Almost $*$-Ricci soliton on paraKenmotsu manifolds

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.

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.

A Ricci soliton is a generalization of an Einstein metric. We reminisce the notion of Ricci soliton according to [15]. On the manifold 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

$$\begin{aligned} (L_Vg)(X,Y)+2\mathrm{{Ric}}(X,Y)=2\lambda g(X,Y), \end{aligned}$$

(1)

where 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:

$$\begin{aligned} \mathrm{{Hess}} f+\mathrm{{Ric}}=\lambda g, \end{aligned}$$

where Hess denotes the Hessian operator $\nabla ^2$ ($\nabla$ denotes the Riemannian connection of 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]

In 2014, Kaimakamis and Panagiotidou [16] 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 Ric in (1) with the $*$-Ricci tensor $\mathrm{{Ric}}^*$. A pseudo-Riemannian metric g on a manifold M is called a $*$-Ricci soliton if there exist a constant $\lambda$ and a vector field V, such that

$$\begin{aligned} (L_Vg)(X,Y)+2 \mathrm{{Ric}}^*(X,Y)=2\lambda g(X,Y), \end{aligned}$$

(2)

for all vector fields 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

$$\begin{aligned} \mathrm{{Hess}} f+\mathrm{{Ric}}^*=\lambda g. \end{aligned}$$

(3)

Note that a $*$-Ricci soliton is trivial if the vector field 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.

A 2n+1-dimensional smooth manifold 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.

From the definition, it pursues that $\varphi \xi =0$, $\eta \circ \varphi =0$ and $\mathrm{{rank}}(\varphi )=2n$. An almost paracontact structure is said to be normal [17] if and only if the (1,2) type torsion tensor $N_\varphi :=[\varphi ,\varphi ]-2d\eta \otimes \xi$ vanishes identically, where $[\varphi ,\varphi ](X,Y)=\varphi ^2[X,Y]+[\varphi X,\varphi Y]-\varphi [\varphi X,\varphi Y]-\varphi [X,\varphi Y]$. If an almost paracontact manifold is endowed with a pseudo-Riemannian metric g, such that

$$\begin{aligned} g(\varphi X,\varphi Y)=-g(X,Y)+\eta (X)\eta (Y), \end{aligned}$$

(4)

where signature of 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.

An almost paracontact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ is called paraKenmotsu manifold if it satisfies (see [28, 38]):

$$\begin{aligned} (\nabla _X\varphi )Y=g(\varphi X, Y)\xi -\eta (Y)\varphi X, \end{aligned}$$

(5)

for any vector fields $X,Y\in TM$. In [38], Zamkovoy proved that $(M,\varphi ,\xi ,\eta ,g)$ is normal but not quasi-paraSasakian and hence not paraSasakian. Also in a paraKenmotsu manifold, we have the following formulas (see [38]):

$$\begin{aligned}&\nabla _X \xi =X-\eta (X)\xi , \end{aligned}$$

(6)

$$\begin{aligned}&(\nabla _X \eta )Y=g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(7)

$$\begin{aligned}&R(X, Y )\xi =\eta (X)Y-\eta (Y)X, \end{aligned}$$

(8)

$$\begin{aligned}&R(X,\xi )Y = g(X,Y)\xi -\eta (Y)X, \end{aligned}$$

(9)

$$\begin{aligned}&\mathrm{{Ric}}(X,\xi )=-\,2n\eta (X), \end{aligned}$$

(10)

$$\begin{aligned}&(L_\xi g)(X,Y)=-\,2\{g(X,Y)-\eta (X)\eta (Y)\}. \end{aligned}$$

(11)

Note that (11) implies that $\xi$ is not Killing in paraKenmotsu manifold. An almost paracontact metric manifold M is said to be $\eta$-Einstein if there exist smooth functions a and b, such that

$$\begin{aligned} \mathrm{{Ric}}(X,Y)=a g(X,Y)+b\eta (X)\eta (Y), \end{aligned}$$

(12)

for all $X,Y\in TM$. If $b=0$, then 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:

$$\begin{aligned} Z(b)-2b\eta (Z) = 0, \end{aligned}$$

for any $Z\in TM$.

