# Smart windows passively driven by greenhouse effect

G. Boudan  Laboratoire Charles Fabry, UMR 8501, Institut d’Optique, CNRS,
Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127 Palaiseau Cedex,
France.  Thales Research and Technology France, 1, Avenue Augustin Fresnel,
F-91767 Palaiseau, Cedex France.  E. Eustache  Thales Research and Technology
France, 1, Avenue Augustin Fresnel, F-91767 Palaiseau, Cedex France.  P.
Garabedian  Thales Research and Technology France, 1, Avenue Augustin Fresnel,
F-91767 Palaiseau, Cedex France.  R. Messina  Laboratoire Charles Fabry, UMR
8501, Institut d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin
Fresnel, 91127 Palaiseau Cedex, France.  P. Ben-Abdallah
<EMAIL_ADDRESS>Laboratoire Charles Fabry, UMR 8501, Institut
d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127
Palaiseau Cedex, France.

(August 29, 2024)

###### Abstract

The rational thermal management of buildings is of major importance for the
reduction of the overall primary energy consumption. Smart windows are
promising systems which could save a significant part of this energy. Here we
introduce a double glazing system made with a thermochromic metal-insulator
transition material and a glass layer separated by an air gap which is able to
switch from its insulating to its conducting phase thanks to the greenhouse
effect occuring in the separation gap. We also show that this passive system
can reduce the incoming heat flux by $30\%$ in comparison with a traditional
double glazing while maintaining the transmittance around $0.35$ over $75\%$
of visible spectrum.

Nowadays buildings consume about $40\%$ of the primary energy used in the
world Ramesh . Therefore their thermal management is a major concern in the
context of current global challenge of energy efficiency. Smart windows could
play a major role to respond to this challenge. These windows can be used to
control heat flux exchanged between the indoor and external environnment. One
way to achieve this control is by tuning the radiative properties of these
systems. Hence, by controlling the overall emissivity/transmittivity of a
window surface it is possible to facilitate or reduce heat exchange with the
environment and therefore to act directly on the indoor temperature. Several
strategies have been proposed so far to perform such a control. For example,
single glazings made of thermochromic materials have been suggested Kim ;
Zheng ; Chang  to tailor the optical properties of windows with respect to
their temperature. Among these materials vanadium dioxide (VO2) a metal-
insulator transition (MIT) material has been largely investigated. Due to
their pronounced change of optical properties in the infrared these phase-
change materials have found numerous applications in the fields of thermal
management pba2013 ; pba2014 ; pba2014_bis ; Reddy2018 ; Kats , energy storage
Kubytskyi  and information treatment pba2016 . However this material undergoes
a reversible structural phase transition at relatively high temperature
($T_{C}\sim 68^{\circ}$C) making its employment challenging in smart windows.
Moreover, to date, there is no real alternative candidate to this material
presenting a pronounced MIT close to the ambient temperature. Electrochromic
materials have also been proposed Cannavale ; Chen  to modify the optical
properties of smart windows using an external bias voltage. Hence, by using
VO2 solid-electrolyte films, the light transmittance of such gate-controlled
smart windows can be dramatically modified by tuning the hydrogen-ion doping
through gating voltage. However, although this optical control can operate at
ambient temperature it requires an external energy supply. Moreover,
electrochromic windows are mainly effective in blocking visible light and not
infrared radiation so that their thermal performances remain today very weak.

In the present letter we introduce a passive smart window based on a double
glazing system (Fig. 1) realized with a VO2 film deposited on the internal
side of the glazing. The two panes of this glazing are separated by a few-
milimeter thick gap filled with noble gas.

Figure 1: Sketch of the passive smart window made with a double glazing of two
SiO2 layers separated by a gap filled of noble gas and a VO2 thin film
deposited on the inner side of one of the two glass panes. The temperature of
the VO2 layer is driven by the greenhouse effect occuring between the two
panes thanks to the external sunshine in the visible range, by the heat flux
radiated in the infrared by the surrounding environnment and the inner wall
and by the convective exchanges.

Following the evolution during the day of external conditions of sunshine
evolve during the day the temperature of this MIT material changes thanks to
the greenhouse effect occuring between the two panes. By optimizing the
geometry of this system we show that we can at the same time control its
transmittance in the visible range and the incoming heat flux. More
specifically, by analyzing the temporal evolution of this thermo-optical
system driven by the external sunshine and the heat flux radiated in the
infrared by the external environment and the inner walls we show that our
smart-window design results in a significant reduction of the heat flux
reaching the inner side, compared to traditional windows, in conjunction with
a low reduction of the transmittance in the visible range of the spectrum.

To start let us describe the physical characteristics of system. The active
part is made of a vanadium dioxide (VO2) film which undergoes a first-order
transition (Mott transition Mott ) from a high-temperature metallic phase to a
low-temperature insulating phase Barker ; Barker2  at $T_{C}$. This film is
deposited on a glass layer which is partially transparent in the infrared
range Palik . As a result of the phase transition both the emissivity and the
reflectivity of this bilayer change radically (Fig. 2) on its two sides both
in the visible and in the infrared. The outside pane of the window is assumed
to be in contact with a thermal bath having temperature $T_{1}(t)$ and is
illuminated by the sun with a time-varying flux $\Phi_{2}(t)$ depending on the
sun trajectory during the day. For the sake of simplicity, the field radiated
by the bath can be assimilated to the field radiated by a blackbody at the
same temperature. On its opposite side the window interacts with the inner
walls of average emisssivity $\epsilon$ at temperature $T_{4}$. Due to the
thermal inertia of walls, $T_{4}$ is assumed here constant during a solar
cycle.

Figure 2: Reflectance spectra (averaged over the two polarization states) of
SiO2-VO2 (left) and VO2-SiO2 (right) bilayer across the MIT process. The glass
layer is $5\,$mm thick while the thickness of VO2 film is $100\,$nm.

The optical properties (emissivity/transmittivity) of the window are driven by
the temperature of each pane and therefore by the heat flux they exchange
between them and with the outdoor and indoor environments. The time evolution
of the thermal state $(T_{2}(t),T_{3}(t))$ of the window under the external
driving is governed by the coupled energy-balance equations

$\begin{split}C_{i}\frac{d{T}_{i}}{dt}&=\mathscr{P}_{i}(T_{1},...,T_{4};t)+\Phi_{i}(t)\\\
&\,+\mathscr{P}_{\text{conv}\rightarrow i}(T_{i-1},T_{i},T_{i+1};t),\quad
i={2,3}.\end{split}$ (1)

Here $\mathscr{P}_{i}$ denotes the total surfacic power received by the
$i^{th}$ element from all other elements of the system, $\Phi_{i}$ the
contribution of solar flux, $\mathscr{P}_{\text{conv}\rightarrow i}$ the
losses/gains due to the convective exchanges and $C_{i}$ the heat capacity per
unit surface for the pane $i$ of the window. The power exchanged per unit
surface by each plate ($i=2,3$) with its surrounding by convection is
calculated from the convective heat transfer coefficients $h_{c}$,
$h_{\text{ext}}$ and $h_{\text{int}}$ (within the cavity formed by the two
panes, between the external medium and the external pane and between the inner
medium and the internal pane) using the linear Newton’s law of cooling

$\mathscr{P}_{\text{conv}\rightarrow
2,3}=h_{c}(T_{3,2}-T_{2,3})+h_{\text{ext,int}}(T_{1,4}-T_{2,3}),$ (2)

which takes into account the convective interactions on all sides of the
layer. We consider here a double glazing with a cavity filled with a noble
gas, such as Argon, and we assume each pane cooled or heated up by natural
convection on its opposite side. With vertical windows and temperature
differences between the panel and its surrounding smaller than $\Delta T\sim
50^{\circ}$C the corresponding heat transfer coefficients can be reasonably
set to $h_{c}=1\,$W$\cdot$m${}^{-2}\cdot$K-1 and
$h_{\text{ext,int}}=5\,$W$\cdot$m${}^{-2}\cdot$K-1 which correspond to the
typical values for vertical panels in these conditions Awbi ; McAdams . Hence,
the power exchanged by convection between each panel and its surrounding is
about $\mathscr{P}_{\text{conv}\rightarrow
2,3}\sim(h_{c}+h_{\text{ext}})\Delta T$ that is a third of the solar
irradiation. In this preliminary study, we do not include the heat exchanges
due to forced convection or turbulent exchanges between each pane and its
surrounding environment. These effects as well geometric effects such as the
window inclination and sizing will be analyzed in future works. The heat flux
across any plane parallel to the interacting surfaces is given by the normal
component of Poynting vector

$\varphi(z)=\langle\mathbf{E}(z)\times\mathbf{H}(z)\rangle\cdot\boldsymbol{e}_{z}.$
(3)

Here $\boldsymbol{e}_{z}$ denotes the unit vector normal to the panes of
windows, $\mathbf{E}$ and $\mathbf{H}$ are the local electric and magnetic
fields which are generated by the randomly fluctuating source currents in each
solid and by the fields radiated by media $1$ and $4$ while $\langle.\rangle$
represents the classical statistical averaging over all field realizations.
Using the Fourier decomposition of electric and magnetic fields $\varphi(z)$
can be recasted in terms of the field correlators
$\mathfrak{C}_{j}^{\phi,\phi^{\prime}}(\omega,\kappa)=\langle
E_{j}^{\phi}(\omega,\kappa)E_{j}^{\phi^{\prime}\dagger}(\omega,\kappa)\rangle$
of local field amplitudes in polarization $j$ RMP ; Latella

$\begin{split}\varphi(z)=2\epsilon_{0}c^{2}\\!\\!\sum\limits_{\underset{\phi=\left\\{+,-\right\\}}{j=s,p}}\int_{0}^{\infty}\\!\\!\frac{d\omega}{2\pi}\underset{\kappa<\omega/c}{\int}\\!\\!\\!\kappa\frac{{\rm
d}\kappa}{2\pi}\frac{\phi
k_{z}}{\omega}\mathfrak{C}_{j}^{\phi,\phi}(\omega,\kappa),\end{split}$ (4)

where $\kappa$ and $k_{z}=\sqrt{\frac{\omega^{2}}{c^{2}}-\kappa^{2}}$ denote
the parallel and normal components of the wavector. The correlators
$\mathfrak{C}_{j}^{++}$ and $\mathfrak{C}_{j}^{--}$ in each region of the
system can be written in term of correlators of fields radiated by each solid
and by the external baths. We summarize below the calculation of one of such
correlators (say $\mathfrak{C}_{j}^{++}$ in the windows cavity), the
calculation of all others correlators being similar. The right and left
propagating fields $E_{c}^{\pm}$ in this cavity are related to the fields in
the region of baths $E_{1,4}^{\pm}$ and to the fields $E_{i}^{\pm}$ ($i=2,3$)
emitted by each pane by the following systems (by omitting to write the
polarization for clarity reasons)

