Special non-lightlike ruled surfaces in Minkowski 3-space

1.   Introduction

Ruled surfaces have a great effect on differential geometry and they are used in many fields, such as architecture, robotics, design and so on^[1]. In real life, ruled surfaces can be seen everywhere, such as in most of the cooling tower structures of thermal power plants, the famous Mobius ring, and saddle-shaped potato chips. Ruled surfaces can be formed by a moving line in continuous motion. There are many kinds of special ruled surfaces. According to the different direction vectors, Kaya and Önder defined three kinds of ruled surfaces, i.e., the osculating-type ruled surface, the generalized normal ruled surface, and the generalized rectifying ruled surface whose base curves are regular curves in the Euclidean 3-space^[2,3,4]. The direction vector of the osculating-type ruled surface is a combination of the tangent and the principal normal vectors. Changing the tangent vector to the binormal vector, we get the generalized normal ruled surface. Changing the principal normal vector to the binormal vector, we get the generalized rectifying ruled surface. Later, framed curves in Euclidean space were defined by Honda and Takahashi in order to research curves with singular points^[5]. Many scholars applied this idea to study the ruled surface of a curve with singular points^[6]. Most of the surfaces that are studied may have singular points. In order to address this matter, in 2019, Fukunaga and Takahashi gave the definition of framed surfaces in the Euclidean space ^[7]. A framed surface is a smooth surface with two unit orthogonal vector fields along it. The first vector is orthogonal to the tangent space of the surface anywhere. The theory of framed surfaces provides more possibilities for the study of singularities. Many researchers have used it. For details see^[8,9,10]. The ruled surfaces have also been studied extensively in the Minkowski space. Kobayashi studied maximal surfaces and the Lorentz surfaces were differentiated by Kim and Yoon ^[11,12]. Ruled surfaces have been studied extensively for a long time and are considered as regular points. See^{[13,14,15,16,17]}. Using the theory of framed surfaces as a tool, we can extend the framed surface with singular points in Euclidean space to Minkowski space.

In the present paper, we give the existence and uniqueness theorems for non-lightlike framed surfaces and define special non-lightlike ruled surfaces with a spacelike regular base curve that has timelike principal normal vectors. The basic knowledge in Minkowski 3-space is reviewed in Section 2. In Section 3, a new special ruled surface is defined and the conditions for identifying cross-caps are given. Besides, the conditions for being cylindrical, stricture curves, as well as the connection between them are obtained. We give the fundamental theory for non-lightlike framed surfaces in Section 4. In Section 5, the ruled surfaces are considered to be non-lightlike framed surfaces. We obtain the fundamental invariants and curvatures. The conditions for being developable, and maximal, as well as the parameter curves on the surface are analyzed. And the relationship between them is found. Finally, we give an example of the special ruled surface with singular points in Section 6.

All maps and manifolds considered here are differentiable for class $C^\infty.$

2.   Preliminaries

Let $\mathbb{R}^{3}$ be the 3-dimensional real vector space. For any $\mathit{\boldsymbol{c}} = ({c}_{1}, {c}_{2}, {c}_{3})$ and $\mathit{\boldsymbol{d}} = ({d}_{1}, {d}_{2}, {d}_{3})\in\mathbb{R}^{3},$ the pseudo inner product is denoted by

$(\mathbb{R}^{3}, \langle, \rangle)$ is called the Minkowski 3-space and is denoted by $\mathbb{R}_{1}^{3}.$

For any nonzero vector $\mathit{\boldsymbol{c}}\in\mathbb{R}_{1}^{3}\backslash\{\mathit{\boldsymbol{0}}\},$ it is called spacelike, timelike or lightlike if $\langle\mathit{\boldsymbol{c}}, \mathit{\boldsymbol{c}}\rangle > 0, \; \langle\mathit{\boldsymbol{c}}, \mathit{\boldsymbol{c}}\rangle < 0,$ or $\langle\mathit{\boldsymbol{c}}, \mathit{\boldsymbol{c}}\rangle = 0,$ respectively. The norm of $\mathit{\boldsymbol{c}}$ is $\parallel\mathit{\boldsymbol{c}}\parallel = \sqrt{\mid\langle\mathit{\boldsymbol{c}}, \mathit{\boldsymbol{c}}\rangle\mid}.$ The pseudo vector product of $\mathit{\boldsymbol{c}}$ and $\mathit{\boldsymbol{d}}$ is

