# Survey On The Biharmonic Hypersurfaces In Terms Of the Induced Metric Of
Tensor Ricci

N. Mosadegh  Department of Mathematics, Azarbaijan Shahid Madani University,
Tabriz 53751 71379, Iran<EMAIL_ADDRESS>E. Abedi  Department of
Mathematics, Azarbaijan Shahid Madani University,
Tabriz 53751 71379, Iran<EMAIL_ADDRESS>
(Date: January 1, 2004)

###### Abstract.

In this article, we study the biharmonic hypersurfaces in the Sasakian space
form with the induced metric of tensor Ricci. We find the existence necessary
and sufficient condition of the biharmonic hypersurfaces there. We show that
the biharmonic Hopf hypersurfaces are minimal where gradient of the mean
curvature is in direction of the structural vector fields. Furthermore, we
prove that does not exist any biharmonic Hopf hypersurfaces when the gradient
of the mean curvature is a principal direction.

###### Key words and phrases:

Biharmonic hypersurfaces, Sasakian space forms.

###### 1991 Mathematics Subject Classification:

Primary 53C42; Secondary 53C43, 53B25

11footnotetext: The first author is as corresponding author.

##  1\. Introduction

Consider an immersion $\psi:(M,g)\rightarrow(N,h)$ between two Riemannian
manifolds, which is known as the critical point of the energy functional

$\displaystyle E:C^{\infty}(M,N)\rightarrow R,$ $\displaystyle
E(\psi)=\int_{M}|\tau(\psi)|^{2}d\vartheta,$

called a biharmonic map and quivalently $M$ is said the biharmonic
submanifold, where $\tau(\psi)=\textsf{trace}\nabla d\psi$ known the tension
field of $\psi$ see ([7, 8]). Vanishing of the tension field characterizes the
harmonic map as well. The Euler-Lagrangian equation associated to the energy
functional, written as

$\displaystyle 0=\tau_{2}(\psi)$ $\displaystyle=$
$\displaystyle-\Delta\tau(\psi)-\textsf{trace}R^{N}(d\psi(.),\tau(\psi))d\psi(.),$
(1.1)

see [5]. Here $\Delta=-\textsf{trace}\nabla^{2}$ stands for the Laplace-
Beltrami operator, where $\nabla$ is a induced connection in the pull back
bundle $\varphi^{-1}(TN)$ and $R^{N}$ is the curvature operator on $N$ in
which defined by $R^{N}(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}$ for all
$X$ and $Y$ tangent to $N$. We recall, it is known that vanishing the tension
field is characterization of harmonic map, therefore, the curiosity is about
nonharmonic biharmonic maps which are called proper biharmonic.

In 1980s, the characterization formula of the biharmonic Riemannian immersion
in the Euclidean space, that is, $\Delta H=0$, where $H$ denotes the mean
curvature vector field, was introduced by Chen [3], independently. Then, it
was shown that every biharmonic hypersurface in the Euclidean space $E^{n}$ is
a minimal hypersurface where $n=3,4,5$ (see[2], [17],[15]). Yu Fu [18] proved
that every biharmonic hypersurface with three distinct principal curvatures in
the Euclidean space is minimal. Also, with respect to the number of distinct
principal curvature concerning the biharmonic hypersurfaces, the authors
reached to the following results. Every biharmonic Hopf hypersurface in the
complex Euclidean space is a minimal one. There exist no proper biharmonic
Ricci saliton hypersurface in the Euclidean space in such away that the
potential vector field is a principal vector in the direction of the gradient
of the mean curvature vector field. Every biharmonic hypersurface equipped
with the recurrent operator in the Euclidean space is minimal as well. The
biharmonic Hopf QR-hypersurfaces are minimal ones in the quaternionic
Euclidean space (see[10, 11, 12, 13]).

Additionally, in space of the nonconstant sectional curvature, several
classification concerning the proper biharmonic has been investigated. For
example, classified all the proper biharmonic Hopf cylinder in $3$-dimensional
Sasakian space forms and all the proper biharmonic homogeneous real
hypersurfaces in the complex projective space [6, 16]. Also, the authors have
already obtained the biharmonic conditions of CMC hypersurfaces in the special
warped product, and the nonexistence result of the biharmonic Hopf
hypersurfaces in the Sasakian space form associated to the Tanaka-Webster
connection, see [9, 14].

TThe aim of this paper is studying about the biharmonic hypersurfaces in the
Sasakian space form, where the ambient manifold is equipped with the induced
metric of the tensor Ricci. We reach the necessary and sufficient condition
about the existence of the biharmonic hypersurfaces there and call them Ricci-
biharmonic hypersurfaces. In the first part of this paper, we present an
example of a biharmonic surface in the Sasakian space form $R^{3}(-3)$ which
is minimal. we obtain the nonexistent result about the Ricci-biharmonic Hopf
hypersurfaces, where the gradH is a principal direction. Also, we show a
Ricci-biharmonic Hopf hypersurface is minimal where the $\mathsf{grad}$H is in
the direction of structural vector fields.

##  2\. Preliminary

Throughout this paper all the objects are considered to be smooth. In this
section we start with the notations of the Sasakian structure that will be
used throughout the paper. Let $\Gamma(TM)$ and $\Gamma(TM^{(p,q)})$ denote
the set of all the vector fields and tensor fields of type $(p,q)$ on manifold
$M$, respectively. Let $\nabla$ stands for the Levi-Civita connection on $M$
and $g\in\Gamma(TM^{(0,2)})$. Take the triple $(\varphi,\xi,\eta)$, which
$\varphi\in\Gamma(TM^{(1,1)}),\eta\in\Gamma(TM^{(1,0)})$ and
$\xi\in\Gamma(TM)$, is known as an almost contact structure on the odd
dimensional manifold $M^{2m+1}$, provided that the following equations hold
for any $X,Y\in\Gamma(TM^{2m+1})$

$\displaystyle\varphi^{2}(X)=-X+\eta(X),\ \ \ \eta(\xi)=1,\ \ \
\varphi(\xi)=0.$ (2.1)

Now, a Riemannian metric $g$ is endowed on $M^{2m+1}$, then
$(\varphi,\xi,\eta,g)$ is called almost contact metric structure if

$\displaystyle\eta(X)=g(\xi,X),\ \ g(\varphi X,\varphi Y)$ $\displaystyle=$
$\displaystyle g(X,Y)-\eta(X)\eta(Y).$ (2.2)

