# Lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space

Shyuichi IZUMIYA

###### Abstract

Anti-de Sitter space is the Lorentzian space form with negative curvature. In
this paper we consider lightlike hypersurfaces along spacelike submanifolds in
anti-de Sitter space with general codimension. In particular, we investigate
the singularities of lightlike hypersurfaces as an application of the theory
of Legendrian singularities.

0002000 Mathematics Subject classification. Primary 53C50; Secondary
58K99000Key Words and Phrases. Anti de Sitter space, lightlike hypersurface,
Legendrian singularities.

##  1 Introduction

Anti-de Sitter space is one of the Lorentz space forms which has rich
geometric properties. It is defined as a pseudo-sphere with negative curvature
in semi-Euclidean space with index 2 which admits the biggest symmetry in
Riemannian or Lorentz space forms. Anti-de Sitter space can be naturally
considered as a Lorentzian version (generalization) of Hyperbolic space.
Recently we discovered interesting geometric properties of submanifolds in
Hyperbolic space as an application of the theory of Legendrian/Lagrangian
singularities[7, 12, 9, 10]. Therefore anti-de Sitter space might have rich
geometric properties comparing with Hyperbolic space. This is one of the
motivations for the investigation of submanifolds in anti-de Sitter space from
a mathematical viewpoint.

On the other hand, anti-de Sitter space plays important roles in theoretical
physics such as the theory of general relativity, the string theory and the
brane world scenario etc [2, 22, 30, 33]. It is well known that Lorentzian
space forms are classified into three types depending on the value of the
scalar curvature. One of them is Lorentz-Minkowski space which has zero
curvature. The Lorentz space form with positive curvature is de Sitter space.
Anti-de Sitter space is a Lorentzian space form with negative curvature.
Recently, submanifolds in Lorentz-Minkowski space or de Sitter space have been
well investigated (cf., [8, 11, 13, 14, 19, 20]). However, there are not so
many results on submanifolds in anti-de Sitter space, in particular from the
viewpoint of singularity theory. The lightlike hypersurfaces (i.e. the light-
sheets in physics) are important objects in theoretical physics because they
provide good models for different types of horizons [4, 26]. A lightlike
hypersurface is generally a ruled hypersurface along a spacelike submanifold
with codimension two whose rulings are lightlike geodesics. In this paper we
consider lightlike hypersurfaces along spacelike submanifolds with general
codimension in anti-de Sitter space. Moreover, the lightlike hypersurface in
anti-de Sitter $5$-space is a very important subject in the brane world
scenario[17, 2, 30].

In the meantime, tools in the theory of singularities have proven to be useful
description of geometrical properties of submanifolds immersed in different
ambient spaces, from both the local and global viewpoint [7, 8, 10, 13, 14,
21]. The natural connection between geometry and singularities relies on the
basic fact that the contacts of a submanifold with the models 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 Legendrian maps ([1, 27, 34]). When working in a Lorentz space
form, the properties associated to the contacts of a given submanifold with
lightcones have a special relevance. In [8, 13, 18], it was constructed a
framework for the study of spacelike submanifolds with codimension two in
Lorentz-Minkowski space or de Sitter space and discovered a Lorentz invariant
concerning their contacts with models related to lightlike hyperplanes. The
geometry described in this framework is called the lightlike geometry of
spacelike submanifolds with codimension two. By using the invariants of
lightlike geometry, the singularities of lightlike hypersurfaces along
spacelike submanifolds with codimension two in Lorentz-Minkowski space or de
Sitter space were investigated in [11, 15, 18]. However, the situation is
rather complicated for the general codimensional case. The main difference
from the Euclidean space (or, Hyperbolic space) case is the fiber of the canal
hypersurface of a spacelike submanifold is neither connected nor compact. In
order to avoid the above difficulty, we arbitrarily choose a timelike future
directed unit normal vector field along the spacelike submanifold which always
exists for an orientable manifold (cf., §3). Then we construct the unit
spherical normal bundle relative to the above timelike unit normal vector
field, which can be considered as a codimension two spacelike canal
submanifold of the ambient space. Therefore, we can apply the idea of the
lightlike geometry of spacelike submanifolds of the ambient space with
codimension two. In this paper we apply the idea of this framework and the
theory of Legendrian singularities to investigate the singularities of
lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space
with general codimension.

In §3 we construct the framework of the lightlike geometry of spacelike
submanifolds with general codimension analogous to [10]. The notion of
lightlike hypersurfaces along spacelike submanifolds is introduced and the
basic properties are investigated in §4. The notion of the anti-de Sitter
height functions families is useful for the study of lightlike hypersurfaces
(cf., §4). The critical value set of the lightlike hypersurface along a
spacelike submanifold is called the lightlike focal set of the submanifold. In
§5 we show that the lightlike focal set of a spacelike submanifold is a point
if and only if the lightlike hypersurface along the submanifold is a subset of
a lightcone (Proposition 5.1). Therefore, an anti-de Sitter lightcone is a
model hypersurface of lightlike hypersurfaces. The geometric meaning of the
singularities of lightlike hypersurface is described by the theory of contact
of submanifolds with model hypersurfaces. Moreover, as an application of the
theory of Legendrian singularities, we show that two lightlike hypersurfaces
are locally diffeomorphic if and only if the types of the contact of spacelike
submanifolds with lightcones are the same in the sense of Montaldi[27] under
some generic conditions (Theorem 5.5). In §6 we describe the case of
codimension two as a special case. We describe the detailed properties of
lightlike focal sets of spacelike surfaces in anti-de Sitter $4$-space. We
also investigate spacelike curves in anti-de Sitter $4$-space as the simplest
case of a higher codimension in §7.

We shall assume throughout the whole paper that all the maps and manifolds are
$C^{\infty}$ unless the contrary is explicitly stated.

##  2 Basic facts and notations on semi-Euclidean space with index 2

In this section we prepare the basic notions on semi-Euclidean (n+2)-space
with index 2. For details of semi-Euclidean geometry, see [29]. Let
${\mathbb{R}}^{n+2}=\\{(x_{-1},x_{0},x_{1},\cdots,x_{n})|x_{i}\in{\mathbb{R}}\
(i=-1,0,\cdots,n)\ \\}$ be an (n+2)-dimensional vector space. For any vectors
$\mbox{\boldmath$x$}=(x_{-1},x_{0},x_{1},\cdots,x_{n})$ and
$\mbox{\boldmath$y$}=(y_{-1},y_{0},y_{1},\cdots,y_{n})$ in
${\mathbb{R}}^{n+2},$ the pseudo scalar product  of $x$ and $y$ is defined to
be
$\langle\mbox{\boldmath$x$},\mbox{\boldmath$y$}\rangle=-x_{-1}y_{-1}-x_{0}y_{0}+\sum_{i=1}^{n}x_{i}y_{i}$.
We call $(\mathbb{R}^{n+2},\langle,\rangle)$ semi-Euclidean (n+2)-space with
index 2 and write $\mathbb{R}^{n+2}_{2}$ instead of
$(\mathbb{R}^{n+2},\langle,\rangle)$. We say that a non-zero vector $x$ in
$\mathbb{R}^{n+2}_{2}$ is spacelike, null or timelike if
$\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle>0,\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle=0$
or $\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle<0$, respectively.
The norm of the vector $\mbox{\boldmath$x$}\in\mathbb{R}^{n+2}_{2}$ is defined
to be
$\|\mbox{\boldmath$x$}\|=\sqrt{|\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle|}$.
We define the signature of $x$ by

${\rm
sign}(\mbox{\boldmath$x$})=\left\\{\begin{array}[]{ccc}1\qquad\quad$\mbox{\boldmath$x$}$\
\mbox{is\ spacelike,}\\\ \mbox{}0\qquad\quad\mbox{\boldmath$x$}\ \mbox{is\
null,}\\\ -1\qquad\mbox{\boldmath$x$}\ \mbox{is\ timelike.}\end{array}\right.$

For a non-zero vector $\mbox{\boldmath$n$}\in\mathbb{R}^{n+2}_{2}$ and a real
number $c$, we define the hyperplane with pseudo-normal $n$ by

$HP(\mbox{\boldmath$n$},c)=\\{\mbox{\boldmath$x$}\in\mathbb{R}^{n+2}_{2}|\langle\mbox{\boldmath$x$},\mbox{\boldmath$n$}\rangle=c\\}.$

We call $HP(\mbox{\boldmath$n$},c)$ a Lorentz hyperplane, a semi-Euclidean
hyperplane with index 2 or a null hyperplane if $n$ is timelike, spacelike or
null respectively.

We now define an anti-de Sitter $(n+1)$-space  (briefly, AdS $(n+1)$-space) by

$AdS^{n+1}=\\{\mbox{\boldmath$x$}\in\mathbb{R}_{2}^{n+2}\ |\
\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle=-1\\}=H^{n+1}_{1},$

a unit pseudo $(n+1)$-sphere with index 2 by

$S^{n+1}_{2}=\\{\mbox{\boldmath$x$}\in{\mathbb{R}}_{2}^{n+2}\ |\
\langle\mbox{\boldmath$x$},\mbox{\boldmath$x$}\rangle=1\\}$

and a (closed) nullcone with vertex $a$ by

$\Lambda_{a}^{n+1}=\\{\mbox{\boldmath$x$}\in{\mathbb{R}}_{2}^{n+2}|\langle\mbox{\boldmath$x$}-\mbox{\boldmath$a$},\mbox{\boldmath$x$}-\mbox{\boldmath$a$}\rangle=0\\}.$

In particular we denote that
$\Lambda^{*}=\Lambda^{n+1}_{0}\setminus\\{\mbox{\boldmath$0$}\\}$ and also
call it the (open) nullcone. Our main subject in this paper is $AdS^{n+1}$.
Since there are timelike closed curves in $AdS^{n+1}$, the causality of
$AdS^{n+1}$ is violated. In order to avoid such a situation, it is usually
considered the universal covering space $\widetilde{AdS}^{n+1}$ of $AdS^{n+1}$
in physics which is called the universal Anti de Sitter space. We remark that
the local structure of these spaces are the same.

For any
$\mbox{\boldmath$x$}_{1},\cdots,\mbox{\boldmath$x$}_{n}\in{\mathbb{R}}^{n+2}_{2}$.
We define a vector
$\mbox{\boldmath$x$}_{1}\wedge\cdots\wedge\mbox{\boldmath$x$}_{n}$ by

$\mbox{\boldmath$x$}_{1}\wedge\cdots\wedge\mbox{\boldmath$x$}_{n}=\begin{vmatrix}-\mbox{\boldmath$e$}_{-1}&-\mbox{\boldmath$e$}_{0}&\mbox{\boldmath$e$}_{1}&\cdots&\mbox{\boldmath$e$}_{n}\vspace{2mm}\\\
x^{1}_{-1}&x^{1}_{0}&x^{1}_{1}&\cdots&x^{1}_{n}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
x^{n}_{-1}&x^{n}_{0}&x^{n}_{1}&\cdots&x^{n}_{n}\end{vmatrix},$

where
$\\{\mbox{\boldmath$e$}_{-1},\mbox{\boldmath$e$}_{0},\mbox{\boldmath$e$}_{1},\cdots,\mbox{\boldmath$e$}_{n}\\}$
is the canonical basis of $\mathbb{R}^{n+2}_{2}$ and
$\mbox{\boldmath$x$}_{i}=(x^{i}_{-1},x^{i}_{0},x^{i}_{1},\cdots,x^{i}_{n})$.
We can easily check that

$\langle\mbox{\boldmath$x$},\
\mbox{\boldmath$x$}_{1}\wedge\cdots\wedge\mbox{\boldmath$x$}_{n}\rangle=\textrm{det}(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_{1},\cdots,\mbox{\boldmath$x$}_{n}),$

so that $\mbox{\boldmath$x$}_{1}\wedge\cdots\wedge\mbox{\boldmath$x$}_{n}$ is
pseudo-orthogonal to any $\mbox{\boldmath$x$}_{i}$ (for $i=1,\cdots,n$).

##  3 Spacelike submanifolds in anti-de Sitter space

We introduce in this section the basic geometrical tools for the study of
spacelike submanifolds in the anti-de Sitter $(n+1)$-space.

Consider the orientation of ${\mathbb{R}}^{n+2}_{2}$ provided by the condition
that
$\textrm{det}(\mbox{\boldmath$e$}_{-1},\mbox{\boldmath$e$}_{0},\mbox{\boldmath$e$}_{1},\cdots,\mbox{\boldmath$e$}_{n})>0.$
This orientation induces the orientation of $x_{-1}x_{0}$-plane, so that it
gives a time orientation on $AdS^{n+1}$. If we consider the universal Anti de
Sitter space $\widetilde{AdS}^{n+1},$ we can determine the future direction.

We consider a spacelike embedding $\mbox{\boldmath$X$}:U\rightarrow AdS^{n+1}$
from an open subset $U\subset{\mathbb{R}}^{s}$ with $s+k=n+1.$ We write
$M=\mbox{\boldmath$X$}(U)$ and identify $M$ and $U$ through the embedding
$\mbox{\boldmath$X$}.$ We say that $X$ is spacelike if the tangent space
$T_{p}M$ consists only spacelike vectors (i.e. spacelike subspace) for any
point $p\in M$. In this case, the pseudo-normal space $N_{p}(M)$ in
${\mathbb{R}}^{n+2}_{2}$ is a $k+1$-dimensional semi-Euclidean space with
index $2$ (cf. [29]). We write $N(M)$ as the pseudo-normal bundle in
${\mathbb{R}}^{n+2}_{2}$ over $M.$ On the pseudo-normal space $N_{p}(M),$ we
have two kinds of pseudo spheres:

$\displaystyle N_{p}(M;-1)$ $\displaystyle=$
$\displaystyle\\{\mbox{\boldmath$v$}\in N_{p}(M)\ |\
\langle\mbox{\boldmath$v$},\mbox{\boldmath$v$}\rangle=-1\ \\}$ $\displaystyle
N_{p}(M;1)$ $\displaystyle=$ $\displaystyle\\{\mbox{\boldmath$v$}\in N_{p}(M)\
|\ \langle\mbox{\boldmath$v$},\mbox{\boldmath$v$}\rangle=1\ \\},$

so that we have two unit spherical normal bundles over $M$:

$N(M;-1)=\bigcup_{p\in M}N_{p}(M;-1)\ \mbox{and}\ N(M;1)=\bigcup_{p\in
M}N_{p}(M;1).$

Then we have the Whitney sum decomposition

$T{\mathbb{R}}^{n+2}_{2}|M=TM\oplus N(M).$

By definition $\mbox{\boldmath$X$}(u)$ is one of the timelike unit normal
vectors of $M$ at $p=\mbox{\boldmath$X$}(u),$ so that $\mbox{\boldmath$X$}\in
N_{p}(M).$ Since $AdS^{n+1}$ is time oriented, we can arbitrarily choose an
adopted unit timelike normal section $\mbox{\boldmath$n$}^{T}(u)\in N_{p}(M)$
pseudo-orthogonal to $\mbox{\boldmath$X$}(u)$ even globally. Here, we say that
$\mbox{\boldmath$n$}^{T}$ is adopted if

$\textrm{det}(\mbox{\boldmath$X$}(u),\mbox{\boldmath$n$}^{T}(u),\mbox{\boldmath$e$}_{1},\dots,\mbox{\boldmath$e$}_{n})>0.$

Therefore we have the pseudo-orthonormal complement
$(\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$n$}^{T}(u)\rangle_{\mathbb{R}})^{\perp}$
in $N_{p}(M)$ which is a $(k-1)$-dimensional subspace of $N_{p}(M).$ We define
a $(k-2)$-dimensional spacelike unit sphere in $N_{p}(M)$ by

$N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}]=\\{\mbox{\boldmath$\xi$}\in N_{p}(M;1)\
|\
\langle\mbox{\boldmath$\xi$},\mbox{\boldmath$n$}^{T}(p)\rangle=\langle\mbox{\boldmath$\xi$},\mbox{\boldmath$X$}(u)\rangle=0,p=\mbox{\boldmath$X$}(u)\
\\}.$

Then we have a spacelike unit $k-2$-spherical bundle over $M$ with respect to
$\mbox{\boldmath$n$}^{T}$ defined by

$N_{1}(M)[\mbox{\boldmath$n$}^{T}]=\bigcup_{p\in
M}N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}].$

Since we have $T_{(p,\xi)}N_{1}(M)[\mbox{\boldmath$n$}^{T}]=T_{p}M\times
T_{\xi}N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}],$ we have the canonical
Riemannian metric on $N_{1}(M)[\mbox{\boldmath$n$}^{T}].$ We denote the
Riemannian metric on $N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ by
$(G_{ij}(p,\mbox{\boldmath$\xi$}))_{1\leqslant i,j\leqslant n-1}.$ We now
arbitrarily choose (at least locally) a unit spacelike normal vector field
$\mbox{\boldmath$n$}^{S}$ with $\mbox{\boldmath$n$}^{S}(u)\in
N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}]$, where $p=\mbox{\boldmath$X$}(u).$ We
call $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ an adopted pair of
normal vector fields along $M.$ Clearly, the vectors
$\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u)$ are null. We define
a mapping

$\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}):U\longrightarrow\Lambda^{*}$

by
$\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u)=\mbox{\boldmath$n$}^{T}(u)+\mbox{\boldmath$n$}^{S}(u).$
We call it the nullcone Gauss image of $M=\mbox{\boldmath$X$}(U)$ with respect
to $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}).$ With the
identification of $M$ and $U$ through $\mbox{\boldmath$X$},$ we have the
linear mapping provided by the derivative of the nullcone Gauss image
$\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ at each point
$p\in M$ as follows:

$d_{p}\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}):T_{p}M\longrightarrow
T_{p}{\mathbb{R}}^{n+1}_{1}=T_{p}M\oplus N_{p}(M).$

Consider the orthogonal projections $\pi^{t}:T_{p}M\oplus N_{p}(M)\rightarrow
T_{p}(M).$ We define

$d_{p}\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})^{t}=\pi^{t}\circ
d_{p}(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S}).$

We call the linear transformation
$S_{p}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})=-d_{p}\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})^{t}$
the $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-shape operator of
$M=\mbox{\boldmath$X$}(U)$ at $p=\mbox{\boldmath$X$}(u).$ Let
$\\{\kappa_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(p)\\}_{i=1}^{s}$
be the eigenvalues of
$S_{p}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$, which are called the
nullcone principal curvatures with respect to
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ at
$p=\mbox{\boldmath$X$}(u)$. Then the nullcone Gauss-Kronecker curvature with
respect to $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ at
$p=\mbox{\boldmath$X$}(u)$ is defined by

$K_{N}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(p)={\rm
det}S_{p}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}).$

We say that a point $p=\mbox{\boldmath$X$}(u)$ is an
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-umbilical point if

$S_{p}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})=\kappa(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(p)1_{T_{p}M}.$

We say that $M=\mbox{\boldmath$X$}(U)$ is totally
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-umbilical if all points on
$M$ are $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-umbilical.
Moreover, $M=\mbox{\boldmath$X$}(U)$ is said to be totally nullcone umbilical
if it is totally $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-umbilical
for any adopted pair $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}).$

We deduce now the nullcone Weingarten formula. Since
$\mbox{\boldmath$X$}_{u_{i}}$ $(i=1,\dots s)$ are spacelike vectors, we have a
Riemannian metric (the first fundamental form ) on $M=\mbox{\boldmath$X$}(U)$
defined by $ds^{2}=\sum_{i=1}^{s}g_{ij}du_{i}du_{j}$, where
$g_{ij}(u)=\langle\mbox{\boldmath$X$}_{u_{i}}(u),\mbox{\boldmath$X$}_{u_{j}}(u)\rangle$
for any $u\in U.$ We also have the nullcone second fundamental invariant with
respect to the normal vector field
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ defined by
$h_{ij}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u)=\langle-(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})_{u_{i}}(u),\mbox{\boldmath$X$}_{u_{j}}(u)\rangle$
for any $u\in U.$ By similar arguments to those in the proof of [13,
Proposition 3.2], we have the following proposition.

