#### Abstract

The purpose of this research is to investigate how a *ρ*-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.

#### 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 by an evolution equation for metrics of the following form:where is the Ricci curvature tensor. The initial metric on satisfies the following equation:where is a vector field on , is a constant, and represents the Lie derivative in the direction of a vector field on . 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 modulo diffeomorphisms and scaling [4]. A Ricci soliton is called shrinking (steady or expanding) if ( or respectively). If or is Killing, then the Ricci soliton is called a trivial Ricci soliton. If is a smooth function and , then the Ricci soliton is described as gradient, is referred to as the potential vector field, and is called the potential function. In this case, equation (2) becomes as follows:where 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 is a smooth function, then we say that is a nearly Ricci soliton manifold [14–16]. A generalization of Einstein soliton has been deduced by considering the Ricci–Bourguignon flows [17–19]:

These manifolds are called -Einstein solitons and are defined as follows: Let be a pseudo-Riemannian manifold, and let , , and . Then, is called a -Einstein soliton if

Likewise, if a smooth function exists such that , then a -Einstein soliton is gradient and denoted by . In this case, equation (5) becomes as follows:

A -Einstein soliton is denoted as steady, shrinking, or expanding, depending on whether has zero, positive, or negative values. The function is called a -Einstein potential of the gradient -Einstein soliton. Later, this perception was circulated in many instructions, such as -quasi Einstein manifolds [20], Ricci–Bourguignon almost solitons [21], and -quasi-Einstein manifolds [22]. Huang got a sufficient condition for a compact gradient shrinking -Einstein soliton to be isometric to a quotient of the round sphere in Reference [23]. Moreover, Mondal and Shaikh proved that a compact gradient -Einstein soliton with a nontrivial conformal vector field is isometric to the Euclidean sphere 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 -Einstein soliton on a Kenmotsu manifold. Some curvature conditions on compact gradient -Einstein soliton are given in Reference [26] to guarantee that is isometric to the Euclidean sphere. In contrast, an integral condition on a noncompact -Einstein soliton is given to ensure the vanishing of the scalar curvature. A splitting theorem of a gradient -Einstein soliton is given in Reference [27]. Accordingly, many characterizations of gradient -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 -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 a become a -Einstein soliton or a gradient -Einstein soliton?(2)What does a factor of a -Einstein soliton or a gradient -Einstein soliton inherit?

To address these problems, first we proved many results on the -Einstein soliton. Then, we investigated necessary and sufficient conditions on a (gradient) -Einstein soliton in order to make its factor (gradient) -Einstein soliton. Additionally, we studied a -Einstein soliton on a 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. -Einstein Solitons on Pseudo-Riemannian Manifolds

If is a conformal vector field with conformal factor in a -Einstein soliton , then

By taking the trace over , we get the following equation:

Since the scalar curvature of Einstein manifolds is constant, the conformal factor is also constant, that is, is homothetic. Moreover, if .

Proposition 1. *Assume that is a conformal vector field on a -Einstein soliton with factor . Then, is homothetic, is Einstein, andwhere . Moreover, if .*

Corollary 1. *Assume that is a Killing vector field on a -Einstein soliton , thenwhere . Moreover, is steady if .**Conversely, assuming that is an Einstein manifold, then**Therefore, is a homothetic vector field on .*

Proposition 2. *In a -Einstein soliton , is a homothetic vector field on if is Einstein. Furthermore, is Killing if .**In local coordinates, a contraction of the defining equation implies that**Thus, the vector field 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 .*

Corollary 2. *The vector field in a -Einstein soliton is divergence-free if and only if .**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 field in a -Einstein soliton are volume-preserving diffeomorphisms if and only if .*

##### 2.2. Warped Product Manifolds

Let denote two -dimensional pseudo-Riemannian manifolds equipped with metric tensors where is the Levi-Civita connection of the metric for . Let be a smooth positive real-valued function. A , denoted by , is the product manifold equipped with the metric tensor (for more details the reader is referred to [32–36] and references therein). Let be a pseudo-Riemannian and for all . Then, the Ricci tensor of is given by,(1),(2),(3), where , and is the Laplacian on .

The scalar curvature a satisfies

Lemma 1. *(see [35]). In a , the Lie derivative with respect to a vector field satisfiesfor any vector fields , where is the Lie derivative on with respect to , for .*

#### 3. -Einstein Soliton Structure on

In this section, we investigate the -Einstein soliton structure on . For the rest of this work, let be a with warping function and let . Also, let be a vector field on . Let be a -Einstein soliton, that is,

Thus, for any vector fields , and on , Lemma 1 implies that

Let , , and , then

Then, is a -Einstein soliton, where

Now, let and , then

Thus,

Then, is a -Einstein soliton, where

Theorem 2. *Let be a -Einstein soliton. Then,*(1)* is a -Einstein soliton if where*(2)* is a -Einstein soliton, where**Let and be two Einstein manifolds with factors and , respectively, and let . Then equation (16) becomes as follows:**Thus,**That is, and are conformal vector fields on and .*

