# The extended quasi-Einstein manifolds with generalised Ricci solitons

Zhiming Huang  Zhiming Huang
School of Mathematics and Physics
Guangxi Minzu University
Guangxi Nanning 530006, China<EMAIL_ADDRESS>,  Weijun Lu  Weijun Lu
School of Mathematics and Physics
Guangxi Minzu University
Guangxi Nanning 530006, China<EMAIL_ADDRESS>and  Fuhong Su  Fuhong Su
School of Mathematics and Physics
Guangxi Minzu University
Guangxi Nanning 530006, China<EMAIL_ADDRESS>
###### Abstract.

As a generalization of Einstein manifolds, the nearly quasi-Einstein manifolds
and pseudo quasi-Einstein manifolds are both interesting and useful in
studying the general relativity. In this paper, we study the extended quasi-
Einstein manifolds which derive from pseudo quasi-Einstein manifolds. After
showing the existence theorem of extended quasi-Einstein manifold, we give
some special geometric properties of such manifolds. At the same time, we also
discuss the extended quasi-Einstein manifolds with certain soliton like
generalised Ricci soliton or Riemann soliton. Furthermore, we construct some
nontrivial example to illustrate these extended quasi-Einstein manifolds.

††footnotetext: 2020 Mathematics Subject Classification: 53C25,
53C35††footnotetext: keywords: nearly quasi-Einstein manifolds, pseudo quasi-
Einstein manifolds, extended quasi-Einstein manifolds, generalised Ricci
soliton, Riemann soliton

##  1\. Introduction

A non-flat Riemannian manifold $({{M}^{n}},g)$ $(n>2)$ is said to be an
Einstein manifold if the following condition

$\text{Ric}(X,Y)=\frac{r}{n}\,g(X,Y),\quad\forall X,Y\in{{C}^{\infty}}(TM),$
(1.1)

hold on $M$, where _r_ and Ric denote the scalar curvature and the Ricci
tensor of $({{M}^{n}},g)$, respectively. According to [8], (1.1) is called the
Einstein metric condition.

In $2008$, Chaki and De [1] introduced the notion of nearly quasi-Einstein
manifolds which is a generalization of Einstein manifold. A non-flat
Riemannian manifold $({{M}^{n}},g)$ $(n\geq 3)$ is said to be a nearly quasi-
Einstein manifold if its Ricci tensor is not identically zero and satisfies
the condition

$\text{Ric}(X,Y)=ag(X,Y)+bE(X,Y),\quad\forall X,Y\in{{C}^{\infty}}(TM),$ (1.2)

where $a$ and $b$ are non-zero scalars and $E$ is a non-zero $(0,2)$ tensor.
We shall call $E$ the associated tensor and $a$ and $b$ as associated scalars.
An $n$ dimensional nearly quasi Einstein manifold will be denoted by
$E^{n}_{\rm{NQ}}$.

In [9, 20], Singh and Pandey consider a type of nearly quasi Einstein
manifold, whose associated tensor $E$ of type $(0,2)$ is given by

$E(X,Y)=A(X)B(Y)+B(X)A(Y),\quad\forall X,Y\in{{C}^{\infty}}(TM),$

where $A$ and $B$ are non-zero $1$-forms associated with orthogonal unit
vector fields $V$ and $U$, i.e.,

$g(X,U)=A(X),g(X,V)=B(X),g(V,U)=0,\quad\forall X\in{{C}^{\infty}}(TM).$ (1.3)

Thus, the equation (1.2), assumes the form

$\text{Ric}(X,Y)=ag(X,Y)+b[A(X)B(Y)+B(X)A(Y)],\quad\forall
X,Y\in{{C}^{\infty}}(TM).$ (1.4)

De and Ghosh gave a similarly definition about this manifolds. They introduced
a type of non-flat Riemannian manifold called mixed quasi-Einstein manifold
[3, 4]. A non-flat Riemannian manifold $({{M}^{n}},g)$ $(n>2)$ is called mixed
quasi-Einstein manifold if its Ricci tensor of type $(0,2)$ is not identically
zero and satisfies the equations (1.3), (1.4) and $b\neq 0$. The vector fields
$U$ and $V$ are called the generators of the manifold. Such a manifolds is
denoted by $E^{n}_{\rm{MQ}}$. More works have been done in the mixed quasi-
Einstein manifolds [12, 17].

A non-flat Riemannian manifold $({{M}^{n}},g)$ $(n\geq 3)$ is called a pseudo
quasi-Einstein manifold if its Ricci tensor Ric of type $(0,2)$ is not
identically zero and satisfies the following [10, 11]:

$\text{Ric}(X,Y)=ag(X,Y)+bA(X)A(Y)+cE(X,Y),\quad\forall
X,Y\in{{C}^{\infty}}(TM),$ (1.5)

where $a$, $b$ and $c$ are scalars of which $c$ is non-zero, and $A$ is a non-
zero $1$-form such that $g(X,U)=A(X)$ for all vector fields $X$ with $U$ being
a unit vector field called the generator of the manifold, $E$ is a symmetric
$(0,2)$ tensor with vanishing trace and satisfying $E(X,U)=0$ for all vector
fields $X$. Also $a$, $b$ and $c$ are called the associated scalars; $A$ is
the associated $1$-form of the manifold and $E$ is called the associated
tensor of the manifold. Such an $n$-dimensional manifold will be denoted by
$E^{n}_{\rm{PQ}}$.

######  Theorem 1.1.

[28](F.Schur) $(M^{n},g)$ $(n>2)$ is a connecting Riemannian manifold. If the
sectional curvature of $(M^{n},g)$ at $p$ with respect to the plane $X\wedge
Y\in T_{p}M$, dependent on the choices of point $p$ not basis $X\wedge Y$,
i.e., $sec_{p}(X\wedge Y)=c(p)$, then $M$ is a manifold of constant curvature.

