#  $2$-Ruled hypersurfaces in a Walker $4$-manifold

Mohamed Ayatola Dramé*, Ameth Ndiaye**, Abdoul Salam Diallo  *  Université
Alioune Diop de Bambey
UFR SATIC, Département de Mathématiques
Équipe de Recherche en Analyse Non Linéaire et Géométrie (ER ANLG)
B. P. 30, Bambey, Sénégal
<EMAIL_ADDRESS>**  Université Cheik Anta Diop de Dakar
FASTEF, Département de Mathématiques
B. P. 880, Dakar, Sénégal
<EMAIL_ADDRESS>***  Université Alioune Diop de Bambey
UFR SATIC, Département de Mathématiques
Équipe de Recherche en Analyse Non Linéaire et Géométrie (ER ANLG)
B. P. 30, Bambey, Sénégal
<EMAIL_ADDRESS>
###### Abstract.

The hypersurface is one of the most important objects in a space. Many authors
studied diffrent geometric aspects of hypersurfaces in a space. In this paper,
we define three types of $2$-ruled hypersurfaces in a Walker $4$-manfold
$\mathbb{E}^{4}_{1}$. We obtain the Gaussian and mean curvatures of the
$2$-ruled hypersurfaces of type-$1$, type-$2$ and type $3$. We give some
characterizations about its minimality. We also deal with the first Laplace-
Beltrami operators of these types of $2$-ruled hypersurfaces in the considered
Walker $4$-manifold.

###### Key words and phrases:

$2$-ruled hypersurface; Walker manifolds.

###### 2010 Mathematics Subject Classification:

14J70, 53A07, 53A10

##  1\. Introduction

The study of hypersurface of a given ambiant space is a naturel interesting
problem which enriches our knowledge and understanding of the geometry of the
space itself. The theory of ruled surfaces in $\mathbb{R}^{3}$ is a classical
subject in diffrential geometry. The study of ruled surfaces of a given
ambiant space is a naturel and interesting problem. A surface $\Sigma$ in $M$
is said to be ruled if every point of $\Sigma$ is on (a open geodesic segment)
in $M$ that lies in $\Sigma$ (see [21]). Locally a ruled surface is made by a
one parameter family of geodesic segments [7]. Ruled surfaces are one-
parameter set of lines and they are one of the important topics of classifical
differential geometry. A ruled surface is defined as

$\displaystyle\varphi(s,t)=\alpha(s)+tX(s),s,t\in I\subset\mathbb{R},$

where the curve $\alpha(s)$ is called base curve and $X(s)$ is called the
ruling of the ruled surface. A lots of studies have been done about different
characterizations of ruled surfaces in $3$-dimensional Euclidean, Minkowskian,
Galilean and pseudo-Galilean space (see [9, 10, 12, 13, 14, 18] and references
therein). In [2], the authors define a quaternionic operator whose scalar part
is a real parameter and vector part is a curve in three dimensional real
vector space $\mathbb{R}^{3}$. They prove that quaternion product of this
operator and a spherical curve represents a ruled surface in $\mathbb{R}^{3}$
if the vector part of the quaternionic operator is perpendicular to the
position vector of the spherical curve. Also in [3], the authors show that the
split quaternion product of a split quaternion operator and a curve, which
lies on Lorentzian unit sphere or on hyperbolic unit sphere, parametrizes a
ruled surface in the $3$-dimensional Minkowski space $\mathbb{E}^{3}_{1}$ if
the vector part of the operator is perpendicular to the position vector of the
spherical curve. Recently, in [19], the authors construct two special families
of ruled surfaces in a three dimensional strict Walker manifold. They show
that the local degeneracy (resp. non-degeneracy) to one of this family has a
strong consequence on the geometry of the ambiant Walker manifold. Ruled
hypersurfaces in higher dimensions have also been studied by many authors [4,
5]. In [6], the intrinsic classification of irreducible ruled hypersurfaces of
$\mathbb{R}^{4}$ has been given. In [16], a new approach to investigating
ruled real hypersurfaces in complex hyperbolic space $\mathbb{CH}^{n}$ is
given. In the paper [15], the authors study ruled real hypersurfaces in the
complex quadric.

A $2$-ruled hypersurface in $\mathbb{R}^{4}$ is a one-parameter family of
planes in $\mathbb{R}^{4}$. This is a generalization of ruled surfaces in
$\mathbb{R}^{3}$. In [25], the author study singularities of $2$-ruled
hypersurfaces in Euclidean $4$-space. After defining a non-degenerate
$2$-ruled hypersurface, he gives a necessary and sufficient condition for such
a map germ to be right-left equivalent to the cross cap $\times$ interval.
Also, the author in [25] discusses the behavior of a generic $2$-ruled
hypersurface map. In [1] the authors obtain the Gauss map (unit normal vector
field) of a $2$-ruled hypersurface in Euclidean $4$-space with the aid of its
general parametric equation. They also obtain Gaussian and mean curvatures of
the $2$-ruled hypersurface and they give some characterizations about its
minimality. Finally, they deal with the first and second Laplace-Beltrami
operators of $2$-ruled hypersurfaces in $\mathbb{E}^{4}$. Recently, in [17]
the authors define three types of $2$-ruled hypersurfaces in the Minkowski
$4$-space $E^{4}_{1}$. They obtain Gaussian and mean curvatures of the
$2$-ruled hypersurfaces of type-$1$ and type-$2$, and some characterizations
about its minimality. They also deal with the first Laplace-Beltrami operators
of these types of $2$-ruled hypersurfaces in $E^{4}_{1}$.

Motivated by the above two works, we study in this paper the $2$-ruled
hypersurfaces in a Walker $4$-manifold. We define three types of $2$-ruled
hypersurfaces and we gives Gaussian and mean curvatures of the $2$-ruled
hypersurface and some characterizations about its minimality. Our paper is
organized as follows. Section 1 introduces the topic. In section 2, we recall
some basics notions on pseudo-Riemannian manifolds. In section 3, we study
$2$-ruled ypersurface on a Walker $4$-manifold.

##  2\. Preliminaries

In this section, we recall some basics notions on pseudo-Riemannian manifolds
taken from the book [11]. We begin with some algebraic preliminaries on non-
degenerate bilinear forms on an $m$-dimensional real vector space $V$.

Let $g:V\times V\to\mathbb{R}$ be a symmetric bilinear form. We say that $g$
is non-degenerate if $g(u,v)=0$ for each $v\in V$ implies $u=0$, otherwise $g$
is called degenerate. A non-degenerate symmetric bilinear form on $V$ is
called a pseudo-Euclidean metric on $V$. It may induce either a non-degenerate
or a degenerate symmetric bilinear form on a subspace $W$ of $V$; then $W$ is
said to be a non-degenerate or a degenerate subspace, respectively. We say
that $g$ is positive (negative) definite provided that $u\neq 0$ implies
$g(u,u)>0(<0)$. If $g$ is non-degenerate, there exists an ordered basis
$(e_{1},e_{2},\ldots,e_{m})$ of $V$ such that:

$\displaystyle g(e_{i},e_{i})$ $\displaystyle=$ $\displaystyle-1,\quad 1\leq
i\leq q,$ $\displaystyle g(e_{i},e_{i})$ $\displaystyle=$ $\displaystyle
1,\quad q+1\leq i\leq m,$ $\displaystyle g(e_{i},e_{j})$ $\displaystyle=$
$\displaystyle 0,\quad i\neq j,$

where $q$ is uniquely determined and $(q,m-q)$ is the signature of $g$.
Obviously, in the case $q=0$ or $s=m$, the first or the second condition has
to be dropped. The integer $q$ is called the index of $g$ on $V$ and it is the
largest dimension of a subspace $W\subset V$ on which the induced metric is
negative definite.

A pseudo-Riemannian metric $g$ on an $m$-dimensional manifold $M$ is a
symmetric tensor field of type $(0,2)$ on $M$ such that for any $p\in M$ the
tensor $g_{p}$ is a non-degenerate symmetric bilinear form on the tangent
space $T_{p}M$ of constant index. We call $(M,g)$ a pseudo-Riemannian
manifold. Frequently, we denote by $M^{m}_{q}$ an $m$-dimensional pseudo-
Riemannian manifold of index $q$. In the particular case $m>2$ and $q=1$, we
call $(M,g)$ a Lorentzian manifold. Obviously, if $q=0,(M,g)$ is a Riemannian
manifold.

Let $N^{n}_{s}$ be a submanifold of a pseudo-Riemannian manifold $M^{m}_{q}$.
If the pseudo-Riemannian metric tensor $g_{M}$ of $M^{m}_{q}$ induces a
pseudo-Riemannian metric tensor, a Riemannian metric tensor or a degenerate
metric tensor $g_{N}$ on $N^{n}_{s}$ , then $N^{n}_{s}$ is called a pseudo-
Riemannian submanifold, a Riemannian submanifold or a degenerate submanifold,
respectively, of $M^{m}_{q}$. Let $M^{m}_{q}$ be an $m$-dimensional pseudo-
Riemannian manifold with pseudo- Riemannian metric tensor $g_{M}$ of index
$q$. Denoting by $\langle,\rangle$ the associated nondegenerate inner product
on $M^{m}_{q}$, a tangent vector $X$ to $M^{m}_{q}$ is said to be spacelike if
$\langle X,X\rangle>0$ ( or $X=0$), timelike if $\langle X,X\rangle<0$ or
lightlike (null) if $\langle X,X\rangle=0$ and $X\neq 0$. The set of null
vectors of $T_{p}M$ is called the null cone at $p\in M$.

Let $M^{m}_{1}(c)$ be an $m$-dimensional Lorentzian space form of constant
curvature $c$, that is, $M^{m}_{1}(c)$ is the de Sitter space-time
$\mathbb{S}^{m}_{1}(c)$, Minkowski space-time $\mathbb{R}^{4}_{1}(c)$ or the
anti-de Sitter space-time $\mathbb{H}^{m}_{1}(c)$ according to $c>0,c=0$ or
$c<0$. For simplicity, we suppose that the constant curvature $c$ of
$M^{m}_{1}(c)$ is equal to $1,0,-1$ according to whether $c>0,c=0,c<0$.

Now, we describe some basic examples of pseudo-Riemannian manifolds. Let
$\mathbb{R}^{m}_{q}$ be an $m$-dimensional pseudo-Euclidean space with metric
tensor given by

$\displaystyle g=-\sum_{i=1}^{q}(du_{i})^{2}+\sum_{i=q+1}^{m}(du_{i})^{2},$

where $(u_{1},\ldots,u_{m})$ is a coordinate system of $\mathbb{R}^{m}_{q}$.
So $(\mathbb{R}^{m}_{q},g)$ is a ﬂat pseudo-Riemannian manifold of index $q$.
Putting:

$\displaystyle\mathbb{S}^{m}_{1}(1)=\\{u\in\mathbb{R}^{m+1}_{1},\langle
u,u\rangle=1\\},$

one obtains an $m$-dimensional pseudo-Riemannian manifold of index $q$ and of
constant curvature $c=1$. In the theory of general relativity,
$\mathbb{S}^{4}_{1}(c)$ is called the de Sitter space-time. Putting:

$\displaystyle\mathbb{H}^{m}_{1}(-1)=\\{u\in\mathbb{R}^{m+1}_{2},\langle
u,u\rangle=-1\\},$

one obtains an $m$-dimensional pseudo-Riemannian manifold of index $q$ and of
constant curvature $c=-1$. $\mathbb{H}^{m}_{1}(-1)$ is called the anti-de
Sitter space. We end this section by the following remark.

######  Remark 2.1.

In contrast to the Riemannian case, there are topological obstructions to the
existence of a Lorentz metric on a manifold $M$. Such a metric exists if
either $M$ is non-compact, or $M$ is compact and has Euler number $\chi(M)=0$.