###### Proposition 3.1

We choose a pseudo-orthonormal frame
$\\{\mbox{\boldmath$X$},\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}_{1},\dots,\mbox{\boldmath$n$}^{S}_{k-1}\\}$
of $N(M)$ with $\mbox{\boldmath$n$}^{S}_{k-1}=\mbox{\boldmath$n$}^{S}.$ Then
we have the following nullcone Weingarten formula :

(a)
$\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})_{u_{i}}=\langle\mbox{\boldmath$n$}^{T}_{u_{i}},\mbox{\boldmath$n$}^{S}\rangle(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})+\sum_{\ell=1}^{k-2}\langle(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})_{u_{i}},\mbox{\boldmath$n$}^{S}_{\ell}\rangle\mbox{\boldmath$n$}^{S}_{\ell}-\sum_{j=1}^{s}h_{i}^{j}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\mbox{\boldmath$X$}_{u_{j}}$

(b)
$\pi^{t}\circ\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})_{u_{i}}=-\sum_{j=1}^{s}h_{i}^{j}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\mbox{\boldmath$X$}_{u_{j}}.$

Here
$\displaystyle{\left(h_{i}^{j}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\right)=\left(h_{ik}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\right)\left(g^{kj}\right)}$
and $\displaystyle{\left(g^{kj}\right)=\left(g_{kj}\right)^{-1}}.$

As a consequence of the above proposition, we have an explicit expression of
the nullcone curvature by

$K_{N}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})=\frac{\displaystyle{{\rm
det}\left(h_{ij}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\right)}}{\displaystyle{{\rm
det}\left(g_{\alpha\beta}\right)}}.$

Since
$\langle-(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})(u),\mbox{\boldmath$X$}_{u_{j}}(u)\rangle=0,$
we have
$h_{ij}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u)=\langle\mbox{\boldmath$n$}^{T}(u)+\mbox{\boldmath$n$}^{S}(u),\mbox{\boldmath$X$}_{u_{i}u_{j}}(u)\rangle.$
Therefore the nullcone second fundamental invariant at a point
$p_{0}=\mbox{\boldmath$X$}(u_{0})$ depends only on the values
$\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$n$}^{S}(u_{0})$ and
$\mbox{\boldmath$X$}_{u_{i}u_{j}}(u_{0})$, respectively. Thus, the nullcone
curvatures also depend only on
$\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$n$}^{S}(u_{0})$,
$\mbox{\boldmath$X$}_{u_{i}}(u_{0})$ and
$\mbox{\boldmath$X$}_{u_{i}u_{j}}(u_{0})$, which are independent of the
derivations of the vector fields $\mbox{\boldmath$n$}^{T}$ and
$\mbox{\boldmath$n$}^{S}.$ We write
$\kappa_{i}(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})(p_{0})$
$(i=1,\dots,s)$ and
$K_{N}(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})(u_{0})$ as the
nullcone curvatures at $p_{0}=\mbox{\boldmath$X$}(u_{0})$ with respect to
$(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})=(\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$n$}^{S}(u_{0})).$
We might also say that a point $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is
$(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})$-umbilical because
the nullcone $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$-shape
operator at $p_{0}$ depends only on the normal vectors
$(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0}).$ So we denote that
$h_{ij}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$\xi$})(u_{0})=h_{ij}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})$
and
$K_{N}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$\xi$})(p_{0})=K_{N}(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})(p_{0})$,
where $\mbox{\boldmath$\xi$}=\mbox{\boldmath$n$}^{S}(u_{0})$ for some local
extension $\mbox{\boldmath$n$}^{T}(u)$ of $\mbox{\boldmath$\xi$}.$
Analogously, we say that a point $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an
$(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})$-parabolic point
of $\mbox{\boldmath$X$}:U\longrightarrow{\mathbb{R}}^{n+1}_{1}$ if
$K_{N}(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})(u_{0})=0.$ We
also say that a point $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is a
$(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})$-flat point  if it
is an $(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})$-umbilical
point and
$K_{N}(\mbox{\boldmath$n$}^{T}_{0},\mbox{\boldmath$n$}^{S}_{0})(u_{0})=0.$

On the other hand, we define a map
$\mathbb{NG}(\mbox{\boldmath$n$}^{T}):N_{1}(M)[\mbox{\boldmath$n$}^{T}]\longrightarrow\Lambda^{*}$
by
$\mathbb{NG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$})=\mbox{\boldmath$n$}^{T}(u)+\mbox{\boldmath$\xi$},$
which we call the nullcone Gauss image of $N_{1}(M)[\mbox{\boldmath$n$}^{T}].$
This map leads us to the notions of curvatures. Let
$T_{(p,\xi)}N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ be the tangent space of
$N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ at $(p,\mbox{\boldmath$\xi$}).$ Under the
canonical identification
$(\mathbb{NG}(\mbox{\boldmath$n$}^{T})^{*}T{\mathbb{R}}^{n+2}_{2})_{(p,\mbox{\scriptsize\boldmath$\xi$})}=T_{(\mbox{\scriptsize\boldmath$n$}^{T}(p)+\mbox{\scriptsize\boldmath$\xi$})}{\mathbb{R}}^{n+2}_{2}\equiv
T_{p}{\mathbb{R}}^{n+2}_{2},$ we have

$T_{(p,\mbox{\scriptsize\boldmath$\xi$})}N_{1}(M)[\mbox{\boldmath$n$}^{T}]=T_{p}M\oplus
T_{\xi}S^{k-2}\subset T_{p}M\oplus N_{p}(M)=T_{p}{\mathbb{R}}^{n+2}_{2},$

where $T_{\xi}S^{k-2}\subset T_{\xi}N_{p}(M)\equiv N_{p}(M)$ and
$p=\mbox{\boldmath$X$}(u).$ Let

$\Pi^{t}:\mathbb{NG}(\mbox{\boldmath$n$}^{T})^{*}T{\mathbb{R}}^{n+2}_{2}=TN_{1}(M)[\mbox{\boldmath$n$}^{T}]\oplus{\mathbb{R}}^{k+1}\longrightarrow
TN_{1}(M)[\mbox{\boldmath$n$}^{T}]$

be the canonical projection. Then we have a linear transformation

$S_{N}(\mbox{\boldmath$n$}^{T})_{(p,\mbox{\scriptsize\boldmath$\xi$})}=-\Pi^{t}_{\mathbb{LG}(n^{T})(p,\xi)}\circ
d_{(p,\xi)}\mathbb{NG}(\mbox{\boldmath$n$}^{T}):T_{(p,\xi)}N_{1}(M)[\mbox{\boldmath$n$}^{T}]\longrightarrow
T_{(p,\xi)}N_{1}(M)[\mbox{\boldmath$n$}^{T}],$

which is called the nullcone shape operator of
$N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ at $(p,\mbox{\boldmath$\xi$}).$ Let
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$ be the
eigenvalues of
$S_{N}(\mbox{\boldmath$n$}^{T})_{(p,\mbox{\scriptsize\boldmath$\xi$})}$,
$(i=1,\dots,n-1)$. Here, we denote
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$,
$(i=1,\dots,s)$ as the eigenvalues belonging to the eigenvectors on $T_{p}M$
and $\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$,
$(i=s+1,\dots n-1)$ as the eigenvalues belonging to the eigenvectors on the
tangent space of the fiber of $N_{1}(M)[\mbox{\boldmath$n$}^{T}].$ Then we
have the following proposition.

###### Proposition 3.2

We choose a (local) pseudo-orthonormal frame
$\\{\mbox{\boldmath$X$},\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}_{1},\dots,\mbox{\boldmath$n$}^{S}_{k-1}\\}$
of $N(M)$ with $\mbox{\boldmath$n$}^{S}_{k-1}=\mbox{\boldmath$n$}^{S}.$ For
$p_{0}=\mbox{\boldmath$X$}(u_{0})$ and
$\mbox{\boldmath$\xi$}_{0}=\mbox{\boldmath$n$}^{S}(u_{0}),$ we have
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0})=\kappa_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})$,
$(i=1,\dots,s)$ and
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0})=-1$,
$(i=s+1,\dots n-1).$

Proof.  Since
$\\{\mbox{\boldmath$X$},\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}_{1},\dots,\mbox{\boldmath$n$}^{S}_{k-1}\\}$
is a pseudo-orthonormal frame of$N(M),$ we have
$\langle\mbox{\boldmath$X$}(u_{0}),\mbox{\boldmath$\xi$}_{0}\rangle=\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$\xi$}_{0}\rangle=\langle\mbox{\boldmath$n$}^{S}_{i}(u_{0}),\mbox{\boldmath$\xi$}_{0}\rangle=0.$
Therefore, we have
$T_{\mbox{\scriptsize\boldmath$\xi$}}S^{k-2}=\langle\mbox{\boldmath$n$}^{S}_{1}(u_{0}),\dots,\mbox{\boldmath$n$}^{S}_{k-2}(u_{0})\rangle.$
By this orthonormal basis of
$T_{\mbox{\scriptsize\boldmath$\xi$}_{0}}S^{k-2},$ the canonical Riemannian
metric $G_{ij}(p_{0},\mbox{\boldmath$\xi$}_{0})$ is represented by

$(G_{ij}(p_{0},\mbox{\boldmath$\xi$}))=\left(\begin{array}[]{cc}g_{ij}(p_{0})&0\\\
0&I_{k-2}\end{array}\right),$

where
$g_{ij}(p_{0})=\langle\mbox{\boldmath$X$}_{u_{i}}(u_{0}),\mbox{\boldmath$X$}_{u_{j}}(u_{0})\rangle$.

On the other hand, by Proposition 3.1, we have

$-\sum_{j=1}^{s}h^{j}_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})\mbox{\boldmath$X$}_{u_{j}}=\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})_{u_{i}}(u_{0})=d_{p_{0}}\mathbb{NG}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})\left(\frac{\partial}{\partial
u_{i}}\right),$

so that we have

$S_{N}(\mbox{\boldmath$n$}^{T})_{(p_{0},\mbox{\boldmath$\xi$}_{0})}\left(\frac{\partial}{\partial
u_{i}}\right)=\sum_{j=1}^{s}h^{j}_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})\mbox{\boldmath$X$}_{u_{j}}.$

Therefore, the representation matrix of
$S_{N}(\mbox{\boldmath$n$}^{T})_{(p_{0},\mbox{\boldmath$\xi$}_{0})}$ with
respect to the basis

$\\{\mbox{\boldmath$X$}_{u_{1}}(u_{0}),\dots,\mbox{\boldmath$X$}_{u_{s}}(u_{0}),\mbox{\boldmath$n$}^{S}_{1}(u_{0}),\dots,\mbox{\boldmath$n$}^{S}_{k-2}(u_{0})\\}$

of $T_{(p_{0},\mbox{\boldmath$\xi$}_{0})}(N_{1}(M)[\mbox{\boldmath$n$}^{T}])$
is of the form

$\left(\begin{array}[]{cc}h^{j}_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})&*\\\
0&-I_{k-2}\end{array}\right).$

Thus, the eigenvalues of this matrix are
$\lambda_{i}=\kappa_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})$,
$(i=1,\dots,s)$ and $\lambda_{i}=-1$ , $(i=s+1,\dots,n-1)$. This completes the
proof. $\Box$

We call $\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$,
$(i=1,\dots,s)$ the nullcone principal curvatures of $M$ with respect to
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$\xi$})$ at $p\in M.$ The nullcone
Lipschitz-Killing curvature of $N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ at
$(p,\mbox{\boldmath$\xi$})$ is defined to be
$K_{N}(\mbox{\boldmath$n$}^{T})(p,\mbox{\boldmath$\xi$})=\det
S_{N}(\mbox{\boldmath$n$}^{T})_{(p,\xi)}.$

##  4 Lightlike hypersurfaces in anti-de Sitter space

We define a hypersurface

$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T}):N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}}\longrightarrow
AdS^{n+1}$

by

$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})((p,\mbox{\boldmath$\xi$}),\mu)=\mbox{\boldmath$X$}(u)+\mu(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$\xi$})(u)=\mbox{\boldmath$X$}(u)+\mu\mathbb{NG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$}),$

where $p=\mbox{\boldmath$X$}(u),$ which is called the lightlike hypersurface
along $M$ relative to $\mbox{\boldmath$n$}^{T}.$ In general, a hypersurface
$H\subset AdS^{n+1}$ is called a lightlike hypersurface if it is tangent to
the lightcone at any regular point. We remark that
$\mathbb{NH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
is a lightlike hypersurface.

We introduce the notion of height functions on spacelike submanifold, which is
useful for the study of singularities of lightlike hypersurfaces. We define a
family of functions

$H:M\times AdS^{n+1}\longrightarrow{\mathbb{R}}$

on a spacelike submanifold $M=\mbox{\boldmath$X$}(U)$ by
$H(p,\mbox{\boldmath$\lambda$})=H(u,\mbox{\boldmath$\lambda$})=\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$\lambda$}\rangle+1,$
where $p=\mbox{\boldmath$X$}(u).$ We call $H$ the anti-de Sitter height
function (briefly, AdS-height function) on the spacelike submanifold $M.$ For
any fixed $\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1},$ we write
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p)=H(p,\mbox{\boldmath$\lambda$}_{0})$
and have the following proposition.

###### Proposition 4.1

Let $M$ be a spacelike submanifold and $H:M\times(AdS^{n+1}\setminus
M)\to{\mathbb{R}}$ the AdS-height function on $M.$ Suppose that
$p_{0}=\mbox{\boldmath$X$}(u_{0})\not=\mbox{\boldmath$\lambda$}_{0}.$ Then we
have the following$:$

(1) $h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p_{0})=\partial
h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial u_{i}(p_{0})=0$,
$(i=1,\dots,s)$ if and only if there exist $\mbox{\boldmath$\xi$}_{0}\in
N_{1}(M)_{p_{0}}[\mbox{\boldmath$n$}^{T}]$ and
$\mu_{0}\in{\mathbb{R}}\setminus\\{0\\}$ such that

$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(u_{0},\mbox{\boldmath$\xi$}_{0})=\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})((p_{0},\mbox{\boldmath$\xi$}_{0}),\mu_{0}).$

(2) $h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p_{0})=\partial
h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial u_{i}(p_{0})={\rm
det}{\mathcal{H}}(h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}})(p_{0})=0$
$(i=1,\dots,s)$ if and only if there exist $\mbox{\boldmath$\xi$}_{0}\in
N_{1}(M)_{p_{0}}[\mbox{\boldmath$n$}^{T}]$ and
$\mu_{0}\in{\mathbb{R}}\setminus\\{0\\}$ such that

$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})((p_{0},\mbox{\boldmath$\xi$}_{0}),\mu_{0})$

and $1/\mu$ is one of the non-zero nullcone principal curvatures
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}),(i=1,\dots,s).$

Here, ${\mathcal{H}}(h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}})(p_{0})$ is
the Hessian matrix of $h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}$ at
$p_{0}.$

(3) With condition (2), ${\rm
rank}\,{\mathcal{H}}(h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}})(p_{0})=0$
if and only if $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is a non-flat
$(\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$\xi$}_{0})$-umbilical point.

Proof.  (1) For $p=\mbox{\boldmath$X$}(u),$ the condition
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p)=\langle\mbox{\boldmath$X$}(u),{\mbox{\boldmath$\lambda$}_{0}}\rangle+1=0$
means that

$\langle\mbox{\boldmath$X$}(u)-\lambda_{0},\mbox{\boldmath$X$}(u)-\mbox{\boldmath$\lambda$}_{0}\rangle=\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$X$}(u)\rangle-2\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$\lambda$}_{0}\rangle+\langle\mbox{\boldmath$\lambda$}_{0},\mbox{\boldmath$\lambda$}_{0}\rangle=-2(1+\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$\lambda$}_{0}\rangle)=0,$

so that
$\mbox{\boldmath$X$}(u)-{\mbox{\boldmath$\lambda$}_{0}}\in\Lambda^{*}.$ Since
$\partial h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial
u_{i}(p)=\langle\mbox{\boldmath$X$}_{u_{i}}(u),{\mbox{\boldmath$\lambda$}_{0}}\rangle$
and $\langle\mbox{\boldmath$X$}_{u_{i}},\mbox{\boldmath$X$}\rangle=0,$ we have
$\langle\mbox{\boldmath$X$}_{u_{i}}(u),\mbox{\boldmath$\lambda$}_{0}\rangle=-\langle\mbox{\boldmath$X$}_{u_{i}}(u)-\mbox{\boldmath$\lambda$}_{0}\rangle$.
Therefore, $\partial h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial
u_{i}(p)=0$ if and only if
$\mbox{\boldmath$X$}(u)-{\mbox{\boldmath$\lambda$}_{0}}\in N_{p}M.$ On the
other hand, the condition
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p)=\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$\lambda$}_{0}\rangle+1=0$
implies that
$\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$X$}(u)-\mbox{\boldmath$\lambda$}_{0}\rangle=0$.
This means that $\mbox{\boldmath$X$}(u)-\mbox{\boldmath$\lambda$}_{0}\in
T_{p}AdS^{n+1}.$ Hence
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p_{0})=\partial
h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial u_{i}((p_{0})=0$
$(i=1,\dots,s)$ if and only if
$\mbox{\boldmath$X$}(u_{0})-{\mbox{\boldmath$\lambda$}_{0}}\in
N_{p_{0}}M\cap\Lambda^{*}\cap T_{p_{0}}AdS^{n+1}.$ Then we denote that
$\mbox{\boldmath$v$}=\mbox{\boldmath$X$}(u_{0})-{\mbox{\boldmath$\lambda$}_{0}}\in
N_{p_{0}}M\cap\Lambda^{*}\cap T_{p_{0}}AdS^{n+1}.$ If
$\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle=0,$ then
$\mbox{\boldmath$n$}^{T}(u_{0})$ belongs to a lightlike hyperplane in the
Lorentz space $T_{p_{0}}AdS^{n+1},$ so that $\mbox{\boldmath$n$}^{T}(u_{0})$
is lightlike or spacelike. This contradiction to the fact that
$\mbox{\boldmath$n$}^{T}(u_{0})$ is a timelike unit vector. Thus,
$\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle\not=0.$ We
set

$\mbox{\boldmath$\xi$}_{0}=\frac{-1}{\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle}\mbox{\boldmath$v$}-\mbox{\boldmath$n$}^{T}(u_{0}).$

Then we have

$\displaystyle\langle\mbox{\boldmath$\xi$}_{0},\mbox{\boldmath$\xi$}_{0}\rangle$
$\displaystyle=$
$\displaystyle-2\frac{-1}{\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle}\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle-1=1$
$\displaystyle\langle\mbox{\boldmath$\xi$}_{0},\mbox{\boldmath$n$}^{T}(u_{0})\rangle$
$\displaystyle=$
$\displaystyle\frac{-1}{\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle}\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle+1=0.$

This means that $\mbox{\boldmath$\xi$}_{0}\in
N_{1}(M)_{p_{0}}(M)[\mbox{\boldmath$n$}^{T}].$ Since
$-\mbox{\boldmath$v$}=\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle(\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$\xi$}_{0}),$
we have
${\mbox{\boldmath$\lambda$}_{0}}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(p_{0},\mbox{\boldmath$\xi$}_{0})$,
where $p_{0}=\mbox{\boldmath$X$}(u_{0})$ and
$\mu_{0}=\langle\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$v$}\rangle.$
For the converse assertion, suppose that
$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(p_{0},\mbox{\boldmath$\xi$}_{0}).$
Then $\mbox{\boldmath$\lambda$}_{0}-\mbox{\boldmath$X$}(u_{0})\in
N_{p_{0}}(M)\cap\Lambda^{*}$ and
$\langle\mbox{\boldmath$\lambda$}_{0}-\mbox{\boldmath$X$}(u_{0}),\mbox{\boldmath$X$}(u_{0})\rangle=\langle\mu_{0}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(p_{0},\mbox{\boldmath$\xi$}_{0}),\mbox{\boldmath$X$}(u_{0})\rangle=0.$
Thus we have $\mbox{\boldmath$\lambda$}_{0}-\mbox{\boldmath$X$}(u_{0})\in
N_{p_{0}}(M)\cap\Lambda^{*}\cap T_{p_{0}}AdS^{n+1}.$ By the previous
arguments, these conditions are equivalent to the condition that
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(p_{0})=\partial
h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}/\partial u_{i}((p_{0})=0$
$(i=1,\dots,s)$.

