Complete classification of surfaces with a canonical principal direction in the Minkowski 3-space

TL;DR

This paper provides a comprehensive classification of all surfaces in Minkowski 3-space that have a canonical principal direction aligned with a fixed space-like or light-like direction, enriching the understanding of their geometric properties.

Contribution

It offers a complete classification of surfaces with canonical principal directions in Minkowski 3-space, considering both space-like and light-like directions, which was not previously fully achieved.

Findings

Complete classification of such surfaces achieved

Characterization of surfaces with respect to space-like directions

Characterization of surfaces with respect to light-like directions

Abstract

In this paper, we characterize and classify all surfaces endowed with canonical principal direction relative to a space-like and light-like, constant direction in Minkowski 3-spaces.

Equations

$$\tilde{g}=\sum _{i=1}^{m-1}d{x}_i^2-d{x}_m^2,$$

$$\begin{array}{rl}{\tilde{\nabla }}_{X}Y=& {\nabla }_{X}Y+h\left(X,Y\right),\\ {\tilde{\nabla }}_{X}N=& -S(X),\end{array}$$

$$({\tilde{\nabla }}_{X}h)(Y,Z)=({\tilde{\nabla }}_{Y}h)(X,Z),$$

$$({\tilde{\nabla }}_{X}h)(Y,Z)={\nabla }_X^{\perp }h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z).$$

$$\displaystyle\displaystyle\begin{split}\mbox{Case I. }S=\left(\begin{array}[]{ccc}k_{1}&\\ &\ddots&\\ &&k_{n}\end{array}\right),&\quad\mbox{Case II. }S=\left(\begin{array}[]{ccccc}k_{1}&1\\ 0&k_{1}&&&\\ &&k_{3}&&\\ &&&\ddots&\\ &&&&k_{n}\end{array}\right),\\ \mbox{Case III. }S=\left(\begin{array}[]{ccccc}k_{1}&\nu&&&\\ -\nu&k_{1}&&&\\ &&k_{3}&&\\ &&&\ddots&\\ &&&&k_{n}\end{array}\right),&\mbox{ Case IV. }S=\left(\begin{array}[]{cccccc}k_{1}&1&1&&&\\ 1&k_{1}&1&&&\\ 1&1&k_{1}&&&\\ &&&k_{4}&&\\ &&&&\ddots&\\ &&&&&k_{n}\end{array}\right)\end{split}$$

$$\displaystyle\langle e_{1},e_{1}\rangle=-1,\langle e_{2},e_{2}\rangle=1,\quad\langle e_{i},e_{j}\rangle=0\mbox{ whenever $\displaystyle i\neq j$}$$

$$\displaystyle\langle e_{A},e_{B}\rangle=\delta_{AB}-1,\quad\langle e_{1},e_{A}\rangle=\langle e_{2},e_{A}\rangle=0,\quad.\langle e_{i},e_{j}\rangle=\delta_{ij}\mbox{whenenver $\displaystyle A,B=1,2$, $\displaystyle i,j>2$}.$$

$$\left|⟨v,w⟩\right|=\Vert v\Vert \Vert w\Vert \cosh \theta .$$

$$\left|⟨v,w⟩\right|=\Vert v\Vert \Vert w\Vert \cos \theta .$$

$$\left|⟨v,w⟩\right|=\Vert v\Vert \Vert w\Vert \cosh \theta .$$

$$\left|⟨v,w⟩\right|=\Vert v\Vert \Vert w\Vert \sinh \theta .$$

$$k=U+⟨N,N⟩⟨k,N⟩N$$

$$k=-⟨k,e_1⟩e_1+⟨k,N⟩N.$$

$$0=-⟨k,{\tilde{\nabla }}_{e_1}{e}_1⟩e_1-⟨k,e_1⟩{\tilde{\nabla }}_{e_1}{e}_1-⟨k,S{e}_1⟩N-⟨k,N⟩S{e}_1.$$

$$k=⟨k,e_1⟩e_1+\epsilon ⟨k,N⟩N,$$

$$k=\varphi (e_1-N),$$

$$k=\cosh \theta e_1+\sinh \theta N$$

$${\nabla }_{e_1}{e}_1={\nabla }_{e_1}{e}_2=0,$$

$${\nabla }_{e_2}{e}_1=\tanh \theta k_2{e}_2,$$

$$S=\left(\begin{array}[]{cc}e_{1}(\theta)&0\\ 0&k_{2}\end{array}\right)$$

$$e_1(k_2)=\tanh \theta k_2(e_1(\theta )-k_2).$$

$$e_2(\theta )=0.$$

$$0=X(\cosh \theta )e_1+\cosh \theta {\nabla }_X{e}_1+\cosh \theta h(e_1,X)-\sinh \theta SX+X(\sinh \theta )N$$

$${\nabla }_{e_1}{e}_1$$

$$e_1(\theta )$$

$${\nabla }_{e_2}{e}_1$$

$$h(e_1,e_1)=-k_{1}N,h(e_1,e_2)=0,h(e_2,e_2)=-k_{2}N.$$

$$g=d{s}^2+m^{2}d{t}^2$$

$$e_1(m)-\tanh \theta k_{2}m=0.$$

$$x(s,t)=\int^{s}{\cosh\theta(\tau)d\tau}\Big{(}1,0,0\Big{)}+\int^{s}{\sinh\theta(\tau)d\tau}\Big{(}0,\sinh t,\cosh t\Big{)}+\gamma(t)$$

$$\gamma (t)=\left(0,{\int }^t\Psi (\tau )\cosh \tau d\tau ,{\int }^t\Psi (\tau )\sinh \tau d\tau \right).$$

$$\displaystyle\displaystyle\begin{split}x(s,t)=&\int^{s}{\cosh\theta(\tau)d\tau}\Big{(}1,0,0\Big{)}+\int^{s}{\sinh\theta(\tau)d\tau}\Big{(}0,\sinh t_{0},\cosh t_{0}\Big{)}\\ &+\Big{(}0,t\cosh t_{0},t\sinh t_{0}\Big{)}.\end{split}$$