$\begin{split}E_{c}^{+}=E_{2}^{+}+\rho_{2}^{+}E_{c}^{-}+\tau_{2}^{+}E_{1}^{+},\\\
E_{c}^{-}=E_{3}^{-}+\rho_{3}^{-}E_{c}^{+}+\tau_{3}^{-}E_{4}^{-},\end{split}$
(5)

where $\rho_{i}^{\pm}$ and $\tau_{i}^{\pm}$ are the reflection and
transmission operators of the layer $i$ toward the right ($+$) and the left
($-$), respectively. By solving this system with respect to the cavity field
we get

$\begin{split}E_{c}^{+}&=U_{23}[E_{2}^{+}+\tau_{2}^{+}E_{1}^{+}+\rho_{2}^{+}(E_{3}^{-}+\tau_{3}^{-}E_{4}^{-})]\\\
E_{c}^{-}&=U_{23}[\rho_{3}^{-}(E_{2}^{+}+\tau_{2}^{+}E_{1}^{+})+E_{3}^{-}+\tau_{3}^{-}E_{4}^{-}]\end{split},$
(6)

where we have set $U_{23}=[1-\rho_{2}^{+}\rho_{3}^{-}]^{-1}$. It follows that
the correlators ($++$) into the cavity reads in term of correlators of fields
radiated by each solid

$\begin{split}\mathfrak{C}_{c}^{+,+}&=\mid
U_{23}\mid^{2}[\mathfrak{C}_{2}^{+,+}+\mid\rho_{2}^{+}\mid^{2}\mathfrak{C}_{3}^{-,-}\\\
&\,+\mid\tau_{2}^{+}\mid^{2}\mathfrak{C}_{1}^{+,+}+\mid\rho_{2}^{+}\mid^{2}\mid\tau_{3}^{-}\mid^{2}\mathfrak{C}_{4}^{-,-}].\end{split},$
(7)

But, according to the fluctuation dissipation theorem Rytov , these
correlators reads

$\begin{split}\mathfrak{C}_{2}^{+,+}&=\mathcal{S}(T_{2})(1-\mid\rho_{2}^{+}\mid^{2}-\mid\tau_{2}^{+}\mid^{2}),\\\
\mathfrak{C}_{3}^{-,-}&=\mathcal{S}(T_{3})(1-\mid\rho_{3}^{-}\mid^{2}-\mid\tau_{3}^{-}\mid^{2}),\\\
\mathfrak{C}_{1}^{+,+}&=\mathcal{S}(T_{1}),\\\
\mathfrak{C}_{4}^{-,-}&=\mathcal{S}(T_{4}),\\\ \end{split}$ (8)

with

$\mathcal{S}(T)=\pi\frac{\omega}{\epsilon_{0}c^{2}}\frac{1}{k_{z}}N(\omega,T),$
(9)

and $N(\omega,T)=\frac{\hbar\omega}{2}\coth(\frac{\hbar\omega}{2k_{B}T})$.
Finally, the radiative power received by the two panes from the rest of the
system (except the sun) reads

$\begin{split}\mathscr{P}_{2}&=\varphi(0)-\varphi(-\delta_{2}),\\\
\mathscr{P}_{3}&=\varphi(d+\delta_{3})-\varphi(d),\end{split}$ (10)

$\delta_{2,3}$ being the panes thickness.

Figure 3: Time evolution during a typical summer day of the external
temperature (a) and that of solar flux (b) at a lattitude of 48.2∘C.

The solar flux plotted in Fig. 3(b) is calculated from the AM1.5 Global
spectrum NREL  over a daily cycle for a window facing south. We also assume a
sinusoidal variation of the outdoor temperature [Fig. 3-(a)] with a minimal
value $T^{\text{min}}_{1}=10^{\circ}$C at $5\,$a.m. and a maximal value
$T^{\text{max}}_{1}=30^{\circ}$C at at $4\,$p.m., corresponding to a typical
variation of this temperature during summer. Finally, in order to demonstrate
the operating mode of the window we assume the inner wall perfectly absorbing
(i.e. $\epsilon=1$). The time evolution of these two primary sources being
very slow in comparison with the thermalization time of glazing ($\sim$ few
seconds to the minute), the energy balance Eq. (1) can be solved in the
adiabatic approximation.

The solutions $(T_{2}(t),T_{3}(t))$ of this balance equation corresponding to
the external conditions (solar flux $\Phi(t)$, outdoor temperatures
$T_{1}(t)$) given the indoor temperature $T_{4}=25^{\circ}$C are plotted in
Fig. 4 for different thicknesses of VO2 film and compared to the thermal state
of a simple glass-glass double glazing. As shown in Figs. 4(a) and (b) the
temperature of the active layer can reach, at the midday, levels significantly
higher than the critical temperature $T_{C}$. The comparison of $T_{1}$ (resp.
$T_{2}$) with the temperatures of a double (glass-glass) glazing of same
thickness demonstrates the presence of a strong greenhouse effect in the
designed window due to the trapping of the infrared light in the separation
gap between the two panes. The magnitude of this effect is directly related to
the thickness of VO2 film. As shown in Figs. 4(a) and (b), the thicker is this
film the greater is the blocked infrared radiation in the separation gap
between the two panes of glazing. The heating induced by this greenhouse
effect in the smart window surpasses the one traditionally observed in the
traditional double-glazing window by more than $100\%$ with VO2 film $100\,$nm
thick in $S_{1}$-type configuration and it is still $55\%$ higher with a film
$10\,$nm in the same configuration. The physical consequences of this heating
are shown in Figs. 4(c) and (d) demonstrating both the thermal and optical
performances of this smart window.

Figure 4: Thermal state and performances of the smart window. (a) Evolution of
panes temperature in a $S_{1}$-type window. (b) Temperatures in a window of
$S_{2}$-type. (c) Incoming net power $\mathscr{P}_{net}$ with respect to time
for a $S_{1}$ and $S_{2}$ window and its comparison with a traditional double
glazing. The glass and VO2 films have thicknesses $d_{\text{SiO}_{2}}=5\,$mm
and $d_{\text{VO}_{2}}=10\,$nm or $d_{\text{VO}_{2}}=100\,$nm, respectively.
(d) Transmittance in the visible range of the smart window ($S_{1}$
configuration) below (resp. beyond) the critical temperature $T_{C}$ and its
comparison with a double glazing SiO2-SiO2.

The net incoming heat flux $\mathscr{P}_{\text{net}}$, with respect to time,
plotted in Fig. 4(c) is drastically reduced in comparison with a traditional
window (double glazing SiO2-SiO2). This reduction is approximately
$400\,\text{W}\cdot\text{m}^{-2}$ at the midday when the VO2 film
$d_{\text{VO}_{2}}=100\,$nm is deposited on the inner side of the external
pane ($S_{1}$ configuration in Fig. 1), a value representing $40\%$ of the
solar irradiation at $48.2^{\circ}$ lattitude and about $55\%$ of the incoming
flux through a traditional window. Such isolating behavior exceeds by far the
performances of all previous smart windows rewiew_Feng ; rewiew_Cui . On the
other hand, when the MIT layer is on the opposite pane ($S_{2}$ configuration)
the incoming net flux becomes, in comparison, more important, the temperature
of this pane being increased by the greenhouse effect. For $10\,$nm thick VO2
the thermal insulation is reduced by a third.

Beside the thermal characteristics of the window we show in Fig. 4(d) the
transmittance

$T(\omega)=\int_{0}^{\pi/2}[t_{s}(\omega,\theta)+t_{p}(\omega,\theta)]\cos\theta\sin\theta
d\theta$ (11)

of the smart window in the visible range for both phases of thermochromic
material. Here $t_{s}$ (resp. $t_{p}$) denotes the directionnal spectral
transmission of the window under an incident angle $\theta$ (with respect to
the normal, see Fig. 1) in $s$ (resp. $p$) polarization. Notice that this
transmittance depends weakly, in the visible, on the crystallographic state of
VO2 film, the main changes taking place mainly in the infrared. Although
relatively small, this transmittance ($\sim 35\%$ for $10\,$nm thick VO2) over
$75\%$ of spectrum is comparable with that of usual double glazing SiO2-SiO2.
This transmittance falls down to $20\%$ with $100\,$nm thick VO2, making the
window partially opaque. Nevertheless, this value could be significantly
improved using a nanostructured MIT layer without significantly affecting its
properties in the infrared, as discussed in Ref. rewiew_Feng ; rewiew_Cui ;
Taylor ; Liu . However, this could also significantly increase the overall
fabrication cost. This problem as well as the optimization of the window will
be addressed in a specific study.

In conclusion we have introduced a smart window based on a double glazing made
with a SiO2-VO2 bilayer which is autonomously thermally and optically
regulated by greenhouse effect. This system overcomes the primary weaknesses
of thermochromic and electrochromic smart windows. Beyond its application in
the developpemnt of smart windows for more sustainable buidlings, the passive
insulation mechanism introduced in the present work could find broader
applications in the development of self-regulated insulating materials for
complex systems.

## Acknowledgments

This research was supported by the French Agence Nationale de la Recherche
(ANR), under grant ANR-19-CE08-0034 (PassiveHEAT). P. B.-A. acknowledges
discussions with E. Blandre and A. Losquin.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

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# Clairaut conformal submersions from Ricci solitons

Murat Polat

###### Abstract.

In the present article, we characterize Clairaut conformal submersions whose
total manifolds admit a Ricci soliton and provide a non-trivial example of
such Clairaut conformal submersions. We firstly calculate scalar curvature and
Ricci tensors of total manifolds of Clairaut conformal submersions and provide
necessary conditions for the fibres of such Clairaut conformal submersions to
be almost Ricci solitons and Einstein. Further, we provide necessary
conditions for the base manifold to be Ricci soliton and Einstein. Then, we
find a necessary condition for vector field $\dot{\zeta}$ to be conformal
vector field and killing vector field. Besides, we indicate that if total
manifolds of Clairaut conformal submersions admit a Ricci soliton with the
potential mean curvature vector field $H$ of $Ker\vartheta_{\ast},$ then the
total manifolds of Clairaut conformal submersions admit a gradient Ricci
soliton. Finally, by solving Poisson equation, we acquire a necessary and
sufficient condition for Clairaut conformal submersions to be harmonic.

Keywords: Ricci soliton, Clairaut conformal submersion, harmonic map,
Riemannian manifold, Riemannian submersion.

AMS. 2020: 53C12; 53B20; 53C43; 53C25.