An almost contact metric structure on $M^{2m+1}$ is a contact metric structure
where $d\eta(X,Y)=g(X,\varphi Y)$. Now, a contact metric manifold is named a
Sasakian manifold if and if

$\displaystyle(\nabla_{X}\varphi)Y=g(X,Y)\xi-g(Y,\xi)X,$ (2.3)

where $X,Y\in\Gamma(TM^{2m+1})$ and $\nabla$ stands for the Levi-Civita
connection on $M^{2m+1}$. For a Sasakian manifold it is known that

$\displaystyle\nabla_{X}\xi=-\varphi X.$ (2.4)

At point $x$ of a Sasakian manifold $M^{2m+1}$ a $\varphi$-section means a
2-plane spanned by $\\{X,\varphi X\\}$ in $T_{x}(M^{2m+1})$, where $X$ is an
unit tangent vector orthogonal to $\xi$. A Sasakian manifold
$(M^{2m+1},\varphi,\xi,\eta,g)$ has constant $\varphi$-sectional curvature
$c\in R$, provided that its curvature tensor satisfies

$\displaystyle R(X,Y)Z$ $\displaystyle=$
$\displaystyle-\frac{c-1}{4}\\{\eta(Z)[\eta(Y)X-\eta(X)Y]$
$\displaystyle+[g(Y,Z)\eta(X)-g(X,Z)\eta(Y)]\xi$ $\displaystyle+g(\varphi
X,Z)\varphi Y+2g(\varphi X,Y)\varphi Z-g(\varphi Y,Z)\varphi X\\}$
$\displaystyle+\frac{c+3}{4}\\{g(Y,Z)X-g(X,Z)Y\\},$

where $X,Y$ and $Z\in\Gamma(TM^{2m+1})$. A Sasakian manifold of constant
$\varphi$-sectional curvature $c$ is called a Sasakian space form and
determined by $\overline{M}^{2m+1}(c)$.

At the end of this section, we construct an example of a Riemannian biharmonic
surface in the Sasakian space form $R^{3}(-3)$, which is harmonic($H=0$) as
well.

######  Example.

Let $R^{3}$ be a hypersurface in the Euclidean space $R^{4}$. Let $J$ be the
standard almost complex structure in $R^{4}$ and set $\xi=-JN$, where $N$ is
an unit normal vector field of $R^{3}$. Define $\varphi$ by $\pi oJ$, where
$\pi$ is the natural projection of the tangent space of $R^{4}$ in to the
tangent space of $R^{3}$. Let $(x,y,z)$ be the Euclidean coordinate in
$R^{3}$, we consider

$\displaystyle\eta=\frac{1}{2}(dz-ydx),\ \
g=\eta\otimes\eta+\frac{1}{4}(dx^{2}+dy^{2}),$ $\displaystyle\
\varphi(X\frac{\partial}{\partial x}+Y\frac{\partial}{\partial
y}+Z\frac{\partial}{\partial z})=Y\frac{\partial}{\partial
x}-X\frac{\partial}{\partial y}+Yy\frac{\partial}{\partial z}$ (2.6)

where $\xi=2\frac{\partial}{\partial z}$. Then $(R^{3},\varphi,\eta,\xi,g)$ is
called a Sasakian space form where its $\varphi$-sectional curvature is
$c=-3$. Let $f\in C^{\infty}(R^{3}(-3))$ defines $f(x,y,z)=z$, then we
consider the level set of $f$ like $M^{2}=f^{-1}(0)=\\{(x,y,z)\in
R^{3};z=0\\}$ which is claimed as a minimal surface (as well as biharmonic) of
$R^{3}(-3)$. In order to show this property, we choose an appropriate
orthonormal frame field on $R^{3}(-3)$ such as

$\displaystyle e_{1}=2(\frac{\partial}{\partial x}+y\frac{\partial}{\partial
z}),\ \ e_{2}=-2\frac{\partial}{\partial y}\ \
,e_{3}=2\frac{\partial}{\partial z}$ (2.7)

then we calculate
$\textsf{grad}f=\sum_{i=1}^{3}e_{i}(f)e_{i}=2(ye_{1}+e_{3})$. So,
$N=\frac{\textsf{grad}f}{|\textsf{grad
}f|}=\frac{1}{\sqrt{1+y^{2}}}(ye_{1}+e_{3})$ is an unit normal vector on
$M^{2}$. Also, $-\varphi N=V=-\frac{y}{\sqrt{1+y^{2}}}e_{2}$ is in
$\Gamma(TM^{2})$. With respect to these last vectors we take an orthonormal
frame field $\\{E_{1}=\frac{V}{|V|}=-e_{2},\ \
E_{2}=-\frac{1}{\sqrt{1+y^{2}}}(e_{1}-ye_{3})\\}$ on $M^{2}$. Some easy
computations show the following bracket relations, which we need to calculate
the Weingarten operator $A$ of $M^{2}$ in the Sasakian Space $R^{3}(-3)$, as
following

$\displaystyle[e_{1},e_{2}]=2e_{3},\ \ [e_{1},e_{3}]=0,\ \ [e_{2},e_{3}]=0$
$\displaystyle\nabla_{e_{1}}e_{2}=-\nabla_{e_{2}}e_{1}=e_{3},\ \
\nabla_{e_{1}}e_{3}=\nabla_{e_{3}}e_{1}=-e_{2},\ \
\nabla_{e_{2}}e_{3}=\nabla_{e_{3}}e_{2}=e_{1}$

where $\nabla$ is the Levi-Civita connection on $R^{3}(-3)$. After all, we can
calculate

$\displaystyle-AE_{1}=\nabla_{E_{1}}N=\frac{1-y^{2}}{1+y^{2}}E_{2},$
$\displaystyle-AE_{2}=\nabla_{E_{2}}N=\frac{1-y^{2}}{1+y^{2}}E_{1}$

then we have

$\displaystyle A=\left(\begin{array}[]{cc}0&\frac{y^{2}-1}{1+y^{2}}\\\
\frac{y^{2}-1}{1+y^{2}}&0\\\ \end{array}\right).$

So, the shape operator matrix of $M^{2}$ presents that the mean curvature
$|H|=0$. In other words, $xy$-plane is a minimal or harmonic surface in the
Sasakian space form $R^{3}(-3)$.

