research article a new class of almost ricci solitons and

Research ArticleA New Class of Almost Ricci Solitons and TheirPhysical Interpretation

K L Duggal

Department of Mathematics and Statistics University of Windsor Windsor ON Canada N9B 3P4

Correspondence should be addressed to K L Duggal yq8uwindsorca

Received 29 September 2016 Accepted 17 November 2016

Academic Editor Antonio Masiello

Copyright copy 2016 K L Duggal This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We establish a link between a connection symmetry called conformal collineation and almost Ricci soliton (in particular Riccisoliton) in reducible Ricci symmetric semi-Riemannian manifolds As a physical application by investigating the kinematic anddynamic properties of almost Ricci soliton manifolds we present a physical model of imperfect fluid spacetimes This model givesa general relation between the physical quantities (119906 120583 119901 120572 120578 120590119894119895) of the matter tensor of the field equations and does not provideany exact solution Therefore we propose further study on finding exact solutions of our viscous fluid physical model for which itis required that the fluid velocity vector 119906 be tilted We also suggest two open problems

1 Introduction

In general there are two categories of geometric quantitiesand their corresponding two kinds of flows intrinsic quan-tities such as the metric of a manifold and the extrinsicquantities such as the embedding of a manifold in someambient space The Ricci flow the Calabi flow and theYamabe flow belong to the first category and the meancurvature flow (MCF) belongs to the second category In1982 Hamilton [1] introduced the concept of Ricci flow forRiemannianmanifoldsThe self-similar solutions to the Ricciflow (evolved purely by homotheties and diffeomorphisms)are called Ricci solitons Since then his work has been used inresolving many longstanding open problems in Riemanniangeometry and 3-dimensional topology Basic details anda collection of research papers on this area of researchare available in [2 3] respectively There are very limitedpapers on Ricci solitons for semi-Riemannian (in particularLorentzian) manifolds (eg see [4 5]) On the other handthe study on MCF is primarily focused on the fixed pointsof a submanifold of minimal volume embedded in somefixed space Active research is going on all the above statedresearch areas on some aspects of geometry and physicsalthoughmost authors still prefer using Riemannian ambientspace Recently Pigola et al [6] have introduced a modifiedconcept of the Ricci solitons equation (called ldquoalmost Ricci

solitonsrdquo) by allowing the soliton constant 120582 to be a variablefunction Φ (see a brief account on this modification inSection 3) In this paper we present a mathematical modelof almost Ricci soliton (ARS) semi-Riemannian manifoldswhich also admit a connection symmetry called ldquoconformalcollineationsrdquo (Definition 1) and a physical model of almostRicci soliton imperfect fluid (in particular viscous fluid)spacetimes of general relativity

2 Conformal Collineations

The notion of symmetry is a very useful tool in geometryand physics A comprehensive account onmetric (ie Killinghomothetic and conformal Killing) and connection (ieaffine and conformal collineations) symmetries in semi-Riemannian (in particular spacetime) manifolds can befound in [7] and references therein In this paper we usethe following conformal collineation symmetry introduced byTashiro [8]

Definition 1 An (119899+1)-dimensional semi-Riemannianman-ifold (119872 119892) admits a conformal collineation symmetrydefined by a vector field 119881 if

m119881Γ119896119894119895 = 120575119896119894 Ψ119895 + 120575119896119895Ψ119894 minus 119892119894119895Ψ119896 (1)

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2016 Article ID 4903520 6 pageshttpdxdoiorg10115520164903520

2 International Scholarly Research Notices

where Γ119896119894119895 denotes Christoffelrsquos symbols Ψ is a function andΨ119895 = 120597119895(Ψ) 119881 is then called an ldquoAffine Conformal Vectorrdquo(ACV) field of119872

Proposition 2 ([9 Sharma-Duggal (1985)]) A vector field 119881on a semi-Riemannian manifold (119872 119892) is an ACV if and onlyif

m119881119892 = 2Ψ119892 + 119870 (2)

where 119870 is a (0 2) covariant constant (nabla119870 = 0) symmetrictensor field

An ACV reduces to a conformal Killing vector brieflydenoted by CKV if 119870 is proportional to 119892 Thus an ACVdeviates from a CKV field if there exists a second-ordercovariant constant symmetric tensor119870 = 119892 On the existenceof an ACV and specific restrictions on the ambient manifold119872 we recall that in 1923 Eisenhart [10] proved that ldquoIf aRiemannian manifold119872 admits such a tensor 119870 independentof 119892 then 119872 is reduciblerdquo This means that 119872 is locally aproduct manifold of the form (119872 = 1198721 times 1198722 119892 = 1198921 oplus 1198922)and there exists a local coordinate system in terms of whichthe distance element of 119892 is given by

1198891199042 = 119892119886119887 (119909119888) 119889119909119886119889119909119887 + 119892119860119861 (119909119862) 119889119909119860119889119909119861119886 119887 119888 = 1 119903 le 119899 119860 119861 119862 = 119903 + 1 119899 + 1

(3)

Thus an irreducible 119872 admits no ACV field In 1951Patterson [11] proved that ldquoIf a semi-Riemannian119872 admittingsuch a tensor 119870 independent of 119892 is reducible then the matrix(119870119894119895) has at least two distinct characteristic roots at any pointof 119872rdquo Therefore for a semi-Riemannian (119872 119892) a generalcharacterization of an ACV still remains open In 1925 Levy[12] proved that ldquoA second order covariant constant non-singular symmetric tensor in a space of constant curvature isproportional to the metric tensorrdquo Thus a semi-Riemannianmanifold of constant curvature admits no ACV other thana CKV Therefore the study on ACV symmetry is restrictedto only manifolds with nonconstant curvature Physically forexample Minkowski de-Sitter or anti-de-Sitter spacetimesdo not admit an ACV

In particular an ACV is called an affine vector brieflydenoted by AV if Ψ = 0 Tashiro [8] has also proved thatldquoa globally defined ACV is necessarily an AV fieldrdquo Thereforefor proper ACV our results in the paper will hold locallyFor basic details (with examples) on ACV see Tashiro [8]Patterson [11] Duggal [13] Mason-Maartens [14] and othersreferred therein

3 Almost Ricci SolitonSemi-Riemannian Manifolds

Let (119872 119892) be an (119899 + 1)-dimensional semi-Riemannianmanifold Recall that the ldquoRicci flowrdquo on 119872 is defined by thefollowing equation

120597119892119894119895120597119905 = minus2119877119894119895 (119894 119895 = 1 119899 + 1) (4)

where 119877119894119895 is the Ricci tensor of 119872 A solution 119892(119905) of theRicci flow equation (4) is called ldquoRicci solitonrdquo if there existsa positive function 120601(119905) and a 1-parameter family of diffeo-morphisms 120595(119905) 119872 rarr 119872 such that 119892(119905) = 120601(119905)120595(119905)lowast119892(0)where 1206011015840(119905) = minus2120582 for a constant 120582 Substituting this data inthe Ricci flow equation we obtain

119892119894119896nabla119895nabla119896 + 119892119896119895nabla119894nabla119896 = m119881119892119894119895 = 2120582119892119894119895 minus 2119877119894119895 (5)

where 119881 is a vector field on 119872 We say that the vector field119881 satisfying the evolution equation (5) is called a ldquoRiccisoliton vectorrdquo briefly denoted by RS vector and (119872 119892 120582 119881)is called a Ricci soliton (RS) manifold which is said tobe shrinking steady or expanding if 120582 is positive zeroor negative respectively The Ricci soliton manifolds arenatural extension of Einstein manifolds and are self-similar(homothetic) solutions to their Ricci flow equation In year2011 Pigola et al [6] introduced a modified class of the Riccisoliton equation (5) by replacing the soliton constant 120582with avariable functionΦ and then (119872 119892Φ 119881) is called an ldquoalmostRicci solitonrdquo manifold which we denote by ARS-manifoldand119881 the ldquoalmost Ricci soliton vectorrdquo briefly denoted by ARSvector such that the evolution equation (5) becomes

m119881119892119894119895 = 2Φ119892119894119895 minus 2119877119894119895 (6)

Similarly we say that the ARS is shrinking steady orexpanding if Φ is positive zero or negative respectivelyFor an example of an ARS-manifold we refer to [15] If 119881is the gradient of a smooth function 119891 up to the additionof a Killing vector field then we can replace 119881 by nabla119891 and(119872 119892Φ 119881) is called a gradient ARS-manifold for which theevolution equation (6) assumes the form

nabla119894nabla119895119891 + 119877119894119895 = Φ119892119894119895 (7)

Also for the ARS solution of Ricci Flow we consider

119892 (119905) = 120601 (119905 119909119886) 120595 (119905)lowast 119892 (0) (8)

where120595(119905) are diffeomorphisms of119872 generated by a family ofvector fields119883(119905) and 120601(119905 119909119886) are pointwise scaling functionsdepending on all the coordinates (119905 119909119886) of points with theinitial condition 119892119894119895(0) = 119892119894119895 120595(0) = 119868 rarr 120601(119905 119909119886) = 1Differentiating (8) with respect to 119905 using the Ricci flowequation (4) we get

( 120597120597119905120601 (119905 119909119886))

|119905=0

119892119894119895 + m119883(0)119892119894119895 = minus2119877119894119895 (9)

Labelling119883(0) = 119881 and ((120597120597119905)120601(119905 119909119886))|119905=0 = minus2Φwe get thealmost Ricci soliton equation (6)

For an ARS-manifold (119872 119892Φ 119881) with dimension ge 3and homothetic 119881 119892 is Einstein for which Φ = 120582 so ARS-manifold is Ricci soliton Also for an ARS-manifold thevector 119881 is conformal if and only if 119892 is Einstein So far wehave references [6 15ndash18] on ARS manifolds

International Scholarly Research Notices 3

4 Ricci Symmetric AlmostRicci Soliton Manifolds

Recall that a Riemannian or semi-Riemannian manifold(119872 119892) is called Ricci recurrent if its Ricci tensor satisfiesnabla119883Ric(119884 119885) = 120572(119883)Ric(119884 119885) where 120572(119883) is a 1-form on119872 In particular 119872 is Ricci symmetric if nablaRic = 0 Fordetails we refer to [19] Nowwe present amathematicalmodelof a class of Ricci symmetric semi-Riemannian manifolds(119872 119892) admitting an ACV vector field which also satisfiesthe evolution equation (6) of an ARS vector field For thispurpose recall that in 1970 Katzin et al [20] proved thata nonflat conformally flat manifold (119872 119892) admits an ACVwhose covariant constant tensor119870 is given by

119870 = ΩRic Ω = nonzero function on 119872 (10)

which also holds for a semi-Riemannian manifold (seePatterson [11]) This result restricts 119872 to Ricci recurrent(nabla119883Ric(119884 119885) = minus120597(logΩ)(119883)Ric(119884 119885)) spaces due to nabla119870 =0 Thus if we set Ω constant = minus2 (ie take 119872 Riccisymmetric) and then substitute this value of 119870 from (8) in(2) we obtain

m119881119892119894119895 = 2Φ119892119894119895 minus 2119877119894119895 (11)

which is exactly the ARS evolution equation (6)We also referto Levine-Katzin [21] who have proved that ldquoA second ordercovariant constant symmetric tensor 119870 in a conformally flatmanifold is a linear combination of the metric tensor and theRicci tensorrdquo To relate this with the ARS evolution equation(6) we let119870119894119895 = 119886119892119894119895 +119887119877119894119895 for some constants 119886 and 119887 Taking119886 = 0 and 119887 = minus2 we recover (6) Moreover Grycak [22] hasproved that ldquoLevine-Katzinrsquos result also holds for a conformallyRicci recurrent manifold with a locally exact recurrent formrdquoFor basic information on conformally recurrent manifoldswe refer to [23] Thus it is possible that a link of ACVsymmetry with ARS may also hold for a variety of Riccisymmetric semi-Riemannian manifolds other than what weknow from above references For this reason we state thefollowing general result

Theorem 3 Let (119872 119892) be an (119899 + 1)-dimensional locallyreducible Ricci symmetric semi-Riemannian manifold of non-constant curvature Suppose119872 admits an ACV field119881 definedby (2) such that its covariant constant tensor119870119894119895 = minus2119877119894119895Then119881 is also anARS vector fieldTherefore (119872 119892Φ 119881) is anARS-manifold

In general an ARS vector field 119881 admits followingcurvature identities (proof is similar to curvature identitieson CKV [24])

m119881119877119898119894119895119896 = 120575119898119894 Φ119896119895 + 120575119898119895 Φ119894119896 + Φ119898119894 119892119895119896 + Φ119898119895119892119894119896m119881119877119894119895 = div (gradΦ)119892119894119895 minus (119899 minus 1)Φ119894119895m119881119903 = 2119899 div (gradΦ) minus 2Φ119903 + 2119903

(12)

An ARS vector 119881 is RS vector if Φ = 120582 a constant for which(12) reduces to

m119881119877119898119894119895119896 = 0 = m119881119877119894119895m119881119903 = 2 (1 minus 120582) 119903

(13)

Corollary 4 Under the hypothesis ofTheorem 3 (a) if119881 is anaffine vector (AV) field then 119872 is a steady RS-manifold (b) Ifa Ricci soliton manifold (119872 119892 120582) is Einstein (119899 ge 2) (119877119894119895 =(119903(119899 + 1))119892119894119895 119903 = 0) then V is Killing 119903 = dim(119872) and(119872 119892 120582) is a steady RS-manifold

The case (a) holds since Φ = 0 if 119881 is an AV For case (b)using (13) and 119903 = 0 we get

m119881119877119894119895 = 119903119899 + 1m119881119892119894119895 = 0 997904rArr

m119881119892119894119895 = 0(14)

Therefore119881 is Killing 119903 = 119899+1 = dim(119872) and119872 is a steadyRS-manifold

Remark 5 Theorem 3 will certainly hold for above referredthree classes of locally reducible Ricci symmetric semi-Riemannian manifolds The reason for our choice of makinggeneral statement is to initiate research on finding more suchcases for whichTheorem 3 may hold

5 Physical Interpretation

In support of Theorem 3 we now show the existence ofphysically meaningful solutions of a class of fluid spacetimesof general relativity Let (119872 119892Φ 119881) be a 4-dimensionalspacetime manifold Results will also hold for higher dimen-sions The set of all integral curves given by a unit nonnullor null vector field 119906 is called the congruence of nonnull ornull curves Here we consider timelike curves also called flowlines The acceleration of the flow lines along 119906 is given bynabla119906119906 The projective tensor ℎ119894119895 = 119892119894119895 + 119906119894119906119895 is used to projecta tangent vector at a point 119901 in 119872 into a spacelike vectororthogonal to 119906 at 119901 The rate of change of the separation offlow lines from a timelike curve tangent to 119906 is given by theexpansion tensor 120579119894119895 = ℎ119896119894 ℎ119898119895 119906(119896119898)The expansion 120579 the sheartensor 120590119894119895 the vorticity tensor 120596119894119895 and the vorticity vector 120596119894are

120579 = div 119906 = 120579119894119895ℎ119894119895120590119894119895 = 120579119894119895 minus 120579

3ℎ119894119895120596119894119895 = ℎ119896119894 ℎ119898119895 119906[119896119898]120596119894 = 1

2120578119894119895119896119898119906119895120596119896119898

(15)


where 120578119894119895119896119898 is the Levi-Civita volume-form The covariantderivative of 119906 satisfies

119906119886119887 = 120596119886119887 + 120590119886119887 + 1205793ℎ119886119887 minus 119906119887 (119906119888119886119906119888) (16)

The rate at which the expansion 120579 changes along 119906 is given byRaychaudhuri equation as follows

119906120579 = 119889120579119889119904 = minus119877119894119895119906119894119906119895 + 21205962 minus 21205902 minus 1

31205792 + div (nabla119906119906) (17)

where 1205962 = (12)120596119894119895120596119894119895 and 1205902 = (12)120590119894119895120590119894119895 are nonnegativeand 119904 is an arc-length parameter

51 Kinematics of ARS-Spacetimes To find the kinematicproperties of ARS-spacetimes satisfying (6) with unit fluidvelocity vector 119906 we set m119881119906119894 = 120601119906119894 + 119908119894 for some functions120601 and 119908119894 where 119908119894119906119894 = 0 Contracting this with 119906119894 and usingm119881119906119894 = m119881(119892119894119895119906119894) m119881(119906119894119906119894) = 0 and the evolution equation(6) we obtain 120601 = 119906119894(2Φ119906119894 minus 2119877119894119895119906119895 + 119892119894119895m119881119906119895)Therefore

120601119906119894 = minus2 (Φ + 119877119894119895119906119894119906119895) minus m119881119906119894 (18)

Substituting this value of 120601119906119894 in m119881119906119894 = 120601119906119894 + 119908119894 we getm119881119906119894 = minus (Φ + 119877119895119896119906119895119906119896) 119906119894 + 119908119894

2 m119881119906119894 = (Φ minus 119877119895119896119906119895119906119896) 119906119894 minus 2119877119894119895119906119894 + 119908119894

2 (19)