There are three special subsets in $\mathbb{R}_{1}^{3}:$

We denote ${H}_{0}^{2} = {H}_{+}^{2}\cup{H}_{-}^{2}.$

Define a set $\bigtriangleup = \{(\nu_{1}, \nu_{2})\in\mathbb{R}_{1}^{3}\times\mathbb{R}_{1}^{3}\mid \parallel\nu_{1}\parallel = \parallel\nu_{2}\parallel = 1, \langle\nu_{1}, \nu_{2}\rangle = 0\}.$

Assume that ${U}$ is an open domain and $\mathit{\boldsymbol{\chi}}:{U}\rightarrow \mathbb{R}_{1}^{3}$ is a surface. We say that $\mathit{\boldsymbol{\chi}}$ is spacelike, timelike or lightlike if the tangent plane is spacelike, timelike or lightlike at any point $({z}, {w})\in{U},$ respectively. We call both the spacelike surface and the timelike surface the non-lightlike surface. In this paper, we consider the non-lightlike surface which can have singularities, and the tangent plane is non-lightlike at its regular point.

We say that $\mathit{\boldsymbol{\gamma}}:I\rightarrow \mathbb{R}_{1}^3$ is a regular spacelike curve with the arclength parameter ${z}$ which has timelike principal normal vectors with the orthonormal frame $\{\mathit{\boldsymbol{T}}(z), \mathit{\boldsymbol{N}}(z), \mathit{\boldsymbol{B}}(z)\}$. The Frenet formulas are as follows

where $\kappa(z)$ and $\tau(z)$ are the curvature and torsion of $\mathit{\boldsymbol{\gamma}}$.

A ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ is given by

where $\mathit{\boldsymbol{\gamma}}:I\subset\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is the base curve and $\mathit{\boldsymbol{q}}:\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is the direction vector. The ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ is cylindrical if only if $\dot{\mathit{\boldsymbol{q}}}(z) = \mathit{\boldsymbol{0}}$. We say that the curve $\mathit{\boldsymbol{\beta}} = \mathit{\boldsymbol{\beta}}(z)$ is the striction curve of $\mathit{\boldsymbol{\chi}}(z, w)$ if it satisfies the condition that $\langle\dot{\mathit{\boldsymbol{\beta}}}(z), \dot{\mathit{\boldsymbol{q}}}(z)\rangle = 0$.

An arbitrary spacelike surface is a developable surface if ${K} = 0$ and maximal if ${H} = 0$.

3.   Ruled surface of regular spacelike curve with timelike principal normal vectors

Definition 3.1. Let $\mathit{\boldsymbol{\gamma}}:I\subset\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ be a spacelike curve with timelike principal normal vectors. The ruled surface $\mathit{\boldsymbol{\chi}}(z, w):I\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is defined by

where $\mathit{\boldsymbol{q}}(z) = \cos\theta(z)\mathit{\boldsymbol{T}}({z})+\sin\theta(z)\mathit{\boldsymbol{B}}({z})$ and $\theta(z)$ is a smooth function.

It is known that $\theta({z}) = k\pi(k\in\mathbb{Z})$ for all ${z}\in{I}$, $\mathit{\boldsymbol{q}} = \pm\mathit{\boldsymbol{T}}$ and $\mathit{\boldsymbol{\chi}} = \mathit{\boldsymbol{\gamma}}\pm{w}\mathit{\boldsymbol{T}}.$ When $\theta({z}) = \frac{\pi}{2}+k\pi(k\in\mathbb{Z})$ for all ${z}\in{I}$, $\mathit{\boldsymbol{q}} = \pm\mathit{\boldsymbol{B}}$ and $\mathit{\boldsymbol{\chi}} = \mathit{\boldsymbol{\gamma}}\pm{v}\mathit{\boldsymbol{B}}.$

Let ${P}:I\subset\mathbb{R}\rightarrow \mathbb{R}$ and ${Q}:I\times\mathbb{R}\rightarrow \mathbb{R}$ be smooth functions that are respectively defined as

Proposition 3.2. The ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ has singular points if and only if ${P} = {Q} = 0.$

Proof. By taking the derivative directly, we have

Then, from the cross product we obtain

and we have that ${P} = {Q} = 0$ if and only if $\mathit{\boldsymbol{\chi}}_{z}\wedge\mathit{\boldsymbol{\chi}}_{w} = \mathit{\boldsymbol{0}}.$ □

Corollary 3.3.