##  3\. Ricci Biharmonic Hypersurfaces

In this paper the tensor Ric(X,Y) is considered a metric in a Sasakian space
form $(\overline{M}^{(2m+1)}(c),\overline{g})$ and written as

$\displaystyle\mathsf{Ric}(X,Y)$ $\displaystyle=$
$\displaystyle\frac{(m+1)c+3m-1}{2}\overline{g}(X,Y)$ (3.1)
$\displaystyle+\frac{(m+1)(1-c)}{2}\eta(X)\eta(Y),$

for $X$ and $Y\in\Gamma(T\overline{M}^{(2m+1)}(c))$.

By the Koszul formula the connection $\nabla^{\ast}$ associated to the induced
metric of the tensor Ricci is characterized here

######  Lemma 3.1.

Let $\overline{M}^{2m+1}(c)$ be a Sasakian space form with induced metric of
the tensor Ricci. Then, the associated connection $\nabla^{*}$ satisfies

$\displaystyle\nabla^{*}_{X}Y=\overline{\nabla}_{X}Y-\frac{k_{1}}{k_{2}}(\eta(X)\varphi
Y+\eta(Y)\varphi X),$ (3.2)

where $\overline{\nabla}$ denotes the Levi-Civita connection on
$\overline{M}^{2m+1}(c)$, $k_{1}=\frac{(m+1)(1-c)}{2}$ and
$k_{2}=\frac{(m+1)c+3m-1}{2}$.

###### Proof.

Suppose that $\overline{g}$ and $\overline{\nabla}$ are the Riemannian metric
and the Levi-Civita connection on $\overline{M}^{2m+1}(c)$, respectively. From
the Koszul formula and the equation $(\ref{7.1})$ we have

$\displaystyle\mathsf{Ric}(\nabla^{*}_{Y}Z,X)$ $\displaystyle=$ $\displaystyle
Y\mathsf{Ric}(Z,X)+Z\mathsf{Ric}(X,Y)+X\mathsf{Ric}(Y,Z)$ (3.3)
$\displaystyle-\mathsf{Ric}(Y,[Z,X])+\mathsf{Ric}(Z,[X,Y])+\mathsf{Ric}(X,[Y,Z])$
$\displaystyle=$ $\displaystyle
k_{1}(\eta(\overline{\nabla}_{Y}Z)\eta(X)-\overline{g}(\varphi
Y,X)\eta(Z)-\overline{g}(\varphi Z,X)\eta(Y))$
$\displaystyle+k_{2}\overline{g}(\overline{\nabla}_{Y}Z,X),$

then the equations $(\ref{7.1})$ and $(\ref{7.2})$ yield

$\displaystyle k_{1}\overline{g}(\nabla^{*}_{Y}Z,\xi)\xi+k_{2}\nabla^{*}_{Y}Z$
$\displaystyle=$ $\displaystyle
k_{1}\big{(}\overline{g}(\overline{\nabla}_{Y}Z,\xi)\xi-\eta(Z)\varphi
Y-\eta(Y)\varphi(Z)\big{)}$ $\displaystyle+k_{2}\overline{\nabla}_{Y}Z,$

which shows

$\displaystyle\overline{g}(\nabla^{*}_{Y}Z,\xi)=\overline{g}(\overline{\nabla}_{Y}Z,\xi),$

then

$\displaystyle k_{2}\nabla^{*}_{Y}Z=-k_{1}(\eta(Z)\varphi Y+\eta(Y)\varphi
Z)+k_{2}\overline{\nabla}_{Y}Z,$

so, we reach the result. ∎

The bitension field of an isometric immersion
$\psi:M^{2m}\rightarrow\overline{M}^{2m+1}$ with the connection $\nabla^{*}$
holds

$\displaystyle\tau_{2}^{\star}(\psi)=-\Delta^{\star}(H)-\textsf{trace}R^{\star}(d\psi(.),H)d\psi(.),$
(3.4)

here $\Delta^{\star}$ stands for the Laplace-Beltrami operator on sections of
the pull back bundle $\psi^{-1}(\Gamma(T\overline{M}^{2m+1}(c)))$ and
$R^{\star}$ denotes the curvature operator on the Sasakian space form
$\overline{M}^{2m+1}(c)$ where the following sign conventions hold

$\displaystyle\Delta^{\star}X=-\textsf{trace}\nabla^{\star^{2}}X,\ \ \ \
\forall X\in\psi^{-1}(\Gamma(T\overline{M}^{2m+1})),$ (3.5) $\displaystyle
R^{\star}(X,Y)=[\nabla^{\star}_{X},\nabla^{\star}_{Y}]-\nabla^{\star}_{[X,Y]}\
\ \ \ X,Y\in{\Gamma(T\overline{M}^{2m+1})}.$ (3.6)

######  Definition 3.2.

A hypersurface $M^{2m}$ in the Sasakian space form $\overline{M}^{2m+1}(c)$
with the induced metric of tensor Ricci is called Ricci-biharmonic
hypersurface if $\tau_{2}^{\star}(\psi)=0$.

Let $(\overline{M}^{2m+1}(c),\varphi,\xi,\eta,\overline{g})$ be a Sasakian
space form. We suppose that $\xi$ and $V=-\varphi N$ are the tangent vector
fields on $M^{2m}$, where $N$ is a local unit normal vector on $M^{2m}$ as
well.

######  Lemma 3.3.

Let $M^{2m}$ be an immersed hypersurface in the Sasakian space form
$\overline{M}^{2m+1}(c)$ with the induced metric of tensor Ricci,
isometrically. Then

$\displaystyle\Delta^{*}H=\Delta
H+\big{(}3\frac{k_{1}}{k_{2}}+(\frac{k_{1}}{k_{2}})^{2}\big{)}H-2\frac{k_{1}}{k_{2}}\eta(\mathsf{grad}|H|)V,$
(3.7)

where $H$ is the mean curvature vector field of $M^{2m}$ in the Sasakian space
form.

