ζ-Conformally Flat LP-Kenmotsu Manifolds and Ricci–Yamabe Solitons

Abstract

In the present paper, we characterize m-dimensional $\zeta$-conformally flat $LP$-Kenmotsu manifolds (briefly, ${\left(LPK\right)}_m$) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of an ${\left(LPK\right)}_m$ admitting an RYS satisfies the Poisson equation $\Delta r=\frac{4(m-1)}{\delta }\{\beta (m-1)+\rho \}+2(m-3)r-4m(m-1)(m-2)$, where $\rho ,\delta (\ne 0)\in \mathbb{R}$. In this sequel, the condition for which the scalar curvature of an ${\left(LPK\right)}_m$ admitting an RYS holds the Laplace equation is established. We also give an affirmative answer for the existence of a GRYS on an ${\left(LPK\right)}_m$. Finally, a non-trivial example of an $LP$-Kenmotsu manifold $\left(LPK\right)$ of dimension four is constructed to verify some of our results.

1. Introduction

The Ricci solitons (RS) and Yamabe solitons (YS) correspond to self-similar solutions of the Ricci flow, $2S+\frac{\partial }{\partial t}g=0$, and the Yamabe flow, $\frac{\partial }{\partial t}g=-rg,g\left(0\right)=g_0$ (where S denotes the Ricci tensor and r is the scalar curvature of the metric g); they are given by [1,2]

$$\begin{array}{c}\hfill {\pounds }_{\mathcal{E}}g+2\rho g+2S=0,\end{array}$$

(1)

and

$$\begin{array}{c}\hfill {\pounds }_{\mathcal{E}}g=-2(\rho -r)g,\end{array}$$

(2)

respectively, where $\rho \in \mathbb{R}$ (set of real numbers) and ${\pounds }_{\mathcal{E}}$ stands for the Lie derivative operator along the smooth vector field $\mathcal{E}$ on a semi-Riemannian manifold M of dimension m.

Recently, a scalar combination of Ricci and Yamabe flows was established by Güler and Crasmareanu [3]. This class of geometric flow was named a Ricci–Yamabe (RY) flow of type $(\beta ,\delta )$ and was defined by

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}g\left(t\right)+2\beta S\left(g\left(t\right)\right)+\delta r\left(t\right)g\left(t\right)=0,g\left(0\right)=g_0\end{array}$$

(3)

for some scalars $\beta$ and $\delta$.

A solution to the RY flow is called a Ricci–Yamabe soliton (RYS) if it depends only on one parameter group of diffeomorphism and scaling. An M is said to admit an RYS if

$$\begin{array}{c}\hfill {\pounds }_{\mathcal{E}}g+2\beta S+(2\rho -\delta r)g=0,\end{array}$$

(4)

where $\beta ,\delta ,\rho \in \mathbb{R}$. If $\mathcal{E}$ is the gradient of a smooth function u on M, then Equation (4) is called a gradient Ricci–Yamabe soliton (GRYS) and then Equation (4) transforms to

$$\begin{array}{c}\hfill {\nabla }^{2}u+\beta S+(\rho -\frac{\delta r}{2})g=0,\end{array}$$

(5)

where ${\nabla }^{2}u$ is the Hessian of u and is denoted by $Hess\left(u\right)=\nabla \nabla u$. Moreover, we note that a RYS of type $(\beta ,0)$ and of type $(0,\delta )$ are known as $\beta$-Ricci soliton and $\delta$-Yamabe soliton, respectively. An RYS is said to be shrinking, steady or expanding if $\rho <0,=0$ or $>0$, respectively. An RYS is said to be a

• Ricci soliton (RS) [4] if $\beta =1,\delta =0$;
• Yamabe soliton (YS) [5] if $\beta =0,\delta =1$;
• Einstein soliton [6] if $\beta =-\delta =1$;
• $\varrho$-Einstein soliton [7] if $\beta =1,\delta =-2\varrho$.

On the other hand, the Lorentzian manifold which is one of the most important subclass of pseudo-Riemannian manifolds plays an important role in the development of the theory of relativity and cosmology [8]. In 1989, Matsumoto [9] introduced the notion of $LP$-Sasakian manifolds, while in 1992, the same notion was independently studied by Mihai and Rosca [10], and they obtained several results on this manifold. Later, such manifolds were studied by many authors. Recently, Haseeb and Prasad defined and studied the Lorentzian para-Kenmotsu manifold [11] as a subclass of Lorentzian paracontact manifold. For more details about the related studies, we recommend the papers [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and the references therein.

As a continuation of this study, we propose a study of the RYS and GRYS in the framework of a $\zeta$-conformally flat ${\left(LPK\right)}_m$. In Section 2, we include some basic results and definitions which are required to study an ${\left(LPK\right)}_m$. Section 3 and Section 4 are concerned with the study of a RYS and a GRYS on a $\zeta$-conformally flat ${\left(LPK\right)}_m$, respectively. In Section 5, we construct a non-trivial example of an ${\left(LPK\right)}_4$ and proved that an ${\left(LPK\right)}_4$ is $\zeta$-conformally flat and that a GRYS on an ${\left(LPK\right)}_4$ is trivial.

2. Preliminaries

A differentiable manifold M (where the dimension of M is m) with the structure $(f,\zeta ,\omega )$ is named a Lorentzian almost paracontact manifold, where f, $\zeta$ and $\omega$ represent a $(1,1)$-type tensor field, a contravariant vector field and a one-form, respectively, on M satisfying [27]

$$\omega \left(\zeta \right)=-1\mathrm{and}{f}^2=\omega \otimes \zeta +I,$$

(6)

which yields

$$f\zeta =0,\omega \circ f=0,\mathrm{rank}\left(f\right)=m-1.$$

(7)

Let the Lorentzian metric g of M fulfill

$$g(·,\zeta )=\omega (·)\mathrm{and}g(f·,f·)=g(·,·)+\omega (·)\omega (·).$$

(8)

Then, the structure $(f,\zeta ,\omega ,g)$ is said to be an almost paracontact structure and M is called an almost paracontact metric manifold.

Define the second fundamental form $\Phi$ as

$$\Phi ({\mathcal{E}}_1,{\mathcal{E}}_2)=\Phi ({\mathcal{E}}_2,{\mathcal{E}}_1)=g({\mathcal{E}}_1,f{\mathcal{E}}_2)$$

(9)

for any vector fields ${\mathcal{E}}_1,{\mathcal{E}}_2\in \mathfrak{X}\left(M\right)$, where $\mathfrak{X}\left(M\right)$ is the Lie algebra of vector fields on M. If

$$d\omega ({\mathcal{E}}_1,{\mathcal{E}}_2)=\Phi ({\mathcal{E}}_1,{\mathcal{E}}_2),$$

(10)

where d is an exterior derivative, then $(M,f,\zeta ,\omega ,g)$ is termed as a paracontact metric manifold.

If the vector field $\zeta$ is a Killing vector field, then the (para)contact structure is called a K-(para)contact. In such a situation, we have

$${\nabla }_{{\mathcal{E}}_1}\zeta =f{\mathcal{E}}_1.$$

(11)

Definition 1. 

A Lorentzian almost paracontact manifold M is called an ${\left(LPK\right)}_m$ if [11]

$$\left({\nabla }_{{\mathcal{E}}_1}f\right){\mathcal{E}}_2=-g(f{\mathcal{E}}_1,{\mathcal{E}}_2)\zeta -\omega \left({\mathcal{E}}_2\right)f{\mathcal{E}}_1$$

(12)

for any ${\mathcal{E}}_1,{\mathcal{E}}_2$ on $M.$

In an ${\left(LPK\right)}_m$, we have

$${\nabla }_{{\mathcal{E}}_1}\zeta +{\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta =0,$$

(13)

$$\left({\nabla }_{{\mathcal{E}}_1}\omega \right){\mathcal{E}}_2+g({\mathcal{E}}_1,{\mathcal{E}}_2)+\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)=0,$$

(14)

where ∇ stands for the Levi–Civita connection with respect to g.

