$N(k)$-contact metric as $\ast$-conformal Ricci soliton

Dibakar Dey and Pradip Majhi

Department of Pure Mathematics,

University of Calcutta, 35 Ballygunge Circular Road,

Kolkata - 700019, West Bengal, India,

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: The aim of this paper is characterize a class of contact metric
manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a
$(2n+1)$-dimensional $N(k)$-contact metric manifold $M$ admits
$\ast$-conformal Ricci soliton or $\ast$-conformal gradient Ricci soliton,
then the manifold $M$ is $\ast$-Ricci flat and locally isometric to the
Riemannian of a flat $(n+1)$-dimensional manifold and an $n$-dimensional
manifold of constant curvature $4$ for $n>1$ and flat for $n=1$. Further, for
the first case, the soliton vector field is conformal and for the
$\ast$-gradient case, the potential function $f$ is either harmonic or satisfy
a Poisson equation. Finally, an example is presented to support the results.

Mathematics Subject Classification 2010: Primary 53D15; Secondary 53A30;
35Q51.

Keywords: $N(k)$-contact metric manifolds, $\ast$-Ricci tensor, Conformal
Ricci soliton, $\ast$-Conformal Ricci soliton.

##  1\.  Introduction

In 2004, Fischer [10] introduced the notion of conformal Ricci flow as a
variation of the classical Ricci flow equation. Let $M$ be an $n$-dimensional
closed, connected, oriented differentiable manifold. Then the conformal Ricci
flow on $M$ is defined by

$\displaystyle\frac{\partial g}{\partial
t}+2(S+\frac{g}{n})=-pg\;\;\mathrm{and}\;\;r=-1,$

where $p$ is a time dependent non-dynamical scalar field, $S$ is the $(0,2)$
symmetric Ricci tensor and $r$ is the scalar curvature of the manifold.

The concept of conformal Ricci soliton was introduced by Basu and
Bhattacharyya [1] on a $(2n+1)$-dimensional Kenmotsu manifold as

$\displaystyle\pounds_{V}g+2S=[2\lambda-(p+\frac{2}{2n+1})]g,$

where $\lambda$ is a constant and $\pounds_{V}$ is the Lie derivative along
the vector field V. This notion was studied by Dey and Majhi [7], Nagaraja and
Venu [13] and many others on several contact metric manifolds.

In 2002, Hamada [11] defined the $\ast$-Ricci tensor on real hypersurfaces of
complex space forms by

$\displaystyle S^{\ast}(X,Y)=g(Q^{\ast}X,Y)=\frac{1}{2}(trace(\phi\circ
R(X,\phi Y)))$

for any vector fields $X$, $Y$ on $M$, where $Q^{\ast}$ is the $(1,1)$
$\ast$-Ricci operator. The $\ast$-scalar curvature $r^{\ast}$ is defined by
$r^{\ast}=trace(Q^{\ast})$. A Riemannian manifold $M$ is called $\ast$-Ricci
flat if $S^{\ast}$ vanishes identically.

Recently, several notions related to the $\ast$-Ricci tensor were introduced.
In 2014, the notion of $\ast$-Ricci soliton [12] was introduced and further
widely studied by several authors. In 2019, the notion of $\ast$-critical
point equation [8] was introduced and further studied by the authors in [9].
In this paper, we study the notion of $\ast$-conformal Ricci soliton defined
as

######  Definition 1.1.

A Riemannian manifold $(M,g)$ of dimension $(2n+1)\geq 3$ is said to admit
$\ast$-conformal Ricci soliton $(g,V,\lambda)$ if

$\displaystyle\pounds_{V}g+2S^{\ast}=[2\lambda-(p+\frac{2}{2n+1})]g,$ (1.1)

where $\lambda$ is a constant, provided $S^{\ast}$ is symmetric.