###### Proof.

Let $\nabla^{\star}$, $\overline{\nabla}$ and $\nabla$ denote the connections
associated to induced metric of the tensor Ricci, Riemannian metric
$\overline{g}$ and the Levi-Civita connection on $\overline{M}^{2m+1}(c)$ and
$M^{2m}$, respectively. We consider a parallel local orthonormal frame field
$\\{e_{\alpha}\\}_{\alpha=1}^{2m}$ at $p\in M^{2m}$, the equation
$(\ref{7.9})$ and the Weingarten operator
$\overline{\nabla}_{e_{\alpha}}H=\nabla^{\perp}_{e_{\alpha}}H-A_{H}e_{\alpha},$
then

$\displaystyle\Delta^{\star}H$ $\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{2m}\nabla^{\star}_{e_{\alpha}}\nabla^{\star}_{e_{\alpha}}H$
(3.8) $\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{2m}\nabla^{\star}_{e_{\alpha}}\big{(}\overline{\nabla}_{e_{\alpha}}H-\frac{k_{1}}{k_{2}}(\varphi(H)\eta(e_{\alpha})+\varphi(e_{\alpha})\eta(H))\big{)}$
$\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{2m}\big{\\{}\overline{\nabla}_{e_{\alpha}}\overline{\nabla}_{e_{\alpha}}H-\frac{k_{1}}{k_{2}}(\varphi(\overline{\nabla}_{e_{\alpha}}H)\eta(e_{\alpha})+\varphi(e_{\alpha})\eta(\overline{\nabla}_{e_{\alpha}}H))$
$\displaystyle-\frac{k_{1}}{k_{2}}\eta(e_{\alpha})(\overline{\nabla}_{e_{\alpha}}\varphi
H-\frac{k_{1}}{k_{2}}\varphi^{2}H\eta(e_{\alpha}))\big{\\}}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{2m}\\{\overline{\nabla}_{e_{\alpha}}\overline{\nabla}_{e_{\alpha}}H+\frac{k_{1}}{k_{2}}\eta(A_{H}e_{\alpha})\varphi
e_{\alpha}\\}$
$\displaystyle-2\frac{k_{1}}{k_{2}}(\eta(\mathsf{grad}|H|)V+\varphi
A_{H}\xi)+(\frac{k_{1}}{k_{2}})^{2}H.$

We consider the following equations

$\displaystyle\sum_{\alpha=1}^{2m}\eta(\overline{\nabla}_{e_{\alpha}}H)\varphi
e_{\alpha}=\sum_{\alpha=1}^{2m}g(\overline{\nabla}_{e_{\alpha}}H,\xi)\varphi
e_{\alpha}=\sum_{\alpha=1}^{2m}g(H,\varphi e_{\alpha})\varphi e_{\alpha},$

on the one hand we have

$\displaystyle\sum_{\alpha=1}^{2m}\eta(\overline{\nabla}_{e_{\alpha}}H)\varphi
e_{\alpha}=\sum_{\alpha=1}^{2m}g(\overline{\nabla}_{e_{\alpha}}H,\xi)\varphi
e_{\alpha}=-\sum_{\alpha=1}^{2m}g(A_{H}e_{\alpha},\xi)\varphi e_{\alpha},$

consequently

$\displaystyle A_{H}\xi=-|H|V.$ (3.9)

where $-\varphi N=V$. By the above we calculate the first and second terms of
$(\ref{7.12})$ and one obtain

$\displaystyle\sum_{\alpha=1}^{2m}g(A_{H}e_{\alpha},\xi)\varphi e_{\alpha}$
$\displaystyle=$
$\displaystyle\sum_{\alpha,\beta=1}^{2m}g(A_{H}e_{\alpha},\xi)(g(\varphi
e_{\alpha},e_{\beta})e_{\beta}+g(\varphi e_{\alpha},N)N)$ (3.10)
$\displaystyle=$
$\displaystyle-|H|\sum_{\alpha,\beta=1}^{2m}g(e_{\alpha},V)(-g(e_{\alpha},\varphi
e_{\beta})e_{\beta}-g(e_{\alpha},\varphi N)N)$ $\displaystyle=$
$\displaystyle-|H|\sum_{\beta=1}^{2m}(g(V,\varphi
e_{\beta})e_{\beta}+g(V,V)N)$ $\displaystyle=$ $\displaystyle-H,$

also

$\displaystyle\overline{\nabla}_{e_{\alpha}}\overline{\nabla}_{e_{\alpha}}H$
$\displaystyle=$
$\displaystyle\nabla^{\perp}_{e_{\alpha}}\nabla^{\perp}_{e_{\alpha}}H$ (3.11)
$\displaystyle-
A_{\nabla^{\perp}_{e_{\alpha}}H}e_{\alpha}-\nabla_{e_{\alpha}}A_{H}(e_{\alpha})-B(e_{\alpha},A_{H}(e_{\alpha})),$

hence, we obtain

$\displaystyle\Delta
H=-\sum_{\alpha=1}^{2m}\overline{\nabla}_{e_{\alpha}}\overline{\nabla}_{e_{\alpha}}H$
$\displaystyle=$ $\displaystyle-\Delta^{\perp}H+\textsf{trace}B(.,A_{H}.)$
(3.12)
$\displaystyle+\textsf{trace}A_{\nabla_{(.)}^{\perp}H}(.)+\textsf{trace}\nabla_{(.)}A_{H}(.).$

After all, take the equations $(\ref{7.18})$, $(\ref{7.17})$ and
$(\ref{7.16})$ then replace them in $(\ref{7.12})$ we get the result. ∎

In the proof of the main results we need the following lemma.

######  Lemma 3.4.

Let $\psi:M^{2m}\longrightarrow\overline{M}^{2m+1}(c)$ be an isometric
immersion of $2m$-dimensional hypersurface $M^{2m}$ in the Sasakian space form
$\overline{M}^{2m+1}(c)$ with induced metric of the tensor Ricci. Then

$\displaystyle\mathsf{trace}R^{*}(d\psi(.),H)d\psi(.)=kH,$ (3.13)

where
$k=-\frac{-9-11m+18m^{2}+c^{2}(3+5m+2m^{2})+2c(3+11m+6m^{2})}{2(-1+c+3m+cm)^{2}}$
and $H$ denotes the mean curvature vector field of $M^{2m}$ in
$\overline{M}^{2m+1}(c)$.