(1) If $\sin\theta = 0, \cos\theta = 1$ and $\mathit{\boldsymbol{\chi}}(z, w) = \mathit{\boldsymbol{\gamma}}(z)+{w}\mathit{\boldsymbol{T}}(z),$ the set of singular points is

(2) If $\sin\theta = 1, \cos\theta = 0$ and $\mathit{\boldsymbol{\chi}}(z, w) = \mathit{\boldsymbol{\gamma}}(z)+{w}\mathit{\boldsymbol{B}}(z),$ it has no singular points.

To identify the condition of being cross caps, singular points are divided into two classes ${U}_{1}$ and ${U}_{2}.$ From Proposition 3.2 singular points satisfy the following:

Let us assume that ${w} = 0$. Then, the class ${U}_{1}$ can be given by

If ${w}\neq0,$ the class ${U}_{2}$ can be given by

Theorem 3.4. Let $\mathit{\boldsymbol{\chi}}(z, w)$ be a ruled surface of the regular spacelike base curve $\mathit{\boldsymbol{\gamma}}(z)$ with timelike principal normal vectors. Then apply the following:

(1) If $({z}_{1}, 0)\in{U}_{1},$ then $\mathit{\boldsymbol{\chi}}(z, w)$ has no cross-cap.

(2) If $({z}_{2}, {w}_{2})\in{U}_{2}$ and $\dot{\theta}({z}_{2})\dot{f}({z}_{2})\neq{0},$ where $f(z) = \kappa(z)\cos\theta(z)+\tau(z)\sin\theta(z),$ then $\mathit{\boldsymbol{\chi}}(z, w)$ has a cross-cap at the point $({z}_{2}, {w}_{2}).$

Proof. If $({z}_{1}, 0)\in{U}_{1},$ we can get

And

But, from cross-cap judging theorem in ^[18], we know that $\mathit{\boldsymbol{\chi}}_{z}(z_{1}, 0) = \mathit{\boldsymbol{T}}({z}_{1})\neq\mathit{\boldsymbol{0}},$ so if $({z}_{1}, 0)\in{U}_{1},$ then $\mathit{\boldsymbol{\chi}}(z, w)$ has no cross-cap.

If $({z}_{2}, {w}_{2})\in{U}_{2},$ we can get

where $f(z) = \kappa(z)\cos\theta(z)+\tau(z)\sin\theta(z).$

And

Then, from cross-cap judging theorem in ^[18], we know that, if $({z}_{2}, {w}_{2})\in{U}_{2}$ and $\dot{\theta}({z}_{2})\dot{f}({z}_{2})\neq{0}, \; \mathit{\boldsymbol{\chi}}(z, w)$ is a cross-cap at the point $({z}_{2}, {w}_{2}).$ □

Proposition 3.5. Let $\mathit{\boldsymbol{\chi}}(z, w)$ be a ruled surface of a regular spacelike base curve $\mathit{\boldsymbol{\gamma}}(z)$ with timelike principal normal vectors and $\sin\theta(z)\cos\theta(z)\neq0,$ for any ${z}\in{I}.$ The following conclusions are obtained:

(1) The ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ is cylindrical if and only if $\mathit{\boldsymbol{\gamma}}(z)$ is a helix.

(2) The base curve $\mathit{\boldsymbol{\gamma}}(z)$ of $\mathit{\boldsymbol{\chi}}(z, w)$ is its striction curve if and only if $\theta(z)$ is constant.

Proof. (1) By differentiating $\mathit{\boldsymbol{q}} = \cos\theta\mathit{\boldsymbol{T}}+\sin\theta\mathit{\boldsymbol{B}},$ we can get

If the ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ is cylindrical, $\dot{\mathit{\boldsymbol{q}}} = \mathit{\boldsymbol{0}}.$ Then

We know that $\theta$ is constant and $\frac{\kappa}{\tau} = -\tan\theta,$ that is $\mathit{\boldsymbol{\gamma}}$ is a helix.

