Closed \mathrm{G}_2-eigenforms and exact \mathrm{G}_2-structures
aa r X i v : . [ m a t h . DG ] J a n CLOSED G -EIGENFORMS AND EXACT G -STRUCTURES MARCO FREIBERT AND SIMON SALAMON
Abstract.
A study is made of left-invariant G -structures with an exact 3-form on a Lie group G whoseLie algebra g admits a codimension-one nilpotent ideal h . It is shown that such a Lie group G cannotadmit a left-invariant closed G -eigenform for the Laplacian and that any compact solvmanifold Γ \ G arising from G does not admit an (invariant) exact G -structure. We also classify the seven-dimensionalLie algebras g with codimension-one ideal equal to the complex Heisenberg Lie algebra which admitexact G -structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras h admitting an exact SL(3 , C )-structure ρ or a half-flat SU(3)-structure( ω, ρ ) with exact ρ , respectively. Introduction
The group G is one of the exceptional cases in Berger’s celebrated list [Be] of restricted holonomygroups of non-locally symmetric irreducible Riemannian manifolds and only occurs in dimension seven.For over 30 years, it was unknown whether such manifolds exist at all until Bryant found local examples[Br2], Bryant and the second author found complete ones [BrSa], and Joyce [J] constructed compactmanifolds with G holonomy.The construction of these examples relies on the fact that the metric is encoded in a certain type ofthree-form, which we shall refer to as a G -structure. More exactly, a G -structure on a seven-dimensionalmanifold M is a three-form ϕ ∈ Ω M on M with pointwise stabilizer conjugate to G ⊆ SO(7) ⊆ GL(7 , R ).The form ϕ induces a Riemannian metric g ϕ , an orientation and a Hodge star operator ⋆ ϕ on M . Theholonomy group of g ϕ is contained in G if the structure is torsion-free, meaning that ϕ is parallel for theLevi-Civita connection, which is the case if and only if ϕ is closed and coclosed [FG].G -structures that are closed but not coclosed constitute a basic intrinsic torsion class in the Fern´andez-Gray classification, and play a natural role in the construction of compact manifolds with holonomy equalto G . Joyce’s examples were found by first constructing closed G -structures on smooth manifolds withsufficiently small intrinsic torsion and then proving analytically that such closed G -structures may bedeformed to torsion-free ones.Closed G -structure are the initial values for the Laplacian flow ˙ ϕ t = ∆ ϕ t ϕ t for one-parameter familiesof closed G -structures ( ϕ t ) t ∈ I introduced by Bryant in [Br3]. The critical points of this flow are preciselythe torsion-free G -structures [LW1], and the hope is to use the Laplacian flow to deform a closed G -structure (without any smallness assumption on the intrinsic torsion) to a torsion-free one for t → ∞ .Short-time existence and uniqueness of the Laplacian flow were established in [BrXu], and other foun-dational properties were proven in series of papers by Lotay and Wei [LW1, LW2, LW3]. However, a lot isstill unknown about long-time behaviour of the flow, and it is important to characterise finite-time singu-larities. One expects that, like for the Ricci flow, these singularities are modeled by self-similar solutionsof the Laplacian flow. The initial values ϕ of these self-similar solutions are called Laplacian solitons ,and a special class of them is given by closed G -eigenforms characterised by∆ ϕ ϕ = µϕ for some µ ∈ R \ { } . Although this equation looks quite easy, no examples of these structures are known.Moreover, compact manifolds cannot admit a closed G -eigenform [LW1].Closed G -eigenforms are also of interest from another point of view: they constitute a special class ofso-called λ -quadratic closed G -structures , λ ∈ R , namely those with λ = 0. In general, quadratic closedG -structures are exactly the closed G -structures for which the exterior derivative dτ of the associatedtorsion two-form τ depends quadratically on τ . These structures have been studied by Ball [Ba1, Ba2]and include many other interesting closed G -structures. For example, the case λ = corresponds to so-called extremally Ricci-pinched (ERP) closed G -structures, and the case λ = is equivalent to theinduced metric being Einstein.By Lauret’s work [L], homogeneous λ -quadratically closed G -structures on homogeneous manifoldscan only exist for λ ∈ { , , } . Homogeneous ERP closed G -structures were classified in [Ba2] usingthe classification of left-invariant such structures on Lie groups in [LN2]. Moreover, [FFM] shows