Generalized Null 2-Type Surfaces in Minkowski 3-Space

Abstract

For the mean curvature vector field $\mathbf{H}$ and the Laplace operator Δ of a submanifold in the Minkowski space, a submanifold satisfying the condition $\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C}$ is known as a generalized null 2-type, where f and g are smooth functions, and $\mathbf{C}$ is a constant vector. The notion of generalized null 2-type submanifolds is a generalization of null 2-type submanifolds defined by B.-Y. Chen. In this paper, we study flat surfaces in the Minkowski 3-space ${\mathbb{L}}^3$ and classify generalized null 2-type flat surfaces. In addition, we show that the only generalized null 2-type null scroll in ${\mathbb{L}}^3$ is a B-scroll.

1. Introduction

Let $x:M⟶{\mathbb{E}}^m$ be an isometric immersion of an n-dimensional connected submanifold M in an m-dimensional Euclidean space ${\mathbb{E}}^m$. Denote by $\mathbf{H}$ and Δ, respectively, the mean curvature vector field and the Laplacian operator with respect to the induced metric on M induced from that of ${\mathbb{E}}^m$. Then, it is well known as

$$\Delta x=-n\mathbf{H}.$$

(1)

By using (1), Takahashi [1] proved that minimal submanifolds of a hypersphere of ${\mathbb{E}}^m$ are constructed from eigenfunctions of Δ with one eigenvalue λ (≠0). In [2,3], Chen initiated the study of submanifolds in ${\mathbb{E}}^m$ that are constructed from harmonic functions and eigenfunctions of Δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:

$$x=x_0+x_q,\Delta x_0=0,\Delta x_q=\lambda x_q$$

(2)

for some non-constant maps $x_0$ and $x_q$, where λ is a nonzero constant. A submanifold satisfying (2) is said to be of null 2-type [3]. From the definition of null 2-type submanifolds and (1), it follows that the mean curvature vector field $\mathbf{H}$ satisfies the following condition:

$$\Delta \mathbf{H}=\lambda \mathbf{H}.$$

(3)

A result from [4] states that a surface in the Euclidean space ${\mathbb{E}}^3$ satisfying (3) is either a minimal surface or an open part of an ordinary sphere or a circular cylinder. Ferrández and Lucas [5] extended it to the Lorentzian case. They proved that the surface satisfying (3) is either a minimal surface or an open part of a Lorentz circular cylinder, a hyperbolic cylinder, a Lorentz hyperbolic cylinder, a hyperbolic space, a de Sitter space or a B-scroll. Afterwards, several authors studied null 2-type submanifolds in the (pseudo-)Euclidean space [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

Now, we will give a generalization of null 2-type submanifolds in the Minkowski space. It is well known that a Lorentz circular cylinder ${\mathbb{S}}^1\left(r\right)\times {\mathbb{R}}_1^1$ is a null 2-type surface in the Minkowski 3-space ${\mathbb{L}}^3$ satisfying $\Delta \mathbf{H}=\frac{1}{r^2}\mathbf{H}$, where ${\mathbb{S}}^1\left(r\right)$ is a circle with radius r and ${\mathbb{R}}_1^1$ is a Lorentz straight line. However, the following surface has another property as follows: a parametrization

$$x(s,t)=\left(\frac{1}{4}{s}^2-\frac{1}{2}\ln s,\frac{1}{4}{s}^2+\frac{1}{2}\ln s,t\right)$$

is a cylindrical surface in ${\mathbb{L}}^3$. On the other hand, the mean curvature vector field $\mathbf{H}$ of the surface is given by

$$\mathbf{H}=\left(-\frac{1}{4}-\frac{1}{4s^2},-\frac{1}{4}+\frac{1}{4s^2},0\right)$$

and the surface satisfies

$$\Delta \mathbf{H}=-\frac{6}{s^2}\left(\mathbf{H}+(\frac{1}{4},\frac{1}{4},0)\right).$$

Next, we consider another surface with a parametrization

$$x(s,t)=\left(\frac{1}{2}{s}^{2}t+t,st,\frac{1}{2}{s}^{2}t\right).$$

The surface is a conical surface in ${\mathbb{L}}^3$, and it satisfies the following equation for the mean curvature vector $\mathbf{H}$

$$\Delta \mathbf{H}=\frac{1}{t^2}\mathbf{H}+\frac{1}{2t^3}(1,0,1).$$

Thus, based on the above examples, we give the definition:

Definition 1.

A submanifold M of the Minkowski space is said to be of generalized null 2-type if it satisfies the condition

$$\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C}$$

(4)

for some smooth functions $f,g$ and a constant vector $\mathbf{C}$. In particular, if the functions f and g are equal to each other in (4), then the submanifold M is called of generalized null 2-type of the first kind and of the second kind otherwise.

In [22], the authors recently classified generalized null 2-type flat surfaces in the Euclidean 3-space. Conical surfaces, cylindrical surfaces or tangent developable surfaces are developable surfaces (or flat surfaces) as ruled surfaces in the Minkowski 3-space ${\mathbb{L}}^3$. In this paper, we study developable surfaces in ${\mathbb{L}}^3$ and completely classify generalized null 2-type developable surfaces, and give some examples. In addition, we investigate null scrolls in the Minkwoski 3-space ${\mathbb{L}}^3$ satisfying the condition (4).

2. Preliminaries

The Minkowski 3-space ${\mathbb{L}}^3$ is a real space ${\mathbb{R}}^3$ with the standard flat metric given by

$$\langle ,\rangle =-d{x}_1^2+d{x}_2^2+d{x}_3^2,$$

where $(x_1,x_2,x_3)$ is a rectangular coordinate system of ${\mathbb{R}}^3$. An arbitrary vector $\mathbf{x}$ of ${\mathbb{L}}^3$ is said to be space-like if $\langle \mathbf{x},\mathbf{x}\rangle >0$ or $\mathbf{x}=0$, time-like if $\langle \mathbf{x},\mathbf{x}\rangle <0$ and null if $\langle \mathbf{x},\mathbf{x}\rangle =0$ and $\mathbf{x}\ne 0$. A time-like or null vector in ${\mathbb{L}}^3$ is said to be causal. Similarly, an arbitrary curve $\gamma =\gamma \left(s\right)$ is space-like, time-like or null if all of its tangent vectors ${\gamma }^{\prime }\left(s\right)$ are space-like, time-like or null, respectively. From now on, the “prime” means the partial derivative with respect to the parameter s unless mentioned otherwise.

We now put a 2-dimensional space form in ${\mathbb{L}}^3$ as follows:

$${\mathbb{Q}}^2\left(\epsilon \right)=\left\{\begin{array}{c}{\mathbb{S}}_1^2=\{(x_1,x_2,x_3)\in {\mathbb{L}}^3|-x_1^2+x_2^2+x_3^2=1\},\mathrm{if}\epsilon =1;\hfill \\ {\mathbb{H}}^2=\{(x_1,x_2,x_3)\in {\mathbb{L}}^3|-x_1^2+x_2^2+x_3^2=-1\},\mathrm{if}\epsilon =-1.\hfill \end{array}\right.$$

We call ${\mathbb{S}}_1^2$ and ${\mathbb{H}}^2$ the de-Sitter space and the hyperbolic space, respectively.

Let $\gamma :I⟶{\mathbb{L}}^3$ be a space-like or time-like curve in the Minkowski 3-space ${\mathbb{L}}^3$ parameterized by its arc-length s. Denote by $\{\mathbf{t},\mathbf{n},\mathbf{b}\}$ the Frenet frame field along $\gamma \left(s\right)$.

