# SOME CLASSIFICATIONS FOR GAUSS MAP OF TUBULAR HYPERSURFACES IN
$\mathbb{E}^{4}_{1}$ CONCERNING LINEARIZED OPERATORS $\mathcal{L}_{k}$

Ahmet Kazan1∗, Mustafa Altın2 and Nurettin Cenk Turgay3 1Department of
Computer Technologies, Doğanşehir Vahap Küçük Vocational School, Malatya
Turgut Özal University, Malatya, Turkey
2Technical Sciences Vocational School, Bingöl University, Bingöl, Turkey
3Department of Mathematics, Faculty of Science and Letters, İstanbul Technical
University, İstanbul, Turkey
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract.

In this study, we deal with the Gauss map of tubular hypersurfaces in
4-dimensional Lorentz-Minkowski space concerning the linearized operators
$\mathcal{L}_{1}$ (Cheng-Yau) and $\mathcal{L}_{2}$. We obtain the
$\mathcal{L}_{1}$ (Cheng-Yau) operator of the Gauss map of tubular
hypersurfaces that are formed as the envelope of a family of pseudo
hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on timelike
or spacelike curves with non-null Frenet vectors in $\mathbb{E}^{4}_{1}$ and
give some classifications for these hypersurfaces which have generalized
$\mathcal{L}_{k}$ 1-type Gauss map, first kind $\mathcal{L}_{k}$-pointwise
1-type Gauss map, second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and
$\mathcal{L}_{k}$-harmonic Gauss map, $k\in\\{1,2\\}$.

###### Key words and phrases:

Tubular hypersurface, $\mathcal{L}_{1}$ (Cheng-Yau) operator,
$\mathcal{L}_{2}$ operator, Generalized $\mathcal{L}_{k}$ 1-type Gauss map,
First and second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map,
$\mathcal{L}_{k}$-harmonic Gauss map.

###### 2010 Mathematics Subject Classification:

14J70, 53A35.

##  1\. Introduction

Let $(M,g)$ be a hypersurface of $(n+1)$-dimensional Minkowski space
$\mathbb{E}^{n+1}_{1}$, $\Delta$ denote its Laplace operator. A smooth mapping
$\phi:M\rightarrow\mathbb{E}^{n+1}_{1}$ is said to be finite type if it can be
expressed as

$\phi=\phi_{0}+\phi_{1}+\cdots+\phi_{k},$

where $\phi_{0}$ is a constant vector and $\phi_{i}$ is an eigenvector of
$\Delta$ corresponding to the eigenvector $\lambda_{i}$ for $i=1,2,\ldots,k$.
More precisely, if $\lambda_{1},\lambda_{2},\ldots,\lambda_{k}$ are distinct,
then $\psi$ is said to be $k$-type ([4, 6, 7]). Several results on the study
of finite type mappings were summed up in a report by B.-Y. Chen in [5] (See
also [8, 22]).

Let $N$ denote the Gauss map of $M$. From the definition above, one can
conclude that $N$ is of 1-type if and only if it satisfies the equation

$\Delta N=\lambda(N+C)$ (1.1)

for a constant $\lambda\in\mathbb{R}$ and a constant vector $C$. However,
Gauss map of some important submanifolds such as catenoid and helicoid of the
Euclidean 3-space $\mathbb{E}^{3}$ satisfies

$\Delta N=f(N+C)$ (1.2)

which is weaker than (1.1), where $f\in C^{\infty}(M)$ is a smooth function,
[10]. These submanifolds whose Gauss map $N$ satisfying (1.2) are said to have
pointwise 1-type Gauss map. Submanifolds with pointwise 1-type Gauss map have
been worked in several papers (cf. [10, 20, 21, 22]).

On the other hand, the Gauss map of some hypersurfaces of semi-Euclidean
spaces satisfies the equation

$\Delta N=f_{1}N+f_{2}C$ (1.3)

for some smooth functions $f_{1}$, $f_{2}$ and a constant vector $C$. A
submanifold is said to have generalized 1-type Gauss map if its Gauss map
satisfies the condition (1.3), [25]. After this definition was given,
hypersurfaces of pseudo-Euclidean spaces have been considered in terms of
having generalized 1-type Gauss map, [17, 19, 25, 26].

In the recent years, the definition of $\mathcal{L}_{k}$-finite type maps has
been obtained by replacing $\Delta$ in the definition above with the sequence
of operators $\mathcal{L}_{0},\ \mathcal{L}_{1},\ \mathcal{L}_{2},\ \ldots,\
\mathcal{L}_{n-1}$, [1, 2]. Note that, by the definition of these operators,
one can obtain $\mathcal{L}_{0}=-\Delta$ and $\mathcal{L}_{1}=\square$ is
called as the Cheng-Yau operator introduced in [9]. By motivating this idea,
notion of $\mathcal{L}_{k}$-pointwise 1-type Gauss map and generalized
$\mathcal{L}_{k}$ 1-type Gauss map was presented in [14] and [18],
respectively (see Definition 1). After the case $k=1$ is studied in these
papers, many result obtained on hypersurfaces with certain type of Gauss map,
[11, 12, 13, 19, 24, 25, 26].

On the other hand, in [3], the general expression of the canal hypersurfaces
that are formed as the envelope of a family of pseudo hyperspheres, pseudo
hyperbolic hyperspheres and null hypercones whose centers lie on a non-null
curve with non-null Frenet vector fields in $\mathbb{E}^{4}_{1}$ has been
given and their some geometric invariants such as unit normal vector felds,
Gaussian curvatures, mean curvatures and principal curvatures have been
obtained. Also, tubular hypersurfaces in $\mathbb{E}^{4}_{1}$ by taking
constant radius function have been studied in [3].

In this paper, we study the tubular hypersurfaces in Lorentz-Minkowski 4-space
$\mathbb{E}^{4}_{1}$ with the aid of $\mathcal{L}_{k}$ operators,
$k\in\\{1,2\\}$. In Sect. 2, we give basic notation, facts and definitions
about hypersurfaces of Minkowski spaces. In Sect. 3 and Sect 4, we consider
some classifications of tubular hypersurfaces by considering their Gauss maps
in terms of their types with respect to the operators $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$.

##  2\. Preliminaries

Let $\mathbb{\mathbb{E}}_{1}^{n+1}$ be the $(n+1)$-dimensional Lorentz-
Minkowski space with the canonical pseudo-Euclidean metric $\left\langle\text{
},\right\rangle$ of index 1 and signature $(-,+,+,...,+)$ given by

$\left\langle\text{
},\right\rangle=-dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+...+dx_{n+1}^{2}$

where $(x_{1},x_{2},...,x_{n+1})$ is a rectangular coordinate system in
$\mathbb{\mathbb{E}}_{1}^{n+1}$.

If $\Gamma:M\longrightarrow\mathbb{E}_{1}^{n+1}$ is an isometric immersion
from an $n$-dimensional orientable manifold $M$ to
$\mathbb{\mathbb{E}}_{1}^{n+1}$, then the induced metric on $M$ by the
immersion $\Gamma$ can be Riemannian or Lorentzian. Let $N$ denotes a unit
normal vector field and put $\left\langle N,N\right\rangle=\varepsilon=\pm 1$,
so that $\varepsilon=1$ or $\varepsilon=-1$ according to $M$ is endowed with a
Lorentzian or Riemannian metric, respectively.

The operator $\mathcal{L}_{k}$ acting on the coordinate functions of the Gauss
map $N$ of the hypersurface $M$ in $(n+1)$-dimensional Lorentz-Minkowski space
$\mathbb{\mathbb{E}}_{1}^{n+1}$ is

$\mathcal{L}_{k}N=-\varepsilon\mathfrak{C}_{k}\left(\nabla
H_{k+1}+\left(nH_{1}H_{k+1}-\left(n-k-1\right)H_{k+2}\right)N\right).$ (2.1)

Here,

$\left(\begin{array}[c]{c}n\\\
k\end{array}\right)H_{k}=\left(-\varepsilon\right)^{k}a_{k},\text{ \
}\left(\left(\begin{array}[c]{c}n\\\
k\end{array}\right)=\frac{n!}{k!(n-k)!}\right),$ (2.2)

such that

$\left.\begin{array}[c]{l}a_{1}=-\sum_{i=1}^{n}\kappa_{i},\\\ \\\
a_{k}=(-1)^{k}\sum_{i_{1}<i_{2}<...<i_{k}}^{n}\kappa_{i_{1}}\kappa_{i_{2}}...\kappa_{i_{k}},~{}k=2,3,...,n\end{array}\right\\}$
(2.3)

and $H_{k}$ is called the $k$-th mean curvature of order $k$ of $M$.

Also, the constant $\mathfrak{C}_{k}$ is given by

$\mathfrak{C}_{k}=\left(\begin{array}[c]{c}n\\\
k+1\end{array}\right)(-\varepsilon)^{k}.$ (2.4)

(For more details about the linearized operator $\mathcal{L}_{k}$, one can see
[16].)

######  Definition 1.

Let $\mathfrak{m}$ and $\mathfrak{n}$ be non-zero smooth functions on $M$,
$C\in\mathbb{E}_{1}^{n+1}$ be a non-zero constant vector and
$k\in\\{0,1,2,...,n\\}$.