Decompose119881119894 as119881119894 = 120572119906119894+120573119894 where120572 = minus119906119894119881119894 and120573119894119906119894 = 0Then using (16) (see details given in [25]) we obtain m119881119906119894 =119906119894 + 120572(119894 + (ln120572minus1)119895ℎ119895119894 ) + 120573119895119895119906119894 + 2120596119894119895120573119895 Using this alongwith the second equation of (19) and contracting with 119906119894 andℎ119894119896 = 119892119894119896 + 119906119894119906119896 we get

Φ = + 119894119881119894 minus 119877119894119895119906119894119906119895119908119894 = 2120596119894119895119881119894 + 120572 (119894 + (ln120572minus1)

119895ℎ119895119894 ) + 2119877119895119896119906119895ℎ119896119894

(20)

With respect to a nonnull ARS vector field 119881 we define theprojection tensor ℎ119894119895 = 119892119894119895 minus 1205981205722119881119894119881119895 where ℎ119894119895119881119894 = 0 and119881 sdot 119881 = 1205981205722 (120598 = plusmn1 120572 gt 0)Proposition 6 Let (119872 119892Φ 119881) be a locally reducible Riccisymmetric ARS-spacetime which satisfies the evolution equa-tion (6) with a nonnull ARS vector field 119881 where 119881 sdot 119881 =1205981205722 (120598 = plusmn1 120572 gt 0) Then the following kinematic relationshold

(1) 120590119894119895 = 120572minus1((13)ℎ119896119898ℎ119894119895 minus ℎ119896119894 ℎ119898119895 )119877119898119896(2) 120579 = 120572minus1(3Φ minus ℎ119894119895119877119894119895)

where 120590119894119895 and 120579 are the shear tensor and the expansion of 119881The proof follows by contracting (6) with ℎ119896119894 ℎ119898119895 minus

(13)ℎ119896119898ℎ119894119895 and ℎ119894119895 respectively Also it follows from Corol-lary 4(b) that under the hypothesis of above proposition anEinstein RS-spacetime (119872 119892) is expansion-free (120579 equiv 0) andshear-free (120590119894119895 equiv 0)

Remark 7 We highlight that Proposition 6 will also hold foran RS-spacetime for which (Φ = 120582) Therefore it followsthat the physical use of our class of ARS or RS-spacetimes(119872 119892) in general relativity has a role in the study of fluidswithnonzero expansion and shear unless119872 reduces to an EinsteinRS-spacetime which is expansion-free and shear-free Itis important to mention that our choice of timelike ARSsymmetry vector field 119881 in this paper is due to the fact thatthe integral curves of such a timelike connection symmetrycan provide a privileged class of observers or test particles ina spacetime However for a spacelike ARS symmetry vector119881 one can use the congruence of spacelike curves introducedby Greenberg [26] and discuss its kinematic properties andwork on its physical use similar to the results presented innext section

52 Dynamics of ARS-Spacetimes On the issue of any possi-ble existence of physically meaningful solutions of our classof fluid ARS-spacetimes we must know how the Einsteinfield equations are related with purely kinematic results ofprevious subsection For this purpose consider the Einsteinfield equations

119866119894119895 equiv 119877119894119895 minus 12119903 119892119894119895 = 119879119894119895 119903 = minus119879119894119894 (21)

where 119866119894119895 is the Einstein tensor Suppose (119872 119892) is an ARS-spacetime with an ARS vector field 119881 and satisfies thehypothesis of Theorem 3 It is known that the invariance(m119881119866 = 0) of 119866 is physically desirable since that amountsto invariance of matter tensor 119879 useful in finding exactsolutions For example we know that if a spacetime admitsa Killing or homothetic symmetry vector 119881 then 119881 leaves119866 (and therefore 119879) invariant but this may not be trueif spacetime admits some higher symmetry vector such asconformal Killing vector (CKV) However we do knowthat there are some physically important exact solutionsof spacetimes which admit CKV symmetry For exampleMaartens-Maharaj [27] have proved that Robertson-Walkerspacetimes admit a 9-parameter group of CKVs In thefollowing we show that our case of the ARS or RS symmetryvector 119881 is similar to the proper CKV symmetry Indeedassuming m119881119866 = 0 and using the curvature identities and theevolution equation (6) implies that 119881 leaves 119866 invariant onlyif there exists a spacetime which admits a covariant constantRicci tensor of the form

119877119894119895 = 119892119894119895 + (119899 minus 1)119903 [div (gradΦ)119892119894119895 + Φ119894119895] 119903 = 0 (22)

Thus in general the ARS or RS symmetry vector 119881 with119903 = 0 does not leave 119866 invariant We therefore propose thefollowing open question

Does a spacetime with covariant constant Riccitensor of the form (22) exist

In particular ifΦ = 120582 then we know fromCorollary 4(b)and (22) that 119872 is a steady Einstein manifold with Killing 119881and 119903 = dim(119872) = 119899 + 1

Nowwe look for physicallymeaningful solutions of a fluidARS-spacetime (119872 119892) We first recall the following concept


of material curves useful in the study of fluid spacetimes Amaterial curve in a fluid is a curve which moves with thefluid as the fluid evolves Material curves play an importantrole in relativistic fluid dynamics In our case we assume thatthe timelike ARS vector field 119881 = 120572119906 means that 119881 is amaterial curve as it maps flow lines into flow linesTherefore119908119894 = 0 in two equations of (19) This also means that 119906 is aneigenvector of 119877119894119895 which does not hold in general Now werecall from a paper of Hall-da Costa [28] that the existence ofa covariant constant second-order tensor field other than themetric tensor must exclude some spacetimes in particularperfect fluids Based on this result and our Remark 7 wechoose an imperfect fluid matter tensor in following physicalmodel

Theorem 8 Let (119872 119892Φ 119881) be a 4-dimensional locallyreducible Ricci symmetric ARS-spacetime of nonconstant cur-vature with the evolution equation (6) for a timelike ARS vector119881 parallel to the fluid velocity vector 119906 Assume 119872 admits thefollowing imperfect fluid Einstein field equations

