Research ArticleConformal Vector Fields on Doubly Warped ProductManifolds and Applications

H K El-Sayied1 Sameh Shenawy2 and Noha Syied2

1Mathematics Department Faculty of Science Tanta University Tanta 31527 Egypt2Modern Academy for Engineering and Technology Maadi 11585 Egypt

Correspondence should be addressed to Noha Syied drnsyiedmailcom

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

Academic Editor Andrei Moroianu

Copyright copy 2016 H K El-Sayied et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 An Introduction

Bishop andOrsquoNeill introduced Riemannian warped productsto construct manifolds with negative sectional curvature[1] Since then warped product structures have been widelystudiedDoublywarped products are generalizations of singlywarped products Beem Ehrilish and Powell noticed thatthere are many exact solutions to Einsteinrsquos field equationin the form of warped product manifolds Since then singlyand doubly warped product manifolds have became moreindispensable to physicians and mathematicians than everIn [2] Beem and Powell studied Lorentzian doubly warpedproduct manifolds Allison studied causal properties pseu-docovexity and hyperbolicity of doubly warped productmanifolds [3 4] Gebarowski considered doubly warpedproducts with harmonic Weyl conformal curvature tensor in[5] and conformally flat and conformally recurrent doublywarped product manifolds in [6 7] Unal studied geodesiccompleteness of Riemannian and Lorentzian doubly warpedproducts [8] He also studied hyperbolicity of generalizedRobertsonndashWalker spacetime with doubly warped productfibre In this paper Unal finally considered some resultsabout conformal vector fields of doubly warped productsDoubly warped product submanifolds have also been studiedby many authors in various settings such as Faghfouri and

Majidi in [9] Olteanu in [10 11] Perktas and Kilic in [12] andmany others Doubly warped spacetime is good example ofLorentzian doublywarped productmanifoldsThis spacetimeis of interest since it produces many exact solutions toEinsteinrsquos field equations

In physics symmetry assumptions are used to understandthe relation between geometry andmatter of spacetime givenby Einsteinrsquos field equation For example the metric tensor of(pseudo-)Riemannian manifold does not change under theflowof aKilling vector field that is the flowof aKilling vectorfield generates spacetime symmetryThe number of indepen-dent Killing vector fields measures the degree of symmetryof a (pseudo-)Riemannian manifold Conformal vector fieldshave also a well-known geometrical and physical inter-pretations and have been studied on (pseudo-)Riemannianmanifolds for a long timeThe existence of a conformal vectorfield on spacetime is especially useful to study its geometryTheflowof a conformal vector consists of conformal transfor-mations of the Riemannian manifold Thus the problems ofexistence and characterization of different types of conformalvector fields in different spaces are important and are widelydiscussed by both mathematicians and physicists (eg see[13ndash18] and further references contained therein)

The aim of the present paper is to study and exploreconformal vector fields on doubly warped product manifolds

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 6508309 11 pageshttpdxdoiorg10115520166508309

2 Advances in Mathematical Physics

as well as doubly warped spacetime We derive many charac-terizations of conformal vector fields on doublywarped prod-uct manifolds and doubly warped spacetime Then we studymatter and Ricci collineation on doubly warped manifoldsOne may notice that after Pereleman used Ricci soliton tosolve the Poincare conjecture posed in 1904 a growing bodyof research has continued to study Ricci soliton Accordinglywe study Ricci solitons on doubly warped product spacetimeadmitting many types of conformal vector fields We getsome partial answers of the following questions What dodoubly warped Ricci soliton factors inherit Andwhat are theconditions under which doubly warped spacetime is a doublywarped Ricci soliton

This article is organized as follows Section 2 representssome connection and curvature related formulas on doublywarped product manifolds In Section 3 we study conformalvector fields on doubly warped product manifolds Thenwe study conformal and concurrent vector fields on doublywarped spacetime in two subsections Finally Section 4 com-prises a study of Ricci soliton on doubly warped spacetimeadmitting these types of vector fields Almost all considera-tions and statements in this work are local

2 Preliminaries

This section represents connection and curvature relatedformulas on doubly warped product manifolds as a general-ization of similar results on singly warped products [1 19]Also we will provide basic definitions and properties ofconformal vector fields

Let (119872119894 119892119894 119863119894) be two (pseudo-)Riemannian manifoldswith metrics 119892119894 and Levi-Civita connections 119863119894 and let 119891119894 119872119894 rarr (0infin) be a positive function where 119894 = 1 2 Alsosuppose that 120587119894 1198721 times 1198722 rarr 119872119894 is the natural projectionmap of the Cartesian product 1198721 times 1198722 onto 119872119894 where119894 = 1 2 The (pseudo-)Riemannian manifolds doubly warpedproduct manifold 119872=11989121198721 times11989111198722 is the product manifold119872 = 1198721 times1198722 furnished with the metric tensor

119892 = (1198912 ∘ 1205872)2 120587lowast1 (1198921) oplus (1198911 ∘ 1205871)2 120587lowast2 (1198922) (1)

where lowast denotes the pull-back operator on tensors Thefunctions 119891119894 119894 = 1 2 are called the warping functionsof the warped product manifold 119872 In particular if forexample 1198912 = 1 then 119872 = 1198721 times11989111198722 is called a (singly)warped productmanifold A singly warped productmanifold1198721times11989111198722 is said to be trivial if the warping function1198911 is alsoconstant [6 8 9 12 20 21] It is clear that the submanifolds1198721 times 119902 and 119901 times 1198722 are homothetic to 1198721 and 1198722respectively for each 119901 isin 1198721 and 119902 isin 1198722 We shall referto these factor submanifolds as1198721 and1198722 The lift 119883(119901119902) ofa tangent vector 119883119901 isin 1198791199011198721 119902 isin 1198722 is the unique vector in119879(119901119902)119872 such that

120587lowast1 (119883(119901119902)) = 119883119901120587lowast2 (119883(119901119902)) = 0 (2)

Similarly if 119883119894 isin X(119872119894) then the lift of 119883119894 to X(1198721 times 1198722)is the unique vector field in X(1198721 times 1198722) that is 120587119894 related

to 119883119894 and 120587119895 related to zero vector field in X(119872119895) 119894 = 119895 thatis a vector field 119883119894 on 119872119894 is identified with the horizontalor the vertical vector field on 1198721 times 1198722 that is 120587119894 related to119883119894 Throughout this article we use the same notation for avector field and for its lift to the productmanifold A function120596119894 on 119872119894 will be identified with 120596119894 ∘ 120587119894 Thus we have twodifferent meanings for the gradient of 120596119894 namely grad (120596119894 ∘120587119894) isin X(1198721 times 1198722) and the lift of the gradient nabla119894120596119894 of 120596119894 toX(1198721 times1198722) In fact we have

119892 (119883119894 grad (120596119894 ∘ 120587119894)) = 119883119894 (120596119894 ∘ 120587119894) = 119883119894 (120596119894) ∘ 120587119894= 11198912119895 119892 (119883119894 nabla

119894120596119894) (3)

Therefore grad (120596119894 ∘ 120587119894) = (11198912119895 )nabla119894120596119894 (note that we usethe same notation for the vector field nabla119894120596119894 and for its lift toX(1198721 times1198722))

Let (119872 119892119863) be a pseudo-Riemannian doubly warpedproduct manifold of (119872119894 119892119894 119863119894) 119894 = 1 2 with dimensions119899119894 where 119899 = 1198991 +1198992 119877 119877119894 and Ric Ric119894 denote the curvaturetensor and Ricci curvature tensor on 119872119872119894 respectivelyMoreover nabla119894119891119894 and119894119891119894 denote gradient and Laplacian of 119891119894on119872119894 and 119891119894 = 119891119894119894119891119894 + (119899119895 minus1)119892119894(nabla119894119891119894 nabla119894119891119894) 119894 = 119895 For theconnection and curvatures formulas of a pseudo-Riemanniandoubly warped product manifold see for example [20 22]

A vector field 120577 on a (pseudo-)Riemannian manifold(119873 ℎ) with metric ℎ is called a conformal vector field withconformal factor 120588 if

L120577ℎ = 120588ℎ (4)

where L120577 is the Lie derivative on 119873 with respect to 120577 If 120588is constant or zero 120577 is called a homothetic or Killing vectorfield on 119873 respectively One can redefine conformal vectorfields using the following identity Let 120577 be a vector field on119872 and then

(L120577ℎ) (119883 119884) = ℎ (119863119883120577 119884) + ℎ (119883119863119884120577) (5)

for any vector fields 119883119884 isin X(119873) A vector field 120577 on amanifold (119873 ℎ) is called a concurrent vector field if

119863119883120577 = 119883 (6)

for any vector field 119883 isin X(119873) [23] Let 120577 be a concurrentvector field and then

(L120577ℎ) (119883 119884) = 2ℎ (119883 119884) (7)

and so 120577 is homothetic with factor 120588 = 2 A zero vector fieldis not concurrent If both 120577 and 120585 are concurrent vector fieldsthen

119863119883 [120577 120585] = 0 (8)

Also both 120577+120585 and120582120577 are not concurrent vector fields Finallya Killing vector field is not concurrent For example a vectorfield 120572120597119909 is a concurrent vector field on (R 1198891199092) if

119863120597119909 (120572120597119909) = 120597119909 (9)

Advances in Mathematical Physics 3

that is 120572 = 119909 + 119886 Thus concurrent vector fields on (R 1198891199092)are of the form (119909 + 119886)120597119909

The following result represents a simple characterizationof Killing vector fields if (119873 ℎ) is a pseudo-Riemannianmanifold with Riemannian connection 119863 A vector field 120577 isinX(119873) is a Killing vector field if and only if

ℎ (119863119883120577 119883) = 0 (10)

for any vector field119883 isin X(119873)The following discussion represents a good tool to char-

acterize Killing vector fields on pseudo-Riemannian warpedproduct manifolds In [24 25] the authors obtained manycharacterizations of Killing vector fields on warped productmanifolds and on standard static spacetime using theseresults Let119872 = 1198721 times1198911198722 be a pseudo-Riemannian warpedproduct manifold with warping function 119891 Let 120577 = 1205771 + 1205772 isinX(119872) be a vector field on119872 Then

119892 (119863119883120577 119883) = 1198921 (119863111988311205771 1198831) + 11989121198922 (119863211988321205772 1198832)+ 1198911205771 (119891) 10038171003817100381710038171198832100381710038171003817100381722

(L120577119892) (119883 119884) = (L112057711198921) (1198831 1198841)+ 1198912 (L212057721198922) (1198832 1198842)+ 21198911205771 (119891) 1198922 (1198832 1198842)

(11)

for any vector field 119883 = 1198831 + 1198832 isin X(119872) where L119894120577119894 is theLie derivative on119872119894 with respect to 120577119894 for 119894 = 1 2

A pseudo-Riemannian manifold 119872 is said to admit aRicci curvature collineation if there is a vector field 120577 isin X(119872)such that

L120577Ric = 0 (12)

where Ric is the Ricci curvature tensor [26] Finally space-time 119872 is said to admit a matter collineation if there is avector field 120577 isin X(119872) such that

L120577119879 = 0 (13)

where 119879 is the energy-momentum tensor [27] Einsteinrsquos fieldequation with cosmological constant 120582 is given by

Ric minus 1199032119892 = 120581119879 minus 120582119892 (14)

where 119903 is the scalar curvature Suppose that 120577 is a Killingvector field and then

L120577119879 = 0 (15)

that is 120577 is amatter collineationwhereas amatter collineationneed not be a Killing vector field Also a Killing vector field isa Ricci curvature collineation The converse is not generallytrue

3 Conformal Vector Fields on DoublyWarped Products

In this section we investigate the relation between conformalvector fields on doubly warped product manifolds and thoseconformal vector fields on the product factors Throughoutthis section let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product manifold with the metric tensor119892 = 11989122 1198921 oplus 11989121 1198922 and 119891119894 119872119894 rarr (0infin) is a smoothfunction where 119894 = 1 2 and (119872119894 119892119894) are pseudo-RiemannianmanifoldsThe following result gives us an important identityto study such relation [8]

Proposition 1 Suppose that 1205771 1198831 1198841 isin X(1198721) and1205772 1198832 1198842 isin X(1198722) and then(L120577119892) (119883 119884) = 11989122 (L112057711198921) (1198831Y1)

+ 11989121 (L212057721198922) (1198832 1198842)+ 211989111205771 (1198911) 1198922 (1198832 1198842)+ 211989121205772 (1198912) 1198921 (1198831 1198841)

