Inextensible Flows of Curves on Lightlike Surfaces

Abstract

In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface.

1. Introduction

It is well known that many nonlinear phenomena in physics, chemistry and biology are described by dynamics of shapes, such as curves and surfaces, and the time evolution of a curve and a surface has significance in computer vision and image processing. The time evolution of a curve and a surface is described by flows, in particular inextensible flows of a curve and a surface. Physically, inextensible flows give rise to motion, for which no strain energy is induced. The swinging motion of a cord of fixed length or of a piece of paper carried by the wind can be described by inextensible flows of a curve and a surface. Furthermore, the flows arise in the context of many problems in computer vision and computer animation [1,2,3,4].

Chirikjian and Burdick [1] studied applications of inextensible flows of a curve. In [5], the authors derived the time evolution equations for an inextensible flow of a space curve and also studied inextensible flows of a developable ruled surface. In [6], the author investigated the general description of the binormal motion of a spacelike and a timelike curve in a three-dimensional de Sitter space and gave some explicit examples of a binormal motion of the curves. Schief and Rogers [4] studied the binormal motions of curves with constant curvature and torsion. Many authors have studied geometric flow problems [7,8,9,10,11].

The outline of the paper is organized as follows: In Section 2, we give some geometric concepts in Minkowski space and present the pseudo-Darboux frames of a spacelike curve and a lightlike curve on a lightlike surface. In Section 3 and Section 4, we study inextensible flows of a spacelike curve and a lightlike curve on a lightlike surface. In the last section, we classify lightlike ruled surfaces and study inextensible flows of lightlike tangent developable surfaces.

2. Preliminaries

The Minkowski three-space ${\mathbb{R}}_1^3$ is a real space ${\mathbb{R}}^3$ with the indefinite inner product $⟨·,·⟩$ defined on each tangent space by:

$$⟨\mathbf{x},\mathbf{y}⟩=-x_0{y}_0+x_1{y}_1+x_2{y}_2,$$

where $\mathbf{x}=(x_0,x_1,x_2)$ and $\mathbf{y}=(y_0,y_1,y_2)$ are vectors in ${\mathbb{R}}_1^3$.

A nonzero vector $\mathbf{x}$ in ${\mathbb{R}}_1^3$ is said to be spacelike, timelike or lightlike if $⟨\mathbf{x},\mathbf{x}⟩>0$, $⟨\mathbf{x},\mathbf{x}⟩<0$ or $⟨\mathbf{x},\mathbf{x}⟩=0$, respectively. Similarly, an arbitrary curve $\gamma =\gamma \left(s\right)$ is spacelike, timelike or lightlike if all of its tangent vectors ${\gamma }^{\prime }\left(s\right)$ are spacelike, timelike or lightlike, respectively. Here “prime” denotes the derivative with respect to the parameter s.

Let M be a lightlike surface in Minkowski three-space ${\mathbb{R}}_1^3$, that is the induced metric of M is degenerate. Then, a curve $\gamma$ on M is spacelike or lightlike.

Case 1: If $\gamma$ is a spacelike curve, we can reparametrize it by the arc length s. Therefore, we have the unit tangent vector $\mathbf{t}\left(s\right)={\gamma }^{\prime }\left(s\right)$ of $\gamma \left(s\right)$. Since M is a lightlike surface, we have a lightlike normal vector $\mathbf{n}$ along $\gamma$. Therefore, we can choose a vector $\mathbf{g}$ satisfying:

$$⟨\mathbf{n},\mathbf{g}⟩=1,⟨\mathbf{t},\mathbf{g}⟩=⟨\mathbf{g},\mathbf{g}⟩=0.$$

Then, we have pseudo-orthonormal frames $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$, which are called the Darboux frames along $\gamma \left(s\right)$. By standard arguments, we have the following Frenet formulae:

$$\frac{d}{ds}\left(\begin{array}{c}\mathbf{t}\left(s\right)\\ \mathbf{n}\left(s\right)\\ \mathbf{g}\left(s\right)\end{array}\right)=\left(\begin{array}{ccc}0& {\kappa }_g\left(s\right)& {\kappa }_n\left(s\right)\\ -{\kappa }_n\left(s\right)& {\tau }_g\left(s\right)& 0\\ -{\kappa }_g\left(s\right)& 0& -{\tau }_g\left(s\right)\end{array}\right)\left(\begin{array}{c}\mathbf{t}\left(s\right)\\ \mathbf{n}\left(s\right)\\ \mathbf{g}\left(s\right)\end{array}\right),$$

