# Lightlike Hypersurfaces and Canal Hypersurfaces of Lorentzian Surfaces

Authors: Jianguo Sun; Donghe Pei
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The lightlike hypersurfaces in semi-Euclidean space are of special interest in Relativity Theory. In particular, the singularities of these lightlike hypersurfaces provide good models for the study of different horizon types. And we obtain some geometrical propositions of the canal hypersurfaces of Lorentzian surfaces. We introduce the notions of flatness for these hypersurfaces and study their singularities.


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## Body

## 1. Introduction

The extrinsic differential geometry of submanifolds in 4-dimensional semi-Euclidean space is of special interest in Relativity Theory. In particular the lightlike hypersurfaces, which can be constructed as lightlike ruled hypersurfaces over Lorentzian surfaces in anti-de Sitter space, provide good models for the study of different horizon types of black holes, such as Kerr black hole, Cauchy black hole, and Schwarzschild black hole [1–8]. Hiscock described that the horizon was constituted by lightlike hypersurfaces and lightlike wave front was lightlike hypersurface [6]; Dąbrowski et al. have studied the null (lightlike) strings form the photon sphere, moving in the single spacetime of general relativity, including lightlike hypersurfaces [1, 3, 4]. The authors gave the null string evolution in Schwarzschild spacetime by the solutions of null string equations, which are also the null geodesic equations of general relativity appended by an additional stringy constant [3, 4]. In the view of geometry, the null string (null curve) in lightlike surfaces is null geodesic [9]. In this sense, the singularities of lightlike hypersurfaces are deeply related to the shapes of horizons.M. Kossowski introduced a Gauss map on its associated spacelike surface, obtaining in this way interesting conclusions on the lightlike hypersurfaces which parallel the known results for surfaces in Euclidean 3-space concerning their contacts with the model surfaces [10]. When working in semi-Euclidean space, we observe that the properties associated with the contacts of a given submanifold with null cone and lightlike hyperplanes have a special relevance from the geometrical viewpoint. In [11–13], the current authors and so forth pursued with this line by describing the invariant geometric properties of Lorentzian surfaces of codimension two in semi-Euclidean space that arise from their contacts with null cone. For this purpose, the task of this paper is to study some local properties of these Lorentzian surfaces in semi-Euclidean (n+1)-space.Canal hypersurfaces, which are generated by surfaces with codimension 2 along fixed direction, are envelopes of families of hyperspheres. In three-dimensional space, canal surfaces were considered in many classical texts on differential geometry [14]. Since the property of a hypersurface to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15, 16]. However, in all these works the authors did not note the singularities of canal hypersurface in semi-Euclidean space. In this paper, we analyze the geometric meaning of the canal hypersurfaces from the view point of singularity. And we obtain the conclusion that the canal hypersurfaces have the similar singularities as Lorentzian surfaces.The remainder of this paper is organized as follows. In Section2, we give some basic notions about Lorentzian surfaces and lightlike hypersurfaces. Meanwhile, the Lorentzian Gauss-Kronecker curvatures of Lorentzian surfaces are also introduced. In Section 3, we describe Lorentzian distance-squared functions, whose discriminant sets and wave front sets are just right of the given lightlike hypersurfaces. In Section 4, we discuss the contact between lightlike hypersurfaces and null cone by Montald’s theorem. We give an example about the classification of singularities to lightlike hypersurfaces generated by Lorentzian surfaces in anti-de Sitter space in Section 5. In the last section, we consider some geometric properties of canal hypersurfaces, which are generated by Lorentzian surfaces in anti-de Sitter 3-space and the conclusion that the types of singularity of canal hypersurfaces are the same as the Lorentzian surfaces.We will assume throughout the whole paper that all manifolds and maps areC∞ unless the contrary is explicitly stated.