##  1\. Introduction

In [22], O’Neill defined that conformal submersions are a generalization of
Riemannian submersions. Ishihara [14] and Fugledge [10] defined horizontally
conformal maps which have applications in computer graphics and medical
imaging (brain imaging). Gundmundsson found the fundamental equations for the
conformal submersions in [11]. In [8], Falcitelli et al. showed that under
what conditions conformal submersions would become Riemannian submersions. In
[2], Baird et al. indicated that a smooth map
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ is
called weakly conformal at $p_{1}\in M_{1}$ if there exists a number
$\sigma^{2}(p_{1})$ such that

(1.1)
$g_{M_{2}}(\vartheta_{\ast}X_{1},\vartheta_{\ast}X_{2})=\sigma^{2}(p_{1})g_{M_{1}}(X_{1},X_{2}),$

for$~{}X_{1},X_{2}\in\Gamma(T_{p_{1}}M_{1}).$

Let $(M_{1}^{d_{1}},g_{M_{1}})$ and $(M_{2}^{d_{2}},g_{M_{2}})$ be Riemannian
manifolds, then
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ is
called horizontally weakly conformal map at $p_{1}\in M_{1}$ if either $i.$
$\vartheta_{\ast p_{1}}=0$, or $ii.$ $\vartheta_{\ast p_{1}}$ maps the
horizontal space $\mathcal{H}_{p_{1}}=(Ker\vartheta_{\ast p_{1}})^{\bot}$
conformally onto $T_{\vartheta(p_{1})}M_{2}$, i.e. $\vartheta_{\ast p_{1}}$ is
satisfies (1.1) for $X_{1},X_{2}\in\mathcal{H}_{p_{1}}$ and is surjective. If
a node $p_{1}$ is of type $i.$ then $p_{1}\in M_{1}$ is said to be critical
node of $\vartheta$ and if node $p_{1}$ is of type $ii.$ then $p_{1}\in M_{1}$
is said to be regular node. The number $\sigma^{2}(p_{1})$ (non-negative) is
called the square dilation (square root $\sigma(p_{1})$ is called the
dilation). If the gradient of its dilation $\sigma$ is vertical, i.e.
$\mathcal{H}(grad~{}\sigma)=0$ then a horizontally weakly conformal map
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ is
called horizontally homothetic. If $\vartheta$ has no critical nodes then
horizontally weakly conformal map $\vartheta$ is called horizontally conformal
submersion (HCS) [2]. Therefore, a Riemannian submersion is a HCS with
dilation identically one.

If $r$ is the distance to the axis of a surface of revolution and $\theta$ is
the angle between a meridian and the velocity vector of a geodesic, then
Clairaut’s relation [7] states that $r\sin\theta$ is constant. In [1], Bishop
gave Clairaut submersions and provided a necessary and sufficient condition
for Riemannian submersions to be Clairaut Riemannian submersions. Further the
generalization of Clairaut Riemannian submersions as Clairaut Riemannian maps
were introduced in [27] and [18]. Recently, the notion of Clairaut conformal
submersion (CCS) is introduced by Meena and Zawadzki in [16]. The authors
related the geometry of Clairaut Riemannian submersions with the CCS through
conformal changes in the metric.

In [13], Hamilton came up with the idea of Ricci soliton to obtain a desired
metric on a Riemannian manifold $(M_{1},g_{M_{1}})$. The Ricci flow is an
evolution equation (it can also be called the heat equation) and can be
formulated as

$\frac{\partial}{\partial s}g_{M_{1}}(s)=-2Ric.$

Furthermore, we can say that Ricci solitons are a generalization of an
Einstein metric and the self-similar solutions of Ricci flow are Ricci
solitons [4]. $(M_{1},g_{M_{1}},\xi,\mu)$ is called Ricci soliton if there
exists a potential vector field (PVF) $\xi$ which satisfies

(1.2) $\frac{1}{2}(L_{\xi}g_{M_{1}})+Ric+\mu g_{M_{1}}=0,$

where $L_{\xi}g_{M_{1}}$ indicates the Lie derivative of $g_{M_{1}}$, $Ric$
indicates the Ricci tensor of Riemannian manifold and $\mu$ indicates a
constant. If $\mu<0,$ $\mu=0$ or $\mu>0$, then $(M_{1},g_{M_{1}},\xi,\mu)$ is
said to be shrinking, steady or expanding respectively. Furthermore, $\xi$ is
said to be conformal vector field [6] if it provides $L_{\xi}g_{M_{1}}=2\beta
g_{M_{1}}$, here $\beta$ is the potential function of PVF.

In [23], Perelman showed that Ricci solitons are very useful to solve the
Poincaré conjecture. In [24], Pigola et al., by considering $\mu$ as a
variable function, gave almost Ricci soliton. In this way, the geometry of the
Ricci soliton became the most important subject of research because of its
wide applications in theoretical physics and its geometric significance.

Lately, Riemannian submersions whose total manifolds admit Ricci soliton,
almost Ricci soliton, almost $\eta$-Ricci-Bourguignon soliton, $\eta$-Ricci
soliton, $\eta$-Ricci-Yamabe soliton, almost Yamabe soliton, conformal
$\eta$-Ricci soliton were investigated in [20], [3], [5], [9], [19], [28] and
[29]. Also, Riemannian maps whose total or base manifolds admit a Ricci
soliton were investigated in [30], [32], [12], [31] and [18]. Recently, in
[17], conformal submersions whose total manifolds admitting a Ricci soliton
studied by Meena and Yadav. Also, Polat studied Clairaut pointwise slant
Submersion from locally product Riemannian manifolds in [25]. We investigate
CCS whose total manifolds admitting a Ricci soliton in this current article.

The article is organized as: in Sect. 2, for the conformal submersion and CCS
which are required for this article, we gather some basic concepts. In Sect.
3, we obtain the scalar curvature and Ricci tensor by using Ricci soliton.
Then we provide necessary conditions for the fibers of CCS $\vartheta$ to be
almost Ricci soliton and Einstein. Moreover, we provide necessary conditions
for the base manifold to be Ricci soliton and Einstein. Also, we discuss the
harmonicity of CCS between Riemannian manifolds whose total space is Ricci
soliton. Finally, we prepare a illustrative example to support the correctness
of the theory of this article.

##  2\. Preliminaries

In this part, we remind a brief review of some basic facts for the notion of
submersion, Riemannian submersion, conformal submersion and CCS between
Riemannian manifolds.

Let $\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$
be a map between Riemannian manifolds $(M_{1}^{d_{1}},g_{M_{1}})$ and
$(M_{2}^{d_{2}},g_{M_{2}}).$ Then $\vartheta$ is called
$C^{\infty}$-submersion if it has maximal rank at every node $p_{1}\in M_{1}$.
The fibers of $\vartheta$ is defined as $\vartheta^{-1}(p_{2}),$ for any
$p_{2}\in M_{2}$. The vectors tangent to fibers build the smooth vertical
distribution denoting $\nu_{p_{1}}=Ker\vartheta_{\ast p_{1}}$ and its
orthogonal complementary is called horizontal distribution denoting
$\mathcal{H}_{p_{1}}$. $\nu$ and $\mathcal{H}$ indicate projections on the
vertical and horizontal distributions, respectively. A vector field $D$ on
$M_{1}$ is said to be projectable if there exists a vector field $\tilde{D}$
on $M_{2}$ such that $\vartheta_{\ast p_{1}}(D)=\tilde{D}_{\vartheta(p_{1})}$.
Then $D$ and $\tilde{D}$ are called $\vartheta$-related. So, $D$ is called the
horizontal lift of $\tilde{D}$.

Let $\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$
be a submersion between Riemannian manifolds $(M_{1}^{d_{1}},g_{M_{1}})$ and
$(M_{2}^{d_{2}},g_{M_{2}}).$ We say that $\vartheta$ is a Riemannian
submersion if $\vartheta_{\ast p_{1}}$ preserves the length of the horizontal
vectors.

Let $\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$
be a Riemannian submersion. If $\vartheta_{\ast}$ restricted to horizontal
distribution of $\vartheta$ is a conformal map, namely,

(2.1)
$g_{M_{2}}(\vartheta_{\ast}X_{1},\vartheta_{\ast}X_{2})=\sigma^{2}(p_{1})g_{M_{1}}(X_{1},X_{2}),$

$\forall X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}~{}$and$~{}p_{1}\in
M_{1}$ for a smooth function $\sigma:M_{1}\rightarrow\mathbb{R}^{+},$ then
$\vartheta$ is said to be conformal submersion. The O’Neill tensors $S$ and
$T$ formulated as [22]

(2.2) $S_{E_{1}}E_{2}=\mathcal{H}\nabla_{\mathcal{H}E_{1}}\nu
E_{2}+\nu\nabla_{\mathcal{H}E_{1}}\mathcal{H}E_{2},$ (2.3)
$T_{E_{1}}E_{2}=\mathcal{H}\nabla_{\nu E_{1}}\nu E_{2}+\nu\nabla_{\nu
E_{1}}\mathcal{H}E_{2},$

for any $E_{1},E_{2}\in\Gamma(TM_{1})$, here $\nabla$ denotes the Levi-Civita
connection of metric tensor $g_{M_{1}}$. $\forall E_{1}\in\Gamma(TM_{1})$,
$T_{E_{1}}$ and $S_{E_{1}}$ are skew-symmetric operators on
$(\Gamma(TM_{1}),g_{M_{1}}).$ We can easily see that $T_{E_{1}}=T_{\nu E_{1}}$
i.e, $T$ is vertical, and $S_{E_{1}}=S_{\mathcal{H}E_{1}}$ i.e, $S$ is
horizontal and $T$ satisfies $T_{U_{1}}U_{2}=T_{U_{2}}U_{1},~{}\forall
U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})$. Now, from (2.2) and (2.3), we have

(2.4) $\nabla_{U_{1}}U_{2}=T_{U_{1}}U_{2}+\nu\nabla_{U_{1}}U_{2},$ (2.5)
$\nabla_{X_{1}}U_{1}=S_{X_{1}}U_{1}+\nu\nabla_{X_{1}}U_{1},$ (2.6)
$\nabla_{X_{1}}X_{2}=S_{X_{1}}X_{2}+\mathcal{H}\nabla_{X_{1}}X_{2},$

where $U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})$ and
$X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$. A conformal submersion
$\vartheta$ with

(2.7)
$T_{U_{1}}U_{2}=g_{M_{1}}(U_{1},U_{2})H~{}\text{or}~{}T_{U_{1}}X_{1}=-g_{M_{1}}(H,X_{1})U_{1},$

is called as totally umbilical fibers [33, 34]. Herein, $\forall
U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})$,
$X_{1}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$ and $H$ denotes the mean
curvature vector field of the fibers.

######  Proposition 1.

[11] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a HCS
such that $(Ker\vartheta_{\ast})^{\bot}$ is integrable. Then the horizontal
space is totally umbilical in $(M_{1},g_{M_{1}})$, i.e.
$S_{X_{1}}X_{2}=g_{M_{1}}(X_{1},X_{2})H^{\prime}~{}\forall
X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$, where $H^{\prime}$ is the
mean curvature vector field of $(Ker\vartheta_{\ast})^{\bot}$ and stated

(2.8)
$H^{\prime}=\left(\nabla_{\nu}\frac{1}{\sigma^{2}}\right)-\frac{\sigma^{2}}{2}.$

The second fundamental form of $\vartheta$ [21] is given by
$(\nabla\vartheta_{\ast})(X_{1},X_{2})={\nabla}_{X_{1}}^{\vartheta}\vartheta_{\ast}X_{2}-\vartheta_{\ast}({\nabla}_{X_{1}}^{M_{1}}X_{2}),~{}\forall
X_{1},X_{2}\in\Gamma(TM_{1})$, or

(2.9)
$(\nabla\vartheta_{\ast})(X_{1},X_{2})={\nabla}_{\vartheta_{\ast}X_{1}}^{M_{2}}\vartheta_{\ast}X_{2}-\vartheta_{\ast}({\nabla}_{X_{1}}^{M_{1}}X_{2}),~{}\forall
X_{1},X_{2}\in\Gamma(TM_{1}).$

######  Lemma 1.

[11] Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
HCS. Then

$\displaystyle\vartheta_{\ast}(\mathcal{H}\nabla_{X_{1}}X_{2})$
$\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\left\\{-g_{M_{1}}(X_{1},X_{2})\vartheta_{\ast}(grad_{\mathcal{H}}~{}(\frac{1}{\sigma^{2}}))+X_{1}(\frac{1}{\sigma^{2}})\vartheta_{\ast}X_{2}+X_{2}(\frac{1}{\sigma^{2}})\vartheta_{\ast}X_{1}\right\\}$
$\displaystyle+\nabla_{\vartheta_{\ast}X_{1}}^{M_{2}}\vartheta_{\ast}X_{2},$

where $\nabla$ Levi-Civita connection on $M_{1}$ and $X_{1},X_{2}$ are basic
vector fields.