If $\gamma \left(s\right)$ is a space-like curve in ${\mathbb{L}}^3$, the Frenet formulae of $\gamma \left(s\right)$ are given by [23]:

$$\begin{array}{cc}\hfill {\gamma }^{\prime }\left(s\right)& =\mathbf{t}\left(s\right),\hfill \\ \hfill {\mathbf{t}}^{\prime }\left(s\right)& =\kappa \left(s\right)\mathbf{n}\left(s\right),\hfill \\ \hfill {\mathbf{n}}^{\prime }\left(s\right)& =-\epsilon \kappa \left(s\right)\mathbf{t}\left(s\right)+\tau \left(s\right)\mathbf{b}\left(s\right),\hfill \\ \hfill {\mathbf{b}}^{\prime }\left(s\right)& =\epsilon \tau \left(s\right)\mathbf{n}\left(s\right),\hfill \end{array}$$

(5)

where $\langle \mathbf{t},\mathbf{t}\rangle =1,\langle \mathbf{n},\mathbf{n}\rangle =\epsilon (=\pm 1),\langle \mathbf{b},\mathbf{b}\rangle =-\epsilon .$ Here, the functions $\kappa \left(s\right)$ and $\tau \left(s\right)$ are the curvature function and the torsion function of a space-like curve $\gamma \left(s\right)$, respectively.

If $\gamma \left(s\right)$ is a time-like curve in ${\mathbb{L}}^3$, the Frenet formulae of $\gamma \left(s\right)$ are given by [23]:

$$\begin{array}{cc}\hfill {\gamma }^{\prime }\left(s\right)& =\mathbf{t}\left(s\right),\hfill \\ \hfill {\mathbf{t}}^{\prime }\left(s\right)& =\kappa \left(s\right)\mathbf{n}\left(s\right),\hfill \\ \hfill {\mathbf{n}}^{\prime }\left(s\right)& =-\kappa \left(s\right)\mathbf{t}\left(s\right)+\tau \left(s\right)\mathbf{b}\left(s\right),\hfill \\ \hfill {\mathbf{b}}^{\prime }\left(s\right)& =-\tau \left(s\right)\mathbf{n}\left(s\right),\hfill \end{array}$$

(6)

where $\langle \mathbf{t},\mathbf{t}\rangle =-1,\langle \mathbf{n},\mathbf{n}\rangle =\langle \mathbf{b},\mathbf{b}\rangle =1.$ Here $\kappa \left(s\right)$ and $\tau \left(s\right)$ are the curvature function and the torsion function of a time-like curve $\gamma \left(s\right)$, respectively.

If $\gamma \left(s\right)$ is a space-like or time-like pseudo-spherical curve parametrized by arc-length s in ${\mathbb{Q}}^2\left(\epsilon \right)$, let $\mathbf{t}\left(s\right)={\gamma }^{\prime }\left(s\right)$ and $\mathbf{g}\left(s\right)=\gamma \left(s\right)\times {\gamma }^{\prime }\left(s\right)$. Then, we have a pseudo-orthonormal frame $\left\{\gamma \right(s),\mathbf{t}(s),\mathbf{g}(s\left)\right\}$ along $\gamma \left(s\right)$. It is called the pseudo-spherical Frenet frame of the pseudo-spherical curve $\gamma \left(s\right)$. If γ is a space-like curve, then the vector $\mathbf{g}$ is time-like when γ is on ${\mathbb{S}}_1^2$, and the vector $\mathbf{g}$ is space-like when γ is on ${\mathbb{H}}^2$. Similarly, if the curve γ is time-like, then the vector $\mathbf{g}$ is space-like. The following theorem can be easily obtained.

Theorem 1.

([24,25]) Under the above notations, we have the following pseudo-spherical Frenet formulae of γ:

 (1)

If γ is a pseudo-spherical space-like curve,

$$\begin{array}{cc}\hfill {\gamma }^{\prime }\left(s\right)& =\mathbf{t}\left(s\right),\hfill \\ \hfill {\mathbf{t}}^{\prime }\left(s\right)& =-\epsilon \gamma \left(s\right)-\epsilon {\kappa }_g\left(s\right)\mathbf{g}\left(s\right),\hfill \\ \hfill {\mathbf{g}}^{\prime }\left(s\right)& =-{\kappa }_g\left(s\right)\mathbf{t}\left(s\right).\hfill \end{array}$$

(7)

 (2)

If γ is a pseudo-spherical time-like curve,

$$\begin{array}{cc}\hfill {\gamma }^{\prime }\left(s\right)& =\mathbf{t}\left(s\right),\hfill \\ \hfill {\mathbf{t}}^{\prime }\left(s\right)& =\gamma \left(s\right)+{\kappa }_g\left(s\right)\mathbf{g}\left(s\right),\hfill \\ \hfill {\mathbf{g}}^{\prime }\left(s\right)& ={\kappa }_g\left(s\right)\mathbf{t}\left(s\right).\hfill \end{array}$$

(8)

The function ${\kappa }_g\left(s\right)$ is called the geodesic curvature of the pseudo-spherical curve γ.

Now, we define a ruled surface M in ${\mathbb{L}}^3$. Let I and J be open intervals in the real line $\mathbb{R}$. Let $\alpha =\alpha \left(s\right)$ be a curve in ${\mathbb{L}}^3$ and $\beta =\beta \left(s\right)$ a vector field along α with ${\alpha }^{\prime }\left(s\right)\times \beta \left(s\right)\ne 0$ for every $s\in J$. Then, a ruled surface M is defined by the parametrization given as follows:

$$x=x(s,t)=\alpha \left(s\right)+t\beta \left(s\right),s\in J,t\in I.$$

For such a ruled surface, α and β are called the base curve and the director curve respectively. In particular, if β is constant, the ruled surface is said to be cylindrical, and if it is not so, it is called non-cylindrical. Furthermore, we have five different ruled surfaces according to the characters of the base curve α and the director curve β as follows: if the base curve α is space-like or time-like, then the ruled surface M is said to be of type $M_{+}$ or type $M_{-}$, respectively. In addition, the ruled surface of type $M_{+}$ can be divided into three types. In the case that β is space-like, it is said to be of type $M_{+}^1$ or $M_{+}^2$ if ${\beta }^{\prime }$ is non-null or null, respectively. When β is time-like, ${\beta }^{\prime }$ is space-like because of the causal character. In this case, M is said to be of type $M_{+}^3$. On the other hand, for the ruled surface of type $M_{-}$, it is also said to be of type $M_{-}^1$ or $M_{-}^2$ if ${\beta }^{\prime }$ is non-null or null, respectively [26].

However, if the base curve α is a light-like curve and the vector field β along α is a light-like vector field, then the ruled surface M is called a null scroll. In particular, a null scroll with Cartan frame is said to be a B-scroll [27]. It is also a time-like surface.

A non-degenerate surface in ${\mathbb{L}}^3$ with zero Gaussian curvature is called a developable surface. The developable surfaces in ${\mathbb{L}}^3$ are the same as in the Euclidean space, and they are planes, conical surfaces, cylindrical surfaces and tangent developable surfaces [13].

3. Generalized Null 2-Type Cylindrical Surfaces

For a surface in the Minkowski 3-space ${\mathbb{L}}^3$, the next lemma is well known and useful.

Lemma 1.

([16]) Let M be an oriented surface of ${\mathbb{L}}^3$. Then, the Laplacian of the mean curvature vector field $\mathbf{H}$ of M is given by

$$\Delta \mathbf{H}=2A(\nabla H)+\epsilon \nabla H^2{+(\Delta H+\epsilon H|A|}^2)N,$$

(9)

where ε is the sign of the unit normal vector N of the surface M and $\nabla H$, A are the gradient of the mean curvature H and the shape operator of M, respectively.

Theorem 2.

All cylindrical surfaces in ${\mathbb{L}}^3$ are of generalized null 2-type.

Proof. 

Let M be a cylindrical ruled surface in the Minkowski 3-space ${\mathbb{L}}^3$ of type $M_{+}^1$, $M_{-}^1$ or $M_{+}^3$. Then, M is parameterized by

$$x(s,t)=\alpha \left(s\right)+t\beta ,$$

where the base curve $\alpha \left(s\right)$, which is a space-like or time-like curve with the arc-length parameter s, lies in a plane with a space-like or time-like unit normal vector β that is the director of M, that is, $\langle \beta ,\beta \rangle ={\epsilon }_1(=\pm 1)$ and $\langle {\alpha }^{\prime }\left(s\right),{\alpha }^{\prime }\left(s\right)\rangle ={\epsilon }_2(\pm 1)$.