119877119894119895 minus 12119903 119892119894119895 = 119879119894119895

= (120583 + 119901) 119906119894119906119895 + 119901119892 + 120587119894119895 + 119902119894119906119895 + 119902119895119906119894119902119894119906119894 = 0

120587119894119895119906119895 = 0

(23)

with 120583 119901 119902 120587119894119895 the density thermodynamic pressure energyvector and the anisotropic pressure tensor respectively Then

(i) Φ = minus (120583 + 3119901)2 119881 = 120572119906(ii) 120587119894119895 = minus120572120590119894119895 119902 = 0(iii) If Φ = 120582 and 119903 is a nonzero constant then 119872 is

shrinking (120582 = 1) RS-spacetime

Proof Using 119903 = minus119879119894119894 = 120583 minus 3119901 we get the following relationsfrom the field equations (23)

119877119894119895119906119894119906119895 = 120583 + 31199012 (24a)

119877119894119895ℎ119894119896ℎ119895119898 = (120583 minus 119901) ℎ119896119898 + 120587119898119896119877119894119895119906119894ℎ119895119896 = minus119901119902119896

(24b)

SinceARS vector119881 is timelike and parallel to 119906we let119881 = 120572119906This means that 119881 maps flow lines into flow lines so 119908119894 = 0Substituting 119908119894 = 0 in (19) and then using (24a) we get Φ = minus ((120583 + 3119901)2) so (i) holds Now using (24b) in item (1)of the Proposition 6 we get (ii) Finally if Φ = 120582 and 119903 isnonzero constant it follows from the second relation of (12)that 120582 = 1 so 119872 is a shrinking RS-spacetime so (iii) holdswhich completes the proof

Viscous Fluid A particular case of imperfect fluid is theviscous fluid with the viscosity coefficient 120578 It is known [29]that viscous fluid is characterized by the imperfect matter

tensor (22) for which 120587119894119895 = minus2120578120590119894119895 such that 120572 = 2120578 gt 0This establishes the relativistic equivalence to the classicalNavier-Stokes theory of fluid mechanics [30]Thus119881 = 2120578119906Consequently by adjusting the magnitude of the ARS vector119881 one can measure the response of the viscous fluid subjectto the amount of viscosity fluctuations of the fluid

6 Discussion

The motivation for this work comes from the use ofa connection symmetry (Definition 1) in the study ofsemi-Riemannian almost Ricci soliton (ARS) manifolds(119872 119892Φ 119881) recently introduced by Pigola et al [6]The ideaoriginated from close similarity of (2) of the affine conformalvector (ACV) field of connection symmetry with the ARSvector filed 119881 of the evolution equation (6) of the ARSmanifolds as stated in Theorem 3 We highlight that use ofsome basic results on ACV symmetry from references [7 1314] in Section 5 under the condition that119872 is reducible andRicci symmetric plays a key role in finding some physicallymeaningful solutions of imperfect Einstein field equations inparticular viscous fluids Also it is clear from Proposition 6that as the fluid revolves parallel to the velocity vector 119906the presence of shear causes distortion in the fluid and thischange in the fluid pattern is governed by the deviation ofthe Ricci tensor 119877119894119895 from the metric tensor 119892 A stage maycome when the ARS-spacetime 119872 is Einstein which it issheer-free and then it follows from (22) that 119877119894119895 = 119892119894119895 Onthe other hand for the case of ARS-spacetime (119877119894119895 = 119892119894119895)one can adjust the magnitude of the ARS vector filed 119881 inresponse of viscous fluid subject to large viscosity fluctuationsin order to describe the leading finite-size correction inscaling needed for a problem under investigation Howeverour physicalTheorem 8 only gives a general relation betweenthe quantities (119906 120583 119901 120572 120578 120590119894119895) and does not provide anyexact solution The reason is that the exact solutions of thefield equations for a viscous fluid require tilting velocityvector and it cannot be determined by the field equationsalone One needs to specify the type of physical model underinvestigation For example Coley and Tupper [29] have usedtilted velocity vector in presenting radiation-like viscous fluidexact solutions for a prescribed FRWmodel In another paperwe will follow the method used in [29] and study some moreworks on viscous fluids for completing our goal of findingphysically acceptable exact solutions based on the results ofthis paper Finally we propose following problem

Open Problem It has been shown in [31] that locally con-formally flat Lorentzian gradient Ricci solitons are locallyisomorphic to a Robertson-Walker warped product if thegradient of the potential function is nonnull and a planewave if the gradient of the potential function is null but forthis later case gradient Ricci solitons are necessarily steadyWe leave it as an open problem to further study in reference[31] satisfying the hypothesis of our Theorem 3

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper


References

[1] R Hamilton ldquoThree-manifolds with positive Ricci curvaturerdquoJournal of Differential Geometry vol 17 no 2 pp 155ndash306 1982

[2] H D Cao B Chow S C Chu and S T Yau Collected Paperson Ricci Flow vol 37 of Series in Geometry and TopologyInternational Press Somerville Mass USA 2003

[3] B Chow and D Knopf The Ricci Flow An Introduction AMSProvidence RI USA 2004

[4] M Crasmareanu ldquoLiouville and geodesic Ricci solitonsrdquoComptes Rendus Mathematique vol 1 no 21-22 pp 1305ndash13082009

[5] K Onda ldquoLorentz Ricci solitons on 3-dimensional Lie groupsrdquoGeometriae Dedicata vol 147 pp 313ndash322 2010

[6] S Pigola M Rigoli M Rimoldi and A G Setti ldquoRicci almostsolitonsrdquo Annali della Scuola Normale Superiore di Pisa Classedi Scienze Serie V vol 10 no 4 pp 757ndash799 2011

[7] K L Duggal and R Sharma Symmetries of Spacetimes andRiemannian Manifolds vol 487 Kluwer Academic 1999