Now we give general information about the gradient $(\mathop{\mathrm{g}rad})$,
divergence $(\mathop{\mathrm{d}iv})$ and Hessian $(Hess)$ [26]. Let
$\beta\in$F$(M_{1})$, then gradient of function $\beta$, indicated by
$grad\beta$ or $\nabla\beta$, stated

(2.10) $g_{M_{1}}(grad~{}\beta,X_{1})=X_{1}(\beta),$

for $X_{1}\in\Gamma(TM_{1}).$ Let $\\{e_{k}\\}_{1\leq k\leq d_{1}}$ be an
orthonormal basis of $T_{p_{1}}M_{1}$ then, we have

(2.11)
$g_{M_{1}}(X_{1},X_{2})=\sum_{k=1}^{d_{1}}g_{M_{1}}(X_{1},e_{k})g_{M_{1}}(X_{2},e_{k}).$

The divergence of $X_{1}$ given by

(2.12) $div(X_{1})=\sum_{k=1}^{d_{1}}g_{M_{1}}(\nabla_{e_{k}}X_{1},e_{k}),$

for any $X_{1}\in\Gamma(TM_{1}).$ The Hessian tensor $h_{\beta}$ of $\beta$ is
given by

$h_{\beta}(X_{1})=\nabla_{X_{1}}\nabla\beta,$

for $X_{1}\in\Gamma(TM_{1}).$ The Hessian form of $\beta$ is given by

(2.13) $Hess\beta(X_{1},X_{2})=g_{M_{1}}(h_{\beta}(X_{1}),X_{2}),$

for any $X_{1},X_{2}\in\Gamma(TM_{1}).$ The Laplacian of $\beta$ is given by

(2.14) $\Delta\beta=div(\nabla\beta).$

######  Lemma 2.

[11] Let $\beta:M_{1}\rightarrow\mathbb{R}$ be a smooth function and
$(M_{1},g_{M_{1}})$ be a Riemannian manifold. Then

$g_{M_{1}}(\nabla_{X_{1}}grad(\beta),X_{2})=g_{M_{1}}(\nabla_{X_{2}}grad(\beta),X_{1}),$

for$~{}X_{1},X_{2}\in\Gamma(TM_{1}).$

######  Proposition 2.

[11] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a HCS
with dilation $\sigma$. Then

$S_{X_{1}}X_{2}=\frac{1}{2}\left\\{\nu[X_{1},X_{2}]-\sigma^{2}g_{M_{1}}(X_{1},X_{2})\left(\nabla_{\nu}\frac{1}{\sigma^{2}}\right)\right\\},$

for any $X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}.$

######  Theorem 1.

[17] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a HCS
with dilation $\sigma$ such that $(Ker\vartheta_{\ast})^{\bot}$ is totally
geodesic. Then $\sigma$ is constant on $Ker\vartheta_{\ast}$.

######  Theorem 2.

[17] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a HCS.
If $S$ is parallel then $\sigma$ is constant on $Ker\vartheta_{\ast}$.

######  Theorem 3.

[17] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a HCS.
Then $\vartheta$ is totally geodesic if and only if fibers of $\vartheta$ are
totally geodesic, $(Ker\vartheta_{\ast})^{\bot}$ is totally geodesic and
$\vartheta$ is homothetic.

######  Lemma 3.

[17] Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
HCS with dilation $\sigma$ such that $(Ker\vartheta_{\ast})^{\bot}$ is
integrable. Then followings are valid:

1\.
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(S_{X_{l}}U_{1},S_{X_{l}}U_{2})=d_{2}^{2}\frac{\sigma^{4}}{4}g_{M_{1}}(\nabla_{\nu}\frac{1}{\sigma^{2}},U_{1})g_{M_{1}}(\nabla_{\nu}\frac{1}{\sigma^{2}},U_{2})$,

2\.
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{U_{1}}S)_{X_{l}}X_{l},U_{2})=d_{2}g_{M_{1}}(\nabla_{U_{1}}H^{\prime},U_{2})$,

3\.
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{1}}S)_{X_{l}}X_{l},U_{1})=d_{2}g_{M_{1}}(\nabla_{X_{1}}H^{\prime},U_{1})$,

4\.
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{l}}S)_{X_{1}}X_{l},U_{1})=g_{M_{1}}(X_{1},X_{l})g_{M_{1}}(\nabla_{X_{l}}H^{\prime},U_{1})$,

5\.
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{U_{k}}S)_{X_{1}}X_{2},U_{k})=dkv(H^{\prime})g_{M_{1}}(X_{1},X_{2})$,

6\.
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(S_{X_{1}}U_{k},S_{X_{2}}U_{k})=\frac{\sigma^{4}}{4}|\nabla_{\nu}\frac{1}{\sigma^{2}}|^{2}g_{M_{1}}(X_{1},X_{2})$,

here $\\{U_{k}\\}_{d_{2}+1\leq k\leq d_{1}}$ is orthonormal bases of
$Ker\vartheta_{\ast}$ and $\\{X_{l}\\}_{1\leq l\leq d_{2}}$ is orthonormal
bases of $(Ker\vartheta_{\ast})^{\bot},$ for
$U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})~{}$and$~{}X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$.

######  Proposition 3.

Let $\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$
be a HCS with dilation $\sigma$. Then

$\displaystyle Ric(U_{1},U_{2})$ $\displaystyle=$ $\displaystyle
Ric^{\nu}(U_{1},U_{2})-(d_{1}-d_{2})g_{M_{1}}(T_{U_{1}}U_{2},H)$
$\displaystyle+\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(S_{X_{l}}U_{1},S_{X_{l}}U_{2})+\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{U_{1}}S)_{X_{l}}X_{l},U_{2})$
$\displaystyle-\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{l}}T)_{U_{1}}X_{l},U_{2})$
$\displaystyle-\frac{\sigma^{4}}{2}d_{2}g_{M_{1}}(U_{1},\nabla_{\nu}\frac{1}{\sigma^{2}})g_{M_{1}}(U_{2},\nabla_{\nu}\frac{1}{\sigma^{2}}),$
$\displaystyle Ric(U_{1},X_{1})$ $\displaystyle=$
$\displaystyle(d_{1}-d_{2})g_{M_{1}}(\nabla_{U_{1}}H,X_{1})$
$\displaystyle-\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{U_{k}}T)_{U_{1}}U_{k},X_{1})+\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{1}}S)_{X_{l}}X_{l},U_{1})$
$\displaystyle-\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{l}}S)_{X_{1}}X_{l},U_{1})-\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(T_{U_{1}}X_{l},\nu[X_{1},X_{l}]),$
$\displaystyle Ric(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{U_{k}}S)_{X_{1}}X_{2},U_{k})$
$\displaystyle-\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{X_{1}}T)_{U_{k}}X_{2},U_{k})+\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(S_{X_{1}}U_{k},S_{X_{2}}U_{k})$
$\displaystyle-\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(T_{U_{k}}X_{1},T_{U_{k}}X_{2})+\sigma^{2}g_{M_{1}}(S_{X_{1}}X_{2},\nabla_{\nu}\frac{1}{\sigma^{2}})$
$\displaystyle+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\tilde{X}_{1},\tilde{X}_{2})+\frac{3}{4}\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(\nu[X_{1},X_{l}],\nu[X_{l},X_{2}])$
$\displaystyle-\frac{(d_{2}-2)}{2}\sigma^{2}g_{M_{1}}(\nabla_{X_{1}}\nabla\frac{1}{\sigma^{2}},X_{2})$
$\displaystyle-\frac{\sigma^{2}}{2}g_{M_{1}}(X_{1},X_{2})\left\\{\Delta^{\mathcal{H}}\frac{1}{\sigma^{2}}-d_{2}\left(H^{\prime}\frac{1}{\sigma^{2}}\right)\right\\}$
$\displaystyle+\frac{d_{2}\sigma^{4}}{4}g_{M_{1}}(X_{1},X_{2})|\nabla\frac{1}{\sigma^{2}}|^{2}+\frac{\sigma^{4}}{4}(d_{2}-2)(X_{1}\frac{1}{\sigma^{2}})(X_{2}\frac{1}{\sigma^{2}}),$

here $\\{U_{k}\\}_{d_{2}+1\leq k\leq d_{1}}$ is orthonormal bases of
$Ker\vartheta_{\ast}$ and $\\{X_{l}\\}_{1\leq l\leq d_{2}}$ is orthonormal
bases of $(Ker\vartheta_{\ast})^{\bot},$ for
$U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})~{}$and$~{}X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$.
Also, $X_{1}$ and $X_{2}$ are the horizontal lift of $\tilde{X}_{1}$ and
$\tilde{X}_{2}$, respectively.

The Clairaut condition for a Riemannian submersion was originally defined by
Bishop in [1]. Further, Clairaut condition was defined for a Riemannian map in
[27] and [18]. Now, we recall the definition CCS [16]. According to
definition, A conformal submersion
$\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ between Riemannian
manifolds with dilation $\sigma$ is said to be CCS if there is a function
$r:M_{1}\rightarrow\mathbb{R}^{+}$ such that for every geodesic $\zeta$ on
$M_{1}$, the function $(r\circ\zeta)\sin\omega(s)$ is constant along $\zeta$,
where for all $s$, $\omega(s)$ is the angle between $\dot{\zeta}(s)$ and the
horizontal space at $\zeta(s)$.

######  Theorem 4.

[16] Let $\vartheta:(M_{1},g_{M_{1}})\rightarrow(M_{2},g_{M_{2}})$ be a
conformal submersion with connected fibers and dilation $\sigma$. Then
$\vartheta$ is a CCS with $r=e^{\beta}$ if and only if $\nabla\beta$ is
horizontal, fibers of $\vartheta$ are totally umbilical with $H=-\nabla\beta$
i.e. $T_{U_{1}}U_{2}=-g_{M_{1}}(U_{1},U_{2})\nabla\beta$ and $\sigma$ is
constant along the fibers of $\vartheta$.

