Generic Properties of Homogeneous Ricci Solitons
GGENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS
LUCA FABRIZIO DI CERBOAbstract.
We discuss the geometry of homogeneous Ricci solitons. Aftershowing the non-existence of compact homogeneous and non-compact steadyhomogeneous solitons, we concentrate on the study of left invariant Ricci soli-tons. We show that, in the unimodular case, the Ricci soliton equation doesnot admit solutions in the set of left invariant vector fields. We prove thata left invariant soliton of gradient type must be a Riemannian product withnontrivial Euclidean de Rham factor. As an application of our results weprove that any generalized metric Heisenberg Lie group is a non-gradient leftinvariant Ricci soliton of expanding type. Introduction
A Ricci soliton is a solution of the Ricci flow ∂g∂t = − Ric g (1) g (0) = g that changes only by diffeomorphisms and scale. More precisely, g ( t ) is calledRicci soliton if there exist a smooth function σ ( t ) and a 1-parameter family ofdiffeomorphisms { ψ t } of M n such that g ( t ) = σ ( t ) ψ ∗ t ( g ) with σ (0) = 1 and ψ = id M n . It is easy to see that this condition is equivalent to requiring that the initialmetric g satisfies the following Einstein like identity − Ric g = 2 λg + L X g , (2)where λ is a real number and X a complete vector field; for the details we refer to[5]. We then say that the soliton is expanding, shrinking, or steady if λ > λ < λ = 0 respectively. Finally, if the vector field X is the gradient field of a smoothfunction f , one says that the soliton is a gradient Ricci soliton.The soliton theories in the compact and complete non-compact cases are drasti-cally different. In the compact case it is easy to prove the non-existence of steadyand expanding solitons, see [5]; while a result of Perelman [28] ensures that anyshrinking soliton must be of gradient type. Moreover, there are no solitons in di-mension two and three as proved by Hamilton and Ivey [11], [12]. Finally, it isinteresting to notice that nontrivial examples of compact gradient Ricci solitonswere actually constructed by Koiso in [18]. These examples start in dimension fourand are K¨ahler. For more details about compact K¨ahler-Ricci solitons we refer to[5].The complete non-compact theory is much richer. Besides the shrinking andexpanding Gaussian solitons on R n and the cylinder soliton on S n × R ( n ≥
2) it isworthwhile to mention the Hamilton “cigar” soliton on R , the radially symmetricsteady Bryant solitons on R n ( n ≥
3) and their generalization by Ivey, who con-structed Ricci solitons on doubly warped products, see [5], [4], [13]. It turns outthat all these examples are of gradient type. *Supported in part by the Simons Foundation. a r X i v : . [ m a t h . DG ] S e p ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS A completely different family of solitons has been discovered by Lauret, see[19] and the more update [21]. In [19], Lauret searches for a notion weakeningthe Einstein condition for a left invariant metric g on a nilpotent Lie group N n .This is motivated by the fact that nilpotent Lie groups do not admit any leftinvariant Einstein metric, see [24]. The Ricci soliton condition (2) clearly providesa substitute candidate and Lauret proves that, given a nilpotent Lie group N n withLie algebra n and a left invariant metric g , then (2) is satisfied iff Ric g = cI + D, (3)for some real number c and D ∈ Der ( n ), where Der ( n ) denotes the Lie algebraof derivations of n . Note that this condition immediately implies the existence ofa symmetric derivation. In particular, as pointed out in [19], we cannot solve thesoliton equation over a characteristically nilpotent Lie group. Recall that a Liealgebra n is characteristically nilpotent iff Der ( n ) is nilpotent. Here we just noticethat this obstruction does not apply to the class of 2-step nilpotent Lie algebras. Infact a characteristically nilpotent Lie algebra is at least 4-step nilpotent, for moredetails see [22]. Moreover there are no examples of characteristically nilpotent Liealgebras in dimension less or equal that six, as follows from [25].It is