Now, we take a local pseudo-orthonormal frame $\{e_1,e_2,e_3\}$ on ${\mathbb{L}}^3$ such that $e_1=\frac{\partial }{\partial t}$ and $e_2=\frac{\partial }{\partial s}$ are tangent to M, and $e_3$ normal to M. It follows that the Levi–Civita connection $\tilde{\nabla }$ of ${\mathbb{L}}^3$ is expressed as

$$\begin{array}{c}{\tilde{\nabla }}_{e_1}{e}_1={\tilde{\nabla }}_{e_1}{e}_2={\tilde{\nabla }}_{e_2}{e}_1=0,{\tilde{\nabla }}_{e_2}{e}_2={\epsilon }_3\kappa \left(s\right)e_3,\hfill \\ {\tilde{\nabla }}_{e_1}{e}_3=0,{\tilde{\nabla }}_{e_2}{e}_3=-{\epsilon }_2\kappa \left(s\right)e_2,\hfill \end{array}$$

(10)

where $\kappa \left(s\right)$ is the curvature function of $\alpha \left(s\right)$ and ${\epsilon }_3(=\pm 1)$ is the sign of $e_3$. From this, the mean curvature vector field $\mathbf{H}$ of M is given by

$$\mathbf{H}=\frac{{\epsilon }_2\kappa \left(s\right)}{2}{e}_3$$

(11)

and the Laplacian $\Delta \mathbf{H}$ of $\mathbf{H}$ is expressed as

$$\Delta \mathbf{H}=\frac{3}{2}{\epsilon }_1{\epsilon }_2\kappa \left(s\right){\kappa }^{\prime }\left(s\right)e_2-\frac{1}{2}({\kappa }^3\left(s\right)+{\epsilon }_1{\kappa }^{″}\left(s\right))e_3.$$

(12)

Suppose that M is of generalized null 2-type. With the help of (4) and (12), we obtain the following equations:

$$g{C}_1=0,$$

(13)

$$\frac{3}{2}{\epsilon }_1{\epsilon }_2\kappa \left(s\right){\kappa }^{\prime }\left(s\right)=g{C}_2,$$

(14)

$$\frac{1}{2}{\epsilon }_1{\epsilon }_2{\kappa }^3\left(s\right)+\frac{1}{2}{\epsilon }_2{\kappa }^{″}\left(s\right)=-\frac{1}{2}{\epsilon }_1\kappa \left(s\right)f+g{C}_3,$$

(15)

where $\mathbf{C}={\epsilon }_1{C}_1{e}_1+{\epsilon }_2{C}_2{e}_2-{\epsilon }_1{\epsilon }_2{C}_3{e}_3$ with $C_1=\langle \mathbf{C},e_1\rangle$, $C_2=\langle \mathbf{C},e_2\rangle$ and $C_3=\langle \mathbf{C},e_3\rangle$. In this case, $C_1$ is a constant, and $C_2$, $C_3$ are functions of the variable s.

If g is identically zero, then, from (14), the curvature $\kappa \left(s\right)$ is constant, and from (15), the function f is constant, say λ. Thus, M satisfies $\Delta \mathbf{H}=\lambda \mathbf{H}$, that is, it is either a Euclidean plane, a Minkowski plane, a Lorentz circular cylinder ${\mathbb{S}}^2\times {\mathbb{R}}_1^1$, a hyperbolic cylinder ${\mathbb{H}}^1\times \mathbb{R}$ or a Lorentz hyperbolic cylinder ${\mathbb{S}}_1^1\times \mathbb{R}$ according to [16].

We now assume that $g\ne 0$. It follows from (13) that $C_1=0$. By using (10), we can show that the component functions of $\mathbf{C}$ satisfy the following equations:

$$\begin{array}{cc}\hfill C_2^{\prime }\left(s\right)+{\epsilon }_1{\epsilon }_2\kappa \left(s\right)C_3\left(s\right)& = 0,\hfill \\ \hfill C_3^{\prime }\left(s\right)+{\epsilon }_2\kappa \left(s\right)C_2\left(s\right)& = 0,\hfill \end{array}$$

(16)

which yield ${\epsilon }_2{C}_2^2\left(s\right)-{\epsilon }_1{\epsilon }_2{C}_3^2\left(s\right)=\eta d_0^2$ for some nonzero constant $d_0$, where $\eta =\langle \mathbf{C},\mathbf{C}\rangle .$

Case 1: If M is of type $M_{+}^3$, then ${\epsilon }_1=-1$, ${\epsilon }_2=1$ and $\eta =1$. We may put from (16)

$$C_2\left(s\right)=d_0\sin \theta \left(s\right),C_3\left(s\right)=d_0\cos \theta \left(s\right),$$

(17)

where $\theta \left(s\right)={\kappa }_0+\int \kappa \left(s\right)ds$ for some constant ${\kappa }_0$. Therefore, the constant vector $\mathbf{C}$ becomes

$$\mathbf{C}=d_0\sin \theta \left(s\right)e_2+d_0\cos \theta \left(s\right)e_3.$$

(18)

Combining (14), (15) and (17), one also gets

$$g=-\frac{3\kappa \left(s\right){\kappa }^{\prime }\left(s\right)}{2d_0}\csc \theta \left(s\right),f=\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}-{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right)\cot \theta \left(s\right).$$

(19)

Thus, the mean curvature vector field $\mathbf{H}$ of the cylindrical surface $M_3^{+}$ satisfies

$$\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C},$$

where $f,g$ and $\mathbf{C}$ are given in (18) and (19), respectively.

Case 2: Let M be of type $M_{+}^1$. In this case, ${\epsilon }_1=1$, ${\epsilon }_2=1$ and the constant vector $\mathbf{C}$ is space-like, time-like or null.

First of all, we consider the constant vector $\mathbf{C}$ is non-null. Then, from (16), we may put

$$\left\{\begin{array}{cc}{C}_2\left(s\right)=d_0\cosh \theta \left(s\right),C_3=d_0\sinh \theta \left(s\right)\hfill & \mathrm{if}\eta =1,\hfill \\ C_2\left(s\right)=d_0\sinh \theta \left(s\right),C_3=d_0\cosh \theta \left(s\right)\hfill & \mathrm{if}\eta =-1,\hfill \end{array}\right.$$

(20)

where $\theta \left(s\right)=-\int \kappa \left(s\right)ds+{\kappa }_0$ with a constant ${\kappa }_0$.

By using (14), (15) and (17), the functions $f\left(s\right)$ and $g\left(s\right)$ are determined by

$$\left\{\begin{array}{ccc}f\left(s\right)=-\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}+{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right)\tanh \theta \left(s\right),\hfill & g\left(s\right)=\frac{3\kappa \left(s\right){\kappa }^{\prime }\left(s\right)}{2d_0\cosh \theta \left(s\right)}\hfill & \mathrm{if}\eta =1,\hfill \\ f\left(s\right)=-\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}-{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right)\coth \theta \left(s\right),\hfill & g\left(s\right)=\frac{3\kappa \left(s\right){\kappa }^{\prime }\left(s\right)}{2d_0\sinh \theta \left(s\right)}\hfill & \mathrm{if}\eta =-1.\hfill \end{array}\right.$$

(21)

Thus, for the non-null constant vector $\mathbf{C}$, the cylindrical surface $M_1^{+}$ is of generalized null 2-type, that is, it satisfies

$$\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C},$$

where $f,g$ and $\mathbf{C}$ are given by (20) and (21), respectively.