[8] Y Tashiro ldquoOn conformal collineationsrdquoMathematical Journalof Okayama University vol 10 pp 75ndash85 1960

[9] R Sharma and K L Duggal ldquoA characterization of affineconformal vector fieldrdquo Comptes Rendus Mathematiques deslrsquoAcademie des Sciences La Societe Royale du Canada vol 7 pp201ndash205 1985

[10] L P Eisenhart ldquoSymmetric tensors of the second order whosefirst covariant derivatives are zerordquoTransactions of theAmericanMathematical Society vol 25 no 2 pp 297ndash306 1923

[11] E M Patterson ldquoOn symmetric recurrent tensors of the secondorderrdquoTheQuarterly Journal of Mathematics vol 2 pp 151ndash1581951

[12] H Levy ldquoSymmetric tensors of the second order whose covari-ant derivatives vanishrdquo The Annals of Mathematics vol 27 no2 pp 91ndash98 1925

[13] K L Duggal ldquoAffine conformal vector fields in semi-Riemannian manifoldsrdquo Acta Applicandae Mathematicae vol23 no 3 pp 275ndash294 1991

[14] D P Mason and R Maartens ldquoKinematics and dynamics ofconformal collineations in relativityrdquo Journal of MathematicalPhysics vol 28 no 9 pp 2182ndash2186 1987

[15] A Barros R Batista and E Ribeiro Jr ldquoCompact almost soli-tons with constant scalar curvature are gradientrdquo Monatsheftefur Mathematik vol 174 no 1 pp 29ndash39 2014

[16] A Barros and J Ribeiro ldquoSome characterizations for compactalmost Ricci solitonsrdquo Proceedings of the American Mathemati-cal Society vol 140 no 3 pp 1033ndash1040 2012

[17] R Sharma ldquoAlmost Ricci solitons and K-contact geometryrdquoMonatshefte fur Mathematik vol 175 no 4 pp 621ndash628 2014

[18] Y Wang ldquoGradient Ricci almost solitons on two classes ofalmost Kenmotsu manifoldsrdquo Journal of the Korean Mathemat-ical Society vol 53 no 5 pp 1101ndash1114 2016

[19] E M Patterson ldquoSome theorems on Ricci-recurrent spacesrdquoJournal of the London Mathematical Society Second Series vol27 pp 287ndash295 1952

[20] G H Katzin J Levine and W R Davis ldquoCurvature collin-eations in conformally at spaces Irdquo Tensor NS vol 21 pp 51ndash61 1970

[21] J Levine and G H Katzin ldquoConformally at spaces admittingspecial quadratic first integrals 1 Symmetric spacesrdquo Tensor NS vol 19 pp 317ndash328 1968

[22] W Grycak ldquoOn affine collineations in conformally recurrentmanifoldsrdquo The Tensor Society Tensor New Series vol 35 no1 pp 45ndash50 1981

[23] T Adati and T Miyazawa ldquoOn Riemannian space with recur-rent conformal curvaturerdquo Tensor vol 18 pp 348ndash354 1967

[24] K Yano Integral Formulas in Riemannian Geometry MarcelDekker New York NY USA 1970

[25] R Maartens D P Mason and M Tsamparlis ldquoKinematic anddynamic properties of conformal Killing vectors in anisotropicfluidsrdquo Journal of Mathematical Physics vol 27 no 12 pp 2987ndash2994 1986

[26] P J Greenberg ldquoThe general theory of space-like congruenceswith an application to vorticity in relativistic hydrodynamicsrdquoJournal of Mathematical Analysis and Applications vol 30 pp128ndash143 1970

[27] R Maartens and S D Maharaj ldquoConformal Killing vectors inRobertson-Walker spacetimesrdquoClassical and QuantumGravityvol 3 no 5 pp 1005ndash1011 1986

[28] G S Hall and J da Costa ldquoAffine collineations in space-timerdquoJournal of Mathematical Physics vol 29 no 11 pp 2465ndash24721988

[29] A A Coley and B O J Tupper ldquoRadiation-like imperfect fluidcosmologiesrdquo Astrophysical Journal vol 288 no 2 part 1 pp418ndash421 1985

[30] L Landau andEM Lifshitz FluidMechanics Addison-WelselyReading Mass USA 1958

[31] M Brozos-Vazquez E Garcıa-Rıo and S Gavino-FernandezldquoLocally conformally flat Lorentzian gradient Ricci solitonsrdquoJournal of Geometric Analysis vol 23 no 3 pp 1196ndash1212 2013

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where Γ119896119894119895 denotes Christoffelrsquos symbols Ψ is a function andΨ119895 = 120597119895(Ψ) 119881 is then called an ldquoAffine Conformal Vectorrdquo(ACV) field of119872

Proposition 2 ([9 Sharma-Duggal (1985)]) A vector field 119881on a semi-Riemannian manifold (119872 119892) is an ACV if and onlyif

m119881119892 = 2Ψ119892 + 119870 (2)

where 119870 is a (0 2) covariant constant (nabla119870 = 0) symmetrictensor field

An ACV reduces to a conformal Killing vector brieflydenoted by CKV if 119870 is proportional to 119892 Thus an ACVdeviates from a CKV field if there exists a second-ordercovariant constant symmetric tensor119870 = 119892 On the existenceof an ACV and specific restrictions on the ambient manifold119872 we recall that in 1923 Eisenhart [10] proved that ldquoIf aRiemannian manifold119872 admits such a tensor 119870 independentof 119892 then 119872 is reduciblerdquo This means that 119872 is locally aproduct manifold of the form (119872 = 1198721 times 1198722 119892 = 1198921 oplus 1198922)and there exists a local coordinate system in terms of whichthe distance element of 119892 is given by

1198891199042 = 119892119886119887 (119909119888) 119889119909119886119889119909119887 + 119892119860119861 (119909119862) 119889119909119860119889119909119861119886 119887 119888 = 1 119903 le 119899 119860 119861 119862 = 119903 + 1 119899 + 1