now interesting to consider the equation in (3) on a non-nilpotent Lie group G , and using the 1-parameter group of diffeomorphisms generated by D show that g is a Ricci soliton. This fact is actually used by Lauret to exhibit some examplesof non-nilpotent solvable Lie groups which admit a Ricci soliton structure. Solitonssatisfying 3 are also called algebraic solitons , for more details see [21], [17] and thebibliography therein.Finally, we point out that the soliton condition on a nilpotent Lie group can beneatly characterized in terms of metric solvable extensions. Theorem 1.1 (Lauret) . Let N n be a simply connected nilpotent Lie group withLie algebra n and left invariant metric g . Then ( N n , g ) admits a Ricci solitonstructure iff ( n , g ) admits a standard metric solvable extension ( s = a ⊕ n , ˜ g ) whosecorresponding simply connected Lie group ( S, ˜ g ) is Einstein. Recall that a metric solvable extension of ( n , g ) is a metric solvable Lie algebra( s = a ⊕ n , ˜ g ) such that [ s , s ] s = n = a ⊥ , [ , ] s | n × n = [ , ] n , ˜ g | n × n = g . The metricsolvable extension is called standard if a is abelian. Although Lauret results arenon-costructive, the characterization given in Theorem 1.1 can be easily used toprovide examples. For instance, any generalized Heisenberg Lie group (H-type) [3]and many 2-step nilpotent Lie groups admit a Ricci soliton structure, see [20] for acomplete list of all the known examples.With what regard to the uniqueness of nilpotent Ricci solitons, the followingtheorem provides a complete answer. Theorem 1.2 (Lauret) . Let N n be a simply connected nilpotent Lie group with Liealgebra n and left invariant metric g . If g and g (cid:48) are left invariant Ricci solitonmetrics, then there exist c > and η ∈ Aut ( n ) such that g (cid:48) = cη ( g ) . Remark 1.3.
Interestingly, in a recent preprint [17] , Jablonski has extended manyof the results of Lauret to the class of solvmanifolds. Remarkably, a Ricci solvsolitonhas to be an algebraic Ricci soliton. Moreover, these solitons enjoy nice uniquenessproperties as in the nilpotent case. For more details the reader is referred to [17] and to the bibliography therein.
The first explicit construction of Lauret solitons has been obtained by Baird andDanielo in [1]. In [1], the authors study 3-dimensional Ricci solitons which projectsvia a semi-conformal mapping to a surfaces and obtain a complete description ofthe soliton structures on Nil and Sol predicted by Lauret. ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS In [23], Lott uses these particular soliton structures to study the long time be-havior of type III Ricci flow solutions and also gives, among many other results,some explicit examples of four dimensional homogeneous soliton solutions, e.g. onNil . It turns out that all these solitons are expanding and of non-gradient type.These are the first known examples of non-gradient Ricci solitons. We also notethat the property to be non-gradient plays a important role in the analytical studyof the linear stability of these solutions, see [9].In this paper, we concentrate on the study of the generic properties of homoge-neous Ricci solitons providing a new point of view on the results found by Danielo,Baird and Lott.2. The scalar curvature of a homogeneous Ricci soliton
In this section we study the evolution of the scalar curvature on a homogeneousRicci soliton. Recall that on a soliton solution to the Ricci flow the scalar curvaturesatisfies the equation R ( t ) = R σ ( t ) = R λt . (4)In particular if the initial scalar curvature R is constant it stays constant duringthe evolution. In this case R t satisfies the simple ODE ∂R∂t = 2( − λ ) R R , R (0) = R . (5)As proved in [11], the scalar curvature of a general Ricci flow solution evolvesaccording the heat type equation ∂R∂t = ∆ R + 2 | Ric | , (6)we conclude that if the soliton has constant initial scalar curvature with λ = − R n then it must be trivial. In particular a nontrivial soliton structure cannot be asso-ciated to a divergence free vector field X , as can be proved by considering the traceof (2). In summary we have: Lemma 2.1.
