aa r X i v : . [ m a t h . DG ] M a r HOMOGENEOUS COTTON SOLITONS
E. CALVI ˜NO-LOUZAO, L. M. HERVELLA, J. SEOANE-BASCOY, R.V ´AZQUEZ-LORENZO
Abstract.
Left-invariant Cotton solitons on homogeneous manifolds are de-termined. Moreover, algebraic Cotton solitons are studied providing examplesof non-invariant Cotton solitons, both in the Riemannian and Lorentzian ho-mogeneous settings. Introduction
The objective of the different geometric evolution equations is to improve a givenmetric by considering a flow associated to the geometric object under consideration.The Ricci, Yamabe and mean curvature flows are examples extensively studied inthe literature. Under suitable conditions the Ricci flow evolves an initial metric toan Einstein metric while the Yamabe flow evolves an initial metric to a new onewith constant scalar curvature within the same conformal class. Cotton, Yamabeand Ricci flows have many physical applications (we refer to [14, 15, 17] and thereferences therein for more information). There are however certain metrics which,instead of evolving by the flow, remain invariant. Such is the case of those solitonsassociated to self-similar solutions of the flow.In the study of conformal geometry in dimensions greater than three the Weyltensor plays a distinguished role, since its nullity characterizes local conformal flat-ness. The three-dimensional case must be studied in a different way due to the factthat the Weyl tensor vanishes identically. Moreover, the whole curvature is com-pletely determined by the Ricci tensor, ρ . Local conformal flatness is characterizedin dimension three by the fact that the Schouten tensor, defined by S ij = ρ ij − τ g ij where τ denotes the scalar curvature, is a Codazzi tensor, or equivalently the Cot-ton tensor C ijk = ( ∇ i S ) jk − ( ∇ j S ) ik , which is the unique conformal invariant indimension three, vanishes. We refer to [3, 16, 25] and the references therein formore information on the usefulness of the Cotton tensor in describing the geometryof three-dimensional manifolds.The Cotton tensor appears naturally in many physical contexts [11, 12], speciallyin Chern-Simons theory [13] or topologically massive gravity [1, 9, 17]. In particular,field equation in topologically massive gravity implies a proportionality betweenthe Einstein and Cotton tensors. The fact that the Einstein tensor consists ofsecond order derivatives on the metric whereas the Cotton tensor is of order threeimplies that an exact solution of this field equation is difficult to find in general.Indeed, most of the solutions for the field equation in topological massive gravityare constructed on homogeneous spaces. Mathematics Subject Classification.
Key words and phrases.
Homogeneous spaces, Lie groups, Cotton solitons.Supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain).
In [2] a new geometric flow based on the conformally invariant Cotton tensorwas introduced. A Cotton flow is a one-parameter family g ( t ) of three-dimensionalmetrics satisfying(1) ∂∂t g ( t ) = − λ C g ( t ) , where C g ( t ) is the (0 , g ( t ), obtainedfrom the (0 , ⋆ -Hodge operator, and given by C ij = 12 √ g C nmi ǫ nmℓ g ℓj , where ǫ ijk denotes the Levi-Civita permutation symbol ( ǫ = 1).When comparing with the Yamabe flow, the Cotton flow has in some sense anopposite behaviour since the Cotton flow changes the conformal class except in theconformally flat case. The genuine fixed points of the Cotton flow are the locallyconformally flat metrics. However there exist other geometric fixed points, whichcorrespond to Cotton solitons. A pseudo-Riemannian manifold ( M, g ) is a
Cottonsoliton if it admits a vector field X such that(2) L X g + C = λg, where L X denotes the Lie derivative in the direction of the vector field X and λ is a real number. A Cotton soliton is said to be shrinking, steady or expanding ,respectively, if λ > λ = 0 or λ <
0. Since there is no ambiguity, we call Cottonsoliton both to the pseudo-Riemannian manifold (
M, g ) and to the vector field X .Cotton solitons are closely related to Ricci and Yamabe solitons, which are de-fined by L X g + ρ = λg and L X g = ( τ − λ ) g , respectively. In particular, if ( M, g ) isa locally conformally flat homogeneous manifold then the class of Cotton solitonscoincides with the class of Yamabe solitons (see for example [8]). In such a casethe Cotton soliton is said to be trivial . On the other hand, if (
M, g ) is a Lorentzianmanifold which satisfies the field equation of a Topologically Massive Gravity spacethen (
M, g ) is a Ricci soliton if and only if it is a Cotton soliton [17].In Riemannian signature, any compact Cotton soliton is locally conformally flat,while compact Lorentzian examples are available in the non locally conformallyflat setting [7]. Moreover, the fact that any left-invariant homogeneous Ricci orYamabe soliton on a three-dimensional Riemannian Lie group is flat, while non-flat examples exist in the Lorentzian signature (see [4, 8]), motivates a study ofhomogeneous Cotton solitons in the Lorentzian setting.Complete and simply connected three-dimensional Lorentzian homogeneous man-ifolds are either symmetric or a Lie group with a left-invariant Lorentzian metric[6]. Since three-dimensional locally symmetric Lorentzian manifolds are locallyconformally flat, the purpose of this paper is twofold. Firstly, to classify invari-ant homogeneous Cotton solitons on Lie groups. Secondly, to determine algebraicCotton solitons on Lie groups and use them in order to obtain new examples ofnon-invariant Cotton solitons on homogeneous manifolds, both in the Riemannianand Lorentzian settings.
Conventions and structure of the paper.
Throughout this paper, (
M, g ) de-notes a three-dimensional pseudo-Riemannian manifold and (
G, g ) denotes a three-dimensional Lie group equipped with a left-invariant metric; as usual, R stands forthe curvature tensor taken with the sign convention R ( X, Y ) = ∇ [ X,Y ] − [ ∇ X , ∇ Y ], OMOGENEOUS COTTON SOLITONS 3 where ∇ denotes the Levi-Civita connection. The Ricci tensor, ρ , and the cor-responding Ricci operator, b ρ , are given by ρ ( X, Y ) = g ( b ρ ( X ) , Y ) = trace { Z R ( X, Z ) Y } , and we denote by τ the scalar curvature. Finally, ( G, g ) is alwaysassumed to be connected and simply connected.We organize this paper as follows. In Section 2 we review the description of allthree-dimensional Lorentzian Lie algebras. We analyze the existence of non-trivialleft-invariant Cotton solitons on three-dimensional Lorentzian Lie groups in Section3. In Section 4 we determine algebraic Cotton solitons on homogeneous manifolds,showing the existence of Cotton solitons which are non-invariant. Finally, in Sec-tion 5 examples of non-invariant shrinking and expanding Lorentzian homogeneousCotton solitons which are non-trivial are constructed on the Heisenberg group andon the E (1 ,
1) group. 2.
