# Scale Invariant Correspondence Analysis

Choulakian V., Université de Moncton, Canada<EMAIL_ADDRESS>
(December 2023)

###### Abstract

Correspondence analysis is a dimension reduction method for visualization of
nonnegative data sets, in particular contingency tables; but it depends on the
marginals of the data set. Two transformations of the data have been proposed
to render correspondence analysis row and column scales invariant: These two
kinds of transformations change the initial form of the data set into a
bistochastic form. The power transorfmation applied by Greenacre (2010) has
one positive parameter. While the transormation applied by Mosteller (1968)
and Goodman (1996) has (I+J) positive parameters, where the raw data is row
and column scaled by the Sinkhorn (RAS or ipf) algorithm to render it
bistochastic. Goodman (1996) named correspondence analsis of a bistochastic
matrix marginal-free correspondence analysis. We discuss these two
transformations, and further generalize Mosteller-Goodman approach.

Key words: Greenacre’s Theorem; Goodman’s marginal-free correspondence
analysis; Sinkhorn (RAS-ipf) algorithm; bistochastic blocks diagonal matrix;
log-linear association.

AMS 2010 subject classifications: 62H25, 62H30

##  1 Introduction

Correspondence analysis (CA) and logratio analysis (LRA) are two popular
methods for the analysis-visualization of a two-way contingency table or a
compositional data set $\mathbf{N}=(n_{ij})$ of size $I\times J$ for
$i=1,...,I$ and $j=1,...,J.$ LRA is considered an ideal desirable method for
analysis-visualization of N, because it is row and column scales invariant,
see Lemma 1; but it can be applied only to positive data $n_{ij}>0.$ While CA
can be applied to nonnegative data $n_{ij}\geq 0$, but it is NOT row and
column scale invariant because it is dependent on its row and column
marginals. This paper presents two transformations of data that render CA row
and column scales invariant, similar to LRA. These two transformations are
proposed by Mosteller (1968) and Goodman (1996, equation (40)), and, by
Greenacre (2010, result 1).

Goodman (1996, equation (46)) in his seminal paper outlined marginal-free
correspondence analysis (mfCA) without details; where his principal aim was to
reconcile Pearson correlation measure on which CA is based with Yule’s
association measure on which LRA is based in the analysis of contingency
tables. Choulakian (2022) showed that Goodman’s mfCA is a particular case of
CA with prespecified marginals studied by Benzécri and his students in the
early 1980’s. Goodman’s aim of reconciliation is perfectly attained by
Greenacre’s Theorem, see section 3 below, which concerns the convergence of CA
of $\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$ for $\alpha\in(0,1]$ as
$\alpha\rightarrow 0$ to uniformly weighted LRA of $\mathbf{N}$ when
$n_{ij}>0$; for further details see Greenacre (2009, 2010 Result 1, 2022) and
Choulakian (2023).

The aim of this paper is to delve deeper from applied point of view into
Goodman (1996, equation (46))’s statement that mfCA ” can be used as a
replacement for the usual CA model (or canonical correlation model) when
marginal-free methods are desired”; here, marginal-free means row and column
scales invariant. This will be done within the framework of bistochastic (row
and column stochastic) matrices that we introduce next.

###  1.1 Bistochastic matrix

In this paper, the notion of bistochastic (or doubly stochastic) matrix plays
an essential role. A nonnegative matrix $\mathbf{D=}(d_{ij})$ for $i=1,...,I$
and $j=1,...,J,$ is named doubly stochastic (or bistochastic) if the following
two conditions are satisfied:
$\frac{1}{I}\sum_{i}d_{ij}=\frac{1}{J}\sum_{j}d_{ij}=1$ for $i=1,...,I$ and
$j=1,...,J;$ see among others Loukaki (2022). It follows that
$\sum_{i,j}d_{ij}=IJ.$ So $\mathbf{Q=D/(}IJ\mathbf{)}$ is a probability
measure with uniform marginals; that is, $\frac{1}{J}=q_{+j}=\sum_{i}q_{ij}$
and $\frac{1}{I}=q_{i+}=\sum_{j}q_{ij};$ Mosteller (1968) and Goodman (1996,
section 6, equations (39) and (40)) named $\mathbf{Q}$ ”the standardized
distribution”. So the concepts of doubly stochastic matrix $\mathbf{D}$ and
and its associated probability measure $\mathbf{Q=D/(}IJ\mathbf{)}$ with
uniform marginal weights are equivalent for nonnegative data sets.

According to Goodman (1996, section 6), the first step for mfCA is to
transform the contingency table $\mathbf{N=}(n_{ij})$ into a bistochastic
matrix D by row and column scalings, where
$\mathbf{D}=(d_{ij}=a_{i}n_{ij}b_{j})$, $a_{i}>0$ and $b_{j}>0$ for
$i=1,...,I$ and $j=1,...,J.$ Scaling a matrix consists of pre- and post-
multiplying the original matrix $\mathbf{N=}(n_{ij})$ by two diagonal matrices
$Diag(a_{i})$ and $Diag(b_{j})$. The computational tool to achieve this will
be the Sinkhorn algorithm, also known as the RAS method or the iterative
proportional fitting (ipf) procedure; for a comprehensive survey see Idel
(2016). We note that often bistochasticity of a matrix is used for square
matrices, see for instance, Le Gorrec et al. (2022).

###  1.2 Contents

Row and column scales invariant CA of a data set N emerges in two distinct
ways:

a) $\mathbf{N}$ is positive, $n_{ij}>0$; thus scalable to a double stochastic
matrix by two different transformations. First, by a power transformation,
which is equivalent to the uniformly weighted LRA association index by
Greenacre (2010, Result 1)’s Theorem. Here, log odds ratios play central role.
Second, by row and column scaling by Sinkhorn (1964)’s algorithm.

b) $\mathbf{N}$ is nonnegative (there is at least one zero cell). Here we can
have 2 cases. Case 1) $\mathit{\mathbf{N}}$ is scalable to a bistochastic
matrix D. This case is described by Goodman (1996, section 6, equations (39))
and CA of D is named mfCA. Case 2: $\mathbf{N}$ is scalable to a matrix D
having the structure of multiple diagonal bistochastic blocks. We name CA of D
generalized (or multiple) mfCA.

This paper is organized as follows: Section 2 presents preliminaries
concerning CA and LRA association indices and their relationship with the
additive and multiplicative double centerings; section 3 discusses Greenacre’s
Theorem; section 4 presents Goodman’s mfCA (cases 1 and 2); section 5 presents
Sinkhorn (RAS-ipf) algorithm; section 6 compares both transformations; section
7 presents three case studies; and we conclude in section 8.

Benzécri (1973, volumes 1 and 2) is the reference book on CA but difficult to
read because of his use of tensor notation. Among others, Greenacre (1984) is
one of the best representative of Benzécri’s approach ; Beh and Lombardo
(2014) present a panoramic review of CA. Taxicab CA is a robust variant of CA,
see among others Choulakian (2006, 2016), Choulakian et al. (2023) and
Choulakian and Allard (2023).

##  2 Preliminaries on the analysis of contingency tables

We consider a two-way contingency table $\mathbf{N}=(n_{ij})$ for $i=1,...,I$
and $j=1,...,J$, and $\mathbf{P=N/}n=(p_{ij})$ of size $I\times J$ the
associated correspondence matrix (probability table) of the contingency table
N. We define as usual $p_{i+}=\sum_{j=1}^{J}p_{ij}$,
$p_{+j}=\sum_{i=1}^{I}p_{ij},$ the vector
$\mathbf{(}p_{i+})\in\mathbb{R}^{I},$ the vector
$\mathbf{(}p_{+j})\in\mathbb{R}^{J}$, and $\mathbf{M}_{I}=Diag(p_{i+})$ the
diagonal matrix having diagonal elements $p_{i+},$ and similarly
$\mathbf{M}_{J}=Diag(p_{+j}).$ We suppose that $\mathbf{M}_{I}$ and
$\mathbf{M}_{J}$ are positive definite metric matrices of size $I\times I$ and
$J\times J$, respectively; this means that the diagonal elements of
$\mathbf{M}_{I}$ and $\mathbf{M}_{J}$ are strictly positive.

First, we review the CA and LRA association indices, then double centerings of
a data set.

###  2.1 Association indices

The CA association index of a probability table P is

$\Delta_{ij}(\mathbf{P})=\frac{p_{ij}}{p_{i+}p_{+j}}-1,$ (1)

where $\frac{p_{ij}}{p_{i+}p_{+j}}$ is named the density function of $p_{ij}$
with respect to the product measure $p_{i+}p_{+j}.$

Assuming $p_{ij}>0,$ we define the LRA association index (named log-linear
association index by Goodman (1996)) with uniform weights to be

$\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)=\log(p_{ij})+\frac{1}{IJ}\sum_{i,j}\log(p_{ij})-(\frac{1}{I}\sum_{i}\log(p_{ij})+\frac{1}{J}\sum_{j}\log(p_{ij})).$
(2)

Equation (2) appears also in compositional data analysis, see Aitchison
(1986).

What’s the difference between the CA and LRA association indices (1) and (2)?
The CA index is marginal-scale dependent, while the LRA index is scale
invariant, that we describe. Let $\tau_{ij}(p_{ij})$ represent an interaction
(association) index, such as $\Delta_{ij}(\mathbf{P})$ or
$\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$ described in equations (1) and
(2) respectively.

Following Yule (1912) and Goodman (1996), we state the following

Definition 1: An interaction index
$\tau_{ij}(p_{ij}=\frac{n_{ij}}{\sum_{i,j}n_{ij}})$ is scale invariant if
$\tau_{ij}(p_{ij})=\tau_{ij}(q_{ij}=\frac{a_{i}n_{ij}b_{j}}{\sum_{i,j}a_{i}n_{ij}b_{j}})$
for arbitrary positive scales $a_{i}>0$ and $b_{j}>0$.

It is evident that the CA association index $\Delta_{ij}(\mathbf{P})$ in (1)
is not scale invariant, because it depends on the data dependent marginal
weights $(p_{i+},p_{+j}).$

Lemma 1: The LRA association index (3) with prespecified weights
($w_{i}^{R},w_{j}^{C})$

$\lambda(p_{ij},w_{i}^{R},w_{j}^{C})=\log(p_{ij})+\sum_{i,j}w_{i}^{R}w_{j}^{C}\log(p_{ij})-(\sum_{i}w_{i}^{R}\log(p_{ij})+\sum_{j}w_{j}^{C}\log(p_{ij}))$
(3)

is scale invariant. In particular the LRA association index (2) with uniform
weights is scale invariant.

For a proof see Choulakian et al. (2023).

###  2.2 Double centering of a data set

The CA association index $\Delta_{ij}(\mathbf{P})$ and the LRA association
index $\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$ are double centered; that
is:

$\displaystyle 0$
$\displaystyle=\sum_{i=1}^{I}p_{i+}p_{+j}\Delta_{ij}(\mathbf{P})=\sum_{i=1}^{I}\frac{1}{I}\frac{1}{J}\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$
$\displaystyle=\sum_{j=1}^{J}p_{i+}p_{+j}\Delta_{ij}(\mathbf{P})=\sum_{j=1}^{J}\frac{1}{I}\frac{1}{J}\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J).$
(4)

According to Tukey (1977, chapter 10), there are two kinds of double
centerings row-PLUS-column and row-TIMES-column; we name them additive and
multiplicative, that we describe.

We consider the triplet $\mathbf{(Y,M}_{I},\mathbf{M}_{J}\mathbf{)}$, where
$y_{ij}=h(p_{ij})$ is the transformed data set by the function $h,$ and
$(\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(w_{i}^{R}),Diag(w_{j}^{C}))$ are the
row an column metric matrices. We compute the three means pertaining to the
row and column marginals and the total:$\
Y_{i+}=\sum_{j=1}^{J}y_{ij}w_{j}^{C},$ $Y_{+j}=\sum_{i=1}^{I}y_{ij}w_{i}^{R}$
and $Y_{++}=\sum_{j=1}^{J}\sum_{i=1}^{I}y_{ij}w_{i}^{R}w_{j}^{C}.$

The multiplicative double centering is

$\tau_{ij}=y_{ij}-\frac{Y_{i+}Y_{+j}}{Y_{++}},$ (5)

where rank($\tau_{ij})=\ $rank$(Y_{ij})-1.$

The additive double centering is

$\tau_{ij}=y_{ij}+Y_{++}-(Y_{i+}+Y_{+j}),$ (6)

where rank($\tau_{ij})=\ $rank$(y_{ij})-1$ or rank$(y_{ij})-2.$

An evident difference between the two types of double centering is that (6) is
invariant to a row or column additive constants: Thus Definition 1 and Lemma 1
become a corollary to this fact.

