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Article

# Generalized Null 2-Type Surfaces in Minkowski 3-Space

by
Dae Won Yoon
1,
Dong-Soo Kim
2,
Young Ho Kim
3 and
Jae Won Lee
1,*
1
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
2
Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
3
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
*
Author to whom correspondence should be addressed.
Symmetry 2017, 9(1), 14; https://doi.org/10.3390/sym9010014
Submission received: 3 December 2016 / Revised: 16 January 2017 / Accepted: 16 January 2017 / Published: 20 January 2017

## Abstract

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

## 1. Introduction

Let $x : M ⟶ E m$ be an isometric immersion of an n-dimensional connected submanifold M in an m-dimensional Euclidean space $E m$. Denote by $H$ and Δ, respectively, the mean curvature vector field and the Laplacian operator with respect to the induced metric on M induced from that of $E m$. Then, it is well known as
$Δ x = − n H .$
By using (1), Takahashi [1] proved that minimal submanifolds of a hypersphere of $E m$ are constructed from eigenfunctions of Δ with one eigenvalue λ (≠0). In [2,3], Chen initiated the study of submanifolds in $E m$ that are constructed from harmonic functions and eigenfunctions of Δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:
$x = x 0 + x q , Δ x 0 = 0 , Δ x q = λ x q$
for some non-constant maps $x 0$ and $x q$, where λ is a nonzero constant. A submanifold satisfying (2) is said to be of null 2-type [3]. From the definition of null 2-type submanifolds and (1), it follows that the mean curvature vector field $H$ satisfies the following condition:
$Δ H = λ H .$
A result from [4] states that a surface in the Euclidean space $E 3$ satisfying (3) is either a minimal surface or an open part of an ordinary sphere or a circular cylinder. Ferrández and Lucas [5] extended it to the Lorentzian case. They proved that the surface satisfying (3) is either a minimal surface or an open part of a Lorentz circular cylinder, a hyperbolic cylinder, a Lorentz hyperbolic cylinder, a hyperbolic space, a de Sitter space or a B-scroll. Afterwards, several authors studied null 2-type submanifolds in the (pseudo-)Euclidean space [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].
Now, we will give a generalization of null 2-type submanifolds in the Minkowski space. It is well known that a Lorentz circular cylinder $S 1 ( r ) × R 1 1$ is a null 2-type surface in the Minkowski 3-space $L 3$ satisfying $Δ H = 1 r 2 H$, where $S 1 ( r )$ is a circle with radius r and $R 1 1$ is a Lorentz straight line. However, the following surface has another property as follows: a parametrization
$x ( s , t ) = 1 4 s 2 − 1 2 ln s , 1 4 s 2 + 1 2 ln s , t$
is a cylindrical surface in $L 3$. On the other hand, the mean curvature vector field $H$ of the surface is given by
$H = − 1 4 − 1 4 s 2 , − 1 4 + 1 4 s 2 , 0$
and the surface satisfies
$Δ H = − 6 s 2 H + ( 1 4 , 1 4 , 0 ) .$
Next, we consider another surface with a parametrization
$x ( s , t ) = 1 2 s 2 t + t , s t , 1 2 s 2 t .$
The surface is a conical surface in $L 3$, and it satisfies the following equation for the mean curvature vector $H$
$Δ H = 1 t 2 H + 1 2 t 3 ( 1 , 0 , 1 ) .$
Thus, based on the above examples, we give the definition:
Definition 1.
A submanifold M of the Minkowski space is said to be of generalized null 2-type if it satisfies the condition
$Δ H = f H + g C$
for some smooth functions $f , g$ and a constant vector $C$. In particular, if the functions f and g are equal to each other in (4), then the submanifold M is called of generalized null 2-type of the first kind and of the second kind otherwise.
In [22], the authors recently classified generalized null 2-type flat surfaces in the Euclidean 3-space. Conical surfaces, cylindrical surfaces or tangent developable surfaces are developable surfaces (or flat surfaces) as ruled surfaces in the Minkowski 3-space $L 3$. In this paper, we study developable surfaces in $L 3$ and completely classify generalized null 2-type developable surfaces, and give some examples. In addition, we investigate null scrolls in the Minkwoski 3-space $L 3$ satisfying the condition (4).

