# Almost \(*\)-Ricci soliton on paraKenmotsu manifolds

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## Abstract

We consider almost \(*\)-Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric *g* of \(\eta \)-Einstein paraKenmotsu manifold is \(*\)Ricci soliton, then *M* is Einstein. Next, we show that if \(\eta \)-Einstein paraKenmotsu manifold admits a gradient almost \(*\)-Ricci soliton, then either *M* is Einstein or the potential vector field collinear with Reeb vector field \(\xi \). Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature \(-1\). An illustrative example is given to support the obtained results.

## Mathematics Subject Classification

53C15 53C25 53B20 53D15## 1 Introduction

On the analogy of almost contact manifolds, Sato [27] introduced the notion of almost paracontact manifolds. An almost contact manifold is always odd dimensional, but an almost paracontact manifold could be of even dimension as well. Takahashi [31] defined almost contact manifolds, in particular, Sasakian manifolds equipped with an associated pseudo-Riemannian metric. Later, Kaneyuki and Williams [17] introduced the notion of an almost paracontact pseudo-Riemannian structure, as a natural odd dimensional counterpart to paraHermitian structure. In [37], Zamkovoy showed that any almost paracontact structure admits a pseudo-Riemannian metric with signature \((n+1, n)\). In recent years, almost paracontact structure has been studied by many authors, particularly since the appearance of [37]. The curvature identity for different classes of almost paracontact geometry was obtained in [9, 35, 37]. The notion of paraKenmotsu manifold was introduced by Welyczko [34]. This structure is an analogy of Kenmotsu manifold [18] in paracontact geometry. ParaKenmotsu (briefly p-Kenmotsu) and special paraKenmotsu (briefly sp-Kenmotsu) manifolds were studied by Sinha and Prasad [29], Blaga [2], Sai Prasad and Satyanarayana [25], Prakasha and Vikas [22], and many others.

*M*, a Ricci soliton is a triple \((g,V,\lambda )\) with

*g*, a psuedo-Riemannian metric,

*V*, a vector field called potential vector field and \(\lambda \), a real scalar, such that

*L*denotes Lie derivative along

*V*and Ric denotes the Ricci tensor. The Ricci soliton is a special self similar solution of the Hamilton’s Ricci flow: \(\frac{\partial }{\partial t}g(t)=-\mathrm{{Ric}}(t)\) with initial condition \(g(0)=g\); and is said to be shrinking, steady, and expanding accordingly, as \(\lambda \) is positive, zero, and negative, respectively. If the vector field

*V*is the gradient of a smooth function

*f*on

*M*, that is, \(V=\nabla f\), then we say that Ricci soliton is gradient and

*f*is potential function. For a gradient Ricci soliton, Eq. (1) takes the form:

*g*). We recommend the reference [8] for more details about the Ricci flow and Ricci soliton. In the context of paracontact geometry, Ricci solitons were first initiated by Calvaruso and Perrone in [6]. Then, these are extensively studied by [1, 2, 5, 10, 26] and many others. In this junction, it is suitable to mention that \(\eta \)-Ricci solitons on paraSasakian manifolds were studied in the paper [19, 23]

*g*on a manifold

*M*is called a \(*\)-Ricci soliton if there exist a constant \(\lambda \) and a vector field

*V*, such that

*X*,

*Y*on

*M*. Moreover, if the vector field

*V*is a gradient of a smooth function

*f*, then we say that \(*\)-Ricci soliton is gradient and equation (2) takes the form