We consider two functional forms of $y_{ij}=h(p_{ij}).$

a) $y_{ij}=h(p_{ij})=\log p_{ij}$ for $p_{ij}>0.$ First, the log-linear
interaction $\tau_{ij}\mathbf{=}\lambda(p_{ij},w_{i}^{R},w_{j}^{C})$ is
obtained from (6). So the rank
$\mathbf{(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))=$ rank($\log p_{ij})-1,$ as in
Goodman (1991, Table 11); or rank
$\mathbf{(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))=\ $rank($\log p_{ij})-2,$ as
in Goodman (1991, Table 10). Second, to our knowledge the log interaction
obtained from (5) has not been applied yet, most probably due to the fact that
it is not row and column scale invariant.

b) $y_{ij}=h(p_{ij})=\frac{p_{ij}}{w_{i}^{R}w_{j}^{C}}$ is the density
function of the joint probability measure $p_{ij}$ with respect to the product
measure $w_{i}^{R}w_{j}^{C}.$ The CA interaction
$(\tau_{ij}\mathbf{)=(}\Delta_{ij}(\mathbf{P}))$ is obtained either from (5)
or from (6), when ($w_{i}^{R},w_{j}^{C})=(p_{i+},p_{+j}).$ So the
rank$\mathbf{(}\Delta_{ij}(\mathbf{P}))=\
$rank($\frac{p_{ij}}{p_{i+}p_{+j}})-1.$ This property of CA seems unique. So
we ask the following question: Under what condition the additive double
centering equals the multiplicative double centering, $(5)=(6)$? The proof of
the next lemma is easy.

Lemma 2: Assuming $y_{ij}\geq 0$, equations $(5)=(6)$ if and only if
$Y_{++}=Y_{i+}$ or $Y_{++}=Y_{+j}$.

Corollary 1:

a) In CA, $Y_{i+}=Y_{++}=Y_{+j}=1;$ see also Goodman (1996, equation (11)).

b) If $\mathbf{Y=D=\mathit{I}Q}\mathit{J}$ is a bistochastic matrix, then
$Y_{i+}=Y_{++}=Y_{+j}=1$, so $(5)=(6)$.

##  3 Greenacre’s Theorem

Greenacre’s Theorem relates both CA and LRA association indices in a very
simple way (that is equations $(1)=(2)$ indirectly): Let
$\mathbf{P}^{(\alpha)}=(p_{ij}^{(\alpha)}=\frac{n_{ij}^{\alpha}}{\sum_{i,j}n_{ij}^{\alpha}})$
for $i=1,...,I$ and $j=1,...,J\ $be the correspondence table of the simple
power transformed data, $p_{i+}^{(\alpha)}=\sum_{j}p_{ij}^{(\alpha)}$ and
$p_{+j}^{(\alpha)}=\sum_{i}p_{ij}^{(\alpha)}$ for $\alpha>0.$ We consider the
CA association index

$\Delta_{ij}(\mathbf{P}^{(\alpha)})=\frac{p_{ij}^{(\alpha)}-p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}.$
(7)

Then we have, see Choulakian (2023) and by Lemma 1,

Greenacre’s Theorem (Greenacre (2010, Result 1)

Under the assumption $n_{ij}>0,$ for $\alpha\rightarrow 0$ and $\alpha>0$

$\displaystyle\lim_{\alpha\rightarrow
0}\frac{\Delta_{ij}(\mathbf{P}^{(\alpha)})}{\alpha}$
$\displaystyle=\lim_{\alpha\rightarrow
0}(\frac{p_{ij}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}-1)/\alpha$
$\displaystyle=\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$
$\displaystyle=\lambda(q_{ij}=a_{i}p_{ij}b_{j},w_{i}^{R}=1/I,w_{j}^{C}=1/J)\text{
for }a_{i},b_{j}>0$ (8) $\displaystyle=\lim_{\alpha\rightarrow
0}\frac{\Delta_{ij}(\mathbf{Q}^{(\alpha)})}{\alpha}.$

Furthermore,

$\lim_{\alpha\rightarrow 0}p_{i+}^{(\alpha)}=\lim_{\alpha\rightarrow
0}q_{i+}^{(\alpha)}=1/I\text{\ \ for }i=1,...,I\text{\ \ and\ \
}\lim_{\alpha\rightarrow 0}p_{+j}^{(\alpha)}=\lim_{\alpha\rightarrow
0}q_{+j}^{(\alpha)}=1/J\ \text{for}\ j=1,...,J.$ (9)

Remark 1: There are two important aspects in Greenacree’s Theorem that we want
to emphasize.

a) First, it shows the conditions under which CA becomes row and column scales
invariant, equation (8).

b) Second, the row and column marginals of $\mathbf{P}^{(\alpha)}$ for
$\alpha>0$ and $\alpha\simeq 0$ are almost uniform, equation (9); so
$\mathbf{P}^{(\alpha)}$ is bistochastic, see subsection 1.1.

Remark 2: The CA decomposition of (8) is

$\displaystyle\lambda(p_{ij},w_{i}^{R}$
$\displaystyle=1/I,w_{j}^{C}=1/J)=\lim_{\alpha\rightarrow
0}\frac{\Delta_{ij}(\mathbf{P}^{(\alpha)})}{\alpha}$
$\displaystyle=\sum_{m=1}^{M}\mu_{mi}\nu_{mj}\rho_{m},$

where $M=rank(\mathbf{P})-1$ and the parameters ($\mu_{mi},\nu_{mj},\rho_{m})$
satisfy, for $s,m=1,...,M$

$1=\sum_{i=1}^{I}\mu_{mi}^{2}/I=\sum_{j=1}^{J}\upsilon_{mj}^{2}/J$
$0=\sum_{i=1}^{I}\mu_{mi}/I=\sum_{j=1}^{J}\upsilon_{mj}/J$
$0=\sum_{i=1}^{I}\mu_{mi}\mu_{si}/I=\sum_{j=1}^{J}\upsilon_{mj}\upsilon_{sj}/J\text{\
\ for }m\neq s.$

The parameter $\rho_{m},$ which is positive, has two interpretations: First,
$\rho_{m}=corr(\mu_{mi},\nu_{mj})$ is the Pearson’s correlation coefficient
within the framework of CA of $\mathbf{P}^{(\alpha)}$; second, it represents
the uniformly weighted intrinsic association in the framework of Goodman’s RC
association models. In the latter case, Goodman (1991, eqn. .3.7) interpreted
it as a function of log-odds ratio

$\displaystyle\log(\frac{p_{ij}p_{i_{1}j_{1}}}{p_{ij_{1}}p_{i_{1}j}}$
$\displaystyle=\sum_{m=1}^{M}(\mu_{mi}-\mu_{mi_{1}})(\upsilon_{mj}-\upsilon_{mj_{1}})\rho_{m}$
$\displaystyle=\lim_{\alpha\rightarrow
0}(\frac{p_{ij}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}+\frac{p_{i_{1}j_{1}}^{(\alpha)}}{p_{i_{1}+}^{(\alpha)}p_{+j_{1}}^{(\alpha)}}-\frac{p_{ij_{1}}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j_{1}}^{(\alpha)}}-\frac{p_{i_{1}j}^{(\alpha)}}{p_{i_{1}+}^{(\alpha)}p_{+j}^{(\alpha)}})/\alpha.$

The last relation is called tetra differences by Rao (1973, p.12); and it
perfectly reconciles Pearson correlation measure $\rho_{m}$ with Yule’s log-
odds ratio measure, an effort attempted by Mosteller (1968) and by Goodman
(1996, section 10) whom we quote:

”Pearson’s approach to the analysis of cross-classified data was based
primarily on the bivariate normal. He assumed that the row and column
classifications arise from underlying continuous random variables having a
bivariate normal distribution, so that the sample contingency table comes from
a discretized bivariate normal; and he then was concerned with the estimation
of the correlation coefficient for the underlying bivariate normal. On the
other hand, Yule felt that, for many kinds of contingency tables, it was not
desirable in scientific work to introduce assumptions about an underlying
bivariate normal in the analysis of these tables; and for such tables, he
used, to a great extent, coefficients based on the odds-ratios (for example,
Yule’s Q and Y), coefficients that did not require any assumptions about
underlying distributions. The Pearson approach and the Yule approach appear to
be wholly different, but a kind of reconciliation of the two perspectives was
obtained in Goodman (1981a)”.

##  4 Goodman’s mfCA

In Greenacre’s Theorem the positivity assumption of the data $p_{ij}>0$ is
essential and there is only one parameter $\alpha.$ Now we relax the
positivity assumption of each entry to nonnegativity $p_{ij}\geq 0$ and
introduce $(I+J)$ scale parameters to transform $\mathbf{P}=(p_{ij})$ into a
nonnegative matrix $\mathbf{Q}=(q_{ij}=a_{i}p_{ij}b_{j})$ as in the right-side
of equation (8). Depending on the distribution of the zeros in $\mathbf{P}$,
we can have two cases, which Sinkhorn and Knopp (1967) and Brualdi et al.
(1966) named them ”fully indecomposable” and ”reducible”.

Equations (10) and (12) are achieved by Sinkhorn (1964) ’s algorithm, also
known as the RAS algorithm or the iterative proportional fitting (ipf)
procedure.

###  4.1 Case 1: sparse fully indecomposable

The transformed matrix $\mathbf{Q=(}q_{ij}=a_{i}p_{ij}b_{j})$ for
$a_{i},b_{j}>0$ is unique and bistochastic; that is

$\mathbf{Q}=(q_{ij}=a_{i}p_{ij}b_{j})\text{ \ such that }q_{i+}=1/I\text{ and
}q_{+j}=1/J\text{ }$ (10)

is obtained when the data set is positive see Sinkhorn (1964), or the data set
is sparse with a structured full diagonal up to a permutation; that is the
matrix $\mathbf{P}$ is fully ”indecomposable”; see, Sinkhorn and Knopp (1967)
and Brualdi et al. (1966) who independently provided a necessary and
sufficient condition for (10) for square nonnegative matrices; Loukaki (2022)
presented further examples for rectangular matrices.

Goodman’s mfCA association index for the bistochastic matrix $\mathbf{Q}$ is

$\displaystyle\Delta_{ij}(\mathbf{Q})$ $\displaystyle=IJq_{ij}-1$
$\displaystyle=IJa_{i}p_{ij}b_{j}-1$ (11)
$\displaystyle=\Delta_{ij}(\mathbf{P},a_{i},b_{j}).\text{ }$

Lemma 3: Goodman’s mfCA association index (11) is scale invariant; that is,
for $p_{ij}=\frac{n_{ij}}{\sum_{i,j}n_{ij}}=n_{ij}/n_{tot}$, and
$p_{ij}^{\ast}=\frac{\varphi_{i}n_{ij}\psi_{j}}{\sum_{i,j}\varphi_{i}n_{ij}\psi_{j}}=\varphi_{i}n_{ij}\psi_{j}/n_{tot}^{\ast}$
for arbitrary positive scales $\varphi_{i}>0$ and $\psi_{j}>0,$ then

$\displaystyle\Delta_{ij}(\mathbf{Q})$ $\displaystyle=IJq_{ij}-1$
$\displaystyle=\Delta_{ij}(\mathbf{P},a_{i},b_{j})=IJa_{i}p_{ij}b_{j}-1$
$\displaystyle=\Delta_{ij}(\mathbf{P}^{\ast},a_{i}^{\ast},b_{j}^{\ast})=IJa_{i}^{\ast}p_{ij}^{\ast}b_{j}^{\ast}-1.$

if and only if $a_{i}^{\ast}=\tau a_{i}/\varphi_{i}$ and $b_{j}^{\ast}=\tau
b_{j}/\psi_{j}$ for $\tau^{2}=n_{tot}^{\ast}/n_{tot}$.