##  3\. $2$-ruled hypersurfaces on a Walker $4$-manifold

Hypersurfaces are one of the important objects in a space. Hypersurfaces in a
manifold of constant curvature have been studied by many authors. Many ambiant
spaces are not always of constant curvature. In this paper we have studied
$2$-ruled hypersurfaces in a Walker $4$-manifold.

A Walker $4$-manifold noted $M$, is a pseudo-Riemannian manifold, which admits
a field of parallel null $2$-planes with signature $(++--)$. This class of
manifold is locally isometric to $(U,g_{f})$ where $U$ is an open of
$\mathbb{R}^{4}$ and $g_{f}$ is the metric given, respectively to the local
coordinates basis by
$\\{\partial_{i}=\frac{\partial}{\partial_{u_{1}}}\\}_{i=1,2,3,4}$ by

$\displaystyle g_{f}(\partial_{1},\partial_{3})$ $\displaystyle=$
$\displaystyle g_{f}(\partial_{2},\partial_{4})=1,$ $\displaystyle
g_{f}(\partial_{i},\partial_{j})$ $\displaystyle=$ $\displaystyle
g_{f_{ij}}(u_{1},u_{2},u_{3},u_{4})\quad\mbox{for}\quad i,j=3,4.$

The pseudo-Riemannian geometry of Walker metrics satisfying $g_{f_{34}}=0$ has
been studied by Chaichi et al. [8]. The purpose of this paper is to
characterize some metrics propertiers of Walker satisfying :
$g_{f_{33}}=g_{f_{44}}=0$. More precesily, we will consider Walker metrics of
the following form:

$\displaystyle g_{f}=\left(\begin{array}[]{cccc}0&0&1&0\cr 0&0&0&1\cr
1&0&0&f\cr 0&1&f&0\end{array}\right),$ (3.5)

where $f=f(u_{3},u_{4})$ denotes a differentiable function defined $U$. We
denote by $f_{3}=\frac{\partial f(u_{3},u_{4})}{\partial u_{3}}$ and
$f_{4}=\frac{\partial f(u_{3},u_{4})}{\partial u_{4}}$ for any function
$f(u_{3},u_{4})$. It follows after some straightforward calculations that the
non zero christoffel symbols of a Walker metric (3.5) are:

$\displaystyle\Gamma_{33}^{2}=f_{3}\quad\mbox{and}\quad\Gamma_{44}^{1}=f_{4}.$

We deduce that the Levita-Civita connection of a Walker metric is given by

$\displaystyle\nabla_{\partial_{3}}\partial_{3}=f_{3}\partial_{2}\quad\mbox{and}\quad\nabla_{\partial_{4}}\partial_{4}=f_{4}\partial_{1}.$

Let
$\mathbb{R}^{4}=\\{(u_{1},u_{2},u_{3},u_{4})|u_{i}\in\mathbb{R},(i=1,2,3,4)\\}$
be an $4$-dimensional Cartesian space. For any
$u=(u_{1},u_{2},u_{3},u_{4}),v=(v_{1},v_{2},v_{3},v_{4})\in\mathbb{R}^{4}$ ,
the pseudo-scalar product of $u$ and $v$ is defined by

$\displaystyle\langle u,v\rangle=-u_{1}v_{1}+\sum_{i=2}^{4}u_{i}v_{i}.$

We call $(\mathbb{R}^{4},\langle\cdot,\cdot\rangle)$ the Minkowski $4$-space.
We shall write $\mathbb{R}^{4}_{1}$ instead of
$(\mathbb{R}^{4},\langle\cdot,\cdot\rangle)$. We say that a non-zero vector
$u\in\mathbb{R}^{4}_{1}$ is spacelike, lightlike or timelike if $\langle
u,u\rangle>0,\langle u,u\rangle=0$ or $\langle u,u\rangle<0$, respectively.

Since we will deal with $2$-ruled hypersurface in Walker $4$-manifold. We now
define the Sitter $3$-space, the Hyperbolic $3$-space and the light cone at
the origin, respectivily, by

$\displaystyle\mathbb{S}^{3}_{1}$ $\displaystyle=$ $\displaystyle\\{x\in
M,\|u\|=1\\},$ (3.6) $\displaystyle\mathbb{H}^{3}_{+}(-1)$ $\displaystyle=$
$\displaystyle\\{u\in M,\|u\|=-1\\},$ (3.7) $\displaystyle\mathcal{LC}$
$\displaystyle=$ $\displaystyle\\{x\in M,\|u\|=0\\},$ (3.8)

where $\|u\|=\sqrt{g_{f}(u,u)}$.

If
$\overrightarrow{u}=(u_{1},u_{2},u_{3},u_{4}),\overrightarrow{v}=(v_{1},v_{2},v_{3},v_{4})$
and $\overrightarrow{w}=(w_{1},w_{2},w_{3},w_{4})$ are three vectors in $M$,
then te vector product are defined by

$\displaystyle\overrightarrow{u}\times_{f}\overrightarrow{v}\times_{f}\overrightarrow{w}=\left[\begin{array}[]{cccc}0&-f&1&0\\\
-f&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\
\right]\det\left[\begin{array}[]{cccc}\partial_{1}&\partial_{2}&\partial_{3}&\partial_{4}\\\
u_{1}&u_{2}&u_{3}&u_{4}\\\ v_{1}&v_{2}&v_{3}&v_{4}\\\
w_{1}&w_{2}&w_{3}&w_{4}\end{array}\right].$ (3.17)

If

$\displaystyle\varphi:I_{1}\times I_{2}\times I_{3}$ $\displaystyle\to$
$\displaystyle M$ $\displaystyle(u_{1},u_{2},u_{3})$ $\displaystyle\mapsto$
$\displaystyle\varphi(u_{1},u_{2},u_{3}),$

with

$\displaystyle\varphi(u_{1},u_{2},u_{3})=(\varphi_{1}(u_{1},u_{2},u_{3}),\varphi_{2}(u_{1},u_{2},u_{3}),\varphi_{3}(u_{1},u_{2},u_{3}),\varphi_{4}(u_{1},u_{2},u_{3}))$
(3.18)

is a hypersurface in $M$ , then the Gauss map (i.e., the unit normal vector
field), the matrix forms of the first and second fundamental forms are

$\displaystyle
G_{f}=\frac{\varphi_{u_{1}}\times_{f}\varphi_{u_{2}}\times_{f}\varphi_{u_{2}}}{\|\varphi_{u_{1}}\times_{f}\varphi_{u_{2}}\times\varphi_{u_{3}}\|},$
(3.19)
$\displaystyle[g_{ij}]=\left[\begin{array}[]{ccc}g_{11}&g_{12}&g_{13}\\\
g_{21}&g_{22}&g_{23}\\\ g_{31}&g_{32}&g_{33}\end{array}\right]$ (3.23)

and

$\displaystyle[h_{ij}]=\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{21}&h_{22}&h_{23}\\\ h_{31}&h_{32}&h_{33}\end{array}\right],$ (3.27)

respectively, where the coefficients
$g_{ij}=g_{f}(\varphi_{{u_{i}}},\varphi_{{u_{j}}}),h_{ij}=g_{f}(\varphi_{u_{i}u_{j}},G_{f})_{i,j\in\\{1,2,3\\}}$,
with
$\varphi_{uv}=\displaystyle\sum_{k=1}^{4}\Big{\\{}\frac{\partial^{2}\varphi_{k}}{\partial
v\partial
u}+\displaystyle\sum_{ij}\Gamma_{ij}^{k}\frac{\partial\varphi_{i}}{\partial
u}\frac{\partial\varphi_{j}}{\partial v}\Big{\\}}\partial_{k}$. Also, the
matrix of shape operator of the hypersurface $\varphi$ (3.18) is

$\displaystyle S_{f}=[s_{ij}]=[g^{ij}]\cdot[h_{ij}],$ (3.28)

where $[g^{ij}]$ is the inverse matrix of $[g_{ij}]$. With aid of
(3.23)-(3.28), the Gaussian curvature and mean curvature of a hypersurface in
$M$ are given by

$\displaystyle K_{f}=\frac{\det[h_{ij}]}{\det[g_{ij}]}$ (3.29)

and

$\displaystyle 3H_{f}=trace(S_{f}),$ (3.30)

respectively.

###  3.1. $2$-ruled hypersurfaces of type-$1$ in $M$

In this section, we give the deﬁnition of $2$-ruled hypersurfaces of type $1$
and state some results on Gaussian and mean curvatures. By a $2$-ruled
hypersurface of type-$1$ in $M$, we mean a map $\varphi:I_{1}\times
I_{2}\times I_{3}\to M$ of the form

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{3}),$
(3.31)

where $\alpha:I_{1}\to\mathbb{R}^{4}_{1},\beta:I_{2}\to\mathbb{S}^{3}_{1}$ and
$\gamma:I_{3}\to\mathbb{S}^{3}_{1}$ are smooth maps, where
$\mathbb{S}^{3}_{1}$ is the Sitter $3$-space of $M$ and $I_{1},I_{2},I_{3}$
are open intervals. We call $\alpha$ a base curve and two curves $\beta$ and
$\gamma$ director curves. The planes
$(u_{2},u_{3})\to\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{1})$ are
called rulings [25].

Putting:

$\displaystyle\left\\{\begin{array}[]{llllll}\alpha(u_{1})&=&\big{(}\alpha_{1}(u_{1}),\alpha_{2}(u_{1}),\alpha_{3}(u_{1}),\alpha_{4}(u_{1})\big{)}\\\
\beta(u_{1})&=&\big{(}\beta_{1}(u_{1}),\beta_{2}(u_{1}),\beta_{3}(u_{1}),\beta_{4}(u_{1})\big{)}\\\
\gamma(u_{1})&=&\big{(}\gamma_{1}(u_{1}),\gamma_{2}(u_{1}),\gamma_{3}(u_{1}),\gamma_{4}(u_{1})\big{)},\end{array}\right.$
(3.35)

then, the equation (3.31) becomes:

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\left(\begin{array}[]{cc}\alpha_{1}(u_{1})+u_{2}\beta_{1}(u_{1})+u_{3}\gamma_{1}(u_{1}),&\alpha_{2}(u_{1})+u_{2}\beta_{2}(u_{1})+u_{3}\gamma_{2}(u_{1}),\\\
\alpha_{3}(u_{1})+u_{2}\beta_{3}(u_{1})+u_{3}\gamma_{3}(u_{1}),&\alpha_{4}(u_{1})+u_{2}\beta_{4}(u_{1})+u_{3}\gamma_{4}(u_{1})\end{array}\right).$
(3.38)

We see that
$\langle\beta_{i},\beta_{i}\rangle=\langle\gamma_{i},\gamma_{i}\rangle=1$ and
we state $\alpha_{i}=\alpha_{i}(u_{1})$, $\beta_{i}=\beta_{i}(u_{1})$,
$\gamma_{i}=\gamma_{i}(u_{1})$, $\varphi_{i}=\varphi_{i}(u_{1},u_{2},u_{3})$,
$f^{\prime}=\frac{\partial f(u_{1})}{\partial u_{1}}$,
$f^{\prime\prime}=\frac{\partial^{2}f(u_{1})}{\partial u_{1}\partial u_{1}}$,
$i\in\\{1,2,3,4\\}$ and $f\in\\{\alpha,\beta,\gamma\\}$. We denote by

$\displaystyle E_{ij}$ $\displaystyle=$
$\displaystyle\gamma_{i}(\alpha^{\prime}_{j}+u_{2}\beta^{\prime}_{j}+u_{3}\gamma^{\prime}_{j})$
(3.39) $\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\beta_{i}(\alpha^{\prime}_{j}+u_{2}\beta^{\prime}_{j}+u_{3}\gamma^{\prime}_{j}).$
(3.40)

Now, let us prove the following theorem which contains the Gauss map of the
$2$-ruled hypersurface of type-$1$ (3.38).

