Abstract

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

1. Introduction

The extrinsic differential geometry of submanifolds in 4-dimensional semi-Euclidean space is of special interest in Relativity Theory. In particular the lightlike hypersurfaces, which can be constructed as lightlike ruled hypersurfaces over Lorentzian surfaces in anti-de Sitter space, provide good models for the study of different horizon types of black holes, such as Kerr black hole, Cauchy black hole, and Schwarzschild black hole [18]. Hiscock described that the horizon was constituted by lightlike hypersurfaces and lightlike wave front was lightlike hypersurface [6]; Dąbrowski et al. have studied the null (lightlike) strings form the photon sphere, moving in the single spacetime of general relativity, including lightlike hypersurfaces [1, 3, 4]. The authors gave the null string evolution in Schwarzschild spacetime by the solutions of null string equations, which are also the null geodesic equations of general relativity appended by an additional stringy constant [3, 4]. In the view of geometry, the null string (null curve) in lightlike surfaces is null geodesic [9]. In this sense, the singularities of lightlike hypersurfaces are deeply related to the shapes of horizons.

M. Kossowski introduced a Gauss map on its associated spacelike surface, obtaining in this way interesting conclusions on the lightlike hypersurfaces which parallel the known results for surfaces in Euclidean 3-space concerning their contacts with the model surfaces [10]. When working in semi-Euclidean space, we observe that the properties associated with the contacts of a given submanifold with null cone and lightlike hyperplanes have a special relevance from the geometrical viewpoint. In [1113], the current authors and so forth pursued with this line by describing the invariant geometric properties of Lorentzian surfaces of codimension two in semi-Euclidean space that arise from their contacts with null cone. For this purpose, the task of this paper is to study some local properties of these Lorentzian surfaces in semi-Euclidean ( )-space.

Canal hypersurfaces, which are generated by surfaces with codimension 2 along fixed direction, are envelopes of families of hyperspheres. In three-dimensional space, canal surfaces were considered in many classical texts on differential geometry [14]. Since the property of a hypersurface to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15, 16]. However, in all these works the authors did not note the singularities of canal hypersurface in semi-Euclidean space. In this paper, we analyze the geometric meaning of the canal hypersurfaces from the view point of singularity. And we obtain the conclusion that the canal hypersurfaces have the similar singularities as Lorentzian surfaces.

The remainder of this paper is organized as follows. In Section 2, we give some basic notions about Lorentzian surfaces and lightlike hypersurfaces. Meanwhile, the Lorentzian Gauss-Kronecker curvatures of Lorentzian surfaces are also introduced. In Section 3, we describe Lorentzian distance-squared functions, whose discriminant sets and wave front sets are just right of the given lightlike hypersurfaces. In Section 4, we discuss the contact between lightlike hypersurfaces and null cone by Montald’s theorem. We give an example about the classification of singularities to lightlike hypersurfaces generated by Lorentzian surfaces in anti-de Sitter space in Section 5. In the last section, we consider some geometric properties of canal hypersurfaces, which are generated by Lorentzian surfaces in anti-de Sitter 3-space and the conclusion that the types of singularity of canal hypersurfaces are the same as the Lorentzian surfaces.

We will assume throughout the whole paper that all manifolds and maps are unless the contrary is explicitly stated.

2. Preliminaries

Einstein formulated general relativity as a theory of space, time, and gravitation in semi-Euclidean space in 1915. However, this subject has remained dormant for much of its history because its understanding requires advanced mathematics knowledge. Since the end of the twentieth century, semi-Euclidean geometry has been an active area of mathematical research, and it has been applied to a variety of subjects related to differential geometry and general relativity. In this section, we illustrated some basic knowledge of semi-Euclidean space.

Let be an -dimensional vector space. For any vectors and in , the pseudoscalar product of and is defined by The space is called semi-Euclidean ( )-dimensional space with index two, denoted by . A vector is called spacelike, lightlike, or timelike, if is positive, zero, or negative, respectively. There exist some special submanifolds in , such as unit pseudo- -sphere , anti de Sitter -space , null cone , and Lorentz torus , which have the same definitions as in [17].

Definition 1. Let be an embedding, where is an open subset; if there exists such that is timelike vector and    is spacelike vector, we call Lorentzian surface in anti-de Sitter space.

Without loss of generality, we only consider ; the other cases are the same. We construct a unit spacelike normal vector and the vectors are lightlike. Since is a basis of , so the system provides a basis for .

