$*$-Ricci solitons and gradient almost $*$-Ricci solitons on Kenmotsu manifolds

TL;DR

This paper investigates $*$-Ricci and gradient almost $*$-Ricci solitons on Kenmotsu manifolds, establishing conditions for soliton constants, curvature, and Einstein properties, with explicit examples provided.

Contribution

It proves new results about the nature of $*$-Ricci solitons on Kenmotsu manifolds, including conditions for Einstein metrics and the behavior of potential vector fields.

Findings

Soliton constant $\\lambda$ is zero for $*$-Ricci solitons on Kenmotsu manifolds.

3-dimensional Kenmotsu manifolds with $*$-Ricci solitons have constant sectional curvature -1.

If potential vector field is collinear with characteristic vector field, the manifold is Einstein.

Abstract

In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $\lambda$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $\xi$, then $M$ is Einstein and soliton vector field is equal to $\xi$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, then either $M$ is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of $M$. We verify our result by constructing examples for both $*$-Ricci soliton and gradient almost $*$-Ricci soliton.

Equations

$${\pounds }_{V}g+2S+2\lambda g=0,$$

$$\text{Hess }f+S+\lambda g=0,$$

$$S^{\ast }(X,Y)=\frac{1}{2}trace(Z\to R(X,\phi Y)\phi Z),$$

$${\pounds }_{V}g+2S^{\ast }+2\lambda g=0.$$

$$\text{Hess }f+S^{\ast }+\lambda g=0.$$

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

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

$${\nabla }_X\xi =X-\eta (X)\xi ,$$

$$R(X,Y)\xi =\eta (X)Y-\eta (Y)X,$$

$$S(X,\xi )=-2n\eta (X),$$

$$({\pounds }_{\xi }g)(X,Y)=2\{g(X,Y)-\eta (X)\eta (Y)\},$$

$$S=\alpha g+\beta \eta \otimes \eta ,$$

$$\alpha +\beta =-2n.$$

$$\alpha =\left(\frac{r}{2n}+1\right),\beta =-\left(\frac{r}{2n}+2n+1\right).$$

$$S=\left(\frac{r}{2n}+1\right)g-\left(\frac{r}{2n}+2n+1\right)\eta \otimes \eta .$$

$$({\nabla }_{X}Q)\xi =-QX-2nX,$$

$$({\nabla }_{\xi }Q)X=-2QX-4nX,$$

$$({\nabla }_{W}R)(X,Y)\xi =-R(X,Y)W+g(X,W)Y-g(Y,W)X.$$

$$g(({\nabla }_{e_i}R)(e_i,Y)\xi ,Z)=S(Y,Z)+2ng(Y,Z).$$

$$g(({\nabla }_{e_i}R)(Z,\xi )Y,e_i)=g(({\nabla }_{Z}Q)\xi ,Y)-g(({\nabla }_{\xi }Q)Z,Y).$$

$$g(({\nabla }_{\xi }Q)Z,Y)=-2S(Y,Z)-4ng(Y,Z),$$

$$S^{\ast }=S+(2n-1)g+\eta \otimes \eta .$$

$$S^{\ast }(X,Y)=\frac{1}{2}trace(Z\to \phi R(X,\phi Y)Z),$$

$$S^{\ast }(X,Y)=\frac{1}{2}trace\{Z\to -\phi R(\phi Y,Z)X\}+\frac{1}{2}trace\{Z\to -\phi R(Z,X)\phi Y\}.$$

$$R(X,Y)\phi Z-\phi R(X,Y)Z=$$

$$+g(X,\phi Z)Y-g(Y,\phi Z)X,$$

$$R(\phi X,\phi Y)Z=R(X,Y)Z$$

$$+g(Y,\phi Z)\phi X-g(X,\phi Z)\phi Y.$$

$$g(R(X,Y)\phi Z,\phi W)-g(R(X,Y)Z,W)=g(Y,Z)g(X,W)-g(X,Z)g(Y,W)$$

$$+g(X,\phi Z)g(Y,\phi W)-g(Y,\phi Z)g(X,\phi W).$$

$$trace\{Z\to -\phi R(Z,X)\phi W\}=S(X,W)+(2n-1)g(X,W)+\eta (X)\eta (W).$$

$$\sum _{i=1}^{2n}g(R(\phi Y,\phi e_i)e_i,W)=S(Y,W)+(2n-1)g(Y,W)+\eta (Y)\eta (W),$$