Furthermore, in an ${\left(LPK\right)}_m$, the following relations hold [11]:

$$g(\mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2){\mathcal{E}}_3,\zeta )=\omega \left(\mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2){\mathcal{E}}_3\right)=g({\mathcal{E}}_2,{\mathcal{E}}_3)\omega \left({\mathcal{E}}_1\right)-g({\mathcal{E}}_1,{\mathcal{E}}_3)\omega \left({\mathcal{E}}_2\right),$$

(15)

$$\mathcal{R}(\zeta ,{\mathcal{E}}_1){\mathcal{E}}_2=-\mathcal{R}({\mathcal{E}}_1,\zeta ){\mathcal{E}}_2=g({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta -\omega \left({\mathcal{E}}_2\right){\mathcal{E}}_1,$$

(16)

$$\mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta =\omega \left({\mathcal{E}}_2\right){\mathcal{E}}_1-\omega \left({\mathcal{E}}_1\right){\mathcal{E}}_2,$$

(17)

$$\mathcal{R}(\zeta ,{\mathcal{E}}_1)\zeta ={\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta ,$$

(18)

$$S({\mathcal{E}}_1,\zeta )=(m-1)\omega \left({\mathcal{E}}_1\right),S(\zeta ,\zeta )=-(m-1),$$

(19)

$$\mathcal{Q}\zeta =(m-1)\zeta ,$$

(20)

for any ${\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3$ on an ${\left(LPK\right)}_m$, where $\mathcal{R}$ is the curvature tensor and $\mathcal{Q}$ is the Ricci operator of ${\left(LPK\right)}_m$.

Definition 2. 

An ${\left(LPK\right)}_m$ is said to be a perfect fluid spacetime if its $(0,2)$-type Ricci tensor $S(\ne 0)$ satisfies the following condition

$$S({\mathcal{E}}_1,{\mathcal{E}}_2)={\sigma }_{1}g({\mathcal{E}}_1,{\mathcal{E}}_2)+{\sigma }_2\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right),$$

(21)

for smooth functions ${\sigma }_1$ and ${\sigma }_2$, where ω is a one-form such that $g({\mathcal{E}}_1,\zeta )=\omega \left({\mathcal{E}}_1\right)$, for all vector field ${\mathcal{E}}_1$, associated to the unit timelike vector field ζ. The one-form ω is called the associated one-form and ζ is called the velocity vector field. For more details, we refer the reader to [28,29,30,31,32,33,34] and the references therein.

An ${\left(LPK\right)}_m$ is said to be $\zeta$-conformally flat if the conformal curvature tensor $\mathcal{C}$ [35] defined by

$$\begin{array}{cc}\hfill \mathcal{C}({\mathcal{E}}_1,{\mathcal{E}}_2){\mathcal{E}}_3& =\mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2){\mathcal{E}}_3-\frac{1}{m-2}\{S({\mathcal{E}}_2,{\mathcal{E}}_3){\mathcal{E}}_1-S({\mathcal{E}}_1,{\mathcal{E}}_3){\mathcal{E}}_2+g({\mathcal{E}}_2,{\mathcal{E}}_3)\mathcal{Q}{\mathcal{E}}_1\hfill \\ \hfill & -g({\mathcal{E}}_1,{\mathcal{E}}_3)\mathcal{Q}{\mathcal{E}}_2\}+\frac{r}{(m-1)(m-2)}\{g({\mathcal{E}}_2,{\mathcal{E}}_3){\mathcal{E}}_1-g({\mathcal{E}}_1,{\mathcal{E}}_3){\mathcal{E}}_2\},\hfill \end{array}$$

(22)

$\forall {\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3$ on the ${\left(LPK\right)}_m$ satisfies the relation $\mathcal{C}({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta =0$.

Setting ${\mathcal{E}}_2={\mathcal{E}}_3=\zeta$ in Equation (22) and then following Equations (6), (8), (17), (19) and (20), we infer that

$$\mathcal{Q}=\left(\frac{r}{m-1}-1\right)I+\left(\frac{r}{m-1}-m\right)\omega \otimes \zeta ,$$

(23)

which yields that an ${\left(LPK\right)}_m$ is a perfect fluid spacetime. Thus, we write

Proposition 1. 

Every ζ-conformally flat ${\left(LPK\right)}_m$ is a perfect fluid spacetime.

Lemma 1. 

In a ζ-conformally flat ${\left(LPK\right)}_m$, we have

$$\zeta \left(r\right)=2(r-m(m-1\left)\right),$$

(24)

$${\mathcal{E}}_1\left(r\right)=-2(r-m(m-1))\omega \left({\mathcal{E}}_1\right),$$

(25)

$$\omega \left({\nabla }_{\zeta }Dr\right)=4(r-m(m-1))$$

(26)

for any ${\mathcal{E}}_1$ on the ${\left(LPK\right)}_m$.

Proof. 

The covariant differentiation of Equation (23) with respect to ${\mathcal{E}}_2$ and the use of Equations (13) and (14) lead to

$$\begin{array}{ccc}\hfill \left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1& =& \frac{{\mathcal{E}}_2\left(r\right)}{m-1}({\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta )\hfill \\ & -& (\frac{r}{m-1}-m)(g({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta +\omega \left({\mathcal{E}}_1\right){\mathcal{E}}_2+2\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\zeta ).\hfill \end{array}$$

(27)

Taking the inner product of Equation (27) with ${\mathcal{E}}_3$, we have

$$\begin{array}{ccc}\hfill g\left(\left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1,{\mathcal{E}}_3\right)& =& \frac{{\mathcal{E}}_2\left(r\right)}{m-1}(g({\mathcal{E}}_1,{\mathcal{E}}_3)+\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_3\right))\hfill \\ \hfill & -& (\frac{r}{m-1}-m)(g({\mathcal{E}}_1,{\mathcal{E}}_2)\omega \left({\mathcal{E}}_3\right)+\omega \left({\mathcal{E}}_1\right)g({\mathcal{E}}_2,{\mathcal{E}}_3)+2\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\omega \left({\mathcal{E}}_3\right)).\hfill \end{array}$$

(28)

Let {${\ell }_1$, ${\ell }_2$, ${\ell }_3$…${\ell }_{m-1}$, ${\ell }_m=\zeta$ } be the orthonormal basis of the tangent space at each point of an ${\left(LPK\right)}_m$. By putting ${\mathcal{E}}_2={\mathcal{E}}_3={\ell }_i$ and taking the summation over $i(1\le i\le m)$, we find

$$\begin{array}{c}\hfill {\mathcal{E}}_1\left(r\right)=\frac{2(m-1)}{m-3}\left\{\frac{\zeta \left(r\right)}{m-1}-(r-m(m-1))\right\}\omega \left({\mathcal{E}}_1\right),\end{array}$$

(29)

where the trace $\{{\mathcal{E}}_2\to \left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1\}=\frac{1}{2}{\mathcal{E}}_1\left(r\right)$ is used.

Replacing ${\mathcal{E}}_1$ by ζ in Equation (29) and using Equation (6) gives Equation (24). Next, by using Equation (24) in Equation (29), we easily obtain Equation (25). By the covariant differentiation of Equation (24) with respect to ζ and using Equation (13), Equation (26) follows. □

Remark 1. 

From the relation (24), it is noticed that if a ζ-conformally flat ${\left(LPK\right)}_m$ has a constant scalar curvature, then $r=m(m-1)$.