###### Proof.

We calculate the curvature tensor $R^{\star}$ associted to the connection
$\nabla^{\star}$. So, for $X,Y$ and $Z\in\Gamma(T\overline{M}^{2m+1}(c))$ we
have

$\displaystyle R^{*}(X,Y)Z$ $\displaystyle=$
$\displaystyle\nabla^{*}_{X}\nabla^{*}_{Y}Z-\nabla^{*}_{Y}\nabla^{*}_{X}Z-\nabla^{*}_{[X,Y]}Z$
(3.14) $\displaystyle=$
$\displaystyle\overline{R}(X,Y)Z-\frac{k_{1}}{k_{2}}\overline{g}(Z,\overline{\nabla}_{X}\xi)\varphi
Y+\frac{k_{1}}{k_{2}}\overline{g}(Z,\overline{\nabla}_{Y}\xi)\varphi X$
$\displaystyle+\frac{k_{1}}{k_{2}}\big{(}\overline{g}(X,\overline{\nabla}_{Y}\xi)-\overline{g}(Y,\overline{\nabla}_{X}\xi)\big{)}\varphi
Z$
$\displaystyle+\frac{k_{1}}{k_{2}}\big{(}\eta(X)\overline{g}(Y,Z)-\eta(Y)\overline{g}(X,Z)\big{)}\xi$
$\displaystyle+(2\frac{k_{1}}{k_{2}}+(\frac{k_{1}}{k_{2}})^{2})(\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y),$

where $\overline{R}$ stands for the curvature tensor of a Sasakian space form
associated to the Riemannian metric $\overline{g}$. Now, we choose an
appropriate orthonormal frame field
$\\{\xi,V\\}\cup\\{e_{\alpha}\\}_{\alpha=1}^{2m-1}$ on
$\overline{M}^{2m+1}(c)$, in such away that
$\\{\xi^{*},V^{*}\\}\cup\\{e_{\alpha}^{*}\\}_{\alpha=1}^{2m-1}$, where
$e^{*}_{\alpha}=\frac{1}{\surd k_{2}}e_{\alpha},\ \
\xi^{*}=\frac{1}{\surd(k_{1}+k_{2})}\xi$ and $V^{*}=\frac{1}{\surd k_{2}}V$,
on $\overline{M}^{2m+1}(c)$ is an orthonormal frame field as well. Then by the
above one obtain

$\displaystyle\textsf{trace}R^{*}((.),H)(.)$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{2m-1}R^{*}(e^{*}_{\alpha},H)e^{*}_{\alpha}+R^{*}(\xi^{*},H)\xi^{*}+R^{*}(V^{*},H)V^{*},$

such that

$\displaystyle\sum_{\alpha=1}^{2m-1}R^{*}(e^{*}_{\alpha},H)e^{*}_{\alpha}=\frac{1}{k_{2}}\sum_{\alpha=1}^{2m-1}\overline{R}(e_{\alpha},H)e_{\alpha}=-\frac{c+3}{4k_{2}}(2m-1)H,$

and

$\displaystyle
R^{*}(\xi^{*},H)\xi^{*}=\frac{1}{k_{1}+k_{2}}(\overline{R}(\xi,H)\xi-((\frac{k_{1}}{k_{2}})^{2}+2\frac{k_{1}}{k_{2}})H),$

and

$\displaystyle
R^{*}(V^{*},H)V^{*}=\frac{1}{k_{2}}\big{(}\overline{R}(V,H)V+3\frac{k_{1}}{k_{2}}\big{)},$

where $\overline{R}(\xi,H)\xi=-H$ and $\overline{R}(V,H)V=-cH$. Then put all
the above terms together, we obtain the result. ∎

Now, we can have the principal result.

######  Theorem 3.5.

Let $\psi:M^{2m}\longrightarrow\overline{M}^{2m+1}(c)$ be an isometric
immersion of $2m-$ dimensional hypersurface $M^{2m}$ in the Sasakian space
form $\overline{M}^{2m+1}(c)$ with the induced metric of the Ricci tensor.
Then $M^{2m}$ is a Ricci-biharmonic hypersurface if and only if

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$\Delta^{\perp}H=(|B|^{2}+l)H$}\\\
\hbox{A
$(\textsf{grad}|H|)-\frac{k_{1}}{k_{2}}\eta(\textsf{grad}|H|)V+m|H|\textsf{grad}|H|=0$,}\end{array}\right.$

where
$l=\frac{-5-27m-2m^{2}+c^{2}(7+13m+6m^{2})+c(-2+30m+28m^{2})}{2(-1+c+3m+cm)^{2}}$
is constant, $A$ and $B$ denote the shape operator and second fundamental form
of $M^{2m}$ in $\overline{M}^{2m+1}(c)$, respectively.

###### Proof.

Take the definition 3.2 the hypersurface $M^{2m}$ is a Ricci-biharmonic
hypersurface if

$\displaystyle
0=\tau_{2}^{\star}(\psi)=-\Delta^{\star}H-\textsf{trace}R^{\star}(d\psi(.),H)d\psi(.),$

then the Lemma 3.3 and Lemma 3.4 follow

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\tau_{2}^{\star}(\psi)$
$\displaystyle=$ $\displaystyle-\Delta
H-(3\frac{k_{1}}{k_{2}}+(\frac{k_{1}}{k_{2}})^{2})H+2\frac{k_{1}}{k_{2}}\eta(grad|H|)V-kH.$

Thus, by the above separate the tangent and normal parts we get the result. ∎

For the Ricci-biharmonic hypersurfaces with the constant mean curvature the
Theorem 3.5 follows

######  Corollary 3.6.

Let $M^{2m}$ be a Ricci-biharmonic hypersurface with the constant mean
curvature. Then the $\varphi$-sectional curvature satisfies
$c\in(-5,\frac{1}{2})$.

###### Proof.

Consider the case of Ricci-biharmonic hypersurfaces where the mean curvature
vector field is constant $\neq 0$, then the Theorem 3.5 leads $|B|^{2}=-l$.
So, if $\varphi$-sectional curvature $c\notin(-5,\frac{1}{2})$ then $l>0$
which is impossible. ∎

##  4\. Ricci-biharmonic pseudo Hopf hypersurfaces

Let $x:M^{2m}\rightarrow\overline{M}^{2m+1}(c)$ be an isometric immersion from
a real hypersurface $M^{2m}$ in a Sasakian space form $\overline{M}^{2m+1}(c)$
with the induced metric of the Ricci tensor and the associated connection
$\nabla^{\star}$. Let $\overline{\nabla}$ and $\nabla$ denote the Levi-Civita
connection on $\overline{M}^{2m+1}(c)$ and $M^{2m}$, respectively. Let $N$ be
a local unit normal vector field on $M^{2m}$ in such away that $\xi$ and
$V=-\varphi N$ are tangent to $M^{2m}$. Then we have $T(M^{2m})=D\oplus
D^{\perp}$, where $D$ is a maximal $\varphi$-invariant distribution and
$D^{\perp}=\textsf{Span}\\{\xi,V\\}$. Suppose that the Weingarten operator $A$
satisfies $AD^{\perp}\subseteq D^{\perp}$ and $AD\subseteq D$. A hypersurface
$M^{2m}$ is called a pseudo-Hopf hypersurface provided that the Weingarten
operator $A$ is invariant on $\textsf{Span}\\{V,\xi\\}$ (see [1]). Also, it
was supposed that $W_{1},W_{2}\in\textsf{Span}\\{\xi,V\\}$ are the
eigenvectors of the Weingarten operator $A$ as $AW_{1}=\gamma_{1}W_{1}$ and
$AW_{2}=\gamma_{2}W_{2}$ such that