## 2. Preliminaries

The Minkowski 3-space $L 3$ is a real space $R 3$ with the standard flat metric given by
$〈 , 〉 = − d x 1 2 + d x 2 2 + d x 3 2 ,$
where $( x 1 , x 2 , x 3 )$ is a rectangular coordinate system of $R 3$. An arbitrary vector $x$ of $L 3$ is said to be space-like if $〈 x , x 〉 > 0$ or $x = 0$, time-like if $〈 x , x 〉 < 0$ and null if $〈 x , x 〉 = 0$ and $x ≠ 0$. A time-like or null vector in $L 3$ is said to be causal. Similarly, an arbitrary curve $γ = γ ( s )$ is space-like, time-like or null if all of its tangent vectors $γ ′ ( s )$ are space-like, time-like or null, respectively. From now on, the “prime” means the partial derivative with respect to the parameter s unless mentioned otherwise.
We now put a 2-dimensional space form in $L 3$ as follows:
$Q 2 ( ε ) = S 1 2 = { ( x 1 , x 2 , x 3 ) ∈ L 3 | − x 1 2 + x 2 2 + x 3 2 = 1 } , if ε = 1 ; H 2 = { ( x 1 , x 2 , x 3 ) ∈ L 3 | − x 1 2 + x 2 2 + x 3 2 = − 1 } , if ε = − 1 .$
We call $S 1 2$ and $H 2$ the de-Sitter space and the hyperbolic space, respectively.
Let $γ : I ⟶ L 3$ be a space-like or time-like curve in the Minkowski 3-space $L 3$ parameterized by its arc-length s. Denote by ${ t , n , b }$ the Frenet frame field along $γ ( s )$.
If $γ ( s )$ is a space-like curve in $L 3$, the Frenet formulae of $γ ( s )$ are given by [23]:
$γ ′ ( s ) = t ( s ) , t ′ ( s ) = κ ( s ) n ( s ) , n ′ ( s ) = − ε κ ( s ) t ( s ) + τ ( s ) b ( s ) , b ′ ( s ) = ε τ ( s ) n ( s ) ,$
where $〈 t , t 〉 = 1 , 〈 n , n 〉 = ε ( = ± 1 ) , 〈 b , b 〉 = − ε .$ Here, the functions $κ ( s )$ and $τ ( s )$ are the curvature function and the torsion function of a space-like curve $γ ( s )$, respectively.
If $γ ( s )$ is a time-like curve in $L 3$, the Frenet formulae of $γ ( s )$ are given by [23]:
$γ ′ ( s ) = t ( s ) , t ′ ( s ) = κ ( s ) n ( s ) , n ′ ( s ) = − κ ( s ) t ( s ) + τ ( s ) b ( s ) , b ′ ( s ) = − τ ( s ) n ( s ) ,$
where $〈 t , t 〉 = − 1 , 〈 n , n 〉 = 〈 b , b 〉 = 1 .$ Here $κ ( s )$ and $τ ( s )$ are the curvature function and the torsion function of a time-like curve $γ ( s )$, respectively.
If $γ ( s )$ is a space-like or time-like pseudo-spherical curve parametrized by arc-length s in $Q 2 ( ε )$, let $t ( s ) = γ ′ ( s )$ and $g ( s ) = γ ( s ) × γ ′ ( s )$. Then, we have a pseudo-orthonormal frame ${ γ ( s ) , t ( s ) , g ( s ) }$ along $γ ( s )$. It is called the pseudo-spherical Frenet frame of the pseudo-spherical curve $γ ( s )$. If γ is a space-like curve, then the vector $g$ is time-like when γ is on $S 1 2$, and the vector $g$ is space-like when γ is on $H 2$. Similarly, if the curve γ is time-like, then the vector $g$ is space-like. The following theorem can be easily obtained.
Theorem 1.
([24,25]) Under the above notations, we have the following pseudo-spherical Frenet formulae of γ:
(1)
If γ is a pseudo-spherical space-like curve,
$γ ′ ( s ) = t ( s ) , t ′ ( s ) = − ε γ ( s ) − ε κ g ( s ) g ( s ) , g ′ ( s ) = − κ g ( s ) t ( s ) .$
(2)
If γ is a pseudo-spherical time-like curve,
$γ ′ ( s ) = t ( s ) , t ′ ( s ) = γ ( s ) + κ g ( s ) g ( s ) , g ′ ( s ) = κ g ( s ) t ( s ) .$
The function $κ g ( s )$ is called the geodesic curvature of the pseudo-spherical curve γ.
Now, we define a ruled surface M in $L 3$. Let I and J be open intervals in the real line $R$. Let $α = α ( s )$ be a curve in $L 3$ and $β = β ( s )$ a vector field along α with $α ′ ( s ) × β ( s ) ≠ 0$ for every $s ∈ J$. Then, a ruled surface M is defined by the parametrization given as follows:
$x = x ( s , t ) = α ( s ) + t β ( s ) , s ∈ J , t ∈ I .$
For such a ruled surface, α and β are called the base curve and the director curve respectively. In particular, if β is constant, the ruled surface is said to be cylindrical, and if it is not so, it is called non-cylindrical. Furthermore, we have five different ruled surfaces according to the characters of the base curve α and the director curve β as follows: if the base curve α is space-like or time-like, then the ruled surface M is said to be of type $M +$ or type $M −$, respectively. In addition, the ruled surface of type $M +$ can be divided into three types. In the case that β is space-like, it is said to be of type $M + 1$ or $M + 2$ if $β ′$ is non-null or null, respectively. When β is time-like, $β ′$ is space-like because of the causal character. In this case, M is said to be of type $M + 3$. On the other hand, for the ruled surface of type $M −$, it is also said to be of type $M − 1$ or $M − 2$ if $β ′$ is non-null or null, respectively [26].
However, if the base curve α is a light-like curve and the vector field β along α is a light-like vector field, then the ruled surface M is called a null scroll. In particular, a null scroll with Cartan frame is said to be a B-scroll [27]. It is also a time-like surface.
A non-degenerate surface in $L 3$ with zero Gaussian curvature is called a developable surface. The developable surfaces in $L 3$ are the same as in the Euclidean space, and they are planes, conical surfaces, cylindrical surfaces and tangent developable surfaces [13].