## 2. Preliminaries

Einstein formulated general relativity as a theory of space, time, and gravitation in semi-Euclidean space in 1915. However, this subject has remained dormant for much of its history because its understanding requires advanced mathematics knowledge. Since the end of the twentieth century, semi-Euclidean geometry has been an active area of mathematical research, and it has been applied to a variety of subjects related to differential geometry and general relativity. In this section, we illustrated some basic knowledge of semi-Euclidean space.LetRn+1={(x0,x1,…,xn)∣xi∈R,i=0,1,…,n} be an (n+1)-dimensional vector space. For any vectors x=(x0,x1,…,xn) and y=(y0,y1,…,yn) in Rn+1, the pseudoscalar product of x and y is defined by
(1)〈x,y〉=-x0y0-x1y1+∑i=2n‍xiyi.
The space (Rn+1,〈,〉) is called semi-Euclidean (n+1)-dimensional space with index two, denoted by R2n+1. A vector x∈R2n+1∖{0} is calledspacelike, lightlike, ortimelike, if 〈x,x〉 is positive, zero, or negative, respectively. There exist some special submanifolds in R2n+1, such as unit pseudo-n-sphere S2n, anti de Sitter n-space H1n, null cone Λ1n, and Lorentz torus St1×Ssn-1, which have the same definitions as in [17].Definition 1.
LetX:U→H1n be an embedding, where U⊂Rn-1 is an open subset; if there exists i such that X(u),Xui(u) is timelike vector and Xuj(u)  (j≠i) is spacelike vector, we call M=X(U) Lorentzian surface in anti-de Sitter space.Without loss of generality, we only consideri=1; the other cases are the same. We construct a unit spacelike normal vector
(2)N(u)=X(u)∧Xu1(u)∧⋯∧Xun-1(u)∥X(u)∧Xu1(u)∧⋯∧Xun-1(u)∥,
and the vectors X(u)±N(u) are lightlike. Since {Xu1(u),…,Xun-1(u)} is a basis of TpM, so the system {X(u),Xu1(u),…,Xun-1(u),N(u)} provides a basis for R2n+1.We define a mapL±:U→St1×Ssn-1 by
(3)L±(u)=X(u)±N(u)~,
which is calledthe lightcone Gauss indicatrix of X(u). We have shown that πt:TpM⊕NpM→TpM and πn:TpM⊕NpM→NpM [11]. Under the identification of U and M, the derivative dX(u0) can be identified to the identity mapping idTpM on the tangent space TpM, where p=X(u0). This means that dL±(u0)=idTpM±dN(u0). Thus, dL±(u0) can be regarded as a linear transformation on the tangent space TpM. We call the linear transformation Sp±(u)=-dp(L±(u)):TpM→TpM theLorentzian shape operator of M at p=X(u0). We denote the eigenvalue of Sp± by k±, which is called a Lorentzian principal curvature of M at point p. TheLorentzian Gauss-Kronecker curvature of M is defined as Kl±(u)(p)=det⁡Sp±(u).Definition 2.
A pointp=X(u) is an umbilic point if all the principal curvatures coincide at p. M is called totally umbilic surface if all points on M are umbilics.SupposingM=X(U) is totally umbilic, we have the following propositions by simple computation.Proposition 3.
IfX(U) is totally umbilic, we have the following classification.(1)
Suppose thatk±≠0.
(a)
If0≤|k±+1|<1, thenMis a part of an anti-de Sitter space. In particular, ifk±=-1,
thenMis a part of a small anti-de Sitter space.(b)
If|k±+1|>1, thenMis a part of unit pseudo-n-sphere.(2)
Supposek±=0. ThenMis a part of hyperhorosphere.Proposition 4.
LetX(U) be a Lorentzian surface in anti-de Sitter space, the lightcone Gauss indicatrix is constant if and only if there exists a unique lightlike hyperplane HP(n,-1) in R2n+1, such that the M=X(U) is a part of H1n∩HP(n,-1), where n=L±(u).SinceXu1 is timelike vector, Xui  (i=2,…,n-1) is spacelike vector, and semi-Riemannian metric on M=X(U) defined by ds2=Σi=1n-1δigijduiduj [18, 19], where gij=〈Xui(u),Xuj(u)〉, δ1=-1, and δi=1, for any i=2,…,n-1, we have a Lorentzian second fundamental invariant with respect to the vectors X(u),  N(u) defined by hij(X,N)(u)=〈-Lui±(u),Xuj(u)〉, for any u∈U.Proposition 5.
The Lorentzian Weingarten formulas with respect toX(u),  N(u) are as follows.(1)
Lui±=-∑j=1n‍δihij±Xuj,(2)
πt∘Lui±=-∑j=1n‍δihij±Xuj, wherehij±=hik±gkj,gkj=gkj-1.Proof.
There exist real numbersλ,  μ,  Γij± such that Lui±=λX(u)+μN(u)+∑j=1n-1‍Γij±Xuj. Since 〈πn∘Lui±,Xuj〉=0, we have
(4)-hik=〈Lui±,Xuk〉=∑j=1n-1‍Γij±〈Xuj,Xuk〉=∑j=1n-1‍Γijδjgjk.
Hence, we have hij±=∑k=1n-1‍hik±gkj=-∑k=1n-1‍∑m=1n-1‍Γim±δigmkgkj=-δiΓij±; the formula (2) follows the conclusion of item (1).As a corollary of the Proposition5, we have an explicit expression of the Lorentzian Gauss-Kronecker curvature by Riemannian metric and the second fundamental invariant.Corollary 6.
Under the same notations as in the above proposition, the Lorentzian Gauss-Kronecker curvature is given byKl±(u)=det⁡(hij±)/det⁡(gkm).Proof.
By the above proposition, the representation matrix of the Lorentzian shape operator with respect to the basis{Xu1,Xu2,…,Xun-1} is hij±(N)=(hiβ±)(gβj). It follows from this fact that
(5)Kl±(u)=det⁡Sp±(X(u),N(u))=det⁡(-dL±)=-det⁡Γij±=det⁡hij±=det⁡hij±gjk=det⁡(hij±)det⁡(gkm).
So we complete the proof.Since〈-L±(u),Xuj(u)〉=0, we have hij±=〈L±(u),Xuiuj(u)〉. Therefore, the Lorentzian second fundamental invariant depends on the values L±(u0),  Xuiuj(u0). By the above corollary, the Lorentzian Gauss-Kronecker curvature depends only on L±(u0),  Xui(u0),  Xuiuj(u0). It is independent on the choice of the normal vector field N(u).Definition 7.
LetM=X(u) be a Lorentzian surface in anti-de Sitter space and let N(u) be its spacelike normal vector; a hypersurface LHM± defined by LHM±(u,μ)=X(u)+μ(X(u)±N(u)~) is called LHM± the lightlike hypersurface along M.