###### Proof.

Let us suppose $R_{1}(W,Z,X,Y)=g(W,X)g(Z,Y)-g(Z,X)g(W,Y)$, we have

$R(W,Z,X,Y)=c(p)R_{1}(W,Z,X,Y).$

Hence

$\displaystyle R(W,Z,X,\nabla_{U}Y)+g(\nabla_{U}R(W,Z)X,Y)$ $\displaystyle=$
$\displaystyle\nabla_{U}R(W,Z,X,Y))=\nabla_{U}cR_{1}(W,Z,X,Y))$
$\displaystyle=$ $\displaystyle\nabla_{U}c[g(W,X)g(Z,Y)-g(Z,X)g(W,Y)]$
$\displaystyle=$
$\displaystyle(Uc)g(Y,g(W,X)Z-g(Z,X)W)+c[R_{1}(\nabla_{U}W,Z,X,Y)+R_{1}(W,\nabla_{U}Z,X,Y)$
$\displaystyle+R_{1}(W,Z,\nabla_{U}X,Y)+R_{1}(W,Z,X,\nabla_{U}Y)]$
$\displaystyle=$
$\displaystyle(Uc)g(Y,g(W,X)Z-g(Z,X)W)+g(R(\nabla_{U}W,Z)X,Y)+g(R(W,\nabla_{U}Z)X,Y)$
$\displaystyle+g(R(W,Z)\nabla_{U}X,Y)+R(W,Z,X,\nabla_{U}Y).$

We get

$\displaystyle\nabla_{U}R(W,Z)X-R(\nabla_{U}W,Z)X-R(W,\nabla_{U}Z)X-R(W,Z)\nabla_{U}X$
(1.6) $\displaystyle=(Uc)(g(W,X)Z-g(Z,X)W).$

From the (1.6), we obtain

$\displaystyle(Uc)(g(W,X)Z-g(Z,X)W)+(Wc)(g(Z,X)U-g(X,U)Z)$ (1.7)
$\displaystyle+(Zc)(g(X,U)W-g(X,W)U)$
$\displaystyle=(\nabla_{U}R)(W,Z)X+(\nabla_{W}R)(Z,U)X+(\nabla_{Z}R)(U,W)X=0.$

Suppose that $W$, $X$, $Z$ are mutually orthogonal $C^{\infty}$ vector fields
of which $Z$ is a unit vector field. Putting $U=X$ in (1.7), we have

$(Zc)W-(Wc)Z=0,\quad\forall Z,W\in{{C}^{\infty}}(TM).$

Then $Zc=0$ and $Wc=0$.

Let $\left\\{x^{i}\right\\}$ be local coordinate systems with respect to
orthonormal basis $\left\\{X_{i}\right\\}$, then $X_{i}c=0$. Hence

$\frac{\partial}{\partial
x^{i}}c=(\sum\limits_{j=1}^{n}{a_{i}^{j}{{X}_{j}})}c=\sum\limits_{j=1}^{n}{a_{i}^{j}({{X}_{j}}}c)=0,\quad{X}_{j}\in{{C}^{\infty}}(TM),$

then $c$ is a local constant. Since $M$ is connecting, $M$ is a manifold of
constant curvature. ∎

######  Corollary 1.2.

A three dimensional Einstein manifold $(M^{3},g)$ is a manifold of constant
sectional curvature.

From F.Schur theorem and three dimensional Einstein manifold has constant
sectional curvature, motivated these facts and the ideas of quasi-Einstein
manifolds with one generator, we extend to consider three generators into
quasi-Einstein manifolds and induce a new notion so called extended quasi-
Einstein manifold, from which we expect to uncover more information about
sectional curvature and Ricci curvature and so on (etc).

The present paper is organized as follows. In section $2$, we prove the
existence theorem of extended quasi-Einstein manifold and some basic geometric
properties of such manifold are obtained. We consider the associated vector
fields as Killing vector fields, parallel vector fields, concurrent vector
fields, respectively. After that we discuss about the nature of the Ricci
tensor satisfies the Codazzi type and Ricci semi-symmetric manifold. In
section $3$, we study generalised Ricci soliton and Riemann soliton on
extended quasi-Einstein manifold and obtain some characterizations. Finally,
we provide an example of extended quasi-Einstein manifolds.

##  2\. Extended quasi-Einstein manifold and basic geometric properties

In this section, we prove the existence theorem of extended quasi-Einstein
manifold and some basic geometric properties of such manifold are obtained.

######  Definition 2.1.

A non-flat Riemannian manifold $({{M}^{n}},g)$ $(n\geq 3)$ is called an
extended quasi-Einstein manifold if its Ricci tensor of type $(0,2)$ is not
identically zero and satisfies the condition

$\text{Ric}(X,Y)=ag(X,Y)+bA(X)A(Y)+c(B(X)D(Y)+D(X)B(Y)),$ (2.1)

for all $X$, $Y$ $\in{{C}^{\infty}}(TM)$. Here $a$, $b$, $c$ are scalars of
which $c\neq 0$, and $A$, $B$, $D$ are non-zero $1$-forms such that

$g(X,U)=A(X),g(X,V)=B(X),g(X,T)=D(X),$ (2.2)

where $U$, $V$, $T$ are mutually orthogonal unit $C^{\infty}$ vector fields by
using $\rm{dim}(M)=n\geq 3$, i.e.,

$g(U,U)=g(V,V)=g(T,T)=1,g(U,V)=g(U,T)=g(V,T)=0.$ (2.3)

$a$, $b$, $c$ are called the associated scalars, $A$, $B$, $D$ are called the
associated $1$-forms and $U$, $V$, $T$ are the generators of the manifold.
Such an $n$-dimensional manifold is denoted by the symbol $E^{n}_{\rm{EQ}}$.
If $c=0$, then become a quasi-Einstein manifolds. If $b=0$, then become a
mixed quasi-Einstein manifolds. If $A=B$ or $A=D$, then become a generalized
quasi-Einstein manifolds [29]. Let $E(X,Y)=B(X)D(Y)+D(X)B(Y)$, then $E$ is a
symmetric $(0,2)$ tensor and satisfying $E(X,U)=0$ for all vector fields $X$.
If $E$ is a symmetric $(0,2)$ tensor with trace free, then become a pseudo
quasi-Einstein manifolds.

###  2.1. Existence theorem

We state and prove the existence theorem of the extended quasi-Einstein
manifold.

######  Theorem 2.2.

If the Ricci tensor of a non-flat Riemannian manifold is non-vanishing and
satisfies the relation

$\displaystyle\text{Ric}(X,W)g(Y,Z)$
$\displaystyle+\text{Ric}(Y,Z)g(X,W)={{a}_{1}}[g(X,W)g(Y,Z)+g(Y,W)g(X,Z)]$
(2.4)
$\displaystyle+{{a}_{2}}[\text{Ric}(X,Z)\text{Ric}^{2}(Y,W)+\text{Ric}(Y,W)\text{Ric}^{2}(X,Z)],$

where ${{a}_{1}}$, ${{a}_{2}}$ are non-zero scalars. Then the manifold is an
extended quasi-Einstein manifold.