Preliminaries
Let × denote the Lorentzian vector product on R induced by the product of thepara-quaternions (i.e., e × e = − e , e × e = e , e × e = e , where { e , e , e } is an orthonormal basis of signature (+ + − )). Then [ Z, Y ] = L ( Z × Y ) defines aLie algebra, which is unimodular if and only if L is a self-adjoint endomorphism of g [23]. Considering the different Jordan normal forms of L , we have the followingfour classes of unimodular three-dimensional Lorentzian Lie algebras. Type Ia . If L is diagonalizable with eigenvalues { α, β, γ } with respect to an or-thonormal basis { e , e , e } of signature (++ − ), then the corresponding Lie algebrais given by( g Ia ) [ e , e ] = − γe , [ e , e ] = − βe , [ e , e ] = αe . Type Ib . Assume L has a complex eigenvalue. Then, with respect to an orthonormalbasis { e , e , e } of signature (+ + − ), one has L = α γ − β β γ , β = 0and thus the corresponding Lie algebra is given by( g Ib ) [ e , e ] = βe − γe , [ e , e ] = − γe − βe , [ e , e ] = αe . Type II . Assume L has a double root of its minimal polynomial. Then, with respectto an orthonormal basis { e , e , e } of signature (+ + − ), one has L = α + β − − + β and thus the corresponding Lie algebra is given by( g II ) [ e , e ] = e − ( β − ) e , [ e , e ] = − ( β + ) e − e , [ e , e ] = αe . E. CALVI˜NO-LOUZAO, L. M. HERVELLA, J. SEOANE-BASCOY, R. V´AZQUEZ-LORENZO
Type III . Assume L has a triple root of its minimal polynomial. Then, with respectto an orthonormal basis { e , e , e } of signature (+ + − ), one has L = α √ √ √ α − √ α and thus the corresponding Lie algebra is given by( g III ) [ e , e ] = − √ e − αe , [ e , e ] = − √ e − αe , [ e , e ] = αe + √ ( e − e ) . Next we treat the non-unimodular case. First of all, recall that a solvable Liealgebra g belongs to the special class S if [ x, y ] is a linear combination of x and y for any pair of elements in g . Any left-invariant metric on S is of constantsectional curvature [19, 20] and hence locally conformally flat. Now, consider theunimodular kernel, u = ker(trace ad : g → R ). It follows from [10] that non-unimodular Lorentzian Lie algebras not belonging to class S are given, with respectto a suitable basis { e , e , e } , by( g IV ) [ e , e ] = 0 , [ e , e ] = αe + βe , [ e , e ] = γe + δe , α + δ = 0 , where one of the following holds:IV.1 { e , e , e } is orthonormal with g ( e , e ) = − g ( e , e ) = − g ( e , e ) = − αγ − βδ = 0.IV.2 { e , e , e } is orthonormal with g ( e , e ) = g ( e , e ) = − g ( e , e ) = 1 andthe structure constants satisfy αγ + βδ = 0.IV.3 { e , e , e } is pseudo-orthonormal with g ( e , e ) = − g ( e , e ) = 1 and thestructure constants satisfy αγ = 0.As a matter of notation, henceforth we will write ∇ e i e j = X k Φ kij e k to represent the Levi-Civita connection corresponding to the left-invariant metricon the Lie group, where { e , e , e } denotes the basis fixed in each case. Moreover,we will denote by X = P x i e i = ( x , x , x ) a generic vector field expressed in thecorresponding basis.Three-dimensional locally conformally flat Lorentzian Lie groups have been stud-ied by Calvaruso in [5]. We translate his classification to our context, in order tofit the notation used throughout this paper. Lemma 1.
A three-dimensional Lorentzian Lie group ( G, g ) is locally conformallyflat if and only if one of the following holds: (i) ( G, g ) is locally symmetric and (i.a) of Type Ia with α = β = γ or any cyclic permutation of α = β , γ = 0 (in any of these cases ( G, g ) is of constant sectional curvature), or (i.b) of Type II with α = β = 0 , and hence flat, or (i.c) of Type IV.1 with constant sectional curvature, or otherwise α = β = γ = 0 and δ = 0 , or β = γ = δ = 0 and α = 0 , or (i.d) of Type IV.2 with constant sectional curvature, or otherwise α = β = γ = 0 and δ = 0 , or β = γ = δ = 0 and α = 0 , or (i.e) of Type IV.3 and flat, or otherwise γ = δ = 0 and α = 0 , or OMOGENEOUS COTTON SOLITONS 5 (i.f) of Type S and therefore of constant sectional curvature. (ii) ( G, g ) is not locally symmetric and (ii.a) of Type Ib with α = − γ and β = ±√ γ , or (ii.b) of Type III with α = 0 , or (ii.c) of Type IV.3 with γ = 0 and αδ ( α − δ ) = 0 . Finally, note that any two vector fields X and X satisfying Equation (2)( L X i g + C = λ i g , i = 1 ,
2) differ in a homothetic vector field since L X − X g − ( λ − λ ) g = L X g − λ g − L X g + λ g = 0 . Conversely, adding a homothetic vector field to any Cotton soliton gives anotherCotton soliton. As a consequence, if a homogeneous Lorentzian manifold admits twodistinct Cotton solitons (i.e., for constants λ = λ ) then it is locally conformallyflat and therefore trivial (see [8, 15] for more information on homogeneous manifoldsadmitting non-Killing homothetic vector fields).3. Left-invariant Cotton solitons on Lorentzian Lie groups
Now we consider the existence of left-invariant solutions of Equation (2) onthe Lie algebras discussed in Section 2. We completely solve the correspondingequations, obtaining a complete description of all non-trivial left-invariant Cottonsolitons.Since the Cotton tensor is trace-free, if a Killing vector field X satisfies Equation(2) then C = 0 and ( M, g ) is locally conformally flat. Conversely, if (
M, g ) is locallyconformally flat and homogeneous, then any Cotton soliton is a homothetic vectorfield and hence it is Killing or otherwise the Ricci operator is two-step nilpotentand (
M, g, X ) is also a Yamabe soliton [8]. In the homogeneous setting a Cottonsoliton is said to be trivial if C = 0. In what follows we will focus on the non-trivialcase and therefore we will use repeatedly Lemma 1 to exclude the case of locallyconformally flat Lie groups.As a consequence of our analysis in this section the following geometric char-acterization of Lorentzian Lie groups admitting invariant Cotton solitons will beobtained. Theorem 2.