(3)

Thus an irreducible 119872 admits no ACV field In 1951Patterson [11] proved that ldquoIf a semi-Riemannian119872 admittingsuch a tensor 119870 independent of 119892 is reducible then the matrix(119870119894119895) has at least two distinct characteristic roots at any pointof 119872rdquo Therefore for a semi-Riemannian (119872 119892) a generalcharacterization of an ACV still remains open In 1925 Levy[12] proved that ldquoA second order covariant constant non-singular symmetric tensor in a space of constant curvature isproportional to the metric tensorrdquo Thus a semi-Riemannianmanifold of constant curvature admits no ACV other thana CKV Therefore the study on ACV symmetry is restrictedto only manifolds with nonconstant curvature Physically forexample Minkowski de-Sitter or anti-de-Sitter spacetimesdo not admit an ACV

In particular an ACV is called an affine vector brieflydenoted by AV if Ψ = 0 Tashiro [8] has also proved thatldquoa globally defined ACV is necessarily an AV fieldrdquo Thereforefor proper ACV our results in the paper will hold locallyFor basic details (with examples) on ACV see Tashiro [8]Patterson [11] Duggal [13] Mason-Maartens [14] and othersreferred therein

3 Almost Ricci SolitonSemi-Riemannian Manifolds

Let (119872 119892) be an (119899 + 1)-dimensional semi-Riemannianmanifold Recall that the ldquoRicci flowrdquo on 119872 is defined by thefollowing equation

120597119892119894119895120597119905 = minus2119877119894119895 (119894 119895 = 1 119899 + 1) (4)

where 119877119894119895 is the Ricci tensor of 119872 A solution 119892(119905) of theRicci flow equation (4) is called ldquoRicci solitonrdquo if there existsa positive function 120601(119905) and a 1-parameter family of diffeo-morphisms 120595(119905) 119872 rarr 119872 such that 119892(119905) = 120601(119905)120595(119905)lowast119892(0)where 1206011015840(119905) = minus2120582 for a constant 120582 Substituting this data inthe Ricci flow equation we obtain

119892119894119896nabla119895nabla119896 + 119892119896119895nabla119894nabla119896 = m119881119892119894119895 = 2120582119892119894119895 minus 2119877119894119895 (5)

where 119881 is a vector field on 119872 We say that the vector field119881 satisfying the evolution equation (5) is called a ldquoRiccisoliton vectorrdquo briefly denoted by RS vector and (119872 119892 120582 119881)is called a Ricci soliton (RS) manifold which is said tobe shrinking steady or expanding if 120582 is positive zeroor negative respectively The Ricci soliton manifolds arenatural extension of Einstein manifolds and are self-similar(homothetic) solutions to their Ricci flow equation In year2011 Pigola et al [6] introduced a modified class of the Riccisoliton equation (5) by replacing the soliton constant 120582with avariable functionΦ and then (119872 119892Φ 119881) is called an ldquoalmostRicci solitonrdquo manifold which we denote by ARS-manifoldand119881 the ldquoalmost Ricci soliton vectorrdquo briefly denoted by ARSvector such that the evolution equation (5) becomes

m119881119892119894119895 = 2Φ119892119894119895 minus 2119877119894119895 (6)

Similarly we say that the ARS is shrinking steady orexpanding if Φ is positive zero or negative respectivelyFor an example of an ARS-manifold we refer to [15] If 119881is the gradient of a smooth function 119891 up to the additionof a Killing vector field then we can replace 119881 by nabla119891 and(119872 119892Φ 119881) is called a gradient ARS-manifold for which theevolution equation (6) assumes the form

nabla119894nabla119895119891 + 119877119894119895 = Φ119892119894119895 (7)

Also for the ARS solution of Ricci Flow we consider

119892 (119905) = 120601 (119905 119909119886) 120595 (119905)lowast 119892 (0) (8)

where120595(119905) are diffeomorphisms of119872 generated by a family ofvector fields119883(119905) and 120601(119905 119909119886) are pointwise scaling functionsdepending on all the coordinates (119905 119909119886) of points with theinitial condition 119892119894119895(0) = 119892119894119895 120595(0) = 119868 rarr 120601(119905 119909119886) = 1Differentiating (8) with respect to 119905 using the Ricci flowequation (4) we get

( 120597120597119905120601 (119905 119909119886))

|119905=0

119892119894119895 + m119883(0)119892119894119895 = minus2119877119894119895 (9)

Labelling119883(0) = 119881 and ((120597120597119905)120601(119905 119909119886))|119905=0 = minus2Φwe get thealmost Ricci soliton equation (6)

For an ARS-manifold (119872 119892Φ 119881) with dimension ge 3and homothetic 119881 119892 is Einstein for which Φ = 120582 so ARS-manifold is Ricci soliton Also for an ARS-manifold thevector 119881 is conformal if and only if 119892 is Einstein So far wehave references [6 15ndash18] on ARS manifolds






m119881119892119894119895 = 2Φ119892119894119895 minus 2119877119894119895 (11)





(12)


m119881119877119898119894119895119896 = 0 = m119881119877119894119895m119881119903 = 2 (1 minus 120582) 119903

(13)



m119881119877119894119895 = 119903119899 + 1m119881119892119894119895 = 0 997904rArr

m119881119892119894119895 = 0(14)





120579 = div 119906 = 120579119894119895ℎ119894119895120590119894119895 = 120579119894119895 minus 120579

3ℎ119894119895120596119894119895 = ℎ119896119894 ℎ119898119895 119906[119896119898]120596119894 = 1

2120578119894119895119896119898119906119895120596119896119898

(15)



119906119886119887 = 120596119886119887 + 120590119886119887 + 1205793ℎ119886119887 minus 119906119887 (119906119888119886119906119888) (16)