###  4.2 Case 2: sparse reducible

The transformed matrix
$\mathbf{Q}^{\ast}\mathbf{=}(q_{\beta(ij)}^{\ast}=a_{i\beta}^{\ast}p_{ij}b_{j\beta}^{\ast})$
for $a_{i\beta}^{\ast},b_{j\beta}^{\ast}>0$ is in general not unique but has
the structure of multiple, say $k$, bistochastic block diagonal submatrices

$\displaystyle\mathbf{Q}^{\ast}$
$\displaystyle=(q_{\beta(ij)}^{\ast}=a_{i\beta}^{\ast}p_{ij}b_{j\beta}^{\ast})$
$\displaystyle=Diag(\mathbf{B}_{1}...\mathbf{B}_{\beta}...\mathbf{B}_{k})^{\prime},$
(12)

where each submatrix $\mathbf{B}_{\beta}$ for $\beta=1,...,k$ is bistochastic
with different marginals. Then CA of $\mathbf{Q}^{\ast}$ is equivalent to $k$
CA of the submatrices $\mathbf{B}_{\beta}$ for $\beta=1,...,k.$

The generalized Goodman’s mfCA association index is

$\Delta_{ij}(\mathbf{P},a_{i\beta}^{\ast},b_{j\beta}^{\ast})=\frac{a_{i\beta}^{\ast}p_{ij}b_{j\beta}^{\ast}}{c_{\beta}r_{\beta}}-1,\text{
}$ (13)

where the terms $c_{\beta},r_{\beta}>0$ represent different uniform marginals
of the blocks $\beta=1,...k$; that is,
$c_{\beta}=q_{\beta(+j)}^{\ast}=\sum_{i}q_{\beta(ij)}^{\ast}$ and
$r_{\beta}=q_{\beta(i+)}^{\ast}=\sum_{j}q_{\beta(ij)}^{\ast}$ if the indices
$(i,j)$ belong to the $\beta$th block. Often this case happens when the data
set $\mathbf{P}$ is sparse and ”reducible”; that is, it has quasi-multiple-
blocks diagonal structure with different marginals.

The following simple 2 blocks dataset is sparse reducible

$\displaystyle\mathbf{N}$
$\displaystyle=\left[\begin{tabular}[c]{cc}{B}${}_{1}$&0\\\
{0}&{B}${}_{2}$\end{tabular}\right]$
$\displaystyle=\left[\begin{tabular}[c]{ccc}1&1&0\\\
0&0&1\end{tabular}\right].$

Note that the blocks B1 and B2 are bistochastic, but N can not be transformed
into a bistochastic matrix; so we call it 2 blocks bistochastic.

###  4.3 Observations

The following results by Benzécri are useful for the Cases 1 and 2:

Theorem 1: (Benzécri 1973, pp188-190): In CA of a nonegative dataset, if $k$
principal dispersion values are $\rho_{i}=1$ for $i=1,...,k$, where
$k<rank(p_{ij}-p_{i+}p_{+j}),$ then the contingency table has $(k+1)$ blocks
diagonal structure up to permutations.

Remark 3: Generalizing Benzécri (1973, p.189-190)’s observation, we state that
for sparse data it is rare to have $\rho_{1}=1;$ but ”not uncommon” to have
$\rho_{1}\geq 0.837;$ then the underlying structure of the contingency table
has a lateral ordering, which could be quasi-multi-blocks diagonal (Case 2) or
quasi-band-diagonal (Case 1), see Benzécri 1973, vol 1, p. 261-2279, Figures
3A and 3B).

##  5 The Sinkhorn (RAS-ipf) algorithm

The Sinkhorn (RAS-ipf) algorithm is a very simple method to adjust a
contingency table to prespecified marginals. Given that the literature on this
topic is vast, we refer the reader to Idel (2016) for a comprehensive survey.

The Sinkhorn (RAS-ipf) algorithm has three variants, clearly described in
Madre (1980): two of them apply alternating adjustments of the marginals,
which is the standard and the most discussed method; for an example see Landa
et al. (2021, page 6, algorithm 2); and the other uses simultaneous adjustment
of the row and column marginals, which we apply in this paper and in the R
code given below.

We use two criteria of convergence of the algorithm that a bistochastic matrix
$\mathbf{D}=(d_{ij})$ satisfies as described in subsection 1.1 and Corollary
1b. The first criterion is $C2_{dist}$ and the second $ratio$ defined by

$C2_{dist}=\sum_{i,j}|(\frac{1}{I}\sum_{i}d_{ij}+\frac{1}{J}\sum_{j}d_{ij})-2|,$
$ratio=\sum_{i,j}d_{ij}/IJ.$

If $\mathbf{D}$ is bistochastic, then $C2_{dist}=0$ and $ratio=1.\vskip 12.0pt
plus 4.0pt minus 4.0pt$

# MADRE alg3 Simultaneous adjustmnts

nRow $<$\- nrow(dataMatrix)

nCol $<$\- ncol(dataMatrix)

rowPrUnif $<$\- rep(1/nRow, nRow)

colPrUnif $<$\- rep(1/nCol, nCol)

nit $<$\- 500 # nit = number of iterations

qij $<$\- dataMatrix/sum(dataMatrix)

for (jj in 1:nit) {

rowPROBA $<$\- apply(qij,1,sum)

colPROBA $<$\- apply(qij,2,sum)

dij $<$\- qij/(rowPROBA %*% t(colPROBA))

qij $<$\- dij/sum(dij)

Gj $<$\- t(rowPrUnif) %*% dij

Gi $<$\- dij %*% colPrUnif

C2 $<$\- ((rep(1, nRow) %*% Gj) +(Gi %*% t(rep(1, nCol))))-2

C2dist = sum(abs(C2))

ratio = sum(dij)/(nRow*nCol)

su = c(jj,C2dist,ratio)

print(su)

}

##  6 Data transformation

The main instigation of this paper is the bistochastic similarity of equations
(9) and (10). In (9) there is one parameter $\alpha\approx 0$, while in (10)
there are $(I+J)$ positive scaling parameters $(a_{i},b_{j}).$ To understand
this similarity, or equivalently the difference between (9) and (10), we
introduce the concept of a transformation that does not change the internal
order structure in a contingency table, and show that Cases 1 and 2 are
transformations that may change dramatically the internal order structure.

Definition 2: A transformation $T$ of a probability measure ($p_{ij})$ keeps
the internal order structure unchanged if it satisfies the following
inequality:

$p_{ij}\leq p_{i_{1}j_{1}}\ \text{then}\ \ T(p_{ij})\leq T(p_{i_{1}j_{1}}).$
(14)

It is evident that the simple power transformation by
$T(p_{ij})=p_{ij}^{(\alpha)}$ for $\alpha>0$ in Greenacre’s Theorem satisfies
(14); while the transformations in Cases 1 and 2 represented in equations (10)
and (12) by $T(p_{ij})=a_{i}p_{ij}b_{j}$ and
$T(p_{ij})=a_{i\beta}^{\ast}p_{ij}b_{j\beta}^{\ast}$ in general do not satisfy
(14), as shown by the following simple contrived example.

Example 1: The application of the Sinkhorn (RAS-ipf) algorithm to the
following table, a variant first studied by Sinkhorn(1964),

$\mathbf{N}=\genfrac{[}{]}{0.0pt}{}{0\ \ \ \ \ 1}{1\ \ \ \ \ 2}$

produces the following transformed matrix after $nit=\ 50$ iterations

$T(\mathbf{N)=D}=\genfrac{[}{]}{0.0pt}{}{0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ 1.98076923}{1.98076923\ \ \ \ \
\epsilon_{nit=50}=0.03921569},$

which does not satisfy (14). Note that $\epsilon_{nit\rightarrow\infty}=0.$

##  7 Examples

We present three examples.

The first data set named CUPS, has positive elements with 0% of sparsity; this
is to compare Greenacre’s Theorem with Goodman’s mfCA. Given that each entry
of the CUPS data set is positive, so by Sinkhorn (1964)’s theorem, Sinkhorn
(RAS-ipf) algorithm always converges to a unique solution.

The other two data sets are sparse and belong to Cases 1 and 2. Table 1
provides the two outputs obtained from the two sparse datasets, where the the
number of iterations $nit=500$ is fixed, and the output of the last 10
iterations are displaced. Table 2 shows that the rodent data set is scalable
to a 4-blocks-bistochastic matrix (Case 2); while the Milazzese archeological
data set is scalable to a unique bistochastic matrix (Case 1).

Table 1: Sinkhorn (RAS-ipf) algorithm iteration results.
---
Sites | rodent |  | Milazzese |
$\mathit{iteration}$ | $C2_{dist}$ | $ratio$ | $C2_{dist}$ | $ratio$
491 | 302.96414 | 1.380883 | 2.042810e-14 | 1
492 | 257.13691 | 1.380883 | 2.042810e-14 | 1
493 | 302.96414 | 1.380883 | 8.082424e-14 | 1
494 | 257.13691 | 1.380883 | 2.930989e-14 | 1
495 | 302.96414 | 1.380883 | 3.330669e-14 | 1
496 | 257.13691 | 1.380883 | 1.465494e-14 | 1
497 | 302.96414 | 1.380883 | 1.689759e-13 | 1
498 | 257.13691 | 1.380883 | 4.685141e-14 | 1
499 | 302.96414 | 1.380883 | 1.776357e-14 | 1
500 | 257.13691 | 1.380883 | 1.361133e-13 | 1

###  7.1 CUPS data set

The archeological CUPS compositional data set of size $47\times 11$, taken
from Baxter et al. (1990), is analyzed by Greenacre and Lewi (2009), and can
be downloaded from the R packag easyCODA by Greenacre (2020); where each of
the 47 cups is described by the percentages by weight of 11 chemical elements.
Figure 1 in Greenacre and Lewi (2009) represents uniformly weighted log-ratio
biplot of the data, which represents CA of the power transformed bistochastic
matrix by Greenacre’s Theorem. The principal map not shown) produced by
Goodman’s mfCA is almost the same as Figure 1 in Greenacre and Lewi (2009).
Here are some statistics concerning both principal maps.

CA of power-transformed bistochastic matrix (in parenthesis corresponding
values of mfCA) :

$\rho_{1}^{2}=0.00833$ $(0.0101)$ ; variance explained $=39.6\%\ (43.6\%)$

$\ \rho_{2}^{2}=0.00638\ (0.00673)$ ; variance explained $=30.0\%$ $(29.1\%)$.

Table 2: Seriated rodent abundance data. |
---|---
$sites$ | rod1 | rod2 | rod6 | rod8 | rod3 | rod5 | rod9 | rod4 | rod7 | $\mu_{1i}\rho_{1}$
24 | 1 |  |  |  |  |  |  |  |  | 3.49
17 | 3 |  |  |  |  |  |  |  |  | 3.49
21 | 2 | 1 |  |  |  |  |  |  |  | 3.35
10 | 1 | 2 |  |  |  |  |  |  |  | 3.21
9 | 3 | 8 |  |  |  |  |  |  |  | 3.18
14 | 1 | 3 |  |  |  |  |  |  |  | 3.17
7 |  | 11 |  |  |  |  |  |  |  | 3.07
11 |  | 9 |  |  |  |  |  |  |  | 3.07
15 |  | 11 |  |  |  |  |  |  |  | 3.07
25 |  | 5 |  |  |  |  |  |  |  | 3.07
22 |  | 3 |  |  |  |  |  |  |  | 3.07
8 |  | 16 |  |  |  |  |  |  |  | 3.07
16 |  | 4 |  |  |  |  |  |  |  | 3.07
1 |  | 13 | 2 |  | 3 | 1 |  | 1 |  | 1.85
20 | 3 |  |  |  | 27 |  |  | 1 |  | -0.4
12 |  | 3 | 16 | 7 | 1 | 5 |  |  |  | -0.5
3 |  | 4 | 9 |  | 36 | 2 |  |  |  | -0.13
13 |  | 4 | 12 |  | 39 | 4 |  |  |  | -0.17
4 |  | 4 | 30 | 18 | 53 | 5 | 3 | 1 |  | -0.28
18 |  | 2 | 14 | 4 | 78 | 10 |  |  |  | -0.34
5 |  | 2 | 16 |  | 63 | 11 |  | 21 |  | -0.37
23 |  |  | 8 | 2 |  | 2 |  |  |  | -0.38
26 |  |  | 11 | 2 | 22 |  |  |  |  | -0.39
27 |  |  | 9 | 1 | 29 | 10 |  |  |  | -0.40
19 |  |  |  |  | 1 |  |  |  |  | -0.41
28 |  |  | 1 |  | 10 |  |  | 1 |  | -0.42
6 |  | 1 | 8 | 2 | 48 | 12 | 2 | 35 | 12 | -0.44
2 |  | 1 | 16 | 2 | 57 | 9 | 3 | 65 | 8 | -0.45
$\nu_{1j}\rho_{1}$ | 3.02 | 2.65 | -0.32 | -0.32 | -0.36 | -0.37 | -0.44 | -0.48 | -0.52 |