######  Theorem 3.1.

The Gauss map of the $2$-ruled hypersurface of type-$1$ of the form (3.38) is
iven by

$\displaystyle
G_{f}(u_{1},u_{2},u_{3})=\frac{G_{1}(u_{1},u_{2},u_{3})\partial_{1}+G_{2}(u_{1},u_{2},u_{3})\partial_{2}+G_{3}(u_{1},u_{2},u_{3})\partial_{3}+G_{4}(u_{1},u_{2},u_{3})\partial_{4}}{A},$
(3.41)

where

$\displaystyle G_{1}(u_{1},u_{2},u_{3})$ $\displaystyle=$
$\displaystyle-f\big{(}\beta_{1}(E_{43}-E_{34})+\beta_{3}(E_{14}-E_{41})+\beta_{4}(E_{31}-E_{13})\big{)}$
$\displaystyle+\beta_{1}(E_{24}-E_{42})+\beta_{2}(E_{41}-E_{14})+\beta_{4}(E_{12}-E_{21}),$
$\displaystyle G_{2}(u_{1},u_{2},u_{3})$ $\displaystyle=$
$\displaystyle-f\big{(}\beta_{2}(E_{34}-E_{43})+\beta_{3}(E_{42}-E_{24})+\beta_{4}(E_{23}-E_{32}\big{)}$
$\displaystyle+\beta_{1}(E_{32}-E_{23})+\beta_{2}(E_{13}-E_{31})+\beta_{3}(E_{21}-E_{12}),$
$\displaystyle G_{3}(u_{1},u_{2},u_{3})$ $\displaystyle=$
$\displaystyle\beta_{2}(E_{34}-E_{43})+\beta_{3}(E_{42}-E_{24})+\beta_{4}(E_{23}-E_{32}),$
$\displaystyle G_{4}(u_{1},u_{2},u_{3})$ $\displaystyle=$
$\displaystyle\beta_{1}(E_{43}-E_{34})+\beta_{3}(E_{14}-E_{41})+\beta_{4}(E_{31}-E_{13}),$
(3.42)

and

$\displaystyle A=\sqrt{2G_{1}G_{3}+2G_{2}G_{4}+2fG_{3}G_{4}},$ (3.43)

wit
$G_{1}=G_{1}(u_{1},u_{2},u_{3}),G_{2}=G_{2}(u_{1},u_{2},u_{3}),G_{3}=G_{3}(u_{1},u_{2},u_{3})$
and $G_{4}=G_{4}(u_{1},u_{2};u_{3})$.

###### Proof.

If we differentiate (3.38), we get:

$\displaystyle{\left\\{\begin{array}[]{ccc}\varphi_{u_{1}}(u_{1},u_{2},u_{3})&=&\big{(}\alpha^{\prime}_{1}+u_{2}\beta^{\prime}_{1}+u_{3}\gamma^{\prime}_{1},\alpha^{\prime}_{2}+u_{2}\beta^{\prime}_{2}+u_{3}\gamma^{\prime}_{2},\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3},\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4}\big{)}\\\
\varphi_{u_{2}}(u_{1},u_{2},u_{3})&=&\big{(}\beta_{1},\beta_{2},\beta_{3},\beta_{4}\big{)}\\\
\varphi_{u_{3}}(u_{1},u_{2},u_{3})&=&\big{(}\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\big{)}.\end{array}\right.}$

By using the vector product define in (3.17), we get:

$\displaystyle\varphi_{u_{1}}\times_{f}\varphi_{u_{2}}\times_{f}\varphi_{u_{3}}$
$\displaystyle=$
$\displaystyle\Big{(}-f\Big{(}\beta_{1}(E_{43}-E_{34})+\beta_{3}(E_{14}-E_{41})+\beta_{4}(E_{31}-E_{13})\Big{)}$
$\displaystyle+\Big{(}\beta_{1}(E_{24}-E_{42})+\beta_{2}(E_{41}-E_{14})+\beta_{4}(E_{12}-E_{21})\Big{)}\Big{)}\partial_{1}$
$\displaystyle+\Big{(}-f\Big{(}\beta_{2}(E_{34}-E_{43})+\beta_{3}(E_{42}-E_{24})+\beta_{4}(E_{23}-E_{32})\Big{)}$
$\displaystyle+\Big{(}\beta_{1}(E_{32}-E_{23})+\beta_{2}(E_{13}-E_{31})+\beta_{3}(E_{21}-E_{12})\Big{)}\Big{)}\partial_{2}$
$\displaystyle+\Big{(}\beta_{2}(E_{43}-E_{34})+\beta_{3}(E_{24}-E_{42})+\beta_{4}(E_{32}-E_{23})\Big{)}\partial_{3}$
$\displaystyle+\Big{(}\beta_{1}(E_{43}-E_{34})+\beta_{3}(E_{14}-E_{41})+\beta_{4}(E_{31}-E_{13})\Big{)}\partial_{4}.$

Now using the unit normal vector formula in (3.19), we get the result. ∎

From (3.23), we obtain the matrix of the first fundamental form:

$\displaystyle[g_{ij}]=\left[\begin{array}[]{ccc}a&b&c\\\ b&1&e\\\
c&e&1\end{array}\right],$ (3.48)

where

$\displaystyle a$ $\displaystyle=$ $\displaystyle
2f(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})$
$\displaystyle+2\sum_{i=1}^{2}(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})(\alpha^{\prime}_{i+2}+u_{2}\beta^{\prime}_{i+2}+u_{3}\gamma^{\prime}_{i+2}),$
$\displaystyle b$ $\displaystyle=$ $\displaystyle
f(F_{34}-F_{43})+\sum_{i=1}^{2}(F_{i(i+2)}+F_{(i+2)i}),$ $\displaystyle c$
$\displaystyle=$ $\displaystyle
f(E_{34}-E_{43})+\sum_{i=1}^{2}(E_{i(i+2)}+E_{(i+2)i)},$ $\displaystyle e$
$\displaystyle=$ $\displaystyle
f(\beta_{3}\gamma_{4}+\beta_{4}\gamma_{3})+\sum_{i=1}^{2}(\beta_{i}\gamma_{i+2}+\beta_{i+2}\gamma_{i}),$
(3.49)

and we obtain the inverse matrix $[g^{ij}]$ of $[g_{ij}]$ as:

$\displaystyle[g^{ij}]=\frac{1}{\det[g_{ij}]}\left[\begin{array}[]{ccc}1-e^{2}&ce-b&be-c\\\
ce-b&a-c^{2}&bc-ae\\\ be-c&bc-ae&a-b^{2}\end{array}\right],$ (3.53)

where

$\displaystyle\det[g_{ij}]=-b^{2}+2cbe-c^{2}-ae^{2}+a=B.$ (3.54)

Furthermore, from (3.27), the matrix form of the second fundamental from of
the $2$-ruled hypersurface (3.38) is obtained by

$\displaystyle[h_{ij}]=\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\cr
h_{21}&h_{22}&h_{23}\cr h_{31}&h_{32}&h_{33}\end{array}\right],$ (3.58)

where

$\displaystyle\left\\{\begin{array}[]{llllll}h_{11}&=&\displaystyle\frac{f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{3}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\beta_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i^{\prime\prime}}+u_{2}\beta_{i^{\prime\prime}}+u_{3}\gamma_{i^{\prime\prime}})}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(u_{1},u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},\cr
h_{12}&=&h_{21}=\displaystyle\frac{f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\beta_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}-u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(u_{1},u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},\cr
h_{13}&=&h_{31}=\displaystyle\frac{f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(u_{1},u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},\cr
h_{22}&=&\frac{f_{3}\beta^{2}_{3}G_{4}+f_{4}\beta^{2}_{4}G_{3}}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(u_{1},u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},\cr
h_{33}&=&\frac{f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(u_{1},u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},\cr
h_{23}&=&\frac{f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}}{\sqrt{2fG_{3}(u_{1},u_{2},u_{3})G_{4}(uç1,u_{2},u_{3})+\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}}.\end{array}\right.$
(3.65)

We can see easily that the
$\det[h_{ij}]=h_{11}h_{22}h_{33}+2h_{12}h_{13}h_{23}-h^{2}_{12}h_{33}-h^{2}_{13}h_{22}-h^{2}_{23}h_{11}\neq
0$.
Then we can give the following theorem by using (3.29)

######  Theorem 3.2.

The $2$-ruled hypersurfaces of type-$1$ defined in (3.38) is no flat.

######  Corollary 3.3.

The $2$-ruled hypersurfaces of type-$1$ defined in (3.38) is flat if $f$ is
nonzero constant.

###### Proof.

From (3.27) the matrix of second fundamental form of the $2$-ruled
hypersurface (3.38) is obtained by

$\displaystyle[h_{ij}]=\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\cr
h_{21}&0&0\cr h_{31}&0&0\end{array}\right],$ (3.69)

where

$\displaystyle h_{11}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})}{\sqrt{\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},$
(3.70) $\displaystyle h_{12}$ $\displaystyle=$ $\displaystyle
h_{21}=\displaystyle\frac{\sum^{2}_{i=1}G_{i+2}\beta^{\prime}_{i}}{\sqrt{\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},$
(3.71) $\displaystyle h_{13}$ $\displaystyle=$ $\displaystyle
h_{31}\displaystyle\frac{\sum^{2}_{i=2}G_{i+2}\gamma^{\prime}_{i}}{\sqrt{\sum^{2}_{i=1}G_{i}(u_{1},u_{2},u_{3})G_{i+2}(u_{1},u_{2},u_{3})}},$
(3.72) $\displaystyle h_{22}$ $\displaystyle=$ $\displaystyle
h_{23}=h_{33}=0.$ (3.73)

So we have $\det(h_{ij})=0$, hence $G_{f}=0$. ∎

Now we will prove the following theorem about the mean curvature.

######  Theorem 3.4.

The 2-ruled hypersurfaces of type-1 defined in (3.38) is minimal, if

$\displaystyle(1-e^{2})\left[f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{2}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\beta_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})\right]$
$\displaystyle+2(ce-b)\left[f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\beta_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}\right]$
$\displaystyle+2(be-c)\left[f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}\right]$
$\displaystyle+2(bc-
ae)\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]+(a-c^{2})\left[f_{3}\beta^{2}_{3}G_{4}+f_{4}\beta^{2}_{4}G_{3}\right]$
$\displaystyle+(a-b^{2})\left[f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.74)

###### Proof.

By (3.28), the matrix of the shape operator is

$\displaystyle S=\left[\begin{array}[]{ccc}1-e^{2}&ce-b&be-c\\\
ce-b&a-c^{2}&bc-ae\\\ be-c&bc-
ae&a-b^{2}\end{array}\right]\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{12}&h_{22}&h_{23}\\\ h_{13}&h_{23}&h_{33}\end{array}\right],$

where $h_{11}$, $h_{12}$, $h_{13}$, $h_{22}$, $h_{23}$, $h_{33}$ are the same
in (3.65). Then we get the coefficients of $S$ by

$\displaystyle S_{11}$ $\displaystyle=$
$\displaystyle(1-e^{2})h_{11}+(ce-b)h_{12}+(be-c)h_{13}$ $\displaystyle
S_{22}$ $\displaystyle=$ $\displaystyle(ce-b)h_{12}+(a-c^{2})h_{22}+(bc-
ae)h_{23}$ $\displaystyle S_{33}$ $\displaystyle=$
$\displaystyle(be-c)h_{13}+(bc-ae)h_{23}+(a-b^{2})h_{33}.$

And using (3.65) and (3.30) we see that the $2$-ruled hypersurfaces is minimal
if