###### Proof.

Let $U$ be a non-null vector fields defined by $g(X,U)=\omega\,(X)$ for all
vector fields $X$. Putting $X=W=U$ in (2.4), we obtain

$\displaystyle\text{Ric}(Y,Z)g(U,U)$
$\displaystyle+\text{Ric}(U,U)g(Y,Z)={{a}_{1}}[g(U,U)g(Y,Z)+g(Y,U)g(Z,U)]$
(2.5)
$\displaystyle+{{a}_{2}}[\text{Ric}(U,Z)\text{Ric}^{2}(Y,U)+\text{Ric}(Y,U)\text{Ric}^{2}(X,U)].$

From above equation (2.5),

$\displaystyle g(U,U)\text{Ric}(Y,Z)$
$\displaystyle=[{{a}_{1}}g(U,U)-\text{Ric}(U,U)]g(Y,Z)+{{a}_{1}}\omega(Y)\omega(Z)$
(2.6)
$\displaystyle+{{a}_{2}}[\omega(QZ)\omega({Q}^{2}Y)+\omega(QY)\omega({Q}^{2}Z)],$

where $\omega(Y)=g(Y,U)$, $\omega(QY)=\text{Ric}(Y,U)$ and
$\omega({Q}^{2}Y)=\text{Ric}^{2}(Y,U)$. $Q$ is the symmetric endomorphism of
the tangent space at each point corresponding to the Ricci tensor, i.e.,
$g(QX,Y)=\text{Ric}(X,Y)$, $\text{Ric}(QX,Y)=\text{Ric}^{2}(X,Y)$. Since $U$
is non-null, we have $g(U,U)\neq 0$. There

$\displaystyle\text{Ric}(Y,Z)$
$\displaystyle=[{{a}_{1}}-\frac{\text{Ric}(U,U)}{g(U,U)}]g(Y,Z)+\frac{{a}_{1}}{g(U,U)}\omega(Y)\omega(Z)$
(2.7)
$\displaystyle+\frac{{a}_{2}}{g(U,U)}[\omega(QZ)\omega({Q}^{2}Y)+\omega(QY)\omega({Q}^{2}Z)].$

Taking $\omega(QY)={\omega_{1}}(Y)$ and $\omega({Q}^{2}Y)={\omega_{2}}(Y)$, we
get

$\displaystyle\text{Ric}(Y,Z)$
$\displaystyle=ag(Y,Z)+b\omega(Y)\omega(Z)+c[{{\omega}_{1}}(Y){{\omega}_{2}}(Z)+{{\omega}_{2}}(Y){{\omega}_{1}}(Z)],$
(2.8)

where $a={{a}_{1}}-\frac{\text{Ric}(U,U)}{g(U,U)}$,
$b=\frac{{a}_{1}}{g(U,U)}$, $c=\frac{{a}_{2}}{g(U,U)}$. This shows that the
manifold is an extended quasi-Einstein manifold. ∎

###  2.2. Relationship between the associated scalars

From (2.3), we have

$A(V)=A(T)=B(U)=B(T)=D(U)=D(V)=0.$ (2.9)

Putting $Y=U$ in (2.1), have

$\text{Ric}(X,U)=(a+b)A(X).$ (2.10)

Putting $Y=V$ in (2.1), we get

$\text{Ric}(X,V)=aB(X)+cD(X).$ (2.11)

Again, putting $Y=T$ in (2.1), we have

$\text{Ric}(X,T)=aD(X)+cB(X).$ (2.12)

Further,

$\text{Ric}(U,U)=a+b,\quad\text{Ric}(V,V)=\text{Ric}(T,T)=a,$ (2.13)
$\text{Ric}(U,V)=\text{Ric}(U,T)=0,\quad\text{Ric}(T,V)=c.$ (2.14)

Contracting $X$ and $Y$ in (2.1), gives

$r=na+b.$ (2.15)

Next, let $Q$ be the symmetric endomorphism of the tangent space at a point
corresponding to the Ricci tensor, then

$g(QX,Y)=\text{Ric}(X,Y),$ (2.16)

for any $X$, $Y$ $\in{{C}^{\infty}}(TM)$. Let ${{l}^{2}}$ be the square of the
length of the Ricci tensor, then

${{l}^{2}}=\sum\limits_{i=1}^{n}{\text{Ric}(Q{{e}_{i}},{{e}_{i}})},$ (2.17)

where $\\{{e}_{i}\\}$, $i=1,2,3,\cdot\cdot\cdot,n$ is an orthogonal basis of
the tangent space at a point.

######  Proposition 2.3.

In an extended quasi-Einstein manifold, if $a\neq 0$, then the associated
scalar $c$ is less than $\frac{1}{\sqrt{2}}l$.

###### Proof.

From (2.1), we have

$\text{Ric}(Q{{e}_{i}},{{e}_{i}})=ag(Q{{e}_{i}},{{e}_{i}})+bA(Q{{e}_{i}})A({{e}_{i}})+c[B(Q{{e}_{i}})D({{e}_{i}})+D(Q{{e}_{i}})B({{e}_{i}})].$
(2.18)

Hence ${{l}^{2}}=(n-1){{a}^{2}}+{{(a+b)}^{2}}+2{{c}^{2}}.$ Since $a\neq 0$,
then $2{{c}^{2}<{{l}^{2}}},$ i.e., $c<\frac{1}{\sqrt{2}}l.$ ∎

###  2.3. The generators as Killing vector fields

A vector field $X={{X}^{i}}\frac{\partial}{\partial{{x}^{i}}}$ is a Killing
vector field if and only if it satisfies

$\Delta{{X}^{i}}+R_{j}^{i}{{X}^{j}}=0,div(X)=0,$

where $X$ is a fixed vector field on $M$.

