Lightlike Hypersurfaces and Canal Hypersurfaces of Lorentzian Surfaces

and Applied Analysis 3 (2) πt ∘ L± ui = −∑ n


Introduction
The extrinsic differential geometry of submanifolds in 4dimensional semi-Euclidean space is of special interest in Relativity Theory.In particular the lightlike hypersurfaces, which can be constructed as lightlike ruled hypersurfaces over Lorentzian surfaces in anti-de Sitter space, provide good models for the study of different horizon types of black holes, such as Kerr black hole, Cauchy black hole, and Schwarzschild black hole [1][2][3][4][5][6][7][8].Hiscock described that the horizon was constituted by lightlike hypersurfaces and lightlike wave front was lightlike hypersurface [6]; Dąbrowski et al. have studied the null (lightlike) strings form the photon sphere, moving in the single spacetime of general relativity, including lightlike hypersurfaces [1,3,4].The authors gave the null string evolution in Schwarzschild spacetime by the solutions of null string equations, which are also the null geodesic equations of general relativity appended by an additional stringy constant [3,4].In the view of geometry, the null string (null curve) in lightlike surfaces is null geodesic [9].In this sense, the singularities of lightlike hypersurfaces are deeply related to the shapes of horizons.
M. Kossowski introduced a Gauss map on its associated spacelike surface, obtaining in this way interesting conclusions on the lightlike hypersurfaces which parallel the known results for surfaces in Euclidean 3-space concerning their contacts with the model surfaces [10].When working in semi-Euclidean space, we observe that the properties associated with the contacts of a given submanifold with null cone and lightlike hyperplanes have a special relevance from the geometrical viewpoint.In [11][12][13], the current authors and so forth pursued with this line by describing the invariant geometric properties of Lorentzian surfaces of codimension two in semi-Euclidean space that arise from their contacts with null cone.For this purpose, the task of this paper is to study some local properties of these Lorentzian surfaces in semi-Euclidean ( + 1)-space.
Canal hypersurfaces, which are generated by surfaces with codimension 2 along fixed direction, are envelopes of families of hyperspheres.In three-dimensional space, canal surfaces were considered in many classical texts on differential geometry [14].Since the property of a hypersurface to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15,16].However, in all these works the authors did not note the singularities of canal hypersurface in semi-Euclidean space.In this paper, we analyze the geometric meaning of the canal hypersurfaces from the view point of singularity.And we obtain the conclusion that the canal hypersurfaces have the similar singularities as Lorentzian surfaces.

Abstract and Applied Analysis
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 distancesquared 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.

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.
Definition 1.Let X :  → H  1 be an embedding, where  ⊂ R −1 is an open subset; if there exists  such that X(), X   () is timelike vector and X   () ( ̸ = ) is spacelike vector, we call  = X() Lorentzian surface in anti-de Sitter space.
Thus, L ± ( 0 ) can be regarded as a linear transformation on the tangent space   .We call the linear transformation  ±  () = −  (L ± ()) :    →    the Lorentzian shape operator of  at  = X( 0 ).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  ±  ()() = det  ±  ().
Definition 2. A point  = X() is an umbilic point if all the principal curvatures coincide at .  is called totally umbilic surface if all points on  are umbilics.
Supposing  = X() is totally umbilic, we have the following propositions by simple computation.
(a) If 0 ≤ | ± + 1| < 1, then  is a part of an anti-de Sitter space.In particular, if  ± = −1, then  is a part of a small anti-de Sitter space.(b) If | ± + 1| > 1, then  is a part of unit pseudo-sphere.
Proposition 4. Let X() 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 (n, −1) in R +1 2 , such that the  = X() is a part of H  1 ∩ (n, −1), where n = L ± ().
Proposition 5.The Lorentzian Weingarten formulas with respect to X(), N() are as follows. ( Proof.There exist real numbers , , Γ Hence, we have ℎ 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 . ( So we complete the proof. Since ⟨−L ± (), X   ()⟩ = 0, we have ℎ ±  = ⟨L ± (), X     ()⟩.Therefore, the Lorentzian second fundamental invariant depends on the values L ± ( 0 ), X     ( 0 ).By the above corollary, the Lorentzian Gauss-Kronecker curvature depends only on L ± ( 0 ), X   ( 0 ), X     ( 0 ).It is independent on the choice of the normal vector field N().Definition 7. Let  = X() be a Lorentzian surface in anti-de Sitter space and let N() be its spacelike normal vector; a hypersurface  ±  defined by  ±  (, ) = X() + ( X() ± N()) is called  ±  the lightlike hypersurface along .