(2) By a straightforward calculation, we have

$\frac{\partial^{2}h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}}{\partial
u_{i}\partial
u_{j}}(u)=\langle\mbox{\boldmath$X$}_{u_{i}u_{j}},\mbox{\boldmath$\lambda$}_{0}\rangle.$

Under the conditions
${\mbox{\boldmath$\lambda$}_{0}}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}(\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$\xi$}_{0})$,
we have

$\frac{\partial^{2}h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}}{\partial
u_{i}\partial
u_{j}}(u_{0})=\langle\mbox{\boldmath$X$}_{u_{i}u_{j}}(u_{0}),\mbox{\boldmath$X$}(u_{0})\rangle+\mu_{0}\langle\mbox{\boldmath$X$}_{u_{i}u_{j}}(u_{0}),(\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$\xi$}_{0})\rangle.$

Since $\langle\mbox{\boldmath$X$}_{u_{i}},\mbox{\boldmath$X$}\rangle=0,$ we
have
$\langle\mbox{\boldmath$X$}_{u_{i}u_{j}},\mbox{\boldmath$X$}\rangle=-\langle\mbox{\boldmath$X$}_{u_{i}},\mbox{\boldmath$X$}_{u_{j}}\rangle.$
Therefore, we have

$\left(\frac{\partial^{2}h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}}{\partial
u_{i}\partial
u_{\ell}}(u_{0})\right)\left(g^{j\ell}(u_{0})\right)=\left(\mu_{0}h^{j}_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})-\delta^{j}_{i}\right),$

where $\mbox{\boldmath$n$}^{S}$ is the local section of
$N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ with
$\mbox{\boldmath$n$}^{S}(u_{0})=\mbox{\boldmath$\xi$}_{0}.$ It follows that
${\rm det}{\mathcal{H}}(g)(p_{0})=0$ if and only if $1/\mu_{0}$ is an
eigenvalue of
$(h^{i}_{j}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})),$ which
is equal to one of the nullcone principal curvatures
$\kappa_{i}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0})=\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}),(i=1,\dots,s)$.

(3) By the above calculation, ${\rm
rank}\,{\mathcal{H}}(h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}})(p_{0})=0$
if and only if

$(h^{i}_{j}(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})(u_{0}))=\frac{1}{\mu_{0}}(\delta^{j}_{i}),$

where
$1/\mu_{0}=\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}),\
(i=1,\dots,s)$. This means that $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an
$(\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$\xi$}_{0})$-umbilical point.
$\Box$

In order to understand the geometric meaning of the assertions of Proposition
4.1, we briefly review the theory of Legendrian singularities. For detailed
expressions, see [1, 34]. Let
$\pi:PT^{*}({\mathbb{R}}^{n+1})\longrightarrow{\mathbb{R}}^{n+1}$ be the
projective cotangent bundle with its canonical contact structure. We next
review the geometric properties of this bundle. Consider the tangent bundle
$\tau:TPT^{*}({\mathbb{R}}^{n+})\rightarrow PT^{*}({\mathbb{R}}^{n+1})$ and
the differential map $d\pi:TPT^{*}({\mathbb{R}}^{n+1})\rightarrow
T{\mathbb{R}}^{n+1}$ of $\pi.$ For any $X\in TPT^{*}({\mathbb{R}}^{n+1}),$
there exists an element $\alpha\in T^{*}({\mathbb{R}}^{n+1}_{1}$ such that
$\tau(X)=[\alpha].$ For an element $V\in T_{x}({\mathbb{R}}^{n+1}),$ the
property $\alpha(V)=0$ does not depend on the choice of representative of the
class $[\alpha].$ Thus we can define the canonical contact structure on
$PT^{*}({\mathbb{R}}^{n+1})$ by

$K=\\{X\in TPT^{*}({\mathbb{R}}^{n+1})\ |\ \tau(X)(d\pi(X))=0\\}.$

We have a trivialization
$PT^{*}({\mathbb{R}}^{n+1})\cong{\mathbb{R}}^{n+1}\times
P^{n}({\mathbb{R}})^{*},$ and call

$((v_{0},v_{1},\dots,v_{n}),[\xi_{0}:\xi_{1}:\cdots:\xi_{n}])$

the homogeneous coordinates of $PT^{*}({\mathbb{R}}^{n+1}),$ where
$[\xi_{0}:\xi_{1}:\cdots:\xi_{n}]$ are the homogeneous coordinates of the dual
projective space $P^{n}({\mathbb{R}})^{*}.$ It is easy to show that $X\in
K_{(x,[\xi])}$ if and only if $\sum_{i=0}^{n}\mu_{i}\xi_{i}=0,$ where
$d\tilde{\pi}(X)=\sum_{i=0}^{n}\mu_{i}\partial/\partial v_{i}.$ An immersion
$i:L\rightarrow PT^{*}({\mathbb{R}}^{n+1})$ is said to be a Legendrian
immersion if $\text{dim}\,L=n$ and $di_{q}(T_{q}L)\subset K_{i(q)}$ for any
$q\in L.$ The map $\pi\circ i$ is also called the Legendrian map and the set
$W(i)=\text{image}\,\pi\circ i$, the wave front set of $i.$ Moreover, $i$ (or,
the image of $i$) is called the Legendrian lift  of $W(i).$

Let
$F:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}},\mbox{\boldmath$0$})$
be a function germ. We say that $F$ is a Morse family of hypersurfaces if the
map germ

$\Delta^{*}F=\left(F,\frac{\partial F}{\partial q_{1}},\dots,\frac{\partial
F}{\partial
q_{k}}\right):({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}}\times{\mathbb{R}}^{k},\mbox{\boldmath$0$})$

is submersive, where
$(q,x)=(q_{1},\dots,q_{k},x_{0},\dots,x_{n})\in({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$}).$
In this case we have a smooth $n$-dimensional submanifold

$\Sigma_{*}(F)=\Bigl{\\{}(q,x)\in({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\
\Bigm{|}\ F(q,x)=\frac{\partial F}{\partial q_{1}}(q,x)=\cdots=\frac{\partial
F}{\partial q_{k}}(q,x)=0\ \Bigr{\\}}$

and the map germ
$\mathscr{L}_{F}:(\Sigma_{*}(F),\mbox{\boldmath$0$})\longrightarrow
PT^{*}{\mathbb{R}}^{n+1}$ defined by

$\mathscr{L}_{F}(q,x)=\left(x,\left[\frac{\partial F}{\partial
x_{0}}(q,x):\cdots:\frac{\partial F}{\partial x_{n}}(q,x)\right]\right)$

is a Legendrian immersion. We call $F$ a generating family of
$\mathscr{L}_{F}(\Sigma_{*}(F)),$ and the wave front set is given by
$W(\mathscr{L}_{F})\\!=\pi_{n}(\Sigma_{*}(F)),$ where
$\pi_{n}:{\mathbb{R}}^{k}\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}}^{n}$
is the canonical projection. In the theory of unfoldings of function germs,
the wave front set $W(\mathscr{L}_{F})$ is called a discriminant set of $F,$
which we also denote $\mathcal{D}_{F}.$ Therefore, Proposition 4.1 asserts
that the discriminant set of the AdS-height function $H$ is given by

${\mathcal{D}}_{H}=\Bigl{\\{}\mbox{\boldmath$\lambda$}\in
AdS^{n+1}\Bigm{|}\mbox{\boldmath$\lambda$}=\mbox{\boldmath$X$}(u)+\mu(\mbox{\boldmath$n$}^{T}\pm\mbox{\boldmath$\xi$})(u),\
p=\mbox{\boldmath$X$}(u)\in M,\mbox{\boldmath$\xi$}\in
N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}],\mu\in{\mathbb{R}}\ \Bigr{\\}},$

which is the image of the lightlike hypersurface along $M$ relative to
$\mbox{\boldmath$n$}^{T}.$

By the assertion (2) of Proposition 4.1, a singular point of the lightlike
hypersurface is a point
$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$\xi$}_{0})(u_{0})$
for $p_{0}=\mbox{\boldmath$X$}(u_{0})$ and
$\mu_{0}=1/\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}),$
$i=1,\dots.s).$ Then we have the following corollary.

###### Corollary 4.2

The critical value of $\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})$ is the point

$\mbox{\boldmath$\lambda$}=\mbox{\boldmath$X$}(u)+\frac{1}{\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})}\mathbb{LG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$}),$

where $p=\mbox{\boldmath$X$}(u)$ and
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})\not=0.$

For a non-zero nullcone principal curvature
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0})\not=0,$
we have an open subset $O_{i}\subset N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ such
that $\kappa_{N}(\mbox{\boldmath$n$}^{T})(p,\mbox{\boldmath$\xi$})\not=0$
Therefore, we have a non-zero nullcone principal curvature function
$\kappa_{N}(\mbox{\boldmath$n$}^{T}):O_{i}\longrightarrow{\mathbb{R}}$. We
define a mapping
$\mathbb{LF}_{\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}}:O_{i}\longrightarrow
AdS^{n+1}$ by

$\mathbb{LF}_{\kappa_{N}(\mbox{\scriptsize\boldmath$n$}^{T})_{i}}(p,\mbox{\boldmath$\xi$})=\mbox{\boldmath$X$}(u)+\frac{1}{\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$}),$

where $p=\mbox{\boldmath$X$}(u).$ We also define

$\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T})=\bigcup\left\\{\mathbb{LF}_{\kappa_{N}(\mbox{\scriptsize\boldmath$n$}^{T})_{i}}(p,\mbox{\boldmath$\xi$})\
|\ (p,\mbox{\boldmath$\xi$})\in N_{1}(M)[\mbox{\boldmath$n$}^{T}]\
\mbox{s.t.}\
\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})\not=0,i=1,\dots,s\right\\}.$

We call $\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T})$ the lightlike focal set of
$M=\mbox{\boldmath$X$}(U)$ relative to $\mbox{\boldmath$n$}^{T}.$ By
definition, the lightlike focal set of $M=\mbox{\boldmath$X$}(U)$ relative to
$\mbox{\boldmath$n$}^{T}$ is the critical values set of the lightlike
hypersurface
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
along $M$ relative to $\mbox{\boldmath$n$}^{T}.$

We can show that the image of the lightlike hypersurface along $M$ is
independent of the choice of the future directed timelike normal vector field
$\mbox{\boldmath$n$}^{T}$ as a corollary of Proposition 4.1.

###### Corollary 4.3

Let $\mbox{\boldmath$n$}^{T}$ and $\overline{\mbox{\boldmath$n$}}^{T}$ be
future directed timelike unit normal fields along $M$. Then we have

$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})=\mathbb{LH}_{M}(\overline{\mbox{\boldmath$n$}}^{T})(N_{1}(M)[\overline{\mbox{\boldmath$n$}}^{T}]\times{\mathbb{R}})\
\mbox{and}\
\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T})=\mathbb{LF}_{M}(\overline{\mbox{\boldmath$n$}}^{T}).$

Proof.  By Proposition 4.1, the images of the lightlike hypersurface along $M$
relative to $\mbox{\boldmath$n$}^{T}$ and $\overline{\mbox{\boldmath$n$}}^{T}$
are the discriminant sets of the AdS-height function $H$ on $M$. Moreover, the
focal set is the critical value set of the lightlike hypersurface along $M$
relative to $\mbox{\boldmath$n$}^{T}.$ Since $H$ is independent of the choice
of $\mbox{\boldmath$n$}^{T},$ we have the assertion. $\Box$

We have the following proposition.

###### Proposition 4.4

Let $H$ be the AdS-height function on $M.$ For any point
$(u,\mbox{\boldmath$\lambda$})\in\Delta^{*}H^{-1}(0),$ the germ of $H$ at
$(u,\mbox{\boldmath$\lambda$})$ is a Morse family of hypersurfaces.

Proof.  We denote that

$\mbox{\boldmath$X$}(u)=(X_{-1}(u),X_{0}(u),X_{1}(u),\dots,X_{n}(u))\ {\rm
and}\
\mbox{\boldmath$\lambda$}=(\lambda_{-1},\lambda_{0},\lambda_{1},\dots,\lambda_{n}).$

We define an open subset $U_{-1}^{+}=\\{\mbox{\boldmath$\lambda$}\in
AdS^{n+1}\ |\ \lambda_{-1}>0\ \\}$. For any $\mbox{\boldmath$\lambda$}\in
U_{-1}^{+},$ we have

$\lambda_{-1}=\sqrt{1-\lambda_{0}^{2}+\lambda_{1}^{2}+\cdots\lambda_{n}^{2}}.$

Thus, we have a local coordinate of $AdS^{n+1}$ given by
$(\lambda_{0},\lambda_{1},\dots,\lambda_{n})$ on $U_{-1}^{+}.$ By definition,
we have

$H(u,\mbox{\boldmath$\lambda$})=-X_{-1}(u)\sqrt{1-\lambda_{0}^{2}+\sum_{i=1}^{n}\lambda_{i}^{2}}-X_{0}(u)\lambda_{0}+X_{1}(u)\lambda_{1}+\cdots+X_{n}(u)\lambda_{n}.$

We now prove that the mapping

$\Delta^{*}H=\left(H,\frac{\partial H}{\partial u_{1}},\dots,\frac{\partial
H}{\partial u_{s}}\right)$

is non-singular at $(u,\mbox{\boldmath$\lambda$})\in\Delta^{*}H^{-1}(0).$
Indeed, the Jacobian matrix of $\Delta^{*}H$ is given by

$\left(\begin{array}[]{ccccc}&X_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0}&-X_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1}&\cdots&-X_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{n}\\\
\smash{\lower 4.30554pt\hbox{\bg
A}}&X_{-1u_{1}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{1}}&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{1}}&\cdots&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{1}}\\\
&\vdots&\vdots&\ddots&\vdots\\\
&X_{-1u_{s}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{s}}&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{s}}&\cdots&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{s}}\end{array}\right),$

where

$\displaystyle\smash{\lower 4.30554pt\hbox{\bg
A}}=\left(\\!\\!\begin{array}[]{ccc}\langle\mbox{\boldmath$X$}_{u_{1}},\mbox{\boldmath$\lambda$}\rangle&\\!\\!\cdots\\!\\!&\langle\mbox{\boldmath$X$}_{u_{s}},\mbox{\boldmath$\lambda$}\rangle\\\
\langle\mbox{\boldmath$X$}_{u_{1}u_{1}},\mbox{\boldmath$\lambda$}\rangle&\\!\\!\cdots\\!\\!&\langle\mbox{\boldmath$X$}_{u_{1}u_{s}},\mbox{\boldmath$\lambda$}\rangle\\\
\vdots&\\!\\!\ddots\\!\\!&\vdots\\\
\langle\mbox{\boldmath$X$}_{u_{s}u_{1}},\mbox{\boldmath$\lambda$}\rangle&\\!\\!\cdots\\!\\!&\langle\mbox{\boldmath$X$}_{u_{s}u_{s}},\mbox{\boldmath$\lambda$}\rangle\end{array}\\!\\!\right).$

We now show that the rank of

$\smash{\lower 4.30554pt\hbox{\bg
B}}=\left(\begin{array}[]{cccc}X_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0}&-X_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1}&\cdots&-X_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{n}\\\
X_{-1u_{1}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{1}}&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{1}}&\cdots&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{1}}\\\
\vdots&\vdots&\ddots&\vdots\\\
X_{-1u_{s}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{s}}&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{s}}&\cdots&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{s}}\end{array}\right)$

is $s+1$ at $(u,\mbox{\boldmath$\lambda$})\in\Sigma_{*}(H).$ Since
$(u,\mbox{\boldmath$\lambda$})\in\Sigma_{*}(H),$ we have

$\mbox{\boldmath$\lambda$}=\mbox{\boldmath$X$}(u)+\mu\left(\mbox{\boldmath$n$}^{T}(u)+\sum_{i=1}^{k-1}\xi_{i}\mbox{\boldmath$n$}_{i}(u)\right)$

with $\sum_{i=1}^{k-1}\xi^{2}_{i}=1,$ where
$\\{\mbox{\boldmath$X$},\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S}_{1},\dots,\mbox{\boldmath$n$}^{S}_{k-1}\\}$
is a pseudo-orthonormal (local) frame of $N(M).$ Without the loss of
generality, we assume that $\mu\not=0$ and $\xi_{k-1}\not=0.$ We denote that

$\mbox{\boldmath$n$}^{T}(u)=^{t}\\!\\!(n^{T}_{-1}(u),n^{T}_{0}(u),\dots
n^{T}_{n}(u)),\
\mbox{\boldmath$n$}_{i}(u)=^{t}\\!\\!(n^{i}_{-1}(u),n_{0}^{i}(u),\dots
n^{i}_{n}(u)).$

It is enough to show that the rank of the matrix

$\smash{\lower 4.30554pt\hbox{\bg
C}}=\left(\begin{array}[]{cccc}X_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0}&-X_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1}&\cdots&-X_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{n}\\\
X_{-1u_{1}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{1}}&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{1}}&\cdots&-X_{-1u_{1}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{1}}\\\
\vdots&\vdots&\ddots&\vdots\\\
X_{-1u_{s}}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0u_{s}}&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+X_{1u_{s}}&\cdots&-X_{-1u_{s}}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-X_{nu_{s}}\\\
n^{T}_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-n^{T}_{0}&-n^{T}_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+n^{T}_{1}&\cdots&-n^{T}_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-n^{T}_{n}\\\
n^{1}_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-n^{1}_{0}&-n^{1}_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+n^{1}_{1}&\cdots&-n^{1}_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-n^{1}_{n}\\\
\vdots&\vdots&\ddots&\vdots\\\
n^{k-2}_{-1}\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-n^{k-2}_{0}&-n^{k-2}_{-1}\displaystyle{\frac{\lambda_{1}}{\lambda_{-1}}}+n^{k-2}_{1}&\cdots&-n^{k-2}_{-1}\displaystyle{\frac{\lambda_{n}}{\lambda_{-1}}}-n^{k-2}_{n}\end{array}\right)$

is $n+1$ at $(u,\mbox{\boldmath$\lambda$})\in\Sigma_{*}(H).$ We denote that

$\mbox{\boldmath$a$}_{i}=^{t}\\!\\!(x_{i}(u),x_{iu_{1}}(u),\dots
x_{iu_{s}}(u),n^{T}_{i}(u),n^{1}_{i}(u),\dots,n^{k-2}_{i}(u)).$

Then we have

$\smash{\lower 4.30554pt\hbox{\bg
C}}=\left(\mbox{\boldmath$a$}_{-1}\frac{\lambda_{0}}{\lambda_{-1}}-\mbox{\boldmath$a$}_{0},-\mbox{\boldmath$a$}_{-1}\frac{\lambda_{1}}{\lambda_{-1}}+\mbox{\boldmath$a$}_{1},\dots,-\mbox{\boldmath$a$}_{-1}\frac{\lambda_{n}}{\lambda_{-1}}+\mbox{\boldmath$a$}_{n}\right).$

It follows that

$\displaystyle\det\smash{\lower 4.30554pt\hbox{\bg C}}\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=\frac{\lambda_{-1}}{\lambda_{-1}}\det(\mbox{\boldmath$a$}_{0},\mbox{\boldmath$a$}_{1},\dots,\mbox{\boldmath$a$}_{n})+\frac{\lambda_{0}}{\lambda_{-1}}\det(\mbox{\boldmath$a$}_{-1}\mbox{\boldmath$a$}_{1},\dots,\mbox{\boldmath$a$}_{n})$
$\displaystyle-\frac{\lambda_{1}}{\lambda_{-1}}(-1)\det(\mbox{\boldmath$a$}_{-1},\mbox{\boldmath$a$}_{0},\mbox{\boldmath$a$}_{2},\dots,\mbox{\boldmath$a$}_{n})-\cdots-\frac{\lambda_{n}}{\lambda_{-1}}(-1)^{n-1}\det(\mbox{\boldmath$a$}_{-1}\mbox{\boldmath$a$}_{0},\mbox{\boldmath$a$}_{1},\dots,\mbox{\boldmath$a$}_{n-1}).$

Moreover, we define
$\delta_{i}=\det(\mbox{\boldmath$a$}_{-1},\mbox{\boldmath$a$}_{0},\mbox{\boldmath$a$}_{1},\dots,\mbox{\boldmath$a$}_{i-1},\mbox{\boldmath$a$}_{i+1},\dots,\mbox{\boldmath$a$}_{n})$
for $i=-1,0,1,\dots,n$ and
$\mbox{\boldmath$a$}=(-\delta_{-1},-\delta_{0},-\delta_{1},(-1)^{2}\delta_{2},\dots,(-1)^{n-1}\delta_{n}).$
Then we have

$\mbox{\boldmath$a$}=\mbox{\boldmath$X$}\wedge\mbox{\boldmath$X$}_{u_{1}}\wedge\cdots\wedge\mbox{\boldmath$X$}_{u_{s}}\wedge\mbox{\boldmath$n$}^{T}\wedge\mbox{\boldmath$n$}_{1}\wedge\cdots\wedge\mbox{\boldmath$n$}_{k-2}.$

