Abstract
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 space with 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.
1. 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 [1–5]. 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 [6–12]. 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 3-space. 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 3-space. When the parameter is fixed, the sections of one-parameter null hypersurfaces are null surfaces. Moreover, the null ruled surfaces in [18] are the sections of one-parameter 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 () 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 one-parameter 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. Preliminaries
Let be a 4-dimensional vector space. For any two vectors and in , their pseudoscalar product is defined by The space is called semi-Euclidean 4-space with index 2 and denoted by .
For three vectors , , and , we define a vector bywhere is the canonical basis of . We have , so is pseudoorthogonal to , and . A nonzero vector is called spacelike, null, or timelike if , , or , respectively. The norm of is defined by , where denotes the signature of which is given by or if is a spacelike, null, or timelike vector, respectively.
Let be a regular curve in (i.e., ), where is an open interval. For any , the curve is called spacelike, null, or timelike if , , or , respectively. We call a nonnull curve if is a spacelike or timelike curve. The arc-length of a nonnull curve measured from is .
The parameter is determined such that for the nonnull curve, where is the unit tangent vector of . Some submanifolds in are as follows.
The anti-de Sitter space is defined by and the lightlike cone by We also define the one-parameter anti-de Sitter space by
Let be a spacelike regular curve; that is, satisfies , . Since the curve is spacelike, we can reparametrize it by the arc-length . Then we have the tangent vector ; obviously . When , we define a unit vector: Let ; then we have a pseudoorthonormal frame of along . By direct calculating, the following Frenet-Serret type is displayed, under the assumption that : Here, is the geodesic curvature, is the geodesic torsion, and . If , we can obtain ; it means that is a geodesic curve in . We consider (i.e., ) in the following sections.
Let be a unit speed spacelike curve; we write and define by We call the one-parameter null hypersurfaces associated with . We also define the following model surface. For any ,
On the other hand, let be a submersion and let be a spacelike curve. We say that and have -point contact at if the function satisfies , . We also have that and have at least -point contact at if the function satisfies .
3. One-Parameter Height Functions and the Singularities of One-Parameter Null Surfaces
In this section we discuss a kind of Lorentzian invariant functions on a spacelike curve in . It is useful to study the null hypersurfaces of the spacelike curve. Let be a unit spacelike curve. We now define a functionby ; we call one-parameter height function on the spacelike curve . We denote that , . Then, we have the following proposition.
Proposition 1. Let be a unit spacelike curve and . Then one has the following:(1) if and only if there exists such that .(2) if and only if (3) if and only if and .(4) if and only if and .
Proof. (1) Since , we can find that with such that . Because we can get ; when , it means that and . Therefore ; the converse direction also holds.
(2) By (1), an easy computation shows that we get ; therefore (3) Under the assumption that we can get ; assertion (3) follows.
(4) Based on the assumption that the relation follows the fact that is equivalent to so . This proves assertion (4).
Now, we do research on some properties of one-parameter null hypersurfaces of the spacelike curve in . 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 be a unit spacelike curve. Then one has the following:(1)The set is the singularities of one-parameter null hypersurfaces .(2)If is a constant vector, one has for any ; at the same time .
Proof. By calculations we have(1) If the above three vectors are linearly dependent, we can get the singularities of if and only if . Therefore, assertion (1) holds.
(2) For any fixed , if is a constant, then Since then We have This completes the proof.
4. 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.
Let be a function germ; . We call an -parameter unfolding of . If for all and , we say that has -singularity at . We also say that has -singularity at if for all . Let be an -parameter unfolding of and has -singularity at ; we define the -jet of the partial derivative at asIf the rank of matrix is , then is called a versal unfolding of , where . The discriminant set of is defined by There has been the following famous result [20].
Theorem 3. Let be an -parameter unfolding of which has -singularity at ; suppose that is a of . Then one has the following:(a)If , then is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .(c)If , then is locally diffeomorphic to .
By Proposition 1, the discriminant set of the timelike height function is given by
Proposition 4. Let be a one-parameter height function on the spacelike curve . If has -singularity at s , then is a versal unfolding of .
Proof. Let and .
Then Let , so The - of at is given bythe - of at is given by(1) By Proposition 1, has the -singularity at if and only if . Since the curve is regular, the rank of is .
(2) We can get that has the -singularity at if and only if and . When has the -singularity at , we require the matrix to have rank 2, which follows from the proof of the next case.
(3) It also follows from Proposition 1 that has the -singularity at if and only if and , .
We require the matrix to have rank 3.
Let matrix We denote then Since is a singular point, then and we have Therefore In summary, is a versal unfolding of ; this completes the proof.
5. Main Result
The main result in this paper is in this section. We now consider the following conditions:(A1)The number of points of , where at have four-point contact with the curve , is finite.(A2)There is no point of , where at have five-point contact or greater with the curve .
Our main result is as follows.
Theorem 5. Let be a unit regular spacelike curve, , and Then one has the following.
(1) and have at least -point contact at .
(2) and have -point contact at if and only if and . Under this condition the germ of image at is diffeomorphic to the cuspidal edge (Figure 1).
(3) and have -point contact at if and only if and . Under this condition the germ of image at is diffeomorphic to the swallowtail (Figure 2).
Here and .
Proof. Let be a spacelike regular curve and . As , we give a function by ; then we assume that Because and 0 is a regular value of , and have -point contact at if and only if has the -singularity at . By Proposition 1, Theorem 3, and Proposition 4, we get the results.
6. Example
In this section, we construct the one-parameter null hypersurfaces associated with a spacelike curve and two sections of 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.
Let , , , be a spacelike curve in , where is the arc-length parameter. ThenWe have
be the one-parameter null hypersurfaces of At the moment, we can calculate that and ; two sections of with , are as follows:
The pictures of -null hypersurface and its singularities can be seen in Figure 3. And the pictures of -null hypersurface and its singularities can be seen in Figure 4.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was partially supported by NSF of China (nos. 11271063 and 11501051) and NCET of China (no. 05-0319). The first author was partially supported by the Project of Science and Technology of Heilongjiang Provincial Education Department of China (no. UNPYSCT-2015103).