Theorem 3. *In a -Einstein soliton , ,*(1)* is conformal vector field on if and is Einstein, and*(2)* is conformal vector field on if 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 that is a conformal vector field on , that is, for some scalar function , then is constant and**This equation implies**If , then**That is, both the base and fiber manifolds are Einstein.*

Theorem 4. *In a -*Einstein soliton , admitting a CVF ,(1)* is Einstein if , and*(2)* is Einstein.**The condition is equivalent to is a concircular vector field. Equation (16) yields the following:**Suppose that is a concircular vector field with factor , that is, , we get**Then, is a -Einstein soliton where*

Corollary 3. *In a -Einstein soliton , assume that is a concircular vector field with factor , then is a -Einstein soliton where**Bang-Yen Chen proved that a Riemannian manifold admitting a concircular vector field is locally a warped product of the form [37]. Thus, the aforementioned warped product manifold becomes a sequential warped product manifold [38].**From Lemma 1, it is clear that are CVFs on with conformal factors , respectively. Then, by employing equation (28) we get the following equation:**where**Also,where*

Theorem 5. *In a -Einstein soliton admitting a CVF with factor ,*(1)* if , where*(2)*, where**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 represents a Ricci collineation if the Ricci tensor is invariant under the Lie dragging through flow lines of . The previous conclusion establishes the shape of the Lie derivative of the Ricci tensor concerning the fields , on , .**Let be a gradient -Einstein soliton with , then**Thus,**Let **where and at a fixed point of . Then, is a gradient -Einstein soliton whereNow, let , then**This yieldswhere at a fixed point of . Then, is a gradient -Einstein soliton where*

Theorem 6. *In a gradient -Einstein soliton ,*(1)* is a gradient -Einstein soliton where*(2)* is a gradient -Einstein soliton where**This theorem provides an inheritance property of the structure of the gradient -Einstein soliton structure to factor manifolds of the warped product manifold.*

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

Let be a generalized Robertson–Walker (GRW) space-time with metric . Then, the Ricci curvature tensor Ric on is as follows:where , see References [38–40].

Lemma 2. *Suppose that and , thenwhere , and .**Let , , be a -Einstein soliton GRW space-time. Then,where and are vector fields on . Thus,**This yields**Thus, is a -Einstein soliton, where*

Theorem 7. *In a -Einstein soliton , where is a GRW space-time, it is*(1)*,*(2)* is a -Einstein soliton, where**In a -Einstein soliton , where is a GRW space-time and is a CVF on , that is, , and is constant (see Section 2), then**Thus,**Thus,By using equation (57) we get the following equation:**Therefore, is an Einstein manifold with factor .*

Theorem 8. *In a -Einstein soliton admitting a CVF , where is a GRW space-time, is an Einstein manifold with factor .**From Lemma 2, we get is a CVF on with conformal factor . Then, by using Theorem 8, we get the following equation:where*

Theorem 9. *In a -Einstein soliton admitting a CVF , where is a GRW space-time,where**In a -Einstein soliton , where is a GRW space-time, it is**Assume that is Einstein, then for any vector fields , and we have get**Then, is a CVF on with conformal factor where*

Theorem 10. *In a -Einstein soliton , where is a GRW space-time, is a CVF on if is Einstein manifold with conformal factor where*

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

A standard static space-time (or -associated SSST) is a Lorentzian warped product manifold furnished with the metric . The Ricci curvature tensor Ric on is as follows:where denotes the Laplacian of on . 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 that and , thenwhere , and .**Let be a -Einstein soliton , thenwhere , and are vector fields on . Then,**Suppose that , thenwhere*

Theorem 11. *If in a -Einstein soliton where is a standard static space-time, then is a -Einstein soliton, where**The condition is equivalent to is a concircular vector field with factor , that is, . Now, one gets**Then, is an -Einstein soliton where*

Corollary 4. *If is a concircular vector field with factor on a -Einstein soliton where is a standard static space-time, then is an -Einstein soliton, where**Now, assume that is a conformal vector field on , that is, , then**Then**Also,**If , then by using equation (79) we get the following equation:**Thus, is an Einstein manifold with factor .*

Theorem 12. *If is a CVF on a -Einstein soliton where is a standard static space-time and , then is an Einstein manifold with factor .**From Lemma 3, we get is a CVF on with conformal factor . Then, by using Theorem 12, we get**Since is a CVF on , is a CVF on with conformal factor , thuswhere*

Theorem 13. *If is a CVF on a -Einstein soliton where is a standard static space-time, thenwhere**In a -Einstein soliton standard static space-time , it is**Assume that is Einstein manifold and , then**Thus, is a conformal vector field on .*

Theorem 14. *In a -Einstein soliton where is a standard static space-time, assume that is Einstein manifold and , then is a conformal vector field on .*

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This project was supported by the Researchers Supporting Project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia.