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

1 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. AdS⁴/AdS⁵, 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 [1-16]. 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±andK~L± 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.

2 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 ℝ³ ={(x₁, x₂, x₃)|x_i ∈ ℝ, i = 1, 2, 3} be a 3-dimensional vector space. For any vectors x = (x₁, x₂, x₃ and y = (y₁, y₂, y₃ in ℝ³, the pseudo scalarproduct of x and y is defined to be 〈x,y〉 = -x₁y₁ + x₂y₂ + x₃y₃. We call (ℝ³, 〈,〉) the Minkowski 3-space and write ℝ₁³ instead of (ℝ³, 〈,〉).

We say that a non-zero vector x in ℝ₁³ is spacelike, lightlike or timelike if 〈x,x〉 > 0, 〈x,x〉 = 0 or 〈x,x〉 > 0 respectively. The norm of the vector x ∈ ℝ₁³ is defined by ∥x∥=|{x;x}|.

For any x = (x₁, x₂, x₃), y = y₁, y₂, y₃ ∈ ℝ₁³, we define a vector x ∧ y by

where {e₁, e₂, e₃} is the canonical basis of ℝ₁³. For any ω ∈ ℝ₁³, we can easily check that

so that x∧y is pseudo-orthogonal to both x and y. Moreover, by a straightforward calculation, we have the following simple lemma.

For a vector ν ∈ ℝ₁³ and a real number c, we define the plane with the pseudo-normal ν by

We call P(ν,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 de Sitter 2-space by

the (open) lightcone at the origin by

We call LC+∗={x∈LC∗|x1>0} the future lightcone. We also define the spacelike lightcone circle by

For any lightlike vector x = (x₁, x₂, x₃) ∈ LC^*, we have

We study the local differential geometry of spacelike curves on a timelike surface as follows. Firstly, let Y : V → ℝ₁³ be a regular surface (i.e., an embedding), where V ⊂ ℝ² is an open subset. We denote W = 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 ℝ₁³. We define a vector N(v) by

By the definition of wedge product, we have 〈N(ν), Y_νi (u) = 0 (for i = 1,2). This means that N(ν) ∈ N_qW, where ν = (ν₁, ν₂) ∈ V, q = Y (ν) ∈ W and N_qW denotes the normal space of W in ℝ₁³ at q. Since the embedding Y is timelike, N is unit spacelike (i.e., N(ν) ( S₁²)$ . Moreover, we define a regular curve on W by γ : I → W, where I ⊂ ℝ is an open interval. We call this curve the spacelike curve, if t(t) = (dγ/dt)(t) is spacelike at any point t Ȉ I, and denote γ(I) = M. Since γ is a regular spacelike curve on W, it may admit the arc length parametrization s = s(t) . Therefore, we can assume that γ is a unit speed spacelike curve, namely, t(s)=γ′(s)=(dγ/ds)(s)∈S12. Throughout the remainder in this paper we assume that M is a spacelike curve on the timelike surface W.

We define a smooth mapping γ¯:I→Vbyγ¯(s)=v. For any s Ȉ I, we have γ(s)=Y(γ¯(s)). It follows that the unit spacelike normal vector field of W along γ can be defined by n(s)=N(γ¯(s)), for any s Ȉ I. We can also define another unit normal vector field e by e(s) = n(s) ∧ t(s). Since t and n are spacelike, e is timelike. Then we have a pseudo-orthonormal frame {t(s), n(s), e(s)} of ℝ₁³ along γ(s). By a straightforward calculation, we arrive at the following Frenet-Serret type formula:

where κn(s)={t′(s),n(s)},κg(s)={t′(s),e(s)},τg(s)={e′(s),n(s)}. We call them the normal curvature, geodesic curvature and geodesic torsion of γ at point p = γ(s) , respectively. We remark that γ is a spacelike geodesic curve on W if and only if κ_g = 0; γ is a spacelike asymptotic curve on W if and only if κ_n = 0; γ is a spacelike principal curve on W if and only if τ_g = 0.

