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Spacelike surfaces in Minkowski 4−4-space with a canonical normal null direction

Victor H. Patty-Yujra
E-mail:  [email protected]
Carrera de Matemática – Instituto de Investigación Matemática
Facultad de Ciencias Puras y Naturales
Universidad Mayor de San Andrés, Bolivia
Abstract

A canonical normal null direction on a spacelike surface in the four dimensional Minkowski space ℝ3,1\mathbb{R}^{3,1} is a parallel vector field ZZ on ℝ3,1\mathbb{R}^{3,1} such that the normal component of ZZ on the surface is a lightlike vector field. We describe the geometric properties of a spacelike surface endowed with a canonical normal null direction and we obtain some characterizations of these surfaces. Moreover, using their Gauss map we study other properties of these surface: the associated ellipse of curvature and their asymptotic directions. Finally, we give two different ways to create these surfaces, one of them involves a nonlinear partial differential equation.

 

2010 Mathematics Subject Classification:

53B25, 53C42.

Keywords:

Spacelike surfaces, Canonical normal null direction, Asymptotic direction.

 

Introduction

Consider the four dimensional Minkowski space ℝ3,1\mathbb{R}^{3,1} defined by ℝ4\mathbb{R}^{4} endowed with the tensor metric of signature (3,1),(3,1),

⟨⋅,⋅⟩=d​x12+d​x22+d​x32−d​x42.\langle\cdot,\cdot\rangle=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-dx_{4}^{2}.

A surface M⊂ℝ3,1M\subset\mathbb{R}^{3,1} is said to be spacelike if the metric ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle induces a Riemannian metric on M,M, thus, at each point pp of a spacelike surface M,M, the Minkowski space is split as ℝ3,1=Tp​M⊕Np​M,\mathbb{R}^{3,1}=T_{p}M\oplus N_{p}M, where the tangent plane Tp​MT_{p}M and the normal plane Np​MN_{p}M at pp are respectively equipped with a metric of signature (2,0)(2,0) and (1,1)(1,1) (see for example [5]).

Definition 1.

We say that a spacelike surface M⊂ℝ3,1M\subset\mathbb{R}^{3,1} has a canonical normal null direction with respect to a parallel vector field ZZ in ℝ3,1\mathbb{R}^{3,1} if the normal part Z⟂Z^{\perp} of ZZ is a lightlike normal vector field on M.M.

The previous definition on the notion of canonical normal null direction is the mean concept in this paper. It makes sense for spacelike submanifolds, not only for surfaces, in the n−n-dimensional Minkowski space. It is inspired in the concept of timelike surfaces with a canonical null direction with respect to a parallel vector field in Minkowski space defined by the principal author and G. Ruiz in [6]: a canonical null direction on a timelike surface is given as the tangent part of the parallel vector field. We can also related with the notion of canonical principal direction on a surface defined by F. Dillen and his collaborators in [2, 3]: they defined a canonical principal direction as a principal direction of the surface given as the tangent part of the parallel vector field. Finally, E. Garnica, G. Ruiz and O. Palmas in [4] investigated the case of hypersurfaces with a canonical principal direction with respect to a closed conformal vector field.

In this paper, we are interested in the description of the geometric properties of a spacelike surface endowed with a canonical normal null direction in the four dimensional Minkowski space.

This paper is organized as follows. In Section 1, we study the fundamental equations which determine a canonical normal null direction on a spacelike surface in Minkowski space and we get properties about their geometry in terms of a differentiable function and a differential 1−1-form on the surface. An important consequence is given in Proposition 1.15: if the spacelike surface is minimal and has a canonical normal null direction then it is flat and has flat normal bundle. We also characterize these surfaces, in some particular cases, we get that they are ruled surfaces.

In Section 2, we describe the Gauss map of a spacelike surface endowed with a canonical normal null direction using bivectors of the Minkowski space and the Grassmannian of the oriented spacelike 2-planes. We also describe the parametrization of the ellipse of curvature associated, we find the mean curvature directions and the asymptotic directions on the surface; for example, we prove that Z⊤Z^{\top} is an asymptotic direction on the surface and the existence of another asymptotic direction depends on the sign of the Gauss curvature K.K.

In Section 3, we give two different forms to building spacelike surfaces endowed with a canonical normal null direction. The first one consists of translation spacelike surfaces in Minkowski space, that is, surfaces given by the sum of curves. The second one being the graphs of differential functions, in this case, we prove in Proposition 3.2 that such surfaces have a canonical normal null direction respect to the vector field e1e_{1} if and only if the coefficients of the first fundamental form satisfy the fully nonlinear partial differential equation on a open set of ℝ2\mathbb{R}^{2}

(1+fy2)​gx2−2​fx​fy​gx​gy−(1−gy2)​fx2=0.(1+f_{y}^{2})g_{x}^{2}-2f_{x}f_{y}g_{x}g_{y}-(1-g_{y}^{2})f_{x}^{2}=0.

Finally, using conformal functions over the Lorentz numbers, we construct some particular solutions of this partial differential equation.

1 Fundamental equations

We consider a spacelike surface MM in ℝ3,1\mathbb{R}^{3,1} with a given canonical normal null direction Z.Z. Suppose that ZZ is a unit spacelike vector field, in this case, using the natural decomposition

Z=Z⊤+Z⟂∈T​M⊕T​N≃ℝ3,1,Z=Z^{\top}+Z^{\perp}\ \in\ TM\oplus TN\simeq\mathbb{R}^{3,1},

and since ⟨Z⟂,Z⟂⟩=0,\langle Z^{\perp},Z^{\perp}\rangle=0, we have that ⟨Z⊤,Z⊤⟩=1.\langle Z^{\top},Z^{\top}\rangle=1. Here and below we denote by ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle the metric on the Minkowski space ℝ3,1,\mathbb{R}^{3,1}, on the tangent bundle T​MTM and on the normal bundle N​M.NM.

We denote by B:T​M×T​M→N​MB:TM\times TM\to NM the second fundamental form of the immersion M⊂ℝ3,1M\subset\mathbb{R}^{3,1} given by

B​(X,Y)=∇¯X​Y−∇XY,B(X,Y)=\overline{\nabla}_{X}Y-\nabla_{X}Y,

where ∇¯\overline{\nabla} and ∇\nabla are the Levi Civita connections of ℝ3,1\mathbb{R}^{3,1} and M,M, respectively. Moreover, if ν∈N​M,\nu\in NM, Aν:T​M→T​MA_{\nu}:TM\to TM stands for the symmetric operator such that

⟨Aν​(X),Y⟩=⟨B​(X,Y),ν⟩,\langle A_{\nu}(X),Y\rangle=\langle B(X,Y),\nu\rangle,

for all X,Y∈T​M.X,Y\in TM. Finally, we denote by ∇⟂\nabla^{\perp} the Levi Civita connection of the normal bundle N​MNM of the surface M.M.

Proposition 1.1.

Let MM be a spacelike surface in ℝ3,1\mathbb{R}^{3,1} with a canonical normal null direction Z,Z, then the following formulas are satisfied

∇XZ⊤=AZ⟂​(X)and∇X⟂Z⟂=−B​(X,Z⊤)\nabla_{X}Z^{\top}=A_{Z^{\perp}}(X)\hskip 21.68121pt\mbox{and}\hskip 21.68121pt\nabla^{\perp}_{X}Z^{\perp}=-B(X,Z^{\top}) (1)

for all X∈T​M.X\in TM.

Proof.

Using the Gauss and Weingarten equations of the immersion, we have that

0=∇¯X​Z\displaystyle 0=\overline{\nabla}_{X}Z =∇¯X​Z⊤+∇¯X​Z⟂\displaystyle=\overline{\nabla}_{X}Z^{\top}+\overline{\nabla}_{X}Z^{\perp}
=[∇XZ⊤+B​(X,Z⊤)]+[−AZ⟂​(X)+∇X⟂Z⟂]\displaystyle=\left[\nabla_{X}Z^{\top}+B(X,Z^{\top})\right]+\left[-A_{Z^{\perp}}(X)+\nabla^{\perp}_{X}Z^{\perp}\right]

for all X∈T​M.X\in TM. We obtain the results by taking the tangent and normal parts of this equality. ∎

Lemma 1.2.

The following identities are satisfied

AZ⟂​(Z⊤)=0and∇Z⊤Z⊤=0.A_{Z^{\perp}}(Z^{\top})=0\hskip 21.68121pt\mbox{and}\hskip 21.68121pt\nabla_{Z^{\top}}Z^{\top}=0.

In particular, we have ⟨B​(X,Z⊤),Z⟂⟩=0,\langle B(X,Z^{\top}),Z^{\perp}\rangle=0, for all X∈T​M.X\in TM.

Proof.

From (1), we have

0=X​⟨Z⊤,Z⊤⟩=2​⟨∇XZ⊤,Z⊤⟩=2​⟨AZ⟂​(X),Z⊤⟩=2​⟨X,AZ⟂​(Z⊤)⟩,0=X\langle Z^{\top},Z^{\top}\rangle=2\langle\nabla_{X}Z^{\top},Z^{\top}\rangle=2\langle A_{Z^{\perp}}(X),Z^{\top}\rangle=2\langle X,A_{Z^{\perp}}(Z^{\top})\rangle,

for all X∈T​M.X\in TM. Thus ∇Z⊤Z⊤=AZ⟂​(Z⊤)=0.\nabla_{Z^{\top}}Z^{\top}=A_{Z^{\perp}}(Z^{\top})=0. As a consequence, we obtain

⟨B​(X,Z⊤),Z⟂⟩=⟨AZ⟂​(X),Z⊤⟩=⟨X,AZ⟂​(Z⊤)⟩=0,\langle B(X,Z^{\top}),Z^{\perp}\rangle=\langle A_{Z^{\perp}}(X),Z^{\top}\rangle=\langle X,A_{Z^{\perp}}(Z^{\top})\rangle=0,

for all X∈T​M.X\in TM. ∎

Let us consider WW a unit spacelike vector field tangent to MM such that ⟨Z⊤,W⟩=0\langle Z^{\top},W\rangle=0 and (Z⊤,W)(Z^{\top},W) is positively oriented.

We define the differential 1−1-form α:T​M→ℝ\alpha:TM\to\mathbb{R} given by

α​(X)=⟨B​(X,W),Z⟂⟩\alpha(X)=\langle B(X,W),Z^{\perp}\rangle (2)

for all X∈T​M.X\in TM. For the particular case X=W,X=W, we denote by a:=α​(W).a:=\alpha(W).

Lemma 1.3.

The Levi Civita connection of MM satisfies the following relations:

∇Z⊤Z⊤=0,∇WZ⊤=a​W,∇Z⊤W=0​and​∇WW=−a​Z⊤.\nabla_{Z^{\top}}Z^{\top}=0,\hskip 14.45377pt\nabla_{W}Z^{\top}=aW,\hskip 14.45377pt\nabla_{Z^{\top}}W=0\hskip 7.22743pt\mbox{and}\hskip 7.22743pt\nabla_{W}W=-aZ^{\top}.

In particular, [Z⊤,W]=−a​W.[Z^{\top},W]=-aW.

Proof.

The first equality was given in Lemma 1.2. Now, since 0=W​⟨Z⊤,Z⊤⟩=2​⟨∇WZ⊤,Z⊤⟩0=W\langle Z^{\top},Z^{\top}\rangle=2\langle\nabla_{W}Z^{\top},Z^{\top}\rangle and ⟨∇WZ⊤,W⟩=⟨AZ⟂​(W),W⟩=⟨B​(W,W),Z⟂⟩=a,\langle\nabla_{W}Z^{\top},W\rangle=\langle A_{Z^{\perp}}(W),W\rangle=\langle B(W,W),Z^{\perp}\rangle=a, therefore

∇WZ⊤=⟨∇WZ⊤,Z⊤⟩​Z⊤+⟨∇WZ⊤,W⟩​W=a​W.\nabla_{W}Z^{\top}=\langle\nabla_{W}Z^{\top},Z^{\top}\rangle Z^{\top}+\langle\nabla_{W}Z^{\top},W\rangle W=aW.

In a similar way, we have 0=Z⊤​⟨W,Z⊤⟩=⟨∇Z⊤W,Z⊤⟩+⟨W,∇Z⊤Z⊤⟩=⟨∇Z⊤W,Z⊤⟩0=Z^{\top}\langle W,Z^{\top}\rangle=\langle\nabla_{Z^{\top}}W,Z^{\top}\rangle+\langle W,\nabla_{Z^{\top}}Z^{\top}\rangle=\langle\nabla_{Z^{\top}}W,Z^{\top}\rangle and 0=Z⊤​⟨W,W⟩=2​⟨∇Z⊤W,W⟩,0=Z^{\top}\langle W,W\rangle=2\langle\nabla_{Z^{\top}}W,W\rangle, thus

∇Z⊤W=⟨∇Z⊤W,Z⊤⟩​Z⊤+⟨∇Z⊤W,W⟩​W=0.\nabla_{Z^{\top}}W=\langle\nabla_{Z^{\top}}W,Z^{\top}\rangle Z^{\top}+\langle\nabla_{Z^{\top}}W,W\rangle W=0.

On the other hand, 0=W​⟨W,W⟩=2​⟨∇WW,W⟩0=W\langle W,W\rangle=2\langle\nabla_{W}W,W\rangle and 0=W​⟨Z⊤,W⟩=⟨∇WZ⊤,W⟩+⟨Z⊤,∇WW⟩,0=W\langle Z^{\top},W\rangle=\langle\nabla_{W}Z^{\top},W\rangle+\langle Z^{\top},\nabla_{W}W\rangle, thus ⟨Z⊤,∇WW⟩=−⟨∇WZ⊤,W⟩=−⟨a​W,W⟩=−a,\langle Z^{\top},\nabla_{W}W\rangle=-\langle\nabla_{W}Z^{\top},W\rangle=-\langle aW,W\rangle=-a, therefore

∇WW=⟨∇WW,Z⊤⟩​Z⊤+⟨∇WW,W⟩​W=−a​Z⊤.\nabla_{W}W=\langle\nabla_{W}W,Z^{\top}\rangle Z^{\top}+\langle\nabla_{W}W,W\rangle W=-aZ^{\top}.

Finally, [Z⊤,W]=∇Z⊤W−∇WZ⊤=−a​W.[Z^{\top},W]=\nabla_{Z^{\top}}W-\nabla_{W}Z^{\top}=-aW. ∎

Now, in the following results we describe the curvature tensors of the surface M.M.

Proposition 1.4.

The curvature tensor RR and the normal curvature tensor R⟂R^{\perp} of the surface M,M, in the basis (Z⊤,W),(Z^{\top},W), are given by

R​(Z⊤,W)​Z⊤=(−Z⊤​(a)−a2)​WandR⟂​(Z⊤,W)​Z⟂=−a​B​(Z⊤,W).R(Z^{\top},W)Z^{\top}=\left(-Z^{\top}(a)-a^{2}\right)W\hskip 21.68121pt\mbox{and}\hskip 21.68121ptR^{\perp}(Z^{\top},W)Z^{\perp}=-aB(Z^{\top},W).
Proof.

Using the equalities in the Lemma 1.3, we get

R​(Z⊤,W)​Z⊤\displaystyle R(Z^{\top},W)Z^{\top} =∇W∇Z⊤⁡Z⊤−∇Z⊤∇W⁡Z⊤+∇[Z⊤,W]Z⊤\displaystyle=\nabla_{W}\nabla_{Z^{\top}}Z^{\top}-\nabla_{Z^{\top}}\nabla_{W}Z^{\top}+\nabla_{[Z^{\top},W]}Z^{\top}
=−∇Z⊤(a​W)+∇(−a​W)Z⊤\displaystyle=-\nabla_{Z^{\top}}(aW)+\nabla_{(-aW)}Z^{\top}
=−Z⊤​(a)​W−a​∇Z⊤W−a​∇WZ⊤\displaystyle=-Z^{\top}(a)W-a\nabla_{Z^{\top}}W-a\nabla_{W}Z^{\top}
=−Z⊤​(a)​W−a​(a​W)\displaystyle=-Z^{\top}(a)W-a(aW)
=(−Z⊤​(a)−a2)​W.\displaystyle=\left(-Z^{\top}(a)-a^{2}\right)W.

On the other hand, by (1) we obtain

R⟂​(Z⊤,W)​Z⟂\displaystyle R^{\perp}(Z^{\top},W)Z^{\perp} =∇W⟂∇Z⊤⟂⁡Z⟂−∇Z⊤⟂∇W⟂⁡Z⟂+∇[Z⊤,W]⟂Z⟂\displaystyle=\nabla^{\perp}_{W}\nabla^{\perp}_{Z^{\top}}Z^{\perp}-\nabla^{\perp}_{Z^{\top}}\nabla^{\perp}_{W}Z^{\perp}+\nabla^{\perp}_{[Z^{\top},W]}Z^{\perp}
=∇W⟂(−B​(Z⊤,Z⊤))−∇Z⊤⟂(−B​(W,Z⊤))+∇(−a​W)⟂Z⟂\displaystyle=\nabla^{\perp}_{W}\left(-B(Z^{\top},Z^{\top})\right)-\nabla^{\perp}_{Z^{\top}}\left(-B(W,Z^{\top})\right)+\nabla^{\perp}_{(-aW)}Z^{\perp}
=−∇W⟂(B​(Z⊤,Z⊤))+∇Z⊤⟂(B​(W,Z⊤))+a​B​(W,Z⊤);\displaystyle=-\nabla^{\perp}_{W}\left(B(Z^{\top},Z^{\top})\right)+\nabla^{\perp}_{Z^{\top}}\left(B(W,Z^{\top})\right)+aB(W,Z^{\top});

by Codazzi equation and the equalities in the Lemma 1.3, we get

−∇W⟂(B​(Z⊤,Z⊤))+∇Z⊤⟂(B​(W,Z⊤))\displaystyle-\nabla^{\perp}_{W}\left(B(Z^{\top},Z^{\top})\right)+\nabla^{\perp}_{Z^{\top}}\left(B(W,Z^{\top})\right) =−(∇~W​B)​(Z⊤,Z⊤)−B​(∇WZ⊤,Z⊤)−B​(Z⊤,∇WZ⊤)\displaystyle=-(\tilde{\nabla}_{W}B)(Z^{\top},Z^{\top})-B(\nabla_{W}Z^{\top},Z^{\top})-B(Z^{\top},\nabla_{W}Z^{\top})
+(∇~Z⊤​B)​(W,Z⊤)+B​(∇Z⊤W,Z⊤)+B​(W,∇Z⊤Z⊤)\displaystyle\ \ +(\tilde{\nabla}_{Z^{\top}}B)(W,Z^{\top})+B(\nabla_{Z^{\top}}W,Z^{\top})+B(W,\nabla_{Z^{\top}}Z^{\top})
=−2​a​B​(W,Z⊤),\displaystyle=-2aB(W,Z^{\top}),

this finishes the proof. ∎

Corollary 1.5.

The Gauss curvature of the surface MM is given by

K=⟨R​(Z⊤,W)​Z⊤,W⟩|Z⊤|2​|W|2−⟨Z⊤,W⟩=−Z⊤​(a)−a2.K=\dfrac{\langle R(Z^{\top},W)Z^{\top},W\rangle}{|Z^{\top}|^{2}|W|^{2}-\langle Z^{\top},W\rangle}=-Z^{\top}(a)-a^{2}.

Another way to compute the Gauss curvature of the surface MM in terms of the differential 1−1-form defined in (2), is given by the following proposition.

Proposition 1.6.

The Gauss curvature of the surface MM is given by

K=−d​α​(Z⊤,W)K=-d\alpha(Z^{\top},W)
Proof.

Using Lemma 1.2, we easily get

d​α​(Z⊤,W)\displaystyle d\alpha(Z^{\top},W) =Z⊤​(α​(W))−W​(α​(Z⊤))−α​([Z⊤,W])\displaystyle=Z^{\top}(\alpha(W))-W(\alpha(Z^{\top}))-\alpha([Z^{\top},W])
=Z⊤​(a)−W​(⟨B​(Z⊤,W),Z⟂⟩)−α​(−a​W)\displaystyle=Z^{\top}(a)-W(\langle B(Z^{\top},W),Z^{\perp}\rangle)-\alpha(-aW)
=Z⊤​(a)+a2;\displaystyle=Z^{\top}(a)+a^{2};

the by Corollary 1.5, we obtain the result. ∎

According to Lemma 1.2 and since Z⟂Z^{\perp} is a lightlike vector field on the surface, there exist a differential 1−1-form β:T​M→ℝ\beta:TM\to\mathbb{R} given by

B​(X,Z⊤)=β​(X)​Z⟂B(X,Z^{\top})=\beta(X)Z^{\perp} (3)

for all X∈T​M.X\in TM. This 1−1-form β\beta allows as to compute the normal curvature of the surface. We consider the lightlike vector field W′W^{\prime} normal to MM such that ⟨Z⟂,W′⟩=1\langle Z^{\perp},W^{\prime}\rangle=1 and

(Z⊤,W,Z⟂+W′2,Z⟂−W′2)\left(Z^{\top},W,\frac{Z^{\perp}+W^{\prime}}{\sqrt{2}},\frac{Z^{\perp}-W^{\prime}}{\sqrt{2}}\right)

is an orthonormal and positively oriented frame of ℝ3,1.\mathbb{R}^{3,1}.

Corollary 1.7.

The normal curvature of the surface MM is given by

KN=−a​β​(W).K_{N}=-a\beta(W).
Proof.

From the Ricci equation in the orthonormal frame (Z⊤,W)(Z^{\top},W) on T​M,TM,

KN\displaystyle K_{N} =⟨(AZ⟂+W′2∘AZ⟂−W′2−AZ⟂−W′2∘AZ⟂+W′2)​(Z⊤),W⟩\displaystyle=\left\langle\left(A_{\frac{Z^{\perp}+W^{\prime}}{\sqrt{2}}}\circ A_{\frac{Z^{\perp}-W^{\prime}}{\sqrt{2}}}-A_{\frac{Z^{\perp}-W^{\prime}}{\sqrt{2}}}\circ A_{\frac{Z^{\perp}+W^{\prime}}{\sqrt{2}}}\right)\left(Z^{\top}\right),W\right\rangle
=−⟨(AZ⟂∘AW′−AW′∘AZ⟂)​(Z⊤),W⟩\displaystyle=-\left\langle\left(A_{Z^{\perp}}\circ A_{W^{\prime}}-A_{W^{\prime}}\circ A_{Z^{\perp}}\right)\left(Z^{\top}\right),W\right\rangle
=⟨R⟂​(Z⊤,W)​Z⟂,W′⟩.\displaystyle=\langle R^{\perp}(Z^{\top},W)Z^{\perp},W^{\prime}\rangle.

Thus we get the desired result by replacing the second equality given in Proposition 1.4 and the definition of the 1−1-form β\beta in (3). ∎

Analogously, the normal curvature of the surface is given by the exterior derivative of the differential 1−1-form β.\beta.

Lemma 1.8.

The differential 1−1-forms α\alpha and β\beta are related by the identity

Z⊤​(β​(W))−W​(β​(Z⊤))=−2​α​(W)​β​(W).Z^{\top}(\beta(W))-W(\beta(Z^{\top}))=-2\alpha(W)\beta(W).
Proof.

In terms of the 1−1-form β,\beta, the Codazzi equation reads as

−∇W⟂(β​(Z⊤)​Z⟂)+∇Z⊤⟂(β​(W)​Z⟂)=−2​a​β​(W)​Z⟂,-\nabla^{\perp}_{W}\left(\beta(Z^{\top})Z^{\perp}\right)+\nabla^{\perp}_{Z^{\top}}\left(\beta(W)Z^{\perp}\right)=-2a\beta(W)Z^{\perp},

(see the last part of the proof of Proposition 1.4), thus we easily get

[Z⊤​(β​(W))−W​(β​(Z⊤))]​Z⟂=−2​a​β​(W)​Z⟂\left[Z^{\top}(\beta(W))-W(\beta(Z^{\top}))\right]Z^{\perp}=-2a\beta(W)Z^{\perp}

which implies the result. ∎

Proposition 1.9.

We have the following formula

KN=d​β​(Z⊤,W)K_{N}=d\beta(Z^{\top},W)
Proof.

From Lemma 1.8, we get

d​β​(Z⊤,W)\displaystyle d\beta(Z^{\top},W) =Z⊤​(β​(W))−W​(β​(Z⊤))−β​([Z⊤,W])\displaystyle=Z^{\top}(\beta(W))-W(\beta(Z^{\top}))-\beta([Z^{\top},W])
=−2​α​(W)​β​(W)−β​(−a​W)\displaystyle=-2\alpha(W)\beta(W)-\beta(-aW)
=−2​a​β​(W)+a​β​(W)\displaystyle=-2a\beta(W)+a\beta(W)
=−a​β​(W),\displaystyle=-a\beta(W),

and combining with the Corollary 1.7, we obtain the result. ∎

An alternative way to obtain the Gauss curvature in terms of the 1−1-form β\beta is given in the following colollary.

Corollary 1.10.

The Gauss curvature of the surface MM is given by

K=a​β​(Z⊤)K=a\beta(Z^{\top})
Proof.

Since a=⟨B​(W,W),Z⟂⟩,a=\langle B(W,W),Z^{\perp}\rangle, using the expression given in Corollary 1.5 for the Gauss curvature, we have

K=−Z⊤​(a)−a2\displaystyle K=-Z^{\top}(a)-a^{2} =−Z⊤​⟨B​(W,W),Z⟂⟩−a2\displaystyle=-Z^{\top}\langle B(W,W),Z^{\perp}\rangle-a^{2}
=−⟨∇W⟂(B​(Z⊤,W)),Z⟂⟩+a2+a​β​(Z⊤)−a2.\displaystyle=-\langle\nabla_{W}^{\perp}(B(Z^{\top},W)),Z^{\perp}\rangle+a^{2}+a\beta(Z^{\top})-a^{2}.

But, since W​⟨B​(Z⊤,W),Z⟂⟩=0,W\langle B(Z^{\top},W),Z^{\perp}\rangle=0, then ⟨∇W⟂(B​(Z⊤,W)),Z⟂⟩=0,\langle\nabla_{W}^{\perp}(B(Z^{\top},W)),Z^{\perp}\rangle=0, which brings us to the desired result. ∎

The geometric interpretation of the 1−1-form β\beta is given in the following proposition.

Proposition 1.11.

Z⟂Z^{\perp} is a parallel normal vector field on MM if and only if β≡0.\beta\equiv 0. In particular, if Z⟂Z^{\perp} is a parallel normal vector field on M,M, then K=KN=0,K=K_{N}=0, i.e., MM is flat and has a flat normal bundle.

Proof.

Using the second equality in (1), and the definition of β,\beta, we obtain

∇X⟂Z⟂=−B​(X,Z⊤)=−β​(X)​Z⟂,\nabla^{\perp}_{X}Z^{\perp}=-B(X,Z^{\top})=-\beta(X)Z^{\perp},

for all X∈T​M,X\in TM, which implies the result. The particular case is a consequence of Corollaries 1.7 - 1.10. ∎

A partial reciprocal assertion of the particular case in the previous proposition is given as a consequence of the following proposition.

Proposition 1.12.

If a=α​(W)≠0,a=\alpha(W)\neq 0, then the 1−1-form β\beta is given by

β​(X)=⟨X,Ka​Z⊤+−KNa​W⟩\beta(X)=\left\langle X,\frac{K}{a}Z^{\top}+\frac{-K_{N}}{a}W\right\rangle

for all X∈T​M.X\in TM. In particular, if a≠0a\neq 0 and K=KN=0K=K_{N}=0 then Z⟂Z^{\perp} is a parallel normal vector field.

Proof.

Since a≠0,a\neq 0, the first assertion is a direct consequence of Corollaries 1.7-1.10. If a≠0a\neq 0 and K=KN=0,K=K_{N}=0, then β≡0,\beta\equiv 0, the previous proposition implies the result. ∎

Remark 1.13.

If we consider the orthonormal frame (Z⊤,W)(Z^{\top},W) on T​M,TM, the mean curvature vector of the immersion M⊂ℝ3,1M\subset\mathbb{R}^{3,1} is given by

H→=12​tr⟨⋅,⋅⟩​B=12​[B​(Z⊤,Z⊤)+B​(W,W)]\vec{H}=\frac{1}{2}\mbox{tr}_{\langle\cdot,\cdot\rangle}B=\frac{1}{2}\left[B(Z^{\top},Z^{\top})+B(W,W)\right] (4)
Lemma 1.14.

We have a=0a=0 if and only if ⟨H→,Z⟂⟩=0.\langle\vec{H},Z^{\perp}\rangle=0.

Proof.

It is not difficult to see that

⟨H→,Z⟂⟩\displaystyle\langle\vec{H},Z^{\perp}\rangle =12​⟨B​(Z⊤,Z⊤),Z⟂⟩+12​⟨B​(W,W),Z⟂⟩\displaystyle=\frac{1}{2}\langle B(Z^{\top},Z^{\top}),Z^{\perp}\rangle+\frac{1}{2}\langle B(W,W),Z^{\perp}\rangle
=12​⟨β​(Z⊤)​Z⟂,Z⟂⟩+12​a\displaystyle=\frac{1}{2}\langle\beta(Z^{\top})Z^{\perp},Z^{\perp}\rangle+\frac{1}{2}a

which implies the result since ⟨Z⟂,Z⟂⟩=0.\langle Z^{\perp},Z^{\perp}\rangle=0. ∎

The principal relation between the Gauss curvature K,K, the normal curvature KNK_{N} and the mean curvature vector H→\vec{H} of MM is given in the following proposition.

Proposition 1.15.

If the surface MM is minimal (i.e. H→=0\vec{H}=0), then MM is flat and has a flat normal bundle (i.e. K=KN=0K=K_{N}=0).

Proof.

From Lemma 1.14 we have a=0.a=0. By Corollaries 1.5 - 1.7 we obtain respectively K=0K=0 and KN=0.K_{N}=0. ∎

The following proposition gives a nice relation between the Gauss curvature and the mean curvature vector.

Proposition 1.16.