We define a map by which is called the lightcone Gauss indicatrix of . We have shown that and [11]. Under the identification of and , the derivative can be identified to the identity mapping on the tangent space , where . This means that . Thus, can be regarded as a linear transformation on the tangent space . We call the linear transformation the Lorentzian shape operator of at . We denote the eigenvalue of by , which is called a Lorentzian principal curvature of at point . The Lorentzian Gauss-Kronecker curvature of is defined as .

Definition 2. A point is an umbilic point if all the principal curvatures coincide at . is called totally umbilic surface if all points on are umbilics.

Supposing is totally umbilic, we have the following propositions by simple computation.

Proposition 3. If   is totally umbilic, we have the following classification.(1)Suppose that . (a)If , then is a part of an anti-de Sitter space. In particular, if , then is a part of a small anti-de Sitter space. (b)If , then is a part of unit pseudo- -sphere.(2)Suppose . Then is a part of hyperhorosphere.

Proposition 4. Let be a Lorentzian surface in anti-de Sitter space, the lightcone Gauss indicatrix is constant if and only if there exists a unique lightlike hyperplane in , such that the is a part of , where .

Since is timelike vector,    is spacelike vector, and semi-Riemannian metric on defined by [18, 19], where , , and , for any , we have a Lorentzian second fundamental invariant with respect to the vectors ,   defined by , for any .

Proposition 5. The Lorentzian Weingarten formulas with respect to ,   are as follows.(1) ,(2) , where ,   .

Proof. There exist real numbers ,   ,   such that . Since , we have Hence, we have ; the formula (2) follows the conclusion of item (1).

As a corollary of the Proposition 5, we have an explicit expression of the Lorentzian Gauss-Kronecker curvature by Riemannian metric and the second fundamental invariant.

Corollary 6. Under the same notations as in the above proposition, the Lorentzian Gauss-Kronecker curvature is given by .

Proof. By the above proposition, the representation matrix of the Lorentzian shape operator with respect to the basis is . It follows from this fact that So we complete the proof.

Since , we have . Therefore, the Lorentzian second fundamental invariant depends on the values ,   . By the above corollary, the Lorentzian Gauss-Kronecker curvature depends only on ,   ,   . It is independent on the choice of the normal vector field .

Definition 7. Let be a Lorentzian surface in anti-de Sitter space and let be its spacelike normal vector; a hypersurface defined by is called the lightlike hypersurface along .

3. Lorentzian Distance-Squared Function

To describe the existence of singularities of lightlike hypersurfaces, we should construct contact functions, whose wave front set is the singularity set of lightlike hypersurfaces. In this section, we introduce some notions of Lorentzian distance-squared functions on Lorentzian surfaces in anti-de Sitter space, which can supply the contact relationship between Lorentzian surfaces and standard spherical surfaces. Meanwhile, we obtain the Lorentzian distance-squared functions as Morse family.

A function on the Lorentzian surface is given by which is called Lorentzian distance-squared function on . For any fixed , we write and have the following propositions by simple computing.

Proposition 8. Let be a Lorentzian surface in anti-de Sitter space and let be Lorentzian distance-squared function on . Suppose that . Then we have(1) if and only if for .(2) if and only if for and .

Proposition 9. Let and let be a Lorentzian surface without any umbilic point satisfying . Then if and only if is an isolated singular value of the lightlike hypersurface and .

Proof. By definition, if and only if , for any , where is the Lorentzian distance-squared function on . It follows from Proposition 8 that there exists a smooth function such that . Therefore, Hence, we have . Moreover, we get that for any , and from above formulas, we can obtain Therefore, we have , since is lightlike, is a timelike vector, and    is spacelike vector. ,   are linearly independent. Therefore, so if and only if under the assumption that . This means that is an isolated singularity of .

Since we only consider local properties, we may assume that . As the definitions in [11], it follows that The set is defined as the wave front set of . Also, we can write as Thus, a singular point of the lightlike hypersurface satisfied .

Definition 10. Let be a Morse family, a map germ , which satisfied , is called Legendrian immersion germ.

Let be the projective cotangent bundle over an -dimensional manifold in . This fibration can be considered as a Legendrian fibration with the canonical contact structure on . Let us consider the tangent bundle and . The property does not depend on the choice of representative of the class . Thus we can define the canonical contact structure on by . For a local coordinate neighbourhood in , we have a trivialization and we call homogeneous coordinates, where are homogeneous coordinates of the dual projective space . It is known that any Legendrian fibration is locally equivalent to [20].

