# ρ-Einstein Solitons on Warped Product Manifolds and Applications

Authors: Nasser Bin Turki; Sameh Shenawy; H. K. EL-Sayied; N. Syied; C. A. Mantica
Journal: Journal of Mathematics
 (2022)

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## Abstract

The purpose of this research is to investigate how aρ-Einstein soliton structure on a warped product manifold affects its base and fiber factor manifolds. Firstly, the pertinent properties of ρ-Einstein solitons are provided. Secondly, numerous necessary and sufficient conditions of a ρ-Einstein soliton warped product manifold to make its factor ρ-Einstein soliton are examined. On a ρ-Einstein gradient soliton warped product manifold, necessary and sufficient conditions for making its factor ρ-Einstein gradient soliton are presented. ρ-Einstein solitons on warped product manifolds admitting a conformal vector field are also considered. Finally, the structure of ρ-Einstein solitons on some warped product space-times is investigated.


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## Body

## 1. An Introduction

Ricci soliton is crucial in the Ricci flow treatment. In References[1, 2], the Ricci flow is defined on a Riemannian manifold E,g by an evolution equation for metrics gt of the following form:(1)∂tgt=−2Ric,where Ric is the Ricci curvature tensor. The initial metric g on E satisfies the following equation:(2)Ric+12ℒζg=λg,where ζ is a vector field on E, λ is a constant, and ℒζ represents the Lie derivative in the direction of a vector field ζ on E. Manifolds admitting such structure are called Ricci soliton [3]. Hamilton first investigated the study of Ricci solitons as fixed points of the Ricci flow in the space of the metrics on E modulo diffeomorphisms and scaling [4]. A Ricci soliton is called shrinking (steady or expanding) if λ>0 (λ=0 or λ<0 respectively). If ζ=0 or is Killing, then the Ricci soliton is called a trivial Ricci soliton. If f is a smooth function and ζ=∇f, then the Ricci soliton is described as gradient, ζ is referred to as the potential vector field, and f is called the potential function. In this case, equation (2) becomes as follows:(3)Ric+Hf=λg,where Hf is the Hessian tensor. Previously, Ricci solitons have been studied in depth for different reasons and in distinct spaces [5–11]. In Reference [12], it is shown that a complete Ricci soliton is gradient. Gradient Ricci solitons are basic generalizations of Einstein manifolds [13]. If λ is a smooth function, then we say that E,g is a nearly Ricci soliton manifold [14–16]. A generalization of Einstein soliton has been deduced by considering the Ricci–Bourguignon flows [17–19]:(4)∂tgt=−2Ric−ρRg.These manifolds are calledρ-Einstein solitons and are defined as follows: Let E,g be a pseudo-Riemannian manifold, and let λ,ρ∈ℝ, ρ≠0, and ζ∈XE. Then, E,g,ζ,λ is called a ρ-Einstein soliton if(5)Ric+12ℒζg=λg+ρRg.Likewise, if a smooth functionf:E⟶ℝ exists such that ζ=∇f, then a ρ-Einstein soliton E,g,ζ,ρ is gradient and denoted by E,g,f,ρ. In this case, equation (5) becomes as follows:(6)Ric+Hessf=λg+ρRg.Aρ-Einstein soliton is denoted as steady, shrinking, or expanding, depending on whether λ has zero, positive, or negative values. The function f is called a ρ-Einstein potential of the gradient ρ-Einstein soliton. Later, this perception was circulated in many instructions, such as m-quasi Einstein manifolds [20], Ricci–Bourguignon almost solitons [21], and E,ρ-quasi-Einstein manifolds [22]. Huang got a sufficient condition for a compact gradient shrinking ρ-Einstein soliton to be isometric to a quotient of the round sphere Sn in Reference [23]. Moreover, Mondal and Shaikh proved that a compact gradient ρ-Einstein soliton with a nontrivial conformal vector field ∇f is isometric to the Euclidean sphere Sn in Reference [24]. Recently, in Reference [21], Dwivedi demonstrated other isometric theories of the gradient Ricci–Bourguignon soliton. In Reference [25], the authors investigated a gradient ρ-Einstein soliton on a Kenmotsu manifold. Some curvature conditions on compact gradient ρ-Einstein soliton M are given in Reference [26] to guarantee that M is isometric to the Euclidean sphere. In contrast, an integral condition on a noncompact ρ-Einstein soliton M is given to ensure the vanishing of the scalar curvature. A splitting theorem of a gradient ρ-Einstein soliton is given in Reference [27]. Accordingly, many characterizations of gradient ρ-Einstein solitons are considered in Reference [28]. The same study was recently extended to Sasakian manifolds in Reference [29]. A study of the lower bound of the diameter of a compact gradient ρ-Einstein soliton is given in Reference [30].To the best of our knowledge, no research has been completed on such a structure on warped product manifolds. In this regard, the research problems from the point of view of warped product manifolds (WPMs) can be summarized into two directions:(1)
Under what conditions does aWPM become a ρ-Einstein soliton or a gradient ρ-Einstein soliton?(2)
What does a factor of aρ-Einstein soliton WPM or a gradient ρ-Einstein soliton WPM inherit?To address these problems, first we proved many results on theρ-Einstein soliton. Then, we investigated necessary and sufficient conditions on a (gradient) ρ-Einstein soliton WPM in order to make its factor (gradient) ρ-Einstein soliton. Additionally, we studied a ρ-Einstein soliton on a WPM admitting a conformal vector field. Finally, we applied our results to generalized Robertson–Walker (GRW) space-times and standard static space-times.