We remark that $\mbox{\boldmath$a$}\not=0$ and
$\mbox{\boldmath$a$}=\pm\|\mbox{\boldmath$a$}\|\mbox{\boldmath$n$}_{k-1}.$ By
the above calculation, we have

$\displaystyle\det\smash{\lower 4.30554pt\hbox{\bg C}}\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=\left\langle\left(\frac{\lambda{-1}}{\lambda_{-1}},\frac{\lambda_{0}}{\lambda_{-1}},\dots,\frac{\lambda_{n}}{\lambda_{-1}}\right),\mbox{\boldmath$a$}\right\rangle=\frac{1}{\lambda_{-1}}\left\langle\mbox{\boldmath$X$}(u)+\mu\left(\mbox{\boldmath$n$}^{T}(u)+\sum_{i=1}^{k-1}\xi_{i}\mbox{\boldmath$n$}_{i}(u)\right),\mbox{\boldmath$a$}\right\rangle$
$\displaystyle=\frac{1}{\lambda_{-1}}\times\pm\mu\xi_{k-1}\|\mbox{\boldmath$a$}\|=\pm\frac{\mu\xi_{k-1}\|\mbox{\boldmath$a$}\|}{\lambda_{-1}}\not=0.$

Therefore the Jacobi matrix of $\Delta^{*}H$ is non-singular at
$(u,\mbox{\boldmath$\lambda$})\in\Delta^{*}H^{-1}(0).$

For other local coordinates of $AdS^{n+1}$, we can apply the same method for
the proof as the above case. This completes the proof. $\Box$

Here we also consider the local coordinate $U^{+}_{-1}$. Since $H$ is a Morse
family of hypersurfaces, we have a Legendrian immersion

$\mathscr{L}_{H}:\Sigma_{*}(H)\longrightarrow PT^{*}(AdS^{n+1})|U^{+}_{-1}$

by the general theory of Legendrian singularities. By definition, we have

$\frac{\partial H}{\partial\lambda_{0}}(u,\mbox{\boldmath$\lambda$})\\\
=X_{-1}(u)\displaystyle{\frac{\lambda_{0}}{\lambda_{-1}}}-X_{0}(u),\
\frac{\partial
H}{\partial\lambda_{i}}(u,\mbox{\boldmath$\lambda$})=-X_{-1}(u)\displaystyle{\frac{\lambda_{i}}{\lambda_{-1}}}+X_{i}(u),\
(i=1,\dots,n).$

It follows that

$\displaystyle\left[\frac{\partial
H}{\partial\lambda_{0}}(u,\mbox{\boldmath$\lambda$}):\frac{\partial
H}{\partial\lambda_{1}}(u,\mbox{\boldmath$\lambda$}):\cdots:\frac{\partial
H}{\partial\lambda_{n}}(u,\mbox{\boldmath$\lambda$})\right]$
$\displaystyle\qquad=[X_{-1}(u)\lambda_{0}-X_{0}(u)\lambda_{-1}:X_{1}(u)\lambda_{-1}-X_{-1}(u)\lambda_{1}:\cdots:X_{n}(u)\lambda_{-1}-X_{-1}(u)\lambda_{n}].$

Therefore, we have

$\mathscr{L}_{H}(u,\mbox{\boldmath$\lambda$})=(\mbox{\boldmath$\lambda$},[X_{-1}(u)\lambda_{0}-X_{0}(u)\lambda_{-1}:X_{1}(u)\lambda_{-1}-X_{-1}(u)\lambda_{1}:\cdots:X_{n}(u)\lambda_{-1}-X_{-1}(u)\lambda_{n}]),$

where

$\Sigma_{*}(H)=\\{(u,\mbox{\boldmath$\lambda$})\ |\
\mbox{\boldmath$\lambda$}=\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(p,\mbox{\boldmath$\xi$},t)\
((p,\mbox{\boldmath$\xi$}),t)\in
N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}}\\}.$

We observe that $H$ is a generating family of the Legendrian submanifold
$\mathscr{L}_{H}(\Sigma_{*}(H))$ whose wave front is
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$.
Therefore we say that the AdS-height function $H$ on $M$ gives an AdS-
canonical generating family for the Legendrian lift of
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$.
For other local coordinates of $AdS^{n+1},$ we have the similar results to the
above case.

##  5 Contact with lightcones

In this section we consider the geometric meaning of the singularities of
lightlike hypersurfaces in Anti-de Sitter space from the view point of the
theory of contact of submanifolds with model hypersurfaces in [27]. We begin
with the following basic observations.

###### Proposition 5.1

Let $\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1}$ and
$M=\mbox{\boldmath$X$}(U)$ a spacelike submanifold without points satisfying
$K_{N}(\mbox{\boldmath$n$}^{T})(p,\mbox{\boldmath$\xi$})=0.$ Then
$M\subset\Lambda^{n+1}_{\lambda_{0}}\cap AdS^{n+1}$ if and only if
$\\{\mbox{\boldmath$\lambda$}_{0}\\}=\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T})$
is the lightcone focal set. In this case we have
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}])\subset\Lambda^{n+1}_{\lambda_{0}}\cap
AdS^{n+1}$ and $M=\mbox{\boldmath$X$}(U)$ is totally nullcone umbilical.

Proof.  By Proposition 3.1,
$K_{N}(\mbox{\boldmath$n$}^{T})(p_{0},\mbox{\boldmath$\xi$}_{0})\not=0$ if and
only if

$\\{(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S}),(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})_{u_{1}},\dots,(\mbox{\boldmath$n$}^{T}+\mbox{\boldmath$n$}^{S})_{u_{s}}\\}$

is linearly independent for $p_{0}=\mbox{\boldmath$X$}(u_{0})\in M$ and
$\mbox{\boldmath$\xi$}_{0}=\mbox{\boldmath$n$}^{S}(u_{0}),$ where
$\mbox{\boldmath$n$}^{S}:U\longrightarrow N_{1}(M)[\mbox{\boldmath$n$}^{T}]$
is a local section. By the proof of the assertion (1) of Proposition 4.1,
$M\subset\Lambda^{n+1}_{\lambda_{0}}\cap AdS^{n+1}$ if and only if
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(u)=0$ for any $u\in U,$ where
$h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(u)=H(u,\mbox{\boldmath$\lambda$}_{0})$
is the AdS-height function on $M.$ It also follows from Proposition 4.1 that
there exists a smooth function $\eta:U\times
N_{1}(M)[\mbox{\boldmath$n$}^{T}]\longrightarrow{\mathbb{R}}$ and section
$\mbox{\boldmath$n$}^{S}:U\longrightarrow N_{1}(M)[\mbox{\boldmath$n$}^{T}]$
such that

$\mbox{\boldmath$X$}(u)=\mbox{\boldmath$\lambda$}_{0}+\eta(u,\mbox{\boldmath$n$}^{S}(u))(\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u)).$

In fact, we have
$\eta(u,\mbox{\boldmath$n$}^{S}(u))=-1/\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$
$i=1,\dots,s$, where $p=\mbox{\boldmath$X$}(u)$ and
$\mbox{\boldmath$\xi$}=\mbox{\boldmath$n$}^{S}(u).$ It follows that
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})=\kappa_{N}(\mbox{\boldmath$n$}^{T})_{j}(p,\mbox{\boldmath$\xi$}),$
so that $M=\mbox{\boldmath$X$}(U)$ is totally nullcone umbilical. Therefore we
have

$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$n$}^{S}(u),\mu)=\mbox{\boldmath$\lambda$}_{0}+(\mu+\eta(u,\mbox{\boldmath$n$}^{S}(u))(\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u)).$

Hence we have
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})\subset\Lambda^{n+1}_{\lambda_{0}}.$
By Corollary 4.2, the critical value set of
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
is the lightlike focal set $\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T}).$
However, it is equal to $\lambda_{0}$ by the previous arguments.

For the converse assertion, suppose that
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LF}_{M}(\mbox{\boldmath$n$}^{T}).$ Then
we have

$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$X$}(u)+\frac{1}{\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(\mbox{\boldmath$X$}(u),\mbox{\boldmath$\xi$})}\mathbb{LG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$}),$

for any $i=1,\dots,s$ and $(p,\mbox{\boldmath$\xi$})\in
N_{1}(M)[\mbox{\boldmath$n$}^{T}],$ where $p=\mbox{\boldmath$X$}(u).$ Thus, we
have

$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(\mbox{\boldmath$X$}(u),\mbox{\boldmath$\xi$})=\kappa_{N}(\mbox{\boldmath$n$}^{T})_{j}(\mbox{\boldmath$X$}(u),\mbox{\boldmath$\xi$})$

for any $i,j=1,\dots,s,$ so that $M$ is totally nullcone umbilical. Since
$\mathbb{LG}(\mbox{\boldmath$n$}^{T})(u,\mbox{\boldmath$\xi$})$ is null, we
have $\mbox{\boldmath$X$}(u)\in\Lambda^{n+1}_{\lambda_{0}}.$ This completes
the proof. $\Box$

According to the above proposition,
$\Lambda^{n+1}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}\cap AdS^{n+1}$ is
regarded as a model lightlike hypersurface in $AdS^{n+1}.$ We define

$T(AdS^{n+1})_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}=\\{\mbox{\boldmath$x$}\in{\mathbb{R}}^{n+2}_{2}\
|\ \ \mbox{\boldmath$x$}-\mbox{\boldmath$\lambda$}_{0}\in
T_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}AdS^{n+1}\ \\},$

where $T_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}AdS^{n+1}$ is the tangent
space of $AdS^{n+1}$ at $\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1}.$ We call
$T(AdS^{n+1})_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}$ a tangent affine
space of $AdS^{n+1}$ at $\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1}.$ It is
easy to show that

$\Lambda^{n+1}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}\cap
AdS^{n+1}=T(AdS^{n+1})_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}\cap
AdS^{n+1}.$

We denote that
$LC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(AdS^{n+1})=\Lambda^{n+1}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}\cap
AdS^{n+1}=T(AdS^{n+1})_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}\cap
AdS^{n+1}$, which is called an AdS-lightcone with the vertex
$\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1}.$ Therefore, the model lightlike
hypersurface is an AdS-lightcone.

We consider the contact of spacelike submanifolds with AdS-lightcones. Let

${\mathcal{H}}:AdS^{n+1}\times AdS^{n+1}\longrightarrow{\mathbb{R}}$

be a function defined by
${\mathcal{H}}(\mbox{\boldmath$x$},\mbox{\boldmath$\lambda$})=\langle\mbox{\boldmath$x$},\mbox{\boldmath$\lambda$}\rangle+1.$
Given $\mbox{\boldmath$\lambda$}_{0}\in AdS^{n+1},$ we denote
${\mathfrak{h}}_{\lambda_{0}}(\mbox{\boldmath$x$})={\mathcal{H}}(\mbox{\boldmath$x$},\mbox{\boldmath$\lambda$}_{0})$,
so that we have
${\mathfrak{h}}_{\lambda_{0}}^{-1}(0)=LC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(AdS^{n+1}).$
For any $p_{0}=\mbox{\boldmath$X$}(u_{0})\in M$, $\mu_{0}\in{\mathbb{R}}$ and
$\mbox{\boldmath$\xi$}_{0}\in N_{1}(M)_{p}[\mbox{\boldmath$n$}^{T}],$ we
consider the point
$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$X$}(u_{0})+\mu_{0}(\mbox{\boldmath$n$}^{T}(u_{0})+\mbox{\boldmath$\xi$}_{0}).$
Then we have

${\mathfrak{h}}_{\lambda_{0}}\circ\mbox{\boldmath$X$}(u_{0}))={\mathcal{H}}\circ(\mbox{\boldmath$X$}\times
1_{AdS^{n+1}})(u_{0},\mbox{\boldmath$\lambda$}_{0})=H(p_{0},\mbox{\boldmath$\lambda$}_{0})=0,$

where
$\mu_{0}=1/\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}),$
$i=1,\dots,s.$ We also have relations

$\frac{\partial{\mathfrak{h}}_{\lambda_{0}}\circ\mbox{\boldmath$X$}}{\partial
u_{i}}(u_{0})=\frac{\partial H}{\partial
u_{i}}(p_{0},\mbox{\boldmath$\lambda$}_{0})=0,\ i=1,\dots,s.$

These imply that the AdS-lightcone
${\mathfrak{h}}_{\lambda_{0}}^{-1}(0)=LC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(AdS^{n+1})$
is tangent to $M=\mbox{\boldmath$X$}(U)$ at
$p_{0}=\mbox{\boldmath$X$}(u_{0}).$ In this case, we call
$LC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(AdS^{n+1})$ a tangent AdS-
lightcone of $M=\mbox{\boldmath$X$}(U)$ at $p_{0}=\mbox{\boldmath$X$}(u_{0}),$
which is denoted by
$TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}.$ Moreover, the
tangent AdS-lightcone
$TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}$ is called an
osculating AdS-lightcone if
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LF}_{\kappa_{N}(\mbox{\scriptsize\boldmath$n$}^{T})_{i}(p_{0},\mbox{\scriptsize\boldmath$\xi$}_{0})}(u_{0})\in\mathbb{LF}_{M},$
for one of the nullcone principal curvature
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0}).$ In
this case, we call $\mbox{\boldmath$\lambda$}_{0}$ the center of the nullcone
principal curvature
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0})(u_{0}).$
Therefore, we can interpret that the lightlike focal set is the locus of the
centers of nullcone principal curvatures. This fact is analogous to the notion
of the focal sets of submanifolds in Euclidean space.

Firstly, we consider a special contact of $M=\mbox{\boldmath$X$}(U)$ with AdS-
lightcones. We say that $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an AdS-lightlike
$k$-ridge point if $h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}$ has the
$A_{k+2}$ singular point at $u_{0}$ for some $k\geq 1,$ where
$\mbox{\boldmath$\lambda$}_{0}\in\mathcal{D}_{H}=\mathbb{LH}_{M}(N_{1}(M)\times{\mathbb{R}}).$
We simply say that $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an AdS-lightlike
ridge point if it is the AdS-lightlike $k$-ridge point for some $k\geq 1.$ For
a function germ
$f:({\mathbb{R}}^{s},\widetilde{u}_{0})\longrightarrow{\mathbb{R}}$, $f$ has
an $A_{k}$ singular point at $\widetilde{u}_{0}$ if $f$ is
$\mathcal{K}$-equivalent to the germ $u^{k+1}_{1}\pm u^{2}_{2}\pm\dots\pm
u^{2}_{s}$. We say that two function germs
$f_{i}:({\mathbb{R}}^{s},\widetilde{u}_{i})\longrightarrow{\mathbb{R}}$
$(i=1,2)$ are ${\mathcal{K}}$-equivalent if there exists a diffeomorphism germ
$\Phi:({\mathbb{R}}^{s},\widetilde{u}_{1})\longrightarrow({\mathbb{R}}^{s},\widetilde{u}_{2})$
and a non-zero function germ
$\rho:({\mathbb{R}}^{s},\widetilde{u}_{1})\longrightarrow{\mathbb{R}}$ such
that $f_{2}\circ\Phi(u)=\rho(u)f_{1}(u).$ We consider the geometric meaning of
AdS-lightlike ridge points. Let $F:AdS^{n+1}\longrightarrow{\mathbb{R}}$ be a
submersion and $\mbox{\boldmath$X$}:U\longrightarrow AdS^{n+1}$ a spacelike
embedding from an open set $U\subset{\mathbb{R}}^{s}.$ We say that
$M=\mbox{\boldmath$X$}(U)$ and $F^{-1}(0)$ have a corank $r$ contact at
$p_{0}=\mbox{\boldmath$X$}(u_{0})$ if the Hessian of the function
$g(u)=F\circ\mbox{\boldmath$X$}(u)$ has corank $r$ at $u_{0}$. We also say
that $M=\mbox{\boldmath$X$}(U)$ and $F^{-1}(0)$ have an $A_{k}$-type contact
at $p_{0}=\mbox{\boldmath$X$}(u_{0})$ if the function
$g(u)=F\circ\mbox{\boldmath$X$}(u)$ has the $A_{k}$ singularity at $u_{0}.$ By
definition, if $M=\mbox{\boldmath$X$}(U)$ and $F^{-1}(0)$ have an $A_{k}$-type
contact at $p_{0}=\mbox{\boldmath$X$}(u_{0}),$ then these have a corank $1$
contact. For a regular curve $\mbox{\boldmath$\gamma$}:I\longrightarrow
AdS^{n+1},$ we say that $\mbox{\boldmath$\gamma$}(I)$ and $F^{-1}(0)$ have a
contact of order $k$ if $F\circ\mbox{\boldmath$\gamma$}$ has the $A_{k}$
singularity at $s_{0}.$ We have the following simple proposition:

###### Proposition 5.2

For any $p_{0}=\mbox{\boldmath$X$}(u_{0})$ and
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LF}_{\kappa_{N}(\mbox{\scriptsize\boldmath$n$}^{T})_{i}(p_{0},\mbox{\scriptsize\boldmath$\xi$}_{0})}(u_{0}),$
there exists an integer $r$ with $1\leq r\leq s$ such that
$M=\mbox{\boldmath$X$}(U)$ and
$TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}$ have corank $r$
contact at $p_{0}=\mbox{\boldmath$X$}(u_{0}).$

By Proposition 4.1, $M=\mbox{\boldmath$X$}(U)$ and the osculating hypersphere
$TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}$ have corank $s$
contact at an
$(\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$\xi$}_{0})$-umbilical point.
Therefore the AdS-lightlike ridge point is not an
$(\mbox{\boldmath$n$}^{T}(u_{0}),\mbox{\boldmath$\xi$}_{0})$-umbilical point.

By the general theory of unfoldings of function germs, the discriminant set
${\mathcal{D}}_{F}$ is non-singular at the origin if and only if the function
$f=F|{\mathbb{R}}^{k}\times\\{0\\}$ has the $A_{1}$-type singularity (i.e.,
the Morse-type singularity). Therefore we have the following proposition:

###### Proposition 5.3

With the same notations as in the previous proposition, the lightlike
hypersurface $\mathbb{LH}_{M}$ is non-singular at
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LH}_{M}((p_{0},\mbox{\boldmath$\xi$}_{0}),\mu_{0})$
if and only if $M=\mbox{\boldmath$X$}(U)$ and
$TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}$ have the
$A_{1}$-type contact at $p_{0}=\mbox{\boldmath$X$}(u_{0}).$

We now consider the general contact of $M=\mbox{\boldmath$X$}(U)$ with AdS-
lightcones as an application of the theory of contact for submanifolds in
Montaldi[27]. Let $X_{i}$ and $Y_{i},$ $i=1,2,$ be submanifolds of
${\mathbb{R}}^{n}$ with ${\rm dim}\,X_{1}={\rm dim}\,X_{2}$ and ${\rm
dim}\,Y_{1}={\rm dim}\,Y_{2}.$ We say that the contact of $X_{1}$ and $Y_{1}$
at $y_{1}$ is same type as the contact of $X_{2}$ and $Y_{2}$ at $y_{2}$ if
there is a diffeomorphism germ
$\Phi:({\mathbb{R}}^{n},y_{1})\longrightarrow({\mathbb{R}}^{n},y_{2})$ such
that $\Phi(X_{1})=X_{2}$ and $\Phi(Y_{1})=Y_{2}.$ In this case we write
$K(X_{1},Y_{1};y_{1})=K(X_{2},Y_{2};y_{2}).$ Since this definition of contact
is local, we can replace ${\mathbb{R}}^{n}$ by arbitrary $n$-manifold.
Montaldi gives in [27] the following characterization of contact by using
${\mathcal{K}}$-equivalence.

###### Theorem 5.4

Let $X_{i}$ and $Y_{i},$ $i=1,2,$ be submanifolds of ${\mathbb{R}}^{n}$ with
${\rm dim}\,X_{1}={\rm dim}\,X_{2}$ and ${\rm dim}\,Y_{1}={\rm dim}\,Y_{2}.$
Let $g_{i}:(X_{i},x_{i})\longrightarrow({\mathbb{R}}^{n},y_{i})$ be immersion
germs and $f_{i}:({\mathbb{R}}^{n},y_{i})\longrightarrow({\mathbb{R}}^{p},0)$
be submersion germs with $(Y_{i},y_{i})=(f_{i}^{-1}(0),y_{i}).$ Then

$K(X_{1},Y_{1};y_{1})=K(X_{2},Y_{2};y_{2})$

if and only if $f_{1}\circ g_{1}$ and $f_{2}\circ g_{2}$ are
${\mathcal{K}}$-equivalent.