(16)

where 120577 = 1205771 + 1205772119883 = 1198831 +1198832 and 119884 = 1198841 +1198842 are elementsin X(119872)

In [8] the author considered a characterization of con-formal vector fields on doubly warped product manifolds Infact it is just a characterization of homothetic vector fieldsThe following theorem represents a new characterization ofconformal vector fields on doubly warped product manifoldsbut the assumption here is less restrictive

Theorem2 A vector field 120577 = 1205771+1205772 on a pseudo-Riemanniandoubly warped product119872=11989121198721 times11989111198722 is a conformal vectorfield with conformal factor 120588 if and only if

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Moreover the conformal factor of 120577 is 120588 = 120588119894 +2120577119895(ln119891119895) 119894 = 119895Before proceeding further one may notice that a doubly

warped product metric 119892 on 119872 can be expressed as aconformal metric to a product metric on1198721 times1198722 as follows

119892 = 1198912111989122 ( 111989121 1198921 +

111989122 1198922) = 1198912111989122 (1198921 + 1198922)

= 1198912111989122 119892(17)

Let us consider the effect of replacing the metric 119892 on 119872 by119892 = 1198921+1198922 A similar discussion on 4-dimensional spacetimeis considered in [26 Chapter 11] Suppose that 120577 = 1205771 + 1205772 is aconformal vector field on (119872 119892) with factor 120588 and then

L120577119892 = [21205772 (ln1198912) + 21205771 (ln1198911) + 120588] 119892 (18)

4 Advances in Mathematical Physics

Therefore 120577 is a conformal vector field on (119872 119892) with factor120588 = 120588+21205772(ln1198912)+21205771(ln1198911) A similar conclusion applies to(119872119894 119892119894) and (119872119894 119892119894) where 120588119894 = 120588 minus 120577119894(ln119891119894) Thus by usingresults in [28 Theorem 1] one can easily get the following

Theorem 3 Let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product equipped with the metric tensor 119892 =11989122 1198921 + 11989121 1198922 and let 119892 = 1198921 + 1198922 where 119892119894 = (11198912119894 )119892119894 Then

(1) a Killing vector field 120577 = 120577119894 on (119872119894 119892119894) for each 119894 = 1 2is a Killing vector field on (119872 119892)

(2) (119872 119892) admits a homothetic vector field if and only if(119872119894 119892119894) admits a homothetic vector field for each 119894 =1 2(3) each conformal vector field on (119872 119892) is a conformal

vector field on (119872 119892)The above theorem together with Theorem 2 implies the

following results

Theorem 4 Let 120577 = 1205771 + 1205772 be a vector field on apseudo-Riemannian doubly warped product119872=11989121198721 times11989111198722equipped with the metric tensor 119892 = 11989122 1198921 + 11989121 1198922 Assumethat 120577119894 is a Killing vector field on (119872119894 119892119894) for each 119894 = 1 2 and1205771(ln1198911) = 1205772(ln1198912) Then 120577 is a conformal vector field on119872

Theorem 5 Let 120577119894 isin X(119872119894) be homothetic vector fields on(119872119894 119892119894) with factors 119886119894 for each 119894 = 1 2 Assume that 1205771(1198911) =1205772(1198912) = 0 Then 120577 = 11988621205771 + 11988611205772 is a homothetic vector fieldon (119872 119892) with factor 11988611198862Corollary 6 The dimension of the conformal group 119862(119872 119892)on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 is at least

dim1198701 (1198721 1198921) + dim1198701 (1198722 1198922) (19)

where119870119894(119872119894 119892119894) is the isometry group of (119872119894 119892119894)Again Theorem 2 together with Lemma 21 in [16] yields

the following result

Theorem 7 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 suchthat

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Then 120577 preserves the Ricci curvature if and only if119867119906 = 0

where 119906 = 1205882 +21205771(ln1198911) Moreover 120577 preserves the conformalclass of the Ricci tensor (ieL120577Ric = 120582119892 for some function 120582)if and only if nabla(div(120577)) is a conformal vector field

Theorem 8 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 120577 hasconstant length along the integral curve120572 of the vector field119883 =1198831 + 1198832 isin X(119872) if one of the following conditions holds

(1) 119883119894(119891119894) = 0 and 120577119894 is parallel along120587119894∘120572 for each 119894 = 1 2

(2) 119883119894(119891119894) = 0 and 120577119894 has a constant length along 120587119894 ∘ 120572 foreach 119894 = 1 2

Proof Let 120577 = 1205771 + 1205772 isin X(119872) be a vector field on119872 Then

119892 (119863119883120577 120577) = 119892 (1198631198831+11988321205771 + 1205772 120577)= 119892 (11986311988311205771 + 11986311988311205772 + 11986311988321205771 + 11986311988321205772 120577)= 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772)

+ 11989111198831 (1198911) 10038171003817100381710038171205772100381710038171003817100381722 + 11989121198832 (1198912) 10038171003817100381710038171205771100381710038171003817100381721

(20)

In both cases119883119894(119891119894) = 0 and hence

119892 (119863119883120577 120577) = 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772) (21)

The first condition implies that 119863119894119883119894120577119894 = 0 and so 119892119894(119863119894119883119894120577119894120577119894) = 0The second condition implies that 119892119894(120577119894 120577119894) is constantand so

0 = 2119892119894 (119863119894119883119894120577119894 120577119894) (22)

and therefore

119892 (119863119883120577 120577) = 0 (23)

that is 120577 has a constant length along the integral curve 120572 ofthe vector field119883Theorem 9 Let 120577 = 1205771 + 1205772 isin X(119872) be a conformal vectorfield on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 along a curve 120572 with unit tangent vector 119879 = 1198811 +1198812 Then

div (120577) = 119899 [11989122 1198921 (119863111988111205771 1198811) + 11989121 1198922 (119863211988121205772 1198812)+ 11989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 11989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722]

(24)

Proof Let 120577 be a conformal vector field with conformal factor120588 Then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (25)

Let119883 = 119884 = 119879 then2119892 (119863119879120577 119879) = 120588119892 (119879 119879) (26)

and then the conformal factor 120588 is given by

120588 = 2119892 (119863119879120577 119879) (27)

Suppose that 120577 = 1205771 + 1205772 and 119879 = 1198811 + 1198812 and then

120588 = 2119892 (11986311988111205771 + 11986311988111205772 + 11986311988121205771 + 11986311988121205772 119879)= 211989122 1198921 (119863111988111205771 1198811) + 211989121 1198922 (119863211988121205772 1198812)

+ 211989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 211989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722 (28)

But the conformal factor is given by

2 div (120577) = 120588119899 (29)

which completes the proof

Advances in Mathematical Physics 5

31 Conformal Vector Fields on Doubly Warped SpacetimeDoubly warped spacetime is a doubly warped product man-ifold 119872=11989121198721times11989111198722 where one of the factors say 1198721 hasa Lorentz signature and the second is Riemannian Ramos etal considered an invariant characterization of 4-dimensionaldoubly warped spacetime [21] Among many other resultsthey obtained necessary and sufficient conditions for (locally)double warped spacetime to be conformally related to 1 + 3or 2 + 2 decomposable spacetime Then they studied theconformal algebra of 2+2 decomposable spacetime in sectionIV and 1 + 3 decomposable spacetime in section V For adetailed discussion of conformally related 1 + 3 and 2 + 2reducible spacetime see [29 30] and for an extensive self-contained study of conformal symmetry of 4-dimensionalspacetime the reader is referred to [26]

We restrict our study of conformal vector fields ondoubly warped spacetime to dim(1198721) le 2 since this casegeneralizes some well-known exact solutions for the Einsteinfield equations and the beginning of this section representssuch study irrespective of the dimension of the factors Forthis case either dim(1198721) = 1 or dim(1198721) = 2 In thefollowing we deal with both subcases separately Let us firstconsider doubly warped spacetime with a 1-dimensionalbase

Let (119872 119892) be a Riemannian manifold and 119868 be anopen connected interval equipped with the metric minus1198891199052Doubly warped spacetime119872=119891 119868 times120590119872 is the product 119868 times119872furnished with the metric

119892 = minus11989121198891199052 oplus 1205902119892 (30)

where 119891 119872 rarr (0infin) and 120590 119868 rarr (0infin) are smoothfunctions 119872 is generalized RobertsonndashWalker spacetime if119891 is constant and standard static spacetime if 120590 is constant

An investigation of 4-dimensional spacetime that isconformally related to 1 + 3 reducible spacetime was carriedout with many examples in the aforementioned references[21 30] A classification of this spacetime according to itsconformal algebra is considered in the first reference whereasa special attention is paid to gradient conformal vector fieldsin the second reference

Theorem 10 A time-like vector field 120577 = ℎ120597119905 is a conformalvector field on doubly warped spacetime 119872=119891 119868 times120590119872 if andonly if ℎ = 119886120590 where 119886 is a nonnegative constant Moreoverthe conformal factor is 120588 = 2ℎProof If ℎ = 0 then 119886 = 0 and the result is obvious Now weassume that ℎ = 0 Using (16) we get that(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= minus21199091199101198912ℎ + 2ℎ120590119892 (119883 119884)= 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884)

(31)

Suppose that ℎ = 119886120590 and then

(L120577119892) (119883 119884) = 2ℎ119892 (119883 119884) (32)

that is 120577 = ℎ120597119905 is a conformal vector field with conformalfactor 120588 = 2ℎ Conversely suppose that 120577 = ℎ120597119905 isin X(119872) is aconformal vector field with factor 120588 and then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (33)

for any vector fields119883119884 isin X(119872) Now by (16) we get that120588119892 (119883 119884) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884) (34)

Let119883 = 119884 = 0 and we get that

120588119892 (119909120597119905 119910120597119905) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) (35)

that is 120588 = 2ℎ Now let 119909 = 119910 = 0 and we get that 120588120590 = 2ℎThese two differential equations imply that ℎ = 119886120590 for

some positive constant 119886Theorem 11 A vector field 120577 = ℎ120597119905 + 120577 is a Killing vector fieldon doubly warped spacetime119872=119891 119868 times120590119872 if and only if one ofthe following conditions holds

(1) 120577 is time-like and ℎ = = 0(2) 120577 is space-like where 120577 is a Killing vector field on119872 and120577(119891) = 0(3) ℎ = minus120577(ln119891) and 120577 is a conformal vector field on 119872

with conformal factor 1205882 = minus2ℎ120590Proof The first assertion is a special case of the abovetheorem For the second assertion let ℎ = 0 in (16) Thus

(L120577119892) (119883 119884) = minus2119909119910119891120577 (119891) + 1205902 (L120577119892) (119883 119884) (36)

Suppose that 120577 is a Killing vector field on 119872 and 120577(119891) = 0and then

(L120577119892) (119883 119884) = 0 (37)

The converse is direct Finally let 120577 be aKilling vector fieldon119872=119891 119868 times120590119872 and then

0 = (L120577119892) (119883 119884)

0 = minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

(38)

Let119883 = 119884 = 0 and then

minus21199091199101198912 [ℎ + 120577 (ln119891)] = 0 (39)

and so ℎ = minus120577(ln119891) Thus

(L119868ℎ120597119905119892119868) (119909120597119905 119910120597119905) = minus2120577 (ln119891) 119892119868 (119909120597119905 119910120597119905) (40)

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

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Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

as well as doubly warped spacetime We derive many charac-terizations of conformal vector fields on doublywarped prod-uct manifolds and doubly warped spacetime Then we studymatter and Ricci collineation on doubly warped manifoldsOne may notice that after Pereleman used Ricci soliton tosolve the Poincare conjecture posed in 1904 a growing bodyof research has continued to study Ricci soliton Accordinglywe study Ricci solitons on doubly warped product spacetimeadmitting many types of conformal vector fields We getsome partial answers of the following questions What dodoubly warped Ricci soliton factors inherit Andwhat are theconditions under which doubly warped spacetime is a doublywarped Ricci soliton

This article is organized as follows Section 2 representssome connection and curvature related formulas on doublywarped product manifolds In Section 3 we study conformalvector fields on doubly warped product manifolds Thenwe study conformal and concurrent vector fields on doublywarped spacetime in two subsections Finally Section 4 com-prises a study of Ricci soliton on doubly warped spacetimeadmitting these types of vector fields Almost all considera-tions and statements in this work are local