*V*is Killing, and in this case, the manifold becomes \(*\)-Einstein. Here, it is suitable to mention that the notion of \(*\)-Ricci tensor was first introduced by Tachibana [30] on almost Hermitian manifolds and further studied by Hamada [13] on real hypersurfaces of non-flat complex space forms. If \(\lambda \) appearing in (2) and (3) 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 [12] first undertook the study of \(*\)-Ricci solitons on almost contact metric manifolds. The case of \(*\)-Ricci soliton in paraSasakian manifold was treated by Prakasha and Veeresha in [24]. Here, they proved that if the metric of paraSasakian manifold is a \(*\)-Ricci Soliton, then it is \(\eta \)-Einstein ( [24], Lemma 5). In this connection, it is suitable to mention that the present authors [33] studied \(*\)-Ricci soliton on \(\eta \)-Einstein Kenmotsu and three-dimensional Kenmotsu manifolds, and proved that if metric of a \(\eta \)-Einstein Kenmotsu manifold is \(*\)-Ricci soliton, then it is Einstein (see [33], Theorem 3.2). For three-dimensional case, it is proved that if *M* admits a \(*\)-Ricci soliton, then it is of constant sectional curvature \(-1\) (see [33], Theorem 3.3). It is mentioned that any three-dimensional paraKenmotsu manifold is \(\eta \)-Einstein (i.e., the Ricci tensor Ric is of the form \(\mathrm{{Ric}}=ag+b\eta \otimes \eta \), where *a*, *b* are known as associated functions). However, in higher dimensions this is not true. We also know (see [38], Proposition 4.1) that for dimension > 3, the associated functions of an \(\eta \)-Einstein paraKenmotsu manifold are not constant, like paraSasakian manifolds [37].

Inspired by above-mentioned works, here, we consider \(*\)-Ricci soliton in the framework of paraKenmotsu manifold. The present paper is organized as follows: In Sect. 2, we reminisce some fundamental formulas and properties of paraKenmotsu manifolds. In Sect. 3, we prove that if \(\eta \)-Einstein paraKenmotsu manifold admits \(*\)-Ricci soliton, then *M* is Einstein. Next, we consider a gradient almost \(*\)-Ricci soliton and show that either *M* is Einstein or potential vector field collinear with Reeb vector field. Also for three-dimensional case, we prove that if three-dimensional paraKenmotsu manifold admits \(*\)-Ricci soliton, then it is of constant negative curvature \(-1\). In Sect. 4, we given an example to verify our main results.

## 2 Preliminaries

In this section, we reminisce some basic notions of almost paracontact metric manifold and refer to [4, 17, 21, 32, 37] for more information and details.

*M*is said to have an almost paracontact structure if it admits a (1,1)-tensor field \(\varphi \), a vector field \(\xi \), and a 1-form \(\eta \) satisfying the following conditions:

- (i)
\(\varphi ^2=I-\eta \otimes \xi \), \(\eta (\xi )=1\).

- (ii)
The tensor field \(\varphi \) induces an almost paracomplex structure on each fiber of \(\mathcal {D}=\mathrm{{ker}}(\eta )\), i.e., the \(\pm 1\)-eigen distributions \(\mathcal {D}^\pm := \mathcal {D}\varphi (\pm )\) of \(\varphi \) have equal dimension

*n*.

*g*, such that

*g*is necessarily \((n+1, n)\) for all \(X,Y\in TM\), then \((M,\varphi ,\xi ,\eta ,g)\) is called an almost paracontact metric manifold. By

*Q*and

*r*, we will indicate the Ricci operator determined by \(S(X,Y)= g(QX,Y)\) and the scalar curvature of the metric

*g*, respectively. The fundamental 2-form \(\Phi \) of an almost paracontact metric structure \((\varphi ,\xi ,\eta ,g)\) is defined by \(\Phi (X,Y)=g(X,\varphi Y)\). If \(\Phi =\mathrm{d}\eta \), then the manifold \((M,\varphi ,\xi ,\eta ,g)\) is called a paracontact metric manifold and

*g*is an associated metric.

*M*is said to be \(\eta \)-Einstein if there exist smooth functions

*a*and

*b*, such that

*M*becomes an Einstein manifold. Following Zamkovoy ( [38], Proposition 4.1), it is showed that if

*M*is an \(\eta \)-Einstein paraKenmotsu manifold of dimension > 3, then we have:

*g*) automatically requires a symmetric property of the \(*\)-Ricci tensor [14].

