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Advances in Mathematical Physics
Volume 2016, Article ID 6508309, 11 pages
http://dx.doi.org/10.1155/2016/6508309
Research Article

Conformal Vector Fields on Doubly Warped Product Manifolds and Applications

1Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt
2Modern Academy for Engineering and Technology, Maadi 11585, Egypt

Received 3 July 2016; Revised 29 August 2016; Accepted 19 September 2016

Academic Editor: Andrei Moroianu

Copyright © 2016 H. K. El-Sayied et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This article aimed to study and explore conformal vector fields on doubly warped product manifolds as well as on doubly warped spacetime. Then we derive sufficient conditions for matter and Ricci collineations on doubly warped product manifolds. A special attention is paid to concurrent vector fields. Finally, Ricci solitons on doubly warped product spacetime admitting conformal vector fields are considered.

1. An Introduction

Bishop and O’Neill introduced Riemannian warped products to construct manifolds with negative sectional curvature [1]. Since then warped product structures have been widely studied. Doubly warped products are generalizations of singly warped products. Beem, Ehrilish, and Powell noticed that there are many exact solutions to Einstein’s field equation in the form of warped product manifolds. Since then singly and doubly warped product manifolds have became more indispensable to physicians and mathematicians than ever. In [2], Beem and Powell studied Lorentzian doubly warped product manifolds. Allison studied causal properties, pseudocovexity, and hyperbolicity of doubly warped product manifolds [3, 4]. Gebarowski considered doubly warped products with harmonic Weyl conformal curvature tensor in [5] and conformally flat and conformally recurrent doubly warped product manifolds in [6, 7]. Ünal studied geodesic completeness of Riemannian and Lorentzian doubly warped products [8]. He also studied hyperbolicity of generalized Robertson–Walker spacetime with doubly warped product fibre. In this paper, Ünal finally considered some results about conformal vector fields of doubly warped products. Doubly warped product submanifolds have also been studied by many authors in various settings such as Faghfouri and Majidi in [9], Olteanu in [10, 11], Perktas and Kilic in [12], and many others. Doubly warped spacetime is good example of Lorentzian doubly warped product manifolds. This spacetime is of interest since it produces many exact solutions to Einstein’s field equations.

In physics, symmetry assumptions are used to understand the relation between geometry and matter of spacetime given by Einstein’s field equation. For example, the metric tensor of (pseudo-)Riemannian manifold does not change under the flow of a Killing vector field; that is, the flow of a Killing vector field generates spacetime symmetry. The number of independent Killing vector fields measures the degree of symmetry of a (pseudo-)Riemannian manifold. Conformal vector fields have also a well-known geometrical and physical interpretations and have been studied on (pseudo-)Riemannian manifolds for a long time. The existence of a conformal vector field on spacetime is especially useful to study its geometry. The flow of a conformal vector consists of conformal transformations of the Riemannian manifold. Thus, the problems of existence and characterization of different types of conformal vector fields in different spaces are important and are widely discussed by both mathematicians and physicists (e.g., see [1318] and further references contained therein).

The aim of the present paper is to study and explore conformal vector fields on doubly warped product manifolds as well as doubly warped spacetime. We derive many characterizations of conformal vector fields on doubly warped product manifolds and doubly warped spacetime. Then, we study matter and Ricci collineation on doubly warped manifolds. One may notice that after Pereleman used Ricci soliton to solve the Poincaré conjecture posed in 1904, a growing body of research has continued to study Ricci soliton. Accordingly, we study Ricci solitons on doubly warped product spacetime admitting many types of conformal vector fields. We get some partial answers of the following questions: What do doubly warped Ricci soliton factors inherit? And what are the conditions under which doubly warped spacetime is a doubly warped Ricci soliton?

This article is organized as follows. Section 2 represents some connection and curvature related formulas on doubly warped product manifolds. In Section 3, we study conformal vector fields on doubly warped product manifolds. Then we study conformal and concurrent vector fields on doubly warped spacetime in two subsections. Finally, Section 4 comprises a study of Ricci soliton on doubly warped spacetime admitting these types of vector fields. Almost all considerations and statements in this work are local.

2. Preliminaries

This section represents connection and curvature related formulas on doubly warped product manifolds as a generalization of similar results on singly warped products [1, 19]. Also, we will provide basic definitions and properties of conformal vector fields.

