# Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

Authors: Xue Song; Zhigang Wang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariantσ. Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant σ of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone ONCv0⁎. Finally, some relevant counterexamples are indicated.


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## Body

## 1. Introduction

Semi-Euclidean 4-space with index 2 has significant physical background, for example, anti–de Sitter spacetimeAdS3, as a subspace of semi-Euclidean 4-space with index 2, is a solution of Einstein’s field equations for an empty universe with a negative cosmological constant. Very recently, some new results concerning the non-Euclidean geometry were established by physicists and geometers [1–11]; for example, S. Izumiya et al. investigated the regular curve in the embedded unit spheres in the de Sitter space or the light cone via the theory of Legendrian duality [6]. L. Kong and D. Pei [8] investigated the singularities of lightlike surfaces and focal surfaces of spacelike curves in hyperbolic spacetime sphere, and some geometry properties of the spacelike curves are obtained. The second author and his collaborators presented abundant research results concerning the singularities of submanifolds in semi-Euclidean space [12–15]. In [13], they investigated the null developables of timelike curves that lie on nullcone in 3-dimensional semi-Euclidean space with index 2 and classified the singularities of the null developables of timelike curves. It is shown that the null developable ND is locally diffeomorphic to the cuspidal edge or swallowtail which are two classes of topologically stable singularities. The study also shows that the differential geometric invariants of timelike curves measured the order of the contact between a timelike curve and a conic CN. As the generalization and the further exploration on previous study [13], increasing the dimension of semi-Euclidean space from three to four, in the current study we devote ourselves to explore the singularities of a class of indefinite hypersurfaces and a class of indefinite surfaces which are generated by timelike curves in nullcone in 4-dimensional semi-Euclidean space with index 2. In comparison with [13], the current study primarily discusses indefinite hypersurfaces and indefinite surfaces instead of null surfaces and the indefinite hypersurfaces and the indefinite surfaces show more kinds of singularities under certain equivalent conditions, for example, BF, C(2,3,4)×R, C(BF). Moreover, the new definitionosculating nullcone is presented and the locus of center points of osculating nullcone is exactly the critical value set of the indefinite surface discussed in this paper. This will undoubtedly give the indefinite surface more geometric meanings.Methods of the unfolding theory in singularity theory which are quite useful for the study of the geometry of hypersurfaces and surfaces are adopted in this paper. To allow a useful study of the singularities of the indefinite hypersurfaces and the indefinite surfaces located in 3-dimensional nullcone, we consider distance squared function denoted byG:I×(R24∖NC0∗)→R;G(s,v)=〈v-γ(s),v-γ(s)〉 on a nullcone timelike curve γ. These functions are the unfolding of these singularities in the local neighborhood of (s0,v0), and these functions only depend on the germs that they are unfolding. In this paper, we create these functions by varying a fixed point v in the distance squared function to obtain a family of functions. We show that these singularities are R+-versally unfolded by the family of the distance squared functions. Noticing that the indefinite hypersurface IHγ​±(s,μ,ω) is exactly the discriminant set of the order one of the four-parameter unfolding, the indefinite surface ISγ​±(s,μ) is the discriminant set of the order two of the four-parameter unfolding, while the locus Cγ of the center points of the osculating nullcone for timelike curve γ is the discriminant set of the order three of the four-parameter unfolding. If the singularity of gv is Ak-type and if the corresponding four-parameter unfolding is R-versal, then, applying Bruce’s theory to it (see [16, 17]), we know that the discriminant set (i.e., the indefinite hypersurface) of order one of the four-parameter unfolding is locally diffeomorphic to C(2,3)×R2, SW×R, or BF under certain conditions, the discriminant set (i.e., the indefinite surface) of order two of the four-parameter unfolding is locally diffeomorphic to {0}×R2, C(2,3,4)×R, or C(BF) under certain conditions, and the discriminant set (i.e., the curve) of order three of the four-parameter unfolding is locally diffeomorphic to a line or (2,3,4,5)-cusp under certain conditions. Thus, we have completed the classification of the singularities of the indefinite hypersurface IHγ​±(s,μ,ω) and the indefinite surface ISγ​±(s,μ).The paper is organized as follows. We begin in Section2 by establishing the theoretical frame of the explicit differential geometry on curves in R24, defining a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves in nullcone in R24, calculating the singular sets of the indefinite hypersurfaces and the indefinite surfaces, and presenting the main results in this paper. In Section 3 we present a distance squared function of a timelike curve that lies on nullcone and establish the equivalent relations between Ak-singularity of the distance squared function and geometric invariant σ which can characterize the types of singularities of the indefinite hypersurfaces and the indefinite surfaces via differential calculations. In Section 4, we give the proof of the main theorem and complete the classifications of singularities of the indefinite hypersurfaces and the indefinite surfaces by using some general results from the singularity theory for families of function germs. In this Section 5, using transversality theorem as the vehicle, we consider generic properties of timelike curves which lie on NC0∗. To better illustrate our results, we give an example of the indefinite hypersurface and the indefinite surface and draw their graphs in 3-dimensional projection space.