Full text

1. Introduction

It is well known that, a helix is a curve whose tangent lines make a constant angle with a fixed vector. After the question ‘Are there any surface making a constant angle with some fixed vector direction?’ was introduced in [5], the concept of constant angle surfaces, called also as helix surfaces, have been studied geometers. Firstly, the applications of concerning surfaces in the theory of liquid crystals and of layered fluids were considered in [1]. They used for their study of surfaces the Hamilton-Jacobi equation, correlating the surface and and the direction field. Further, Munteanu and Nistor gave another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction in [17]. Moreover, the study of constant angle surfaces was extended in different ambient spaces, e.g. in $\displaystyle\mathbb{S}^{2}\times\mathbb{R}$ [5] and $\displaystyle\mathbb{H}^{2}\times\mathbb{R}$ [6], in $\displaystyle\mathbb{E}^{3}_{1}$ [14, 12, 10]. In higher dimensional Euclidean space, hypersurfaces whose tangent space makes constant angle with a fixed direction are studied and a local description of how these hypersurfaces are constructed is given. They are called helix hypersurfaces, [3].

One of common geometrical properties of this type of surfaces is the following. If we denote by $\displaystyle U^{T}$ the projection of the fixed direction $\displaystyle k$ on the tangent plane of the surface, then $\displaystyle U^{T}$ is a principal direction of the surface with the corresponding principal curvature 0. Because of this reason, a recent natural problem that appears in the context of constant angle surfaces is to study those surfaces for which $\displaystyle U^{T}$ remains a principal direction but the corresponding principal curvature is different from zero.

Let $\displaystyle N$ be a (semi-)Riemannian manifold, $\displaystyle M$ a hypersurface of $\displaystyle N$ and $\displaystyle X$ a vector field tangent to $\displaystyle N$. $\displaystyle M$ is said to have a canonical principal direction (CPD) relative to $\displaystyle X$ if the tangential projection of $\displaystyle X$ to $\displaystyle M$ gives a principal direction, [11]. One of the most common examples of hypersurfaces with CPD is rotational hypersurfaces in Euclidean spaces which have canonical principal direction relative to a vector field parallel to its rotation axis. We also want to note that a hypersurface in an Euclidean space with CPD relative to its position vector is said to be a generalized constant ratio hypersurface, [8, 9].

The problems of classifying hypersurfaces with CPD relative to a fixed direction $\displaystyle k$ have been studied by some authors recently. For example, in [4], this problem was studied in $\displaystyle\mathbb{S}^{2}\times\mathbb{R}$ by Dillen et. al. Further, surfaces with CPD in $\displaystyle\mathbb{H}^{2}\times\mathbb{R}$ was studied in [7]. On these two papers $\displaystyle k$ was chosen to be a unit vector tangent to the second factor. On the other hand, classification results on surfaces in semi-Euclidean spaces with CPD to a chosen relative direction was studied in [10, 18, 19]. Before we proceed, we also would like to note that when the codimension of the submanifold is more than one, a generalization of this notion was given by Tojeiro in [21] and a further study appear in [16].

In the present paper, we would like to move the study of CPD hypersurfaces in Euclidean spaces initiated in [18] into semi-Euclidean spaces by obtaining partial classification of CPD surfaces in Minkowski 3-space studied in [19, 10]. This paper is organized as follows. In Sect. 2, we introduce the notation that we will use and give a brief summary of basic definitions in theory of submanifolds of semi-Euclidean spaces. In Sect. 3, we obtain some new characterizations and the complete classification of space-like and Lorentzian CPD surfaces relative to a space-like and light-like, constant direction in the Minkowski 3-space.

2. CPD Hypersurfaces in Minkowski spaces

In this section after we give some basic equations and facts on hypersurfaces in Minkowski spaces, we would like to consider geometrical properties of hypersurfaces in a Minkowski space $\displaystyle\mathbb{E}^{3}_{1}$ endowed with a canonical principal direction.

2.1. Basic facts and definitions

First, we would like to give a brief summary of basic definitions, facts and equations in the theory of submanifolds of pseudo-Euclidean space (see for detail, [20, 2]).

Let $\displaystyle\mathbb{E}^{m}_{1}$ denote the Minkowski $\displaystyle m$-space with the canonical Lorentzian metric tensor given by

[TABLE]

where $\displaystyle x_{1},x_{2},\ldots,x_{m}$ are rectangular coordinates of the points of $\displaystyle\mathbb{E}^{m}_{1}$. We denote the Levi-Civita connection of $\displaystyle\mathbb{E}_{n+1}^{1}$ by and $\displaystyle\widetilde{\nabla}$.

The causality of a vector in a Minkowski space is defined as following. A non-zero vector $\displaystyle v$ in $\displaystyle\mathbb{E}^{m}_{1}$ is said to be space-like, time-like and light-like (null) regarding to $\displaystyle\left\langle v,v\right\rangle>0$ , $\displaystyle\left\langle v,v\right\rangle<0$ and $\displaystyle\left\langle v,v\right\rangle=0$, respectively. Note that $\displaystyle v$ is said to be causal if it is not space-like.

Let $\displaystyle M$ be an oriented hypersurface in $\displaystyle\mathbb{E}_{n+1}^{1}$, $\displaystyle N$ and $\displaystyle\nabla$ its unit normal vector associated with its orientation and Levi-Civita connection, respectively. Then, Gauss and Weingarten formulas are given by

[TABLE]

respectively whenever $\displaystyle X,Y$ are tangent to $\displaystyle M$, where $\displaystyle h$ and $\displaystyle S$ are the second fundamental form and the shape operator (or Weingarten map) of $\displaystyle M$. Note that $\displaystyle M$ is said to be space-like (resp. time-like) if the induced metric $\displaystyle g=\left.\widetilde{g}\right|_{M}$ of $\displaystyle M$ is Riemannian (resp. Lorentzian). This is equivalent to being time-like (resp. space-like) of $\displaystyle N$ at each point of $\displaystyle M$.