The mean curvature vector and the Gauss curvature of the surface MM satisfy the following identity

4​|H→|2−2​K=|B​(W,W)|24|\vec{H}|^{2}-2K=|B(W,W)|^{2}
Proof.

By a direct computation

4​|H→|2\displaystyle 4|\vec{H}|^{2} =4​⟨H→,H→⟩\displaystyle=4\langle\vec{H},\vec{H}\rangle
=|B​(Z⊤,Z⊤)|2+2​⟨B​(Z⊤,Z⊤),B​(W,W)⟩+|B​(W,W)|2\displaystyle=|B(Z^{\top},Z^{\top})|^{2}+2\langle B(Z^{\top},Z^{\top}),B(W,W)\rangle+|B(W,W)|^{2}
=|β​(Z⊤)​Z⟂|2+2​⟨β​(Z⊤)​Z⟂,B​(W,W)⟩+|B​(W,W)|2\displaystyle=|\beta(Z^{\top})Z^{\perp}|^{2}+2\langle\beta(Z^{\top})Z^{\perp},B(W,W)\rangle+|B(W,W)|^{2}
=|β​(Z⊤)|2​|Z⟂|2+2​β​(Z⊤)​⟨Z⟂,B​(W,W)⟩+|B​(W,W)|2\displaystyle=|\beta(Z^{\top})|^{2}|Z^{\perp}|^{2}+2\beta(Z^{\top})\langle Z^{\perp},B(W,W)\rangle+|B(W,W)|^{2}
=0+2​β​(Z⊤)​a+|B​(W,W)|2.\displaystyle=0+2\beta(Z^{\top})a+|B(W,W)|^{2}.

Since Z⟂Z^{\perp} is a lightlike vector and the definition of α\alpha (see (2)); by Corollary 1.10 we obtain the result. ∎

To finish this section, we prove a result that let us permits study some characterizations of a spacelike surface endowed with a canonical normal null direction.

Proposition 1.17.

We have K=KN=0K=K_{N}=0 if and only if a=0a=0 or β=0.\beta=0.

Proof.

If a=0a=0 or β=0,\beta=0, by Corollaries 1.7 - 1.10 we follow K=KN=0.K=K_{N}=0. Conversely, if K=0,K=0, by Corollary 1.10, a=0a=0 or β​(Z⊤)=0.\beta(Z^{\top})=0. On the other hand, if KN=0,K_{N}=0, by Corollary 1.7, we have a=0a=0 or β​(W)=0,\beta(W)=0, thus a=0a=0 or β=0.\beta=0. ∎

As a consequence of the Proposition 1.17, the following special characterization of a spacelike surface with a canonical normal null direction is given.

Theorem 1.18.

Suppose that MM is a spacelike surface in ℝ3,1\mathbb{R}^{3,1} with a canonical normal null direction ZZ such that a=0a=0 and β=0.\beta=0. Then the surface MM can be parametrized by

ψ​(x,y)=α​(y)+x​Z0⊤,\psi(x,y)=\alpha(y)+xZ_{0}^{\top}, (5)

where α\alpha is a curve in ℝ3,1\mathbb{R}^{3,1} with α′​(y)\alpha^{\prime}(y) orthogonal to the constant vector field Z0⊤.Z_{0}^{\top}.

Proof.

Since a=0,a=0, from Lemma 1.3 we have [Z⊤,W]=0,[Z^{\top},W]=0, thus there is a parametrization (x,y)→ψ​(x,y)(x,y)\to\psi(x,y) of MM such that

∂ψ∂x​(x,y)=Z⊤​(ψ​(x,y))and∂ψ∂y​(x,y)=W​(ψ​(x,y)).\dfrac{\partial\psi}{\partial x}(x,y)=Z^{\top}(\psi(x,y))\hskip 21.68121pt\mbox{and}\hskip 21.68121pt\dfrac{\partial\psi}{\partial y}(x,y)=W(\psi(x,y)).

Since β=0,\beta=0, we have B​(X,Z⊤)=β​(X)​Z⟂=0,B(X,Z^{\top})=\beta(X)Z^{\perp}=0, for all X∈T​M;X\in TM; on the other hand, since ∇Z⊤=0,\nabla Z^{\top}=0, we follow

d​Z⊤​(X)=∇¯X​Z⊤=∇XZ⊤+B​(X,Z⊤)=0dZ^{\top}(X)=\overline{\nabla}_{X}Z^{\top}=\nabla_{X}Z^{\top}+B(X,Z^{\top})=0

for all X∈T​M,X\in TM, thus

Z⊤​(ψ​(x,y))\displaystyle Z^{\top}(\psi(x,y)) =Z⊤​(ψ​(0,y))+∫0x∂∂u​Z⊤​(ψ​(u,y))​𝑑u\displaystyle=Z^{\top}(\psi(0,y))+\int_{0}^{x}\dfrac{\partial}{\partial u}Z^{\top}(\psi(u,y))du
=Z⊤​(ψ​(0,y))+∫0x𝑑Z⊤​(∂ψ∂u​(u,y))​𝑑u\displaystyle=Z^{\top}(\psi(0,y))+\int_{0}^{x}dZ^{\top}\left(\dfrac{\partial\psi}{\partial u}(u,y)\right)du
=Z⊤​(ψ​(0,y)).\displaystyle=Z^{\top}(\psi(0,y)).

Similarly, Z⊤​(ψ​(x,y))=Z⊤​(ψ​(x,0)),Z^{\top}(\psi(x,y))=Z^{\top}(\psi(x,0)), therefore Z⊤​(ψ​(x,y))=Z0⊤Z^{\top}(\psi(x,y))=Z_{0}^{\top} is constant, which in turn implies

ψ​(x,y)\displaystyle\psi(x,y) =ψ​(0,y)+∫0x∂ψ∂u​(u,y)​𝑑u\displaystyle=\psi(0,y)+\int_{0}^{x}\dfrac{\partial\psi}{\partial u}(u,y)du
=ψ​(0,y)+∫0xZ⊤​(ψ​(u,y))​𝑑u\displaystyle=\psi(0,y)+\int_{0}^{x}Z^{\top}(\psi(u,y))du
=ψ​(0,y)+∫0xZ0⊤​𝑑u\displaystyle=\psi(0,y)+\int_{0}^{x}Z_{0}^{\top}du
=ψ​(0,y)+x​Z0⊤.\displaystyle=\psi(0,y)+xZ_{0}^{\top}.

Writing α​(y)=ψ​(0,y),\alpha(y)=\psi(0,y), we get the characterization (5) of ψ.\psi. ∎

With the same ideas we can prove an other similar characterization.

Theorem 1.19.

Suppose that MM is a spacelike surface in ℝ3,1\mathbb{R}^{3,1} with a canonical normal null direction ZZ such that a=0a=0 and β​(Z⊤)=0.\beta(Z^{\top})=0. Then the surface MM can be parametrized by

ψ​(x,y)=α​(y)+x​Z⊤​(y),\psi(x,y)=\alpha(y)+xZ^{\top}(y),

where α\alpha is a curve in ℝ3,1\mathbb{R}^{3,1} and Z⊤​(y)Z^{\top}(y) denotes the restriction of Z⊤Z^{\top} on α.\alpha.

2 The Gauss map of a surface with a canonical normal null direction

2.1 The Grassmannian of the spacelike planes

Consider Λ2​ℝ3,1,\Lambda^{2}\mathbb{R}^{3,1}, the vector space of bivectors of the Minkowski 4−4-space ℝ3,1\mathbb{R}^{3,1} endowed with its natural tensor metric of signature (3,3).(3,3).

The Grassmannian of the oriented spacelike 2-planes (which passes through the origin) in ℝ3,1\mathbb{R}^{3,1} is identified with the submanifold of unit and simple bivectors

𝒬={η∈Λ2​ℝ3,1∣⟨η,η⟩=1,η∧η=0},\mathcal{Q}=\left\{\eta\in\Lambda^{2}\mathbb{R}^{3,1}\mid\langle\eta,\eta\rangle=1,\ \eta\wedge\eta=0\right\}, (6)

and the oriented Gauss map of a spacelike surface in ℝ3,1\mathbb{R}^{3,1} with the map G:M→𝒬G:M\to\mathcal{Q} such that

G​(p)=u1∧u2,G(p)=u_{1}\wedge u_{2}, (7)

where (u1,u2)(u_{1},u_{2}) is an oriented orthonormal basis for the tangent space Tp​M.T_{p}M.

The Hodge star operator ⋆:Λ2ℝ3,1→Λ2ℝ3,1\star:\Lambda^{2}\mathbb{R}^{3,1}\to\Lambda^{2}\mathbb{R}^{3,1} is defined by the relation

⟨⋆η,η′⟩=η∧η′\langle\star\eta,\eta^{\prime}\rangle=\eta\wedge\eta^{\prime} (8)

for all η,η′∈Λ2​ℝ3,1,\eta,\eta^{\prime}\in\Lambda^{2}\mathbb{R}^{3,1}, where we identify Λ4​ℝ3,1≃ℝ\Lambda^{4}\mathbb{R}^{3,1}\simeq\mathbb{R} using the canonical volume element e1∧e2∧e3∧e4≃1.e_{1}\wedge e_{2}\wedge e_{3}\wedge e_{4}\simeq 1. This operator satisfies ⋆2=−IdΛ2​ℝ3,1,\star^{2}=-Id_{\Lambda^{2}\mathbb{R}^{3,1}}, and thus i:=−⋆i:=-\star defines a complex structure on Λ2​ℝ3,1.\Lambda^{2}\mathbb{R}^{3,1}.