## 3. Generalized Null 2-Type Cylindrical Surfaces

For a surface in the Minkowski 3-space $L 3$, the next lemma is well known and useful.
Lemma 1.
([16]) Let M be an oriented surface of $L 3$. Then, the Laplacian of the mean curvature vector field $H$ of M is given by
$Δ H = 2 A ( ∇ H ) + ε ∇ H 2 + ( Δ H + ε H | A | 2 ) N ,$
where ε is the sign of the unit normal vector N of the surface M and $∇ H$, A are the gradient of the mean curvature H and the shape operator of M, respectively.
Theorem 2.
All cylindrical surfaces in $L 3$ are of generalized null 2-type.
Proof.
Let M be a cylindrical ruled surface in the Minkowski 3-space $L 3$ of type $M + 1$, $M − 1$ or $M + 3$. Then, M is parameterized by
$x ( s , t ) = α ( s ) + t β ,$
where the base curve $α ( s )$, which is a space-like or time-like curve with the arc-length parameter s, lies in a plane with a space-like or time-like unit normal vector β that is the director of M, that is, $〈 β , β 〉 = ε 1 ( = ± 1 )$ and $〈 α ′ ( s ) , α ′ ( s ) 〉 = ε 2 ( ± 1 )$.
Now, we take a local pseudo-orthonormal frame ${ e 1 , e 2 , e 3 }$ on $L 3$ such that $e 1 = ∂ ∂ t$ and $e 2 = ∂ ∂ s$ are tangent to M, and $e 3$ normal to M. It follows that the Levi–Civita connection $∇ ˜$ of $L 3$ is expressed as
$∇ ˜ e 1 e 1 = ∇ ˜ e 1 e 2 = ∇ ˜ e 2 e 1 = 0 , ∇ ˜ e 2 e 2 = ε 3 κ ( s ) e 3 , ∇ ˜ e 1 e 3 = 0 , ∇ ˜ e 2 e 3 = − ε 2 κ ( s ) e 2 ,$
where $κ ( s )$ is the curvature function of $α ( s )$ and $ε 3 ( = ± 1 )$ is the sign of $e 3$. From this, the mean curvature vector field $H$ of M is given by
$H = ε 2 κ ( s ) 2 e 3$
and the Laplacian $Δ H$ of $H$ is expressed as
$Δ H = 3 2 ε 1 ε 2 κ ( s ) κ ′ ( s ) e 2 − 1 2 ( κ 3 ( s ) + ε 1 κ ″ ( s ) ) e 3 .$
Suppose that M is of generalized null 2-type. With the help of (4) and (12), we obtain the following equations:
$g C 1 = 0 ,$
$3 2 ε 1 ε 2 κ ( s ) κ ′ ( s ) = g C 2 ,$
$1 2 ε 1 ε 2 κ 3 ( s ) + 1 2 ε 2 κ ″ ( s ) = − 1 2 ε 1 κ ( s ) f + g C 3 ,$
where $C = ε 1 C 1 e 1 + ε 2 C 2 e 2 − ε 1 ε 2 C 3 e 3$ with $C 1 = 〈 C , e 1 〉$, $C 2 = 〈 C , e 2 〉$ and $C 3 = 〈 C , e 3 〉$. In this case, $C 1$ is a constant, and $C 2$, $C 3$ are functions of the variable s.
If g is identically zero, then, from (14), the curvature $κ ( s )$ is constant, and from (15), the function f is constant, say λ. Thus, M satisfies $Δ H = λ H$, that is, it is either a Euclidean plane, a Minkowski plane, a Lorentz circular cylinder $S 2 × R 1 1$, a hyperbolic cylinder $H 1 × R$ or a Lorentz hyperbolic cylinder $S 1 1 × R$ according to [16].
We now assume that $g ≠ 0$. It follows from (13) that $C 1 = 0$. By using (10), we can show that the component functions of $C$ satisfy the following equations:
which yield $ε 2 C 2 2 ( s ) − ε 1 ε 2 C 3 2 ( s ) = η d 0 2$ for some nonzero constant $d 0$, where $η = 〈 C , C 〉 .$
Case 1: If M is of type $M + 3$, then $ε 1 = − 1$, $ε 2 = 1$ and $η = 1$. We may put from (16)
$C 2 ( s ) = d 0 sin θ ( s ) , C 3 ( s ) = d 0 cos θ ( s ) ,$
where $θ ( s ) = κ 0 + ∫ κ ( s ) d s$ for some constant $κ 0$. Therefore, the constant vector $C$ becomes
$C = d 0 sin θ ( s ) e 2 + d 0 cos θ ( s ) e 3 .$
Combining (14), (15) and (17), one also gets
$g = − 3 κ ( s ) κ ′ ( s ) 2 d 0 csc θ ( s ) , f = κ ″ ( s ) κ ( s ) − κ 2 ( s ) + 3 κ ′ ( s ) cot θ ( s ) .$
Thus, the mean curvature vector field $H$ of the cylindrical surface $M 3 +$ satisfies
$Δ H = f H + g C ,$
where $f , g$ and $C$ are given in (18) and (19), respectively.
Case 2: Let M be of type $M + 1$. In this case, $ε 1 = 1$, $ε 2 = 1$ and the constant vector $C$ is space-like, time-like or null.
First of all, we consider the constant vector $C$ is non-null. Then, from (16), we may put
$C 2 ( s ) = d 0 cosh θ ( s ) , C 3 = d 0 sinh θ ( s ) if η = 1 , C 2 ( s ) = d 0 sinh θ ( s ) , C 3 = d 0 cosh θ ( s ) if η = − 1 ,$
where $θ ( s ) = − ∫ κ ( s ) d s + κ 0$ with a constant $κ 0$.
By using (14), (15) and (17), the functions $f ( s )$ and $g ( s )$ are determined by
$f ( s ) = − κ ″ ( s ) κ ( s ) + κ 2 ( s ) + 3 κ ′ ( s ) tanh θ ( s ) , g ( s ) = 3 κ ( s ) κ ′ ( s ) 2 d 0 cosh θ ( s ) if η = 1 , f ( s ) = − κ ″ ( s ) κ ( s ) − κ 2 ( s ) + 3 κ ′ ( s ) coth θ ( s ) , g ( s ) = 3 κ ( s ) κ ′ ( s ) 2 d 0 sinh θ ( s ) if η = − 1 .$
Thus, for the non-null constant vector $C$, the cylindrical surface $M 1 +$ is of generalized null 2-type, that is, it satisfies
$Δ H = f H + g C ,$
where $f , g$ and $C$ are given by (20) and (21), respectively.
Next, let the constant vector $C$ be null, that is, $η = 0$. Then, we get
$C 2 ( s ) = ± C 3 ( s ) .$
We will consider the case $C 2 ( s ) = C 3 ( s )$. It follows from (16) $C 2 ( s ) = e θ ( s )$, where $θ ( s ) = − ∫ κ ( s ) d s + κ 0$ for some constant $κ 0$. In this case, we have
$f ( s ) = − κ ″ ( s ) κ ( s ) − κ 2 ( s ) + 3 κ ′ ( s ) , g ( s ) = 3 2 e − θ ( s ) κ ( s ) κ ′ ( s )$
and, for the null constant vector $C$, the surface satisfies the condition $Δ H = f H + g C$.
Case 3: Let M be of type $M − 1$, that is, $ε 1 = 1$, $ε 2 = − 1$. In this case, the constant vector $C$ is space-like, time-like or null.
Applying the same method as in Case 2, the functions $f ( s )$ and $g ( s )$ are determined by
$f ( s ) = κ ″ ( s ) κ ( s ) + κ 2 ( s ) + 3 κ ′ ( s ) coth θ ( s ) , g ( s ) = 3 κ ( s ) κ ′ ( s ) 2 d 0 sinh θ ( s ) if η = 1 , f ( s ) = κ ″ ( s ) κ ( s ) + κ 2 ( s ) + 3 κ ′ ( s ) tanh θ ( s ) , g ( s ) = 3 κ ( s ) κ ′ ( s ) 2 d 0 cosh θ ( s ) if η = − 1 , f ( s ) = κ ″ ( s ) κ ( s ) + κ 2 ( s ) + 3 κ ′ ( s ) , g ( s ) = 3 2 e − θ ( s ) κ ( s ) κ ′ ( s ) if η = 0 ,$
and the component functions of $C$ are given by
$C 2 ( s ) = d 0 sinh θ ( s ) , C 3 ( s ) = d 0 cosh θ ( s ) , if η = 1 , C 2 ( s ) = d 0 cosh θ ( s ) , C 3 ( s ) = d 0 sinh θ ( s ) , if η = − 1 , C 2 ( s ) = ± C 3 ( s ) , if η = 0 ,$
where $θ ( s ) = ∫ κ ( s ) d s + κ 0$ for some constant $κ 0$.
Thus, from Cases 1, 2 and 3, Theorem 2 is proved.  ☐
Example 1.
We consider a surface defined by
$x ( s , t ) = 1 4 s 2 − 1 2 ln s , 1 4 s 2 + 1 2 ln s , t .$
This parametrization is a cylindrical ruled surface of type $M + 1$. In this case, the mean curvature vector field $H$ of the surface is given by
$H = − 1 4 − 1 4 s 2 , − 1 4 + 1 4 s 2 , 0 .$
By a direct computation, the Laplacian $Δ H$ of the mean curvature vector field $H$ becomes
$Δ H = 3 2 s 4 , − 3 2 s 4 , 0 ,$
and it can be rewritten in terms of the mean curvature vector field $H$ and a constant vector $C$ as follows:
$Δ H = − 6 s 2 ( H + C ) ,$
where $C = ( 1 4 , 1 4 , 0 )$ is a null vector. Thus, the cylindrical ruled surface defined by (25) is a generalized null 2-type surface of the first kind.
Remark 1.
A cylindrical surface in $L 3$ generated by the base curve $α ( s )$ with the curvature $κ ( s ) = 1 s$ and a constant director β is a generalized null 2-type surface of the first kind if the constant vector $C$ is null.