On the other hand, we now return to the review on the theory of Legendrian
singularities. We introduce a natural equivalence relation among Legendrian
submanifold germs. Let
$F,G:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}},0)$
be Morse families of hypersurfaces. Then we say that
$\mathscr{L}_{F}(\Sigma_{*}(F))$ and $\mathscr{L}_{G}(\Sigma_{*}(G))$ are
Legendrian equivalent if there exists a contact diffeomorphism germ
$H:(PT^{*}{\mathbb{R}}^{n+1},z)\longrightarrow(PT^{*}{\mathbb{R}}^{n+1},z^{\prime})$
such that $H$ preserves fibers of $\pi$ and that
$H(\mathscr{L}_{F}(\Sigma_{*}(F)))=\mathscr{L}_{G}(\Sigma_{*}(G))$, where
$z=\mathscr{L}_{F}(0),z^{\prime}=\mathscr{L}_{G}(0).$ By using the Legendrian
equivalence, we can define the notion of Legendrian stability for Legendrian
submanifold germs by the ordinary way (see, [1, Part III]). We can interpret
the Legendrian equivalence by using the notion of generating families. We
denote ${\cal E}_{k}$ as the local ring of function germs
$({\mathbb{R}}^{k},\mbox{\boldmath$0$})\longrightarrow{\mathbb{R}}$ with the
unique maximal ideal $\mathfrak{M}_{k}=\\{h\in{\cal E}_{k}\ |\ h(0)=0\ \\}.$
Let
$F,G:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}},\mbox{\boldmath$0$})$
be function germs. We say that $F$ and $G$ are $P$-${\cal K}$-equivalent if
there exists a diffeomorphism germ
$\Psi:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})$
of the form $\Psi(x,u)=(\psi_{1}(q,x),\psi_{2}(x))$ for
$(q,x)\in({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})$ such
that $\Psi^{*}(\langle F\rangle_{{\cal E}_{k+n+1}})=\langle G\rangle_{{\cal
E}_{k+n+1}}$. Here $\Psi^{*}:{\cal E}_{k+n+1}\longrightarrow{\cal E}_{k+n+1}$
is the pull back ${\mathbb{R}}$-algebra isomorphism defined by
$\Psi^{*}(h)=h\circ\Psi$. We say that $F$ is an infinitesimally ${\cal
K}$-versal deformation of
$f=F|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\}$ if

${\cal E}_{k}=T_{e}({\cal K})(f)+\left\langle\frac{\partial F}{\partial
x_{1}}|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\},\dots,\frac{\partial
F}{\partial
x_{n+1}}|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\}\right\rangle_{\mathbb{R}},$

where

$T_{e}({\cal K})(f)=\left\langle\frac{\partial f}{\partial
q_{1}},\dots,\frac{\partial f}{\partial q_{k}},f\right\rangle_{{\cal E}_{k}}$

(see, [23]). The main result in the theory of Legendrian singularities ([1],
§20.8 and [34], THEOREM 2) is the following:

###### Theorem 5.5

Let
$F,G:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}},0)$
be Morse families of hypersurfaces. Then we have the following assertions:

(1) $\mathscr{L}_{F}(\Sigma_{*}(F))$ and $\mathscr{L}_{G}(\Sigma_{*}(G))$ are
Legendrian equivalent if and only if $F$ and $G$ are $P$-${\cal
K}$-equivalent,

(2) $\mathscr{L}_{F}(\Sigma_{*}(F))$ is Legendrian stable if and only if $F$
is an infinitesimally ${\cal K}$-versal deformation of
$f=F|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\}.$

Since $F$ and $G$ are function germs on the common space germ
$({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$}),$ we do not
need the notion of stably $P$-${\cal K}$-equivalences in this situation [34,
page 27]. For any map germ
$f:({\mathbb{R}}^{k},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}}^{p},\mbox{\boldmath$0$}),$
we define the local ring of $f$ by $Q_{r}(f)={\cal
E}_{k}/(f^{*}(\mathfrak{M}_{p}){\cal E}_{n}+\mathfrak{M}_{k}^{r+1}).$ We have
the following classification result of Legendrian stable germs (cf. [7,
Proposition A.4]) which is the key for the purpose in this section.

###### Proposition 5.6

Let
$F,G:({\mathbb{R}}^{k}\times{\mathbb{R}}^{n+1},\mbox{\boldmath$0$})\longrightarrow({\mathbb{R}},0)$
be Morse families of hypersurfaces and
$f=F|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\},g=G|{\mathbb{R}}^{k}\times\\{\mbox{\boldmath$0$}\\}$.
Suppose that $\mathscr{L}_{F}(\Sigma_{*}(F))$ and
$\mathscr{L}_{G}(\Sigma_{*}(G))$ are Legendrian stable. The the following
conditions are equivalent:

(1) $(W(\mathscr{L}_{F}),\mbox{\boldmath$0$})$ and
$(W(\mathscr{L}_{G}),\mbox{\boldmath$0$})$ are diffeomorphic as set germs,

(2) $\mathscr{L}_{F}(\Sigma_{*}(F))$ and $\mathscr{L}_{G}(\Sigma_{*}(G))$ are
Legendrian equivalent,

(3) $Q_{n+2}(f)$ and $Q_{n+2}(g)$ are isomorphic as ${\mathbb{R}}$-algebras.

We now describe the contacts of spacelike submanifolds in $AdS^{n+1}$ with
AdS-lightcones. We denote $Q(\mbox{\boldmath$X$},u_{0})$ as the local ring of
the function germ
$\widetilde{h}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}:(U,u_{0})\longrightarrow{\mathbb{R}},$
where
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LC}_{M}(u_{0},\mbox{\boldmath$\xi$}_{0},\mu_{0}).$
We remark that we can explicitly write the local ring as follows:

$Q_{n+2}(\mbox{\boldmath$X$},u_{0})=\frac{C^{\infty}_{u_{0}}(U)}{\displaystyle{\langle\langle\mbox{\boldmath$X$}(u),\mbox{\boldmath$\lambda$}_{0}\rangle+1\rangle_{C^{\infty}_{u_{0}}(U)}}+\mathfrak{M}_{u_{0}}(U)^{n+2}},$

where $C^{\infty}_{u_{0}}(U)$ is the local ring of function germs on $U$ at
$u_{0}.$

Let
$\mathbb{LH}_{M_{i}}(\mbox{\boldmath$n$}^{T}_{i}):(N_{1}(M_{i})[\mbox{\boldmath$n$}^{T}_{i}]\times{\mathbb{R}},(p_{i},\mbox{\boldmath$\xi$}_{i},\mu_{i}))\longrightarrow(AdS^{n+1},\mbox{\boldmath$\lambda$}_{i}),$
$(i=1,2)$ be two lightlike hypersurface germs of spacelike submanifold germs
$\mbox{\boldmath$X$}_{i}:(U,u^{i})\longrightarrow(AdS^{n+1},p_{i}).$ Let
$H_{i}:(U\times
AdS^{n+1},(u^{i},\mbox{\boldmath$\lambda$}_{i}))\longrightarrow{\mathbb{R}}$
be the AdS-height function germ of $\mbox{\boldmath$X$}_{i}.$ Then we have the
following theorem:

###### Theorem 5.7

Let $\mbox{\boldmath$X$}_{i}:(U,u^{i})\longrightarrow(AdS^{n+1},p_{i}),$
$i=1,2,$ be spacelike submanifold germs such that the corresponding Legendrian
submanifold germs $\mathscr{L}_{H_{i}}(\Sigma_{*}(H_{i}))$ are Legendrian
stable. We denote that $\mbox{\boldmath$X$}_{i}(U)=M_{i}.$ Then the following
conditions are equivalent$:$

(1)
$(\mathbb{LH}_{M_{1}}(N_{1}(M_{1})[\mbox{\boldmath$n$}^{T}_{1}]\times{\mathbb{R}}),\mbox{\boldmath$\lambda$}_{1})$
and
$(\mathbb{LH}_{M_{2}}(N_{1}(M_{2})[\mbox{\boldmath$n$}^{T}_{2}]\times{\mathbb{R}}),\mbox{\boldmath$\lambda$}_{2})$
are diffeomorphic,

(2) $(\mathscr{L}_{H_{1}}(\Sigma_{*}(H_{1})),(u^{1},\lambda_{1})$ and
$(\mathscr{L}_{H_{2}}(\Sigma_{*}(H_{2})),(u^{2},\lambda_{2})$ are Legendrian
equivalent,

(3) $H_{1}$ and $H_{2}$ are $P$-${\mathcal{K}}$-equivalent,

(4) $h_{1,\lambda_{1}}$ and $h_{2,\lambda_{2}}$ are ${\cal K}$-equivalent,

(5)
$K(M_{1},TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{1}}(M_{1})_{p_{1}},p_{1})=K(M_{2},TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{2}}(M_{2})_{p_{2}},p_{2}).$

(6) $Q_{n+1}(\mbox{\boldmath$X$}_{1},u^{1})$ and
$Q_{n+1}(\mbox{\boldmath$X$}_{2},u^{2})$ are isomorphic as
${\mathbb{R}}$-algebras.

Proof.  By Proposition 5.6, the conditions (1), (2) and (6) are equivalent.
This condition is also equivalent to that two generating families $H_{1}$ and
$H_{2}$ are $P$-${\cal K}$-equivalent by Theorem 5.3. If we denote
$h_{i,\mbox{\scriptsize\boldmath$\lambda$}_{i}}(u)=H_{i}(u,\mbox{\boldmath$\lambda$}_{i}),$
then we have
$h_{i,\mbox{\scriptsize\boldmath$\lambda$}_{i}}(u)=\mathfrak{h}_{\mbox{\scriptsize\boldmath$\lambda$}_{i}}\circ\mbox{\boldmath$X$}_{i}(u).$
By Theorem 5.2,
$K(\mbox{\boldmath$X$}_{1}(U),LC_{\mbox{\scriptsize\boldmath$\lambda$}_{1}},p_{1})=K(\mbox{\boldmath$x$}_{2}(U),LC{\lambda_{2}},p_{2})$
if and only if $\widetilde{h}_{1,\mbox{\scriptsize\boldmath$\lambda$}_{1}}$
and $\widetilde{h}_{2,\mbox{\scriptsize\boldmath$\lambda$}_{2}}$ are
${\mathcal{K}}$-equivalent. This means that (4) and (5) are equivalent. By
definition, (3) implies (4). The uniqueness of the infinitesimally
$\mathcal{K}$-versal deformation of
$h_{i,\mbox{\scriptsize\boldmath$\lambda$}_{i}}$ (cf., [23]) leads that (4)
implies (3). This completes the proof. $\Box$

For a spacelike embedding germ
$\mbox{\boldmath$X$}:(U,u_{0})\longrightarrow(AdS^{n+1},p_{0})$, we consider a
set germ
$(\mbox{\boldmath$X$}^{-1}(TLC_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}),u_{0}),$
which is called the AdS-tangent lightcone indicatrix germ of
$\mbox{\boldmath$X$},$ where
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LH}_{M}(p_{0},\mbox{\boldmath$\xi$}_{0},\mu_{0})$
and
$\mu_{0}=-1/\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p_{0},\mbox{\boldmath$\xi$}_{0})(i=1,\dots
s).$ We have the following corollary of Theorem 5.7.

###### Corollary 5.8

With the assumptions of Theorem 5.7, if the lightlike hypersurface germs

$(\mathbb{LH}_{M_{1}}(N_{1}(M_{1})[\mbox{\boldmath$n$}^{T}_{1}]\times{\mathbb{R}}),\mbox{\boldmath$\lambda$}_{1})\
and\
(\mathbb{LH}_{M_{2}}(N_{1}(M_{2})[\mbox{\boldmath$n$}^{T}_{2}]\times{\mathbb{R}}),\mbox{\boldmath$\lambda$}_{2})$

are diffeomorphic as set germs, then AdS-tangent lightcone indicatrix germs

$(\mbox{\boldmath$X$}_{1}^{-1}(TLC_{\lambda_{1}}(M_{1}))_{p_{1}},u^{1})\ and\
(\mbox{\boldmath$X$}_{2}^{-1}(TLC_{\lambda_{2}}(M_{2})_{p_{2}}),u^{2})$

are diffeomorphic as set germs.

Proof.  We remark that the tangent lightcone indicatrix germ of
$\mbox{\boldmath$X$}_{i}$ is the zero level set of $h_{i,\lambda_{i}}.$ Since
${\mathcal{K}}$-equivalence among function germs preserves the zero-level sets
of function germs, the assertion follows from Theorem 5.7. $\Box$

On the other hand, we consider generic properties of lightlike hypersurfaces
along spacelike submanifolds. Let ${\rm Emb}_{\rm sp}\,(U,AdS^{n+1})$ be the
space of spacelike embeddings with the $C^{\infty}$-topology for an open set
$U\subset{\mathbb{R}}^{s}.$ We consider the function
${\mathcal{H}}:AdS^{n+1}\times AdS^{n+1}\longrightarrow{\mathbb{R}}$ again. We
claim that $\mathfrak{h}_{\mbox{\scriptsize\boldmath$\lambda$}}$ is a
submersion at $\mbox{\boldmath$x$}\not=\mbox{\boldmath$\lambda$}$ for any
$\mbox{\boldmath$\lambda$}\in AdS^{n+1}.$ For any $\mbox{\boldmath$X$}\in{\rm
Emb}_{\rm sp}\,(U,AdS^{n+1})$, we have
$H=\mathcal{H}\circ(\mbox{\boldmath$X$}\times 1_{AdS^{n+1}}).$ We have the
$r$-jet extension $j^{r}_{1}H:U\times AdS^{n+1}\longrightarrow
J^{r}(U,{\mathbb{R}})$ defined by
$j^{r}_{1}H(u,\mbox{\boldmath$\lambda$})=j^{r}h_{\mbox{\scriptsize\boldmath$\lambda$}}(u),$
where $J^{k}(U,{\mathbb{R}})$ is the $k$-jet space of functions on $U.$ We
consider the trivialization $J^{r}(U,{\mathbb{R}})\equiv
U\times{\mathbb{R}}\times J^{r}(s,1).$ For any submanifold $Q\subset
J^{r}(s,1),$ we denote that $\widetilde{Q}=U\times{\mathbb{R}}\times Q.$ As an
application of [32, Lemma 6], the set

$T_{Q}=\\{\mbox{\boldmath$X$}\in{\rm Emb}_{\rm sp}\,(U,AdS^{n+1})\ |\
j^{r}_{1}G\ \mbox{is transversal to}\ \widetilde{Q}\ \\}$

is a residual set of ${\rm Emb}_{\rm sp}\,(U,AdS^{n+1}).$ Moreover, if $Q$ is
a closet subset , then $T_{Q}$ is open. It is known [6] that there exists a
semi-algebraic set $W^{r}(s,1)\subset J^{k}(s,1)$ and a stratification
$\mathcal{A}^{r}(s,1)$ of $J^{k}(s,1)\setminus W^{r}(s,1)$ such that
$\lim_{k\mapsto\infty}{\rm cod}\,W^{r}(s,1)=+\infty.$ The stratification
$\mathcal{A}^{r}(s,1)$ is called the canonical stratification. We define a
stratification $\mathcal{A}^{r}(U,{\mathbb{R}})$ of
$J^{r}(U,{\mathbb{R}})\setminus W^{r}(U,{\mathbb{R}})$ by

$U\times({\mathbb{R}}\setminus\\{0\\})\times(J^{r}(s,1)\setminus W^{r}(s,1)),\
U\times\\{0\\}\times\mathcal{A}^{r}(s,1),$

where $W^{r}(U,{\mathbb{R}})=U\times{\mathbb{R}}\times W^{r}(s,1).$ In [31],
it was shown that if $j^{r}_{1}H(U\times AdS^{n+1})\cap
W^{r}(U,{\mathbb{R}})=\emptyset$ and $j^{r}_{1}H$ is transversal to
$\mathcal{A}^{r}(U,{\mathbb{R}})$, then the map
$\pi|H^{-1}(0):H^{-1}(0)\longrightarrow AdS^{n+1}$ is MT-stable map-germ at
each point, where $\pi:U\times{\mathbb{R}}^{n+1}_{1}\longrightarrow AdS^{n+1}$
is the canonical projection. Here, a map germ is said to be MT-stable if the
jet extension is transversal to the canonical stratification of the jet space
of sufficiently higher order (cf., [6, 25]). The main result of the theory of
Topological stability of Mather is that MT-stability implies topological
stability. By Proposition 4.1, the lightlike hypersurface
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
is the discriminant set of $H$, which is equal to the critical value set of
$\pi|H^{-1}(0).$ Since ${\rm cod}\,W^{r}(U,{\mathbb{R}})>s+n+1$ for
sufficiently large $k,$ the set

$\mathcal{O}_{1}=\\{\mbox{\boldmath$X$}\in{\rm Emb}_{\rm
sp}\,(U,AdS^{n+1}_{1})\ |\ j^{r}_{1}H(U\times AdS^{n+1})\cap
W^{r}(U,{\mathbb{R}})=\emptyset\ \\}$

is a residual set. It follows that the set

$\mathcal{O}=\\{\mbox{\boldmath$X$}\in\mathcal{O}_{1}\ |\ j^{r}_{1}H\ \mbox{is
transversal to}\ \mathcal{A}^{r}(U,{\mathbb{R}})\ \\}$

is a residual set. Therefore, we have the following theorem.

###### Theorem 5.9

There exists a residual set $\mathcal{O}\subset{\rm Emb}_{\rm
sp}\,(U,AdS^{n+1})$ such that for any $\mbox{\boldmath$X$}\in\mathcal{O}$, the
germ of the lightlike hypersurface
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
at any point is a germ of the critical value set of an MT-stable map germ.

In the case when $n\leq 5,$ by the classification results of the
$\mathcal{K}$-equivalence among function germs, the canonical stratification
$\mathcal{A}^{k}(s,1)$ is given by the finite collection of the
$\mathcal{K}$-orbits. Moreover, if $j^{r}_{1}H$ is transversal to the
$\mathcal{K}$-orbit of
$j^{r}h_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(u_{0})$ for sufficiently
large $r,$ then $H$ is an infinitesimally $\mathcal{K}$-versal deformation of
$h_{\mbox{\scriptsize\boldmath$\lambda$}}$ at
$(u_{0},\mbox{\boldmath$\lambda$}_{0})$ [23]. By Theorem 5.5, we have the
following theorem.

###### Theorem 5.10

Suppose that $n\leq 5.$ Then there exists a residual set
${\mathcal{O}}\subset{\rm Emb}_{\rm sp}\,(U,AdS^{n+1})$ such that for any
$\mbox{\boldmath$X$}\in{\mathcal{O}},$ the germ of the lightlike hypersurface
$\mathbb{LH}_{M}(\mbox{\boldmath$n$}^{T})(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})$
at any point is the germ of the wave front set of a stable Legendrian
submanifold germ $\mathscr{L}_{H}(\Sigma_{*}(H)).$

##  6 Spacelike submanifolds with codimension two

In the case when $s=n-1,$ $N_{1}(M)[\mbox{\boldmath$n$}^{T}]$ is a double
covering of $M=\mbox{\boldmath$X$}(U).$ We can construct a spacelike unit
normal section $\mbox{\boldmath$n$}^{S}(u)\in N_{p}(M)$ by

$\mbox{\boldmath$n$}^{S}(u)=\frac{\mbox{\boldmath$X$}(u)\wedge\mbox{\boldmath$n$}^{T}(u)\wedge\mbox{\boldmath$X$}_{u_{1}}(u)\wedge\cdots\wedge\mbox{\boldmath$X$}_{u_{n-1}}(u)}{\|\mbox{\boldmath$X$}(u)\wedge\mbox{\boldmath$n$}^{T}(u)\wedge\mbox{\boldmath$X$}_{u_{1}}(u)\wedge\cdots\wedge\mbox{\boldmath$X$}_{u_{n-1}}(u)\|}.$

Then
$\mbox{\boldmath$\sigma$}^{\pm}(u)=(\mbox{\boldmath$X$}(u),\pm\mbox{\boldmath$n$}^{S}(u))$
are sections of $N_{1}(M)[\mbox{\boldmath$n$}^{T}]$. We call
$(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ a adopted normal frame
along $M=\mbox{\boldmath$X$}(U).$ The vectors
$\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u)$ are null. Since
$\\{\mbox{\boldmath$X$}_{u_{1}}(u),\dots,\mbox{\boldmath$X$}_{u_{n-1}}(u)\\}$
is a basis of $T_{p}M,$ the system
$\\{\mbox{\boldmath$X$}(u),\mbox{\boldmath$n$}^{T}(u),\mbox{\boldmath$n$}^{S}(u),\mbox{\boldmath$X$}_{u_{1}}(u),\dots,\mbox{\boldmath$X$}_{u_{n-1}}(u)\\}$
provides a basis for $T_{p}{\mathbb{R}}^{n+2}_{2}$ such that
$\\{\mbox{\boldmath$n$}^{T}(u),\mbox{\boldmath$n$}^{S}(u)\\}$ is a pseudo-
orthonormal frame of the normal timelike plane $N_{p}(M)\cap T_{p}AdS^{n+1}$
in $AdS^{n+1}.$

###### Lemma 6.1

Given two adopted unit timelike normal sections
$\mbox{\boldmath$n$}^{T}(u),\bar{\mbox{\boldmath$n$}}^{T}(u)\in N_{p}(M),$ the
corresponding nullcone normal sections
$\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u),\bar{\mbox{\boldmath$n$}}^{T}(u)\pm\bar{\mbox{\boldmath$n$}}^{S}(u)$
are parallel.