###  7.2 Rodent ecological abunance data

We consider the rodent data set N of size $28\times 9$ found in the R package
TaxicabCA by Allard and Choulakian (2019), taken from Quinn and Keough (2002).
This is an abundance data set of 9 species of rats in 28 cities in California.
The percentage of zero counts in N is $66.27\%$. Choulakian (2017) analyzed it
by comparing the CA and TCA maps; using Benzécri’s observation noted in Remark
3, Choulakian (2021) showed that it has quasi 2-blocks diagonal structure (or
partly decomposable) as shown in Table 2, where the rows and the columns are
ordered following the 1st factor CA scores $(\mu_{1i}\rho_{1},\
\upsilon_{1i}\rho_{1}).$ Furthermore Choulakian (2022), using the R packag
ipfr by Ward and Macfarlane (2020), compared Goodman’s marginal-free CA and
marginal-free TCA methods. It is important to note that the package(ipfr)
replaces the zero-valued cells by a very small positive value, so that the
Sinkhorn algorithm converges to a unique solution. In this paper, the zero
valued cells are unchanged before and after application of the algorithm, that
is why the algorithm oscillates and we get two different solutions as shown in
Table 2; so the scaled solution is a 4-blocks diagonal matrix (Case 2) as
displayed in Table 3.

Table 3: Row and column uniform marginals of 4 blocks of rodent data.
---
iteration 500 (499) |  | $C2_{dist}=257.14\ (302.96)$
100*colPROBA$\mathit{(c}_{\beta})$ | 36.23(4.76) | 29.96(6.71) | 5.25(13.13) | 3.76(11.45)
columns | 1 | 2 | 3,4,5,6,8 | 7,9
100*rowPROBA$\mathit{(r}_{\beta})$ | 6.04(0.79) | 4.28(0.96) | 2.51(7.63) | 2.19(5.47)
rows | 24,21,17, | 25,22,16,15 | 6,4,2 | 28,27,26,23
| 14,10,9 | 11,8,7 |  | 20,19,18,13
|  |  |  | 12,5,3,1

Note that the 2 solutions produced the same 4 blocks, but with different
uniform row $\mathit{(r}_{\beta})$ and column $\mathit{(c}_{\beta})$ marginals
as seen in Table 3. By Theorem 1 due to Benzécri this is confirmed by
calculating the CA coefficients of correlation of the factor scores of the
matrices $dataM=d_{ij},$ $\sigma_{1}=$ $\sigma_{2}=\sigma_{3}=1$, displayed
below by the R package ca by Greenacre et al.(2022).

iter = 500 ca(dataM)$sv

[1] 1.0000 1.0000 1.0000 0.8052 0.7174 0.6336 0.4936 0.2558

iter = 499 ca(dataM)$sv

[1] 1.0000 1.0000 1.0000 0.8052 0.7174 0.6336 0.4936 0.2558

By Remark 3, we see that the original dataset $\mathbf{N}$ has quasi 2-blocks
diagonal structure ; that is $\mathbf{N}$ is ”reducible”, for
$\rho_{1}=0.8639>0.837$ as displayed in Table 2$.\vskip 12.0pt plus 4.0pt
minus 4.0pt$

ca(dataMatrix)$sv # dataMatrix = N

[1] 0.8639 0.6776 0.5362 0.3909 0.1889 0.1568 0.1065

###  7.3 Milazzese archeological data

The Milazzese data set N is of size $31\times 19;$ the source of the data is
Alberti (2013a,b), but it can be easily downloaded from the R package
TaxicabCA. The percentage of zero counts in N is $58.06\%$, and it is scalable
to a bistochastic matrix, as seen in the R output in Table 1.

For the iteration number 500 the row and column marginals of the probability
measure $Q=(q_{ij}=\frac{d_{ij}}{IJ}=\frac{d_{ij}}{589})$ are uniform:
$q_{i+}=\frac{1}{I}=0.03225806$ and $q_{+j}=\frac{1}{J}=0.05263158.$

Below are the outputs of the correlation coefficients (sv) of the factor
scores obtained by CA of the original dataMatrix=N and the scaled bistochastic
$dataM=d_{ij}$; where we note that the scaling ($q_{ij}=a_{i}p_{ij}b_{j})\
$has changed the correlation coefficients, in particular the first one: from
shape of the original matrix ($p_{ij})$ into quasi-band-diagonal structure
($\rho_{1}=0.8499>0.837),$ see Remark 3.

$>$ ca(Q)$sv

[1] 0.8499 0.7979 0.7698 0.7590 0.6701 0.6654 0.6279 0.5914

$>$ ca(dataMatrix)$sv # dataMatrix = N

[1] 0.5879 0.4374 0.3898 0.3668 0.3459 0.3352 0.3232

###  7.4 Map interpretation of the Milazzese data set

CA is a dimension reduction method for exploratory visualization of non-
negative data, such as counts or abundances. CA has body and soul: the body
resides in the production of maps, mostly two-dimensional; the soul resides in
the interpretation of the maps. We consider the maps and their interpretation
as the essential aspects of CA.

Given that interpretability of a map is an essential aspect in CA, we classify
it into two kinds: objective and subjective. An objective interpretation of a
map is that it reflects meta-information; that is, an external covariate (or
covariates) which is (or are) not included in the computations.

The Punta Milazzese of Panarea data set represents the abundances of 31
artifacts found in 19 huts, excavated at the Middle Bronze age settlement at
Punta Milazzese on the island of Panarea (Aeolian Archipelago, Italy).
Furthermore, two complementary pieces of meta-information (2 external
covariates) on the huts are also available from Alberti (2013a). First, their
geographic locations displayed in Figure 1. Second, their surface areas in
$m^{2}$, which we include in the labelling of the huts; for example, 10M1
means hut number 1 has $10m^{2}$ of surface area, and M14? means the surface
area of hut number 14 is unknown and ? means it has very low marginal count (a
rare observation less than 5 counts). Furthermore, Alberti (2013a) suggested
two functional uses of the huts: residential (surface area larger than
15$m^{2}$ is sufficient for a couple) or utilitarian.

The aim of using CA and TCA is to help discover meaningful patterns and
clusters of the huts and the artifacts, from which some useful information may
be inferred on the past activities of the settlement. Given that 58% of the
cells are zero counts, TCA produced more interpretable results than CA (not
shown) of the original counts data N as shown in Figure 2 copied from Gauthier
and Choulakian (2015). TCA map of the huts admits an objective interpretation:
Globally it reflects the two complementary meta-information provided by
Alberti (2013a) stated above: the geographical locations east-west and the
surface areas; all huts on the right side of Figure 2 have surface areas less
than $15m^{2}.$

Figure 3 displays mfCA: The Guttman effect (parabola) is evident, and which
reflects the meta-information: on the right-side the huts have surface areas
less than 15m${}^{2}.$ However, the second dimension is dominated by the row
28 which has only 1 count and which intersects the column 17M3. So, mfCA is
also sensitive to rare points.

Note that TCA plot (not shown) of the bistochastic matrix Q is not objectively
interpretable.

##  8 Conclusion

Two kinds of transformations render CA scale-marginal free, the power
transformation for positive data as in Greenacre’s Theorem and by double-
scaling for positive and non-negative data first studied quite in detail by
Mosteller (1968) in his presidential address, and later on outlined by Goodman
(1996). We highlighted the important notion of bistochasticity of the two
transformations.

We think that generalized or multiple mfCA of extremely sparse and large data
sets, such as textual or networks data, can be useful: This will be pursued
on.

    Acknowledgements.

Choulakian’s research has been supported by NSERC of Canada.

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00footnotetext: $\bf{2020\ Mathematics\ Subject\ Classification\>}.$ 53D15,
53C25.
Key words and phrases: Almost Co-Kähler manifold; $(\kappa,\mu)$-almost co-
Kähler manifold; compact manifold; $(m,\rho)$-quasi-Einstein soliton.

# almost co-Kähler manifolds and $(m,\rho)$-quasi-Einstein solitons

Krishnendu De, Mohammad Nazrul Islam Khan∗ and Uday Chand De  Department of
Mathematics, Kabi Sukanta Mahavidyalaya, The University of Burdwan.
Bhadreswar, P.O.-Angus, Hooghly, Pin 712221, West Bengal, India. ORCID iD:
https://orcid.org/0000-0001-6520-4520<EMAIL_ADDRESS>Department of
Computer Engineering, College of Computer, Qassim University, Buraydah 51452,
Saudi Arabia<EMAIL_ADDRESS>Department of Pure Mathematics, University
of Calcutta, West Bengal, India. ORCID iD:
https://orcid.org/0000-0002-8990-4609<EMAIL_ADDRESS>
###### Abstract.

The present paper aims to investigate $(m,\rho)$-quasi-Einstein metrices on
almost co-Kähler manifolds $\mathcal{M}$. It is proven that if a
$(\kappa,\mu)$-almost co-Kähler manifold with $\kappa<0$ is $(m,\rho)$-quasi-
Einstein manifold, then $\mathcal{M}$ represents a $N(\kappa)$-almost co-
Kähler manifold and the manifold is locally isomorphic to a solvable non-
nilpotent Lie group. Next, we study the three dimensional case and get the
above mentioned result along with the manifold $\mathcal{M}^{3}$ becoming an
$\eta$-Einstein manifold. We also show that there does not exist
$(m,\rho)$-quasi-Einstein structure on a compact $(\kappa,\mu)$-almost co-
Kähler manifold of dimension greater than three with $\kappa<0$. Further, we
prove that an almost co-Kähler manifold satisfying $\eta$-Einstein condition
with constant coefficients reduces to a $K$-almost co-Kähler manifold,
provided $ma_{1}\neq(2n-1)b_{1}$ and $m\neq 1$. We also characterize perfect
fluid spacetime whose Lorentzian metric is equipped with $(m,\rho)$-quasi
Einstein solitons and acquired that the perfect fluid spacetime has vanishing
vorticity, or it represents dark energy era under certain restriction on the
potential function. Finally, we construct an example of an almost co-Kähler
manifold with $(m,\rho)$-quasi-Einstein solitons.

∗ Corresponding author

##  1\. Introduction

In [5], Boothby and Wang investigated an odd-dimensional differentiable
manifold in 1958 with the help of almost contact and contact structure and
investigated its features from a topological perspective. Making use of tensor
calculus, Sasaki [31] described the characteristics of a differentiable
manifold which is odd-dimensional, with contact structures in 1960. Such
manifolds were referred to as contact manifolds. After that, other researchers
have discovered various types of contact manifolds and investigated their
characteristics. The Kenmotsu, the Sasakian, and the co-Kähler manifolds are
Tanno’s divisions of the almost contact metric manifolds [33], whose
automorphism groups have the highest dimensions. It should be noted that the
cosymplectic manifolds that Blair [4] introduced and Goldberg and Yano [24]
examined are nothing but the co-Kähler manifolds. The co-Kähler manifolds can
be thought of from some topological perspectives as the analogue of Kähler
manifolds in odd-dimension, that is why the new terminology has been
introduced. The characteristics of almost co-Kähler manifolds, which are an
extension of co-Kähler manifolds, were examined by several researchers.
Perrone [29] provided a thorough classification of the three-dimensional
homogeneous almost co-Kähler manifolds as well as a local characterization of
these manifolds under the assumption of local symmetry. We advise readers to
read ([12], [32], [34]-[36]) and the references therein for more information
regarding almost co-Kähler manifolds.

In both mathematics and physics, Einstein manifolds are crucial. In Riemannian
and Semi-Riemannian geometry, it is interesting to investigate Einstein
manifolds and their generalizations. Numerous generalizations of Einstein
manifolds have been developed recently, including quasi-Einstein manifolds
[19], generalized quasi-Einstein manifolds [9], $m$-quasi-Einstein manifolds
[35], $(m,\rho)$-quasi-Einstein manifolds [25], and many more.