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 distancesquared 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  :  × R +1 2 → R on the Lorentzian surface is given by which is called Lorentzian distance-squared function on .
For any fixed  0 ∈ R +1 2 , we write   0 () = (,  0 ) and have the following propositions by simple computing.Proposition 8. Let  be a Lorentzian surface in anti-de Sitter space and let  : ( and let  be a Lorentzian surface without any umbilic point satisfying  ±  () ̸ = 0. Then  ⊂ Λ  1 0 if and only if  0 is an isolated singular value of the lightlike hypersurface   and   ⊂ Λ  1 0 .

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 K-equivalent and P-Kequivalent.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   ( = 1, 2) be submanifolds in R  with dim  1 = dim  2 and dim  1 = dim  2 .We say that the contact of  1 and  1 at  1 is of the same type as the contact of  2 and  2 at  2 if there is a diffeomorphism germ  : (R  ,  1 ) → (R  ,  2 ) such that ( 1 ) =  2 and ( 1 ) =  2 .In this case, we write ( 1 ,  1 ;  1 ) = ( 2 ,  2 ;  2 ).In his paper [21], Montaldi gives a characterization of the notion of contact by using the terminology of singularity theory.
Theorem 14 (see [22]).Let ,  : (R  × R  , 0) → (R, 0) be Morse families.Then ( 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  : (R  , 0) → (R  , 0), we give the local ring of  by () =   /(M  ).
Proposition 15.Let  and  : (R  × R  , 0) → (R, 0) be Morse families.Suppose that L  and L  are Legendrian stable.Then the following conditions are equivalent.
(2) L  and L  are Legendrian equivalent.
(3) () and () are isomorphic as R-algebras, where if and only if  1 1 and  2 2 are K-equivalent.Therefore, we can denote the local ring of the function g 0 :  → R, we remark that we can explicitly write the local ring as follows: where  ∞  0 () is the local ring of function germs with the maximal ideal M() in [12].
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 R-versal deformation of    ; by the uniqueness result of the R-versal deformation, condition (2) implies condition (3).By definition, we know condition (3) implies condition (2).It follows from Theorem 13 that conditions (3) and ( 4) are equivalent.As the same way, we can obtain conditions ( 5) and (1) as equivalent by Proposition 15, so we complete the proof.
Proof.The tangent indicatrix germ of X  is the zero level set of  ,  Since K-equivalent among function germs preserves the zero-level sets of function germs, the assertion follows Theorem 16.
We denote by X the natural embedding of  in R 4 2 and by (, N()) the point X() + N() ∈ .From Looijenga's theorem [15], there is a residual subset of embeddings X :  → R 4  2 , for which the family of height functions  :  × Λ 3  1 → R by (, k) = ⟨X(), k⟩ is locally stable as a family of function on  with parameters on Λ 3 1 .Moreover, the corresponding family ℎ( X) on the canal hypersurface is also generic.In fact the singularities of ℎ(X) and ℎ( X) are tightly related [16].
Lemma 23 (see [15]).Given a critical point (, v) ∈  of the height function ℎ V , we have the following.
(1)  is a nondegenerate critical point of ℎ V if and only if (, k) is a regular point of L ± .
(2)  is a degenerate critical point of ℎ V if and only if (, k) is singular point of L ± .
Let K  :  → R be the Gaussian curvature function on .The parabolic set, K  (0) of  is the singular set of L ± .It can be shown that for a generic embedding of , K  (0) is a regular surface except by a finite number of points (, k), which are singularities of type Σ 2,0 of L ± or equivalently umbilic points ( ± 4 ) of ℎ V [16].Let p :  →  be the natural projection of  onto (i.e., p(, k) = ).The image of the set of parabolic points K  (0) by  is the set Δ ≤ 0.
(3) If Δ() = 0, then there is a unique vector w ∈  ()  such that  is a degenerate critical point of ℎ  .
When  is a degenerate critical point of ℎ V , the hyperplane H V , orthogonal to k, has a higher order contact with  at X().Therefore, we will say that k is a binormal vector of  at X() and H V can be an osculating hyperplane [20].
At each point of K −1  (0) − Σ 2 (L ± ), there is a unique principal direction of zero curvature for .This direction is tangent to the surface K −1  (0) on a curve made of points of type Σ 1,1 (L ± ).This curve is in turn tangent to a zero principal direction of curvature at isolated points [16].
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.

( 1 )
L  and L  are Legendrian equivalent if and only if  and  are P-K-equivalent.

Theorem 19 .
There exists an open dense subset O ⊂   (, H 3 1 ) such that for any X ⊂ O, 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 O ⊂   (, H 3 1 ) such that for any X ⊂ O, the germ of the corresponding lightlike hypersurface  ±  at any point (, , ) ∈  × R is A-equivalent to one of the map germs   (1 ≤  ≤ 4) or  ± 4 , where