Next, let the constant vector $\mathbf{C}$ be null, that is, $\eta =0$. Then, we get

$$C_2\left(s\right)=\pm C_3\left(s\right).$$

We will consider the case $C_2\left(s\right)=C_3\left(s\right)$. It follows from (16) $C_2\left(s\right)=e^{\theta \left(s\right)}$, where $\theta \left(s\right)=-\int \kappa \left(s\right)ds+{\kappa }_0$ for some constant ${\kappa }_0$. In this case, we have

$$f\left(s\right)=-\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}-{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right),g\left(s\right)=\frac{3}{2}{e}^{-\theta \left(s\right)}\kappa \left(s\right){\kappa }^{\prime }\left(s\right)$$

(22)

and, for the null constant vector $\mathbf{C}$, the surface satisfies the condition $\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C}$.

Case 3: Let M be of type $M_{-}^1$, that is, ${\epsilon }_1=1$, ${\epsilon }_2=-1$. In this case, the constant vector $\mathbf{C}$ is space-like, time-like or null.

Applying the same method as in Case 2, the functions $f\left(s\right)$ and $g\left(s\right)$ are determined by

$$\left\{\begin{array}{ccc}f\left(s\right)=\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}+{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right)\coth \theta \left(s\right),\hfill & g\left(s\right)=\frac{3\kappa \left(s\right){\kappa }^{\prime }\left(s\right)}{2d_0\sinh \theta \left(s\right)}\hfill & \mathrm{if}\eta =1,\hfill \\ f\left(s\right)=\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}+{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right)\tanh \theta \left(s\right),\hfill & g\left(s\right)=\frac{3\kappa \left(s\right){\kappa }^{\prime }\left(s\right)}{2d_0\cosh \theta \left(s\right)}\hfill & \mathrm{if}\eta =-1,\hfill \\ f\left(s\right)=\frac{{\kappa }^{″}\left(s\right)}{\kappa \left(s\right)}+{\kappa }^2\left(s\right)+3{\kappa }^{\prime }\left(s\right),\hfill & g\left(s\right)=\frac{3}{2}{e}^{-\theta \left(s\right)}\kappa \left(s\right){\kappa }^{\prime }\left(s\right)\hfill & \mathrm{if}\eta =0,\hfill \end{array}\right.$$

(23)

and the component functions of $\mathbf{C}$ are given by

$$\left\{\begin{array}{cc}{C}_2\left(s\right)=d_0\sinh \theta \left(s\right),C_3\left(s\right)=d_0\cosh \theta \left(s\right),\hfill & \mathrm{if}\eta =1,\hfill \\ C_2\left(s\right)=d_0\cosh \theta \left(s\right),C_3\left(s\right)=d_0\sinh \theta \left(s\right),\hfill & \mathrm{if}\eta =-1,\hfill \\ C_2\left(s\right)=\pm C_3\left(s\right),\hfill & \mathrm{if}\eta =0,\hfill \end{array}\right.$$

(24)

where $\theta \left(s\right)=\int \kappa \left(s\right)ds+{\kappa }_0$ for some constant ${\kappa }_0$.

Thus, from Cases 1, 2 and 3, Theorem 2 is proved.  ☐

Example 1.

We consider a surface defined by

$$x(s,t)=\left(\frac{1}{4}{s}^2-\frac{1}{2}\ln s,\frac{1}{4}{s}^2+\frac{1}{2}\ln s,t\right).$$

(25)

This parametrization is a cylindrical ruled surface of type $M_{+}^1$. In this case, the mean curvature vector field $\mathbf{H}$ of the surface is given by

$$\mathbf{H}=\left(-\frac{1}{4}-\frac{1}{4s^2},-\frac{1}{4}+\frac{1}{4s^2},0\right).$$

By a direct computation, the Laplacian $\Delta \mathbf{H}$ of the mean curvature vector field $\mathbf{H}$ becomes

$$\Delta \mathbf{H}=\left(\frac{3}{2s^4},-\frac{3}{2s^4},0\right),$$

and it can be rewritten in terms of the mean curvature vector field $\mathbf{H}$ and a constant vector $\mathbf{C}$ as follows:

$$\Delta \mathbf{H}=-\frac{6}{s^2}(\mathbf{H}+\mathbf{C}),$$

where $\mathbf{C}=(\frac{1}{4},\frac{1}{4},0)$ is a null vector. Thus, the cylindrical ruled surface defined by (25) is a generalized null 2-type surface of the first kind.

Remark 1.

A cylindrical surface in ${\mathbb{L}}^3$ generated by the base curve $\alpha \left(s\right)$ with the curvature $\kappa \left(s\right)=\frac{1}{s}$ and a constant director β is a generalized null 2-type surface of the first kind if the constant vector $\mathbf{C}$ is null.

4. Generalized Null 2-Type Non-Cylindrical Flat Surfaces

In this section, we classify non-cylindrical flat surfaces satisfying

$$\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C}.$$

(26)

It is well-known that a non-cylindrical flat surface in the Minkowski 3-space ${\mathbb{L}}^3$ is an open part of a conical surface or a tangent developable surface.

First of all, we consider a conical surface M in ${\mathbb{L}}^3$. Then, we may give the parametrization of M by

$$x(s,t)={\alpha }_0+t\beta \left(s\right),s\in I,t>0,$$

such that $\langle {\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right)\rangle ={\epsilon }_1$ and $\langle \beta \left(s\right),\beta \left(s\right)\rangle ={\epsilon }_2$, where ${\alpha }_0$ is a constant vector. We take the orthonormal tangent frame $\{e_1,e_2\}$ on M such that $e_1=\frac{1}{t}\frac{\partial }{\partial s}$ and $e_2=\frac{\partial }{\partial t}$. The unit normal vector of M is given by $e_3=e_1\times e_2$. By the Gauss and Weingarten formulas, we have

$$\begin{array}{cc}& {\tilde{\nabla }}_{e_1}{e}_1=-\frac{{\epsilon }_1{\epsilon }_2}{t}{e}_2+\frac{{\epsilon }_1{\epsilon }_2{\kappa }_g\left(s\right)}{t}{e}_3,{\tilde{\nabla }}_{e_1}{e}_2=\frac{1}{t}{e}_1,{\tilde{\nabla }}_{e_2}{e}_1={\tilde{\nabla }}_{e_2}{e}_2=0,\hfill \\ & {\tilde{\nabla }}_{e_1}{e}_3=\frac{{\epsilon }_1{\kappa }_g\left(s\right)}{t}{e}_1,{\tilde{\nabla }}_{e_2}{e}_3=0,\hfill \end{array}$$

(27)

where ${\kappa }_g\left(s\right)=\langle \beta \left(s\right),{\beta }^{\prime }\left(s\right)\times {\beta }^{″}\left(s\right)\rangle$, which is the geodesic curvature of the pseudo-spherical curve $\beta \left(s\right)$ in ${\mathbb{Q}}^2\left(\epsilon \right)$. From (27), the mean curvature vector field $\mathbf{H}$ of M is given by

$$\mathbf{H}=-\frac{{\epsilon }_1{\kappa }_g\left(s\right)}{2t}{e}_3,$$

(28)

and the Laplacian $\Delta \mathbf{H}$ of the mean curvature vector field $\mathbf{H}$ is expressed as

$$\Delta \mathbf{H}=\frac{3{\epsilon }_1}{2t^3}{\kappa }_g\left(s\right){\kappa }_g^{\prime }\left(s\right)e_1-\frac{{\epsilon }_2}{2t^3}{\kappa }_g^2\left(s\right)e_2+\left(\frac{1}{2t^3}{\kappa }_g^{″}\left(s\right)+\frac{{\epsilon }_2}{2t^3}{\kappa }_g^3\left(s\right)+\frac{{\epsilon }_1{\epsilon }_2}{2t^3}{\kappa }_g\left(s\right)\right)e_3.$$

(29)

Suppose that ${\kappa }_g$ is constant. If ${\kappa }_g=0$, by a rigid motion, the pseudo-spherical curve $\beta \left(s\right)$ in ${\mathbb{Q}}^2\left(\epsilon \right)$ lies on $yz$-plane or $xz$-plane. Thus M is an open part of a Euclidean plane or a Minkowski plane. If ${\kappa }_g$ is a non-zero constant, from (27), we can obtain by a straightforward computation

$${\beta }^{‴}\left(s\right)={\epsilon }_2({\kappa }_g^2\left(s\right)-{\epsilon }_1){\beta }^{\prime }\left(s\right).$$

(30)

Case 1: ${\epsilon }_2({\kappa }_g^2\left(s\right)-{\epsilon }_1)=k^2$ for some real number k.