2 Preliminaries

This section represents connection and curvature relatedformulas on doubly warped product manifolds as a general-ization of similar results on singly warped products [1 19]Also we will provide basic definitions and properties ofconformal vector fields

Let (119872119894 119892119894 119863119894) be two (pseudo-)Riemannian manifoldswith metrics 119892119894 and Levi-Civita connections 119863119894 and let 119891119894 119872119894 rarr (0infin) be a positive function where 119894 = 1 2 Alsosuppose that 120587119894 1198721 times 1198722 rarr 119872119894 is the natural projectionmap of the Cartesian product 1198721 times 1198722 onto 119872119894 where119894 = 1 2 The (pseudo-)Riemannian manifolds doubly warpedproduct manifold 119872=11989121198721 times11989111198722 is the product manifold119872 = 1198721 times1198722 furnished with the metric tensor

119892 = (1198912 ∘ 1205872)2 120587lowast1 (1198921) oplus (1198911 ∘ 1205871)2 120587lowast2 (1198922) (1)

where lowast denotes the pull-back operator on tensors Thefunctions 119891119894 119894 = 1 2 are called the warping functionsof the warped product manifold 119872 In particular if forexample 1198912 = 1 then 119872 = 1198721 times11989111198722 is called a (singly)warped productmanifold A singly warped productmanifold1198721times11989111198722 is said to be trivial if the warping function1198911 is alsoconstant [6 8 9 12 20 21] It is clear that the submanifolds1198721 times 119902 and 119901 times 1198722 are homothetic to 1198721 and 1198722respectively for each 119901 isin 1198721 and 119902 isin 1198722 We shall referto these factor submanifolds as1198721 and1198722 The lift 119883(119901119902) ofa tangent vector 119883119901 isin 1198791199011198721 119902 isin 1198722 is the unique vector in119879(119901119902)119872 such that

120587lowast1 (119883(119901119902)) = 119883119901120587lowast2 (119883(119901119902)) = 0 (2)

Similarly if 119883119894 isin X(119872119894) then the lift of 119883119894 to X(1198721 times 1198722)is the unique vector field in X(1198721 times 1198722) that is 120587119894 related

to 119883119894 and 120587119895 related to zero vector field in X(119872119895) 119894 = 119895 thatis a vector field 119883119894 on 119872119894 is identified with the horizontalor the vertical vector field on 1198721 times 1198722 that is 120587119894 related to119883119894 Throughout this article we use the same notation for avector field and for its lift to the productmanifold A function120596119894 on 119872119894 will be identified with 120596119894 ∘ 120587119894 Thus we have twodifferent meanings for the gradient of 120596119894 namely grad (120596119894 ∘120587119894) isin X(1198721 times 1198722) and the lift of the gradient nabla119894120596119894 of 120596119894 toX(1198721 times1198722) In fact we have

119892 (119883119894 grad (120596119894 ∘ 120587119894)) = 119883119894 (120596119894 ∘ 120587119894) = 119883119894 (120596119894) ∘ 120587119894= 11198912119895 119892 (119883119894 nabla

119894120596119894) (3)

Therefore grad (120596119894 ∘ 120587119894) = (11198912119895 )nabla119894120596119894 (note that we usethe same notation for the vector field nabla119894120596119894 and for its lift toX(1198721 times1198722))

Let (119872 119892119863) be a pseudo-Riemannian doubly warpedproduct manifold of (119872119894 119892119894 119863119894) 119894 = 1 2 with dimensions119899119894 where 119899 = 1198991 +1198992 119877 119877119894 and Ric Ric119894 denote the curvaturetensor and Ricci curvature tensor on 119872119872119894 respectivelyMoreover nabla119894119891119894 and119894119891119894 denote gradient and Laplacian of 119891119894on119872119894 and 119891119894 = 119891119894119894119891119894 + (119899119895 minus1)119892119894(nabla119894119891119894 nabla119894119891119894) 119894 = 119895 For theconnection and curvatures formulas of a pseudo-Riemanniandoubly warped product manifold see for example [20 22]

A vector field 120577 on a (pseudo-)Riemannian manifold(119873 ℎ) with metric ℎ is called a conformal vector field withconformal factor 120588 if

L120577ℎ = 120588ℎ (4)

where L120577 is the Lie derivative on 119873 with respect to 120577 If 120588is constant or zero 120577 is called a homothetic or Killing vectorfield on 119873 respectively One can redefine conformal vectorfields using the following identity Let 120577 be a vector field on119872 and then

(L120577ℎ) (119883 119884) = ℎ (119863119883120577 119884) + ℎ (119883119863119884120577) (5)

for any vector fields 119883119884 isin X(119873) A vector field 120577 on amanifold (119873 ℎ) is called a concurrent vector field if

119863119883120577 = 119883 (6)

for any vector field 119883 isin X(119873) [23] Let 120577 be a concurrentvector field and then

(L120577ℎ) (119883 119884) = 2ℎ (119883 119884) (7)

and so 120577 is homothetic with factor 120588 = 2 A zero vector fieldis not concurrent If both 120577 and 120585 are concurrent vector fieldsthen

119863119883 [120577 120585] = 0 (8)

Also both 120577+120585 and120582120577 are not concurrent vector fields Finallya Killing vector field is not concurrent For example a vectorfield 120572120597119909 is a concurrent vector field on (R 1198891199092) if

119863120597119909 (120572120597119909) = 120597119909 (9)

Advances in Mathematical Physics 3

that is 120572 = 119909 + 119886 Thus concurrent vector fields on (R 1198891199092)are of the form (119909 + 119886)120597119909

The following result represents a simple characterizationof Killing vector fields if (119873 ℎ) is a pseudo-Riemannianmanifold with Riemannian connection 119863 A vector field 120577 isinX(119873) is a Killing vector field if and only if

ℎ (119863119883120577 119883) = 0 (10)

for any vector field119883 isin X(119873)The following discussion represents a good tool to char-

acterize Killing vector fields on pseudo-Riemannian warpedproduct manifolds In [24 25] the authors obtained manycharacterizations of Killing vector fields on warped productmanifolds and on standard static spacetime using theseresults Let119872 = 1198721 times1198911198722 be a pseudo-Riemannian warpedproduct manifold with warping function 119891 Let 120577 = 1205771 + 1205772 isinX(119872) be a vector field on119872 Then

119892 (119863119883120577 119883) = 1198921 (119863111988311205771 1198831) + 11989121198922 (119863211988321205772 1198832)+ 1198911205771 (119891) 10038171003817100381710038171198832100381710038171003817100381722

(L120577119892) (119883 119884) = (L112057711198921) (1198831 1198841)+ 1198912 (L212057721198922) (1198832 1198842)+ 21198911205771 (119891) 1198922 (1198832 1198842)

(11)

for any vector field 119883 = 1198831 + 1198832 isin X(119872) where L119894120577119894 is theLie derivative on119872119894 with respect to 120577119894 for 119894 = 1 2

A pseudo-Riemannian manifold 119872 is said to admit aRicci curvature collineation if there is a vector field 120577 isin X(119872)such that

L120577Ric = 0 (12)

where Ric is the Ricci curvature tensor [26] Finally space-time 119872 is said to admit a matter collineation if there is avector field 120577 isin X(119872) such that

L120577119879 = 0 (13)

where 119879 is the energy-momentum tensor [27] Einsteinrsquos fieldequation with cosmological constant 120582 is given by

Ric minus 1199032119892 = 120581119879 minus 120582119892 (14)

where 119903 is the scalar curvature Suppose that 120577 is a Killingvector field and then

L120577119879 = 0 (15)

that is 120577 is amatter collineationwhereas amatter collineationneed not be a Killing vector field Also a Killing vector field isa Ricci curvature collineation The converse is not generallytrue

3 Conformal Vector Fields on DoublyWarped Products

In this section we investigate the relation between conformalvector fields on doubly warped product manifolds and thoseconformal vector fields on the product factors Throughoutthis section let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product manifold with the metric tensor119892 = 11989122 1198921 oplus 11989121 1198922 and 119891119894 119872119894 rarr (0infin) is a smoothfunction where 119894 = 1 2 and (119872119894 119892119894) are pseudo-RiemannianmanifoldsThe following result gives us an important identityto study such relation [8]

Proposition 1 Suppose that 1205771 1198831 1198841 isin X(1198721) and1205772 1198832 1198842 isin X(1198722) and then(L120577119892) (119883 119884) = 11989122 (L112057711198921) (1198831Y1)

+ 11989121 (L212057721198922) (1198832 1198842)+ 211989111205771 (1198911) 1198922 (1198832 1198842)+ 211989121205772 (1198912) 1198921 (1198831 1198841)

(16)

where 120577 = 1205771 + 1205772119883 = 1198831 +1198832 and 119884 = 1198841 +1198842 are elementsin X(119872)

In [8] the author considered a characterization of con-formal vector fields on doubly warped product manifolds Infact it is just a characterization of homothetic vector fieldsThe following theorem represents a new characterization ofconformal vector fields on doubly warped product manifoldsbut the assumption here is less restrictive

Theorem2 A vector field 120577 = 1205771+1205772 on a pseudo-Riemanniandoubly warped product119872=11989121198721 times11989111198722 is a conformal vectorfield with conformal factor 120588 if and only if

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Moreover the conformal factor of 120577 is 120588 = 120588119894 +2120577119895(ln119891119895) 119894 = 119895Before proceeding further one may notice that a doubly

warped product metric 119892 on 119872 can be expressed as aconformal metric to a product metric on1198721 times1198722 as follows

119892 = 1198912111989122 ( 111989121 1198921 +

111989122 1198922) = 1198912111989122 (1198921 + 1198922)

= 1198912111989122 119892(17)

Let us consider the effect of replacing the metric 119892 on 119872 by119892 = 1198921+1198922 A similar discussion on 4-dimensional spacetimeis considered in [26 Chapter 11] Suppose that 120577 = 1205771 + 1205772 is aconformal vector field on (119872 119892) with factor 120588 and then

L120577119892 = [21205772 (ln1198912) + 21205771 (ln1198911) + 120588] 119892 (18)

4 Advances in Mathematical Physics

Therefore 120577 is a conformal vector field on (119872 119892) with factor120588 = 120588+21205772(ln1198912)+21205771(ln1198911) A similar conclusion applies to(119872119894 119892119894) and (119872119894 119892119894) where 120588119894 = 120588 minus 120577119894(ln119891119894) Thus by usingresults in [28 Theorem 1] one can easily get the following

Theorem 3 Let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product equipped with the metric tensor 119892 =11989122 1198921 + 11989121 1198922 and let 119892 = 1198921 + 1198922 where 119892119894 = (11198912119894 )119892119894 Then

(1) a Killing vector field 120577 = 120577119894 on (119872119894 119892119894) for each 119894 = 1 2is a Killing vector field on (119872 119892)

(2) (119872 119892) admits a homothetic vector field if and only if(119872119894 119892119894) admits a homothetic vector field for each 119894 =1 2(3) each conformal vector field on (119872 119892) is a conformal

vector field on (119872 119892)The above theorem together with Theorem 2 implies the

following results

Theorem 4 Let 120577 = 1205771 + 1205772 be a vector field on apseudo-Riemannian doubly warped product119872=11989121198721 times11989111198722equipped with the metric tensor 119892 = 11989122 1198921 + 11989121 1198922 Assumethat 120577119894 is a Killing vector field on (119872119894 119892119894) for each 119894 = 1 2 and1205771(ln1198911) = 1205772(ln1198912) Then 120577 is a conformal vector field on119872

Theorem 5 Let 120577119894 isin X(119872119894) be homothetic vector fields on(119872119894 119892119894) with factors 119886119894 for each 119894 = 1 2 Assume that 1205771(1198911) =1205772(1198912) = 0 Then 120577 = 11988621205771 + 11988611205772 is a homothetic vector fieldon (119872 119892) with factor 11988611198862Corollary 6 The dimension of the conformal group 119862(119872 119892)on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 is at least

dim1198701 (1198721 1198921) + dim1198701 (1198722 1198922) (19)

where119870119894(119872119894 119892119894) is the isometry group of (119872119894 119892119894)Again Theorem 2 together with Lemma 21 in [16] yields

the following result

Theorem 7 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 suchthat

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Then 120577 preserves the Ricci curvature if and only if119867119906 = 0

where 119906 = 1205882 +21205771(ln1198911) Moreover 120577 preserves the conformalclass of the Ricci tensor (ieL120577Ric = 120582119892 for some function 120582)if and only if nabla(div(120577)) is a conformal vector field