##  3\. Clairaut conformal submersions whose total manifolds admit a Ricci
soliton

In this part, we conceive a CCS whose total space is a Ricci soliton and study
its geometry.

######  Proposition 4.

Let $\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$
be a CCS with dilation $\sigma$. Then the Ricci tensor

$\displaystyle Ric(U_{1},U_{2})$ $\displaystyle=$ $\displaystyle
Ric^{\nu}(U_{1},U_{2})-(d_{1}-d_{2})g_{M_{1}}(U_{1},U_{2})|\nabla\beta|^{2}-g_{M_{1}}(U_{1},U_{2})div(\nabla\beta),$
$\displaystyle Ric(U_{1},X_{1})$ $\displaystyle=$
$\displaystyle(d_{1}-d_{2}+1)g_{M_{1}}(S_{X_{1}}U_{1},\nabla\beta)-\sum\limits_{l=1}^{d_{2}}\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(\nabla_{X_{l}}S_{X_{1}}X_{l},U_{k}),$
$\displaystyle Ric(X_{1},X_{2})$ $\displaystyle=$ $\displaystyle
div(S_{X_{1}}X_{2})-2\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(S_{X_{1}}U_{k},S_{X_{2}}U_{k})$
$\displaystyle-(d_{1}-d_{2})g_{M_{1}}(X_{2},\nabla_{X_{1}}\nabla\beta)-X_{1}(\beta)X_{2}(\beta)(d_{1}-d_{2})$
$\displaystyle+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\tilde{X}_{1},\tilde{X}_{2})-\frac{(d_{2}-2)}{2}\sigma^{2}g_{M_{1}}(\nabla_{X_{1}}\nabla\frac{1}{\sigma^{2}},X_{2})$
$\displaystyle-\frac{\sigma^{2}}{2}g_{M_{1}}(X_{1},X_{2})\Delta^{\mathcal{H}}\frac{1}{\sigma^{2}}+\frac{d_{2}\sigma^{4}}{4}g_{M_{1}}(X_{1},X_{2})|\nabla\frac{1}{\sigma^{2}}|^{2}$
$\displaystyle+\frac{\sigma^{4}}{4}(d_{2}-2)(X_{1}\frac{1}{\sigma^{2}})(X_{2}\frac{1}{\sigma^{2}}),$

here $\\{U_{k}\\}_{d_{2}+1\leq k\leq d_{1}}$ is orthonormal bases of
$Ker\vartheta_{\ast}$ and $\\{X_{l}\\}_{1\leq l\leq d_{2}}$ is orthonormal
bases of $(Ker\vartheta_{\ast})^{\bot},$ for
$U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})~{}$and$~{}X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$.
Also, $X_{1}$ and $X_{2}$ are the horizontal lift of $\tilde{X}_{1}$ and
$\tilde{X}_{2}$, respectively.

###### Proof.

Since $\vartheta$ is CCS, utilizing Theorem 4, we obtain

(3.4) $g_{M_{1}}(T_{U_{1}}U_{2},H)=g_{M_{1}}(U_{1},U_{2})|\nabla\beta|^{2},$

and

(3.5)
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{l}}T)_{U_{1}}X_{l},U_{2})=g_{M_{1}}(U_{1},U_{2})div(\nabla\beta).$

Also, by using Theorem 4 in Lemma 3, we have

(3.6) $\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(S_{X_{l}}U_{1},S_{X_{l}}U_{2})=0,$

and

(3.7)
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{U_{1}}S)_{X_{l}}X_{l},U_{2})=0.$

Then using (3.4)-(3.7) in (3), we get (4).

Also, by using Theorem 4, we obtain

(3.8)
$g_{M_{1}}(\nabla_{U_{1}}H,X_{1})=-g_{M_{1}}(\nabla_{U_{1}}\nabla\beta,X_{1})$
(3.9)
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{U_{k}}T)_{U_{1}}U_{k},X_{1})=g_{M_{1}}(S_{X_{1}}U_{1},\nabla\beta)$
(3.10)
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{1}}S)_{X_{l}}X_{l},U_{1})=0$
$\displaystyle\sum\limits_{l=1}^{d_{2}}g_{M_{1}}((\nabla_{X_{l}}S)_{X_{1}}X_{l},U_{1})$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{d_{2}}\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{X_{l}}S)_{X_{1}}X_{l},U_{k})g_{M_{1}}(U_{k},U_{k})$
(3.12)
$\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(T_{U_{1}}X_{l},\nu[X_{1},X_{l}])=-2g_{M_{1}}(S_{X_{1}}U_{1},\nabla\beta).$

Then using (3.8)-(3.12) in (3), we get (4). Finally,

(3.13)
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{U_{k}}S)_{X_{1}}X_{2},U_{k})=div(S_{X_{1}}X_{2})$
(3.14)
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(S_{X_{1}}U_{k},S_{X_{2}}U_{k})=0$
(3.15)
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}((\nabla_{X_{1}}T)_{U_{k}}X_{2},U_{k})=(d_{1}-d_{2})g_{M_{1}}(X_{2},\nabla_{X_{1}}\nabla\beta)$
(3.16)
$\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(T_{U_{k}}X_{1},T_{U_{k}}X_{2})=g_{M_{1}}\left(\nabla_{X_{1}}\mathcal{H}\nabla\frac{1}{\sigma^{2}},X_{2}\right).$

Then using (3.13)-(3.16) in (3), we get (4).

######  Theorem 5.

Let $(M_{1},g_{M_{1}},\xi,\mu)$ be a Ricci soliton with the PVF
$\xi\in\Gamma(TM_{1})$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
CCS between Riemannian manifolds. Then following options are valid:

(i) any fiber of $\vartheta$ is an almost Ricci soliton if
$\xi=W\in\Gamma(Ker\vartheta_{\ast})$,

(ii) any fiber of $\vartheta$ is an Einstein if
$\xi=X_{1}\in\Gamma(Ker\vartheta_{\ast})^{\bot}$.

###### Proof.

Since $(M_{1},g_{M_{1}},\xi,\mu)$ is a Ricci soliton, then using (1.2), we
obtain

(3.17) $\frac{1}{2}(L_{\xi}g_{M_{1}})(U_{1},U_{2})+Ric(U_{1},U_{2})+\mu
g_{M_{1}}(U_{1},U_{2})=0,~{}$

for$~{}U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast}).$ Using (4) in (3.17), we
have

$\displaystyle\frac{1}{2}(L_{\xi}g_{M_{1}})(U_{1},U_{2})+Ric^{\nu}(U_{1},U_{2})$
$\displaystyle+g_{M_{1}}(U_{1},U_{2})(\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(\nabla\beta))$
$\displaystyle=$ $\displaystyle 0.$

If $\xi=W\in\Gamma(Ker\vartheta_{\ast})$, then from (3), we acquire

$\frac{1}{2}(L_{W}g_{M_{1}})(U_{1},U_{2})+Ric^{\nu}(U_{1},U_{2})+\mu^{\prime}g_{M_{1}}(U_{1},U_{2})=0,$

where $\mu^{\prime 2}-div(\nabla\beta)$ is a smooth function on $M_{1}$. This
implies $(i)$. Now, if $\xi=X_{1}\in\Gamma(Ker\vartheta_{\ast})^{\bot},$ then
(3) implies

$\displaystyle\frac{1}{2}\\{g_{M_{1}}(\nabla_{U_{1}}X_{1},U_{2})+g_{M_{1}}(\nabla_{U_{2}}X_{1},U_{1})\\}$
$\displaystyle+Ric^{\nu}(U_{1},U_{2})+g_{M_{1}}(U_{1},U_{2})(\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(\nabla\beta))$
$\displaystyle=$ $\displaystyle 0,$

which implies

$\displaystyle-
g_{M_{1}}(\mathcal{H}\nabla_{U_{1}}U_{2},X_{1})+Ric^{\nu}(U_{1},U_{2})$
$\displaystyle+g_{M_{1}}(U_{1},U_{2})(\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(\nabla\beta))$
$\displaystyle=$ $\displaystyle 0.$

Using (2.4) and Theorem 4 in above equation, we get

$Ric^{\nu}(U_{1},U_{2})+g_{M_{1}}(U_{1},U_{2})(\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(H)+X_{1}(\beta))=0.$

which implies $(ii)$.

######  Theorem 6.

Let $(M_{1},g_{M_{1}},\xi,\mu)$ be a Ricci soliton with the PVF
$\xi\in\Gamma(TM_{1})$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
totally geodesic CCS between Riemannian manifolds. Then the following options
are valid:

(i) $M_{2}$ is a Ricci soliton if
$\xi=Z\in\Gamma(Ker\vartheta_{\ast})^{\bot}$,

(ii) $M_{2}$ is an Einstein if $\xi=W\in\Gamma(Ker\vartheta_{\ast})$.

###### Proof.

Since $(M_{1},g_{M_{1}},\xi,\mu)$ is a Ricci soliton then using (1.2), we
obtain

$\frac{1}{2}(L_{\xi}g_{M_{1}})(X_{1},X_{2})+Ric(X_{1},X_{2})+\mu
g_{M_{1}}(X_{1},X_{2})=0,~{}$

which can be written as

$\displaystyle\frac{1}{2}\left\\{g_{M_{1}}(\nabla_{X_{1}}\xi,X_{2})+g_{M_{1}}(\nabla_{X_{2}}\xi,X_{1})\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0,$

for$~{}X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}.$ If
$\xi=Z\in\Gamma(Ker\vartheta_{\ast})^{\bot},$ then utilizing (1.1) in (3)

$\displaystyle\frac{1}{2\sigma^{2}}\left\\{g_{M_{2}}(\vartheta_{\ast}(\nabla_{X_{1}}Z),\vartheta_{\ast}X_{2})+g_{M_{2}}(\vartheta_{\ast}(\nabla_{X_{2}}Z),\vartheta_{\ast}X_{1})\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0.$

Using (2.8) in (3), we have

$\displaystyle\frac{1}{2\sigma^{2}}\left\\{\begin{array}[]{c}g_{M_{2}}(\nabla_{X_{1}}^{M_{2}}\bar{Z}-(\nabla\vartheta_{\ast})(X_{1},Z),\tilde{X}_{2})\\\
+g_{M_{2}}(\nabla_{X_{2}}^{M_{2}}\bar{Z}-(\nabla\vartheta_{\ast})(X_{2},Z),\bar{X}_{1})\end{array}\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\frac{\mu}{2\sigma^{2}}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})$
$\displaystyle=$ $\displaystyle 0,$

where $X_{1},X_{2}$ and $Z$ are the horizontal lift of
$\bar{X}_{1},\tilde{X}_{2}$ and $\bar{Z},$ respectively. Since $\vartheta$ is
totally geodesic, then $(\nabla\vartheta_{\ast})(X_{1},Z)=0$ and
$(\nabla\vartheta_{\ast})(X_{2},Z)=0.$ Using Theorem 1, Theorem 2 and Theorem
3 in (4) then (3) can be written as

$\displaystyle\frac{1}{2\sigma^{2}}\left\\{g_{M_{2}}(\nabla_{X_{1}}^{M_{2}}\bar{Z},\tilde{X}_{2})+g_{M_{2}}(\nabla_{\tilde{X}_{2}}^{M_{2}}\bar{Z},\bar{X}_{1})\right\\}$
$\displaystyle+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})+\frac{\mu}{\sigma^{2}}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})$
$\displaystyle=$ $\displaystyle 0,$

which completes proof of (i).