$$\frac{1}{2}trace\{Z\to -\phi R(\phi Y,Z)X\}$$

$$=\frac{1}{2}\sum _{i=1}^{2n}g(R(\phi Y,\phi e_i)e_i,X).$$

$$({\pounds }_{V}g)(X,Y)=-2S(X,Y)-2(2n-1+\lambda )g(X,Y)-2\eta (X)\eta (Y).$$

Full text

The second author (D.M.N.) is grateful to University Grants Commission, New Delhi (Ref. No.:20/12/2015(ii)EU-V) for financial support in the form of Junior Research Fellowship.

1. Introduction

A Ricci soliton on a Riemannian manifold $(M,g)$ is defined by

[TABLE]

where $\pounds$ denotes the Lie derivative operator, $\lambda$ is a constant and $S$ is the Ricci tensor of the metric $g$. Ricci soliton is a natural generalization of the Einstein metric (that is, $S=ag$, for some constant $a$), and is a special self similar solution of the Hamilton’s Ricci flow (see [8]) $\frac{\partial}{\partial t}g(t)=-2S(t)$ with initial condition $g(0)=g$. We say that the Ricci soliton is shrinking when $\lambda<0$, steady when $\lambda=0$, and expanding when $\lambda>0$. If the vector field $V$ is the gradient of a smooth function $f$ (denoted by $Df$, where $D$ indicates the gradient operator), then $g$ is called a gradient Ricci soliton and in such a case (1.1) becomes

[TABLE]

where $\text{Hess }f$ is the Hessian of the smooth function $f$. Equations (1.1) and (1.2) are respectively called almost Ricci soliton and gradient almost Ricci soliton, if $\lambda$ is a variable smooth function on $M$. We recommend the reference [2] for more details about the Ricci flow and Ricci soliton. In this connection, we mention that within the framework of contact geometry Ricci solitons were first considered by Sharma in [12].

The notion of $*$-Ricci tensor was first introduced by Tachibana [13] on almost Hermitian manifolds and Hamada [7] apply this notion of $*$-Ricci tensor to almost contact manifolds defined by

[TABLE]

for any $X,Y\in\mathfrak{X}(M)$ (where $\mathfrak{X}(M)$ is the Lie algebra of all vector fields on $M$). In 2014, Kaimakamis and Panagiotidou [9] introduced the concept of $*$-Ricci solitons within the framework of real hypersurfaces of a complex space form, where they essentially modified the definition of Ricci soliton by replacing the Ricci tensor $S$ in (1.1) with the $*$-Ricci tensor $S^{*}$. More precisely, a Riemannian metric $g$ on a manifold $M$ is called a $*$-Ricci soliton if there exists a constant $\lambda$ and a vector field $V$ such that

[TABLE]

Moreover, if the vector field $V$ is a gradient of a smooth function $f$, then we say that it is gradient $*$-Ricci soliton and (1.3) becomes

[TABLE]

Note that $*$-Ricci soliton is trivial if the vector field $V$ is Killing, and in this case the manifold becomes $*$-Einstein (that is, $S^{*}=\alpha g$, for some function $\alpha$). If $\lambda$ appearing in (1.3) and (1.4) is a variable smooth function on $M$, then $g$ is called almost $*$-Ricci soliton and gradient almost $*$-Ricci soliton respectively.

Very recently in 2018, Ghosh and Patra [6] first undertook the study of $*$-Ricci solitons on almost contact metric manifolds. In their paper, the authors proved that if the metric of Sasakian manifold is a $*$-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Furthermore, they also proved that if a complete Sasakian metric is a gradient almost $*$-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere $S^{2n+1}$. Here we also mention the works of Prakasha and Veeresha [11] within the frame-work of paracontact geometry. This motivates the present authors to consider Kenmotsu manifolds whose metric as a $*$-Ricci soliton and gradient almost $*$-Ricci soliton.