A Lorentzian Lie group ( G, g ) admits a non-trivial left-invariant Cot-ton soliton if and only if the Cotton operator is nilpotent. Previous result remarks the difference between the Riemannian and Lorentziansettings, since any nilpotent self-adjoint operator vanishes identically in the Rie-mannian category.3.1.
Unimodular case.
In this subsection we consider the existence of left-invariantCotton solitons on three-dimensional unimodular Lorentzian Lie groups whose cor-responding Lie algebras were introduced in Section 2.3.1.1.
Type Ia.
In this first case the Levi-Civita connection is determined byΦ = Φ = ( α − β − γ ) , Φ = Φ = ( α − β + γ ) , Φ = − Φ = ( α + β − γ ) , E. CALVI˜NO-LOUZAO, L. M. HERVELLA, J. SEOANE-BASCOY, R. V´AZQUEZ-LORENZO expressions which, for a left-invariant vector field X = P x i e i , allow us to calculatethe non-zero terms in the Lie derivative of the metric, L X g , given by(3) ( L X g ) = ( α − β ) x , ( L X g ) = ( γ − α ) x , ( L X g ) = ( β − γ ) x , and also to determine the non-zero components of the Cotton tensor, C , as(4) C = (cid:0) α − α ( β + γ ) − ( β − γ ) ( β + γ ) (cid:1) ,C = (cid:0) β − β ( α + γ ) − ( α − γ ) ( α + γ ) (cid:1) ,C = − (cid:0) γ − γ ( α + β ) − ( α − β ) ( α + β ) (cid:1) . In this first case we show that left-invariant Cotton solitons are necessarily trivial.
Theorem 3.
If a Type Ia unimodular Lorentzian Lie group is a left-invariantCotton soliton then it is necessarily trivial.Proof.
From Equations (3) and (4), a left-invariant vector field X = ( x , x , x ) isa Cotton soliton if and only if(5) x ( β − γ ) = x ( γ − α ) = x ( α − β ) = 0 ,α ( β + γ ) + ( β − γ ) ( β + γ ) − α + 2 λ = 0 ,β ( α + γ ) + ( α − γ ) ( α + γ ) − β + 2 λ = 0 ,γ ( α + β ) + ( α − β ) ( α + β ) − γ + 2 λ = 0 . Note that the first equation in (5) implies that the existence of non-zero Cottonsolitons is possible only if α = β , or α = γ , or β = γ . Next assume that α = β (the study of the other two cases is analogous). Under this condition, again thefirst equation in (5) implies that either β = γ or x = x = 0. In the first case, if β = γ , the Lie group is of constant sectional curvature − α . In the second case, if x = x = 0, Equation (5) reduces to γ ( β − γ ) − λ = 0 , γ ( β − γ ) + λ = 0 , and it follows that if β = γ then necessarily γ = 0 and the Lie group is flat. Weconclude that, in any case, the Cotton soliton is trivial. (cid:3) Type Ib.
The Levi-Civita connection is determined byΦ = Φ = ( α − γ ) , Φ = − Φ = − Φ = − Φ = − β, Φ = Φ = Φ = − Φ = α , and a straightforward calculation shows that the Lie derivative of the metric isdetermined by(6) ( L X g ) = x β + x ( α − γ ) , ( L X g ) = x β + x ( γ − α ) , ( L X g ) = ( L X g ) = − βx , while the Cotton tensor is characterized by(7) C = − C = 2 C = α − α γ + 4 β γ,C = β (cid:0) α + 4 β − γ + 4 αγ (cid:1) . OMOGENEOUS COTTON SOLITONS 7
As in the Type Ia unimodular case, we do not obtain non-trivial left-invariantCotton solitons in this second case.
Theorem 4.
A Type Ib unimodular Lorentzian Lie group does not admit any non-zero left-invariant Cotton soliton.Proof.
For a left-invariant vector field X = ( x , x , x ), Equations (6) and (7) implythat X is a Cotton soliton if and only if(8) x β + x ( α − γ ) = 0 ,x β + x ( γ − α ) = 0 ,β (cid:0) α + 4 αγ + 4 β − γ (cid:1) = 0 ,α − α γ + 4 β γ − λ = 0 ,α − α γ + 4 β γ + 2 λ + 4 βx = 0 ,α − α γ + 4 β γ + 2 λ − βx = 0 . Note that β = 0. Hence, the first equation in (8) implies that x = x ( γ − α ) β and, asa consequence, the second equation in (8) is equivalent to x ( β + ( γ − α ) ) = 0.Therefore, it follows that x = 0 and hence x = 0. Finally, from the last twoequations in (8) one obtains x = 0, and this ends the proof. (cid:3) Type II.
In this case, the Levi-Civita connection is determined byΦ = Φ = ( α − β ) , Φ = − Φ = − Φ = − Φ = − , Φ = Φ = ( α − , Φ = − Φ = ( α + 1) , and thus the Lie derivative of the metric is characterized by(9) ( L X g ) = ( x + (2 α − β − x ) , ( L X g ) = ( x − (2 α − β + 1) x )( L X g ) = ( L X g ) = − ( L X g ) = − x . Moreover, the Cotton tensor is determined by(10) C = α ( α − β ) ,C = − (cid:0) α − β + 4 αβ − α (2 β − (cid:1) ,C = ( α − β + 4 αβ ) ,C = (cid:0) α + 8 β − αβ − α (2 β + 1) (cid:1) . Next we determine the left-invariant Cotton solitons in the Type II unimodularcase.
Theorem 5.