Let be two (pseudo-)Riemannian manifolds with metrics and Levi-Civita connections and let be a positive function, where . Also, suppose that is the natural projection map of the Cartesian product onto , where . The (pseudo-)Riemannian manifolds doubly warped product manifold is the product manifold furnished with the metric tensor where denotes the pull-back operator on tensors. The functions , are called the warping functions of the warped product manifold . In particular, if, for example, , then is called a (singly) warped product manifold. A singly warped product manifold is said to be trivial if the warping function is also constant [6, 8, 9, 12, 20, 21]. It is clear that the submanifolds and are homothetic to and , respectively, for each and . We shall refer to these factor submanifolds as and . The lift of a tangent vector , , is the unique vector in such that Similarly, if , then the lift of to is the unique vector field in , that is, related to and related to zero vector field in , ; that is, a vector field on is identified with the horizontal or the vertical vector field on , that is, related to . Throughout this article we use the same notation for a vector field and for its lift to the product manifold. A function on will be identified with . Thus, we have two different meanings for the gradient of , namely, grad and the lift of the gradient of to . In fact, we have Therefore, grad (note that we use the same notation for the vector field and for its lift to ).

Let be a pseudo-Riemannian doubly warped product manifold of , , with dimensions , where . and , denote the curvature tensor and Ricci curvature tensor on , respectively. Moreover, and denote gradient and Laplacian of on and . For the connection and curvatures formulas of a pseudo-Riemannian doubly warped product manifold see, for example, [20, 22].

A vector field on a (pseudo-)Riemannian manifold with metric is called a conformal vector field with conformal factor if where is the Lie derivative on with respect to . If is constant or zero, is called a homothetic or Killing vector field on , respectively. One can redefine conformal vector fields using the following identity. Let be a vector field on , and thenfor any vector fields . A vector field on a manifold is called a concurrent vector field if for any vector field [23]. Let be a concurrent vector field, and then and so is homothetic with factor . A zero vector field is not concurrent. If both and are concurrent vector fields, then Also both and are not concurrent vector fields. Finally, a Killing vector field is not concurrent. For example, a vector field is a concurrent vector field on if that is, . Thus concurrent vector fields on are of the form .

The following result represents a simple characterization of Killing vector fields, if is a pseudo-Riemannian manifold with Riemannian connection . A vector field is a Killing vector field if and only iffor any vector field .

The following discussion represents a good tool to characterize Killing vector fields on pseudo-Riemannian warped product manifolds. In [24, 25], the authors obtained many characterizations of Killing vector fields on warped product manifolds and on standard static spacetime using these results. Let be a pseudo-Riemannian warped product manifold with warping function . Let be a vector field on . Thenfor any vector field , where is the Lie derivative on with respect to , for .

A pseudo-Riemannian manifold is said to admit a Ricci curvature collineation if there is a vector field such that where is the Ricci curvature tensor [26]. Finally, spacetime is said to admit a matter collineation if there is a vector field such that where is the energy-momentum tensor [27]. Einstein’s field equation with cosmological constant is given by where is the scalar curvature. Suppose that is a Killing vector field, and then that is, is a matter collineation whereas a matter collineation need not be a Killing vector field. Also, a Killing vector field is a Ricci curvature collineation. The converse is not generally true.

3. Conformal Vector Fields on Doubly Warped Products

In this section we investigate the relation between conformal vector fields on doubly warped product manifolds and those conformal vector fields on the product factors. Throughout this section, let be a pseudo-Riemannian doubly warped product manifold with the metric tensor and is a smooth function, where and are pseudo-Riemannian manifolds. The following result gives us an important identity to study such relation [8].

Proposition 1. Suppose that and , and then where , , and are elements in .

In [8], the author considered a characterization of conformal vector fields on doubly warped product manifolds. In fact, it is just a characterization of homothetic vector fields. The following theorem represents a new characterization of conformal vector fields on doubly warped product manifolds but the assumption here is less restrictive.

Theorem 2. A vector field on a pseudo-Riemannian doubly warped product is a conformal vector field with conformal factor if and only if(1) is a conformal vector field on with conformal factor ,(2).Moreover, the conformal factor of is .

Before proceeding further, one may notice that a doubly warped product metric on can be expressed as a conformal metric to a product metric on as follows: Let us consider the effect of replacing the metric on by . A similar discussion on -dimensional spacetime is considered in [26, Chapter 11]. Suppose that is a conformal vector field on with factor , and then Therefore, is a conformal vector field on with factor . A similar conclusion applies to and , where . Thus, by using results in [28, Theorem  1], one can easily get the following.

Theorem 3. Let be a pseudo-Riemannian doubly warped product equipped with the metric tensor and let , where . Then,(1)a Killing vector field on , for each , is a Killing vector field on ,(2) admits a homothetic vector field if and only if admits a homothetic vector field for each ,(3)each conformal vector field on is a conformal vector field on .

The above theorem together with Theorem 2 implies the following results.

Theorem 4. Let be a vector field on a pseudo-Riemannian doubly warped product equipped with the metric tensor . Assume that is a Killing vector field on for each and . Then is a conformal vector field on .

Theorem 5. Let be homothetic vector fields on with factors for each . Assume that . Then, is a homothetic vector field on with factor .