$\displaystyle S_{11}+S_{22}+S_{33}=0,$

then that end the proof. ∎

######  Corollary 3.5.

If the curves $\beta$ and $\gamma$ are orthogonal then the $2$-ruled
hypersurfaces of type-1 defined in (3.38) is minimal if

$\displaystyle\left[f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{3}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})\right]$
$\displaystyle-2b\left[f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{4})+f_{4}\beta_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}\right]$
$\displaystyle-2c\left[f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}\right]$
$\displaystyle+2bc\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]+(a-c^{2})\left[f_{3}\beta^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]$
$\displaystyle+(a-b^{2})\left[f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.76)

The Laplace-Beltrami operator of a smooth function
$\varphi=\varphi(u_{1},u_{2},u_{3})$ of class $C^{3}$ with respect to the
first fundamental form of a hypersurface is defined as follows:

$\displaystyle\Delta\varphi=\frac{1}{\sqrt{\det[g_{ij}]}}\sum_{i,j}^{3}\frac{\partial}{\partial
u_{i}}\left(\sqrt{\det[g_{ij}]}g^{ij}\frac{\partial\varphi}{\partial
u_{j}}\right).$ (3.77)

Using (3.77), we get the Laplace-Beltrami operator of the $2$-ruled
hypersurface of type-$1$ (3.35) by

$\displaystyle\Delta\varphi=(\Delta\varphi_{1},\Delta\varphi_{2},\Delta\varphi_{3},\Delta\varphi_{4}),$

where

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{B}}\left[\begin{aligned}
&\frac{\partial}{\partial
u_{1}}\left(\frac{(1-e^{2})\varphi_{iu_{1}}+(ce-b)\varphi_{iu_{2}}+(be-c)\varphi_{iu_{3}}}{\sqrt{\det[g_{ij}]}}\right)\\\
&+\frac{\partial}{\partial
u_{2}}\left(\frac{(ce-b)\varphi_{iu_{1}}+(a-c^{2})\varphi_{iu_{2}}+(bc-
ae)\varphi_{iu_{3}}}{\sqrt{\det[g_{ij}]}}\right)\\\ &+\frac{\partial}{\partial
u_{3}}\left(\frac{(be-c)\varphi_{iu_{1}}+(bc-
ae)\varphi_{iu_{2}}+(a-b^{2})\varphi_{iu_{3}}}{\sqrt{\det[g_{ij}]}}\right)\end{aligned}\right].$
(3.78)

That is

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{B}}\left[\begin{aligned}
&\frac{\partial}{\partial
u_{1}}\left(\frac{(1-e^{2})(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(ce-b)\beta_{i}+(be-c)\gamma_{i}}{\sqrt{\det[g_{ij}]}}\right)\\\
&+\frac{\partial}{\partial
u_{2}}\left(\frac{(ce-b)(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(a-c^{2})\beta_{i}+(bc-
ae)\gamma_{i}}{\sqrt{\det[g_{ij}]}}\right)\\\ &+\frac{\partial}{\partial
u_{3}}\left(\frac{(be-c)(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(bc-
ae)\beta_{i}+(a-b^{2})\gamma_{i}}{\sqrt{\det[g_{ij}]}}\right)\end{aligned}\right].$
(3.79)

If we suppose that $\beta$ and $\gamma$ are orthogonal, then the Laplace-
Beltrami operator of the $2$-ruled hypersuface of type-1 is given by

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{a-b^{2}-c^{2}}}\left[\begin{aligned}
&\frac{\partial}{\partial
u_{1}}\left(\frac{(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})-b\beta_{i}-c\gamma_{i}}{\sqrt{a-b^{2}-c^{2}}}\right)\\\
&+\frac{\partial}{\partial
u_{2}}\left(\frac{-b(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(a-c^{2})\beta_{i}+bc\gamma_{i}}{\sqrt{a-b^{2}-c^{2}}}\right)\\\
&+\frac{\partial}{\partial
u_{3}}\left(\frac{-c(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+bc\beta_{i}+(a-b^{2})\gamma_{i}}{\sqrt{a-b^{2}-c^{2}}}\right)\end{aligned}\right].$
(3.80)

######  Theorem 3.6.

The components of the Laplace-Beltrami operator of the $2$-ruled hypersurface
of type-$1$ are

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{W}}\left[\begin{aligned}
&\frac{(\alpha^{\prime\prime}_{i}+u_{2}\beta^{\prime\prime}_{i}+u_{3}\gamma^{\prime\prime}_{i}-(b\beta_{i})_{u_{1}}-(c\gamma_{i})_{u_{1}})W-V_{1}(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{2}\gamma^{\prime}_{i}-b\beta_{i}-c\gamma_{i})}{W^{\frac{3}{2}}}\\\
&+\frac{(-b\beta^{\prime}_{i}+((a-c^{2})\beta_{i})_{u_{2}}+(bc\gamma_{i})_{u_{2}})W-V_{2}(-b(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(a-c^{2})\beta_{i}+bc\gamma_{i})}{W^{\frac{3}{2}}}\\\
&+\frac{(-c\gamma^{\prime}_{i}+(bc\beta_{i})_{u_{3}}+((a-b^{2})\gamma_{i})_{u_{3}})W-V_{3}(-c(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+bc\beta_{i}+(a-b^{2})\gamma_{i})}{W^{\frac{3}{2}}}\end{aligned}\right],$
(3.81)

where $i=1,2,3,4$; $\beta$ and $\gamma$ are orthogonal; $W=a-b^{2}-c^{2}$,
$V_{1}=a_{u_{1}}-2bb_{u_{1}}-2cc_{u_{1}}$,
$V_{2}=a_{u_{2}}-2bb_{u_{2}}-2cc_{u_{2}}$,
$V_{3}=a_{u_{3}}-2bb_{u_{3}}-2cc_{u_{3}}$.

###  3.2. 2-Ruled hypersurfaces of type-2 in $M$

A $2$-ruled hypersurface of type-2 in $M$ means (the image of) a map
$\varphi:I_{1}\times I_{2}\times I_{3}\to M$ of the form

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{1}),$
(3.82)

where $\alpha:I_{1}\to M$, $\beta:I_{2}\to H^{3}_{+}(-1)$, $\gamma:I_{3}\to
H^{3}_{+}(-1)$ are smooth maps, $H^{3}_{+}(-1)$ is the hyperbolic $3$-space of
$M$ and $I_{1},I_{2},I_{3}$ are open intervals. We call $\alpha$ a base curve,
$\beta$ and $\gamma$ director curves. The planes
$(u_{2},u_{3})\mapsto\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{1})$ are
called rulings. So, if we take

$\displaystyle\left\\{\begin{array}[]{llllll}\alpha(u_{1})&=&\big{(}\alpha_{1}(u_{1}),\alpha_{2}(u_{1}),\alpha_{3}(u_{1}),\alpha_{4}(u_{1})\big{)}\\\
\beta(u_{1})&=&\big{(}\beta_{1}(u_{1}),\beta_{2}(u_{1}),\beta_{3}(u_{1}),\beta_{4}(u_{1})\big{)}\\\
\gamma(u_{1})&=&\big{(}\gamma_{1}(u_{1}),\gamma_{2}(u_{1}),\gamma_{3}(u_{1}),\gamma_{4}(u_{1})\big{)}\end{array}\right.$
(3.86)

in (3.82), then we can write the $2$-ruled hypersurface of type-$2$ as

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\left(\begin{array}[]{cc}\alpha_{1}(u_{1})+u_{2}\beta_{1}(u_{1})+u_{3}\gamma_{1}(u_{1}),&\alpha_{2}(u_{1})+u_{2}\beta_{2}(u_{1})+u_{3}\gamma_{2}(u_{1}),\\\
\alpha_{3}(u_{1})+u_{2}\beta_{3}(u_{1})+u_{2}\gamma_{3}(u_{1}),&\alpha_{4}(u_{1})+u_{2}\beta_{4}(u_{1})+u_{3}\gamma_{4}(u_{1})\end{array}\right).$
(3.89)

We see that
$\langle\beta_{i},\beta_{i}\rangle=\langle\gamma_{i},\gamma_{i}\rangle=-1$ and
we state $\alpha_{i}=\alpha_{i}(u_{1})$, $\beta_{i}=\beta_{i}(u_{1})$,
$\gamma_{i}=\gamma_{i}(u_{1})$, $\varphi_{i}=\varphi_{i}(u_{1},u_{2},u_{3})$,
$f^{\prime}=\frac{\partial f(u_{1})}{\partial u_{1}}$,
$f^{\prime\prime}=\frac{\partial^{2}f(u_{1})}{\partial u_{1}\partial u_{1}}$,
$i\in\\{1,2,3,4\\}$ and $f\in\\{\alpha,\beta,\gamma\\}$.

From (3.23), we obtain the matrix of the first fundamental form

$\displaystyle[g_{ij}]=\left[\begin{array}[]{ccc}a&b&c\\\ b&-1&e\\\
c&e&-1\end{array}\right].$ (3.93)

And we obtain the inverse matrix $[g^{ij}]$ of $[g_{ij}]$ as

$\displaystyle[g^{ij}]=\frac{1}{\det[g_{ij}]}\left[\begin{array}[]{ccc}1-e^{2}&ce+b&be+c\\\
ce+b&-a-c^{2}&bc-ae\\\ be+c&bc-ae&-a-b^{2}\end{array}\right].$ (3.97)

where $a$, $b$, $c$ and $e$ are the same in (3.1) and

$\displaystyle\det[g_{ij}]=b^{2}+2cbe+c^{2}-ae^{2}+a=C.$ (3.98)

Furthermore, from (3.27), the matrix form of the second fundamental from of
the $2$-ruled hypersurface (3.89) is the same given in (3.58) and (3.65). And
we have the following theorem since the $\det[h_{ij}]\neq 0$.

######  Theorem 3.7.

The $2$-ruled hypersurfaces of type-$2$ defined in (3.89) is not flat.

######  Corollary 3.8.

The $2$-ruled hypersurfaces of type-$2$ defined in (3.89) is flat if $f$ is
nonzero constant.