A Riemannian metric $g$ of an almost co-Kähler manifold is named a Ricci
soliton [8] if there is a $\lambda_{1}\in\mathbb{R}=$ constant, $\mathbb{R}$
is the set of all real numbers and a smooth vector field $X$ such that

$Ric+\frac{1}{2}\pounds_{X}g=\lambda_{1}g,$ (1.1)

in which $Ric$ indicates the Ricci tensor of the metric tensor $g$ and
$\pounds$ denotes the Lie-derivative. If $\lambda_{1}$ is a smooth function,
then the above soliton is called an almost Ricci soliton [21].

If we choose a smooth function $\omega:\mathcal{M}\to\mathbb{R}$ with
$X=D\omega$, $D$ being the gradient operator of $g$, then it is named a
gradient Ricci soliton. Hence, the equation (1.1) reduces to

$Ric+Hess\,\omega=\lambda_{1}g,$ (1.2)

where $Hess$ is the Hessian operator.

A Ricci soliton is nothing but a natural generalization of Einstein metric
[1]. In this paper we study $(m,\rho)$-quasi-Einstein solitons which are the
generalization of Einstein solitons and gradient Ricci solitons.

In an almost co-Kähler manifold the metric $g$, is named a generalized quasi-
Einstein soliton if there exist $\omega$, $\alpha$ and $\beta$ (smooth
functions) such that

$Ric+Hess\,\omega-\alpha d\omega\otimes d\omega=\beta g.$ (1.3)

In particular, if $\beta\in\mathbb{R}$ and $\alpha=0$ , then the foregoing
soliton reduces to a gradient Ricci soliton and $m$-quasi-Einstein soliton
[2], if $\alpha=\frac{1}{m}$ and $\beta\in\mathbb{R}$, where $m\in\mathbb{N}$.
Catino [9] presented the idea of a generalized quasi-Einstein soliton, and
Huang and Wei [25] further proposed taking into consideration the idea of a
$(m,\rho)$-quasi-Einstein-soliton as its special case.

######  Definition 1.1.

In an almost co-Kähler manifold the metric $g$ is named $(m,\rho)$-quasi-
Einstein soliton if there exists a smooth function
$\omega:\mathcal{M}^{n}\to\mathcal{R}$ and $m,\rho,\lambda_{1}\in\mathbb{R}=$
constant with $0<m\leq\infty$ such that

$Ric+Hess\,\omega-\frac{1}{m}d\omega\otimes d\omega=\beta
g=(\rho\tau+\lambda_{1})g,$ (1.4)

where $\tau$ denotes the scalar curvature.

Obviously, a $(\infty,0)$-quasi-Einstein soliton is a gradient Ricci soliton.
In [15], De and De investigated $(m,\rho)$-quasi-Einstein solitons in the
frame-work of paracontact manifolds. Demirbe$\breve{{\rm g}}$ and Güler [20]
studied rigidity of $(m,\rho)$-quasi-Einstein manifolds. As the study of
almost co-Kähler manifolds with $(m,\rho)$-quasi-Einstein solitons is still
pending, we want to fill this gap in this paper. Precisely, we establish the
subsequent theorems:

######  Theorem 1.1.

If the metric of a $(\kappa,\mu)$-almost co-Kähler manifold
$\mathcal{M}^{2n+1}$, $n>1$, with $\kappa<0$ is a $(m,\rho)$-quasi-Einstein
metric, then $\mathcal{M}$ represents a $N(\kappa)$-almost co-Kähler manifold.
Also, the manifold is locally isomorphic to a solvable non-nilpotent Lie-group
$\mathcal{G}_{\sigma}$ with the almost co-Kähler structure
$(\eta,\zeta,\varphi,g)$, where $\sigma=\sqrt{-\kappa}$.

######  Theorem 1.2.

Let the metric of a $(\kappa,\mu)$-almost co-Kähler manifold $\mathcal{M}^{3}$
with $\kappa<0$ is a $(m,\rho)$-quasi-Einstein metric. Then $\mathcal{M}$
represents a $N(\kappa)$-almost co-Kähler manifold. Also, $\mathcal{M}^{3}$ is
locally isomorphic to $\mathcal{G}_{\sigma}$ with the almost co-Kähler
structure, where $\sigma=\sqrt{-\kappa}$ and $\mathcal{M}^{3}$ becomes an
$\eta$-Einstein manifold.

######  Theorem 1.3.

There does not exist $(m,\rho)$-quasi-Einstein structures
$(g,\omega,\lambda_{1})$ with $D\omega=(\zeta\omega)\zeta$ on a compact
$(\kappa,\mu)$-almost co-Kähler manifold with $\kappa<0$.

######  Theorem 1.4.

If an almost co-Kähler manifold $\mathcal{M}^{2n+1}$ satisfying
$\eta$-Einstein condition $Ric=a_{1}g+b_{1}\eta\otimes\eta,\,b_{1}\neq 0$ with
constant coefficients is $(m,\rho)$-quasi-Einstein manifold and $m\neq 1$,
then $\mathcal{M}^{2n+1}$ is a $K$-almost co-Kähler manifold, provided
$ma_{1}\neq(2n-1)b_{1}$.

Solitons are actually waves that physically propagate with some energy loss
and maintain their speed and shape after colliding with another wave of a
similar kind. Solitons are crucial in the solution of initial-value problems
in nonlinear partial differential equations describing wave propagation.
Additionally, it describes the Fermi-Pasta-Ulam system’s recurrence [26].

Interest in studying Ricci solitons and their generalizations in different
geometrical contexts has also considerably increased due to their connection
to general relativity. Recently, in perfect fluid spacetimes, many authors
investigated many type of solitons like Ricci solitons [17], gradient Ricci
solitons ([16], [17]), Yamabe solitons [18], gradient Yamabe solitons[16],
gradient m-quasi Einstein solitons[16], gradient $\eta$-Einstein
solitons([17]), gradient Schouten solitons([17]), respectively. Also, physical
applications of almost Ricci solitons are investigated in ([21], [22]).
Motivated by the above studies, in this article, we intend to investigate the
$(m,\rho)$-quasi-Einstein solitons in perfect fluid spacetimes. As a result,
we establish the following theorems.

######  Theorem 1.5.

If a perfect fluid spacetime of dimension $n\geq 4$ admits a $(m,\rho)$-quasi
Einstein soliton with the scalars $\alpha_{1}$, $\beta$ and the potential
function $\omega$ remain invariant under the velocity vector field $\rho$,
then either the perfect fluid spacetime has vanishing vorticity or, the
spacetime represents dark energy era.

##  2\.  almost co-Kähler manifolds

$\mathcal{M}^{2n+1}$, a smooth manifold is named an almost contact metric
manifold if there exist a vector field $\zeta$, a tensor field $\varphi$, a
one-form $\eta$ and the Riemannian metric $g$ satisfying

$\varphi^{2}+I=\eta\otimes\zeta,\quad\eta(\zeta)=1,$ (2.1)

$g(E_{1},F_{1})=g(\varphi E_{1},\varphi F_{1})+\eta(E_{1})\eta(F_{1})$ (2.2)

for all vector fields $E_{1},F_{1}$ on $\mathcal{M}^{2n+1}$. From (2.1), we
get $\eta\circ\varphi=0$ and $\varphi\zeta=0$.

To each $\mathcal{M}^{2n+1}$ there is an associated 2-form
$\Phi(E_{1},F_{1})=g(E_{1},\varphi F_{1})$ for all vector fields $E_{1},F_{1}$
on $\mathcal{M}^{2n+1}$. $\eta\wedge\Phi^{n}$ is non-vanishing everywhere and
hence $\mathcal{M}^{2n+1}$ is orientable.

The Riemannian curvature tensor $K$ is described by

$K(E_{1},F_{1})=[\nabla_{E_{1}},\nabla_{F_{1}}]-\nabla_{[E_{1},F_{1}]},$ (2.3)

in which $\nabla$ denotes the Riemannian connection.

The manifold $\mathcal{M}$ is named an almost co-Kähler manifold [24] if
$d\eta=0$ and $d\Phi=0$. In particular, an almost co-Kähler manifold is called
co-Kähler if it is normal, that is, $\nabla\Phi=0$. An almost co-Kähler
structure is said to be strictly almost co-Kähler if it is not a co-Kähler
structure.

On an almost co-Kähler manifold $\mathcal{M}^{2n+1}(\eta,\zeta,\varphi,g)$, we
set $2h=\pounds_{\zeta}\varphi$ and $h^{\prime}=h\circ\varphi$. It is known
that $h$ and $h^{\prime}$ are symmetric $(1,1)$-tensor fields and satisfy the
following relations ([27], [28], [29])

$h\zeta=0,\quad\varphi h+h\varphi=0,\quad{\rm tr}\,h=0={\rm tr}\,h^{\prime},$
(2.4) $\nabla_{\zeta}\varphi=0,\quad\nabla\zeta=h^{\prime},$ (2.5)
$\nabla_{\zeta}h=-h^{2}\varphi-\varphi\ell,$ (2.6)
$\varphi\ell\varphi-\ell=2h^{2},$ (2.7) $Ric(\zeta,\zeta)+{\rm tr}\,h^{2}=0,$
(2.8)

where $\ell=K(.,\zeta)\zeta$ indicates the Jacobi operator and ${\rm tr}$
denotes the trace.

######  Definition 2.1.

An almost co-Kähler manifold is named a $K$-almost co-Kähler manifold if the
characteristic vector field $\zeta$ is Killing.

It is known that, any co-Kähler manifold is a $K$-almost co-Kähler manifold,
but the converse is not true, in general. But it is true in 3-dimensinal
manifold.

######  Lemma 2.1.

[24] Any 3-dimensional almost co-Kähler manifold is co-Kähler if and only if
$\zeta$ is Killing.

##  3\.  $(\kappa,\mu)$\- almost co-Kähler manifolds

Let $\mathcal{M}^{2n+1}$ be an almost co-Kähler manifold. Then
$\mathcal{M}^{2n+1}$ is named a $(\kappa,\mu)$-almost co-Kähler manifold if
$\xi$ belongs to the $(\kappa,\mu)$-nullity distribution, that is,

$K(E_{1},F_{1})\xi=\kappa(\eta(F_{1})E_{1}-\eta(E_{1})F_{1})+\mu(\eta(F_{1})hE_{1}-\eta(E_{1})hF_{1})$
(3.1)

for all vector fields $E_{1},F_{1}$ on $\mathcal{M}^{2n+1}$ and
$(\kappa,\mu)\in\mathcal{R}^{2}$. Such a manifold was first presented by Endo
[23] and were generalized by Dacko and Olszak [14] to
$(\kappa,\mu,\nu)$-spaces. Replacing $F_{1}$ by $\xi$ in (3.1), we acquire
$\ell=-\kappa\varphi^{2}+\mu h$. Putting the value of $\ell$ in (2.7) and
using second equation of (2.4), we obtain

$h^{2}=\kappa\varphi^{2}.$ (3.2)

By (3.2), we easily see that $\kappa=0$ and $\kappa\leq 0$ if and only if
$\mathcal{M}^{2n+1}$ is a $K$-almost co-Kähler manifold. If $\kappa<0$, then
$\mathcal{M}^{2n+1}$ is a strictly almost co-Kähler manifold. In particular,
if $\mu=0$, then $\xi$ belongs to the $\kappa$-nullity distribution
$N(\kappa)$ described by

$\displaystyle N(\kappa):p\to N_{p}(\kappa)$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\{G_{1}\in
T_{p}(\mathcal{M}^{2n+1}):K(E_{1},F_{1})G_{1}$
$\displaystyle=\kappa(g(F_{1},G_{1})E_{1}-g(E_{1},G_{1})F_{1}),\,\forall\,E_{1},F_{1}\in
T_{p}(\mathcal{M}^{2n+1})\\},$

where $T_{p}(\mathcal{M}^{2n+1})$ is the tangent space of $\mathcal{M}^{2n+1}$
at $p\in\mathcal{M}^{2n+1}$ and investigated by Dacko [13]. According to
Theorem 4 of [13], we have

######  Theorem 3.1.

Let $\mathcal{M}^{2n+1}$ be an almost co-Kähler manifold with $\kappa<0$. Then
$\mathcal{M}^{2n+1}$ is locally isomorphic to a solvable non-nilpotent Lie
group $\mathcal{G}_{\sigma}$ with the almost co-Kähler structure
$(\eta,\xi,\varphi,g)$, where $\sigma=\sqrt{-\kappa}$ and underlying manifold
of $\mathcal{G}_{\sigma}$ is the space $\mathbb{R}^{2n+1}$.