that nosolvable Lie group can admit a left-invariant closed Einstein G -structure. Since the strong Alekseevskyconjecture is true in dimension seven (i.e. any simply-connected homogeneous 7-Einstein manifold ofnegative scalar curvature is isometric to a left-invariant metric on a simply-connected solvable Lie group)[AL], there are no closed homogeneous Einstein G -structures. So the homogeneous case is settled for λ ∈ { , } leaving open the case λ = 0, where nothing at all is known.We shall fill this gap as follows. Let G be a 7-dimensional Lie group with Lie algebra g . We prove that G cannot admit a left-invariant closed G -eigenform if g is almost nilpotent , i.e. it admits a codimension-onenilpotent ideal. We are led to focus on ideals of two types, n and n , with the former of step 4, andthe latter of step 2 and isomorphic to the real Lie algebra underlying the complex Heisenberg group. Itis striking that our non-existence proof is at the limit of, but just within, the realm of computations thatcan be checked by hand. This fact has enabled us to complement our conclusions with more positive onesrelating to n , mentioned below.We are naturally led to the class of almost nilpotent Lie algebras by the following facts:Lauret and Nicolini [LN1] showed that any Lie algebra g that possesses a closed G -structure has acodimension-one unimodular ideal h . Hence, it is quite natural to start with those for which h is nilpotent.Motivation is also provided by the results of Podest`a and Raffero [PR] on closed G structures onseven-manifolds with a transtive reductive group of symmetries.Moreover, a closed G -eigenform ϕ is always (cohomologically) exact, and the existence problem of exactG -structures on a restricted class of almost nilpotent Lie algebras has been studied in [FFR]: there areno exact G -structures on strongly unimodular Lie algebras g with b ( g ) = b ( g ) = 0. The latter impliesthat g is almost nilpotent [MaSw], whereas ‘strongly unimodular’ is a technical condition, necessary forthe existence of a cocompact lattice in the associated simply-connected Lie group G .We also answer negatively the existence problem for exact G -structures on strongly unimodular almostnilpotent Lie algebras, and so on compact almost nilpotent (completely solvable) solvmanifolds, therebygeneralising the result of [FFR]. It is not known if there exists any compact manifold with an exact G -structure, though it is known that (in contrast to other situations) nilmanifolds cannot serve as examples.We do succeed in classifying all almost nilpotent Lie algebras admitting an exact G -structure for whichthe codimension-one nilpotent ideal is isomorphic to n . For such almost nilpotent Lie algebras, we alsoclassify those that admit exact G -structures with special torsion of positive or negative type, a notionintroduced by Ball in [Ba2].To prove our results, we split our almost nilpotent Lie algebra g as a vector space into g = h ⊕ R e with h being the codimension-one nilpotent ideal and e ∈ h ⊥ of norm one. Then the equations determininga closed G -eigenform or an exact G -structure can be encoded into conditions on the induced SU(3)-structure ( ω, ρ ) on h . In particular, for an exact G -structure, the SL(3 , C )-structure ρ has to be exactand for a closed G -eigenform, ( ω, ρ ) has to be half-flat with ρ being the exterior derivative of a primitive(1 , ν . The extra equation ν ∧ ω = 0 turns out to be of crucial importance in enabling us to ruleout solutions to the eigenform equations.We show that exactly five out of 34 six-dimensional nilpotent Lie algebras admit an exact SL(3 , C )-structure and that exactly two of them admit a half-flat SU(3)-structure ( ω, ρ ) with ρ exact, namely n and n . Both results have independent interest because special kinds of closed and exact SL(3 , C )-structureson six-dimensional nilpotent Lie algebras have been studied in [FS], and the six-dimensional nilpotent Liealgebras admitting a half-flat SU(3)-structure ( ω, ρ ) with dω = ρ were determined in [FR]. Moreover,the result on exact SL(3 , C )-structures implies that if an almost nilpotent Lie algebra g admits an exactG -structure, then the codimension-one nilpotent ideal h has to be one of the five Lie algebras. We provideexamples of exact G -structures on almost nilpotent Lie algebras with codimension-one nilpotent ideal h for all possible nilpotent Lie algebras h except when h equals the nilpotent Lie algebra called n .This leaves open the