The Codazzi equations is given by

[TABLE]

where $\displaystyle R$ is the curvature tensor associated with the connection $\displaystyle\nabla$ and $\displaystyle\widetilde{\nabla}h$ is defined by

[TABLE]

If $\displaystyle M$ is space-like, then its shape operator $\displaystyle S$ is diagonalizable, i.e., there exists a local orthonormal frame field $\displaystyle\{e_{1},e_{2}\}$ of the tangent bundle of $\displaystyle M$ such that $\displaystyle Se_{i}=k_{i}e_{i},\quad i=1,2,\ldots,n$. In this case, the vector field $\displaystyle e_{i}$ and smooth function $\displaystyle k_{i}$ are called a principal direction and a principal curvature of $\displaystyle M$.

On the other hand, if $\displaystyle M$ is time-like, then by choosing an appropriated frame field of the tangent bundle of $\displaystyle M$, $\displaystyle S$ can be assumed to have one of the following three matrix representations

[TABLE]

for some smooth functions $\displaystyle k_{1},k_{2},\ldots,k_{n},\nu$ (see for example [15]). We would like to note that in Case I and Case III of (2.2), the frame field $\displaystyle\{e_{1},e_{2}\}$ is orthonormal, i.e.

[TABLE]

and it is pseudo-orthonormal in Case II and Case IV with

[TABLE]

Now, let $\displaystyle M$ be a surface in the Minkowski 3-space. Then, its mean curvature and Gaussian curvature are defined by $\displaystyle H=\mathrm{trace}\,S$ and $\displaystyle H=\mathrm{det}\,S$, respectively. $\displaystyle M$ is said to be flat if $\displaystyle K$ vanishes identically. On the other hand, if $\displaystyle H=0$ and $\displaystyle M$ is space-like, then it is called maximal while a time-like surface with identically vanishing mean curvature is said to be a minimal surface.

Before we proceed to next subsection, we would like to notice the notion of angle in the Minkowski 3-space (see for example [8]):

Definition 2.1**.**

Let $\displaystyle v$ and $\displaystyle w$ be future pointing (past pointing) time-like vectors in $\displaystyle\mathbb{E}_{1}^{3}$. Then, there is a unique non-negative real number $\displaystyle\theta$ such that

[TABLE]

The real number $\displaystyle\theta$ is called the Lorentzian time-like angle between $\displaystyle v$ and $\displaystyle w$.

Definition 2.2**.**

Let $\displaystyle v$ and $\displaystyle w$ be a space-like vectors in $\displaystyle\mathbb{E}_{1}^{3}$ that span a space-like vector subspace. Then, we have $\displaystyle\left|\left\langle v,w\right\rangle\right|\leq\left\|v\right\|\left\|w\right\|$ and hence, there is a unique real number $\displaystyle\theta\in[0,\pi/2]$ such that

[TABLE]

The real number $\displaystyle\theta$ is called the Lorentzian space-like angle between $\displaystyle v$ and $\displaystyle w$.

Definition 2.3**.**

Let $\displaystyle v$ and $\displaystyle w$ be a space-like vectors in $\displaystyle\mathbb{E}_{1}^{3}$ that span a time-like vector subspace. Then, we have $\displaystyle\left|\left\langle v,w\right\rangle\right|>\left\|v\right\|\left\|w\right\|$ and hence, there is a unique positive real number $\displaystyle\theta$ such that

[TABLE]

The real number $\displaystyle\theta$ is called the Lorentzian time-like angle between $\displaystyle v$ and $\displaystyle w$.

Definition 2.4**.**

Let $\displaystyle v$ be a space-like vectors and $\displaystyle w$ a future pointing time-like vector in $\displaystyle\mathbb{E}_{1}^{3}$. Then, there is a unique non-negative real number $\displaystyle\theta$ such that

[TABLE]

The real number $\displaystyle\theta$ is called the Lorentzian time-like angle between $\displaystyle v$ and $\displaystyle w$.

2.2. A characterization of CPD hypersurfaces

First, we would like to recall the following definition (See for example[10, 19, 11]).

Definition 2.5**.**

Let $\displaystyle M$ be a non-degenerated hypersurface in $\displaystyle\mathbb{E}^{n+1}_{1}$ and $\displaystyle\zeta$ a vector field in $\displaystyle\mathbb{E}^{n+1}_{1}$. $\displaystyle M$ is said to be endowed with CPD relative to $\displaystyle\zeta$ if its tangential component is a principle direction, i.e., $\displaystyle S(\zeta^{T})=k_{1}\zeta^{T}$ for a smooth function $\displaystyle k_{1}$, where $\displaystyle\zeta^{T}$ denotes the tangential component of $\displaystyle\zeta$. In particular if $\displaystyle X=k$ for a fixed direction $\displaystyle k$ in $\displaystyle\mathbb{E}^{n+1}_{t}$, we will say that $\displaystyle M$ is a CPD-hypersurface.

On the other hand, a surface $\displaystyle M$ in $\displaystyle\mathbb{E}^{3}$ is said to be a constant angle surface (CAS) if its unit normal makes a constant angle with a fixed vector, [17] (see also [5, 6, 10]. Later, in [12, 14] this definition is extended to surfaces in Minkowski spaces with obvious restrictions on the causality of the fixed vector and the normal vector because of the definition of ‘angle’ in the Minkowski space (See Definition 2.1- Definition 2.4).

Remark 2.6*.*

In fact, if the ambient space is pseudo-Euclidean, then a CAS surface is a CPD surface with corresponding principle curvature $\displaystyle k_{1}=0$ (see [12, 14, 17]). Thus, we will exclude this case. Therefore, after this point, we will locally assume that the principle curvature $\displaystyle k_{1}$ corresponding to the principle direction of tangential part of $\displaystyle k$ is a non-vanishing function.