## 2. Preliminaries

### 2.1.ρ-Einstein Solitons on Pseudo-Riemannian Manifolds

Ifζ is a conformal vector field with conformal factor 2ω in a ρ-Einstein soliton E,g,ζ,λ, then(7)RicU,V+12ℒζgU,V=λgU,V+ρRgU,V,RicU,V+ωgU,V=λgU,V+ρRgU,V,RicU,V=λ−ω+ρRgU,V.By taking the trace overU,V, we get the following equation:(8)Rn=λ−ω+ρR,R=λ−ωn1−nρ.Since the scalar curvature of Einstein manifolds is constant, the conformal factor is also constant, that is,ζ is homothetic. Moreover, λ=ω if ρ=1/n.Proposition 1.
Assume thatζ is a conformal vector field on a ρ-Einstein soliton E,g,ζ,λ with factor 2ω. Then, ζ is homothetic, E,g is Einstein, and(9)R=λ−ωn1−nρ,where ρ≠1/n. Moreover, λ=ω if ρ=1/n.Corollary 1.
Assume thatζ is a Killing vector field on a ρ-Einstein soliton E,g,ζ,λ, then(10)R=nλ1−nρ,where ρ≠1/n. Moreover, E,g,ζ,λ is steady if ρ=1/n.
Conversely, assuming thatE,g is an Einstein manifold, then(11)RngU,V+12ℒζgU,V=λgU,V+ρRgU,V,ℒζgU,V=λ−Rn+ρRgU,V.
Therefore, ζ is a homothetic vector field on E.Proposition 2.
In aρ-Einstein soliton E,g,ζ,λ, ζ is a homothetic vector field on E if E,g is Einstein. Furthermore, ζ is Killing if λ=1/n−ρR.
In local coordinates, a contraction of the defining equation implies that(12)Rij+12∇iζj+∇jζi=λgij+ρRgij,∇iζi=nλ+nρ−1R.
Thus, the vector field ζ is divergence-free. The conservative laws in physics usually arise from the vanishing of the divergence of a tensor field. Here is a simple characterization of the vanishing of the divergence of ζ.Corollary 2.
The vector fieldζ in a ρ-Einstein soliton E,g,ζ,λ is divergence-free if and only if nλ+nρ−1R=0.
It is also known that the flow lines of a divergence-free vector field are volume-preserving diffeomorphisms ([31], Chapter 3). This discussion leads to the following result.Theorem 1.
The flow lines of the vector fieldζ in a ρ-Einstein soliton E,g,ζ,λ are volume-preserving diffeomorphisms if and only if nλ+nρ−1R=0.


### 2.2. Warped Product Manifolds

LetEi,gi,Di,i=1,2 denote two ni-dimensional C∞ pseudo-Riemannian manifolds equipped with metric tensors gi where Di is the Levi-Civita connection of the metric gi for i=1,2. Let f1:E1⟶0,∞ be a smooth positive real-valued function. A WPM, denoted by E=E1×fE2, is the product manifold E1×E2 equipped with the metric tensor g=g1⊕f2g2 (for more details the reader is referred to [32–36] and references therein). Let E=E1×fE2 be a pseudo-Riemannian WPM and Ui,Vi∈XEi for all i=1,2. Then, the Ricci tensor Ric of E is given by,(1)
RicU1,V1=Ric1U1,V1−n2/fHfU1,V1,(2)
RicU1,U2=0,(3)
RicU2,V2=Ric2U2,V2−f∘g2U2,V2, where f∘=fΔf+n2−1∇f12, and Δ is the Laplacian on E1.The scalar curvature aWPM satisfies(13)R=R1+1f2R2−2nΔff−nn−11f2g1∇f,∇f.Lemma 1.
(see [35]). In a WPME1×fE2, the Lie derivative with respect to a vector field ζ=ζ1+ζ2 satisfies(14)ℒζgU,V=ℒζ11g1U1,V1+f2ℒζ22g2U2,V2+2fζ1fg2U2,V2,for any vector fields U=U1+U2,V=V1+V2, where ℒζii is the Lie derivative on Ei with respect to ζi, for i=1,2.