######  Lemma 3.1.

[10] Let $\mathcal{M}^{2n+1}$ be a $(\kappa,\mu)$-almost co-Kähler manifold
with $\kappa<0$. Then the Ricci operator $Q$ is described by

$Q=2n\kappa\eta\otimes\xi+\mu h.$ (3.3)

Thought the paper we assume $(\kappa,\mu)$-almost co-Kähler manifolds in the
sense of Endo [23], that is $\kappa$ and $\mu$ are constant.

##  4\.  Proof of the results

Naturally, there arises a question regarding the existence of Theorem 3.1 for
$(\kappa,\mu)$-almost co-Kähler manifolds with $(m,\rho)$-quasi-Einstein
solitons. The affirmative answer of the above question is given by Theorem
1.1.

Proof of the Theorem 1.1. From (3.3), we infer that $\tau=2n\kappa={\rm
constant}$.
By virtue of (1.4) and (3.3), we obtain

$\nabla_{E_{1}}D\omega=(2n\kappa\rho+\lambda_{1})E_{1}-\mu
hE_{1}-2n\kappa\eta(E_{1})\zeta+\frac{1}{m}g(E_{1},D\omega)D\omega$ (4.1)

for all vector field $E_{1}$ on $\mathcal{M}^{2n+1}$. Taking covariant
derivative of (4.1) along $F_{1}$ and using (2.5), we get

$\displaystyle\nabla_{F_{1}}\nabla_{E_{1}}D\omega$ $\displaystyle=$
$\displaystyle(2n\kappa\rho+\lambda_{1})\nabla_{F_{1}}E_{1}-\mu\nabla_{F_{1}}(hE_{1})$
(4.2) $\displaystyle-$ $\displaystyle
2n\kappa(F_{1}\eta(E_{1}))\zeta-2n\kappa\eta(E_{1})h^{\prime}F_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{m}[g(\nabla_{F_{1}}E_{1},D\omega)D\omega+g(E_{1},\nabla_{F_{1}}D\omega)D\omega$
$\displaystyle+$ $\displaystyle g(E_{1},D\omega)\nabla_{F_{1}}D\omega].$

Interchanging $E_{1}$ and $F_{1}$ in the above equation, we have

$\displaystyle\nabla_{E_{1}}\nabla_{F_{1}}D\omega$ $\displaystyle=$
$\displaystyle(2n\kappa\rho+\lambda_{1})\nabla_{E_{1}}F_{1}-\mu\nabla_{E_{1}}(hF_{1})$
(4.3) $\displaystyle-$ $\displaystyle
2n\kappa(E_{1}\eta(F_{1}))\zeta-2n\kappa\eta(F_{1})h^{\prime}E_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{m}[g(\nabla_{E_{1}}F_{1},D\omega)D\omega+g(F_{1},\nabla_{E_{1}}D\omega)D\omega$
$\displaystyle+$ $\displaystyle g(F_{1},D\omega)\nabla_{E_{1}}D\omega].$

Using (4.1), (4.2) and (4.3) in (2.3), we infer that

$\displaystyle K(E_{1},F_{1})D\omega$ $\displaystyle=$
$\displaystyle-\mu(\nabla_{E_{1}}h)F_{1}+\mu(\nabla_{F_{1}}h)E_{1}$ (4.4)
$\displaystyle-$ $\displaystyle
2n\kappa(\eta(F_{1})h^{\prime}E_{1}-\eta(E_{1})h^{\prime}F_{1})$
$\displaystyle+$
$\displaystyle\frac{2n\kappa\rho+\lambda_{1}}{m}((F_{1}\omega)E_{1}-(E_{1}\omega)F_{1})$
$\displaystyle-$
$\displaystyle\frac{\mu}{m}((F_{1}\omega)hE_{1}-(E_{1}\omega)hF_{1})$
$\displaystyle-$
$\displaystyle\frac{2n\kappa}{m}((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1}))\zeta.$

Taking inner product of the previous equation with $\zeta$ and utilizing
(3.2), we obtain

$\displaystyle g(K(E_{1},F_{1})D\omega,\zeta)=2\mu\kappa g(E_{1},\varphi
F_{1})+\frac{2n\kappa(\rho-1)+\lambda_{1}}{m}((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1})).$
(4.5)

From (3.1) and (4.5), it follows that

$\displaystyle\kappa((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1}))+\mu(g(hF_{1},D\omega)\eta(E_{1})-g(hE_{1},D\omega)\eta(F_{1}))$
(4.6) $\displaystyle=$ $\displaystyle 2\mu\kappa g(E_{1},\varphi
F_{1})+\frac{2n\kappa(\rho-1)+\lambda_{1}}{m}((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1})).$

Replacing $E_{1}$ by $\varphi E_{1}$ and $F_{1}$ by $\varphi F_{1}$ in the
foregoing equation, we get

$0=2\mu\kappa g(E_{1},\varphi F_{1})$

which implies that $\mu=0$, because $\kappa<0$.

If $\mu=0$, the $(\kappa,\mu)$-almost co-Kähler manifold reduces to a
$N(\kappa)$-almost co-Kähler manifold.

Here $\zeta$ belongs to $\kappa$-nullity distribution with $\kappa<0$. By
Theorem 3.1 $\mathcal{M}^{2n+1}$ is locally isomorphic to a solvable non-
nilpotent Lie group $\mathcal{G}_{\sigma}$, where $\sigma=\sqrt{-\kappa}$.

Thus the proof is completed.

It is well known that on any 3-dimensional manifold, the curvature tensor is
described by

$\displaystyle K(E_{1},F_{1})G_{1}$ $\displaystyle=$ $\displaystyle
Ric(F_{1},G_{1})E_{1}-Ric(E_{1},G_{1})F_{1}+g(F_{1},G_{1})QE_{1}-g(E_{1},G_{1})QF_{1}$
(4.7) $\displaystyle-$
$\displaystyle\frac{\tau}{2}(g(F_{1},G_{1})E_{1}-g(E_{1},G_{1})F_{1})$

for all vector fields $E_{1},F_{1},G_{1}$ on $\mathcal{M}^{3}$.

Taking trace of (3.1), we get $Q\zeta=2\kappa\zeta$. Putting
$F_{1}=G_{1}=\zeta$ in (4.7) and using $\ell=-\kappa\varphi^{2}+\mu h$, we
obtain

$Q=\left(\frac{\tau}{2}-\kappa\right)I+\left(3\kappa-\frac{\tau}{2}\right)\eta\otimes\zeta+\mu
h.$ (4.8)

This is the expression of the Ricci operator $Q$ on any 3-dimensional
$(\kappa,\mu)$-almost co-Kähler manifold.

Proof of the Theorem 1.2. Using (4.8) in (1.4), we get

$\displaystyle\nabla_{E_{1}}D\omega$ $\displaystyle=$
$\displaystyle\left(\rho\tau+\lambda_{1}-\frac{\tau}{2}+\kappa\right)E_{1}+\left(\frac{\tau}{2}-3\kappa\right)\eta(E_{1})\zeta$
$\displaystyle-$ $\displaystyle\mu hE_{1}+\frac{1}{m}g(E_{1},D\omega)D\omega$

for all $E_{1}$ on $\mathcal{M}^{3}$. By direct computation, we have

$\displaystyle K(E_{1},F_{1})D\omega$ $\displaystyle=$
$\displaystyle\left(\rho-\frac{1}{2}\right)(E_{1}\tau)F_{1}-\left(\rho-\frac{1}{2}\right)(F_{1}\tau)E_{1}$
(4.9) $\displaystyle+$
$\displaystyle\frac{1}{2}(E_{1}\tau)\eta(F_{1})\zeta-\frac{1}{2}(F_{1}\tau)\eta(E_{1})\zeta$
$\displaystyle+$
$\displaystyle\left(\frac{\tau}{2}-3\kappa\right)(\eta(F_{1})h^{\prime}E_{1}-\eta(E_{1})h^{\prime}F_{1})$
$\displaystyle-$
$\displaystyle-\mu(\nabla_{E_{1}}h)F_{1}+\mu(\nabla_{F_{1}}h)E_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{m}\left(\rho\tau+\lambda_{1}-\frac{\tau}{2}+\kappa\right)((F_{1}\omega)E_{1}-(E_{1}\omega)F_{1})$
$\displaystyle+$
$\displaystyle\frac{1}{m}\left(\frac{\tau}{2}-3\kappa\right)((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1}))$
$\displaystyle-$
$\displaystyle\frac{\mu}{m}((F_{1}\omega)h(E_{1})-(E_{1}\omega)h(F_{1}))$

for all $E_{1},F_{1}$ on $\mathcal{M}^{3}$. Taking inner product of (4.9) with
$\zeta$, we obtain

$\displaystyle g(K(E_{1},F_{1})Df,\zeta)$ $\displaystyle=$
$\displaystyle\rho(E_{1}\tau)\eta(F_{1})-\rho(F_{1}\tau)\eta(E_{1})+2\mu\kappa
g(E_{1},\varphi F_{1})$ (4.10) $\displaystyle+$
$\displaystyle\frac{1}{m}(\rho\tau+\lambda_{1}-2\kappa)((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1})),$

where we have used (3.2). By virtue of (3.1) and (4.10), we have

$\displaystyle\kappa((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1}))+\mu(g(hF_{1},Df)\eta(E_{1})-g(hE_{1},Df)\eta(F_{1}))$
(4.11) $\displaystyle=$
$\displaystyle\rho(E_{1}\tau)\eta(F_{1})-\rho(F_{1}\tau)\eta(E_{1})+2\mu\kappa
g(E_{1},\varphi F_{1})$ $\displaystyle+$
$\displaystyle\frac{1}{m}(\rho\tau+\lambda_{1}-2\kappa)((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1})).$

Now, replacing $E_{1}$ and $F_{1}$ by $\varphi E_{1}$ and $\varphi F_{1}$,
respectively, in (4.11) we get

$0=2\mu\kappa g(E_{1},\varphi F_{1}),$

which implies that $\mu=0$, because $\kappa<0$.

If $\mu=0$, the $(\kappa,\mu)$-almost co-Kähler manifold turns into a
$N(\kappa)$-almost co-Kähler manifold.

By Theorem 3.1 $\mathcal{M}^{3}$ is locally isometric to a solvable non-
nilpotent Lie group $\mathcal{G}_{\sigma}$, where $\sigma=\sqrt{-\kappa}$.
Since $\mu=0$, from (4.8) it follows that $\mathcal{M}^{3}$ is an
$\eta$-Einstein manifold.

Hence the proof.

Proof of the Theorem 1.3. From (3.3), we acquire that the scalar curvature
$\tau=2n\kappa$. Differentiating $D\omega=(\zeta\omega)\zeta$ along the
arbitrary vector field $E_{1}$, we get

$\nabla_{E_{1}}D\omega=(E_{1}(\zeta\omega))\zeta+(\zeta\omega)h^{\prime}E_{1}.$
(4.12)

By (3.3) and (4.12), equation (1.4) takes the form

$\displaystyle 2n\kappa\eta(E_{1})\eta(F_{1})+\mu
g(hE_{1},F_{1})+(E_{1}(\zeta\omega))\eta(F_{1})+(\zeta\omega)g(h^{\prime}E_{1},F_{1})$
(4.13) $\displaystyle-$
$\displaystyle\frac{1}{m}(\zeta\omega)\eta(E_{1})\eta(F_{1})=(2n\kappa\rho+\lambda_{1})g(E_{1},F_{1})$

for all vector fields $E_{1},F_{1}$ on $\mathcal{M}^{2n+1}$. Replacing $E_{1}$
by $\varphi E_{1}$ and $F_{1}$ by $\varphi F_{1}$, we obtain

$-\mu g(hE_{1},F_{1})-(\zeta
f)g(h^{\prime}E_{1},F_{1})=(2n\kappa\rho+\lambda_{1})g(\varphi E_{1},\varphi
F_{1}).$

Contracting the preceding equation, we get

$2n\kappa\rho+\lambda_{1}=0.$ (4.14)

Setting $E_{1}=F_{1}=\zeta$ in (4.13) and using (4.14), we have

$2n\kappa+\zeta(\zeta\omega)-\frac{1}{m}(\zeta\omega)^{2}=0.$ (4.15)

Contracting in (4.12), we find

$\Delta\omega+\zeta(\zeta\omega)=0,$ (4.16)

where $\Delta=-{\rm div}\,D$ is the Laplacian operator. Using (4.16) in
(4.15), we obtain

$\Delta\omega=2n\kappa-\frac{1}{m}(\zeta\omega)^{2}.$

By divergence theorem

$\frac{1}{m}\int_{\mathcal{M}}(\zeta\omega)^{2}d\mathcal{M}=2n\kappa\int_{\mathcal{M}}d\mathcal{M}.$

$d\mathcal{M}$ represents the volume form of the manifold and is positive,
since $\mathcal{M}$ is orientable. Therefore, the right hand side is negative,
because $\kappa<0$. Thus, the above relation does not hold.