We also define the map H:Λ2​ℝ3,1×Λ2​ℝ3,1⟶ℂH:\Lambda^{2}\mathbb{R}^{3,1}\times\Lambda^{2}\mathbb{R}^{3,1}\longrightarrow\mathbb{C} by

H​(η,η′)=⟨η,η′⟩+i​η∧η′H(\eta,\eta^{\prime})=\langle\eta,\eta^{\prime}\rangle+i\ \eta\wedge\eta^{\prime} (9)

for all η,η′∈Λ2​ℝ3,1.\eta,\eta^{\prime}\in\Lambda^{2}\mathbb{R}^{3,1}. This map is a ℂ−\mathbb{C}-bilinear map on Λ2​ℝ3,1,\Lambda^{2}\mathbb{R}^{3,1}, and the Grassmannian (6) remains as

𝒬={η∈Λ2​ℝ3,1∣H​(η,η)=1}.\mathcal{Q}=\left\{\eta\in\Lambda^{2}\mathbb{R}^{3,1}\mid H(\eta,\eta)=1\right\}. (10)

The bivectors

{E1:=e1∧e2,E2:=e2∧e3,E3:=e3∧e1}\left\{E_{1}:=e_{1}\wedge e_{2},\hskip 14.45377ptE_{2}:=e_{2}\wedge e_{3},\hskip 14.45377ptE_{3}:=e_{3}\wedge e_{1}\right\} (11)

form an orthonormal basis, with respect to the form HH of Λ2​ℝ3,1\Lambda^{2}\mathbb{R}^{3,1} as a complex 3−3-space with signature (+,+,+).(+,+,+). Using this basis of Λ2​ℝ3,1,\Lambda^{2}\mathbb{R}^{3,1}, the Grassmannian (10) is identified with a complex sphere

𝒬={(z1,z2,z3)∈ℂ3∣z12+z22+z32=1}\mathcal{Q}=\left\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}\mid z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\right\} (12)

2.2 Spacelike surfaces with a canonical normal null direction

We consider a spacelike surface MM in ℝ3,1\mathbb{R}^{3,1} endowed with a canonical normal null direction Z,Z, with |Z⊤|2=1|Z^{\top}|^{2}=1 and such that a=⟨B​(W,W),Z⟂⟩≠0.a=\langle B(W,W),Z^{\perp}\rangle\neq 0. We recall that W′W^{\prime} is a lightlike vector field normal to MM such that ⟨Z⟂,W′⟩=1.\langle Z^{\perp},W^{\prime}\rangle=1. If we write B​(W,W):=b​Z⟂+a​W′,B(W,W):=bZ^{\perp}+aW^{\prime}, the vectors

e1=Z⊤,e2=W,e3=(a−b)​Z⟂+B​(W,W)2​a,ande4=(a+b)​Z⟂−B​(W,W)2​ae_{1}=Z^{\top},\ \ e_{2}=W,\ \ e_{3}=\dfrac{(a-b)Z^{\perp}+B(W,W)}{\sqrt{2}a},\hskip 7.22743pt\mbox{and}\hskip 7.22743pt\ e_{4}=\dfrac{(a+b)Z^{\perp}-B(W,W)}{\sqrt{2}a} (13)

form an oriented and orthonormal basis of ℝ3,1\mathbb{R}^{3,1} adapted to the immersion M⊂ℝ3,1M\subset\mathbb{R}^{3,1}; therefore, we can define the orthonormal basis (11) of Λ2​ℝ3,1.\Lambda^{2}\mathbb{R}^{3,1}.

Lemma 2.1.

The Gauss map of MM is given by G=Z⊤∧WG=Z^{\top}\wedge W and its differential satisfies

d​G​(Z⊤)=β​(Z⊤)​Z⟂∧W+β​(W)​Z⊤∧Z⟂andd​G​(W)=β​(W)​Z⟂∧W+Z⊤∧B​(W,W).dG(Z^{\top})=\beta(Z^{\top})Z^{\perp}\wedge W+\beta(W)Z^{\top}\wedge Z^{\perp}\hskip 14.45377pt\mbox{and}\hskip 14.45377ptdG(W)=\beta(W)Z^{\perp}\wedge W+Z^{\top}\wedge B(W,W).
Proof.

Clearly G=Z⊤∧W.G=Z^{\top}\wedge W. The differential of GG is given by

d​G​(u)=(∇uZ⊤+B​(Z⊤,u))∧W+Z⊤∧(∇uW+B​(W,u))dG(u)=(\nabla_{u}Z^{\top}+B(Z^{\top},u))\wedge W+Z^{\top}\wedge(\nabla_{u}W+B(W,u))

for all u∈Tp​M;u\in T_{p}M; using the identities of Lema 1.3 and the definition of the 1−1-form β\beta (see Remark 3) we conclude the result. ∎

We describe now the differential of the Gauss map in terms of the orthonormal basis defined in (11) of Λ2​ℝ3,1.\Lambda^{2}\mathbb{R}^{3,1}.

Proposition 2.2.

The differential of the Gauss map GG satisfies

d​G​(Z⊤)=−K+i​KN2​a​E2+KN−i​K2​a​E3dG(Z^{\top})=-\frac{K+iK_{N}}{\sqrt{2}a}E_{2}+\frac{K_{N}-iK}{\sqrt{2}a}E_{3}

and

d​G​(W)=KN−i​a​(a−b)2​a​E2−a​(a+b)−i​KN2​a​E3dG(W)=\frac{K_{N}-ia(a-b)}{\sqrt{2}a}E_{2}-\frac{a(a+b)-iK_{N}}{\sqrt{2}a}E_{3}
Proof.

From Lemma 2.1, the definitions of the frame adapted to the immersion (13), and Corollaries 1.7, 1.10, we easily get the result. ∎

The pull-back of the form HH by the Gauss map.

The pull-back of the form H,H, defined in (9), by the Gauss map G:M→𝒬⊂Λ2​ℝ3,1G:M\to\mathcal{Q}\subset\Lambda^{2}\mathbb{R}^{3,1} lets us to define, for all p∈M,p\in M, the complex quadratic form G∗​Hp:Tp​M→ℂG^{*}H_{p}:T_{p}M\to\mathbb{C} given by

G∗​Hp​(u):=H​(d​G​(u),d​G​(u)).G^{*}H_{p}(u):=H(dG(u),dG(u)).

This form is analogous to the third fundamental form in the classical theory of surfaces in Euclidean 3−3-space. We will describe some properties of this quadratic form for a spacelike surface with a canonical normal null direction.

Lemma 2.3.

If a≠0,a\neq 0, the complex quadratic form G∗​HG^{*}H satisfies the following identities:

  1. 1.

    H​(d​G​(Z⊤),d​G​(Z⊤))=0,H(dG(Z^{\top}),dG(Z^{\top}))=0,

  2. 2.

    H​(d​G​(W),d​G​(W))=2​(2​|H→|2−K)−i​(2​KN),H(dG(W),dG(W))=2(2|\vec{H}|^{2}-K)-i(2K_{N}), and

  3. 3.

    H​(d​G​(Z⊤),d​G​(W))=−KN+i​K.H(dG(Z^{\top}),dG(W))=-K_{N}+iK.

Proof.

The proof of these identities are obtained by a direct computation using the formulas of d​G​(Z⊤)dG(Z^{\top}) and d​G​(W)dG(W) given in Proposition 2.2. ∎

Proposition 2.4.

If a≠0,a\neq 0, the discriminant of the complex quadratic form G∗​HG^{*}H satisfies

disc​G∗​H:=−detG∗​H=−(K+i​KN)2.\mbox{\sf disc}\ G^{*}H:=-\det G^{*}H=-(K+iK_{N})^{2}.
Proof.

Using the identities of Lemma 2.3, by a direct computation we get

detG∗​H=−(KN+i​K)2=(K+i​KN)2\det G^{*}H=-(K_{N}+iK)^{2}=(K+iK_{N})^{2}

which implies the result. ∎

An other direct consequence of Lemma 2.3 is the following result.

Proposition 2.5.

If a≠0,a\neq 0, the complex quadratic form G∗​HG^{*}H in null at every point of MM if and only if K=KN=|H→|2=0K=K_{N}=|\vec{H}|^{2}=0 on M,M, i.e. MM is flat, has flat normal bundle, and its mean curvature vector is a lightlike vector.

The interpretation of the condition G∗​H≡0G^{*}H\equiv 0 is the following: for all p∈M,p\in M, the space d​Gp​(Tp​M)dG_{p}(T_{p}M) belongs to

G​(p)+{ξ∈Λ2​ℝ3,1∣H​(G​(p),ξ)=0=H​(ξ,ξ)}⊂TG​(p)​𝒬;G(p)+\left\{\xi\in\Lambda^{2}\mathbb{R}^{3,1}\mid H(G(p),\xi)=0=H(\xi,\xi)\right\}\ \subset\ T_{G(p)}\mathcal{Q};

this set is the union of two complex lines through G​(p)G(p) in the Grassmannian 𝒬\mathcal{Q} of the oriented spacelike planes of ℝ3,1;\mathbb{R}^{3,1}; explicitly, this complex lines are given by

G​(p)+ℂ​E2andG​(p)+ℂ​E3.G(p)+\mathbb{C}E_{2}\hskip 21.68121pt\mbox{and}\hskip 21.68121ptG(p)+\mathbb{C}E_{3}.

In particular, the first normal space in p∈Mp\in M is 1−1-dimensional, i.e. the osculator space of the surface is degenerate at every point p∈M.p\in M.

2.3 The curvature ellipse.

The curvature ellipse associated to the second fundamental form BB of a spacelike surface MM in ℝ3,1\mathbb{R}^{3,1} is defined as the subset on Np​MN_{p}M

Ep:={B​(u,u)∣u∈Tp​M,|u|=1}E_{p}:=\left\{B(u,u)\mid u\in T_{p}M,\ |u|=1\right\} (14)

Suppose that the surface MM has a canonical normal null direction Z.Z. Recall that a=⟨B​(W,W),Z⟂⟩.a=\langle B(W,W),Z^{\perp}\rangle. Thus, in order to describe the ellipse of curvature of MM we have two cases to consider: a≠0a\neq 0 and a=0.a=0.

Proposition 2.6.

If a≠0,a\neq 0, the curvature ellipse is not degenerated, at the basis (Z⟂,W′)(Z^{\perp},W^{\prime}) for the normal bundle, and origin in H→,\vec{H}, the curvature ellipse is parametrized by the equations

x=cos⁡(2​θ)​[K−|H→|2a]+sin⁡(2​θ)​[−KNa]andy=cos⁡(2​θ)​[−a2]x=\cos(2\theta)\left[\frac{K-|\vec{H}|^{2}}{a}\right]+\sin(2\theta)\left[-\frac{K_{N}}{a}\right]\hskip 14.45377pt\mbox{and}\hskip 14.45377pty=\cos(2\theta)\left[-\frac{a}{2}\right]
Proof.