question to be studied in future work: Is there an almost nilpotent Lie algebrawith codimension-one nilpotent ideal isomorphic to n that admits an exact G -structure? LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 3 The paper is organised as follows.In Section 2, we summarise basic facts about SL(3 , C )-, SU(3)- and G -structures that are relevant toour investigation. In Section 3, we show how one can reduce the existence problem of a closed G -eigenformor an exact G -structure on a seven-dimensional Lie algebra g to the existence problem of SL(3 , C )- orSU(3)-structures satisfying certain equations on a six-dimensional ideal h in g . Next, in Section 4, weprove our results on exact SL(3 , C )-structures and on half-flat SU(3)-structures ( ω, ρ ) with exact ρ . Weuse these results to prove in Section 5 that no strongly unimodular almost nilpotent Lie algebra, and soalso no compact almost nilpotent (completely solvable) solvmanifold, can admit an exact G -structure.Finally, in Section 6, we carried out a detailed analysis of the respective cases n and n in order to showthat no almost nilpotent Lie algebra can admit a closed G -eigenform. Moreover, we prove the mentionedclassification results of almost nilpotent Lie algebras with a codimension ideal isomorphic to n admittingexact G -structures. 2. Preliminaries G -structures in six and seven-dimensions. In this subsection, we define three different types of G -structures in six and seven dimensions, and recall some of their basic properties. Proofs of the relevantfacts and more information may be found, for example, in [Br2, Br3, Hi].In all cases, the G -structure is defined by one or two differential forms which are pointwise isomorphicto one or two ‘model’ forms on R n , n = 6 or n = 7, whose GL( n, R )-stabiliser is G . Here, pointwiseisomorphic means that for each p ∈ M there is a vector space isomorphism u : T p M → R n that identifiesthe differential forms at the point p ∈ M with the model forms on R n . Definition 2.1. • An SL(3 , C ) -structure on an oriented six-dimensional manifold is a three-form ρ ∈ Ω M which ispointwise isomorphic to ρ := e − e − e − e ∈ Λ ( R ) ∗ . • An SU(3) -structure on a six-dimensional manifold is a pair ( ω, ρ ) of a two-form ω ∈ Ω M and athree-form ρ ∈ Ω M which is pointwise isomorphic to ( ω , ρ ) with ω := e + e + e ∈ Λ ( R ) ∗ . • A G -structure on a seven-dimensional manifold M is a three-form ϕ ∈ Ω M which is pointwiseisomorphic to ϕ := ω ∧ e + ρ ∈ Λ ( R ) ∗ . In all three cases, if u : T p M → R n is one of the pointwise isomorphisms, then the basis ( u − ( e ) , . . . , u − ( e n ))of T p M is called an adapted basis for the G -structure in question. Sometimes, we will also call the dualbasis of ( u − ( e ) , . . . , u − ( e n )) an adapted basis for the G -structure in question.Since SL(3 , C ) ⊆ GL(3 , C ) and GL(3 , C )-structures are almost complex structures, an SL(3 , C )-structure ρ has to induce an almost complex structure J ρ . Explicitly, J ρ is obtained as follows: Definition 2.2.
Let ρ be an SL(3 , C )-structure on an oriented six-dimensional manifold M . Then ρ induces an almost complex structure J = J ρ on M defined in p ∈ M to be the unique endomorphism J p of T p M satisfying J p f i − = − f i , J p f i = f i − . for one, and so any, adapted oriented basis ( f , . . . , f ) of T p M .Moreover, set ˆ ρ := J ∗ ρ ∈ Ω M . Thenˆ ρ p = f − f − f − f , where ( f , . . . , f ) is the dual basis of the adapted basis ( f , . . . , f ) at p ∈ M . Furthermore, Ψ := ρ + i ˆ ρ ∈ Ω ( M, C ) is a non-zero (3 , , C )-structure and for this have to introduce a quarticinvariant λ of a three-form on a vector space: MARCO FREIBERT AND SIMON SALAMON
Definition 2.3.
Let V be a six-dimensional vector space. Let κ : Λ V ∗ → V ⊗ Λ V ∗ be the natural GL( V )-equivariant isomorphism, i.e. κ − ( v ⊗ ν ) = v y ν . Next, let ρ ∈ Λ V ∗ and define K ρ ∈ End( V ) ⊗ Λ V ∗ by K ρ ( v ) = κ (( v y ρ ) ∧ ρ )and finally set λ ( ρ ) := tr (cid:0) K ρ (cid:1) ∈ (Λ V ∗ ) ⊗ . It makes sense to say that λ ( ρ ) >
0, meaning that λ ( ρ ) is the square of some element in Λ V ∗ . Thus,one may also speak of λ ( ρ ) <
0. Using this notation, Hitchin [Hi] gives the following characterisation ofan SL(3 , C )-structure: Lemma 2.4.
Let ρ ∈ Ω M be a three-form on an oriented six-dimensional manifold M . Then ρ is an SL(3 , C ) -structure if and only if λ ( ρ p ) < for all p ∈ M . We will also need the following technical statements in the sequel:
Lemma 2.5.