## 2. Preliminaries

LetR24 denote the 4-dimensional semi-Euclidean space with index 2, that is to say, the manifold R4 with a flat metric 〈·,·〉 of signature (-,-,+,+), for any vectors x=(x1,x2,x3,x4) and y=(y1,y2,y3,y4) in R4,(1)x,y=-x1y1-x2y2+x3y3+x4y4.We also define thepseudovector product of x, y, and z as follows:(2)x∧y∧z=-e1-e2e3e4x1x2x3x4y1y2y3y4z1z2z3z4,where (e1,e2,e3,e4) is the canonical basis of R24. We state that a vector x∈R24∖{0} isspacelike, null, ortimelike if 〈x,x〉 is positive, zero, or negative, respectively. Thenorm of a vector x∈R24 is defined as x=x,x, and we call x aunit vector if x=1. For a vector v∈R24 and a real number c, we define the hyperplane with pseudonormal vector v by(3)HPv,c=x∈R24:x,v=c. We call HP(v,c) a Lorentz hyperplane, a semi-Euclidean hyperplane with index 2, or a null hyperplane if v is timelike, spacelike, or null, respectively.Letγ:I→R24 be a smooth curve in R24 (i.e., γ′(t)≠0 for any t∈I), where I is an open interval. For any t∈I, the curve γ is calledspacelike curve,null curve, ortimelike curve if all its velocities are 〈γ′(t),γ′(t)〉>0, 〈γ′(t),γ′(t)〉=0, or 〈γ′(t),γ′(t)〉<0, respectively. We call γ thenonnull curve if γ is a timelike curve or a spacelike curve.The arc-length of a nonnull curveγ, measured from γ(t0)(t0∈I), is s(t)=∫t0tγ′(t)dt. Then the parameter s is determined such that γ′(s)=1 for the nonnull curve, where γ′(s)=(dγ/ds)(s). Therefore, we say that a nonnull curve γ is parameterized by arc-length; throughout the remainder in this paper, we denote the parameter s of γ as the arc-length parameter.Let’s define a three-dimensional submanifold in semi-Euclidean 4-space(4)NCp=x∈R24∣x-p,x-p=0,where p=(p1,p2,p3,p4); we call NCp∗=NCp∖{p} a nullcone at the vertex p.There exist spacelike curves, timelike curves, and null curves inNC0∗; among them, the null curves had been investigated by the second author [15]; in this paper, we will restrict our attention to the case of timelike curve in NC0∗; the case of space curve would be discussed in future work. Let γ:I→NC0∗ be a timelike regular curve; we consider a basis E={γ(s),γ′(s)=ξ(s),γ′′(s),γ′′′(s)} for all s, such that 〈γ′(s),γ′(s)〉=-1. We choose the γ(s),ξ(s)=γ′(s), and then there exists a null transversal vector γ¯ such that γ¯=〈γ′′(s),γ′′(s)〉γ(s)-2γ′′(s). Moreover, defining e(s)=(γs∧ξs∧γ¯s)/2, we have a Frenet type frame F={γ(s),ξ(s),γ¯(s),e(s)} of R24 along γ(s) satisfying(5)γs,γs=γ¯s,γ¯s=0,γs,γ¯s=-2,ξs,ξs=-1,es,es=1,γs,ξs=γs,γ¯s=γs,es=γ¯s,ξs=0,γ¯s,es=es,ξs=0and(6)γs∧ξs∧es=γs,12γ¯s∧γs∧es=ξs,ξs∧γ¯s∧es=γ¯s,12γs∧ξs∧γ¯s=es.It is easy to see that F and E have the same positive orientation. In order to study the timelike curve that lies on nullcone, we have the following fundamental formulae to use:(7)γ′s=ξs,γ¯′s=-ksξs+τses,ξ′s=ks2γs-12γ¯s,e′s=τs2γs,and we call (6) the Frenet type equations, where(8)ks=γ′′s,γ′′s,τs=-2det⁡γs,γ′s,γ′′s,γ′′′sand k(s) and τ(s) are calledcone curvature function of the curve γ(s). The Frenet type frame F={γ(s),γ¯(s),ξ(s),e(s)} is also called theasymptotic orthonormal frame on R24 along the curve γ(s) in NC0∗.Before stating the main theorem we define several geometric objects involved in this paper. Letγ:I→NC0∗ be a unit speed timelike curve with τ2(s)k(s)-k′2(s)≠0; firstly, we define an indefinite hypersurface in R24 along γ(I) in NC0∗:(9)IHγ​±:I×R×R⟶R24such that(10)IHγ​±s,μ,ω=ω+1γs+μγ¯s±2ωμeswith ωμ>0. Next, we also define an indefinite surface ISγ:I×R→R24 given by(11)ISγ​±s,μ=μksγs+γ¯s±2ksμ-1μeswith (k(s)μ-1)μ>0. Last but not least, let’s define a curve Cγ:I×R→R24 given by(12)Cγs=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.By applying the formulae expressed in (7), we can calculate(13)∂IHγ​±∂ss,μ,ω=ω-μks+1ξs±ωμτsγs+μτses,(14)∂IHγ​±∂μs,μ,ω=γ¯s±ωωμes,∂IHγ​±∂ωs,μ,ω=γs±μωμes.At each of the regular points