Lightlike developables in Minkowski 3-space

We say that a surface in Minkowski 3-space is a lightlike developable if any pseudo-normal vector of the regular part of the surface is lightlike. We show that such a surface is a part of a lightlike plane, the lightcone, the tangent surface of a spacelike curve in a lightlike plane, the tangent surface of a lightlike curve or the glue of such surfaces. The most interesting surfaces in such the class of surfaces is the tangent surface of a lightlike curve. We give a classiﬁcation of the singularities for the tangent surface of a generic lightlike curve. As a consequence, the H 3 type singularity appears in generic.


Introduction
A surface in Euclidean space whose Gaussian curvature vanishes on the regular part is called a developable surface. It has been known that a developable surface is a part of a conical surface, a cylindrical surface, the tangent surface of a space curve or the glue of such surfaces. Developable surfaces have singularities in general. The tangent surface of a space curve has the most interesting singularities in the above three kinds of surfaces. In fact Cleave [1] shown that the germ of the tangent surface of a generic space curve is locally diffeomorphic to the cuspidal edge C × R or the cuspidal cross cap CCR. Here, cuspdialedge cuspidal cross cap In this paper we consider the developable surfaces in Minkowski 3-space. In [9] Pei introduced the RP 2 -valued Gauss map for the study of Lorentzian geometric properties of surfaces in Minkowski 3-space. We say that a surface is a developable surface in the Minkowski sense if the RP 2 -valued Gauss map is singular at any point analogous to the definition of developable surfaces in the Euclidean sense. We can show that the developable surfaces in the Minkowski sense are the nothing but the developable surfaces as in the Euclidean sense (cf., Theorem 3.1). Of course the notion of the developable surfaces is independent of the Euclidean structure. However it might be specially interesting subject if we assume that any pseudo-normal is lightlike. We call such the developable surface a lightlike developable. We can show that a lightlike developable is a part of a lightlike plane, a part of a lightcone, a part of the tangent surface of a spacelike curve in a lightlike plane, a part of the tangent surface of a lightlike curve or the glue of such four kinds of surfaces (Theorem 5.1). The most interesting case is the tangent surface of a lightlike space curve. We can show that the germ of the tangent surface of a generic lightlike curve at a singular point is locally diffeomorphic to the cuspdialedge C×R, the Scherbak surface SB or the swallowtail SW (Theorems 5.2, 5.3). Here,

Scherbak surface swallowtail
The space of lightlike curves will be described in §5, so that the exact meaning of genericity of the lightlike curve will be established. We remark that Scherbak [10] shown that SB is given as the irregular orbit of the finite reflection group H 3 on C 3 . We also remark that any lightlike developable is obtained as a one parameter family of lightlike lines along a spacelike curve. In [5] we gave a classification of singularities of the lightlike developable along a generic spacelike curve. As a consequence, only C × R or SW appear as generic singularities. The results in [5] is different from the result in this paper, because the space of spacelike curves is different from the space of lightlike curves. The classification of the singularities in this paper is generic for lightlike curves (Theorems 5.2 and 5.3). We shall assume throughout the whole paper that all the maps and manifolds are C ∞ unless the contrary is explicitly stated.

Developable surfaces in Euclidean space
In this section we briefly review the results on developable surfaces in Euclidean space. Let x : U −→ R 3 be an embedding from an open region U ⊂ R 2 . We call x or the image S = x(U ) a regular surface in R 3 .For any regular surface x : U −→ R 3 , we define the first fundamental invariants: where a · b denotes the Euclidean scaler product of a, b. We define the unit normal vector where a × b is the vector product of a, b. Then we define the second fundamental invariants by The Gauss curvature K(u, v) is defined by We say that a surface x : If the surface has singularities, we say that it is a developable surfaces if the Gauss curvature of the regular part of the surface vanishes. Since the Gauss curvature is the determinant of the differential of the Gauss map, S = x(U ) is a developable surface if and only if the Gauss map of the surface is singular at any point of S. It has been known that a developable surface is a ruled surface [12]. A ruled surface in R 3 is a surface given by a oneparameter family of lines [6,12]. It is locally defined as a mapping F (γ,δ) : Then we have the following well-known classification theorem of developable surfaces [12].
Theorem 2.1 A developable surface is one of the following: (1) A part of a cylindrical surface.
(2) A part of a conical surface.
(3) A part of a tangent developable surface.
(4) A glue of the above three surfaces.
We remark that once we have the above classification theorem, the notion of the developable surfaces is independent of the metric structure of R 3 . We only need the affine structure on R 3 for defining the developable surfaces. In the reminder of the paper, we say that a surface is a developable surface if it is one of the four surfaces in the above theorem. In general, developable surfaces have singularities. The tangent surface has the most interesting singularities of the surfaces in the above theorem. Therefore there are many articles concerning the singularities of tangent surfaces. Let γ : I −→ R 3 be a smooth curve and denote that under a suitable Affine coordinate transformation of R 3 around γ(t 0 ) and a parameter transformation. In this case we say that A = (a 1 , a 2 , a 3 ) is the type of γ at γ(t 0 ) and denote that A(γ t 0 ). We say that a type We have the following theorem [1,3,4,7,8,11].

Theorem 2.2 The type A of a smooth curve germ in R 3 is deterministic if and only if A is one of the following types:
(1) A = (1, 2, 2 + r), r = 1, 2, 3 . . . , We can recognize the type of a smooth curve germ by using the following simple calculations.