## 3. Lorentzian Distance-Squared Function

To describe the existence of singularities of lightlike hypersurfaces, we should construct contact functions, whose wave front set is the singularity set of lightlike hypersurfaces. In this section, we introduce some notions of Lorentzian distance-squared functions on Lorentzian surfaces in anti-de Sitter space, which can supply the contact relationship between Lorentzian surfaces and standard spherical surfaces. Meanwhile, we obtain the Lorentzian distance-squared functions as Morse family.A functionG:M×R2n+1→R on the Lorentzian surface is given by
(6)G(u,λ)=〈X(u)-λ,X(u)-λ〉,
which is calledLorentzian distance-squared function on M. For any fixed λ0∈R2n+1, we write gλ0(u)=G(u,λ0) and have the following propositions by simple computing.Proposition 8.
LetM be a Lorentzian surface in anti-de Sitter space and let G:M×R2n+1→R be Lorentzian distance-squared function on M. Suppose that λ0≠p0=X(u0). Then we have(1)
gλ0(p0)=∂gλ0/∂u0=⋯=∂gλ0/∂un-1=0if and only ifp0-λ0=μ(X(u0)±N(u0)~)forμ∈R∖{0}.(2)
gλ0(p0)=∂gλ0/∂u0=⋯=∂gλ0/∂un-1=det⁡Hess(gλ0)(u0)=0if and only ifp0-λ0=μ(X(u0)±N(u0)~)forμ∈R∖{0}andKl±(u0)=0.Proposition 9.
Letλ0∈R2n+1 and let M be a Lorentzian surface without any umbilic point satisfying Kl±(u)≠0. Then M⊂Λ1λ0n if and only if λ0 is an isolated singular value of the lightlike hypersurface LHM and LHM⊂Λ1λ0n.Proof.
By definition,M⊂Λ1λ0n if and only if gλ0(u)=0, for any u∈U, where gλ0(u)=G(u,λ0) is the Lorentzian distance-squared function on M. It follows from Proposition 8 that there exists a smooth function ρ:U→R such that X(u)=λ0+ρ(u)(X(u)±N(u)~). Therefore,
(7)LHM(μ,ρ)=λ0+(μ+ρ(u))(X(u)±N(u)~).
Hence, we have LHM(μ,ρ)⊂Λ1λ0n. Moreover, we get that
(8)∂LHM∂μ=X(u)±N(u)~,∂LHM∂ui=ρui(X(u)±N(u)~)+(ρ+μ)(X(u)+N(u)~)ui
for any i=1,2…,n-1, and from above formulas, we can obtain
(9)∂LHM∂ρ∧∂LHM∂u1∧⋯∧∂LHM∂un-1=(ρ+μ)n-1(X(u)±N(u)~)∧(X(u)±N(u)~)u1∧⋯∧(X(u)±N(u)~)un-1.
Therefore, we have X(u)-λ0=ρ(u)(X(u)±N(u)~), since X(u)-λ0 is lightlike, Xu1 is a timelike vector, and Xui  (i=2,…,n-1) is spacelike vector. X(u)-λ0,  Xu1,…,Xun-1 are linearly independent. Therefore,
(10)(X(u)-λ0)∧Xu1∧⋯∧Xun-1=μn(u)(X(u)±N(u)~)∧(X(u)±N(u)~)u1∧⋯∧(X(u)±N(u)~)un-1≠0,
so ∂LHM/∂ρ∧∂LHM/∂u1∧⋯∧∂LHM/∂un-1=0 if and only if μ+ρ(u)=0 under the assumption that Kl(u)≠0. This means that λ0 is an isolated singularity of LHM±.Since we only consider local properties, we may assume thatM=Rn. As the definitions in [11], it follows that