(1)

where ${\kappa }_n=⟨{\mathbf{t}}^{\prime }\left(s\right),\mathbf{n}\left(s\right)⟩,{\kappa }_g=⟨{\mathbf{t}}^{\prime }\left(s\right),\mathbf{g}\left(s\right)⟩$ and ${\tau }_g=-⟨\mathbf{n}\left(s\right),{\mathbf{g}}^{\prime }\left(s\right)⟩.$

Case 2: Let $\gamma$ be a lightlike curve parametrized by a pseudo arc length parameter s on a lightlike surface M in ${\mathbb{R}}_1^3$. Since a normal vector $\mathbf{n}$ of a lightlike surface M is lightlike, we can choose a vector $\mathbf{g}$ such that:

$$⟨\mathbf{g},\mathbf{g}⟩=1,⟨\mathbf{t},\mathbf{g}⟩=⟨\mathbf{g},\mathbf{n}⟩=0.$$

Furthermore, we consider:

$$⟨\mathbf{t},\mathbf{n}⟩=1.$$

Then, we have pseudo-orthonormal Darboux frames $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ along a nongeodesic lightlike curve $\gamma \left(s\right)$ on M and get the following Frenet formulae:

$$\frac{d}{ds}\left(\begin{array}{c}\mathbf{t}\left(s\right)\\ \mathbf{n}\left(s\right)\\ \mathbf{g}\left(s\right)\end{array}\right)=\left(\begin{array}{ccc}{\kappa }_n\left(s\right)& 0& {\kappa }_g\left(s\right)\\ 0& -{\kappa }_n\left(s\right)& {\tau }_g\left(s\right)\\ -{\tau }_g\left(s\right)& -{\kappa }_g\left(s\right)& 0\end{array}\right)\left(\begin{array}{c}\mathbf{t}\left(s\right)\\ \mathbf{n}\left(s\right)\\ \mathbf{g}\left(s\right)\end{array}\right),$$

(2)

where ${\kappa }_n=⟨{\mathbf{t}}^{\prime }\left(s\right),\mathbf{n}\left(s\right)⟩,{\kappa }_g=⟨{\mathbf{t}}^{\prime }\left(s\right),\mathbf{g}\left(s\right)⟩$ and ${\tau }_g=-⟨\mathbf{n}\left(s\right),{\mathbf{g}}^{\prime }\left(s\right)⟩.$

3. Inextensible Flows of a Spacelike Curve

We assume that $\gamma :[0,l]\times [0,w]\to M\subset {\mathbb{R}}_1^3$ is a one-parameter family of the smooth spacelike curve on a lightlike surface in ${\mathbb{R}}_1^3$, where l is the arc length of the initial curve. Let u be the curve parametrization variable, $0\le u\le l.$ We put $v=\left|\right|\frac{\partial \gamma }{\partial u}\left|\right|$, from which the arc length of $\gamma$ is defined by $s\left(u\right)={\int }_0^{u}vdu.$ Furthermore, the operator $\frac{\partial }{\partial s}$ is given in terms of u by $\frac{\partial }{\partial s}=\frac{1}{v}\frac{\partial }{\partial u}$, and the arc length parameter is given by $ds=vdu.$

On the Darboux frames $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ of the spacelike curve $\gamma$ on a lightlike surface M in ${\mathbb{R}}_1^3$, any flow of $\gamma$ can be given by:

$$\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g},$$

(3)

where $f_1,f_2,f_3$ are scalar speeds of the spacelike curve $\gamma$ on a lightlike surface M, respectively. We put $s(u,t)={\int }_0^{u}vdu$; it is called the arc length variation of $\gamma$. From this, the requirement that the curve is not subject to any elongation or compression can be expressed by the condition:

$$\frac{\partial }{\partial t}s(u,t)={\int }_0^u\frac{\partial v}{\partial t}du=0$$

(4)

for all $u\in [0,l].$

Definition 1.

A curve evolution $\gamma (u,t)$ and its flow $\frac{\partial \gamma }{\partial t}$ of a spacelike curve in ${\mathbb{R}}_1^3$ are said to be inextensible if:

$$\frac{\partial }{\partial t}\left|\left|\frac{\partial \gamma }{\partial u}\right|\right|=0.$$

Now, we give the arc length preserving condition for curve flows.

Theorem 1.

Let M be a lightlike surface in Minkowski three-space ${\mathbb{R}}_1^3$ and $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ be the Darboux frames of a spacelike curve γ on M. If $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ is a flow of γ on a lightlike surface M in ${\mathbb{R}}_1^3$, then we have the following equation:

$$\frac{\partial v}{\partial t}=\frac{\partial f_1}{\partial u}-v{f}_2{\kappa }_n-v{f}_3{\kappa }_g.$$

(5)

Proof. 