(2) The expression of the striction curve of $\mathit{\boldsymbol{\chi}}(z, w)$ is obtained as follows:

It is clear that $\mathit{\boldsymbol{\gamma}}(z)$ of $\mathit{\boldsymbol{\chi}}(z, w)$ is its striction curve if and only if $\theta(z)$ is constant.□

Proposition 3.6. Let the base curve $\boldsymbol{\gamma}(z)$ be a striction curve of $\mathit{\boldsymbol{\chi}}(z, w)$ and $\sin\theta(z)\cos\theta(z)\neq0,$ for any ${z}\in{I}. \; {\mathit{\boldsymbol{\chi}}(z, w)}$ is a developable surface if and only if ${\mathit{\boldsymbol{\gamma}}(z)}$ is a helix.

Proof. We get

$\mathit{\boldsymbol{\chi}}(z, w)$ is developable if and only if $\mathrm{det}(\dot{\mathit{\boldsymbol{\gamma}}}, \mathit{\boldsymbol{q}}, \dot{\mathit{\boldsymbol{q}}}) = 0.$ Then

If the base curve $\gamma(z)$ is a striction curve, $\theta(z)$ is constant. Finally, we can get the result. □

Corollary 3.7. Let the base curve $\boldsymbol{\gamma}(z)$ be a striction curve of $\mathit{\boldsymbol{\chi}}(u, w)$ and $\sin\theta(z)\cos\theta(z)\neq0,$ for any ${z}\in{I}. \; {\mathit{\boldsymbol{\chi}}(z, w)}$ is a developable surface if and only if ${\mathit{\boldsymbol{\chi}}(z, w)}$ is cylindrical.

Corollary 3.8. If the ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ is cylindrical, its parameter expression is

where ${c}\in(-1, 1)$ is constant.

Proof. Let the ruled surface $\mathit{\boldsymbol{\chi}}(z, w)$ of the base curve $\mathit{\boldsymbol{\gamma}}(z)$ be cylindrical. According to Proposition 3.5, $\theta(z)$ is constant. We can write $\sin\theta(z) = {c}$ and $\cos\theta(z) = \sqrt{1-{c}^{2}}.$ So $\mathit{\boldsymbol{\chi}}(z, w) = \mathit{\boldsymbol{\gamma}}(z)+{w}({c}\mathit{\boldsymbol{T}}(z)+\sqrt{1-{c}^{2}}\mathit{\boldsymbol{B}}(z)).$ □

Proposition 3.9. The trajectory of the singular points of $\mathit{\boldsymbol{\chi}}(z, w)$ can be expressed as follows:

(1) If the singular points belongs to ${U}_{1},$ the trajectory of the singular points of $\mathit{\boldsymbol{\chi}}(z, w)$ is $\mathit{\boldsymbol{\gamma}}(z).$

(2) If the singular points belongs to ${U}_{2},$ the trajectory of the singular points of $\mathit{\boldsymbol{\chi}}(z, w)$ can be expressed as $\boldsymbol{\rho}(z) = \mathit{\boldsymbol{\gamma}}(z)+\frac{\sin\theta(z)}{\dot{\theta}(z)}(\cos\theta(z)\mathit{\boldsymbol{T}}(z)+\sin\theta(z)\mathit{\boldsymbol{B}}(z)).$

4.   Non-lightlike framed surface

Definition 4.1. Let ${U}$ be an open domain of $\mathbb{R}^{2}.$ We call $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}):{U}\rightarrow \mathbb{R}_{1}^{3}\times\bigtriangleup$ a non-lightlike framed surface if $\langle\mathit{\boldsymbol{\chi}}_{z}(z, w), \mathit{\boldsymbol{\psi}}_{1}(z, w)\rangle = \langle\mathit{\boldsymbol{\chi}}_{w}(z, w), \mathit{\boldsymbol{\psi}}_{1}(z, w)\rangle = 0$ for all $({z}, {w})\in{U},$ where $\mathit{\boldsymbol{\chi}}_{z} = \frac{\partial\mathit{\boldsymbol{\chi}}}{\partial{z}}$ and $\mathit{\boldsymbol{\chi}}_{w} = \frac{\partial\mathit{\boldsymbol{\chi}}}{\partial{w}}.$ A surface $\mathit{\boldsymbol{\chi}}:{U}\rightarrow \mathbb{R}_{1}^{3}$ is a non-lightlike framed base surface if there exists $(\mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}):{U}\rightarrow \triangle$ such that $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ is a non-lightlike framed surface.