p=(s,μ,ω) of IHγ​±; that is, the formula(15)∂IHγ​±∂ss,μ,ω∧∂IHγ​±∂μs,μ,ω∧∂IHγ​±∂ωs,μ,ω≠0holds at p; the tangent hyperplane TIHp=Span{(∂IHγ​±/∂s)(s,μ,ω), (∂IHγ​±/∂μ)(s,μ,ω), (∂IHγ​±/∂ω)(s,μ,ω)} can be Lorentz, semi-Euclidean, or null. Similar calculation shows that the tangent plane of ISγ​± can also be Lorentz, semi-Euclidean, or null. This is the cause why we call IHγ​± an indefinite hypersurface and ISγ​± an indefinite surface.It is obvious that ifτ(s)=0, then the four vectors γ(s),γ′(s),γ′′(s),γ′′′(s) are linearly dependent; this situation is trivial. We shall assume throughout the whole paper that all the maps and manifolds are C∞ and τ(s)≠0 unless the contrary is explicitly stated. In this paper, one of our main tasks is to study the singularities of IHγ±, ISγ±; we state the following proposition.Proposition 1.
Letγ:I→NC0∗ be a regular timelike curve in the nullcone NC0∗ with τ2(s)k(s)-k′2(s)≠0. One can state the following:
(i)(s,μ,ω) is a singularity of IHγ± if and only if ω=k(s)μ-1.
(ii)(s,μ) is a singularity of ISγ± if and only if ±(k(s)μ-1)μ=-(k′s/τs)μ and u=τ2(s)/(τ2sks-k′2s).Proof.
( i ). It has been calculated in (13) and (14) that(16)∂IHγ​±∂ss,μ,ω=ω-μks+1ξs±ωμτsγs+μτses,∂IHγ​±∂μs,μ,ω=γ¯s±ωωμes,∂IHγ​±∂ωs,μ,ω=γs±μωμes.The three vectors above are linearly dependent if and only if (17)∂IHγ​±∂ss,μ,ω∧∂IHγ​±∂μs,μ,ω∧∂IHγ​±∂ωs,μ,ω=0,that is, if and only if ω=k(s)μ-1; thus, assertion (i) is true.
( i i ) . We calculate(18)∂ISγ​±∂ss,μ=μk′s±τsksμ-1μγs+μμk′s±τsksμ-1μksμ-1μes,∂ISγ​±∂μs,μ=ksγs+γ-s±2μks-1ksμ-1μes.The two vectors above are linearly dependent if and only if μk′(s)±τ(s)(k(s)μ-1)μ=0, that is, if and only if u=τ2(s)/(τ2sks-k′2s) and ±(k(s)μ-1)μ=-(k′s/τs)μ. Thus, assertion (ii) is proven.We remark the relationships betweenISγ​±, IHγ​±, and Cγ as follows.Remark 2.
By Proposition1, the images of ISγ​± are the critical value sets of IHγ​±, and the images of Cγ are the critical value sets of ISγ​±.Definition 3.
LetF:NC0∗→R be a submersion and γ→NC0∗ be a timelike curve with τ2(s)k(s)-k′2(s)≠0. One says that γ and F-1(0) have k-point contact for t=t0 if the function g(t)=F∘γ(t) satisfies g(t0)=g′(t0)=⋯=gk-1(t0)=0, gk(t0)≠0. One also says that γ and F-1(0) have at least k-point contact for t=t0 if the function g(t)=F∘γ(t) satisfies g(t0)=g′(t0)=⋯=gk-1(t0)=0.In order to characterize the singularities of the indefinite hypersurfaces, the indefinite surfaces, and the curves, we introduce a nullcone invariantσ(s) defined to be(19)σs=-τ3s+2τsk′′s-2k′sτ′s.The main result in the paper is as follows.Theorem 4.
Letγ:I→NC0∗ be a unit speed timelike curve that lies on nullcone with τ2(s)k(s)-k′2(s)≠0. For v0=IHγ​±(s0,μ0,ω0) and the nullcone NCv0∗=u∈NC0∗∣v0-u,v0-u=0,v0,v0≠0, one has the following:
(1)γ(s) and NCv0∗ have at least 2-point contact for s0.
(2)γ(s) and NCv0∗ have 3-point contact for s0 if and only if(20)v0=μksγs+γ¯s±2ksμ-1μes.
Under the condition, the hypersurfaceIHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to C(2,3)×R2, and ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to {0}×R2.
(3)γ(s) and NCv0∗ have 4-point contact for s0 if and only if σ(s)=-τ3(s)+2τsk′′(s)-2k′sτ′(s)≠0 and(21)v0=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Under the condition, the hypersurface IHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to SW×R, ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to C(2,3,4)×R, and Cγ at Cγ(s0) is locally diffeomorphic to a line.
(4)γ(s) and NCv0∗ have 5-point contact for s0 if and only if σ(s)=0, σ′(s)≠0, and(22)v0=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Under the condition, the hypersurface IHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to BF, ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to C(BF), and Cγ at Cγ(s0) is locally diffeomorphic to (2,3,4,5)-cusp.