Proof.  We consider the orientation and the timelike orientation on the normal
space $N_{p}(M)$ induced by the orientation and the timelike orientation of
${\mathbb{R}}^{n+1}_{1}$ and
$\\{\mbox{\boldmath$X$}_{u_{1}}(u),\dots,\mbox{\boldmath$X$}_{u_{n-1}}(u)\\}.$
By the construction, both the pseudo-orthogonal basis
$\\{\mbox{\boldmath$n$}^{T}(u),\mbox{\boldmath$n$}^{S}(u)\\}$ and
$\\{\bar{\mbox{\boldmath$n$}}^{T}(u),\bar{\mbox{\boldmath$n$}}^{S}(u)\\}$ of
$N_{p}(M)\cap T_{p}AdS^{n}$ correspond to the same orientation and the same
timelike orientation on $N_{p}(M)\cap T_{p}AdS^{n}.$ Since both of
$\mbox{\boldmath$n$}^{T}(u)$ and $\bar{\mbox{\boldmath$n$}}^{T}(u)$ are
adopted and
$\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u),\bar{\mbox{\boldmath$n$}}^{T}(u)\pm\bar{\mbox{\boldmath$n$}}^{S}(u)$
are null in the Lorentz plane $N_{p}(M)\cap T_{p}AdS^{n+1}$,
$\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u)$ and
$\bar{\mbox{\boldmath$n$}}^{T}(u)\pm\bar{\mbox{\boldmath$n$}}^{S}(u)$ are
parallel. This completes the proof. $\Box$

Therefore, the null cone Gauss images of $M=\mbox{\boldmath$X$}(U)$ with
respect to $(\mbox{\boldmath$n$}^{T},\mbox{\boldmath$n$}^{S})$ are given by
$\mathbb{NG}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)=\mbox{\boldmath$n$}^{T}(u)\pm\mbox{\boldmath$n$}^{S}(u).$
Since $N_{p}(M)[\mbox{\boldmath$n$}^{T}]$ is a spacelike line in $N_{p}(M),$
we have $\mbox{\boldmath$\xi$}=\mbox{\boldmath$n$}^{S}(u)$ or
$\mbox{\boldmath$\xi$}=-\mbox{\boldmath$n$}^{S}(u)$ for any
$(\mbox{\boldmath$X$}(u),\mbox{\boldmath$\xi$})\in
N_{1}(M)[\mbox{\boldmath$n$}^{T}].$ We denote that
$\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)$,
$i=1,\dots,n-1$ instead of
$\kappa_{N}(\mbox{\boldmath$n$}^{T})_{i}(p,\mbox{\boldmath$\xi$})$,
$i=1,\dots,n-1$ for $\mbox{\boldmath$\xi$}=\pm\mbox{\boldmath$n$}^{S}(u)$ and
$p=\mbox{\boldmath$X$}(u).$ Then we have the following decomposition of the
lightlike hypersurface along $M=\mbox{\boldmath$X$}(U)$:

$\mathbb{LH}_{M}(N_{1}(M)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})=\mathbb{LH}_{M}^{+}(U\times{\mathbb{R}})\cup\mathbb{LH}_{M}^{-}(U\times{\mathbb{R}}),$

where

$\mathbb{LH}_{M}^{\pm}(u,\mu)=\mbox{\boldmath$X$}(u)+\mu(\mbox{\boldmath$n$}^{T}\pm\mbox{\boldmath$n$}^{S})(u).$

In this case the critical value of $\mathbb{LH}_{M}^{\pm}$ is the point where
$\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(p)\not=0$ and

$\mbox{\boldmath$\lambda$}^{\pm}=\mbox{\boldmath$X$}(u)+\frac{1}{\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)}(\mbox{\boldmath$n$}^{T}\pm\mbox{\boldmath$n$}^{S})(u).$

For each $i=1,\dots,n-1,$ we have a mapping
$\mathbb{LE}_{\kappa_{i}(\mbox{\scriptsize\boldmath$n$}^{T},\pm\mbox{\scriptsize\boldmath$n$}^{S})}:O_{i}\longrightarrow
AdS^{n+1}$ defined by

$\mathbb{LE}_{\kappa_{i}(\mbox{\scriptsize\boldmath$n$}^{T},\pm\mbox{\scriptsize\boldmath$n$}^{S})}(u)=\mbox{\boldmath$X$}(u)+\frac{1}{\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)}(\mbox{\boldmath$n$}^{T}\pm\mbox{\boldmath$n$}^{S})(u),$

where $O_{i}=\\{u\in U\ |\
\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)\not=0\ \\}.$
Then we define

$\mathbb{LE}^{\pm}_{M}=\bigcup\left\\{\mathbb{LE}_{\kappa_{i}(\mbox{\scriptsize\boldmath$n$}^{T},\pm\mbox{\scriptsize\boldmath$n$}^{S})}(u)\
|\ u\in U\ \mbox{such\ that}\
\kappa_{i}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u)\not=0,i=1,\dots,n-1.\right\\}.$

By the above arguments, we know that $\mathbb{LE}^{\pm}_{M}$ is nothing but
the AdS-lightlike focal set of $M=\mbox{\boldmath$X$}(U)$ relative to
$\mbox{\boldmath$n$}^{T}.$ However, we call it the AdS-lightlike evolute of
$M=\mbox{\boldmath$X$}(U)$ in the case when ${\rm codim}\,M=2.$ For any
$p_{0}=\mbox{\boldmath$X$}(u_{0}),$ we have the tangent AdS-lightcones
$TLC^{\pm}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}},$ where
$\mbox{\boldmath$\lambda$}^{\pm}_{0}=\mathbb{LE}_{\kappa_{i}(\mbox{\scriptsize\boldmath$n$}^{T},\pm\mbox{\scriptsize\boldmath$n$}^{S})}(u_{0}).$
In the general codimension case, it depends on the choice of
$\mbox{\boldmath$n$}^{S}$, so that there are infinitely many tangent AdS-
lightcones of $M$ at $p_{0}=\mbox{\boldmath$X$}(u_{0})$. However, the
codimension two case, $\mbox{\boldmath$n$}^{S}$ is uniquely determined by
$\mbox{\boldmath$n$}^{T}$. Therefore, we have two tangent AdS-lightcones of
$M$ at $p_{0}=\mbox{\boldmath$X$}(u_{0}).$ In this case, each one of the
tangent AdS-lightcones
$TLC^{\pm}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(M)_{p_{0}}$ is called an
osculating AdS-lightcone of $M=\mbox{\boldmath$x$}(U)$. The AdS-lightlike
evolutes are the loci of the vertices of osculating AdS-lightcones of $M.$
Analogous to the case of hypersurfaces in Euclidean space, we call
$\mathbb{LE}^{\pm}_{M}$ the AdS-lightlike evolute of
$M=\mbox{\boldmath$X$}(U).$

We now consider the low dimensions cases in the following subsections.

###  6.1 Spacelike curves in $AdS^{3}$

We consider spacelike curves in $AdS^{3}$ as a simplest case of the
codimension two case. Let $\mbox{\boldmath$\gamma$}:I\longrightarrow AdS^{3}$
be a unit speed spacelike curve with
$\langle\mbox{\boldmath$\gamma$}^{\prime\prime}(s),\gamma^{\prime\prime}(s)\rangle\not=-1$,
where $I$ is an open interval. We denote that $C=\gamma(I).$ Then we define
$\mbox{\boldmath$t$}(s)=\mbox{\boldmath$\gamma$}^{\prime}(s)$ and call
$\mbox{\boldmath$t$}(s)$ a unit tangent vector  of $\gamma$ at $s.$ The
curvature  of $\gamma$ at $s$ is defined to be
$\kappa_{g}(s)=\sqrt{|\langle\mbox{\boldmath$\gamma$}(s)-\mbox{\boldmath$\gamma$}^{\prime\prime}(s),\mbox{\boldmath$\gamma$}(s)-\mbox{\boldmath$\gamma$}^{\prime\prime}(s)\rangle|}.$
Since $\kappa_{g}(s)\not=0,$ the unit principal normal vector
$\mbox{\boldmath$n$}(s)$ of the curve $\gamma$ at $s$ is defined by
$\mbox{\boldmath$\gamma$}^{\prime\prime}(s)-\mbox{\boldmath$\gamma$}(s)=\kappa_{g}(s)\mbox{\boldmath$n$}(s).$
We denote that $\delta{(\mbox{\boldmath$\gamma$}(s))}={\rm
sign}(\mbox{\boldmath$n$}(s))$. The unit vector
$\mbox{\boldmath$b$}(s)=\mbox{\boldmath$\gamma$}(s)\wedge\mbox{\boldmath$t$}(s)\wedge\mbox{\boldmath$n$}(s)$
is called a unit binormal vector of the curve $\gamma$ at $s.$ Since
$\mbox{\boldmath$\gamma$}(s)$ is timelike and $\mbox{\boldmath$t$}(s)$ is
spacelike, we have
$\langle\mbox{\boldmath$b$}(s),\mbox{\boldmath$b$}(s)\rangle=-\delta{(\mbox{\boldmath$\gamma$}(s))}$
and ${\rm sign}\,({\mbox{\boldmath$\gamma$}}^{\prime}(s))=1$ Then the
following Frenet-Serret type formulae hold:

$\left\\{\begin{array}[]{l}\mbox{\boldmath$\gamma$}^{\prime}(s)=\mbox{\boldmath$t$}(s)\\\
\mbox{\boldmath$t$}^{\prime}{(s)}=\kappa_{g}(s)\mbox{\boldmath$n$}(s)+\mbox{\boldmath$\gamma$}(s),\\\
\mbox{\boldmath$n$}^{\prime}{(s)}=\delta(\mbox{\boldmath$\gamma$}(s))(-\kappa_{g}(s)\mbox{\boldmath$t$}(s)+\tau_{g}{(s)}\mbox{\boldmath$b$}(s)),\\\
\mbox{\boldmath$b$}^{\prime}{(s)}=\delta(\mbox{\boldmath$\gamma$}(s))\tau_{g}{(s)}\mbox{\boldmath$n$}(s),\end{array}\right.$

where
$\tau_{g}{(s)}=\langle\mbox{\boldmath$b$}^{\prime}(s),\mbox{\boldmath$n$}(s)\rangle$
is the torsion of the curve $\gamma$ at $s$. In [5] the singularities of
lightlike hypersurfaces along spacelike curves in $AdS^{3}$ are classified. We
define an invariant
$\sigma^{\pm}(s)=\kappa_{g}^{\prime}(s)\mp\kappa_{g}(s)\tau_{g}(s).$ Then we
have the following theorem.

######  Theorem 6.2 ([5])

Let $\mbox{\boldmath$\gamma$}:I\longrightarrow AdS^{3}$ be a unit speed
spacelike curve with
$\langle\mbox{\boldmath$\gamma$}^{\prime\prime}(s),\gamma^{\prime\prime}(s)\rangle\not=-1$.
Then we have the following:

(1) The AdS-lightlike evolute of $\gamma$ is

$\mathbb{LE}_{C}=\left\\{\mbox{\boldmath$\lambda$}=\mbox{\boldmath$\gamma$}(s)+\frac{1}{\delta(\mbox{\boldmath$\gamma$}(s))\kappa_{g}(s)}(\mbox{\boldmath$n$}(s)\pm\mbox{\boldmath$b$}(s))\right\\}.$

(2) The germ of $\mathbb{LH}_{M}$ at
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LE}_{\kappa_{g}}(s_{0})$ is
diffeomorphic to the cuspidal edge $C(2,3)\times{\mathbb{R}}$ if and only if
$\sigma^{\pm}(s_{0})\not=0.$

(3) The germ of $\mathbb{LH}_{M}$ at
$\mbox{\boldmath$\lambda$}_{0}=\mathbb{LE}_{\kappa_{g}}(s_{0})$ is
diffeomorphic to the swallowtail $SW$ if and only if $\sigma^{\pm}(s_{0})=0$
and $(\sigma^{\pm})^{\prime}(s_{0})\not=0.$

Here, $C(2,3)\times{\mathbb{R}}=\\{(x_{1},x_{2})\
|x_{1}^{2}-x_{2}^{3}=0\\}\times{\mathbb{R}}$ is the cuspidal edge and
$SW=\\{(x_{1},x_{2},x_{3})\
|x_{1}=3u^{2}+u^{2}v,x_{2}=4u^{3}+2uv,x_{3}=v,(u,v)\in({\mathbb{R}}^{2},0)\\}$
is the swallowtail.

It has been also shown the following geometric characterizations of the
singularities of AdS-lightlike hypersurfaces in [5].

###### Proposition 6.3

Let $\mbox{\boldmath$\gamma$}:I\longrightarrow AdS^{3}$ be a unit speed
spacelike curve with
$\langle\mbox{\boldmath$\gamma$}^{\prime\prime}(s),\gamma^{\prime\prime}(s)\rangle\not=-1$.
Then we have the followings:

(1) $\kappa_{g}(s_{0})\not=0$ and $\sigma^{\pm}(s_{0})\not=0$ if and only if
$C=\mbox{\boldmath$\gamma$}(I)$ and the osculating AdS-lightcone
$TLC^{\pm}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(C)_{p_{0}}$ have contact
of order $2.$

(2) $\kappa_{g}(s_{0})\not=0$, $\sigma^{\pm}(s_{0})=0$ and
$(\sigma^{\pm})^{\prime}(s_{0})\not=0$ if and only if
$C=\mbox{\boldmath$\gamma$}(I)$ and the osculating AdS-lightcone
$TLC^{\pm}_{\mbox{\scriptsize\boldmath$\lambda$}_{0}}(C)_{p_{0}}$ have contact
of order $3.$

###  6.2 Spacelike surfaces in $AdS^{4}$

We consider spacelike surfaces in $AdS^{4}$ here. Let
$\mbox{\boldmath$X$}:U\longrightarrow AdS^{4}$ be a spacelike embedding from
an open subset $U\subset{\mathbb{R}}^{2}.$ As a corollary of Theorem 5.10, we
have the following generic classification theorem. We say that two map germs
$f,g:({\mathbb{R}}^{n},0)\longrightarrow({\mathbb{R}}^{p},0)$ are
$\mathcal{A}$-equivalent if there exists diffeomorphism germs
$\phi:({\mathbb{R}}^{n},0)\longrightarrow({\mathbb{R}}^{n},0)$ and
$\psi:({\mathbb{R}}^{p},0)\longrightarrow({\mathbb{R}}^{p},0)$ such that
$f\circ\phi=\psi\circ g.$

###### Theorem 6.4

There exists an open dense subset ${\cal O}\subset{\rm Emb}_{\rm
sp}\,(U,AdS^{4})$ such that for any $\mbox{\boldmath$X$}\in{\cal O},$ the germ
of the corresponding lightlike hypersurfaces $\mathbb{LH}^{\pm}_{M}$ at any
point $(u_{0},\mu_{0})\in U\times{\mathbb{R}}$ is ${\mathcal{A}}$-equivalent
to one of the map germs $A_{k}$ $(1\leq k\leq 4)$ or $D_{4}^{\pm}:$ where,
$A_{k},\ D^{\pm}_{4}$-map germ
$f:({\mathbb{R}}^{3},0)\longrightarrow({\mathbb{R}}^{4},0)$ are given by

$A_{1};\ f(u_{1},u_{2},u_{3})=(u_{1},u_{2},u_{3},0),$

$A_{2};\ f(u_{1},u_{2},u_{3})=(3u_{1}^{2},2u_{1}^{3},u_{2},u_{3}),$

$A_{3};\
f(u_{1},u_{2},u_{3})=(4u^{3}_{1}+2u_{1}u_{2},3u^{4}_{1}+u_{2}u_{1}^{2},u_{2},u_{3}),$

$A_{4};\
f(u_{1},u_{2},u_{3})=(5u_{1}^{4}+3u_{2}u_{1}^{2}+2u_{1}u_{3},4u_{1}^{5}+2u_{2}u_{1}^{3}+u_{3}u_{1}^{2},u_{2},u_{3}),$

$D^{+}_{4};\
f(u_{1},u_{2},u_{3})=(2(u_{1}^{3}+u_{2}^{3})+u_{1}u_{2}u_{3},3u_{1}^{2}+u_{2}u_{3},3u_{2}^{2}+u_{1}u_{3},u_{3}),$

$D^{-}_{4};\
\displaystyle{f(u_{1},u_{2},u_{3})=\left(\left(\frac{u_{1}^{3}}{3}-u_{1}u_{2}^{2}\right)+(u_{1}^{2}+u_{2}^{2})u_{3},u_{2}^{2}-u_{1}^{2}-2u_{1}u_{3},2(u_{1}u_{2}-u_{2}u_{3}),u_{3}\right).}$

Proof.  By Theorems 5.5 and 5.10, the AdS-height function $H$ on $M$ is a
${\mathcal{K}}$-versal deformation of $h_{\lambda_{0}}$ at each
$(u_{0},\mbox{\boldmath$\lambda$}_{0})\in U\times AdS^{4}.$ Therefore we can
apply the classification of ${\mathcal{K}}$-versal deformations
$F(x,y,\mbox{\boldmath$\lambda$})$ of function germs up to $4$-parameters [1].
For any $F(x,y,\mbox{\boldmath$\lambda$}),$ we have

$\Sigma_{*}(F)=\Bigl{\\{}(x,y,\mbox{\boldmath$\lambda$})\in({\mathbb{R}}^{2}\times{\mathbb{R}}^{4},0)\
\Bigm{|}\ F(x,y,\mbox{\boldmath$\lambda$})=\frac{\partial F}{\partial
x}(x,y,\mbox{\boldmath$\lambda$})=\frac{\partial F}{\partial
y}(x,y,\mbox{\boldmath$\lambda$})=0\Bigr{\\}}.$

The normal forms are given by

$\displaystyle A_{k};$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!F(x,y,\mbox{\boldmath$\lambda$})=x^{k+1}\pm
y^{2}+\lambda_{1}+\lambda_{2}x+\cdots+\lambda_{k-1}x^{k-1},\ 1\leq k\leq 4,$
$\displaystyle D^{+}_{4};$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!F(x,y,\mbox{\boldmath$\lambda$})=x^{3}+y^{3}+\lambda_{1}+\lambda_{2}x+\lambda_{3}y+\lambda_{4}xy,$
$\displaystyle D^{-}_{4};$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!F(x,y,\mbox{\boldmath$\lambda$})=\frac{x^{3}}{3}-xy^{2}+\lambda_{1}+\lambda_{2}x+\lambda_{3}y+\lambda_{4}(x^{2}+y^{2}).$

For example, if we consider the germ given by

$F(x,y,\mbox{\boldmath$\lambda$})=x^{3}+y^{3}+\lambda_{1}+\lambda_{2}x+\lambda_{3}y+\lambda_{4}xy.$

Then we get

$\Sigma_{*}(F)=\\{(x,y,2(x^{3}+y^{3})+\lambda_{4}xy,-3x^{2}-\lambda_{4}y,-3y^{2}-\lambda_{4}x,\lambda_{4})\
|\ (x,y,\lambda_{4})\in{\mathbb{R}}^{3}\\}.$

Therefore the corresponding Legendrian map germ is

$f(u_{1},u_{2},u_{3})=(2(u_{1}^{2}+u_{2}^{2})+u_{1}u_{2}u_{3},3u_{1}^{2}+u_{2}u_{3},3u_{2}^{2}+u_{1}u_{3},u_{3})\quad(D^{+}_{4}).$

The other cases follow from similar arguments, so that we omit the details.
$\Box$

As a corollary of the above theorem, we have the following generic local
classification of AdS-lightlike evolutes along spacelike surfaces. We define
that $C(2,3,4)=\\{(u_{1}^{2},u_{1}^{3},u_{1}^{4})\ |\ u_{1}\in{\mathbb{R}}\\}$
which is called a $(2,3,4)$-cusp. We also define that
$C(BF)=\\{(10u_{1}^{3}+3u_{2}u_{1},5u_{1}^{4}+u_{2}u_{1}^{2},6u_{1}^{5}+u_{2}u_{1}^{3},u_{2})\
|\ (u_{1},u_{2})\in{\mathbb{R}}^{2}\\}$. We call $C(BF)$ a C-butterfly (i.e.,
the critical value set of the butterfly). Finally we define that where
$C(2,3,4,5)=\\{(u_{1}^{2},u_{1}^{3},u_{1}^{4},u_{1}^{5})\ |\
u_{1}\in{\mathbb{R}}\\}$ which is called a $(2,3,4,5)$-cusp.