Let ρ ∈ Ω M be an SL(3 , C ) -structure on a seven-dimensional manifold. Let p ∈ M and v, w ∈ T p M and set J := J ρ . Then: (a) ρ p ( v, w, · ) = 0 if and only if v , w are C -linearly dependent. (b) If v = 0 , then the two-forms ω := ρ p ( v, · , · ) ∈ Λ T ∗ p M and ω := ρ p ( J p v, · , · ) ∈ Λ T ∗ p M satisfy ker( ω i ) = span( v, J p v ) , ω i ∧ ω j = δ ij ω for all i, j = 1 , .Proof. (a) First, let v, w are C -linearly dependent. Without loss of generality, we may assume that w = cJv for some c ∈ R . Since Ψ is a (3 , v, Jv, u ) = − Ψ( Jv, J v, u ) =Ψ( Jv, v, u ) = − Ψ( v, Jv, u ), i.e. Ψ( v, Jv, u ) = 0 for any u ∈ T p M . As ρ = Re(Ψ) this implies ρ ( v, w, · ) = cρ ( v, Jv, · ) = 0.Next, assume that v and w are C -linearly independent. Then we may extend v and w to a C -basis of ( T p M, J ) by an element u ∈ T p M . Then v C := v − iJv , w C := w − iJw and u C := u − iJu form a basis of ( T p M ) , and so 0 = Ψ( v C , w C , u C ) since Ψ is a (3 , z − iJz, · , · ) = Ψ( z, · , · ) − i Ψ( Jz, · , · ) = Ψ( z, · , · ) − i Ψ( z, · , · ) = 2Ψ( z, · , · )for any z ∈ T p M and so Ψ( v C , w C , u C ) = 8Ψ( v, w, u ). Consequently, Ψ( v, w, u ) = 0. Thus, ρ ( v, w, u ) = 0 or ˆ ρ ( v, w, u ) = 0. In the latter case, we do have ρ ( v, w, Ju ) + i ˆ ρ ( v, w, Ju ) = Ψ( v, w, Ju ) = i Ψ( v, w, u ) = i ˆ ρ ( v, w, u ) = − ˆ ρ ( v, w, u ) = 0 , i.e. ρ ( v, w, Ju ) = 0.(b) Since all equations are invariant under non-zero rescalings, we may assume that v has normone. Now SU(3) acts transitively on the six-sphere S . Consequently, there is an adapted basis( f , . . . , f ) of ρ at p ∈ M with f = v , and so f = J p f = J p v . But so ω = v y ρ p = f y (cid:0) f − f − f − f (cid:1) = f − f ,ω = ( J p v ) y ρ p = f y (cid:0) f − f − f − f (cid:1) = − f − f , Then a straightforward computation shows that ω and ω have the desired properties. (cid:3) Similarly, one knows that SU(3) ⊆ SO(6) and so an SU(3)-structure induces a Riemannian metric g asfollows: Definition 2.6.
Let ( ω, ρ ) be an SU(3)-structure on a six-dimensional manifold M . Define g = g ( ω,ρ ) to bethe Riemannian metric on M for which any adapted basis ( f , . . . , f ) at any point p ∈ M is orthonormal.Now ω is a volume form on M and so ω induces an orientation on M . We get an induced almost complexstructure J ρ , which relates g to ω by the equation g = ω ( J ρ · , · ) . Hence, ( g, J, ω ) is an almost Hermitian structure on M . Moreover, Ψ = ρ + i ˆ ρ is of constant length.Next, we turn to the special class of SU(3)-structures defined in [ChSa]: LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 5 Definition 2.7.
An SU(3)-structure ( ω, ρ ) on a six-dimensional manifold M is called half-flat if dω = 0and dρ = 0.Finally, we turn to G -structures and use that G ⊆ SO(7) ⊆ GL + (7 , R ): Definition 2.8.
Let ϕ ∈ Ω M be a G -structure on a seven-dimensional manifold. Define g = g ϕ to bethe Riemannian metric on M for which any adapted basis ( f , . . . , f ) at any point p ∈ M is orthonormal.Similarly, define an orientation on M by requiring that any adapted basis ( f , . . . , f ) at any point p ∈ M is oriented. We get an induced Hodge star operator ⋆ ϕ and( ⋆ ϕ ϕ ) p = f + f + f + f + f + f − f for any p ∈ M and any adapted basis ( f , . . . , f ) at p with dual basis ( f , . . . , f ). Moreover, ⋆ ϕ ϕ ispointwise isomorphic to ⋆ ϕ ϕ = ω + e ∧ ˆ ρ ∈ Λ ( R ) ∗ . A G -structure on a seven-dimensional vector space V is simply a constant G -structure on the manifold V , or said more directly, a three-form ϕ ∈ Λ V ∗ for which there exists an vector space isomorphism u : V → R with u ∗ ϕ = ϕ . So G -structures ϕ ∈ Ω M on seven-dimensional manifolds M are those forwhich ϕ p is a G -structure on the seven-dimensional vector space T p M for all p ∈ M .Similarly, we define an SU(3) -structure on a six-dimensional vector space V and get the following result: Lemma 2.9.
