00footnotetext: $\mathbf{AMS\;2010\hskip
5.0ptMathematics\;Subject\;Classification\;}:$ 53C25,53D15.
Key words and phrases: Almost Kenmotsu manifold, Ricci soliton.

# Non-existence of Ricci solitons in almost Kenmotsu manifolds

U. C. De
Department of Pure Mathematics, University of Calcutta, $35$, Ballygunge
Circular Road, Kol-$700019$, West Bengal, INDIA<EMAIL_ADDRESS>
###### Abstract.

In this short note we prove that there does not exist Ricci soliton $(g,\xi)$
in an almost Kenmotsu manifold.

##  1\. Introduction

A Ricci soliton is a generalization of an Einstein metric. In a Riemannian
manifold $(M,g)$ a Ricci soliton is a triplet $(g,V,\lambda)$, with $g$, a
Riemannian metric, $V$ a smooth vector field (called the potential vector
field) and $\lambda$ a constant such that

(1.1) $\pounds_{V}g+2S-2\lambda g=0,$

where $\pounds_{V}g$ is the Lie derivative of $g$ along a vector field $V$ and
$S$ is the Ricci tensor of type $(0,2)$. If $V$ is zero or Killing, the the
soliton is an Einstein metric. The Ricci soliton is said to be shrinking,
steady or expanding according as $\lambda$ is positive, zero or negative,
respectively. Compact Ricci solitons are the fixed points of the Ricci flow
$\frac{\partial}{\partial t}g=-2S$ projected from the space of metrics onto
its quotient modulo diffeomorphisms and scalings. They often arise as blow-up
limits for the Ricci flow on compact manifolds. Metrics satisfying (1.1) are
very useful in physics. Theoretical physicists have also been looking into the
equation of Ricci soliton in relation with string theory. A Ricci soliton on a
compact manifold has constant curvature in dimension $2$ (Hamilton [14]) and
also in dimension $3$ (Ivey [15]).

Recently Cho [5] obtained some results about Ricci solitons in almost contact
and contact geometry. Also Ricci solitons have been studied by several authors
such as Bejan and Crasmareanu [1], Calin and Crasmareanu [4], Chow and Knopf
[6], De et al. [7], De and Mandal [10], Wang and Liu [19] and many others.

The purpose of this paper is to study Ricci soliton in an almost Kenmotsu
manifold. The present paper is organized as follows: Section 2 contains some
preliminary results of almost Kenmotsu manifolds. Finally, we prove that there
does not exist Ricci soliton in an almost Kenmotsu manifold.

##  2\. Almost Kenmotsu manifolds

A differentiable $(2n+1)$-dimensional manifold $M$ is said to have a
$(\phi,\xi,\eta)$-structure or an almost contact structure, if it admits a
$(1,1)$ tensor field $\phi$, a characteristic vector field $\xi$ and a 1-form
$\eta$ satisfying ([2],[3])

(2.1) $\phi^{2}=-I+\eta\otimes\xi,\;\eta(\xi)=1,$

where $I$ denote the identity endomorphism. Here also $\phi\xi=0$ and
$\eta\circ\phi=0$; both can be derived from $(2.1)$ easily.
If a manifold $M$ with a $(\phi,\xi,\eta)$-structure admits a Riemannian
metric $g$ such that

$g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y),$

for any vector fields $X$, $Y$ of $T_{p}M^{2n+1}$, then $M$ is said to have an
almost contact structure $(\phi,\xi,\eta,g)$. The fundamental 2-form $\Phi$ on
an almost contact metric manifold is defined by $\Phi(X,Y)=g(X,\Phi Y)$ for
any $X$, $Y$ of $T_{p}M^{2n+1}$. The condition for an almost contact metric
manifold being normal is equivalent to vanishing of the $(1,2)$-type torsion
tensor $N_{\phi}$, defined by $N_{\phi}=[\phi,\phi]+2d\eta\otimes\xi$, where
$[\phi,\phi]$ is the Nijenhuis torsion of $\phi$ [2]. Recently in
([11],[12],[13]), almost contact metric manifold such that $\eta$ is closed
and $d\Phi=2\eta\wedge\Phi$ are studied and they are called almost Kenmotsu
manifolds. Obviously, a normal almost Kenmotsu manifold is a Kenmotsu
manifold. Also Kenmotsu manifolds can be characterized by

$(\nabla_{X}\phi)Y=g(\phi X,Y)\xi-\eta(Y)\phi X,$

for any vector fields $X$, $Y$. It is well known [16] that a Kenmotsu manifold
$M^{2n+1}$ is locally a warped product $I\times_{f}N^{2n}$ where $N^{2n}$ is a
Kähler manifold, $I$ is an open interval with coordinate $t$ and the warping
function $f$, defined by $f=ce^{t}$ for some positive constant c. Let us
denote the distribution orthogonal to $\xi$ by $\mathcal{D}$ and defined by
$\mathcal{D}=Ker(\eta)=Im(\phi)$. In an almost Kenmotsu manifold, since $\eta$
is closed, $\mathcal{D}$ is an intregrable distribution. Let $M^{2n+1}$ be an
almost Kenmotsu manifold. We denote by $h=\frac{1}{2}\pounds_{\xi}\phi$ and
$l=R(\cdot,\xi)\xi$ on $M^{2n+1}$. The tensor fields $l$ and $h$ are symmetric
operators and satisfy the following relations :