A complete vector field X associated to a nontrivial constant curva-ture Ricci soliton satisfies div ( X ) = k , where k is a nonzero constant. We can now use Lemma 2.1 to obtain a complete description of the compactcase.
Theorem 2.2.
There are no non-trivial compact solitons with constant scalar cur-vature.Proof.
Stokes’ theorem. (cid:3)
With regard to the compact homogeneous case it is interesting to notice thatTheorem 2.2 can also be derived from the fact that there is a natural scaling in-variant quantity that is increasing along the flow.
Lemma 2.3.
Let ( M n , g ( t )) be a compact homogeneous solution to the Ricci Flow,then R ( t ) V ( t ) n is monotonically increasing unless the initial manifold is Einstein.Proof. Because of the diffeomorphism invariance of the Ricci tensor and the shorttime existence and uniqueness for the Ricci flow on compact manifolds [10], theflow preserves the isometries of the initial Riemannian manifold. We conclude thata compact homogeneous Riemannian manifold remains homogeneous during the
ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS flow. Using the standard variation formulas for the volume and the scalar curvaturefunctions [10] we have ddt R ( t ) V ( t ) n = R (cid:48) ( t ) V ( t ) n + 2 n V ( t ) − nn V (cid:48) ( t ) R ( t )= 2 | Ric | V ( t ) n − n R V ( t ) n = 2 (cid:12)(cid:12)(cid:12)(cid:12) Ric − Rn g (cid:12)(cid:12)(cid:12)(cid:12) V ( t ) n . (cid:3) Note that, because of the existence of the solitons discovered by Lauret, there isno analogue of Lemma 2.3 in the noncompact homogeneous case. Nevertheless, wecan use the monotonicity property of the scalar curvature to derive the followingpartial generalization of Theorem 2.2.
Theorem 2.4.
There are no steady non-compact Ricci solitons with constant scalarcurvature.Proof.
By the formula dR ( t ) dt = 2 | Ric | (7)we have that the scalar curvature is increasing unless the initial metric is Ricciflat. (cid:3) It is interesting to note that Theorems 2.2 and 2.4 immediately extend to ruleout the existence of compact homogeneous and non-compact homogeneous steadybreathers. For the definition of breather see [5]. This observation implies thefollowing remark.
Remark 2.5.
As pointed out in [10] , the Ricci flow equation on a given homoge-neous spaces reduces to an ODE on the finite dimensional moduli space of homo-geneous metrics. Now the nonexistence of steady breathers then implies that theseinteresting geometrical ODEs have no periodic solutions, at least for those initialdata that admit geometrical interpretation. For a detailed study of the Ricci flowODEs on three and four dimensional homogeneous spaces we refer to [14] and [15] . Finally, we notice that the sign of the scalar curvature determines if the solitonis expanding or shrinking.
Lemma 2.6.
Any constant scalar curvature soliton of expanding or shrinking typemust have respectively negative or positive initial scalar curvature.
For more results concerning homogeneous shrinking solitons with positive scalarcurvature the reader is referred to Remark 3.2 below.3.
Left invariant Ricci solitons
In this section we concentrate on the study of the geometrical properties of leftinvariant Ricci solitons. First, we explain why all the known explicit examplesare of expanding type, then we study the possibility to solve the soliton equationwithin the set of left invariant vector fields and then we show that in “general” a leftinvariant soliton cannot be of gradient type. We point out that we usually work onsimply connected Lie group since a non-trivial soliton structure on a Lie group withnon-trivial fundamental group can clearly be transported to its universal cover.
ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS The sign of the soliton constant.
As noticed in Lemma 2.6 the sign of thescalar curvature determines the type of the soliton. The sign of the scalar curvatureassociated to a left invariant metric has been extensively studied by many authors,see in particular [24] and [2]. We can then state the following.
Proposition 3.1.