## 2.1.ρ-Einstein Solitons on Pseudo-Riemannian Manifolds

Ifζ is a conformal vector field with conformal factor 2ω in a ρ-Einstein soliton E,g,ζ,λ, then(7)RicU,V+12ℒζgU,V=λgU,V+ρRgU,V,RicU,V+ωgU,V=λgU,V+ρRgU,V,RicU,V=λ−ω+ρRgU,V.By taking the trace overU,V, we get the following equation:(8)Rn=λ−ω+ρR,R=λ−ωn1−nρ.Since the scalar curvature of Einstein manifolds is constant, the conformal factor is also constant, that is,ζ is homothetic. Moreover, λ=ω if ρ=1/n.Proposition 1.
Assume thatζ is a conformal vector field on a ρ-Einstein soliton E,g,ζ,λ with factor 2ω. Then, ζ is homothetic, E,g is Einstein, and(9)R=λ−ωn1−nρ,where ρ≠1/n. Moreover, λ=ω if ρ=1/n.Corollary 1.
Assume thatζ is a Killing vector field on a ρ-Einstein soliton E,g,ζ,λ, then(10)R=nλ1−nρ,where ρ≠1/n. Moreover, E,g,ζ,λ is steady if ρ=1/n.
Conversely, assuming thatE,g is an Einstein manifold, then(11)RngU,V+12ℒζgU,V=λgU,V+ρRgU,V,ℒζgU,V=λ−Rn+ρRgU,V.
Therefore, ζ is a homothetic vector field on E.Proposition 2.
In aρ-Einstein soliton E,g,ζ,λ, ζ is a homothetic vector field on E if E,g is Einstein. Furthermore, ζ is Killing if λ=1/n−ρR.
In local coordinates, a contraction of the defining equation implies that(12)Rij+12∇iζj+∇jζi=λgij+ρRgij,∇iζi=nλ+nρ−1R.
Thus, the vector field ζ is divergence-free. The conservative laws in physics usually arise from the vanishing of the divergence of a tensor field. Here is a simple characterization of the vanishing of the divergence of ζ.Corollary 2.
The vector fieldζ in a ρ-Einstein soliton E,g,ζ,λ is divergence-free if and only if nλ+nρ−1R=0.
It is also known that the flow lines of a divergence-free vector field are volume-preserving diffeomorphisms ([31], Chapter 3). This discussion leads to the following result.Theorem 1.
The flow lines of the vector fieldζ in a ρ-Einstein soliton E,g,ζ,λ are volume-preserving diffeomorphisms if and only if nλ+nρ−1R=0.


## 2.2. Warped Product Manifolds

LetEi,gi,Di,i=1,2 denote two ni-dimensional C∞ pseudo-Riemannian manifolds equipped with metric tensors gi where Di is the Levi-Civita connection of the metric gi for i=1,2. Let f1:E1⟶0,∞ be a smooth positive real-valued function. A WPM, denoted by E=E1×fE2, is the product manifold E1×E2 equipped with the metric tensor g=g1⊕f2g2 (for more details the reader is referred to [32–36] and references therein). Let E=E1×fE2 be a pseudo-Riemannian WPM and Ui,Vi∈XEi for all i=1,2. Then, the Ricci tensor Ric of E is given by,(1)
RicU1,V1=Ric1U1,V1−n2/fHfU1,V1,(2)
RicU1,U2=0,(3)
RicU2,V2=Ric2U2,V2−f∘g2U2,V2, where f∘=fΔf+n2−1∇f12, and Δ is the Laplacian on E1.The scalar curvature aWPM satisfies(13)R=R1+1f2R2−2nΔff−nn−11f2g1∇f,∇f.Lemma 1.
(see [35]). In a WPME1×fE2, the Lie derivative with respect to a vector field ζ=ζ1+ζ2 satisfies(14)ℒζgU,V=ℒζ11g1U1,V1+f2ℒζ22g2U2,V2+2fζ1fg2U2,V2,for any vector fields U=U1+U2,V=V1+V2, where ℒζii is the Lie derivative on Ei with respect to ζi, for i=1,2.