For the mean curvature we have:

######  Theorem 3.9.

The $2$-ruled hypersurfaces of type-$2$ defined in (3.89) is minimal in $M$,
if

$\displaystyle(1-e^{2})\left[f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{3}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\gamma_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})\right]$
$\displaystyle+2(ce+b)\left[f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\beta_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}+\right]$
$\displaystyle+2(be+c)\left[f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}\right]$
$\displaystyle+2(bc-
ae)\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]+(-a-c^{2})\left[f_{3}\beta^{2}_{3}G_{4}+f_{4}\beta^{2}_{4}G_{3}\right]$
$\displaystyle+(-a-b^{2})\left[f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.99)

###### Proof.

By (3.28), the matrix of the shape operator is

$\displaystyle S=\left[\begin{array}[]{ccc}1-e^{2}&ce+b&be+c\\\
ce+b&-a-c^{2}&bc-ae\\\ be+c&bc-
ae&-a-b^{2}\end{array}\right]\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{12}&h_{22}&h_{23}\\\ h_{13}&h_{23}&h_{33}\end{array}\right],$

where $h_{11}$, $h_{12}$, $h_{13}$, $h_{22}$, $h_{23}$, $h_{33}$ are the same
in (3.65). Then we get the coefficients of $S$ by

$\displaystyle S_{11}$ $\displaystyle=$
$\displaystyle(1-e^{2})h_{11}+(ce+b)h_{12}+(be+c)h_{13},$ $\displaystyle
S_{22}$ $\displaystyle=$ $\displaystyle(ce+b)h_{12}+(-a-c^{2})h_{22}+(bc-
ae)h_{23},$ $\displaystyle S_{33}$ $\displaystyle=$
$\displaystyle(be+c)h_{13}+(bc-ae)h_{23}+(-a-b^{2})h_{33}.$

And using (3.65) and (3.30), we see that the $2$-ruled hypersurfaces of
type-$2$ is minimal if

$\displaystyle S_{11}+S_{22}+S_{33}=0,$

then that end the proof. ∎

######  Corollary 3.10.

If the curves $\beta$ and $\gamma$ are orthogonal then the $2$-ruled
hypersurfaces of type-$2$ defined in (3.89) is minimal if

$\displaystyle\left[f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{3}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\beta_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})\right]$
$\displaystyle+2b\left[f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\beta_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\beta_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}\right]$
$\displaystyle+2c\left[f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{2}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}\right]$
$\displaystyle+2bc\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]+(-a-c^{2})\left[f_{3}\beta^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]$
$\displaystyle+(-a-b^{2})\left[f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.101)

To end this section, we will give the operator of Laplace-Beltrami in the
following theorem:

######  Theorem 3.11.

The components of the Laplace-Beltrami operator of the $2$-ruled hypersurface
of type-$2$ are

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{T}}\left[\begin{aligned}
&\frac{(\alpha^{\prime\prime}_{i}+u_{2}\beta^{\prime\prime}_{i}+u_{3}\gamma^{\prime\prime}_{i})+(b\beta_{i})_{u_{1}}+(c\gamma_{i})_{u_{1}})T-R_{1}(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i}+b\beta_{i}+c\gamma_{i})}{T^{\frac{3}{2}}}\\\
&+\frac{(b\beta^{\prime}_{i}+((-a-c^{2})\beta_{i})_{u_{2}}+(bc\gamma_{i})_{u_{2}})T-R_{2}(b(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(-a-c^{2})\beta_{i}+bc\gamma_{i})}{R^{\frac{3}{2}}}\\\
&+\frac{(c\gamma^{\prime}_{i}+(bc\beta_{i})_{u_{3}}+((-a-b^{2})\gamma_{i})_{u_{3}})T-R_{3}(c(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+bc\beta_{i}+(-a-b^{2})\gamma_{i})}{R^{\frac{3}{2}}}\end{aligned}\right],$
(3.102)

where $i=1,2,3,4$; $\beta$ and $\gamma$ are orthogonal; $T=a+b^{2}+c^{2}$,
$R_{1}=a_{u_{1}}+2bb_{u_{2}}+2cc_{u_{1}}$,
$R_{2}=a_{u_{2}}+2bb_{u_{2}}+2cc_{u_{2}}$,
$R_{3}=a_{u_{3}}+2bb_{u_{3}}+2cc_{u_{3}}$.

###  3.3. 2-Ruled hypersurfaces of type-3 in $M$

A $2$-ruled hypersurface of type-$3$ in $M$ means (the image of) a map
$\varphi:I_{1}\times I_{2}\times I_{3}\to M$ of the form

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{1}),$
(3.103)

where $\alpha:I_{1}\to M$, $\beta:I_{2}\to\mathcal{LC}$,
$\gamma:I_{3}\to\mathcal{LC}$ are smooth maps, $\mathcal{LC}$ is the light
cone of $M$ and $I_{1},I_{2},I_{3}$ are open intervals. We call $\alpha$ a
base curve, $\beta$ and $\gamma$ director curves. The planes
$(u_{2},u_{3})\mapsto\alpha(u_{1})+u_{2}\beta(u_{1})+u_{3}\gamma(u_{1})$ are
called rulings. So, if we take

$\displaystyle\left\\{\begin{array}[]{llllll}\alpha(u_{1})&=&\big{(}\alpha_{1}(u_{1}),\alpha_{2}(u_{1}),\alpha_{3}(u_{1}),\alpha_{4}(u_{1})\big{)},\\\
\beta(u_{1})&=&\big{(}\beta_{1}(u_{1}),\beta_{2}(u_{1}),\beta_{3}(u_{1}),\beta_{4}(u_{1})\big{)},\\\
\gamma(u_{1})&=&\big{(}\gamma_{1}(u_{1}),\gamma_{2}(u_{1}),\gamma_{3}(u_{1}),\gamma_{4}(u_{1})\big{)},\end{array}\right.$
(3.107)

in (3.103), then we can write the $2$-ruled hypersurface of type-$3$ as

$\displaystyle\varphi(u_{1},u_{2},u_{3})=\left(\begin{array}[]{cc}\alpha_{1}(u_{1})+u_{2}\beta_{1}(u_{1})+u_{3}\gamma_{1}(u_{1}),&\alpha_{2}(u_{1})+u_{2}\beta_{2}(u_{1})+u_{3}\gamma_{2}(u_{1}),\\\
\alpha_{3}(u_{1})+u_{2}\beta_{3}(u_{1})+u_{2}\gamma_{3}(u_{1}),&\alpha_{4}(u_{1})+u_{2}\beta_{4}(u_{1})+u_{3}\gamma_{4}(u_{1})\end{array}\right).$
(3.110)

We see that
$\langle\beta_{i},\beta_{i}\rangle=\langle\gamma_{i},\gamma_{i}\rangle=-1$ and
we state $\alpha_{i}=\alpha_{i}(u_{1})$, $\beta_{i}=\beta_{i}(u_{1})$,
$\gamma_{i}=\gamma_{i}(u_{1})$, $\varphi_{i}=\varphi_{i}(u_{1},u_{2},u_{3})$,
$f^{\prime}=\frac{\partial f(u_{1})}{\partial u_{1}}$,
$f^{\prime\prime}=\frac{\partial^{2}f(u_{1})}{\partial u_{1}\partial u_{1}}$,
$i\in\\{1,2,3,4\\}$ and $f\in\\{\alpha,\beta,\gamma\\}$.

From (3.23), we obtain the matrix of the first fundamental form

$\displaystyle[g_{ij}]=\left[\begin{array}[]{ccc}a&b&c\\\ b&0&e\\\
c&e&0\end{array}\right].$ (3.114)

And we obtain the inverse matrix $[g^{ij}]$ of $[g_{ij}]$ as

$\displaystyle[g^{ij}]=\frac{1}{\det[g_{ij}]}\left[\begin{array}[]{ccc}-e^{2}&ce&be\\\
ce&-c^{2}&bc-ae\\\ be&bc-ae&-b^{2}\end{array}\right],$ (3.118)

where $a$, $b$, $c$ and $e$ are the same in (3.1) and

$\displaystyle\det[g_{ij}]=2bce-ae^{2}=D.$ (3.119)

Furthermore, from (3.27), the matrix form of the second fundamental from of
the $2$-ruled hypersurface (3.110) is the same given in (3.58) and (3.65). And
we have the following theorem since the $\det[h_{ij}]\neq 0$.

######  Theorem 3.12.

The $2$-ruled hypersurfaces of type-$3$ defined in (3.110) is no flat.

######  Corollary 3.13.

The $2$-ruled hypersurfaces of type-$3$ is flat if $f$ is non zero constant.

For the mean curvature we have

######  Theorem 3.14.

The $2$-ruled hypersurfaces of type-$3$ defined in (3.110) is minimal in $M$,
if

$\displaystyle-e^{2}\left[f_{3}G_{4}(\alpha_{3}^{\prime}+u_{2}\beta_{3}^{\prime}+u_{3}\gamma_{3}^{\prime})+f_{4}G_{3}(\alpha_{4}^{\prime}+u_{2}\gamma_{4}^{\prime}+u_{3}\gamma_{4}^{\prime})+\sum^{2}_{i=1}G_{i+2}(\alpha_{i}^{\prime\prime}+u_{2}\beta_{i}^{\prime\prime}+u_{3}\gamma_{i}^{\prime\prime})\right]$
$\displaystyle+2ce\left[f_{3}\beta_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\beta_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\beta^{\prime}_{i}+\right]$
$\displaystyle+2be\left[f_{3}\gamma_{3}G_{4}(\alpha^{\prime}_{3}+u_{2}\beta^{\prime}_{3}+u_{3}\gamma^{\prime}_{3})+f_{4}\gamma_{4}G_{3}(\alpha^{\prime}_{4}+u_{2}\beta^{\prime}_{4}+u_{3}\gamma^{\prime}_{4})+\sum_{i=1}^{2}G_{i+2}\gamma^{\prime}_{i}\right]$
$\displaystyle+2(bc-
ae)\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]-c^{2}\left[f_{3}\beta^{2}_{3}G_{4}+f_{4}\beta^{2}_{4}G_{3}\right]$
$\displaystyle-b^{2}\left[h_{3}\gamma^{2}_{3}G_{4}+h_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.120)

###### Proof.

By (3.28) the matrix of the shape operator is

$\displaystyle S=\left[\begin{array}[]{ccc}-e^{2}&ce&be\\\ ce&-c^{2}&bc-ae\\\
be&bc-
ae&-b^{2}\end{array}\right]\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{12}&h_{22}&h_{23}\\\ h_{13}&h_{23}&h_{33}\end{array}\right],$

where $h_{11}$, $h_{12}$, $h_{13}$, $h_{22}$, $h_{23}$, $h_{33}$ are the same
in (3.65). Then we get the coefficients of $S$ by

$\displaystyle S_{11}$ $\displaystyle=$
$\displaystyle-e^{2}h_{11}+ceh_{12}+beh_{13},$ $\displaystyle S_{22}$
$\displaystyle=$ $\displaystyle ceh_{12}-c^{2}h_{22}+(bc-ae)h_{23},$
$\displaystyle S_{33}$ $\displaystyle=$ $\displaystyle be+h_{13}+(bc-
ae)h_{23}-b^{2}h_{33}.$

And using (3.65) and (3.30), we see that the $2$-ruled hypersurfaces of
type-$3$ is minimal if

$\displaystyle S_{11}+S_{22}+S_{33}=0,$

then that end the proof. ∎

######  Corollary 3.15.

If the curves $\beta$ and $\gamma$ are orthogonal then the $3$-ruled
hypersurfaces of type-$3$ defined in (3.110) is minimal if

$\displaystyle+2bc\left[f_{3}\beta_{3}\gamma_{3}G_{4}+f_{4}\beta_{4}\gamma_{4}G_{3}\right]-c^{2}\left[f_{3}\beta^{2}_{3}G_{4}+h_{4}\gamma^{2}_{4}G_{3}\right]-b^{2}\left[f_{3}\gamma^{2}_{3}G_{4}+f_{4}\gamma^{2}_{4}G_{3}\right]=0.$
(3.122)

To end this section, we will give the operator of Laplace-Beltrami in the
following theorem:

######  Theorem 3.16.

The components of the Laplace-Beltrami operator of the $2$-ruled hypersurface
of type-$3$ are

$\displaystyle\Delta\varphi_{i}=\frac{1}{\sqrt{L}}\left[\begin{aligned}
&\frac{(\alpha^{\prime\prime}_{i}+u_{2}\beta^{\prime\prime}_{i}+u_{3}\gamma^{\prime\prime}_{i})+(b\beta_{i})_{u_{1}}+(c\gamma_{i})_{u_{1}})L-J_{1}(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i}+b\beta_{i}+c\gamma_{i})}{L^{\frac{3}{2}}}\\\
&+\frac{(b\beta^{\prime}_{i}+((-a-c^{2})\beta_{i})_{u_{2}}+(bc\gamma_{i})_{u_{2}})L-J_{2}(b(\alpha^{\prime}_{i}+u_{2}\beta^{\prime}_{i}+u_{3}\gamma^{\prime}_{i})+(-a-c^{2})\beta_{i}+bc\gamma_{i})}{L^{\frac{3}{2}}}\\\
&+\frac{(c\gamma^{\prime}_{i}+(bc\beta_{i})_{u_{3}}+((-a-b^{2})\gamma_{i})_{u_{3}})L-J_{3}(c(\alpha^{\prime}_{i}+y\beta^{\prime}_{i}+z\gamma^{\prime}_{i})+bc\beta_{i}+(-a-b^{2})\gamma_{i})}{L^{\frac{3}{2}}}\end{aligned}\right],$
(3.123)

where $i=1,2,3,4$; ; $L=2bce-ae^{2}2$,
$J_{1}=2b_{u_{1}}ce+2bc_{u_{1}}e+2bce_{u_{1}}-2aa_{u_{1}}$,
$J_{2}=2b_{u_{2}}ce+2bc_{u_{2}}e+2bce_{u_{2}}-2aa_{u_{2}}$,
$J_{3}=2b_{u_{3}}ce+2bc_{u_{3}}e+2bce_{u_{3}}-2aa_{u_{3}}$.