If the Gauss map $N$ of an oriented submanifold $M$ in $\mathbb{E}^{4}_{1}$
satisfies

i:

$\mathcal{L}_{k}N=\mathfrak{m}N+\mathfrak{n}C,$ then $M$ has generalized
$\mathcal{L}_{k}$ 1-type Gauss map;

ii:

$\mathcal{L}_{k}N=\mathfrak{m}N,$ then $M$ has first kind
$\mathcal{L}_{k}$-pointwise 1-type Gauss map;

iii:

$\mathcal{L}_{k}N=\mathfrak{m}(N+C),$ then $M$ has second kind
$\mathcal{L}_{k}$-pointwise 1-type Gauss map;

iv:

$\mathcal{L}_{k}N=0$, then $N$ is called $\mathcal{L}_{k}$-harmonic.

In this study, we will deal with Gauss maps of tubular hypersurfaces in
4-dimensional Lorentz-Minkowski space $\mathbb{E}^{4}_{1}$ concerning
linearized operators $L_{1}$ and $L_{2}$. So, let us give some notions in
$\mathbb{E}^{4}_{1}$.

Let $\overrightarrow{u}=(u_{1},u_{2},u_{3},u_{4})$,
$\overrightarrow{v}=(v_{1},v_{2},v_{3},v_{4})$ and
$\overrightarrow{w}=(w_{1},w_{2},w_{3},w_{4})$ be three vectors in
$\mathbb{E}^{4}_{1}$. The inner product and vector product are defined by

$\left\langle\overrightarrow{u},\overrightarrow{v}\right\rangle=-u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}+u_{4}v_{4}$
(2.5)

and

$\overrightarrow{u}\times\overrightarrow{v}\times\overrightarrow{w}=\det\left[\begin{array}[c]{cccc}-e_{1}&e_{2}&e_{3}&e_{4}\\\
u_{1}&u_{2}&u_{3}&u_{4}\\\ v_{1}&v_{2}&v_{3}&v_{4}\\\
w_{1}&w_{2}&w_{3}&w_{4}\end{array}\right],$ (2.6)

respectively. Here $e_{i},(i=1,2,3,4)$ are standard basis vectors.

A vector $\overrightarrow{u}\in E_{1}^{4}-\\{0\\}$ is called spacelike,
timelike or lightlike (null) if
$\left\langle\overrightarrow{u},\overrightarrow{u}\right\rangle>0$ (or
$\overrightarrow{u}=0$),
$\left\langle\overrightarrow{u},\overrightarrow{u}\right\rangle<0$ or
$\left\langle\overrightarrow{u},\overrightarrow{u}\right\rangle=0$,
respectively. A curve $\beta(s)$ in $\mathbb{E}^{4}_{1}$ is spacelike,
timelike or lightlike (null), if all its velocity vectors $\beta^{\prime}(s)$
are spacelike, timelike or lightlike, respectively and a non-null (i.e.
timelike or spacelike) curve has unit speed if
$\left\langle\beta^{\prime},\beta^{\prime}\right\rangle=\mp 1$. Also, the norm
of the vector $\overrightarrow{u}$ is
$\left\|\overrightarrow{u}\right\|=\sqrt{\left|\left\langle\overrightarrow{u},\overrightarrow{u}\right\rangle\right|}$
[15].

Let $F_{1},$ $F_{2},$ $F_{3},$ $F_{4}$ be unit tangent vector field, principal
normal vector field, binormal vector field, trinormal vector field of a
timelike or spacelike curve $\beta(s)$, respectively and
$\\{F_{1},F_{2},F_{3},F_{4}\\}$ be the moving Frenet frame along $\beta(s)$ in
$\mathbb{E}^{4}_{1}$. The Frenet equations can be given according to the
causal characters of non-null Frenet vector fields $F_{1},$ $F_{2},$ $F_{3}$
and $F_{4}$ as follows [23]:

If the curve $\beta(s)$ is timelike, i.e. $\left\langle
F_{1},F_{1}\right\rangle=-1,$ $\left\langle F_{i},F_{i}\right\rangle=1$
($i=2,3,4$), then

$\left.\begin{array}[c]{l}F_{1}^{\prime}=k_{1}F_{2},\\\
F_{2}^{\prime}=k_{1}F_{1}+k_{2}F_{3},\\\
F_{3}^{\prime}=-k_{2}F_{2}+k_{3}F_{4},\\\
F_{4}^{\prime}=-k_{3}F_{3};\end{array}\right\\}$ (2.7)

if the curve $\beta(s)$ is spacelike with timelike principal normal vector
field $F_{2}$, i.e. $\left\langle F_{2},F_{2}\right\rangle=-1,$ $\left\langle
F_{i},F_{i}\right\rangle=1$ ($i=1,3,4$), then

$\left.\begin{array}[c]{l}F_{1}^{\prime}=k_{1}F_{2},\\\
F_{2}^{\prime}=k_{1}F_{1}+k_{2}F_{3},\\\
F_{3}^{\prime}=k_{2}F_{2}+k_{3}F_{4},\\\
F_{4}^{\prime}=-k_{3}F_{3};\end{array}\right\\}$ (2.8)

if the curve $\beta(s)$ is spacelike with timelike binormal vector field
$F_{3}$, i.e. $\left\langle F_{3},F_{3}\right\rangle=-1,$ $\left\langle
F_{i},F_{i}\right\rangle=1$ ($i=1,2,4$), then

$\left.\begin{array}[c]{l}F_{1}^{\prime}=k_{1}F_{2},\\\
F_{2}^{\prime}=-k_{1}F_{1}+k_{2}F_{3},\\\
F_{3}^{\prime}=k_{2}F_{2}+k_{3}F_{4},\\\
F_{4}^{\prime}=k_{3}F_{3};\end{array}\right\\}$ (2.9)

if the curve $\beta(s)$ is spacelike with timelike trinormal vector field
$F_{4}$, i.e. $\left\langle F_{4},F_{4}\right\rangle=-1,$ $\left\langle
F_{i},F_{i}\right\rangle=1$ ($i=1,2,3$), then

$\left.\begin{array}[c]{l}F_{1}^{\prime}=k_{1}F_{2},\\\
F_{2}^{\prime}=-k_{1}F_{1}+k_{2}F_{3},\\\
F_{3}^{\prime}=-k_{2}F_{2}+k_{3}F_{4},\\\
F_{4}^{\prime}=k_{3}F_{3}.\end{array}\right\\}$ (2.10)

Here $k_{1},k_{2},k_{3}$ are the first, second and third curvatures of the
non-null curve $\beta(s).$

Also, if $p$ is a fixed point in $\mathbb{E}^{4}_{1}$ and $r$ is a positive
constant, then the pseudo-Riemannian hypersphere and the pseudo-Riemannian
hyperbolic space are defined by

$S_{1}^{3}(p,r)=\\{x\in\mathbb{E}^{4}_{1}:\left\langle
x-p,x-p\right\rangle=r^{2}\\}$

and

$H_{0}^{3}(p,r)=\\{x\in\mathbb{E}^{4}_{1}:\left\langle
x-p,x-p\right\rangle=-r^{2}\\},$

respectively.

If $M$ is an oriented hypersurface in $E_{1}^{4},$ then the gradient of a
smooth function $f(s,t,w)$, which is defined in $M$, can be obtained by

$\nabla
f=\frac{1}{\mathfrak{g}}\left(\begin{array}[c]{c}\left(\left(g_{23}^{2}-g_{22}g_{33}\right)f_{s}+\left(-g_{13}g_{23}+g_{12}g_{33}\right)f_{t}+\left(g_{13}g_{22}-g_{12}g_{23}\right)f_{w}\right)\partial
s\\\
+\left(\left(-g_{13}g_{23}+g_{12}g_{33}\right)f_{s}+\left(g_{13}^{2}-g_{11}g_{33}\right)f_{t}+\left(-g_{12}g_{13}+g_{11}g_{23}\right)f_{w}\right)\partial
t\\\
+\left(\left(g_{13}g_{22}-g_{12}g_{23}\right)f_{s}+\left(-g_{12}g_{13}+g_{11}g_{23}\right)f_{t}+\left(g_{12}^{2}-g_{11}g_{22}\right)f_{w}\right)\partial
w\end{array}\right),$ (2.11)

where

$\mathfrak{g=}g_{13}^{2}g_{22}-2g_{12}g_{13}g_{23}+g_{11}g_{23}^{2}+g_{12}^{2}g_{33}-g_{11}g_{22}g_{33};$

$\\{s,t,w\\}$ is a local coordinat system of $M$; $f_{s}$, $f_{t}$, $f_{w}$
are the partial derivatives of $f$ and $g_{11}=\left\langle\partial s,\partial
s\right\rangle,$ $g_{12}=\left\langle\partial s,\partial t\right\rangle,$
$g_{13}=\left\langle\partial s,\partial w\right\rangle,$
$g_{22}=\left\langle\partial t,\partial t\right\rangle,$
$g_{23}=\left\langle\partial t,\partial w\right\rangle,$
$g_{33}=\left\langle\partial w,\partial w\right\rangle$.

##  3\. Some Classifications for Tubular Hypersurfaces Generated by Timelike
Curves with $L_{k}$ Operators in $\mathbb{E}^{4}_{1}$

In this section, we obtain the $\mathcal{L}_{1}$ (Cheng-Yau) and
$\mathcal{L}_{2}$ operators of the Gauss map of the tubular hypersurfaces
$\mathcal{T}(s,t,w)$ that are formed as the envelope of a family of pseudo
hyperspheres whose centers lie on a timelike curve with non-null Frenet
vectors in $\mathbb{E}^{4}_{1}$ and give some classifications for these
hypersurfaces which have generalized $\mathcal{L}_{k}$ 1-type Gauss map, first
kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and second kind
$\mathcal{L}_{k}$-pointwise 1-type Gauss map and $\mathcal{L}_{k}$-harmonic
Gauss map, $k\in\\{1,2\\}$.

The tubular hypersurfaces $\mathcal{T}(s,t,w)$ that are formed as the envelope
of a family of pseudo hyperspheres whose centers lie on a timelike curve with
non-null Frenet vectors in $\mathbb{E}^{4}_{1}$ can be parametrized by

$\mathcal{T}(s,t,w)=\beta(s)+r(\cos t\cos wF_{2}(s)+\sin t\cos wF_{3}(s)+\sin
wF_{4}(s)).$ (3.1)

The unit normal vector field of (3.1) is

$N={-}\left({\cos t\cos wF}_{2}+{\sin t\cos wF}_{3}+{\sin wF}_{4}\right)$
(3.2)

and so,

$\left\langle N,N\right\rangle=1.$ (3.3)

The coefficients of the first fundamental form of (3.1) are

$\left.\begin{array}[c]{l}g_{11}={\small-(1+rk}_{1}\cos{\small t}\cos{\small
w)}^{2}{\small+(rk}_{2}\cos{\small t}\cos{\small w-rk}_{3}\sin{\small
w)}^{2}{\small+r}^{2}\left({\small
k}_{2}^{2}{\small+k}_{3}^{2}\right)\sin^{2}{\small t}\cos^{2}{\small w,}\\\
g_{12}=g_{21}=r^{2}(k_{2}\cos w-k_{3}\cos t\sin w)\cos w,\text{
}g_{22}=r^{2}\cos^{2}w,\\\ g_{13}=g_{31}=r^{2}k_{3}\sin t,\text{
}g_{23}=g_{32}=0,\text{ }g_{33}=r^{2}.\end{array}\right\\}$ (3.4)

The principal curvatures of (3.1) are

$\kappa_{1}=\kappa_{2}=\frac{1}{r},\text{ }\kappa_{3}=\frac{k_{1}\cos t\cos
w}{1+rk_{1}\cos t\cos w}.$ (3.5)

For more details about these hypersurfaces, one can see [3].

###  3.1. Some Classifications for Tubular Hypersurfaces Generated by Timelike
Curves with $\mathcal{L}_{1}$ (Cheng-Yau) Operator in $\mathbb{E}^{4}_{1}$

The functions $a_{k}$ of the tubular hypersurfaces (3.1) in
$\mathbb{E}^{4}_{1}$ are obtained from (2.3) and (3.5) by

$a_{1}=\frac{-2-3rk_{1}\cos t\cos w}{r(1+rk_{1}\cos t\cos w)},\text{
}a_{2}=\frac{1+3rk_{1}\cos t\cos w}{r^{2}(1+rk_{1}\cos t\cos w)},\text{
}a_{3}=-\frac{k_{1}}{r^{2}(rk_{1}+\sec t\sec w)}.$ (3.6)

Also, from (2.11), (3.4) and (3.6), we have

$\displaystyle\nabla a_{2}$ $\displaystyle={-\frac{2\left(k_{1}^{\prime}\cos
t+k_{1}k_{2}\sin t\right)\cos w}{r(1+rk_{1}\cos t\cos
w)^{3}}}F_{1}{-\frac{k_{1}\left(2\cos^{2}t\cos(2w)+\cos(2t)-3\right)}{2r^{2}(1+rk_{1}\cos
t\cos w)^{2}}}F_{2}$ $\displaystyle{-\frac{2k_{1}\sin t\cos
t\cos^{2}w}{r^{2}(1+rk_{1}\cos t\cos w)^{2}}}F_{3}{-\frac{2k_{1}\cos t\sin
w\cos w}{r^{2}(1+rk_{1}\cos t\cos w)^{2}}}F_{4}.$ (3.7)

So, from (2.1), (2.2), (3.2), (3.3), (3.6) and (3.7), we reach that

$\displaystyle\mathcal{L}_{1}N$ $\displaystyle={-\frac{2\left(k_{1}k_{2}\sin
t+k_{1}^{\prime}\cos t\right)\cos w}{r(1+rk_{1}\cos t\cos w)^{3}}}F_{1}$
$\displaystyle{-\frac{2\left(rk_{1}\left(3rk_{1}\cos^{3}t\cos^{3}w+2\cos^{2}t\cos(2w)+\cos(2t)\right)+\cos
t\cos w\right)}{r^{3}(1+rk_{1}\cos t\cos w)^{2}}}F_{2}$
$\displaystyle{-\frac{2(1+3rk_{1}\cos t\cos w)\sin t\cos w}{r^{3}(1+rk_{1}\cos
t\cos w)}}F_{3}{-\frac{2(3rk_{1}\cos t\cos w+1)\sin w}{r^{3}(1+rk_{1}\cos
t\cos w)}}F_{4}.$ (3.8)

Now, let us give some classifications for the tubular hypersurfaces (3.1)
which have generalized $\mathcal{L}_{1}$ 1-type Gauss map, first kind
$\mathcal{L}_{1}$-pointwise 1-type Gauss map and second kind
$\mathcal{L}_{1}$-pointwise 1-type Gauss map and $\mathcal{L}_{1}$-harmonic
Gauss map.

Let the tubular hypersurfaces $\mathcal{T}(s,t,w)$ have generalized
$\mathcal{L}_{1}$ (Cheng-Yau) 1-type Gauss map, i.e.,
$\mathcal{L}_{1}N=\mathfrak{m}N+\mathfrak{n}C,$ where
$C=C_{1}F_{1}+C_{2}F_{2}+C_{3}F_{3}+C_{4}F_{4}$ is a constant vector. Here, by
taking derivatives of the constant vector $C$ with respect to $s,$ from (2.7)
we obtain that

$\left.\begin{array}[c]{l}C_{1}^{\prime}+C_{2}k_{1}=0,\\\
C_{2}^{\prime}+C_{1}k_{1}-C_{3}k_{2}=0,\\\
C_{3}^{\prime}+C_{2}k_{2}-C_{4}k_{3}=0,\\\
C_{4}^{\prime}+C_{3}k_{3}=0.\end{array}\right\\}$ (3.9)

Also, by taking derivatives the constant vector $C$ with respect to $t$ and
$w$ separately, one can see that the functions $C_{i}$ depend only on $s.$

Firstly, let us classificate the tubular hypersurfaces $\mathcal{T}(s,t,w)$
which have generalized $\mathcal{L}_{1}$ (Cheng-Yau) 1-type Gauss map.

From (3.2) and (3.8), we get

$\left.\begin{array}[c]{l}{-\frac{2\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r(1+rk_{1}\cos t\cos w)^{3}}}=\mathfrak{n}C_{1},\\\ \\\
{-\frac{2\left(rk_{1}\left(3rk_{1}\cos^{3}t\cos^{3}w+2\cos^{2}t\cos(2w)+\cos(2t)\right)+\cos
t\cos w\right)}{r^{3}(1+rk_{1}\cos t\cos w)^{2}}}=\mathfrak{m}\left({-\cos
t\cos w}\right)+\mathfrak{n}C_{2},\\\ \\\ {-\frac{2(1+3rk_{1}\cos t\cos w)\sin
t\cos w}{r^{3}(1+rk_{1}\cos t\cos w)}}=\mathfrak{m}\left(-{\sin t\cos
w}\right)+\mathfrak{n}C_{3},\\\ \\\ {-\frac{2(1+3rk_{1}\cos t\cos w)\sin
w}{r^{3}(1+rk_{1}\cos t\cos w)}}=\mathfrak{m}\left(-{\sin
w}\right)+\mathfrak{n}C_{4}.\end{array}\right\\}$ (3.10)

Now let us investigate the non-zero functions $\mathfrak{m}(s,t,w)$ and
$\mathfrak{n}(s,t,w)$ from the above four equations.

Firstly, let us assume that $C_{1}\neq 0$.

