Singularities for One-Parameter Null Hypersurfaces of Anti-de Sitter Spacelike Curves in Semi-Euclidean Space

We consider one-parameter null hypersurfaces associated with spacelike curves. The spacelike curves are in anti-de Sitter 3-space while one-parameter null hypersurfaces lie in 4-dimensional semi-Euclidean spacewith index 2.We classify the generic singularities of the hypersurfaces, which are cuspidal edges and swallowtails. And we reveal the geometric meanings of the singularities of such hypersurfaces by the singularity theory.


Introduction
Semi-Euclidean space is a vector space with pseudoscalar product which is different from Euclidean space.The study of semi-Euclidean space has produced fruitful results; please see [1][2][3][4][5].It is well known that there exist spacelike submanifolds, timelike submanifolds, and null submanifolds in semi-Euclidean space.Null submanifolds appear in many physics papers.For example, the null submanifolds are of interest because they provide models of different horizon types such as event horizons of Kerr black holes, Cauchy horizons, isolated horizons, Kruskal horizons, and Killing horizons [6][7][8][9][10][11][12].Null submanifolds are also studied in the theory of electromagnetism.
Anti-de Sitter space is a maximally symmetric semi-Riemannian manifold with constant negative scalar curvature.This space is a very important subject in physics; it is also one of the vacuum solutions of Einstein's field equation in the theory of relativity.There is a conjecture in physics that the classical gravitation theory on anti-de Sitter space is equivalent to the conformal field theory on the ideal boundary of anti-de Sitter space.It is called the AdS/CFT correspondence or the holographic principle by E. Witten.In mathematics this conjecture is that the extrinsic geometric properties on submanifolds in anti-de Sitter space have corresponding Gauge theoretic geometric properties in its ideal boundary.Therefore, it is necessary to investigate the submanifolds in anti-de Sitter space.During the last four decades, singularity theory has enjoyed rapid development.The French mathematician R. Thom (Fields medallist) first put forward the philosophical idea of applying singularity theory to the study of differential geometry.Porteous applied the thoughts of Thom to the study of Euclidean geometry [13].The first attempts to apply the singularity theory to non-Euclidean geometry were undertaken by S. Izumiya, the second author, and T. Sano et al.
Recently there appear several results on submanifolds in anti-de Sitter space from the viewpoint of singularity theory.The timelike hypersurfaces are studied in the anti-de Sitter space from the viewpoint of Lagrangian singularity theory [14].In the study of submanifolds, the null submanifolds happen to be the most interesting subjects, both from the viewpoint of singularity theory and the theory of relativity [15,16].Fusho and Izumiya have discussed the spacelike curves in de Sitter 3-space [17]; they define the null surfaces of spacelike curves.The spacelike curves have degenerate contact with null cones at the singularities of the null surfaces.In [18], L. Chen, Q. Han, the second author, and W. Sun consider null ruled surfaces along spacelike curves in anti-de Sitter 3space.They give the classifications of singularities of the ruled surfaces which are the codimensional two submanifolds in semi-Euclidean space with index 2. Null surfaces have been studied in preceding literature.As we all know, the horizon of the black hole is a null hypersurface or a part.However, to the best of the authors' knowledge, no literature exists regarding the singularities of one-parameter null hypersurfaces as they relate to spacelike curves in anti-de Sitter 3-space.Thus, the current study hopes to serve such a need.Therefore, in this paper, we stick to the one-parameter null hypersurfaces, which are generated by spacelike curves in anti-de Sitter 3space.When the parameter is fixed, the sections of oneparameter null hypersurfaces are null surfaces.Moreover, the null ruled surfaces in [18] are the sections of oneparameter null hypersurfaces.And the one-parameter null hypersurfaces can be taken as the most elementary case for the study of the lowest codimensional submanifolds in semi-Euclidean space with index 2.
A singularity is a point at which a function blows up.It is a point at which a function is at a maximum/minimum or a submanifold is no longer smooth and regular.In [19], we have discussed the singularities of normal hypersurface associated with a timelike curve.In this paper we first consider spacelike curves in anti-de Sitter 3-space and then define the one-parameter null hypersurfaces which are bundles along spacelike curves whose fibres are null lines or timelike curves.We also define the one-parameter height functions on spacelike curves and apply the versal unfolding theory of functions to discuss them; the functions can be used to investigate the geometric properties of one-parameter null hypersurfaces.In fact, one-parameter null hypersurfaces are the discriminant sets of these functions (the discriminant sets of one-parameter height functions are precisely the wavefronts of spacelike curves); the singularities of null hypersurfaces are   -singularities ( ≥ 2) of these functions.The main result in this paper is Theorem 5.This theorem characterizes the contact of spacelike curves with null cones in semi-Euclidean space with index 2.
A brief description of the organization of this paper is as follows.In Section 2, we review the concepts of submanifolds in semi-Euclidean space with index 2.In Section 3, we give one-parameter height functions of a spacelike curve, by which we can obtain several geometric invariants of the spacelike curve.We also get the singularities of oneparameter null hypersurfaces, and the geometric meaning of Theorem 5 is described in this section.The preparations for the proof of Theorem 5 are in Section 4. We give the proof of Theorem 5 in Section 5.In Section 6, we give an example to illustrate the results of Theorem 5.
We will assume throughout the whole paper that all manifolds and maps are  ∞ unless the contrary is stated.
(2) By ( 1 (3) Under the assumption that we can get () =    () ∓   ()  () = 0; assertion (3) follows.(4) Based on the assumption that the relation follows the fact that ℎ (4) V, () = 0 is equivalent to so   = 0.This proves assertion (4).Now, we do research on some properties of oneparameter null hypersurfaces of the spacelike curve in  3  1 .As we can know the functions   (),   (), and () have particular meanings.Here, we consider the case when the one-parameter null hypersurfaces have the most degenerate singularities.We have the following proposition.Proposition 2. Let  :  →  3  1 be a unit spacelike curve.Then one has the following: ) is a constant vector, one has   () ∈  V 0 () for any  ∈ ; at the same time () = 0.
Proof.By calculations we have (1) If the above three vectors are linearly dependent, we can get the singularities of  ±  (, ) if and only if sinh  −   () = 0. Therefore, assertion (1) holds.

Unfoldings of One-Parameter Height Functions
In this section we classify singularities of the one-parameter null hypersurfaces along  as an application of the unfolding theory of functions.
(32) the one-parameter null hypersurfaces.The two sections are null surfaces and they are also the wavefronts of spacelike curves.By calculating, we get the singularities of null surfaces.
It is useful to understand the one-parameter null hypersurfaces.