###### Corollary 6.5

There exists an open dense subset ${\cal O}\subset{\rm Emb}_{\rm
sp}\,(U,AdS^{4})$ such that for any $\mbox{\boldmath$X$}\in{\cal O},$ the germ
of the corresponding AdS-lightlike evolute $\mathbb{LE}^{\pm}_{M}$ at any
point $(u_{0},\mu_{0})\in U\times{\mathbb{R}}$ is diffeomorphic to one of the
following set germs at the origin in ${\mathbb{R}}^{4}$:

$A_{2};\ \\{(0,0)\\}\times{\mathbb{R}}^{2},$

$A_{3};\ C(2,3,4)\times{\mathbb{R}},$

$A_{4};\ C(BF),$

$D^{+}_{4};\
\\{(2(u_{1}^{3}+u_{2}^{3})+u_{1}u_{2}u_{3},3u_{1}^{2}+u_{2}u_{3},3u_{2}^{2}+u_{1}u_{3},u_{3})\
|\ u_{3}^{2}=36u_{1}u_{2}\\},$

$D^{-}_{4};\
\displaystyle{\left\\{\left(\left(\frac{u_{1}^{3}}{3}-u_{1}u_{2}^{2}\right)+(u_{1}^{2}+u_{2}^{2})u_{3},u_{2}^{2}-u_{1}^{2}-2u_{1}u_{3},2(u_{1}u_{2}-u_{2}u_{3}),u_{3}\right)\
\Bigl{|}\ u_{3}^{2}=u_{1}^{2}+u_{2}^{2}\right\\}.}$

Proof.  For $A_{3}$, we can calculate the Jacobi matrix of the normal form $f$
in Theorem 6.4:

$J_{f}=\left(\begin{array}[]{ccc}12u^{2}_{1}+2u_{1}&2u_{1}&0\\\
12u^{3}_{1}+2u_{1}u_{2}&u^{2}_{1}&0\\\ 0&1&0\\\ 0&0&1\end{array}\right),$

so that ${\rm rank}\,J_{f}<3$ if and only if $6u^{2}_{1}+u_{2}=0.$ Thus, the
critical value set of $f$ is

$C(f)=\\{(-8u^{3}_{1},-3u^{4}_{1},-6u^{2}_{1},u_{3})\ |\
(u_{1},u_{3})\in{\mathbb{R}}^{2}\\}.$

It is $C(2,3,4)\times{\mathbb{R}}.$ By the similar calculation, we can show
that the germ of $A_{4}$ is diffeomorphic to $C(BF).$ For $D^{+}_{4},$ we can
calculate the Jacobi matrix o the normal form $f$:

$J_{f}=\left(\begin{array}[]{ccc}6u^{2}_{1}+u_{2}u_{3}&6u_{2}^{2}+u_{1}u_{3},u_{1}u_{2}&0\\\
6u_{1}&u_{3}&u_{2}\\\ u_{3}&6u_{2}&u_{1}\\\ 0&0&1\end{array}\right).$

Therefore, ${\rm rank}\,J_{f}<3$ if and only if

$\left|\begin{array}[]{cc}6u^{2}_{1}+u_{2}u_{3}&6u_{2}^{2}+u_{1}u_{3},u_{1}u_{2}\\\
6u_{1}&u_{3}\\\
\end{array}\right|=\left|\begin{array}[]{cc}6u^{2}_{1}+u_{2}u_{3}&6u_{2}^{2}+u_{1}u_{3},u_{1}u_{2}\\\
u_{3}&6u_{2}\\\ \end{array}\right|=\left|\begin{array}[]{cc}6u_{1}&u_{3}\\\
u_{3}&6u_{2}\\\ \end{array}\right|=0,$

which is equivalent to the condition that $u_{3}^{2}=36u_{1}u_{2}.$ For
$D^{-}_{4},$ by the similar calculation to the above, we have the condition
that $u_{3}^{2}=u_{1}^{2}+u_{2}^{2}.$ This completes the proof. $\Box$

In the list of the above corollary, one of the projections of the image of
$D^{+}_{4}$ into ${\mathbb{R}}^{3}$ is
$PU=\\{(3u_{1}^{2}+u_{2}u_{3},3u_{2}^{2}+u_{1}u_{3},u_{3})\ |\
u_{3}^{2}=36u_{1}u_{2}\\}$ which is called a purse and one of the projection
of the image of $D^{-}_{4}$ is called a pyramid given by

$PY=\left\\{\left(u_{2}^{2}-u_{1}^{2}-2u_{1}u_{3},2(u_{1}u_{2}-u_{2}u_{3}),u_{3}\right)\
\Bigl{|}\ u_{3}^{2}=u_{1}^{2}+u_{2}^{2}\right\\}.$

We can draw these pictures as follows: We denote that $\widetilde{PU}$ as the
set of $D^{+}_{4}$ and $\widetilde{PY}$ as the set of $D^{-}_{4}$ in the list
of Corollary 6.5, respectively. We also have a classification of the
singularities of $\mathbb{LE}^{\pm}_{M}$ as a corollary of Theorem 6.4 and
Corollary 6.5. The set of singularities of $\mathbb{LE}^{\pm}_{M}$ is denoted
by $\Sigma(\mathbb{LE}^{\pm}_{M}).$

###### Corollary 6.6

There exists an open dense subset ${\cal O}\subset{\rm Emb}_{\rm
sp}\,(U,AdS^{4})$ such that for any $\mbox{\boldmath$X$}\in{\cal O},$ the germ
of the pair $(\mathbb{LE}^{\pm}_{M},\Sigma(\mathbb{LE}^{\pm}_{M}))$ at any
point $\mbox{\boldmath$\lambda$}_{0}\in\Sigma(\mathbb{LE}^{\pm}_{M})$ is
diffeomorphic to one of the following pairs of set germs at the origin in
${\mathbb{R}}^{4}$:

$A_{3};\ (C(2,3,4)\times{\mathbb{R}},\\{(0,0,0)\\}\times{\mathbb{R}}),$

$A_{4};\ (C(BF),C(2,3,4,5)),$

$D^{+}_{4};\ (\widetilde{PU},\Sigma(\widetilde{PU})),$ where
$\Sigma(\widetilde{PU})=\\{(5u^{3}_{3}/108,u^{2}_{3}/4,u^{2}_{3}/4,u_{3})\ |\
u_{3}\in{\mathbb{R}}\\},$

$D^{-}_{4};\ (\widetilde{PY},\Sigma(\widetilde{PY})),$ where

$\Sigma(\widetilde{PY}))=\\{\left(4u^{3}_{3}/3,-3u^{2}_{3},0,u_{3}\right)\ |\
u_{3}\in{\mathbb{R}}\\}\cup\\\
\hfill\\{(4u^{3}_{3}/3,3u^{2}_{3}/2,-3\sqrt{3}u^{2}_{3}/2,u_{3})\ |\
u_{3}\in{\mathbb{R}}\\}\cup\\{(4u^{3}_{3}/3,3u^{2}_{3}/2,3\sqrt{3}u^{2}_{3}/2,u_{3})\
|\ u_{3}\in{\mathbb{R}}\\}.$

Proof.  For $A_{3}$, in order to detect the singularities, we calculate the
Jacobi matrix of $f(u_{1},u_{2})=(u_{1}^{2},u_{1}^{3},u_{1}^{4},u_{2}).$ Then
we have

$J_{f}=\left(\begin{array}[]{cc}2u_{1}&0\\\ 3u_{1}^{2}&0\\\ 4u_{1}^{3}&0\\\
0&1\end{array}\right),$

so that ${\rm rank}\,J_{f}<2$ if and only if $u_{1}=0.$ This means that
$\Sigma(C(2,3,4)\times{\mathbb{R}})=\\{(0,0,0)\\}\times{\mathbb{R}}.$ Similar
to the case $A_{3},$ we calculate the Jacobi matrix of
$f(u_{1},u_{2})=(10u_{1}^{3}+3u_{2}u_{1},5u_{1}^{4}+u_{2}u_{1}^{2},6u_{1}^{5}+u_{2}u_{1}^{3},u_{2})$.
Then we have

$J_{f}=\left(\begin{array}[]{cc}30u_{1}^{2}+3u_{2}&3u_{1}\\\
20u_{1}^{3}+2u_{2}u_{2}&u_{1}^{2}\\\ 30u_{1}^{4}+3u_{2}u_{1}^{2}&u_{1}^{3}\\\
0&1\end{array}\right),$

so that ${\rm rank}\,J_{f}<2$ if and only if $u_{2}=-10u_{1}^{2}.$ Therefore,
we have

$f(u_{1},-10u_{1}^{2})=(-20u_{1}^{3},-5u_{1}^{4},-4u_{1}^{5},-10u_{1}^{2}).$

This means that $\Sigma(C(BF))=C(2,3,4,5).$ For $D^{+}_{4},$ we consider the
following parameter transformation:

$u_{1}+u_{2}=\frac{u_{3}}{3}\cosh\phi,\ u_{1}-u_{2}=\frac{u_{3}}{3}\sinh\phi.$

Then $\widetilde{PU}$ is parametrized by

$f(\phi,u_{3})=\left(\frac{4}{6^{3}}u_{3}^{3}\left(\cosh
3\phi+\frac{1}{4}\right),\frac{u_{3}^{2}}{6^{2}}\left(3e^{2\phi}+6e^{-\phi}\right),\frac{u_{3}^{2}}{6^{2}}\left(3e^{-2\phi}+e^{\phi}\right),u_{3}\right).$

Thus , the Jacobi matrix is

$J_{f}=\left(\begin{array}[]{cc}\displaystyle{\frac{1}{18}u_{3}^{3}\sinh
3\phi}&\displaystyle{\frac{1}{18}u_{3}^{2}(\cosh 3\phi+\frac{1}{4})}\\\
\displaystyle{\frac{u_{3}^{2}}{6}(e^{2\phi}-e^{-\phi})}&\displaystyle{\frac{1}{6}(e^{2\phi}+2e^{-\phi})}\\\
\displaystyle{\frac{u_{3}^{2}}{6}(-e^{-2\phi}+e^{\phi})}&\displaystyle{\frac{1}{6}(e^{-2\phi}+2e^{\phi})}\\\
0&1\end{array}\right),$

so that ${\rm rank}\,J_{f}<2$ if and only if $\phi=0$ or $u_{3}=0.$ This means
that $u_{1}=u_{3}/6,\ u_{2}=u_{3}/6.$ Therefore, we have

$\Sigma(\widetilde{PU})=\left\\{\left(\frac{5}{108}u^{3}_{3},\frac{1}{4}u^{2}_{3},\frac{1}{4}u^{2}_{3},u_{3}\right)\
\Bigl{|}\ u_{3}\in{\mathbb{R}}\right\\}.$

For $D^{-}_{4},$ we also consider the following parameter transformation:

$u_{1}=u_{3}\cos\theta,\ u_{2}=u_{3}\sin\theta.$

Then $\widetilde{PY}$ is parametrized by

$f(\theta,u_{3})=\left(\frac{1}{3}u_{3}^{3}\left(\cos
3\theta+1\right),-u_{3}^{2}(\cos 2\theta+2\cos\theta),2u_{3}^{2}(\sin
2\theta-2\sin\theta),u_{3}\right).$

Thus , the Jacobi matrix is

$J_{f}=\left(\begin{array}[]{cc}-u_{3}^{3}\sin 3\theta&u_{3}^{2}(\cos
3\theta+1)\\\ 2u_{3}^{2}(\sin 2\theta+\sin\theta)&-2u_{3}(\cos
2\theta+2\cos\theta)\\\ 2u_{3}^{2}(\cos 2\theta-\cos\theta)&2u_{3}(\sin
2\theta-2\sin\theta)\\\ 0&1\end{array}\right),$

so that ${\rm rank}\,J_{f}<2$ if and only if $\theta=0,\ 2\pi/3,5\pi/3$ or
$u_{3}=0.$ If $\theta=0,$ then we have $u_{1}=u_{3},\ u_{2}=0,$ so that we
have
$\\{\left(4u^{3}_{3}/3,-3u^{2}_{3},0,u_{3}\right)|u_{3}\in{\mathbb{R}}\\}.$ If
$\theta=2\pi/3,$ then we have $u_{1}=-u_{3}/2,\ u_{2}=\sqrt{3}u_{3}/2,$ so
that we have
$\\{(4u^{3}_{3}/3,3u^{2}_{3}/2,-3\sqrt{3}u^{2}_{3}/2,u_{3}|u_{3}\in{\mathbb{R}}\\}.$
Finally, if $\theta=5\pi/3,$ then we have $u=-u_{3}/2,\
u_{2}=-\sqrt{3}u_{3}/2,$ so that we have
$\\{(4u^{3}_{3}/3,3u^{2}_{3}/2,3\sqrt{3}u^{2}_{3}/2,u_{3})|u_{3}\in{\mathbb{R}}\\}.$
It follows that $\Sigma(\widetilde{PY}))$ is the union of these three set
germs. This completes the proof. $\Box$

We can interpret the geometric meaning of the above normal forms of the
singularities of lightlike hypersurfaces as the following theorem shows.

###### Theorem 6.7

Let $\mathcal{O}\subset{\rm Emb}_{\rm sp}\,(U,AdS^{4})$ be the open dense
subset given in the above theorem. For any $\mbox{\boldmath$X$}\in{\cal O},$
the germ of the corresponding lightlike hypersurfaces
$(\mathbb{LH}^{\pm}_{M},\mbox{\boldmath$\lambda$}_{0})$ at a point
$(u_{0},\mu_{0})\in U\times{\mathbb{R}}$ is characterized as follows:

(1) $(A_{1})$ if and only if
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LH}^{+}_{M}(U\times{\mathbb{R}})\cup\mathbb{LH}^{-}_{M}(U\times{\mathbb{R}})$
is a non-singular point.

(2) $(A_{2})$ if and only if $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is not a AdS-
lightlike ridge point and
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LE}^{+}_{M}\cup\mathbb{LE}^{-}_{M}.$

(3) $(A_{3})$ if and only if $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an AdS-
lightlike $1$-ridge point and
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LE}^{+}_{M}\cup\mathbb{LE}^{-}_{M}.$

(4) $(A_{4})$ if and only if $p_{0}=\mbox{\boldmath$X$}(u_{0})$ is an AdS-
lightlike $2$-ridge point,
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LE}^{+}_{M}\cup\mathbb{LE}^{-}_{M}$
and $u_{0}\in U$ satisfies the following condition: For sufficiently small
$\varepsilon>0$, there exist two different AdS-lightlike $1$-ridge points
$u^{1},u^{2}\in U$ such that $|u_{0}-u^{i}|<\varepsilon$, $(i=1,2)$.

(5) $(D_{4}^{+})$ if and only if
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LE}^{+}_{M}\cup\mathbb{LE}^{-}_{M}$,
$p_{0}=\mbox{\boldmath$X$}(u_{0})$ has corank two contact with the osculating
lightcone and $u_{0}\in U$ satisfies the following condition:

(a)
$\kappa_{1}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})=\kappa_{2}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})$.

(b) For sufficiently small $\varepsilon>0$, there exist two different AdS-
lightlike $1$-ridge points $u^{1},u^{2}\in U$ such that
$|u_{0}-u^{i}|<\varepsilon$, $(i=1,2)$ and
$\kappa_{1}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{i})\not=\kappa_{2}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{i})$.

(6) $(D_{4}^{-})$ if and only if
$\mbox{\boldmath$\lambda$}_{0}\in\mathbb{LE}_{M}$,
$p_{0}=\mbox{\boldmath$X$}(u_{0})$ has corank two contact with the the
osculating lightcone and $u_{0}\in U$ satisfies the following condition:

(a)
$\kappa_{1}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})=\kappa_{2}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})$.

(b) For sufficiently small $\varepsilon>0$, there exist three different AdS-
lightlike $1$-ridge points $u^{i}\in U$, $(i=1,2,3)$ such that
$|u_{0}-u^{i}|<\varepsilon$ and
$\kappa_{1}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{i})\not=\kappa_{2}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{i})$.

Proof.  By the normal form $(A_{1})$ and $(A_{2})$ in Theorem 6.4, the
assertions (1) and (2) are trivial. For the normal form $(A_{3})$, the AdS-
height function has the $A_{3}$ singularity at
$p_{0}=\mbox{\boldmath$X$}(u_{0})$, so that it is an AdS-lightlike $1$-ridge
point.