3. RYS on a $\zeta$-Conformally Flat ${\left(\mathbf{LPK}\right)}_{\mathbf{m}}$

Let the metric of a $\zeta$-conformally flat ${\left(LPK\right)}_m$ be an RYS, then, in view of Equation (23), Equation (4) takes the form

$$\begin{array}{ccc}\hfill \left({\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_1,{\mathcal{E}}_2)& =& -2\{\beta (\frac{r}{m-1}-1)+(\rho -\frac{\delta r}{2})\}g({\mathcal{E}}_1,{\mathcal{E}}_2)\hfill \\ \hfill & & -2\beta (\frac{r}{m-1}-m)\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\hfill \end{array}$$

(30)

for any ${\mathcal{E}}_1$, ${\mathcal{E}}_2$ on ${\left(LPK\right)}_m$.

Taking the covariant derivative of Equation (30) with respect to ${\mathcal{E}}_3$, we find

$$\begin{array}{ccc}\hfill \left({\nabla }_{{\mathcal{E}}_3}{\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_1,{\mathcal{E}}_2)& =& -2(\frac{\beta }{m-1}-\frac{\delta }{2}){\mathcal{E}}_3\left(r\right)g({\mathcal{E}}_1,{\mathcal{E}}_2)-\frac{2\beta }{m-1}{\mathcal{E}}_3\left(r\right)\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\hfill \\ & & +2\beta (\frac{r}{m-1}-m)(g({\mathcal{E}}_1,{\mathcal{E}}_3)\omega \left({\mathcal{E}}_2\right)+g({\mathcal{E}}_2,{\mathcal{E}}_3)\omega \left({\mathcal{E}}_1\right)\hfill \\ & & +2\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\omega \left({\mathcal{E}}_3\right)).\hfill \end{array}$$

(31)

Since $\nabla g=0$, then the formula [36]

$$({\pounds }_{\mathcal{E}}{\nabla }_{{\mathcal{E}}_1}g-{\nabla }_{{\mathcal{E}}_1}{\pounds }_{\mathcal{E}}g-{\nabla }_{[\mathcal{E},{\mathcal{E}}_1]}g)({\mathcal{E}}_2,{\mathcal{E}}_3)=-g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_2),{\mathcal{E}}_3)-g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_3),{\mathcal{E}}_2)$$

(32)

becomes

$$\left({\nabla }_{{\mathcal{E}}_1}{\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_2,{\mathcal{E}}_3)=g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_2),{\mathcal{E}}_3)+g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_3),{\mathcal{E}}_2).$$

(33)

Moreover, since ${\pounds }_{\mathcal{E}}\nabla$ is symmetric, then we have

$$2g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_2),{\mathcal{E}}_3)=\left({\nabla }_{{\mathcal{E}}_1}{\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_2,{\mathcal{E}}_3)+\left({\nabla }_{{\mathcal{E}}_2}{\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_1,{\mathcal{E}}_3)-\left({\nabla }_{{\mathcal{E}}_3}{\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_1,{\mathcal{E}}_2).$$

(34)

By using Equation (31) in the the last equation, we arrive at

$$\begin{array}{ccc}\hfill 2g(({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_2),{\mathcal{E}}_3)& =& -{\mathcal{E}}_1\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2})g({\mathcal{E}}_2,{\mathcal{E}}_3)+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\omega \left({\mathcal{E}}_3\right)\}\hfill \\ \hfill & & -{\mathcal{E}}_2\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2})g({\mathcal{E}}_1,{\mathcal{E}}_3)+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}\hfill \\ \hfill & & +{\mathcal{E}}_3\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2})g({\mathcal{E}}_1,{\mathcal{E}}_2)+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}\hfill \\ \hfill & & +4\beta (\frac{r}{m-1}-m)\{\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\omega \left({\mathcal{E}}_3\right)+g({\mathcal{E}}_1,{\mathcal{E}}_2)\omega \left({\mathcal{E}}_3\right)\},\hfill \end{array}$$

(35)

and from Equation (35), it follows that

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill 2({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_2)& =& -{\mathcal{E}}_1\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2}){\mathcal{E}}_2+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\zeta \}\hfill \\ & & -{\mathcal{E}}_2\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2}){\mathcal{E}}_1+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\zeta \}\hfill \\ & & +D\left(r\right)\{2(\frac{\beta }{m-1}-\frac{\delta }{2})g({\mathcal{E}}_1,{\mathcal{E}}_2)+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}\hfill \\ & & +4\beta (\frac{r}{m-1}-m)\{g({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta +\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\zeta \}.\hfill \end{array}\end{array}$$

(36)

Putting ${\mathcal{E}}_1=\zeta$ in Equation (36), then using Equations (6), (8) and (24), we find

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill 2({\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_2,\zeta )& =& \delta g(Dr,{\mathcal{E}}_2)\zeta -\delta D\left(r\right)\omega \left({\mathcal{E}}_2\right)\hfill \\ & & -2(r-m(m-1))\{2(\frac{\beta }{m-1}-\frac{\delta }{2}){\mathcal{E}}_2+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\zeta \}.\hfill \end{array}\end{array}$$

(37)

The covariant differentiation of Equation (37) along ${\mathcal{E}}_1$ and the use of Equations (6), (8) and (37) give

$$\begin{array}{ccc}\hfill 2({\nabla }_{{\mathcal{E}}_1}{\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_2,\zeta )& =& -3g(Dr,{\mathcal{E}}_1)\{2(\frac{\beta }{m-1}-\frac{\delta }{2}){\mathcal{E}}_2+\frac{2\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\zeta \}\hfill \\ & & -\frac{2\beta }{m-1}g(Dr,{\mathcal{E}}_2)({\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta )+\frac{2\beta }{m-1}D\left(r\right)\{g({\mathcal{E}}_1,{\mathcal{E}}_2)\hfill \\ & & +\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}+\frac{4(r-m(m-1\left)\right)}{m-1}\{\beta \omega \left({\mathcal{E}}_2\right){\mathcal{E}}_1\hfill \\ & & -(\beta -\frac{\delta (m-1)}{2})\omega \left({\mathcal{E}}_1\right){\mathcal{E}}_2+2\beta g({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta +2\beta \omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\zeta \}\hfill \\ & & +\delta g({\nabla }_{{\mathcal{E}}_1}Dr,{\mathcal{E}}_2)\zeta -\delta \left({\nabla }_{{\mathcal{E}}_1}Dr\right)\omega \left({\mathcal{E}}_2\right).\hfill \end{array}$$

(38)

Again, from [36], we have

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{E}}\mathcal{R}\right)({\mathcal{E}}_1,{\mathcal{E}}_2){\mathcal{E}}_3=({\nabla }_{{\mathcal{E}}_1}{\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_2,{\mathcal{E}}_3)-({\nabla }_{{\mathcal{E}}_2}{\pounds }_{\mathcal{E}}\nabla )({\mathcal{E}}_1,{\mathcal{E}}_3).\end{array}$$

(39)