Now, we recall the notion of $*$-Ricci tensor. This is defined by (see [3, 11]):

$$\begin{aligned} \mathrm{{Ric}}^*(X,Y)=\sum _i g(R(X,e_i)\varphi e_i,\varphi Y)=\frac{1}{2}\sum _i g(\varphi R(X,\varphi Y)e_i,e_i), \end{aligned}$$

(13)

where ${e_i}$ is a local orthonormal frame and the last equality follows from the first Bianchi identity. It should be remarked that $\mathrm{{Ric}}^*$ is not symmetric, in general. Thus, the condition $*$-Einstein (that is, $\mathrm{{Ric}}^*$ is a constant multiple of the metric 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

A $*$-Ricci tensor on a $(2n+1)$-dimensional paraKenmotsu manifold $(M,\varphi ,\xi ,\eta ,g)$ is given by:

$$\begin{aligned} \mathrm{{Ric}}^*(X,Y)=-\mathrm{{Ric}}(X,Y)-(2n-1)g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(14)

for any vector fields X, Y on M.

Proof

In a paraKenmotsu manifold, the following formula is known (see [38])

$$\begin{aligned} R(X,Y)\varphi Z-\varphi R(X,Y)Z&=g(Y,Z)\varphi X-g(X,Z)\varphi Y\nonumber \\&\quad -g(Y,\varphi Z)X+g(X,\varphi Z)Y. \end{aligned}$$

(15)

From (15), it follows that:

$$\begin{aligned} R(X,Y,\varphi Z,\varphi W)-g(\varphi R(X,Y)Z,\varphi W)&=g(Y,Z)g(\varphi X,\varphi W)-g(X,Z)g(\varphi Y,\varphi W)\\&\quad -g(Y,\varphi Z)g(X,\varphi W)+g(X,\varphi Z)g(Y,\varphi W). \end{aligned}$$

Making use of (4) with the above equation takes the form:

$$\begin{aligned} R(X,Y,\varphi Z,\varphi W)&=-R(X,Y,Z,W)-g(Y,Z)g(X,W)+g(X,Z)g(Y,W)\nonumber \\&\quad -g(Y,\varphi Z)g(X,\varphi W)+g(X,\varphi Z)g(Y,\varphi W). \end{aligned}$$

(16)

Contracting (16) over 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

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

$$\begin{aligned} a+b=-\,2n. \end{aligned}$$

(17)

Contracting (12) gives the scalar curvature $r=(2n+1)a+b$. Combining this with (17) yields $a=(1+\frac{r}{2n})$ and $b=-\{(2n+1)+\frac{r}{2n}\}$. Thus, Eq. (12) takes the form:

$$\begin{aligned} \mathrm{{Ric}}(X,Y)=\left( 1+\frac{r}{2n}\right) g(X,Y)-\left\{ (2n+1)+\frac{r}{2n}\right\} \eta (X)\eta (Y). \end{aligned}$$

(18)

In view of (14) and (18), Eq. (2) can be written as:

$$\begin{aligned} (L_Vg)(Y,Z)=\left\{ 2(2n+\lambda )+\frac{r}{n}\right\} g(Y,Z)-\left\{ 4n+\frac{r}{n}\right\} \eta (Y)\eta (Z). \end{aligned}$$

(19)

Differentiating (19) along an arbitrary vector field X and using (7), we obtain:

$$\begin{aligned} (\nabla _X L_V g)(Y,Z)&=\frac{Xr}{n}g(Y,Z)-\frac{Xr}{n}\eta (Y)\eta (Z)-\left( 4n+\frac{r}{n}\right) \left\{ g(X,Y)\eta (Z)\right. \nonumber \\&\qquad \left. +g(X,Z)\eta (Y)-2\eta (X)\eta (Y)\eta (Z)\right\} . \end{aligned}$$

(20)