Let ${\epsilon }_1=1$. Without loss of generality, we may assume ${\beta }^{\prime }\left(0\right)=(0,1,0)$. Thus, ${\beta }^{‴}\left(s\right)=k^2{\beta }^{\prime }\left(s\right)$ implies

$${\beta }^{\prime }\left(s\right)=(B_1\sinh ks,\cosh ks+B_2\sinh ks,B_3\sinh ks)$$

for some constants $B_1,B_2$ and $B_3$. Since ${\epsilon }_1=1$, we have $B_1^2-B_3^2=1$ and $B_2=0$. From this, we can obtain

$$\beta \left(s\right)=\left(\frac{B_1}{k}\cosh ks+D_1,\frac{1}{k}\sinh ks,\frac{B_3}{k}\cosh ks+D_3\right)$$

(31)

for some constants $D_1,D_3$ satisfying $D_3^2-D_1^2=\frac{1}{k^2}+{\epsilon }_2$, $B_1{D}_1=B_3{D}_3$ and $B_1^2-B_3^2=1$. We now change the coordinates by $\overline{x},\overline{y},\overline{z}$ such that $\overline{x}=B_{1}x-B_{3}z$, $\overline{y}=y$, $\overline{z}=-B_{3}x+B_{1}z$, that is,

$$\left(\begin{array}{c}\overline{x}\\ \overline{y}\\ \overline{z}\end{array}\right)=\left(\begin{array}{ccc}{B}_1& 0& -B_3\\ 0& 1& 0\\ -B_3& 0& B_1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right).$$

With respect to the coordinates $(\overline{x},\overline{y},\overline{z})$, $\beta \left(s\right)$ turns into

$$\beta \left(s\right)=\left(\frac{1}{k}\cosh ks,\frac{1}{k}\sinh ks,D\right)$$

(32)

for a constant $D=B_1{D}_3-B_3{D}_1$ with $D^2=\frac{1}{k^2}+{\epsilon }_2$. Thus, up to a rigid motion M has the parametrization of the form

$$x(s,t)={\alpha }_0+t\left(\frac{1}{k}\cosh ks,\frac{1}{k}\sinh ks,D\right).$$

We call such a surface a hyperbolic conical surface of the first kind, and it satisfies

$$\Delta \mathbf{H}=\left(\frac{{\epsilon }_2(1-D^2-k^2)}{k^2{t}^2}\right)\mathbf{H}+\left(\frac{{\epsilon }_{2}D(1-D^2{k}^2)}{2k^4{t}^3}\right)(0,0,1).$$

Next, let $({\epsilon }_1,{\epsilon }_2)=(-1,1)$. We now consider a initial condition ${\beta }^{\prime }\left(0\right)=(1,0,0)$ of the ordinary differential equation (ODE) (30). Quite similarly as we did, we obtain

$$\beta \left(s\right)=\left(\frac{1}{k}\sinh ks,\frac{B_2}{k}\cosh ks+D_2,\frac{B_3}{k}\cosh ks+D_3\right),$$

satisfying $B_2^2+B_3^2=1,B_2{D}_2+B_3{D}_3=0$ and $D_2^2+D_3^2=1-\frac{1}{k^2}$.

If we adopt the coordinates’ transformation,

$$\left(\begin{array}{c}\overline{x}\\ \overline{y}\\ \overline{z}\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& B_2& B_3\\ 0& -B_3& B_2\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right).$$

With respect to the new coordinates $(\overline{x},\overline{y},\overline{z})$, the vector $\beta \left(s\right)$ becomes

$$\beta \left(s\right)=\left(\frac{1}{k}\sinh ks,\frac{1}{k}\cosh ks,D\right),$$

(33)

where $D=B_2{D}_3-B_3{D}_2$ with $D^2=1-\frac{1}{k^2}$. We call such a surface generated by (33) a hyperbolic conical surface of the second kind and it satisfies

$$\Delta \mathbf{H}=\left(\frac{1+D^2+k^2}{k^2{t}^2}\right)\mathbf{H}+\left(\frac{D(1+k^2{D}^2)}{2k^4{t}^3}\right)(0,0,1).$$

Case 2: ${\epsilon }_2({\kappa }_g^2\left(s\right)-{\epsilon }_1)=-k^2$ for some real number k.

Let ${\epsilon }_1=1$. We may give the initial condition by ${\beta }^{\prime }\left(0\right)=(0,1,0)$ for the differential equation ${\beta }^{‴}\left(s\right)+k^2{\beta }^{\prime }\left(s\right)=0$. Under such an initial condition, a vector field $\beta \left(s\right)$ is given by

$$\beta \left(s\right)=\left(-\frac{B_1}{k}\cos ks+D_1,\frac{1}{k}\sin ks,-\frac{B_3}{k}\cos ks+D_3\right),$$

where $B_1,B_3,D_1$ and $D_3$ are some constants satisfying $B_3^2-B_1^2=1$, $B_1{D}_1=B_3{D}_3$ and $D_1^2-D_3^2=\frac{1}{k^2}-{\epsilon }_2$. If we take another coordinate system $(\overline{x},\overline{y},\overline{z})$ such that

$$\overline{x}=-B_{3}x+B_{1}z,\overline{y}=y,\overline{z}=B_{1}x-B_{3}z,$$

then a vector $\beta \left(s\right)$ takes the form

$$\beta \left(s\right)=\left(D,\frac{1}{k}\sin ks,\frac{1}{k}\cos ks\right),$$

(34)

where $D=B_1{D}_3-B_3{D}_1$ satisfying $D^2=\frac{1}{k^2}-{\epsilon }_2$. We call such a surface generated by (34) an elliptic conical surface and it satisfies

$$\Delta \mathbf{H}=\left(\frac{{\epsilon }_1-{\epsilon }_1{D}^2-k^2}{k^2{t}^2}\right)\mathbf{H}-\left(\frac{D(k^2{D}^2-1)}{2k^4{t}^3}\right)(1,0,0).$$

Case of ${\epsilon }_1=-1$ gives ${\epsilon }_2=-1$. It is impossible by the causal character of Lorentz geometry.

Case 3: ${\kappa }_g^2\left(s\right)-{\epsilon }_1=0.$

In this case, ${\kappa }_g^2\left(s\right)=1$, in other words, ${\epsilon }_1=1$, which implies by using (27) $\langle {\beta }^{″}\left(s\right),{\beta }^{″}\left(s\right)\rangle =0$. Since ${\beta }^{″}\left(s\right)$ is a constant vector by (30), we may put ${\beta }^{″}\left(s\right)=(d_1,d_2,d_3)$ for some constants $d_1$, $d_2$, $d_3$ satisfying $-d_1^2+d_2^2+d_3^2=0$ and so ${\beta }^{\prime }\left(s\right)=(d_{1}s+k_1,d_{2}s+k_2,d_{3}s+k_3)$ for some constants $k_1,k_2$ and $k_3$. Since $\langle {\beta }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right)\rangle ={\epsilon }_1=1$, we may set $(k_1,k_2,k_3)=(0,1,0)$ up to an isometry and hence $\beta \left(s\right)=(\frac{d_1}{2}{s}^2+c_1,\frac{d_2}{2}{s}^2+s+c_2,\frac{d_3}{2}{s}^2+c_3)$ for some constants $c_1,c_2$ and $c_3$. However, $\langle \beta \left(s\right),\beta \left(s\right)\rangle ={\epsilon }_2$ implies $d_2=c_2=0$ and $d_1^2=d_3^2$, $-c_1^2+c_3^2={\epsilon }_2$, $-d_1{c}_1+d_3{c}_3+1=0$. Thus, $\beta \left(s\right)$ takes the form

$$\beta \left(s\right)=\left(\frac{d_1}{2}{s}^2+c_1,s,\frac{d_3}{2}{s}^2+c_3\right).$$

(35)

We call such a surface generated by (35) a quadric conical surface.