## 3 Main results

Before entering our main results, first, we find the expression of \(*\)-Ricci tensor in paraKenmotsu manifolds

### Lemma 3.1

*X*,

*Y*on

*M*.

### Proof

*Y*and

*Z*and by definition of \(*\)-Ricci tensor, we obtain (14). This completes the proof. \(\square \)

In view of recent results on paraSasakian manifold [24] and \(\eta \)-Einstein Kenmotsu manifold [33], a natural question arises whether there exists paraKenmotsu manifold admits a \(*\)-Ricci soliton. For this, we consider an \(\eta \)-Einstein paraKenmotsu manifold; such a manifold is in general not paraSasakian. Now, we prove the following.

### Theorem 3.2

If the metric of \(\eta \)-Einstein paraKenmotsu manifold of dimension \(>3\) is a \(*\)-Ricci soliton, then it is Einstein manifold.

### Proof

*M*is \(\eta \)-Einstein, taking \(Y=\xi \) in (12) and making use of (10), we have:

*X*and using (7), we obtain:

*X*,

*Y*,

*Z*on

*M*. Since

*g*is parallel with respect to Levi-Civita connection \(\nabla \), the above relation becomes:

*Z*and

*D*is the gradient operator of

*g*. Setting \(X=Y=e_i\) (where \(\{e_i: i = 1, 2, \ldots , 2n + 1\}\) is an orthonormal frame) in (23) and summing over

*i*, we find:

*X*, we obtain \((\nabla _XL_V g)(Y,Z)=-2(\nabla _X\mathrm{{Ric}}^*)(Y,Z)\). Substituting this in (22), we have:

*Z*and then using (7), we get:

*X*and

*Y*by \(e_i\) in (27) and summing over

*i*, we find:

*Y*and using (6) and (30), we find:

*Z*by \(\xi \) in (32) and taking into account of (31), we obtain:

*X*and noting that \(\mathrm{D}r=(\xi r)\xi \), we have \((L_V \mathrm{{Ric}})(Y,\xi )=0\). Next, taking Lie derivative of (10) along

*V*, making use of last equation and (18), we have:

*Y*and

*Z*in (19) gives \(\eta (L_V\xi )=0\). Thus, making use of \(\lambda =0\) and \(\eta (L_V\xi )=0\), Eq. (34) becomes:

*M*is Einstein.

*M*. Then, on \(\mathcal {O}\), \(L_V\xi =0\). This together with (6) yields:

*Y*by \(\xi \) in the well-known formula (see [36]):

*r*is constant. Thus, (29) implies that \(r=-2n(2n+1)\) on \(\mathcal {O}\). This contradicts our assumption. This establish the proof. \(\square \)

Now, we consider gradient almost \(*\)-Ricci soliton in \(\eta \)-Einstein paraKenmotsu manifolds and prove the following;

### Theorem 3.3

Let *M* be a \((2n+1)\)-dimensional \(\eta \)-Einstein paraKenmotsu manifold. If *g* represents a gradient almost \(*\)-Ricci soliton, then either *M* is Einstein or the potential vector field is pointwise colinear with the Reeb vector field \(\xi \).

### Proof

*g*of a \(\eta \)-Einstein paraKenmotsu manifold is gradient almost \(*\)-Ricci soliton, then from (14) and (3), we obtain:

*X*on

*M*. Taking covariant differentiation of (36) in the direction of an arbitrary vector field

*Y*on

*M*yields:

*Y*and using (7), we obtain:

*Y*by \(\xi \) in the resulting equation, we obtain:

*d*is the exterior derivative. This means that \(\lambda -f\) is invariant along the distribution \(\mathcal {D}\) (where \(\mathcal {D}\) is \(Ker\eta \)); that is, \(\lambda -f\) is constant for all vector field \(X\in \mathcal {D}\).