By putting ${\mathcal{E}}_3=\zeta$ and using Equation (38), Equation (39) takes the form

$$\begin{array}{ccc}\hfill 2\left({\pounds }_{\mathcal{E}}\mathcal{R}\right)({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta & =& g(Dr,{\mathcal{E}}_1)\{(3\delta -\frac{4\beta }{m-1}){\mathcal{E}}_2-\frac{4\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\zeta \}\hfill \\ & & -g(Dr,{\mathcal{E}}_2)\{(3\delta -\frac{4\beta }{m-1}){\mathcal{E}}_1-\frac{4\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\zeta \}\hfill \\ & & -\frac{4(r-m(m-1\left)\right)}{m-1}\{(2\beta -\frac{\delta (m-1)}{2})\omega \left({\mathcal{E}}_1\right){\mathcal{E}}_2\hfill \\ & & -(2\beta -\frac{\delta (m-1)}{2})\omega \left({\mathcal{E}}_2\right){\mathcal{E}}_1\}+\delta g({\nabla }_{{\mathcal{E}}_1}Dr,{\mathcal{E}}_2)\zeta \hfill \\ & & -\delta g({\nabla }_{{\mathcal{E}}_2}Dr,{\mathcal{E}}_1)\zeta +\delta \left({\nabla }_{{\mathcal{E}}_2}Dr\right)\omega \left({\mathcal{E}}_1\right)-\delta \left({\nabla }_{{\mathcal{E}}_1}Dr\right)\omega \left({\mathcal{E}}_2\right).\hfill \end{array}$$

(40)

Taking the inner product of Equation (40) with ${\mathcal{E}}_4$, we have

$$\begin{array}{ccc}\hfill 2g\left(\left({\pounds }_{\mathcal{E}}\mathcal{R}\right)({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta ,{\mathcal{E}}_4\right)& =& g(Dr,{\mathcal{E}}_1)\{(3\delta -\frac{4\beta }{m-1})g({\mathcal{E}}_2,{\mathcal{E}}_4)-\frac{4\beta }{m-1}\omega \left({\mathcal{E}}_2\right)\omega \left({\mathcal{E}}_4\right)\}\hfill \\ \hfill & & -g(Dr,{\mathcal{E}}_2)\{(3\delta -\frac{4\beta }{m-1})g({\mathcal{E}}_1,{\mathcal{E}}_4)-\frac{4\beta }{m-1}\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_4\right)\}\hfill \\ \hfill & & -\frac{4(r-m(m-1\left)\right)}{m-1}\{(2\beta -\frac{\delta (m-1)}{2})\omega \left({\mathcal{E}}_1\right)g({\mathcal{E}}_2,{\mathcal{E}}_4)\hfill \\ \hfill & & -(2\beta -\frac{\delta (m-1)}{2})\omega \left({\mathcal{E}}_2\right)g({\mathcal{E}}_1,{\mathcal{E}}_4)\}+\delta g({\nabla }_{{\mathcal{E}}_1}Dr,{\mathcal{E}}_2)\omega \left({\mathcal{E}}_4\right)\hfill \\ \hfill & & -\delta g({\nabla }_{{\mathcal{E}}_2}Dr,{\mathcal{E}}_1)\omega \left({\mathcal{E}}_4\right)+\delta g(\left({\nabla }_{{\mathcal{E}}_2}Dr\right),{\mathcal{E}}_4)\omega \left({\mathcal{E}}_1\right)\hfill \\ \hfill & & -\delta g(\left({\nabla }_{{\mathcal{E}}_1}Dr\right),{\mathcal{E}}_4)\omega \left({\mathcal{E}}_2\right).\hfill \end{array}$$

(41)

Let {${\ell }_1$, ${\ell }_2$, ${\ell }_3$…${\ell }_{m-1}$, ${\ell }_m=\zeta$ } be the orthonormal basis of the tangent space at each point of ${\left(LPK\right)}_m$. By putting ${\mathcal{E}}_1={\mathcal{E}}_4={\ell }_i$ and taking the summation over $i(1\le i\le m)$, we find

$$\begin{array}{ccc}\hfill 2\left({\pounds }_{\mathcal{E}}S\right)({\mathcal{E}}_2,\zeta )& =& \{\frac{4\beta (m-2)}{m-1}-3(m-1)\delta \}{\mathcal{E}}_2\left(r\right)\hfill \\ & & +2(r-m(m-1))\{\frac{4\beta (m-2)}{m-1}-(m-1)\delta \}\omega \left({\mathcal{E}}_2\right)\hfill \\ & & +\delta g({\nabla }_{\zeta }Dr,{\mathcal{E}}_2)-\delta (\Delta r)\omega \left({\mathcal{E}}_2\right),\hfill \end{array}$$

(42)

where Equation (24) is used and $\Delta$ appears for the Laplacian of g. By putting ${\mathcal{E}}_2=\zeta$ in Equation (42), then using Equations (6), (24) and (26), we find

$$\begin{array}{ccc}\hfill 2\left({\pounds }_{\mathcal{E}}S\right)(\zeta ,\zeta )& =& -4(m-2)(r-m(m-1\left)\right)\delta +\delta (\Delta r).\hfill \end{array}$$

(43)

Taking the Lie derivative of Equation (19) along $\mathcal{E}$, we have

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{E}}S\right)(\zeta ,\zeta )=-2(m-1)\omega \left({\pounds }_{\mathcal{E}}\zeta \right).\end{array}$$

(44)

By putting ${\mathcal{E}}_2=\zeta$ in Equation (30), we have

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{E}}g\right)({\mathcal{E}}_1,\zeta )=-\{2\beta (m-1)+2\rho -\delta r\}\omega \left({\mathcal{E}}_1\right).\end{array}$$

(45)

The Lie derivative of $g(\zeta ,\zeta )+1=0$ leads to

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{E}}g\right)(\zeta ,\zeta )=-2\omega \left({\pounds }_{\mathcal{E}}\zeta \right)\end{array}$$

(46)

Now combining Equations (43)–(46), we deduce

$$\begin{array}{c}\hfill \Delta r=\Psi ,\end{array}$$

(47)

where $\Psi =\frac{4(m-1)}{\delta }\{\beta (m-1)+\rho \}+2(m-3)r-4m(m-1)(m-2)$, $\delta \ne 0$.

An M of dimension m satisfies Poisson’s equation if $\Delta \vartheta =\Psi$ holds for smooth functions $\vartheta$ and $\Psi$ on M. Poisson’s equation reduces to the Laplace equation if $\Psi =0$.

This definition, together with Equation (47), states the following:

Theorem 1. 

Let the metric of a ζ-conformally flat ${\left(LPK\right)}_m$ be an RYS $(g,\mathcal{E},\rho ,\beta ,\delta )$. Then, the scalar curvature of ${\left(LPK\right)}_m$ satisfies the Poisson Equation (47).

Corollary 1. 

The scalar curvature of a ζ-conformally flat ${\left(LPK\right)}_m,m(>3)$ admitting an RYS $(g,\mathcal{E},\rho ,\beta ,\delta )$ satisfies the Laplace equation if and only if $r=\frac{2(m-1)}{m-3}[m(m-2)-\frac{\left(\beta \right(m-1)+\rho )}{\delta }]$.

Let a $\zeta$-conformally flat ${\left(LPK\right)}_m,m(>3)$ admit an RYS $(g,\mathcal{E},\rho ,\beta ,\delta )$. If r satisfies the Laplace equation, then $r=\frac{2(m-1)}{m-3}[m(m-2)-\frac{\left(\beta \right(m-1)+\rho )}{\delta }]=\mathrm{constant}$. This equation together with Remark 1 gives $\rho =(m-1)(\frac{\delta m}{2}-\beta )$. Thus, we state:

Corollary 2. 

Let the metric of a ζ-conformally flat ${\left(LPK\right)}_m,m(>$3)be an RYS $(g,\mathcal{E},\rho ,\beta ,\delta )$ and suppose that its scalar curvature satisfies the Laplace equation. Then, we have:

Values of $\beta ,\delta$	Soliton type	Conditions for $(g,\mathcal{E},\rho ,\beta ,\delta )$ to be expanding, shrinking or steady
$\beta =0$, $\delta =1$	Yamabe soliton	$(g,\mathcal{E},\rho ,\beta ,\delta )$ is expanding.
$\beta =1$, $\delta =-1$	Einstein soliton	$(g,\mathcal{E},\rho ,\beta ,\delta )$ is shrinking.