For (2.1), if the associated scalars $a$, $b$, $c$ are constants, then

$\displaystyle({{\nabla}_{X}}\text{Ric})(Y,Z)=$ $\displaystyle
b[({{\nabla}_{X}}A)(Y)A(Z)+A(Y)({{\nabla}_{X}}A)(Z)]$ (2.19)
$\displaystyle+c[({{\nabla}_{X}}B)(Y)D(Z)+B(Y)({{\nabla}_{X}}D)(Z)$
$\displaystyle+({{\nabla}_{X}}D)(Y)B(Z)+D(Y)({{\nabla}_{X}}B)(Z)],$
$\displaystyle({{\nabla}_{Y}}\text{Ric})(Z,X)=$ $\displaystyle
b[({{\nabla}_{Y}}A)(Z)A(X)+A(Z)({{\nabla}_{Y}}A)(X)]$
$\displaystyle+c[({{\nabla}_{Y}}B)(Z)D(X)+B(Z)({{\nabla}_{Y}}D)(X)$
$\displaystyle+({{\nabla}_{Y}}D)(Z)B(X)+D(Z)({{\nabla}_{Y}}B)(X)],$
$\displaystyle({{\nabla}_{Z}}\text{Ric})(X,Y)=$ $\displaystyle
b[({{\nabla}_{Z}}A)(X)A(Y)+A(X)({{\nabla}_{Z}}A)(Y)]$
$\displaystyle+c[({{\nabla}_{Z}}B)(X)D(Y)+B(X)({{\nabla}_{Z}}D)(Y)$
$\displaystyle+({{\nabla}_{Z}}D)(X)B(Y)+D(X)({{\nabla}_{Z}}B)(Y)],$

for any $X$, $Y$, $Z$ $\in{{C}^{\infty}}(TM).$

######  Theorem 2.4.

If the generators $U$, $V$ and $T$ are Killing vector field, then the extended
quasi-Einstein manifold satisfies cyclic parallel Ricci tensor.

###### Proof.

Let us suppose that the generator $U$ of the manifold is a Killing vector
field, then we have

$({{\mathcal{L}}_{U}}g)(X,Y)=0,$

where $\mathcal{L}$ denotes the Lie derivative. From which we get

$g({{\nabla}_{X}}U,Y)+g(X,{{\nabla}_{Y}}U)=0.$ (2.20)

Again since $g({{\nabla}_{X}}U,Y)=({{\nabla}_{X}}A)(Y)$, we get that

$({{\nabla}_{X}}A)(Y)+({{\nabla}_{Y}}A)(X)=0,$ (2.21)

for any $X$, $Y$ $\in{{C}^{\infty}}(TM)$. Similarly, we have

$({{\nabla}_{X}}A)(Z)+({{\nabla}_{Z}}A)(X)=0,\quad({{\nabla}_{Z}}A)(Y)+({{\nabla}_{Y}}A)(Z)=0.$
(2.22)

Further, we suppose that the generator $V$, $T$ of the manifold are Killing
vector field, too, then we have

$\displaystyle({{\nabla}_{X}}B)(Y)+({{\nabla}_{Y}}B)(X)=0,\quad({{\nabla}_{X}}D)(Y)+({{\nabla}_{Y}}D)(X)=0,$
(2.23)
$\displaystyle({{\nabla}_{X}}B)(Z)+({{\nabla}_{Z}}B)(X)=0,\quad({{\nabla}_{X}}D)(Z)+({{\nabla}_{Z}}D)(X)=0,$
$\displaystyle({{\nabla}_{Y}}B)(Z)+({{\nabla}_{Z}}B)(Y)=0,\quad({{\nabla}_{Y}}D)(Z)+({{\nabla}_{Z}}D)(Y)=0.$

Using (2.21), (2.22), (2.23) and (2.19), we get

$({{\nabla}_{X}}\text{Ric})(Y,Z)+({{\nabla}_{Y}}\text{Ric})(Z,X)+({{\nabla}_{Z}}\text{Ric})(X,Y)=0.$
(2.24)

∎

###  2.4. The generators as parallel vector fields

A Riemannian manifold ${{M}^{n}}$ is said to be Ricci-recurrent [5], if the
Ricci tensor Ric is non-zero and satisfying the condition

$({{\nabla}_{X}}\text{Ric})(Y,Z)=\alpha(X)\text{Ric}(Y,Z),$ (2.25)

for any $X$, $Y$, $Z$ $\in{{C}^{\infty}}(TM)$, where $\alpha$ is a non-zero
$1$-form.

######  Theorem 2.5.

If the generators $U$, $V$, $T$ are parallel vector field and $B\neq D$, then
the extended quasi-Einstein manifold is Ricci-recurrent manifold.

###### Proof.

We consider $\nabla U=0$, $\nabla V=0$, $\nabla T=0$, then

$\text{Ric}(X,U)=0,\quad\text{Ric}(X,V)=0,\quad\text{Ric}(X,T)=0.$ (2.26)

From (2.10), (2.11), (2.12) and (2.26), we have

$(a+b)A(X)=0,\quad(a-c)(B(X)-D(X))=0.$ (2.27)

$A$ is a non-zero $1$-form, then $a=-b$. Since $a=c$, the equation (2.1) can
be expressed as

$\text{Ric}(X,Y)=c[g(X,Y)-A(X)A(Y)+B(X)D(Y)+D(X)B(Y)].$ (2.28)

So

$\displaystyle({{\nabla}_{Z}}\text{Ric})(X,Y)$
$\displaystyle=({{\nabla}_{Z}}c)[g(X,Y)-A(X)A(Y)+B(X)D(Y)+D(X)B(Y)]$ (2.29)
$\displaystyle-c[-({{\nabla}_{Z}}A)(X)A(Y)-A(X)({{\nabla}_{Z}}A)(Y)+({{\nabla}_{Z}}B)(X)D(Y)$
$\displaystyle+B(X)({{\nabla}_{Z}}D)(Y)+({{\nabla}_{Z}}D)(X)B(Y)+D(X)({{\nabla}_{Z}}B)(Y)].$

Since $U$, $V$, $T$ are parallel vector field, we have

$({{\nabla}_{Z}}A)X=0,\quad({{\nabla}_{Z}}B)X=0,\quad({{\nabla}_{Z}}D)X=0.$

Putting the above formula in (2.29), we get

$({{\nabla}_{Z}}\text{Ric})(X,Y)=({{\nabla}_{Z}}c)[g(X,Y)-A(X)A(Y)+B(X)D(Y)+D(X)B(Y)].$

Let $F(Z)=\frac{{{\nabla}_{Z}}c}{c}$, then

$({{\nabla}_{Z}}\text{Ric})(X,Y)=F(Z)\text{Ric}(X,Y).$

∎

###  2.5. The generators as concurrent vector fields

A vector field ${\xi}$ is said to be concurrent [2] if
${{\nabla}_{X}}{\xi}=\alpha X,$ where $\alpha$ is a nonzero constant. If
$\alpha=0$, the vector field reduces to a parallel vector field.

######  Theorem 2.6.

If the associated vector fields of a $E^{n}_{\rm{EQ}}$ are concurrent vector
fields and the associated scalars $a$, $b$ are constants, then the manifold
reduces to a mixed generalized quasi-Einstein manifold [27].