(2.2) $h\xi=0,\;l\xi=0,\;tr(h)=0,\;tr(h\phi)=0,\;h\phi+\phi h=0,$ (2.3)
$\nabla_{X}\xi=-\phi^{2}X-\phi hX(\Rightarrow\nabla_{\xi}\xi=0),$ (2.4) $\phi
l\phi-l=2(h^{2}-\phi^{2}),$ (2.5) $R(X,Y)\xi=\eta(X)(Y-\phi hY)-\eta(Y)(X-\phi
hX)+(\nabla_{Y}\phi h)X-(\nabla_{X}\phi h)Y,$

for any vector fields $X$, $Y$. The $(1,1)$-type symmetric tensor field
$h^{\prime}=h\circ\phi$ is anticommuting with $\phi$ and $h^{\prime}\xi=0$.
Also it is clear that ([11], [18])

(2.6) $h=0\Leftrightarrow h^{\prime}=0,\;\;h^{\prime
2}=(k+1)\phi^{2}(\Leftrightarrow h^{2}=(k+1)\phi^{2}).$

Almost Kenmotsu manifolds have been studied by several authors such as Dileo
and Pastore ([11], [12], [13]), Wang and Liu ([17], [18], [20]), De and Mandal
([8], [9]) and many others.

Here we state a lemma which will be used latter.

######  Lemma 2.1.

[5] If $(g,V)$ is a Ricci soliton of a Riemannian manifold, then we have

(2.7) $\displaystyle\frac{1}{2}||\pounds_{V}g||^{2}=dr(V)+2div(\lambda V-QV),$

where $r$ denotes the scalar curvature and $Q$ denotes the Ricci operator
defined by $S(X,Y)=g(QX,Y)$.

##  3\. Ricci solitons

In this section we characterize almost Kenmotsu manifolds admitting a Ricci
soliton. Now suppose that an almost Kenmotsu manifold admits a Ricci soliton.
Here we consider the Ricci soliton whose potential vector field is the Reeb
vector field. Hence

(3.1) $\displaystyle\pounds_{\xi}g+2S-2\lambda g=0.$

i.e.,

(3.2)
$\displaystyle\frac{1}{2}[g(\nabla_{X}\xi,Y)+g(\nabla_{Y}\xi,X)]+S(X,Y)-\lambda
g(X,Y)=0.$

Using $(\ref{b2})$ and $(\ref{g1})$ in $(\ref{1})$ yields

$\displaystyle\frac{1}{2}[g(X-\eta(X)\xi-\phi hX,Y)+g(Y-\eta(Y)\xi-\phi
hY,X)]$ (3.3) $\displaystyle+S(X,Y)-\lambda g(X,Y)=0.$

Using $(\ref{b1})$ in $(\ref{2})$ implies

$\displaystyle\frac{1}{2}[2g(X,Y)-2\eta(X)\eta(Y)-2g(\phi
hX,Y)]+S(X,Y)-\lambda g(X,Y)=0,$

that is,

(3.4) $\displaystyle g(X,Y)-\eta(X)\eta(Y)-g(\phi hX,Y)+S(X,Y)-\lambda
g(X,Y)=0.$

From the above equation it follows that

(3.5) $\displaystyle QX=(\lambda-1)X+\eta(X)\xi+\phi hX.$

Replacing $X$ by $\xi$ in $(\ref{4})$ we have

(3.6) $\displaystyle Q\xi=\lambda\xi.$

Let $\\{e_{1},e_{2},e_{3},...,e_{2n+1}\\}$ be a local orthonormal basis of the
tangent space at a point of the manifold $M$. Then by putting $X=Y=e_{i}$ in
$(\ref{3})$ and taking summation over $i$, $1\leqslant i\leqslant(2n+1)$, we
have

(3.7) $r=\lambda(2n+1)-2n.$

Therefore the scalar curvature is constant as $\lambda$ is constant.
Since

(3.8) $\displaystyle\frac{1}{2}||\pounds_{V}g||^{2}=dr(V)+2div(\lambda V-QV),$

so replacing $V$ by $\xi$ and using the fact the sclar curvature $r$ is
constant, we get

(3.9) $\displaystyle\frac{1}{2}||\pounds_{\xi}g||^{2}=2div(\lambda\xi-Q\xi).$

Making use of $(\ref{5})$ in $(\ref{7})$ we obtain

$||\pounds_{\xi}g||^{2}=0,$

which implies that $\xi$ is a Killing vector field. But in an almost Kenmotsu
manifold $\xi$ can never be a Killing vector field.

This leads to the following:

######  Theorem 3.1.

There does not exist Ricci soliton $(g,\xi)$ in an almost Kenmotsu manifold.

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