A $\ast$-conformal Ricci soliton is said to be $\ast$-conformal gradient Ricci
soliton if the vector field $V$ is gradient of some smooth function $f$ on
$M$. In this case, the $\ast$-conformal gradient Ricci soliton is given by

$\displaystyle\nabla^{2}f+S^{\ast}=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]g,$
(1.2)

where $(\nabla^{2}f)(X,Y)=Hessf(X,Y)=g(\nabla_{X}Df,Y)$ is the Hessian of $f$
and $D$ is the gradient operator.

Note that, the $\ast$-Ricci tensor is not symmetric in general. Hence, for a
non-symmetric $\ast$-Ricci tensor of a manifold, the above notion is
incosistant. In a $N(k)$-contact metric manifold, $S^{\ast}$ is symmetric
(given later) and hence, the above definition is well defined on
$N(k)$-contact metric manifolds.

The present paper is organized as follows: In section 2, we recall some
preliminary results from the literature of $N(k)$-contact metric manifolds.
Section 3 deals with $N(k)$-contact metric manifolds admitting
$\ast$-conformal Ricci soliton and $\ast$-conformal gradient Ricci soliton. In
the final section, we present an example to verify our results.

##  2\. Preliminaries

A $(2n+1)$-dimensional almost contact metric manifold $M$ is a smooth manifold
together with a structure $(\phi,\xi,\eta,g)$ satisfying

$\displaystyle\phi^{2}X=-X+\eta(X)\xi,\;\;\eta(\xi)=1,\;\;\phi\xi=0,\;\;\eta\circ\phi=0,$
(2.1) $\displaystyle g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y)$ (2.2)

for any vector fields $X$, $Y$ on $M$, where $\phi$ is a $(1,1)$ tensor field,
$\xi$ is a unit vector field, $\eta$ is a one form defined by
$\eta(X)=g(X,\xi)$ and $g$ is the Riemannian metric. Using (2.2), we can
easily see that $\phi$ is skew-symmetric, that is,

$\displaystyle g(\phi X,Y)=-g(X,\phi Y).$ (2.3)

An almost contact metric structure becomes a contact metric structure if
$g(\phi X,Y)=d\eta(X,Y)$ for all vector fields $X$, $Y$ on $M$. On a contact
metric manifold, the $(1,1)$-tensor field $h$ is defined as
$h=\frac{1}{2}\pounds_{\xi}\phi$. The tensor field $h$ is symmetric and
satisfies

$\displaystyle h\phi=-\phi h,\;\;trace(h)=trace(\phi h)=0,\;\;h\xi=0.$ (2.4)

Also on a contact metric manifold, we have

$\displaystyle\nabla_{X}\xi=-\phi X-\phi hX.$ (2.5)

In [16], Tanno introduced the notion of $k$-nullity distribution on a
Riemannian manifold as

$\displaystyle N(k)=\\{Z\in T(M):R(X,Y)Z=k[g(Y,Z)X-g(X,Z)Y]\\},$

$k$ being a real number and $T(M)$ is the Lie algebra of all vector fields on
$M$. If the characteristic vector field $\xi\in N(k)$, then we call a contact
metric manifold as $N(k)$-contact metric manifold [16]. However, for a
$(2n+1)$-dimensional $N(k)$-contact metric manifold, we have ( see [2], [4])

$\displaystyle h^{2}=(k-1)\phi^{2},$ (2.6) $\displaystyle
R(X,Y)\xi=k[\eta(Y)X-\eta(X)Y],$ (2.7) $\displaystyle
R(\xi,X)Y=k[g(X,Y)\xi-\eta(Y)X],$ (2.8)
$\displaystyle(\nabla_{X}\eta)Y=g(X+hX,\phi Y),$ (2.9)
$\displaystyle(\nabla_{X}\phi)Y=g(X+hX,Y)\xi-\eta(Y)(X+hX),$ (2.10)
$\displaystyle(\nabla_{X}\phi h)Y$ $\displaystyle=$
$\displaystyle[g(X,hY)+(k-1)g(X,-Y+\eta(Y)\xi)]\xi$ (2.11)
$\displaystyle+\eta(Y)[hX+(k-1)(-X+\eta(X)\xi)]$

for any vector fields $X$, $Y$ on $M$, where $R$ is the Riemann curvature
tensor. For further details on $N(k)$-contact metric manifolds, we refer the
reader to go through thereferences ([5], [14], [15]) and references therein.

##  3\. $\ast$-Conformal Ricci soliton

In this section, we study the notion of $\ast$-conformal Ricci soliton in the
framework of $N(k)$-contact metric manifolds. To prove the main theorems, we
need the following lemmas:

######  Lemma 3.1.

([3]) A contact metric manifold $M^{2n+1}$ satisfying the condition
$R(X,Y)\xi=0$ for all $X$, $Y$ is locally isometric to the Riemannian product
of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of
positive curvature $4$, i.e., $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat
for $n=1$.

######  Lemma 3.2.

([8]) A $(2n+1)$-dimensional $N(k)$-contact metric manifold is
$\ast$-$\eta$-Einstein and the $\ast$-Ricci tensor is given by

$\displaystyle S^{\ast}(X,Y)=-k[g(X,Y)-\eta(X)\eta(Y)].$ (3.1)

######  Note 3.3.

We observe from lemma 3.2 that the $\ast$-Ricci tensor $S^{\ast}$ of a
$N(k)$-contact metric manifold is symmetric, i.e.,
$S^{\ast}(X,Y)=S^{\ast}(Y,X)$. Hence, the notion of $\ast$-conformal Ricci
soliton is consistent in this setting.

######  Lemma 3.4.

On a $(2n+1)$-dimensional $N(k)$-contact metric manifold $M$, the $\ast$-Ricci
tensor $S^{\ast}$ satisfies the following relation:

$\displaystyle(\nabla_{Z}S^{\ast})(X,Y)-(\nabla_{X}S^{\ast})(Y,Z)-(\nabla_{Y}S^{\ast})(X,Z)$
$\displaystyle=-2k[\eta(Y)g(\phi X,Z)+\eta(Y)g(\phi X,hZ)+\eta(X)g(\phi
Y,Z)+\eta(X)g(\phi Y,hZ)]$

for any vector fields $X$, $Y$ and $Z$ on $M$.

###### Proof.

Differentiating (3.1) covariantly along any vector field $Z$, we have

$\displaystyle\nabla_{Z}S^{\ast}(X,Y)=-k[\nabla_{Z}g(X,Y)-(\nabla_{Z}\eta(X))\eta(Y)-(\nabla_{Z}\eta(Y))\eta(X)].$
(3.2)

Now,

$\displaystyle(\nabla_{Z}S^{\ast})(X,Y)=\nabla_{Z}S^{\ast}(X,Y)-S^{\ast}(\nabla_{Z}X,Y)-S^{\ast}(X,\nabla_{Z}Y).$

Using (3.1) and (3.2) in the foregoing equation, we obtain

$\displaystyle(\nabla_{Z}S^{\ast})(X,Y)=k[((\nabla_{Z}\eta)X)\eta(Y)+((\nabla_{Z}\eta)Y)\eta(X)].$
(3.3)

Using (2.9) in (3.3), we infer that

$\displaystyle(\nabla_{Z}S^{\ast})(X,Y)=k[\eta(Y)g(Z+hZ,\phi
X)+\eta(X)g(Z+hZ,\phi Y)].$ (3.4)

In a similar manner, we get

$\displaystyle(\nabla_{X}S^{\ast})(Y,Z)=k[\eta(Z)g(X+hX,\phi
Y)+\eta(Y)g(X+hX,\phi Z)].$ (3.5)
$\displaystyle(\nabla_{Y}S^{\ast})(X,Z)=k[\eta(Z)g(Y+hY,\phi
X)+\eta(X)g(Y+hY,\phi Z)].$ (3.6)

With the help of (3.4)-(3.6), we complete the proof by using (2.3) and (2.4).
∎

######  Theorem 3.5.

Let $M$ be a $(2n+1)$-dimensional $N(k)$-contact metric manifold admitting
$\ast$-conformal Ricci soliton $(g,V,\lambda)$, then

  * $(1)$

The manifold $M$ is locally isometric to $E^{n+1}(0)\times S^{n}(4)$ for $n>1$
and flat for $n=1$.

  * $(2)$

The manifold $M$ is $\ast$-Ricci flat.

  * $(3)$

The vector field V is conformal,

provided $\lambda\neq\frac{p}{2}+\frac{1}{2n+1}$.

###### Proof.

From (1.1), we have

$\displaystyle(\pounds_{V}g)(X,Y)+2S^{\ast}(X,Y)=[2\lambda-(p+\frac{2}{2n+1})]g(X,Y).$
(3.7)

Differentiating the above equation covariantly along any vector field $Z$, we
get

$\displaystyle(\nabla_{Z}\pounds_{V}g)(X,Y)=-2(\nabla_{Z}S^{\ast})(X,Y).$
(3.8)

It is well known that (see [17])

$(\pounds_{V}\nabla_{X}g-\nabla_{X}\pounds_{V}g-\nabla_{[V,X]}g)(Y,Z)=-g((\pounds_{V}\nabla)(X,Y),Z)-g((\pounds_{V}\nabla)(X,Z),Y).$

Since $\nabla g=0$, then the above relation becomes

$\displaystyle(\nabla_{X}\pounds_{V}g)(Y,Z)=g((\pounds_{V}\nabla)(X,Y),Z)+g((\pounds_{V}\nabla)(X,Z),Y).$
(3.9)

Since $\pounds_{V}\nabla$ is symmetric, then it follows from (3.9) that

$\displaystyle g((\pounds_{V}\nabla)(X,Y),Z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\nabla_{X}\pounds_{V}g)(Y,Z)+\frac{1}{2}(\nabla_{Y}\pounds_{V}g)(X,Z)$
(3.10) $\displaystyle-\frac{1}{2}(\nabla_{Z}\pounds_{V}g)(X,Y).$

Using (3.8) in (3.10) we have

$g((\pounds_{V}\nabla)(X,Y),Z)=(\nabla_{Z}S^{\ast})(X,Y)-(\nabla_{X}S^{\ast})(Y,Z)-(\nabla_{Y}S^{\ast})(X,Z).$

Now, using Lemma 3.4 in the foregoing equation yields

$\displaystyle g((\pounds_{V}\nabla)(X,Y),Z)$ $\displaystyle=$ $\displaystyle
2k[\eta(Y)g(\phi X,Z)+\eta(Y)g(\phi X,hZ)$ $\displaystyle+\eta(X)g(\phi
Y,Z)+\eta(X)g(\phi Y,hZ)],$

which implies

$\displaystyle(\pounds_{V}\nabla)(X,Y)=2k[\eta(Y)\phi X+\eta(Y)h\phi
X+\eta(X)\phi Y+\eta(X)h\phi Y].$ (3.11)

Substituting $Y=\xi$ in (3.11), we get

$\displaystyle(\pounds_{V}\nabla)(X,\xi)=2k[\phi X+h\phi X].$ (3.12)

Differentiating (3.12) along any vector field $Y$, we obtain

$\displaystyle\nabla_{Y}(\pounds_{V}\nabla)(X,\xi)=2k[\nabla_{Y}\phi
X+\nabla_{Y}h\phi X].$ (3.13)

Now,

$\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,\xi)=\nabla_{Y}(\pounds_{V}\nabla)(X,\xi)-(\pounds_{V}\nabla)(\nabla_{Y}X,\xi)-(\pounds_{V}\nabla)(X,\nabla_{Y}\xi).$