Theorem 8 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 120577 hasconstant length along the integral curve120572 of the vector field119883 =1198831 + 1198832 isin X(119872) if one of the following conditions holds

(1) 119883119894(119891119894) = 0 and 120577119894 is parallel along120587119894∘120572 for each 119894 = 1 2

(2) 119883119894(119891119894) = 0 and 120577119894 has a constant length along 120587119894 ∘ 120572 foreach 119894 = 1 2

Proof Let 120577 = 1205771 + 1205772 isin X(119872) be a vector field on119872 Then

119892 (119863119883120577 120577) = 119892 (1198631198831+11988321205771 + 1205772 120577)= 119892 (11986311988311205771 + 11986311988311205772 + 11986311988321205771 + 11986311988321205772 120577)= 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772)

+ 11989111198831 (1198911) 10038171003817100381710038171205772100381710038171003817100381722 + 11989121198832 (1198912) 10038171003817100381710038171205771100381710038171003817100381721

(20)

In both cases119883119894(119891119894) = 0 and hence

119892 (119863119883120577 120577) = 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772) (21)

The first condition implies that 119863119894119883119894120577119894 = 0 and so 119892119894(119863119894119883119894120577119894120577119894) = 0The second condition implies that 119892119894(120577119894 120577119894) is constantand so

0 = 2119892119894 (119863119894119883119894120577119894 120577119894) (22)

and therefore

119892 (119863119883120577 120577) = 0 (23)

that is 120577 has a constant length along the integral curve 120572 ofthe vector field119883Theorem 9 Let 120577 = 1205771 + 1205772 isin X(119872) be a conformal vectorfield on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 along a curve 120572 with unit tangent vector 119879 = 1198811 +1198812 Then

div (120577) = 119899 [11989122 1198921 (119863111988111205771 1198811) + 11989121 1198922 (119863211988121205772 1198812)+ 11989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 11989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722]

(24)

Proof Let 120577 be a conformal vector field with conformal factor120588 Then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (25)

Let119883 = 119884 = 119879 then2119892 (119863119879120577 119879) = 120588119892 (119879 119879) (26)

and then the conformal factor 120588 is given by

120588 = 2119892 (119863119879120577 119879) (27)

Suppose that 120577 = 1205771 + 1205772 and 119879 = 1198811 + 1198812 and then

120588 = 2119892 (11986311988111205771 + 11986311988111205772 + 11986311988121205771 + 11986311988121205772 119879)= 211989122 1198921 (119863111988111205771 1198811) + 211989121 1198922 (119863211988121205772 1198812)

+ 211989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 211989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722 (28)

But the conformal factor is given by

2 div (120577) = 120588119899 (29)

which completes the proof

Advances in Mathematical Physics 5

31 Conformal Vector Fields on Doubly Warped SpacetimeDoubly warped spacetime is a doubly warped product man-ifold 119872=11989121198721times11989111198722 where one of the factors say 1198721 hasa Lorentz signature and the second is Riemannian Ramos etal considered an invariant characterization of 4-dimensionaldoubly warped spacetime [21] Among many other resultsthey obtained necessary and sufficient conditions for (locally)double warped spacetime to be conformally related to 1 + 3or 2 + 2 decomposable spacetime Then they studied theconformal algebra of 2+2 decomposable spacetime in sectionIV and 1 + 3 decomposable spacetime in section V For adetailed discussion of conformally related 1 + 3 and 2 + 2reducible spacetime see [29 30] and for an extensive self-contained study of conformal symmetry of 4-dimensionalspacetime the reader is referred to [26]

We restrict our study of conformal vector fields ondoubly warped spacetime to dim(1198721) le 2 since this casegeneralizes some well-known exact solutions for the Einsteinfield equations and the beginning of this section representssuch study irrespective of the dimension of the factors Forthis case either dim(1198721) = 1 or dim(1198721) = 2 In thefollowing we deal with both subcases separately Let us firstconsider doubly warped spacetime with a 1-dimensionalbase

Let (119872 119892) be a Riemannian manifold and 119868 be anopen connected interval equipped with the metric minus1198891199052Doubly warped spacetime119872=119891 119868 times120590119872 is the product 119868 times119872furnished with the metric

119892 = minus11989121198891199052 oplus 1205902119892 (30)

where 119891 119872 rarr (0infin) and 120590 119868 rarr (0infin) are smoothfunctions 119872 is generalized RobertsonndashWalker spacetime if119891 is constant and standard static spacetime if 120590 is constant

An investigation of 4-dimensional spacetime that isconformally related to 1 + 3 reducible spacetime was carriedout with many examples in the aforementioned references[21 30] A classification of this spacetime according to itsconformal algebra is considered in the first reference whereasa special attention is paid to gradient conformal vector fieldsin the second reference

Theorem 10 A time-like vector field 120577 = ℎ120597119905 is a conformalvector field on doubly warped spacetime 119872=119891 119868 times120590119872 if andonly if ℎ = 119886120590 where 119886 is a nonnegative constant Moreoverthe conformal factor is 120588 = 2ℎProof If ℎ = 0 then 119886 = 0 and the result is obvious Now weassume that ℎ = 0 Using (16) we get that(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= minus21199091199101198912ℎ + 2ℎ120590119892 (119883 119884)= 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884)

(31)

Suppose that ℎ = 119886120590 and then

(L120577119892) (119883 119884) = 2ℎ119892 (119883 119884) (32)

that is 120577 = ℎ120597119905 is a conformal vector field with conformalfactor 120588 = 2ℎ Conversely suppose that 120577 = ℎ120597119905 isin X(119872) is aconformal vector field with factor 120588 and then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (33)

for any vector fields119883119884 isin X(119872) Now by (16) we get that120588119892 (119883 119884) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884) (34)

Let119883 = 119884 = 0 and we get that

120588119892 (119909120597119905 119910120597119905) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) (35)

that is 120588 = 2ℎ Now let 119909 = 119910 = 0 and we get that 120588120590 = 2ℎThese two differential equations imply that ℎ = 119886120590 for

some positive constant 119886Theorem 11 A vector field 120577 = ℎ120597119905 + 120577 is a Killing vector fieldon doubly warped spacetime119872=119891 119868 times120590119872 if and only if one ofthe following conditions holds

(1) 120577 is time-like and ℎ = = 0(2) 120577 is space-like where 120577 is a Killing vector field on119872 and120577(119891) = 0(3) ℎ = minus120577(ln119891) and 120577 is a conformal vector field on 119872

with conformal factor 1205882 = minus2ℎ120590Proof The first assertion is a special case of the abovetheorem For the second assertion let ℎ = 0 in (16) Thus

(L120577119892) (119883 119884) = minus2119909119910119891120577 (119891) + 1205902 (L120577119892) (119883 119884) (36)

Suppose that 120577 is a Killing vector field on 119872 and 120577(119891) = 0and then

(L120577119892) (119883 119884) = 0 (37)

The converse is direct Finally let 120577 be aKilling vector fieldon119872=119891 119868 times120590119872 and then

0 = (L120577119892) (119883 119884)

0 = minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

(38)

Let119883 = 119884 = 0 and then

minus21199091199101198912 [ℎ + 120577 (ln119891)] = 0 (39)

and so ℎ = minus120577(ln119891) Thus

(L119868ℎ120597119905119892119868) (119909120597119905 119910120597119905) = minus2120577 (ln119891) 119892119868 (119909120597119905 119910120597119905) (40)

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

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Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

that is 120572 = 119909 + 119886 Thus concurrent vector fields on (R 1198891199092)are of the form (119909 + 119886)120597119909

The following result represents a simple characterizationof Killing vector fields if (119873 ℎ) is a pseudo-Riemannianmanifold with Riemannian connection 119863 A vector field 120577 isinX(119873) is a Killing vector field if and only if

ℎ (119863119883120577 119883) = 0 (10)

for any vector field119883 isin X(119873)The following discussion represents a good tool to char-

acterize Killing vector fields on pseudo-Riemannian warpedproduct manifolds In [24 25] the authors obtained manycharacterizations of Killing vector fields on warped productmanifolds and on standard static spacetime using theseresults Let119872 = 1198721 times1198911198722 be a pseudo-Riemannian warpedproduct manifold with warping function 119891 Let 120577 = 1205771 + 1205772 isinX(119872) be a vector field on119872 Then

119892 (119863119883120577 119883) = 1198921 (119863111988311205771 1198831) + 11989121198922 (119863211988321205772 1198832)+ 1198911205771 (119891) 10038171003817100381710038171198832100381710038171003817100381722

(L120577119892) (119883 119884) = (L112057711198921) (1198831 1198841)+ 1198912 (L212057721198922) (1198832 1198842)+ 21198911205771 (119891) 1198922 (1198832 1198842)

(11)

for any vector field 119883 = 1198831 + 1198832 isin X(119872) where L119894120577119894 is theLie derivative on119872119894 with respect to 120577119894 for 119894 = 1 2

A pseudo-Riemannian manifold 119872 is said to admit aRicci curvature collineation if there is a vector field 120577 isin X(119872)such that

L120577Ric = 0 (12)

where Ric is the Ricci curvature tensor [26] Finally space-time 119872 is said to admit a matter collineation if there is avector field 120577 isin X(119872) such that

L120577119879 = 0 (13)

where 119879 is the energy-momentum tensor [27] Einsteinrsquos fieldequation with cosmological constant 120582 is given by

Ric minus 1199032119892 = 120581119879 minus 120582119892 (14)

where 119903 is the scalar curvature Suppose that 120577 is a Killingvector field and then

L120577119879 = 0 (15)

that is 120577 is amatter collineationwhereas amatter collineationneed not be a Killing vector field Also a Killing vector field isa Ricci curvature collineation The converse is not generallytrue

3 Conformal Vector Fields on DoublyWarped Products

In this section we investigate the relation between conformalvector fields on doubly warped product manifolds and thoseconformal vector fields on the product factors Throughoutthis section let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product manifold with the metric tensor119892 = 11989122 1198921 oplus 11989121 1198922 and 119891119894 119872119894 rarr (0infin) is a smoothfunction where 119894 = 1 2 and (119872119894 119892119894) are pseudo-RiemannianmanifoldsThe following result gives us an important identityto study such relation [8]

Proposition 1 Suppose that 1205771 1198831 1198841 isin X(1198721) and1205772 1198832 1198842 isin X(1198722) and then(L120577119892) (119883 119884) = 11989122 (L112057711198921) (1198831Y1)

+ 11989121 (L212057721198922) (1198832 1198842)+ 211989111205771 (1198911) 1198922 (1198832 1198842)+ 211989121205772 (1198912) 1198921 (1198831 1198841)

(16)

where 120577 = 1205771 + 1205772119883 = 1198831 +1198832 and 119884 = 1198841 +1198842 are elementsin X(119872)

In [8] the author considered a characterization of con-formal vector fields on doubly warped product manifolds Infact it is just a characterization of homothetic vector fieldsThe following theorem represents a new characterization ofconformal vector fields on doubly warped product manifoldsbut the assumption here is less restrictive

Theorem2 A vector field 120577 = 1205771+1205772 on a pseudo-Riemanniandoubly warped product119872=11989121198721 times11989111198722 is a conformal vectorfield with conformal factor 120588 if and only if

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Moreover the conformal factor of 120577 is 120588 = 120588119894 +2120577119895(ln119891119895) 119894 = 119895Before proceeding further one may notice that a doubly

warped product metric 119892 on 119872 can be expressed as aconformal metric to a product metric on1198721 times1198722 as follows

119892 = 1198912111989122 ( 111989121 1198921 +

111989122 1198922) = 1198912111989122 (1198921 + 1198922)

= 1198912111989122 119892(17)

Let us consider the effect of replacing the metric 119892 on 119872 by119892 = 1198921+1198922 A similar discussion on 4-dimensional spacetimeis considered in [26 Chapter 11] Suppose that 120577 = 1205771 + 1205772 is aconformal vector field on (119872 119892) with factor 120588 and then

L120577119892 = [21205772 (ln1198912) + 21205771 (ln1198911) + 120588] 119892 (18)

4 Advances in Mathematical Physics

Therefore 120577 is a conformal vector field on (119872 119892) with factor120588 = 120588+21205772(ln1198912)+21205771(ln1198911) A similar conclusion applies to(119872119894 119892119894) and (119872119894 119892119894) where 120588119894 = 120588 minus 120577119894(ln119891119894) Thus by usingresults in [28 Theorem 1] one can easily get the following