Note that the hypersurfaces constructed in this paper are not flat. Unlike
Euclidean and Monkowskian spaces, where the ruled hypersurfaces are flat

##  4\. Conclusion

We end this work by giving some applications of ruled surfaces and $2$-ruled
hypersurfaces as eneralisations of the first one. Ruled surfaces have been
applied in different areas such as CAD, electric discharge machining [1, 24].
The authors [23] present an elementary introduction to the theory of Betrand
pairs of curves and ruled surfaces. Bertrand pairs of ruled surfaces are
introduced as offsets in the context of line geometry. Also, the ruled
surfaces has an important application area on kinematics [1, 22].
Additionally, ruled surfaces have an important application area in arcitecture
[1]. For lightweight structures in the field of architecture and civil
engineering, concrete shells with negative Gaussian curvature are frequently
used. One class of such surfaces are the skew ruled surfaces [20].

## Acknowledgments

The authors would like to thank the referee for his/her valuable suggestions
and comments that helped them improve the paper.

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#  Implementation of Q Learning and Deep Q Network For Controlling a Self
Balancing Robot Model

MD Muhaimin Rahman1,a, SM Hasanur Rashid2,a and M.M Hossain b
a Department of Mechanical Engineering, bDepartment of Electrical and
Electronic Engineering
Bangladesh University of Engineering & Technology
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

In this paper, the implementation of two Reinforcement learnings namely, Q
Learning and Deep Q Network(DQN) on a Self Balancing Robot Gazebo model has
been discussed. The goal of the experiments is to make the robot model learn
the best actions for staying balanced in an environment. The more time it can
stay within a specified limit , the more reward it accumulates and hence more
balanced it is. Different experiments with different learning parameters on Q
Learning and DQN are conducted and the plots of the experiments are shown.

##  I INTRODUCTION

Control systems is one of the most important aspects of Robotics Research.
Gazebo is one of the most robust multi robot simulators at present. The
ability to use Robot Operating System (ROS) with Gazebo makes it more
powerful. But there are very few documentations on how to use ROS and Gazebo
for Controllers development. In our previous paper, [1], we attempted to
demonstrate and document the use of PID, Fuzzy logic and LQR controllers using
ROS and Gazebo on a self balancing robot model. Later on, we have worked on
Reinforcement learning. In this paper, implementation of Q Learning and Deep Q
Network on the same model is discussed. The paper is structured as follows .
Section II shows the related works on the subject. Section III discusses the
Robot Model . Section IV shows the Implementation of Q Learning and DQN as
controllers . Finally, section VI is the conclusion.

##  II Related Works

Lei Tai and Ming Liu [2], had worked on Mobile Robots Exploration using CNN
based reinforcement learning. They trained and developed turtlebot Gazebo
simulation to develop an exploration strategy based on raw sensor value from
RGB-D sensor. The company ErleRobotics have extended OpenAI environment to
Gazebo [3]. They have deployed Q-learning and Sarsa algorithms for various
exploratory environments. Loc Tran et al [4] developed training model for an
Unmanned aerial vehicle to explore with static obstacles in both Gazebo and
real world. Their proposed Reinforcement learning is unclear from the paper.
Volodymyr Sereda [5] used Q - learning on a custom Gazebo model using ROS in
for exploration strategy. Rowan Border [6] used Q-learning with neural network
presentation for robo search and rescue using ROS and Turtlebot.

##  III Robot Model

Figure 1: Simple Block Diagram of the Model Figure 2: Gazebo Model

The Robot Model is described in paper [1]. It has one chassis and two wheels.
The task of the model is to keep the robot balanced i.e. keeping it’s pitch
angle in between $\pm 5^{0}$. The more it remains in between the limits , the
more it gets the reward. The Fig. 1 shows the block diagram and the Fig.2
shows the Gazebo model.

###  III-A Controller

The robot’s IMU sensor measures the roll, pitch and yaw angles of the chassis
every second and sends it to the controller. The controller calculates optimum
action value to make the chassis tilt according to set point. Fig. 3 shows the
control system of the robot.

Figure 3: Controller Block Diagram

##  IV Reinforcement Learning Methods as Controllers

In [1], we worked on traditional Controllers like PID, Fuzzy PD,PD+I & LQR .
The biggest problem with those methods is that , they need to be tuned
manually. So, reaching optimal values of Controllers depends on many trials
and errors. Many a times optimum values aren’t reached at all. The biggest
benefit of Reinforcement learning algorithms as Controllers is that , the
model tunes itself to reach the Optimum values. The following two sections
discuss Q Learning and Deep Q Network.

###  IV-A Q Learning

Q- learning was developed by Christopher John Cornish Hellaby Watkins [7].
According to Watkins, ”it provides agents with the capability of learning to
act optimally in Markovian domains by experiencing the consequences of
actions, without requiring them to build maps of the domains.” [8]. In a
Markovian domain, $Q$ function- the model to be generated using the algorithm-
calculates the expected utility for a given finite state $s$ and every
possible finite action $a$. The agent - which is the robot in this case-
selects the optimum action $a$ having the highest value of $Q(s,a)$ , this
action choosing rule is also called Policy. [8] . Initially, the $Q(s,a)$
function values are assumed to be zero. After every training step , the values
are updated according to the following equation

$Q(s,a_{t})\leftarrow Q(s,a_{t})+\alpha(r+\gamma maxQ(s_{t+1},a))$ (1)

####  IV-A1 Algorithm

The objective for the Model in our project is to keep it within limits i.e.
$\pm 5^{0}$ . At first , the robot model, $Q$ matrix, Policy $\pi$ are
initialized . There are some interesting points to make. The states are not
finite. Within limit range , hundreds and thousands of pitch angles are
possible. Having thousands of columns is not possible. So, the state values
were discretized. We discretized the values to 20 discrete state angles from
$-10^{0}$ to $10^{0}$. For action value, we chose 10 different velocities.
They are $[-200,-100,-50,-25,-10,10,25,50,100,200]ms^{-1}$. The $Q$ matrix had
20 columns , each column representing a state and 10 rows each representing
every action. Initially ,the Q -values were assumed to be $0$ and random
actions were specified for every state in the policy $\pi$ . The training was
done for 1500 episodes and in each episode, the training was iterated $2000$
times . At the beginning of each episode, the simulation was refreshed.
Whenever the robot’s state exceeded the limit it was penalized by assigning
reward to $-100$ . The $Q$ Table is updated at each step according to equation
1. The Algorithm 1 shows the full algorithm.

####  IV-A2 Result and Discussion

The simulation was run for three different $\alpha$ values $(0.7,0.65,0.8)$ ,
with $\gamma$ value of $(0.999)$. The Fig. 4 shows the Rewards vs Episodes for
those $\alpha$s. It is evident that, the robot couldn’t reach the target
rewards within the training period for those learning rates. We see that, for
the $\alpha$ values $0.7$ and $0.8$, the robot reaches maximum possible
accumulated rewards, 200, within 400 episodes. The curve with $\alpha$ value
$0.7$ is less stable compared to that of $0.8$. But The curve with $\alpha$
value $0.65$ never reaches the maximum accumulated reward.

Figure 4: Rewards for different $\alpha$

Initialize Robot;

Initialize Q Matrix $Q$;

Initialize Policy $\pi$;

Initialize Penalty Reward $pen$;

for  _number of episodes_ do

Reset simulation ;

Wait for 1 second ;

Pause simulation ;

Read the pitch angle $\phi$ of the robot ;

$state\leftarrow\phi$ ;

Unpause simulation ;

for  _number of iterations_ do

Generate a random number $rand$;

if  _ $rand\leq\epsilon$ _ then

take random action ;

end if

else

take action based on $\pi$ ;

end if

$state_{new}\leftarrow\phi$;

Pause simulation;

if  _absolute value of $state_{new}\geq limit$_ then

if  _ $reward_{total}\leq Target$_ then

$reward\leftarrow pen$;

Update $Q$ ;

Update $\pi$;

end if

Break ;

else

Print Passed ;

Break ;

end if

end if

else

$reward\leftarrow$ 1;

Update $Q$;

Update $\pi$ $state\leftarrow state_{new}$

end if

end for

end for

Algorithm 1 Q Learning Algorithm as applied in the system

###  IV-B Deep Q Network (DQN)

V Mnih et al [9] first used Deep Learning as a variant of Q Learning algorithm
to play six games of Atari 2600 , which outperformed all other previous
algorithms. . In their paper, two unique approaches were used.

  * •

Experience Replay

  * •

Derivation of Q Values in one forward pass

####  IV-B1 Experience Replay

The technique of Experience Replay, experiences of an agent , i.e.
$(state,reward,action,state_{new})$ are stored over many episodes . In the
learning period, after each episode random batches of data from experience are
used to update the model. [9]. There are several benefits of such approach.
According to the paper,

  * •

It allows greater data efficiency as each step of experience can be used in
many weight updates

  * •

Randomizing batches breaks correlations between samples

  * •

Behaviour distribution is averaged over many of its previous states

####  IV-B2  Derivation of Q Values in one Forward Pass

In the classical Q learning approach, one has to give state and action as an
input resulting in $Q$ value for that state and action. Replicating this
approach in Neural Network is problematic. Because in that case one has to
give state and action for each possible action of the agent to the Model. It
will lead to many forward passes in the same model. Instead , they designed
the model in such a way that it will predict Q values for each action for a
given state . As a result, only one forward pass is required. Figure 5 shows a
sample architecture for one state with two actions

Figure 5:  Sample Deep Q Network Architecture

####  IV-B3 Implementation on the Robot Model

Initialize Robot;

Initialize model $M$;

Initialize Penalty Reward $pen$;

for  _number of episodes_ do

Reset simulation ;

Wait for 1 second ;

Pause simulation ;

Read the pitch angle $\phi$ of the robot ;

$state\leftarrow\phi$ ;

Unpause simulation ;

for  _number of iterations_ do

Generate a random number $rand$;

if  _ $rand\leq\epsilon$ _ then

take random action ;

end if

else

$Q\leftarrow M(state)$;

$action\leftarrow actionformax(Q)$;

end if

$state_{new}\leftarrow\phi$;

Pause simulation;

if  _absolute value of $state_{new}\geq limit$_ then

if  _ $reward_{total}\leq Target$_ then

$reward\leftarrow pen$;

$experience\leftarrow(state,reward,action,state_{new})$;

Add Experience to Memory;

end if

Break ;

else

Print Passed ;

Break ;

end if

end if

else

$reward\leftarrow$ 1;

$experience\leftarrow(state,reward,action,state_{new})$;

Add Experience to Memory;

$state\leftarrow state_{new}$

end if

end for

Take random minibatch of Experience;

if  _$reward==pen$_ then

$Q_{pred}\leftarrow reward$;

end if

else

$Q_{pred}\leftarrow reward+\gamma max(Q(state_{new},action))$

end if

;

Train the model according to loss
$abs(Q_{pred}(state,action)-Q_{pred}(state,action))$;

end for

Algorithm 2 DQN Algorithm as applied in the system

The implementation of the DQN on our Robot model is similar to $Q$ Learning
Method. However, there are some exceptions. At first, a model was initialized
instead of Initializing $Q$ matrix. In the $\epsilon$ greedy policy , instead
of choosing action based on policy $\pi$ , $Q$ values were calculated
according to the model. At the end of every episode, the model was trained
using random minibatches of Experience. At first, an architecture with 2
hidden Relu layers of 20 units was selected. The last layer was a Linear Dense
layer with 10 units. With the $\gamma$ 0.999 and $\alpha$ of $(0.65,0.7,0.8)$
. Algorithm 2 shows the DQN algorithm as implemented on the robot model.