Proposition 11. All Legendrian submanifold germs in are constructed by the above method.

Proposition 12. Let be the Lorentzian distance-squared function on . For any point , is a Morse family around .

Proof. Denote and . By definition, we have We now prove that the mapping is nonsingular at . Indeed, the Jacobian matrix of is given by where the matrix is given by and . Since is an immersion, the rank of the matrix is equal to and is lightlike, so that it is linearly independent of tangent vector . This means that the rank of is equal to , where Therefore, the Jacobi matrix of is nonsingularity at .

4. Contact with Null Cone

In this section, we gave the singularities of lightlike hypersurfaces are stable, whose types are not changed with small disturbance under the view of -equivalent and -equivalent. Before we start to consider the contact between lightlike hypersurfaces and null cone, we briefly review the theory of contact due to Montaldi [21, 22]. Let and    be submanifolds in with and . We say that the contact of and at is of the same type as the contact of and at if there is a diffeomorphism germ such that and . In this case, we write . In his paper [21], Montaldi gives a characterization of the notion of contact by using the terminology of singularity theory.

Theorem 13 (see [21]). Let and    be submanifolds of with and . Let be immersion germs and let be submersion germs with . Then if and only if and are -equivalent.

For the -equivalent among smooth map germs, considering the function by and denoting , we have . For , we can take the vector . Then and the relations are    . This means that the lightcone is tangent to at . In this case, we call each a tangent null cone of at . We denote by the local ring of function germs with the unique maximal ideal

Let ,   be function germs. We say that ,   are -equivalent if there exists a diffeomorphism germ of the form for such that , where is the pullback -algebra isomorphism defined by .

We apply the tools for the study of the contact theory. Let be two null cone Legendrian Gauss map germs of Lorentzian surface germs    . We say that and are -equivalent if there exist diffeomorphism germs and such that .

Let be a function germ, is -versal deformation of if , where . The main result in the theory [22] is as follows.

Theorem 14 (see [22]). Let ,   be Morse families. Then(1) and are Legendrian equivalent if and only if and are -equivalent.(2) is Legendrian stable if and only if is a -versal deformation of .

Since and are function germs on the common space, by the uniqueness result of the versal deformation of a function germ, we have the following classification results of Legendrian stable germs. For a map germ , we give the local ring of by .

Proposition 15. Let and be Morse families. Suppose that and are Legendrian stable. Then the following conditions are equivalent.(1) and are diffeomorphic as germs. (2) and are Legendrian equivalent. (3) and are isomorphic as -algebras, where and .

Let be the Lorentzian distance-squared function germs of    . We denote , then . By Theorem 13, we know if and only if and are -equivalent. Therefore, we can denote the local ring of the function , we remark that we can explicitly write the local ring as follows: where is the local ring of function germs with the maximal ideal in [12].

Theorem 16 (see [12]). Let    be Lorentzian surface germs such that the corresponding Legendrian lift germs are Legendrian stable. Then the following conditions are equivalent.(1)The lightlike hypersurface germs and are -equivalent.(2) and are -equivalent.(3) and are -equivalent.(4) .(5) and are isomorphic as -algebras.

Proof. Since the Lorentzian distance-squared function is a Morse family of functions, conditions (1) and (2) are equivalent. Moreover, is Lagrangian stable, is the -versal deformation of ; by the uniqueness result of the -versal deformation, condition (2) implies condition (3). By definition, we know condition (3) implies condition (2). It follows from Theorem 13 that conditions (3) and (4) are equivalent. As the same way, we can obtain conditions (5) and (1) as equivalent by Proposition 15, so we complete the proof.

Given a Lorentzian surface , we call the tangent indicatrix germ of , where and    .

Corollary 17. The lightlike hypersurface germs and are -equivalent, then tangent indicatrix germs and are diffeomorphic as set germs.

Proof. The tangent indicatrix germ of is the zero level set of   Since -equivalent among function germs preserves the zero-level sets of function germs, the assertion follows Theorem 16.

5. Singularities of Lightlike Hypersurfaces in

In this section, we study the classification of singularities of 3-dimensional lightlike hypersurfaces, which are generated by Lorentzian surface in anti-de Sitter 3-space, also, we consider the space of Lorentzian embeddings with Whitney -topology, where is an open subset. As the choose of the standard arguments in [11], we consider a function by and claim that is a submersion at     for any fixed . Given , we have . We have the -jet extension defined by . Consider the trivialization . For any submanifold , we denote . Then we have the following proposition [12, 17].