Let $\displaystyle M$ be a hypersurface in a Minkowski space $\displaystyle\mathbb{E}^{n+1}_{1}$ and $\displaystyle k$ be a fixed direction in $\displaystyle\mathbb{E}^{n+1}_{1}$ . The fixed vector $\displaystyle k$ can be expressed as

[TABLE]

for a tangent vector $\displaystyle U$. We would like to give the following new characterization of CPD surfaces different from given in [19, Theorem 2.1] and [10, Theorem 3.7 and Theorem 4.5].

Proposition 2.7**.**

Let $\displaystyle M$ be an oriented hypersurface in the Minkowski space $\displaystyle\mathbb{E}^{n+1}_{1}$ and $\displaystyle k$ be a fixed vector on the tangent plane to the surface. Consider a unit tangent vector field $\displaystyle e_{1}$ along $\displaystyle U$. Then, $\displaystyle M$ is a CPD hypersurface if and only if a curve $\displaystyle\alpha$ is a geodesic of $\displaystyle M$ whenever it is an integral curve of $\displaystyle e_{1}$.

Proof.

We will consider three cases seperately subject to causality of $\displaystyle U$.

Case I. Let $\displaystyle e_{1}$ is time-like. Thus, we have

[TABLE]

Since $\displaystyle\widetilde{\nabla}_{e_{1}}k=0$, this equation yields

[TABLE]

The tangential part of this equation yields $\displaystyle Se_{1}=k_{1}e_{1}$ if and only if $\displaystyle\nabla_{e_{1}}e_{1}=0$ which is equivalent to being geodesic of all integral curves of $\displaystyle e_{1}$.

Case II. Let $\displaystyle e_{1}$ is space-like. Thus, we have

[TABLE]

where $\displaystyle\varepsilon$ is either 1 or -1 regarding to being time-like or space-like of $\displaystyle M$, respectively.

Similar to Case I, we obtain $\displaystyle Se_{1}=k_{1}e_{1}$ if and only if $\displaystyle\nabla_{e_{1}}e_{1}=0$.

Case III. Let $\displaystyle e_{1}$ is light-like. In this case, $\displaystyle k$ can be decompose as

[TABLE]

for a non-constant function $\displaystyle\phi$.

Similar to the other case, we obtain $\displaystyle Se_{1}=k_{1}e_{1}$ if and only if $\displaystyle\nabla_{e_{1}}e_{1}=0$. ∎

3. Classifications of CPD Surfaces in $\displaystyle\mathbb{E}^{3}_{1}$

In this section, we want to complete classification of CPD surfaces in $\displaystyle\mathbb{E}^{3}_{1}$. We would like to note that the complete classification of surfaces endowed with canonical principal direction relative to a time-like constant direction $\displaystyle k=(0,0,1)$ was obtained in [10, 19].

3.1. CPD surfaces relative to a space-like, constant direction.

In this subsection, we consider surfaces endowed with CPD relative to a space-like, constant direction $\displaystyle k$. In this case, up to a linear isometry of $\displaystyle\mathbb{E}^{3}_{1}$, we may assume that $\displaystyle k=(1,0,0)$.

First, we will assume that $\displaystyle M$ is a space-like surface endowed with CPD relative to $\displaystyle k=(1,0,0)$. In this case, $\displaystyle N$ is time-like and (2.3) becomes

[TABLE]

for a smooth function $\displaystyle\theta$. Let $\displaystyle e_{2}$ be a unit tangent vector field satisfying $\displaystyle\langle e_{1},e_{2}\rangle=0$. By a simple computation considering (3.1) we obtain the following lemma.

Lemma 3.1**.**

The Levi-Civita connection $\displaystyle\nabla$ of $\displaystyle M$ is given by

[TABLE]

and the matrix representation shape operator $\displaystyle S$ of $\displaystyle M$ with respect to $\displaystyle\{e_{1},e_{2}\}$ is

[TABLE]

for a function $\displaystyle k_{2}$ satisfying

[TABLE]

Furthermore, $\displaystyle\theta$ satisfies

[TABLE]

Proof.

By considering (3.1), one can get

[TABLE]

whenever $\displaystyle X$ is tangent to $\displaystyle M$. (3.6) for $\displaystyle X=e_{1}$ gives

[TABLE]

while (3.6) for $\displaystyle X=e_{2}$ is giving

[TABLE]

where $\displaystyle e_{2}$ is the other principle direction of $\displaystyle M$ with $\displaystyle k_{2}$ is the principle curvature $\displaystyle k_{2}$ corresponding to $\displaystyle e_{2}$. Thus, we have (3.2) and (3.4) and (3.5) and the second fundamental form of $\displaystyle M$ becomes

[TABLE]

By considering the Codazzi equation, we obtain (3.4). ∎

Remark 3.2*.*

Because of (3.7), if $\displaystyle e_{1}(\theta)\equiv 0$ implies $\displaystyle k_{1}=0$. We will not consider this case because of Remark 2.6).

Now, we consider a point $\displaystyle p\in M$ at which $\displaystyle e_{1}(\theta)$ does not vanish. First, we would like to prove the following lemma.

Lemma 3.3**.**

There exists a local coordinate system $\displaystyle(s,t)$ defined in a neighborhood $\displaystyle\mathcal{N}_{p}$ of $\displaystyle p$ such that the induced metric of $\displaystyle M$ is

[TABLE]

for a function satisfying

[TABLE]

Furthermore, the vector fields $\displaystyle e_{1},e_{2}$ described above become $\displaystyle e_{1}=\partial_{s}$, $\displaystyle\displaystyle e_{2}=\frac{1}{m}\partial_{t}$ in $\displaystyle\mathcal{N}_{p}$.

Proof.

Because of (3.2) we have $\displaystyle[e_{1},e_{2}]=-\tanh\theta k_{2}e_{2}$ because of (3.2). Thus, if $\displaystyle m$ is a non-vanishing smooth function on $\displaystyle M$ satisfying (3.9), then we have $\displaystyle\displaystyle\left[e_{1},me_{2}\right]=0$. Therefore, there exists a local coordinate system $\displaystyle(s,t)$ such that $\displaystyle e_{1}=\partial_{s}$ and $\displaystyle\displaystyle e_{2}=\frac{1}{m}\partial_{t}$. Thus, the induced metric of $\displaystyle M$ is as given in (3.8) ∎

Now, we are ready to obtain the classification theorem.