We write each tangent vector u∈Tp​Mu\in T_{p}M as u=cos⁡θ​Z⊤+sin⁡θ​W,u=\cos\theta Z^{\top}+\sin\theta W, with 0≤θ≤2​π.0\leq\theta\leq 2\pi. By a direct computation we have

B​(u,u)=H→+cos⁡(2​θ)​[B​(Z⊤,Z⊤)−H→]+sin⁡(2​θ)​B​(Z⊤,W),B(u,u)=\vec{H}+\cos(2\theta)\left[B(Z^{\top},Z^{\top})-\vec{H}\right]+\sin(2\theta)B(Z^{\top},W), (15)

writing these normal vectors at the basis (Z⟂,W′)(Z^{\perp},W^{\prime}) and using the definition of the 1−1-form β\beta (see Remark 3), the identities of Corollaries 1.7, 1.10, and Proposition 1.16, we easily get the result. ∎

The curvature ellipse in a point p∈Mp\in M such that a>0,a>0, K−|H→|2>0,K-|\vec{H}|^{2}>0, and KN<0K_{N}<0 is given in the following figure:

[Uncaptioned image]
Remark 2.7.

The normal vector H→∗:=B​(Z⊤,Z⊤)−H→\vec{H}^{*}:=B(Z^{\top},Z^{\top})-\vec{H} satisfies |H→∗|2=2​(|H→|2−K).|\vec{H}^{*}|^{2}=2(|\vec{H}|^{2}-K). Moreover, since a≠0,a\neq 0, we have that H→∗\vec{H}^{*} and Z⟂Z^{\perp} are linearly independent. Thus, if |H→|2≠K,|\vec{H}|^{2}\neq K, using the relation (15), we obtain the following characterization: in (H→,H→∗,Z⟂),(\vec{H},\vec{H}^{*},Z^{\perp}), the equation of the ellipse is given by

x22​||H→|2−K|+y2KN2a2=1.\frac{x^{2}}{2||\vec{H}|^{2}-K|}+\frac{y^{2}}{\frac{K_{N}^{2}}{a^{2}}}=1.

We can also describe the curvature ellipse in the degenerate case where a=0a=0 as in the following proposition. For briefness, we omit the proof.

Proposition 2.8.

If a=0,a=0, the curvature ellipse degenerates the segment

[H→+h​Z⟂,H→−h​Z⟂]\left[\vec{H}+hZ^{\perp},\vec{H}-hZ^{\perp}\right]

where h=max⁡{±⟨H→∗,W′⟩,±β​(W)}.h=\max\{\pm\langle\vec{H}^{*},W^{\prime}\rangle,\pm\beta(W)\}.

Mean curvature directions and asymptotic directions.

If pp is a point on a spacelike surface MM in ℝ3,1,\mathbb{R}^{3,1}, a mean curvature direction in the tangent plane Tp​MT_{p}M is defined as the inverse image by the second fundamental form of the points in the ellipse of curvature where the line defined by the mean curvature vector intersects the ellipse.

For all p∈Mp\in M and u∈Tp​M,u\in T_{p}M, the condition that determines the mean curvature direction is

[H→,B​(u,u)]=0,[\vec{H},B(u,u)]=0,

where the brackets stand for the mixed product in Np​MN_{p}M (the determinant in a positively oriented Lorentzian basis).

Lemma 2.9.

If a≠0,a\neq 0, in the basis (Z⟂,W′)(Z^{\perp},W^{\prime}) for the normal bundle, we have

H→=|H→|2a​Z⟂+a2​W′andB​(u,u)=[u12​β​(Z⊤)+2​u1​u2​β​(W)+u22​b]​Z⟂+(a​u22)​W′\vec{H}=\frac{|\vec{H}|^{2}}{a}Z^{\perp}+\frac{a}{2}W^{\prime}\hskip 14.45377pt\mbox{and}\hskip 14.45377ptB(u,u)=\left[u_{1}^{2}\beta(Z^{\top})+2u_{1}u_{2}\beta(W)+u_{2}^{2}b\right]Z^{\perp}+(au_{2}^{2})W^{\prime}

for all u=u1​Z⊤+u2​W∈Tp​M.u=u_{1}Z^{\top}+u_{2}W\in T_{p}M.

Proof.

We have the following:

H→=12​[B​(Z⊤,Z⊤)+B​(W,W)]=12​[β​(Z⊤)​Z⟂+b​Z⟂+a​W′]=β​(Z⊤)+b2​Z⟂+a2​W′,\vec{H}=\frac{1}{2}\left[B(Z^{\top},Z^{\top})+B(W,W)\right]=\frac{1}{2}\left[\beta(Z^{\top})Z^{\perp}+bZ^{\perp}+aW^{\prime}\right]=\frac{\beta(Z^{\top})+b}{2}Z^{\perp}+\frac{a}{2}W^{\prime},

but, using Proposition 1.16, we have

β​(Z⊤)+b2=K2+b2=K+a​b2​a=|H→|2a.\frac{\beta(Z^{\top})+b}{2}=\frac{\frac{K}{2}+b}{2}=\frac{K+ab}{2a}=\frac{|\vec{H}|^{2}}{a}.

The expression for B​(u,u)B(u,u) is obtained from a direct calculation. ∎

Proposition 2.10.

If a≠0,a\neq 0, the mean curvature directions are given by

K​Z⊤+(−KN±K2+KN2)​W.KZ^{\top}+\left(-K_{N}\pm\sqrt{K^{2}+K_{N}^{2}}\right)W.

If a=0,a=0, every tangent vector u∈Tp​Mu\in T_{p}M defines a mean curvature direction.

Proof.

We suppose that u=u1​Z⊤+u2​Wu=u_{1}Z^{\top}+u_{2}W is a mean curvature direction, using the expressions of the Lemma 2.9 and Proposition 1.16, we obtain

0=[H→,B​(u,u)]\displaystyle 0=[\vec{H},B(u,u)] =u22​[2​|H→|2−a​b2]−u12​K2+u1​u2​KN\displaystyle=u_{2}^{2}\left[\frac{2|\vec{H}|^{2}-ab}{2}\right]-u_{1}^{2}\frac{K}{2}+u_{1}u_{2}K_{N}
=u22​K2−u12​K2+u1​u2​KN;\displaystyle=u_{2}^{2}\frac{K}{2}-u_{1}^{2}\frac{K}{2}+u_{1}u_{2}K_{N};

solving these equation we get

u2=(−KN±K2+KN2)​u1K,u_{2}=\left(-K_{N}\pm\sqrt{K^{2}+K_{N}^{2}}\right)\frac{u_{1}}{K},

taking u1=Ku_{1}=K we obtain the result. ∎

An asymptotic direction at Tp​MT_{p}M is defined as the inverse image of the second fundamental form of a point where the line that contains the origin is tangent to the ellipse of curvature.

For all p∈M,p\in M, we consider the real quadratic form

δ:Tp​M⟶ℝ,u⟼d​Gp​(u)∧d​Gp​(u),\delta:T_{p}M\longrightarrow\mathbb{R},\hskip 14.45377ptu\longmapsto dG_{p}(u)\wedge dG_{p}(u),

where Λ4​ℝ3,1\Lambda^{4}\mathbb{R}^{3,1} is identified with ℝ\mathbb{R} by means of the volume element e1∧e2∧e3∧e4≃1.e_{1}\wedge e_{2}\wedge e_{3}\wedge e_{4}\simeq 1. A non-zero vector u∈Tp​Mu\in T_{p}M defines an asymptotic direction at pp if δ​(u)=0.\delta(u)=0. The opposite of the determinant of δ,\delta, with respect to the metric in M,M,

Δ:=−detδ,\Delta:=-\det\delta,

is a second order invariant of the surface. There exist asymptotic directions if and only if Δ≥0\Delta\geq 0; moreover, Δ>0\Delta>0 if and only if the surface admits two distinct asymptotic directions at every point. We refer to [1, Section 4] for a complete description of the asymptotic directions of a spacelike surface in ℝ3,1.\mathbb{R}^{3,1}.

In the following proposition, we will compute the invariant Δ\Delta and describe the asymptotic directions of a spacelike surface with a canonical normal null direction.

Proposition 2.11.

At every point of MM we have Δ=K2,\Delta=K^{2}, where KK is the Gauss curvature of M.M. In particular, there are asymptotic directions at every point of M.M.

Proof.

Since δ​(u)=ℑ⁡m​H​(d​G​(u),d​G​(u))\delta(u)=\Im m\ H(dG(u),dG(u)) (δ\delta is the imaginary part of the quadratic form G∗​HG^{*}H) for all u∈Tp​M,u\in T_{p}M, using the equalities of Lemma 2.3 we have δ​(Z⊤)=0\delta(Z^{\top})=0 and

d​G​(Z⊤)∧d​G​(W)=ℑ⁡m​H​(d​G​(Z⊤),d​G​(W))=K.dG(Z^{\top})\wedge dG(W)=\Im m\ H(dG(Z^{\top}),dG(W))=K.

By a direct computation we get Δ=[d​G​(Z⊤)∧d​G​(W)]2−δ​(Z⊤)​δ​(W)=K2.\Delta=[dG(Z^{\top})\wedge dG(W)]^{2}-\delta(Z^{\top})\delta(W)=K^{2}. ∎

Proposition 2.12.

At every point of M,M, Z⊤Z^{\top} is an asymptotic direction. Moreover, WW is an asymptotic direction if and only if MM has a flat normal bundle.

Proof.

Since δ​(u)=ℑ⁡m​H​(d​G​(u),d​G​(u))\delta(u)=\Im m\ H(dG(u),dG(u)) for all u∈Tp​M,u\in T_{p}M, by Lemma 2.3 we have

δ​(Z⊤)=0andδ​(W)=−2​KN\delta(Z^{\top})=0\hskip 14.45377pt\mbox{and}\hskip 14.45377pt\delta(W)=-2K_{N}

which implies the result. ∎

According to Proposition 2.11, if the Gauss curvature KK is not zero, there exists two distinct asymptotic directions at every point of the surface. From Proposition 2.12, Z⊤Z^{\top} is an asymptotic direction; by a direct computation, in the following proposition we describe the other asymptotic direction.