A left invariant soliton or breather structure on a solvable Liegroup is necessarily expanding.Proof.
As proved in [24], any left invariant metric g over a solvable Lie group G iseither flat or has strictly negative scalar curvature. If the scalar curvature negative,by Lemma 2.6 we have that the left invariant soliton structure (if any) must be ofexpanding type. (cid:3) This simple result explains why the examples explicitly constructed by Baird,Danielo and Lott are of expanding type. Moreover it ensures that every nilsolitonpredicted by Lauret must be of expanding type. The above lemma can actually beimproved. In [24], Milnor studies the problem of which Lie groups endowed with leftinvariant metric admit positive scalar curvature. Using the Iwasawa decompositiontheorem Wallach was able to give a sufficient condition, namely that the universalcovering of the Lie group is not homeomorphic to an Euclidean space, for the detailssee [24]. In [24] it is actually conjectured that this condition is also necessary. Thisconjecture turned to be true as shown by B´erard-Bergery in [2]. We then have thata Lie group whose universal cover is homeomorphic to a Euclidean space can admitonly expanding left invariant soliton structures. We also have that a non-compactshrinking left invariant soliton must be homeomorphic to a product of a compactLie group with some Euclidean space.
Remark 3.2.
It follows from results of Naber [26] and Petersen-Wylie [29] thatany shrinking homogeneous Ricci soliton must be trivial. More precisely, it must beisometric to a product of a compact positive Einstein manifold with some Euclideanspace. In fact, by Theorem 1.2. in [26] a shrinking homogeneous soliton must be ofgradient type. Finally, by Theorem 1.1. in [29] such a soliton must be trivial. Formore details see also Remark 3.8.
Nonsolvability for left invariant vector fields.
We derive some general-ities about the Ricci soliton equation over a metric g that is left invariant. Recallthat a metric g is called left invariant if L ∗ h g = g for all h ∈ G where L h is theleft translation by h . Recall also that a vector field X is called left invariant iff L ∗ h X = X for all h ∈ G . Consider now the Ricci soliton equation (2) and take thepull back by a left translation, we then get − L ∗ h Ric g = L ∗ h ( L X g ) + 2 λL ∗ h g − Ric L ∗ h g = L L ∗ h X ( L ∗ h g ) + 2 λg − Ric g = L L ∗ h X g + 2 λg, which implies L ( L ∗ h X − X ) g = 0 . We conclude that L ∗ h X − X is a Killing vector field for all h ∈ G , we then say that X is left invariant modulo Killing fields or simply a left-Killing field. Following thisterminology a trivial left-Killing field is just an ordinary left invariant vector field.At this stage of the theory we cannot be sure of the existence of nontrivial left-Killing fields, indeed one may think to consider the usual left invariant vector fieldsin order to reduce the soliton condition to an algebraic set of equations on the Liealgebra of the group. It turns out that this is not possible in general. Lemma 3.3.
Any left invariant vector field over an unimodular Lie group is di-vergence free.
ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS Proof.
First, we notice that any left invariant vector field has constant divergence.Let X be a left invariant vector field over a Lie group G n equipped with a leftinvariant metric g and let { e i } ni =1 be a left invariant global orthonormal frame. Wecan then compute the divergence as follows div ( X ) = (cid:88) i g ( ∇ e i X, e i )and notice that is constant since the {∇ e i X } are left invariant. Recall now thata Lie group is called unimodular if its left invariant Haar measure ω is also rightinvariant. Recall also that given a left invariant vector field X its flow can beexplicitly written in terms of right translations and the exponential map, namelyfor any x ∈ G n we have F t ( x ) = R exp(tX) ( x ) . Let dω the left invariant volume form associated to the Haar measure. Since weare assuming dω to be also right invariant we clearly have L X dω = 0. We concluderecalling the identity L X dω = div ( X ) dω . (cid:3) We can now combine the above result with Lemma 2.1 to obtain the followingproposition.
Proposition 3.4.