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

In this section, we classify non-cylindrical flat surfaces satisfying
$Δ H = f H + g C .$
It is well-known that a non-cylindrical flat surface in the Minkowski 3-space $L 3$ is an open part of a conical surface or a tangent developable surface.
First of all, we consider a conical surface M in $L 3$. Then, we may give the parametrization of M by
$x ( s , t ) = α 0 + t β ( s ) , s ∈ I , t > 0 ,$
such that $〈 β ′ ( s ) , β ′ ( s ) 〉 = ε 1$ and $〈 β ( s ) , β ( s ) 〉 = ε 2$, where $α 0$ is a constant vector. We take the orthonormal tangent frame ${ e 1 , e 2 }$ on M such that $e 1 = 1 t ∂ ∂ s$ and $e 2 = ∂ ∂ t$. The unit normal vector of M is given by $e 3 = e 1 × e 2$. By the Gauss and Weingarten formulas, we have
$∇ ˜ e 1 e 1 = − ε 1 ε 2 t e 2 + ε 1 ε 2 κ g ( s ) t e 3 , ∇ ˜ e 1 e 2 = 1 t e 1 , ∇ ˜ e 2 e 1 = ∇ ˜ e 2 e 2 = 0 , ∇ ˜ e 1 e 3 = ε 1 κ g ( s ) t e 1 , ∇ ˜ e 2 e 3 = 0 ,$
where $κ g ( s ) = 〈 β ( s ) , β ′ ( s ) × β ″ ( s ) 〉$, which is the geodesic curvature of the pseudo-spherical curve $β ( s )$ in $Q 2 ( ε )$. From (27), the mean curvature vector field $H$ of M is given by
$H = − ε 1 κ g ( s ) 2 t e 3 ,$
and the Laplacian $Δ H$ of the mean curvature vector field $H$ is expressed as
$Δ H = 3 ε 1 2 t 3 κ g ( s ) κ g ′ ( s ) e 1 − ε 2 2 t 3 κ g 2 ( s ) e 2 + 1 2 t 3 κ g ″ ( s ) + ε 2 2 t 3 κ g 3 ( s ) + ε 1 ε 2 2 t 3 κ g ( s ) e 3 .$
Suppose that $κ g$ is constant. If $κ g = 0$, by a rigid motion, the pseudo-spherical curve $β ( s )$ in $Q 2 ( ε )$ lies on $y z$-plane or $x z$-plane. Thus M is an open part of a Euclidean plane or a Minkowski plane. If $κ g$ is a non-zero constant, from (27), we can obtain by a straightforward computation
$β ‴ ( s ) = ε 2 ( κ g 2 ( s ) − ε 1 ) β ′ ( s ) .$
Case 1: $ε 2 ( κ g 2 ( s ) − ε 1 ) = k 2$ for some real number k.
Let $ε 1 = 1$. Without loss of generality, we may assume $β ′ ( 0 ) = ( 0 , 1 , 0 )$. Thus, $β ‴ ( s ) = k 2 β ′ ( s )$ implies
$β ′ ( s ) = ( B 1 sinh k s , cosh k s + B 2 sinh k s , B 3 sinh k s )$
for some constants $B 1 , B 2$ and $B 3$. Since $ε 1 = 1$, we have $B 1 2 − B 3 2 = 1$ and $B 2 = 0$. From this, we can obtain
$β ( s ) = B 1 k cosh k s + D 1 , 1 k sinh k s , B 3 k cosh k s + D 3$
for some constants $D 1 , D 3$ satisfying $D 3 2 − D 1 2 = 1 k 2 + ε 2$, $B 1 D 1 = B 3 D 3$ and $B 1 2 − B 3 2 = 1$. We now change the coordinates by $x ¯ , y ¯ , z ¯$ such that $x ¯ = B 1 x − B 3 z$, $y ¯ = y$, $z ¯ = − B 3 x + B 1 z$, that is,
$x ¯ y ¯ z ¯ = B 1 0 − B 3 0 1 0 − B 3 0 B 1 x y z .$
With respect to the coordinates $( x ¯ , y ¯ , z ¯ )$, $β ( s )$ turns into
$β ( s ) = 1 k cosh k s , 1 k sinh k s , D$
for a constant $D = B 1 D 3 − B 3 D 1$ with $D 2 = 1 k 2 + ε 2$. Thus, up to a rigid motion M has the parametrization of the form
$x ( s , t ) = α 0 + t 1 k cosh k s , 1 k sinh k s , D .$
We call such a surface a hyperbolic conical surface of the first kind, and it satisfies
$Δ H = ε 2 ( 1 − D 2 − k 2 ) k 2 t 2 H + ε 2 D ( 1 − D 2 k 2 ) 2 k 4 t 3 ( 0 , 0 , 1 ) .$
Next, let $( ε 1 , ε 2 ) = ( − 1 , 1 )$. We now consider a initial condition $β ′ ( 0 ) = ( 1 , 0 , 0 )$ of the ordinary differential equation (ODE) (30). Quite similarly as we did, we obtain
$β ( s ) = 1 k sinh k s , B 2 k cosh k s + D 2 , B 3 k cosh k s + D 3 ,$
satisfying $B 2 2 + B 3 2 = 1 , B 2 D 2 + B 3 D 3 = 0$ and $D 2 2 + D 3 2 = 1 − 1 k 2$.
If we adopt the coordinates’ transformation,
$x ¯ y ¯ z ¯ = 1 0 0 0 B 2 B 3 0 − B 3 B 2 x y z .$
With respect to the new coordinates $( x ¯ , y ¯ , z ¯ )$, the vector $β ( s )$ becomes
$β ( s ) = 1 k sinh k s , 1 k cosh k s , D ,$
where $D = B 2 D 3 − B 3 D 2$ with $D 2 = 1 − 1 k 2$. We call such a surface generated by (33) a hyperbolic conical surface of the second kind and it satisfies
$Δ H = 1 + D 2 + k 2 k 2 t 2 H + D ( 1 + k 2 D 2 ) 2 k 4 t 3 ( 0 , 0 , 1 ) .$
Case 2: $ε 2 ( κ g 2 ( s ) − ε 1 ) = − k 2$ for some real number k.
Let $ε 1 = 1$. We may give the initial condition by $β ′ ( 0 ) = ( 0 , 1 , 0 )$ for the differential equation $β ‴ ( s ) + k 2 β ′ ( s ) = 0$. Under such an initial condition, a vector field $β ( s )$ is given by
$β ( s ) = − B 1 k cos k s + D 1 , 1 k sin k s , − B 3 k cos k s + D 3 ,$
where $B 1 , B 3 , D 1$ and $D 3$ are some constants satisfying $B 3 2 − B 1 2 = 1$, $B 1 D 1 = B 3 D 3$ and $D 1 2 − D 3 2 = 1 k 2 − ε 2$. If we take another coordinate system $( x ¯ , y ¯ , z ¯ )$ such that
$x ¯ = − B 3 x + B 1 z , y ¯ = y , z ¯ = B 1 x − B 3 z ,$
then a vector $β ( s )$ takes the form
$β ( s ) = D , 1 k sin k s , 1 k cos k s ,$
where $D = B 1 D 3 − B 3 D 1$ satisfying $D 2 = 1 k 2 − ε 2$. We call such a surface generated by (34) an elliptic conical surface and it satisfies
$Δ H = ε 1 − ε 1 D 2 − k 2 k 2 t 2 H − D ( k 2 D 2 − 1 ) 2 k 4 t 3 ( 1 , 0 , 0 ) .$