$\displaystyle W_{1}=\xi\cos\theta+V\sin\theta,\ \ $ $\displaystyle
W_{2}=-\xi\sin\theta+V\cos\theta,$ (4.1)

for some $0<\theta<\frac{\pi}{2}$, where $\gamma_{1}=-\tan\theta$ and
$\gamma_{2}=\cot\theta$. Let, $AV=\alpha\xi+\beta V$, then we have $\alpha=-1$
and $\beta=\frac{\cos 2\theta}{\cos\theta\sin\theta}$.

######  Lemma 4.1.

Let $(M^{2m},g)$ be a hypersurface in the Sasakian space form
$(\overline{M}^{2m+1}(c),\overline{g})$ where the ambient manifold is equipped
with the induced metric of the tensor Ricci. Then the Codazzi equation holds

$\displaystyle g((\nabla_{X}A)Y-(\nabla_{Y}A)X,Z)$ (4.2) $\displaystyle=$
$\displaystyle-k_{2}\frac{c-1}{4}\overline{g}\big{(}\overline{g}(\varphi
X,Z)Y+2\overline{g}(\varphi X,Y)Z-\overline{g}(\varphi Y,Z)X,V\big{)},$

where $X,Y$ and $Z$ tangent on $M^{2m}$.

###### Proof.

Let $N$ be a local unit normal vector field on $M^{2m}$ in the Sasakian space
form $\overline{M}^{2m+1}(c)$ and $\xi\in\Gamma(TM^{2m})$, then from the
equations $(\ref{7.15})$ and $(\ref{7.1})$ we have

$\displaystyle\mathsf{Ric}(\overline{R}(X,Y)Z,N)$ $\displaystyle=$
$\displaystyle
k_{1}\eta(\overline{R}(X,Y)Z)\eta(N)+k_{2}\overline{g}(\overline{R}(X,Y)Z,N)$
$\displaystyle=$ $\displaystyle-
k_{2}\frac{c-1}{4}\overline{g}\big{(}\overline{g}(\varphi
X,Z)Y+2\overline{g}(\varphi X,Y)Z-\overline{g}(\varphi Y,Z)X,V\big{)},$
$\displaystyle=$ $\displaystyle g((\nabla_{X}A)Y-(\nabla_{Y}A)X,Z),$

where $\overline{R}$ is the curvature tensor of the ambient manifold. ∎

######  Lemma 4.2.

Let $M^{2m}$ be a pseudo Hopf hypersurface in the Sasakian space form
$\overline{M}^{2m+1}(c)$. If the Weingarten operator $A$ for some $X\in D$
satisfies $AX=\lambda X$, then $\varphi X$ is an eigenvector of the Weingarten
operator corresponding to the eigenvalue in such a way that

$\displaystyle A\varphi X=\frac{\beta\lambda+2-n}{2\lambda-\beta}\varphi X,$
(4.3)

where $n=-k_{2}\frac{c-1}{2}$.

###### Proof.

Let $X,Y\in D$ be the eigenvectors of the Weingarten operator $A$. We consider
$AV=-\xi+\beta V$ and take the covariant derivative of both sides of this,
then

$\displaystyle(\nabla_{X}A)V+A\nabla_{X}V=X(\beta)V+\beta\nabla_{X}V+\varphi
X,$

where $\nabla$ denotes the Levi-Civita connection on $M^{2m}$ and
$\nabla_{X}V=\varphi AX$. Hence, we get

$\displaystyle g((\nabla_{X}A)V,Y)+g(A\varphi AX,Y)=\beta g(\varphi
AX,Y)+g(\varphi X,Y),$

similarly

$\displaystyle g((\nabla_{Y}A)V,X)+g(A\varphi AY,X)=\beta g(\varphi
AY,X)+g(\varphi Y,X).$

Since $\varphi$ is skew-symmetric, therefore

$\displaystyle g((\nabla_{X}A)Y-(\nabla_{Y}A)X,V)+2g(A\varphi AX,Y)=$
$\displaystyle\beta g(\varphi AX,Y)+\beta g(A\varphi X,Y)+2g(\varphi X,Y),$

then, from the Codazzi equation in $(\ref{7.23})$ we obtain

$\displaystyle-k_{2}\frac{c-1}{2}g(\varphi X,Y)+2g(A\varphi AX,Y)=$
$\displaystyle\beta g(\varphi AX,Y)+\beta g(A\varphi X,Y)+2g(\varphi X,Y),$

consequently, we have

$\displaystyle(2\lambda-\beta)g(A\varphi
X,Y)=(\beta\lambda+2+k_{2}\frac{c-1}{2})g(\varphi X,Y),$

as it was claimed. ∎

In the rest of this section the Ricci-biharmonic pseudo Hopf hypersurfaces
discuss in more details.

######  Theorem 4.3.

There exists no Ricci-biharmonic pseudo Hopf hypersurface, where
$\textsf{grad}|H|$ is a principal direction.