Theorem 3.4**.**

Let $\displaystyle M$ be an oriented space-like surface in $\displaystyle\mathbb{E}^{3}_{1}$. Then, $\displaystyle M$ is endowed with a canonical principal direction relative to a space-like constant direction if and only if it is congruent to the surface given by one of the followings

1. (1)

A surface given by

[TABLE] 2. (2)

A flat surface given by

[TABLE]

for a constant $\displaystyle t_{0}$.

Proof.

In order to proof the necessary condition, we assume that $\displaystyle M$ is endowed with a CPD relative to $\displaystyle k=(1,0,0)$ with the isometric immersion $\displaystyle x:M\rightarrow\mathbb{E}^{3}_{1}$. Let $\displaystyle\{e_{1},e_{2};N\}$ is the local orthonormal frame field described before Lemma 3.1, $\displaystyle k_{1},k_{2}$ principal curvatures of $\displaystyle M$ and $\displaystyle(s,t)$ a local coordinate system given in Lemma 3.3.

Note that (3.9) and (3.4) become

[TABLE]

respectively and $\displaystyle e_{2}(\theta)=0$ implies $\displaystyle\theta=\theta(s)$. Moreover, we have

[TABLE]

By combining (3.12) with (3.3), we obtain the shape operator $\displaystyle S$ of $\displaystyle M$ as

[TABLE]

where ′ denotes ordinary differentiation with respect to the appropriated variable.

By combining (3.12) and (3.13) we obtain

[TABLE]

whose general solution is

[TABLE]

for some smooth functions $\displaystyle\Psi_{1},\Psi_{2}$. Therefore, by re-defining $\displaystyle t$ properly, we may assume either

[TABLE]

Case 1. $\displaystyle m$ satisfies (3.16a). In this case, by considering the equation (3.2) with $\displaystyle m$ given in (3.16a), we get the Levi-Civita connection of $\displaystyle M$ satisfies

[TABLE]

By combining this equation with (3.15) and using Gauss formula, we obtain

[TABLE]

On the other hand, from the decomposition (3.1), we have $\displaystyle\left\langle x_{s},k\right\rangle=\cosh\theta$ and $\displaystyle\left\langle x_{t},k\right\rangle=0.$ By considering these equations, we see that $\displaystyle x$ has the form of

[TABLE]

for a $\displaystyle\mathbb{E}^{3}_{1}$-valued smooth function $\displaystyle\gamma=\left(0,\gamma_{2},\gamma_{3}\right)$. On the other hand, by combining (3.14) and (3.17) with (3.1), we yield

[TABLE]

By considering (3.18) and $\displaystyle\left\langle x_{s},x_{s}\right\rangle=1$, we solve (3.19) to obtain

[TABLE]

for a smooth function $\displaystyle\varphi$. Note that (3.20) implies

[TABLE]

and because of $\displaystyle\langle x_{s},x_{t}\rangle=0$ we have $\displaystyle(0,\gamma_{2}^{\prime},\gamma_{3}^{\prime})=h(\sinh\varphi,\cosh\varphi)$ for a smooth function $\displaystyle h=h(t)$. Therefore, (3.21) turns into

[TABLE]

By combining this equation with $\displaystyle\langle x_{t},x_{t}\rangle=m^{2}$ and using (3.16a), we obtain $\displaystyle\varphi(t)=t$ and $\displaystyle h(t)=\Psi(t)$ which gives (3.10b). In addition, $\displaystyle\varphi(t)=t$ and yields (3.10b). Thus, we have the Case (1) of the theorem.

Case 2. $\displaystyle m$ is given as (3.16b). In this case, the induced metric of $\displaystyle M$ becomes $\displaystyle g=ds^{2}+dt^{2}$, the Levi Civita connection of $\displaystyle M$ satisfies

[TABLE]

and (3.15) gives

[TABLE]

Therefore, $\displaystyle x$ and $\displaystyle N$ satisfies

[TABLE]

A straightforward computation yields that $\displaystyle M$ is congruent to the surface given in Case (2) of the theorem. Hence, the proof for the necessary condition is obtained.

The poof of sufficient condition follows from a direct computation. ∎

As a direct result of Theorem 3.4, we obtain the following classification of maximal CPD surfaces.

Proposition 3.5**.**

A maximal surface in $\displaystyle\mathbb{E}^{3}_{1}$ endowed with CPD relative to a constant, space-like direction is either an open part of a plane or congruent to the surface given by

[TABLE]

for a non-zero constant $\displaystyle c$.

In this case the angle function $\displaystyle\theta$ is

[TABLE]

Proof.

Let $\displaystyle M$ be a space-like CPD surface and assume that it is not an open part of a plane. If $\displaystyle M$ is maximal, then Theorem 3.4 yields that $\displaystyle M$ is congruent to the surface given by (3.10). Note that the shape operator $\displaystyle S$ of $\displaystyle M$ is (3.15) for the function $\displaystyle m$ satisfying (3.16a). Considering the maximality condition $\displaystyle\mathrm{tr}\,S=0$ and (3.3), we have

[TABLE]

Solving this equation, we get

[TABLE]

for a non-zero constant $\displaystyle c$. Furthermore, one can conclude from (3.26) that the function $\displaystyle m$ depends only on $\displaystyle s$. So (3.16a) implies $\displaystyle\Psi(t)=0$ which yields $\displaystyle m(s)=\int^{s}{\sinh\theta(\tau)d\tau}$ and $\displaystyle\gamma(t)=(0,0,0).$ Therefore, (3.26) becomes

[TABLE]

By solving this equation, we get the expression (3.25). By a further computation, we obtain (3.24). Thus, we complete the proof of theorem. ∎

In the remaining part of this section, we will assume that $\displaystyle M$ is a Lorentzian surface in the Minkowski 3-space endowed with CPD relative to $\displaystyle k=(1,0,0)$.