We know the following commutation formula (see [36]):

$$\begin{aligned}&(L_V \nabla _X g-\nabla _X L_V g-\nabla _{[V,X]}g)(Y,Z)\\&\quad =-\,g((L_V\nabla )(X,Y),Z)-g((L_V\nabla )(X,Z),Y), \end{aligned}$$

for all vector fields X, Y, Z on M. Since g is parallel with respect to Levi-Civita connection $\nabla$, the above relation becomes:

$$\begin{aligned} (\nabla _X L_V g)(Y,Z)=g((L_V\nabla )(X,Y),Z)+g((L_V\nabla )(X,Z),Y). \end{aligned}$$

(21)

We know that $L_V\nabla$ is a symmetric tensor of type (1, 2) and so it follows from (21) that

$$\begin{aligned} g((L_V\nabla )(X,Y),Z)=\frac{1}{2}\left\{ (\nabla _XL_Vg)(Y,Z)+(\nabla _YL_Vg)(Z,X)-(\nabla _ZL_Vg)(X,Y)\right\} . \end{aligned}$$

(22)

By a straightforward combinatorial computation, and keeping in mind that $L_V\nabla$ is a symmetric operator, the foregoing equation gives:

$$\begin{aligned} 2n(L_V\nabla )(X,Y)&=(Xr)Y-(Xr)\eta (Y)\xi +(Yr)X \nonumber \\&\quad -(Yr)\eta (X)\xi -g(X,Y)Dr+\eta (X)\eta (Y)Dr \nonumber \\&\quad -2(4n^2+r)\left\{ g(X,Y)\xi -\eta (X)\eta (Y)\xi \right\} , \end{aligned}$$

(23)

for all vector fields 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:

$$\begin{aligned} n\sum _{i=1}^{2n+1}\varepsilon _i(L_V\nabla )(e_i,e_i)=(1-n)Dr-(\xi r)\xi -2n(4n^2+r)\xi , \end{aligned}$$

(24)

where $\varepsilon _i=g(e_i,e_i)$. Now, taking covariant differentiation of $*$-Ricci soliton Eq. (2) along a vector field X, we obtain $(\nabla _XL_V g)(Y,Z)=-2(\nabla _X\mathrm{{Ric}}^*)(Y,Z)$. Substituting this in (22), we have:

$$\begin{aligned} g((L_V\nabla )(X,Y),Z)=(\nabla _Z\mathrm{{Ric}}^*)(X,Y)-(\nabla _X\mathrm{{Ric}}^*)(Y,Z)-(\nabla _Y\mathrm{{Ric}}^*)(X,Z). \end{aligned}$$

(25)

Again, taking covariant differentiation of (14) with respect to Z and then using (7), we get:

$$\begin{aligned} (\nabla _Z\mathrm{{Ric}}^*)(X,Y)&=-(\nabla _Z\mathrm{{Ric}})(X,Y)-\{g(Z,X)\eta (Y)\nonumber \\&\quad +g(Z,Y)\eta (X)-2\eta (X)\eta (Y)\eta (Z)\}. \end{aligned}$$

(26)

Combining (26) with (25) yields:

$$\begin{aligned} g((L_V\nabla )(X,Y),Z)&=-\,(\nabla _Z\mathrm{{Ric}})(X,Y)+(\nabla _X\mathrm{{Ric}})(Y,Z)+(\nabla _Y\mathrm{{Ric}})(Z,X)\nonumber \\&\quad +2\{g(X,Y)\eta (Z)-\eta (X)\eta (Y)\eta (Z)\}. \end{aligned}$$

(27)

Replacing X and Y by $e_i$ in (27) and summing over i, we find:

$$\begin{aligned} \sum _{i=1}^{2n+1}\varepsilon _i(L_V\nabla )(e_i,e_i)=4n\xi . \end{aligned}$$

(28)

In view of (28) and (24), we at once obtain:

$$\begin{aligned} (n-1)\mathrm{D}r+(\xi r)\xi +2n\{2n(2n+1)+r\}\xi =0. \end{aligned}$$

(29)