Theorem 3 Let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product equipped with the metric tensor 119892 =11989122 1198921 + 11989121 1198922 and let 119892 = 1198921 + 1198922 where 119892119894 = (11198912119894 )119892119894 Then

(1) a Killing vector field 120577 = 120577119894 on (119872119894 119892119894) for each 119894 = 1 2is a Killing vector field on (119872 119892)

(2) (119872 119892) admits a homothetic vector field if and only if(119872119894 119892119894) admits a homothetic vector field for each 119894 =1 2(3) each conformal vector field on (119872 119892) is a conformal

vector field on (119872 119892)The above theorem together with Theorem 2 implies the

following results

Theorem 4 Let 120577 = 1205771 + 1205772 be a vector field on apseudo-Riemannian doubly warped product119872=11989121198721 times11989111198722equipped with the metric tensor 119892 = 11989122 1198921 + 11989121 1198922 Assumethat 120577119894 is a Killing vector field on (119872119894 119892119894) for each 119894 = 1 2 and1205771(ln1198911) = 1205772(ln1198912) Then 120577 is a conformal vector field on119872

Theorem 5 Let 120577119894 isin X(119872119894) be homothetic vector fields on(119872119894 119892119894) with factors 119886119894 for each 119894 = 1 2 Assume that 1205771(1198911) =1205772(1198912) = 0 Then 120577 = 11988621205771 + 11988611205772 is a homothetic vector fieldon (119872 119892) with factor 11988611198862Corollary 6 The dimension of the conformal group 119862(119872 119892)on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 is at least

dim1198701 (1198721 1198921) + dim1198701 (1198722 1198922) (19)

where119870119894(119872119894 119892119894) is the isometry group of (119872119894 119892119894)Again Theorem 2 together with Lemma 21 in [16] yields

the following result

Theorem 7 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 suchthat

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Then 120577 preserves the Ricci curvature if and only if119867119906 = 0

where 119906 = 1205882 +21205771(ln1198911) Moreover 120577 preserves the conformalclass of the Ricci tensor (ieL120577Ric = 120582119892 for some function 120582)if and only if nabla(div(120577)) is a conformal vector field

Theorem 8 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 120577 hasconstant length along the integral curve120572 of the vector field119883 =1198831 + 1198832 isin X(119872) if one of the following conditions holds

(1) 119883119894(119891119894) = 0 and 120577119894 is parallel along120587119894∘120572 for each 119894 = 1 2

(2) 119883119894(119891119894) = 0 and 120577119894 has a constant length along 120587119894 ∘ 120572 foreach 119894 = 1 2

Proof Let 120577 = 1205771 + 1205772 isin X(119872) be a vector field on119872 Then

119892 (119863119883120577 120577) = 119892 (1198631198831+11988321205771 + 1205772 120577)= 119892 (11986311988311205771 + 11986311988311205772 + 11986311988321205771 + 11986311988321205772 120577)= 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772)

+ 11989111198831 (1198911) 10038171003817100381710038171205772100381710038171003817100381722 + 11989121198832 (1198912) 10038171003817100381710038171205771100381710038171003817100381721

(20)

In both cases119883119894(119891119894) = 0 and hence

119892 (119863119883120577 120577) = 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772) (21)

The first condition implies that 119863119894119883119894120577119894 = 0 and so 119892119894(119863119894119883119894120577119894120577119894) = 0The second condition implies that 119892119894(120577119894 120577119894) is constantand so

0 = 2119892119894 (119863119894119883119894120577119894 120577119894) (22)

and therefore

119892 (119863119883120577 120577) = 0 (23)

that is 120577 has a constant length along the integral curve 120572 ofthe vector field119883Theorem 9 Let 120577 = 1205771 + 1205772 isin X(119872) be a conformal vectorfield on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 along a curve 120572 with unit tangent vector 119879 = 1198811 +1198812 Then

div (120577) = 119899 [11989122 1198921 (119863111988111205771 1198811) + 11989121 1198922 (119863211988121205772 1198812)+ 11989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 11989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722]

(24)

Proof Let 120577 be a conformal vector field with conformal factor120588 Then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (25)

Let119883 = 119884 = 119879 then2119892 (119863119879120577 119879) = 120588119892 (119879 119879) (26)

and then the conformal factor 120588 is given by

120588 = 2119892 (119863119879120577 119879) (27)

Suppose that 120577 = 1205771 + 1205772 and 119879 = 1198811 + 1198812 and then

120588 = 2119892 (11986311988111205771 + 11986311988111205772 + 11986311988121205771 + 11986311988121205772 119879)= 211989122 1198921 (119863111988111205771 1198811) + 211989121 1198922 (119863211988121205772 1198812)

+ 211989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 211989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722 (28)

But the conformal factor is given by

2 div (120577) = 120588119899 (29)

which completes the proof

Advances in Mathematical Physics 5

31 Conformal Vector Fields on Doubly Warped SpacetimeDoubly warped spacetime is a doubly warped product man-ifold 119872=11989121198721times11989111198722 where one of the factors say 1198721 hasa Lorentz signature and the second is Riemannian Ramos etal considered an invariant characterization of 4-dimensionaldoubly warped spacetime [21] Among many other resultsthey obtained necessary and sufficient conditions for (locally)double warped spacetime to be conformally related to 1 + 3or 2 + 2 decomposable spacetime Then they studied theconformal algebra of 2+2 decomposable spacetime in sectionIV and 1 + 3 decomposable spacetime in section V For adetailed discussion of conformally related 1 + 3 and 2 + 2reducible spacetime see [29 30] and for an extensive self-contained study of conformal symmetry of 4-dimensionalspacetime the reader is referred to [26]

We restrict our study of conformal vector fields ondoubly warped spacetime to dim(1198721) le 2 since this casegeneralizes some well-known exact solutions for the Einsteinfield equations and the beginning of this section representssuch study irrespective of the dimension of the factors Forthis case either dim(1198721) = 1 or dim(1198721) = 2 In thefollowing we deal with both subcases separately Let us firstconsider doubly warped spacetime with a 1-dimensionalbase

Let (119872 119892) be a Riemannian manifold and 119868 be anopen connected interval equipped with the metric minus1198891199052Doubly warped spacetime119872=119891 119868 times120590119872 is the product 119868 times119872furnished with the metric

119892 = minus11989121198891199052 oplus 1205902119892 (30)

where 119891 119872 rarr (0infin) and 120590 119868 rarr (0infin) are smoothfunctions 119872 is generalized RobertsonndashWalker spacetime if119891 is constant and standard static spacetime if 120590 is constant

An investigation of 4-dimensional spacetime that isconformally related to 1 + 3 reducible spacetime was carriedout with many examples in the aforementioned references[21 30] A classification of this spacetime according to itsconformal algebra is considered in the first reference whereasa special attention is paid to gradient conformal vector fieldsin the second reference

Theorem 10 A time-like vector field 120577 = ℎ120597119905 is a conformalvector field on doubly warped spacetime 119872=119891 119868 times120590119872 if andonly if ℎ = 119886120590 where 119886 is a nonnegative constant Moreoverthe conformal factor is 120588 = 2ℎProof If ℎ = 0 then 119886 = 0 and the result is obvious Now weassume that ℎ = 0 Using (16) we get that(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= minus21199091199101198912ℎ + 2ℎ120590119892 (119883 119884)= 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884)

(31)

Suppose that ℎ = 119886120590 and then

(L120577119892) (119883 119884) = 2ℎ119892 (119883 119884) (32)

that is 120577 = ℎ120597119905 is a conformal vector field with conformalfactor 120588 = 2ℎ Conversely suppose that 120577 = ℎ120597119905 isin X(119872) is aconformal vector field with factor 120588 and then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (33)

for any vector fields119883119884 isin X(119872) Now by (16) we get that120588119892 (119883 119884) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884) (34)

Let119883 = 119884 = 0 and we get that

120588119892 (119909120597119905 119910120597119905) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) (35)

that is 120588 = 2ℎ Now let 119909 = 119910 = 0 and we get that 120588120590 = 2ℎThese two differential equations imply that ℎ = 119886120590 for

some positive constant 119886Theorem 11 A vector field 120577 = ℎ120597119905 + 120577 is a Killing vector fieldon doubly warped spacetime119872=119891 119868 times120590119872 if and only if one ofthe following conditions holds

(1) 120577 is time-like and ℎ = = 0(2) 120577 is space-like where 120577 is a Killing vector field on119872 and120577(119891) = 0(3) ℎ = minus120577(ln119891) and 120577 is a conformal vector field on 119872

with conformal factor 1205882 = minus2ℎ120590Proof The first assertion is a special case of the abovetheorem For the second assertion let ℎ = 0 in (16) Thus

(L120577119892) (119883 119884) = minus2119909119910119891120577 (119891) + 1205902 (L120577119892) (119883 119884) (36)

Suppose that 120577 is a Killing vector field on 119872 and 120577(119891) = 0and then

(L120577119892) (119883 119884) = 0 (37)

The converse is direct Finally let 120577 be aKilling vector fieldon119872=119891 119868 times120590119872 and then

0 = (L120577119892) (119883 119884)

0 = minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

(38)

Let119883 = 119884 = 0 and then

minus21199091199101198912 [ℎ + 120577 (ln119891)] = 0 (39)

and so ℎ = minus120577(ln119891) Thus

(L119868ℎ120597119905119892119868) (119909120597119905 119910120597119905) = minus2120577 (ln119891) 119892119868 (119909120597119905 119910120597119905) (40)

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

Therefore 120577 is a conformal vector field on (119872 119892) with factor120588 = 120588+21205772(ln1198912)+21205771(ln1198911) A similar conclusion applies to(119872119894 119892119894) and (119872119894 119892119894) where 120588119894 = 120588 minus 120577119894(ln119891119894) Thus by usingresults in [28 Theorem 1] one can easily get the following

Theorem 3 Let 119872=11989121198721 times11989111198722 be a pseudo-Riemanniandoubly warped product equipped with the metric tensor 119892 =11989122 1198921 + 11989121 1198922 and let 119892 = 1198921 + 1198922 where 119892119894 = (11198912119894 )119892119894 Then

(1) a Killing vector field 120577 = 120577119894 on (119872119894 119892119894) for each 119894 = 1 2is a Killing vector field on (119872 119892)

(2) (119872 119892) admits a homothetic vector field if and only if(119872119894 119892119894) admits a homothetic vector field for each 119894 =1 2(3) each conformal vector field on (119872 119892) is a conformal

vector field on (119872 119892)The above theorem together with Theorem 2 implies the

following results

Theorem 4 Let 120577 = 1205771 + 1205772 be a vector field on apseudo-Riemannian doubly warped product119872=11989121198721 times11989111198722equipped with the metric tensor 119892 = 11989122 1198921 + 11989121 1198922 Assumethat 120577119894 is a Killing vector field on (119872119894 119892119894) for each 119894 = 1 2 and1205771(ln1198911) = 1205772(ln1198912) Then 120577 is a conformal vector field on119872

Theorem 5 Let 120577119894 isin X(119872119894) be homothetic vector fields on(119872119894 119892119894) with factors 119886119894 for each 119894 = 1 2 Assume that 1205771(1198911) =1205772(1198912) = 0 Then 120577 = 11988621205771 + 11988611205772 is a homothetic vector fieldon (119872 119892) with factor 11988611198862Corollary 6 The dimension of the conformal group 119862(119872 119892)on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 is at least

dim1198701 (1198721 1198921) + dim1198701 (1198722 1198922) (19)

where119870119894(119872119894 119892119894) is the isometry group of (119872119894 119892119894)Again Theorem 2 together with Lemma 21 in [16] yields

the following result

Theorem 7 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 suchthat

(1) 120577119894 is a conformal vector field on 119872119894 with conformalfactor 120588119894 119894 = 1 2

(2) 1205881 + 21205772(ln1198912) = 1205882 + 21205771(ln1198911)Then 120577 preserves the Ricci curvature if and only if119867119906 = 0

where 119906 = 1205882 +21205771(ln1198911) Moreover 120577 preserves the conformalclass of the Ricci tensor (ieL120577Ric = 120582119892 for some function 120582)if and only if nabla(div(120577)) is a conformal vector field

Theorem 8 Let 120577 = 1205771 + 1205772 be a vector field on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 120577 hasconstant length along the integral curve120572 of the vector field119883 =1198831 + 1198832 isin X(119872) if one of the following conditions holds