## 3. Unfoldings of Functions of One-Variable andAk-Singularity

In this section, let us first recall the following definition (see [16] and references therein): Let F:(R×Rr,(s0,x0))→R be the germ of a function. One calls F anr-parameter unfolding of f, where f(s)=Fx0(s,x0). One says that f(s) has Ak-singularity at s0 if f(p)(s0)=0 for all 1≤p≤k and if f(k+1)(s0)≠0; One also says that f(s) has A≥k-singularity at s0 if f(p)(s0)=0 for all 1≤p≤k.Furthermore, we define a family of functions on a timelike curves which is useful to the study of geometric invariants of timelike curve. We also study the geometric properties of the indefinite hypersurfaces and the indefinite surfaces of a timelike curve in nullcone. We define the distance squared function of a timelike curve that lies on nullcone by(23)G:I×R24∖NC0∗⟶R,Gs,v=v-γs,v-γs,where v is independent of γ(s); we denote that gv(s)=G(s,v) for any fixed vector v0 in R24∖NC0∗. Using (7) and making a simple calculation, we can state the following facts.Proposition 5.
Suppose that one has a unit speed timelike curveγ:I→NC0∗ that lies on nullcone with τ2(s)k(s)-k′2(s)≠0; then
(i)gv(s)=0 if and only if there exist some ω,μ,ε,η∈R such that -4ωμ-ε2+η2=0 and(24)v-γs=ωγs+μγ¯s+εξs+ηes.
(ii)gv(s)=gv′(s)=0 if and only if there exist ε=0 and ωμ>0 such that(25)v-γs=ωγs+μγ¯s±2ωμes.
(iii)gv(s)=gv′(s)=gv′′(s)=0 if and only if ω=μk(s)-1 and ωμ>0 such that(26)v=μksγs+γ¯s±2ksμ-1μes.
(iv)gv(s)=gv′(s)=gv′′(s)=gv′′′(s)=0 if and only if(27)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτsτ2sks-k′2ses.
(v)gv(s)=gv′(s)=gv′′(s)=gv′′′(s)=gv(4)(s)=0 if and only if σ(s)=-τ3(s)+2τsk′′(s)-2k′sτ′(s)=0 and(28)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.
(vi)gv(s)=gv′(s)=gv′′(s)=gv′′′(s)=gv(4)(s)=gv(5)(s)=0 if and only if σ(s)=σ′(s)=0 and(29)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Proof.
( i ) Because v∈span{γ(s),γ¯(s),ξ(s),e(s)}, from gv(s)=〈v-γ(s),v-γ(s)〉=0, there exist ω,μ,ε,η∈R with -4ωμ-ε2+η2=0 such that v-γ(s)=ωγ(s)+μγ¯(s)+εξ(s)+ηe(s). Thus, assertion (i) holds.
( i i ) Under this condition gv(s)=0, if gv′(s)=〈-ξ(s),v-γ(s)〉=〈-ξ(s),ωγ(s)+μγ¯(s)+εξ(s)+ηe(s)=0, then we have ε=0 such that v-γ(s)=ωγ(s)+μγ¯(s)+ηe(s) and η=±2ωμ; thus, assertion (ii) holds.
( i i i ) When gv(s)=gv′(s)=0, we compute that(30)gv′′s=-ξ′s,v-γs+-ξs,-ξs=-ks2γs+12γ¯s,ωγs+μγ¯s+ηes-1=0implies that -ω+μk(s)-1=0; hence, ω=k(s)μ-1 and v=μk(s)γ(s)+μγ¯(s)+ηe(s), and, in combination with -4ωμ+η2=0, it is easy to deduce that gv(s)=gv′(s)=gv′′(s)=0 if and only if ω=μk(s)-1 and ωμ>0 such that v=μ(k(s)γ(s)+γ¯(s))±2(k(s)μ-1)μe(s); that is, assertion (iii) holds.
( i v ) From gv(s)=gv′(s)=gv′′(s)=0, we calculate that(31)gv′′′s=-ξ′′s,v-γs+-ξ′s,-ξs=-k′s2γs-ksξs+τs2es,μks-1γs+μγ¯s±2ksμ-1μes=k′sμ±τsksμ-1μ.If gv′′′(s)=0, then ±(k(s)μ-1)μ=-k′(s)μ/τ(s) and μ=τ2(s)/(τ2sks-k′2s); hence, v=(τ2s/τ2sks-k′2s)(k(s)γ(s)+γ¯(s)-(2k′s/τs)e(s)). Thus, assertion (iv) holds.
( v ) Suppose that gvs=gv′s=gv′′(s)=gv′′′(s)=0 holds; we notice that(32)gv4s=-ξ′′′s,v-γs+-ξ′′s,-ξs=τ2s-2k′′s-2k2s4γs-3k′s2ξs+ks2γ¯s+τ′s2es,1τ2sks-k′2sk′2sγs+τ2sγ¯s-2k′sτses-ks=τs-τ3s+2τsk′′s-2k′sτ′s2τ2sks-k′2s.Hence, gv(s)=gv′(s)=gv′′(s)=gv′′′(s)=gv4(s)=0 if and only if σ(s)=-τ3(s)+2τsk′′(s)-2k′sτ′(s)=0 and(33)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Thus, assertion (vi) holds.
( v i ) Assume that gv(s)=gv′(s)=gv′′(s)=gv′′′(s)=gv4(s)=0 holds; we calculate that(34)gv5s=-ξ4s,v-γs+-ξ′′′s,-ξs-k′s=-2k′′′s+7ksk′s-3τsτ′s4γs-8k′′s+4k2s-τ2s4ξs+5k′s4γ¯s+ksτs+τ′′s2es,1τ2sks-k′2sk′2sγs+τ2sγ¯s-2k′sτses-3k′s2-k′s=τs2τsk′′′s-3τ2sτ′s-2τ′′sk′s2τ2sks-k′2s.
Thus, assertion(vi) holds.Corollary 6.
Suppose that we have a unit speed timelike curveγ:I→NC0∗ that lies on nullcone with τ2(s)k(s)-k′2(s)≠0; then
(i)gv(s) has A1-singularity at s0 if and only if there exist ε=0, ωμ>0, and ω≠μk(s)-1 such that(35)v=ω+1γs+μγ¯s±2ωμes.
(ii)gv(s) has A2-singularity at s0 if and only if ω=μk(s)-1, ωμ>0, and μ≠τ2(s)/(τ2sks-k′2s) such that(36)v=μksγs+γ¯s±2ksμ-1μes.
(iii)gv(s) has A3-singularity at s0 if and only if σ(s)=-τ3(s)+2τsk′′(s)-2k′sτ′(s)≠0 and(37)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.
(iv)gv(s) has A4-singularity at s0 if and only if σ(s)=0, σ′(s)≠0, and(38)v=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.