As we mentioned in the previous subsection, the shape operator $\displaystyle S$ of $\displaystyle M$ can be non-diagonalizable. In this case, we can choose a pseudo-orthonormal frame field $\displaystyle\{e_{1},e_{2}\}$ of the tangent bundle such that $\displaystyle S$ has the matrix representation

[TABLE]

In this case, (2.3) becomes

[TABLE]

By a simple computation we obtain $\displaystyle k_{1}=0$. Thus $\displaystyle M$ is a flat, minimal B-scroll. It is well known that it must be congruent to the surface given by

[TABLE]

(See for example [13]). Hence, we have the following result.

Proposition 3.6**.**

Let $\displaystyle M$ be an oriented Lorentzian surface in $\displaystyle\mathbb{E}^{3}_{1}$ with non-diagonalizable shape operator. If $\displaystyle M$ is a surface endowed with a canonical principal direction relative to a space-like constant direction, then it is congruent to the surface given by (3.31)

Now, assume that $\displaystyle M$ is time-like and its shape operator $\displaystyle S$ is diagonalizable. Let $\displaystyle\{e_{1},e_{2}\}$ be a local orthonormal frame field of the tangent bundle of $\displaystyle M$ and $\displaystyle e_{1}$ is proportional to $\displaystyle U$. Since $\displaystyle N$ is space-like we have two cases for subject to casuality of $\displaystyle e_{1}$.

Case A. $\displaystyle e_{1}$ is a space-like vector. In this case, (2.3) implies

[TABLE]

Case B. $\displaystyle e_{1}$ is a time-like vector. In this case, (2.3) implies

[TABLE]

We have the following lemma which is the analogous of Lemma 3.1.

Lemma 3.7**.**

Let $\displaystyle M$ be a Lorentzian surface endowed with CPD relative to $\displaystyle k=(1,0,0)$ and $\displaystyle\{e_{1},e_{2}\}$ its principle directions such that $\displaystyle\langle k,e_{2}\rangle=0$. Then we have the following statements.

1. (1)

If $\displaystyle e_{1}$ is space-like, then the Levi-Civita connection $\displaystyle\nabla$ of $\displaystyle M$ is given by

[TABLE]

for a function $\displaystyle k_{2}$ satisfying

[TABLE] 2. (2)

If $\displaystyle e_{1}$ is time-like, then the Levi-Civita connection $\displaystyle\nabla$ of $\displaystyle M$ is given by

[TABLE]

and for a function $\displaystyle k_{2}$ satisfying

[TABLE] 3. (3)

In both cases $\displaystyle\theta$ satisfies (3.5) and the matrix representation shape operator $\displaystyle S$ is

[TABLE]

Proof.

We use exactly same way with the proof of Lemma 3.1. By considering (3.32) and (3.33), we get the statement (1) and (2) of the lemma, respectively and obtain (3.5) and (3.38) for both cases. ∎

The proof of the following lemma is similar to the proof of Lemma 3.3.

Lemma 3.8**.**

Let $\displaystyle M$ be a Lorentzian surface endowed with CPD relative to $\displaystyle k=(1,0,0)$ and $\displaystyle\{e_{1},e_{2}\}$ its principle directions such that $\displaystyle\langle k,e_{2}\rangle=0$. Then there exists a neighborhood $\displaystyle\mathcal{N}_{p}$ of $\displaystyle p$ on which $\displaystyle e_{1}=\partial_{s}$ and $\displaystyle\displaystyle e_{2}=\frac{1}{m}\partial_{t}$ for a smooth function $\displaystyle m$. Moreover, if $\displaystyle e_{1}$ is space-like then the induced metric of $\displaystyle\mathcal{N}_{p}$ becomes

[TABLE]

and $\displaystyle m$ satisfies

[TABLE]

On the other hand, if if $\displaystyle e_{1}$ is time-like then the induced metric of $\displaystyle\mathcal{N}_{m}$ becomes

[TABLE]

and $\displaystyle m$ satisfies

[TABLE]

Theorem 3.9**.**

Let $\displaystyle M$ be an oriented Lorentzian surface in $\displaystyle\mathbb{E}^{3}_{1}$ with diagonalizable shape operator. Then, $\displaystyle M$ is endowed with a canonical principal direction relative to a space-like, constant direction if and only if it is congruent to the surface given by one of the followings

1. (1)

A surface given by

[TABLE] 2. (2)

A surface given by

[TABLE]

for a constant $\displaystyle t_{0}$;

[TABLE] 3. (3)

A surface given by

[TABLE]

where $\displaystyle\gamma$ is

[TABLE]

for a function $\displaystyle\Psi\in C^{\infty}(M)$; 4. (4)

A surface given by

[TABLE]

for a constant $\displaystyle t_{0}$.

Proof.

In order to prove the necessary condition, we assume that $\displaystyle M$ is endowed with CPD relative to $\displaystyle k=(1,0,0)$. Let $\displaystyle x:M\rightarrow\mathbb{E}^{3}_{1}$ be an isometric immersion, $\displaystyle\{e_{1},e_{2};N\}$ the local orthonormal frame field described before Lemma 3.7, $\displaystyle k_{1},k_{2}$ principal curvatures of $\displaystyle M$ and $\displaystyle(s,t)$ a local coordinate system given in Lemma 3.8. We will consider two cases described above seperately.

Case A. $\displaystyle e_{1}$ is a space-like vector. In this case, we have (3.34)-(3.35), (3.39) and (3.40). Note that (3.40) and (3.35) turns into

[TABLE]

respectively.