Case of $ε 1 = − 1$ gives $ε 2 = − 1$. It is impossible by the causal character of Lorentz geometry.
Case 3: $κ g 2 ( s ) − ε 1 = 0 .$
In this case, $κ g 2 ( s ) = 1$, in other words, $ε 1 = 1$, which implies by using (27) $〈 β ″ ( s ) , β ″ ( s ) 〉 = 0$. Since $β ″ ( s )$ is a constant vector by (30), we may put $β ″ ( s ) = ( d 1 , d 2 , d 3 )$ for some constants $d 1$, $d 2$, $d 3$ satisfying $− d 1 2 + d 2 2 + d 3 2 = 0$ and so $β ′ ( s ) = ( d 1 s + k 1 , d 2 s + k 2 , d 3 s + k 3 )$ for some constants $k 1 , k 2$ and $k 3$. Since $〈 β ′ ( s ) , β ′ ( s ) 〉 = ε 1 = 1$, we may set $( k 1 , k 2 , k 3 ) = ( 0 , 1 , 0 )$ up to an isometry and hence $β ( s ) = ( d 1 2 s 2 + c 1 , d 2 2 s 2 + s + c 2 , d 3 2 s 2 + c 3 )$ for some constants $c 1 , c 2$ and $c 3$. However, $〈 β ( s ) , β ( s ) 〉 = ε 2$ implies $d 2 = c 2 = 0$ and $d 1 2 = d 3 2$, $− c 1 2 + c 3 2 = ε 2$, $− d 1 c 1 + d 3 c 3 + 1 = 0$. Thus, $β ( s )$ takes the form
$β ( s ) = d 1 2 s 2 + c 1 , s , d 3 2 s 2 + c 3 .$
We call such a surface generated by (35) a quadric conical surface.
As shown in the Introduction, a quadric conical surface is of generalized null 2-type of the first kind. Let us suppose that $κ g$ is a non-constant, i.e., $κ g ′ ≠ 0$ on an open interval. Suppose that M is of generalized null 2-type, that is, M satisfies the condition (4). Then, we have the following equations:
$3 κ g ( s ) κ g ′ ( s ) 2 t 3 = g C 1 ,$
$− κ g 2 ( s ) 2 t 3 = g C 2 ,$
$− 1 2 t 3 ( ε 1 ε 2 κ g ″ ( s ) + ε 1 κ g 3 ( s ) + κ g ( s ) ) = ε 2 κ g ( s ) 2 t f + g C 3 ,$
where $C = ε 1 C 1 e 1 + ε 2 C 2 e 2 − ε 1 ε 2 C 3 e 3$ with $C 1 = 〈 C , e 1 〉$, $C 2 = 〈 C , e 2 〉$ and $C 3 = 〈 C , e 3 〉$. Since $e 1 = β ′ ( s )$, $e 2 = β ( s )$ and $e 3 = β ′ ( s ) × β ( s )$, the component functions $C i$ $( i = 1 , 2 , 3 )$ of $C$ depend only on variable s. Let us differentiate $C 1$, $C 2$ and $C 3$ covariantly with respect to $e 1$. Then, from (27), we have the following equations:
$C 1 ′ ( s ) + ε 1 ε 2 C 2 ( s ) − ε 1 ε 2 κ g ( s ) C 3 ( s ) = 0 ,$
$C 2 ′ ( s ) − C 1 ( s ) = 0 ,$
$C 3 ′ ( s ) − ε 1 κ g ( s ) C 1 ( s ) = 0 .$
Combining (36) and (37), and using (40), we have
$C 1 = − 3 c κ g ′ κ g 4 and C 2 = c κ g 3 ,$
where c is a constant of integration.
Together with (37) and (42), we can find
$g = − κ g 5 2 c t 3 .$
Substituting (42) into (39), we get
$C 3 = c ( κ g 2 − 3 ε 1 ε 2 κ g κ g ″ + 12 ε 1 ε 2 κ g ′ 2 ) κ g 6 .$
Then, (38) and (44) lead to
$f = − ε 2 t 2 κ g 2 ( 4 ε 1 ε 2 κ g κ g ″ − 12 ε 1 ε 2 κ g ′ 2 + ε 1 κ g 4 ) .$
Furthermore, it follows from (41) and (42) that
$C 3 ′ = − 3 ε 1 c κ g ′ κ g 3$
and its solution is given by
$C 3 = 3 ε 1 c 2 κ g 2 + a 1$
for some constant $a 1$.
Combining (44) and (46), the geodesic curvature $κ g$ satisfies the following equation:
$κ g ″ − 4 κ g κ g ′ 2 − 1 3 ε 1 ε 2 κ g + 1 2 ε 2 κ g 3 + a 1 3 c ε 1 ε 2 κ g 5 = 0 .$
To solve the ODE, we put $p = κ g ′$. Then, (47) can be written of the form
$d p d κ g − 4 κ g p = 1 p 1 3 ε 1 ε 2 κ g − 1 2 ε 2 κ g 3 − a 1 3 c ε 1 ε 2 κ g 5 ,$
and it is a Bernoulli differential equation. Thus, the solution is given by
$p = ± κ g a 2 κ g 6 + a 1 3 c ε 1 ε 2 κ g 4 + 1 4 ε 2 κ g 2 − 1 9 ε 1 ε 2 1 2 ,$
which is equivalent to
$κ g − 1 a 2 κ g 6 + a 1 3 c ε 1 ε 2 κ g 4 + 1 4 ε 2 κ g 2 − 1 9 ε 1 ε 2 − 1 2 d κ g = ± d s$
for some constant $a 2$. If we put
$F ( v ) = ∫ ψ ( v ) d v ,$
where
$ψ ( v ) = v − 1 a 2 v 6 + a 1 3 c ε 1 ε 2 v 4 + 1 4 ε 2 v 2 − 1 9 ε 1 ε 2 − 1 2 ,$
and then we have
$F ( κ g ) = ± s + a 3$
for some constant $a 3$. Thus, the geodesic curvature $κ g$ is given by
$κ g ( s ) = F − 1 ( ± s + a 3 ) .$
Furthermore, the constant vector $C$ can be expressed as
$C = − 3 c κ g ′ κ g 4 e 1 + c κ g 3 e 2 + c ( κ g 2 − 3 ε 1 ε 2 κ g κ g ″ + 12 ε 1 ε 2 κ g ′ 2 ) κ g 6 e 3 .$
Conversely, for some constants $a 1 , a 2$ and c such that the function
$ψ ( v ) = v − 1 a 2 v 6 + a 1 3 c ε 1 ε 2 v 4 + 1 4 ε 2 v 2 − 1 9 ε 1 ε 2 − 1 2$
is well-defined on an open interval $J ⊂ ( 0 , ∞ )$, we take an indefinite integral $F ( v )$ of the function $ψ ( v )$. Let I be the image of the function F. We can take an open subinterval $J 1 ⊂ J$ such that $F : J 1 → I$ is a strictly increasing function with $F ′ ( v ) = ψ ( v )$. Let us consider the function φ defined by $φ ( s ) = F − 1 ( ± s + a 3 )$ for some constant $a 3$. Then, the function φ satisfies $F ( φ ) = ± s + a 3$.
For any unit speed pseudo-spherical curve $β ( s )$ in $Q 2 ( ε )$ with geodesic curvature $κ g ( s ) = φ ( s )$, we consider the conical surface M in $L 3$ parametrized by
$x ( s , t ) = α 0 + t β ( s ) , s ∈ I , t > 0 ,$
where $α 0$ is a constant vector. Given any nonzero constant c, we put f and g the functions, respectively, given by