Taking inner product of (29) with $\xi$ yields $(\xi r)+2(2n(2n+1)+r)=0$. Making use of this in (29) provides $Dr=(\xi r)\xi$, as $n>1$. Next, substituting $Y=\xi$ in (23), it follows that

$$\begin{aligned} 2n(L_V\nabla )(X,\xi )=(\xi r)\varphi ^2 X. \end{aligned}$$

(30)

Differentiating (30) along an arbitrary vector field Y and using (6) and (30), we find:

$$\begin{aligned}&2n(\nabla _YL_V\nabla )(X,\xi )+2n(L_V\nabla )(X,Y)=(Y(\xi r))\varphi ^2X\nonumber \\&\quad -(\xi r)\{g(X,Y)\xi +\eta (X)Y-\eta (Y)X-\eta (X)\eta (Y)\xi \}. \end{aligned}$$

(31)

According to Yano [36], we have the following well-known commutation formula:

$$\begin{aligned} (L_V R)(X,Y)Z=(\nabla _XL_V\nabla )(Y,Z)-(\nabla _YL_V\nabla )(X,Z). \end{aligned}$$

(32)

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

$$\begin{aligned} 2n(L_V R)(X,Y)\xi =(X(\xi r))\varphi ^2 Y-(Y(\xi r))\varphi ^2 X+2(\xi r)\left\{ \eta (X)Y-\eta (Y)X\right\} . \end{aligned}$$

(33)

Contracting this over 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:

$$\begin{aligned}&\left( 1+\frac{r}{2n}\right) g(Y,L_V\xi )-\left\{ (2n+1)+\frac{r}{2n}\right\} \eta (Y)\eta (L_V\xi )\nonumber \\&\quad =-\,4n\lambda \eta (Y)-2ng(Y,L_V\xi ). \end{aligned}$$

(34)

Taking $Y=\xi$ in (34), we have $\lambda =0$. Furthermore, substituting $\xi$ for 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:

$$\begin{aligned} \{2n(2n+1)+r\}L_V\xi =0. \end{aligned}$$

(35)

Now, if $r=-2n(2n+1)$, then it follows from (18) that M is Einstein.

Suppose that we assume that $r\ne -\,2n(2n+1)$ in some open set $\mathcal {O}$ of M. Then, on $\mathcal {O}$, $L_V\xi =0$. This together with (6) yields:

$$\begin{aligned} \nabla _\xi V=V-\eta (V)\xi . \end{aligned}$$

Taking $Y=\xi$ in (19) and using $\lambda =0$, we have $(L_V g)(X,\xi )=0$. From this, we have $L_V\eta =0$. Replacing Y by $\xi$ in the well-known formula (see [36]):

$$\begin{aligned} (L_V\nabla )(X,Y)=L_V\nabla _X Y-\nabla _XL_V Y-\nabla _{[V,X]} Y, \end{aligned}$$

and by virtue of (6), (30), $L_V\xi =0$ and $L_V\eta =0$, we obtain $(\xi r)=0$. Since $Dr=(\xi r)\xi$, we see that 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

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

$$\begin{aligned} \nabla _X \mathrm{D}f=QX+(2n-1+\lambda )X+\eta (X)\xi , \end{aligned}$$

(36)

for any vector field X on M. Taking covariant differentiation of (36) in the direction of an arbitrary vector field Y on M yields:

$$\begin{aligned} \nabla _Y\nabla _X \mathrm{D}f&=(\nabla _Y Q)X+Q\nabla _Y X+(2n-1+\lambda )\nabla _Y X+(Y\lambda )X\nonumber \\&\quad +(\nabla _Y\eta )(X)\xi +\eta (\nabla _Y X)\xi +\eta (X)\nabla _Y\xi . \end{aligned}$$

(37)

Making use of (36) and (37) in the well-known expression of curvature tensor $R(X,Y)=\nabla _X\nabla _Y-\nabla _Y\nabla _X-\nabla _{[X,Y]}$, we deduce:

$$\begin{aligned} R(X,Y)\mathrm{D}f=(\nabla _X Q)Y-(\nabla _Y Q)X+(X\lambda )Y-(Y\lambda )X+\eta (Y)X-\eta (X)Y. \end{aligned}$$