In this case, from the first equation of (3.10) it’s easy to see that

$\mathfrak{n}(s,t,w)={-\frac{2\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r(1+rk_{1}\cos t\cos w)^{3}C_{1}}.}$ (3.11)

Here, when the equation (3.11) is successively substituted into the second,
third and fourth equations of (3.10), we obtain

$\left.\begin{array}[c]{l}\mathfrak{m}(s,t,w)=\frac{2\left(\begin{array}[c]{l}\left({\small
rk}_{1}\left(\begin{array}[c]{l}{\small 3rk}_{1}\cos^{3}{\small
t}\cos^{3}{\small w}\\\ {\small+2}\cos^{2}{\small
t}\cos{\small(2w)+}\cos{\small(2t)}\end{array}\right){\small+}\cos{\small
t}\cos{\small w}\right){\small C}_{1}{\small(1+rk}_{1}\cos{\small
t}\cos{\small w)}\\\ \\\ {\small-C}_{2}{\small r}^{2}\left(k_{1}^{\prime}\cos
t+k_{1}k_{2}\sin t\right)\cos w\end{array}\right)}{C_{1}r^{3}(1+rk_{1}\cos
t\cos w)^{3}\cos t\cos w},\\\ \\\
\mathfrak{m}(s,t,w)=\frac{2\left(C_{1}(1+rk_{1}\cos t\cos w)^{2}(1+3rk_{1}\cos
t\cos w)-C_{3}r^{2}\left(k_{1}^{\prime}\cot
t+k_{1}k_{2}\right)\right)}{C_{1}r^{3}(1+rk_{1}\cos t\cos w)^{3}},\\\ \\\
\mathfrak{m}(s,t,w)=\frac{2\left(C_{1}(1+rk_{1}\cos t\cos w)^{2}(1+3rk_{1}\cos
t\cos w)-C_{4}r^{2}\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin t\right)\cot
w\right)}{C_{1}r^{3}(1+rk_{1}\cos t\cos w)^{3}}.\end{array}\right.$

When we equate the functions $\mathfrak{m}(s,t,w)$ found above to each other,
we arrive at the following equations:

$\displaystyle(C_{3}-C_{4}\sin t\cot w)\left(k_{1}^{\prime}\cot
t+k_{1}k_{2}\right)=0,$ (3.12) $\displaystyle{\small
k}_{1}{\small(C}_{1}\sec{\small t}\sec{\small
w+k}_{2}r{\small(C}_{2}\tan{\small t-C}_{4}\sin{\small t}\cot{\small
w))}+{\small r(C}_{1}{\small k}_{1}^{2}-{\small
k}_{1}^{\prime}{\small(C}_{4}\cos{\small t}\cot{\small w-C}_{2}{\small))=0,}$
(3.13) $\displaystyle k_{1}(rk_{2}(C_{3}-C_{2}\tan t)-C_{1}\sec t\sec
w)-C_{1}rk_{1}^{2}+rk_{1}^{\prime}(C_{3}\cot t-C_{2})=0.$ (3.14)

In the equation (3.12), it holds that $k_{1}^{\prime}\cot t+k_{1}k_{2}\neq 0$.
This is because, when $k_{1}^{\prime}\cot t+k_{1}k_{2}=0,$ the function
$\mathfrak{n}(s,t,w)$ in the first equation of (3.10) becomes zero. This, in
turn, contradicts the definition of the function $\mathfrak{n}(s,t,w)$ in our
classification as $\mathcal{L}_{1}N=\mathfrak{m}N+\mathfrak{n}C$. So, from the
equation (3.12) and $k_{1}^{\prime}\cot t+k_{1}k_{2}\neq 0,$ we have
$C_{3}=C_{4}=0.$ When $C_{3}=C_{4}=0$, substituting this into the equation
(3.14) yields

$\left(C_{1}rk_{1}^{2}+C_{2}rk_{1}^{\prime}\right)\cos t+C_{1}k_{1}\sec
w+C_{2}rk_{1}k_{2}\sin t=0.$

Thus, we have

$C_{1}k_{1}^{2}+C_{2}k_{1}^{\prime}=C_{1}k_{1}=C_{2}k_{1}k_{2}=0$

and so $C_{1}=C_{2}=0.$ This is a contradiction.

Secondly, let us assume that $C_{1}=0$.

In this case, from the first equation of the set of equations (3.9) it’s easy
to see that

$C_{2}k_{1}=0{.}$ (3.15)

If $k_{1}=0$ in (3.15), then from the second, third and fourth equations of
(3.10), it is calculated as

$\left.\begin{array}[c]{l}C_{2}r^{3}\mathfrak{n}(s,t,w)=\left(\mathfrak{m}(s,t,w)r^{3}-2\right)\cos
t\cos w,\\\ \\\
C_{3}r^{3}\mathfrak{n}(s,t,w)=\left(\mathfrak{m}(s,t,w)r^{3}-2\right)\sin
t\cos w,\\\ \\\
C_{4}r^{3}\mathfrak{n}(s,t,w)=\left(\mathfrak{m}(s,t,w)r^{3}-2\right)\sin
w,\end{array}\right\\}$ (3.16)

respectively. Since the functions $C_{i}$ depend only on $s$, there is no
solution for functions $\mathfrak{n}(s,t,w)$ in (3.16).

Now, let us assume that $C_{2}=0$ in (3.15). In this case, from the second
equation of (3.10), it’s easy to see that

$\mathfrak{m}(s,t,w)=\frac{2\left(\cos t\cos
w+rk_{1}\left(\cos(2t)+3rk_{1}\cos^{3}t\cos^{3}w+2\cos^{2}t\cos(2w)\right)\right)}{r^{3}(1+rk_{1}\cos
t\cos w)^{2}\cos t\cos w}{.}$ (3.17)

Here, when the equation (3.17) is successively substituted into the third and
fourth equations of (3.10), we obtain

$\left.\begin{array}[c]{l}\mathfrak{n}(s,t,w)C_{3}=\frac{-2k_{1}}{r^{2}\left(1+r\cos
t\cos wk_{1}\right)^{2}}\tan t,\\\ \\\
\mathfrak{n}(s,t,w)C_{4}=\frac{-2k_{1}}{r^{2}\left(1+rk_{1}\cos t\cos
w\right)^{2}}\sec t\tan w.\end{array}\right.$

Here, there is no solution for functions $\mathfrak{n}(s,t,w)$.

Hence, we can state the following theorem:

######  Theorem 1.

There are no tubular hypersurfaces (3.1), obtained by pseudo hyperspheres
whose centers lie on a timelike curve in $\mathbb{E}^{4}_{1}$, with
generalized $\mathcal{L}_{1}$ 1-type Gauss map.

Now, let us classificate the tubular hypersurfaces $\mathcal{T}(s,t,w)$ which
have second kind $\mathcal{L}_{1}$-pointwise 1-type Gauss map, i.e.,
$\mathcal{L}_{1}N=\mathfrak{m}\left(N+C\right).$

From (3.2) and (3.8), we get

$\left.\begin{array}[c]{l}-\frac{2\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r(1+rk_{1}\cos t\cos w)^{3}}=\mathfrak{m}C_{1},\\\ \\\
-\frac{2\left(rk_{1}\left(3rk_{1}\cos^{3}t\cos^{3}w+2\cos^{2}t\cos(2w)+\cos(2t)\right)+\cos
t\cos w\right)}{r^{3}(1+rk_{1}\cos t\cos w)^{2}}=\mathfrak{m}\left({-\cos
t\cos w+}C_{2}\right),\\\ \\\ -\frac{2(1+3rk_{1}\cos t\cos w)\sin t\cos
w}{r^{3}(1+rk_{1}\cos t\cos w)}=\mathfrak{m}\left(-{\sin t\cos
w+}C_{3}\right),\\\ \\\ -\frac{2(1+3rk_{1}\cos t\cos w)\sin
w}{r^{3}(1+rk_{1}\cos t\cos w)}=\mathfrak{m}\left(-{\sin
w+}C_{4}\right).\end{array}\right\\}$ (3.18)

Here, from the fourth equation of (3.18) it’s easy to see that

$\mathfrak{m}(s,t,w)={-\frac{2\left(1+3rk_{1}\cos t\cos w\right)\sin
w}{r^{3}(1+rk_{1}\cos t\cos w)\left(-{\sin w+}C_{4}\right)}.}$ (3.19)

When the equation (3.19) is successively substituted into the second and third
equations of (3.18), we obtain

$\displaystyle 3C_{2}r^{2}k_{1}^{2}\cos^{2}t\cos^{2}w+4C_{2}rk_{1}\cos t\cos
w+C_{2}-rk_{1}=0,$ $\displaystyle C_{4}\sin t\cos w-C_{3}\sin w=0.$

So, we have

$k_{1}=C_{2}=C_{3}=C_{4}=0.$ (3.20)

Now, when the components of the equation (3.20) is substituted into the second
or third equations of (3.18), we calculated

$\mathfrak{m}(s,t,w)={\frac{2}{r^{3}}.}$ (3.21)

Also, from the first equation of (3.18) and (3.21), we have $C_{1}=0.$

From the calculations made above for classificate the tubular hypersurfaces
$\mathcal{T}(s,t,w)$ which have second kind $\mathcal{L}_{1}$-pointwise 1-type
Gauss map, i.e., $\mathcal{L}_{1}N=\mathfrak{m}\left(N+C\right)$, we can give
the following theorem:

######  Theorem 2.

There are no tubular hypersurfaces (3.1), obtained by pseudo hyperspheres
whose centers lie on a timelike curve in $\mathbb{E}^{4}_{1}$, with second
kind $\mathcal{L}_{1}$-pointwise 1-type Gauss map.

Moreover, if the function $m$ is constant in Definition 1 (ii or iii), then we
say $M$ has first or second kind $\mathcal{L}_{k}$-(global) pointwise 1-type
Gauss map. Thus, we can state the following theorem:

######  Theorem 3.

The tubular hypersurfaces (3.1), obtained by pseudo hyperspheres whose centers
lie on a timelike curve in $\mathbb{E}^{4}_{1}$, have first kind
$\mathcal{L}_{1}$-(global) pointwise 1-type Gauss map, i.e.,
$\mathcal{L}_{1}N=\mathfrak{m}N$ if and only if $k_{1}=0,$ where
$\mathfrak{m}(s,t,w)={\frac{2}{r^{3}}.}$

Finally, in the equation (3.8), since the coefficients of
$F_{1},~{}F_{2},~{}F_{3}$ and $F_{4}$ cannot all be zero, we can give the
following theorem:

######  Theorem 4.

The tubular hypersurfaces (3.1), obtained by pseudo hyperspheres whose centers
lie on a timelike curve in $\mathbb{E}^{4}_{1}$, cannot have
$\mathcal{L}_{1}$-harmonic Gauss map.