4. GRYS on a $\zeta$-Conformally Flat ${\left(\mathbf{LPK}\right)}_{\mathbf{m}}$

Let the metric of a $\zeta$-conformally flat ${\left(LPK\right)}_m$ be a GRYS. Then Equation (5) can be written as

$$\begin{array}{c}\hfill {\nabla }_{{\mathcal{E}}_1}Du+\beta \mathcal{Q}{\mathcal{E}}_1+(\rho -\frac{\delta r}{2}){\mathcal{E}}_1=0\end{array}$$

(48)

for all ${\mathcal{E}}_1$ on ${\left(LPK\right)}_m$, where D appears for the gradient operator of g. The covariant derivative of Equation (48) along ${\mathcal{E}}_2$ leads to

$$\begin{array}{c}\hfill {\nabla }_{{\mathcal{E}}_2}{\nabla }_{{\mathcal{E}}_1}Du=-\beta \{\left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1+\mathcal{Q}\left({\nabla }_{{\mathcal{E}}_2}{\mathcal{E}}_1\right)\}+\delta \frac{{\mathcal{E}}_2\left(r\right)}{2}{\mathcal{E}}_1-(\rho -\frac{\delta r}{2}){\nabla }_{{\mathcal{E}}_2}{\mathcal{E}}_1.\end{array}$$

(49)

Interchanging ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ in Equation (49), we have

$$\begin{array}{c}\hfill {\nabla }_{{\mathcal{E}}_1}{\nabla }_{{\mathcal{E}}_2}Du=-\beta \{\left({\nabla }_{{\mathcal{E}}_1}\mathcal{Q}\right){\mathcal{E}}_2+\mathcal{Q}\left({\nabla }_{{\mathcal{E}}_1}{\mathcal{E}}_2\right)\}+\delta \frac{{\mathcal{E}}_1\left(r\right)}{2}{\mathcal{E}}_2-(\rho -\frac{\delta r}{2}){\nabla }_{{\mathcal{E}}_1}{\mathcal{E}}_2.\end{array}$$

(50)

On account of Equations (49) and (50), we easily find

$$\begin{array}{c}\hfill R({\mathcal{E}}_1,{\mathcal{E}}_2)Du=\beta \{\left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1-\left({\nabla }_{{\mathcal{E}}_1}\mathcal{Q}\right){\mathcal{E}}_2\}+\frac{\delta }{2}\{{\mathcal{E}}_1\left(r\right){\mathcal{E}}_2-{\mathcal{E}}_2\left(r\right){\mathcal{E}}_1\}.\end{array}$$

(51)

Taking the inner product of Equation (51) with ${\mathcal{E}}_3$, we have

$$\begin{array}{ccc}\hfill g(R({\mathcal{E}}_1,{\mathcal{E}}_2)Du,{\mathcal{E}}_3)& =& \beta \{g(\left({\nabla }_{{\mathcal{E}}_2}\mathcal{Q}\right){\mathcal{E}}_1,{\mathcal{E}}_3)-g(\left({\nabla }_{{\mathcal{E}}_1}\mathcal{Q}\right){\mathcal{E}}_2,{\mathcal{E}}_3)\}\hfill \\ \hfill & & +\frac{\delta }{2}\{{\mathcal{E}}_1\left(r\right)g({\mathcal{E}}_2,{\mathcal{E}}_3)-{\mathcal{E}}_2\left(r\right)g({\mathcal{E}}_1,{\mathcal{E}}_3)\}.\hfill \end{array}$$

(52)

Let {${\ell }_1$, ${\ell }_2$, ${\ell }_3$…${\ell }_{m-1}$, ${\ell }_m=\zeta$ } be the orthonormal basis of the tangent space at each point of an ${\left(LPK\right)}_m$. By putting ${\mathcal{E}}_1={\mathcal{E}}_3={\ell }_i$ and taking the summation over $i(1\le i\le m)$, we find

$$\begin{array}{c}\hfill S({\mathcal{E}}_2,Du)=\left\{\frac{\beta -(m-1)\delta }{2}\right\}{\mathcal{E}}_2\left(r\right).\end{array}$$

(53)

From Equation (23), we can write

$$\begin{array}{c}\hfill S({\mathcal{E}}_2,Du)=(\frac{r}{m-1}-1){\mathcal{E}}_2\left(u\right)+(\frac{r}{m-1}-m)\omega \left(u\right)\zeta \left(u\right).\end{array}$$

(54)

From Equations (53) and (54), we find

$$\begin{array}{c}\hfill (\beta -(m-1)\delta ){\mathcal{E}}_2\left(r\right)=2(\frac{r}{m-1}-1){\mathcal{E}}_2\left(u\right)+2(\frac{r}{m-1}-m)\omega \left({\mathcal{E}}_2\right)\zeta \left(u\right).\end{array}$$

(55)

By putting ${\mathcal{E}}_2=\zeta$ in Equation (55), then using Equations (6) and (24), we find

$$\begin{array}{c}\hfill \zeta \left(u\right)=\left\{\frac{\beta -(m-1)\delta }{m-1}\right\}(r-m(m-1)).\end{array}$$

(56)

By the use of Equation (56) in Equation (55), we have

$$\begin{array}{ccc}\hfill (\beta -(m-1)\delta ){\mathcal{E}}_2\left(r\right)& =& 2(\frac{r}{m-1}-1){\mathcal{E}}_2\left(u\right)\hfill \\ \hfill & & +2(\beta -(m-1)\delta ){(\frac{r}{m-1}-m)}^2\omega \left({\mathcal{E}}_2\right).\hfill \end{array}$$

(57)

Taking the covariant derivative of Equation (57) along ${\mathcal{E}}_1$, we find

$$\begin{array}{c}(\beta -(m-1)\delta )g({\nabla }_{{\mathcal{E}}_1}Dr,{\mathcal{E}}_2)\hfill \\ =\frac{2{\mathcal{E}}_1\left(r\right)}{m-1}{\mathcal{E}}_2\left(u\right)+2(\frac{r}{m-1}-1)g({\nabla }_{{\mathcal{E}}_1}Du,{\mathcal{E}}_2)\hfill \\ +\frac{2(r-m(m-1\left)\right)(\beta -(m-1\left)\delta \right)}{{(m-1)}^2}{\mathcal{E}}_1\left(r\right)\omega \left({\mathcal{E}}_2\right)\hfill \\ -\frac{2{(r-m(m-1))}^2(\beta -(m-1)\delta )}{{(m-1)}^2}\{g({\mathcal{E}}_1,{\mathcal{E}}_2)+\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}\hfill \end{array}$$

(58)

Interchanging ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ in Equation (58), we have

$$\begin{array}{c}(\beta -(m-1)\delta )g({\nabla }_{{\mathcal{E}}_2}Dr,{\mathcal{E}}_1)\hfill \\ =\frac{2{\mathcal{E}}_2\left(r\right)}{m-1}{\mathcal{E}}_1\left(u\right)+2(\frac{r}{m-1}-1)g({\nabla }_{{\mathcal{E}}_2}Du,{\mathcal{E}}_1)\hfill \\ +\frac{2(r-m(m-1\left)\right)(\beta -(m-1\left)\delta \right)}{{(m-1)}^2}{\mathcal{E}}_2\left(r\right)\omega \left({\mathcal{E}}_1\right)\hfill \\ -\frac{2{(r-m(m-1))}^2(\beta -(m-1)\delta )}{{(m-1)}^2}\{g({\mathcal{E}}_1,{\mathcal{E}}_2)+\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)\}\hfill \end{array}$$

(59)