For any unimodular Lie group the homogeneous soliton equationcannot be solved within the set of left invariant vector fields.
Note that Proposition 3.4 implies the existence of nontrivial left-Killing fields.Recall that, in terms of the Lie algebra g , the unimodular condition is equivalentto requiring that the linear transformation ad x has trace zero for every x ∈ g . Weconclude that any nilpotent Lie group is unimodular which implies, by Proposi-tion 3.4, the existence of at least one complete nontrivial left-Killing field on anynilsoliton.We now briefly study the same question on non-unimodular Lie groups. Noticethat on a non-unimodular Lie group the divergence of a left invariant vector field canbe different from zero. Let ( g , (cid:104) , (cid:105) ) be a three dimensional non-unimodular metricLie algebra. As shown in Section 6 of [24], we can always find an orthonormal basis e , e , e so that [ e , e ] = αe + βe [ e , e ] = γe + δe with [ e , e ] = 0, α + δ (cid:54) = 0 and αγ + βδ = 0. Following the literature we refer tothis special frame as a Milnor frame (M-frame). The divergence of the elements inthe M-frame is easily computed: div ( e ) = g ( ∇ e e , e ) + g ( ∇ e e , e ) + g ( ∇ e e , e )= 0 + 12 { g ([ e , e ] , e ) − g ([ e , e ] , e ) + g ([ e , e ] , e ) } + 12 { g ([ e , e ] , e ) − g ([ e , e ] , e ) + g ([ e , e ] , e ) } = − ( α + δ );an analogous compututation shows that div ( e ) = div ( e ) = 0. In summary forany orthonormal M-frame e has nonzero divergence while e and e are divergencefree. Assume now the soliton equation can be solved within the set of left invariantvector fields. Let ( g , (cid:104) , (cid:105) ) our soliton metric Lie algebra, and let e , e , e theassociated M-frame. Notice that the left invariant vector field that solve the solitoncondition must have a nonzero component in the direction of e . Using Lemma 6.5. ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS in [24], it is easy to express the Ricci tensor in terms of the constants of structureof the Lie algebra. Using this fact and performing analogous computations for theLie derivative of the metric we reduce the soliton condition − Ric g = 2 λg + L X g ,where X = ae + be + ce with a (cid:54) = 0, to the following set of algebraic equations − − α − δ − ( β + γ ) − α ( α + δ ) + ( γ − β ) 00 0 − δ ( α + δ ) + ( β − γ ) = 2 λ + a − α − ( γ + β )0 − ( γ + β ) − δ + bα + cδ bβ + cδbα + cδ bβ + cδ . This system is easily reduced to2( α + δ ) = 2 λ α ( α + δ ) = 2 λ − aα δ ( α + δ ) = 2 λ − aδ. Thus, λ = α + δ which implies, under the assumption δ (cid:54) = α , a = − ( α + δ ).Substituting this value in the second of the equations above we obtain δ = α = 0that is a contradiction. In the remaining case δ = α an easy computation showsthat a = 0 which clearly implies the triviality of the soliton. In summary we provedthe following proposition. Proposition 3.5.
For any three dimensional non-unimodular Lie algebra the soli-ton equation cannot be solved within the set of left invariant vector fields.
We don’t know if a similar restriction holds in dimension greater than three.3.3.
Left invariant gradient solitons.