*Y*, we obtain:

*X*by \(\xi \) in the above equation, we have:

*Y*by \(\xi \) in (41) and taking inner product with

*Y*yield:

*X*in (46) is orthogonal to \(\xi \). Keeping in mind that \(\lambda -f\) is constant along \(\mathcal {D}\) and making use of (54) and (55), one gets \(\{r+2n(2n+1)\}(Xf)=0\), for all \(X\in \mathcal {D}\). This implies that

*M*is Einstein. If \(r\ne -2n(2n+1)\), then we have \(\mathrm{D}f=(\xi f)\xi \). This shows that potential vector field is collinear with \(\xi \), and this completes the proof. \(\square \)

Now, we study \(*\)-Ricci soliton in three-dimensional paraKenmotsu manifold and prove the following;

### Theorem 3.4

If the metric *g* of three-dimensional paraKenmotsu manifold is a \(*\)-Ricci soliton, then it is of constant curvature \(-1\).

### Proof

*V*and making use of (19) give:

*X*and using (58), we have:

*M*is of constant curvature \(-1.\)

*M*. Then, Eq. (62) can be written as:

*Y*by \(\xi \) in the well-known commutation formula [36]:

*X*,

*Y*in the foregoing equation and recalling the Poincare lemma: \(g(\nabla _X Dr,Y)=g(X,\nabla _Y Dr)\), we find:

*Y*by \(\xi \) in (65) and making use of (29), we obtain:

**Case 1:**First, we assume that \(Dr=(\xi r)\xi \). By virtue of this, Eq. (64) can be written as \(L_V\xi =0\) on \(\mathcal {O}\). This together with (6) gives:

*Y*by \(\xi \) in the well-known formula [36]

**Case 2:** Now, assume that \(\xi f=2f\). This together with \(f=-\frac{2}{r+6}\), we have \(\xi r=-2(r+6)\), and consequently, we get \(g(Dr,\xi )=-2(r+6)\). Since \(r\ne -6\) in some open set \(\mathcal {O}\), so the last equation implies that \(Dr=f\xi \), for some smooth function *f*. In fact, we have \(Dr=(\xi r)\xi \), and so, by Case 1, we get a contradiction. This establishes the proof of the theorem. \(\square \)

As we know, in differential geometry, symmetric spaces play an important role. In the late 20s, Cartan [7] initiated Riemannian symmetric spaces and obtained a classification of those spaces. If the Riemannian curvature tensor of a Riemannian manifold satisfies the condition \(\nabla R=0\), then this manifold is called locally symmetric [7]. For every point of this manifold, this symmetry condition is equivalent to the fact that the local geodesic symmetry is an isometry [20]. The class of Riemannian symmetric manifolds is very natural generalization of the class of manifolds of constant curvature.

### Definition 3.5

It is known that a three-dimensional paraKenmotsu manifold is locally \(\varphi \)-symmetric if and only if the scalar curvature is constant [38]. Therefore, by Theorem 3.4, we state the following.

### Corollary 3.6

A three-dimensional paraKenmotsu manifold admitting \(*\)-Ricci soliton is locally \(\varphi \)-symmetric.

### Remark 3.7

Corollary 3.6 builds the connection between \(*\)-Ricci soliton and symmetry of the manifold. The symmetry of a manifold is vital, because it is connected with the curvature of the manifold. The curvature has important physical significance in the theory of gravitation.

## 4 Example

In this section, we give an example of \(*\)-Ricci solitons in three-dimensional paraKenmotsu manifold which verifies Theorem 3.4 and Corollary 3.6.

### Example 4.1

*x*,

*y*,

*z*) and the vector fields:

*M*with characteristic vector field \(\xi =\partial _3=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\). Let

*g*be a pseudo-Riemannian metric defined by:

*V*is not a Killing vector field. Now, the \(*\)-Ricci soliton equation

*M*satisfies:

*g*is a \(*\)-Ricci soliton. Using (70), we have constant scalar curvature as follows:

*M*is an Einstein manifold. It is easy to verify that the manifold is locally \(\varphi \)-symmetric. Hence, the results of Theorem 3.4 and Corollary 3.6 are verified.

## Notes

### Acknowledgements

The authors would like to thank the anonymous referee for his or her valuable suggestions that have improved the original paper.

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