(11)Σ*(G)={(u,λ)∈(U×Rn+1,0)∣∂G∂uk0000000000G(u,λ)=∂G∂u1(u,λ)=⋯=∂G∂uk(u,λ)=0}.
The set Σ*(G) is defined as the wave front set of G. Also, we can write Σ*(G) as
(12)DG={X(u)+μL±(u),μ∈R}.
Thus, a singular point of the lightlike hypersurface satisfied λ0=X(u0)+μ0(X(u0)±N(u0)~).Definition 10.
LetG be a Morse family, a map germ LG:(Σ*(G),0)→PT*R2n+1, which satisfied LG(q,x)=(x,[(∂G/∂q1)(q,x):⋯:(∂G/∂qk)(q,x)]), is calledLegendrian immersion germ.Letπ:PT*R2n+1→R2n+1 be the projective cotangent bundle over an n-dimensional manifold in R2n+1. This fibration can be considered as a Legendrian fibration with the canonical contact structure K on PT*R2n+1. Let us consider the tangent bundle τ:TPT*R2n+1→PT*R2n+1 and dπ:TPT*R2n+1→TR2n+1. The property α(V)=0 does not depend on the choice of representative of the class [α]. Thus we can define the canonical contact structure on PT*R2n+1 by K={X∈TPT*R2n+1∣τ(X)(dπ(X))=0}. For a local coordinate neighbourhood in R2n+1, we have a trivialization PT*U≅U×P(Rn)* and we call ((x1,…,xn),[ξ1:⋯:ξn])homogeneous coordinates, where [ξ1:⋯:ξn] are homogeneous coordinates of the dual projective space P(Rn)*. It is known that any Legendrian fibration is locally equivalent to π:PT*R2n+1→R2n+1 [20].Proposition 11.
All Legendrian submanifold germs inPT*R2n+1 are constructed by the above method.Proposition 12.
LetG be the Lorentzian distance-squared function on M. For any point (u,λ)∈G-1(0), G is a Morse family around (u,λ).Proof.
DenoteX(u)=(X0(u),X1(u),…,Xn(u)) and λ=(λ0,λ1,…,λn). By definition, we have
(13)G(u,λ)=-(X0(u)-λ0)2-(X1(u)-λ1)2+∑i=2n‍(Xi(u)-λi)2.
We now prove that the mapping Δ*G={G,∂G/∂u0,…,∂G/∂un-1} is nonsingular at (u,λ)∈G-1(0). Indeed, the Jacobian matrix of Δ*G is given by (14)(2(X0(u)-λ0)2(X1(u)-λ1)⋯-2(Xn(u)-λn)2X0u0(u)2X1u0(u)⋯-2Xnu0(u)A⋮⋮⋮⋮2X0un-1(u)2X1un-1(u)⋯-2Xnun-1(u)),where the matrix A is given by(15)(2〈X-λ,Xu0〉⋯2〈X-λ,Xun〉2(〈Xu0,Xu0〉+〈X-λ,Xu0u0〉)⋯2(〈Xu0,Xun〉+〈X-λ,Xunu0〉)⋮⋮⋮2(〈Xun,Xu0〉+〈X-λ,Xu0un〉)⋯2(〈Xun,Xun〉+〈X-λ,Xunun〉)),and Xuiuj(u)=∂2X(u)/∂ui∂uj(u). Since X(u) is an immersion, the rank of the matrix(16)(2X0u0(u)2X1u0(u)-2X2u0(u)⋯-2Xnu0(u)2X0u1(u)2X1u1(u)-2X2u1(u)⋯-2Xnu1(u)⋮⋮⋮⋮2X0un-1(u)2X1un-1(u)-2X2un-1(u)⋯-2Xnun-1(u))is equal to n-1 and X(u)-λ is lightlike, so that it is linearly independent of tangent vector Xu0,…,Xun-1. This means that the rank of B is equal to n, where(17)B=(2(X0(u)-λ0)2(X1(u)-λ1)⋯-2(Xn(u)-λn)2X0u0(u)2X1u0(u)⋯-2Xnu0(u)⋮⋮⋮⋮2X0un-1(u)2X1un-1(u)⋯-2Xnun-1(u)).Therefore, the Jacobi matrix of Δ*G is nonsingularity at (u,λ)∈G-1(0).