(1) 119883119894(119891119894) = 0 and 120577119894 is parallel along120587119894∘120572 for each 119894 = 1 2

(2) 119883119894(119891119894) = 0 and 120577119894 has a constant length along 120587119894 ∘ 120572 foreach 119894 = 1 2

Proof Let 120577 = 1205771 + 1205772 isin X(119872) be a vector field on119872 Then

119892 (119863119883120577 120577) = 119892 (1198631198831+11988321205771 + 1205772 120577)= 119892 (11986311988311205771 + 11986311988311205772 + 11986311988321205771 + 11986311988321205772 120577)= 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772)

+ 11989111198831 (1198911) 10038171003817100381710038171205772100381710038171003817100381722 + 11989121198832 (1198912) 10038171003817100381710038171205771100381710038171003817100381721

(20)

In both cases119883119894(119891119894) = 0 and hence

119892 (119863119883120577 120577) = 11989122 1198921 (119863111988311205771 1205771) + 11989121 1198922 (119863211988321205772 1205772) (21)

The first condition implies that 119863119894119883119894120577119894 = 0 and so 119892119894(119863119894119883119894120577119894120577119894) = 0The second condition implies that 119892119894(120577119894 120577119894) is constantand so

0 = 2119892119894 (119863119894119883119894120577119894 120577119894) (22)

and therefore

119892 (119863119883120577 120577) = 0 (23)

that is 120577 has a constant length along the integral curve 120572 ofthe vector field119883Theorem 9 Let 120577 = 1205771 + 1205772 isin X(119872) be a conformal vectorfield on a pseudo-Riemannian doubly warped product 119872=11989121198721 times11989111198722 along a curve 120572 with unit tangent vector 119879 = 1198811 +1198812 Then

div (120577) = 119899 [11989122 1198921 (119863111988111205771 1198811) + 11989121 1198922 (119863211988121205772 1198812)+ 11989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 11989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722]

(24)

Proof Let 120577 be a conformal vector field with conformal factor120588 Then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (25)

Let119883 = 119884 = 119879 then2119892 (119863119879120577 119879) = 120588119892 (119879 119879) (26)

and then the conformal factor 120588 is given by

120588 = 2119892 (119863119879120577 119879) (27)

Suppose that 120577 = 1205771 + 1205772 and 119879 = 1198811 + 1198812 and then

120588 = 2119892 (11986311988111205771 + 11986311988111205772 + 11986311988121205771 + 11986311988121205772 119879)= 211989122 1198921 (119863111988111205771 1198811) + 211989121 1198922 (119863211988121205772 1198812)

+ 211989121205772 (1198912) 10038171003817100381710038171198811100381710038171003817100381721 + 211989111205771 (1198911) 10038171003817100381710038171198812100381710038171003817100381722 (28)

But the conformal factor is given by

2 div (120577) = 120588119899 (29)

which completes the proof

Advances in Mathematical Physics 5

31 Conformal Vector Fields on Doubly Warped SpacetimeDoubly warped spacetime is a doubly warped product man-ifold 119872=11989121198721times11989111198722 where one of the factors say 1198721 hasa Lorentz signature and the second is Riemannian Ramos etal considered an invariant characterization of 4-dimensionaldoubly warped spacetime [21] Among many other resultsthey obtained necessary and sufficient conditions for (locally)double warped spacetime to be conformally related to 1 + 3or 2 + 2 decomposable spacetime Then they studied theconformal algebra of 2+2 decomposable spacetime in sectionIV and 1 + 3 decomposable spacetime in section V For adetailed discussion of conformally related 1 + 3 and 2 + 2reducible spacetime see [29 30] and for an extensive self-contained study of conformal symmetry of 4-dimensionalspacetime the reader is referred to [26]

We restrict our study of conformal vector fields ondoubly warped spacetime to dim(1198721) le 2 since this casegeneralizes some well-known exact solutions for the Einsteinfield equations and the beginning of this section representssuch study irrespective of the dimension of the factors Forthis case either dim(1198721) = 1 or dim(1198721) = 2 In thefollowing we deal with both subcases separately Let us firstconsider doubly warped spacetime with a 1-dimensionalbase

Let (119872 119892) be a Riemannian manifold and 119868 be anopen connected interval equipped with the metric minus1198891199052Doubly warped spacetime119872=119891 119868 times120590119872 is the product 119868 times119872furnished with the metric

119892 = minus11989121198891199052 oplus 1205902119892 (30)

where 119891 119872 rarr (0infin) and 120590 119868 rarr (0infin) are smoothfunctions 119872 is generalized RobertsonndashWalker spacetime if119891 is constant and standard static spacetime if 120590 is constant

An investigation of 4-dimensional spacetime that isconformally related to 1 + 3 reducible spacetime was carriedout with many examples in the aforementioned references[21 30] A classification of this spacetime according to itsconformal algebra is considered in the first reference whereasa special attention is paid to gradient conformal vector fieldsin the second reference

Theorem 10 A time-like vector field 120577 = ℎ120597119905 is a conformalvector field on doubly warped spacetime 119872=119891 119868 times120590119872 if andonly if ℎ = 119886120590 where 119886 is a nonnegative constant Moreoverthe conformal factor is 120588 = 2ℎProof If ℎ = 0 then 119886 = 0 and the result is obvious Now weassume that ℎ = 0 Using (16) we get that(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= minus21199091199101198912ℎ + 2ℎ120590119892 (119883 119884)= 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884)

(31)

Suppose that ℎ = 119886120590 and then

(L120577119892) (119883 119884) = 2ℎ119892 (119883 119884) (32)

that is 120577 = ℎ120597119905 is a conformal vector field with conformalfactor 120588 = 2ℎ Conversely suppose that 120577 = ℎ120597119905 isin X(119872) is aconformal vector field with factor 120588 and then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (33)

for any vector fields119883119884 isin X(119872) Now by (16) we get that120588119892 (119883 119884) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884) (34)

Let119883 = 119884 = 0 and we get that

120588119892 (119909120597119905 119910120597119905) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) (35)

that is 120588 = 2ℎ Now let 119909 = 119910 = 0 and we get that 120588120590 = 2ℎThese two differential equations imply that ℎ = 119886120590 for

some positive constant 119886Theorem 11 A vector field 120577 = ℎ120597119905 + 120577 is a Killing vector fieldon doubly warped spacetime119872=119891 119868 times120590119872 if and only if one ofthe following conditions holds

(1) 120577 is time-like and ℎ = = 0(2) 120577 is space-like where 120577 is a Killing vector field on119872 and120577(119891) = 0(3) ℎ = minus120577(ln119891) and 120577 is a conformal vector field on 119872

with conformal factor 1205882 = minus2ℎ120590Proof The first assertion is a special case of the abovetheorem For the second assertion let ℎ = 0 in (16) Thus

(L120577119892) (119883 119884) = minus2119909119910119891120577 (119891) + 1205902 (L120577119892) (119883 119884) (36)

Suppose that 120577 is a Killing vector field on 119872 and 120577(119891) = 0and then

(L120577119892) (119883 119884) = 0 (37)

The converse is direct Finally let 120577 be aKilling vector fieldon119872=119891 119868 times120590119872 and then

0 = (L120577119892) (119883 119884)

0 = minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

(38)

Let119883 = 119884 = 0 and then

minus21199091199101198912 [ℎ + 120577 (ln119891)] = 0 (39)

and so ℎ = minus120577(ln119891) Thus

(L119868ℎ120597119905119892119868) (119909120597119905 119910120597119905) = minus2120577 (ln119891) 119892119868 (119909120597119905 119910120597119905) (40)

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

31 Conformal Vector Fields on Doubly Warped SpacetimeDoubly warped spacetime is a doubly warped product man-ifold 119872=11989121198721times11989111198722 where one of the factors say 1198721 hasa Lorentz signature and the second is Riemannian Ramos etal considered an invariant characterization of 4-dimensionaldoubly warped spacetime [21] Among many other resultsthey obtained necessary and sufficient conditions for (locally)double warped spacetime to be conformally related to 1 + 3or 2 + 2 decomposable spacetime Then they studied theconformal algebra of 2+2 decomposable spacetime in sectionIV and 1 + 3 decomposable spacetime in section V For adetailed discussion of conformally related 1 + 3 and 2 + 2reducible spacetime see [29 30] and for an extensive self-contained study of conformal symmetry of 4-dimensionalspacetime the reader is referred to [26]

We restrict our study of conformal vector fields ondoubly warped spacetime to dim(1198721) le 2 since this casegeneralizes some well-known exact solutions for the Einsteinfield equations and the beginning of this section representssuch study irrespective of the dimension of the factors Forthis case either dim(1198721) = 1 or dim(1198721) = 2 In thefollowing we deal with both subcases separately Let us firstconsider doubly warped spacetime with a 1-dimensionalbase

Let (119872 119892) be a Riemannian manifold and 119868 be anopen connected interval equipped with the metric minus1198891199052Doubly warped spacetime119872=119891 119868 times120590119872 is the product 119868 times119872furnished with the metric

119892 = minus11989121198891199052 oplus 1205902119892 (30)

where 119891 119872 rarr (0infin) and 120590 119868 rarr (0infin) are smoothfunctions 119872 is generalized RobertsonndashWalker spacetime if119891 is constant and standard static spacetime if 120590 is constant

An investigation of 4-dimensional spacetime that isconformally related to 1 + 3 reducible spacetime was carriedout with many examples in the aforementioned references[21 30] A classification of this spacetime according to itsconformal algebra is considered in the first reference whereasa special attention is paid to gradient conformal vector fieldsin the second reference

Theorem 10 A time-like vector field 120577 = ℎ120597119905 is a conformalvector field on doubly warped spacetime 119872=119891 119868 times120590119872 if andonly if ℎ = 119886120590 where 119886 is a nonnegative constant Moreoverthe conformal factor is 120588 = 2ℎProof If ℎ = 0 then 119886 = 0 and the result is obvious Now weassume that ℎ = 0 Using (16) we get that(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= minus21199091199101198912ℎ + 2ℎ120590119892 (119883 119884)= 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884)

(31)

Suppose that ℎ = 119886120590 and then

(L120577119892) (119883 119884) = 2ℎ119892 (119883 119884) (32)

that is 120577 = ℎ120597119905 is a conformal vector field with conformalfactor 120588 = 2ℎ Conversely suppose that 120577 = ℎ120597119905 isin X(119872) is aconformal vector field with factor 120588 and then

(L120577119892) (119883 119884) = 120588119892 (119883 119884) (33)

for any vector fields119883119884 isin X(119872) Now by (16) we get that120588119892 (119883 119884) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) + 2ℎ

120590 1205902119892 (119883 119884) (34)

Let119883 = 119884 = 0 and we get that

120588119892 (119909120597119905 119910120597119905) = 2ℎ1198912119892119868 (119909120597119905 119910120597119905) (35)

that is 120588 = 2ℎ Now let 119909 = 119910 = 0 and we get that 120588120590 = 2ℎThese two differential equations imply that ℎ = 119886120590 for

some positive constant 119886Theorem 11 A vector field 120577 = ℎ120597119905 + 120577 is a Killing vector fieldon doubly warped spacetime119872=119891 119868 times120590119872 if and only if one ofthe following conditions holds

(1) 120577 is time-like and ℎ = = 0(2) 120577 is space-like where 120577 is a Killing vector field on119872 and120577(119891) = 0(3) ℎ = minus120577(ln119891) and 120577 is a conformal vector field on 119872

with conformal factor 1205882 = minus2ℎ120590Proof The first assertion is a special case of the abovetheorem For the second assertion let ℎ = 0 in (16) Thus

(L120577119892) (119883 119884) = minus2119909119910119891120577 (119891) + 1205902 (L120577119892) (119883 119884) (36)

Suppose that 120577 is a Killing vector field on 119872 and 120577(119891) = 0and then

(L120577119892) (119883 119884) = 0 (37)

The converse is direct Finally let 120577 be aKilling vector fieldon119872=119891 119868 times120590119872 and then

0 = (L120577119892) (119883 119884)

0 = minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

(38)

Let119883 = 119884 = 0 and then

minus21199091199101198912 [ℎ + 120577 (ln119891)] = 0 (39)

and so ℎ = minus120577(ln119891) Thus

(L119868ℎ120597119905119892119868) (119909120597119905 119910120597119905) = minus2120577 (ln119891) 119892119868 (119909120597119905 119910120597119905) (40)