If $\xi=W\in\Gamma(Ker\vartheta_{\ast})$ then (3) becomes

$\displaystyle\frac{1}{2}\left\\{g_{M_{1}}(\nabla_{X_{1}}W,X_{2})+g_{M_{1}}(\nabla_{X_{2}}W,X_{1})\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0,$

which can be written as

$\displaystyle-\frac{1}{2}\left\\{g_{M_{1}}(\nabla_{X_{1}}X_{2},W)+g_{M_{1}}(\nabla_{X_{2}}X_{1},W)\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0.$

Using (2.1) in (2.6), we have

$\displaystyle-\frac{1}{2}\left\\{g_{M_{1}}(S_{X_{1}}X_{2},W)+g_{M_{1}}(S_{X_{2}}X_{1},W)\right\\}$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0.$

Since $\vartheta$ is totally geodesic, then using (4), Proposition 2 and
Theorem 3 in (3), we obtain

$\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})+\frac{\mu}{\sigma^{2}}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})=0,$

where $X_{1}$ and $X_{2}$ are the horizontal lift of $\bar{X}_{1}$ and
$\tilde{X}_{2},$ respectively. This completes the proof of (ii).

######  Corollary 1.

Let $(M_{1},g_{M_{1}},\xi,\mu)$ be a Ricci soliton with the PVF
$W\in\Gamma(Ker\vartheta_{\ast})$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
totally geodesic CCS between Riemannian manifolds. If
$Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})=-\mu
g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2}),$ then $W$ is a killing vector field on
$M_{2}$.

######  Theorem 7.

Let $\zeta$ be a geodesic curve on $M_{1}$ and
$(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ be a Ricci soliton with the PVF
$\dot{\zeta}\in\Gamma(TM_{1}).$ Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
CCS with $r=e^{\beta}$ between Riemannian manifolds such that fibers of
$\vartheta$ are Einstein. Then $\dot{\zeta}$ is conformal vector field on
$Ker\vartheta_{\ast}.$

###### Proof.

Since $(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ is a Ricci soliton followings are
obtained:

(3.25)
$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(U_{1},U_{2})+Ric(U_{1},U_{2})+\mu
g_{M_{1}}(U_{1},U_{2})=0,~{}$

for$~{}U_{1},U_{2}\in\Gamma(Ker\vartheta_{\ast})$ and
$\dot{\zeta}\in\Gamma(TM_{1}).$ Using (4) in (3.25), we have

$\displaystyle\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(U_{1},U_{2})+Ric^{v}(U_{1},U_{2})-(d_{1}-d_{2})g_{M_{1}}(U_{1},U_{2})\left\|\nabla\beta\right\|^{2}$
$\displaystyle-g_{M_{1}}(U_{1},U_{2})\mathop{\mathrm{d}iv}(\nabla\beta)+\mu
g_{M_{1}}(U_{1},U_{2})$ $\displaystyle=$ $\displaystyle 0.$

Since fibers of $\vartheta$ is Einstein, using $Ric^{v}(U_{1},U_{2})=-\mu
g_{M_{1}}(U_{1},U_{2})$ we get

$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(U_{1},U_{2})-\left\\{(d_{1}-d_{2})\left\|\nabla\beta\right\|^{2}+\mathop{\mathrm{d}iv}(\nabla\beta)\right\\}g_{M_{1}}(U_{1},U_{2})=0,$

which can be written as

$(L_{\dot{\zeta}}g_{M_{1}})(U_{1},U_{2})=2\beta_{1}g_{M_{1}}(U_{1},U_{2}),$

where
$\beta_{1}=(d_{1}-d_{2})\left\|\nabla\beta\right\|^{2}+\mathop{\mathrm{d}iv}(\nabla\beta)$
is a smooth function on $M_{1}.$ Thus $\dot{\zeta}$ is conformal vector field
on $Ker\vartheta_{\ast}.$

######  Corollary 2.

Let $\zeta$ be a geodesic curve on $M_{1}$ and
$(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ be a Ricci soliton with the PVF
$\dot{\zeta}\in\Gamma(TM_{1}).$ Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
CCS with $r=e^{\beta}$ between Riemannian manifolds. If fibers of $\vartheta$
are totally geodesic then, $\dot{\zeta}$ is killing vector field on
$Ker\vartheta_{\ast}.$

######  Theorem 8.

Let $\zeta$ be a geodesic curve on $M_{1}$ and
$(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ be a Ricci soliton with the PVF
$\dot{\zeta}\in\Gamma(TM_{1}).$ Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
homothetic CCS with $r=e^{\beta}$ from a Riemannian manifold $M_{1}$ to an
Einstein manifold $M_{2}$ such that fibers of $\vartheta$ are totally geodesic
and $(Ker\vartheta_{\ast})^{\bot}$ is integrable. Then $\dot{\zeta}$ is
conformal vector field on $(Ker\vartheta_{\ast})^{\bot}.$

###### Proof.

Since $(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ is a Ricci soliton then by (1.2) we
get

$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+Ric(X_{1},X_{2})+\mu
g_{M_{1}}(X_{1},X_{2})=0,~{}$

for$~{}X_{1},X_{2}\in\Gamma(Ker\vartheta_{\ast})^{\bot}.$ Using Theorem 1,
Theorem 2 and Theorem 3 in (4) then we have

$\displaystyle\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})-\frac{\sigma^{2}}{2}g_{M_{1}}(X_{1},X_{2})\Delta^{\mathcal{H}}\frac{1}{\sigma^{2}}$
$\displaystyle+\frac{d_{2}\sigma^{4}}{4}g_{M_{1}}(X_{1},X_{2})|\nabla\frac{1}{\sigma^{2}}|^{2}+\mu
g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$ $\displaystyle 0.$

Using (2.1) in the above equation

$\displaystyle\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})-\frac{1}{2}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})\Delta^{\mathcal{H}}\frac{1}{\sigma^{2}}$
$\displaystyle+\frac{d_{2}\sigma^{4}}{4}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})|\nabla\frac{1}{\sigma^{2}}|^{2}+\frac{\mu}{\sigma^{2}}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})$
$\displaystyle=$ $\displaystyle 0,$

which can be written as

$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})+\beta_{1}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})=0,$

or

(3.26)
$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+\frac{1}{\sigma^{2}}\left\\{Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})+\beta_{2}g_{M_{2}}(\bar{X}_{1},\tilde{X}_{2})\right\\}=0,$

where
$\beta_{1}=-\frac{1}{2}\Delta^{\mathcal{H}}\frac{1}{\sigma^{2}}+\frac{d_{2}\sigma^{4}}{4}|\nabla\frac{1}{\sigma^{2}}|^{2}+\frac{\mu}{\sigma^{2}}$
and $\beta_{2}=\sigma^{2}\beta_{1}$ Since $M_{2}$ is an Einstein, putting
$Ric^{M_{2}}(\tilde{X}_{1},\tilde{X}_{2})=\beta_{2}g_{M_{2}}(\tilde{X}_{1},\tilde{X}_{2})$
in (3.26), we obtain

$\frac{1}{2}(L_{\dot{\zeta}}g_{M_{1}})(X_{1},X_{2})+2\beta_{1}g_{M_{1}}(X_{1},X_{2})=0,$

which implies that $\dot{\zeta}$ is conformal vector field on
$(Ker\vartheta_{\ast})^{\bot}$.

######  Theorem 9.

[17]Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
totally geodesic HCS with dilation $\sigma.$ Then

$s=s^{v}+\frac{1}{\sigma^{2}}s^{M_{2}},$

where $s,s^{v}$ and $s^{M_{2}}$ indicate the scalar curvatures of
$M_{1},Ker\vartheta_{\ast}$ and $M_{2},$ respectively.

Using Theorem 9, the following result can be written

######  Theorem 10.

Let $(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ be a Ricci soliton with the PVF
$\dot{\zeta}\in\Gamma(TM_{1}).$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
totally geodesic CCS with $r=e^{\beta}$ between Riemannian manifolds such that
$(Ker\vartheta_{\ast})^{\bot}$ is integrable. Then $M_{1}$ has constant scalar
curvature by $-\mu d_{1}.$

###### Proof.

Since $(M_{1},g_{M_{1}},\dot{\zeta},\mu)$ is a Ricci soliton then by (1.2) we
have

$\displaystyle\frac{1}{2}(g_{M_{1}}(\nabla_{X_{1}}\dot{\zeta},X_{2})+g_{M_{1}}(\nabla_{X_{2}}\dot{\zeta},X_{1}))$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0,$

for $X_{1},X_{2},\dot{\zeta}\in\Gamma(TM_{1}).$ Now, we decompose
$X_{1},X_{2}$ and $\dot{\zeta}$ such that $X_{1}=\nu
X_{1}+\mathcal{H}X_{1},X_{2}=\nu X_{2}+\mathcal{H}X_{2}$ and $\dot{\zeta}=\nu
W+\mathcal{H}Z$. Then (3) stated

$\displaystyle\frac{1}{2}\left\\{\begin{array}[]{c}g_{M_{1}}(\nabla_{\nu
X_{1}+\mathcal{H}X_{1}}\nu W+\mathcal{H}Z,\nu X_{2}+\mathcal{H}X_{2})\\\
+g_{M_{1}}(\nabla_{\nu X_{2}+\mathcal{H}X_{2}}\nu W+\mathcal{H}Z,\nu
X_{1}+\mathcal{H}X_{1})\end{array}\right\\}$
$\displaystyle+Ric(vX_{1},vX_{2})+Ric(\mathcal{H}X_{1},\mathcal{H}X_{2})+Ric(vX_{1},\mathcal{H}X_{2})$
$\displaystyle+Ric(\mathcal{H}X_{1},vX_{2})+\mu\left\\{g_{M_{1}}(vX_{1},vX_{2})+(\mathcal{H}X_{1},\mathcal{H}X_{2})\right\\}$
$\displaystyle=$ $\displaystyle 0.$

Taking trace of (3), we have

$\displaystyle\frac{1}{2}\left\\{2\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(\nabla_{X_{l}}X_{l},X_{l})+2\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(\nabla_{Uk}U_{k},U_{k})\right\\}$
$\displaystyle+\sum\limits_{k=d_{2}+1}^{d_{1}}Ric(U_{k},U_{k})+\sum\limits_{l=1}^{d_{2}}Ric(X_{l},X_{l})+2\sum\limits_{k,l}Ric(U_{k},X_{l})$
$\displaystyle+\mu\left\\{\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(X_{l},X_{l})+\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(U_{k},U_{k})\right\\}$
$\displaystyle=$ $\displaystyle 0,$

where $\\{U_{k}\\}_{d_{2}+1\leq k\leq d_{1}}$ is orthonormal bases of
$Ker\vartheta_{\ast}$ and $\\{X_{l}\\}_{1\leq l\leq d_{2}}$ is orthonormal
bases of $(Ker\vartheta_{\ast})^{\bot}$. Since $\vartheta$ is totally geodesic
and $(Ker\vartheta_{\ast})^{\bot}$ is integrable, then using Theorem 1,
Theorem 2 and Theorem 3 in (4), (4) and (4), we can write (3) as

$\displaystyle\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(\nabla_{Uk}U_{k},U_{k})+\sum\limits_{l=1}^{d_{2}}g_{M_{1}}(\nabla_{X_{l}}X_{l},X_{l})$
$\displaystyle+Ric^{v}(U_{k},U_{k})+\frac{1}{\sigma^{2}}Ric^{M_{2}}(\bar{X}_{1},\tilde{X}_{2})+\mu(d_{1}-d_{2}+d_{2})$
$\displaystyle=$ $\displaystyle 0,$

where $Ric^{v}$ and $Ric^{M_{2}}$ indicate the scalar curvatures of
$Ker\vartheta_{\ast}$ and $M_{2},$ respectively. From Theorem 9, (3) and since
$\nabla$ is metric connection on $M_{1},$ then we have

$\displaystyle\frac{1}{2}\left\\{\sum\limits_{l=1}^{d_{2}}\nabla_{X_{l}}g_{M_{1}}(X_{l},X_{l})+\sum\limits_{k=d_{2}+1}^{d_{1}}\nabla_{U_{k}}g_{M_{1}}(U_{k},U_{k})\right\\}$
$\displaystyle+Ric^{v}+\frac{1}{\sigma^{2}}Ric^{M_{2}}+\mu d_{1}$
$\displaystyle=$ $\displaystyle 0.$

Thus we obtain $s+\mu d_{1}=0,$ where
$s=Ric^{v}+\frac{1}{\sigma^{2}}Ric^{M_{2}}$ is the scalar curvature on
$M_{1}$.

######  Theorem 11.

Let $(M_{1},g_{M_{1}},-H,\mu)$ be a Ricci soliton with the PVF
$-H\in(Ker\vartheta_{\ast})^{\bot}$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
CCS with $r=e^{\beta}$ between Riemannian manifolds. Then the following
options are true:

(i) $M_{1}$ admits a gradient Ricci soliton,

(ii) The mean curvature vector field of $Ker\vartheta_{\ast}$ is constant.

###### Proof.

Since $(M_{1},g_{M_{1}},-H,\mu)$ is a Ricci soliton then using (1.2), we
obtain

$\displaystyle-\frac{1}{2}(g_{M_{1}}(\nabla_{X_{1}}H,X_{2})+g_{M_{1}}(\nabla_{X_{2}}H,X_{1}))$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0,$

for $X_{1},X_{2}\in\Gamma(TM_{1})$ and
$-H\in\Gamma(Ker\vartheta_{\ast})^{\bot}.$ Using Theorem 4, then (3) can be
written as

$\displaystyle\frac{1}{2}(g_{M_{1}}(\nabla_{X_{1}}\nabla\beta,X_{2})+g_{M_{1}}(\nabla_{X_{2}}\nabla\beta,X_{1}))$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0,$

which is equal to

$\displaystyle\frac{1}{2}(g_{M_{1}}(g_{M_{2}\beta}(X_{1}),X_{2})+g_{M_{1}}(g_{M_{2}\beta}(X_{2}),X_{1}))$
$\displaystyle+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle 0.$

Since $h_{\beta}$ is self-adjoint, (3) can be written as

$g_{M_{1}}(h_{\beta}(X_{1}),X_{2})+Ric(X_{1},X_{2})+\mu
g_{M_{1}}(X_{1},X_{2})=0.$

Using (2.13) in above equation

(3.35) $Hess\beta(X_{1},X_{2})+Ric(X_{1},X_{2})+\mu g_{M_{1}}(X_{1},X_{2})=0,$

which completes the proof of (i). If we taking trace (3.35), from (2.14) we
have

$\Delta\beta+s+\mu d_{1}=0.$

Thus, the Poisson equation on $(M_{1},g_{M_{1}})$ is

(3.36) $\Delta\beta=\mathop{\mathrm{d}iv}(\nabla\beta)=-s-\mu d_{1},$

where $\Delta$ is the Laplace operator, $\beta$ is a smooth function on
$M_{1}.$ To determine its solution using Theorem 10 in (3.36), we obtain

$\Delta\beta=\mathop{\mathrm{d}iv}(\nabla\beta)=0,i.e.\nabla(H)=0,$

which means $H$ is constant. This completes the proof of (ii).

######  Definition 1.

[26] Let
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
smooth map between Riemannian manifolds. Then $\vartheta$ is harmonic if and
only if the tension field $\tau(\vartheta)$ of $\vartheta$ vanishes at every
node $p_{1}\in M_{1}.$

Considering this definition, the following theorem can be written:

######  Theorem 12.

Let $(M_{1},g_{M_{1}},W,\mu)$ be a Ricci soliton with the PVF
$W\in\Gamma(Ker\vartheta_{\ast})$ and
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ be a
totally geodesic CCS with $r=e^{\beta}$ between Riemannian manifolds such that
$(Ker\vartheta_{\ast})^{\bot}$ is totally geodesic. Then $\vartheta$ is
harmonic if and only if the scalar curvature of $Ker\vartheta_{\ast}$ is
$-\mu(d_{1}-d_{2}).$

###### Proof.

Since $\vartheta$ is a conformal submersion then we have

(3.37)
$\tau(\vartheta)=(d_{1}-d_{2})\vartheta_{\ast}(\nabla\beta)+(d_{2}-2)\frac{\sigma^{2}}{2}\vartheta_{\ast}(\nabla_{\mathcal{H}}\frac{1}{\sigma^{2}}).$

Now, putting $\xi=W$ in (3), we have

$\displaystyle\frac{1}{2}\left\\{g_{M_{1}}(\nabla_{U_{1}}W,U_{2})+g_{M_{1}}(\nabla_{U_{2}}W,U_{1})\right\\}+Ric^{\nu}(U_{1},U_{2})$
$\displaystyle+g_{M_{1}}(U_{1},U_{2})\left\\{\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(\nabla\beta)\right\\}$
$\displaystyle=$ $\displaystyle 0.$

Taking trace, we have

$\displaystyle\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(\nabla_{U_{k}}U_{k},U_{k})+\sum\limits_{k=d_{2}+1}^{d_{1}}Ric^{v}(U_{k},U_{k})$
$\displaystyle-\left\\{\mu-(d_{1}-d_{2})|\nabla\beta|^{2}-div(\nabla\beta)\right\\}\sum\limits_{k=d_{2}+1}^{d_{1}}g_{M_{1}}(U_{k},U_{k})$
$\displaystyle=$ $\displaystyle 0,$

herein $\left\\{U_{k}\right\\}_{d_{2}+1\leq k\leq d_{1}}$ is an orthonormal
bases of $Ker\vartheta_{\ast}.$ Since $\nabla$ is metric connection, then (3)
can be written as

(3.39)
$s^{Ker\vartheta_{\ast}}-\left\\{\mu-(d_{1}-d_{2})|\nabla\beta|^{2}(d_{1}-d_{2})-div(\nabla\beta)\right\\}=0,$

where $s^{Ker\vartheta_{\ast}}$ is the scalar curvature of
$Ker\vartheta_{\ast}.$ Since $s^{Ker\vartheta_{\ast}}=-\mu(d_{1}-d_{2})$ then
(3.39) implies

(3.40)
$(d_{1}-d_{2})^{2}|\nabla\beta|^{2}+div(\nabla\beta)(d_{1}-d_{2})=0\Leftrightarrow\nabla\beta=0.$

Thus from (3.40) and Lemma 1 we obtain
$\tau(\vartheta)=0\Leftrightarrow\nabla\beta=0.$ This completes the proof.

######  Example 1.

Let $M_{1}=\left\\{(u_{1},u_{2},u_{3})\in\mathbb{R}^{3}:u_{1}\neq 0\right\\}$
be a Riemannian manifold with Riemannian metric $g_{M_{1}}$ on $M_{1}$ stated
$g_{M_{1}}=e^{-2u_{1}}du_{1}^{2}+e^{-2u_{1}}du_{2}^{2}+e^{-2u_{1}}du_{3}^{2}.$
Let $M_{2}=\left\\{(v_{1},v_{2})\in\mathbb{R}^{2}\right\\}$ be a Riemannian
manifold with Riemannian metric $g_{M_{2}}$ on $M_{2}$ stated
$g_{M_{2}}=e^{2u_{1}}dv_{1}^{2}+e^{2u_{1}}dv_{2}^{2}.$ Define a map
$\vartheta:(M_{1}^{d_{1}},g_{M_{1}})\rightarrow(M_{2}^{d_{2}},g_{M_{2}})$ by

$\vartheta(u_{1},u_{2},u_{3})=(u_{1},u_{2}).$

by direct computations

$Ker\vartheta_{\ast}=Span\left\\{V=e_{3}\right\\},$

and

$(Ker\vartheta_{\ast})^{\bot}=Span\left\\{X_{1}=e_{1},X_{2}=e_{2}\right\\},$

herein $\left\\{e_{1}=e^{u_{1}}\frac{\partial}{\partial
u_{1}},e_{2}=e^{u_{1}}\frac{\partial}{\partial
u_{2}},e_{3}=e^{u_{1}}\frac{\partial}{\partial u_{3}}\right\\}$ are bases of
$T_{p_{1}}M_{1}$ and $\left\\{e_{1}^{\ast}=e^{u_{1}}\frac{\partial}{\partial
v_{1}},e_{2}^{\ast}=e^{u_{1}}\frac{\partial}{\partial v_{2}}\right\\}$ are
bases of $T_{\vartheta(p_{1})}M_{2},$ for any $p_{1}\in M_{1}.$ By direct
computations, one can see that
$\vartheta_{\ast}X_{1}=e_{1}^{\ast},\vartheta_{\ast}X_{2}=e_{2}^{\ast},\vartheta_{\ast}V=0$
and
$g_{M_{2}}(\vartheta_{\ast}X_{i},\vartheta_{\ast}X_{j})=\sigma^{2}g_{M_{1}}(X_{i},X_{j}),$
for $\sigma=e^{2u_{1}}.$ Thus $\vartheta$ is conformal submersion with
dilation $\sigma=e^{u_{1}}.$ Now, we will find a smooth function $\beta$ on
$M_{1}$ satisfying $T_{V}V=-g_{M_{1}}(V,V)\nabla\beta$ for
$V\in\Gamma(Ker\vartheta_{\ast}).$ we can simply calculate that