Corollary 6. The dimension of the conformal group on a pseudo-Riemannian doubly warped product is at least where is the isometry group of .

Again Theorem 2 together with Lemma  2.1 in [16] yields the following result.

Theorem 7. Let be a vector field on a pseudo-Riemannian doubly warped product such that(1) is a conformal vector field on with conformal factor ,(2).Then, preserves the Ricci curvature if and only if , where . Moreover, preserves the conformal class of the Ricci tensor (i.e., for some function ) if and only if is a conformal vector field.

Theorem 8. Let be a vector field on a pseudo-Riemannian doubly warped product . has constant length along the integral curve of the vector field if one of the following conditions holds:(1) and is parallel along for each .(2) and has a constant length along for each .

Proof. Let be a vector field on . Then, In both cases , and hence The first condition implies that and so . The second condition implies that is constant and so and therefore that is, has a constant length along the integral curve of the vector field

Theorem 9. Let be a conformal vector field on a pseudo-Riemannian doubly warped product along a curve with unit tangent vector . Then,

Proof. Let be a conformal vector field with conformal factor . Then, Let , then and then the conformal factor is given by Suppose that and , and then But the conformal factor is given by which completes the proof.

3.1. Conformal Vector Fields on Doubly Warped Spacetime

Doubly warped spacetime is a doubly warped product manifold , where one of the factors, say , has a Lorentz signature and the second is Riemannian. Ramos et al. considered an invariant characterization of -dimensional doubly warped spacetime [21]. Among many other results, they obtained necessary and sufficient conditions for (locally) double warped spacetime to be conformally related to or decomposable spacetime. Then they studied the conformal algebra of decomposable spacetime in section IV and decomposable spacetime in section V. For a detailed discussion of conformally related and reducible spacetime see [29, 30] and, for an extensive self-contained study of conformal symmetry of -dimensional spacetime, the reader is referred to [26].

We restrict our study of conformal vector fields on doubly warped spacetime to since this case generalizes some well-known exact solutions for the Einstein field equations and the beginning of this section represents such study irrespective of the dimension of the factors. For this case, either or . In the following we deal with both subcases separately. Let us first consider doubly warped spacetime with a -dimensional base.

Let be a Riemannian manifold and be an open connected interval equipped with the metric . Doubly warped spacetime is the product furnished with the metric where and are smooth functions. is generalized Robertson–Walker spacetime if is constant and standard static spacetime if is constant.

An investigation of -dimensional spacetime that is conformally related to reducible spacetime was carried out with many examples in the aforementioned references [21, 30]. A classification of this spacetime according to its conformal algebra is considered in the first reference whereas a special attention is paid to gradient conformal vector fields in the second reference.

Theorem 10. A time-like vector field is a conformal vector field on doubly warped spacetime if and only if , where is a nonnegative constant. Moreover, the conformal factor is .

Proof. If , then and the result is obvious. Now, we assume that . Using (16), we get that Suppose that , and then that is, is a conformal vector field with conformal factor . Conversely, suppose that is a conformal vector field with factor , and then for any vector fields . Now, by (16), we get that Let , and we get that that is, . Now, let , and we get that .
These two differential equations imply that for some positive constant .

Theorem 11. A vector field is a Killing vector field on doubly warped spacetime if and only if one of the following conditions holds:(1) is time-like and (2) is space-like where is a Killing vector field on and (3) and is a conformal vector field on with conformal factor .

Proof. The first assertion is a special case of the above theorem. For the second assertion, let in (16). Thus, Suppose that is a Killing vector field on and , and then The converse is direct. Finally, let be a Killing vector field on , and then Let , and then and so . Thus, that is, is a conformal vector field on with conformal factor . Now let , and then that is, is a conformal vector field on with conformal factor .
Conversely, suppose that and is a conformal vector field on with conformal factor , and then Thus, that is, is a Killing vector field on .

Corollary 12. Let be vector field on doubly warped spacetime obeying Einstein’s field equation. Then,(1) admits a time-like matter collineation if ,(2) admits a space-like matter collineation if is a Killing vector field on and ,(3) admits a matter collineation if and is a conformal vector field on with conformal factor .

The following result is a direct consequence of Theorem 9.

Corollary 13. Let be a conformal vector field on doubly warped spacetime along a curve with unit tangent vector . Then the conformal factor of is given by

In the sequel, we present doubly warped spacetime with a -dimensional base. In this subcase, doubly warped spacetime is considered to be doubly warped product manifold with a -dimensional pseudo-Riemannian base and -dimensional Riemannian fibre; doubly warped spacetime is clearly conformal to a product manifold. In [21, 29] and references therein, the Lie conformal algebra of conformally related reducible spacetime is extensively studied. Many interesting results and examples are given there. For example, in [29], Carot and Tupper considered an invariant characterization that imposes conditions on the conformal factor and on two null vectors. Moreover, Van den Bergh considered non-conformally flat perfect fluids spacetime which is conformally decomposable spacetime with factor spaces of constant curvature [31].