This completes the proof.

$\eta$-Einstein almost co-Kähler manifolds:

An almost co-Kähler manifold $(\mathcal{M}^{2n+1},g)$ is named an
$\eta$-Einstein manifold if

$Ric=a_{1}g+b_{1}\eta\otimes\eta,$ (4.17)

where $a_{1}$ and $b_{1}$ are smooth functions. For $b_{1}=0$, it reduces to
an Einstein manifold. In particular, if both $a_{1}$ and $b_{1}$ are constant,
then we say that $\mathcal{M}^{2n+1}$ satisfies $\eta$-Einstein condition with
constant coefficients. From (2.8), we have

$a_{1}+b_{1}=-{\rm tr}\,h^{2}.$ (4.18)

The $\eta$-Einstein condition on an almost co-Kähler manifold implies that
$\zeta$ is a harmonic vector field [30].

Proof of the Theorem 1.4. From (4.17), we infer that scalar curvature
$\tau=(2n+1)a_{1}+b_{1}={\rm constant}$. By virtue of (4.17), we obtain from
(1.4)

$\nabla_{E_{1}}D\omega=(\rho\tau+\lambda_{1}-a_{1})E_{1}-b_{1}\eta(E_{1})\zeta+\frac{1}{m}g(E_{1},D\omega)D\omega$
(4.19)

for all vector field $E_{1}$ on $\mathcal{M}^{2n+1}$. Using this we compute

$\displaystyle K(E_{1},F_{1})D\omega$ $\displaystyle=$ $\displaystyle-
b_{1}\eta(F_{1})h^{\prime}E_{1}+b_{1}\eta(E_{1})h^{\prime}F_{1}$ (4.20)
$\displaystyle+$
$\displaystyle\frac{1}{m}(\rho\tau+\lambda_{1}-a_{1})((F_{1}\omega)E_{1}-(E_{1}\omega)F_{1})$
$\displaystyle-$
$\displaystyle\frac{b_{1}}{m}((F_{1}\omega)\eta(E_{1})-(E_{1}\omega)\eta(F_{1}))\zeta.$

Contracting (4.20), we get

$Ric(F_{1},D\omega)=\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}(F_{1}\omega)-\frac{b_{1}}{m}((F_{1}\omega)-(\zeta\omega)\eta(F_{1})),$
(4.21)

where we have used ${\rm tr}\,h^{\prime}=0$. From (4.17) and (4.21), it can be
written as

$a_{1}(F_{1}\omega)+b_{1}(\zeta\omega)\eta(F_{1})=\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}(F_{1}\omega)-\frac{b_{1}}{m}((F_{1}\omega)-(\zeta\omega)\eta(F_{1})).$
(4.22)

Replacing $F_{1}$ by $\zeta$ in (4.22), we have

$\left(a_{1}+b_{1}-\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}\right)(\zeta\omega)=0.$

From the preceding equation, we infer either
$a_{1}+b_{1}-\displaystyle\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}=0$, or
$\zeta\omega=0$.

If $a_{1}+b_{1}-\displaystyle\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}=0$, from
(4.22) we can find

$\frac{b_{1}(1-m)}{m}((F_{1}\omega)-(\zeta\omega)\eta(F_{1}))=0,$

which implies $D\omega=(\zeta\omega)\zeta$, since $b_{1}\neq 0$ and $m\neq 1$.
Putting $D\omega=(\zeta\omega)\zeta$ in (4.19) and using (2.5), we get

$(E_{1}(\zeta\omega))\zeta+(\zeta\omega)h^{\prime}E_{1}=\frac{m(a_{1}+b_{1})}{2n}E_{1}-b_{1}\eta(E_{1})\zeta+\frac{1}{m}(\zeta\omega)^{2}\eta(E_{1})\zeta.$
(4.23)

Taking trace of (4.23), we have

$\zeta(\zeta\omega)=\frac{m(a_{1}+b_{1})}{2n}(2n+1)-b_{1}+\frac{1}{m}(\zeta\omega)^{2}.$
(4.24)

Setting $E_{1}=\zeta$ in (4.23) and after that taking inner product, we
acquire

$\zeta(\zeta\omega)=\frac{m(a_{1}+b_{1})}{2n}-b_{1}+\frac{1}{m}(\zeta\omega)^{2}.$
(4.25)

The foregoing two equations imply that $a_{1}+b_{1}=0$. From (4.18), ${\rm
tr}\,h^{2}=0$. Since $h$ is a symmetric tensor, $h=0$. Hence $\zeta$ is a
Killing vector field. Consequently, $\mathcal{M}^{2n+1}$ is a $K$-almost co-
Kähler manifold.

On the other hand if $\zeta\omega=0$, then from (4.22) we get

$\left(a_{1}-\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}+\frac{b_{1}}{m}\right)(F_{1}\omega)=0.$
(4.26)

If $F_{1}\omega=0$, then $\omega$ is a constant and from (1.4) we see that the
manifold is Einstein, which is a contradiction because $b_{1}\neq 0$.
Therefore, from (4.26) we observed that

$a_{1}-\frac{2n(\rho\tau+\lambda_{1}-a_{1})}{m}+\frac{b_{1}}{m}=0.$ (4.27)

Putting $E_{1}=\zeta$ in (4.19) and then taking inner product with $\zeta$, we
get

$\rho\tau+\lambda_{1}-a_{1}-b_{1}=0,$ (4.28)

where we have used $\nabla_{\zeta}\zeta=0$. From (4.27) and (4.28), it follows
that $ma_{1}=(2n-1)b_{1}$.

Thus the proof is finished.

Using the Lemma 2.1, we can state that

######  Corollary 4.1.

If an almost co-Kähler manifold $\mathcal{M}^{3}$ satisfies $\eta$-Einstein
condition $Ric=a_{1}g+b_{1}\eta\otimes\eta$, $b_{1}\neq 0$ with constant
coefficients is a $(m,\rho)$-quasi-Einstein manifold and $m\neq 1$, then
$\mathcal{M}^{3}$ is a co-Kähler manifold, provided $ma_{1}\neq b_{1}$.

By Lemma 5 of [13], if $\mathcal{M}^{2n+1}$ is an almost co-Kähler manifold
whose structure vector field $\zeta$ belongs to the $\kappa$-nullity
distribution with $\kappa<0$, then $\mathcal{M}$ satisfies $\eta$-Einstein
condition with constant coefficients with $a_{1}=0$ and $b_{1}=2n\kappa\neq
0$. Also $ma_{1}\neq(2n-1)b_{1}$. Thus, from Theorem 1.4, we can state that

######  Corollary 4.2.

There does not exist $(m,\rho)$-quasi-Einstein structure with $m\neq 1$ on an
almost co-Kähler manifold whose structure vector field belongs to
$\kappa$-nullity distribution with $\kappa<0$.

##  5\.  Applications to physics

A Lorentzian manifold $\mathcal{N}^{n}$ is called a perfect fluid spacetime if
its non-vanishing Ricci tensor $Ric$ obeys

$Ric=\alpha_{1}g+\beta_{1}A\otimes A,$ (5.1)

where $\alpha_{1}$, $\beta_{1}$ are scalar (not simultaneously zero) and for
all $E_{1}$, $g(E_{1},\rho)=A(E_{1})$, $\rho$ being the velocity vector field.
We know that in a perfect fluid spacetime, the vector field $\rho$ is unit
timelike, hence, we have $g(\rho,\rho)=-1$.

From (5.1), we get

$QE_{1}=\alpha_{1}E_{1}+\beta_{1}A(E_{1})\rho,$ (5.2)

where $Q$ is the Ricci operator defined by $g(QE_{1},F_{1})=Ric(E_{1},F_{1})$
and contracting $E_{1}$ in (5.2), we infer

$r=\sum_{i}\epsilon_{i}QE_{i}=n\alpha_{1}-\beta_{1},$ (5.3)

where $\\{E_{i}\\}$ is an orthonormal basis of the tangent space at each point
of the spacetime and $\epsilon_{i}=g(E_{i},E_{i})=\pm 1$. The covariant
derivative of (5.2) gives

$(\nabla_{E_{1}}Q)(F_{1})=E_{1}(\alpha_{1})F_{1}+E_{1}(\beta_{1})A(F_{1})\rho+\beta_{1}(\nabla_{E_{1}}A)(F_{1})\rho+\beta_{1}A(F_{1})\nabla_{E_{1}}\rho.$
(5.4)

If $k$ denotes the gravitational constant and $T$ indicates the energy
momentum tensor, then Einstein’s field equations in the absence of
cosmological constant have the structure

$Ric-\frac{r}{2}g=kT.$ (5.5)

In case of perfect fluid spacetime if $p$ indicates isotropic pressure and
$\sigma$ is the energy density, then $T$ is defined as

$T=(p+\sigma)A\otimes A+pg.$ (5.6)

The necessary and sufficient condition for the constant scalar curvature of a
perfect fluid spacetime is that $nE_{1}(\alpha_{1})=E_{1}(\beta_{1}).$
Combining the equations (5.1), (5.5) and (5.6), we infer that

$\alpha_{1}=\frac{k(p-\sigma)}{2-n},\;\beta_{1}=\kappa(p+\sigma).$ (5.7)

Furthermore, characterizing the particular sort of perfect fluid spacetimes,
$\sigma$ and $p$ are interconnected by the relation $p=\Omega\sigma$ in which
$\Omega$ is named the equation of state (EOS) parameter. If the state equation
is of the form $p=p(\sigma)$, then the perfect fluid spacetime is called
isentropic. Also, if $p=\sigma$, then the perfect fluid spacetime represents
stiff matter fluid[11]. The equation of state of stiff matter fluid was
introduced by Zeldovich [37] to describe a cold gas of baryons, and used in
his cosmological model.

A dust solution, in general relativity, is one in which the gravitational
field is totally produced by the mass, momentum, and stress density of a
perfect fluid with positive mass density but vanishing pressure. This is a
form of exact solution to the Einstein field equation. If $p=0$, perfect fluid
spacetime represents the dust matter fluid whereas the perfect fluid spacetime
represents the radiation era with $p-\frac{\sigma}{3}=0$ and the dark energy
era with $p=-\sigma$ [11].

Moreover, Dark energy era that satisfies the equation of state with
$\Omega<-1$ is known as phantom regime or phantom energy. It has positive
energy density but negative pressure, such that $p+\sigma<0$. The physical
consequences are explored in ([6], [7]).

Proof of the Theorem 1.5 .

Let the perfect fluid spacetime of dimension $n\geq 4$ admit a
$(m,\rho)$-quasi Einstein metric and at first, we prove the following result

######  Lemma 5.1.

Every perfect fluid spacetime of dimension $n\geq 4$ obeying $(m,\rho)$-quasi
Einstein soliton satisfies the following:

$\displaystyle
K(E_{1},F_{1})D\omega=(\nabla_{F_{1}}Q)E_{1}-(\nabla_{E_{1}}Q)F_{1}+\frac{\beta}{m}\\{F_{1}(\omega)E_{1}-E_{1}(\omega)F_{1}\\}$
$\displaystyle+\frac{1}{m}\\{E_{1}(\omega)QF_{1}-F_{1}(\omega)QE_{1}\\}+\\{(E_{1}\beta)F_{1}-(F_{1}\beta)E_{1}\\},$
(5.8)

for all $E_{1},\,F_{1}$.

