Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

Abstract In this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.


Introduction
This paper is written as a part of our research project on the study of Lorentz pairs in semi-Euclidean space with index 2 from the viewpoint of Lagrangian/Legendrian singularity theory. Our aim is to investigate the geometric properties of different Lorentzian pairs by constructing a unified way. A Lorentzian pair consists of a Lorentzian hypersurface W in semi-Euclidean space with index two and a timelike hypersurface M in W . AdS 4 =AdS 5 , for example, is a Lorentzian pair which is one of the space-time models in physics. As the first step of this research, we consider the simplest Lorentzian pair, i.e., a timelike curve on a Lorentzian surface in semi-Euclidean 3-space with index 2. However, a Lorentz-Minkowski 3-space is diffeomorphic to a semi-Euclidean 3-space with index 2, although the causalities of these two spaces are different. For the geometric properties, we can investigate a spacelike curve on a timelike surface in Lorentz-Minkowski 3-space instead of a timelike curve on a Lorentzian surface in semi-Euclidean 3-space with index 2.
On the other hand, singularity theory tools are useful in the investigation of geometric properties of submanifolds immersed in different ambient spaces, from both the local and global viewpoint [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The natural connection between geometry and singularities relies on the basic fact that the contacts of a submanifold with the models (invariant under the action of a suitable transformation group) of the ambient space can be described by means of the analysis of the singularities of appropriate families of contact functions, or equivalently, of their associated Lagrangian and/or Legendrian maps. This is our main motivation for the investigation of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space from the viewpoint of singularity theory.
The organization of the paper is as follows. We construct the framework of local differential geometry of spacelike curves on a timelike surface in Section 2. We give the Frenet-Serret type formula corresponding to the spacelike curves. Moreover, we define the lightcone Gauss image and the normalized Gauss map. We also define new invariants KL and Q KL and call them lightcone Gauss-Kronecker curvature and normalized lightcone Gauss-Kronecker curvature, respectively. We investigate their relations. We can prove that these two Gauss-Kronecker curvature functions have the same zero sets. In Section 3 we introduce the notion of height functions on spacelike curves on a timelike surface which are useful to show that the normalized lightcone Gauss map has a singular point if and only if the lightcone Gauss-Kronecker curvature vanishes at such point. According to the general results on the singularity theory for families of function germs (cf. [17]), we study the relationship of these height functions (cf. Theorem 3.5). In the last section, we define a curve in the lightcone, named the lightcone pedal, as a tool to study the geometric properties of singularities of the normalized lightcone Gauss map from the contact viewpoint.

The local differential geometry of spacelike curves on a timelike surface
In this section, we investigate the basic ideas on semi-Euclidean (n+1)-space with index two and the local differential geometry of Lorentzian pairs in semi-Euclidean (n+1)-space. For details about semi-Euclidean geometry, see [18].
Let R 3 D f.x 1 ; x 2 ; x 3 /jx i 2 R; i D 1; 2; 3g be a 3-dimensional vector space. For any vectors x D .x 1 ; x 2 ; x 3 / and y D .y 1 ; y 2 ; y 3 / in R 3 ; the pseudo scalar product of x and y is defined to be hx; yi D x 1 y 1 C x 2 y 2 C x 3 y 3 . We call .R 3 ; h; i/ the Minkowski 3-space and write R 3 1 instead of .R 3 ; h; i/. We say that a non-zero vector x in R 3 1 is spacelike, lightlike or timelike if hx; xi > 0; hx; xi D 0 or hx; xi < 0 respectively. The norm of the vector x 2 R 3 1 is defined by kxk D p jhx; xij. For any x D .x 1 ; x 2 ; x 3 /; y D .y 1 ; y 2 ; y 3 / 2 R 3 1 , we define a vector x^y by x^y Dˇ where fe 1 ; e 2 ; e 3 g is the canonical basis of R 3 1 . For any w 2 R 3 1 , we can easily check that hw; x^yi D det.w; x; y/; so that x^y is pseudo-orthogonal to both x and y. Moreover, by a straightforward calculation, we have the following simple lemma.
Lemma 2.1. For any non-zero vectors x; y 2 R 3 1 , we assume that hx; yi D 0 and x^y D z. Then we have the following assertions: (1) If x is a timelike vector, y is a spacelike vector, then z^x D y; y^z D x: (2) If x is a spacelike vector and y is a timelike vector, then z^x D y; y^z D x: (3) If both x and y are spacelike vectors, then z^x D y; y^z D x: For a vector v 2 R 3 1 and a real number c, we define the plane with the pseudo-normal v by We call P .v; c/ a timelike plane, spacelike plane or lightlike plane if v is spacelike, timelike or lightlike, respectively. We define the hyperbolic 2-space by the (open) lightcone at the origin by LC D fx 2 R 3 1 n f0gj hx; xi D 0g: We call LC C D fx 2 LC jx 1 > 0g the future lightcone. We also define the spacelike lightcone circle by For any lightlike vector x D .x 1 ; x 2 ; x 3 / 2 LC , we have e x D .1; We study the local differential geometry of spacelike curves on a timelike surface as follows. Firstly, let Y W V ! R 3 1 be a regular surface (i.e., an embedding), where V R 2 is an open subset. We denote W D Y .V / and identify W with V via the embedding Y . The embedding Y is said to be timelike if the induced metric I of W is Lorentzian. Throughout the remainder of this paper we assume that W is a timelike surface in R 3 1 . We define a vector N .v/ by By the definition of wedge product, we have hN .v/; R is an open interval. We call this curve the spacelike curve, if t.t / D .d =dt /.t / is spacelike at any point t 2 I , and denote .I / D M . Since is a regular spacelike curve on W , it may admit the arc length parametrization s D s.t /. Therefore, we can assume that is a unit speed spacelike curve, namely, t.s/ D 0 .s/ D .d =ds/.s/ 2 S 2 1 . Throughout the remainder in this paper we assume that M is a spacelike curve on the timelike surface W .
We define a smooth mapping W I ! V by .s/ D v. For any s 2 I , we have .s/ D Y . .s//. It follows that the unit spacelike normal vector field of W along can be defined by n.s/ D N . .s//, for any s 2 I . We can also define another unit normal vector field e by e.s/ D n.s/^t.s/. Since t and n are spacelike, e is timelike. Then we have a pseudo-orthonormal frame ft.s/; n.s/; e.s/g of R 3 1 along .s/. By a straightforward calculation, we arrive at the following Frenet-Serret type formula: where Ä n .s/ D ht 0 .s/; n.s/i; Ä g .s/ D ht 0 .s/; e.s/i; g .s/ D he 0 .s/; n.s/i. We call them the normal curvature, geodesic curvature and geodesic torsion of at point p D .s/, respectively. We remark that is a spacelike geodesic curve on W if and only if Ä g Á 0; is a spacelike asymptotic curve on W if and only if Ä n Á 0; is a spacelike principal curve on W if and only if g Á 0.
On the other hand, suppose that e GL is constant. We assume that v D e GL . Then h .s/; vi 0 D 0. This means that h .s/; vi D c, where c 2 R is a constant real number. Therefore, is a part of P .v; c/ \ W . Moreover, we assume that there exists a constant lightlike vector v 2 S 1 C such that Im s/ for any s 2 I . It follows that the conditions (2) and (3) are equivalent. This completes the proof.
As an application of the above proposition, we have the following corollary.