As shown in the Introduction, a quadric conical surface is of generalized null 2-type of the first kind. Let us suppose that ${\kappa }_g$ is a non-constant, i.e., ${\kappa }_g^{\prime }\ne 0$ on an open interval. Suppose that M is of generalized null 2-type, that is, M satisfies the condition (4). Then, we have the following equations:

$$\frac{3{\kappa }_g\left(s\right){\kappa }_g^{\prime }\left(s\right)}{2t^3}=g{C}_1,$$

(36)

$$-\frac{{\kappa }_g^2\left(s\right)}{2t^3}=g{C}_2,$$

(37)

$$-\frac{1}{2t^3}({\epsilon }_1{\epsilon }_2{\kappa }_g^{″}\left(s\right)+{\epsilon }_1{\kappa }_g^3\left(s\right)+{\kappa }_g\left(s\right))=\frac{{\epsilon }_2{\kappa }_g\left(s\right)}{2t}f+g{C}_3,$$

(38)

where $\mathbf{C}={\epsilon }_1{C}_1{e}_1+{\epsilon }_2{C}_2{e}_2-{\epsilon }_1{\epsilon }_2{C}_3{e}_3$ with $C_1=\langle \mathbf{C},e_1\rangle$, $C_2=\langle \mathbf{C},e_2\rangle$ and $C_3=\langle \mathbf{C},e_3\rangle$. Since $e_1={\beta }^{\prime }\left(s\right)$, $e_2=\beta \left(s\right)$ and $e_3={\beta }^{\prime }\left(s\right)\times \beta \left(s\right)$, the component functions $C_i$ $(i=1,2,3)$ of $\mathbf{C}$ depend only on variable s. Let us differentiate $C_1$, $C_2$ and $C_3$ covariantly with respect to $e_1$. Then, from (27), we have the following equations:

$$C_1^{\prime }\left(s\right)+{\epsilon }_1{\epsilon }_2{C}_2\left(s\right)-{\epsilon }_1{\epsilon }_2{\kappa }_g\left(s\right)C_3\left(s\right)=0,$$

(39)

$$C_2^{\prime }\left(s\right)-C_1\left(s\right)=0,$$

(40)

$$C_3^{\prime }\left(s\right)-{\epsilon }_1{\kappa }_g\left(s\right)C_1\left(s\right)=0.$$

(41)

Combining (36) and (37), and using (40), we have

$$C_1=-\frac{3c{\kappa }_g^{\prime }}{{\kappa }_g^4}\mathrm{and}{C}_2=\frac{c}{{\kappa }_g^3},$$

(42)

where c is a constant of integration.

Together with (37) and (42), we can find

$$g=-\frac{{\kappa }_g^5}{2c{t}^3}.$$

(43)

Substituting (42) into (39), we get

$$C_3=\frac{c({\kappa }_g^2-3{\epsilon }_1{\epsilon }_2{\kappa }_g{\kappa }_g^{″}+12{\epsilon }_1{\epsilon }_2{{\kappa }_g^{\prime }}^2)}{{\kappa }_g^6}.$$

(44)

Then, (38) and (44) lead to

$$f=-\frac{{\epsilon }_2}{t^2{\kappa }_g^2}(4{\epsilon }_1{\epsilon }_2{\kappa }_g{\kappa }_g^{″}-12{\epsilon }_1{\epsilon }_2{{\kappa }_g^{\prime }}^2+{\epsilon }_1{\kappa }_g^4).$$

(45)

Furthermore, it follows from (41) and (42) that

$$C_3^{\prime }=-\frac{3{\epsilon }_{1}c{\kappa }_g^{\prime }}{{\kappa }_g^3}$$

and its solution is given by

$$C_3=\frac{3{\epsilon }_{1}c}{2{\kappa }_g^2}+a_1$$

(46)

for some constant $a_1$.

Combining (44) and (46), the geodesic curvature ${\kappa }_g$ satisfies the following equation:

$${\kappa }_g^{″}-\frac{4}{{\kappa }_g}{{\kappa }_g^{\prime }}^2-\frac{1}{3}{\epsilon }_1{\epsilon }_2{\kappa }_g+\frac{1}{2}{\epsilon }_2{\kappa }_g^3+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{\kappa }_g^5=0.$$

(47)

To solve the ODE, we put $p={\kappa }_g^{\prime }$. Then, (47) can be written of the form

$$\frac{dp}{d{\kappa }_g}-\frac{4}{{\kappa }_g}p=\frac{1}{p}\left(\frac{1}{3}{\epsilon }_1{\epsilon }_2{\kappa }_g-\frac{1}{2}{\epsilon }_2{\kappa }_g^3-\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{\kappa }_g^5\right),$$

(48)

and it is a Bernoulli differential equation. Thus, the solution is given by

$$p=\pm {\kappa }_g{\left(a_2{\kappa }_g^6+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{\kappa }_g^4+\frac{1}{4}{\epsilon }_2{\kappa }_g^2-\frac{1}{9}{\epsilon }_1{\epsilon }_2\right)}^{\frac{1}{2}},$$

which is equivalent to

$${\kappa }_g^{-1}{\left(a_2{\kappa }_g^6+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{\kappa }_g^4+\frac{1}{4}{\epsilon }_2{\kappa }_g^2-\frac{1}{9}{\epsilon }_1{\epsilon }_2\right)}^{-\frac{1}{2}}d{\kappa }_g=\pm ds$$

for some constant $a_2$. If we put

$$F(v)=\int \psi (v)dv,$$

where

$$\psi \left(v\right)=v^{-1}{\left(a_2{v}^6+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{v}^4+\frac{1}{4}{\epsilon }_2{v}^2-\frac{1}{9}{\epsilon }_1{\epsilon }_2\right)}^{-\frac{1}{2}},$$

and then we have

$$F\left({\kappa }_g\right)=\pm s+a_3$$

(49)

for some constant $a_3$. Thus, the geodesic curvature ${\kappa }_g$ is given by

$${\kappa }_g\left(s\right)=F^{-1}(\pm s+a_3).$$

(50)

Furthermore, the constant vector $\mathbf{C}$ can be expressed as

$$\mathbf{C}=-\frac{3c{\kappa }_g^{\prime }}{{\kappa }_g^4}{e}_1+\frac{c}{{\kappa }_g^3}{e}_2+\frac{c({\kappa }_g^2-3{\epsilon }_1{\epsilon }_2{\kappa }_g{\kappa }_g^{″}+12{\epsilon }_1{\epsilon }_2{{\kappa }_g^{\prime }}^2)}{{\kappa }_g^6}{e}_3.$$

(51)

Conversely, for some constants $a_1,a_2$ and c such that the function

$$\psi \left(v\right)=v^{-1}{\left(a_2{v}^6+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{v}^4+\frac{1}{4}{\epsilon }_2{v}^2-\frac{1}{9}{\epsilon }_1{\epsilon }_2\right)}^{-\frac{1}{2}}$$

(52)

is well-defined on an open interval $J\subset (0,\infty )$, we take an indefinite integral $F\left(v\right)$ of the function $\psi \left(v\right)$. Let I be the image of the function F. We can take an open subinterval $J_1\subset J$ such that $F:J_1\to I$ is a strictly increasing function with $F^{\prime }\left(v\right)=\psi \left(v\right)$. Let us consider the function φ defined by $\phi \left(s\right)=F^{-1}(\pm s+a_3)$ for some constant $a_3$. Then, the function φ satisfies $F\left(\phi \right)=\pm s+a_3$.