From the definition of a spacelike curve $\gamma$, we have $v^2=⟨\frac{\partial \gamma }{\partial u},\frac{\partial \gamma }{\partial u}⟩.$ Since u and t are independent coordinates, $\frac{\partial }{\partial u}$ and $\frac{\partial }{\partial t}$ commute. Therefore, by differentiating $v^2$, we have:

$$\begin{array}{cc}\hfill 2v\frac{\partial v}{\partial t}& =\frac{\partial }{\partial t}⟨\frac{\partial \gamma }{\partial u},\frac{\partial \gamma }{\partial u}⟩\hfill \\ & =2⟨\frac{\partial \gamma }{\partial u},\frac{\partial }{\partial u}(\frac{\partial \gamma }{\partial t})⟩\hfill \\ & =2⟨\frac{\partial \gamma }{\partial u},\frac{\partial }{\partial u}(f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g})⟩\hfill \\ & =2v⟨\mathbf{t},(\frac{\partial f_1}{\partial u}-v{f}_2{\kappa }_n-v{f}_3{\kappa }_g)\mathbf{t}+(\frac{\partial f_2}{\partial u}+v{f}_1{\kappa }_g+v{f}_2{\tau }_g)\mathbf{n}+(\frac{\partial f_3}{\partial u}+v{f}_1{\kappa }_n-v{f}_3{\tau }_g)\mathbf{g}⟩\hfill \\ & =2v\left(\frac{\partial f_1}{\partial u}-v{f}_2{\kappa }_n-v{f}_3{\kappa }_g\right).\hfill \end{array}$$

This completes the proof. □

Corollary 1.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a spacelike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. If the curve γ is a geodesic curve or an asymptotic curve, then the following equation holds, respectively:

$$\frac{\partial v}{\partial t}=\frac{\partial f_1}{\partial u}-v{f}_2{\kappa }_n$$

or:

$$\frac{\partial v}{\partial t}=\frac{\partial f_1}{\partial u}-v{f}_3{\kappa }_g.$$

Theorem 2.

(Necessary and sufficient condition for an inextensible flow)

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a spacelike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. Then, the flow is inextensible if and only if:

$$\frac{\partial f_1}{\partial s}=f_2{\kappa }_n+f_3{\kappa }_g.$$

(6)

Proof. 

Suppose that the flow of a spacelike curve $\gamma$ on M is inextensible. From (4) and (5), we have:

$$\frac{\partial }{\partial t}s(u,t)={\int }_0^u\frac{\partial v}{\partial t}du={\int }_0^u\left(\frac{\partial f_1}{\partial u}-v{f}_2{\kappa }_n-v{f}_3{\kappa }_g\right)du=0.$$

It follows that:

$$\frac{\partial f_1}{\partial u}=v{f}_2{\kappa }_n+v{f}_3{\kappa }_g.$$

Since $\frac{\partial }{\partial s}=\frac{1}{v}\frac{\partial }{\partial u}$, we can obtain (6).

Conversely, by following a similar way as above, the proof is completed. □

Theorem 3.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a spacelike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. If the flow is inextensible, then a time evolution of the Darboux frame $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ along a curve γ on a lightlike surface M is given by:

$$\frac{d}{dt}\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{g}\end{array}\right)=\left(\begin{array}{ccc}0& {\phi }_1& {\phi }_2\\ -{\phi }_2& {\phi }_3& 0\\ -{\phi }_1& 0& -{\phi }_3\end{array}\right)\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{g}\end{array}\right),$$

(7)

where:

$$\begin{array}{cc}\hfill {\phi }_1& =\frac{\partial f_2}{\partial s}+f_1{\kappa }_g+f_2{\tau }_g,\hfill \\ \hfill {\phi }_2& =\frac{\partial f_3}{\partial s}+f_1{\kappa }_n-f_3{\tau }_g,\hfill \\ \hfill {\phi }_3& =⟨\frac{\partial \mathbf{n}}{\partial t},\mathbf{g}⟩.\hfill \end{array}$$

(8)

Proof. 

Noting that:

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{t}}{\partial t}& =\frac{\partial }{\partial t}\left(\frac{\partial \gamma }{\partial s}\right)=\frac{\partial }{\partial s}(f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g})\hfill \\ & =\left(\frac{\partial f_2}{\partial s}+f_1{\kappa }_g+f_2{\tau }_g\right)\mathbf{n}+\left(\frac{\partial f_3}{\partial s}+f_1{\kappa }_n-f_3{\tau }_g\right)\mathbf{g}.\hfill \end{array}$$

(9)