## 4. Contact with Null Cone

In this section, we gave the singularities of lightlike hypersurfaces are stable, whose types are not changed with small disturbance under the view ofK-equivalent and P-K-equivalent. Before we start to consider the contact between lightlike hypersurfaces and null cone, we briefly review the theory of contact due to Montaldi [21, 22]. Let Xi and Yi  (i=1,2) be submanifolds in Rn with dim⁡X1=dim⁡X2 and dim⁡Y1=dim⁡Y2. We say that thecontact of X1 and Y1 at y1 is of the same type as the contact of X2 and Y2 at y2 if there is a diffeomorphism germ ϕ:(Rn,y1)→(Rn,y2) such that ϕ(X1)=X2 and ϕ(Y1)=Y2. In this case, we write K(X1,Y1;y1)=K(X2,Y2;y2). In his paper [21], Montaldi gives a characterization of the notion of contact by using the terminology of singularity theory.Theorem 13 (see [21]).
LetXi and Yi  (i=1,2) be submanifolds of Rn with dim⁡X1=dim⁡X2 and dim⁡Y1=dim⁡Y2. Let gi:(Xi,xi)→(Rn,yi) be immersion germs and let fi:(Rn,yi)→(Rp,0) be submersion germs with (Yi,yi)=(fi-1(0),yi). Then K(X1,Y1;y1)=K(X2,Y2;y2) if and only if f1∘g1 and f2∘g2 are K-equivalent.For theK-equivalent among smooth map germs, considering the function G:R2n+1×R2n+1→R by G(X,λ)=〈X-λ,X-λ〉 and denoting gλ0(X)=G(X,λ0), we have gλ0-1(0)=Λ1n. For p0=X(u0), we can take the vector λ0=X(u0)+μ0(X(u0)±N(u0)~). Then gλ0∘X(u0)=G∘(X×idR2n+1)=G(u0,λ0)=0 and the relations are ((∂gλ0∘X)/∂ui)(p0)=(∂G/∂ui)(p0,λ0)=0  (i=1,2,…,n-1). This means that the lightcone gλ0-1(0)=Λ1λ0n is tangent to M at p0. In this case, we call each Λ1λ0n a tangent null cone of M at p0. We denote by En the local ring of function germs (Rn,0)→(R,0) with the unique maximal ideal
(18)Mn={h∈En∣h(0)=0}.LetF,  G:(Rk×Rn,0)→(R,0) be function germs. We say that F,  G are P-K-equivalent if there exists a diffeomorphism germ ψ:(Rk×Rn,0)→(Rk×Rn,0) of the form ψ(q,x)=(ψ1(q,x),ψ2(x)) for (q,x)∈(Rk×Rn,0) such that ψ*(〈F〉Ek+n)=〈G〉Ek+n, where ψ*:Ek+n→Ek+n is the pullback R-algebra isomorphism defined by ψ*(h)=h∘ψ.We apply the tools for the study of the contact theory. LetLi±:U→Λ1n be two null cone Legendrian Gauss map germs of Lorentzian surface germs Xi:U→H1n  (i=1,2). We say that L1± and L2± are A-equivalent if there exist diffeomorphism germs φ:(U,(u11,…,u1n))→(U,(u21,…,u2n)) and ϕ:(R2n+1,λ1)→(R2n+1,λ2) such that ϕ∘L1±=L2±∘φ.LetF:(Rk×Rn,0)→(R,0) be a function germ, F is K-versal deformation of f=F∣Rk×{0} if εk=Te(K)(f)+〈(∂F/∂x1)∣Rk×{0},…,(∂F/∂xn)∣Rk×{0}〉, where Te(K)(f)=〈∂f/∂q1,…,∂f/∂qk〉εk. The main result in the theory [22] is as follows.Theorem 14 (see [22]).
LetF,  G:(Rk×Rn,0)→(R,0) be Morse families. Then(1)
LF and LG are Legendrian equivalent if and only if F and G are P-K-equivalent.(2)