###### Proof.

We consider the vector fields $U$, $V$ and $T$ corresponding to the associated
$1$-forms $A$, $B$ and $D$, respectively, are concurrent. Then

$({{\nabla}_{X}}A)(Y)=\alpha g(X,Y),$ (2.30) $({{\nabla}_{X}}B)(Y)=\beta
g(X,Y),$ (2.31)

and

$({{\nabla}_{X}}D)(Y)=\gamma g(X,Y),$ (2.32)

where $\alpha$, $\beta$ and $\gamma$ are nonzero constants.

Using (2.30), (2.31), (2.32) to

$\displaystyle({{\nabla}_{X}}\text{Ric})(Y,Z)=$ $\displaystyle
b[({{\nabla}_{X}}A)(Y)A(Z)+A(Y)({{\nabla}_{X}}A)(Z)]$
$\displaystyle+c[({{\nabla}_{X}}B)(Y)D(Z)+B(Y)({{\nabla}_{X}}D)(Z)$
$\displaystyle+({{\nabla}_{X}}D)(Y)B(Z)+D(Y)({{\nabla}_{X}}B)(Z)],$

we get

$\displaystyle({{\nabla}_{X}}\text{Ric})(Y,Z)=$ $\displaystyle b[\alpha
g(X,Y)A(Z)+\alpha g(X,Z)A(Y)]$ (2.33) $\displaystyle+c[\beta g(X,Y)D(Z)+\gamma
g(X,Z)B(Y)$ $\displaystyle+\gamma g(X,Y)B(Z)+\beta g(X,Z)D(Y)].$

Contracting (2.33) over $Y$ and $Z$ , we obtain

$dr(X)=2b\alpha A(X)+2c[\beta D(X)+\gamma B(X)],$ (2.34)

where $r$ is the scalar curvature of the manifold. Since $a$, $b$
$\in\mathbb{R}$, we obtain $dr(X)=0$, for all $X$. Then

$b\alpha A(X)+c[\beta D(X)+\gamma B(X)]=0.$

Since $b$ and $\alpha$ are nonzero constants, we have

$A(X)=-\frac{c\beta}{b\alpha}D(X)-\frac{c\gamma}{b\alpha}B(X).$ (2.35)

Using (2.35) in (2.1), we obtain

$\text{Ric}(X,Y)=a_{1}g(X,Y)+a_{2}B(X)B(Y)+a_{3}D(X)D(Y)+a_{4}[B(X)D(Y)+D(X)B(Y)],$
(2.36)

where $a_{1}=a$, $a_{2}=\frac{(c\gamma)^{2}}{(b\alpha)^{2}}$,
$a_{3}=\frac{(c\beta)^{2}}{(b\alpha)^{2}}$ and
$a_{4}=c+\frac{c^{2}\beta\gamma}{(b\alpha)^{2}}$. The manifold reduces to a
mixed generalized quasi-Einstein manifold. ∎

###  2.6. Codazzi type of Ricci tensor

A Riemannian manifold $({{M}^{n}},g)$ is said to satisfy Codazzi type of Ricci
tensor [18], if its Ricci tensor Ric satisfies the following condition

$({{\nabla}_{X}}\text{Ric})(Y,Z)=({{\nabla}_{Y}}\text{Ric})(X,Z),$ (2.37)

for any $X$, $Y$ $\in{{C}^{\infty}}(TM)$.

######  Proposition 2.7.

If an extended quasi-Einstein manifold satisfies the Codazzi type of Ricci
tensor and the generator $U$ is a concurrent vector field, then the associated
$1$-form $A$ is closed.

###### Proof.

An extended quasi-Einstein manifold satisfies the Codazzi type of Ricci
tensor, then the Ricci tensor satisfies (2.37). Putting $Z=U$ in (2.37), we
have

$\displaystyle
b[({{\nabla}_{X}}A)(Y)]+c[B(Y)({{\nabla}_{X}}D)(U)+D(Y)({{\nabla}_{X}}B)(U)]$
$\displaystyle=b[({{\nabla}_{Y}}A)(X)]+c[B(X)({{\nabla}_{Y}}D)(U)+D(X)({{\nabla}_{Y}}B)(U)],$

where the associated scalars $a$, $b$, $c$ are constants. Therefore

$\displaystyle b[({{\nabla}_{X}}A)(Y)-({{\nabla}_{Y}}A)(X)]$ (2.38)
$\displaystyle+c[B(X)D({{\nabla}_{Y}}U)+D(X)B({{\nabla}_{Y}}U)-B(Y)D({{\nabla}_{X}}U)-D(Y)B({{\nabla}_{X}}U)]=0.$

Next, let the generator $U$ is a concurrent vector field [2], then

${{\nabla}_{X}}U=\alpha X,$ (2.39)

where $\alpha\ $ is a non-zero constant. Using (2.39) in (2.38), we have

$\displaystyle b[({{\nabla}_{X}}A)(Y)-({{\nabla}_{Y}}A)(X)]$
$\displaystyle+\alpha c[B(X)D(Y)+D(X)B(Y)-B(Y)D(X)-D(Y)B(X)]=0.$

So

$b[({{\nabla}_{X}}A)(Y)-({{\nabla}_{Y}}A)(X)]=0.$

Since $b\neq 0$, we have $({{\nabla}_{X}}A)(Y)-({{\nabla}_{Y}}A)(X)=0,$ i.e.,
$dA(X,Y)=0.$ ∎

###  2.7. Ricci-semi symmetric manifold

A Riemannian manifold $({{M}^{n}},g)$ is said to be Ricci-semi symmetric [6],
if the riemannian tensor and Ricci tensor satisfying the condition

$R\cdot\text{Ric}=0.$ (2.40)

The condition $R\cdot\text{Ric}$ can express as

$\displaystyle(R(X,Y)\cdot\text{Ric})(Z,W)=$
$\displaystyle-\text{Ric}(R(X,Y)Z,W)-\text{Ric}(Z,R(X,Y)W)$ (2.41)
$\displaystyle=$ $\displaystyle-b[A(R(X,Y)Z)A(W)+A(Z)A(R(X,Y)W)]$
$\displaystyle-c[B(R(X,Y)Z)D(W)+D(R(X,Y)Z)B(W)$
$\displaystyle+B(Z)D(R(X,Y)W)+D(Z)B(R(X,Y)W)],$

for any $X$, $Y$, $Z$, $W$ $\in{{C}^{\infty}}(TM)$.

######  Proposition 2.8.

If an extended quasi-Einstein manifold is a Ricci-semi symmetric manifold,
then $R(X,Y,V,T)=0$.