The equality of Equations (58) and (59) yields

$$\begin{array}{c}(m-1){\mathcal{E}}_1\left(r\right){\mathcal{E}}_2\left(u\right)+2(r-m(m-1))(\beta -(m-1)\delta ){\mathcal{E}}_1\left(r\right)\omega \left({\mathcal{E}}_2\right)\\ -(m-1){\mathcal{E}}_2\left(r\right){\mathcal{E}}_1\left(u\right)-2(r-m(m-1))(\beta -(m-1)\delta ){\mathcal{E}}_2\left(r\right)\omega \left({\mathcal{E}}_1\right)=0,\end{array}$$

(60)

from which, by substituting ${\mathcal{E}}_2=\zeta$ and following Equations (6), (24) and (56), from Equation (60), we infer

$$\begin{array}{c}\hfill (r-m(m-1))\{(\beta -(m-1)\delta ){\mathcal{E}}_1\left(r\right)+2(m-1){\mathcal{E}}_1\left(u\right)\\ \hfill +4(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)\}=0.\end{array}$$

(61)

Thus, we have either $(\beta -(m-1)\delta ){\mathcal{E}}_1\left(r\right)+2(m-1){\mathcal{E}}_1\left(u\right)+4(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)=0$, or $r=m(m-1)$. For the second case $r=m(m-1)$, Equations (56) and (57) yield that u is constant and hence the GRYS on a $\zeta$-conformally flat ${\left(LPK\right)}_m$ is trivial. Moreover, a $\zeta$-conformally flat ${\left(LPK\right)}_m$ is an Einstein manifold and its scalar curvature is constant. On the other hand, if r is non-constant, that is, $r\ne m(m-1)$ and $(\beta -(m-1)\delta ){\mathcal{E}}_1\left(r\right)=-2(m-1){\mathcal{E}}_1\left(u\right)-4(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)$, in view of Equation (57), it becomes

$$\begin{array}{c}\hfill (r+(m-1)(m-2))\{(m-1){\mathcal{E}}_1\left(u\right)+(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)\}=0.\end{array}$$

(62)

From Equation (62), it follows that either $(m-1){\mathcal{E}}_1\left(u\right)+(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)=0$ or $r=-(m-1)(m-2)=\mathrm{constant}$, which is inadmissible (by hypothesis).

Thus, we have,

$${\mathcal{E}}_1\left(u\right)=-\frac{1}{m-1}(r-m(m-1))(\beta -(m-1)\delta )\omega \left({\mathcal{E}}_1\right)⟺$$

$$\begin{array}{c}\hfill Du=-\frac{1}{m-1}(r-m(m-1))(\beta -(m-1)\delta )\zeta =-\zeta \left(u\right)\zeta .\end{array}$$

(63)

This shows that the gradient of u is pointwise collinear with the velocity vector field $\zeta$.

Now, taking the covariant derivative of Equation (63) with respect to ${\mathcal{E}}_1$, then using Equations (13) and (48), we find

$$\begin{array}{c}\hfill \beta \mathcal{Q}{\mathcal{E}}_1+(\rho -\frac{\delta r}{2}){\mathcal{E}}_1={\mathcal{E}}_1\left(\zeta \left(u\right)\right)\zeta -\zeta \left(u\right)({\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta ),\end{array}$$

(64)

which forms a perfect fluid spacetime.

Now, by replacing ${\mathcal{E}}_1$ by $\zeta$ in Equation (64), then using Equations (6), (20) and (56), we find

$$\begin{array}{c}\hfill \rho =\frac{(4\beta -3(m-1\left)\delta \right)r}{2(m-1)}-\{(3m-1)\beta -2m(m-1)\delta \},\end{array}$$

(65)

which yields that the scalar curvature of the ${\left(LPK\right)}_m$ is constant. This contradicts our hypothesis that r is non-constant. Thus, the only possibility is $r=m(m-1)$. By considering the above facts, we have the following results:

Theorem 2. 

An ${\left(LPK\right)}_m$ admitting a GRYS is an Einstein spacetime and the GRYS is trivial.

Corollary 3. 

If the metric of an ${\left(LPK\right)}_m$ is a gradient Ricci soliton, then the ${\left(LPK\right)}_m$ has a constant scalar curvature.

Equations (23) and (64) together with Theorem 2 reduce to

$$\begin{array}{c}\hfill \mathcal{Q}=(m-1)I,\end{array}$$

(66)

and

$$\begin{array}{c}\hfill \beta \mathcal{Q}+\rho -\frac{\delta m(m-1)}{2}=\omega \otimes \zeta .\end{array}$$

(67)

The above Equation (66) and Equation (67) lead to $\rho =\frac{m-1}{2}\{\delta m-2\beta -\frac{2}{m-1}\}$. Thus, the GRYS on the manifold is expanding, shrinking or steady if $m\delta >2\beta +\frac{2}{m-1},m\delta <2\beta +\frac{2}{m-1}$ or $m\delta =2\beta +\frac{2}{m-1}$. Now, we state:

Corollary 4. 

A GRYS on an ${\left(LPK\right)}_m$ is either expanding or shrinking or steady if either $m\delta >2\beta +\frac{2}{m-1}$, $m\delta <2\beta +\frac{2}{m-1}$ or $m\delta =2\beta +\frac{2}{m-1}$.

Corollary 5. 

Let the metric of an ${\left(LPK\right)}_m$ be a GRYS $(g,Du,\rho ,\beta ,\delta )$. Then, we have

Values of $\beta ,\delta$	Soliton type	Soliton constant	$(g,Du,\rho ,\beta ,\delta )\mathit{tobeexpanding},\mathit{shrinkingorsteady}$
$\beta =1$, $\delta =0$	Ricci soliton	$\rho =-m$	$(g,Du,\rho ,\beta ,\delta )\mathit{isshrinking}$
$\beta =0$, $\delta =1$	Yamabe soliton	$\rho =\frac{(m-2)(m+1)}{2},providedm>2$	$(g,Du,\rho ,\beta ,\delta )\mathit{isexpanding}$
$\beta =1$, $\delta =-1$	Einstein soliton	$\rho =-\frac{1}{2}m(m-3),providedm>3$	$(g,Du,\rho ,\beta ,\delta )\mathit{isshrinking}$
$\beta =1$, $\delta =-2\rho$	$\rho$-Einstein soliton	$\rho =-m(m-1)[\varrho +\frac{1}{m-1}]$	$\left(i\right)(g,Du,\rho ,\beta ,\delta )\mathit{isshrinkingif}$ $\varrho (m-1)+1>0$ $\left(ii\right)(g,Du,\rho ,\beta ,\delta )\mathit{isexpandingif}$ $\varrho (m-1)+1<0$ $\left(iii\right)(g,Du,\rho ,\beta ,\delta )\mathit{isshrinkingif}$ $\varrho (m-1)+1=0$

5. Example of Lorentzian Para-Kenmotsu Manifold

Let $M^4=\left\{(x,y,z,\kappa )\in R^4:\kappa >0\right\}$ be a manifold of dimension four, where $(x,y,z,\kappa )$ are the standard coordinates in $R^4$. Let ${\ell }_1,$${\ell }_2$, ${\ell }_3$ and ${\ell }_4$ be the vector fields on $M^4$ given by

$${\ell }_1=\kappa \frac{\partial }{\partial x},{\ell }_2=\kappa \frac{\partial }{\partial y},{\ell }_3=\kappa \frac{\partial }{\partial z},{\ell }_4=\kappa \frac{\partial }{\partial \kappa }=\zeta ,$$

(68)

which are linearly independent at each point of $M^4$. Let g be the Lorentzian metric defined by

$$g({\ell }_i,{\ell }_j)=\left\{\begin{array}{c}1,1\le i=j\le 3,\hfill \\ -1,i=j=4,\hfill \\ 0,1\le i\ne j\le 4.\hfill \end{array}\right.$$