Using (2.1)-(2.5) and (3.11)-(3.13) in the foregoing equation, we obtain

$\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,\xi)$ $\displaystyle=$
$\displaystyle 2k[(\nabla_{Y}\phi)X+(\nabla_{Y}h\phi)X+(k-2)\eta(X)\xi$ (3.14)
$\displaystyle-(k-2)\eta(X)\eta(Y)\xi-2\eta(X)hY].$

Now, using (2.10) and (2.11) in (3.14), we get

$\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,\xi)$ $\displaystyle=$
$\displaystyle 2k[kg(X,Y)\xi+(k-2)\eta(Y)X$ (3.15)
$\displaystyle+(k-2)\eta(X)Y-2\eta(X)hY$
$\displaystyle-2\eta(Y)hX-(3k-4)\eta(X)\eta(Y)\xi].$

Due to Yano [17], it is known that

$\displaystyle(\pounds_{V}R)(X,Y)Z=(\nabla_{X}\pounds_{V}\nabla)(Y,Z)-(\nabla_{Y}\pounds_{V}\nabla)(X,Z),$

Using the equation (3.15) in the above formula, we obtain

$\displaystyle(\pounds_{V}R)(X,\xi)\xi=(\nabla_{X}\pounds_{V}\nabla)(\xi,\xi)-(\nabla_{\xi}\pounds_{V}\nabla)(X,\xi)=0.$
(3.16)

Now, substituting $Y=\xi$ in (3.7), we have

$\displaystyle(\pounds_{V}g)(X,\xi)=[2\lambda-(p+\frac{2}{2n+1})]\eta(X),$
(3.17)

which implies

$\displaystyle(\pounds_{V}\eta)X-g(X,\pounds_{V}\xi)=[2\lambda-(p+\frac{2}{2n+1})]\eta(X).$
(3.18)

From (3.18), after putting $X=\xi$, we can easily obtain that

$\displaystyle\eta(\pounds_{V}\xi)=-[\lambda-(\frac{p}{2}+\frac{1}{2n+1})].$
(3.19)

Now, from (2.7), we have

$\displaystyle R(X,\xi)\xi=k(X-\eta(X)\xi).$ (3.20)

With the help of (3.18)-(3.20) and (2.7)-(2.8), we obtain

$\displaystyle(\pounds_{V}R)(X,\xi)\xi=k[2\lambda-(p+\frac{2}{2n+1})](X-\eta(X)\xi).$
(3.21)

Equating (3.16) and (3.21) and then taking inner product with $Y$ yields

$\displaystyle k[2\lambda-(p+\frac{2}{2n+1})](g(X,Y)-\eta(X)\eta(Y))=0,$

which implies $k=0$, since by hypothesis,
$\lambda\neq\frac{p}{2}+\frac{1}{2n+1}$. Therefore, from (3.1), we have
$S^{\ast}=0$, i.e., the manifold is $\ast$-Ricci flat. Again from (2.7), we
have $R(X,Y)\xi=0$ and hence, from lemma 3.1, it follows that the manifold $M$
is locally isometric to $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for
$n=1$.
Now, we know that a vector field $X$ on a Riemannian manifold $M$ is said to
be conformal if there is a smooth function $\sigma$ on $M$ such that
$\pounds_{X}g=2\sigma g$. Using $S^{\ast}=0$ in (3.7), we get
$\pounds_{V}g=2[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]g$ and hence $V$ is a
conformal vector field. ∎

######  Note 3.6.

If $\lambda=(\frac{p}{2}+\frac{1}{2n+1})$, then from (1.1), we can say that
the $\ast$-conformal Ricci soliton reduces to a steady $\ast$-Ricci soliton.
We need the following well known definition to discuss about this further.