$f ( s , t ) = − ε 2 t 2 φ 2 ( 4 ε 1 ε 2 φ φ ″ − 12 ε 1 ε 2 φ ′ 2 + ε 1 φ 4 ) , g ( s , t ) = − φ 5 2 c t 3 .$
For a nonzero constant c and the pseudo-orthonormal frame ${ e 1 , e 2 , e 3 }$ on $L 3$ such that $e 1 = 1 t ∂ ∂ s$ and $e 2 = ∂ ∂ t$ are tangent to M and $e 3$ normal to M, we put
$C = − 3 c φ ′ φ 4 e 1 + c φ 3 e 2 + c ( φ 2 − 3 ε 1 ε 2 φ φ ″ + 12 ε 1 ε 2 φ ′ 2 ) φ 6 e 3 .$
Note that it follows from the definition of φ that the function φ satisfies (47). Hence, using (27), it is straightforward to show that
$∇ ˜ e 1 C = ∇ ˜ e 2 C = 0 ,$
which implies that $C$ is a constant vector. Furthermore, the same argument as in the first part of this subsection yields the mean curvature vector field $H$ of the conical surface M satisfies
$Δ H = f H + g C ,$
where $f , g$ and $C$ are given in (54) and (55), respectively. This shows that the conical surface is of generalized null 2-type.
Thus, we have the following:
Theorem 3.
Let M be a conical surface in the Minkowski 3-space $L 3$. Then, M is of generalized null 2-type if and only if it is an open part of one of the following surfaces:
(1)
a Euclidean plane;
(2)
a Minkowski plane;
(3)
a hyperbolic conical surface of the first kind;
(4)
a hyperbolic conical surface of the second kind;
(5)
an elliptic conical surface;
(6)
(7)
a conical surface parameterized by
$x ( s , t ) = α 0 + t β ( s ) ,$
where $α 0$ is a constant vector and $β ( s )$ is a unit speed pseudo-spherical curve in $Q 2 ( ε )$ with the non-constant geodesic curvature $κ g$ which is, for some indefinite integral $F ( v )$ of the function
$ψ ( v ) = v − 1 a 2 v 6 + a 1 3 c ε 1 ε 2 v 4 + 1 4 ε 2 v 2 − 1 9 ε 1 ε 2 − 1 2$
with $a 1 , a 2 , c ∈ R$, given by
$κ g ( s ) = F − 1 ( ± s + a 3 ) ,$
where $a 3$ is constant.
Next, we study tangent developable surfaces in the Minkowski 3-space $L 3$.
Theorem 4.
Let M be a tangent developable surface in the Minkowski 3-space $L 3$. Then, M is of generalized null 2-type if and only if M is an open part of a Euclidean plane or a Minkowski plane.
Proof.
Let $α ( s )$ be a curve parameterized by arc-length s in $L 3$ with non-zero curvature $κ ( s )$. Then, a non-degenerate tangent developable surface M in $L 3$ is defined by
$x ( s , t ) = α ( s ) + t α ′ ( s ) , t ≠ 0 .$
In the case, we can take the pseudo-orthonormal frame ${ e 1 , e 2 , e 3 }$ of $L 3$ such that $e 1 = ∂ ∂ t$ and $e 2 = ε 2 t κ ( s ) ∂ ∂ s − ∂ ∂ t$ are tangent to M and $e 3$ is normal to M. By a direct calculation, we obtain
$∇ ˜ e 1 e 1 = ∇ ˜ e 1 e 2 = 0 , ∇ ˜ e 2 e 1 = 1 t e 2 , ∇ ˜ e 2 e 2 = − ε 1 ε 2 t e 1 − ε 1 τ ( s ) t κ ( s ) e 3 , ∇ ˜ e 1 e 3 = 0 , ∇ ˜ e 2 e 3 = ε 2 τ ( s ) t κ ( s ) e 2 ,$
where $〈 e 1 , e 1 〉 = ε 1 ( = ± 1 )$, $〈 e 2 , e 2 〉 = ε 2 ( = ± 1 )$ and $τ ( s )$ is the torsion of $α ( s )$. Therefore, the mean curvature vector field $H$ of M is given by
$H = τ ( s ) 2 t κ ( s ) e 3 .$
By a long computation, the Laplacian $Δ H$ of the mean curvature vector field $H$ turns out to be
$Δ H = − ε 1 τ 2 2 κ 2 t 3 e 1 + 1 2 κ 4 t 4 3 κ τ 2 − 2 κ ′ τ 2 t + 3 κ τ τ ′ t e 2 + 1 2 κ 4 t 5 − ε 1 κ 3 τ t 2 + ε 1 κ ′ τ t − 3 ε 2 κ τ − ( ε 1 κ 2 t + ε 2 κ κ ′ t 2 ) ( τ κ ) ′ − ε 2 κ 2 ( τ κ ) ″ t 2 − ε 1 τ 3 κ t 2 e 3 .$
Suppose that M is of generalized null 2-type, that is, M satisfies $Δ H = f H + g C$ for some smooth functions $f , g$ and a constant vector $C$. With the help of (57) and (58), (4) can be written in the form
$g C 1 = − τ 2 2 κ 2 t 3 , g C 2 = ε 2 2 κ 4 t 4 3 κ τ 2 − 2 κ ′ τ 2 t + 3 κ τ τ ′ t , − ε 1 ε 2 τ 2 κ t f + g C 3 = − ε 1 ε 2 2 κ 4 t 5 − ε 1 κ 3 τ t 2 + ε 1 κ ′ τ t − 3 ε 2 κ τ − ( ε 1 κ 2 t + ε 2 κ κ ′ t 2 ) ( τ κ ) ′ − ε 2 κ 2 ( τ κ ) ″ t 2 − ε 1 τ 3 κ t 2 ,$
where $C = ε 1 C 1 e 1 + ε 2 C 2 e 2 − ε 1 ε 2 C 3 e 3$ with $C 1 = 〈 C , e 1 〉$, $C 2 = 〈 C , e 2 〉$ and $C 3 = 〈 C , e 3 〉$. In this case, the components $C i$ of $C$ are functions of only s. It follows from (56) that we have
$C 1 ′ − ε 2 κ C 2 = 0 ,$
$C 2 ′ + ε 1 κ C 1 + ε 1 ε 2 τ C 3 = 0 ,$
$C 3 ′ + ε 2 τ C 2 = 0 .$
By combining the first and second equations of (59), we get
$3 ε 2 κ τ 2 C 1 + ( 3 ε 2 κ τ τ ′ C 1 − 2 ε 2 κ ′ τ 2 C 1 + κ 2 τ 2 C 2 ) t = 0 .$
This shows that we obtain
$3 ε 2 κ τ 2 C 1 = 0 , 3 ε 2 κ τ τ ′ C 1 − 2 ε 2 κ ′ τ 2 C 1 + κ 2 τ 2 C 2 = 0 .$
Consider the open set $O = { p ∈ M | τ ( p ) ≠ 0 }$. Suppose that $O$ is a non-empty set. (63) shows that $C 1 = 0$ and $C 2 = 0$, and it follows from (61) that $C 3 = 0$. That is, $C = 0$ on $O$. In addition, (59) gives $τ = 0$, and it is a contradiction. Thus, the open set $O$ is empty and τ is identically zero. Therefore, $α ( s )$ is a plane curve, and the surface M is an open part of a Euclidean plane or a Minkowski plane.
The converse of Theorem 4 follows a straightforward calculation.  ☐