Proposition 2.13.

If a≠0,a\neq 0, we have two cases to consider: when K≠0,K\neq 0, there exists two different asymptotic directions given by

Z⊤andKNK​Z⊤+W;Z^{\top}\hskip 21.68121pt\mbox{and}\hskip 21.68121pt\dfrac{K_{N}}{K}Z^{\top}+W;

when K=0,K=0, there exists a double asymptotic direction given by Z⊤.Z^{\top}.

If a=0,a=0, every tangent vector u∈Tp​Mu\in T_{p}M defines an asymptotic direction.

Proof.

We suppose that the other asymptotic direction is given by r​Z⊤+s​WrZ^{\top}+sW for some r,s∈ℝ,r,s\in\mathbb{R}, with s≠0.s\neq 0. We have

0=δ​(r​Z⊤+s​W)\displaystyle 0=\delta(rZ^{\top}+sW) =d​G​(r​Z⊤+s​W)∧d​G​(r​Z⊤+s​W)\displaystyle=dG(rZ^{\top}+sW)\wedge dG(rZ^{\top}+sW)
=r2​δ​(Z⊤)+s2​δ​(W)+2​(r​s)​d​G​(Z⊤)∧d​G​(W)\displaystyle=r^{2}\delta(Z^{\top})+s^{2}\delta(W)+2(rs)dG(Z^{\top})\wedge dG(W)
=s2​(−2​KN)+2​(r​s)​K\displaystyle=s^{2}(-2K_{N})+2(rs)K
=2​s​(−s​KN+r​K)\displaystyle=2s(-sK_{N}+rK)

(see the proof of Propositions 2.11-2.12), thus r=KNK​s;r=\frac{K_{N}}{K}s; taking s=1s=1 we obtain the result. ∎

3 Construction of spacelike surfaces with a canonical normal null direction

In this section, we construct some spacelike surfaces with a cononical normal null direction in the four dimensional Minskowski space ℝ3,1.\mathbb{R}^{3,1}.

3.1 Translation Spacelike surfaces in ℝ3,1\mathbb{R}^{3,1}

Let us consider the (translation) surface MM in ℝ3,1\mathbb{R}^{3,1} parametrized by

ψ​(x,y)=α​(x)+δ​(y)\psi(x,y)=\alpha(x)+\delta(y)

where α\alpha and δ\delta are two regular spacelike orthogonal curves in ℝ3,1\mathbb{R}^{3,1} such that α′\alpha^{\prime} and δ\delta lies in the hyperplane orthogonal to canonical vector field e1=(1,0,0,0).e_{1}=(1,0,0,0).

In this case, the components of the induced metric ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle in MM are given by

E:=⟨ψx,ψx⟩=⟨α′​(x),α′​(x)⟩=|α′​(x)|2,F=⟨ψx,ψy⟩=⟨α′​(x),δ′​(y)⟩=0E:=\langle\psi_{x},\psi_{x}\rangle=\langle\alpha^{\prime}(x),\alpha^{\prime}(x)\rangle=|\alpha^{\prime}(x)|^{2},\hskip 14.45377ptF=\langle\psi_{x},\psi_{y}\rangle=\langle\alpha^{\prime}(x),\delta^{\prime}(y)\rangle=0

and

G=⟨ψy,ψy⟩=⟨δ′​(y),δ′​(y)⟩=|δ′​(y)|2,G=\langle\psi_{y},\psi_{y}\rangle=\langle\delta^{\prime}(y),\delta^{\prime}(y)\rangle=|\delta^{\prime}(y)|^{2},

thus, the determinant of this metric is

det⟨⋅,⋅⟩=E​G−F2=|α′​(x)|2​|δ′​(y)|2,\det\langle\cdot,\cdot\rangle=EG-F^{2}=|\alpha^{\prime}(x)|^{2}|\delta^{\prime}(y)|^{2},

in particular, since det⟨⋅,⋅⟩>0,\det\langle\cdot,\cdot\rangle>0, MM is spacelike (translation) surface in ℝ3,1.\mathbb{R}^{3,1}.

Now we study the conditions on α\alpha and δ\delta such that the canonical vector field e1e_{1} on ℝ3,1\mathbb{R}^{3,1} induce a canonical normal null direction on the surface M.M. Writing e1=e1⊤+e1⟂,e_{1}=e_{1}^{\top}+e_{1}^{\perp}, we have

e1⊤=c​ψx+d​ψy=c​α′​(x)+d​δ′​(y),e_{1}^{\top}=c\psi_{x}+d\psi_{y}=c\alpha^{\prime}(x)+d\delta^{\prime}(y),

for some functions cc and dd on M.M. So, we have

⟨e1⊤,α′​(x)⟩=c​|α′​(x)|2and0=⟨e1,δ′​(y)⟩=⟨e1⊤,δ′​(y)⟩=d​|δ′​(y)|2,\langle e_{1}^{\top},\alpha^{\prime}(x)\rangle=c|\alpha^{\prime}(x)|^{2}\hskip 14.45377pt\mbox{and}\hskip 14.45377pt0=\langle e_{1},\delta^{\prime}(y)\rangle=\langle e_{1}^{\top},\delta^{\prime}(y)\rangle=d|\delta^{\prime}(y)|^{2},

therefore d=0,d=0, and thus

e1⊤=⟨e1⊤,α′​(x)⟩|α′​(x)|2​α′​(x).e_{1}^{\top}=\frac{\langle e_{1}^{\top},\alpha^{\prime}(x)\rangle}{|\alpha^{\prime}(x)|^{2}}\alpha^{\prime}(x).
Proposition 3.1.

The canonical vector field e1e_{1} induces a canonical normal null direction on the spacelike (translation) surface MM if and only if

⟨e1⊤,α′​(x)⟩2=|α′​(x)|2.\langle e_{1}^{\top},\alpha^{\prime}(x)\rangle^{2}=|\alpha^{\prime}(x)|^{2}.
Proof.

This result is a consequence of |e1⊤|2=1.|e_{1}^{\top}|^{2}=1. ∎

Suppose that MM has a canonical normal null direction with respect e1.e_{1}. We can prove that e1⊤e_{1}^{\top} only depends on x.x. Writing λ​(x)=⟨e1⊤,α′​(x)⟩|α′​(x)|2,\lambda(x)=\frac{\langle e_{1}^{\top},\alpha^{\prime}(x)\rangle}{|\alpha^{\prime}(x)|^{2}}, we have e1⊤=λ​(x)​α′​(x),e_{1}^{\top}=\lambda(x)\alpha^{\prime}(x), by Lemma 1.3, we follow ∇e1⊤e1⊤=0,\nabla_{e_{1}^{\top}}e_{1}^{\top}=0, thus

∇α′​(x)α′​(x)=−λ′​(x)λ​(x)​α′​(x).\nabla_{\alpha^{\prime}(x)}\alpha^{\prime}(x)=-\frac{\lambda^{\prime}(x)}{\lambda(x)}\alpha^{\prime}(x). (16)

Analogously, since W:=δ′​(y)|δ′​(y)|W:=\frac{\delta^{\prime}(y)}{|\delta^{\prime}(y)|} is a unit spacelike vector field tangent to MM and orthogonal to e1⊤,e_{1}^{\top}, from Lemma 1.3 we have ∇e1⊤W=0,\nabla_{e_{1}^{\top}}W=0, thus λ​(x)​∇α′​(x)δ′​(y)|δ′​(y)|=0\lambda(x)\nabla_{\alpha^{\prime}(x)}\frac{\delta^{\prime}(y)}{|\delta^{\prime}(y)|}=0 i.e.,

(1|δ′​(y)|)x​δ′​(y)+1|δ′​(y)|​∇α′​(x)δ′​(y)=0,\left(\frac{1}{|\delta^{\prime}(y)|}\right)_{x}\delta^{\prime}(y)+\frac{1}{|\delta^{\prime}(y)|}\nabla_{\alpha^{\prime}(x)}\delta^{\prime}(y)=0,

therefore

∇α′​(x)δ′​(y)=0.\nabla_{\alpha^{\prime}(x)}\delta^{\prime}(y)=0. (17)

On the other hand, since ∇We1⊤=a​W,\nabla_{W}e_{1}^{\top}=aW, we obtain

∇δ′​(y)|δ′​(y)|λ​(x)​α′​(x)=a​δ′​(y)|δ′​(y)|\nabla_{\frac{\delta^{\prime}(y)}{|\delta^{\prime}(y)|}}\lambda(x)\alpha^{\prime}(x)=a\frac{\delta^{\prime}(y)}{|\delta^{\prime}(y)|}

or

0=λy​(x)​α′​(x)+λ​(x)​∇δ′​(y)α′​(x)=a​δ′​(y),0=\lambda_{y}(x)\alpha^{\prime}(x)+\lambda(x)\nabla_{\delta^{\prime}(y)}\alpha^{\prime}(x)=a\delta^{\prime}(y),

thus we conclude that a=0.a=0. In particular, from Corollaries 1.7 - 1.10 we have that MM is flat and has a flat normal bundle.

Furthermore, by the mean curvature vector H→,\vec{H}, given in (4), the surface MM is minimal (i.e. H→=0\vec{H}=0 ) if and only if B​(e1⊤,e1⊤)=B​(W,W)=0.B(e_{1}^{\top},e_{1}^{\top})=B(W,W)=0. Indeed, this vectors depends only on xx and y,y, respectively. Using (16), we derive

B​(e1⊤,e1⊤)=λ2​(x)​α′′​(x)+λ​(x)​λ′​(x)​α′​(x)≠0B\left(e_{1}^{\top},e_{1}^{\top}\right)=\lambda^{2}(x)\alpha^{\prime\prime}(x)+\lambda(x)\lambda^{\prime}(x)\alpha^{\prime}(x)\neq 0

if and only if λ​α′\lambda\alpha^{\prime} is not a constant function.