######  Definition 3.7.

On an almost contact metric manifold $M$, a vector field $V$ is said to be
Killing if $\pounds_{V}g=0$ and an infinitesimal contact transformation if
$\pounds_{V}\eta=f\eta$ for some smooth function $f$ on $M$. In particular, if
$f=0$, then $V$ is said to be strict infinitesimal contact transformation.

######  Remark 3.8.

If $k\neq 0$ and $\lambda=(\frac{p}{2}+\frac{1}{2n+1})$, then from (3.18), we
have $(\pounds_{V}\eta)X=g(X,\pounds_{V}\xi)$. Thus $V$ will be an
infinitesimal contact transformation if $\pounds_{V}\xi=f\xi$ for some smooth
function $f$ on $M$. But in view of (3.19), we have $\eta(\pounds_{V}\xi)=0$,
which implies $\pounds_{V}\xi\bot\xi$. Hence $\pounds_{V}\xi\not=f\xi$ for any
smooth function $f$ on $M$, unless $f=0$ identically. Hence, $V$ cannot be an
infinitesimal contact transformation on $M$ but it can be a strict
infinitesimal contact transformation if $\pounds_{V}\xi=0$.

######  Remark 3.9.

If $k=0$ and $\lambda=(\frac{p}{2}+\frac{1}{2n+1})$, then from (3.7), we have
$\pounds_{V}g=0$. Hence $V$ is a Killing vector field.

To prove our next theorem regarding $\ast$-conformal gradient Ricci soliton,
we first state and prove the following lemma:

######  Lemma 3.10.

Let $M$ be $(2n+1)$-dimensional $N(k)$-contact metric manifold admitting
$\ast$-conformal gradient Ricci soliton $(g,V,\lambda)$. Then the curvature
tensor $R$ can be expressed as

$R(X,Y)Df=k[2g(\phi X,Y)\xi-\eta(X)(\phi Y+\phi hY)+\eta(Y)(\phi X+\phi hX)]$
(3.22)

for any vector fields $X$ and $Y$ on $M$, where $V=Df$.

###### Proof.

Equation (1.2) can be written as

$\displaystyle\nabla_{X}Df=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]X-Q^{\ast}X.$
(3.23)

Differentiating (3.23) along any vector field $Y$, we obtain

$\displaystyle\nabla_{Y}\nabla_{X}Df=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]\nabla_{Y}X-\nabla_{Y}Q^{\ast}X.$
(3.24)

Interchanging $X$ and $Y$ in the above equation, we get

$\displaystyle\nabla_{X}\nabla_{Y}Df=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]\nabla_{X}Y-\nabla_{X}Q^{\ast}Y.$
(3.25)

Again from (3.23), we have

$\displaystyle\nabla_{[X,Y]}Df=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})](\nabla_{X}Y-\nabla_{Y}X)-Q^{\ast}(\nabla_{X}Y-\nabla_{Y}X).$
(3.26)

It is well known that

$\displaystyle
R(X,Y)Df=\nabla_{X}\nabla_{Y}Df-\nabla_{Y}\nabla_{X}Df-\nabla_{[X,Y]}Df.$

Substituting (3.24)-(3.26) in the foregoing equation, we obtain

$\displaystyle R(X,Y)Df=(\nabla_{Y}Q^{\ast})X-(\nabla_{X}Q^{\ast})Y.$ (3.27)

Now, from (3.5) and (3.6), we can write

$\displaystyle(\nabla_{X}Q^{\ast})Y=k[g(X+hX,\phi Y)\xi-\eta(Y)(\phi X+\phi
hX)].$ (3.28) $\displaystyle(\nabla_{Y}Q^{\ast})X=k[g(Y+hY,\phi
X)\xi-\eta(X)(\phi Y+\phi hY)].$ (3.29)

We now complete the proof by substituting (3.28) and (3.29) in (3.27). ∎

######  Theorem 3.11.

Let $M$ be a $(2n+1)$-dimensional $N(k)$-contact metric manifold admitting
$\ast$-conformal gradient Ricci soliton $(g,V,\lambda)$, where $V=Df$ for some
smooth function $f$ on $M$, then

  * $(1)$

The Manifold $M$ is locally isometric to $E^{n+1}(0)\times S^{n}(4)$ for $n>1$
and flat for $n=1$.