(3.41) $\displaystyle\Gamma_{11}^{1}$ $\displaystyle=$
$\displaystyle-1,\Gamma_{11}^{2}=\Gamma_{11}^{3}=0,$
$\displaystyle\Gamma_{21}^{2}$ $\displaystyle=$
$\displaystyle-1,\Gamma_{21}^{1}=\Gamma_{21}^{3}=0,$
$\displaystyle\Gamma_{12}^{2}$ $\displaystyle=$
$\displaystyle-1,\Gamma_{12}^{1}=\Gamma_{12}^{3}=0,$
$\displaystyle\Gamma_{22}^{1}$ $\displaystyle=$ $\displaystyle
1,\Gamma_{22}^{2}=\Gamma_{22}^{3}=0,$ $\displaystyle\Gamma_{33}^{1}$
$\displaystyle=$ $\displaystyle 1,\Gamma_{33}^{2}=\Gamma_{33}^{3}=0,$
$\displaystyle\Gamma_{31}^{3}$ $\displaystyle=$
$\displaystyle-1,\Gamma_{31}^{1}=\Gamma_{31}^{2}=0,$
$\displaystyle\Gamma_{13}^{3}$ $\displaystyle=$
$\displaystyle-1,\Gamma_{13}^{1}=\Gamma_{13}^{2}=0,$
$\displaystyle\Gamma_{32}^{1}$ $\displaystyle=$
$\displaystyle\Gamma_{32}^{2}=\Gamma_{32}^{3}=0,$
$\displaystyle\Gamma_{23}^{1}$ $\displaystyle=$
$\displaystyle\Gamma_{23}^{2}=\Gamma_{23}^{3}=0.$

By using (3.41), we get

(3.42) $\displaystyle\nabla_{e_{1}}e_{1}$ $\displaystyle=$ $\displaystyle
0,\nabla_{e_{2}}e_{2}=e^{2u_{1}}\frac{\partial}{\partial
u_{1}},\nabla_{e_{3}}e_{3}=e^{2u_{1}}\frac{\partial}{\partial u_{1}},$
$\displaystyle\nabla_{e_{1}}e_{2}$ $\displaystyle=$ $\displaystyle
0,\nabla_{e_{2}}e_{1}=-e^{2u_{1}}\frac{\partial}{\partial
u_{2}},\nabla_{e_{3}}e_{1}=-e^{2u_{1}}\frac{\partial}{\partial u_{3}},$
$\displaystyle\nabla_{e_{3}}e_{2}$ $\displaystyle=$
$\displaystyle\nabla_{e_{1}}e_{3}=\nabla_{e_{2}}e_{3}=0.$

Using (3.42), we get

$\nabla_{V}V=e^{u_{1}}X_{1}.$

Its mean that $\mathcal{H}\nabla_{V}V=e^{u_{1}}X_{1},v\nabla_{V}V=0.$ By
(2.4), we have

$T_{V}V=e^{u_{1}}X_{1}.$

Similarly, if we take $X=\lambda_{1}X_{1}+\lambda_{2}X_{2}$ for
$\lambda_{1},\lambda_{2}\in\mathbb{R}$ then

$\nabla_{V}X=-\lambda_{1}e^{u_{1}}V.$

Its mean that $\mathcal{H}\nabla_{V}X=0,v\nabla_{V}X=-\lambda_{1}e^{u_{1}}V.$
Also,

$\nabla_{X}V=0,$

i.e. $\mathcal{H}\nabla_{X}V=0,v\nabla_{X}V=0.$ Then by (2.5), we have

(3.43) $S_{X}V=0,$

and

$\nabla_{X}X=\lambda_{2}^{2}e^{u_{1}}X_{1}-\lambda_{1}\lambda_{2}e^{u_{1}}X_{2},$

i.e.
$\mathcal{H}\nabla_{X}V=\lambda_{2}^{2}e^{u_{1}}X_{1}-\lambda_{1}\lambda_{2}e^{u_{1}}X_{2},v\nabla_{X}X=0.$
Then by (2.6), we have

(3.44) $S_{X}X=0.$

For any smooth function on $M_{1},$ the gradient of $\beta$ with respect to
the metric $g_{M_{1}}$ is given by
$\nabla\beta=\sum\limits_{i,j=1}^{3}g_{M_{1}}^{ij}\frac{\partial\beta}{\partial
u_{i}}\frac{\partial}{\partial u_{j}}.$ Therefore,

$\nabla\beta=e^{2u_{1}}\frac{\partial\beta}{\partial
u_{1}}\frac{\partial}{\partial u_{1}}+e^{2u_{1}}\frac{\partial\beta}{\partial
u_{2}}\frac{\partial}{\partial u_{2}}+e^{2u_{1}}\frac{\partial\beta}{\partial
u_{3}}\frac{\partial}{\partial u_{3}}.$

Hence,

$\displaystyle\nabla\beta$ $\displaystyle=$
$\displaystyle-e^{2u_{1}}\frac{\partial}{\partial u_{1}}$ $\displaystyle=$
$\displaystyle-e^{u_{1}}X_{1},$

for the function $\beta=-u_{1}.$ On the other hand, we have

$g_{M_{1}}(V,V)=1.$

Then it is easy to verify that $T_{V}V=-g_{M_{1}}(V,V)\nabla\beta.$ Thus, by
(2.7) and Theorem 4, we see that this map is CCS with $r=e^{-u_{1}}.$ Now, we
will show that $M_{1}$ admits a Ricci soliton, i.e.

(3.46) $\frac{1}{2}(L_{E}g_{M_{1}})(F,G)+Ric(F,G)+\mu g_{M_{1}}(F,G)=0,$

for any $E,F,G\in\Gamma(TM_{1}).$ Since here the dimension of
$Ker\vartheta_{\ast}$ is one and the dimension of
$(Ker\vartheta_{\ast})^{\bot}$ is two, therefore we can decompose $E,F$ and
$G$ such that
$F=\lambda_{1}V+\lambda_{2}X_{1}+\lambda_{3}X_{2},G=\lambda_{4}V+\lambda_{5}X_{1}+\lambda_{6}X_{2}$
and $E=\lambda_{7}V+\lambda_{8}X_{1}+\lambda_{9}X_{2},$ where $V$ denotes for
component of $Ker\vartheta_{\ast}$ and $X_{1},X_{2}$ denote for component of
$(Ker\vartheta_{\ast})^{\bot}$ and $\left\\{\lambda_{i}\right\\}_{1\leq i\leq
9}\in\mathbb{R}$ are some scalars. Now, since

$\frac{1}{2}(L_{E}g_{M_{1}})(F,G)=\frac{1}{2}\left\\{g_{M_{1}}(\nabla_{F}E,G)+g_{M_{1}}(\nabla_{G}E,F)\right\\},$

which is equal to

$\displaystyle\frac{1}{2}(L_{E}g_{M_{1}})(F,G)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\begin{array}[]{c}g_{M_{1}}(\nabla_{\lambda_{1}V+\lambda_{2}X_{1}+\lambda_{3}X_{2}}\lambda_{7}V+\lambda_{8}X_{1}+\lambda_{9}X_{2},\lambda_{4}V+\lambda_{5}X_{1}+\lambda_{6}X_{2})\\\
+g_{M_{1}}(\nabla_{\lambda_{4}V+\lambda_{5}X_{1}+\lambda_{6}X_{2}}\lambda_{7}V+\lambda_{8}X_{1}+\lambda_{9}X_{2},\lambda_{1}V+\lambda_{2}X_{1}+\lambda_{3}X_{2})\end{array}\right\\},$

which implies

(3.48)
$\frac{1}{2}(L_{E}g_{M_{1}})(F,G)=\frac{1}{2}\left\\{\begin{array}[]{c}e^{u_{1}}(\lambda_{1}\lambda_{5}\lambda_{7}-2\lambda_{1}\lambda_{4}\lambda_{8}\\\
-2\lambda_{3}\lambda_{6}\lambda_{8}+\lambda_{3}\lambda_{5}\lambda_{9}+\lambda_{2}\lambda_{4}\lambda_{7}+\lambda_{2}\lambda_{6}\lambda_{9})\end{array}\right\\}.$

Also,

(3.49)
$g_{M_{1}}(F,G)=\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6},$

and

$\displaystyle Ric(F,G)$ $\displaystyle=$
$\displaystyle\lambda_{1}\lambda_{4}Ric(V,V)+\lambda_{1}\lambda_{5}Ric(V,X_{1})+\lambda_{1}\lambda_{6}Ric(V,X_{2})$
$\displaystyle+\lambda_{2}\lambda_{4}Ric(X_{1},V)+\lambda_{2}\lambda_{5}Ric(X_{1},X_{1})+\lambda_{2}\lambda_{6}Ric(X_{1},X_{2})$
$\displaystyle+\lambda_{3}\lambda_{4}Ric(X_{2},V)+\lambda_{3}\lambda_{5}Ric(X_{2},X_{1})+\lambda_{3}\lambda_{6}Ric(X_{2},X_{2}).$

Since $\dim(Ker\vartheta_{\ast})=1,$ so $Ric^{v}(V,V)=0$ and using (3.43),
(3.44) and (1) in (4), we get

(3.51) $Ric(V,V)=e^{2u_{1}}.$

From (3.43) and (3.44) we have

(3.52) $Ric(V,X_{1})=Ric(V,X_{2})=0.$

From (4), we have

(3.53) $Ric(X_{1},X_{1})=e^{2u_{1}}$ (3.54) $Ric(X_{2},X_{2})=e^{2u_{1}}.$

Using (3.51), (3.52), (3.53) and (3.54) in (1), we have

(3.55)
$Ric(F,G)=e^{2u_{1}}\left\\{\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6}\right\\}.$

Now, by using (3.48), (3.49) and (3.55) in (3.46), we have

$\displaystyle\frac{1}{2}\left\\{\begin{array}[]{c}e^{u_{1}}(\lambda_{1}\lambda_{5}\lambda_{7}-2\lambda_{1}\lambda_{4}\lambda_{8}-2\lambda_{3}\lambda_{6}\lambda_{8}\\\
+\lambda_{3}\lambda_{5}\lambda_{9}+\lambda_{2}\lambda_{4}\lambda_{7}+\lambda_{2}\lambda_{6}\lambda_{9})\end{array}\right\\}$
$\displaystyle+e^{2u_{1}}\left\\{\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6}\right\\}+\mu(\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6})$
$\displaystyle=$ $\displaystyle 0.$

So, $(M_{1},g_{M_{1}})$ admit a Ricci soliton for

$\mu=-e^{2u_{1}}-\frac{e^{u_{1}}(\lambda_{1}\lambda_{5}\lambda_{7}-2\lambda_{1}\lambda_{4}\lambda_{8}-2\lambda_{3}\lambda_{6}\lambda_{8}+\lambda_{3}\lambda_{5}\lambda_{9}+\lambda_{2}\lambda_{4}\lambda_{7}+\lambda_{2}\lambda_{6}\lambda_{9})}{\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6}},$

where
$\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{5}+\lambda_{3}\lambda_{6}\neq
0.$Ricci soliton $(M_{1},g_{M_{1}})$ becomes expanding, steady or shrinking
for some values of $\lambda_{i}$’s with respect to $\mu>0,$ $\mu=0$ or
$\mu<0,$ respectively.

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M. Polat

Department of Mathematics, Faculty of Sci. Dicle University, 21280, Sur,
Diyarbakır-Turkey.
E-mail<EMAIL_ADDRESS>{M. Polat}