(38)

In view of (18), we have:

$$\begin{aligned} QX=\left( 1+\frac{r}{2n}\right) X-\left\{ (2n+1)+\frac{r}{2n}\right\} \eta (X)\xi . \end{aligned}$$

(39)

Differentiating the foregoing equation along an arbitrary vector field Y and using (7), we obtain:

$$\begin{aligned} (\nabla _Y Q)X=\frac{Y r}{2n}\left\{ X-\eta (X)\xi \right\} -\left\{ (2n+1)+\frac{r}{2n}\right\} \left\{ g(X,Y)\xi -\eta (X)Y\right\} . \end{aligned}$$

(40)

In view of (40), we get from (38) that

$$\begin{aligned} R(X,Y)\mathrm{D}f&=\frac{Xr}{2n}(Y-\eta (Y)\xi )-\frac{Yr}{2n}(X-\eta (X)\xi )+ \bigg \{(2n+2) \nonumber \\&\quad +\frac{r}{2n}\bigg \}(\eta (Y)X-\eta (X)Y)+(X\lambda )Y-(Y\lambda )X. \end{aligned}$$

(41)

By virtue of above equation, we can easily see that

$$\begin{aligned} g(R(X,Y)\mathrm{D}f,\xi )=(X\lambda )\eta (Y)-(Y\lambda )\eta (X). \end{aligned}$$

(42)

Also, we have from (8) that

$$\begin{aligned} g(R(X,Y)\xi ,\mathrm{D}f)=(Yf)\eta (X)-(Xf)\eta (Y). \end{aligned}$$

(43)

Comparing (42) with (43) and substituting Y by $\xi$ in the resulting equation, we obtain:

$$\begin{aligned} d(\lambda -f)=\xi (\lambda -f)\eta , \end{aligned}$$

(44)

where 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 $X\in \mathcal {D}$.

Contracting (38) over Y, we obtain:

$$\begin{aligned} S(X,\mathrm{D}f)=-\frac{1}{2}Xr-2n(X\lambda )+2n\eta (Y). \end{aligned}$$

(45)

In view of (39), the foregoing equation gives:

$$\begin{aligned} \left( 1+\frac{r}{2n}\right) (Xf)-\left\{ (2n+1)+\frac{r}{2n}\right\} \eta (X)(\xi f)+\frac{1}{2}(Xr)+2n(X\lambda )-2n\eta (X)=0, \end{aligned}$$

(46)

for all $X\in TM$. Replacing X by $\xi$ in the above equation, we have:

$$\begin{aligned} 2n \xi (\lambda -f)+\frac{1}{2}(\xi r)-2n=0. \end{aligned}$$

(47)

Now, plugging Y by $\xi$ in (41) and taking inner product with Y yield:

$$\begin{aligned} g(R(X,\xi )\mathrm{D}f,Y)=&-\frac{(\xi r)}{2n}(g(X,Y)-\eta (X)\eta (Y))+\left\{ (2n+2)+\frac{r}{2n}\right\} (g(X,Y)\nonumber \\&-\eta (X)\eta (Y))+(X\lambda )\eta (Y)-(\xi \lambda )g(X,Y). \end{aligned}$$

(48)

By virtue of (9), we have:

$$\begin{aligned} g(R(X,\xi )Y,\mathrm{D}f)=(\xi f)g(X,Y)-(X f)\eta (Y). \end{aligned}$$

(49)

Comparing (48) with (49), one can get:

$$\begin{aligned}&-\frac{(\xi r)}{2n}(g(X,Y)-\eta (X)\eta (Y))+\left\{ (2n+2)+\frac{r}{2n}\right\} (g(X,Y) -\eta (X)\eta (Y))\nonumber \\&\quad +X(\lambda -f)\eta (Y)-\xi (\lambda -f)g(X,Y)=0. \end{aligned}$$