###### Proof.

Let $\psi:M^{2m}\rightarrow\overline{M}^{2m+1}(c)$ be an isometric immersion
of a Ricci-biharmonic pseudo Hopf hypersurface in the Sasakian space form
$\overline{M}^{2m+1}(c)$. Suppose that grad$|H|$ is in the direction of
vectors in $D$. Then, by applying Theorem 3.5 directly, we get

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$\Delta^{\perp}H=(|B|^{2}+L)H$,}\\\
\hbox{A $\textsf{grad}|H|+m|H|\textsf{grad}|H|=0$,}\end{array}\right.$

in other words in this case grad$|H|$ is an eigenvector of the Weingarten
operator $A$ corresponding to the eigenvalue $-m|H|$. The Lemma 4.2 yields we
can choose a suitable orthogonal frame field
$\big{\\{}e_{1},...,e_{m-1},e_{m}=\varphi e_{1},...,e_{2m-2}=\varphi
e_{m-1},e_{2m-1}=W_{1},e_{2m}=W_{2}\big{\\}}$ on $M^{2m}$ in which

$\displaystyle Ae_{i}=\lambda_{i}e_{i},\ \ i=1,...,m-1$ $\displaystyle
A\varphi e_{i}=\overline{\lambda_{i}}\varphi e_{i},\ \ i=1,...,m-1$ (4.6)
$\displaystyle AW_{1}=-\gamma_{1}W_{1},\ \ \ AW_{2}=\gamma_{2}W_{2}$

where, $\lambda_{i}$ and
$\overline{\lambda_{i}}=\frac{\beta\lambda_{i}+2-n}{2\lambda_{i}-\beta}$ are
the eigenvalues corresponding to the eigenvectors $e_{i}$ and $\varphi e_{i}$,
respectively. We recall that $\gamma_{1}=-\tan\theta$ and
$\gamma_{2}=\cot\theta$, consequently we get $\gamma_{1}\gamma_{2}=-1$. Let
$e_{1}=\frac{\textsf{grad}|H|}{|\textsf{grad}|H||}$ such that grad$|H|$ is
given by $\textsf{grad}|H|=\sum_{i=1}^{2m}e_{i}(|H|)e_{i}$. Then

$\displaystyle{e_{1}(|H|)\neq 0,\ \ \ \ \ \ \ \ \ e_{i}(|H|)=0,\ \ \ \ \ \ \ \
\ i=2,...,2m.}$ (4.7)

Also, it is written

$\displaystyle\nabla_{e_{i}}e_{j}=\sum_{k=1}^{2m}\omega_{ij}^{k}e_{k},$ (4.8)

where $\nabla$ denotes the Levi-Civita connection on $M^{2m}$ and
$\omega_{ij}^{k}$ is known the connection forms. Computing the compatibility
condition $\nabla_{e_{k}}\langle e_{i},e_{j}\rangle=0,$ yields

$\displaystyle\omega_{ki}^{i}$ $\displaystyle=$ $\displaystyle 0,\ \ i=j$
$\displaystyle\omega_{ki}^{j}+\omega_{kj}^{i}$ $\displaystyle=0$
$\displaystyle,\ \ i\neq j,\ \ i,j,k=1,...,2m.$

Moreovere, take $(\ref{6.21})$ and $(\ref{6.23})$ the Codazzi equation implies

$\displaystyle
e_{k}(\lambda_{i})e_{i}+(\lambda_{i}-\lambda_{j})\omega_{ki}^{j}e_{j}=e_{i}(\lambda_{k})e_{k}+(\lambda_{k}-\lambda_{j})\omega_{ik}^{j}e_{j},$

which follows

$\displaystyle e_{i}(\lambda_{j})$ $\displaystyle=$
$\displaystyle(\lambda_{i}-\lambda_{j})\omega_{ji}^{j}$ (4.9)
$\displaystyle(\lambda_{i}-\lambda_{j})\omega_{ki}^{j}$ $\displaystyle=$
$\displaystyle(\lambda_{k}-\lambda_{j})\omega_{ik}^{j},$ (4.10)

for distinct $i,j,k=1,...,2m$. From $\lambda_{1}=-m|H|$ and $(\ref{6.22})$ we
obtain

$\displaystyle e_{1}(\lambda_{1})\neq 0,\ \ \ \ \ \ \ \ e_{i}(\lambda_{1})=0,\
\ \ \ \ i=2,...,2m,$ (4.11)

and

$\displaystyle
0=[e_{i},e_{j}]\lambda_{1}=(\nabla_{e_{i}}e_{j}-\nabla_{e_{j}}e_{i})\lambda_{1},\
\ \ 2\leq i,j\leq 2m,i\neq j.$ (4.12)

Thus

$\displaystyle\omega_{ij}^{1}=\omega_{ji}^{1},$ (4.13)

for distinct $i,j=2,...,2m$. It is claimed that, $\lambda_{j}\neq\lambda_{1}$
for $j=2,...,2m$. Since, if $\lambda_{j}=\lambda_{1}$ for $j\neq 1$, utilize
$(\ref{6.27})$ and put $i=1$

$\displaystyle
0=(\lambda_{1}-\lambda_{j})\omega_{j1}^{j}=e_{1}(\lambda_{j})=e_{1}(\lambda_{1}),$

which contradicts $(\ref{6.29})$. For $j=1$ and $k,i\neq 1$ the equation
$(\ref{6.28})$ implies

$\displaystyle(\lambda_{i}-\lambda_{1})\omega_{ki}^{1}=(\lambda_{k}-\lambda_{1})\omega_{ik}^{1},$

then by the above and $(\ref{6.30})$ it follows

$\displaystyle\omega_{ij}^{1}=0,\ \ \ \ \ \ \ \ \ \ i\neq j,\ \ \ \ \ \ \
i,j=2,...,2m,$ (4.14)

take $(\ref{6.26})$, we get $\omega_{i1}^{j}=0$, for $i\neq j$ ,
$i,j=1,...,2m$. Now to continue our approach we compute

$\displaystyle\nabla_{e_{m}}W_{2}$ $\displaystyle=$
$\displaystyle\nabla_{e_{m}}(-\xi\sin\theta+V\cos\theta)$ (4.15)
$\displaystyle=$ $\displaystyle-
e_{m}(\sin\theta)\xi-\sin(\theta)\nabla_{e_{m}}\xi+e_{m}(\cos\theta)V+\cos(\theta)\nabla_{e_{m}}V$
$\displaystyle=$ $\displaystyle-
e_{m}(\sin\theta)(W_{1}\sin\theta+W_{2}\cos\theta)-\sin\theta(-\varphi e_{m})$
$\displaystyle+$ $\displaystyle e_{m}(\cos\theta)(W_{1}\cos\theta-
W_{2}\sin\theta)+\cos\theta(\varphi Ae_{m})$ $\displaystyle=$
$\displaystyle(-e_{m}(\sin\theta)\sin\theta+e_{m}(\cos\theta)\cos\theta)W_{1}-(\sin\theta+\overline{\lambda_{1}}\cos\theta)e_{1}$
$\displaystyle-$
$\displaystyle(e_{m}(\sin\theta)\cos\theta+e_{m}(\cos\theta)\sin\theta)W_{2}.$