For any unit speed pseudo-spherical curve $\beta \left(s\right)$ in ${\mathbb{Q}}^2\left(\epsilon \right)$ with geodesic curvature ${\kappa }_g\left(s\right)=\phi \left(s\right)$, we consider the conical surface M in ${\mathbb{L}}^3$ parametrized by

$$x(s,t)={\alpha }_0+t\beta \left(s\right),s\in I,t>0,$$

(53)

where ${\alpha }_0$ is a constant vector. Given any nonzero constant c, we put f and g the functions, respectively, given by

$$f(s,t)=-\frac{{\epsilon }_2}{t^2{\phi }^2}(4{\epsilon }_1{\epsilon }_2\phi {\phi }^{″}-12{\epsilon }_1{\epsilon }_2{{\phi }^{\prime }}^2+{\epsilon }_1{\phi }^4),g(s,t)=-\frac{{\phi }^5}{2c{t}^3}.$$

(54)

For a nonzero constant c and the pseudo-orthonormal frame $\{e_1,e_2,e_3\}$ on ${\mathbb{L}}^3$ such that $e_1=\frac{1}{t}\frac{\partial }{\partial s}$ and $e_2=\frac{\partial }{\partial t}$ are tangent to M and $e_3$ normal to M, we put

$$\mathbf{C}=-\frac{3c{\phi }^{\prime }}{{\phi }^4}{e}_1+\frac{c}{{\phi }^3}{e}_2+\frac{c({\phi }^2-3{\epsilon }_1{\epsilon }_2\phi {\phi }^{″}+12{\epsilon }_1{\epsilon }_2{{\phi }^{\prime }}^2)}{{\phi }^6}{e}_3.$$

(55)

Note that it follows from the definition of φ that the function φ satisfies (47). Hence, using (27), it is straightforward to show that

$${\tilde{\nabla }}_{e_1}\mathbf{C}={\tilde{\nabla }}_{e_2}\mathbf{C}=0,$$

which implies that $\mathbf{C}$ is a constant vector. Furthermore, the same argument as in the first part of this subsection yields the mean curvature vector field $\mathbf{H}$ of the conical surface M satisfies

$$\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C},$$

where $f,g$ and $\mathbf{C}$ are given in (54) and (55), respectively. This shows that the conical surface is of generalized null 2-type.

Thus, we have the following:

Theorem 3.

Let M be a conical surface in the Minkowski 3-space ${\mathbb{L}}^3$. Then, M is of generalized null 2-type if and only if it is an open part of one of the following surfaces:

(1) 

a Euclidean plane;

(2) 

a Minkowski plane;

(3) 

a hyperbolic conical surface of the first kind;

(4) 

a hyperbolic conical surface of the second kind;

(5) 

an elliptic conical surface;

(6) 

a quadric conical surface;

(7) 

a conical surface parameterized by

$$x(s,t)={\alpha }_0+t\beta \left(s\right),$$

where ${\alpha }_0$ is a constant vector and $\beta \left(s\right)$ is a unit speed pseudo-spherical curve in ${\mathbb{Q}}^2\left(\epsilon \right)$ with the non-constant geodesic curvature ${\kappa }_g$ which is, for some indefinite integral $F\left(v\right)$ of the function

$$\psi \left(v\right)=v^{-1}{\left(a_2{v}^6+\frac{a_1}{3c}{\epsilon }_1{\epsilon }_2{v}^4+\frac{1}{4}{\epsilon }_2{v}^2-\frac{1}{9}{\epsilon }_1{\epsilon }_2\right)}^{-\frac{1}{2}}$$

with $a_1,a_2,c\in R$, given by

$${\kappa }_g\left(s\right)=F^{-1}(\pm s+a_3),$$

where $a_3$ is constant.

Next, we study tangent developable surfaces in the Minkowski 3-space ${\mathbb{L}}^3$.

Theorem 4.

Let M be a tangent developable surface in the Minkowski 3-space ${\mathbb{L}}^3$. Then, M is of generalized null 2-type if and only if M is an open part of a Euclidean plane or a Minkowski plane.

Proof. 

Let $\alpha \left(s\right)$ be a curve parameterized by arc-length s in ${\mathbb{L}}^3$ with non-zero curvature $\kappa \left(s\right)$. Then, a non-degenerate tangent developable surface M in ${\mathbb{L}}^3$ is defined by

$$x(s,t)=\alpha \left(s\right)+t{\alpha }^{\prime }\left(s\right),t\ne 0.$$

In the case, we can take the pseudo-orthonormal frame $\{e_1,e_2,e_3\}$ of ${\mathbb{L}}^3$ such that $e_1=\frac{\partial }{\partial t}$ and $e_2=\frac{{\epsilon }_2}{t\kappa \left(s\right)}\left(\frac{\partial }{\partial s}-\frac{\partial }{\partial t}\right)$ are tangent to M and $e_3$ is normal to M. By a direct calculation, we obtain

$$\begin{array}{cc}& {\tilde{\nabla }}_{e_1}{e}_1={\tilde{\nabla }}_{e_1}{e}_2=0,{\tilde{\nabla }}_{e_2}{e}_1=\frac{1}{t}{e}_2,{\tilde{\nabla }}_{e_2}{e}_2=-\frac{{\epsilon }_1{\epsilon }_2}{t}{e}_1-\frac{{\epsilon }_1\tau \left(s\right)}{t\kappa \left(s\right)}{e}_3,\hfill \\ & {\tilde{\nabla }}_{e_1}{e}_3=0,{\tilde{\nabla }}_{e_2}{e}_3=\frac{{\epsilon }_2\tau \left(s\right)}{t\kappa \left(s\right)}{e}_2,\hfill \end{array}$$

(56)

where $\langle e_1,e_1\rangle ={\epsilon }_1(=\pm 1)$, $\langle e_2,e_2\rangle ={\epsilon }_2(=\pm 1)$ and $\tau \left(s\right)$ is the torsion of $\alpha \left(s\right)$. Therefore, the mean curvature vector field $\mathbf{H}$ of M is given by

$$\mathbf{H}=\frac{\tau \left(s\right)}{2t\kappa \left(s\right)}{e}_3.$$

(57)

By a long computation, the Laplacian $\Delta \mathbf{H}$ of the mean curvature vector field $\mathbf{H}$ turns out to be

$$\begin{array}{cc}\hfill \Delta \mathbf{H}& =-\frac{{\epsilon }_1{\tau }^2}{2{\kappa }^2{t}^3}{e}_1+\frac{1}{2{\kappa }^4{t}^4}\left(3\kappa {\tau }^2-2{\kappa }^{\prime }{\tau }^{2}t+3\kappa \tau {\tau }^{\prime }t\right)e_2\hfill \\ & +\frac{1}{2{\kappa }^4{t}^5}\left(-{\epsilon }_1{\kappa }^3\tau t^2+{\epsilon }_1{\kappa }^{\prime }\tau t-3{\epsilon }_2\kappa \tau -({\epsilon }_1{\kappa }^{2}t+{\epsilon }_2\kappa {\kappa }^{\prime }{t}^2){(\frac{\tau }{\kappa })}^{\prime }\right.\hfill \\ & \left.-{\epsilon }_2{\kappa }^2{(\frac{\tau }{\kappa })}^{″}{t}^2-{\epsilon }_1{\tau }^3\kappa t^2\right)e_3.\hfill \end{array}$$

(58)