## 3.ρ-Einstein Soliton Structure on WPMs

In this section, we investigate theρ-Einstein soliton structure on WPMs. For the rest of this work, let E=E1×fE2 be a WPM with warping function f and let g=g1⊕f2g2. Also, let ζ=ζ1+ζ2 be a vector field on E. Let E,g,ζ,λ be a ρ-Einstein soliton, that is,(15)RicU,V+12ℒζgU,V=λgU,V+ρRgU,V.Thus, for any vector fieldsU=U1+U2,V=V1+V2, and ζ=ζ1+ζ2 on E=E1×fE2, Lemma 1 implies that(16)Ric1U1,V1−n2fHfU1,V1+Ric2U2,V2−f∘g2U2,V2+12ℒζ11g1U1,V1+12f2ℒζ22g2U2,V2+fζ1fg2U2,V2=λg1U1,V1+λf2g2U2,V2+ρRg1U1,V1+ρRf2g2U2,V2.LetU=U1, V=V1, and Hf=σg, then(17)Ric1U1,V1+12ℒζ11g1U1,V1=λ1g1U1,V1+−λ1+λ+n2fσ+ρRg1U1,V1=λ1g1U1,V1+ρ1R1g1U1,V1.Then,E1,g1,ζ1,λ1 is a ρ1-Einstein soliton, where(18)ρ1R1+λ1=ρR+n2fσ+λ.Now, letU=U2 and V=V2, then(19)Ric2U2,V2−f∘g2U2,V2+12f2ℒζ22g2U2,V2+fζ1fg2U2,V2=λf2g2U2,V2+ρRf2g2U2,V2.Thus,(20)Ric2U2,V2+12f2ℒζ22g2U2,V2=λf2+f°−fζ1f+ρRf2g2U2,V2=λ2g2U2,V2+−λ2+λf2+f°−fζ1f+ρRf2g2U2,V2=λ2g2U2,V2+ρ2R2g2U2,V2.Then,E2,g2,f2ζ2,λ2 is a ρ2-Einstein soliton, where(21)ρ2R2+λ2=ρRf2+λf2+f∘−fζ1f.Theorem 2.
LetE,g,ζ,λ,ρ be a ρ-Einstein soliton. Then,(1)
E1,g1,ζ1,λ1 is a ρ1-Einstein soliton if Hf=σg where(22)ρ1R1+λ1=ρR+n2fσ+λ.(2)
E2,g2,f2ζ2,λ2 is a ρ2-Einstein soliton, where(23)ρ2R2+λ2=ρRf2+λf2+f∘−fζ1f.
LetE1,g1 and E2,g2 be two Einstein manifolds with factors μ1 and μ2, respectively, and let Hf=σg. Then equation (16) becomes as follows:(24)μ1g1U1,V1+μ2g2U2,V2−n2fσg1U1,V1−f∘g2U2,V2+12ℒζ11g1U1,V1+12f2ℒζ22g2U2,V2+fζ1fg2U2,V2=λg1U1,V1+λf2g2U2,V2+ρRg1U1,V1+ρRf2g2U2,V2.
Thus,(25)ℒζ11g1U1,V1=2λ+n2fσ−μ1+ρRg1U1,V1,ℒζ22g2U2,V2=2f2f∘−μ2−fζ1f+λf2+ρRf2g2U2,V2.
That is, ζ1 and ζ2 are conformal vector fields on E1 and E2.Theorem 3.
In aρ-Einstein soliton E,g,ζ,λ, E=E1×fE2,(1)
ζ1 is conformal vector field on E1 if Hf=σg and E1,g1 is Einstein, and(2)
ζ2 is conformal vector field on E2 if E2,g2 is Einstein.
The symmetry assumptions induced by Killing vector fields (KVFs) are widely used in general relativity to gain a better understanding of the relationship between matter and the geometry of a space-time. In this case, the metric tensor does not change along the flow lines of a KVF. Such symmetry is measured by the number of independent KVFs. Manifolds of constant curvature admit the maximum number of independent KVFs. Similarly, conformal vector fields (CVFs) play a crucial role in the study of space-time physics. The flow lines of a CVF are conformal transformations of the ambient space. Thus, the existence and characterization of CVFs in pseudo-Riemannian manifolds are essential and therefore are extensively discussed by both mathematicians and physicists.
Now, assume thatζ is a conformal vector field on E, that is, ℒζg=2ωg for some scalar function ω, then ω is constant and(26)RicU,V=λ−ω+ρRgU,V.
This equation implies(27)Ric1U1,V1−n2fHfU1,V1+Ric2U2,V2−f∘g2U2,V2=λ−ω+ρRg1U1,V1+λ−ω+ρRf2g2U2,V2.
IfHf=σg, then(28)Ric1U1,V1=λ−ω+ρR+n2fσg1U1,V1,Ric2U2,V2=f∘+λf2−ωf2+ρRf2g2U2,V2.
That is, both the base and fiber manifolds are Einstein.Theorem 4.
In aρ-Einstein soliton E,g,ζ,λ, E=E1×fE2 admitting a CVF ζ=ζ1+ζ2,(1)