  * $(2)$

The manifold $M$ is $\ast$-Ricci flat.

  * $(3)$

The potential function $f$ is either harmonic or satisfy a physical Poisson
equation.

###### Proof.

Putting $X=\xi$ in (3.22), we obtain

$\displaystyle R(\xi,Y)Df=-k[\phi Y+\phi hY].$

Taking inner product of the foregoing equation yields

$\displaystyle g(R(\xi,Y)Df,X)=-k[g(\phi X,Y)+g(\phi hX,Y)].$ (3.30)

Since $g(R(\xi,Y)Df,X)=-g(R(\xi,Y)X,Df)$, then using (2.8), we obtain

$\displaystyle g(R(\xi,Y)Df,X)=-kg(X,Y)(\xi f)+k\eta(X)(Yf).$ (3.31)

Equating (3.30) and (3.31) and then antisymmetrizing yields

$\displaystyle k\eta(X)(Yf)-k\eta(Y)(Xf)-2kg(\phi X,Y)=0.$

Putting $X=\xi$ in the above equation, we obtain

$\displaystyle k[(Yf)-(\xi f)\eta(Y)]=0,$

which implies

$\displaystyle k[Df-(\xi f)\xi]=0.$

Hence, either $k=0$ or $Df=(\xi f)\xi$.

Case 1: If $k=0$, then (3.1) implies $S^{\ast}=0$. Also from (2.7),
$R(X,Y)\xi=0$. Using lemma 3.1, we can say that $M$ is locally isometric to
$E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$. Again using
$S^{\ast}=0$ in (1.2) and then tracing yields $\Delta
f=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})](2n+1)$, where $\Delta$ is the Laplace
operator. Hence, $f$ satisfies a Poisson equation.

Case 2: If $Df=(\xi f)\xi$, then differentiating this along any vector field
$X$, we obtain

$\displaystyle\nabla_{X}Df=(X(\xi f))\xi+(\xi f)(-\phi X-\phi hX).$ (3.32)

Equating (3.23) and (3.32), we get

$\displaystyle Q^{\ast}X=[\lambda-(\frac{p}{2}+\frac{1}{2n+1})]X-(X(\xi
f))\xi+(\xi f)(\phi X+\phi hX).$ (3.33)

Comparing the coefficients of $X$, $\xi$ and $\phi X$ from (3.1) and (3.33),
we obtain the followings

$\displaystyle\lambda-(\frac{p}{2}+\frac{1}{2n+1})=-k.$ (3.34) $\displaystyle
X(\xi f)=k\eta(X).$ (3.35) $\displaystyle(\xi f)=0.$ (3.36)

Using (3.36) in (3.35), we obtain $k=0$. Hence, from (3.34),
$\lambda=\frac{p}{2}+\frac{1}{2n+1}$. Since $k=0$, by the same argument as in
case 1, the manifold $M$ is $\ast$-Ricci flat and locally isometric to
$E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$. Again using
$S^{\ast}=0$ and $\lambda=\frac{p}{2}+\frac{1}{2n+1}$ in (1.2), we obtain
$\nabla^{2}f=0$, which implies $\Delta f=0$. Therefore, $f$ is harmonic. This
completes the proof. ∎

##  4\. Example

In [6], the authors presented an example of a $3$-dimensional
$N(1-\alpha^{2})$-contact metric manifold. Using the expressions of the
curvature tensor and several values of the linear connection, we can easily
calculate the followings:

$S^{\ast}(e_{1},e_{1})=0,\;\;S^{\ast}(e_{2},e_{2})=S^{\ast}(e_{3},e_{3})=-(1-\alpha^{2}).$
$(\pounds_{e_{1}}g)(X,Y)=0\;\;\mathrm{for\;all}\;\;X,\;Y\in\\{e_{1},e_{2},e_{3}\\}.$