Define $\mathit{\boldsymbol{\psi}}_{3} = \mathit{\boldsymbol{\psi}}_{1}\wedge\mathit{\boldsymbol{\psi}}_{2}. \; \{\mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}, \mathit{\boldsymbol{\psi}}_{3}\}$ is a moving frame on the surface $\mathit{\boldsymbol{\chi}}$ and the Frenet type formulas are

where $\sigma = {\rm sign}(\mathit{\boldsymbol{\psi}}_{1}(z, w)), \; \delta = {\rm sign}(\mathit{\boldsymbol{\psi}}_{2}(z, w)).$

The functions $a_i, b_i, e_i, f_i, g_i : U \rightarrow \mathbb{R} \; (i = 1, 2)$ are called the basic invariants of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}).$

Remark 4.2. $\mathit{\boldsymbol{\chi}}$ is singular at $(z_0, w_0) \in U$ if and only if the determinant ${\rm det}\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} = 0.$

Because $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ is smooth, $\mathit{\boldsymbol{\chi}}_{zw} = \mathit{\boldsymbol{\chi}}_{wz}, \mathit{\boldsymbol{\psi}}_{1zw} = \mathit{\boldsymbol{\psi}}_{1wz}$ and $\mathit{\boldsymbol{\psi}}_{2zw} = \mathit{\boldsymbol{\psi}}_{2wz}.$ So we can get the following integrability conditions

Theorem 4.3 (Existence). Give two groups of smooth functions $a_i, b_i, e_i, f_i, g_i : U \rightarrow \mathbb{R} \; (i = 1, 2),$ which satisfy the integrability conditions (4.4) and (4.5). Then, there exists a non-lightlike framed surface $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}) : U \rightarrow \mathbb{R}_1^3 \times \Delta,$ whose invariants are $a_i, b_i, e_i, f_i, g_i : U \rightarrow \mathbb{R}, \; i = 1, 2.$

Proof. Since $a_i, b_i, e_i, f_i, g_i, \; i = 1, 2$ are smooth and (4.5) holds, there exists a pseudo-orthogonal moving frame $\{\mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}, \mathit{\boldsymbol{\psi}}_{3}\}$ that satisfies (4.2) and (4.3). And (4.4) holds; then, there exists a surface $\mathit{\boldsymbol{\chi}} : U \rightarrow \mathbb{R}_1^3$ that satisfies (4.1).

So $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}) : U \rightarrow \mathbb{R}_1^3 \times \Delta$ is a non-lightlike framed surface and its invariants are $a_i, b_i, e_i, f_i, g_i, i = 1, 2.$ □

Definition 4.4. Let $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}) : U \rightarrow \mathbb{R}_1^3 \times \Delta$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2}) : U \rightarrow \mathbb{R}_1^3 \times \Delta$ be two non-lightlike framed surfaces. $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2})$ are congruent through a Lorentz motion if there exists a matrix $A$ and a constant vector $\mathit{\boldsymbol{c}} \in \mathbb{R}_1^3$

for any $(z, w) \in U,$ where $A$ satisfies the following:

Proposition 4.5. If two non-lightlike framed surfaces $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2})$ are congruent through a Lorentz motion, then their basic invariants are equal, that is $(a_i, b_i, e_i, f_i, g_i) = (\tilde{a}_i, \tilde{b}_i, \tilde{e}_i, \tilde{f}_i, \tilde{g}_i), i = 1, 2.$

Theorem 4.6 (Uniqueness). Let $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2})$ be two non-lightlike framed surfaces which have the same basic invariants $a_i, b_i, e_i, f_i, g_i : U \rightarrow \mathbb{R} \; (i = 1, 2).$ Assume that two surfaces have the same time orientation. Then $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2})$ are congruent through a Lorentz motion.