E1,g1 is Einstein if Hf=σg, and(2)
E2,g2 is Einstein.
The condition Hf=σg is equivalent to ∇f is a concircular vector field. Equation (16) yields the following:(29)Ric1U1,V1−n2fHfU1,V1+12ℒζ11g1U1,V1=λg1U1,V1+ρRg1U1,V1.
Suppose that∇f is a concircular vector field with factor γ, that is, DU1∇f=γU1, we get(30)Ric1U1,V1+12ℒζ11g1U1,V1=λg1U1,V1+γn2f+ρRg1U1,V1=λ1g1U1,V1+−λ1+λ+γn2f+ρRg1U1,V1=λ1g1U1,V1+ρ1R1g1U1,V1.
Then, E1,g1,ζ1,λ1 is a ρ1-Einstein soliton where(31)ρ1R1+λ1=γn2f+ρR.Corollary 3.
In aρ-Einstein soliton E,g,ζ,λ,ρ, assume that ∇f is a concircular vector field with factor γ, then E1,g1,ζ1,λ1 is a ρ1-Einstein soliton where(32)ρ1R1+λ1=γn2f+ρR.
Bang-Yen Chen proved that a Riemannian manifold admitting a concircular vector field is locally a warped product of the formI×φE¯1 [37]. Thus, the aforementioned warped product manifold becomes a sequential warped product manifold [38].
From Lemma1, it is clear that ζ1,ζ2 are CVFs on E1,E2 with conformal factors η1,η2, respectively. Then, by employing equation (28) we get the following equation:(33)ℒζ11Ric1U1,V1=n2fσ+λ−ω+ρRℒζ11g1U1,V1+ζ1n2fσ+λ−ω+ρRg1U1,V1.ℒζ11Ric1U1,V1=n2fσ+λ−ω+ρRη1+ζ1n2fσ+λ−ω+ρRg1U1,V1=φ1g1U1,V1.
where(34)φ1=n2fσ+λ−ω+ρRη1+ζ1n2fσ+λ−ω+ρR.
Also,(35)ℒζ22Ric2U2,V2=f∘f2+λ−ωf2+ρRf2ℒζ22g2U2,V2,ℒζ22Ric2U2,V2=f2f∘f2+λ−ω+ρRη2g2U2,V2=φ2g2U2,V2,where(36)φ2=f2f∘f2+λ−ω+ρRη2.Theorem 5.
In aρ-Einstein soliton E,g,ζ,λ admitting a CVF ζ with factor ω,(1)
ℒζ11Ric1U1,V1=φ1g1U1,V1 if Hf=σg, where(37)φ1=n2fσ+λ−ω+ρRη1+ζ1n2fσ+λ−ω+ρR,(2)
ℒζ22Ric2U2,V2=φ2g2U2,V2, where(38)φ2=f2f∘f2+λ−ω+ρRη2.
The KVFs provide the isometries of space-time, whereas the symmetry of the energy-momentum tensor is given by the Ricci collineation. A vector field ζ represents a Ricci collineation if the Ricci tensor is invariant under the Lie dragging through flow lines of ζ. The previous conclusion establishes the shape of the Lie derivative of the Ricci tensor concerning the fields ζi, on Mi, i=1,2.
LetE,g,ζ,λ,ρ be a gradient ρ-Einstein soliton with ζ=∇u, then(39)Ric+Hu=λg+ρRg.
Thus,(40)RicU1+U2,V1+V2+HuU1+U2,V1+V2=λgU1+U2,V1+V2+ρRgU1+U2,V1+V2.
LetU=U1,V=V1(41)Ric1U1,V1−n2fHfU1,V1+H1u1U1,V1=λg1U1,V1+ρRg1U1,V1,Ric1U1,V1+H1ϕ1U1,V1=λ1g1U1,V1+−λ1+λ+ρRg1U1,V1=λ1g1U1,V1+ρ1R1g1U1,V1,
whereϕ1=u1−u2lnf and u1=u at a fixed point of E2. Then, E1,g1,ζ1,ρ1 is a gradient ρ1-Einstein soliton where(42)ρ1R1+λ1=λ+ρR.Now, let U=U2,V=V2, then(43)Ric2U2,V2−f∘g2U2,V2+H2ϕ2U2,V2=λf2g2U2,V2+ρRf2g2U2,V2.
This yields(44)Ric2U2,V2+H2ϕ2U2,V2=λ2g2U2,V2+−λ2+λf2+f∘+ρRf2g2U2,V2=λ2g2U2,V2+ρ2R2g2U2,V2,where u2=u at a fixed point of E1. Then, E2,g2,ζ2,ρ2 is a gradient ρ2-Einstein soliton where(45)ρ2R2+λ2=λf2+f∘+ρRf2.Theorem 6.
In a gradientρ-Einstein soliton E,g,ζ,λ,(1)
E1,g1,ζ1,λ1 is a gradient ρ1-Einstein soliton where(46)ρ1R1+λ1=λ+ρR,(2)
E2,g2,ζ2,λ2 is a gradient ρ2-Einstein soliton where(47)ρ2R2+λ2=λf2+f∘+ρRf2.
This theorem provides an inheritance property of the structure of the gradient ρ-Einstein soliton structure to factor manifolds of the warped product manifold.