####  IV-B4 Result and Discussion

From figure 6, we see that the total rewards for $\alpha$ $0.65$ is
significantly higher. It starts approximately from 1750 and reaches the
maximum total rewards, 2000 within 200th episode. But the accumulated rewards
with $\alpha$ values of $0.7$and $0.8$ are very low. They have accumulated
rewards approximately 50-60 for the whole time. Later , the architecture was
changed to 2 hidden layers of 40 Relu Units . The $\gamma$ was selected to be
$0.9$ . Figure 7 shows that the both curves reached highest accumulated
rewards within 200 episodes in the new configuration.

Figure 6: ”Rewards for three different $\alpha$s with $\gamma$ 0.999” Figure
7: Rewards vs Episodes for New Architecture

##  V Comparison to traditional methods

In our previous paper, [1] , we compared PID, Fuzzy Logic and LQR . The Fig. 8
shows the performance curves for different controllers. It shows that LQR and
Fuzzy controllers were not so stable like PID. Although we had to tune all of
them manually. But the Fig. 7 shows that DQN learning is more stable after
some iterations.

Figure 8: Performance curve for PID, Fuzzy logic and LQR

##  VI CONCLUSION and FUTURE WORK

The implementation of $Q$ Learning and Deep $Q$ Network as a controller in the
Gazebo Robot Model was shown in this paper. It showed the details of the
algorithms. However, some further improvments can be done. Like, It was
assumed that the robot will work on Markovian State space , which generally
not the case. In general Inverted pendulum models are Non-markovian models .
So there must exist some kind of dependancies among the states. So In future,
Recurrent Neural Network have a great possibility. Moreover, 10 predefined
values of velocities for action were used. In the real world application,
action values have continous range. So for more complex models, this method
may not work. In that case, deep reinforcment learning algorithms with
continuous action space like Actro Critic Reinforcement Learning algorithm
[10] can be used. Finally, this work should be improved for real world
scenarios.

##  VII Declarations

###  VII-A Competing Interests

The Authors declare that they have no competing interests

###  VII-B Author’s contribution

The original project is this paper and [1] . The Contributions of MD Muhaimin
Rahman is the simulations and writing of this paper. The contributions of SM
Hasanur Rashid and M.M. Hossain is reviewing both papers.

###  VII-C Funding

The paper has no external source of funding.

## References

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  * [10] V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu, “Asynchronous methods for deep reinforcement learning,” in _Proceedings of The 33rd International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, M. F. Balcan and K. Q. Weinberger, Eds., vol. 48. New York, New York, USA: PMLR, 20–22 Jun 2016, pp. 1928–1937. [Online]. Available: http://proceedings.mlr.press/v48/mniha16.html


# Handover Management through Reconfigurable Intelligent Surfaces for VLC
under Blockage Conditions

Kapila W. S. Palitharathna1, Anna Maria Vegni2, Panagiotis D. Diamantoulakis3,
Himal A. Suraweera4, Ioannis Krikidis1  1Department of Electrical and Computer
Engineering, University of Cyprus, Nicosia, Cyprus
2Department of Industrial, Electronic and Mechanical Engineering, Roma Tre
University, Rome, Italy
3Department of Applied Informatics, University of Macedonia, Thessaloniki,
Greece
4Department of Electrical and Electronic Engineering, University of
Peradeniya, Peradeniya, Sri Lanka

###### Abstract

In this paper, we consider an indoor visible light communication (VLC) system
with multiple “white” light emitting diodes serving to form overlapping
wireless communication cells. In order to maintain seamless connectivity to
mobile users, a handover procedure should be implemented. In particular,
practical conditions such as blockages due to obstacles inside the room
environment and the mobility of users can affect direct VLC connectivity. The
use of reconfigurable intelligent surfaces (RISs) in optical wireless systems
allows to exploit non-direct connectivity links, thus providing efficient
communication links. In this paper, we present a proactive handover mechanism
that exploits the presence of a RIS, in order to redirect the communication
links in case of blockages. The proposed approach has been implemented both in
hard and soft modes and assessed in terms of achievable data rate and handover
latency for a user walking in a given reference room at different user speeds
and blockage conditions. Our presented results and comparisons with
conventional handover methods (i.e., without RIS) are helpful in showing the
superiority of the presented algorithm.

##  I Introduction

In next generation wireless networks, the use of alternative technology
bearers than radio frequency (RF) spectrum is strongly needed by a set of RF
drawbacks and limitations arisen so far. Due to the huge amount of mobile
devices, the RF spectrum is suffering and cannot guarantee enough bandwidth to
support such a large demand of data traffic, thus causing the well-known
“spectrum crunch problem” [1].

Visible light communication (VLC) is a candidate technology for beyond 5G
(B5G) networks, thanks to its high frequency bandwidth that provides high data
rates by modulating the optical intensity generated by light emitting diode
(LED) based luminaires [2]. VLC is the unique technology that allows
simultaneously data transmission and illumination, thus representing a green
technology bearer. Furthermore, due to the high directivity of line-of-sight
(LoS) links, security can be guaranteed for a point-to-point data
transmission. However, this condition strongly limits data propagation in case
of mobile users, since the alignment between a LED and a photodetector (PD)
should be maintained for effective data transmission. It follows that any
blockage laying between a LED and the PD, or shadow dimming the LED or the PD,
can affect the LoS condition and then reduce the quality-of-service (QoS)
level of the data transmission [3].

Handover management techniques should be considered in order to guarantee an
ubiquitous and seamless LoS connectivity. For this aim, overlapping lighting
cells is a condition necessary to manage handover and switch LoS connectivity
from one LED to another or several other LEDs. However, this requires that
multiple LEDs should populate the reference environment, and that is a
condition not always achievable, especially in large rooms or office.
Furthermore, a handover from a serving LED to a candidate one is valid since
still considering LoS condition, which cannot be relaxed [4, 5]. Extending the
assumption to non LoS (NLoS) components due to multiple reflections of the
wireless optical signal does not provide enough power at the receiver.

The use of reconfigurable intelligent surfaces (RISs) results as a benefit for
optical wireless handover management. The RIS concept has been arisen from RF
technology, allowing to control the wireless channel by manipulating the
impinging RF signal. In case of optical wireless systems, RISs can be
exploited in indoor environments to steer the reflected signal toward a
desired receiver. Specifically in the visible frequency range, RIS can be
physically implemented by mirror-based array or metasurfaces [6, 7]. The RIS
device can be deployed both at the transmitter/receiver side and in the
wireless environment and different positions of the RIS allow to achieve
different features [8].

In this paper, we are interested in the RIS deployment for handover management
in an indoor environment. The RIS acts as a reflecting surface that redirects
the impinging wireless optical signals to the PD. If direct LoS connectivity
is not supported due to blockages, the RIS can act as a bridge, where the
optical signal from the LED is steered to the RIS and then redirected to the
PD, providing seamless connectivity to the final user.

Related Work: Handover mechanisms in VLC networks have been largely
investigated by the research community, mainly for indoor systems where VLC
access points coexist in conjunction with RF technology. In [9], Wang and Haas
investigate a dynamic load balancing technique for hybrid VLC/RF systems, that
is able to mitigate unwanted and unnecessary handovers, but forcing mobile
users to be served by RF access points. In [10], an experimental demonstration
of a real-time RF/VLC data streaming system is presented. Handover occurs
based on signal strength measurement, and RF technology is exploited mainly
for mobile users, while almost-static ones are preferably connected to narrow
beam VLC coverage.

Focusing on pure VLC systems, handover solutions become more complex since
coverage is not always guaranteed and performance are limited to the
availability of LoS connectivity links [4]. Dinc et al. [11] investigate soft
handover methods for VLC, as well Demir et al. [12] present a coordinated
multipoint transmission technique in case of soft handover. Vegni and
Diamantoulakis [5] investigate a proactive handover mechanism for mitigation
of blockages. The proposed approach allows handovers trigged according to the
presence of blockages. Both soft and hard handover occurrences are assumed,
showing the best performance are achieved in case of soft handover. However,
in all previous works it is assumed the existence of overlapping lighting
cells that guarantee the handover procedure, but can cause interference.

Motivation and Contribution: In such scenario, the use of RISs for handover
management can represent a suitable and effective solution. However, for the
best of authors’ knowledge, no contribution exists yet addressing this topic
for pure VLC scenario. The idea is similar to the work in [13], but it is
investigated for mmWave systems. It was observed that, under various channel
blockage conditions, it is possible to reduce the handover overhead by jointly
adjusting beamformers and controlling the RIS.

Leveraging on the idea of using RIS for handover management in VLC systems
under blockage conditions, this paper represents an improved version of a
previous work in [5], where RISs are introduced for enhanced performance. We
present an enhanced system comprised of a RIS device able to assist the
handover procedure. In our vision, RIS acts as a relay node that redirects the
optical intensity toward the PD, in case of blockages incurring between the
transmitting LED and the PD. Differently from [5], where handovers occur only
assuming that overlapping wireless signals are available in each position of
the environment thus strongly limiting the effectiveness of handover
management, in this paper using RIS it is possible to guarantee a real
seamless connectivity to the final user. Indeed, scattered and reflected
signals are exploited and directed from the RIS to the PD, even if no
overlapping lighting cells are available. This allows to reduce the number of
LEDs, with a reduction of energy and deployment costs.

##  II System model

Let us consider an indoor environment where a set of $N$ LEDs that act as
access points (APs) are randomly deployed on the ceiling while guaranteeing a
sufficient level of illuminance and data rate constraints all over the room.
The user is walking randomly while connected to his own mobile device,
equipped with a PD to receive information from APs. The LED APs are assigned
with an identification code that is represented by a direct-sequence optical
code division multiple access (DS-OCDMA) [14]. Moreover, the communication is
assisted by a RIS mounted on the wall, which includes $M$ number of mirror
elements each having a height of $h_{m}$ [m] and a width of $w_{m}$ [m]. Each
mirror element can be controlled using micro motors and the yaw and roll angle
of the $j$-th mirror element (i.e., $\alpha_{j}$ and $\beta_{j}$) can be
controlled to focus light signals on the receiver resulting in a better
performance [7, 6]. The mid-point of $j$-th RIS element is $(x_{j},y_{j})$ on
the RIS plane.

Figure 1: Schematic of (a) hard and (b) soft handovers, for a mobile user in
an overlapping lighting cells indoor scenario with the help of RIS mounted on
a wall in the reference scenario.

Overlapping light coverage may occur in this setup. Thanks to the
orthogonality of the codes, we assume that the $N$ APs do not cause
interference with each other. In a given position, the receiver is then able
to collect the power levels from different APs by considering all the possible
sequences of a given spreading factor. Specifically, the APs transmit
orthogonal sequences by employing Walsh-Hadamard codes so as to grant
orthogonality. In this context, we consider the special requirements for VLC
transmission, particularly the condition that the transmitted symbols should
be real and positive. We then adopt a VLC-adapted version of the DS-CDMA, in
which a fixed DC offset is added to convert the bipolar signal into a unipolar
signal before modulating the light source.