Proof. Fix a point $(z_0, w_0) \in U$; then, there exists a Lorentz motion $A \in \mathbb{R}^{3 \times 3}$ such that $\mathit{\boldsymbol{\tilde{\psi}}}_{1}(z_0, w_0) = A \mathit{\boldsymbol{\psi}}_{1}(z_0, w_0)$ and $\mathit{\boldsymbol{\tilde{\psi}}}_{2}(z_0, w_0) = A \mathit{\boldsymbol{\psi}}_{2}(z_0, w_0).$ Then we have that $\mathit{\boldsymbol{\tilde{\psi}}}_{3}(z_0, w_0) = A \mathit{\boldsymbol{\psi}}_{3}(z_0, w_0).$

Note that $\{\mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}, \mathit{\boldsymbol{\psi}}_{3}\}$ and $\{\mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2}, \mathit{\boldsymbol{\tilde{\psi}}}_{3}\}$ are both solutions of (4.2) and (4.3), so $\{\mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}, \mathit{\boldsymbol{\psi}}_{3}\} = \{\mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2}, \mathit{\boldsymbol{\tilde{\psi}}}_{3}\}.$

Take $\mathit{\boldsymbol{c}} = \tilde{\mathit{\boldsymbol{\chi}}}(z_0, w_0) - A\mathit{\boldsymbol{\chi}}(z_0, w_0).$ Since $\tilde{\mathit{\boldsymbol{\chi}}}(z, w)$ and $A\mathit{\boldsymbol{\chi}}(z, w) + \mathit{\boldsymbol{c}}$ both satisfy the Eq (4.1), then $\tilde{\mathit{\boldsymbol{\chi}}}(z, w) = A\mathit{\boldsymbol{\chi}}(z, w) + \mathit{\boldsymbol{c}}.$ Thus $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and $(\mathit{\boldsymbol{\tilde{\chi}}}, \mathit{\boldsymbol{\tilde{\psi}}}_{1}, \mathit{\boldsymbol{\tilde{\psi}}}_{2})$ are congruent through a Lorentz motion. □

Definition 4.7. Let $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}):{U}\rightarrow \mathbb{R}_{1}^{3}\times\bigtriangleup$ be a non-lightlike framed surface. Define the following functions

We call ${E}, {F}, {G}$ the first fundamental invariants of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ and ${L}, {M}, {N}$ the second fundamental invariants of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}).$

Definition 4.8. Let $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}):{U}\rightarrow \mathbb{R}_{1}^{3}\times\bigtriangleup$ be a non-lightlike framed surface. Define

We call $({J}_{\chi}, {K}_{\chi}, {H}_{\chi})$ the curvature of the non-lightlike framed surface $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}).$

Proposition 4.9. Let $\mathit{\boldsymbol{\chi}}:{U}\rightarrow \mathbb{R}_{1}^{3}$ be a non-lightlike regular surface. The Gauss curvature and the mean curvature of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ are respectively denoted by

5.   Special spacelike framed ruled surface

Definition 5.1. Assume that $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}):U\rightarrow \mathbb{R}_{1}^{3}\times\bigtriangleup$ is a non-lightlike framed surface. $\boldsymbol{\gamma}(s) = \mathit{\boldsymbol{\chi}}(z(s), w(s))$ is a curve on $\mathit{\boldsymbol{\chi}}$.

(1) If the principal normal vector of $\boldsymbol{\gamma}$ is parallel to $\mathit{\boldsymbol{\psi}}_{1}$, $\boldsymbol{\gamma}$ is called the geodesic curve.

(2) If the principal normal vector of $\boldsymbol{\gamma}$ is perpendicular to $\mathit{\boldsymbol{\psi}}_{1}$, $\boldsymbol{\gamma}$ is called the asymptotic curve.

(3) If the tangent vector of the curve $\boldsymbol{\gamma}$ is parallel to $\frac{d}{ds}\mathit{\boldsymbol{\psi}}_{1}$, $\boldsymbol{\gamma}$ is called the line of curvature.