Moreover, from $(\ref{6.31})$ the connection form $\omega_{m2m}^{1}=0$. Then
$(\ref{6.41})$ follows

$\displaystyle
0=\omega_{m2m}^{1}=\sin\theta+\overline{\lambda_{1}}\cos\theta,$

which yields

$\displaystyle\overline{\lambda_{1}}=-\tan\theta,$ (4.16)

where $\overline{\lambda}_{1}$ is an eigenvalue of the Weingarten operator
corresponding to the eigenvector $e_{m}=\varphi e_{1}$. Similarly, we compute
$\nabla_{e_{m}}W_{1}$ and apply $(\ref{6.31})$ then

$\displaystyle
0=\omega_{m2m-1}^{1}=\cos\theta-\overline{\lambda_{1}}\sin\theta,\ \ $

which yields

$\displaystyle\overline{\lambda_{1}}=\cot\theta.$ (4.17)

Finally, $(\ref{2.12})$ and $(\ref{2.13})$ show a contradiction. Hence, Ricci-
biharmonic pseudo Hopf hypersurfaces do not exist in the Sasakian space forms
where grad$|H|$ is an eigenvector of the Weingarten operator. ∎

In the remainder of the studying Ricci-biharmonic pseudo Hopf hypersurfaces
are considered, where grad$|H|\in D^{\perp}$.

######  Proposition 4.4.

Let $M^{2m}$ be a Ricci-biharmonic pseudo Hopf hypersurface where
grad$|H|=\alpha\xi+\beta V$. Then

  * •

$M^{2m}$ is a hypersurface with the constant mean curvature, if $\alpha=0$.

  * •

does not exist any Ricci-biharmonic pseudo Hopf hypersurface, where
$\alpha\neq 0$ and the dimensional of hypersurface is big enough for the
$\varphi$-sectional curvature $c>1$.

  * •

Ricci-biharmonic pseudo Hopf hypersurface has the mean curvature that holds

$\displaystyle|H|=\frac{1}{2}(-\frac{-1+m(\gamma_{1}+\gamma_{2})}{m^{2}}\pm\sqrt{\frac{-\frac{4(-1+c)m^{2}(1+m)}{(-1+c+3m+cm)}+(-1+m(\gamma_{1}+\gamma_{2}))^{2}}{m^{4}}}),$

for $\alpha\neq 0$ and the eigenvalues $(\gamma_{1}+\gamma_{2})\neq 0$ where
$\varphi$-sectional curvature $c<\frac{(1-3m)}{(1+m)}$ or $1<c$.

###### Proof.

Take the assumption $\textsf{grad}|H|=\alpha\xi+\beta V$ then apply the
Theorem 3.5 follows

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$\Delta^{\perp}H=(|B|^{2}+l)H$
,}\\\ \hbox{A $(\alpha\xi+\beta V)-\frac{k_{1}}{k_{2}}\alpha
V+m|H|(\alpha\xi+\beta V)=0$}.\end{array}\right.$ (4.20)

By the above and $(\ref{6.00})$ we have

$\displaystyle AW_{1}=\gamma_{1}W_{1}=\cos\theta A\xi+\sin\theta AV,$

so

$\displaystyle AV=-\xi+(\gamma_{1}+\gamma_{2})V,$ (4.21)

then, the second term of the equations $(\ref{7.24})$ yields

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$-\alpha(|H|+\frac{k_{1}}{k_{2}})+\beta((\gamma_{1}+\gamma_{2})+m|H|)=0$}\\\
\hbox{$-\beta+m|H|\alpha=0$}\end{array}\right.$ (4.24)

hence

$\displaystyle\alpha
m^{2}\big{(}|H|^{2}+\frac{(\gamma_{1}+\gamma_{2})m-1}{m^{2}}|H|-\frac{(m+1)(1-c)}{(m+1)c+3m-1}\big{)}=0.$
(4.25)

Suppose that $\alpha=0$, then the equation $(\ref{7.26})$ yields $\beta=0$.
Then, the assumption shows $\textsf{grad}|H|=0$. It follows $|H|$ is constant.
Nevertheless, where $\alpha\neq 0$, a solution of the equation (4.25) presents
that $(\gamma_{1}+\gamma_{2})<\frac{1}{m}$. Here,
$(\gamma_{1}+\gamma_{2})\rightarrow 0$, where the dimension of hypersurfaces
is big enough. It implies $\theta=\frac{\pi}{4}$. So, under these
circumstances the equation (4.25) yields that there is no Ricci- biharmonic
pseudo Hopf hypersurface, where $\varphi$\- sectional curvature $c>1$ because
of $|H|^{2}\geq 0$. If $\alpha\neq 0$ and the dimension of hypersurface is
appropriate, then $(\ref{7.27})$ shows that the boundary for
$\varphi$-sectional curvature and the mean curvature hold as the claimed. ∎

######  Corollary 4.5.

Let $M^{2m}$ be a Ricci-biharmonic pseudo Hopf hypersurface where
$\textsf{grad}|H|=\alpha\xi$. Then, $M^{2m}$ is a minimal hypersurface.

###### Proof.

In this case applying the Theorem 3.5 follows

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$\Delta^{\perp}H=(|B|^{2}+l)H$;}\\\
\hbox{A$(\alpha\xi)-\frac{k_{1}}{k_{2}}\alpha
V+m|H|\alpha\xi=0$.}\end{array}\right.$ (4.27)

since $\textsf{grad}|H|=\alpha\xi$. Then from (3.9) and (4.21) the second
terms of the above implies

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$-\alpha(|H|+\frac{(m+1)(1-c)}{(m+1)c+3m-1})=0$,}\\\
\hbox{$m|H|\alpha=0.$}\end{array}\right.$

So, if $\alpha\neq 0$ it shows $|H|=0$. ∎

######  Corollary 4.6.

A Ricci-biharmonic pseudo Hopf hypersurface has the constant mean curvature
where $\textsf{grad}|H|=\beta V$.

###### Proof.

Applying the Theorem 3.5 implies

$\displaystyle\left\\{\begin{array}[]{ll}\hbox{$\beta(\gamma_{1}+\gamma_{2}+m|H|)=0$,}\\\
\hbox{$\beta=0,$}\end{array}\right.$ (4.31)

where $\textsf{grad}|H|=\beta V$. It presents that $\beta=0$ and $|H|$ is
constant. ∎

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