## 4. Singularities of Indefinite Hypersurfaces, Indefinite Surfaces, and Curves

In this section, we use some general results from the singularity theory for families of function germs ([16, 17]). Let F be an unfolding of f, where f(s) has Ak(k≥1) at s0. We denote the (k-1)-jet of the partial derivative ∂F/∂xi at s0 by jk-1((∂F/∂xi)(s,x0))(s0)=∑j=1k-1αjisj, for i=1,…,r. Then, F is called a R+-versal unfoldingif the (k-1)×r matrix of coefficients (αji) has rank k-1(k-1≤r). Under the same conditions as above, F is called a R-versal unfoldingif the k×r matrix of coefficients (α0i,αji) has rank k(k≤r), where α0i=(∂F/∂xi)(s0,x0).We now introduce the following important sets concerning the unfolding:(39)DFl=x∈Rr,Fs,x=∂F∂s=⋯=∂lF∂sl=0,which is called adiscriminant set of orderl. Of course, DF1=DF, and DF2 is the set of singular points of DF. In order to understand the geometric properties of the discriminant set of order l, we introduce an equivalence relationship among the unfoldings of functions. Let F and G be the r-parameter unfoldings of f(s) and g(s), respectively. We state that F and G are P-R-equivalent if there exists a diffeomorphism germ Φ:(R×Rr,(s0,x0))→(R×Rr,(s0′,x0′)) of the form Φ(s,x)=(Φ1(s,x),ϕ(x)) such that G∘Φ=F. Straightforward calculations yield the following proposition.Proposition 7.
LetF and G be the r-parameter unfoldings of f(s) and g(s), respectively. If F and G are P-R-equivalent by a diffeomorphism germ Φ:(R×Rr,(s0,x0))→(R×Rr,(s0′,x0′)) of the form Φ(s,x)=(Φ1(s,x),ϕ(x)), then ϕ(DFl)=DGl are set germs.Theorem 8.
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, F is P-R-equivalent to one of the following unfoldings:
(a)k=1:±s2+x1.
(b)k=2:±s3+x1+sx2.
(c)k=3:±s4+x1+sx2+s2x3.
(d)k=4:±s5+x1+sx2+s2x3+s3x4.We review the following result [16].Theorem 9.
LetF:(R×Rr,(s0,x0))→R be an r-parameter unfolding of f(s), which has Ak singularity at s0. Suppose that F is a versal unfolding.
(i) Ifk=1, then DF is locally diffeomorphic to {0}×Rr-1 and DF2=⌀.
(ii) Ifk=2, then DF is locally diffeomorphic to C(2,3)×Rr-2, DF2 is locally diffeomorphic to {0}×Rr-2, DF3=⌀.
(iii) Ifk=3, then DF is locally diffeomorphic to SW×Rr-3, DF2 is locally diffeomorphic to C(2,3,4)×Rr-3, DF3 is locally diffeomorphic to {0}×Rr-3, and DF4=⌀.
(iv) 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, DF4 is locally diffeomorphic to {0}×Rr-4, and DF5=⌀.We remark that all of diffeomorphisms in the above assertions are diffeomorphism germs. We, respectively, callC(2,3)=x1,x2∣x1=u2,x2=u3 a (2,3)-cusp (see Figure 1), C(2,3,4)=x1,x2,x3∣x1=u2,x2=u3,x3=u4 a (2,3,4)-cusp (see Figure 2), C(2,3,4,5)=x1,x2,x3,x4∣x1=u2,x2=u3,x3=u4,x4=u5 a (2,3,4,5)-cusp (see Figure 3), C(2,3)×R=x1,x2∣x1=u2,x2=u3×R acuspidal edge (see Figure 4), SW=x1,x2,x3∣x1=3u4+u2v,x2=4u3+2uv,x3=v aswallowtail (see Figure 5), C(2,3)=x1,x2∣x1=u2,x2=u3 a (2,3)-cusp (see Figure 5), BF=x1,x2,x3,x4∣x1=5u4+3vu2+2wu,x2=4u5+2vu3+wu2,x3=u,x4=v abutterfly, and C(BF)=x1,x2,x3,x4∣x1=6u5+u3v,x2=25u4+9u2v,x3=10u3+3uv,x4=v a c-butterfly (i.e., the critical value set of the butterfly) (see Figure 6). We have the following key proposition for G.Figure 1
(2,3)-cusp.Figure 2
(2,3,4)-cusp.Figure 3
(2,3,4,5)-cusp.Figure 4
Cuspidal edge.Figure 5
Swallowtail.Figure 6
Projection of butterfly onx1x2x3-space.Proposition 10.
For a unit speed timelike curveγ:I→NC0∗ with τ2(s)k(s)-k′2(s)≠0 and G is the distance squared function of γ with gv(s)=G(s,v). If gv(s) has Ak-singularity at s0(k=1,2,3,4,), then G is a R-versal unfolding of gv(s).Proof.
Let(40)γs=x0s,x1s,x2s,x3sinNC0∗andv=v0,v1,v2,v3∈R24∖NC0∗.By the definition of distance squared function, we obtain that(41)Gs,v=-v0-x0s2-v1-x1s2+v2-x2s2+v3-x3s2,∂G∂v0s,v=-2vi-xisi=0,1,∂G∂vis,v=2vi-xisi=2,3,∂2G∂s∂vis,v=2xi′si=0,1,∂2G∂s∂vis,v=-2xi′si=2,3,∂3G∂s2∂v0s,v=2x0′′si=0,1,∂3G∂s2∂vis,v=-2xi′′si=2,3,∂4G∂s3∂v0s,v=2x0′′′si=0,1,∂4G∂s3∂vis,v=-2xi′′′si=2,3.For a fixed v0=(v0,v1,v2,v3), let j3(∂G/∂vi)(s,v0)(s0) be the 3-jet of (∂G/∂vi)(s,v) at s0 (i=0,1,2,3) such that(42)∂G∂vis0,v0+j3∂G∂vis,v0s0=∂G∂vis0,v0+∂∂s∂G∂vis0,v0s-s0+12∂∂s2∂G∂vis0,v0s-s02+16∂∂s3∂G∂vis0,v0s-s03=α0,i+α1,is-s0+12α2,is-s02+16α3,is-s03.
Wheng(s) has A1-singularity at s0, we have(43)A=α0,1α0,2α0,3α0,4=-2v0-x0s-2v1-x1s2v2-x2s2v3-x3s.It is easy to see that the rank of A is 1 since v-γ(s)≠0.