###  3.2. Some Classifications for Tubular Hypersurfaces Generated by Timelike
Curves with $\mathcal{L}_{2}$ Operator in $\mathbb{E}^{4}_{1}$

Firstly, it is calculated from (2.11), (3.4) and (3.6) as

$\displaystyle\nabla a_{3}=$ $\displaystyle\frac{\left(k_{1}^{\prime}\cos
t+k_{1}k_{2}\sin t\right)\cos w}{r^{2}(1+rk_{1}\cos t\cos
w)^{3}}{}F_{1}+\frac{k_{1}\left(2\cos^{2}t\cos(2w)+\cos(2t)-3\right)}{4r^{3}(1+rk_{1}\cos
t\cos w)^{2}}{}F_{2}$ $\displaystyle+\frac{k_{1}\sin t\cos
t\cos^{2}w}{r^{3}(1+rk_{1}\cos t\cos w)^{2}}{}F_{3}+\frac{k_{1}\cos t\sin
w\cos w}{r^{3}(1+rk_{1}\cos t\cos w)^{2}}{}F_{4}.$ (3.22)

So, from (2.1), (2.2), (3.2), (3.3), (3.6) and (3.22), we have

$\displaystyle\mathcal{L}_{2}N$ $\displaystyle=\frac{\left(k_{1}^{\prime}\cos
t+k_{1}k_{2}\sin t\right)\cos w}{r^{2}(1+rk_{1}\cos t\cos w)^{3}}{}F_{1}$
$\displaystyle+\frac{k_{1}\left(\begin{array}[c]{l}\frac{rk_{1}}{2}\left(\begin{array}[c]{l}24rk_{1}\cos^{4}t\cos^{4}w+12\cos^{3}t\cos(3w)\\\
+19\cos t\cos w+9\cos(3t)\cos w\end{array}\right)\\\
+6\cos^{2}t\cos(2w)+3\cos(2t)-1\end{array}\right)}{4r^{3}(1+rk_{1}\cos t\cos
w)^{3}}{}F_{2}$ (3.27) $\displaystyle+\frac{3k_{1}\sin t\cos
t\cos^{2}w}{r^{3}(1+rk_{1}\cos t\cos w)}{}F_{3}+\frac{3k_{1}\cos t\sin w\cos
w}{r^{3}(1+rk_{1}\cos t\cos w)}{}F_{4}.$ (3.28)

Now, let us give some classifications for the tubular hypersurfaces (3.1)
which have generalized $\mathcal{L}_{2}$ 1-type Gauss map, first kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map and second kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map and $\mathcal{L}_{2}$-harmonic
Gauss map.

Now, let us classificate the tubular hypersurfaces $\mathcal{T}(s,t,w)$ which
have generalized $\mathcal{L}_{2}$ 1-type Gauss map. From (3.2) and (3.28), we
get

$\left.\begin{array}[c]{l}\frac{\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r^{2}(1+rk_{1}\cos t\cos w)^{3}}=\mathfrak{n}C_{1},\\\ \\\
\frac{k_{1}\left(\begin{array}[c]{l}\frac{rk_{1}}{2}\left(\begin{array}[c]{l}24rk_{1}\cos^{4}t\cos^{4}w+12\cos^{3}t\cos(3w)\\\
+19\cos t\cos w+9\cos(3t)\cos w\end{array}\right)\\\
+6\cos^{2}t\cos(2w)+3\cos(2t)-1\end{array}\right)}{4r^{3}(1+rk_{1}\cos t\cos
w)^{3}}=\mathfrak{m}\left({-\cos t\cos w}\right)+\mathfrak{n}C_{2},\\\ \\\
\frac{3k_{1}\sin t\cos t\cos^{2}w}{r^{3}(1+rk_{1}\cos t\cos
w)}=\mathfrak{m}\left(-{\sin t\cos w}\right)+\mathfrak{n}C_{3},\\\ \\\
\frac{3k_{1}\cos t\sin w\cos w}{r^{3}(1+rk_{1}\cos t\cos
w)}=\mathfrak{m}\left(-{\sin w}\right)+\mathfrak{n}C_{4}.\end{array}\right\\}$
(3.29)

Firstly, let us assume that $C_{1}\neq 0$.

In this case, from the first equation of (3.29) it’s easy to see that

$\mathfrak{n}(s,t,w)=\frac{\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r^{2}(1+rk_{1}\cos t\cos w)^{3}C_{1}}{.}$ (3.30)

Here, when the equation (3.30) is successively substituted into the second,
third and fourth equations of (3.29), we obtain

$\left.\begin{array}[c]{l}\mathfrak{m}(s,t,w)=\frac{\left(\begin{array}[c]{l}2k_{1}(-3C_{1}\cos
t\cos w+C_{1}\sec t\sec w+C_{2}rk_{2}\tan t)+2C_{2}rk_{1}^{\prime}\\\
-6C_{1}r^{2}k_{1}^{3}\cos^{3}t\cos^{3}w-C_{1}rk_{1}^{2}\left(6\cos^{2}t\cos(2w)+3\cos(2t)+1\right)\end{array}\right)}{2C_{1}r^{3}(1+rk_{1}\cos
t\cos w)^{3}},\\\ \\\ \mathfrak{m}(s,t,w)=\frac{C_{3}r\left(k_{1}^{\prime}\cot
t+k_{1}k_{2}\right)-3C_{1}k_{1}(1+rk_{1}\cos t\cos w)^{2}\cos t\cos
w}{C_{1}r^{3}(1+rk_{1}\cos t\cos w)^{3}},\\\ \\\
\mathfrak{m}(s,t,w)=\frac{C_{4}r\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cot w-3C_{1}k_{1}(1+rk_{1}\cos t\cos w)^{2}\cos t\cos
w}{C_{1}r^{3}(1+rk_{1}\cos t\cos w)^{3}}.\end{array}\right.$

When we equate the functions $\mathfrak{m}(s,t,w)$ found above to each other,
we arrive at the following equations:

$\displaystyle(C_{4}\sin t\cos w-C_{3}\sin w)\left(k_{1}^{\prime}\cos
t+k_{1}k_{2}\sin t\right)=0,$ (3.31) $\displaystyle k_{1}(C_{1}\sec t\sec
w+rk_{2}(C_{2}\tan t-C_{4}\sin t\cot
w))+C_{1}rk_{1}^{2}+rk_{1}^{\prime}(C_{2}-C_{4}\cos t\cot w){\small=0,}$
(3.32) $\displaystyle k_{1}(C_{1}\sec t\sec w+rk_{2}(C_{2}\tan
t-C_{3}))+C_{1}rk_{1}^{2}+rk_{1}^{\prime}(C_{2}-C_{3}\cot t)=0.$ (3.33)

In the equation (3.31), it holds that $k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\neq 0$. This is because when $k_{1}^{\prime}\cos t+k_{1}k_{2}\sin t=0,$ the
function $\mathfrak{n}(s,t,w)$ in the first equation of (3.29) becomes zero.
This, in turn, contradicts the definition of the function
$\mathfrak{n}(s,t,w)$ in our classification as
$\mathcal{L}_{2}N=\mathfrak{m}N+\mathfrak{n}C$. So, from the equation (3.31)
and $k_{1}^{\prime}\cos t+k_{1}k_{2}\sin t\neq 0,$ we have $C_{3}=C_{4}=0.$
When $C_{3}=C_{4}=0$, substituting this into the equation (3.33) yields

$r\left(C_{1}k_{1}^{2}+C_{2}k_{1}^{\prime}\right)\cos t+C_{2}rk_{1}k_{2}\sin
t+C_{1}k_{1}\sec w=0.$

Thus, we have

$C_{1}k_{1}^{2}+C_{2}k_{1}^{\prime}=C_{1}k_{1}=C_{2}k_{1}k_{2}=0$

and so $C_{1}=C_{2}=0.$ This is a contradiction.

Secondly, let us assume that $C_{1}=0$.

In this case, from the first equation of the set of equations (3.9) it’s easy
to see that

$C_{2}k_{1}=0{.}$ (3.34)

If $k_{1}=0$ in (3.34), then from the second, third and fourth equations of
(3.29), it is calculated as

$\left.\begin{array}[c]{l}\mathfrak{m}(s,t,w)\cos t\cos
w=\mathfrak{n}(s,t,w)C_{2},\\\ \mathfrak{m}(s,t,w)\sin t\cos
w=\mathfrak{n}(s,t,w)C_{3},\\\ \mathfrak{m}(s,t,w)\sin
w=\mathfrak{n}(s,t,w)C_{4},\end{array}\right\\}$ (3.35)

respectively. Since the functions $C_{i}$ depend only on $s$, there is no
solution for functions $\mathfrak{m}(s,t,w)$ and $\mathfrak{n}(s,t,w)$ in
(3.35).