Here, we give a remark on the classification of $\mathcal{K}$-simple
singularities of function germs. In the list of the classification, we say
that a class of singularities $L$ is adjacent to a class $K$ (notation:
$K\leftarrow L$) if every function germ $f\in L$ can be deformed into a
function of $K$ by an arbitrarily small perturbation. For the class of
$A_{k},D^{\pm}_{k}$ of the $\mathcal{K}$-classification are adjacent to each
other as follows [1, Page 243]:

$\begin{array}[]{cccccccc}A_{1}&\leftarrow&A_{2}&\leftarrow&A_{3}&\leftarrow&A_{4}&\leftarrow\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&\uparrow&{}\hfil&\uparrow&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&D^{\pm}_{4}&\leftarrow&D^{\pm}_{5}&\leftarrow\\\
\end{array}$

By the normal form $(A_{4})$, the singularities of the AdS-lightcone evolute
is a $(2,3,4,5)$-cusp. Therefore, two singular loci approach to the
$(2,3,4,5)$-cusp point. Since $A_{4}$ is adjacent to $A_{3}$, such the
singular loci consist of AdS-lightlike $1$-ridge points except the origin.
Thus, the assertion (4) holds. For $(D^{+}_{4})$, the singularities of the
AdS-lightcone evolute is $\Sigma(\widetilde{PU})$. By the normal form of the
generating family, the corank of the AdS-height function at $u^{0}\in U$ is
two, so that it is an
$(\mbox{\boldmath$n$}^{T}(u_{0}),\pm\mbox{\boldmath$n$}^{S}(u_{0}))$-umbilical
point. Therefore, we have
$\kappa_{1}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})=\kappa_{2}(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})(u^{0})$.
By the normal form of $(D^{+}_{4})$ in Corollary 6.6, two singular loci
approach to the origin from both the positive and the negative side of the
parameter $u_{3}$. Since $D^{+}_{4}$ is adjacent to $A_{3}$, such the singular
loci consist of AdS-lightlike $1$-ridge points except at the origin. Of
course, an AdS-lightlike $1$-ridge point is not an
$(\mbox{\boldmath$n$}^{T},\pm\mbox{\boldmath$n$}^{S})$-umbilical point. So
that two nullcone principal curvatures are different at an AdS-lightlike
$1$-ridge point. For $(D^{-}_{4})$, we have the assertion by the similar
arguments to the case $(D^{+}_{4}).$ This completes the proof. $\Box$

##  7 Spacelike curves in $AdS^{4}$

In this section we consider lightlike hypersurfaces along spacelike curves in
$AdS^{4}$ as the simplest case of higher codimension. Let
$\mbox{\boldmath$\gamma$}:I\longrightarrow AdS^{4}$ be a spacelike curve with
$\|\mbox{\boldmath$\gamma$}^{\prime\prime}(s)\|\neq-1.$ In this case we write
$C=\mbox{\boldmath$\gamma$}(I)$ instead of $M=\mbox{\boldmath$\gamma$}(I).$
Since $\|\mbox{\boldmath$\gamma$}^{\prime}(s)\|>0,$ we can reparameterize it
by the arc-length s. So we have the unit tangent vector
$\mbox{\boldmath$t$}(s)=\mbox{\boldmath$\gamma$}^{\prime}(s)$ of
$\mbox{\boldmath$\gamma$}(s).$ Moreover we have two unit normal vectors
$\displaystyle{\mbox{\boldmath$n$}_{1}(s)=\frac{\mbox{\boldmath$\gamma$}^{\prime\prime}(s)-\mbox{\boldmath$\gamma$}(s)}{\|\mbox{\boldmath$\gamma$}^{\prime\prime}(s)-\mbox{\boldmath$\gamma$}(s)\|}},$
$\displaystyle{\mbox{\boldmath$n$}_{2}(s)=\frac{\mbox{\boldmath$n$}^{\prime}_{1}(s)+\delta\kappa_{1}(s)\mbox{\boldmath$t$}(s)}{\|\mbox{\boldmath$n$}^{\prime}_{1}(s)+\delta\kappa_{1}(s)\mbox{\boldmath$t$}(s)\|}}$
under the conditions that
$\kappa_{1}(s)=\|\mbox{\boldmath$\gamma$}^{\prime\prime}(s)-\mbox{\boldmath$\gamma$}(s)\|\neq
0,$ $\kappa_{2}(s)=\|\mbox{\boldmath$n$}^{\prime}_{1}(s)+\delta
k_{1}(s)\mbox{\boldmath$t$}(s)\|\neq 0,$ where $\delta_{i}={\rm
sign}(\mbox{\boldmath$n$}_{i}(s))$ and ${\rm
sign}(\mbox{\boldmath$n$}_{i}(s))$ is the signature of
$\mbox{\boldmath$n$}_{i}(s)$ $(i=1,2,3).$ Then we have another unit normal
vector field $\mbox{\boldmath$n$}_{3}(s)$ defined by
$\mbox{\boldmath$n$}_{3}(s)=\mbox{\boldmath$\gamma$}(s)\wedge\mbox{\boldmath$t$}(s)\wedge\mbox{\boldmath$n$}_{1}(s)\wedge\mbox{\boldmath$n$}_{2}(s).$
Therefore we can construct a pseudo-orthogonal frame
$\\{\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$t$}(s),\mbox{\boldmath$n$}_{1}(s),\mbox{\boldmath$n$}_{2}(s),\mbox{\boldmath$n$}_{3}(s)\\},$
which satisfies the Frenet-Serret type formulae:

$\left\\{\begin{array}[]{ll}\mbox{\boldmath$\gamma$}^{\prime}(s)=\mbox{\boldmath$t$}(s),\\\
\mbox{\boldmath$t$}^{\prime}(s)=\mbox{\boldmath$\gamma$}(s)+\kappa_{1}(s)\mbox{\boldmath$n$}_{1}(s),\\\
\mbox{\boldmath$n$}_{1}^{\prime}(s)=-\delta_{1}\kappa_{1}(s)\mbox{\boldmath$t$}(s)+\kappa_{2}(s)\mbox{\boldmath$n$}_{2}(s),\\\
\mbox{\boldmath$n$}_{2}^{\prime}(s)=\delta_{3}\kappa_{2}(s)\mbox{\boldmath$n$}_{1}(s)+\kappa_{3}(s)\mbox{\boldmath$n$}_{3}(s),\\\
\mbox{\boldmath$n$}_{3}^{\prime}(s)=\delta_{1}\kappa_{3}(s)n_{2}(s),\end{array}\right.$

where
$\kappa_{2}(s)=\delta_{2}\langle\mbox{\boldmath$n$}_{1}^{\prime}(s),\mbox{\boldmath$n$}_{2}(s)\rangle$
and
$\kappa_{3}(s)=\delta_{3}\langle\mbox{\boldmath$n$}_{2}^{\prime}(s),\mbox{\boldmath$n$}_{3}(s)\rangle.$
Since $\mbox{\boldmath$\gamma$}(s)$ is timelike and $\mbox{\boldmath$t$}(s)$
is spacelike, we distinguish the following three cases:

Case 1: $\mbox{\boldmath$n$}_{1}(s)$ is timelike, that is, $\delta_{1}=-1$ and
$\delta_{2}=\delta_{3}=1.$

Case 2: $\mbox{\boldmath$n$}_{2}(s)$ is timelike, that is, $\delta_{2}=-1$ and
$\delta_{1}=\delta_{3}=1.$

Case 3: $\mbox{\boldmath$n$}_{3}(s)$ is timelike, that is, $\delta_{3}=-1$ and
$\delta_{1}=\delta_{2}=1.$

We consider the lightlike hypersurface along $C,$ and calculate the anti-de
Sitter height function on $C$ which is useful for the study the singularities
of lightlike hypersurfaces in the each case.

###  7.1 Case 1

Suppose that $\mbox{\boldmath$n$}_{1}(s)$ is timelike. In this case we adopt
$\mbox{\boldmath$n$}^{T}(s)=\mbox{\boldmath$n$}_{1}(s)$ and denote that
$\mbox{\boldmath$b$}_{1}(s)=\mbox{\boldmath$n$}_{2}(s),\mbox{\boldmath$b$}_{2}(s)=\mbox{\boldmath$n$}_{3}(s).$
Then we have the pseudo-orthogonal frame

$\\{\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$t$}(s),\mbox{\boldmath$n$}^{T}(s),\mbox{\boldmath$b$}_{1}(s),\mbox{\boldmath$b$}_{2}(s)\\},$

$\delta_{1}=-1$ and $\delta_{2}=\delta_{3}=1$, which satisfies the following
Frenet-Serret type formulae:

$\left\\{\begin{array}[]{ll}\mbox{\boldmath$\gamma$}^{\prime}(s)=\mbox{\boldmath$t$}(s),\\\
\mbox{\boldmath$t$}^{\prime}(s)=\mbox{\boldmath$\gamma$}(s)+\kappa_{1}(s)\mbox{\boldmath$n$}^{T}(s),\\\
{\mbox{\boldmath$n$}^{T}}^{\prime}(s)=\kappa_{1}(s)\mbox{\boldmath$t$}(s)+\kappa_{2}(s)\mbox{\boldmath$b$}_{1}(s),\\\
\mbox{\boldmath$b$}_{1}^{\prime}(s)=\kappa_{2}(s)\mbox{\boldmath$n$}^{T}(s)+\kappa_{3}(s)\mbox{\boldmath$b$}_{2}(s),\\\
\mbox{\boldmath$b$}_{2}^{\prime}(s)=-\kappa_{3}(s)\mbox{\boldmath$b$}_{1}(s).\end{array}\right.$

Since $N_{1}(C)[\mbox{\boldmath$n$}^{T}]$ is parametrized by

$N_{1}(C)[\mbox{\boldmath$n$}^{T}]=\\{(\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$\xi$})\in\mbox{\boldmath$\gamma$}^{*}T{\mathbb{R}}^{5}_{1}\
|\
\mbox{\boldmath$\xi$}=\cos\theta\mbox{\boldmath$b$}_{1}(s)+\sin\theta\mbox{\boldmath$b$}_{2}(s)\in
N_{\mbox{\scriptsize\boldmath$\gamma$}(s)}(C),\ s\in I\\},$

the nullcone Gauss image of $N_{1}(C)_{p}[\mbox{\boldmath$n$}^{T}]$ is given
by

$\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s,\theta)=\mbox{\boldmath$n$}^{T}(s)+\cos\theta\mbox{\boldmath$b$}_{1}(s)+\sin\theta\mbox{\boldmath$b$}_{2}(s).$

Then we have the lightlike hypersurface along $C$

$\mathbb{LH}_{C}((s,\theta),\mu)=\mbox{\boldmath$\gamma$}(s)+\mu(\mbox{\boldmath$n$}^{T}(s)+\cos\theta\mbox{\boldmath$b$}_{1}(s)+\sin\theta\mbox{\boldmath$b$}_{2}(s))=\mbox{\boldmath$\gamma$}(s)+\mu\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s,\theta).$

We remark that the image of this lightlike hypersurface along $C$ is
independent of the choice of the future directed timelike normal vector field
$\mbox{\boldmath$n$}^{T}$ by Corollary 4.3.

Now we investigate the anti- de Sitter height functions $H:I\times
AdS^{4}\longrightarrow{\mathbb{R}}$ on a spacelike curve
$C=\mbox{\boldmath$\gamma$}(I)$ defined by

$H(p,\mbox{\boldmath$\lambda$})=H(s,\mbox{\boldmath$\lambda$})=\langle\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$\lambda$}\rangle+1,$

where $p=\mbox{\boldmath$\gamma$}(s).$ For any fixed
$\mbox{\boldmath$\lambda$}_{0}\in AdS^{4},$ we write
$h(p)=H_{\lambda_{0}}(p)=H(p,\mbox{\boldmath$\lambda$}_{0}).$

By Proposition 4.1, the discriminant set of the anti-de Sitter height function
$H$ is given by

${\mathcal{D}}_{H}=\mathbb{LH}_{C}(N_{1}(C)[\mbox{\boldmath$n$}^{T}]\times{\mathbb{R}})=\Bigl{\\{}\mbox{\boldmath$\lambda$}=\mbox{\boldmath$\gamma$}(s)+\mu\mathbb{NG}(s,\theta)\Bigm{|}\
\theta\in[0,2\pi),s\in I,\ \mu\in{\mathbb{R}}\ \Bigr{\\}},$

which is the image of the lightlike hypersurface along $C.$ We also calculate
that
$h^{\prime\prime}(p)=\langle\mbox{\boldmath$\gamma$}^{\prime\prime}(s),\lambda_{0}\rangle=-\mu\kappa_{1}-1.$
Then $h^{\prime\prime}(p)=0$ if and only if $\mu=-1/\kappa_{1}(s).$ It means
that a singular point of the lightlike hypersurface is a point
$\mbox{\boldmath$\lambda$}_{0}=\mbox{\boldmath$\gamma$}(s_{0})+\mu_{0}\mathbb{NG}(\theta_{0},s_{0})$
for $\mu_{0}=-1/\kappa_{1}(s_{0}).$ Therefore, the lightlike focal surface is

$\mathbb{LF}_{C}=\Bigl{\\{}\mbox{\boldmath$\lambda$}=\mbox{\boldmath$\gamma$}(s)-\frac{1}{\kappa_{1}(s)}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s,\theta)\Bigm{|}\
s\in I,\ \theta\in[0,2\pi)\ \Bigr{\\}}.$

Moreover, if we calculate the third, 4th and 5th derivatives of $h(s),$ we
have the following proposition.

###### Proposition 7.1

Let $C$ be a spacelike curve and $H:C\times(AdS^{4}\setminus
C)\to{\mathbb{R}}$ the anti-de Sitter height function on $C.$ Suppose that
$p_{0}=\mbox{\boldmath$\gamma$}(s_{0})\not=\mbox{\boldmath$\lambda$}_{0}.$
Then we have the followings:

(1) $h(p_{0})=h^{\prime}(p_{0})=0$ if and only if there exist
$\theta_{0}\in[0,2\pi)$ and $\mu\in{\mathbb{R}}\setminus\\{0\\}$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\mu\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0}).$

(2) $h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=0$ if and only if
there exists $\theta_{0}\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=-\frac{1}{\kappa_{1}(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0}).$

(3)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=0$
if and only if there exists $\theta_{0}\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=-\frac{1}{\kappa_{1}(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0})$

and
$\kappa_{1}^{\prime}(s_{0})-\cos\theta_{0}\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0,$
so that we can write $\theta_{0}=\theta(s_{0}).$

(4)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=h^{(4)}(p_{0})=0$
if and only if there exists $\theta_{0}=\theta(s_{0})\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=-\frac{1}{\kappa_{1}(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta(s_{0})),$

$\kappa_{1}^{\prime}(s_{0})-\cos\theta(s_{0})\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0$
and
$(2\kappa_{1}^{\prime}(s_{0})\kappa_{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}^{\prime}(s_{0}))\cos\theta(s_{0})-\kappa_{1}^{\prime\prime}(s_{0})-\kappa_{1}(s_{0})\kappa_{2}^{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}(s_{0})\kappa_{3}(s_{0})\sin\theta(s_{0})=0.$

(5)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=h^{(4)}(p_{0})=h^{(5)}(p_{0})=0$
if and only if there exists $\theta_{0}=\theta(s_{0})\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=-\frac{1}{\kappa_{1}(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta(s_{0})),$

$\kappa_{1}^{\prime}(s_{0})-\cos\theta(s_{0})\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0$,
$(2\kappa_{1}^{\prime}(s_{0})\kappa_{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}^{\prime}(s_{0}))\cos\theta(s_{0})-\kappa_{1}^{\prime\prime}(s_{0})-\kappa_{1}(s_{0})\kappa_{2}^{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}(s_{0})\kappa_{3}(s_{0})\sin\theta(s_{0})=0$
and
$((2\kappa_{1}^{\prime}(s_{0})\kappa_{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}^{\prime}(s_{0}))\cos\theta(s_{0})-\kappa_{1}^{\prime\prime}(s_{0})-\kappa_{1}(s_{0})\kappa_{2}^{2}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}(s_{0})\kappa_{3}(s_{0})\sin\theta(s_{0}))^{\prime}=0.$

Taking account of the above proposition, we denote that
$\rho_{1}(s,\theta)=\kappa_{1}^{\prime}(s)-\cos\theta\kappa_{1}(s)\kappa_{2}(s)$
and
$\eta_{1}(s,\theta)=(2\kappa_{1}^{\prime}(s)\kappa_{2}(s)+\kappa_{1}(s)\kappa_{2}^{\prime}(s))\cos\theta-\kappa_{1}^{\prime\prime}(s)-\kappa_{1}(s)\kappa_{2}^{2}(s)+\kappa_{1}(s)\kappa_{2}(s)\kappa_{3}(s)\sin\theta,$
which might be important invariants of $C=\mbox{\boldmath$\gamma$}(I).$ Then
we can show that $\rho_{1}(s,\theta)=\eta_{1}(s,\theta)=0$ if and only if
$\rho_{1}(s,\theta)=\sigma_{1}(s)=0$, where

$\sigma_{1}(s)=\left[\kappa_{1}\kappa_{2}(\kappa_{1}^{\prime\prime}+\kappa_{1}\kappa_{2}^{2})-\kappa_{1}^{\prime}(2\kappa_{1}^{\prime}\kappa_{2}+\kappa_{1}\kappa^{\prime}_{2})\mp\kappa_{1}\kappa_{2}\kappa_{3}\sqrt{(\kappa_{1}\kappa_{2})^{2}-(\kappa^{\prime}_{1})^{2}}\right](s).$

###  7.2 Case 2

Suppose that $\mbox{\boldmath$n$}_{2}(s)$ is timelike. Then we adopt
$\mbox{\boldmath$n$}^{T}(s)=\mbox{\boldmath$n$}_{2}(s)$ and denote that
$\mbox{\boldmath$b$}_{1}(s)=\mbox{\boldmath$n$}_{1}(s),\mbox{\boldmath$b$}_{2}(s)=\mbox{\boldmath$n$}_{3}(s).$
We have a pseudo-orthogonal frame
$\\{\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$t$}(s),\mbox{\boldmath$n$}^{T}(s),\mbox{\boldmath$b$}_{1}(s),\mbox{\boldmath$b$}_{2}(s)\\}$,
$\delta_{2}=-1$ and $\delta_{1}=\delta_{3}=1,$ which satisfies the following
Frenet-Serret type formulae:

$\left\\{\begin{array}[]{ll}\mbox{\boldmath$\gamma$}^{\prime}(s)=\mbox{\boldmath$t$}(s),\\\
\mbox{\boldmath$t$}^{\prime}(s)=\mbox{\boldmath$\gamma$}(s)+\kappa_{1}(s)\mbox{\boldmath$b$}_{1}(s),\\\
\mbox{\boldmath$b$}_{1}^{\prime}(s)=-\kappa_{1}(s)\mbox{\boldmath$t$}(s)+\kappa_{2}(s)\mbox{\boldmath$n$}^{T}(s),\\\
{\mbox{\boldmath$n$}^{T}}^{\prime}(s)=\kappa_{2}(s)\mbox{\boldmath$b$}_{1}(s)+\kappa_{3}(s)\mbox{\boldmath$b$}_{2}(s),\\\
\mbox{\boldmath$b$}_{2}^{\prime}(s)=\kappa_{3}(s)\mbox{\boldmath$n$}^{T}(s),\end{array}\right.$

Here, $N_{1}(C)[\mbox{\boldmath$n$}^{T}]$ is parametrized by

$N_{1}(C)[\mbox{\boldmath$n$}^{T}]=\\{(\mbox{\boldmath$\gamma$}(s),\mbox{\boldmath$\xi$})\in\mbox{\boldmath$\gamma$}^{*}T{\mathbb{R}}^{5}_{1}\
|\
\mbox{\boldmath$\xi$}=\cos\theta\mbox{\boldmath$b$}_{1}(s)+\sin\theta\mbox{\boldmath$b$}_{2}(s)\in
N_{\mbox{\scriptsize\boldmath$\gamma$}(s)}(C),\ s\in I\\},$

so that we have the lightlike hypersurface along
$C=\mbox{\boldmath$\gamma$}(I)$:

$\mathbb{LH}_{C}((s,\theta),t)=\mbox{\boldmath$\gamma$}(s)+\mu\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s,\theta).$

We consider the anti-de Sitter height function $H:I\times
AdS^{4}\longrightarrow{\mathbb{R}}$ on a spacelike curve
$C=\mbox{\boldmath$\gamma$}(I)$. Under the similar notations to the case 1),
we have the following proposition:

###### Proposition 7.2

Let $C$ be a spacelike curve and $H:C\times(AdS^{4}\setminus
C)\to{\mathbb{R}}$ the anti-de Sitter height function on $C.$ Suppose that
$p_{0}\not=\mbox{\boldmath$\lambda$}_{0}.$ Then we have the following$:$

(1) $h(p_{0})=h^{\prime}(p_{0})=0$ if and only if there exist
$\theta_{0}\in[0,2\pi)$ and $\mu\in{\mathbb{R}}\setminus\\{0\\}$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\mu\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0}).$

(2) $h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=0$ if and only if
there exists $\theta_{0}\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\frac{1}{\kappa_{1}(s_{0})\cos\theta_{0}}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0}).$

(3)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=0$
if and only if there exists $\theta_{0}\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\frac{1}{\kappa_{1}(s_{0})\cos\theta_{0}}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta_{0})$

and
$\kappa_{1}^{\prime}(s_{0})\cos\theta_{0}-\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0,$
so that we can write $\theta_{0}=\theta(s_{0}).$

(4)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=h^{(4)}(p_{0})=0$
if and only if there exists $\theta_{0}=\theta(s_{0})\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\frac{1}{\kappa_{1}(s_{0})\cos\theta(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta(s_{0})),$

$\kappa_{1}^{\prime}(s_{0})\cos\theta(s_{0})-\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0$
and
$(\kappa_{1}^{\prime\prime}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}^{2}(s_{0}))\cos\theta(s_{0})-2\kappa_{1}^{\prime}(s_{0})\kappa_{2}(s_{0})-\kappa_{1}(s_{0})\kappa_{2}^{\prime}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}(s_{0})\kappa_{3}(s_{0})\sin\theta(s_{0})=0$.

(5)
$h(p_{0})=h^{\prime}(p_{0})=h^{\prime\prime}(p_{0})=h^{\prime\prime\prime}(p_{0})=h^{(4)}(p_{0})=h^{(5)}(p_{0})=0$
if and only if there exists $\theta_{0}=\theta(s_{0})\in[0,2\pi)$ such that

$\mbox{\boldmath$\gamma$}(s_{0})-\mbox{\boldmath$\lambda$}_{0}=\frac{1}{\kappa_{1}(s_{0})\cos\theta(s_{0})}\mathbb{NG}(\mbox{\boldmath$n$}^{T})(s_{0},\theta(s_{0})),$

$\kappa_{1}^{\prime}(s_{0})\cos\theta(s_{0})-\kappa_{1}(s_{0})\kappa_{2}(s_{0})=0$,
$(\kappa_{1}^{\prime\prime}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}^{2}(s_{0}))\cos\theta(s_{0})-2\kappa_{1}^{\prime}(s_{0})\kappa_{2}(s_{0})-\kappa_{1}(s_{0})\kappa_{2}^{\prime}(s_{0})+\kappa_{1}(s_{0})\kappa_{2}(s_{0})\kappa_{3}(s_{0})\sin\theta(s_{0})=0$
and