A unimodular Lorentzian Lie group ( G, g ) of Type II admits non-trivial left-invariant Cotton solitons if and only if one of the following conditionsholds: (i) α = β = 0 , and then G = O (1 , or G = SL (2 , R ) , (ii) α = 0 = β , and then G = E (1 , .Moreover, in these cases, the Cotton solitons are always steady and respectivelygiven by: (i) X = β e + κ ( e + e ) , where κ ∈ R . E. CALVI˜NO-LOUZAO, L. M. HERVELLA, J. SEOANE-BASCOY, R. V´AZQUEZ-LORENZO (ii) X = 2 β e .Proof. Considering Equations (9) and (10), the Cotton soliton condition for a left-invariant vector field X = ( x , x , x ) can be expressed as(11) x + x (2 α − β −
1) = 0 ,x − x (2 α − β + 1) = 0 ,α − β + 4 αβ + 4 x = 0 ,α ( α − β ) − λ = 0 , α − β + 4 αβ − α (2 β −
1) + 4 x + 4 λ = 0 , α + 8 β − αβ − α (2 β + 1) − x + 4 λ = 0 . Adding the first and the second equations in (11) we obtain ( α − β )( x − x ) = 0.Hence, either α = β or x = x . Suppose first that α = β ; in this case, thefourth equation in (11) implies that λ = 0. Now, the first equation in (11) leads to x = x and finally the last equation in (11) implies that x = β . With this setof conditions Equation (11) holds and the Cotton soliton is non-trivial whenever α = β = 0, which shows (i).Next we analyze the case x = x , with α = β . In this case, the first equationin (11) implies that x ( α − β ) = 0 and hence x = 0. On the other hand, fromthe fourth equation in (11) we have λ = α ( α − β ). Moreover, from the last twoequations in (11) one easily obtains α ( α − β ) = 0, and therefore α = 0. Finally,Equation (11) reduces to x = 2 β , which shows (ii). (cid:3) Remark 6.
Left-invariant Cotton solitons on a unimodular Lorentzian Lie groupof Type II are gradient if and only if α = 0 = β . To show this, we analyze the twodifferent cases obtained in Theorem 5. First, assume that α = β = 0; in this case,left-invariant Cotton solitons are of the form X = β e + κ ( e + e ), κ ∈ R , andhence the dual form X b is given by X b = 34 β e + κ ( e − e ) . A straightforward calculation shows that dX b = − βκ ( e ∧ e − e ∧ e ) − β e ∧ e and therefore the Cotton soliton is never gradient. Now suppose that α = 0 = β ;in this second case, the left-invariant Cotton soliton is given by X = 2 β e . Thedual form is X b = 2 β e and, as a consequence, dX b = 0. Thus we conclude that there exists a smoothfunction f such that X = ∇ f .3.1.4. Type III.
In this last unimodular case, the Levi-Civita connection is deter-mined by Φ = − Φ = − Φ = − Φ = Φ = Φ = Φ = Φ = √ , Φ = Φ = − Φ = − Φ = − Φ = Φ = − α , OMOGENEOUS COTTON SOLITONS 9 and hence the Lie derivative of the metric is given by(12) ( L X g ) = −√ x + x ) , ( L X g ) = ( L X g ) = x √ , ( L X g ) = √ x , ( L X g ) = x − x √ , ( L X g ) = −√ x , while the Cotton tensor is characterized by(13) C = C = α √ , C = C = C = 3 α. For this case, we get the following.
Theorem 7.
A unimodular Lorentzian Lie group ( G, g ) of Type III admits non-trivial left-invariant Cotton solitons if and only if α = 0 , and then G = O (1 , or G = SL (2 , R ) . Moreover, in such a case, the Cotton soliton is steady and given by X = − α e + α √ ( e − e ) .Proof. For a left-invariant vector field X = ( x , x , x ), Equations (12) and (13)imply that X is a Cotton soliton if and only if(14) x + 3 α = 0 , √ x + √ x + λ = 0 , √ α − x + x = 0 , α + √ x − λ = 0 , α − √ x + λ = 0 . If α = 0, then it follows from Equation (14) that the Lie group does not admitany non-zero left-invariant Cotton soliton. Next suppose that α = 0. The firstequation in (14) implies that x = − α and the second equation in (14) impliesthat x = − x − λ √ . Now, from the last two equations in (14) we get λ = 0 and x = − α √ , from where the result follows. (cid:3) Remark 8.
A unimodular Lorentzian Lie group of Type III does not admit anyleft-invariant gradient Cotton soliton. Indeed, considering the Cotton soliton de-termined in Theorem 7, the dual form X b is given by X b = − α e + 3 α √ e + e )and a straightforward calculation shows that dX b = 3 α √ (cid:16) e ∧ e + e ∧ e + √ αe ∧ e (cid:17) . Hence, the Cotton soliton X is not gradient.3.2. Non-unimodular case.
In this subsection the existence of left-invariant Cot-ton solitons on three-dimensional non-unimodular Lorentzian Lie groups is consid-ered; the corresponding Lie algebras were introduced in Section 2. We exclude thestudy of Type S , since any Lie group of that type is of constant sectional curvatureand it does not admit any non-zero Cotton soliton. Next we show that in any othernon-unimodular case the Cotton solitons are trivial. Type IV.1.
In this case, the Levi-Civita connection is determined byΦ = Φ = α, Φ = − Φ = Φ = Φ = − β − γ , Φ = − Φ = − δ, Φ = Φ = − β + γ , and thus the Lie derivative of the metric is characterized by(15) ( L X g ) = − αx , ( L X g ) = ( β − γ ) x , ( L X g ) = αx + γx , ( L X g ) = 2 δx , ( L X g ) = − βx − δx . Moreover, the Cotton tensor is determined by(16) C = − (cid:0) β − γ − α β + βγ + 2 γδ − βδ (cid:1) ,C = δ ( α ( α − δ ) − β ( β − γ )) ,C = − (cid:0) β − γ − α β − β γ + γδ + αβδ (cid:1) ,C = − ( β + γ )( α + β − γ − δ )( α − β + γ − δ ) . We show the non-existence of non-trivial left-invariant Cotton solitons in thiscase in the following result.
Theorem 9.
If a Type IV.1 non-unimodular Lorentzian Lie group is a left-invariantCotton soliton then it is necessarily trivial.Proof.