Now, if we consider $\alpha=1$, then the curvature tensor $R$ vanishes and
also $S^{\ast}=0$. Tracing (1.1), we get $\lambda=\frac{p}{2}+\frac{1}{3}$.
Thus $(g,e_{1},\lambda)$ is a $\ast$-conformal Ricci soliton on this
$N(0)$-contact metric manifold. Here, note that $e_{1}$ is a Killing vector
field. This verifies our theorem 3.5 and remark 3.9.
Again if $e_{1}=Df$ for some smooth function $f$, then tracing (1.2) and
considering $\alpha=1$, we have $\Delta
f=[\lambda-(\frac{p}{2}+\frac{1}{3})]3$, which is a Poisson equation. Also if
$\lambda=\frac{p}{2}+\frac{1}{3}$, then $\Delta f=0$ and therefore, $f$ is
harmonic. This verifies our theorem 3.11.

### Acknowledgements

The author Dibakar Dey is thankful to the Council of Scientific and Industrial
Research, India (File No. 09/028(1010)/2017-EMR-1) for their assistance.

## References

  * [1] N. Basu and A. Bhattacharyya, Conformal Ricci solition in Kenmotsu manifold, Glob. J. Adv. Res. Class. Mod. Geom. 4(2015), 15-21.
  * [2] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes on Mathematics, 509, Springer, Berlin, 1976.
  * [3] D. E. Blair, Two remarks on contact metric structure, Tohoku Math. J. 29(1977), 319-324.
  * [4] D. E. Blair, Riemannian geometry on contact and symplectic manifolds, Progress in Mathematics, 203, Birkh$\ddot{a}$user, Boston, 2010.
  * [5] A. De and J. B. Jun, On $N(k)$-contact metric manifolds satisfying certain curvature conditions, Kyungpook Math. J. 51(2011), 457-468.
  * [6] U. C. De, A. Yildiz and S. Ghosh, On a class of $N(k)$-contact metric manifolds, Math. Reports 16(2004), 207-217.
  * [7] D. Dey and P. Majhi, Almost Kenmotsu metric as conformal Ricci soliton, Conform. Geom. Dyn. 23(2019), 105-116.
  * [8] D. Dey and P. Majhi, $\ast$-Critical point equation on $N(k)$-contact manifolds, Bull. Transliv. Univ. Brasov. Ser. III 12(2019), 275-282.
  * [9] D. Dey and P. Majhi, $\ast$-Critical point equation on a class of almost Kenmotsu manifolds, J. Geom. 111(2020), no. 1, paper no. 16.
  * [10] A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav. 21(2004), 171-218.
  * [11] T. Hamada Real hypersurfaces of complex space forms in terms of Ricci $\ast$-tensor, Tokyo J. Math. 25(2002), 473-483.
  * [12] G. Kaimakanois and K. Panagiotidou, $\ast$-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86(2014), 408-413.
  * [13] H. G. Nagaraja and K. Venu, $f$-Kenmotsu metric as conformal Ricci soliton, An. Univ. Vest. Timis. Ser. Mat.-Inform. 55(2017), 119-127.
  * [14] C. $\ddot{O}$zg$\ddot{u}r$, Contact metric manifolds with cyclic-parallel Ricci tensor, Diff. Geom. Dynamical systems, 4(2002), 21-25.
  * [15] C. $\ddot{O}$zg$\ddot{u}$r and S. Sular, On $N(k)$-contact metric manifolds satisfying certain curvature conditions, SUT J. Math. 44(2008), 89-99.
  * [16] S. Tanno, Ricci curvature of contact Riemannian manifolds, Tohoku Math. J. 40(1988), 441-448.
  * [17] K. Yano Integral formulas in Riemannian Geometry, Marcel Dekker, New York, 1970.