Let ϕ ∈ Λ V ∗ be a G -structure on a seven-dimensional vector space V . Moreover, let v ∈ V be of norm one with respect to g ϕ and let W := v ⊥ gϕ . Then there is a unique SU(3) -structure ( ω, ρ ) ∈ Λ W ∗ × Λ W ∗ on W such that ϕ = ω ∧ α + ρ, ⋆ ϕ ϕ = ω + α ∧ ˆ ρ, with α ∈ V ∗ uniquely defined by α ( W ) = 0 and α ( v ) = 1 , and W ∗ identified with the annihilator of v .Proof. G acts transitively on the unit sphere in R . Hence, we may assume that ϕ has an adapted basis( f , . . . , f ) with v = f and so α = f . Since ( f , . . . , f ) is an orthonormal basis of V , we have W =span( f , . . . , f ) and the statements follow from the relations ϕ = ω ∧ e + ρ and ⋆ ϕ ϕ = ω + e ∧ ˆ ρ between the model forms of a G - and an SU(3)-structure. (cid:3) Lemma 2.9 yields the following non-degeneracy condition for G -structures, whose proof we omit: Lemma 2.10.
Let ϕ ∈ Λ V ∗ be a G -structure on a seven-dimensional vector space V and let v, w ∈ V be linearly independent. Then ϕ ( v, w, · ) = 0 . Next, we consider some representation theory of G . Consider the G -representation Λ k ( R ) ∗ for k ∈ { , . . . , } . We decompose this representation into its irreducible components, which are by now well-known. Since the Hodge star operator ⋆ ϕ is an isomorphim of G -representations between Λ k ( R ) ∗ andΛ − k ( R ) ∗ , it suffices to do this for k ∈ { , . . . , } . Obviously, Λ ( R ) ∗ is trivial, and Λ ( R ) ∗ ∼ = R is alsoirreducible. For k = 2 ,
3, we have: Λ ( R ) ∗ = Λ ⊕ Λ , Λ ( R ) ∗ = R ϕ ⊕ Λ ⊕ Λ with Λ = (cid:8) v y ϕ (cid:12)(cid:12) v ∈ R (cid:9) , Λ = (cid:8) ν ∈ Λ ( R ) ∗ (cid:12)(cid:12) ν ∧ ϕ = − ⋆ ϕ ν (cid:9) , Λ = (cid:8) v y ⋆ ϕ ϕ (cid:12)(cid:12) v ∈ R (cid:9) , Λ = (cid:8) γ ∈ Λ ( R ) ∗ (cid:12)(cid:12) γ ∧ ϕ = 0 , γ ∧ ⋆ ϕ ϕ = 0 (cid:9) . The subscript denotes the dimension of the irreducible representation. For example, Λ is isomorphicto the adjoint representation g , as can be seen by using the metric g ϕ to identify a two-form ν ∈ Λ with an endomorphism of R . The decompositions of Λ k ( R ) ∗ into irreducible G -representations giverise to corresponding decompositions of Ω k ( M ). In particular, Ω ( M ) = (cid:8) ν ∈ Ω ( M ) | ν ∧ ϕ = ⋆ ϕ ν (cid:9) is a C ∞ ( M )-submodule of Ω ( M ). MARCO FREIBERT AND SIMON SALAMON
Closed G -structures. In this subsection, we consider the following situation.
Definition 2.11.