## 5. Null Scrolls

Let $α = α ( s )$ be a null curve in $L 3$ and $β = β ( s )$ a null vector field along α satisfying $〈 α ′ , β 〉 = − 1$. Then, the null scroll M is parameterized by
$x ( s , t ) = α ( s ) + t β ( s ) .$
Furthermore, without loss of generality, we may choose $α ( s )$ as a null geodesic of M, i.e, $〈 α ′ ( s ) , β ′ ( s ) 〉 = 0$ for all s. By putting $γ ( s ) = α ′ ( s ) × β ( s )$, then ${ α ′ ( s ) , β ( s ) , γ ( s ) }$ is a pseudo-orthonormal frame along $α ( s )$ in $L 3$. We define the smooth functions k and u by
$k ( s ) = 〈 α ″ ( s ) , γ ( s ) 〉 , u ( s ) = 〈 β ( s ) , γ ′ ( s ) 〉 .$
On the other hand, the induced Lorentz metric on M is given by $g 11 = u ( s ) 2 t 2$, $g 12 = − 1$ and $g 22 = 0$. Since M is a non-degenerate surface, $u ( s ) t$ is non-vanishing everywhere.
In terms of the pseudo-orthonormal frame, we have
$α ″ ( s ) = k ( s ) γ ( s ) , β ′ ( s ) = − u ( s ) γ ( s ) , γ ′ ( s ) = − u ( s ) α ′ ( s ) + k ( s ) β ( s ) .$
The mean curvature vector field $H$ of M is given by
$H = − u 2 t β + u γ ,$
and its Laplacian $Δ H$ is expressed as
$Δ H = ( − 4 u u ′ − 2 u 4 t ) β + 2 u 3 γ .$
Suppose that M is a generalized null 2-type surface. Then, we have
$4 u u ′ + 2 u 4 t = u 2 t f − g C 2 , g C 1 = 0 , 2 u 3 = u f + g C 3 ,$
for a constant vector $C = C 1 α ′ + C 2 β + C 3 γ$ with $C 1 = − 〈 C , β 〉$, $C 2 = − 〈 C , α ′ 〉$ and $C 3 = 〈 C , γ 〉$.
Suppose that g is identically zero. By combining the first and third Equations in (67), we see that u is constant, say $u 0$. In this case, we have $f = 2 u 0 2$. Thus, M is a B-scroll, and it satisfies $Δ H = 2 u 0 2 H$ (see [16]).
Consider the open set $O = { p ∈ M | g ( p ) ≠ 0 }$. Suppose that $O$ is a non-empty set. Then, from (67), we find $C 1 = 0$ on a component $O 0$ on $O$. Let us differentiate $C 1$ with respect to s and use (65). Then, $C 3 = 0$ on $O 0$. Since
$α ′ × β ′ = − u α ′ , α ″ × β = k β ,$
by differentiating the equation $C 3 = 0$ with respect to s, we can obtain
$k C 1 − u C 2 = 0 .$
It follows that $C 2 = 0$ on $O 0$ because $C 1 = 0$ and $u ≠ 0$. Since $C$ is a constant vector, it is a zero vector. From the first and third Equations in (67), u is a non-zero constant, say $u 0$, and $f = 2 u 0 2$ on M. Thus, M is of null 2-type and it is a B-scroll.
Consequently, we have
Theorem 5.
Let M be a null scroll in the Minkowski 3-space $L 3$. Then, M is of generalized null 2-type if and only if M is an open piece of a B-scroll.
We now propose an open problem.
Problem 1.
Classify all generalized null 2-type surfaces in the Euclidean space or pseudo-Euclidean space.