(69)

Let the one-form $\omega$ be defined by $\omega \left({\mathcal{E}}_1\right)=g({\mathcal{E}}_1,{\ell }_4)=g({\mathcal{E}}_1,\zeta )$ for all ${\mathcal{E}}_1\in \mathfrak{X}\left(M^4\right)$, and let f be the $(1,1)$-tensor field defined by

$$f{\ell }_1=-{\ell }_1,f{\ell }_2=-{\ell }_2,f{\ell }_3=-{\ell }_3,f{\ell }_4=0.$$

(70)

By using the linearity of f and g, we have

$$\omega \left(\zeta \right)=g(\zeta ,\zeta )=-1,f^2{\mathcal{E}}_1={\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta \mathrm{and}g(f{\mathcal{E}}_1,f{\mathcal{E}}_2)=g({\mathcal{E}}_1,{\mathcal{E}}_2)+\omega \left({\mathcal{E}}_1\right)\omega \left({\mathcal{E}}_2\right)$$

(71)

for all ${\mathcal{E}}_1,{\mathcal{E}}_2\in \mathfrak{X}\left(M^4\right)$. Thus, for ${\ell }_4=\zeta$, the structure $(f,\zeta ,\omega ,g)$ defines a Lorentzian almost paracontact metric structure on $M^4$.

Then, we have

$$[{\ell }_i,{\ell }_4]=\left\{\begin{array}{c}-{\ell }_i,\mathrm{for}1\le i\le 3,\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

(72)

By using Koszul’s formula, we can easily find

$${\nabla }_{{\ell }_1}{\ell }_1=-{\ell }_4,{\nabla }_{{\ell }_1}{\ell }_2=0,{\nabla }_{{\ell }_1}{\ell }_3=0,{\nabla }_{{\ell }_1}{\ell }_4=-{\ell }_1,$$

$${\nabla }_{{\ell }_2}{\ell }_1=0,{\nabla }_{{\ell }_2}{\ell }_2=-{\ell }_4,{\nabla }_{{\ell }_2}{\ell }_3=0,{\nabla }_{{\ell }_2}{\ell }_4=-{\ell }_2,$$

(73)

$${\nabla }_{{\ell }_3}{\ell }_1=0,{\nabla }_{{\ell }_3}{\ell }_2=0,{\nabla }_{{\ell }_3}{\ell }_3=-{\ell }_4,{\nabla }_{{\ell }_3}{\ell }_4=-{\ell }_3,$$

$${\nabla }_{{\ell }_4}{\ell }_1=0,{\nabla }_{{\ell }_4}{\ell }_2=0,{\nabla }_{{\ell }_4}{\ell }_3=0,{\nabla }_{{\ell }_4}{\ell }_4=0.$$

Moreover, one can easily verify that

$${\nabla }_{{\mathcal{E}}_1}\zeta +{\mathcal{E}}_1+\omega \left({\mathcal{E}}_1\right)\zeta =0\mathrm{and}\left({\nabla }_{{\mathcal{E}}_1}f\right){\mathcal{E}}_2+g(f{\mathcal{E}}_1,{\mathcal{E}}_2)\zeta +\omega \left({\mathcal{E}}_2\right)f{\mathcal{E}}_1=0.$$

(74)

Therefore, $M^4$ is an $LP$-Kenmotsu manifold.

The non-vanishing components of $\mathcal{R}$ are obtained as follows:

$$\mathcal{R}({\ell }_1,{\ell }_2){\ell }_1=-{\ell }_2,\mathcal{R}({\ell }_1,{\ell }_2){\ell }_2={\ell }_1,\mathcal{R}({\ell }_1,{\ell }_3){\ell }_1=-{\ell }_3,\mathcal{R}({\ell }_1,{\ell }_3){\ell }_3={\ell }_1,$$

$$\mathcal{R}({\ell }_1,{\ell }_4){\ell }_1=-{\ell }_4,\mathcal{R}({\ell }_1,{\ell }_4){\ell }_4=-{\ell }_1,\mathcal{R}({\ell }_2,{\ell }_3){\ell }_2=-{\ell }_3,\mathcal{R}({\ell }_2,{\ell }_3){\ell }_3={\ell }_2,$$

(75)

$$\mathcal{R}({\ell }_2,{\ell }_4){\ell }_2=-{\ell }_4,\mathcal{R}({\ell }_2,{\ell }_4){\ell }_4=-{\ell }_2,\mathcal{R}({\ell }_3,{\ell }_4){\ell }_3=-{\ell }_4,\mathcal{R}({\ell }_3,{\ell }_4){\ell }_4=-{\ell }_3.$$

Moreover, we calculate $\mathcal{S}$ as follows:

$$S({\ell }_1,{\ell }_1)=3=S({\ell }_2,{\ell }_2)=S({\ell }_3,{\ell }_3),S({\ell }_4,{\ell }_4)=-3.$$

(76)

Therefore, we have

$$r=S({\ell }_1,{\ell }_1)+S({\ell }_2,{\ell }_2)+S({\ell }_3,{\ell }_3)-S({\ell }_4,{\ell }_4)=12.$$

(77)

Let ${\ell }_1,{\ell }_2,$ and ${\ell }_3$ be the vector fields given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{E}}_1=a_1{\ell }_1+a_2{\ell }_2+a_3{\ell }_3+a_4{\ell }_4,\hfill \\ {\mathcal{E}}_2=b_1{\ell }_1+b_2{\ell }_2+b_3{\ell }_3+b_4{\ell }_4,\hfill \\ {\mathcal{E}}_3=c_1{\ell }_1+c_2{\ell }_2+c_3{\ell }_3+c_4{\ell }_4,\hfill \end{array}\right.\end{array}$$

(78)

where $a_i,b_i,c_i\in \mathbb{R}$, for all $i=1,2,3,4$.

Putting ${\mathcal{E}}_3=\zeta$ and $n=4$ in Equation (22), we have

$$\begin{array}{c}\mathcal{C}({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta =\mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta -\frac{1}{2}\{S({\mathcal{E}}_2,\zeta ){\mathcal{E}}_1-S({\mathcal{E}}_1,\zeta ){\mathcal{E}}_2+g({\mathcal{E}}_2,\zeta )\mathcal{Q}{\mathcal{E}}_1\\ -g({\mathcal{E}}_1,\zeta )\mathcal{Q}{\mathcal{E}}_2\}+\frac{r}{6}\{g({\mathcal{E}}_2,{\mathcal{E}}_3){\mathcal{E}}_1-g({\mathcal{E}}_1,{\mathcal{E}}_3){\mathcal{E}}_2\}.\end{array}$$

(79)

By using the above listed values of $\mathcal{R}$, $\mathcal{S}$ and r, we have

$$\begin{array}{c}\hfill \mathcal{R}({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta =(a_4{b}_1-a_1{b}_4){\ell }_1+(a_4{b}_2-a_2{b}_4){\ell }_2+(a_4{b}_3-a_3{b}_4){\ell }_3,\end{array}$$

(80)

$$\begin{array}{c}\hfill \mathcal{S}({\mathcal{E}}_2,\zeta ){\mathcal{E}}_1=-3(a_1{b}_4{\ell }_1+a_2{b}_4{\ell }_2+a_3{b}_4{\ell }_3+a_4{b}_4{\ell }_4),\end{array}$$

(81)

$$\begin{array}{c}\hfill \mathcal{S}({\mathcal{E}}_1,\zeta ){\mathcal{E}}_2=-3(b_1{a}_4{\ell }_1+b_2{a}_4{\ell }_2+b_3{a}_4{\ell }_3+b_4{a}_4{\ell }_4),\end{array}$$

(82)

$$\begin{array}{c}\hfill g({\mathcal{E}}_1,\zeta )=-a_4,g({\mathcal{E}}_2,\zeta )=-b_4,\end{array}$$

(83)

$$\begin{array}{c}\hfill g({\mathcal{E}}_2,\zeta )\mathcal{Q}{\mathcal{E}}_1=-3(a_1{b}_4{\ell }_1+a_2{b}_4{\ell }_2+a_3{b}_4{\ell }_3-a_4{b}_4{\ell }_4),\end{array}$$

(84)

$$\begin{array}{c}\hfill g({\mathcal{E}}_1,\zeta )\mathcal{Q}{\mathcal{E}}_2=-3(b_1{a}_4{\ell }_1+b_2{a}_4{\ell }_2+b_3{a}_4{\ell }_3-b_4{a}_4{\ell }_4).\end{array}$$

(85)

It can be easily seen that $C({\mathcal{E}}_1,{\mathcal{E}}_2)\zeta =0$. Thus, an ${\left(LPK\right)}_4$ is $\zeta$-conformally flat.