A G -structure ϕ ∈ Ω M on a seven-dimensional manifold M is called closed if dϕ = 0.If ϕ is a closed G -structure, then d ⋆ ϕ ϕ = τ ∧ ϕ for a unique two-form τ ∈ Ω ( M ). The two-form τ is called the torsion two-form of ϕ and encodes theintrinsic torsion of ϕ .The notion of a closed G -structure with special torsion of positive or negative type was introduced byBall in [Ba2] in an attempt to study so-called quadratic closed G -structures , as explained in the nextsubsection.To motivate the definition of closed G -structure with special torsion, let ϕ ∈ Ω M be a closed G -structure with associated torsion two-form τ . Then τ pointwise lies in the adjoint representation g . Theadjoint action of G on g has three different types of orbits, distinguished by the conjugacy classes of theG -stabiliser at some point in the orbit:There is the ‘generic’ case, where the stabiliser is a maximal torus T inside of G .Then there are two exceptional orbits, where the stabiliser is some copy of U(2), but the two copiesU(2) + and U(2) − are not conjugate to each other. The first orbit G / U(2) + is the twistor space of theWolf space G /SO (4), whereas G / U(2) − can be regarded as a twistor space of S ∼ = G / SU(3) and assuch is biholomorphic to a complex quadric [Br1].We say that ϕ has special torsion of positive type or negative type , respectively, if the pointwise stabiliserof τ is in each point conjugate to U(2) + or U(2) − , respectively. As shown in [Ba2], these conditions canbe characterised by properties of τ : Definition 2.12.
Let ϕ ∈ Ω M be a closed G -structure on a seven-dimensional manifold M withassociated torsion two-form τ ∈ Ω M . Then ϕ has special torsion of positive type if τ = 0, and ϕ has special torsion of negative type if (cid:12)(cid:12) τ (cid:12)(cid:12) ϕ = | τ | ϕ .2.3. Closed G -eigenforms for the Laplacian. In this subsection, we discuss properties of closedG -eigenforms (from now on, we shall omit the words ‘for the Laplacian’), and their relation to otherstructures. We repeat the definition: Definition 2.13.
A G -structure ϕ on a seven-dimensional manifold M is called a closed G -eigenform if dϕ = 0 and there exists some µ ∈ R \ { } such that∆ ϕ ϕ = µϕ. Remark . • Any closed G -structure ϕ (on a connected manifold) with ∆ ϕ ϕ = f ϕ for some f ∈ C ∞ ( M ), f = 0, is a closed G -eigenform. For, differentiation gives df ∧ ϕ = d ( f ϕ ) = d ∆ ϕ ϕ = d δ ϕ ϕ = 0 , and so df = 0 since wedging with ϕ injects Ω M into Ω M . Hence, f is constant. • A closed G -eigenform ϕ is an exact G -structure since ϕ = µ ∆ ϕ ϕ = d (cid:16) δ ϕ ϕµ (cid:17) . • A closed G -eigenform ϕ is an example of a Laplacian soliton , i.e. a soliton for the
Laplacian flow(of closed G -structures) given by ˙ ϕ t = ∆ ϕ t ϕ t . More generally, a
Laplacian soliton ϕ is a closed G -structure satisfying∆ ϕ ϕ = µϕ + L X ϕ for some X ∈ X ( M ), µ ∈ R . • Lotay and Wei showed in [LW1] that a compact seven-dimensional manifold cannot support anyLaplacian soliton ϕ with X = 0 unless ϕ is torsion-free. In particular, there do not exist any closedG -eigenforms on compact manifolds. • In [LW1] it is also shown that for a closed G -eigenform ϕ ∈ Ω M with ∆ ϕ ϕ = µϕ we must have µ > LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 7 Let ϕ be a closed G -eigenform on a seven-dimensional manifold M . By the last remark, we then have∆ ϕ ϕ = µϕ for some µ >
0. Hence, by scaling ϕ by an appropriate non-zero factor, we may and willassume for the rest of the article (unless stated otherwise) that µ = 1, i.e. that(2.1) ∆ ϕ ϕ = ϕ In the last subsection, we introduced the torsion 2-form τ ∈ Ω ( M ) uniquely defined by d ⋆ ϕ ϕ = τ ∧ ϕ. Since τ ∈ Ω ( M ), we do have τ ∧ ϕ = − ⋆ ϕ τ and so τ = ⋆ ϕ ⋆ ϕ τ = − ⋆ ϕ ( τ ∧ ϕ ) = − ⋆ ϕ d ⋆ ϕ ϕ = δ ϕ ϕ, which yields(2.2) ∆ ϕ ϕ = dδ ϕ ϕ = dτ Remark . Let ϕ be a closed G -eigenform. Then dτ = ∆ ϕ ϕ = µϕ for some µ >