Now, let us assume that $C_{2}=0$ in (3.34). In this case, from the second
equation of (3.29), it’s easy to see that

$\mathfrak{m}(s,t,w)=-\frac{k_{1}\left(\begin{array}[c]{l}rk_{1}\left(\begin{array}[c]{l}24rk_{1}\cos^{4}t\cos^{4}w+12\cos^{3}t\cos(3w)\\\
+19\cos t\cos w+9\cos(3t)\cos w\end{array}\right)\\\
+12\cos^{2}t\cos(2w)+6\cos(2t)-2\end{array}\right)}{8r^{3}(1+rk_{1}\cos t\cos
w)^{3}\cos t\cos w}{.}$ (3.36)

Here, when the equation (3.36) is successively substituted into the third and
fourth equations of (3.29), we obtain

$\left.\begin{array}[c]{l}\mathfrak{n}(s,t,w)C_{3}=\frac{k_{1}(rk_{1}\sin
t\cos w+\tan t)}{r^{3}(1+rk_{1}\cos t\cos w)^{3}},\\\ \\\
\mathfrak{n}(s,t,w)C_{4}=\frac{k_{1}(rk_{1}\sin w+\sec t\tan
w)}{r^{3}(1+rk_{1}\cos t\cos w)^{3}}.\end{array}\right.$

Here, there is no solution for functions $\mathfrak{n}(s,t,w)$.

Therefore, we can give the following theorem:

######  Theorem 5.

There are no tubular hypersurfaces (3.1), obtained by pseudo hyperspheres
whose centers lie on a timelike curve in $\mathbb{E}^{4}_{1}$, with
generalized $\mathcal{L}_{2}$ 1-type Gauss map.

Now, let us classificate the tubular hypersurfaces $\mathcal{T}(s,t,w)$ which
have second kind $\mathcal{L}_{2}$-pointwise 1-type Gauss map, i.e.,
$\mathcal{L}_{2}N=\mathfrak{m}\left(N+C\right).$

From (3.2) and (3.28), we get

$\left.\begin{array}[c]{l}\frac{\left(k_{1}^{\prime}\cos t+k_{1}k_{2}\sin
t\right)\cos w}{r^{2}(1+rk_{1}\cos t\cos w)^{3}}=\mathfrak{m}C_{1},\\\ \\\
\frac{k_{1}\left(\begin{array}[c]{l}\frac{rk_{1}}{2}\left(\begin{array}[c]{l}24rk_{1}\cos^{4}t\cos^{4}w+12\cos^{3}t\cos(3w)\\\
+19\cos t\cos w+9\cos(3t)\cos w\end{array}\right)\\\
+6\cos^{2}t\cos(2w)+3\cos(2t)-1\end{array}\right)}{4r^{3}(1+rk_{1}\cos t\cos
w)^{3}}=\mathfrak{m}\left({-\cos t\cos w+}C_{2}\right),\\\ \\\
\frac{3k_{1}\sin t\cos t\cos^{2}w}{r^{3}(1+rk_{1}\cos t\cos
w)}=\mathfrak{m}\left(-{\sin t\cos w+}C_{3}\right),\\\ \\\ \frac{3k_{1}\cos
t\sin w\cos w}{r^{3}(1+rk_{1}\cos t\cos w)}=\mathfrak{m}\left(-{\sin
w+}C_{4}\right).\end{array}\right\\}$ (3.37)

Here, from the last equation of (3.37) it’s easy to see that

$\mathfrak{m}(s,t,w)=\frac{3k_{1}\cos t\sin w\cos w}{r^{3}(1+rk_{1}\cos t\cos
w)\left(-{\sin w+}C_{4}\right)}{.}$ (3.38)

Here, when the equation (3.38) is substituted into the second equation of
(3.37), we obtain

$-1+3C_{2}\cos t\cos w+3C_{2}rk_{1}\cos^{2}t\cos^{2}w=0.$

Since the last equality is never zero, we can give the following theorem:

######  Theorem 6.

There are no tubular hypersurface (3.1), obtained by pseudo hyperspheres whose
centers lie on a timelike curve in $\mathbb{E}^{4}_{1}$, with second kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map.

Now, let us classificate the tubular hypersurfaces $\mathcal{T}(s,t,w)$ which
have first kind $\mathcal{L}_{2}$-pointwise 1-type Gauss map, i.e.,
$\mathcal{L}_{2}N=\mathfrak{m}N.$

From (3.2) and (3.28), we get

$\left.\begin{array}[c]{l}\frac{\cos w\left(\cos
tk_{1}^{\prime}+k_{1}k_{2}\sin t\right)}{r^{2}(1+rk_{1}\cos t\cos
w)^{3}}=0,\\\ \\\
\frac{k_{1}\left(\begin{array}[c]{l}\frac{1}{2}rk_{1}\left(\begin{array}[c]{l}24rk_{1}\cos^{4}t\cos^{4}w+12\cos^{3}t\cos(3w)\\\
+19\cos t\cos w+9\cos(3t)\cos w\end{array}\right)\\\
+6\cos^{2}t\cos(2w)+3\cos(2t)-1\end{array}\right)}{4r^{3}(1+rk_{1}\cos t\cos
w)^{3}}=\mathfrak{m}\left({-\cos t\cos w}\right),\\\ \\\ \frac{3k_{1}\sin
t\cos t\cos^{2}w}{r^{3}(1+rk_{1}\cos t\cos w)}=\mathfrak{m}\left(-{\sin t\cos
w}\right),\\\ \\\ \frac{3k_{1}\cos t\sin w\cos w}{r^{3}(1+rk_{1}\cos t\cos
w)}=\mathfrak{m}\left(-{\sin w}\right).\end{array}\right\\}$ (3.39)

Here, from the last equation of (3.39) it’s easy to see that

$\mathfrak{m}(s,t,w)=\frac{-3k_{1}\cos t\cos w}{r^{3}(1+rk_{1}\cos t\cos
w)}{.}$ (3.40)

Here, when the equation (3.40) is substituted into the second equation of
(3.37), we obtain

$k_{1}(rk_{1}+\sec t\sec w)=0.$

Since the last equality is never zero, we can give the following theorem:

######  Theorem 7.

There are no tubular hypersurfaces (3.1), obtained by pseudo hyperspheres
whose centers lie on a timelike curve in $\mathbb{E}^{4}_{1}$, with first kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map.

Finally, since the coefficients $F_{1},~{}F_{2},~{}F_{3}$ and $F_{4}$ in
equation (3.28) are all zero only for $k_{1}=0$, we can give the following
theorem:

######  Theorem 8.

The tubular hypersurfaces (3.1), obtained by pseudo hyperspheres whose centers
lie on a timelike curve in $\mathbb{E}^{4}_{1}$, have
$\mathcal{L}_{2}$-harmonic Gauss map if and only if $k_{1}=0$.

##  4\. Some Classifications for Tubular Hypersurfaces Generated by Spacelike
Curves with $\mathcal{L}_{k}$ Operators in $\mathbb{E}^{4}_{1}$

In this section, we give the general formulas for $\mathcal{L}_{1}$ (Cheng-
Yau) and $\mathcal{L}_{2}$ operators of the Gauss maps of the six types of
tubular hypersurfaces $\mathcal{T}^{\\{j,\lambda\\}}(s,t,w)$ that are formed
as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic
hyperspheres whose centers lie on spacelike curves $\beta(s)$ with non-null
Frenet vectors in $\mathbb{E}^{4}_{1}$ and give some classifications for these
hypersurfaces which have generalized $\mathcal{L}_{k}$ 1-type Gauss map, first
kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and second kind
$\mathcal{L}_{k}$-pointwise 1-type Gauss map and $\mathcal{L}_{k}$-harmonic
Gauss map, $k\in\\{1,2\\}$.

The tubular hypersurfaces $\mathcal{T}^{\\{j,\lambda\\}}(s,t,w)$ that are
formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic
hyperspheres whose centers lie on a spacelike curve with non-null Frenet
vectors $F_{i}$ in $\mathbb{E}^{4}_{1}$ can be parametrized by

$\left.\begin{array}[c]{l}\mathcal{T}^{\\{2,1\\}}(s,t,w)=\beta(s)+r(\cosh
t\sinh wF_{2}(s)+\cosh wF_{3}(s)+\sinh t\sinh wF_{4}(s)),\\\
\mathcal{T}^{\\{2,-1\\}}(s,t,w)=\beta(s)+r(\cosh t\cosh wF_{2}(s)+\sinh
wF_{3}(s)+\sinh t\cosh wF_{4}(s)),\\\
\mathcal{T}^{\\{3,1\\}}(s,t,w)=\beta(s)+r(\sinh t\sinh wF_{2}(s)+\cosh t\sinh
wF_{3}(s)+\cosh wF_{4}(s)),\\\
\mathcal{T}^{\\{3,-1\\}}(s,t,w)=\beta(s)+r(\sinh t\cosh wF_{2}(s)+\cosh t\cosh
wF_{3}(s)+\sinh wF_{4}(s)),\\\ \mathcal{T}^{\\{4,1\\}}(s,t,w)=\beta(s)+r(\cosh
wF_{2}(s)+\sinh t\sinh wF_{3}(s)+\cosh t\sinh wF_{4}(s)),\\\
\mathcal{T}^{\\{4,-1\\}}(s,t,w)=\beta(s)+r(\sinh wF_{2}(s)+\sinh t\cosh
wF_{3}(s)+\cosh t\cosh wF_{4}(s)),\end{array}\right\\}$ (4.1)

respectively. Here, we suppose for $\mathcal{T}^{\\{j;\lambda\\}}(s,t,w)$ that

i) $\left\langle F_{j},F_{j}\right\rangle=-1=\varepsilon_{j}$ and for $i\neq
j,$ $\left\langle F_{i},F_{i}\right\rangle=1=\varepsilon_{i},$
$i,j\in\\{1,2,3,4\\},$

ii) if the tubular hypersurface is foliated by pseudo hyperspheres or pseudo
hyperbolic hyperspheres, then $\lambda=1$ or $\lambda=-1,$ respectively (for
more details, one can see [3]).