As mentioned in the introduction, allthe known explicit constructions of left invariant Ricci solitons are of non-gradienttype. Remarkably these are the first examples of non-gradient Ricci solitons. Wenow show that this property is indeed generic for this class of solitons.Recall that the gradient Ricci soliton equation is given by
Ric g + ∇∇ f + λ g = 0 . (8)We then say that g is an expanding, shrinking, or steady soliton if λ > λ < λ = 0 respectively. By standard manipulations in (8) we can derive the followinguseful identities R + ∆ f + λ n = 0(9) dR = 2 Ric ( ∇ f, · )(10) ∆ R = g ( ∇ R, ∇ f ) − λR − | Ric | , (11)for a proof see [5]. Using identity (9), we notice that if R is constant then ∆ f isconstant. We can then derive another proof of theorem 2.2. In fact, by Perelman’sresult on compact Ricci solitons we can restrict our attention to shrinking gradientsolitons. Thus, using the nonexistence of sub/superharmonic functions on compact ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS manifolds we can rule out again the constant scalar curvature case. Combining nowidentities (9) and (11), we have2 n ( R + ∆ f ) R = 2 | Ric | that implies 2 n ∆ f R = 2 (cid:12)(cid:12)(cid:12)(cid:12) Ric − Rn g (cid:12)(cid:12)(cid:12)(cid:12) . (12)We conclude that on a gradient shrinking soliton with constant scalar curvature∆ f and R have the same sign. Finally, using (10) we can derive a nice rigidityproperty for constant scalar curvature solitons. Indeed, using the identity0 = dR = 2 Ric ( ∇ f, · ) , (13)we have that if the Ricci curvature has definite sign then the soliton must necessarilybe Einstein. More precisely we can state the following: Proposition 3.6.
A Riemannian manifold with constant scalar curvature and nondegenerate Ricci tensor cannot be a gradient Ricci soliton.
We conclude our study showing that Proposition 3.6 can actually be improvedif we restrict to left invariant solitons. Formula (13) combined with the Bochneridentity for |∇ f | gives∆ |∇ f | = 2 |∇∇ f | + 2 Ric ( ∇ f, ∇ f ) + 2 g ( ∇ f, ∇ ∆ f )= 2 |∇∇ f | , then if we further assume |∇ f | = c , with c a constant, we derive that the solitonis trivial. Thus, since L ∗ h ∇ g f = ∇ L ∗ h g ( f ◦ L h ) = ∇ g ( f ◦ L h ) , (14)we have that for at least some h ∈ G the vector field ∇ g ( f ◦ L h − f ) is a nontrivialKilling field. Now, a gradient Killing field must be parallel. We conclude thatour group splits locally as a Riemannian product with an interval. The simpleobservation that a gradient soliton structure on Lie group lifts to its universalRiemannian cover immediately implies the following theorem. Theorem 3.7.
Any metric Lie algebra ( g , (cid:104) , (cid:105) ) with trivial Euclidean de Rhamfactor cannot be a gradient Ricci soliton. Remark 3.8.
It follows from a rigidity result of Petersen-Wylie [29] that any ho-mogeneous gradient Ricci soliton must have a non-trivial Euclidean deRham factor.The reader should compare Theorem 3.7 with the more general Theorem 1.1. in [29] .The first version of this work and the preprint of [29] appeared independently onthe arXiv in October 2007.
We can now apply Proposition 3.6 to the class of nonsingular 2-step nilpotentLie algebras.
Theorem 3.9.
A nonsingular 2-step nilpotent Lie algebra cannot admit a left in-variant gradient soliton structure.Proof.
Recall that a 2-step nilpotent Lie algebra n is called nonsingular if the map ad x : n → z is surjective for all x ∈ n − z , where z is the center of n . Now, on ageneral 2-step nilpotent Lie algebra equipped with a positive definite inner product (cid:104) , (cid:105) , denoted by v the orthogonal complement of z in n , we can define for each z ∈ z a skew symmetric linear transformation j ( z ) : v → v by j ( z ) x = ( ad x ) ∗ z for all x ∈ v ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS where ( ad x ) ∗ denotes the adjoint of ad x . As extensively shown by many authors,see in particular [7] and [3], most of the geometry of the metric Lie algebra { n , (cid:104) , (cid:105)} is given by the properties of the maps j ( z ). In particular the kernel of the Riccitensor, seen as a symmetric linear transformation on the Lie algebra, is given bythe following subspace of the center { z ∈ v | j ( z ) = 0 } , see Proposition 2.5 in [7].Being n nonsingular, an easy argument shows that j ( z ) : v → v is nonsingular forany nonzero z ∈ z . We conclude that for any nonsingular 2-step nilpotent metricLie algebra { n , (cid:104) , (cid:105)} the associated Ricci tensor is non-degenerate. The claim is nowa consequence of Theorem 3.7. (cid:3) We then have the following theorem for generalized Heisenberg Lie groups (H-type).