For instance, the functions α​(x)=(cos⁡x,sin⁡x,0,0)\alpha(x)=(\cos x,\sin x,0,0) and δ​(y)=(0,0,sinh⁡y,cosh⁡y)\delta(y)=(0,0,\sinh y,\cosh y) are spacelike orthogonal curves in Minkowski space ℝ3,1,\mathbb{R}^{3,1}, such that δ\delta lies in the hyperplane orthogonal to e1.e_{1}. By the previous arguments, the surface MM parametrized by

ψ​(x,y)=(cos⁡x,sin⁡x,sinh⁡y,cosh⁡y),\psi(x,y)=(\cos x,\sin x,\sinh y,\cosh y),

is a spacelike surface in ℝ3,1\mathbb{R}^{3,1} with a canonical normal null direction induced by e1;e_{1}; in this case we have

e1⊤=(sin2⁡x,−sin⁡x​cos⁡x,0,0),e_{1}^{\top}=(\sin^{2}x,-\sin x\cos x,0,0),

which is not constant, therefore, MM is flat and has flat normal bundle, but it’s not a minimal surface.

3.2 Spacelike surfaces in ℝ3,1\mathbb{R}^{3,1} as a graph of a function

We will study the situation where a spacelike surface is given as the graph of a smooth function.

Let f,g:U⊂ℝ2→ℝf,g:U\subset\mathbb{R}^{2}\to\mathbb{R} be two smooth functions and consider the surface

M:={(x,y,f​(x,y),g​(x,y))∈ℝ3,1∣(x,y)∈U}M:=\left\{(x,y,f(x,y),g(x,y))\in\mathbb{R}^{3,1}\mid(x,y)\in U\right\} (18)

given as the graph of the function (x,y)↦(f​(x,y),g​(x,y)).(x,y)\mapsto(f(x,y),g(x,y)). A global parametrization of the surface MM is given by ψ:U⊂ℝ2→ℝ3,1\psi:U\subset\mathbb{R}^{2}\to\mathbb{R}^{3,1} which satisfies

ψ​(x,y)=(x,y,f​(x,y),g​(x,y)).\psi(x,y)=(x,y,f(x,y),g(x,y)).

The tangent vectors to the surface MM are ψx=(1,0,fx,gx)\psi_{x}=(1,0,f_{x},g_{x}) and ψy=(0,1,fy,gy)\psi_{y}=(0,1,f_{y},g_{y}) and the components of the induced metric ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle in MM are given by

E:=⟨ψx,ψx⟩=1+fx2−gx2,F:=⟨ψx,ψy⟩=fx​fy−gx​gyE:=\langle\psi_{x},\psi_{x}\rangle=1+f_{x}^{2}-g_{x}^{2},\hskip 28.90755ptF:=\langle\psi_{x},\psi_{y}\rangle=f_{x}f_{y}-g_{x}g_{y}

and

G:=⟨ψy,ψy⟩=1+fy2−gy2.G:=\langle\psi_{y},\psi_{y}\rangle=1+f_{y}^{2}-g_{y}^{2}.

The determinant of this metric is

det⟨⋅,⋅⟩=E​G−F2=(1+|∇f|2)​(1−|∇g|2)+⟨∇f,∇g⟩2,\det\langle\cdot,\cdot\rangle=EG-F^{2}=\left(1+|\nabla f|^{2}\right)\left(1-|\nabla g|^{2}\right)+\langle\nabla f,\nabla g\rangle^{2},

where the right hand side is calculated on ℝ2\mathbb{R}^{2} with its standard Riemannian flat metric; in particular, MM is a spacelike surface if and only if det⟨⋅,⋅⟩>0.\det\langle\cdot,\cdot\rangle>0.

Proposition 3.2.

Let MM be a spacelike surface in ℝ3,1\mathbb{R}^{3,1} given as in (18). Then MM has a canonical normal null direction with respect to e1e_{1} if and only if E​G−F2=G.EG-F^{2}=G. In this case we have

e1⊤=ψx−FG​ψye_{1}^{\top}=\psi_{x}-\frac{F}{G}\psi_{y}
Proof.

We need to compute the tangent part of e1e_{1} along M.M. Writing

e1⊤=a​ψx+b​ψy∈Tp​M,e_{1}^{\top}=a\psi_{x}+b\psi_{y}\ \in\ T_{p}M,

we get

1=⟨e1,ψx⟩=⟨e1⊤,ψx⟩=a​E+b​Fand0=⟨e1,ψy⟩=⟨e1⊤,ψy⟩=a​F+b​G.1=\langle e_{1},\psi_{x}\rangle=\langle e_{1}^{\top},\psi_{x}\rangle=aE+bF\hskip 21.68121pt\mbox{and}\hskip 21.68121pt0=\langle e_{1},\psi_{y}\rangle=\langle e_{1}^{\top},\psi_{y}\rangle=aF+bG.

Thus,

a=GE​G−F2andb=−FE​G−F2,a=\frac{G}{EG-F^{2}}\hskip 21.68121pt\mbox{and}\hskip 21.68121ptb=\frac{-F}{EG-F^{2}},

therefore

e1⊤=GE​G−F2​ψx+−FE​G−F2​ψy.e_{1}^{\top}=\frac{G}{EG-F^{2}}\psi_{x}+\frac{-F}{EG-F^{2}}\psi_{y}.

Note that e1⟂e_{1}^{\perp} is a lightlike normal vector field if and only if e1⊤e_{1}^{\top} is a unit spacelike vector field along M,M, thus, from the last equality we obtain

⟨e1⊤,e1⊤⟩\displaystyle\langle e_{1}^{\top},e_{1}^{\top}\rangle =G2(E​G−F2)2​E−2​F​G(E​G−F2)2​F+F2(E​G−F2)2​G\displaystyle=\frac{G^{2}}{(EG-F^{2})^{2}}E-2\frac{FG}{(EG-F^{2})^{2}}F+\frac{F^{2}}{(EG-F^{2})^{2}}G
=GE​G−F2,\displaystyle=\frac{G}{EG-F^{2}},

that is, e1⟂e_{1}^{\perp} is a lightlike normal vector field along MM if and only if E​G−F2=G.EG-F^{2}=G. ∎

Remark 3.3.

In a similar way, if MM is a spacelike surface in ℝ3,1\mathbb{R}^{3,1} given as in (18), then MM has a canonical normal null direction with respect to e2e_{2} if and only if E​G−F2=E.EG-F^{2}=E. In this case we have e2⊤=−FG​ψx+ψy.e_{2}^{\top}=-\frac{F}{G}\psi_{x}+\psi_{y}.

Corollary 3.4.

With the same hypothesis as in Proposition 3.2, MM has a canonical normal null direction with respect to e1e_{1} if and only if

(1+fy2)​gx2−2​fx​fy​gx​gy−(1−gy2)​fx2=0.(1+f_{y}^{2})g_{x}^{2}-2f_{x}f_{y}g_{x}g_{y}-(1-g_{y}^{2})f_{x}^{2}=0. (19)
Proof.

The condition E​G−F2=GEG-F^{2}=G is equivalent to the equation (19). ∎

3.2.1 Particular solutions of the PDE (19)

In order to find some particular solutions of the PDE (19) we use conformal functions over the Lorentz numbers 𝒜={x+σ​y∣x,y∈ℝ,σ∉ℝ,σ2=1},\mathcal{A}=\{x+\sigma y\mid x,y\in\mathbb{R},\sigma\notin\mathbb{R},\sigma^{2}=1\}, see for example [7, 8].

We consider the función h:ℝ2→𝒜,h:\mathbb{R}^{2}\to\mathcal{A}, over the Lorentz numbers 𝒜\mathcal{A} given by

h​(x,y)=f​(x,y)+σ​g​(x,y),h(x,y)=f(x,y)+\sigma g(x,y),

where ff and gg are the same functions of previous section; writing z=x+σ​y,z=x+\sigma y, z¯=x−σ​y,\overline{z}=x-\sigma y, |z|2=z​z¯|z|^{2}=z\overline{z} and the operators

∂∂z=12​(∂∂x+σ​∂∂y)and∂∂z¯=12​(∂∂x−σ​∂∂y);\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}+\sigma\frac{\partial}{\partial y}\right)\hskip 21.68121pt\mbox{and}\hskip 21.68121pt\frac{\partial}{\partial\overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}-\sigma\frac{\partial}{\partial y}\right);

we easily get that the equation (19) is equivalent to

(|∂h∂z|2−|∂h∂z¯|2)2=|∂h∂z+∂h∂z¯|2.\left(\Big{|}\frac{\partial h}{\partial z}\Big{|}^{2}-\Big{|}\frac{\partial h}{\partial\overline{z}}\Big{|}^{2}\right)^{2}=\Big{|}\frac{\partial h}{\partial z}+\frac{\partial h}{\partial\overline{z}}\Big{|}^{2}.

In particular, if h:ℝ2→𝒜h:\mathbb{R}^{2}\to\mathcal{A} is a conformal function, i.e., ∂h∂z¯=0,\frac{\partial h}{\partial\overline{z}}=0, we get

|∂h∂z|2=0or|∂h∂z|2=1;\Big{|}\frac{\partial h}{\partial z}\Big{|}^{2}=0\hskip 28.90755pt\mbox{or}\hskip 28.90755pt\Big{|}\frac{\partial h}{\partial z}\Big{|}^{2}=1;

in the first case, the components ff and gg of the function hh are given by

f​(x,y)=α​(x+y)+k2andg​(x,y)=α​(x+y)−k2,f(x,y)=\frac{\alpha(x+y)+k}{2}\hskip 28.90755pt\mbox{and}\hskip 28.90755ptg(x,y)=\frac{\alpha(x+y)-k}{2},

where α\alpha is some real function and kk is a constant. In the second case, the components ff and gg of the function hh are given by

f​(x,y)=α2​(x+y)+x2−y2+∫c​(x+y)​d​(x+y)+k​(x−y)2​α​(x+y)f(x,y)=\frac{\alpha^{2}(x+y)+x^{2}-y^{2}+\int c(x+y)d(x+y)+k(x-y)}{2\alpha(x+y)}

and

g​(x,y)=α2​(x+y)−x2+y2−∫c​(x+y)​d​(x+y)−k​(x−y)2​α​(x+y),g(x,y)=\frac{\alpha^{2}(x+y)-x^{2}+y^{2}-\int c(x+y)d(x+y)-k(x-y)}{2\alpha(x+y)},

for some real función α,c\alpha,c and k.k.

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