It is well-known that each -dimensional manifold is conformally flat. Thus we may simply take the base manifold as , where . Let be -dimensional doubly warped product spacetime furnished with the metric .

Proposition 14. Suppose that , , and , and thenwhere and are elements in .

Corollary 15. A vector field is a Killing vector field if one of the following conditions holds:(1), , , and .(2) is a Killing vector field on , , and

3.2. Concurrent Vector Fields on Doubly Warped Spacetime

In this subsection, we study concurrent vector fields on doubly warped spacetime with a -dimensional base. One can extend most of the results to doubly warped spacetime with a -dimensional base. Throughout this subsection, let be doubly warped spacetime equipped with the metric tensor .

Theorem 16. A vector field on doubly warped spacetime is a concurrent vector field if(1) and are concurrent vector fields on and , respectively,(2)both and are constant.

Proof. Suppose that is any vector field on , and then Now suppose that and are concurrent vector fields; then and . If both and are constant, then that is, is concurrent.

It is well-known that a homothetic vector field is a matter collineation. Thus the above theorem yields the following result.

Corollary 17. A vector field on doubly warped spacetime is a matter collineation if(1) and are concurrent vector fields on and , respectively,(2)both and are constant.

Theorem 18. Let be a concurrent vector field on doubly warped spacetime equipped with the metric tensor . Then, is a concurrent vector field on if one of the following conditions holds:(1), or(2) is a constant; that is, is standard static spacetime.Moreover, condition (1) implies condition (2) and the converse is true if is not constant.

Proof. From the above proof we havefor any and . Let , and then Thus is concurrent if .
If , then If then which is a contradiction and so ; that is, is standard static spacetime.
If , then which implies that for a nonconstant function .

Theorem 19. Let be a concurrent vector field on , where . Then and are concurrent vector fields on and , respectively, if . In this case is also constant.

Example 20. Table 1 summarizes all three cases of concurrent vector fields on the -dimensional doubly warped spacetime of the form equipped with the metric . For more details see Appendix.

Table 1

4. Ricci Soliton on Doubly Warped Spacetime

A smooth vector field on a Riemannian manifold is said to define a Ricci soliton if where denotes the Lie derivative of the metric tensor , is the Ricci curvature, and is a constant [3236].

Theorem 21. Let be a Ricci soliton where is doubly warped spacetime and . Then

Proof. Let be a Ricci soliton, and then where and are vector fields on . Then Let , and we get and so Now, let us put , and then and so

The following corollaries are consequences of the above theorem.

Corollary 22. Let be a Ricci soliton where is doubly warped spacetime and . Then(1) is a conformal vector field on with factor ,(2) is a Ricci soliton if ,(3) is a Ricci soliton if and .

Theorem 23. Let be a Ricci soliton where is doubly warped spacetime and is a conformal vector field on with factor . Then is Einstein manifold with factor if is constant.

Proof. Let be a Ricci soliton where is doubly warped spacetime and is a conformal vector field on . Then, Let , and then This equation implies that and so

Corollary 24. Let be a Ricci soliton where is doubly warped spacetime and is a homothetic vector field on with factor . Then,

Proof. Let be a Ricci soliton and be a homothetic vector field on , and then for any vector fields and . Let us take , and then Then, and the proof is complete.

Theorem 25. Let be a Ricci soliton where is doubly warped spacetime and is a concurrent vector field on . Then,(1) is Einstein manifold with factor if is constant,(2).

Let , and then Thus, Suppose that and is a Ricci soliton on , and then

Therefore, is a Ricci soliton where . This discussion leads us to the following result.

Theorem 26. Let be doubly warped spacetime and be a vector field on . Then is a Ricci soliton if(1) is a Ricci soliton on ,(2),(3).

Let , be a conformal vector field with factor , and be Einstein with factor , and thenthat is, is a Ricci soliton if

Theorem 27. Let be doubly warped spacetime and be a vector field on . Then, is a Ricci soliton if(1) is Einstein with factor ,(2), and is conformal with factor ,(3).In this case, .

Appendix

Concurrent Vector Fields on Doubly Spacetime

Let us now consider an example. Let be -dimension doubly warped spacetime equipped with the metric . Then, A vector field is a concurrent vector field ifThe first equation implies that and soAlso, (A.3) implies that and soBy solving (A.6) and (A.8), we get . Thus, or . In both cases, ; that is, or . This discussion shows that we have the following cases using (A.5) and (A.9).

Case 1. and : then and and so and Therefore, , where both and are positive.

Case 2. and : then and and so and similarly where both and are positive.

Case 3. and : then and and so and .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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