By considering (3.47a) we obtain $\displaystyle k_{2}=\tan\theta\frac{m_{s}}{m}.$ Thus, (3.38) becomes

[TABLE]

Furthermore, by differentiating (3.47a) with respect to $\displaystyle s$ and using (3.47), we obtain

[TABLE]

Therefore, $\displaystyle m$ satisfies either

[TABLE]

Case A1. $\displaystyle m$ satisfies (3.49a). In this case, similar to the Case (1) in the proof of Theorem 3.4, we consider (3.34) and (3.48) to get

[TABLE]

Furthermore, considering (3.32) we have $\displaystyle\left\langle e_{1},k\right\rangle=\left\langle x_{s},k\right\rangle=\sin\theta$ and $\displaystyle\left\langle x_{t},k\right\rangle=0$. So we get

[TABLE]

for a $\displaystyle\mathbb{E}^{3}_{1}$-valued smooth function $\displaystyle\gamma=\left(0,\gamma_{2},\gamma_{3}\right)$. Also (3.32) and (3.50a) imply

[TABLE]

By considering (3.51) and $\displaystyle\langle x_{s},x_{s}\rangle=1$, we solve (3.52) and obtain

[TABLE]

for a smooth function $\displaystyle\varphi$. By a similar way in the Case (1) in the proof of Theorem 3.4, we could get $\displaystyle\varphi(t)=t$ and (3.43b) by considering (3.39) and (3.53). Furthermore, considering $\displaystyle\varphi(t)=t$ and (3.43b) in (3.53) we get (3.43a). Hence, we get the classification of surface in the case (1) of the Theorem 3.9.

Case A2. $\displaystyle m$ satisfies (3.49b). In this case,(3.39) turns into $\displaystyle g=ds^{2}-dt^{2}$, and (3.15) gives (3.23). Therefore, $\displaystyle x$ and $\displaystyle N$ satisfies

[TABLE]

A straightforward computation yields that $\displaystyle M$ is congruent to the surface given in Case (2) of the Theorem 3.9. Hence, the proof for the necessary condition is obtained.

Now, we would like to get the case (3) and the case (4) of the Theorem 3.9.

Case B. $\displaystyle e_{1}$ is a time-like vector. In this case, we have (3.36)-(3.37), (3.41) and (3.42). By a similar way to Case A we obtain

[TABLE]

Similar to the Case A, we obtain

[TABLE]

which yields that $\displaystyle m$ satisfies either

[TABLE]

for a smooth function $\displaystyle\Psi$ or (3.49b).

If $\displaystyle m$ satisfies (3.55), we use exactly the same way that we did in the Case A1 and obtain the Case (3) of the theorem. On the other hand, if $\displaystyle m(s,t)=1$, then we get the Case (4) of the theorem. Hence, the proof of the necessary condition is completed.

The proof of sufficient condition follows from a direct computation. ∎

Proposition 3.10**.**

A minimal surface in $\displaystyle\mathbb{E}^{3}_{1}$ endowed with CPD relative to a constant, space-like direction is either an open part of a plane or congruent to one of following two surface given below

1. (1)

A surface given by

[TABLE]

for a non-zero constant $\displaystyle c$. In this case, the angle function $\displaystyle\theta$ is

[TABLE] 2. (2)

A surface given by

[TABLE]

for a non-zero constant $\displaystyle c$. In this case, the angle function $\displaystyle\theta$ is

[TABLE]

Proof.

Let $\displaystyle M$ be a Lorentzian CPD surface and assume that it is not an open part of a plane. If $\displaystyle M$ is minimal, then Theorem 3.9 yields that $\displaystyle M$ is congruent to the one of surfaces given by (3.43) and (3.45).

Case 1. $\displaystyle M$ is congruent to the surface given by (3.43). Note that the shape operator $\displaystyle S$ of $\displaystyle M$ is (3.48) for the function $\displaystyle m$ satisfying (3.49a). Then the minimality condition $\displaystyle\mathrm{tr}S=0$ and (3.48) give

[TABLE]

which implies

[TABLE]

for a non-zero constant $\displaystyle c$ and $\displaystyle m=m(s)$. Therefore, (3.49a) give $\displaystyle\Psi=0$. So,

[TABLE]

By combining this equation with (3.60) we obtain (3.57). By a further computation, we obtain (3.56).

Case 2. $\displaystyle M$ is congruent to the surface given by (3.45). Note that the shape operator $\displaystyle S$ of $\displaystyle M$ is (3.54) for the function $\displaystyle m$ satisfying (3.55). In this case, the minimality condition $\displaystyle\mathrm{tr}S=0$ and (3.48) give

[TABLE]

By a similar way to Case 1, we obtain (3.58) and (3.59). ∎

3.2. CPD surfaces relative to a light-like, constant direction.

In this subsection we will consider surfaces endowed with CPD relative to the fixed vector $\displaystyle k=(1,0,1)$ which is light-like.

Theorem 3.11**.**

Let $\displaystyle M$ be an oriented surface in $\displaystyle\mathbb{E}^{3}_{1}$ with diagonalizable shape operator. Then, $\displaystyle M$ is endowed with a canonical principal direction relative to a light-like, constant direction if and only if it is congruent to the surface given by

[TABLE]

for some smooth functions $\displaystyle b,\gamma_{0}$, some constants $\displaystyle s_{0},t_{0}$ and $\displaystyle\varepsilon\in\{-1,1\}$ and a non-vanishing function $\displaystyle\phi$ whose derivative does not vanish. In this case, the tangential vector field $\displaystyle(1,0,1)^{T}=\phi(s)e_{1}$ is a principle direction of $\displaystyle M$ for a vector field $\displaystyle\langle e_{1},e_{1}\rangle=\varepsilon.$

Proof.

Let $\displaystyle N$ be the unit normal vector field of $\displaystyle M$ associated with its orientation and $\displaystyle x:M\rightarrow\mathbb{E}^{3}_{1}$ an isometric immersion. We put $\displaystyle\varepsilon=-\langle N,N\rangle$. Assume that $\displaystyle e_{1}$ is the unit tangent normal vector field proportional to tangential part of $\displaystyle k=(1,0,1)$ and $\displaystyle e_{2}$ is a unit space-like tangent vector field with $\displaystyle\langle e_{1},e_{2}\rangle=0$. Then, we have

[TABLE]

for a smooth function $\displaystyle\phi.$ Note that we have $\displaystyle\langle e_{1},e_{1}\rangle=\varepsilon$.