On the other hand,

$$\begin{array}{cc}\hfill 0& =\frac{\partial }{\partial t}⟨\mathbf{t},\mathbf{n}⟩=⟨\frac{\partial \mathbf{t}}{\partial t},\mathbf{n}⟩+⟨\mathbf{t},\frac{\partial \mathbf{n}}{\partial t}⟩=\frac{\partial f_3}{\partial s}+f_1{\kappa }_n-f_3{\tau }_g+⟨\mathbf{t},\frac{\partial \mathbf{n}}{\partial t}⟩\hfill \\ \hfill 0& =\frac{\partial }{\partial t}⟨\mathbf{t},\mathbf{g}⟩=⟨\frac{\partial \mathbf{t}}{\partial t},\mathbf{g}⟩+⟨\mathbf{t},\frac{\partial \mathbf{g}}{\partial t}⟩=\frac{\partial f_2}{\partial s}+f_1{\kappa }_g+f_2{\tau }_g+⟨\mathbf{t},\frac{\partial \mathbf{n}}{\partial t}⟩\hfill \end{array}$$

because of $⟨\mathbf{n},\mathbf{n}⟩=⟨\mathbf{g},\mathbf{g}⟩=0$ and $⟨\mathbf{n},\mathbf{g}⟩=1$.

Thus, we have:

$$\frac{\partial \mathbf{n}}{\partial t}=-\left(\frac{\partial f_3}{\partial s}+f_1{\kappa }_n-f_3{\tau }_g\right)\mathbf{t}+{\phi }_3\mathbf{n},$$

(10)

$$\frac{\partial \mathbf{g}}{\partial t}=-\left(\frac{\partial f_2}{\partial s}+f_1{\kappa }_g+f_2{\tau }_g\right)\mathbf{t}-{\phi }_3\mathbf{g},$$

(11)

where ${\phi }_3=⟨\frac{\partial \mathbf{n}}{\partial t},\mathbf{g}⟩$. This completes the proof. □

Now, by using Theorem 3, we give the time evolution equations of the geodesic curvature, the normal curvature and the geodesic torsion of a spacelike curve on a lightlike surface.

Theorem 4.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a spacelike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. Then, the time evolution equations of the functions ${\kappa }_g$, ${\kappa }_n$ and ${\tau }_g$ for the inextensible spacelike curve γ are given by:

$$\begin{array}{cc}\hfill \frac{\partial {\kappa }_g}{\partial t}& =\frac{\partial {\phi }_1}{\partial s}+{\phi }_1{\tau }_g-{\phi }_3{\kappa }_g,\hfill \\ \hfill \frac{\partial {\kappa }_n}{\partial t}& =\frac{\partial {\phi }_2}{\partial s}-{\phi }_2{\tau }_g+{\phi }_3{\kappa }_n\hfill \\ \hfill \frac{\partial {\tau }_g}{\partial t}& =\frac{\partial {\phi }_3}{\partial s}+{\phi }_1{\kappa }_n-{\phi }_2{\kappa }_g+2{\phi }_3{\tau }_g.\hfill \end{array}$$

(12)

Proof. 

It is well known that the arc length and time derivatives commute. This implies the inextensibility of $\gamma$. Accordingly, the compatibility conditions are $\frac{\partial }{\partial s}\left(\frac{\partial \mathbf{t}}{\partial t}\right)=\frac{\partial }{\partial t}\left(\frac{\partial \mathbf{t}}{\partial s}\right)$, etc. On the other hand,

$$\begin{array}{cc}\hfill \frac{\partial }{\partial s}\left(\frac{\partial \mathbf{t}}{\partial t}\right)& =\frac{\partial }{\partial s}({\phi }_1\mathbf{n}+{\phi }_2\mathbf{g})\hfill \\ & =(-{\phi }_1{\kappa }_n-{\phi }_2{\kappa }_g)\mathbf{t}+(\frac{\partial {\phi }_1}{\partial s}+{\phi }_1{\tau }_g)\mathbf{n}+(\frac{\partial {\phi }_2}{\partial s}-{\phi }_2{\tau }_g)\mathbf{g},\hfill \end{array}$$

and:

$$\begin{array}{cc}\hfill \frac{\partial }{\partial t}\left(\frac{\partial \mathbf{t}}{\partial s}\right)& =\frac{\partial }{\partial t}({\kappa }_g\mathbf{n}+{\kappa }_n\mathbf{g})\hfill \\ & =(-{\phi }_1{\kappa }_n-{\phi }_2{\kappa }_g)\mathbf{t}+(\frac{\partial {\kappa }_g}{\partial t}+{\phi }_3{\kappa }_g)\mathbf{n}+(\frac{\partial {\kappa }_n}{\partial t}-{\phi }_3{\kappa }_n)\mathbf{g}.\hfill \end{array}$$

Comparing the two equations, we find:

$$\begin{array}{cc}\hfill \frac{\partial {\kappa }_g}{\partial t}& =\frac{\partial {\phi }_1}{\partial s}+{\phi }_1{\tau }_g-{\phi }_3{\kappa }_g,\hfill \\ \hfill \frac{\partial {\kappa }_n}{\partial t}& =\frac{\partial {\phi }_2}{\partial s}-{\phi }_2{\tau }_g+{\phi }_3{\kappa }_n.\hfill \end{array}$$

It follows from (8) that we can obtain the first and the second equation of (12).