The present paper is organized as follows: In Section 2, we recall some fundamental definitions related to Kenmotsu manifolds. Section 3 is devoted to the study of Kenmotsu manifolds whose metric is a $*$-Ricci soliton. We prove that if a Kenmotsu manifold $M$ admits a $*$-Ricci soliton, then the soliton constant $\lambda=0$. Moreover, we show that if $M$ is an $\eta$-Einstein manifold of dimension $>3$ admitting $*$-Ricci soliton, then $M$ is Einstein. For Kenmotsu 3-manifold admitting $*$-Ricci soliton, we prove that $M$ is of constant negative curvature $-1$. At the end of this section, we give an example of Kenmotsu manifold which admits a $*$-Ricci soliton, and verify our results. The final section deals with Kenmotsu manifold $M$ admitting gradient almost $*$-Ricci soliton, and we show that in such a case either $M$ is Einstein or the soliton vector field is collinear with characteristic vector field on an open set of $M$. The section ends with an example of a gradient almost $*$-Ricci soliton on a Kenmotsu 3-manifold.

2. Preliminaries

A $(2n+1)$-dimensional smooth manifold $M$ is said to have an almost contact structure if it admits a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$ (called the characteristic vector field or Reeb vector field), and a 1-form $\eta$ such that

[TABLE]

An immediate consequence of (2.1) is that $\varphi\xi=0$ and $\eta\circ\varphi=0$. It is well known that a $(2n+1)$ dimensional smooth manifold $M$ admits an almost contact structure if and only if the structure group of the tangent bundle of $M$ reduces to $U(n)\times 1$. For more details, we refer to [1].

If $M$ with $(\varphi,\xi,\eta)$-structure admits a Riemannian metric $g$ such that $g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),$ for all $X,Y\in\mathfrak{X}(M)$, then $(M,\varphi,\xi,\eta,g)$ is called an almost contact metric manifold. We consider the sign convension of the Riemannian curvature tensor as $R(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}$.

An almost contact metric manifold $(M,\varphi,\xi,\eta,g)$ is said to be Kenmotsu (see [10]) if

[TABLE]

for any $X,Y\in\mathfrak{X}(M)$. For a Kenmotsu manifold, we also have (see [10])

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$. Note that (2.6) implies that $\xi$ is not Killing in Kenmotsu manifold. We say $M$ is $\eta$-Einstein if the Ricci tensor satisfy

[TABLE]

for certain smooth function $\alpha$ and $\beta$. If $\beta=0$, then $M$ becomes an Einstein manifold. From (2.7) and (2.5), we have

[TABLE]

Contracting (2.7) and using (2.8), we get

[TABLE]

Thus, a Kenmotsu manifold $M$ is $\eta$-Einstein if and only if

[TABLE]

3. *-Ricci soliton on Kenmotsu manifolds

First we need the following lemmas.

Lemma 3.1**.**

A $(2n+1)$ dimensional Kenmotsu manifold satisfies

[TABLE]

where $Q$ is the Ricci operator defined by $S(X,Y)=g(QX,Y)$.

Proof.

Note that (2.5) implies $Q\xi=-2n\xi$. Differentiation this, and recalling (2.3) provides (3.1).

Now differentiating (2.4) along $W$ leads to

[TABLE]

Let $\{e_{i}\}_{i=1}^{2n+1}$ be a local orthonormal basis on $M$. Taking inner product of the above equation with $Z$ and then plugging $X=Y=e_{i}$ and summing over $i$ shows that

[TABLE]

From second Bianchi identity, one can easily obtain

[TABLE]

Fetching (3.4) in (3.3) and using (3.1), we obtain

[TABLE]

which proves (3.2). ∎

Now we derive the expression of $*$-Ricci tensor on Kenmotsu manifolds.

Lemma 3.2**.**

On a Kenmotsu manifold, the $*$-Ricci tensor is given by

[TABLE]

Proof.