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

that is ℎ120597119905 is a conformal vector field on 119868 with conformalfactor minus2120577(ln119891) Now let 119909 = 119910 = 0 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (41)

that is 120577 is a conformal vector field on 119872 with conformalfactor 120588 = minus2ℎ120590

Conversely suppose that ℎ = minus120577(ln119891) and 120577 is a con-formal vector field on 119872 with conformal factor 120588 = minus2ℎ120590 and then

(L120577119892) (119883 119884) = minus2ℎ120590 119892 (119883 119884) (42)

Thus

(L120577119892) (119883 119884) = minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)= 0

(43)

that is 120577 = ℎ120597119905+120577 is a Killing vector field on119872=119891 119868 times120590119872

Corollary 12 Let 120577 = ℎ120597119905 +120577 be vector field on doubly warpedspacetime119872=119891 119868 times120590119872 obeying Einsteinrsquos field equationThen

(1) 119872 admits a time-like matter collineation 120577 = ℎ120597119905 if ℎ = = 0(2) 119872 admits a space-like matter collineation 120577 = 120577 if 120577 is

a Killing vector field on119872 and 120577(119891) = 0(3) 119872 admits a matter collineation 120577 = ℎ120597119905 + 120577 if ℎ =minus120577(ln119891) and 120577 is a conformal vector field on 119872 with

conformal factor 1205882 = minus2ℎ120590The following result is a direct consequence ofTheorem9

Corollary 13 Let 120577 = ℎ120597119905 + 120577 isin X(119872) be a conformal vectorfield on doubly warped spacetime 119872=119891 119868 times120590119872 along a curve120572 with unit tangent vector 119881 = V120597119905 + 119881 Then the conformalfactor 120588 of 120577 is given by

120588 = 2ℎ + 21205902119892 ([120577 119881] 119881) + 2 (ℎ120590 minus ℎ1205902) 119892 (119881 119881) (44)

In the sequel we present doubly warped spacetimewith a 2-dimensional base In this subcase 2 + 119899 doublywarped spacetime is considered to be doubly warped productmanifold with a 2-dimensional pseudo-Riemannian baseand 119899-dimensional Riemannian fibre 2 + 119899 doubly warpedspacetime is clearly conformal to a product manifold In[21 29] and references therein the Lie conformal algebra ofconformally related 2 + 2 reducible spacetime is extensivelystudied Many interesting results and examples are giventhere For example in [29] Carot and Tupper consideredan invariant characterization that imposes conditions on theconformal factor and on two null vectors Moreover Vanden Bergh considered non-conformally flat perfect fluids

spacetimewhich is conformally 2+2decomposable spacetimewith factor spaces of constant curvature [31]

It is well-known that each 2-dimensional manifold isconformally flat Thus we may simply take the base manifoldas (R2 1198891199042) where 1198891199042 = minus1198891199052 + 1198891199092 Let 119872=119891R2times120590119872be (2 + 119899)-dimensional doubly warped product spacetimefurnished with the metric 119892 = 11989121198891199042 + 1205902119892Proposition 14 Suppose that ℎ(119905)120597119905 + 119906(119909)120597119909 ℎ119894(119905)120597119905 +119906119894(119909)120597119909 isin X(R2) 119894 = 1 2 and 120577 1198831 1198832 isin X(119872) and then

(L120577119892) (119883 119884) = 21198912 (minusℎ1ℎ2ℎ + 119906111990621199061015840)

+ 1205902 (L120577119892) (119883 119884)+ 2120590 (ℎ120590119905 + 119906120590119909) 119892 (119883 119884)+ 2119891120577 (119891) (minusℎ1ℎ2 + 11990611199062)

(45)

where 120577 = ℎ120597119905 +119906120597119909 +120577 and119883119894 = ℎ119894120597119905 +119906119894120597119909 +119883119894 are elementsin X(119872)Corollary 15 A vector field 120577 = ℎ(119905)120597119905 + 119906(119909)120597119909 + 120577 isin X(119872)is a Killing vector field if one of the following conditions holds

(1) 120577 = 0 ℎ = 119886 119906 = 119887 and 119886120590119905 + 119887120590119909 = 0(2) 120577 is aKilling vector field on119872 ℎ = 119906 = 0 and 120577(119891) = 0

32 Concurrent Vector Fields on Doubly Warped SpacetimeIn this subsection we study concurrent vector fields ondoubly warped spacetime with a 1-dimensional base Onecan extend most of the results to doubly warped spacetimewith a 2-dimensional base Throughout this subsection let119872=119891 119868 times120590119872 be doubly warped spacetime equipped with themetric tensor 119892 = minus11989121198891199052 oplus 1205902119892Theorem 16 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a concurrent vector field if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Proof Suppose that 119883 = 119909120597119905 + 119883 isin X(119872) is any vector fieldon119872 and then

119863119883120577 = 119863119909120597119905ℎ120597119905 + 119863119909120597119905120577 + 119863119883ℎ120597119905 + 119863119883120577= 119909ℎ120597119905 + 119909ℎ119891

1205902 nabla119891 + 119909(ln120590 )120577 + 120577 (ln119891) 119909120597119905+ ℎ(ln120590 )119883 + 119883 (ln119891) ℎ120597119905 + 119863119883120577minus 1205901198912 119892 (119883 120577) 120597119905

= [119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)] 120597119905

+ 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(46)

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

Table 1

Case 120577 120590 119891ℎ = 0 = 0 120577 = (119909 + 119886)120597119909 Constant 119903(119909 + 119886)119896 = 0 1198911015840 = 0 120577 = (119905 + 119886)120597119905 119903(119905 + 119886) Constant1198911015840 = 0 = 0 120577 = (119905 + 119886)120597119905 + (119909 + 119886)120597119909 Constant Constant

Now suppose that ℎ120597119905 and 120577 are concurrent vector fields thenℎ = 1 and119863119883120577 = 119883 If both 120590 and 119891 are constant then

119863119883120577 = 119909ℎ120597119905 + 119863119883120577 = 119883 (47)

that is 120577 is concurrentIt is well-known that a homothetic vector field is a matter

collineation Thus the above theorem yields the followingresult

Corollary 17 A vector field 120577 = ℎ120597119905 + 120577 on doubly warpedspacetime119872=119891 119868 times120590119872 is a matter collineation if

(1) 120577 and ℎ120597119905 are concurrent vector fields on 119872 and 119868respectively

(2) both 119891 and 120590 are constant

Theorem 18 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector fieldon doubly warped spacetime 119872=119891 119868 times120590119872 equipped with themetric tensor 119892 = minus11989121198891199052 oplus1205902119892 Then 120577 is a concurrent vectorfield on119872 if one of the following conditions holds

(1) ℎ = 0 or(2) 120590 is a constant that is119872 is standard static spacetime

Moreover condition (1) implies condition (2) and theconverse is true if 119891 is not constant

Proof From the above proof we have

119909 = 119909ℎ + 120577 (ln119891) 119909 + 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = 119909ℎ1198911205902 nabla119891 + 119909(ln120590 )120577 + ℎ(ln120590 )119883 + 119863119883120577

(48)

for any 119909 and119883 Let 119909 = 0 and then

0 = 119883 (ln119891) ℎ minus 1205901198912 119892 (119883 120577)

119883 = ℎ(ln120590 )119883 + 119863119883120577119863119883120577 = [1 minus ℎ

120590 ]119883(49)

Thus 120577 is concurrent if ℎ = 0If ℎ = 0 then

1205901198912 119892 (119883 120577) = 0 (50)

If 119892(119883 120577) = 0 then 120577 = 0 which is a contradiction and so = 0 that is119872 is standard static spacetime

If = 0 then119883(119891) ℎ = 0 (51)

which implies that ℎ = 0 for a nonconstant function 119891Theorem 19 Let 120577 = ℎ120597119905 + 120577 be a concurrent vector field on119872 where ℎ = 0 Then 120577 and ℎ120597119905 are concurrent vector fields on119872 and 119868 respectively if = 0 In this case 119891 is also constant

Example 20 Table 1 summarizes all three cases of concurrentvector fields on the 2-dimensional doubly warped spacetimeof the form 119872=119891 119868 times120590R equipped with the metric 119892 =minus11989121198891199052 oplus 12059021198891199092 For more details see Appendix

4 Ricci Soliton on Doubly Warped Spacetime

A smooth vector field 120577 on a Riemannian manifold (119872 119892) issaid to define a Ricci soliton if

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (52)

where L120577 denotes the Lie derivative of the metric tensor 119892Ric is the Ricci curvature and 120582 is a constant [32ndash36]

Theorem 21 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(53)

Proof Let119872=119891 119868 times120590119872 be a Ricci soliton and then

12 (L120577119892) (119883 119884) + Ric (119883 119884) = 120582119892 (119883 119884) (54)

where 119883 = 119909120597119905 + 119883 and 119884 = 119910120597119905 + 119884 are vector fields on119872Then

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 (L120577119892) (119883 119884)

+ Ric (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

+ Ric (119909120597119905 119910120597119905) + Ric (119909120597119905 119884) + Ric (119883 119910120597119905)+ Ric (119883 119884)

minus 1205821199091199101198912 + 1205821205902119892 (119883 119884) = 12 [minus21199091199101198912 [ℎ + 120577 (ln119891)]

+ 1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)] + (119899 minus 1)sdot (119909120590 )119884 (ln119891) + (119899 minus 1) (119910120590 )119883 (ln119891) + 119899

120590119909119910

+ 1199091199101198911205902 + Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(55)

Let119883 = 119884 = 0 and we get

1199091199101198912 [ℎ + 120577 (ln119891)] minus 119899120590119909119910 minus 1199091199101198911205902 minus 1205821198912119909119910 = 0

119909119910[ℎ1198912 + 119891120577 (119891) minus 119899120590 minus

1198911205902 minus 1205821198912] = 0

(56)

and so

ℎ1198912 = 1205821198912 minus 119891120577 (119891) + 119899120590 +

1198911205902

ℎ = 11198912 (1205821198912 minus 119891120577 (119891) + 119899

120590 +1198911205902 )

(57)

Now let us put 119909 = 119910 = 0 and then

1205821205902119892 (119883 119884) = 12 [1205902 (L120577119892) (119883 119884) + 2ℎ120590119892 (119883 119884)]+ Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

minus 1205901198912 119892 (119883 119884)

(58)

and so

121205902 (L120577119892) (119883 119884) + Ric (119883 119884) minus 1

119891119867119891 (119883 119884)

= (1205821205902 minus ℎ120590 + 1205901198912 )119892 (119883 119884)

(59)

The following corollaries are consequences of the abovetheorem

Corollary 22 Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872)Then

(1) ℎ120597119905 is a conformal vector field on 119868 with factor(21198912)(1205821198912 minus 119891120577(119891) + (119899120590) + 1198911205902)

(2) (119872 119892 120577 120582) is a Ricci soliton if 119891 = 120590 = 1(3) (119872 119892 120577 120582) is a Ricci soliton if 120590 = 1 and119867119891 = 0

Theorem 23 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a conformal vector field on 119872 with factor 2120588 Then(119872 119892) is Einstein manifold with factor 120583 = (120582 minus 120588)1205902 + 1205901198912if 119891 is constant

Proof Let (119872 119892 120577 120582) be a Ricci soliton where 119872=119891 119868 times120590119872is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isin X(119872) is aconformal vector field on119872 Then

Ric (119883 119884) = (120582 minus 120588) 119892 (119883 119884) (60)

Let 119909 = 119910 = 0 and then

Ric (119883 119884) = (120582 minus 120588) 1205902119892 (119883 119884) (61)

This equation implies that

Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

= (120582 minus 120588) 1205902119892 (119883 119884)(62)

and so

Ric (119883 119884) minus 1119891119867119891 (119883 119884)

= [(120582 minus 120588) 1205902 + 1205901198912 ]119892 (119883 119884)

(63)

Corollary 24 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a homothetic vector field on119872 with factor 2119888 Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (64)

Proof Let (119872 119892 120577 120582) be a Ricci soliton and 120577 = ℎ120597119905 + 120577 isinX(119872) be a homothetic vector field on119872 and then

(120582 minus 119888) 119892 (119883 119884) = Ric (119883 119884) (65)

for any vector fields119883 = 119909120597119905 +119883 and 119884 = 119910120597119905 +119884 Let us take119883 = 119884 = 0 and then

Ric (119909120597119905 119910120597119905) = (120582 minus 119888) 119892 (119909120597119905 119910120597119905)1199091199101198911205902 + 119909119910119899120590 = minus119909119910 (120582 minus 119888) 1198912