Theorem 3.10.
Any simply connected metric H-type Lie group admits an expand-ing non-gradient left invariant Ricci soliton structure.Proof.
Recall that a 2-step nilpotent metric lie algebra { n , (cid:104) , (cid:105)} is of Heisenberggeneralized type if j ( z ) = −| z | id v for all z ∈ z . This clearly implies that for any z ∈ z the linear map j ( z ) is nonsingular. Now a routine linear algebra argumentshows that the Lie algebra n must be nonsingular. By Theorem 3.9 we concludethat a H-type Lie group cannot be a gradient Ricci soliton. For what regard theexistence we follow Lauret. The standard metric solvable extension of the H-typeLie groups are exactly the harmonic Damek-Ricci spaces that are solvable andEinstein, see for example [3]. By Theorem 1.1 we conclude that these groups arealgebraic Ricci solitons. Finally, using Proposition 3.1 we have that these structuresmust be of expanding type. (cid:3) Final remarks
Theorem 3.10 provides the first example of a countably infinite family of non-gradient Ricci solitons. Recall that for any n ∈ N there exist a countably infinitenumber of non-isomorphic H-type Lie algebras with dim z = n , see again [3]. It isalso interesting to notice that any H-type group admits a lattice, i.e., a cocompactdiscrete subroup. This result is easily derived from the existence of an integralstructure on any H-type Lie algebra, see [6]. In fact a well known result by Malcev,see for example [30], ensures that a simply connected nilpotent Lie group admitsa lattice iff its Lie algebra admits a rational structure. Thus, let N be a H-typeLie group and Γ a lattice. We then have that the left-Killing vector field X is notpreserved by the action of the subroup Γ otherwise we could project the solitonstructure on ( N, g ) to a well defined homogeneous soliton structure on the compactmanifold ( N/ Γ , ˜ g ). Thus, Theorem 3.10 also provides an infinite family of pseu-dosolitons , i.e., compact manifold with no soliton structure that acquire one whenlifted to the universal cover, see [9] for the definition of pseudosolitons.As a final application we derive that the 3-dimensional non-product geometriesNil , Sol and SL(2 , R ) do not admit any left invariant gradient soliton structure,since any left invariant metric on these groups is de Rham indecomposable, see [31].Note how a similar result cannot be obtained by using Proposition 3.6. In fact, asproved in [24], the signature of the Ricci form on both Sol and SL(2 , R ) can beeither (+ , − , − ) or (0 , , − ). More in general Theorem 3.7 provides a satisfactoryobstruction in the case of irreducible left invariant solitons.It is interesting to note that very recently Jablonski in [17] has finally rule outthe existence of a homogeneous Ricci soliton structure on SL(2 , R ). This result, ina sense, closes up the connection between the theory of homogeneous Ricci solitonsand the special geometries of Thurston. ENERIC PROPERTIES OF HOMOGENEOUS RICCI SOLITONS Finally, I would like to point out that since the appearance of first version of thiswork on the arXiv, the theory of homogeneous Ricci solitons has vigorously grown.The interested reader should for example consult [8], [16], [27], [20], [15], [17], [29],[26], [21].
Acknowledgements . I would like to thank Jorge Lauret for his kind interestin this work and for several constructive comments. Moreover, I would like tothank Professor John Milnor for several useful discussions regarding [24]. I alsowould like to thank the referee for guiding me through the most recent literatureon homogeneous solutions of the Ricci flow and for pertinent comments on thismanuscript.
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