Now, in order to proof the necessary condition, we assume that $\displaystyle e_{1}$ is a principle direction of $\displaystyle M$ with corresponding principle curvature $\displaystyle k_{1}$. By a simple computation considering (3.62) we obtain

[TABLE]

whenever $\displaystyle X$ is tangent to $\displaystyle M$. Note that (3.63) for $\displaystyle X=e_{1}$ gives

[TABLE]

where $\displaystyle e_{2}$ is the other principle direction of $\displaystyle M$ with corresponding principle curvature $\displaystyle k_{2}$ and $\displaystyle\langle e_{2},e_{2}\rangle=1$. In addition, the second fundamental form of $\displaystyle M$ becomes

[TABLE]

Therefore, the Codazzi equation gives

[TABLE]

Note that, because of Remark 2.6, (3.64b) implies that $\displaystyle e_{1}(\phi)$ does not vanish on $\displaystyle M$.

Let $\displaystyle p\in M$. First, we would like to prove the following claim.

Claim 3.11.1**.**

There exists a neighborhood $\displaystyle\mathcal{N}_{p}$ of $\displaystyle p$ on which the induced metric of $\displaystyle M$ becomes

[TABLE]

for some smooth functions $\displaystyle a,b$ such that $\displaystyle e_{1}=\phi\partial_{s}$, $\displaystyle\displaystyle e_{2}=\frac{1}{a(t)s+b(t)}\partial_{t}$ and

[TABLE]

Proof of Claim 3.11.1. Note that we have $\displaystyle[e_{1},e_{2}]=k_{2}e_{2}$ because of (3.64a) and (3.64c). Therefore, (3.64d) implies $\displaystyle\displaystyle\left[\frac{1}{\phi}e_{1},Ge_{2}\right]=0$ for any function $\displaystyle G$ satisfying

[TABLE]

Therefore, there exists a local coordinate system $\displaystyle(s,t)$ such that $\displaystyle e_{1}=\phi\partial_{s}$ and $\displaystyle\displaystyle e_{2}=\frac{1}{G}\partial_{t}$. Thus, the induced metric of $\displaystyle M$ is

[TABLE]

Note that we have $\displaystyle k_{1}=k_{1}(s)$ and (3.68) because of (3.64b), (3.64d) and (3.66). In addition, the first equation in (3.66) and (3.69) give

[TABLE]

and

[TABLE]

respectively. Now, getting derivative of (3.69) implies

[TABLE]

By combining (3.72), (3.68) and (3.71) with (3.73), we obtain $\displaystyle\phi G_{ss}=0$ which yields $\displaystyle G=a(t)s+b(t)$ for some smooth functions $\displaystyle a,b$. Therefore, (3.70) becomes (3.67).

Hence, the proof of the Claim 3.11.1 is completed. $\displaystyle\blacksquare$

Now, let $\displaystyle s,t$ be local coordinates described in the Claim 3.11.1. Note that we have

[TABLE]

Moreover, (3.64a) and (3.65) imply

[TABLE]

from which we get

[TABLE]

By combining (3.74) and (3.75) with (3.62) we get

[TABLE]

which yields

[TABLE]

By integrating this equation and considering (3.74), we get

[TABLE]

for an $\displaystyle\mathbb{E}^{3}_{1}$-valued smooth function $\displaystyle\gamma$. As $\displaystyle\langle x_{s},x_{s}\rangle=\frac{\varepsilon}{\phi^{2}}$, we have

[TABLE]

Since $\displaystyle\phi$ is not constant, the above equation implies $\displaystyle\langle(1,0,1),\gamma(t)\rangle=\varepsilon$ and $\displaystyle\langle\gamma(t),\gamma(t)\rangle=0.$ By considering these equations, we obtain

[TABLE]

for a smooth function $\displaystyle\gamma_{0}$. Therefore (3.76) becomes

[TABLE]

(3.74) and (3.75) imply

[TABLE]

By integrating (3.77), we obtain

[TABLE]

for a smooth $\displaystyle\mathbb{E}^{3}_{1}$-valued function $\displaystyle\Gamma$. By combining (3.79) and (3.78c), we get

[TABLE]

from which we conclude

[TABLE]

and

[TABLE]

By combining the last equation with (3.79), we obtain (3.61). Hence, the proof of the necessary condition is completed.

Conversely, consider the surface $\displaystyle M$ given by (3.61) whose derivative does not vanish. A direct computation yields that unit normal of $\displaystyle M$ is

[TABLE]

and the principle curvatures of $\displaystyle M$ are

[TABLE]

Moreover, we have $\displaystyle\langle e_{2},(1,0,1)\rangle=0$ which yields that $\displaystyle(1,0,1)^{T}$ is a principle direction. Hence the proof of sufficient condition is completed. ∎

By considering the surface proof of Theorem 3.11, we obtain the following proposition.

Proposition 3.12**.**

Let $\displaystyle M$ be the surface given by (3.61). Then, the matrix representation of the shape operator $\displaystyle S$ of $\displaystyle M$ with respect to $\displaystyle\{e_{1},e_{2}\}$ is

[TABLE]

where $\displaystyle e_{1},\ e_{2}$ are vector fields given by (3.80)

From Proposition 3.12 we conclude the following characterization results.

Corollary 3.13**.**

A flat surface with diagonalizable shape operator in $\displaystyle\mathbb{E}^{3}_{1}$ endowed with CPD relative to a light-like direction is congruent to the surface given by

[TABLE]

for some constants $\displaystyle s_{0},\varepsilon\in\{-1,1\}$ and a non-vanishing function $\displaystyle\phi$ whose derivative does not vanish.

Corollary 3.14**.**

A minimal (resp. maximal) surface with diagonalizable shape operator in $\displaystyle\mathbb{E}^{3}_{1}$ endowed with CPD relative to a light-like direction is congruent to the surface given by

[TABLE]

for some constants $\displaystyle c_{2}>0,c_{1}$ with $\displaystyle\varepsilon=-1$ (resp. $\displaystyle\varepsilon=1$).

Acknowledgments

This paper is a part of PhD thesis of the first named author who is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) as a PhD scholar.