CHARACTERISTICS OF CONFORMAL RICCI SOLITON ON WARPED PRODUCT SPACES

By

Dipen Ganguly111The first author D. Ganguly is the corresponding author and is
thankful to the National Board for Higher Mathematics (NBHM), India(Ref No:
0203/11/2017/RD-II/10440) for their financial support to carry on this
research work.,Nirabhra Basu2 and Arindam Bhattacharyya3

1,3Department of Mathematics

Jadavpur University,

Kolkata-700032, India.

E-mail<EMAIL_ADDRESS>
E-mail<EMAIL_ADDRESS>
2Department of Mathematics

Bhawanipur Education Society College,

Kolkata-700020, India.

E-mail<EMAIL_ADDRESS>
Abstract

Conformal Ricci solitons are self similar solutions of the conformal Ricci
flow equation. This paper deals with the study of conformal Ricci solitons
within the framework of warped product manifolds which extends the notion of
usual Riemannian product manifolds. First, we prove that if a warped product
manifold admits conformal Ricci soliton then the base and the fiber also share
the same property. In the next section the characterization of conformal Ricci
solitons on warped product manifolds in terms of Killing and conformal vector
fields has been studied. Next, we prove that a warped product manifold
admitting conformal Ricci soliton with concurrent potential vector field is
Ricci flat. Finally, an application of conformal Ricci soliton on a class of
warped product spacetimes namely, generalized Robertson-Walker spacetimes has
been discussed.

Key words : Ricci flow, Ricci soliton, conformal Ricci flow, conformal Ricci
soliton, Warped product, Spacetimes, Killing fields, concurrent vector fields.

2010 Mathematics Subject Classification : 53C15, 53C25, 53C44.

##  1\. Introduction

Hamilton’s [15],[8] theory of Ricci flow reached to a highest magnitude and
popularity after G.Perelman [20],[21] successfully applied it in solving the
Poincaré conjecture. The study of Ricci solitons was also introduced by
Hamilton as fixed or stationary points of the Ricci flow in the space of the
parameterized metrics ${g(t)}$ on $M$ modulo diffeomorphisms and scaling.
Since then, both the topics have been studied by many mathematicians like
Brendle [3], H. D. Cao [5], B.Y. Chen [7] etc. and many others
[24],[18],[19],[4].

A smooth manifold $M$ equipped with a Riemannian metric $g$ is said to be a
Ricci soliton, if for some constant $\lambda$, there exist a smooth vector
field $X$ on $M$ satisfying the equation

$Ric+\frac{1}{2}\mathcal{L}_{X}g=\lambda g,$

where $\mathcal{L}_{X}$ denotes the Lie derivative and $Ric$ is the Ricci
tensor. The Ricci soliton is called shrinking if $\lambda>0$, steady if
$\lambda=0$ and expanding if $\lambda<0$.

A. E. Fischer [12] has introduced conformal Ricci flow in 2004 as a modified
Ricci flow. It preserves the constant scalar curvature of the evolving
metrics. Because of the role of conformal geometry plays in maintaing scalar
curvature constant such a modified Ricci flow was named as conformal Ricci
flow.

The conformal Ricci flow equation on a smooth closed connected oriented
n-manifold, $n\geq 3$, is given by

$\frac{\partial g}{\partial
t}+2(Ric+\frac{g}{n})=-pg~{}~{}~{}and~{}~{}~{}r(g)=-1,$

where $p$ is a non-dynamical(time dependent) scalar field and $r(g)$ is the
scalar curvature of the manifold.The term $-pg$ acts as the constraint force
to maintain the scalar curvature constraint. Thus these evolution equations
are analogous to famous Navier-Stokes equations where the constraint is
divergence free. For this reason $p$ is also called the conformal pressure.

In 2015, Basu and Bhattacharyya [1] introduced the notion of conformal Ricci
soliton equation as

$\mathcal{L}_{X}g+2Ric=[2\lambda-(p+\frac{2}{n})]g,$ (1.1)

where $\lambda$ is constant and $p$ is the conformal pressure. The equation is
the generalization of the Ricci soliton equation and it satisfies the
conformal Ricci flow equation.

After N. Basu’s paper [1], many authors have studied conformal Ricci solitons
and obtained interesting results. D. Ganguly and A. Bhattacharyya studied the
conformal Ricci soliton within the framework of almost co-Kähler manifolds
[13] and $(LCS)_{n}$-manifolds [14] and obtained some interesting results. T.
Dutta et. al. in [10] showed that if a Lorentzian $\alpha$-Sasakian manifold
admits conformal Ricci soliton and is Weyl conformally semi-symmetric, then
the manifold is $\eta$-Einstein. Again in [9] it has been proved that a
$(k,\mu)^{{}^{\prime}}$-almost Kenmotsu manifold admitting conformal Ricci
soliton is either locally isometric to
$\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or the soliton is expanding, steady
or shrinking according to some conditions on the conformal pressure $p$. S.K.
Hui et. al. [16] proved that a conformal Ricci soliton on a three-dimensional
$f$-Kenmotsu manifold with torqued potential vector field is an almost quasi-
Einstein manifold. In [23] M. D. Siddiqui et al., established that a conformal
Ricci soliton, on a perfect fluid spacetime with torse-forming vector field
and without cosmological constant, is expanding. So, motivated by the above
studies, in this paper we investigate the nature of conformal Ricci soliton on
warped product spaces.

The concept of warped product was introduced by Bishop and O’Neill [2]. They
have given the examples of complete Riemannian manifolds with negative
sectional curvature. After that the study of warped product have been of great
interest among both mathematicians and physicists. Considering two Riemannian
manifolds $(B,g_{B})$ and $(F,g_{F})$ as well as a positive smooth function
$f$ on $B$,we define on the product manifold $B\times F$, the metric

$g=\pi^{*}g_{B}+(f\circ\pi)^{2}\sigma^{*}g_{F},$ (1.2)

where $\pi$ and $\sigma$ are the natural projections on $B$ and $F$
respectively. Under this conditions the product manifold is said to be the
warped product of $B$ and $F$; it is denoted by $M=B\times_{f}F$. Here the
manifold $B$ is called the base manifold and $F$ is called the fiber. The
function $f$ is called the warping function.

In our study, we shall focus on the conditions that makes the warped product
to a conformal Ricci soliton. To begin with, let us first recall a very
important result (for details see [2]) which will be required for our purpose
in later sections.

######  Lemma 1.1.

Let $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with dim$B=m$ and dim$F=n$.
Then for all $X,Y\in\mathfrak{X}(B)$ and $U,V\in\mathfrak{X}(F)$

  1. (1)

$D_{X}U=D_{U}X=\frac{X(f)}{f}U$,

  2. (2)

$Ric(X,U)=0$,

  3. (3)

$Ric(X,Y)=Ric^{B}(X,Y)-\frac{n}{f}H^{f}(X,Y)$,

  4. (4)

$Ric(U,V)=Ric^{F}(U,V)-(\frac{\Delta f}{f}+(n-1)\frac{\|\nabla
f\|^{2}}{f^{2}})g(U,V)$,

where $D_{X}Y$ is the lift of $\nabla_{X}Y$ on $B$ and $Ric^{B}$, $Ric^{F}$
are the lifts of the Ricci tensors on the base $B$ and the fiber $F$
respectively.

##  2\. Conformal Ricci soliton on warped product manifolds

This section deals with the investigations of conformal Ricci soliton on
warped product manifolds. Basically here we want to study if a warped product
manifold admits a conformal Ricci soliton then how its effect is on the base
manifold and on the fiber i.e; we try to find out under which conditions they
become conformal Ricci soliton. So, for this purpose let us first suppose that
$(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with dim$B=m$ and dim$F=n$.
Now let $(M,g,\mu,\xi)$ be a conformal Ricci soliton, where
$\mu=[2\lambda-(p+\frac{2}{n})]$. Then from equation $(1.1)$ we get

$\mathcal{L}_{\xi}g+2Ric=[2\lambda-(p+\frac{2}{n})]g=\mu g$ (2.1)

where $\mu=[2\lambda-(p+\frac{2}{n})]$. Again note that from the lemma $1.1$
of the previous section (for details see [2]) the following two well-known
formulas for warped product manifolds can be easily deduced

$\displaystyle\mathcal{L}_{\xi}g$ $\displaystyle=$
$\displaystyle{\mathcal{L}}^{B}_{\xi_{B}}g_{B}+f^{2}{\mathcal{L}}^{F}_{\xi_{F}}g_{F}+2f\xi_{B}(f)g_{F},$
(2.2) $\displaystyle Ric$ $\displaystyle=$ $\displaystyle
Ric^{B}-\frac{n}{f}H^{f}+Ric^{F}-\tilde{f}g_{F},$ (2.3)

where $\tilde{f}=f\Delta f+(n-1)\|\nabla f\|^{2}_{B}$. Now in equation $(2.1)$
using the definition of the warped metric from equation $(1.2)$ and then
applying the values from the above two equations $(2.2)$ and $(2.3)$ we have

$\displaystyle\mu(g_{B}+f^{2}g_{F})$ $\displaystyle=$ $\displaystyle\mu g$
(2.4) $\displaystyle=$ $\displaystyle\mathcal{L}_{\xi}g+2Ric$ $\displaystyle=$
$\displaystyle{\mathcal{L}}^{B}_{\xi_{B}}g_{B}+f^{2}{\mathcal{L}}^{F}_{\xi_{F}}g_{F}+2f\xi_{B}(f)g_{F}+2Ric^{B}$
$\displaystyle-2\frac{n}{f}H^{f}+2Ric^{F}-2\tilde{f}g_{F},$

Again for all $U,V\in\mathfrak{X}(B)$, using the definition of Lie derivation
we can write

$({\mathcal{L}}^{B}_{\xi_{B}}g_{B})(U,V)=g_{B}(D^{B}_{U}\xi_{B},V)+g_{B}(U,D^{B}_{V}\xi_{B}).$
(2.5)

Now from the definition of Hessian and the above equation $(2.5)$ we have

$({\mathcal{L}}^{B}_{\xi_{B}}g_{B}-2\frac{n}{f}H^{f})(U,V)=g_{B}(D^{B}_{U}\xi_{B},V)+g_{B}(U,D^{B}_{V}\xi_{B})-2\frac{n}{f}g_{B}(D^{B}_{U}\nabla^{B}f,V).$

The above equation can be rewritten as

$\displaystyle({\mathcal{L}}^{B}_{\xi_{B}}g_{B}-2\frac{n}{f}H^{f})(U,V)$
$\displaystyle=$
$\displaystyle(g_{B}(D^{B}_{U}\xi_{B},V)-\frac{n}{f}g_{B}(D^{B}_{U}\nabla^{B}f,V))$
(2.6)
$\displaystyle+(g_{B}(U,D^{B}_{V}\xi_{B})-\frac{n}{f}g_{B}(D^{B}_{U}\nabla^{B}f,V))$
$\displaystyle=$ $\displaystyle g_{B}(D^{B}_{U}(\xi_{B}-n\nabla^{B}\ln f),V)$
$\displaystyle+g_{B}(U,D^{B}_{V}(\xi_{B}-n\nabla^{B}\ln f)).$

Using the definition of Lie derivative again equation $(2.6)$ becomes

$({\mathcal{L}}^{B}_{\xi_{B}}g_{B}-2\frac{n}{f}H^{f})(U,V)=({\mathcal{L}}^{B}_{\xi_{B}-n\nabla^{B}\ln
f}g_{B})(U,V),\forall U,V\in\mathfrak{X}(B).$

Since the above equation is true for all $U,V\in\mathfrak{X}(B)$, in operator
notation we can write

${\mathcal{L}}^{B}_{\xi_{B}}g_{B}-2\frac{n}{f}H^{f}={\mathcal{L}}^{B}_{\xi_{B}-n\nabla^{B}\ln
f}g_{B}$ (2.7)

Now using the value from equation $(2.7)$, the equation $(2.4)$ finally yields

$({\mathcal{L}}^{B}_{\xi_{B}-n\nabla^{B}\ln
f}g_{B}+2Ric^{B})+(f^{2}{\mathcal{L}}^{F}_{\xi_{F}}g_{F}+2Ric^{F})\\\ =\mu
g_{B}+(\mu f^{2}-2f\xi_{B}(f)+2\tilde{f})g_{F}).$ (2.8)

Hence from the above discussion and equation $(2.8)$ we have the following
theorem

######  Theorem 2.1.

Let us consider that $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an
warped product of two Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with
warping function $f$, dim$B=m$ and dim$F=n$. If $(M,g,\mu,\xi)$ be a conformal
Ricci soliton, then the base $(B,g_{B},\mu,\xi_{B}-n\nabla^{B}\ln f)$ and the
fiber $(F,g_{F},\mu f^{2}-2f\xi_{B}(f)+2\tilde{f},f^{2}\xi_{F})$ are both
conformal Ricci solitons; where $\tilde{f}=f\Delta f+(n-1)\|\nabla
f\|^{2}_{B}$, $\mu=[2\lambda-(p+\frac{2}{n})]$, $\lambda$ is the soliton
constant and $p$ is the conformal pressure.

Now we study a special case when the soliton vector field $\xi$ of the
conformal Ricci soliton $(M,g,\mu,\xi)$ becomes gradient of some smooth
function $\phi$ i.e; when $\xi=grad\phi=\nabla\phi$. In this case we call the
soliton a conformal gradient Ricci soliton and the function $\phi$ is then
called the potential function of the soliton. Also for notational purpose
without any confusion we denote a conformal gradient Ricci soliton as
$(M,g,\mu,\phi)$, where the last term specifies the potential function of the
soliton.

Let us assume that $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped
product of two Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with dim$B=m$
and dim$F=n$. Then if $(M,g,\mu,\phi)$ be a conformal gradient Ricci soliton,
for any vector fields $X,Y\in\mathfrak{X}(M)$, equation $(1.1)$ implies

$2H^{\phi}(X,Y)+2Ric(X,Y)=[2\lambda-(p+\frac{2}{n})]g(X,Y)=\mu g(X,Y).$ (2.9)

Now if we take $X=X_{B}$ and $Y=Y_{B}$, where $X_{B},Y_{B}$ are the lifts of
the vector fields $X,Y$ in $\mathfrak{X}(B)$, then the equation $(2.9)$ gives
us

$2H^{\phi}(X_{B},Y_{B})+2Ric(X_{B},Y_{B})=\mu g(X_{B},Y_{B}).$

Using the value of the Ricci tensor for the base manifold from lemma $1.1$,
the above equation becomes

$2H^{\phi_{B}}_{B}(X_{B},Y_{B})+2Ric^{B}(X_{B},Y_{B})-2\frac{n}{f}H^{f}_{B}(X_{B},Y_{B})=\mu
g_{B}(X_{B},Y_{B}),$

where $\phi_{B}=\phi$ at a fixed point of the fiber $F$. Finally using the
properties of Hessian in the above equation we get

$2H^{\phi_{B}-n\ln f}_{B}(X_{B},Y_{B})+2Ric^{B}(X_{B},Y_{B})=\mu
g_{B}(X_{B},Y_{B}).$ (2.10)

This shows that $(B,g_{B},\mu,\phi_{B}-n\ln f)$ is a conformal gradient Ricci
soliton.

Again taking $X=X_{F}$ and $Y=Y_{F}$, where $X_{F},Y_{F}$ are the lifts of the
vector fields $X,Y$ in $\mathfrak{X}(F)$, then the equation $(2.9)$ gives us

$2H^{\phi}(X_{F},Y_{F})+2Ric(X_{F},Y_{F})=\mu g(X_{F},Y_{F}).$

Using equation $(2.3)$ and lemma $1.1$ the above equation becomes

$2H^{\phi_{F}}_{F}(X_{F},Y_{F})+2Ric^{F}(X_{F},Y_{F})-\tilde{f}g_{F}(X_{F},Y_{F})=\mu
f^{2}g_{F}(X_{F},Y_{F}),$

where $\phi_{F}=\phi$ at a fixed point of the base $B$ and $\tilde{f}=f\Delta
f+(n-1)\|\nabla f\|^{2}_{B}$. Thus finally we get from the above equation

$2H^{\phi_{F}}_{F}(X_{F},Y_{F})+2Ric^{F}(X_{F},Y_{F})=(\mu
f^{2}+\tilde{f})g_{F}(X_{F},Y_{F}).$

Therefore if the warping function $f$ is constant, the term $\tilde{f}=f\Delta
f+(n-1)\|\nabla f\|^{2}_{B}$ vanishes from the right hand side of the above
equation and we get the following

$2H^{\phi_{F}}_{F}(X_{F},Y_{F})+2Ric^{F}(X_{F},Y_{F})=\mu
f^{2}g_{F}(X_{F},Y_{F}).$ (2.11)

Thus $(F,g_{F},\mu f^{2},\phi_{F})$ is a conformal gradient Ricci soliton.
Hence from the above observations and equations $(2.10)$ and $(2.11)$ we can
state the following

######  Theorem 2.2.

Let $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function $f$,
dim$B=m$ and dim$F=n$. If $(M,g,\mu,\phi)$ be a conformal gradient Ricci
soliton, then

  1. (1)

the base $(B,g_{B},\mu,\phi_{B}-n\ln f)$ is a conformal gradient Ricci soliton
with $\phi_{B}=\phi$ at a fixed point of the fiber $F$.

  2. (2)

the fiber $(F,g_{F},\mu f^{2},\phi_{F})$ is a conformal gradient Ricci soliton
with $\phi_{F}=\phi$ at a fixed point of the base $B$, provided the warping
function $f$ is constant.

##  3\. Effect of certain special types of vector fields on conformal Ricci
soliton on warped product manifolds

The main purpose of this section is to study the effects of some special types
of smooth vector fields on conformal Ricci solitons on warped product spaces.
In particular, we will focus on Killing vector fileds, conformal vector fields
and concurrent vector fields. So, We have included some necessary definitions
before proceeding further.

######  Definition 3.1.

A smooth vector field $X$ on a Riemannian manifold $(M,g)$ is called

  1. (1)

a Killing vector field or an infinitesimal isometry, if the local
$1$-parameter group of transformations generated by $X$ in a neighbourhood of
each point of $M$ consists of local isometries, or in otherwords, if $X$
satisfies $\mathcal{L}_{X}g=0$ and

  2. (2)

a conformal vector field if $X$ satisfies $\mathcal{L}_{X}g=\rho g$,

where $\rho$ is a smooth function on the manifold $M$ and $\mathcal{L}_{X}g$
denotes the Lie differentiation of the Riemannian metric $g$ in the direction
of the vector field $X$.

So, based on the above definition our first result of this section is the
following

######  Proposition 3.1.

Let $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function $f$,
dim$B=m$ and dim$F=n$. If $(M,g,\mu,\xi)$ is a conformal Ricci soliton and any
one of the following conditions hold

  1. (1)

$\xi=\xi_{B}$ and $\xi_{B}$ is a Killing vector field on the base $B$.

  2. (2)

$\xi=\xi_{F}$ and $\xi_{F}$ is a Killing vector field on the fiber $F$.

Then the manifold $(M,g)$ becomes an Einstein manifold.

###### Proof.

As per our assumption $(M,g,\mu,\xi)$ being a conformal Ricci soliton, it
satisfies equation $(1.1)$ and we get

$\mathcal{L}_{\xi}g+2Ric=\mu g,$ (3.1)

Now let, $\xi=\xi_{B}$, and $\xi_{B}$ is Killing on $B$, we get
${\mathcal{L}}^{B}_{\xi_{B}}g_{B}=0$. Then using it in equation $(2.2)$ we
have $\mathcal{L}_{\xi}g=0$. Therefore equation $(3.1)$ gives us
$Ric=\frac{\mu}{2}g$ and this implies $(M,g)$ is Einstein manifold.

Again if $\xi=\xi_{F}$ and $\xi_{F}$ is a Killing vector field on the fiber
$F$, ${\mathcal{L}}^{F}_{\xi_{F}}g_{F}=0$. Then using equations $(2.2)$ and
$(3.1)$ and proceeding similarly as the first part of the proof, it can be
easily shown that in this case also $(M,g)$ is Einstein.

∎

######  Theorem 3.1.

Let $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function $f$,
dim$B=m$ and dim$F=n$. If $(M,g,\mu,\xi)$ is a conformal Ricci soliton and
$\xi_{B}$ is Killing vector field on the base $B$; then the base
$(B,g_{B},\mu,-n\ln f)$ is a conformal gradient Ricci soliton; where $\xi_{B}$
is the lift of the vector field $\xi$ to $\mathfrak{X}(B)$.

###### Proof.

Since it is given that $(M,g,\mu,\xi)$ is a conformal Ricci soliton from
theorem $2.1$ it follows that the base $(B,g_{B},\mu,\xi_{B}-n\nabla^{B}\ln
f)$ is also a conformal Ricci soliton and hence it satisfies equation $1.1$.
Thus we can write

${\mathcal{L}}^{B}_{\xi_{B}-n\nabla^{B}\ln f}g_{B}+2Ric^{B}=\mu g_{B}.$ (3.2)

Again using equation $(2.7)$ the above equation $(3.2)$ becomes

${\mathcal{L}}^{B}_{\xi_{B}}g_{B}-2\frac{n}{f}H^{f}+2Ric^{B}=\mu g_{B}.$

Now, as $\xi_{B}$ is Killing vector field on the base $B$, we have
${\mathcal{L}}^{B}_{\xi_{B}}g_{B}=0$. Thus with the help of this, the above
equation gives us

$-2\frac{n}{f}H^{f}+2Ric^{B}=\mu g_{B}.$

Thus using the properties of Hessian, the above equation finally yields

$2H^{-n\ln f}+2Ric^{B}=\mu g_{B}.$ (3.3)

Hence compairing the above equation $(3.3)$ with the conformal gradient Ricci
soliton equation $(2.9)$ completes the proof. ∎

We conclude this portion of study of Killing vector fields on conformal Ricci
soliton warped product manifolds with the following result

######  Theorem 3.2.

Assume that $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product
of two Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function
$f$, dim$B=m$ and dim$F=n$. Let $(M,g,\mu,\xi)$ be a conformal Ricci soliton
and both the lifts $\xi_{B}$ and $\xi_{F}$ are Killing on the base $B$ and the
fiber $F$ respectively. Then the manifold $(M,g)$ is Einstein if
$\xi_{B}(f)=0$.

###### Proof.

Since it is given that both $\xi_{B}$ and $\xi_{F}$ are Killing, we have
${\mathcal{L}}^{B}_{\xi_{B}}g_{B}=0$ and ${\mathcal{L}}^{F}_{\xi_{F}}g_{F}=0$.
Then using these values in equation $(2.2)$ we get

$\mathcal{L}_{\xi}g=2f\xi_{B}(f)g_{F}.$ (3.4)

Again as per our hypothesis $(M,g,\mu,\xi)$ being a conformal Ricci soliton,
from equation $(1.1)$ we get

$\mathcal{L}_{\xi}g+2Ric=\mu g.$

Now using equation $(3.4)$ in the above equation, gives us

$2f\xi_{B}(f)g_{F}+2Ric=\mu g.$ (3.5)

Thus if $\xi_{B}(f)=0$, the above equation $(3.5)$ yields
$Ric=\frac{\mu}{2}g$, which implies the manifold $(M,g)$ is Einstein and this
completes the proof. ∎

Now we shall focus on the effect of conformal vector fields on warped product
manifolds admitting conformal Ricci solitons. In this direction a very
immediate result is the following

######  Proposition 3.2.

Let $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of two
Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function $f$,
dim$B=m$ and dim$F=n$. Let $(M,g,\mu,\xi)$ is a conformal Ricci soliton. Then
the manifold $(M,g)$ becomes an Einstein manifold with factor
$(\frac{\mu}{2}-\rho)$ if and only if the vector field $\xi$ is conformal with
factor $2\rho$.

###### Proof.

$(M,g,\mu,\xi)$ being a conformal Ricci soliton, from $(1.1)$ we can write

$\mathcal{L}_{\xi}g+2Ric=\mu g$ (3.6)

Since the vector field $\xi$ is conformal with factor $2\rho$, by definition
$3.1$ we have $\mathcal{L}_{X}g=2\rho g$, where $\rho$ is a smooth function.
Thus using this value in equation $(3.6)$ finally we get

$Ric=(\frac{\mu}{2}-\rho)g.$ (3.7)

This implies $(M,g)$ is an Einstein manifold. Similarly by reverse calculation
process it can be shown that if $(M,g)$ is an Einstein manifold with factor
$(\frac{\mu}{2}-\rho)$ then $\xi$ becomes conformal with factor $2\rho$. This
completes the proof. ∎

It is to be noted that in the above result we have discussed on conformal
Ricci solitons with the vector field $\xi$ is taken conformal. So it is
natural to ask whether it is necessary to consider $\xi$ conformal as a whole,
or is there a weaker condition than this. The following theorem could put some
light on it.

######  Theorem 3.3.

Assume that $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product
of two Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function
$f$, dim$B=m$ and dim$F=n$. Let $(M,g,\mu,\xi)$ be a conformal Ricci soliton
and both the lifts $\xi_{B}$ and $\xi_{F}$ are conformal on the base $B$ and
the fiber $F$ with factors $2\rho_{B}$ and $2\rho_{F}$ respectively; where
$\rho_{B}$ and $\rho_{F}$ are two smooth functions. Then the manifold $(M,g)$
is Einstein provided $\rho_{B}=\rho_{F}+\xi_{B}(\ln f)$.

###### Proof.

Since $\xi_{B}$ is conformal on the base $B$ with factor $2\rho_{B}$, we have
${\mathcal{L}}^{B}_{\xi_{B}}g_{B}=2\rho_{B}g_{B}$. Also $\xi_{F}$ being
conformal with factor $2\rho_{F}$, we get
${\mathcal{L}}^{F}_{\xi_{F}}g_{F}=2\rho_{F}g_{F}$. Then using these two values
in equation $(2.2)$ we get

$\mathcal{L}_{\xi}g=2(\rho_{B}g_{B}+f^{2}\rho_{F}g_{F}+f\xi_{B}(f)g_{F}).$
(3.8)

Again $(M,g,\mu,\xi)$ being a conformal Ricci soliton, from equation $(1.1)$
and the above equation $(3.8)$ we have

$2(\rho_{B}g_{B}+f^{2}\rho_{F}g_{F}+f\xi_{B}(f)g_{F}+Ric)=\mu g.$

The above equation can be rewritten as

$Ric=\frac{\mu}{2}g-\rho_{B}g_{B}-f^{2}(\rho_{F}+\xi_{B}(\ln f))g_{F}.$ (3.9)

Hence if $\rho_{B}=\rho_{F}+\xi_{B}(\ln f)$, and using equation $(1.2)$, the
above equation $(3.9)$ finally gives us $Ric=(\frac{\mu}{2}-\rho_{B})g$. This
implies $(M,g)$ is Einstein and thus completes the proof. ∎

We end this section with our last theorem, which actually gives the converse
part of the previous theorem. In the previous result we characterised the
conformal Ricci soliton $(M,g,\mu,\xi)$ whereas our next result gives
conditions under which a warped product manifold $(M,g)$ admits a conformal
Ricci soliton.

######  Theorem 3.4.

Let $(B,g_{B},\mu,\xi_{B})$ be a conformal Ricci soliton and $(F,g_{F})$ be an
Einstein manifold with factor $\beta$, where dim$B=m$ and dim$F=n$. Let
$(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product of
$(B,g_{B})$ and $(F,g_{F})$ with warping function $f$ and $\xi_{F}$ is
conformal vector field with factor $2\rho$. Then $(M,g,\mu,\xi)$ is a
conformal Ricci soliton if $H^{f}=0$ and the warping function $f$ satisfies
the quadratic equation

$(2\rho-\mu)f^{2}+2f\xi_{B}(f)+2\beta+2(1-n)k^{2}=0,$

where $k^{2}=\|\nabla f\|^{2}_{B}=g_{B}(\nabla f,\nabla f)$ for some real
number $k$.

###### Proof.

$(B,g_{B},\mu,\xi_{B})$ being a conformal Ricci soliton, from equation $(1.1)$
we get

$\mathcal{L}^{B}_{\xi_{B}}g_{B}+2Ric^{B}=\mu g_{B}.$ (3.10)

Again as $(F,g_{F})$ is an Einstein manifold with factor $\beta$, the Ricci
tensor is given by $Ric^{F}=\beta g_{F}$. Using this value in equation $(2.3)$
gives us

$Ric=Ric^{B}-\frac{n}{f}H^{f}+\beta g_{F}-\tilde{f}g_{F},$ (3.11)

where $\tilde{f}=f\Delta f+(n-1)\|\nabla f\|^{2}_{B}$. Now, using equation
$(3.10)$ in the equation $(2.2)$ we get

$\mathcal{L}_{\xi}g=\mu
g_{B}-2Ric^{B}+f^{2}{\mathcal{L}}^{F}_{\xi_{F}}g_{F}+2f\xi_{B}(f)g_{F}.$
(3.12)

Multiplying both sides of the equation $(3.11)$ by $2$ and then adding it with
equation $(3.12)$ yields

$\mathcal{L}_{\xi}g+2Ric=\mu
g_{B}+f^{2}{\mathcal{L}}^{F}_{\xi_{F}}g_{F}+2f\xi_{B}(f)g_{F}+2(-\frac{n}{f}H^{f}+\beta
g_{F}-\tilde{f}g_{F}).$

Now since the vector field $\xi_{F}$ is conformal with factor $2\rho$ i.e;
${\mathcal{L}}^{F}_{\xi_{F}}g_{F}=2\rho g_{F}$, the above equation becomes

$\mathcal{L}_{\xi}g+2Ric=\mu g_{B}+2f^{2}\rho
g_{F}+2f\xi_{B}(f)g_{F}+2(-\frac{n}{f}H^{f}+\beta g_{F}-\tilde{f}g_{F}).$
(3.13)

As it is given that $H^{f}=0$, then it implies that $\Delta f=0$ and hence
$\tilde{f}=f\Delta f+(n-1)\|\nabla f\|^{2}_{B}$ becomes
$\tilde{f}=(n-1)\|\nabla f\|^{2}_{B}=(n-1)k^{2}$, where $k^{2}=\|\nabla
f\|^{2}_{B}=g_{B}(\nabla f,\nabla f)$ for some real number $k$. Therefore
using these results in the above equation $(3.13)$ we get

$\displaystyle\mathcal{L}_{\xi}g+2Ric$ $\displaystyle=$ $\displaystyle\mu
g_{B}+2f^{2}\rho g_{F}+2f\xi_{B}(f)g_{F}+2(\beta g_{F}-(n-1)k^{2}g_{F})$
$\displaystyle=$ $\displaystyle\mu(g_{B}+f^{2}g_{F})+(2f^{2}\rho-\mu
f^{+}2f\xi_{B}(f)+2(\beta-(n-1)k^{2}))g_{F}.$

Thus if $(2f^{2}\rho-\mu f^{2}+2f\xi_{B}(f)+2(\beta-(n-1)k^{2}))=0$ i.e; if
$f$ satisfies the quadratic equation
$(2\rho-\mu)f^{2}+2f\xi_{B}(f)+2\beta+2(1-n)k^{2}=0$; the above equation
finally becomes

$\mathcal{L}_{\xi}g+2Ric=\mu(g_{B}+f^{2}g_{F})=\mu g$ (3.14)

Therefore from equation $(3.14)$ we can conclude that $(M,g,\mu,\xi)$ is a
conformal Ricci soliton and this completes the proof. ∎

##  4\. Warped product manifolds admitting conformal Ricci soliton with
concurrent vector field

K. Yano [25] introduced concircular vector fields to study concircular
mappings, which are basically conformal mappings that preserve geodesic
circles. In Mathematical Physics and General Relativity concircular vector
fields have many applications. B.Y. Chen in [6] proved that a Loretzian
manifold is a generalised Robertson-Walker spacetime if and only if it admits
a timellike concircular vector field. A vector field $\xi$ on a Riemannian
manifold $M$ satisfying

$\nabla_{X}\xi=\alpha X,$ (4.1)

for all vector fields $X\in\mathfrak{X}(M)$, is called a concircular vector
field [7], where $\alpha$ is a non-trivial function on $M$. In particular, if
the function $\alpha$ is constant one then the vector field $\xi$ is called a
concurrent vector field. Thus we have the following definition [7]

######  Definition 4.1.

A vector field $\xi$ on a Riemannian manifold $M$ is called a concurrent
vector field if, for all vector fields $X\in\mathfrak{X}(M)$, the vector field
$\xi$ satisfies the following equation

$\nabla_{X}\xi=X.$ (4.2)

Based on the above definition, in this section we study conformal Ricci
solitons with the soliton vector field $\xi$ being concircular (also,
concurrent) vector field. So in this direction our first result is as follows

######  Theorem 4.1.

Let $(M,g,\mu,\xi)$ be a conformal Ricci soliton on an $n$-dimensional
Riemannian manifold $(M,g)$ and the soliton vector field $\xi$ is concircular
with factor $\alpha$, then

  1. (1)

the manifold $(M,g)$ is an Einstein manifold with factor $(\mu-2\alpha)$ and

  2. (2)

the soliton is expanding, steady or shrinking according as
$(p+2\alpha+\frac{1}{n})<0$, $(p+2\alpha+\frac{1}{n})=0$ or
$(p+2\alpha+\frac{1}{n})>0$ respectively.

###### Proof.

As per our assumption the soliton vector field $\xi$ is concircular with
factor $\alpha$, then from equation $(4.1)$ we get $\nabla_{X}\xi=\alpha X$.
Then using it in the definition of Lie differentiation we get

$\displaystyle(\mathcal{L}_{\xi}g)(X,Y)$ $\displaystyle=$ $\displaystyle
g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y}\xi)$ (4.3) $\displaystyle=$ $\displaystyle
g(\alpha X,Y)+g(X,\alpha Y)$ $\displaystyle=$ $\displaystyle 2\alpha g(X,Y)$

for all vector fields $X,Y\in\mathfrak{X}(M)$. Again, $(M,g,\mu,\xi)$ being a
conformal Ricci soliton, using the value from $(4.3)$ in the equation $(1.1)$
we get

$Ric(X,Y)=(\mu-2\alpha)g(X,Y),$ (4.4)

for all vector fields $X,Y\in\mathfrak{X}(M)$, and
$\mu=[2\lambda-(p+\frac{2}{n})]$. Thus equation $(4.4)$ proves that $(M,g)$
Einstein with factor $(\mu-2\alpha)$ and this completes the first part of the
theorem.

Again we know that for conformal Ricci flow, the scalar curvature $r(g)=-1$.
So taking an orthonormal basis $\\{e_{i}:1\leq i\leq n\\}$ of the manifold $M$
and summing over $1\leq i\leq n$ in both sides of the equation $(4.4)$ gives
us

$-1=r(g)=n(\mu-2\alpha).$

Finally using the value $\mu=[2\lambda-(p+\frac{2}{n})]$ in the above equation
and after simplification we get

$\lambda=\alpha+\frac{p}{2}+\frac{1}{2n}.$ (4.5)

We know that the soliton is expanding steady or shrinking if $\lambda<0$,
$\lambda=0$ or $\lambda>0$, thus applying it in equation $(4.5)$ completes the
proof. ∎

Next, we have a result on concurrent vector field, which immediately follows
from the above theorem.

######  Corollary 4.1.

$(M,g,\mu,\xi)$ be a conformal Ricci soliton with the soliton vector field
$\xi$ is concurrent, then

  1. (1)

the manifold $(M,g)$ is an Einstein manifold with factor $(\mu-2)$ and

  2. (2)

the soliton is expanding, steady or shrinking according as
$(p+2+\frac{1}{n})<0$, $(p+2+\frac{1}{n})=0$ or $(p+2+\frac{1}{n})>0$
respectively.

###### Proof.

Proceeding similarly as theorem $4.1$ and then putting $\alpha=1$ in the
equations $(4.4)$ and $(4.5)$ completes the proof. ∎

We conclude this section with the following theorem on concurrent vector field

######  Theorem 4.2.

Assume that $(M,g)=(B\times_{f}F,g_{B}\oplus f^{2}g_{F})$ be an warped product
of two Riemannian manifolds $(B,g_{B})$ and $(F,g_{F})$ with warping function
$f$, dim$B=m$ and dim$F=n$. Let $(M,g,\mu,\xi)$ be a conformal Ricci soliton
with concurrent vector field $\xi$. If $f$ is constant and both the lifts
$\xi_{B}$ and $\xi_{F}$ are concurrent on the base $B$ and the fiber $F$ then

  1. (1)

the soliton $(M,g,\mu,\xi)$ is expanding, steady or shrinking according as
$(\frac{p}{2}+\frac{1}{n}+1)<0$, $(\frac{p}{2}+\frac{1}{n}+1)=0$ or
$(\frac{p}{2}+\frac{1}{n}+1)>0$ respectively,

  2. (2)

all the three manifolds $M,B$ and $F$ are Ricci flat manifolds and

  3. (3)

all the three manifolds $M,B$ and $F$ admit conformal gradient Ricci solitons.

###### Proof.

Since $(M,g,\mu,\xi)$ is a conformal Ricci soliton on $M$ with concurrent
vector field $\xi$, from first part of the corollary $4.1$ we can write

$Ric(X,Y)=(\mu-2)g(X,Y),$ (4.6)

for all vector fields $X,Y\in\mathfrak{X}(M)$.
Now if we set $X=X_{F}$ and $Y=Y_{F}$, then from lemma $1.1$ and equation
$(2.3)$ we get

$Ric(X_{F},Y_{F})=Ric^{F}(X_{F},Y_{F})-\tilde{f}g_{F}(X_{F},Y_{F}),$ (4.7)

where $\tilde{f}=f\Delta f+(n-1)\|\nabla f\|^{2}_{B}$. Now using equation
$(4.6)$ and $(1.2)$, in the above equation $(4.7)$ yields

$Ric^{F}(X_{F},Y_{F})=\tilde{f}g_{F}(X_{F},Y_{F})+(\mu-2)f^{2}g_{F}(X_{F},Y_{F}),$

where $\tilde{f}=f\Delta f+(n-1)\|\nabla f\|^{2}_{B}$. Since it is given that
$f$ is constant, say $f=c$ for some constant $c$, then it implies that
$\tilde{f}=0$ and thus the above equation becomes

$Ric^{F}(X_{F},Y_{F})=c^{2}(\mu-2)g_{F}(X_{F},Y_{F}),$ (4.8)

for all vector fields $X_{F},Y_{F}\in\mathfrak{X}(F)$. Thus from the above
equation $(4.8)$ we can say that $F$ is Einstein. Now as the equation $(4.8)$
is true for any vector field in $\mathfrak{X}(F)$, by putting
$X_{F}=Y_{F}=\xi_{F}$ in above we get

$\displaystyle Ric^{F}(\xi_{F},\xi_{F})$ $\displaystyle=$ $\displaystyle
c^{2}(\mu-2)g_{F}(\xi_{F},\xi_{F})$ (4.9) $\displaystyle=$ $\displaystyle
c^{2}(\mu-2)\|\xi_{F}\|^{2}_{F}.$

Let $\\{\xi_{F},e_{1},e_{2},e_{3},....,e_{n-1}\\}$ be an orthonormal basis of
$\mathfrak{X}(F)$. Then the curvature tensor of the manifold $F$ is given by

$R^{F}(\xi_{F},e_{i},\xi_{F},e_{i})=g_{F}(R^{F}(\xi_{F},e_{i})\xi_{F},e_{i}).$

Using the well-known formula for curvature, the above equation can be
rewritten as

$R^{F}(\xi_{F},e_{i},\xi_{F},e_{i})=g_{F}(\nabla^{F}_{\xi_{F}}\nabla^{F}_{e_{i}}\xi_{F}-\nabla^{F}_{e_{i}}\nabla^{F}_{\xi_{F}}\xi_{F}-\nabla^{F}_{[\xi_{F},e_{i}]}\xi_{F},e_{i}).$
(4.10)

Also since $\xi_{F}$ is concurrent vector field, from equation $(4.2)$ we have
$\nabla_{X}\xi_{F}=X$, for all $X\in\mathfrak{X}(F)$ and using this in
equation $(4.10)$ we get

$R^{F}(\xi_{F},e_{i},\xi_{F},e_{i})=g_{F}(\nabla^{F}_{\xi_{F}}{e_{i}}-\nabla^{F}_{e_{i}}\xi_{F}-[\xi_{F},e_{i}],e_{i})=0.$

This implies $Ric^{F}(\xi_{F},\xi_{F})=0$ and then from equation $(4.9)$ we
get $\mu=2$, i.e; $\mu=[2\lambda-(p+\frac{2}{n})]=2$. After simplification
this gives $\lambda=(\frac{p}{2}+\frac{1}{n}+1)$ and the soliton is shrinking,
steady or expanding according as $\lambda>0$, $\lambda=0$ or $\lambda<0$. This
proves the first part of the theorem.

Now, using this value $\mu=2$ in equations $(4.8)$ and $(4.6)$ we have
$Ric=Ric^{F}=0$. This proves that both the manifolds $M$ and $F$ are Ricci
flat.
Again if we set $X=X_{B}$ and $Y=Y_{B}$, then from lemma $1.1$ we can write

$Ric(X_{B},Y_{B})=Ric^{B}(X_{B},Y_{B})-\frac{n}{f}H^{f}(X_{B},Y_{B}),$

for all $X_{B},Y_{B}\in\mathfrak{X}(B)$. Now since we just proved $Ric=0$, the
above equation becomes

$Ric^{B}(X_{B},Y_{B})=\frac{n}{f}H^{f}(X_{B},Y_{B}).$ (4.11)

Since we assumed that $f$ is constant, it implies $H^{f}=0$ and thus the above
equation $(4.11)$ finally gives us $Ric^{B}(X_{B},Y_{B})=0$, for all
$X_{B},Y_{B}\in\mathfrak{X}(B)$. Therefore we get $Ric^{B}=0$ and this proves
that the manifold $B$ is Ricci flat. This completes the proof of the second
part of the theorem.

To prove the last part of the theorem, let us assume that
$\phi=\frac{1}{2}g(\xi,\xi)$. Then

$g(X,grad\phi)=X(\phi)=g(\nabla_{X}\xi,\xi),$ (4.12)

for all $X\in\mathfrak{X}(M)$. Again $\xi$ being concurrent, from equation
$(4.2)$ we have $\nabla_{X}\xi=X$ and using this value in equation $(4.12)$ we
get

$g(X,grad\phi)=g(X,\xi)$

for all $X\in\mathfrak{X}(M)$. Since the above equation is true for any vector
field $X\in\mathfrak{X}(M)$, we can conclude that $\xi=grad\phi$. Hence
$(M,g)$ admits a conformal gradient Ricci soliton.
Again taking $\phi_{B}=\frac{1}{2}g(\xi_{B},\xi_{B})$ and
$\phi_{F}=\frac{1}{2}g(\xi_{F},\xi_{F})$ and proceeding similarly we can show
that $\xi_{B}=grad\phi_{B}$ and $\xi_{F}=grad\phi_{F}$. Also from theorem
$2.1$ we know that since $(M,g)$ is conformal Ricci soliton, $B$ and $F$ both
are conformal Ricci soliton. Hence can conclude that both the manifolds $B$
and $F$ admit conformal gradient Ricci soliton. ∎

##  5\. Application of conformal Ricci soliton on generalized Robertson-Walker
spacetimes

This section deals with the study of conformal Ricci solitons on a very well-
known warped spacetime called a generalized Robertson-Walker spacetime which
is an extension of the classical Robertson-Walker spacetimes. It is to be
noted that, generalized Robertson-Walker spacetimes also obey the Weyl
hypothesis, i.e; the world lines should be everywhere orthogonal to a family
of spacelike slices. M. Sánchez [22] characterized generalized Robertson-
Walker spacetimes in terms of timelike and spatially conformal conformal
vector fields. Also, a characterization of generalized Robertson-Walker
spacetimes in terms of timelike concircular vector field has been studied by
B.Y. Chen [6].

######  Definition 5.1.

A generalized Robertson-Walker spacetime is a warped product manifold
$M=I\times_{f}F$ endowed with the Lorentzian metric

$g=-dt^{2}\oplus f^{2}g_{F},$ (5.1)

where the base is an open interval $I$ of $\mathbb{R}$ with its usual metric
reversed $(I,-dt^{2})$, the fiber is an $n$-dimensional Riemannian manifold
$(F,g_{F})$ and the warping function is any positive function $f>0$ on $I$.

Based on the above definiton we will consider a generalized Robertson-Walker
spacetime and study the effect of conformal Ricci soliton on it. Our main
result of this section is the following

######  Theorem 5.1.

Let $M=I\times_{f}F$ be a generalized Robertson-Walker spacetime endowed with
the metric $g=-dt^{2}\oplus f^{2}g_{F}$ and let $\phi=\int_{c}^{t}f(z)dz$, for
some constant $c\in I$. If $(M,g,\mu,\phi)$ admits a conformal gradient Ricci
soliton, then

  1. (1)

the generalized Robertson-Walker spacetime $(M,g)$ becomes Ricci flat if the
soliton constant $\lambda$ satisfies the relation
$\lambda=\dot{f}+\frac{p}{2}+\frac{1}{n}$ and

  2. (2)

the generalized Robertson-Walker spacetime $(M,g)$ is an Einstein manifold if
the warping function $f$ is of the form $f(t)=at+b$, where $a,b$ are
constants.

###### Proof.

As per our assumption, $(M,g,\mu,\phi)$ being a conformal gradient Ricci
soliton, setting $\xi=grad\phi$, from equation $(1.1)$ we can write

$(\mathcal{L}_{\xi}g)(X,Y)+2Ric(X,Y)=\mu
g(X,Y)=[2\lambda-(p+\frac{2}{n})]g(X,Y),$ (5.2)

for all $X,Y\in\mathfrak{X}(M)$.
Again since $\phi=\int_{c}^{t}f(z)dz$, then $\xi=grad\phi$ implies that
$\xi=f(t)\frac{\partial}{\partial t}$ and it can be seen that the vector field
$\xi$ is orthogonal to the manifold $F$.
Let us assume that $\\{\frac{\partial}{\partial t},\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}},....,\frac{\partial}{\partial
x_{n}}\\}$ be an orthonormal basis of $\mathfrak{X}(M)$. Then the Hessian of
the function $\phi$ is given by

$H^{\phi}(X,Y)=g(\nabla_{X}grad\phi,Y).$ (5.3)

Now, we consider the following three cases.

Case 1: First let us consider $X=Y=\frac{\partial}{\partial t}$.
Then from equation $(5.3)$ we get

$\displaystyle H^{\phi}(\frac{\partial}{\partial t},\frac{\partial}{\partial
t})$ $\displaystyle=$ $\displaystyle g(\nabla_{\frac{\partial}{\partial
t}}grad\phi,\frac{\partial}{\partial t})$ (5.4) $\displaystyle=$
$\displaystyle\dot{f}g(\frac{\partial}{\partial t},\frac{\partial}{\partial
t}).$

Case 2: Next we consider $X=\frac{\partial}{\partial t}$ and
$Y=\frac{\partial}{\partial x_{i}}$ for $i=1,2,....,n$.
Then in this case equation $(5.3)$ implies

$\displaystyle H^{\phi}(\frac{\partial}{\partial t},\frac{\partial}{\partial
x_{i}})$ $\displaystyle=$ $\displaystyle g(\nabla_{\frac{\partial}{\partial
t}}grad\phi,\frac{\partial}{\partial x_{i}})$ (5.5) $\displaystyle=$
$\displaystyle\dot{f}g(\frac{\partial}{\partial t},\frac{\partial}{\partial
x_{i}}).$

Case 3: Finally we consider $X=\frac{\partial}{\partial x_{i}}$ and
$Y=\frac{\partial}{\partial x_{j}}$ for $1\leq i,j\leq n$.
Then from equation $(5.3)$ we have

$\displaystyle H^{\phi}(\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial x_{j}})$ $\displaystyle=$ $\displaystyle
g(\nabla_{\frac{\partial}{\partial x_{i}}}grad\phi,\frac{\partial}{\partial
x_{j}})$ (5.6) $\displaystyle=$ $\displaystyle
fg(\nabla_{\frac{\partial}{\partial x_{i}}}\frac{\partial}{\partial
t},\frac{\partial}{\partial x_{j}})$ $\displaystyle=$ $\displaystyle
fg(\frac{\dot{f}}{f}\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial
x_{j}})$ $\displaystyle=$ $\displaystyle\dot{f}g(\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial x_{j}}).$

Therefore combining equations $(5.4)$, $(5.5)$ and $(5.6)$ and using it in
$(5.3)$ we get

$H^{\phi}(X,Y)=\dot{f}g(X,Y).$ (5.7)

Now, since $\xi=grad\phi$, using the definition of Lie differentiation we can
write

$\displaystyle(\mathcal{L}_{\xi}g)(X,Y)$ $\displaystyle=$ $\displaystyle
g(\nabla_{X}grad\phi,Y)+g(X,\nabla_{Y}grad\phi)$ $\displaystyle=$
$\displaystyle 2H^{\phi}(X,Y).$

Thus using equation $(5.7)$, the above equation becomes

$(\mathcal{L}_{\xi}g)(X,Y)=2\dot{f}g(X,Y).$ (5.8)

Using the value of equation $(5.8)$ in the equation $(5.2)$ and after
simplification we get

$Ric(X,Y)=[\lambda-\dot{f}-\frac{p}{2}-\frac{1}{n}]g(X,Y).$ (5.9)

Thus if $\lambda=\dot{f}+\frac{p}{2}+\frac{1}{n}$, from equation $(5.9)$, it
implies that $(M,g)$ is Ricci flat. This completes the first part of the
theorem.

Again if $\dot{f}$ is a constant, say $\dot{f}=a$, i.e; if $df=adt$, i.e; if
$f=at+b$ for some arbitrary constant $b$, then from equation $(5.9)$ we can
conclude that $(M,g)$ is Einstein. This completes the proof. ∎

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