Furthermore by using $\frac{\partial }{\partial s}\left(\frac{\partial \mathbf{n}}{\partial t}\right)=\frac{\partial }{\partial t}\left(\frac{\partial \mathbf{n}}{\partial s}\right)$ and following a similar way as above, we can obtain the third equation of (12). The proof is completed. □

Remark 1.

As applications of inextensible flows of a spacelike curve on a lightlike surface, we can consider geometric phases of the repulsive-type nonlinear Schödinger equation ($NL{S}^{-}$) (cf. [12]).

4. Inextensible Flows of a Lightlike Curve

Let $\gamma$ be a lightlike curve on a lightlike surface M in ${\mathbb{R}}_1^3$. We note that a lightlike curve $\gamma \left(u\right)$ satisfies $⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩\ge 0$. We say that a lightlike curve $\gamma \left(u\right)$ is parametrized by the pseudo arc length if $⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩=1.$ If a lightlike curve $\gamma \left(u\right)$ satisfies $⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩\ne 0$, then $⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩>0$, and:

$$s\left(u\right)={\int }_0^u{⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩}^{\frac{1}{4}}du$$

becomes the pseudo arc length parameter. Let us consider a lightlike curve $\gamma \left(u\right)$ on a lightlike surface M in ${\mathbb{R}}_1^3$ with $⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩\ne 0$.

Let $\gamma :[0,l]\times [0,w]\to M\subset {\mathbb{R}}_1^3$ be a one-parameter family of smooth lightlike curves on a lightlike surface in ${\mathbb{R}}_1^3$, where l is the arc length of the initial curve. We put $v^4=⟨{\gamma }^{″}\left(u\right),{\gamma }^{″}\left(u\right)⟩$, from which the pseudo arc length of $\gamma$ is defined by $s\left(u\right)={\int }_0^{u}vdu.$ Furthermore, the operator $\frac{\partial }{\partial s}$ is given in terms of u by $\frac{\partial }{\partial s}=\frac{1}{v}\frac{\partial }{\partial u}$, and the pseudo arc length parameter is given by $ds=vdu.$

On the other hand, a flow $\frac{\partial \gamma }{\partial t}$ of $\gamma$ can be given by:

$$\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$$

(13)

in terms of the Darboux frames $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ of the lightlike curve $\gamma$ on a lightlike surface M in ${\mathbb{R}}_1^3$, where $f_1,f_2,f_3$ are scalar speeds of the lightlike curve $\gamma$, respectively. We put $s(u,t)={\int }_0^{u}vdu$, it is called the pseudo arc length variation of $\gamma$. From this, we have the following condition:

$$\frac{\partial }{\partial t}s(u,t)={\int }_0^u\frac{\partial v}{\partial t}du=0$$

(14)

for all $u\in [0,l].$

Definition 2.

A curve evolution $\gamma (u,t)$ and its flow $\frac{\partial \gamma }{\partial t}$ of a lightlike curve γ in ${\mathbb{R}}_1^3$ are said to be inextensible if:

$$\frac{\partial }{\partial t}{⟨\frac{{\partial }^2\gamma }{\partial u^2},\frac{{\partial }^2\gamma }{\partial u^2}⟩}^{\frac{1}{4}}=0.$$

Theorem 5.

Let M be a lightlike surface in Minkowski three-space ${\mathbb{R}}_1^3$ and $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ be the Darboux frames along a lightlike curve γ on M. If $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ is a flow of γ on a lightlike surface M, then we have the following equation:

$$\frac{\partial v}{\partial t}=\frac{1}{2v^3}\left[(\frac{\partial v}{\partial u}+v^2{\kappa }_n)\left(\frac{\partial {\Phi }_2}{\partial u}-v{\kappa }_n{\Phi }_2-v{\kappa }_g{\Phi }_3\right)+v^2{\kappa }_g\left(\frac{\partial {\Phi }_3}{\partial u}+v{\kappa }_g{\Phi }_1+v{\tau }_g{\Phi }_2\right)\right],$$

(15)

where:

$$\begin{array}{cc}\hfill {\Phi }_1& =\frac{\partial f_1}{\partial u}+v{f}_1{\kappa }_n-v{f}_3{\tau }_g,\hfill \\ \hfill {\Phi }_2& =\frac{\partial f_2}{\partial u}-v{f}_2{\kappa }_n-v{f}_3{\kappa }_g,\hfill \\ \hfill {\Phi }_3& =\frac{\partial f_1}{\partial u}+v{f}_1{\kappa }_g+v{f}_2{\tau }_g.\hfill \end{array}$$

Proof. 