Wheng(s) has A2-singularity at s0, we get(44)B=α0,1α0,2α0,3α0,4α1,1α1,2α1,3α1,4=-2v0-x0s0-2v1-x1s02v1-x2s02v3-x3s02x0′s02x1′s0-2x2′s0-2x3′s0.
Becausev-γ(s)∈NC0∗ is a null vector and ξ(s) is a timelike vector for each s∈I, v-γ(s) is independent of ξ(s) implying that the rank of B is 2.
Assume thatg(s) has Ak-singularity at s0 (k=3,4); we want to verify the rank of the matrix(45)C=α0,1α0,2α0,3α0,4α1,1α1,2α1,3α1,4α2,1α2,2α2,3α2,4α3,1α3,2α3,3α3,4=-2v0-x0s0-2v1-x1s02v1-x2s02v3-x3s02x0′s02x1′s0-2x2′s0-2x3′s02x0′′s02x1′′s0-2x2′′s0-2x3′′s02x0′′′s02x1′′′s0-2x2′′′s0-2x3′′′s0is four. In fact, we can use an elementary transformation such that matrix B becomes matrix B1, where(46)C1=v0-x0s0v1-x1s0v2-x2s0v3-x3s0x0′s0x1′s0x2′s0x3′s0x0′′s0x1′′s0x2′′s0x3′′s0x0′′′s0x1′′′s0x2′′′s0x3′′′s0.It is obvious that the rank of C is equal to the rank of C1. In Corollary 6, it has been shown that when gv(s) has Ak-singularity at s(k=3,4), v-γ(s)=(1/(τ2sks-k′2s))(k′2(s)γ(s)+τ2(s)γ¯(s)-2k′(s)τ(s)e(s)); in combination with formulae (6), we calculate the determinant of B1(47)det⁡C1=v-γs∧γ′s∧γ′′s,γ′′′s=1τ2sks-k′2sk′2sγs+τ2sγ¯s-2k′sτses∧ξs∧ks2γs-12γ¯s,k′s2γs+ksξs-τs2es=-τs2.Noticing that τ(s)≠0, the rank of matrix C1 is 4. Thus, we can conclude that when gv(s) has Ak-singularity at s(k=1,2,3,4), G is a R-versal unfolding of gv. This completes the proof.Just as the definition of osculating sphere in Euclidean 3-space, we define the osculating nullcone as follows.Definition 11.
Letγ be a timelike curve in R24. Then the nullcone having at least 4-point contact with γ is called the osculating nullcone of γ. One denotes the osculating nullcone with vertex p as ONCp∗.
As a corollary of Proposition5, we give the following statements.Corollary 12.
Letγ:I→NC0∗ be a regular timelike curve with τ2(s)k(s)-k′2(s)≠0 in NC0∗. The center points (or vertex points) of the osculating nullcone at a point v0 are(48)v0=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.In other words, the critical value set Cγ(s) of ISγ​± is the locus of center points (or vertex point) of the osculating nullcone.Remark 13.
Based on Definition11 and Corollary 12, we can rewrite the asserts (3) and (4) of Theorem 4 as follows:
( 3 )   γ ( s ) and osculating nullcone ONCv0∗ have 4-point contact for s0 if and only if σ(s0)=-τ3(s0)+2τs0k′′(s0)-2k′s0τ′(s0)≠0 and(49)v0=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Under the condition, the indefinite hypersurface IHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to SW×R, the indefinite surface ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to C(2,3,4)×R, and the locus Cγ of the center points of the osculating nullcone at Cγ(s0) is locally diffeomorphic to a line.
( 4 )   γ ( s ) and osculating nullcone ONCv0∗ have 5-point contact for s0 if and only if σ(s0)=0, σ′(s0)≠0, and(50)v0=τ2sτ2sks-k′2sksγs+γ¯s-2k′sτses.Under the condition, the indefinite hypersurface IHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to BF, the indefinite surface ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to C(BF), and the locus Cγ of the center points of the osculating nullcone at Cγ(s0) is locally diffeomorphic to (2,3,4,5)-cusp.Proof of Theorem4.
In Corollary6, some equivalent conditions are presented when gv(s) has Ak-singularity (k=1,2,3,4). Moreover, it is noticed that the indefinite hypersurface IHγ​±(s,μ,ω) is exactly the discriminant set of the order one of the four-parameter unfolding, the indefinite surface ISγ​±(s,μ) is the discriminant set of the order two of the four-parameter unfolding, and the locus Cγ of the center points of the osculating nullcone for timelike curve γ is the discriminant set of the order three of the four-parameter unfolding. Proposition 10 shows that when gv(s) has Ak-singularity (k=1,2,3,4), G is a R-versal unfolding of gv. Applying Theorem 9, it follows that under equivalent conditions corresponding to the assertions that gv(s) has Ak-singularity (k=1,2,3,4), the indefinite hypersurface IHγ​±(s,μ,ω), the indefinite surface IHγ​±(s,μ,ω), and the locus Cγ of the center points of the osculating nullcone exhibit different characteristics of singularities; that is, the assertions in Theorem 4 are supported.