###### Proof.

Now we suppose that an extended quasi-Einstein manifold is Ricci-semi
symmetric manifold, i.e., such manifold satisfies (2.40). Then

$\displaystyle b[A(R(X,Y)Z)A(W)+A(Z)A(R(X,Y)W)]+c[B(R(X,Y)Z)D(W)$ (2.42)
$\displaystyle+D(R(X,Y)Z)B(W)+B(Z)D(R(X,Y)W)+D(Z)B(R(X,Y)W)]=0.$

Putting $W=Z=V$ in (2.42), then we have

$2c[D(R(X,Y)V)]=0.$

Putting $W=Z=T$ in (2.42), yields

$2c[B(R(X,Y)T)]=0.$

Since $c\neq 0$,

$D(R(X,Y)V)=0,B(R(X,Y)T)=0.$ (2.43)

Due to (2.43), then $R(X,Y,V,T)=0.$ ∎

##  3\. Two solitons structures

In 2014, Nurowski and Randall introduced the concept of the generalised Ricci
soliton equations [14]. These equations depend on three real parameters.
Siddiqi MD study generalised Ricci solitons on trans sasakian manifolds [13].
As a generalied of Ricci soliton, Hiricu and Udriste introduced and studied
Riemann soliton [16]. Riemann solitons are generalized fixed points of the
Riemann flow. In 2020, Venkatesha study Riemann solitons and almost Rieman
solitons on almost Kenmotsu manifolds [15].

A generalised Ricci soliton is a (pseudo)-Riemannian manifold $({{M}^{n}},g)$
admitting a smooth vector field $X$, such that

${{\mathcal{L}}_{X}}g+2{{c}_{1}}{{X}^{b}}\odot{{X}^{b}}=2{{c}_{2}}\text{Ric}+2\lambda
g,$ (3.1)

for arbitrary real constant ${c}_{1}$, ${c}_{2}$ and $\lambda$. Here
${{\mathcal{L}}_{X}}g$ is the Lie derivative of the metric $g$ with respect to
$X$, ${{X}^{b}}$ is a non-zero $1$-form such that ${{X}^{b}}(Y)=g(X,Y)$, Ric
is the Ricci tensor of $g$. We call (3.1) the generalised Ricci soliton
equations. A pair $(g,X)$ is called a generalised Ricci soliton if (3.1) is
satisfied.

A Riemann soliton are defined by a smooth vector field $V$ and a real constant
$\lambda$ which satisfy the following equation

$R+\frac{1}{2}{{\mathcal{L}}_{V}}g\wedge g=\frac{\lambda}{2}g\wedge g,$ (3.2)

where ${{\mathcal{L}}_{V}}g$ denotes the Lie derivative of $g$ and $\wedge$ is
the Kulkarni-Nomizu product.

A vector field $\varphi$ on a Riemannian manifold $({{M}^{n}},g)$ is said to
be a $\varphi(\text{Ric})$-vector field [19] if it satisfies

${{\nabla}_{X}}\varphi=\mu QX,$ (3.3)

where $\mu$ is a constant and $Q$ is the ricci operator defined by
$\text{Ric}(X,Y)=g(QX,Y)$. If $\mu\neq 0$, then the vector field $\varphi$ is
called a proper $\varphi(\text{Ric})$-vector field.

Next, we consider an extended quasi-Einstein manifold with a generalised Ricci
soliton and Riemannian soliton, respectively.

###  3.1. Generalised Ricci soliton

######  Proposition 3.1.

Let $M$ be an extended quasi-Einstein manifold with a generalised Ricci
soliton $(g,U)$ such that the vector field $U$ is the generator of $M$. Then
the integral curves of $U$ are geodesic on $M$.

###### Proof.

Since $(g,U)$ is a generalised Ricci soliton on $M$, from (3.1) we have

$({{\mathcal{L}}_{U}}g)(Y,Z)+2{{c}_{1}}g(U,Y)g(U,Z)=2{{c}_{2}}\text{Ric}(Y,Z)+2\lambda
g(Y,Z),$ (3.4)

for any $Y$, $Z$ $\in{{C}^{\infty}}(TM)$. Putting $Y=U$ in (3.4) and from
(2.10) gives

$g({{\nabla}_{U}}U,Z)=[2{{c}_{2}}(a+b)+2\lambda-2{{c}_{1}}]A(Z).$ (3.5)

Putting $Z=U$ in (3.5), we have

$\lambda={{c}_{1}}-{{c}_{2}}(a+b).$ (3.6)

Therefore, from (3.5) and (3.6) we obtain

$g({{\nabla}_{U}}U,Z)=0,$ (3.7)

which implies ${{\nabla}_{U}}U=0$, the integral curves of the vector field $U$
are geodesic. ∎

######  Proposition 3.2.

Let $M$ be an extended quasi-Einstein manifold with a generalised Ricci
soliton $(g,V)$ such that the vector field $V$ is the generator of $M$. If
${{c}_{2}}\neq 0$, then the integral curves of $V$ are geodesic on $M$ if and
only if the manifold $M$ is a quasi-Einstein manifold.

###### Proof.

Since $(g,V)$ is a generalised Ricci soliton on $M$, from (3.1) we have

$({{\mathcal{L}}_{V}}g)(Y,Z)+2{{c}_{1}}g(V,Y)g(V,Z)=2{{c}_{2}}\text{Ric}(Y,Z)+2\lambda
g(Y,Z),$ (3.8)

for any $Y$, $Z$ $\in{{C}^{\infty}}(TM)$. Putting $Y=V$ in (3.8) and from
(2.11) yields

$g({{\nabla}_{V}}V,Z)=2[a{{c}_{2}}+\lambda-{{c}_{1}}]B(Z)+2c{{c}_{2}}D(Z).$
(3.9)

Putting $Z=V$ in (3.9), we obtain

$\lambda={{c}_{1}}-a{{c}_{2}}.$ (3.10)

From (3.9) and (3.10), we obtain

$g({{\nabla}_{V}}V,Z)=2c{{c}_{2}}D(Z).$ (3.11)

Let us suppose that the integral curves of $V$ are geodesic on $M$. Then,
(3.11) becomes

$2c{{c}_{2}}D(Z)=0.$ (3.12)

Taking $Z=T$ in (3.12) and ${{c}_{2}}\neq 0$, we find $c=0$. This implies that
$M$ is a quasi-Einstein manifolds.