Blockages can present due to shadows along the LoS connectivity link from the
LED to the PD. To model the blockage due to humans inside the room, we use a
more realistic model based on the mobility of users and a geometrical approach
to determine the blockage as in [15] and [16]. According to this model, it is
assumed that $N_{B}$ number of humans walk inside the room with random
starting points, ending points, and speeds. While mobility, each of them can
create a blockage in the LoS link from the $i$-th LED to the receiver.

The complete optical channel gain from the $i$-th LED to the receiver can be
written as [17]

$\displaystyle
h_{i}=I_{i}h_{i}^{\text{LoS}}+\sum_{j=1}^{M}h_{i,j}(\alpha_{j},\beta_{j}),$
(1)

where $I_{i}=\prod_{b=1}^{N_{B}}I_{i,b}$ is the blockage from the $i$-th LED
to the receiver, $h_{i}^{\text{LoS}}$ is the LoS channel gain without
blockage, and $h_{i,j}(\alpha_{j},\beta_{j})$ is the non-LoS channel gain due
to the $j$-th RIS element, and $N_{B}$ is the number of blockages. $I_{i,b}$
is the blockage along the link from the $i$-th LED to the receiver due to the
presence of the $b$-th human inside the room. In case of any blockage, then
$I_{i}=0$, and the LoS channel gain will be null. If there is no blockage,
$I_{i}=1$ and LoS channel gain will exist. $h_{i}^{\text{LoS}}$ is evaluated
with the help of the well-known Lambertian radian pattern equation [15]. The
RIS channel gain $h_{i,j}(\alpha_{j},\beta_{j})$ for a given
$(\alpha_{j},\beta_{j})$ can be modeled with the light reflection properties
of a mirror and geometry similar to [18] and [16].

The exact capacity of VLC is not known. Hence, we use an achievable rate for
the VLC link, which is expressed as [19]

$\displaystyle R=B\log_{2}\left(1+\frac{e}{2\pi}\eta\right),$ (2)

where $B$ [Hz] is the bandwidth of the signal, and $\eta$ is the signal-to-
noise ratio (SNR). In the case of overlapping cells, the receiver combines
signals coming from multiple $L_{0}$ LEDs (i.e., $L_{0}\geq 2$), which are
transmitting the same information, and hence, the achievable data rate,
$R_{0}$ can be written by replacing the parameter $\eta$ in Eq. (2) with
$\sum_{i=1}^{L_{0}}\eta_{i}$ where $\eta_{i}$ is the SNR for the signal
received from the $i$-th AP [5].

##  III Proactive Handover Mechanism

We introduce a proactive handover mechanism, with the aid of RIS, under the
LoS link blockage. First, the method of identifying the blockage is
introduced. Let $\xi_{i}$ be the blockage degree from the $i$-th LED, defined
as a value in the range $[0,1]$ that represents the extent to which the
channel blockage affects the channel quality. If $\xi_{i}=0$, no blockage
occurs in the $i$-th LoS channel, while $\xi_{i}=1$ stands for a complete
blockage. More in details, from [5] the blockage probability $p_{b,i}$ of the
$i$-th LED is defined as $p_{b,i}=\Pr[I_{elec,i}<\mathcal{I}]$ where
$\mathcal{I}$ is a given threshold and $I_{elec,i}$ is the electric current
obtained from the gathered photons at the PD, expressed as [20]

$\vspace{-1mm}I_{elec,i}=(1-\xi_{i})rh^{LoS}_{i}P_{i},$ (3)

with $r$ [A/W] as the PD responsivity, $P_{i}$ [Watt] is the transmitted
optical power from the $i$-th LED, and $h^{LoS}_{i}$ are the channel
coefficients of LoS link from the $i$-th LED to the receiver. From Eq. (3) we
can easily approximate the blockage probability as follows:

$\vspace{-2mm}p_{b,i}=\begin{cases}1&{\rm{if}}\;\;\xi_{i}\geq 0.5\\\
0&\text{otherwise}\\\ \end{cases}$ (4)

The handover decision is taken according to the blockage probability and
availability of the RIS. At a given receiver position $P_{k}=(x_{k},y_{k})$,
the blockage probability $p_{b,i}$ is calculated for the $i$-th LED. In the
case of overlapping cell environments, if all the LEDs are in blockage, i.e.,
$\xi_{i}=1$, $\forall i$, the angles of all RIS elements are set to focus
light signals coming from the closest LED to the receiver that ensures no
connectivity hole. Hence with the use of RIS, the probability of having a
connectivity hole is very low in a large area of the room.

On the other hand, we compute the set of overlapping LEDs that are not in
blockage for the considered position, $\mathcal{L}$. If the cardinality of
$\mathcal{L}$ is equal to $1$, a hard handover will occur, while it is a soft
handover in the case of $\mathcal{L}\geq 2$. In particular, the hard handover
will take place when there is a link break to the serving cell, and then a
link is created to a candidate cell ensuring a connectivity is established
only with one AP. Moreover, the RIS angles are set in such a way that all the
elements focus light signal coming from the connected AP to the receiver. For
this scenario, RIS angles can be found with a geometrical method presented in
[21] once the connected AP location and the receiver location are known. It
can be noted that when an RIS is used, the probability of handover becomes
lower since the RIS can support the existing link even under blockage
conditions. At the same time, the use of RIS increases the data rate due to
high channel gain. A soft handover occurs when connectivity with more than one
AP is possible. It allows the receiver to connect with more than one AP at the
same time, maintaining connectivity with both the serving cell and the
candidate cell.

Figure 1 depicts both hard and soft handovers with the assistance of RISs. In
general, the data rate is higher in the case of soft handover, since the
receiver is connected to several LEDs. In the case of soft handover, the RIS
elements support several links and the RIS element assignment needs to be done
carefully to further improve the performance. Due to the complexity of
traditional RIS assignment algorithms, we use a training-based machine
learning approach as explained in Subsection III-A. Algorithm 1 presents the
key steps involved in the handover mechanism.

Algorithm 1 : Proactive Handover Using RIS

1:Inputs: $N$, $M$, $(x_{i},y_{i})$, $(x_{j},y_{j})$

2:Outputs: $R_{h/s}$, $\delta_{h/s}$, $N_{h/s}$, $(\alpha_{j},\beta_{j})$

3:for each $P_{k}=(x_{k},y_{k})$ do

4: Compute $I_{elec,i}$ using Eq. (3)

5: Compute $p_{b,i}$ using Eq. (4)

6: if $\xi_{i}=0,\forall i$ then

7: Assign all RIS elements closest LED

8: $R_{h}=R$ $\triangleright$ $R$ calculated only with RIS channels

9: $N_{h}=N_{h}+1$ $\triangleright$ Hard handover executions

10: else

11: Compute set of LEDs not in blockage ($\mathcal{L}$)

12: if $|\mathcal{L}|=1$ then

13: Assign all RIS elements to the selected LED

14: $R_{h}=R$ $\triangleright$ Hard handover rate with RIS

15: Compute $\delta_{h}$ $\triangleright$ Hard handover latency

16: $N_{h}=N_{h}+1$ $\triangleright$ Hard handover executions

17: else

18: Assign each RIS elements among LEDs in the set $\mathcal{L}$ using the ANN
architecture

19: $R_{s}=R_{0}$ $\triangleright$ Soft handover with RIS

20: Compute $\delta_{s}$ $\triangleright$ Soft handover latency

21: $N_{s}=N_{s}+1$ $\triangleright$ Soft handover executions

22: end if

23: end if

24:end for

###  III-A ANN-based RIS Assignment

The RIS element assignment to different APs is crucial for achieving better
performance, especially in the soft handover case. We use a well-trained
lightweight artificial neural network (ANN) for this purpose. The use of a
lightweight ANN is motivated due to its ability to learn blockage patterns
induced by human mobility and the low complexity of the design that avoids
additional latency to handover when RISs are used.

In this ANN architecture, the estimated blockage values, $\xi_{i}$, $\forall
i$, and user position are used as inputs to the system while the outputs are
RIS association values, $X_{j}$ for $j=(1,2,\ldots,M)$. The value of $X_{j}$
is selected from the set that includes the AP number $\\{1,2,\ldots,N\\}$. The
ANN includes several layers as described below. First, the input is fed into a
layer with $N+2$ number of nodes. It is followed by three fully connected
layers with $N_{1}$, $N_{2}$, and $N_{3}$ number of nodes and the activation
function is the rectified linear unit (ReLU). Finally, a softmax layer is used
to obtain the $M$ number of RIS associations. This model is trained with
$10^{7}$ blockage instances and corresponding RIS association vectors obtained
from a brute-force simulation.

The presented RIS-assisted handover management technique is compared with the
existing handover management technique presented in [5] in which no RIS is
used. In this method, connectivity holes can present when all links are
blocked. When there is only one AP that is not in the blockage, a hard
handover will occur while a soft handover will take place in the case of
multiple APs are not blocked.

##  IV Numerical Results

Numerical results are presented to illustrate the superior performance gains
of the presented RIS-assisted handover mechanism under different system and
channel parameters. A room dimension of $5\text{m}\times 5\text{m}\times
3\text{m}$ is assumed. The APs are assumed to be randomly placed on the
ceiling. Further, simulation parameters are selected similarly to [5].

Figure 2 shows the average handover data rate versus the number of APs in the
cases of low mobility, high mobility, hard handover, and soft handover.
Moreover, results are compared with the case of no RISs being involved. Our
results verify that the use of RIS results in data rate improvement in almost
all cases. Especially, in the case of a small number of APs, the percentage
improvement of the data rate is comparably high (i.e., upto $60\%$) compared
to the case of a large number of APs. This is expected since RIS avoids
connectivity holes that can be present when a low number of APs are used. Both
soft and hard handovers show a saturating nature in the high APs regime, due
to zero connectivity holes. In general, the soft handover results in better
performance than the hard handover, and especially, up to $80\%$ data rate
improvement can be observed. The performance improvement with the RIS in the
soft handover is higher than in the hard handover scenario as a result of
focusing light from several APs to the receiver using the ANN method.

Figure 2: Average handover data rate vs. number of APs. Figure 3: Average
handover latency vs. number of APs.

Figure 3 shows the average handover latency versus the number of APs. The
handover latency shows a decreasing nature when the number of APs is increased
which is a counter-trend to the data rate since it is inversely proportional
to the data rate. The latency saturates around $1\mu s$ and $2\mu s$ at the
cases of soft and hard handover cases. The use of RIS guarantees a low latency
compared to the case without RISs. Especially, latency is almost constant for
different numbers of APs when RIS is used. For the considered setup, the
handover latency lies between $1\mu s$ and $4.5\mu s$ for all the cases.
Though it is illustrated, the use of RIS results in a decrease in the number
of handovers in all cases compared to no RIS scenario. The handover decrease
is high for soft handover compared to hard handover. Specifically up to $12\%$
handover reduction is expected in soft handover case compared to hard
handover.

##  V Conclusions

In this paper, we presented an RIS-assisted proactive handover mechanism for
indoor VLC systems under blockage conditions. Especially, under blockage
conditions induced by human movements, the use of mirror-based RIS is a
promising approach to improving the handover performance. We present an
algorithm to perform the hard and soft handover and RIS assignment for the
considered scenario and is supported by an ANN architecture. Our results show
that the use of RIS improves the handover performance and helps to achieve
high data rates and low handover latency. Specifically, data rate improvement
up to $60\%$ is expected. The ANN-based RIS assignment results in a very low
amount of connectivity holes and guarantees a smooth handover. It is noted
that soft handover results in better performance compared to hard handover in
most cases. Specifically, upto $12\%$ of handovers can be reduced using soft
handover compared to hard handover for high mobility.

## Acknowledgment

This article is partially based on work from COST Action NEWFOCUS CA19111,
supported by COST (European Cooperation in Science and Technology) and has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement No.
819819).

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