Proposition 5.2. Let $\mathit{\boldsymbol{\gamma}}(z)$ be a spacelike base curve with timelike principal normal vectors. $\mathit{\boldsymbol{\chi}}(z, w)$ is the ruled surface of $\mathit{\boldsymbol{\gamma}}(z).$ If there exist two smooth functions $\varphi, \phi:{I}\times\mathbb{R}\rightarrow \mathbb{R}$ satisfying $\langle\mathit{\boldsymbol{\chi}}_{z}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0, \; \langle\mathit{\boldsymbol{\chi}}_{w}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0$ and

then $\mathit{\boldsymbol{\chi}}:{I}\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is a spacelike framed base surface, where

Proof. According to Proposition 3.2,

Since $\mathit{\boldsymbol{\psi}}_{1}$ is parallel to $\mathit{\boldsymbol{\chi}}_{z}\wedge\mathit{\boldsymbol{\chi}}_{w}$ at the regular points, we obtain

By calculation,

Finally, we get the proposition.□

Remark 5.3. If there exist two smooth functions $\varphi, \phi:{I}\times\mathbb{R}\rightarrow \mathbb{R}$ satisfying $\langle\mathit{\boldsymbol{\chi}}_{z}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0, \; \langle\mathit{\boldsymbol{\chi}}_{w}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0$ and

$\mathit{\boldsymbol{\chi}}:{I}\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is a timelike framed base surface. It is similar to the spacelike framed base surface in the context of the differential geometry property. So we take the spacelike framed surface as an example.

After this part, for convenience, let

Proposition 5.4. The basic invariants of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ can be given by

Proposition 5.5. The curvature $({J}_{\chi}, {K}_{\chi}, {H}_{\chi})$ of $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2})$ is obtained

Proposition 5.6. Let $\mathit{\boldsymbol{\chi}}:{I}\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ be a regular surface. The coefficients of the fundamental forms are calculated as follows:

Proposition 5.7. The Gauss curvature and the mean curvature of the regular surface $\mathit{\boldsymbol{\chi}}(z, w)$ can be respectively expressed as

Corollary 5.8. The regular surface $\mathit{\boldsymbol{\chi}}(z, w):{I}\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is developable if and only if $\varphi_{w} = 0.$

Proof. Because $\boldsymbol{\chi}(z, w)$ is smooth, we obtain

So ${K} = -\varphi_{w}^{2}.$ Therefore, $\boldsymbol{\chi}(z, w)$ is developable if and only if $\varphi_{w} = 0.$ □

Corollary 5.9. The regular surface $\mathit{\boldsymbol{\chi}}(z, w):{I}\times\mathbb{R}\rightarrow \mathbb{R}_{1}^{3}$ is a maximal surface if and only if $\varphi_{w}\cos\theta-\tau\cos\theta+\kappa\sin\theta-\varphi_{z} = 0.$

Proposition 5.10. Assume that ${\Pi}$ is a spacelike framed base surface. The base curve $\mathit{\boldsymbol{\gamma}}(z)$ of ${\Pi}$ is a geodesic curve if and only if $\sinh\varphi = 0$ or $\kappa = 0.$

Proof. According to the definition of a geodesic curve in Definition 5.1, $\ddot{\mathit{\boldsymbol{\gamma}}}\wedge\mathit{\boldsymbol{\psi}}_{1} = 0$ is a sufficient and necessary condition of a geodesic curve:

Then, we get the result.□

Proposition 5.11. Assume that ${\Pi}$ is a spacelike framed base surface. The base curve $\mathit{\boldsymbol{\gamma}}(z)$ of ${\Pi}$ is an asymptotic curve if and only if $\kappa = 0.$

Proof. According to the definition of an asymptotic curve in Definition 5.1, $\langle\ddot{\mathit{\boldsymbol{\gamma}}}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0$ is a sufficient and necessary condition of the asymptotic curve:

So $\mathit{\boldsymbol{\gamma}}(z)$ is an asymptotic curve if and only if $\kappa = 0.$ □

Proposition 5.12. Let ${\Pi}$ be a spacelike framed base surface. The results are as follows.

(1) The ${z}$-parameter curve of ${\Pi}$ is an asymptotic curve if and only if

(2) The ${w}$-parameter curve of ${\Pi}$ is always an asymptotic curve.

Proof. (1) According to the definition of an asymptotic curve in Definition 5.1, $\langle\mathit{\boldsymbol{\chi}}_{zz}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0$ is a sufficient and necessary condition for the ${z}$-parameter curve to be an asymptotic curve. We can get

Then, we get the result.