## Acknowledgments

We would like to thank the referee for the careful review and the valuable comments, which really improved the paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01060046).

## Author Contributions

Dae Won Yoon gave the idea to establish generalized null finite type surfaces on Minkowski space. Dong-Soo Kim, Yong Ho Kim and Jaewon Lee checked and polished the draft.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Takahashi, T. Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 1966, 18, 380–385. [Google Scholar] [CrossRef]
2. Chen, B.-Y. Total Mean Curvature and Submanifolds of Finite Type; World Scientific Publishing: Singapore, 1984. [Google Scholar]
3. Chen, B.-Y. Null 2-type surfaces in $E$3 are circular cylinders. Kodai Math. J. 1988, 11, 295–299. [Google Scholar] [CrossRef]
4. Chen, B.-Y. Null 2-Type Surfaces in Euclidean Space. In Proceedings of the Algebra, Analysis and Geometry, Taipei, Taiwan, 27–29 June 1988; World Science Publishing: Teaneck, NJ, USA; pp. 1–18.
5. Ferrández, A.; Lucas, P. On surfaces in the 3-dimensional Lorentz-Minkowski space. Pac. J. Math. 1992, 152, 93–100. [Google Scholar] [CrossRef]
6. Chen, B.-Y. Submanifolds in de Sitter space-time satisfying ∇H = λH. Israel J. Math. 1995, 89, 373–391. [Google Scholar] [CrossRef]
7. Chen, B.-Y.; Fu, Y. δ(3)-ideal null 2-type hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 2015, 40, 43–56. [Google Scholar] [CrossRef]
8. Chen, B.-Y.; Garay, O.J. δ(2)-ideal null 2-type hypersurfaces of Euclidean space are spherical cylinders. Kodai Math. J. 2012, 35, 382–391. [Google Scholar] [CrossRef]
9. Chen, B.-Y.; Song, H.Z. Null 2-type surfaces in Minkowski space-time. Algebras Groups Geom. 1989, 6, 333–352. [Google Scholar]
10. Dursun, U. Null 2-type submanifolds of the Euclidean space $E$5 with non-parallel mean curvature vector. J. Geom. 2006, 86, 73–80. [Google Scholar] [CrossRef]
11. Dursun, U. Null 2-type space-like submanifolds of $E t 5$ with normalized parallel mean curvature vector. Balkan J. Geom. Appl. 2006, 11, 61–72. [Google Scholar]
12. Dursun, U. Null 2-type submanifolds of the Euclidean space $E$5 with parallel normalized mean curvature vector. Kodai Math. J. 2005, 28, 191–198. [Google Scholar] [CrossRef]
13. Dursun, U.; Coşkun, E. Flat surfaces in the Minkowski space $E 1 3$ with pointwise 1-type Gauss map. Turk. J. Math. 2012, 36, 613–629. [Google Scholar]
14. Ferrández, A.; Lucas, P. Null 2-type hypersurfaces in a Lorentz space. Can. Math. Bull. 1992, 35, 354–360. [Google Scholar]
15. Fu, Y. Biharmonic submanifolds with parallel mean curvature vector in pseudo-Euclidean 5-space. J. Geom. Phys. 2013, 16, 331–344. [Google Scholar] [CrossRef]
16. Fu, Y. Null 2-type hypersurfaces with at most three distinct principal curvatures in Euclidean space. Taiwanese J. Math. 2015, 19, 519–533. [Google Scholar] [CrossRef]
17. Kim, D.-S.; Kim, Y.H. Spherical submanifolds of null 2-type. Kyungpook Math. J. 1996, 36, 361–369. [Google Scholar]
18. Kim, D.-S.; Kim, Y.H. Null 2-type surfaces in Minkowski 4-space. Houston J. Math. 1996, 22, 279–296. [Google Scholar]
19. Kim, D.-S.; Kim, Y.H. B -scrolls with non-diagonalizable shape operators. Rocky Mt. J. Math. 2003, 33, 175–190. [Google Scholar] [CrossRef]
20. Li, S.J. Null 2-type Chen surfaces. Glasgow Math. J. 1995, 37, 233–242. [Google Scholar] [CrossRef]
21. Li, S.J. Null 2-type surfaces in $E m$with parallel normalized mean curvature vector. Math. J. Toyama Univ. 1994, 17, 23–30. [Google Scholar]
22. Lee, J.W.; Kim, D.-S.; Kim, Y.H.; Yoon, D.W. Generalized null 2-type immersions in Euclidean space. Adv. Geom. 2016. accepted for publication. [Google Scholar]
23. Walrave, J. Curves and Surfaces in Minkowski Space. Doctor’s Thesis, Fac. of Science, K. U. Leuven, Leuven, Belgium, 1995. [Google Scholar]
24. Babaarslan, M.; Yayli, Y. Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space. Proc. Natl. Acad. Sci. India Sect. A 2014, 84, 535–540. [Google Scholar] [CrossRef]
25. Babaarslan, M.; Yayli, Y. On Space-Like Constant Slope Surfaces and Bertrand Curves in Minkowski 3-Space. Ann. Alexandru Ioan Cuza Univ. Math. 2015. [Google Scholar] [CrossRef]
26. Kim, Y.H.; Yoon, D.W. Ruled surfaces with pointwise 1-type Gauss map. J. Geom. Phys. 2000, 34, 191–205. [Google Scholar] [CrossRef]
27. Graves, L.K. Codimension one isometric immersions between Lorentz spaces. Trans. Am. Math. Soc. 1979, 252, 367–392. [Google Scholar] [CrossRef]

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Yoon, D.W.; Kim, D.-S.; Kim, Y.H.; Lee, J.W. Generalized Null 2-Type Surfaces in Minkowski 3-Space. Symmetry 2017, 9, 14. https://doi.org/10.3390/sym9010014

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Yoon DW, Kim D-S, Kim YH, Lee JW. Generalized Null 2-Type Surfaces in Minkowski 3-Space. Symmetry. 2017; 9(1):14. https://doi.org/10.3390/sym9010014

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Yoon, Dae Won, Dong-Soo Kim, Young Ho Kim, and Jae Won Lee. 2017. "Generalized Null 2-Type Surfaces in Minkowski 3-Space" Symmetry 9, no. 1: 14. https://doi.org/10.3390/sym9010014

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