### 3.1.ρ¯-Einstein Solitons on GRW Space-Times

LetE¯=I×fE be a generalized Robertson–Walker (GRW) space-time with metric g¯=−dt2⊕f2g. Then, the Ricci curvature tensor Ric on E is as follows:(48)R¯ic∂t,∂t=−nf¨f,R¯icU,∂t=0,R¯icU,V=RicU,V−f⋄gU,V,where f⋄=−ff¨−n−1f˙2, see References [38–40].Lemma 2.
Suppose thath∂t,u∂t,v∂t∈XI and ζ,U,V∈XE, then(49)ℒ¯ζ¯g¯U¯,V¯=−2h˙uv+f2ℒζgU,V+2hff˙gU,V,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ.
LetE¯,g¯,ζ¯,λ¯, E¯=I×fE, be a ρ¯-Einstein soliton GRW space-time. Then,(50)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯,where U¯=u∂t+U,V¯=v∂t+V and ζ¯=h∂t+ζ are vector fields on E¯. Thus,(51)−nf¨fuv+RicU,V−f⋄gU,V−h˙uv+12f2ℒζgU,V+hff˙gU,V=−λ¯uv+f2λ¯gU,V−ρ¯R¯uv+ρ¯R¯f2gU,V.
This yields(52)nf¨=fλ¯−h˙+ρ¯R¯f,RicU,V+12f2ℒζgU,V=λ¯f2gU,V+ρ¯R¯f2gU,V+f⋄gU,V−hff˙gU,V.
Thus, E,g,f2ζ,ρ is a ρ-Einstein soliton, where(53)ρR+λ=λ¯+ρ¯R¯f2+f⋄−hff˙.Theorem 7.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, it is(1)
nf¨=fλ¯−h˙+ρ¯R¯f,(2)
E,g,f2ζ,λ is a ρ-Einstein soliton, where(54)ρR+λ=λ¯+ρ¯R¯f2.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time and ζ¯=h∂t+ζ is a CVF on E¯, that is, ℒ¯ζ¯g¯=ω¯g¯, and ω¯ is constant (see Section 2), then(55)R¯icU¯,V¯=λ¯−ω¯+ρ¯R¯g¯U¯,V¯.
Thus,(56)−nf¨fuv+RicU,V−f⋄gU,V=−λ¯−ω¯+ρ¯R¯uv+λ¯−ω¯+ρ¯R¯f2gU,V.
Thus,(57)nf¨f=λ¯−ω¯+ρ¯R¯,(58)RicU,V=f⋄+λ¯−ω¯+ρ¯R¯f2gU,V.By using equation (57) we get the following equation:(59)RicU,V=n−1ff¨−f˙2gU,V.
Therefore, E,g is an Einstein manifold with factor μ=n−1ff¨−f˙2.Theorem 8.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ admitting a CVF ζ¯=h∂t+ζ, where E¯=I×fE is a GRW space-time, E,g is an Einstein manifold with factor μ=n−1ff¨−f˙2.
From Lemma2, we get ζ is a CVF on E with conformal factor η. Then, by using Theorem 8, we get the following equation:(60)ℒζRicU,V=n−1ff¨−f˙2ℒζgU,V=n−1ff¨−f˙2ηgU,V=φgU,V,where(61)φ=n−1ff¨−f˙2η.Theorem 9.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ admitting a CVF ζ¯=h∂t+ζ, where E¯=I×fE is a GRW space-time,(62)ℒζRicU,V=φgU,V,where(63)φ=n−1ff¨−f˙2η.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, it is(64)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯.
Assume thatE,g is Einstein, then for any vector fields U¯=U,V¯=V, and ζ¯=h∂t+ζ we have get(65)ℒζgU,V=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯gU,V=ηgU,V.
Then, ζ is a CVF on E with conformal factor η where(66)η=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯.Theorem 10.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, ζ is a CVF on E if E,g is Einstein manifold with conformal factor η where(67)η=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯.


### 3.2.ρ¯-Einstein Solitons on a Standard Static Space-Times

A standard static space-time (orf-associated SSST) is a Lorentzian warped product manifold E¯=If×E furnished with the metric g¯=−f2dt2⊕g. The Ricci curvature tensor Ric on E is as follows:(68)R¯ic∂t,∂t=fΔf,R¯icU,∂t=0,R¯icU,V=RicU,V−1fHfU,V,where Δf denotes the Laplacian of f on E. This space-time is a generalization of several notable classical space-times. The Einstein static universe and Minkowski space-time are good examples of standard static space-times [13].Lemma 3.
Suppose thath∂t,u∂t,v∂t∈XI and ζ,U,V∈XE, then(69)ℒ¯ζ¯g¯U¯,V¯=ℒζgU,V−2uvf2h˙+ζlnf,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ.
LetE¯=If×E be a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, then(70)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ are vector fields on E¯. Then,(71)−Δf+fh˙+ζf=λ¯+ρ¯R¯f,RicU,V+12ℒζgU,V=λ¯gU,V+ρ¯R¯gU,V+1fHfU,V.
Suppose thatHfU,V=σg, then(72)RicU,V+12ℒζgU,V=λgU,V+ρRgU,V,where(73)ρR+λ=λ¯+σf+ρ¯R¯.Theorem 11.
IfHfU,V=σg in a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then E,g,ζ,λ is a ρ-Einstein soliton, where(74)ρR+λ=λ¯+σf+ρ¯R¯.
The condition Hf=σg is equivalent to ∇f is a concircular vector field with factor γ, that is, DU∇f=γU. Now, one gets(75)RicU,V−γfgU,V+12ℒζgU,V=λgU,V+−λ+λ¯+γf+ρ¯R¯gU,V=λgU,V+ρRgU,V.
Then, E,g is an ρ-Einstein soliton where(76)ρR+λ=λ¯+γf+ρ¯R¯.Corollary 4.
If∇f is a concircular vector field with factor σ on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then E,g,ζ,λ is an ρ-Einstein soliton, where(77)ρR+λ=λ¯+γf+ρ¯R¯.
Now, assume thatζ¯=h∂t+ζ is a conformal vector field on E¯, that is, ℒ¯ζ¯g¯=ωg¯, then(78)R¯icU¯,V¯=λ¯−ω¯+ρ¯R¯g¯U¯,V¯.
Then(79)−Δff=λ¯−ω¯+ρ¯R¯.
Also,(80)RicU,V−1fHfU,V=λ¯−ω¯+ρ¯R¯gU,V.
IfHfU,V=σg, then by using equation (79) we get the following equation:(81)RicU,V=1fσ−ΔfgU,V.
Thus, E,g is an Einstein manifold with factor μ=1/fσ−Δf.Theorem 12.
Ifζ¯=h∂t+ζ is a CVF on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time and HfU,V=σg, then E,g is an Einstein manifold with factor μ=1/fσ−Δf.
From Lemma3, we get ζ is a CVF on E with conformal factor η. Then, by using Theorem 12, we get(82)ℒζRicU,V=1fσ−ΔfℒζgU,V.
Sinceζ¯=h∂t+ζ is a CVF on E¯, ζ is a CVF on E with conformal factor η, thus(83)ℒζRicU,V=1fσ−ΔfηgU,V=φgU,V,where(84)φ=1fσ−Δfη.Theorem 13.
Ifζ¯=h∂t+ζ is a CVF on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then(85)ℒζRicU,V=φgU,V,where(86)φ=1fσ−Δfη.
In aρ¯-Einstein soliton standard static space-time E¯,g¯,ζ¯,λ¯,ρ, it is(87)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯.
Assume thatE,g is Einstein manifold and HfU,V=σg, then(88)ℒζgU,V=2σf−μ+λ¯+ρ¯R¯gU,V.
Thus, ζ is a conformal vector field on E.Theorem 14.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, assume that E,g is Einstein manifold and HfU,V=σg, then ζ is a conformal vector field on E.