## 5. Generic Properties of Timelike Curves

In this section we consider generic properties of timelike curves which lie onNC0∗. The main tool is transversality theorem. Let Embsp(I,NC0∗) be the space of timelike embedding γ:I→NC0∗ equipped with Whitney C∞-topology. We also consider the function Gv:NC0∗×(R24∖NC0∗) defined by G(u,v)=〈v-u,v-u〉. We claim that Gv is a submersion for any v∈R24, where Gv=〈v-u,v-u〉. For any γ∈Embsp(I,NC0∗), we have G=G∘(γ×id(R24∖NC0∗)). We also have the l-jet extension(51)j1lG:I×R24∖NC0∗⟶JlI,Rdefined by jlG(s,v)=jlgv(s,v). 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. We have the following proposition as a corollary of Lemma 6 in Wassermann [18].Proposition 14.
Let Q be submanifolds ofJl(1,1). Then the set(52)TQ=γ∈EmbspI,NC0∗∣j1lGis transversal toQ~is a residual subset of Embsp(I,NC0∗). If Q is closed subset, then TQ is open.
Letf:(R,0)→(R,0) be a function germ which has an 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 an Ak-singularity, then the codimension of the orbit is k. There is another characterization of versal unfoldings as follows.Proposition 15.
LetF:(R×Rr,0)→(R,0) be 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 jlF is transversal to the orbit Ll(jlf(0)~) for l≥k+1. Here, j1lF:(R×Rr,0)→Jl(R,R) is the l-jet extension of F given by j1lF(s,x)=jlFx(s).We can prove the following generic classification theorem.Theorem 16.
There exists an open and dense subsetTLkl⊂Embsp(I,NC0∗) such that, for any γ∈TLkl, the indefinite hypersurface IHγ​± is locally diffeomorphic to C(2,3)×R2, SW×R, or BFCγ, the indefinite surface ISγ​± is locally diffeomorphic to C(2,3,4)×R or C(BF) at its singularities, and the locus Cγ of the center point of the osculating nullcone at Cγ(s0) is locally diffeomorphic to (2,3,4,5)-cusp if the point is singular.Proof.
Forl≥5, we consider the decomposition of the jet space Jl(1,1) into Ll(1) orbits. We now define a semialgebraic set by(53)Σl=z=jlf0∈Jl1,1∣fhas anA≥5-singularity.Then the codimension of Σl is 5. Therefore, the codimension of Σl~=I×0×Σl is 6. We have the orbit decomposition of Jl(1,1)-Σl into(54)Jl1,1-Σl=L0l∪L1l∪L2l∪L3l∪L4l,where Lkl is the orbit through an Ak-singularity. Thus, the codimension of Lkl~ is k+1. We consider the l-jet extension j1lG of the function G. By Proposition 14, there exists an open and dense subset TLkl⊂Embsp(I,NC0∗) such that j1lG is transversal to Lkl~ (k=0,1,2,3,4) and the orbit decomposition of Σ~l. This means that j1lG(I×(R24∖NC0∗))∩Σl~=⌀ and G is an R-versal unfolding of g at any point (s0,v0). By Theorem 9, the indefinite hypersurface IHγ​± (the discriminant set of order 1 of G) is locally diffeomorphic to C(2,3)×R2, SW×R, or BF at its singularities, the indefinite surface ISγ​± (the discriminant set of order 2 of G) is locally diffeomorphic to C(2,3,4)×R or C(BF) at its singularities, and the locus Cγ (the discriminant set of order 3 of G) of the center point of the osculating nullcone is locally diffeomorphic to (2,3,4,5)-cusp at its singularities.