0, which we do notassume to be equal to 1 in this remark. Since wedging with ⋆ ϕ ϕ is pointwise G -equivariant from Ω ( M )to Ω M and Ω M is pointwise isomorphic to the G -representation R , we must have τ ∧ ⋆ ϕ ϕ = 0. Thus,7vol ϕ = ϕ ∧ ⋆ ϕ ϕ = µ dτ ∧ ⋆ ϕ ϕ = − µ τ ∧ d ⋆ ϕ ϕ = − µ τ ∧ ϕ = µ τ ∧ ⋆ ϕ τ = | τ | ϕ µ vol ϕ , i.e. µ = | τ | ϕ , and so dτ = | τ | ϕ ϕ. In particular, | τ | ϕ is constant. Moreover, closed G -eigenforms are special kinds of so-called λ -quadraticclosed G -structures , which are closed G -structures ϕ ∈ Ω M fulfilling dτ = | τ | ϕ ϕ + λ (cid:16) | τ | ϕ ϕ + ⋆ ϕ ( τ ∧ τ ) (cid:17) for λ ∈ R , namely those for λ = 0.Note that λ (cid:0) | τ | ϕ ϕ + ⋆ ϕ ( τ ∧ τ ) (cid:1) lies in Ω ( M ), so the above decomposition can be seen as one of dτ ∈ Ω M into the three components Ω ( M ) := C ∞ ( M ) · ϕ , Ω ( M ) and Ω ( M ) of Ω M with theΩ ( M )-component being zero.More generally, for any closed G -structure ϕ , the Ω ( M )-part of dτ equals | τ | ϕ ϕ and the Ω ( M )-partof dτ vanishes, i.e. we always have dτ = | τ | ϕ ϕ + γ for some γ ∈ Ω ( M ).One can show that λ -quadratic closed G -structure are exactly those closed G -structures for which γ ∈ Ω ( M ), and so the entire three-form dτ , depends quadratically on τ , explaining the naming of thesestructures.We restrict now to left-invariant G -structures on seven-dimensional Lie groups G . These will from nowon be identified with the corresponding structures on the associated seven-dimensional Lie algebra g .As stated already, it is well known that a seven-dimensional nilpotent Lie algebra cannot admit anexact G -structure, see e.g. [CF], and so we have to look for exact G -structures on the more general classof solvable Lie algebras. We will give now a new proof of this fact, and in the process prove a slightlystronger result. Fist, we recall the following. Definition 2.16.
Let k be a Lie algebra.The descending central series k , k , . . . of k is defined by k := k , k := [ k , k ] and inductively by k k :=[ k , k k − ] for all k ∈ N .The ascending central series k , k , . . . of k is defined by k := { } , k := z ( k ) and inductively by k k := (cid:8) X ∈ k | [ X, k ] ⊆ k k − (cid:9) for all k ∈ N .Note that k is nilpotent if and only if k r = { } for some r ∈ N or, equivalently, if and only if k s = k forsome s ∈ N (and then r = s ).This allows us to prove: Proposition 2.17.
Let g be a seven-dimensional Lie algebra. If g admits an exact G -structure, then dim( z ( g )) ≤ and g = z ( g ) . In particular, if g is nilpotent, or, more generally, if g = a ⊕ b is a directsum of Lie algebras a , b with b nilpotent and dim( b ) ≥ , then it cannot admit any exact G -structure. MARCO FREIBERT AND SIMON SALAMON
Proof.
Assume that ϕ ∈ Λ g ∗ is an exact G -structure on g , i.e. dχ = ϕ for some χ ∈ Λ g ∗ . Let X, Y ∈ z ( g ). Then 0 = dχ ( X, Y, Z ) = ϕ ( X, Y, Z )for any Z ∈ g . Hence, by Lemma 2.10, the vectors X and Y have to be linearly dependent. Thus,dim( z ( g )) ≤ z ( g )) = 0, then trivially g = z ( g ). So let us assume that dim( z ( g )) = 1 and take X ∈ z ( g ) \ { } .If g = z ( g ), then there exists Y ∈ g linearly independent of X and we get [ Y, Z ] ∈ span( X ) and so0 = − χ ([ Y, Z ] , X ) = dχ ( X, Y, Z ) = ϕ ( X, Y, Z )for any Z ∈ g , which contradicts Lemma 2.10. Thus, g = z ( g ).So g certainly cannot be nilpotent nor can it be of the form g = a ⊕ b with b being nilpotent anddim( b ) ≥ (cid:3) Reduction to six dimensions
In this section, we reduce the existence problem of closed G -eigenforms or exact G -structures on aseven-dimensional Lie algebra g to the existence of SU(3)-structures of a certain type on a codimension-oneideal h satisfying specific equations. By [LN1, Proposition 3.2], such an ideal h always exists: Proposition 3.1.
Let g be a seven-dimensional Lie algebra admitting a closed G -structure ϕ ∈ Λ g ∗ .Then g admits a unimodular codimension-one ideal h . To obtain the reduction from seven to six dimensions, we need to recall what a derivation of a Liealgebra h is and how an endomorphism of h acts on Λ ∗ h ∗ : Definition 3.2.