Throughout the proof recall that αγ − βδ = 0 and α + δ = 0. Equations(15) and (16) imply that a left-invariant vector field X = ( x , x , x ) is a Cottonsoliton if and only if(17) αx + γx = 0 ,βx + δx = 0 , ( β + γ )( α + β − γ − δ )( α − β + γ − δ ) + 2 λ = 0 ,δ ( α ( α − δ ) − β ( β − γ )) + ( β − γ ) x = 0 ,β − γ − α β + βγ + 2 δ γ − βδ + 4 αx − λ = 0 , β − γ − α β − β γ + γδ + αβδ − δx + 2 λ = 0 . We analyze first the case α = 0; in this case, β = 0 and δ = 0. The second equationin (17) implies that x = 0. On the other hand, the fourth equation in (17) reducesto γx = 0 and therefore either γ = 0 or x = 0. If γ = 0 then the Lie group islocally conformally flat and hence the Cotton soliton is trivial. If x = 0 and γ = 0then the last two equations in (17) imply that ( γ − δ )( γ + δ ) = 0, which leads tothe constancy of the sectional curvature equals to − δ ; therefore the Cotton solitonmust be trivial.Assume now that α = 0; in this case, γ = βδα and the first equation in (17)implies that x = − βδx α , while the second equation in (17) reduces to(18) (cid:0) α − β (cid:1) δx = 0 . If α − β = 0 then α = εβ and γ = εδ , where ε = 1; under these conditionsthe Lie group is of constant sectional curvature − ( β + εδ ) and hence the Cottonsoliton is trivial.Next we consider the case δ = 0 in Equation (18) and therefore γ = 0. Thefourth equation in (17) is equivalent to βx = 0. Now, for β = 0 the Lie group is OMOGENEOUS COTTON SOLITONS 11 locally conformally flat and hence the Cotton soliton is trivial. On the other hand,if x = 0 and β = 0 then Equation (17) reduces to β − α β − λ = 0 , β − α β + λ = 0 . Therefore α − β = 0, which shows that the Lie group is of constant sectionalcurvature and the Cotton soliton is again trivial.Finally, we analyze the case x = 0 in Equation (18); we assume δ = 0 and α − β = 0 to avoid the previous cases. Note that, in this case, x = 0 and thefourth equation in (17) transforms into( α − δ ) (cid:0) δ (cid:0) α − β (cid:1) + βx (cid:1) = 0 . If α = δ then Equation (17) reduces to λ = 0 and δx = 0; hence x = 0 and theCotton soliton is zero. Therefore, we assume α = δ and we have δ (cid:0) α − β (cid:1) + βx =0. Note that β must be non-zero since we are assuming δ ( α − β ) = 0. Thus, x = − δ ( α − β ) β and from the third equation in (17) we have λ = − β (cid:0) α − β (cid:1) ( α − δ ) ( α + δ )2 α and Equation (17) reduces to ( α β − β δ + 2 αβ δ + 4 α = 0 , α β − β δ − αβ δ − α δ = 0 . Now, it follows that (cid:0) α + β (cid:1) (cid:0) α − δ (cid:1) = 0, which is a contradiction, and thisends the proof. (cid:3) Type IV.2.
In this case, the Levi-Civita connection and the Lie derivative ofthe metric are determined byΦ = Φ = α, Φ = Φ = Φ = Φ = β + γ , Φ = Φ = δ, Φ = − Φ = − β − γ , and(19) ( L X g ) = 2 αx , ( L X g ) = ( β + γ ) x , ( L X g ) = − αx − γx , ( L X g ) = 2 δx , ( L X g ) = − βx − δx , respectively. Now, the Cotton tensor is given by(20) C = (cid:0) β + 2 γ + α β + βγ + 2 γδ + βδ (cid:1) ,C = δ ( α ( α − δ ) + β ( β + γ )) ,C = − (cid:0) β + γ + 2 α β + β γ + γδ − αβδ (cid:1) ,C = − ( β − γ ) (cid:0) ( α − δ ) + ( β + γ ) (cid:1) . For this case we have the following.
Theorem 10.
If a Type IV.2 non-unimodular Lorentzian Lie group is a left-invariant Cotton soliton then it is necessarily trivial.
Proof.
Throughout the proof recall that αγ + βδ = 0 and α + δ = 0. For a left-invariant vector field X = ( x , x , x ), Equations (19) and (20) imply that X is aCotton soliton if and only if(21) αx + γx = 0 ,βx + δx = 0 , ( β − γ ) (cid:0) ( α − δ ) + ( β + γ ) (cid:1) − λ = 0 ,δ ( α ( α − δ ) + β ( β + γ )) + ( β + γ ) x = 0 ,β + 2 γ + α β + βγ + 2 δ γ + βδ + 4 αx − λ = 0 , β + γ + 2 α β + β γ + γδ − αβδ − δx + 2 λ = 0 . Assume first that α = 0. In this case, β = 0 and δ = 0, and from the third andfifth equations in (21) one obtains γ (cid:0) γ + δ (cid:1) = 0. Therefore γ = 0 and the Liegroup is locally conformally flat; thus the left-invariant Cotton soliton is trivial.Next suppose that α = 0. Hence γ = − βδα and the first equation in (21) impliesthat x = βδx α . Now, the second equation in (21) is equivalent to(22) (cid:0) α + β (cid:1) δx = 0 , and therefore either δ = 0 or x = 0. If δ = 0 then the fourth equation in (21)reduces to βx = 0. Note that if x = 0 then Equation (21) reduces to β + α β + λ = 0 , β + α β − λ = 0 , which implies that β = 0. Hence, in any case, necessarily β = 0, and it follows thatthe Lie group is locally conformally flat. Thus the Cotton soliton is trivial.Finally, we analyze the case x = 0 in Equation (22); we assume δ = 0 to avoidthe previous case. Note that, in this case, x = 0 and the fourth equation in (21)is equivalent to ( α − δ )( δ ( α + β ) + βx ) = 0 . If α = δ then Equation (21) reduces to λ = 0 and δx = 0; hence x = 0 and theCotton soliton is zero. Now, if α = δ then δ ( α + β ) + βx = 0. Note that β mustbe non-zero since we are assuming that δ = 0. Thus, x = − δ ( α + β ) β and the thirdequation in (21) implies that λ = β (cid:0) α + β (cid:1) ( α − δ ) ( α + δ )2 α . Now, Equation (21) reduces to ( α − α β − αβ δ + 3 β δ = 0 ,α (cid:0) β + 4 δ (cid:1) − αβ δ − β δ = 0 , and it follows that ( α − β )( α + β )( α − δ )( α + δ ) = 0. Since we are assuming that α = δ and, moreover, α + δ = 0, then we have α = ± β and in such a case the abovesystem of equations has no solution (since δ = 0). This ends the proof. (cid:3) OMOGENEOUS COTTON SOLITONS 13
Type IV.3.