(50)

Contracting the above equation, we get:

$$\begin{aligned} -(\xi r)+2n\left( 2n+2+\frac{r}{2n}\right) -2n\xi (\lambda -f)=0. \end{aligned}$$

(51)

Making use of (51) in (47), we easily obtain:

$$\begin{aligned} (\xi r)=2\{r+2n(2n+1)\}. \end{aligned}$$

(52)

By virtue of (52), we can easily find that

$$\begin{aligned} \xi (\lambda -f)=-\left( \frac{r}{2n}+2n\right) . \end{aligned}$$

(53)

Making use of (53) in (44), we obtain that

$$\begin{aligned} d(\lambda -f)=-\left( \frac{r}{2n}+2n\right) \eta . \end{aligned}$$

(54)

Applying the well-known Poincare lemma and using the fact $d\eta =0$ on the above equation, we obtain $-dr\wedge \eta =0$, and making use of (52), we have:

$$\begin{aligned} Dr=2\{r+2n(2n+1)\}\xi . \end{aligned}$$

(55)

Suppose that 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

$$\begin{aligned} \{r+2n(2n+1)\}(\mathrm{D}f-(\xi f)\xi )=0. \end{aligned}$$

(56)

Suppose $r=-2n(2n+1)$, then using this relation in (39), we see that $QX=-2n X$, and hence. 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

It is known that for any three-dimensional pseudo-Riemannian manifold, we have the following well-known expression:

$$\begin{aligned} R(X,Y)Z&=g(Y,Z)QX-g(X,Z)QY+\mathrm{{Ric}}(Y,Z)X-\mathrm{{Ric}}(X,Z)Y\nonumber \\&\quad -\frac{r}{2}\{g(Y,Z)X-g(X,Z)Y\}. \end{aligned}$$

(57)

Setting $Y=Z=\xi$ in the above relation and making use of (8) and (10) give:

$$\begin{aligned} QX=\left( 1+\frac{r}{2}\right) X-\left( 3+\frac{r}{2}\right) \eta (X)\xi , \end{aligned}$$

which is equivalent to

$$\begin{aligned} \mathrm{{Ric}}(X,Y)=\left( 1+\frac{r}{2}\right) g(X,Y)-\left( 3+\frac{r}{2}\right) \eta (X)\eta (Y). \end{aligned}$$

(58)

Proceeding in the similar manner as in proof of Theorem 3.2. In dimension 3, that is, for $n=1$ all Eqs. (19)–(33) holds true. Thus, (33) becomes:

$$\begin{aligned} 2(L_VR)(X,Y)\xi&=(X(\xi r))\varphi ^2Y-(Y(\xi r))\varphi ^2X\nonumber \\&\quad +2(\xi r)\{\eta (X)Y-\eta (Y)X\}. \end{aligned}$$

(59)

Lie differentiating (8) along V and making use of (19) give:

$$\begin{aligned} (L_VR)(X,Y)\xi +R(X,Y)L_V\xi&=2\lambda \{\eta (X)Y-\eta (Y)X\}\nonumber \\&+g(X,L_V\xi )Y-g(Y,L_V\xi )X. \end{aligned}$$

(60)

In view of (59) and (60), we have:

$$\begin{aligned}&(X(\xi r))\varphi ^2Y-(Y(\xi r))\varphi ^2X+2(\xi r)\{\eta (X)Y-\eta (Y)X\}+2R(X,Y)L_V\xi \\&\quad =4\lambda \{\eta (X)Y-\eta (Y)X\}+2\{g(X,L_V\xi )Y-g(Y,L_V\xi )X\}. \end{aligned}$$

Contracting the foregoing equation over X and using (58), we have:

$$\begin{aligned}&(r+6)g(Y,L_V\xi )-(r+6)\eta (Y)\eta (L_V\xi )\nonumber \\&\quad =Y(\xi r)+\{\xi (\xi r)+4(\xi r)-8\lambda \}\eta (Y). \end{aligned}$$

(61)