From the definition of a lightlike curve $\gamma$, we have $v^4=⟨\frac{{\partial }^2\gamma }{\partial u^2},\frac{{\partial }^2\gamma }{\partial u^2}⟩.$ By differentiating $v^4$, we have:

$$4v^3\frac{\partial v}{\partial t}=\frac{\partial }{\partial t}⟨\frac{{\partial }^2\gamma }{\partial u^2},\frac{{\partial }^2\gamma }{\partial u^2}⟩=2⟨\frac{{\partial }^2\gamma }{\partial u^2},\frac{{\partial }^2}{\partial u^2}(\frac{\partial \gamma }{\partial t})⟩.$$

(16)

On the other hand,

$$\frac{{\partial }^2\gamma }{\partial u^2}=\frac{\partial }{\partial u}\left(\frac{\partial \gamma }{\partial u}\right)=\frac{\partial }{\partial u}\left(v\mathbf{t}\right)=\left(\frac{\partial v}{\partial u}+v^2{\kappa }_n\right)\mathbf{t}+v^2{\kappa }_g\mathbf{g}$$

and:

$$\begin{array}{c}\frac{{\partial }^2}{\partial u^2}(\frac{\partial \gamma }{\partial t})=\frac{{\partial }^2}{\partial u^2}(f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g})\hfill \\ =\left[\frac{\partial {\Phi }_1}{\partial u}+v{\kappa }_n{\Phi }_1-v{\tau }_g{\Phi }_3\right]\mathbf{t}+\left[\frac{\partial {\Phi }_2}{\partial u}-v{\kappa }_n{\Phi }_2-v{\kappa }_g{\Phi }_3\right]\mathbf{n}+\left[\frac{\partial {\Phi }_3}{\partial u}+v{\kappa }_g{\Phi }_1+v{\tau }_g{\Phi }_2\right]\mathbf{g}.\hfill \end{array}$$

Thus, (16) implies (15). This completes the proof. □

Theorem 6.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a lightlike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. Then, the flow is inextensible if and only if:

$$\left(\frac{\partial v}{\partial s}+v{\kappa }_n\right)\frac{\partial {\Phi }_2}{\partial s}+v{\kappa }_g\frac{\partial {\Phi }_3}{\partial s}=\left(\frac{\partial v}{\partial s}+v{\kappa }_n\right)({\kappa }_n{\Phi }_2+{\kappa }_g{\Phi }_3)-v{\kappa }_g({\kappa }_g{\Phi }_1+{\tau }_g{\Phi }_2).$$

(17)

Proof. 

Suppose that the flow of a lightlike curve $\gamma$ on M is inextensible. By using (15) and $\frac{\partial }{\partial s}=\frac{1}{v}\frac{\partial }{\partial u}$, (14) gives (17). Conversely, by following a similar way as above, the proof is completed. □

Next, we give the time evolution equations of the Darboux frame of a lightlike curve on a lightlike surface.

Theorem 7.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a lightlike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. If the flow is inextensible, then a time evolution of the Darboux frame $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ along a curve γ on a lightlike surface M is given by:

$$\frac{d}{dt}\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{g}\end{array}\right)=\left(\begin{array}{ccc}\frac{{\Phi }_1}{v}& 0& \frac{{\Phi }_3}{v}\\ 0& -\frac{{\Phi }_1}{v}& \Theta \\ -\Theta & -\frac{{\Phi }_3}{v}& 0\end{array}\right)\left(\begin{array}{c}\mathbf{t}\\ \mathbf{n}\\ \mathbf{g}\end{array}\right),$$

(18)

where $\Theta =⟨\frac{\partial \mathbf{n}}{\partial t},\mathbf{g}⟩.$

Proof. 

The proof can be obtained by using a similar method of proof of Theorem 3. □

Theorem 8.