###### Proof.

By hypothesis, the perfect fluid spacetime is endowed with $(m,\rho)$-quasi
Einstein metric. Therefore, using (1.4) we can write

$\nabla_{E_{1}}D\omega+QE_{1}=\frac{1}{m}g(E_{1},D\omega)D\omega+\beta E_{1}.$
(5.9)

Differentiating the equation (5.9) covariantly along $F_{1}$, we get

$\displaystyle\nabla_{F_{1}}\nabla_{E_{1}}D\omega$ $\displaystyle=$
$\displaystyle-\nabla_{F_{1}}QE_{1}+\frac{1}{m}\nabla_{F_{1}}g(E_{1},D\omega)D\omega$
(5.10)
$\displaystyle+\frac{1}{m}g(E_{1},D\omega)\nabla_{F_{1}}D\omega+\beta\nabla_{F_{1}}E_{1}+(F_{1}\beta)E_{1}.$

Exchanging $E_{1}$ and $F_{1}$ in (5.10), we lead

$\displaystyle\nabla_{E_{1}}\nabla_{F_{1}}D\omega$ $\displaystyle=$
$\displaystyle-\nabla_{E_{1}}QF_{1}+\frac{1}{m}\nabla_{E_{1}}g(F_{1},D\omega)D\omega$
(5.11)
$\displaystyle+\frac{1}{m}g(F_{1},D\omega)\nabla_{E_{1}}D\omega+\beta\nabla_{E_{1}}F_{1}+(E_{1}\beta)F_{1}$

and

$\nabla_{[E_{1},F_{1}]}D\omega=-Q[E_{1},F_{1}]+\frac{1}{m}g([E_{1},F_{1}],D\omega)D\omega+\beta[E_{1},F_{1}].$
(5.12)

Using (5.9)-(5.12) and the relation
$K(E_{1},F_{1})D\omega=\nabla_{E_{1}}\nabla_{F_{1}}D\omega-\nabla_{F_{1}}\nabla_{E_{1}}D\omega-\nabla_{[E_{1},F_{1}]}D\omega$,
we acquire

$\displaystyle K(E_{1},F_{1})D\omega$ $\displaystyle=$
$\displaystyle(\nabla_{F_{1}}Q)E_{1}-(\nabla_{E_{1}}Q)F_{1}+\frac{\beta}{m}\\{F_{1}(\omega)E_{1}-E_{1}(\omega)F_{1}\\}$
$\displaystyle+\frac{1}{m}\\{E_{1}(\omega)QF_{1}-F_{1}(\omega)QE_{1}\\}+\\{(E_{1}\beta)F_{1}-(F_{1}\beta)E_{1}\\}.$

∎

Utilizing the previous Lemma and the equations (5.2), (5.4), we infer

$\displaystyle
K(E_{1},F_{1})D\omega=E_{1}(\alpha_{1})F_{1}-F_{1}(\alpha_{1})E_{1}+\\{E_{1}(\beta_{1})A(F_{1})-F_{1}(\beta_{1})A(E_{1})\\}\rho$
$\displaystyle+\beta_{1}\\{(\nabla_{E_{1}}A)(F_{1})\rho+A(F_{1})\nabla_{E_{1}}\rho-\nabla_{F_{1}}A)(E_{1})\rho-A(E_{1})\nabla_{F_{1}}\rho\\}$
$\displaystyle+\frac{\beta}{m}\\{F_{1}(\omega)E_{1}-E_{1}(\omega)F_{1}\\}+\frac{1}{m}\\{\alpha_{1}E_{1}(\omega)F_{1}+\beta_{1}E_{1}(\omega)A(F_{1})\rho$
$\displaystyle-\alpha_{1}F_{1}(\omega)E_{1}-\beta_{1}F_{1}(\omega)A(E_{1})\rho\\}+\\{(E_{1}\beta)F_{1}-(F_{1}\beta)E_{1}\\}.$
(5.13)

Taking a set of orthonormal frame field and executing contraction of the
equation (5), we get

$\displaystyle Ric(E_{1},D\omega)$ $\displaystyle=$
$\displaystyle(1-n)E_{1}(\alpha_{1})+E_{1}(\beta_{1})+\rho(\beta_{1})A(E_{1})$
(5.14)
$\displaystyle+\beta_{1}\\{(\nabla_{E_{1}}A)(F_{1})-\nabla_{E_{1}}A)(\rho)+A(E_{1})div\rho\\}+\frac{\beta}{m}(n-1)(E_{1}\omega)$
$\displaystyle+\frac{1}{m}\\{\alpha_{1}E_{1}(\omega)+\beta_{1}\rho(\omega)A(E_{1})-n\alpha_{1}E_{1}(\omega)+\beta_{1}E_{1}(\omega)\\}$
$\displaystyle+(1-n)E_{1}(\beta).$

Setting $E_{1}=\rho$ in the last equation and (5.1), and then comparing both
the equations, we find

$\\{\frac{m}{1-n}(\alpha_{1}-\beta_{1})+\beta-\alpha_{1}\\}\rho(\omega)=m\\{(1-n)[\rho(\alpha_{1})+\rho(\beta)-\beta_{1}div\rho\\}.$
(5.15)

Let $\alpha_{1}$, $\beta$ and $\omega$ are invariant under the velocity vector
field $\rho$ which mean $\rho(\alpha_{1})=0$, $\rho(\beta)=0$ and
$\rho(\omega)=0$. Hence, the foregoing equation yields either, $div\rho=0$, or
$\beta_{1}=0$, since $m\neq 0$.

If $div\rho=0$, then the velocity vector field is conservative. Since a
conservative vector field is always irrotational, we get the vorticity of the
perfect fluid is zero.

If $\beta_{1}=0$, then we obtain that $\sigma+p=0$. This represents dark
energy era.

This finishes the proof.

##  6\. Example

Let $\mathcal{M}^{3}=\\{(x,y,z)\in\mathcal{R}^{3}:z>0\\}$, where $(x,y,z)$ are
the standard co-ordinate of $\mathcal{R}^{3}$. We choose the Riemannian metric
$g$ on $\mathcal{M}^{3}$ described by

$g=dx\otimes dx+dy\otimes dy+\frac{x^{2}+y^{2}+e^{-2z}}{4z}dz\otimes
dz-\frac{y}{\sqrt{z}}dx\otimes dz-\frac{x}{\sqrt{z}}dy\otimes dz.$

Let

$\delta_{1}=\frac{\partial}{\partial
x},\quad\delta_{2}=\frac{\partial}{\partial
y},\quad\delta_{3}=ye^{z}\frac{\partial}{\partial
x}+xe^{z}\frac{\partial}{\partial y}+2\sqrt{z}e^{z}\frac{\partial}{\partial
z}.$

Then, the vector fields $\delta_{1},\delta_{2},\delta_{3}$ are orthonormal.

We have

$[\delta_{1},\delta_{2}]=0,\quad[\delta_{1},\delta_{3}]=e^{z}\delta_{2},\quad[\delta_{2},\delta_{3}]=e^{z}\delta_{1}.$

We define $\eta$, $\zeta$ and $\varphi$ by

$\eta=\frac{e^{-z}}{2\sqrt{z}}\,dz,\quad\zeta=\delta_{3},\quad\varphi\delta_{1}=\delta_{2},\quad\varphi\delta_{2}=-\delta_{1},\quad\varphi\delta_{3}=0.$

We can easily verify that, $\eta$ and $\Phi$ are closed. Hence,
$\mathcal{M}^{3}$ is an almost co-Kähler manifold.

Using Koszul’s formula we get

$\nabla_{\delta_{1}}\delta_{1}=0,\quad\nabla_{\delta_{1}}\delta_{2}=-e^{z}\delta_{3},\quad\nabla_{\delta_{1}}\delta_{3}=e^{z}\delta_{2},$
$\nabla_{\delta_{2}}\delta_{1}=-e^{z}\delta_{3},\quad\nabla_{\delta_{2}}\delta_{2}=0,\quad\nabla_{\delta_{2}}\delta_{3}=e^{z}\delta_{1},$
$\nabla_{\delta_{3}}\delta_{1}=0,\quad\nabla_{\delta_{3}}\delta_{2}=0,\quad\nabla_{\delta_{3}}\delta_{3}=0.$

The components of the curvature tensor $K$ are given by

$K(\delta_{1},\delta_{2})\delta_{1}=-e^{2z}\delta_{2},\quad
K(\delta_{1},\delta_{2})\delta_{2}=e^{2z}\delta_{1},\quad
K(\delta_{1},\delta_{2})\delta_{3}=0,$
$K(\delta_{1},\delta_{3})\delta_{1}=e^{2z}\delta_{3},\quad
K(\delta_{1},\delta_{3})\delta_{2}=2\sqrt{z}e^{2z}\delta_{3},$
$K(\delta_{1},\delta_{3})\delta_{3}=-e^{2z}\delta_{1}-2\sqrt{z}e^{2z}\delta_{2},\quad
K(\delta_{2},\delta_{3})\delta_{1}=2\sqrt{z}e^{2z}\delta_{3},$
$K(\delta_{2},\delta_{3})\delta_{2}=e^{2z}\delta_{3},\quad
K(\delta_{2},\delta_{3})\delta_{3}=-2\sqrt{z}e^{2z}\delta_{1}-e^{2z}\delta_{2}.$

Utilizing the foregoing expressions of $K$, we compute the Ricci operator $Q$
by

$Q\delta_{1}=-2\sqrt{z}e^{2z}\delta_{2},\quad
Q\delta_{2}=-2\sqrt{z}e^{2z}\delta_{1},\quad Q\delta_{3}=-2e^{2z}\delta_{3}.$
(6.1)

The tensor field $h$ is given by

$h\delta_{1}=-e^{z}\delta_{1},\quad h\delta_{2}=e^{z}\delta_{2},\quad
h\delta_{3}=0.$ (6.2)

Suppose that $\omega=z$. Then, $D\omega=2\sqrt{z}e^{z}\delta_{3}$. By directed
computation, we have

$\begin{cases}\nabla_{\delta_{1}}D\omega=2\sqrt{z}e^{2z}\delta_{2},\\\
\nabla_{\delta_{2}}D\omega=2\sqrt{z}e^{2z}\delta_{1},\\\
\nabla_{\delta_{3}}D\omega=2e^{2z}(1+2z)\delta_{3}.\end{cases}$ (6.3)

From (6.1) and (6.3), we can verify that

$\begin{cases}Q\delta_{1}+\nabla_{\delta_{1}}D\omega-g(\delta_{1},D\omega)D\omega=0,\\\
Q\delta_{2}+\nabla_{\delta_{2}}D\omega-g(\delta_{2},D\omega)D\omega=0,\\\
Q\delta_{3}+\nabla_{\delta_{3}}D\omega-g(\delta_{3},D\omega)D\omega=0.\end{cases}$

Thus for $m=1,\,\rho=\lambda_{1}=0$, equation (1.4) is satisfied. Hence $M$ is
an $(m,\rho)$-quasi-Einstein manifold.

##  7\. Conclusion

Einstein solitons are essential in both mathematics and physics. It is
intriguing to look into Einstein solitons and their generalizations in
Riemannian and Semi-Riemannian geometry. Recent years have seen the
development of many generalizations of Einstein solitons, including quasi-
Einstein solitons, generalized quasi-Einstein solitons, $m$-quasi-Einstein
solitons, $(m,\rho)$-quasi-Einstein solitons. In this study, we investigate
the $(m,\rho)$-quasi-Einstein solitons in almost co-Kähler manifolds.

Here we show that an almost co-Kähler manifold admitting $(m,\rho)$-quasi-
Einstein solitons is locally isomorphic to a solvable non-nilpotent Lie group.
We also establish that there does not exist $(m,\rho)$-quasi-Einstein
structure on a compact $(\kappa,\mu)$-almost co-Kähler manifold. Finally, we
construct an example of an almost co-Kähler manifold with $(m,\rho)$-quasi-
Einstein solitons.

Further study of $(m,\rho)$-quasi-Einstein solitons in the context of Hopf
manifolds, as well as in the general theory of relativity and cosmology, is
possible in the near future.

##  8\. Acknowledgement

We would like to thank the Referees and the Editor for reviewing the paper
carefully and their valuable comments to improve the quality of the paper.

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