Height functions on spacelike curves
In this section we define three families of functions on the spacelike curve on W which are helpful for investigating the geometric properties of the spacelike curve. Let W I ! W be a spacelike curve on W . We first define a function H W I S     On the other hand, we will introduce some general results on the singularity theory for families of function germs as follows. Let F W .R R r ; .s 0 ; x 0 // ! R be a function germ. We call F an r-parameter unfolding of f , where f .s/ D F .s; x 0 /. We say that f has an A k -singularity at s 0 if f .p/ .s 0 / D 0 for all 1 Ä p Ä k, and f .kC1/ .s 0 / ¤ 0. We also say that f has an A k -singularity at s 0 if f .p/ .s 0 / D 0 for all 1 Ä p Ä k. Let F be an unfolding of f and f .s/ have an A k -singularity (k 1) at s 0 . We denote the .k 1/-jet of the partial derivative @F @x i at s 0 by j k 1 . @F @x i .s; x 0 //.s 0 / D P k 1 j D1˛j i s j for i D 1; ; r. Then F is called an R C -versal unfolding if and only if the .k 1/ r matrix of coefficients .˛j i / has rank k 1 (k 1 Ä r). Moreover, we call F an R-versal unfolding if and only if the k r matrix of coefficients .˛0 i ;˛j i / has rank k (k Ä r), where˛0 i D @F @x i .s 0 ; x 0 /. We now introduce important sets concerning the unfoldings relative to the above notions. The discriminant set of F is the set D F D fx 2 R r j there exists s with F D @F @s D 0 at .s; x/g. The catastrophe set of F is the set The bifurcation set of F is the set B F D fx 2 R r j there exists s with @F @s D @ 2 F @s 2 D 0 at .s; x/g. Then we have the following well-known result (cf. [17]).

Lightcone pedal curves
We now define a mapping PL W I ! LC by PL .s/ D h .s/; e GL .s/i e GL .s/. We call it the lightcone pedal curve. We also define another mapping CPL W I ! We also say that the order of contact is k. Dropping the condition g .k/ .t 0 / ¤ 0 we say that there is at least k-point contact (cf. [17]). Then we have the following result.    Proof. We first consider (A). According to Corollary 2.3, the assertions (1) and (2)  .s/ ¤ 0. Therefore, the assertions (2) and (5) are equivalent. On the other hand, we consider (B). By Proposition 3.1 (2), the assertions (1) and (3)   .s 0 / D 0. Therefore, the assertions (2) and (3) are equivalent. Moreover, if we consider the mapping H W W ! R defined in the proof of (A), then we obtain that the assertions (3) and (4)  The previous discussion shows that CPL  (2), (3) and (5) are equivalent. By Proposition 3.5, we have the assertions (3), (6) and (7) are equivalent. This completes the proof.