(2) $\langle\mathit{\boldsymbol{\chi}}_{ww}, \mathit{\boldsymbol{\psi}}_{1}\rangle = 0$ is a sufficient and necessary condition for the ${w}$-parameter curve to be an asymptotic curve. We can get

Then we get the result. □

Proposition 5.13. Let ${\Pi}$ be a spacelike framed base surface; the ${z}$-parameter curve and the ${w}$-parameter curve of $\Pi$ are lines of curvature if and only if

(i) $\cos\theta = 0, \varphi_{w} = 0,$

or

(ii) $\cos\theta = 0, \cosh\varphi\sin\theta+w\tau\sinh\varphi\sin\theta = 0.$

Proof. According to the definition of a line of curvature in Definition 5.1, $\langle\mathit{\boldsymbol{\chi}}_{z}, \mathit{\boldsymbol{\chi}}_{w}\rangle = \langle\mathit{\boldsymbol{\chi}}_{z}, \mathit{\boldsymbol{\psi}}_{1w}\rangle = 0$ is a sufficient and necessary condition for the parameter curves to be lines of curvature. That means that ${F} = {M} = 0.$ We know, from in Proposition 5.6, that

Then, we get the result.□

Theorem 5.14. Let ${\Pi}$ be a developable spacelike framed base surface given by $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}). \; {\Pi}$ is a maximal surface if and only if the ${z}$-parameter curve of ${\Pi}$ is an asymptotic curve.

Proof. From Proposition 5.6, we know that ${N} = 0$ and ${G} = 1.$ So according to the expression of a developable surface, if $\Pi$ is developable, then ${M} = 0.$ The ${z}$-parameter curve of $\Pi$ is an asymptotic curve; then, ${L} = 0.$ Thus, ${H} = 0.$ Finally, we can get the result. □

Proposition 5.15. Let the ${z}$-parameter curve and the ${w}$-parameter curves of $\Pi$ be lines of curvature of the spacelike framed base surface given by $(\mathit{\boldsymbol{\chi}}, \mathit{\boldsymbol{\psi}}_{1}, \mathit{\boldsymbol{\psi}}_{2}).$ The ${z}$-parameter curve of $\Pi$ is an asymptotic curve if and only if $\Pi$ is a maximal surface.

Proof. According to the definition of a line of curvature, ${F} = {M} = 0$ is a sufficient and necessary condition for parameter curves to be lines of curvature. And we know that ${N} = 0$ and ${G} = 1.$ Assume that the ${z}$-parameter curve of $\Pi$ is an asymptotic curve, which means that ${L} = 0.$ Thus, ${H} = 0.$ Finally, we can get the result. □

6.   Example

This special ruled surface which we consider in Section 5 may have singular points; we will give an example. See Figure 1.

Example 6.1. Let $\mathit{\boldsymbol{\gamma}}:[0, 2\pi)\rightarrow \mathbb{R}_{1}^{3}$ be a regular spacelike curve defined by

Through calculation, we obtain

and

respectively. And $\kappa = \tau = \frac{\sqrt{2}}{2}.$ Let us assume that $\cos\theta(z) = \cos{z}$ and $\sin\theta(z) = \sin{z}.$ So $\mathit{\boldsymbol{\chi}}(z, w) = \mathit{\boldsymbol{\gamma}}(z)+w\mathit{\boldsymbol{q}}(z),$ where

(1) If $w = 0$ and $\sin{z}_{1} = 0,$ the surface is singular at $(0, 0)$ and $(\pi, 0).$

(2) $\kappa = \tau = \frac{\sqrt{2}}{2}$ is constant and ${\sin\theta-{w}\dot{\theta}\neq0}$. So for any ${({z}_{2}, {w}_{2})\in{U}_{2}}$ is not a singular point, where ${{U}_{2} = \{(z_{2}, w_{2})\in{I}\times\mathbb{R}\mid\kappa\cos\theta+\tau\sin\theta = 0, \sin\theta-{w}\dot{\theta} = 0, {w}\neq0\}}.$

7.   Conclusions

In this paper, we have investigated the singular properties of a special ruled surface which is generated by spacelike straight lines and given the basic theory of non-lightlike framed surfaces. Regarding the ruled surface as a non-lightlike framed base surface, we have discussed its differential geometric properties.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11671070).

Conflict of interest

The authors declare that there are no conflicts of interest that may influence the publication of this work.