Proposition 18. Let be a submanifold of .
Consider is a residual subset of . If is a closed subset, then is open.

On the other hand, we have a stratification given by the set of -orbits in (for the definition of and additional properties refer to [12]).

Theorem 19. There exists an open dense subset such that for any , the germ of the Legendrian lift of the corresponding lightlike hypersurface at each point is Legendrian stable.

Proposition 20. There exists an open dense subset such that for any , the germ of the corresponding lightlike hypersurface at any point is -equivalent to one of the map germs    or , where (embedding), (cuspidal edge) (Figure 1), (swallowtail) (Figure 2), (butterfly), (purse) (Figure 3), (pyramid) (Figure 4).

By using the generic normal forms of generating families and Corollary 17, we have the following corollary.

Corollary 21. There exists an open dense subset such that for any , the germ of the corresponding tangent indicatrix at any point is diffeomorphic to one of the germs in the following lists.(1) (ordinary cusp); (Figure 5),(2) (tacnode or point); (Figure 6),(3) (rhamphoid cusp); (Figure 7),(4) (three lines); (Figure 8),(5) (a line).

6. Canal Hypersurface of Lorentzian Surface

Canal hypersurfaces, which are generated by surfaces with codimension 2 along fixed direction, are envelopes of families of hyperspheres. Since the property of a hypersurface is to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15, 16]. In this section, we mainly consider the canal hypersurfaces in semi-Euclidean space with index 2. Let be a Lorentzian surface; the Mongle form is as follows: The second fundamental form of is characterized by two quadratic forms. Their functional coefficients will be denoted by and , respectively [15].

We have the following function: The Gaussian curvature of is and the matrix is where ,   ,   ,   ,   , and .

Definition 22. Let be a Lorentzian surface in anti-de Sitter space and let be its spacelike normal vector; a hypersurface defined as is called canal hypersurface of , where is a sufficiently small positive real number chosen such that is embedded in .

We denote by the natural embedding of in and by the point . From Looijenga’s theorem [15], there is a residual subset of embeddings , for which the family of height functions by is locally stable as a family of function on with parameters on . Moreover, the corresponding family on the canal hypersurface is also generic. In fact the singularities of and are tightly related [16].

Thus, for a generic , those may be one of the following types: Morse , fold , cusp , swallowtail , and elliptic or hyperbolic umbilic . Moreover, the singularities of the lightcone Gauss indicatrix can be described in terms as follows [15].

Lemma 23 (see [15]). Given a critical point of the height function , we have the following.(1) is a nondegenerate critical point of if and only if is a regular point of .(2) is a degenerate critical point of if and only if is singular point of .

Let be the Gaussian curvature function on . The parabolic set, of is the singular set of . It can be shown that for a generic embedding of , is a regular surface except by a finite number of points , which are singularities of type of or equivalently umbilic points of [16].

Let be the natural projection of onto   . The image of the set of parabolic points by is the set .

Theorem 24. (1) If , then is a nondegenerate critical point of for any .
(2) If , then there are exactly two vectors , such that is a degenerate critical point of ,   .
(3) If , then there is a unique vector such that is a degenerate critical point of .

Proof. Let be the local expression of the embedding in Monge’s form and let the height function in -direction be where . If is a critical point of the height function , then and the determinant of the Hessian matrix of at is given by where ,   are the above coefficients. Now, and the equation has two, one, or zero solutions as , , or , respectively.

When is a degenerate critical point of , the hyperplane , orthogonal to , has a higher order contact with at . Therefore, we will say that is a binormal vector of at and can be an osculating hyperplane [20].

At each point of , there is a unique principal direction of zero curvature for . This direction is tangent to the surface on a curve made of points of type . This curve is in turn tangent to a zero principal direction of curvature at isolated points [16].

Proposition 25. The image of zero principal directions of curvature in under are asymptotic directions on .

Proof. For the curvature vector is given by where is the mean curvature vector; we can choose local coordinates for such that This choice will imply that is the zero curvature direction and . Then, it follows easily that and are parallel.

Therefore, we can have the singularities of canal hypersurfaces in the following theorem.

Theorem 26. The canal hypersurfaces have the same singularities as Lorentzian surfaces in anti-de Sitter apace, so we can easily obtain the singularities of canal hypersurfaces as in Section 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express thank to the referee for his/her valuable suggestions. NSFC no. 11271063 partially supported the second author and FRFCU no. 14CX02147A partially supported the first author.