Let n(s) = (n₁(s), n₂(s), n₃(s)) and e(s) = (e₁(s), e₂(s), e₃(s)). Since n(s) ± e(s)Ȉ LC^*, we can define a mapping GL±:I→LC∗byGL±(s)=n(s)±e(s). We call it the lightcone Gauss image. Moreover, we define another mapping GL±:I→S+1byGL±(s)=n(s)±e(s)=1ξ±(s)GL±(s), where ξ±(s)=n1(s)±e1(s). We call it the normalized lightcone Gauss map. By a straightforward calculation, we have

Let πT:TpM⊕NpM→TpM. It follows that −πT∘GL±′(s)=(κn(s)±κg(s))t(s). Moreover, we also have −πT∘GL±′(s)=1ξ±(s)(κn(s)±κg(s))t(s). According to the above calculations, we can define new invariants KL± and K~L± by KL±(s)=κn(s)±κg(s) and K¯L±(s)=1ξ±(s)(κn(s)±κg(s)), respectively. We call them the lightcone Gauss-Kronecker curvature (or, lightcone G-K curvature) of M and normalized lightcone Gauss-Kronecker curvature (or, normalized lightcone G-K curvature) of M, respectively. By the definitions, we have KL±(s)=0 if and only if K~L±(s)=0, for sȈ I.

For v∈S+1,c∈R we denote SL(v,c)=P(v,c)∩W and call it the lightlike slice. Then we have the following proposition.

3 Height functions on spacelike curves

In this section we define three families of functions on the spacelike curve γ on W which are helpful for investigating the geometric properties of the spacelike curve.

Let γ : I → W be a spacelike curve on W. We first define a functionH:I×S+1→RbyH(s,v)=〈γ(s),v〉. We call it the lightcone circle height function on γ. For any fixed ν ∈ S₊¹, we denote h_v(s)= H(s,v). Then we have the following proposition.

By a straightforward calculation, we have

It follows that

For s=s₀, we suppose that −r2′(s0)sin⁡θ0+r3′(s)cos⁡θ0=0 and t(s0)=(r1′(s0),r2′(s0),r3′(s0)). Since 〈t(s0),v0〉=−r1′(s0)+r2′(s0)cos⁡θ0+r3′(s0)sin⁡θ0=0,, we have −r1′(s0)sin⁡θ0+r3′(s0)=0. Moreover, we also have −r1′(s0)cos⁡θ0+r2′(s0)=0. Therefore, t(s0)=r1′(s0)(1;cos⁡θ0;sin⁡θ0). It follows that t(s) is a unit spacelike vector for any s ∈ I, so we have a contradiction. Thus, (∂2H/∂s∂θ)(s0,θ0)≠0 Therefore, the rank of the matrix (−r2′(s0)sin⁡θ0+r3′(s)cos⁡θ0)is one. We have thus shown that the assertions (1) and (2) are equivalent

On the other hand, for extended lightcone circle height function H¯, we have

Thus,

It follows that

Suppose that

Since (∂2H/∂s∂θ)(s0,θ0)≠0, we have (∂2H¯/∂s∂θ)(s0,θ0,r0)≠0. This means that rankA =2. Therefore, the assertions (1) and (3) are equivalent. This completes the proof.

4 Lightcone pedal curves

We now define a mapping PL±:I→LC∗byPL±(s)=〈γ(s),G~L±(s)〉G~L±(s).

We call it the lightcone pedal curve. We also define another mapping CPL±:I→S+1×R∗byCPL±(s)=(G~L±(s),〈γ(s),G~L±(s)〉), where ℝ ^* = ℝ \ {0}. We call CP_L^± the cylindrical lightcone pedal curve. By definitions, we have {PL±(s)|s∈I}=DH~ and {CPL±(s)|s∈I}=DH¯. In order to investigate the relationship between the lightcone pedal curve and the cylindrical lightcone pedal, we define a mapping ϕ:S+1×R∗→LC∗byϕ(v,r)=rv. It is easy to check that Φ is a diffeomorphism and ϕ(CPL±(s))=PL±(s), this means that CP_L^± and P_L^± are diffeomorphic. Therefore, the singular sets of DH~andDH¯ are diffeomorphic.

On the other hand, let F : W → ℝ be a submersion and γ : I → W be a spacelike curve on W. We say that γ and F⁻¹(0) have k-point contact for t=t₀ provided the function g defined by g(t)=F(γ(t)) satisfies

We also say that the order of contact is k. Dropping the condition g^(k) (t₀) ≠ 0 we say that there is at least k-point contact (cf. [17]). Then we have the following result.