## 3.1.ρ¯-Einstein Solitons on GRW Space-Times

LetE¯=I×fE be a generalized Robertson–Walker (GRW) space-time with metric g¯=−dt2⊕f2g. Then, the Ricci curvature tensor Ric on E is as follows:(48)R¯ic∂t,∂t=−nf¨f,R¯icU,∂t=0,R¯icU,V=RicU,V−f⋄gU,V,where f⋄=−ff¨−n−1f˙2, see References [38–40].Lemma 2.
Suppose thath∂t,u∂t,v∂t∈XI and ζ,U,V∈XE, then(49)ℒ¯ζ¯g¯U¯,V¯=−2h˙uv+f2ℒζgU,V+2hff˙gU,V,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ.
LetE¯,g¯,ζ¯,λ¯, E¯=I×fE, be a ρ¯-Einstein soliton GRW space-time. Then,(50)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯,where U¯=u∂t+U,V¯=v∂t+V and ζ¯=h∂t+ζ are vector fields on E¯. Thus,(51)−nf¨fuv+RicU,V−f⋄gU,V−h˙uv+12f2ℒζgU,V+hff˙gU,V=−λ¯uv+f2λ¯gU,V−ρ¯R¯uv+ρ¯R¯f2gU,V.
This yields(52)nf¨=fλ¯−h˙+ρ¯R¯f,RicU,V+12f2ℒζgU,V=λ¯f2gU,V+ρ¯R¯f2gU,V+f⋄gU,V−hff˙gU,V.
Thus, E,g,f2ζ,ρ is a ρ-Einstein soliton, where(53)ρR+λ=λ¯+ρ¯R¯f2+f⋄−hff˙.Theorem 7.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, it is(1)
nf¨=fλ¯−h˙+ρ¯R¯f,(2)
E,g,f2ζ,λ is a ρ-Einstein soliton, where(54)ρR+λ=λ¯+ρ¯R¯f2.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time and ζ¯=h∂t+ζ is a CVF on E¯, that is, ℒ¯ζ¯g¯=ω¯g¯, and ω¯ is constant (see Section 2), then(55)R¯icU¯,V¯=λ¯−ω¯+ρ¯R¯g¯U¯,V¯.
Thus,(56)−nf¨fuv+RicU,V−f⋄gU,V=−λ¯−ω¯+ρ¯R¯uv+λ¯−ω¯+ρ¯R¯f2gU,V.
Thus,(57)nf¨f=λ¯−ω¯+ρ¯R¯,(58)RicU,V=f⋄+λ¯−ω¯+ρ¯R¯f2gU,V.By using equation (57) we get the following equation:(59)RicU,V=n−1ff¨−f˙2gU,V.
Therefore, E,g is an Einstein manifold with factor μ=n−1ff¨−f˙2.Theorem 8.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ admitting a CVF ζ¯=h∂t+ζ, where E¯=I×fE is a GRW space-time, E,g is an Einstein manifold with factor μ=n−1ff¨−f˙2.
From Lemma2, we get ζ is a CVF on E with conformal factor η. Then, by using Theorem 8, we get the following equation:(60)ℒζRicU,V=n−1ff¨−f˙2ℒζgU,V=n−1ff¨−f˙2ηgU,V=φgU,V,where(61)φ=n−1ff¨−f˙2η.Theorem 9.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ admitting a CVF ζ¯=h∂t+ζ, where E¯=I×fE is a GRW space-time,(62)ℒζRicU,V=φgU,V,where(63)φ=n−1ff¨−f˙2η.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, it is(64)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯.
Assume thatE,g is Einstein, then for any vector fields U¯=U,V¯=V, and ζ¯=h∂t+ζ we have get(65)ℒζgU,V=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯gU,V=ηgU,V.
Then, ζ is a CVF on E with conformal factor η where(66)η=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯.Theorem 10.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, where E¯=I×fE is a GRW space-time, ζ is a CVF on E if E,g is Einstein manifold with conformal factor η where(67)η=21f2−μ+f⋄−hff˙+f2λ¯+ρ¯R¯.