In this case, the Levi-Civita connection is determined byΦ = Φ = α, Φ = − Φ = Φ = Φ = − Φ = − Φ = γ , Φ = Φ = − β, Φ = − Φ = − δ. Hence, the Lie derivative of the metric and the Cotton tensor are characterized by(23) ( L X g ) = 2 αx , ( L X g ) = γx , ( L X g ) = − αx − γx − βx , ( L X g ) = − δx , ( L X g ) = 2( βx + δx ) , and(24) C = γ , C = γ , C = βγ . Next we show that left-invariant Cotton solitons reduce to left-invariant Yamabesolitons in this case.
Theorem 11.
If a Type IV.3 non-unimodular Lorentzian Lie group is a left-invariant Cotton soliton then it is necessarily trivial.Proof.
Using Equations (23) and (24) it follows that a left-invariant vector field X = ( x , x , x ) is a Cotton soliton if and only if(25) γx = 0 ,αx + γx + βx = 0 , αx + γ − λ = 0 ,γ − δx + 2 λ = 0 ,βγ + 4 βx + 4 δx = 0 . Recall that αγ = 0 and α + δ = 0. First, if α = γ = 0 then the Lie group is flat andhence the left-invariant Cotton soliton is trivial. Now, if α = 0 and γ = 0 then thefirst equation in (25) implies that x = 0, and therefore from the third and fourthequations in (25) we easily obtain that γ = 0, which is a contradiction. Finally, if γ = 0 and α = 0 then the Lie group is locally conformally flat and therefore theleft-invariant Cotton soliton is trivial. (cid:3) Remark 12.
In the Riemannian setting, the unimodular Lie groups correspondto Type Ia (just considering the usual cross product induced by the quaternions),while the non-unimodular case corresponds to Type S and Type IV.2 previouslydiscussed [19].With regard to Types Ia and IV.2, the behaviour is exactly the same in boththe Riemannian and the Lorentzian cases, since Theorems 3 and 10 remain true inthe Riemannian setting. Thus, since locally symmetric spaces and Lie groups ofType S are locally conformally flat, one has that three-dimensional homogeneousRiemannian manifolds do not admit any non-trivial left-invariant Cotton soliton.Hence, any left-invariant Riemannian Cotton soliton is a Yamabe soliton, and inRiemannian signature this implies the flatness of the manifold.Let H be the Heisenberg group and consider on H the left-invariant metricgiven by g = dx + dy + ( dz − xdy ) . Proceeding as in Section 5 it is obtained that( H , g ) is a shrinking non-invariant Riemannian Cotton soliton which is not trivial. Proof of Theorem 2.
A careful examination of the cases obtained in Theorem 5 andin Theorem 7 shows that the corresponding Cotton operator is two-step nilpotentfor unimodular Lorentzian Lie groups of Type II with α = β = 0 or α = 0 = β ,while the degree of nilpotency is three for unimodular Lorentzian Lie groups of TypeIII with α = 0. Conversely, a case by case examination of the Cotton operator inthe different unimodular and non-unimodular Lorentzian Lie groups shows, after along but straightforward calculation, that the cases with nilpotent Cotton operatorare precisely those obtained in Theorems 5 and 7. (cid:3) Algebraic Lorentzian Cotton solitons
Cotton solitons are fixed points of the Cotton flow up to diffeomorphisms andrescaling. The specific behaviour of a homogenous manifold allows us to considera stronger condition than the Cotton soliton one. More precisely, we can considersoliton solutions for the Cotton flow up to automorphisms instead of diffeomor-phisms; in such a case the soliton is referred to as an algebraic Cotton soliton. Let(
G, g ) be a simply connected Lie group equipped with a left-invariant Lorentzianmetric g and let g denote the Lie algebra of the Lie group G . Following the seminalwork of Lauret [18], ( G, g ) is said to be an algebraic Cotton soliton if it satisfies(26) b C = λ Id + D where b C stands for the Cotton operator ( g ( b C ( X ) , Y ) = C ( X, Y )), λ is a realconstant and D ∈ Der( g ), i.e., D [ X, Y ] = [
DX, Y ] + [
X, DY ] for all
X, Y ∈ g . Next we show that the algebraic Cotton soliton condition is stronger than theCotton soliton one.
Proposition 13.
Let ( G, g ) be a simply connected Lie group endowed with a left-invariant Lorentzian metric g . If ( G, g ) is an algebraic Cotton soliton, i.e. itsatisfies Equation (26), then it is a Cotton soliton such that a vector field solvingEquation (2) is given by X = ddt | t =0 ϕ t ( p ) , with dϕ t | e = exp (cid:18) t D (cid:19) , where e denotes the identity element of G .Proof. Let (
G, g ) be an algebraic Cotton soliton, i.e. b C = λ Id + D , let { e i } denotea pseudo-orthonormal basis and define a one-parameter family of automorphisms ϕ t by setting dϕ t | e = exp (cid:18) t D (cid:19) . Considering the vector field X given by X = ddt | t =0 ϕ t ( p ) a direct calculation shows that the Lie derivative in the direction of X is given by( L X g ) ( e i , e j ) = ddt | t =0 ϕ ∗ t g ( e i , e j ) = 12 ( g ( De i , e j ) + g ( e i , De j )) , OMOGENEOUS COTTON SOLITONS 15 for i, j ∈ { , , } . Then, C ( e i , e j ) = (cid:16) g ( b C ( e i ) , e j ) + g ( e i , b C ( e j )) (cid:17) = ( g (( λ Id + D )( e i ) , e j ) + g ( e i , ( λ Id + D )( e j )))= λ g ( e i , e j ) + ( L X g ) ( e i , e j ) , which shows that ( G, g, X ) is a Cotton soliton (cid:3)
In view of the proposition above it seems natural to ask whether a Cotton solitonon a Lie group comes from an algebraic Cotton soliton or not. The remaining ofthis section is devoted to clarify this question. In this sense, the next theorem showshow algebraic Cotton solitons allow constructing non-invariant Cotton solitons.
Theorem 14.