From the definition of *-Ricci tensor we also have

[TABLE]

which by virtue of first Bianchi identity gives

[TABLE]

We now recall the following identities (see [10]):

[TABLE]

for any $X,Y,Z\in\mathfrak{X}(M)$. Taking inner product of (3.7) with $\varphi W$, and then making use of skew-symmetry of $\varphi$ and (2.1), we obtain

[TABLE]

Let $\{e_{i}\}_{i=1}^{2n+1}$ with $e_{2n+1}=\xi$ be a local orthonormal basis on $M$. Then putting $Y=Z=e_{i}$ in the above equation and summing over $i$ gives

[TABLE]

On the other hand it follows from (3.8) that

[TABLE]

where we used (2.4). Note that, if $\{e_{i}\}_{i=1}^{2n+1}$ with $e_{2n+1}=\xi$ is an orthonormal basis of vector fields on $M$, then $\{\varphi e_{i},\xi\}_{i=1}^{2n}$ is also a local orthonormal basis on $M$. Thus, as $\varphi\xi=0$, it follows that

[TABLE]

Now use of equations (3), (3.10) and (3.9) in (3.6) gives (3.5). ∎

Note that due to the presence of some extra terms in the expression of $*$-Ricci tensor, the defining condition of the $*$-Ricci soliton is different from Ricci soliton.

Theorem 3.1**.**

If the metric of a Kenmotsu manifold is a $*$-Ricci soliton, then the soliton constant $\lambda=0$.

Proof.

Feeding the expression of $*$-Ricci tensor as given by (3.5) into the $*$-Ricci soliton equation (1.3), it follows that

[TABLE]

Taking covariant derivative of (3.12) along an arbitrary vector field $Z$, we get

[TABLE]

From Yano [14], we know the following well known commutation formula:

[TABLE]

for all $X,Y,Z\in\mathfrak{X}(M)$. Since $\nabla g=0$, the above equation gives

[TABLE]

for all $X,Y,Z\in\mathfrak{X}(M)$. As $\pounds_{V}\nabla$ is a symmetric, it follows from (3.14) that

[TABLE]

Making use of (3) in (3) we have

[TABLE]

Plugging $Y=\xi$ in the above equation and using (3.1) and (3.2), we have

[TABLE]

Differentiating the above equation along $Y$ and using (2.3), we obtain

[TABLE]

Feeding the above obtained expression into the following well known formula (see Yano [14])

[TABLE]

and using the symmetry of $\pounds_{V}\nabla$, we immediately obtain

[TABLE]

Substituting $Y=\xi$ in the above equation, we get

[TABLE]

Now taking the Lie-derivative of $R(X,\xi)\xi=-X+\eta(X)\xi$ along $V$ gives

[TABLE]

which by virtue of (3.18) becomes

[TABLE]

With the help of (2.5), the equation (3.12) takes the form

[TABLE]

Substituting $X=\xi$ in the above equation gives

[TABLE]

Now Lie-differentiating $\eta(X)=g(X,\xi)$ yields $(\pounds_{V}\eta)(X)=(\pounds_{V}g)(X,\xi)+g(X,\pounds_{V}\xi)$. Using this and (3.21) in (3.19) provides $\lambda(nX-\eta(X)\xi)=0.$ Tracing the previous equation yield $\lambda=0$. ∎

Lemma 3.3**.**

If the metric of a Kenmotsu manifold is a $*$-Ricci soliton, then the Ricci tensor satisfy

[TABLE]

Proof.

Contracting the equation (3) with respect to $X$ and recalling the following well-known formulas

[TABLE]

we easily obtain

[TABLE]

Substituting $Y=\xi$, we have $(\pounds_{V}S)(\xi,\xi)=-\xi(r)-2\{r+2n(2n+1)\}$. On the other hand, contracting (3.18) gives $(\pounds_{V}S)(\xi,\xi)=0.$ Using this in the previous equation leads to

[TABLE]

Hence (3.24) and (3.23) gives (3.22). ∎

Lemma 3.4**.**

The scalar curvature $r$ of an $\eta$-Einstein Kenmotsu manifold $M$ of dimension $>3$ satisfies

[TABLE]

Proof.

Since $M$ is $\eta$-Einstein, from (2.9) we have

[TABLE]

Using (3.26), (2.5) and (2.4), one can easily verify that

[TABLE]

which also gives

[TABLE]

Putting $Y=\xi$ in (3) and then differentiating it along $W$ and using (3), we get

[TABLE]

Taking inner product of above equation with $Y$ and contracting with respect to $X$ and $W$ yields

[TABLE]

where $\{e_{i}\}$ is a local orthonormal basis on $M$. From second Bianchi identity we easily obtain

[TABLE]

Then from (3) and (3.30) and noting that $n>1$ we get

[TABLE]

Since $\nabla Q$ is symmetric, the above equation becomes

[TABLE]

On the other hand from (3.1) and (3.2), the left hand side of above equation vanishes. Then (3.31) leads to $Yr=\eta(Y)\xi(r)$ which gives (3.25). ∎

Theorem 3.2**.**

Let $(M,\varphi,\xi,\eta,g)$ be an $\eta$-Einstein Kenmotsu manifold of dimension $>3$. If $g$ is a $*$-Ricci soliton, then $M$ is Einstein.

Proof.

Making use of (3.25) in (3.22), we have $(\pounds_{V}S)(X,\xi)=0$. Now, Lie-differentiating (2.5) along $V$, using (2.9), (3.20), $\lambda=0$ and $\eta(\pounds_{V}\xi)=0$, we obtain

[TABLE]

Suppose if $r=-2n(2n+1)$, then (2.9) shows that $M$ is Einstein.

So that we assume $r\neq-2n(2n+1)$ in some open set $\mathcal{O}$ of $M$. Therefore on $\mathcal{O}$, we have $\pounds_{V}\xi=0$, and so it follows from (2.3) that

[TABLE]

Clearly, (3.20) shows that $(\pounds_{V}g)(X,\xi)=0$ for any $X\in TM$. This together with (3.32), it follows that

[TABLE]

From Duggal and Sharma [3], we know the following formula

[TABLE]

Setting $Y=\xi$ in the above equation and by virtue of (2.3), (2.4), (3.32) and (3.33), we have $\xi r=0$, which with the help of (3.24) shows that $r=-2n(2n+1)$. This leads to a contradiction and completes the proof. ∎

Now we consider Kenmotsu 3-manifolds which admits $*$-Ricci solitons.

Theorem 3.3**.**

If the metric of a Kenmotsu 3-manifold is a $*$-Ricci soliton, then $M$ is of constant negative curvature $-1$.

Proof.

It is well known that the Riemannian curvature tenor of $3$-dimensional Riemannian manifold is given by

[TABLE]

Putting $Y=Z=\xi$ in (3) and using (2.4) and (2.5) gives

[TABLE]

Making use of above equation in (3) gives

[TABLE]

Replacing $Y$ by $\xi$ in the above equation and comparing it with (3.18), we obtain

[TABLE]

Contracting the above equation with respect to $X$ gives $\xi(r)=0$, and consequently it follows from (3.24) that $r=-6$. Then from (3.35) we have $QX=-2X$, and substituting this in (3) shows that $M$ is of constant negative curvature $-1$. ∎

Remark 1**.**

In [4], Ghosh proved that, if $(M,g)$ is a Kenmotsu manifold of dimension $3$ and if $g$ is a Ricci soliton, then $M$ is of constant negative curvature $-1$, and here we have the same conclusion when $g$ is a $*$-Ricci soliton. Note that our approach and technique to obtain the result is different to that of Ghosh.

According to Kenmotsu [10], the warped product space $\mathbb{R}\times_{f}K$, where $f(t)=ce^{t}$ on the real line $R$ and $K$ is Kähler manifold, admits a Kenmotsu structure. Consequently, we have the following result.

Corollary 3.3.1**.**

Consider the warped product $M=\mathbb{R}\times_{f}K$, where $f(t)=ce^{t}$ and $K$ is a 2-dimensional Kähler manifold. If $M$ admits a $*$-Ricci soliton, then $M$ locally a hyperbolic space $\mathbb{H}^{3}(-1)$.

We know that Kenmotsu manifold $M$ do not admit a Ricci soliton whose soliton vector field is equal to the characteristic vector field. Because, if $V=\xi$, then from (2.6) the Ricci soliton equation (1.1) would become

[TABLE]

which means $M$ is $\eta$-Einstein. But according to Ghosh (see Theorem 1 of [5] and Theorem 1 of [4]) $M$ must Einstein, and this will be a contradiction to equation (3.37). Thus in our study an interesting case arises when the soliton vector field $V$ of the *-Ricci soliton is equal to $\xi$, moreover, when $V$ is pointwise collinear with $\xi$. We consider this here and we prove the following.

Theorem 3.4**.**

*If the metric of a Kenmotsu manifold is a -Ricci soliton with a non-zero potential vector field $V$ pointwise collinear with the characteristic vector field $\xi$, then $M$ is Einstein and $V=\xi$.

Proof.

Observing the equation (3.12), keeping in mind that $\lambda=0$ gives

[TABLE]

Putting $V=a\xi$, $a$ being a smooth function on $M$, in the above equation we have

[TABLE]

Substituting $Y=\xi$ and using (2.3) and (2.5), the above equation becomes

[TABLE]

This means $a$ is invariant along the distribution $Ker\eta$, that is, $X(a)=0$ for all $X\in Ker\eta$. Replacing $X$ by $\xi$ in the above equation, we have

[TABLE]

Thus $a$ is constant, and consequently we have

[TABLE]

Making use of (2.6) in the above equation gives

[TABLE]

which means $M$ is $\eta$-Einstein. Then it follows from Theorem 3.2 and Theorem 3.3 that $M$ is Einstein. Hence (3.39) gives $a=1$ and the result follows. ∎

Now we provide an example of a Kenmotsu 3-manifold which admits a $*$-Ricci soliton and verify our results.

Example** 1****.**

Let $M=N\times I$, where $N$ is an open connected subset of $\mathbb{R}^{2}$ and $I$ is an open interval in $\mathbb{R}$. Let $(x,y,z)$ be the Cartesian coordinates in it. Define the structure $(\varphi,\xi,\eta,g)$ on $M$ as follows:

[TABLE]

Now from Koszul’s formula, Levi-Civita connection $\nabla$ is given by

[TABLE]

where $\partial_{x}=\frac{\partial}{\partial x},\partial_{y}=\frac{\partial}{\partial y}$ and $\partial_{z}=\frac{\partial}{\partial z}$. From (3.40), one can easily verify (2.2), and so $(M,\varphi,\xi,\eta,g)$ is a Kenmotsu manifold.

With the help of (3.40), we find the following:

[TABLE]

Let $e_{1}=e^{-z}\partial_{x},e_{2}=e^{-z}\partial_{y}$ and $e_{3}=\xi=\partial_{z}$. Clearly, $\{e_{1},e_{2},e_{3}\}$ forms an orthonormal $\varphi$-basis of vector fields on $M$. Making use of (3.41) one can easily show that $M$ is Einstein, that is, $S(X,Y)=-2ng(X,Y)$, for any $X,Y\in\mathfrak{X}(M)$. Also from the definition of *-Ricci tensor it is not hard to see that

[TABLE]

for any $X,Y\in\mathfrak{X}(M)$.

Let us consider the vector field

[TABLE]

where $a\neq 1$ is a constant. Using (3.40) one can easily verify that

[TABLE]

for any $X,Y\in\mathfrak{X}(M)$. Combining (3.44) and (3.42), we obtain that $g$ is a $*$-Ricci soliton, that is, (1.3) holds true with $V$ as in (3.43) and $\lambda=0$. Further (3.41) shows that $R(X,Y)Z=-\{g(Y,Z)X-g(X,Z)Y\}$ for any $X,Y,Z\in\mathfrak{X}(M)$, which means $M$ is of constant negative curvature $-1$ and this verifies the Theorem 3.3.

4. Gradient almost $*$-Ricci solitons on Kenmotsu manifolds

First, we prove the following result.

Lemma 4.1**.**

If the metric of a Kenmotsu manifold is a gradient almost $*$-Ricci soliton, then the Riemannian curvature tensor $R$ can be expressed as

[TABLE]

for any $X,Y\in\mathfrak{X}(M)$.

Proof.