## 3.2.ρ¯-Einstein Solitons on a Standard Static Space-Times

A standard static space-time (orf-associated SSST) is a Lorentzian warped product manifold E¯=If×E furnished with the metric g¯=−f2dt2⊕g. The Ricci curvature tensor Ric on E is as follows:(68)R¯ic∂t,∂t=fΔf,R¯icU,∂t=0,R¯icU,V=RicU,V−1fHfU,V,where Δf denotes the Laplacian of f on E. This space-time is a generalization of several notable classical space-times. The Einstein static universe and Minkowski space-time are good examples of standard static space-times [13].Lemma 3.
Suppose thath∂t,u∂t,v∂t∈XI and ζ,U,V∈XE, then(69)ℒ¯ζ¯g¯U¯,V¯=ℒζgU,V−2uvf2h˙+ζlnf,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ.
LetE¯=If×E be a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯, then(70)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯,where U¯=u∂t+U,V¯=v∂t+V, and ζ¯=h∂t+ζ are vector fields on E¯. Then,(71)−Δf+fh˙+ζf=λ¯+ρ¯R¯f,RicU,V+12ℒζgU,V=λ¯gU,V+ρ¯R¯gU,V+1fHfU,V.
Suppose thatHfU,V=σg, then(72)RicU,V+12ℒζgU,V=λgU,V+ρRgU,V,where(73)ρR+λ=λ¯+σf+ρ¯R¯.Theorem 11.
IfHfU,V=σg in a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then E,g,ζ,λ is a ρ-Einstein soliton, where(74)ρR+λ=λ¯+σf+ρ¯R¯.
The condition Hf=σg is equivalent to ∇f is a concircular vector field with factor γ, that is, DU∇f=γU. Now, one gets(75)RicU,V−γfgU,V+12ℒζgU,V=λgU,V+−λ+λ¯+γf+ρ¯R¯gU,V=λgU,V+ρRgU,V.
Then, E,g is an ρ-Einstein soliton where(76)ρR+λ=λ¯+γf+ρ¯R¯.Corollary 4.
If∇f is a concircular vector field with factor σ on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then E,g,ζ,λ is an ρ-Einstein soliton, where(77)ρR+λ=λ¯+γf+ρ¯R¯.
Now, assume thatζ¯=h∂t+ζ is a conformal vector field on E¯, that is, ℒ¯ζ¯g¯=ωg¯, then(78)R¯icU¯,V¯=λ¯−ω¯+ρ¯R¯g¯U¯,V¯.
Then(79)−Δff=λ¯−ω¯+ρ¯R¯.
Also,(80)RicU,V−1fHfU,V=λ¯−ω¯+ρ¯R¯gU,V.
IfHfU,V=σg, then by using equation (79) we get the following equation:(81)RicU,V=1fσ−ΔfgU,V.
Thus, E,g is an Einstein manifold with factor μ=1/fσ−Δf.Theorem 12.
Ifζ¯=h∂t+ζ is a CVF on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time and HfU,V=σg, then E,g is an Einstein manifold with factor μ=1/fσ−Δf.
From Lemma3, we get ζ is a CVF on E with conformal factor η. Then, by using Theorem 12, we get(82)ℒζRicU,V=1fσ−ΔfℒζgU,V.
Sinceζ¯=h∂t+ζ is a CVF on E¯, ζ is a CVF on E with conformal factor η, thus(83)ℒζRicU,V=1fσ−ΔfηgU,V=φgU,V,where(84)φ=1fσ−Δfη.Theorem 13.
Ifζ¯=h∂t+ζ is a CVF on a ρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, then(85)ℒζRicU,V=φgU,V,where(86)φ=1fσ−Δfη.
In aρ¯-Einstein soliton standard static space-time E¯,g¯,ζ¯,λ¯,ρ, it is(87)R¯icU¯,V¯+12ℒ¯ζ¯g¯U¯,V¯=λ¯g¯U¯,V¯+ρ¯R¯g¯U¯,V¯.
Assume thatE,g is Einstein manifold and HfU,V=σg, then(88)ℒζgU,V=2σf−μ+λ¯+ρ¯R¯gU,V.
Thus, ζ is a conformal vector field on E.Theorem 14.
In aρ¯-Einstein soliton E¯,g¯,ζ¯,λ¯ where E¯=If×E is a standard static space-time, assume that E,g is Einstein manifold and HfU,V=σg, then ζ is a conformal vector field on E.



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