Developable surfaces in Minkowski 3-space
We now prepare basic notions on Minkowski space. Let be a 3-dimensional vector space. For any vectors x = (x 0 , x 1 , x 2 ), y = (y 0 , y 1 , y 2 ) ∈ R 3 , the pseudo scalar product of x and y is defined by x, y = −x 0 y 0 +x 1 y 1 +x 2 y 2 . The space (R 3 , , ) is called Minkowski 3-space (or,Lorentz-Minkowski 3-space) and denoted by R 3 1 . We say that a vector x in R 3 1 is spacelike, lightlike or timelike if x, x > 0, = 0 or < 0 respectively. We remark that the zero vector is considered to be lightlike in this paper. The norm of the vector x ∈ R 3 1 is defined by x = | x, x |. Given a vector n ∈ R 3 1 and a real number c, the plane with pseudo normal n is given by We say that P (n, c) is a spacelike , timelike or lightlike hyperplane if n is timelike, spacelike or lightlike respectively. For any point p ∈ R 3 1 , the lightcone with the vertex p is defined by We also define the lightcone circle by For any non zero lightlike vector x = (x 0 , x 1 , x 2 ), we denote that Moreover, the following hypersurface is called de Sitter sphere: For any x = (x 0 , x 1 , x 2 ), y = (y 0 , y 1 , y 2 ) ∈ R 3 1 , the pseudo vector product of x and y is defined as follows: In We call G M the Minkowski Gauss map of S = x(U ). We consider a surface in Minkowski 3-space such that the Minkowski Gauss map is singular at any point of the surface. We can show that such surfaces are developable surfaces. Proof. We consider the canonical Euclidean scalar product on R 3 1 : x · y = x 0 y 0 + x 1 y 1 + x 2 y 2 .
For any x = (x 0 , x 1 , x 2 ) ∈ R 3 1 , we denote that x = (−x 0 , x 1 , x 2 ). It follows that x and y are pseudo-orthogonal by the Minkowski scalar product if and only if x and y are orthogonal by the canonical Euclidean scalar product. We define a map G E : U −→ S 2 by where a E is the Euclidean norm of a. Then G E is the Gauss map of S = x(U ) in the Euclidean sense.
On the other hand we consider a mapping C : singular at a point p = x(u, v) if and only if G E is singular at p. This completes the proof. Since vectors in R 3 1 are classified into three kinds of vectors, RP 2 is a disjoint union of the disk D 2 , the circle S 1 and the Moebius strip M B such that x R ∈ D 2 if x is timelike,

Lightlike developables in Minkowski 3-space
In this section we study a special class of developable surfaces in Minkowski 3-space. By Theorem 3.1, if the Minkowski Gauss map is singular at any point of a surface, then the surface is a developable surface. The most interesting developable surfaces in Minkowski 3-space are surfaces whose pseudo normal field x u ∧ x v is always lightlike. We call such a surface a lightlike developable surface. Of course the lightlike developable surface is a developable surface, so that we can apply the classification theorem. Proof. Let x : U −→ R 3 1 be a lightlike developable surface. If the Minkowskian Gauss map G M is a point, then x(U ) is a part of a lightlike plane. We now assume that the image of the Minkowski Gauss map G M is a non-singular curve. By Theorem 2.1, a developable surface is a conical surface, a cylindrical surface, a tangent surface of a space curve or a glue of these three surfaces. We distinguish three cases.
(1) Suppose that a surface is a cylindrical surface x(t, u) = γ(t) + ue, where e is a constant vector. The pseudo normal vector is given by Suppose thatγ(t) ∧ e is lightlike. If the smooth curve γ(t) ∧ e is not a line, there exist three points t 0 , t 1 , t 2 ∈ R such that two pairsγ(t 0 ) ∧ e,γ(t 1 ) ∧ eγ(t 0 ) ∧ e,γ(t 2 ) ∧ e are consist of linearly independent vectors. Therefore we have two different lines Since γ(t) ∧ e, e = 0, we have e ∈ LP (γ(t) ∧ e, 0) for any t. However,we have This is a contradiction. Thereforeγ(t) ∧ e has a constant direction v. Sinceγ(t) ∈ LP (v, 0), we have γ(t), v = c. It follows from the fact e ∈ LP (v, 0) that x(t, u) ∈ LP (v, c), so that a lightlike cylindrical surface is a part of a lightlike plane.
(2) Suppose that a surface is a conical surface x(t, u) = a + ue(t), where a is a constant vector. The pseudo normal vector is given by Suppose thatė(t) ∧ e(t) is lightlike. We remark that the surface x(t, u) is the envelope of the family of tangent planes On the other hand, we consider a lightcone defined by We also consider a function F (X, t) = X − a,ė(t) ∧ e(t) . Then we have .
If we have derivative with respect to t, we have Therefore the lightcone x(t, v) is also the envelope of the same families of lightlike planes LP (ė(t) ∧ e(t), c(t)), so that the surface x(t, u) is a part of a lightcone.
Theorem 5.2 Let γ : I −→ R 3 1 be a smooth curve such thatγ ∈ L(I, R 3 1 ). Then we have the followings: Proof. We now calculate the type A of γ at t 0 under the above three conditions.
Therefore, the type of γ at t 0 is (2,3,4), so that the tangent surface germ F (γ, ė γ) (I × R) at (t 0 , 0) is diffeomorphic to the swallowtail. 2 By the standard jet transversality theorem (cf., [2], Theorem 4.9) we have the following proposition.  is open and dense in L(I, R 3 1 ). Therefore Theorem 5.2 gives a classification of singularities of the tangent surface of a generic lightlike curve.