## 6. Example

In this section, to better illustrate our results, we give an example of a regular timelike curve on nullcone, and then we want to show further the indefinite hypersurfacesIHγ​±, the indefinite surface ISγ​±, and the curve Cγ involved in the timelike curve. We consider a unit speed timelike curve γ:I→NC0∗ defined by(55)γs=sin⁡s2-1,cos⁡s2-1,sin⁡ln⁡s+u2-1,cos⁡ln⁡s+s2-1,where I is the open interval (1.3,6). We calculate(56)γ′s=scos⁡s2-1u2-1,-ssin⁡s2-1s2-1,cos⁡ln⁡s+s2-1s2-1,-sin⁡ln⁡s+s2-1s2-1.Let γ′′(s)=(γ1′′(s),γ2′′(s),γ3′′(s),γ4′′(s)) and γ′′′(s)=(γ1′′′(s),γ2′′′(s),γ3′′′(s),γ4′′′(s)), and we get (57)γ1′′s=-cos⁡s2-1s2-13/2-s2sin⁡s2-1s2-1,γ2′′s=-s2cos⁡s2-1s2-1+sin⁡s2-1s2-13/2,γ3′′s=-scos⁡ln⁡s+s2-1s2-13/2-sin⁡ln⁡s+s2-1s2-1,γ4′′s=-cos⁡ln⁡s+s2-1s2-1+ssin⁡ln⁡s+s2-1s2-13/2and(58)γ1′′′s=s3+s2-s4cos⁡s2-1+31+s2sin⁡s2-1s2-15/2,γ2′′′s=s31+s2cos⁡s2-1-3+s2-s4sin⁡s2-1s2-15/2,γ3′′′s=2+s2cos⁡ln⁡s+s2-1+3ss2-1sin⁡ln⁡s+s2-1s2-15/2,γ4′′′s=3ss2-1cos⁡ln⁡s+s2-1-2+s2sin⁡ln⁡s+s2-1s2-15/2.Using the Frenet equations (7), we obtain(59)ks=γ′′s,γ′′s=2-s4s2-12,τs=-4s+s3s2-12.Therefore, we have(60)γ¯s=-2γ′′s+ksγs=γ¯1s,γ¯2s,γ¯3s,γ¯4s,where (61)γ¯1s=2s2-1coss2-1+s2-1s4-2s2+2sins2-1s2-15/2,γ¯2s=s2-1s4-2s2+2cos⁡s2-1-2s2-1sin⁡s2-1s2-15/2,γ¯3s=2ss2-1cos⁡ln⁡s+s2-1+s2s2-1s2-2sin⁡lns+s2-1s2-15/2,γ¯4s=-s2s2-1s2-2cos⁡ln⁡s+s2-1-2ss2-1sin⁡ln⁡s+s2-1s2-15/2.Moreover, we obtain e(s)=(1/2)(γ(s)∧ξ(s)∧γ¯(s))=(e1(s),e2(s),e3(s),e4(s)), where (62)e1s=s2-1cos⁡s2-1+sin⁡s2-1s2-1,e2s=cos⁡s2-1-s2-1sin⁡s2-1s2-1,e3s=ss2-1cos⁡ln⁡s+s2-1+sin⁡ln⁡s+s2-1s2-1,e4s=cos⁡ln⁡s+s2-1-ss2-1sin⁡ln⁡s+s2-1s2-1.Thus, the surface ISγ​±(s,μ)=μ(k(s)γ(s)+γ¯(s))±2(k(s)μ-1)μe(s) is given by(63)ISγ​±s,μ=x1u,s,x2u,s,x3u,s,x4u,s,where(64)x1s,μ=μ2-s4sin⁡s2-1s2-12+2s2-1cos⁡s2-1+s2-1s4-2s2+2sin⁡s2-1s2-15/2±22-s4s2-12μ-1μs2-1cos⁡s2-1+sin⁡s2-1s2-1,x2s,μ=μ2-s4cos⁡s2-1s2-12+s2-1s4-2s2+2cos⁡s2-1-2s2-1sin⁡s2-1s2-15/2±22-s4s2-12μ-1μcos⁡s2-1-s2-1sin⁡s2-1s2-1,x3s,μ=μ2-s4sin⁡ln⁡s+s2-1s2-12+2ss2-1cos⁡ln⁡s+s2-1+s2s2-1s2-2sin⁡ln⁡s+s2-1s2-15/2±22-s4s2-12μ-1μss2-1cos⁡ln⁡s+s2-1+sin⁡ln⁡s+s2-1s2-1,x4s,μ=μ2-s4cos⁡ln⁡s+s2-1s2-12+-s2s2-1s2-2cos⁡ln⁡s+s2-1-2ss2-1sin⁡ln⁡s+s2-1s2-15/2±22-s4s2-12μ-1μcos⁡ln⁡s+s2-1-μs2-1sin⁡ln⁡s+s2-1s2-1.Thus, the hypersurface IHγ​±(s,μ,ω)=ω+1γs+μγ¯(s)±2ωμe(s) is given by(65)IHγ​±s,μ,ω=y1s,μ,ω,y2s,μ,ω,y3s,μ,ω,y4s,μ,ω,where(66)y1s,μ,ω=ω+1sin⁡s2-1+μ2s2-1cos⁡s2-1+s2-1s4-2s2+2sins2-1s2-15/2±2ωμs2-1cos⁡s2-1+sin⁡s2-1s2-1,y2s,μ,ω=ω+1cos⁡s2-1+μs2-1s4-2s2+2cos⁡s2-1-2s2-1sin⁡s2-1s2-15/2±2ωμcos⁡s2-1-s2-1sin⁡s2-1s2-1,y3s,μ,ω=ω+1sin⁡ln⁡s+s2-1±2ωμss2-1cos⁡ln⁡s+s2-1+sin⁡ln⁡s+s2-1s2-1+μ2ss2-1cos⁡ln⁡s+s2-1+s2s2-1s2-2sin⁡ln⁡s+s2-1s2-15/2,y4s,μ,ω=ω+1cos⁡ln⁡s+s2-1±2ωμcos⁡ln⁡s+s2-1-μs2-1sin⁡lns+s2-1s2-1+μ-s2s2-1s2-2cos⁡ln⁡s+s2-1-2ss2-1sin⁡ln⁡s+s2-1s2-15/2.The curve Cγ(s)=τ2s/τ2sks-k′2s(k(s)γ(s)+γ¯(s)-(2k′s/τs)e(s)) is given by(67)Cγs=c1s,c2s,c3s,c4s,where(68)c1s=-s4-12s8+2s6-8s2+22-s4sin⁡s2-1s2-12+2s2-1cos⁡s2-1s2-1s4-2s2+2sin⁡s2-1s2-15/2+2s2-2s4-1s2-1cos⁡s2-1+sin⁡s2-1s2-1,c2s=-s4-12s8+2s6-8s2+22-s4cos⁡s2-1s2-12+s2-1s4-2s2+2sin⁡s2-1-2s2-1cos⁡s2-1s2-15/2+2s2-2s4-1sin⁡s2-1-s2-1cos⁡s2-1s2-1,c3s=-s4-12s8+2s6-8s2+22-s4sin⁡ln⁡s+s2-1s2-12+2s2-2s4-1ss2-1cos⁡ln⁡s+s2-1+sin⁡ln⁡s+s2-1s2-1+2ss2-1cos⁡ln⁡s+s2-1-s2s2-1s2-2sin⁡ln⁡s+s2-1s2-15/2,c4s=-s4-12s8+2s6-8s2+22-s4cos⁡ln⁡s+s2-1s2-12+2s2-2s4-1cos⁡ln⁡s+s2-1-ss2-1sin⁡ln⁡s+s2-1s2-1+-s2s2-1s2-2sin⁡ln⁡s+s2-1-2ss2-1cos⁡ln⁡s+s2-1s2-15/2.We calculate the geometric invariant(69)σs=64s32-s2+4s4+s6s2-16.It can be verified easily that there are no real roots when σ(s)=0 for I=(1.3,6); that is, σ(s)≠0 for any s∈I⊂R. Hence, according to Theorem 4, we have the following: the indefinite hypersurface IHγ​± at IHγ​±(s0,μ0,ω0) is locally diffeomorphic to SW×R for any s0∈I⊂R,μ0∈R,ω0∈R, the indefinite surface ISγ​± at ISγ​±(s0,μ0) is locally diffeomorphic to C(2,3,4)×R for any s0∈I⊂R,μ0∈R, and Cγ at Cγ(s0) is locally diffeomorphic to a line for any s0∈I⊂R.We draw the projections ofIHγ​±(s0,μ0,ω0), ISγ​±(s0,μ0), and Cγ(s0) to 3-space in Figures 7–14. It is shown clearly that Cγ(s0) is the intersection of ISγ​+(s0,μ0) and ISγ​-(s0,μ0) as well as the critical value set of ISγ​±. There is a good agreement between the results of Theorem 4 and these intuitive figures in terms of this example.Figure 7
Projection ofISγ​±(s,μ) (in green and in gray) and Cγ(s) (in red) on x1x2x3-space.Figure 8
Projection ofISγ​±(s,μ) (in green and in gray) and Cγ(s) (in red) on x1x2x4-space.Figure 9
Projection ofISγ​±(s,μ) (in green and in gray) and Cγ(s) (in red) on x1x3x4-space.Figure 10
Projection ofISγ​±(s,μ) (in green and in gray) and Cγ(s) (in red) on x2x3x4-space.Figure 11
Projection of the intersection ofIHγ​±(s,μ,ω) and ω=0 on x1x2x3-space.Figure 12
Projection of the intersection ofIHγ​±(s,μ,ω) and ω=0 on x1x2x4-space.Figure 13
Projection of the intersection ofIHγ​±(s,μ,ω) and ω=0 on x1x3x4-space.Figure 14
Projection of the intersection ofIHγ​±(s,μ,ω) and ω=0 on x2x3x4-space.



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