Conversely, we assume that $M$ is a quasi-Einstein manifolds. Then, we have
$c=0$. Hence, we get

$g({{\nabla}_{V}}V,Z)=0.$

which implies that ${{\nabla}_{V}}V=0$. ∎

######  Proposition 3.3.

An extended quasi-Einstein manifold $M$ admitting a generalised Ricci soliton
$(g,V)$, then $divV=n(2r-a){{c}_{2}}$.

###### Proof.

By the suitable contraction of (3.8), we obtain

$\lambda=\frac{1}{n}divV+{{c}_{1}}-2{{c}_{2}}.$ (3.13)

Using (3.10) and (3.13), we get

$divV=n(2r-a){{c}_{2}},$

where $r$ is scalar curvature. ∎

######  Theorem 3.4.

Let $M$ be an extended quasi-Einstein manifold with a generalised Ricci
soliton $(g,V)$ and the vector field $V$ is a $V$(Ric)-vector field. Then, $M$
is either a quasi-Einstein manifold or the generalised Ricci soliton reduces
to a steady Ricci soliton.

###### Proof.

We consider $V$ and $T$ are mutually orthogonal unit vector fields, i.e.,
$t=0$. It follows from the Lie-derivative and from (3.3), we have

$({{\mathcal{L}}_{V}}g)(X,Y)=2\mu\text{Ric}(X,Y),$ (3.14)

for any $X$, $Y$ $\in{{C}^{\infty}}(TM)$. $M$ is an extended quasi-Einstein
manifold with generalised Ricci soliton $(g,V)$. From (3.1) and (3.14) we find

$(\mu-{{c}_{2}})\text{Ric}(X,Y)=\lambda g(X,Y)-{{c}_{1}}B(X)B(Y).$ (3.15)

Taking $X=Y=U$ in (3.15) gives

$(a+b)(\mu-{{c}_{2}})=\lambda.$ (3.16)

Taking $X=Y=V$ in (3.15) gives

$a(\mu-{{c}_{2}})=\lambda-{{c}_{1}}.$ (3.17)

Taking $X=V$, $Y=T$ in (3.15) gives

$c(\mu-{{c}_{2}})=0.$ (3.18)

which implies that $\mu={{c}_{2}}$ or $c=0$.

If $c=0$, then $M$ is a quasi-Einstein manifold. If $\mu={{c}_{2}}$, then from
(3.16) and (3.17) we have $\lambda={{c}_{1}}=0$. This means that the
generalised Ricci soliton reduces to a steady Ricci soliton. ∎

###  3.2. Riemann soliton

######  Proposition 3.5.

Let $M$ be an extended quasi-Einstein manifold with a Riemann soliton $(g,V)$.
Then, the integral curves of $V$ are geodesic on $M$ if and only if the
manifold M is a quasi-Einstein manifold.

###### Proof.

The Riemannian soliton equation (3.2) can be expressed as

$\displaystyle 2R(X,Y,Z,W)$
$\displaystyle+\\{g(X,W)({{\mathcal{L}}_{V}}g)(Y,Z)+g(Y,Z)({{\mathcal{L}}_{V}}g)(X,W)$
(3.19)
$\displaystyle-g(X,Z)({{\mathcal{L}}_{V}}g)(Y,W)-g(Y,W)({{\mathcal{L}}_{V}}g)(X,Z)\\}$
$\displaystyle=2\lambda\\{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\\}.$

Contracting (3.19) over $X$ and $W$, we obtain

$({{\mathcal{L}}_{V}}g)(Y,Z)+\frac{2}{n-2}\text{Ric}(Y,Z)-\frac{2}{n-2}[(n-1)\lambda-\text{div}V]g(Y,Z)=0.$
(3.20)

Putting $Y=V$ in (3.20), we obtain

$\displaystyle g({{\nabla}_{V}}V,Z)+\frac{2}{n-2}[aB(Z)+cD(Z)]$ (3.21)
$\displaystyle-\frac{2}{n-2}[(n-1)\lambda-\text{div}V]B(Z)$ $\displaystyle=0.$

Putting $Z=V$ in (3.21), we have $a=(n-1)\lambda-\text{div}V$. Then, (3.21)
become

$g({{\nabla}_{V}}V,Z)+\frac{2}{n-2}cD(Z)=0.$ (3.22)

Now, let us suppose that the integral curves of $V$ are geodesic on $M$. Then,
(3.22) becomes

$\frac{2}{n-2}cD(Z)=0.$ (3.23)

Taking $Z=T$ in (3.23), we find $c=0$. This implies that $M$ is a quasi-
Einstein mainfold.

Conversely, we assume that $M$ is a quasi-Einstein manifold. Then, we have
$c=0$. Hence, we get

$g({{\nabla}_{V}}V,Z)=0,$

which implies that ${{\nabla}_{V}}V=0$. ∎

######  Proposition 3.6.

Let $M$ be an extended quasi-Einstein manifold with a Riemann soliton $(g,U)$
such that the vector field $U$ is the generator of $M$. Then the integral
curves of $U$ are geodesic on $M$.

###### Proof.

From (3.20), we have

$({{\mathcal{L}}_{U}}g)(Y,Z)+\frac{2}{n-2}\text{Ric}(Y,Z)-\frac{2}{n-2}[(n-1)\lambda-\text{div}U]g(Y,Z)=0.$
(3.24)

Substituting $Y=U$ into (3.24), yields

$g({{\nabla}_{U}}U,Z)+\frac{2}{n-2}[(a+b)A(Z)]-\frac{2}{n-2}[(n-1)\lambda-\text{div}U]A(Z)=0.$
(3.25)

Putting $Z=U$ in (3.25), we have $a+b=(n-1)\lambda-\text{div}U$. Then, (3.25)
become

$g({{\nabla}_{U}}U,Z)=0,$ (3.26)

which implies ${{\nabla}_{U}}U=0$, the integral curves of the vector field $U$
are geodesic. ∎

######  Proposition 3.7.

Let $M$ be an extended quasi-Einstein manifold admitting a Riemann soliton
$(g,V)$ such that the vector field $V$ is a $V$(Ric)-vector field. Then $V$ is
a proper $V$(Ric)-vector field, and $\mu=-\frac{1}{1-2n}$.

