#### 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 [511]. 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 [1416]. A generalization of Einstein soliton has been deduced by considering the Ricci–Bourguignon flows [1719]:

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, thenTherefore, 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 thatThus, 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 [3236] 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, whereLet 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 andThis equation impliesIf , thenThat 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 getThen, 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 whereBang-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:whereAlso,where

Theorem 5. In a -Einstein soliton admitting a CVF with factor ,(1) if , where(2), whereThe 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 , thenThus,Let where and at a fixed point of . Then, is a gradient -Einstein soliton whereNow, let , thenThis 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 whereThis 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 [3840].

Lemma 2. Suppose that and , thenwhere , and .
Let , , be a -Einstein soliton GRW space-time. Then,where and are vector fields on . Thus,This yieldsThus, 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, whereIn a -Einstein soliton , where is a GRW space-time and is a CVF on , that is, , and is constant (see Section 2), thenThus,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,whereIn a -Einstein soliton , where is a GRW space-time, it isAssume that is Einstein, then for any vector fields , and we have getThen, 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, whereThe condition is equivalent to is a concircular vector field with factor , that is, . Now, one getsThen, 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, whereNow, assume that is a conformal vector field on , that is, , thenThenAlso,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 getSince 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, thenwhereIn a -Einstein soliton standard static space-time , it isAssume that is Einstein manifold and , thenThus, 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.