LF is Legendrian stable if and only if F is a K-versal deformation of F∣Rk×{0}.SinceF and G are function germs on the common space, by the uniqueness result of the versal deformation of a function germ, we have the following classification results of Legendrian stable germs. For a map germ f:(Rn,0)→(Rp,0), we give the local ring of f by Q(f)=εn/f(Mp).Proposition 15.
LetF and G:(Rk×Rn,0)→(R,0) be Morse families. Suppose that LF and LG are Legendrian stable. Then the following conditions are equivalent.(1)
(W(LF),0)and(W(LG),0)are diffeomorphic as germs.(2)
LFandLGare Legendrian equivalent.(3)
Q(f)andQ(g)are isomorphic asR-algebras, wheref=F∣Rk×{0}andg=G∣Rk×{0}.LetGi:U×R2n+1→R be the Lorentzian distance-squared function germs of Xi  (i=1,2). We denote giλi(u)=Gi(u,λi), then giλi=gλi∘Xi. By Theorem 13, we know K(X1,Λ1λ1n;λ1)=K(X2,Λ1λ2n;λ2) if and only if g1λ1 and g2λ2 are K-equivalent. Therefore, we can denote the local ring of the function g~λ0:U→R, we remark that we can explicitly write the local ring as follows:
(19)Q(X,(u1,…,un))=Cu0∞(U)(〈X(u),X(u0)+N(u0)~〉)Cu0∞(U)-1,
where Cu0∞(U) is the local ring of function germs with the maximal ideal M(u) in [12].Theorem 16 (see [12]).
LetXi:Ui→H1n  (i=1,2) be Lorentzian surface germs such that the corresponding Legendrian lift germs are Legendrian stable. Then the following conditions are equivalent.(1)
The lightlike hypersurface germsLHM1± and LHM2± are A-equivalent.(2)
G1 and G2 are P-K-equivalent.(3)
g1λ1 and g2λ2 are K-equivalent.(4)
K(X1,Λ1λ1n;λ1)=K(X2,Λ1λ2n;λ2).(5)
Q(X1) and Q(X2) are isomorphic as R-algebras.Proof.
Since the Lorentzian distance-squared function is a Morse family of functions, conditions (1) and (2) are equivalent. Moreover,LHMi± is Lagrangian stable, Gi is the R-versal deformation of giλi; by the uniqueness result of the R-versal deformation, condition (2) implies condition (3). By definition, we know condition (3) implies condition (2). It follows from Theorem 13 that conditions (3) and (4) are equivalent. As the same way, we can obtain conditions (5) and (1) as equivalent by Proposition 15, so we complete the proof.Given a Lorentzian surfaceX:U→H1n, we call (X-1(Λ1λn),u0) the tangent indicatrix germ of X, where λ=X(u0)+μ0(X(u0)±N(u0)~) and μ0=∓(1/ki)  (i=1,2,…,n-1).Corollary 17.
The lightlike hypersurface germsLHM1 and LHM2 are A-equivalent, then tangent indicatrix germs (X1-1,u1) and (X2-1,u2) are diffeomorphic as set germs.Proof.
The tangent indicatrix germ ofXi is the zero level set of gi,λi  Since K-equivalent among function germs preserves the zero-level sets of function germs, the assertion follows Theorem 16.


## 5. Singularities of Lightlike Hypersurfaces inR24

In this section, we study the classification of singularities of 3-dimensional lightlike hypersurfaces, which are generated by Lorentzian surface in anti-de Sitter 3-space, also, we consider the space of Lorentzian embeddingsEmbL(U,H13) with Whitney C∞-topology, where U⊂R2 is an open subset. As the choose of the standard arguments in [11], we consider a function G:R24×R24→R by G(v,λ)=〈v-λ,v-λ〉 and claim that Gλ(v) is a submersion at  v≠λ  for any fixed λ∈R24. Given X∈EmbL(U,H13), we have G=G∘(X×idR24). We have the l-jet extension j1lG:U×R24→Jl(U,R) defined by j1lG(u,λ)=jlgλ(u). Consider the trivialization Jl(U,R)=U×R×Jl(2,1). For any submanifold Q⊂Jl(2,1), we denote Q~=U×R×Q. Then we have the following proposition [12, 17].Proposition 18.
LetQ be a submanifold of Jl(2,1).
Consider(20)TQ={X∈EmbL(U,H13)∣j1lGistransversaltoQ~}
is a residual subset of EmbL(U,H13). If Q is a closed subset, then TQ is open.On the other hand, we have a stratification given by the set ofK-orbits in Jl(2,1)∖Wl(2,1) (for the definition of Wl(2,1) and additional properties refer to [12]).Theorem 19.
There exists an open dense subsetO⊂EmbL(U,H13) such that for any X⊂O, the germ of the Legendrian lift of the corresponding lightlike hypersurface LHM± at each point is Legendrian stable.Proposition 20.
There exists an open dense subsetO⊂EmbL(U,H13) such that for any X⊂O, the germ of the corresponding lightlike hypersurface LHM± at any point (x,y,u)∈U×R is A-equivalent to one of the map germs Ak  (1≤k≤4) or D4±, where (A1)
f(u1,u2,u3)=(u1,u2,u3,0)(embedding),(A2)
f(u1,u2,u3)=(3u12,2u13,u2,u3)(cuspidal edge) (Figure1),(A3)
f(u1,u2,u3)=(4u13+2u1u2,3u14+u12u2,u2,u3)(swallowtail) (Figure2),(A4)
f(u1,u2,u3)=(5u14+3u2u12+2u1u3,4u15+2u13u2+u12u3,u1,u2)(butterfly),(D4+)
f(u1,u2,u3)=(2u12+2u22+u1u2u3,3u12+u2u3,3u22+u1u3,u3)(purse) (Figure3),(D4-)
f(u1,u2,u3)=(2u13-2u1u22+u12u3+u22u3,u22-3u12-2u1u3,u1u3-u2u3,u3)(pyramid) (Figure4).Figure 1
Cuspidal edge.Figure 2
Swallowtail.Figure 3
Purse.Figure 4
Pyramid.By using the generic normal forms of generating families and Corollary17, we have the following corollary.Corollary 21.
There exists an open dense subsetO⊂EmbL(U,H13) such that for any X⊂O, the germ of the corresponding tangent indicatrix at any point (x0,y0)∈U is diffeomorphic to one of the germs in the following lists.(1)
{(x,y)∈(R2,0)∣x3+y2=0} (ordinary cusp); (Figure 5),(2)
{(x,y)∈(R2,0)∣x4±y2=0} (tacnode or point); (Figure 6),(3)
{(x,y)∈(R2,0)∣x5+y2=0} (rhamphoid cusp); (Figure 7),(4)
{(x,y)∈(R2,0)∣x3-xy2=0} (three lines); (Figure 8),(5)
{(x,y)∈(R2,0)∣x3+y3=0} (a line).Figure 5
Ordinary cusp.Figure 6
Tacnode.Figure 7
Rhamphoid cusp.Figure 8
Three lines.


## 6. Canal Hypersurface of Lorentzian Surface

Canal hypersurfaces, which are generated by surfaces with codimension 2 along fixed direction, are envelopes of families of hyperspheres. Since the property of a hypersurface is to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15, 16]. In this section, we mainly consider the canal hypersurfaces in semi-Euclidean space with index 2. Let X(U) be a Lorentzian surface; the Mongle form is as follows:
(21)X(u1,u2)={u1,u2,f1(u1,u2),f2(u1,u2)}.
The second fundamental form of M=X(U) is characterized by two quadratic forms. Their functional coefficients will be denoted by (a,b,c) and (e,f,g), respectively [15].We have the following function:(22)Δ(u)=14det⁡(-a-2bc0e2fg00a2bc0e2fg)(u).
The Gaussian curvature of M is
(23)K(u)=(ac-b2+eg-f2)(u),
and the matrix is
(24)α(u)=(abcefg)(u),
where a=∂2f1/∂u1∂u1,  b=∂2f1/∂u1∂u2,  c=∂2f1/∂u2∂u2,  e=∂2f2/∂u1∂u1,  f=∂2f2/∂u1∂u2, and g=∂2f2/∂u2∂u2.Definition 22.
LetX(U) be a Lorentzian surface in anti-de Sitter space and let N(u) be its spacelike normal vector; a hypersurface defined as CM=X(u)+εvN(u)∈R24 is called canal hypersurface of X(U), where ε is a sufficiently small positive real number chosen such that CM is embedded in R24.We denote byX^ the natural embedding of CM in R24 and by (u,N(u)) the point X(u)+εN(u)∈CM. From Looijenga’s theorem [15], there is a residual subset of embeddings X:U↪R24, for which the family of height functions H:U×Λ13→R by H(u,v)=〈X(u),v〉 is locally stable as a family of function on M with parameters on Λ13. Moreover, the corresponding family h(X^) on the canal hypersurface is also generic. In fact the singularities of h(X) and h(X^) are tightly related [16].Thus, for a genericX, those may be one of the following types: Morse (A1), fold (A2), cusp (A3), swallowtail (A4), and elliptic or hyperbolic umbilic (D4±). Moreover, the singularities of the lightcone Gauss indicatrix L±:CM→St1×Ssn-1 can be described in terms as follows [15].Lemma 23 (see [15]).
Given a critical point(u,v)∈CM of the height function hv, we have the following.(1)
u is a nondegenerate critical point of hv if and only if (u,v) is a regular point of L±.(2)
u is a degenerate critical point of hv if and only if (u,v) is singular point of L±.LetKc:CM→R be the Gaussian curvature function on CM. The parabolic set, Kc(0) of CM is the singular set of L±. It can be shown that for a generic embedding of M, Kc(0) is a regular surface except by a finite number of points (u,v), which are singularities of type Σ2,0 of L± or equivalently umbilic points (D4±) of hv [16].Letp:CM→M be the natural projection of CM onto  (i.e.,p(u,v)=u). The image of the set of parabolic points Kc(0) by p is the set Δ≤0.Theorem 24.
(1) IfΔ(u)>0, then u is a nondegenerate critical point of hv for any v∈NX(u)M.
(2) IfΔ(u)<0, then there are exactly two vectors v1,v2∈NX(u)M, such that u is a degenerate critical point of hvi,  i=1,2.
(3) IfΔ(u)=0, then there is a unique vector w∈NX(u)M such that u is a degenerate critical point of hw.Proof.
LetX(u1,u2)={u1,u2,f1(u1,u2),f2(u1,u2)} be the local expression of the embedding in Monge’s form and let the height function in v-direction be
(25)hv(u1,u2)=v1u1+v2u2+v3f1(u1,u2)+v4f2(u1,u2),
where v=(v1,v2,v3,v4)∈R24. If (0,0) is a critical point of the height function hv, then v=(0,0,v3,v4) and the determinant of the Hessian matrix of hv at (0,0) is given by
(26)det⁡H(hv)(0,0)=(ac-b2)v32+(ag+ce-2bf)v3v4+(eg-f2)v42,
where (a,b,c),  (e,f,g) are the above coefficients. Now,
(27)Δ=(ac-b2)(eg-f2)-14(ag+ce-2bf)2
and the equation H(hv)(0,0)=0 has two, one, or zero solutions as Δ<0, Δ=0, or Δ>0, respectively.Whenu is a degenerate critical point of hv, the hyperplane Hv, orthogonal to v, has a higher order contact with M at X(u). Therefore, we will say that v is a binormal vector of M at X(u) and Hv can be an osculating hyperplane [20].At each point ofKc-1(0)-Σ2(L±), there is a unique principal direction of zero curvature for CM. This direction is tangent to the surface Kc-1(0) on a curve made of points of type Σ1,1(L±). This curve is in turn tangent to a zero principal direction of curvature at isolated points [16].Proposition 25.
The image of zero principal directions of curvature inKc-1(0)-Σ2(L±) under p∣Kc-1(0)-Σ2(L±) are asymptotic directions on M.Proof.
For the curvature vectorη(θ) is given by
(28)η(θ)=(12(a-c)cos⁡2θ+bsin2θ)e3+(12(e-g)cos⁡2θ+fsin2θ)e4+H,
where H=(1/2)(a+c)e3+(1/2)(e+g)e4 is the mean curvature vector; we can choose local coordinates for M such that
(29)α(u)=(abc001)(u).
This choice will imply that e1=(1,0,0,0)∈TCM is the zero curvature direction and T∘p(u,v)·e1=e1∈TX(u)M. Then, it follows easily that η(0) and (∂N(u)/∂θ)(0) are parallel.Therefore, we can have the singularities of canal hypersurfaces in the following theorem.Theorem 26.
The canal hypersurfaces have the same singularities as Lorentzian surfaces in anti-de Sitter apace, so we can easily obtain the singularities of canal hypersurfaces as in Section5.



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