A three-dimensional Lie group ( G, g ) equipped with a left-invariantLorentzian metric is an algebraic Cotton soliton if and only if one of the followingconditions holds: (i) ( G, g ) is of Type Ia with α = β = 0 and γ = 0 , or any cyclic permutation.In this case, λ = − γ and G = H is the Heisenberg group. (ii) ( G, g ) is of Type Ib with α = 0 and γ = ε √ β . In this case, λ = 2 ε √ β with ε = 1 , and G = E (1 , .Proof. We analyze the existence of algebraic Cotton solitons on a three-dimensionalLorentzian Lie algebra. To do that, it is sufficient to study when an operator D ofthe form D = b C − λ Idis a derivation on a Lie algebra g .We start with Type Ia. From Equation (4) it is easy to get that D = b C − λ Idis a derivation if and only if(27) γ (cid:0) α − α β − α (cid:0) β − γ (cid:1) + β + βγ − γ − λ (cid:1) = 0 ,β (cid:0) α − α γ − α (cid:0) γ − β (cid:1) − β + β γ + γ − λ (cid:1) = 0 ,α (cid:0) α − α ( β + γ ) − β + β γ + βγ − γ + λ (cid:1) = 0 . If all the structure constants vanish then the Lie algebra is abelian and hence thealgebraic Cotton soliton is trivial. Hence, we assume that at least one of them isnon-zero, for instance γ = 0. Thus, from the first equation in (27) we obtain that λ = γ ( α + β ) + ( α − β ) ( α + β ) − γ . Then, Equation (27) becomes(28) ( β ( β − γ ) (cid:0) α + 2 α ( β + γ ) − β − βγ − γ (cid:1) = 0 ,α ( α − γ ) (cid:0) α + 2 α ( γ − β ) − ( β − γ )( β + 3 γ ) (cid:1) = 0 . A first non-trivial solution is obtained taking α = β = 0 and λ = − γ . For α = 0and β = 0 a straightforward calculation shows that α = γ , and in this case themanifold is locally conformally flat. The same conclusion is obtained for α = 0 and β = 0. Finally, assume α = 0 = β . If α = γ or β = γ or α = β Equation (28)implies that α = β = γ and therefore the sectional curvature is constant; thus themanifold is locally conformally flat. If α = β = γ = α the Equation (28) reduces to(29) ( α + 2 α ( β + γ ) − β − βγ − γ = 0 , α + 2 α ( γ − β ) − ( β − γ )( β + 3 γ ) = 0 . From this last equation we get ( α − β ) + 3 γ = 0, which is a contradiction. Pro-ceeding in an analogous way for α = 0 or β = 0 (i) is obtained.We consider now the Type Ib. From Equation (7), and taking into account that β = 0, it follows that D = b C − λ Id is a derivation if and only if(30) α + 4 αγ + 8 β γ − γ − λ = 0 ,α + 4 αγ + 4 β − γ = 0 ,αλ = 0 . The last equation in (30) implies that either α = 0 or λ = 0. First, suppose that α = 0. Thus, Equation (30) becomes(31) ( β γ − γ − λ = 0 ,β − γ = 0 , from where (ii) is obtained. We assume now that λ = 0 and α = 0. Then, Equation(30) becomes(32) ( γ − α − αγ − β γ = 0 ,α + 4 αγ + 4 β − γ = 0 . A long but straightforward calculation shows that α = − γ and β = ±√ γ , and inthis case the manifold is locally conformally flat.The remaining types of three-dimensional Lorentzian Lie algebras do not providenon-trivial algebraic Cotton solitons. The proof is obtained by proceeding as in theprevious cases. We omit the details for sake of brevity since the calculations arestandard. (cid:3) Remark 15.
Case (i) in Theorem 14 is the Lie algebra associated to the Heisen-berg group, while case (ii) corresponds to the Lie algebra associated to the solvableLie group E (1 , Remark 16.
Theorem 14 highlights that left-invariant Cotton solitons do not comefrom algebraic Cotton solitons.5.
Non-invariant Lorentzian Cotton solitons
In this section, motivated by the existence of algebraic Cotton solitons, we show the existence of non-invariant Cotton solitons on three-dimensional homogeneousLorentzian manifolds which are non-trivial . To do this, we consider the algebraicCotton solitons obtained in Theorem 14. Note that Theorem 2 implies that anyLie group admitting a left-invariant Cotton soliton has necessarily nilpotent Cottonoperator and this condition does not hold for the Lie groups obtained in Theorem14. Thus, the Lie groups admitting algebraic Cotton solitons provide non-invariantexamples of Cotton solitons.
OMOGENEOUS COTTON SOLITONS 17
Non-invariant Cotton solitons on the Heisenberg group.