119909119910(1198911205902 +119899120590 + (120582 minus 119888) 1198912) = 0

(66)

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

Then

120582 = 119888 minus 11198912 (

119899120590 +

1198911205902 ) (67)

and the proof is complete

Theorem 25 Let (119872 119892 120577 120582) be a Ricci soliton where119872=119891 119868 times120590119872 is doubly warped spacetime and 120577 = ℎ120597119905 + 120577 isinX(119872) is a concurrent vector field on119872 Then

(1) (119872 119892) is Einstein manifold with factor 120583 = (120582minus2)1205902 +1205901198912 if 119891 is constant(2) 120582 = 2 minus (11198912)((119899120590) + 1198911205902)Let 120577 = ℎ120597119905 + 120577 isin X(119872) and then

(L120577119892) (119883 119884)

= minus21199091199101198912 [ℎ + 120577 (ln119891)] + 1205902 (L120577119892) (119883 119884)+ 2ℎ120590119892 (119883 119884)

Ric (119883 119884)= 119909119910(119899120590 + 119891

1205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(68)

Thus

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus1199091199101198912 [ℎ + 120577 (ln119891)] + 121205902 (L120577119892) (119883 119884)

+ ℎ120590119892 (119883 119884) + 119909119910(119899120590 + 1198911205902 )

+ (119899 minus 1) (119909120590 119884 (ln119891) + 119910120590 119883 (ln119891))

+ Ric (119883 119884) minus 1119891119867119891 (119883 119884) minus

1205901198912 119892 (119883 119884)

(69)

Suppose that 119891 = 120590 = 1 and (119872 119892 120577 ℎ) is a Ricci soliton on119872 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + ℎ119892 (119883 119884) = ℎ119892 (119883 119884) (70)

Therefore (119872 119892 120577 120582) is a Ricci soliton where 120582 = ℎ Thisdiscussion leads us to the following result

Theorem 26 Let 119872=119891 119868 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892 120577 ℎ) is a Ricci soliton on119872(2) 119891 = 120590 = 1(3) 120582 = ℎLet 119891 = 1 120577 be a conformal vector field with factor 2120588

and119872 be Einstein with factor 120583 and then

12 (L120577119892) (119883 119884) + Ric (119883 119884)

= minus119909119910ℎ + 1205881205902119892 (119883 119884) + ℎ120590119892 (119883 119884) + 119909119910(119899120590 )+ (120583 minus 120590) 119892 (119883 119884)

= minus119909119910(ℎ minus 119899120590 )

+ (120583 minus 1205901205902 + 120588 + ℎ

120590) 1205902119892 (119883 119884)

(71)

that is (119872 119892 120577 120582) is a Ricci soliton if

ℎ minus 119899120590 = 120583 minus 120590

1205902 + 120588 + ℎ120590

(ℎ minus 120588) 1205902 = 120583 + (119899 minus 1) (120590 minus 2) + ℎ120590(72)

Theorem 27 Let 119872 = 119868119891 times120590119872 be doubly warped spacetimeand 120577 = ℎ120597119905 + 120577 isin X(119872) be a vector field on 119872 Then(119872 119892 120577 120582) is a Ricci soliton if

(1) (119872 119892) is Einstein with factor 120583(2) 119891 = 1 and 120577 is conformal with factor 2120588(3) (ℎ minus 120588)1205902 = 120583 + (119899 minus 1)(120590 minus 2) + ℎ120590

In this case 120582 = ℎ minus 119899120590

Appendix

Concurrent Vector Fields onDoubly Spacetime

Let us now consider an example Let 119872=119891 119868 times120590R be 2-dimension doubly warped spacetime equipped with themetric 119892 = minus11989121198891199052 oplus 12059021198891199092 Then

119863120597119905120597119905 = 11989111989110158401205902 120597119909

119863120597119909120597119905 = 1198911015840119891 120597119905 +

120590120597119909

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

119863120597119905120597119909 = 119863120597119909120597119905119863120597119909120597119909 = minus1205901198912 120597119905

(A1)

A vector field 120577 = ℎ120597119905 + 119896120597119909 isin X(119872) is a concurrent vectorfield if

119863120597119905120577 = 120597119905 (A2)

119863120597119909120577 = 120597119909 (A3)

The first equation implies that

119863120597119905 (ℎ120597119905 + 119896120597119909) = 120597119905ℎ120597119905 + ℎ1198911198911015840

1205902 120597119909 + 119896(1198911015840119891 120597119905 + 120590120597119909) = 120597119905

(A4)

and so

ℎ119891 + 1198961198911015840 = 119891 (A5)

ℎ1198911198911015840 + 119896120590 = 0 (A6)

Also (A3) implies that

119863120597119909 (ℎ120597119905 + 119896120597119909) = 120597119909ℎ(1198911015840119891 120597119905 +

120590120597119909) + 1198961015840120597119909 + 119896(minus1205901198912 120597119905) = 120597119909(A7)

and so

ℎ1198911198911015840 minus 119896120590 = 0 (A8)

ℎ + 1198961015840120590 = 120590 (A9)

By solving (A6) and (A8) we get ℎ1198911198911015840 = 0 Thus ℎ = 0 or1198911015840 = 0 In both cases 119896120590 = 0 that is 119896 = 0 or = 0 Thisdiscussion shows that we have the following cases using (A5)and (A9)

Case 1 ℎ = 0 and = 0 then 1198961198911015840 = 119891 and 1198961015840120590 = 120590 and so119896 = 119909 + 119886 = 0 and1198911015840119891 = 1

119909 + 119886 (A10)

Therefore 119891 = 119903(119909+119886) where both 119903 and (119909+119886) are positiveCase 2 1198911015840 = 0 and 119896 = 0 then ℎ119891 = 119891 and ℎ = 120590 and soℎ = 119905 + 119886 = 0 and similarly 120590 = 119903(119905 + 119886) where both 119903 and(119905 + 119886) are positiveCase 3 1198911015840 = 0 and = 0 then ℎ119891 = 119891 and 1198961015840120590 = 120590 and soℎ = 119905 + 119886 and 119896 = 119909 + 119887Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R L Bishop and B OrsquoNeill ldquoManifolds of negative curvaturerdquoTransactions of the American Mathematical Society vol 145 pp1ndash49 1969

[2] J K Beem and T G Powell ldquoGeodesic completeness andmaximality in Lorentzian warped productsrdquo Tensor vol 39 pp31ndash36 1982

[3] D E Allison ldquoGeodesic completeness in static space-timesrdquoGeometriae Dedicata vol 26 no 1 pp 85ndash97 1988

[4] D Allison ldquoPseudoconvexity in Lorentzian doubly warpedproductsrdquoGeometriae Dedicata vol 39 no 2 pp 223ndash227 1991

[5] A Gebarowski ldquoDoubly warped products with harmonic Weylconformal curvature tensorrdquo Colloquium Mathematicum vol67 pp 73ndash89 1993

[6] A Gebarowski ldquoOn conformally flat doubly warped productsrdquoSoochow Journal of Mathematics vol 21 no 1 pp 125ndash129 1995

[7] A Gebarowski ldquoOn conformally recurrent doubly warpedproductsrdquo Tensor vol 57 no 2 pp 192ndash196 1996

[8] B Unal ldquoDoubly warped productsrdquo Differential Geometry andIts Applications vol 15 no 3 pp 253ndash263 2001

[9] M Faghfouri and A Majidi ldquoOn doubly warped productimmersionsrdquo Journal of Geometry vol 106 no 2 pp 243ndash2542015

[10] A Olteanu ldquoA general inequality for doubly warped productsubmanifoldsrdquo Mathematical Journal of Okayama Universityvol 52 pp 133ndash142 2010

[11] A Olteanu ldquoDoubly warped products in S-space formsrdquo Roma-nian Journal of Mathematics and Computer Science vol 4 no 1pp 111ndash124 2014

[12] S Y Perktas and E Kilic ldquoBiharmonic maps between doublywarped product manifoldsrdquo Balkan Journal of Geometry and ItsApplications vol 15 no 2 pp 159ndash170 2010

[13] V N Berestovskii and Yu G Nikonorov ldquoKilling vectorfields of constant length on Riemannian manifoldsrdquo SiberianMathematical Journal vol 49 no 3 pp 395ndash407 2008

[14] S Deshmukh and F R Al-Solamy ldquoConformal vector fields ona Riemannian manifoldrdquo Balkan Journal of Geometry and ItsApplications vol 19 no 2 pp 86ndash93 2014

[15] S Deshmukh and F Al-Solamy ldquoA note on conformal vectorfields on a Riemannian manifoldrdquo Colloquium Mathematicumvol 136 no 1 pp 65ndash73 2014

[16] W Kuhnel and H-B Rademacher ldquoConformal vector fieldson pseudo-Riemannian spacesrdquo Differential Geometry and ItsApplications vol 7 no 3 pp 237ndash250 1997

[17] M Sanchez ldquoOn the geometry of generalized Robertson-Walker spacetimes curvature and Killing fieldsrdquo Journal ofGeometry and Physics vol 31 no 1 pp 1ndash15 1999

[18] M Steller ldquoConformal vector fields on spacetimesrdquo Annals ofGlobal Analysis and Geometry vol 29 no 4 pp 293ndash317 2006

[19] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press Limited London UK 1983

[20] Y Agaoka I-B Kim B H Kim and D J Yeom ldquoOn doublywarped productmanifoldsrdquoMemoirs of the Faculty of IntegratedArts and Sciences Hiroshima University Series IV vol 24 pp 1ndash10 1998

[21] M P Ramos E G Vaz and J Carot ldquoDouble warped space-timesrdquo Journal ofMathematical Physics vol 44 no 10 pp 4839ndash4865 2003

[22] B Unal Doubly warped products [PhD thesis] University ofMissouri Columbia Mo USA 2000

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

[23] B-Y Chen and S Deshmukh ldquoRicci solitons and concurrentvector fieldsrdquo Balkan Journal of Geometry and Its Applicationsvol 20 no 1 pp 14ndash25 2015

[24] F Dobarro and B Unal ldquoCharacterizing Killing vector fields ofstandard static space-timesrdquo Journal of Geometry and Physicsvol 62 no 5 pp 1070ndash1087 2012

[25] S Shenawy and B Unal ldquo2-Killing vector fields on warpedproduct manifoldsrdquo International Journal of Mathematics vol26 no 9 Article ID 1550065 17 pages 2015

[26] G S Hall Symmetries and Curvature Structure in GeneralRelativity World Scientific Singapore 2004

[27] J Carot J da Costa and E G Vaz ldquoMatter collineations theinverse lsquosymmetry inheritancersquo problemrdquo Journal of Mathemat-ical Physics vol 35 no 9 pp 4832ndash4838 1994

[28] P S Apostolopoulos and J G Carot ldquoConformal symmetries inwarped manifoldsrdquo Journal of Physics Conference Series vol 8pp 28ndash33 2005

[29] J Carot and B O J Tupper ldquoConformally reducible 2 + 2spacetimesrdquo Classical and Quantum Gravity vol 19 no 15 pp4141ndash4166 2002

[30] J Carot A J Keane and B O J Tupper ldquoConformally reducible1+3 spacetimesrdquo Classical and Quantum Gravity vol 25 no 5Article ID 055002 2008

[31] N Van den Bergh ldquoConformally 2 + 2-decomposable perfectfluid spacetimes with constant curvature factor manifoldsrdquoClassical and Quantum Gravity vol 29 no 23 Article ID235003 2012

[32] A Barros and E Ribeiro Jr ldquoSome characterizations forcompact almost Ricci solitonsrdquo Proceedings of the AmericanMathematical Society vol 140 no 3 pp 1033ndash1040 2012

[33] A Barros J N Gomes and E Ribeiro Jr ldquoA note on rigidity ofthe almost Ricci solitonrdquo Archiv der Mathematik vol 100 no 5pp 481ndash490 2013

[34] M Fernandez-Lopez and E Garcıa-Rıo ldquoRigidity of shrinkingRicci solitonsrdquo Mathematische Zeitschrift vol 269 no 1-2 pp461ndash466 2011

[35] O Munteanu and N Sesum ldquoOn gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 2 pp 539ndash561 2013

[36] P Petersen and W Wylie ldquoRigidity of gradient Ricci solitonsrdquoPacific Journal ofMathematics vol 241 no 2 pp 329ndash345 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of