Setting $Y=\xi$ in (61) and using (29), we have $\lambda =0$. In view of (19), we obtain $(L_Vg)(Y,\xi )=0$, which implies $\eta (L_V\xi )=0$. Thus, using $\lambda =0$, $\eta (L_V\xi )=0$ and (29), Eq. (61) reduces: to

$$\begin{aligned} (r+6)g(Y,L_V\xi )=-2\{Yr-(\xi r)\eta (Y)\}. \end{aligned}$$

(62)

Suppose that $r=-6$, then from (58), we can see that it is Einstein and $QX=-2X$. This together with (57) gives:

$$\begin{aligned} R(X,Y)Z=g(X,Z)Y-g(Y,Z)X, \end{aligned}$$

(63)

showing that M is of constant curvature $-1.$

On the other hand, suppose that $r\ne -6$ in some open set $\mathcal {O}$ of M. Then, Eq. (62) can be written as:

$$\begin{aligned} L_V\xi =f\{Dr-(\xi r)\xi \}, \end{aligned}$$

(64)

where $f=-\frac{2}{r+6}$. Replacing Y by $\xi$ in the well-known commutation formula [36]:

$$\begin{aligned} (L_V\nabla )(X,Y)=L_V\nabla _X Y-\nabla _XL_VY-\nabla _{[V,X]}Y, \end{aligned}$$

and using (6), (30), and (64), one can easily get:

$$\begin{aligned}&f\left\{ (Xr)\eta (Y)+(Yr)\eta (X)-2(\xi r)\eta (X)\eta (Y)+g(\nabla _XDr,Y)-(X(\xi r))\eta (Y)\right\} \\&\quad +(X f)\left\{ Yr-(\xi r)\eta (Y)\right\} +\left\{ f(\xi r)-\frac{1}{2}(\xi r)\right\} g(\varphi X,\varphi Y)=0. \end{aligned}$$

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

$$\begin{aligned}&f\{(Y(\xi r))\eta (X)-(X(\xi r))\eta (Y)\}+(Xf)\{Yr-(\xi r)\eta (Y)\}\nonumber \\&\quad -(Yf)\{Xr-(\xi r)\eta (X)\}=0. \end{aligned}$$

(65)

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

$$\begin{aligned} (2f-\xi f)(Xr-(\xi r)\eta (X))=0, \end{aligned}$$

which implies that

$$\begin{aligned} (2f-\xi f)(Dr-(\xi r)\xi )=0. \end{aligned}$$

(66)

From (66), we have either $Dr=(\xi r)\xi$ or $\xi f=2f$.

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:

$$\begin{aligned} \nabla _\xi V=V-\eta (V)\xi . \end{aligned}$$

(67)

By virtue of $(L_Vg)(X,\xi )=0$ and (67), one gets:

$$\begin{aligned} g(\nabla _X, V)=-g(X,\nabla _\xi V)=-g(X,V)+\eta (X)\eta (V). \end{aligned}$$

(68)

Replacing Y by $\xi$ in the well-known formula [36]

$$\begin{aligned} (L_V\nabla )(X,Y)=\nabla _X\nabla _Y V-\nabla _{\nabla _X Y}V+R(V,X)Y, \end{aligned}$$

(69)

and making use of (6), (8), (30), (67), and (68), we find $\xi r=0$. Hence from (29), it follows that $r=-6$ on $\mathcal {O}$, which yields a contradiction.

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

An almost paracontact metric manifold is said to be locally $\varphi$-symmetric if

$$\begin{aligned} \varphi ^2(\nabla _W R)(X,Y)Z=0, \end{aligned}$$

for all vector fields W,X,Y,Z orthogonal to $\xi$.