We start withthe Lorentzian Heisenberg group. In [24] it is shown that this Lie group can beendowed with three different left-invariant Lorentzian metrics, up to isometry andscaling, given by g = − dx + dy + ( xdy + dz ) ,g = dx + dy − ( xdy + dz ) ,g = dx + ( xdy + dz ) − ((1 − x ) dy − dz ) . Metric g is flat [20] and hence a trivial Cotton soliton, and metrics g and g areshrinking non-gradient Ricci solitons [21, 22]. Next we analyze metrics g and g by separate. Starting with the metric g , for a vector field X = A ( x, y, z ) ∂ x + B ( x, y, z ) ∂ y + C ( x, y, z ) ∂ z the Lie derivative of the metric is given by(33) ( L X g )( ∂ x , ∂ x ) = − A x , ( L X g )( ∂ x , ∂ y ) = −A y + (cid:0) x + 1 (cid:1) B x + x C x , ( L X g )( ∂ x , ∂ z ) = −A z + x B x + C x , ( L X g )( ∂ y , ∂ y ) = 2 (cid:0) x A + (cid:0) x + 1 (cid:1) B y + x C y (cid:1) , ( L X g )( ∂ y , ∂ z ) = A + x B y + (cid:0) x + 1 (cid:1) B z + C y + x C z , ( L X g )( ∂ z , ∂ z ) = 2 ( x B z + C z ) , and the (0 , C ( ∂ x , ∂ x ) = − , C ( ∂ y , ∂ y ) = − x , C ( ∂ y , ∂ z ) = − x, C ( ∂ z , ∂ z ) = − . Now, Equations (33) and (34) imply that X is a Cotton soliton if and only if(35) A x + 1 − λ = 0 , x B z + 2 C z − − λ = 0 , A z − x B x − C x = 0 , A y − ( x + 1) B x − x C x = 0 , A + x B y + (cid:0) x + 1 (cid:1) B z + C y + x C z − x − xλ = 0 , x A + 4( x + 1) B y + 4 x C y − x + 1 − x + 1) λ = 0 . We start integrating first and second equations in (35) to get A ( x, y, z ) = A ( y, z ) + (2 λ − x, C ( x, y, z ) = C ( x, y ) − B ( x, y, z ) x + ( λ + 1) z. Now, the third equation in (35) transforms into B ( x, y, z ) = ( C ) x ( x, y ) − ( A ) z ( y, z )and hence differentiating the fourth equation in (35) with respect to x and z weobtain ( A ) zz = 0, and thus A ( y, z ) = A ( y ) z + A ( y ) . Next, differentiating the fifth equation in (35) with respect to z we get A ( y ) = 0and the fourth equation in (35) transforms into ( C ) xx ( x, y ) − A ′ ( y ) = 0, fromwhere it follows that C ( x, y ) = A ′ ( y ) x + C ( y ) x + C ( y ) . At this point, the fifth equation in (35) reduces to4 A ( y ) + 4 C ′ ( y ) + (2 A ′′ ( y ) x + 4 C ′ ( y ) − x = 0and from this equation a straightforward calculation shows that A ( y ) = κ y + κ , C ( y ) = y + κ , C ( y ) = − κ y − κ y + κ , where κ i ∈ R , i = 1 , . . . ,
4. Thus, Equation (35) finally reduces to λ − H , g ) is a Cotton soliton if and only if it is shrinking with λ = 2and X given by X = ( κ y + x + κ ) ∂ x + ( κ x + y + κ ) ∂ y − ( κ ( x + y ) + κ y − z − κ ) ∂ z . Note that as an special case of the Cotton solitons obtained above, one has that X = x∂ x + y∂ y + z∂ z defines a complete Cotton soliton on ( H , g ).Now, it is easy to see that the associated Cotton operator, b C , is characterizedby b C ( ∂ x ) = ∂ x , b C ( ∂ y ) = ∂ y − x∂ z , b C ( ∂ z ) = − ∂ z . As a consequence, it diagonalizes with eigenvalues {− , , } and hence the Cottonsoliton cannot be left-invariant (see Theorem 2). Moreover, the Ricci operator, b ρ ,is given by b ρ ( ∂ x ) = ∂ x , b ρ ( ∂ y ) = ∂ y − x∂ z , b ρ ( ∂ z ) = − ∂ z , and thus it does not have any zero eigenvalue, which implies that the soliton is notgradient [7]. Remark 17.
A completely analogous study can be developed with the metric g to obtain that ( H , g ) is a Cotton soliton if and only if it is expanding with λ = − X = ( κ y − x + κ ) ∂ x − ( κ x + y − κ ) ∂ y + ( κ ( x − y ) − κ y − z + κ ) ∂ z , where κ i ∈ R , i = 1 , . . . ,
4. Again the Cotton soliton is not left-invariant and it isnot gradient.5.2.
Non-invariant Cotton solitons on E (1 , . The Lie group E (1 ,
1) admitstwo kinds of algebraic Cotton solitons (up to some sign ε = ±
1, see Theorem 14)and therefore it can be equipped with a left-invariant metric with non-nilpotentCotton operator admitting some Cotton soliton. Moreover, these Cotton solitonswill be necessarily non-invariant (see Theorem 2).For our purpose we consider the frame [22] E = ∂ x , E = 12 (cid:0) e x ∂ y + e − x ∂ z (cid:1) , E = 12 (cid:0) e x ∂ y − e − x ∂ z (cid:1) and the associated coframe given by E = dx, E = e − x dy + e x dz, E = e − x dy − e x dz. Henceforth we consider left-invariant metrics corresponding to Theorem 14-(ii).First, we take the metric g given by g = 23 ( E ) − ( E ) + 2 √ E E ) − E ) , or in local coordinates g = 23 dx + (2 √ − e − x dy − (2 √ e x dz + 4 dydz. OMOGENEOUS COTTON SOLITONS 19
Now, for a vector field X = A ( x, y, z ) ∂ x + B ( x, y, z ) ∂ y + C ( x, y, z ) ∂ z the Lie derivativeof the metric is given by(36) ( L X g )( ∂ x , ∂ x ) = A x , ( L X g )( ∂ x , ∂ y ) = (cid:0) A y + 3 (cid:0) √ − (cid:1) e − x B x + 3 C x (cid:1) , ( L X g )( ∂ x , ∂ z ) = (cid:0) A z + 3 B x − (cid:0) √ (cid:1) e x C x (cid:1) , ( L X g )( ∂ y , ∂ y ) = 4 C y − (cid:0) √ − (cid:1) e − x ( A − B y ) , ( L X g )( ∂ y , ∂ z ) = 2 (cid:0)(cid:0) √ − (cid:1) e − x B z + B y + C z − (cid:0) √ (cid:1) e x C y (cid:1) , ( L X g )( ∂ z , ∂ z ) = 4 B z − (cid:0) √ (cid:1) e x ( A + C z )and a straightforward calculation shows that X = y √ ∂ y + z √ ∂ z is a shrinkingCotton soliton with λ = 2 √
2. Moreover, due to the non-existence of homotheticvector fields on E (1 ,
1) it follows that this is the unique shrinking Cotton solitonup to Killing vector fields [8].Next we consider the Lorentzian metric g on E (1 ,
1) given by g = 23 ( E ) − E ) + 2 √ E E ) − ( E ) , or in local coordinates g = 23 dx + (2 √ − e − x dy − (2 √ e x dz − dydz. Proceeding as in the previous case we get that X = − y √ ∂ y − z √ ∂ z is a expandingCotton soliton for λ = − √
2. Furthermore, it is the unique expanding Cottonsoliton up to Killing vector fields due to the results in [8].
Remark 18.
As a final remark, it is worth pointing out once again that the non-invariant Cotton solitons constructed in this section come from algebraic Cottonsolitons obtained in Theorem 14. Moreover, let us emphasize that algebraic Cottonsolitons have an opposite behaviour with respect to invariant Cotton solitons onLie groups since no left-invariant Cotton soliton can be obtained from an algebraicone.
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Clas-sical Quantum Gravity (2008), 035007. Department of Geometry and Topology, Faculty of Mathematics, University of San-tiago de Compostela, 15782 Santiago de Compostela, Spain
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