Now, by taking $Du=\left({\ell }_{1}u\right){\ell }_1+\left({\ell }_{2}u\right){\ell }_2+\left({\ell }_{3}u\right){\ell }_3+\left({\ell }_{4}u\right){\ell }_4$, we have

$$\begin{array}{c}\hfill {\nabla }_{{\ell }_1}Du=({\ell }_1\left({\ell }_{1}u\right)-\left({\ell }_{4}u\right)){\ell }_1+({\ell }_1\left({\ell }_{2}u\right)){\ell }_2+({\ell }_1\left({\ell }_{3}u\right)){\ell }_3+({\ell }_1\left({\ell }_{4}u\right)-\left({\ell }_{1}u\right)){\ell }_4,\end{array}$$

(86)

$$\begin{array}{c}\hfill {\nabla }_{{\ell }_2}Du=\left({\ell }_2\left({\ell }_{1}u\right)\right){\ell }_1+({\ell }_2\left({\ell }_{2}u\right)-\left({\ell }_{4}u\right)){\ell }_2+({\ell }_2\left({\ell }_{3}u\right)){\ell }_3+({\ell }_2\left({\ell }_{4}u\right)-\left({\ell }_{2}u\right)){\ell }_4,\end{array}$$

(87)

$$\begin{array}{c}\hfill {\nabla }_{{\ell }_3}Du=\left({\ell }_3\left({\ell }_{1}u\right)\right){\ell }_1+\left({\ell }_3\left({\ell }_{2}u\right)\right){\ell }_2+({\ell }_3\left({\ell }_{3}u\right)-\left({\ell }_{4}u\right)){\ell }_3+({\ell }_3\left({\ell }_{4}u\right)-\left({\ell }_{3}u\right)){\ell }_4,\end{array}$$

(88)

$$\begin{array}{c}\hfill {\nabla }_{{\ell }_4}Du=\left({\ell }_4\left({\ell }_{1}u\right)\right){\ell }_1+\left({\ell }_4\left({\ell }_{2}u\right)\right){\ell }_2+\left({\ell }_4\left({\ell }_{3}u\right)\right){\ell }_3+\left({\ell }_4\left({\ell }_{4}u\right)\right){\ell }_4.\end{array}$$

(89)

Thus, by virtue of Equation (48), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\ell }_1\left({\ell }_{1}u\right)-{\ell }_{4}u=-(\rho +3\beta -6\delta ),\hfill \\ {\ell }_2\left({\ell }_{2}u\right)-{\ell }_{4}u=-(\rho +3\beta -6\delta ),\hfill \\ {\ell }_3\left({\ell }_{3}u\right)-{\ell }_{4}u=-(\rho +3\beta -6\delta ),\hfill \\ {\ell }_4\left({\ell }_{4}u\right)=-(\rho +3\beta -6\delta ),\hfill \\ {\ell }_1\left({\ell }_{2}u\right)={\ell }_1\left({\ell }_{3}u\right)=0,\hfill \\ {\ell }_2\left({\ell }_{1}u\right)={\ell }_2\left({\ell }_{3}u\right)=0,\hfill \\ {\ell }_3\left({\ell }_{1}u\right)={\ell }_3\left({\ell }_{2}u\right)=0,\hfill \\ {\ell }_4\left({\ell }_{1}u\right)={\ell }_4\left({\ell }_{2}u\right)={\ell }_4\left({\ell }_{3}u\right)=0,\hfill \\ {\ell }_1\left({\ell }_{4}u\right)-\left({\ell }_{1}u\right)={\ell }_2\left({\ell }_{4}u\right)-\left({\ell }_{2}u\right)=0,\hfill \\ {\ell }_3\left({\ell }_{4}u\right)-\left({\ell }_{3}u\right)=0.\hfill \end{array}\right.\end{array}$$

(90)

Thus, the equations in Equation (90) are, respectively, equal to

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial x^2}-k\frac{\partial u}{\partial k}=-(\rho +3\beta -6\delta ),\end{array}$$

(91)

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial y^2}-k\frac{\partial u}{\partial k}=-(\rho +3\beta -6\delta ),\end{array}$$

(92)

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial z^2}-k\frac{\partial u}{\partial k}=-(\rho +3\beta -6\delta ),\end{array}$$

(93)

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial k^2}+k\frac{\partial u}{\partial k}=-(\rho +3\beta -6\delta ),\end{array}$$

(94)

$$\begin{array}{c}\hfill \frac{{\partial }^{2}u}{\partial x\partial y}=\frac{{\partial }^{2}u}{\partial y\partial z}=\frac{{\partial }^{2}u}{\partial x\partial z}=0,\end{array}$$

(95)

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial k\partial x}+k\frac{\partial u}{\partial x}=k^2\frac{{\partial }^{2}u}{\partial k\partial y}+k\frac{\partial u}{\partial y}=k^2\frac{{\partial }^{2}u}{\partial k\partial z}+k\frac{\partial u}{\partial z}=0.\end{array}$$

(96)

$$\begin{array}{c}\hfill k^2\frac{{\partial }^{2}u}{\partial x\partial k}-k\frac{\partial u}{\partial x}=k^2\frac{{\partial }^{2}u}{\partial y\partial k}-k\frac{\partial f}{\partial y}=k^2\frac{{\partial }^{2}u}{\partial z\partial k}-k\frac{\partial u}{\partial z}=0.\end{array}$$

(97)

From the above equations, it is observed that u is constant for $\rho =-3\beta +6\delta$. Hence, Equation (48) is satisfied. Thus, g is a GRYS with the soliton vector field $\mathcal{E}=Du$, where u is constant and $\rho =-3\beta +6\delta$. This verifies Theorem 2 and Corollary 3.

6. Conclusions

The Ricci flow has been applied as a tool to prove the Poincaré conjecture, geometrization conjecture, differentiable sphere conjecture, uniformalization theorem, etc. It can also be applied to study cancer invasion, avascular tumor growth and decay control, brain surface conformal parameterization, medical imaging (such as the parameterization of a surface, the matching of a surface, splines of a manifold and the formation of a geometric structure on general surfaces), computer graphics, geometric modeling, computer vision, wireless sensor networking, mathematics and physics, etc. It is well known that the Laplace operator is used to study celestial mechanics and measure the flux density of the gradient flow of a function [37]. Several differential equations are expressed in terms of the Laplacian, used to explain various physical problems. The Laplacian appears in problems of computer vision and image processing, electrical and gravitational potentials, the diffusion equation for fluid and heat flow, the de Rham cohomology, the Hodge theory, etc. This manuscript dealt with the study of the Laplacian, and the equations of Poisson and Laplace. We also addressed the existence of a proper gradient Ricci–Yamabe soliton on an ${\left(LPK\right)}_m$.