Now, let us write the following lemma which states the general parametric
expressions of 6 different types of tubular hypersurfaces given by (4.1) and
obtained by pseudo hyperspheres and pseudo hyperbolic hyperspheres whose
centers lie on a spacelike curve with non-null Frenet vector fields in
$\mathbb{E}^{4}_{1}$.

######  Lemma 1.

The general expression of the tubular hypersurfaces
$\mathcal{T}^{\\{j,\lambda\\}}(s,t,w)$ that are formed as the envelope of a
family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers
lie on a spacelike curve $\beta(s)$ with non-null Frenet vectors $F_{i}(s)$ in
$\mathbb{E}^{4}_{1}$ can be given by

$\mathcal{T}^{\\{j,\lambda\\}}(s,t,w)=\beta(s)+r\left({\textstyle\sum_{i=2}^{4}}\mu_{i}^{\lambda}(s,t,w)F_{i}(s)\right),$
(4.2)

where

$\mu_{5}^{\lambda}(s,t,w)=\mu_{2}^{\lambda}(s,t,w),\text{
}\mu_{6}^{\lambda}(s,t,w)=\mu_{3}^{\lambda}(s,t,w)$

and for $j=2,3,4$

$\left.\begin{array}[c]{l}\mu_{j}^{\lambda}(s,t,w)=\left(\sinh
w\right)^{\frac{1+\lambda}{2}}\left(\cosh w\right)^{\frac{1-\lambda}{2}}\cosh
t,\\\ \\\ \mu_{j+1}^{\lambda}(s,t,w)=\left(\sinh
w\right)^{\frac{1-\lambda}{2}}\left(\cosh w\right)^{\frac{1+\lambda}{2}},\\\
\\\ \mu_{j+2}^{\lambda}(s,t,w)=\left(\sinh
w\right)^{\frac{1+\lambda}{2}}\left(\cosh w\right)^{\frac{1-\lambda}{2}}\sinh
t.\end{array}\right.\text{ }$

Here, if the canal hypersurface is foliated by pseudo hyperspheres or pseudo
hyperbolic hyperspheres, then $\lambda=1$ or $\lambda=-1,$ respectively.

Here, we can give the general parametric expressions of the unit normal vector
fields, the coefficients of the first fundamental forms and the principal
curvatures of the tubular hypersurfaces $\mathcal{T}^{\\{j,\lambda\\}}$
parametrized by (4.2).

The unit normal vector fields $N^{\\{j,\lambda\\}}$ $(j=2,3,4)$ of (4.2) are

$N^{\\{j,\lambda\\}}=-(-1)^{(4-j)!}\lambda^{j}{\textstyle\sum_{i=2}^{4}}\mu_{i}^{\lambda}F_{i}.$
(4.3)

The coefficients of the first fundamental forms $g_{ik}^{\\{j,\lambda\\}}$
$(j=2,3,4)$ of (4.2) are

$\left.\begin{array}[c]{l}g_{11}^{\\{j,\lambda\\}}=1+r^{2}(k_{2})^{2}\left(-(-1)^{(4-j)!}\left(\mu_{3}^{\lambda}\right)^{2}+(-1)^{j}\left(\mu_{2}^{\lambda}\right)^{2}\right)\\\
\text{ \ \ \ \ \ \ \ \ \
}+r^{2}(k_{3})^{2}\left((-1)^{(5-j)!}\left(\mu_{3}^{\lambda}\right)^{2}+(-1)^{j}\left(\mu_{4}^{\lambda}\right)^{2}\right)\\\
\text{ \ \ \ \ \ \ \ \ \
}+2(-1)^{(4-j)!}rk_{1}\mu_{2}^{\lambda}+r^{2}(k_{1})^{2}\left(\mu_{2}^{\lambda}\right)^{2}-2(-1)^{(5-j)!}r^{2}k_{2}k_{3}\mu_{2}^{\lambda}\mu_{4}^{\lambda},\\\
\\\
g_{12}^{\\{j,\lambda\\}}=g_{21}^{\\{j,\lambda\\}}=r^{2}(\mu_{j+1}^{\lambda})_{w}\left((-1)^{j}k_{3}\left(\mu_{2}^{\lambda}\right)_{w}-(-1)^{(4-j)!}k_{2}\left(\mu_{4}^{\lambda}\right)_{w}\right),\\\
\\\
g_{22}^{\\{j,\lambda\\}}=r^{2}\left((\mu_{j+1}^{\lambda})_{w}\right)^{2},\\\
\\\ g_{13}^{\\{2,\lambda\\}}=g_{31}^{\\{2,\lambda\\}}=\lambda
r^{2}\left(-k_{2}\cosh t+k_{3}\sinh t\right),\\\
g_{13}^{\\{3,\lambda\\}}=g_{31}^{\\{3,\lambda\\}}=-\lambda r^{2}k_{3}\cosh
t,\\\ g_{13}^{\\{4,\lambda\\}}=g_{31}^{\\{4,\lambda\\}}=\lambda
r^{2}k_{2}\sinh t,\\\ \\\
g_{23}^{\\{j,\lambda\\}}=g_{32}^{\\{j,\lambda\\}}=0,\\\ \\\
g_{33}^{\\{j,\lambda\\}}=-\lambda r^{2}.\end{array}\right\\}$ (4.4)

The principal curvatures $\kappa_{i}^{\\{j,\lambda\\}}$ $(j=2,3,4)$ of (4.2)
are

$\left.\begin{array}[c]{l}\kappa_{1}^{\\{j,\lambda\\}}=\kappa_{2}^{\\{j,\lambda\\}}=\frac{(-1)^{(4-j)!}\lambda^{j}}{r},\\\
\\\
\kappa_{3}^{\\{j,\lambda\\}}=\frac{k_{1}\mu_{2}^{\lambda}}{\lambda^{j}\left(1+(-1)^{(4-j)!}rk_{1}\mu_{2}^{\lambda}\right)}.\end{array}\right\\}$
(4.5)

From Lemma 1, (4.3), (4.4) and (4.5), we get

$\displaystyle\mathcal{L}_{1}N^{\\{j,\lambda\\}}$
$\displaystyle=\frac{2\left(-(-1)^{(5-j)!}k_{1}k_{2}\mu_{3}^{\lambda}+k_{1}^{\prime}\mu_{2}^{\lambda}\right)}{r\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)^{3}}F_{1}$
$\displaystyle+\frac{-2\lambda\left(\mu_{2}^{\lambda}+3r^{2}(\mu_{2}^{\lambda})^{3}\left(k_{1}\right)^{2}+k_{1}\left(\lambda
r+4(-1)^{(4-j)!}r(\mu_{2}^{\lambda})^{2}\right)\right)}{r^{3}\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)^{2}}F_{2}$
$\displaystyle+\frac{-2\lambda(-1)^{(4-j)!}\mu_{3}^{\lambda}\left(1+3(-1)^{(4-j)!}rk_{1}\mu_{2}^{\lambda}\right)}{r^{3}\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)}F_{3}+\frac{-2\lambda\mu_{4}^{\lambda}\left((-1)^{(4-j)!}+3rk_{1}\mu_{2}^{\lambda}\right)}{r^{3}\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)}F_{4}.$
(4.6)

Let $\mathcal{T}^{\left\\{j,\lambda\right\\}}(s,t,w)$ have generalized $L_{1}$
(Cheng-Yau) 1-type Gauss map, i.e.,
$\mathcal{L}_{1}N^{\\{j,\lambda\\}}=\mathfrak{m}N^{\\{j,\lambda\\}}+\mathfrak{n}C,$
where $C=C_{1}F_{1}+C_{2}F_{2}+C_{3}F_{3}+C_{4}F_{4}$ is a constant vector.
Here, by taking derivatives of the constant vector $C$ with respect to $s,$
from (2.8)-(2.10) we obtain for $\mathcal{T}^{\left\\{j,\lambda\right\\}}$
that

$\left.\begin{array}[c]{l}C_{1}^{\prime}+(-1)^{(4-j)!}C_{2}k_{1}=0,\\\
C_{2}^{\prime}+C_{1}k_{1}+(-1)^{(5-j)!}C_{3}k_{2}=0,\\\
C_{3}^{\prime}+C_{2}k_{2}-(-1)^{(4-j)!}C_{4}k_{3}=0,\\\
C_{4}^{\prime}+C_{3}k_{3}=0.\end{array}\right\\}$ (4.7)

Also, by taking derivatives the constant vector $C$ with respect to$~{}t,~{}w$
separately, one can see that the functions $C_{i}$ depend only on $s.$

So, with similar procedure in Subsection 3.1, we can give the following
theorems:

######  Theorem 9.

There are no tubular hypersurfaces (4.2), obtained by pseudo hyperspheres and
pseudo hyperbolic hyperspheres whose centers lie on a spacelike curve with
non-null Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$, with
generalized $\mathcal{L}_{1}$ 1-type Gauss map in $\mathbb{E}^{4}_{1}$.