Suppose that M is of generalized null 2-type, that is, M satisfies $\Delta \mathbf{H}=f\mathbf{H}+g\mathbf{C}$ for some smooth functions $f,g$ and a constant vector $\mathbf{C}$. With the help of (57) and (58), (4) can be written in the form

$$\begin{array}{cc}\hfill g{C}_1& =-\frac{{\tau }^2}{2{\kappa }^2{t}^3},\hfill \\ \hfill g{C}_2& =\frac{{\epsilon }_2}{2{\kappa }^4{t}^4}\left(3\kappa {\tau }^2-2{\kappa }^{\prime }{\tau }^{2}t+3\kappa \tau {\tau }^{\prime }t\right),\hfill \\ \hfill -\frac{{\epsilon }_1{\epsilon }_2\tau }{2\kappa t}f+g{C}_3& =-\frac{{\epsilon }_1{\epsilon }_2}{2{\kappa }^4{t}^5}\left(-{\epsilon }_1{\kappa }^3\tau t^2+{\epsilon }_1{\kappa }^{\prime }\tau t-3{\epsilon }_2\kappa \tau -({\epsilon }_1{\kappa }^{2}t+{\epsilon }_2\kappa {\kappa }^{\prime }{t}^2){(\frac{\tau }{\kappa })}^{\prime }\right.\hfill \\ & \left.-{\epsilon }_2{\kappa }^2{(\frac{\tau }{\kappa })}^{″}{t}^2-{\epsilon }_1{\tau }^3\kappa t^2\right),\hfill \end{array}$$

(59)

where $\mathbf{C}={\epsilon }_1{C}_1{e}_1+{\epsilon }_2{C}_2{e}_2-{\epsilon }_1{\epsilon }_2{C}_3{e}_3$ with $C_1=\langle \mathbf{C},e_1\rangle$, $C_2=\langle \mathbf{C},e_2\rangle$ and $C_3=\langle \mathbf{C},e_3\rangle$. In this case, the components $C_i$ of $\mathbf{C}$ are functions of only s. It follows from (56) that we have

$$C_1^{\prime }-{\epsilon }_2\kappa C_2=0,$$

(60)

$$C_2^{\prime }+{\epsilon }_1\kappa C_1+{\epsilon }_1{\epsilon }_2\tau C_3=0,$$

(61)

$$C_3^{\prime }+{\epsilon }_2\tau C_2=0.$$

(62)

By combining the first and second equations of (59), we get

$$3{\epsilon }_2\kappa {\tau }^2{C}_1+(3{\epsilon }_2\kappa \tau {\tau }^{\prime }{C}_1-2{\epsilon }_2{\kappa }^{\prime }{\tau }^2{C}_1+{\kappa }^2{\tau }^2{C}_2)t=0.$$

This shows that we obtain

$$\begin{array}{cc}\hfill 3{\epsilon }_2\kappa {\tau }^2{C}_1& =0,\hfill \\ \hfill 3{\epsilon }_2\kappa \tau {\tau }^{\prime }{C}_1-2{\epsilon }_2{\kappa }^{\prime }{\tau }^2{C}_1+{\kappa }^2{\tau }^2{C}_2& =0.\hfill \end{array}$$

(63)

Consider the open set $\mathcal{O}=\{p\in M|\tau \left(p\right)\ne 0\}$. Suppose that $\mathcal{O}$ is a non-empty set. (63) shows that $C_1=0$ and $C_2=0$, and it follows from (61) that $C_3=0$. That is, $\mathbf{C}=0$ on $\mathcal{O}$. In addition, (59) gives $\tau =0$, and it is a contradiction. Thus, the open set $\mathcal{O}$ is empty and τ is identically zero. Therefore, $\alpha \left(s\right)$ is a plane curve, and the surface M is an open part of a Euclidean plane or a Minkowski plane.

The converse of Theorem 4 follows a straightforward calculation.  ☐

5. Null Scrolls

Let $\alpha =\alpha \left(s\right)$ be a null curve in ${\mathbb{L}}^3$ and $\beta =\beta \left(s\right)$ a null vector field along α satisfying $\langle {\alpha }^{\prime },\beta \rangle =-1$. Then, the null scroll M is parameterized by

$$x(s,t)=\alpha \left(s\right)+t\beta \left(s\right).$$

(64)

Furthermore, without loss of generality, we may choose $\alpha \left(s\right)$ as a null geodesic of M, i.e, $\langle {\alpha }^{\prime }\left(s\right),{\beta }^{\prime }\left(s\right)\rangle =0$ for all s. By putting $\gamma \left(s\right)={\alpha }^{\prime }\left(s\right)\times \beta \left(s\right)$, then $\{{\alpha }^{\prime }\left(s\right),\beta \left(s\right),\gamma \left(s\right)\}$ is a pseudo-orthonormal frame along $\alpha \left(s\right)$ in ${\mathbb{L}}^3$. We define the smooth functions k and u by

$$k\left(s\right)=\langle {\alpha }^{″}\left(s\right),\gamma \left(s\right)\rangle ,u\left(s\right)=\langle \beta \left(s\right),{\gamma }^{\prime }\left(s\right)\rangle .$$

On the other hand, the induced Lorentz metric on M is given by $g_{11}=u{\left(s\right)}^2{t}^2$, $g_{12}=-1$ and $g_{22}=0$. Since M is a non-degenerate surface, $u\left(s\right)t$ is non-vanishing everywhere.

In terms of the pseudo-orthonormal frame, we have

$$\begin{array}{cc}\hfill {\alpha }^{″}\left(s\right)& =k\left(s\right)\gamma \left(s\right),\hfill \\ \hfill {\beta }^{\prime }\left(s\right)& =-u\left(s\right)\gamma \left(s\right),\hfill \\ \hfill {\gamma }^{\prime }\left(s\right)& =-u\left(s\right){\alpha }^{\prime }\left(s\right)+k\left(s\right)\beta \left(s\right).\hfill \end{array}$$

(65)

The mean curvature vector field $\mathbf{H}$ of M is given by

$$\mathbf{H}=-u^{2}t\beta +u\gamma ,$$

and its Laplacian $\Delta \mathbf{H}$ is expressed as

$$\Delta \mathbf{H}=(-4u{u}^{\prime }-2u^{4}t)\beta +2u^3\gamma .$$

(66)

Suppose that M is a generalized null 2-type surface. Then, we have

$$\begin{array}{cc}\hfill 4u{u}^{\prime }+2u^{4}t& =u^{2}tf-g{C}_2,\hfill \\ \hfill g{C}_1& =0,\hfill \\ \hfill 2u^3& =uf+g{C}_3,\hfill \end{array}$$

(67)

for a constant vector $\mathbf{C}=C_1{\alpha }^{\prime }+C_2\beta +C_3\gamma$ with $C_1=-\langle \mathbf{C},\beta \rangle$, $C_2=-\langle \mathbf{C},{\alpha }^{\prime }\rangle$ and $C_3=\langle \mathbf{C},\gamma \rangle$.

Suppose that g is identically zero. By combining the first and third Equations in (67), we see that u is constant, say $u_0$. In this case, we have $f=2u_0^2$. Thus, M is a B-scroll, and it satisfies $\Delta \mathbf{H}=2u_0^2\mathbf{H}$ (see [16]).

Consider the open set $\mathcal{O}=\{p\in M|g\left(p\right)\ne 0\}$. Suppose that $\mathcal{O}$ is a non-empty set. Then, from (67), we find $C_1=0$ on a component ${\mathcal{O}}_0$ on $\mathcal{O}$. Let us differentiate $C_1$ with respect to s and use (65). Then, $C_3=0$ on ${\mathcal{O}}_0$. Since

$${\alpha }^{\prime }\times {\beta }^{\prime }=-u{\alpha }^{\prime },{\alpha }^{″}\times \beta =k\beta ,$$

by differentiating the equation $C_3=0$ with respect to s, we can obtain

$$k{C}_1-u{C}_2=0.$$

It follows that $C_2=0$ on ${\mathcal{O}}_0$ because $C_1=0$ and $u\ne 0$. Since $\mathbf{C}$ is a constant vector, it is a zero vector. From the first and third Equations in (67), u is a non-zero constant, say $u_0$, and $f=2u_0^2$ on M. Thus, M is of null 2-type and it is a B-scroll.

Consequently, we have

Theorem 5.

Let M be a null scroll in the Minkowski 3-space ${\mathbb{L}}^3$. Then, M is of generalized null 2-type if and only if M is an open piece of a B-scroll.

We now propose an open problem.

Problem 1.

Classify all generalized null 2-type surfaces in the Euclidean space or pseudo-Euclidean space.