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

We consider three-dimensional manifold $M=\{(x,y,z)\in \mathcal {R}^3,z\ne 0\}$ with the Cartesian coordinates (x, y, z) and the vector fields:

$$\begin{aligned} \partial _1=\varphi \partial _2,\quad \partial _2=\varphi \partial _1,\quad \varphi \partial _3=0, \end{aligned}$$

where

$$\begin{aligned} \partial _1=\frac{\partial }{\partial x},\quad \partial _2=\frac{\partial }{\partial y},\quad \partial _3=x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+\frac{\partial }{\partial z}. \end{aligned}$$

The 1-form $\eta =dz$ defines an almost paracontact structure on 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:

$$\begin{aligned} {[}g(\partial _i,\partial _j)]=\left[ \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right] , \end{aligned}$$

with respect to the basis $\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}$.

Using Koszul formula, we have:

$$\begin{aligned}&\nabla _{\partial _1}\partial _1=-\partial _3,\quad \nabla _{\partial _1}\partial _2=0,\quad \nabla _{\partial _1}\partial _3=\partial _1,\\&\nabla _{\partial _2}\partial _1=0, \quad \nabla _{\partial _2}\partial _2=\partial _3,\quad \nabla _{\partial _2}\partial _3=\partial _2,\\&\nabla _{\partial _3}\partial _1=0, \quad \nabla _{\partial _3}\partial _2=0, \quad \nabla _{\partial _3}\partial _3=0. \end{aligned}$$

It is not hard to verify that the conditions (5) and (6) for paraKenmotsu manifold are satisfied. Hence, the manifold under consideration is a paraKenmotsu manifold. The components of the curvature tensor are:

$$\begin{aligned} R(\partial _1,\partial _2)\partial _1&=\partial _2, \quad R(\partial _1,\partial _2)\partial _2=\partial _1, \quad R(\partial _1,\partial _2)\partial _3=0,\\ R(\partial _1,\partial _3)\partial _1&=\partial _3, \quad R(\partial _1,\partial _3)\partial _2=0, \quad R(\partial _1,\partial _3)\partial _3=-\partial _1,\\ R(\partial _2,\partial _3)\partial _1&=0, \quad R(\partial _2,\partial _3)\partial _2=-\,\partial _3, \quad R(\partial _2,\partial _3)\partial _3=-\,\partial _2. \end{aligned}$$

The components of Ricci tensor and $*$-Ricci tensor are:

$$\begin{aligned} \mathrm{{Ric}}(\partial _1,\partial _1)&=-2,\quad \mathrm{{Ric}}(\partial _2,\partial _2)=2,\quad \mathrm{{Ric}}(\partial _3,\partial _3)=-2,\nonumber \\ \mathrm{{Ric}}^*(\partial _1,\partial _1)&=1, \quad \mathrm{{Ric}}^*(\partial _2,\partial _2)=-1,\quad \mathrm{{Ric}}^*(\partial _3,\partial _3)=0. \end{aligned}$$

(70)

If we choose $V=\partial _1-\partial _3$, then we see that

$$\begin{aligned} (L_V g)(\partial _1,\partial _1)&=Vg(\partial _1,\partial _1)-2g(L_V \partial _1,\partial _1)\nonumber \\&=-\,2g([\partial _1-\partial _3,\partial _1],\partial _1)=-2. \end{aligned}$$

(71)

Thus, V is not a Killing vector field. Now, the $*$-Ricci soliton equation

$$\begin{aligned} (L_Vg)(\partial _1,\partial _1)+2\mathrm{{Ric}}^*(\partial _1,\partial _1)=2\lambda g(\partial _1,\partial _1), \end{aligned}$$

gives $\lambda =0$ (we know that if paraKenmotsu manifold admits $*$-Ricci soliton, then $\lambda =0$). Similarly, we can check the other components and verify that M satisfies:

$$\begin{aligned} (L_V g)(X,Y)+2\mathrm{{Ric}}^*(X,Y)=2\lambda g(X,Y). \end{aligned}$$

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

$$\begin{aligned} r=\mathrm{{Ric}}(\partial _1,\partial _1)- \mathrm{{Ric}}(\partial _2,\partial _2)+\mathrm{{Ric}}(\partial _3,\partial _3)=-6. \end{aligned}$$

(72)

Because of scalar curvature $r=-6$, from Theorem 3.4, we can conclude that 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.