###### Proof.

Taking covariant derivative of the equation (3.20) gives

$({{\nabla}_{X}}{{\mathcal{L}}_{V}}g)(Y,Z)+\frac{2}{2n-1}({{\nabla}_{X}}\text{Ric})(Y,Z)=0,$
(3.27)

where $V$ has a constant divergence. Similarly, taking covariant derivative of
the equation (3.14), have

$({{\nabla}_{X}}{{\mathcal{L}}_{V}}g)(Y,Z)=2\mu({{\nabla}_{X}}\text{Ric})(Y,Z).$
(3.28)

From (3.26) and (3.28), we have $\mu=-\frac{1}{1-2n}$, which implies $V$ is a
proper $V$(Ric)-vector field. ∎

##  4\. Example

We consider a Riemannian manifold $({{M}^{4}},g)$ endowed with the Riemannian
metric $g$ defined by

$d{{s}^{2}}={{g}_{ij}}d{{x}^{i}}d{{x}^{j}}={{x}^{2}}{{(d{{x}^{1}})}^{2}}+{{x}^{1}}{{(d{{x}^{2}})}^{2}}+{{(d{{x}^{3}})}^{2}}+{{(d{{x}^{4}})}^{2}},$
(4.1)

where $i,j=1,2,3,4$. The non-vanishing components of Christoffel symbols are

$\Gamma_{12}^{1}=\frac{1}{2{{x}^{2}}},\quad\Gamma_{22}^{1}=-\frac{1}{2{{x}^{2}}},\quad\Gamma_{11}^{2}=-\frac{1}{2{{x}^{1}}},\quad\Gamma_{12}^{2}=\frac{1}{2{{x}^{1}}}.$

The non-zero curvature tensor are

${{R}_{1212}}=\frac{1}{4{{x}^{1}}}-\frac{1}{4{{x}^{2}}}.$

The non-vanishing components of Ricci tensor are

${{\text{Ric}}_{11}}=\frac{1}{4{{x}^{1}}{{x}^{2}}}-\frac{1}{4{({{x}^{1}})^{2}}},\quad{{\text{Ric}}_{22}}=\frac{1}{4{({{x}^{2}})^{2}}}-\frac{1}{4{{x}^{1}}{{x}^{2}}},$

and the scalar curvature is

$r=\frac{1}{2{{x}^{1}}{({{x}^{2}})^{2}}}-\frac{1}{2{{x}^{2}}{({{x}^{1}})^{2}}}.$

which is non zero and non-constant. We shall now show that this ${{M}^{4}}$ is
an extended quasi-Einstein manifold, i.e., it satisfies the defining relation
(2.1).

Now, we take the associated scalars as follows:

$a=\frac{1}{{{x}^{1}}{{x}^{2}}},\quad b=-\frac{2}{{{x}^{1}}{{x}^{2}}},\quad
c=-\frac{\sqrt{10}}{5{{x}^{1}}{{x}^{2}}[\frac{1}{{{x}^{2}}}-\frac{1}{{{x}^{1}}}]}.$

We take the 1-forms as follows:

${{A}_{i}}(x)=\left\\{\begin{aligned} &\frac{\sqrt{2{{x}^{2}}}}{2},\quad
for\,i=1\\\ &\frac{\sqrt{2{{x}^{1}}}}{2},\quad for\,i=2\\\ &0,\quad\quad\quad
for\,i=3,4\end{aligned}\right.,\quad{{B}_{i}}(x)=\left\\{\begin{aligned}
&\frac{\sqrt{2{{x}^{2}}}}{4},\quad\quad\,for\,i=1\\\
&-\frac{\sqrt{2{{x}^{1}}}}{4},\quad for\,i=2\\\
&0,\quad\quad\quad\quad\,for\,i=3\\\
&\frac{\sqrt{3}}{2},\quad\quad\quad\,for\,i=4\\\ \end{aligned}\right.$

and

${{D}_{i}}(x)=\left\\{\begin{aligned} &-\frac{\sqrt{5{{x}^{2}}}}{4},\quad
for\,i=1\\\ &\frac{\sqrt{5{{x}^{1}}}}{4},\quad\quad for\,i=2\\\
&\frac{\sqrt{6}}{6},\quad\quad\quad for\,i=3\\\
&\frac{\sqrt{30}}{12},\,\quad\quad\,\,for\,i=4\end{aligned},\right.$

at any point $x\in M$. In $({{M}^{4}},g)$, (2.1) reduces with these associated
scalars and $1$-forms to the following equation:

${\text{Ric}_{11}}=a{{g}_{11}}+b{{A}_{1}}{{A}_{1}}+c[{{B}_{1}}{{D}_{1}}+{{D}_{1}}{{B}_{1}}],$
(4.2)
${\text{Ric}_{22}}=a{{g}_{22}}+b{{A}_{2}}{{A}_{2}}+c[{{B}_{2}}{{D}_{2}}+{{D}_{2}}{{B}_{2}}].$
(4.3)

It can prove that (4.2), (4.3) is true. We shall now show that the 1-forms are
unit,

${{g}^{ij}}{{A}_{i}}{{A}_{j}}=1,{{g}^{ij}}{{B}_{i}}{{B}_{j}}=1,{{g}^{ij}}{{D}_{i}}{{D}_{j}}=1,{{g}^{ij}}{{A}_{i}}{{B}_{j}}=0,{{g}^{ij}}{{A}_{i}}{{D}_{j}}=0,{{g}^{ij}}{{B}_{i}}{{D}_{j}}=0.$

So manifold is an extended quasi-Einstein manifold.

##  5\. Discussions and further questions

In this final section, we give some related questions which deserve to be
studied further.

  1. (1)

The concepts of quasi conformally curvature tensor, conformally curvature
tensor, projective curvature tensor and conharmonic curvature tensor are
introduced in [21, 22, 23, 24]. In the next step of our research, we will
consider an extended quasi-Einstein manifolds with some flat condition.
Further, we study some symmetric condition on the extended quasi-Einstein
manifold, such as, semi-symmetric, conformally semi-symmetric, projectively
semi-symmetric, concircularly semi-symmetric, conharmonically semi-symmetric
[7].

  2. (2)

We study Deszcz pseudo symmetric conditions on the extended quasi-Einstein
manifold base on the study of equivalence of geometric structures in [25] by
considering different conditions into various groups or classes.

  3. (3)

Imposing an additional circulant structure on four-dimensional Riemannian
manifolds and the components of the circulant structure in local coordinates
are circulant matrices, such additional structures have been widely studied
[26]. We will impose a skew-circulant structure on the extended quasi-Einstein
manifolds, and study geometric properties of such manifold.

Acknowledgement. W. J. Lu gratefully acknowledges support by the Natural
Science Foundation of China($\\#$. 12061014) and the National Science
Foundation of Guangxi Province($\\#$. 2019GXNSFAA245043)

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