃L1l⋃L2l⋃L3l⋃L4l,where Lkl is the orbit going through the Ak-singularity. Thus, the codimension of Lkl~ is k+1. We consider the l-jet extension j1lG of the function G. By Proposition 7, there exists a dense and open subset TLkl⊂Embt(I,R24) such that j1lG is transversal to Lkl~(k=0,1,2,3,4) and the orbit decomposition of ∑l~. This means that j1lG(I×R24)∩∑l~=∅ and G is a versal unfolding of g at any point (s0,v0). By Theorem 5, the discriminant set of G (i.e., the lightlike hypersurface of γ) is locally diffeomorphic to cuspidal edge, swallowtail, or butterfly if the point is singular.



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*Source: 502789-2015-03-31.xml*