Let $\frac{\partial \gamma }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$ be a flow of a lightlike curve γ on a lightlike surface M in ${\mathbb{R}}_1^3$. Then, the time evolution equations of the functions ${\kappa }_g$, ${\kappa }_n$ and ${\tau }_g$ for the inextensible spacelike curve γ are given by:

$$\begin{array}{cc}\hfill \frac{\partial {\kappa }_g}{\partial t}& =\frac{\partial }{\partial s}(\frac{1}{v}{\Phi }_3)+\frac{1}{v}\left({\kappa }_g{\Phi }_1-{\kappa }_n{\Phi }_3\right),\hfill \\ \hfill \frac{\partial {\kappa }_n}{\partial t}& =\frac{\partial }{\partial s}(\frac{1}{v}{\Phi }_1)+{\kappa }_g\Theta -\frac{1}{v}{\tau }_g{\Phi }_3,\hfill \\ \hfill \frac{\partial {\tau }_g}{\partial t}& =\frac{\partial \Theta }{\partial s}+{\kappa }_n\Theta -\frac{1}{v}{\tau }_g{\Phi }_1.\hfill \end{array}$$

(19)

Proof. 

The proof can be obtained by using a similar method of proof of Theorem 4. □

5. Lightlike Ruled Surfaces

In this section, we investigate inextensible flows of ruled surfaces, in particular lightlike ruled surfaces in Minkowski three-space ${\mathbb{R}}_1^3$.

Let I be an open interval on the real line $\mathbb{R}$. Let $\alpha$ be a curve in ${\mathbb{R}}_1^3$ defined on I and $\beta$ a transversal vector field along $\alpha$. For an open interval J of $\mathbb{R}$, we have the parametrization for M:

$$X(u,v)=\alpha \left(u\right)+v\beta \left(u\right),u\in I,v\in J.$$

Here, $\alpha$ is called a base curve and $\beta$ a director vector field. In particular, the director vector field $\beta$ can be naturally chosen so that it is orthogonal to $\alpha ,$ that is $⟨{\alpha }^{\prime },\beta ⟩=0.$ It is well known that the ruled surface is developable if $det\left({\alpha }^{\prime }\beta {\beta }^{\prime }\right)$ is identically zero. A developable surface is a surface whose Gaussian curvature of the surface is everywhere zero.

On the other hand, the tangent vectors are given by:

$$X_u=\frac{\partial X}{\partial u}={\alpha }^{\prime }\left(u\right)+v{\beta }^{\prime }\left(u\right),X_v=\frac{\partial X}{\partial v}=\beta \left(u\right),$$

which imply that the coefficients of the first fundamental form of the surface are given by:

$$\begin{array}{cc}\hfill E=⟨X_u,X_u⟩& =⟨{\alpha }^{\prime },{\alpha }^{\prime }⟩+2v⟨{\alpha }^{\prime },{\beta }^{\prime }⟩+v^2⟨{\beta }^{\prime },{\beta }^{\prime }⟩,\hfill \\ \hfill F=⟨X_u,X_v⟩& =0,\hfill \\ \hfill G=⟨X_v,X_v⟩& =⟨\beta ,\beta ⟩.\hfill \end{array}$$

Suppose that the ruled surface is lightlike. Then, we get $E=0$ or $G=0$.

First of all, we consider $E=0$; it implies that:

$$⟨{\alpha }^{\prime },{\alpha }^{\prime }⟩=0,⟨{\alpha }^{\prime },{\beta }^{\prime }⟩=0,⟨{\beta }^{\prime },{\beta }^{\prime }⟩=0.$$

(20)

Thus, a base curve $\alpha$ is lightlike, and a director vector $\beta$ is constant or ${\beta }^{\prime }$ is lightlike.

Case 1: If $\beta$ is constant, from $⟨{\alpha }^{\prime },\beta ⟩=0$, $\beta$ is a lightlike vector or a spacelike vector. If $\beta$ is lightlike, there exists a smooth function k such that $\beta =k{\alpha }^{\prime }$. This is a contradiction because $G=0$. If $\beta$ is spacelike as a constant vector, then the lightlike cylindrical ruled surface is parametrized by:

$$X(u,v)=\alpha \left(u\right)+v\beta ,$$

where $\alpha$ is a lightlike curve and $\beta$ is a constant spacelike vector.

Case 2: Let ${\beta }^{\prime }$ be a lightlike vector. Since $⟨{\alpha }^{\prime },{\beta }^{\prime }⟩=0$, there exists a smooth function k such that ${\beta }^{\prime }=k{\alpha }^{\prime }$. Thus, a lightlike non-cylindrical ruled surface is parametrized by:

$$X(u,v)=\alpha \left(u\right)+v\beta \left(u\right),$$

(21)

where $\alpha$ and $\beta$ satisfy the condition (20).

Next, we consider $G=⟨\beta ,\beta ⟩=0$, since $\beta \ne \mathbf{0}$, a director vector $\beta$ must be lightlike. Furthermore, since $⟨{\alpha }^{\prime },\beta ⟩=0$, $\alpha$ is a spacelike curve or a lightlike curve.