Using the expression of $*$-Ricci tensor as given in (3.5) into the definition of gradient almost *-Ricci soliton, we get

[TABLE]

Differentiating the above equation along an arbitrary vector field $Y$, we have

[TABLE]

Then applying the above equation in the expression of Riemannian curvature tensor, we obtain the desired result. ∎

Now we prove the following fruitful result.

Theorem 4.1**.**

If the metric of a Kenmotsu manifold $M$ is a gradient almost $*$-Ricci soliton, then either $M$ is Einstein or the soliton vector field $V$ is pointwise collinear with the characteristic vector field $\xi$ on an open set $\mathcal{O}$ of $M$.

Proof.

Taking inner product of (4.1) with $\xi$, with the help of (3.1), we get

[TABLE]

Now taking inner product of (2.4) with $Df$ gives

[TABLE]

Comparing the last two equations and then plugging $Y=\xi$ yields $X(f+\lambda)=\xi(f+\lambda)\eta(X)$, from which we have

[TABLE]

which means $f+\lambda$ is invariant along the distribution $Ker\eta$, that is, $X(f+\lambda)=0$ for all $X\in Ker\eta$.

Now putting $X=\xi$ in (4.1) and then taking inner product with $Z$ gives

[TABLE]

On the other hand, from (2.4) we have

[TABLE]

Combining the above two equations, one can easily obtain

[TABLE]

which means $M$ is $\eta$-Einstein. Contracting the above equation, one immediately obtain

[TABLE]

Using the above equation in (4.3) one can easily have (2.9). Now contracting (4.1) over $X$, we obtain

[TABLE]

Comparing the above equation with (2.9) shows that

[TABLE]

for all $Y\in\mathfrak{X}(M)$. Substituting $Y=\xi$ in the above equation, it follows from (4.4) that

[TABLE]

Operating (4.2) by $d$, and since $d^{2}=0$ and $d\eta=0$, we obtain $dr\wedge\eta=0$, that is,

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$. After replacing $Y$ by $\xi$ in the above equation and using (4.6), we have $X(r)=-2(r+2n(2n+1))\eta(X)$, which means

[TABLE]

Let $X\in Ker\eta$ be arbitrary. Then with the help of (4.7), the equation (4.5) becomes

[TABLE]

Using (4.2) and (4.4) in the above equation, we have $(r+2n(2n+1))X(f)=0$. This implies that

[TABLE]

If $r=-2n(2n+1)$, then it follows from the $\eta$-Einstein condition (2.9) that $S=-2ng$, and hence $M$ is Einstein. Suppose if $r\neq-2n(2n+1)$ on some open set $\mathcal{O}$ of $M$, then we have $Df=\xi(f)$, and this completes the proof. ∎

Corollary 4.1.1**.**

If the metric of a Kenmotsu 3-manifold is a gradient almost $*$-Ricci soliton, then either $M$ is of constant curvature $-1$ or the soliton vector field $V$ is pointwise collinear with the characteristic vector field $\xi$ on an open set $\mathcal{O}$ of $M$.

Proof.

From Theorem 4.1 we have $M$ is Einstein, that is, $QX=-2nX$. Using this in (3) shows that $M$ is of constant negative curvature $-1$. ∎

Now we provide an example of a gradient almost $*$-Ricci soliton on a Kenmotsu 3-manifold.

Example** 2****.**

Let $M=N\times I$, where $N$ is an open connected subset of $\mathbb{R}^{2}$ and $I$ is an open interval in $\mathbb{R}$. Let $(x,y,z)$ be the Cartesian coordinates in it. Define the Kenmotsu structure $(\varphi,\xi,\eta,g)$ on $M$ as in Example 1.

Let $f:M\to\mathbb{R}$ be a smooth function defined by

[TABLE]

Then the gradient of $f$ with respect to the metric $g$ is given by

[TABLE]

With the help of (3.40) one can easily verify that

[TABLE]

Using (3.42) in the above equation, we have

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$. Thus $g$ is a gradient almost $*$-Ricci soliton with the soliton vector field $V=Df$, $f$ as given in (4.8) and $\lambda=xe^{z}$. As shown in Example 1, $M$ is Einstein and is of constant negative curvature $-1$. This verifies Theorem 4.1 and Corollary 4.1.1