31205792 + div (nabla119906119906) (17)



120601119906119894 = minus2 (Φ + 119877119894119895119906119894119906119895) minus m119881119906119894 (18)


2 m119881119906119894 = (Φ minus 119877119895119896119906119895119906119896) 119906119894 minus 2119877119894119895119906119894 + 119908119894

2 (19)


Φ = + 119894119881119894 minus 119877119894119895119906119894119906119895119908119894 = 2120596119894119895119881119894 + 120572 (119894 + (ln120572minus1)

119895ℎ119895119894 ) + 2119877119895119896119906119895ℎ119896119894

(20)









119877119894119895 = 119892119894119895 + (119899 minus 1)119903 [div (gradΦ)119892119894119895 + Φ119894119895] 119903 = 0 (22)








119877119894119895 minus 12119903 119892119894119895 = 119879119894119895

= (120583 + 119901) 119906119894119906119895 + 119901119892 + 120587119894119895 + 119902119894119906119895 + 119902119895119906119894119902119894119906119894 = 0

120587119894119895119906119895 = 0

(23)





119877119894119895119906119894119906119895 = 120583 + 31199012 (24a)

119877119894119895ℎ119894119896ℎ119895119898 = (120583 minus 119901) ℎ119896119898 + 120587119898119896119877119894119895119906119894ℎ119895119896 = minus119901119902119896

(24b)




6 Discussion



Competing Interests



References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














m119881119892119894119895 = 2Φ119892119894119895 minus 2119877119894119895 (11)





(12)


m119881119877119898119894119895119896 = 0 = m119881119877119894119895m119881119903 = 2 (1 minus 120582) 119903

(13)



m119881119877119894119895 = 119903119899 + 1m119881119892119894119895 = 0 997904rArr

m119881119892119894119895 = 0(14)





120579 = div 119906 = 120579119894119895ℎ119894119895120590119894119895 = 120579119894119895 minus 120579

3ℎ119894119895120596119894119895 = ℎ119896119894 ℎ119898119895 119906[119896119898]120596119894 = 1

2120578119894119895119896119898119906119895120596119896119898

(15)



119906119886119887 = 120596119886119887 + 120590119886119887 + 1205793ℎ119886119887 minus 119906119887 (119906119888119886119906119888) (16)



31205792 + div (nabla119906119906) (17)



120601119906119894 = minus2 (Φ + 119877119894119895119906119894119906119895) minus m119881119906119894 (18)


2 m119881119906119894 = (Φ minus 119877119895119896119906119895119906119896) 119906119894 minus 2119877119894119895119906119894 + 119908119894

2 (19)


Φ = + 119894119881119894 minus 119877119894119895119906119894119906119895119908119894 = 2120596119894119895119881119894 + 120572 (119894 + (ln120572minus1)

119895ℎ119895119894 ) + 2119877119895119896119906119895ℎ119896119894

(20)









119877119894119895 = 119892119894119895 + (119899 minus 1)119903 [div (gradΦ)119892119894119895 + Φ119894119895] 119903 = 0 (22)








119877119894119895 minus 12119903 119892119894119895 = 119879119894119895

= (120583 + 119901) 119906119894119906119895 + 119901119892 + 120587119894119895 + 119902119894119906119895 + 119902119895119906119894119902119894119906119894 = 0

120587119894119895119906119895 = 0

(23)





119877119894119895119906119894119906119895 = 120583 + 31199012 (24a)

119877119894119895ℎ119894119896ℎ119895119898 = (120583 minus 119901) ℎ119896119898 + 120587119898119896119877119894119895119906119894ℎ119895119896 = minus119901119902119896

(24b)




6 Discussion



Competing Interests



References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906119886119887 = 120596119886119887 + 120590119886119887 + 1205793ℎ119886119887 minus 119906119887 (119906119888119886119906119888) (16)



31205792 + div (nabla119906119906) (17)



120601119906119894 = minus2 (Φ + 119877119894119895119906119894119906119895) minus m119881119906119894 (18)


2 m119881119906119894 = (Φ minus 119877119895119896119906119895119906119896) 119906119894 minus 2119877119894119895119906119894 + 119908119894

2 (19)


Φ = + 119894119881119894 minus 119877119894119895119906119894119906119895119908119894 = 2120596119894119895119881119894 + 120572 (119894 + (ln120572minus1)

119895ℎ119895119894 ) + 2119877119895119896119906119895ℎ119896119894

(20)









119877119894119895 = 119892119894119895 + (119899 minus 1)119903 [div (gradΦ)119892119894119895 + Φ119894119895] 119903 = 0 (22)








119877119894119895 minus 12119903 119892119894119895 = 119879119894119895

= (120583 + 119901) 119906119894119906119895 + 119901119892 + 120587119894119895 + 119902119894119906119895 + 119902119895119906119894119902119894119906119894 = 0

120587119894119895119906119895 = 0

(23)





119877119894119895119906119894119906119895 = 120583 + 31199012 (24a)

119877119894119895ℎ119894119896ℎ119895119898 = (120583 minus 119901) ℎ119896119898 + 120587119898119896119877119894119895119906119894ℎ119895119896 = minus119901119902119896

(24b)




6 Discussion



Competing Interests



References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119877119894119895 minus 12119903 119892119894119895 = 119879119894119895

= (120583 + 119901) 119906119894119906119895 + 119901119892 + 120587119894119895 + 119902119894119906119895 + 119902119895119906119894119902119894119906119894 = 0

120587119894119895119906119895 = 0

(23)





119877119894119895119906119894119906119895 = 120583 + 31199012 (24a)

119877119894119895ℎ119894119896ℎ119895119898 = (120583 minus 119901) ℎ119896119898 + 120587119898119896119877119894119895119906119894ℎ119895119896 = minus119901119902119896

(24b)




6 Discussion



Competing Interests



References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new class of almost ricci solitons and

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