######  Theorem 10.

There are no tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and
pseudo hyperbolic hyperspheres whose centers lie on a spacelike curve with
non-null Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ with second kind
$\mathcal{L}_{1}$-pointwise 1-type Gauss map in $\mathbb{E}^{4}_{1}$.

######  Theorem 11.

The tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and pseudo
hyperbolic hyperspheres whose centers lie on a spacelike curve with non-null
Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ have first kind
$\mathcal{L}_{1}$-(global) pointwise 1-type Gauss map, i.e.,
$\mathcal{L}_{1}N^{\\{j,\lambda\\}}=\mathfrak{m}N^{\\{j,\lambda\\}}$ in
$\mathbb{E}^{4}_{1}$ if and only if $k_{1}=0,$ where
$\mathfrak{m}(s,t,w)={\frac{2\lambda^{j+1}(-1)^{(4-j)!}}{r^{3}}.}$

######  Theorem 12.

The tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and pseudo
hyperbolic hyperspheres whose centers lie on a spacelike curve with non-null
Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ cannot have
$\mathcal{L}_{1}$-harmonic Gauss map.

Also, from Lemma 1, (4.3), (4.4) and (4.5), we get

$\displaystyle\mathcal{L}_{2}N^{\\{j,\lambda\\}}$
$\displaystyle=\frac{\lambda^{j}\left((-1)^{j}\mu_{3}^{\lambda}k_{1}k_{2}-(-1)^{(4-j)!}\mu_{2}^{\lambda}k_{1}^{\prime}\right)}{r^{2}\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)^{3}}F_{1}$
$\displaystyle+\frac{-\lambda^{j+1}k_{1}\left(2\lambda(-1)^{(4-j)!}-3(-1)^{j}\left(\mu_{4}^{\lambda}\right)^{2}-3(-1)^{(5-j)!}\left(\mu_{3}^{\lambda}\right)^{2}-3(-1)^{(4-j)!}rk_{1}\left(\mu_{2}^{\lambda}\right)^{3}\right)}{r^{3}\left((-1)^{(4-j)!}+rk_{1}\mu_{2}^{\lambda}\right)^{2}}F_{2}$
$\displaystyle+\frac{\lambda^{j+1}\mu_{3}^{\lambda}\left(3(-1)^{(5-j)!}rk_{1}\mu_{2}^{\lambda}\right)}{r^{4}\left((-1)^{(5-j)!}+(-1)^{j}rk_{1}\mu_{2}^{\lambda}\right)}F_{3}+\frac{\lambda^{j+1}\mu_{4}^{\lambda}\left(3(-1)^{(5-j)!}rk_{1}\mu_{2}^{\lambda}\right)}{r^{4}\left((-1)^{(5-j)!}+(-1)^{j}rk_{1}\mu_{2}^{\lambda}\right)}F_{4}.$
(4.8)

Thus, with similar procedure in Subsection 3.2, we can give the following
theorems:

######  Theorem 13.

There are no tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and
pseudo hyperbolic hyperspheres whose centers lie on a spacelike curve with
non-null Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ with generalized
$\mathcal{L}_{2}$ 1-type Gauss map in $\mathbb{E}^{4}_{1}$.

######  Theorem 14.

There are no tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and
pseudo hyperbolic hyperspheres whose centers lie on a spacelike curve with
non-null Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ with second kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map in $\mathbb{E}^{4}_{1}$.

######  Theorem 15.

There are no tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and
pseudo hyperbolic hyperspheres whose centers lie on a spacelike curve with
non-null Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ with first kind
$\mathcal{L}_{2}$-pointwise 1-type Gauss map in $\mathbb{E}^{4}_{1}$.

######  Theorem 16.

The tubular hypersurfaces (4.2) obtained by pseudo hyperspheres and pseudo
hyperbolic hyperspheres whose centers lie on a spacelike curve with non-null
Frenet vector fields $F_{i}$ in $\mathbb{E}^{4}_{1}$ have
$\mathcal{L}_{2}$-harmonic Gauss map if and only if $k_{1}=0$.

## References

  * [1] Alias, L., Ferrández, A. and Lucas, P., Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying $\Delta x=Ax+B$, Pacific J. Math. 156(2) (1992), 201-208.
  * [2] Alias, L.J. and Gürbüz, N., An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata, 121(1) (2006), 113-127.
  * [3] Altın, M., Kazan, A. and Yoon, D.W., Canal Hypersurfaces generated by Non-null Curves in Lorentz-Minkowski 4-Space, Bull. Korean Math. Soc. 60(5) (2023), 1299-1320.
  * [4] Chen, B.-Y., _Total Mean Curvature and Submanifold of Finite Type_ , World Scientific Publisher, 1984.
  * [5] Chen, B.-Y., _A report on submanifolds of finite type_ , Soochow J. Math., 22(2) (1996), 117-337.
  * [6] Chen, B.-Y., Morvan, J.-M. and Nore, T., _Energy, tension and finite type maps_ , Kodai Math. J., 9(3) (1986), 406–418.
  * [7] Chen, B.-Y. and Petrovic, M., On spectral decomposition of immersions of finite type, Bull. Aust. Math. Soc. 44(1) (1991), 117-129.
  * [8] Chen, B.-Y. and Piccinni, P. _Submanifolds with Finite Type Gauss Map_ , Bull. Austral. Math. Soc., 35(2) (1987), 161-186.
  * [9] Cheng, S.-Y. and Yau, S.-T., Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195-204.
  * [10] Choi, M. and Kim, Y.H., _Characterization of the helicoid as ruled surface with pointwise 1-type Gauss map_ , Bull. Korean Math. Soc., 38(4) (2001), 753-761.
  * [11] Kazan, A., Altın, M. and Turgay, N.C., Rotational hypersurfaces in $\mathbb{E}^{4}_{1}$ with Generalized $L_{k}$ 1-Type Gauss Map, arXiv:2403.19671v1, (2024).
  * [12] Kelleci, A. ,Rotational surfaces with Cheng-Yau operator in Galilean 3-spaces, Hacet. J. Math. Stat. 50(2) (2021), 365-376.
  * [13] Kim, D-S., Kim, J.R. and Kim, Y.H., Cheng-Yau Operator and Gauss Map of Surfaces of Revolution, Bull. Malays. Math. Sci. Soc. 39(4) (2016), 1319-1327.
  * [14] Kim, Y.H. and Turgay, N.C., _On the surfaces in $\mathbb{E}^{3}$ with $L_{1}$ pointwise 1-type Gauss map_, (submitted).
  * [15] Kuhnel, W.: Differential geometry: curves-surfaces-manifolds, American Mathematical Soc., Braunschweig, Wiesbaden, 1999.
  * [16] Lucas, P. and Ramírez-Ospina, H.F., Hypersurfaces in the Lorentz-Minkowski space satisfying $L_{k}\psi=A\psi+b$, Geom. Dedicata 153(1) (2011), 151-175.
  * [17] Qian, J., Fu, X. and Jung, S.D., _Dual associate null scrolls with generalized 1-type Gauss maps_ , Mathematics 8(7) (2020), 1111.
  * [18] Qian, J., Fu, X., Tian, X. and Kim, Y.H., _Surfaces of Revolution and Canal Surfaces with Generalized Cheng-Yau 1-Type Gauss Maps_ , Mathematics 8(10) (2020), 1728.
  * [19] Qian, J., Su, M., Kim, Y.H., _Canal surfaces with generalized 1-type Gauss map_ , Rev. Union Mat. Argent. 62(1) (2021), 199-211.
  * [20] Yeğin Şen, R., Dursun, U., _On submanifolds with 2-type pseudo-hyperbolic Gauss map in pseudo-hyperbolic space_ , Mediterr. J. Math. 14(1) (2017), 1-20.
  * [21] Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18(4) (1966), 380-385.
  * [22] Yoon, D.W., _Rotation surfaces with finite type Gauss Map in $E^{4}$_, Indian J. Pure. Appl. Math. 32(12) (2001), 1803-1808.
  * [23] Walrave, J., Curves and surfaces in Minkowski space, Dissertation, K. U. Leuven, Fac. of Science, Leuven, 1995.
  * [24] Yang, B. and Liu, X.,  Hypersurfaces satisfying $L_{r}x=Rx$ in sphere $S^{n+1}$ or hyperbolic space $H^{n+1}$, Proc. Indian Acad. Sci. (Math. Sci.) 119 (2009), no. 4, 487-499.
  * [25] Yoon, D.W., Kim, D.S., Kim, Y.H. and Lee, J.W., _Hypersurfaces with generalized 1-type Gauss maps_ , Mathematics 6(8) (2018), 130.
  * [26] Yoon, D.W., Kim, D.S., Kim, Y.H. and Lee, J.W., _Classifications of flat surfaces with generalized 1-type Gauss map in $\mathbb{L}^{3}$_, Mediterr. J. Math. 15 (2018), 78.