Case 1: If $\alpha$ is a spacelike curve, then a lightlike non-cylindrical ruled surface is parametrized by:

$$X(u,v)=\alpha \left(u\right)+v\beta \left(u\right),$$

(22)

where $\alpha$ is a spacelike curve and $\beta$ is a lightlike vector.

Case 2: Let $\alpha$ be a lightlike curve. Then, there exists a smooth function k such that ${\beta }^{\prime }=k{\alpha }^{\prime }$, and a lightlike ruled surface as a tangent developable surface is parametrized by:

$$X(u,v)=\alpha \left(u\right)+vk{\alpha }^{\prime }\left(u\right),$$

(23)

where $\alpha$ and ${\alpha }^{″}$ are a lightlike curve and a spacelike vector, respectively.

In [5], the authors gave the following:

Definition 3.

A surface evolution $X(u,v,t)$ and its flow $\frac{\partial X}{\partial t}$ are said to be inextensible if the coefficients of the first fundamental form of the surface satisfy:

$$\frac{\partial E}{\partial t}=\frac{\partial F}{\partial t}=\frac{\partial G}{\partial t}=0.$$

This definition states that the surface $X(u,v,t)$ is, for all time t, the isometric image of the original surface $X(u,v,t_0)$ defined at some initial time $t_0$.

Now, we study inextensible flows of a lightlike tangent developable surface in Minkowski three-space.

Consider a lightlike tangent developable surface parametrized by:

$$X(u,v)=\alpha \left(u\right)+v{\alpha }^{\prime }\left(u\right),$$

(24)

where $\alpha$ is a lightlike curve. Suppose that the parameter u is a pseudo-arc length of $\alpha$. In this case, we get $E=v^2\left|\right|{\alpha }^{″}{\left|\right|}^2$ and $F=G=0$.

Thus, we have:

Theorem 9.

Let $X(u,v)$ be a lightlike tangent developable surface given by (24). The surface evolution $X(u,v,t)=\alpha (u,t)+v{\alpha }^{\prime }(u,t)$ is inextensible if and only if:

$$\frac{\partial }{\partial t}\left|\right|{\alpha }^{″}{\left|\right|}^2=0.$$

As a consequence, we have the following results:

Theorem 10.

Let $X(u,v,t)=\alpha (u,t)+v{\alpha }^{\prime }(u,t)$ be a surface evolution of a lightlike tangent developable surface given by (24) in ${\mathbb{R}}_1^3$. Then, we have the following statements:

(1)$\alpha (u,t)$ is an inextensible evolution of a lightlike curve $\alpha \left(u\right)$ in ${\mathbb{R}}_1^3$.

(2)An inextensible evolution of a lightlike tangent developable surface can be completely characterized by the inextensible evolutions of a lightlike curve $\alpha \left(u\right)$ in ${\mathbb{R}}_1^3$.

Proof. 

In fact, $0=\frac{\partial }{\partial t}\left|\right|{\alpha }^{″}{\left|\right|}^2=2\left|\right|{\alpha }^{″}\left|\right|\frac{\partial }{\partial t}\left|\right|{\alpha }^{″}\left|\right|$ and ${\alpha }^{″}\ne \mathbf{0}$, and we get $\frac{\partial }{\partial t}\left|\right|{\alpha }^{″}\left|\right|=0$; it implies $\frac{\partial }{\partial t}\left|\right|{\alpha }^{″}{\left|\right|}^{\frac{1}{2}}=0$. This means that $\alpha (u,t)$ satisfies the condition for Definition 2. □

Theorem 11.

Let $X(u,v,t)=\alpha (u,t)+v{\alpha }^{\prime }(u,t)$ be a surface evolution of a lightlike tangent developable surface given by (24) in ${\mathbb{R}}_1^3$, and $\frac{\partial \alpha }{\partial t}=f_1\mathbf{t}+f_2\mathbf{n}+f_3\mathbf{g}$, where $\mathbf{t},\mathbf{n},\mathbf{g}$ are the Darboux frames along a lightlike curve α on a lightlike surface. If the surface evolution $X(u,v,t)$ is inextensible, then $f_1,f_2,f_3$ satisfy Equation (19).

6. Conclusions

We study an inextensible flow of a spacelike or a lightlike curve on a lightlike surface in Minkowski three-space and investigate a time evolution of the Darboux frame $\{\mathbf{t},\mathbf{n},\mathbf{g}\}$ (see Theorems 3 and 7) and the functions ${\kappa }_n,{\kappa }_g$ and ${\tau }_g$ (see Theorems 4 and 8). Furthermore, in Theorems 2 and 6, we give a necessary and sufficient condition of inextensible flows of a spacelike curve and a lightlike curve on a lightlike surface in terms of a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we completely classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface (see Theorems 9 and 10).