Let h be a Lie algebra, f ∈ End( h ) be an (vector space) endomorphism of h and α ∈ Λ k h ∗ be a k -form on h . Then the k -form f.α ∈ Λ k h ∗ is defined by( f.α )( X , . . . , X k ) := − ( α ( f ( X ) , X , . . . , X k ) + . . . + α ( X , . . . , X k − , f ( X k ))) . A derivation of h is a (vector space) endomorphism f ∈ End( h ) of h such that f ([ X, Y ]) = [ f ( X ) , Y ] +[ X, f ( Y )] for all X, Y ∈ h . Remark . Let h be a Lie algebra, α ∈ Λ k h ∗ , β ∈ Λ l h ∗ and f, g ∈ End( h ). Then: • Due to the global minus sign in the definition of f.α , we have [ f, g ] .α = f. ( g.α ) − g. ( f.α ), i.e.End( h ) ∋ f f. ∈ End(Λ k h ∗ ) is a representation of the Lie algebra End( h ) on Λ k h ∗ . • Moreover, we have f. ( α ∧ β ) = f.α ∧ β + α ∧ f.β and so f. ( α ∧ α ) = 2 α ∧ f.α if k is even. • f is a derivation if and only if f.dγ = d ( f.γ ) for all one-forms γ ∈ h ∗ on h and then the same formulaholds for forms of arbitrary degree on h . Moreover, the vector space Der( h ) of all derivations of h is a subalgebra of the Lie algebra End( h ) of all (vector space) endomorphisms of h and it is theLie algebra of the Lie group Aut( h ) of all Lie algebra automorphisms of h , i.e. ofAut( h ) := { F ∈ GL( h ) | F ([ X, Y ]) = [ F ( x ) , F ( Y )] for all X, Y ∈ h } ⊆ GL( h ) . Let us begin with the reduction to six dimensions:For this, let ϕ be a closed G -eigenform on a seven-dimensional Lie algebra g and h be a codimension-oneideal. Choose e ∈ g of norm one in the orthogonal complement h ⊥ gϕ of h in h . Split now g = h ⊕ span( e )and, similarly, g ∗ = h ∗ ⊕ span( e ), where e is the unique element in the annihilator of h with e ( e ) = 1and h ∗ is identified with the annihilator of e . Set f := ad( e ) | h and note that f is a derivation of h . Then dα = d h α + e ∧ f.α, d ( α ∧ e ) = d h α ∧ e , de = 0for any α ∈ Λ k h ∗ , where d h is the differential of h . Next, decompose ϕ according to the splitting, i.e. write(3.1) ϕ = ω ∧ e + ρ with ω ∈ Λ h ∗ , ρ ∈ Λ h ∗ . Then ( ω, ρ ) is an SU(3)-structure on h by Lemma 2.9 and one has ⋆ ϕ ϕ = ω + e ∧ ˆ ρ. LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 9 We do the same for the torsion-two form τ ∈ Λ g ∗ of ϕ , i.e. we write(3.2) τ = ν + α ∧ e with ν ∈ Λ h ∗ and α ∈ h ∗ . Now the G -representation Λ g ∗ splits as SU(3)-representations into Λ g ∗ = h ∗ ∧ e ⊕ [Λ , h ∗ ] as Λ g ∗ is the adjoint representation of G and [Λ , h ∗ ] is the adjoint representation ofSU(3). Thus, ν ∈ [Λ , h ∗ ] and so ν ∧ ρ = 0. Consequently, d h ( ω ) + e ∧ ( ω ∧ f.ω − d h ˆ ρ ) = d ⋆ ϕ ϕ = τ ∧ ϕ = ( ν + α ∧ e ) ∧ ( ω ∧ e + ρ )= e ∧ ( ω ∧ ν − α ∧ ρ ) , i.e. d h ( ω ) = 0 , ω ∧ f.ω − d h ˆ ρ = ω ∧ ν − α ∧ ρ. Moreover, d h ν + e ∧ ( f.ν − d h α ) = dτ = ϕ = ω ∧ e + ρ, i.e. d h ν = ρ, f.ν − d h α = ω, Hence, d h ρ = 0 and d h ( ω ) = 0 and so the SU(3)-structure ( ω, ρ ) on h is half-flat with exact ρ . Moreover,we necessarily have α = 0. For this, note that τ ∈ Ω M = { β ∈ Ω M | ⋆ ϕ β = − β ∧ ϕ } and so τ = ⋆ ϕ τ = − ⋆ ϕ ( τ ∧ ϕ ) = − ⋆ ϕ (cid:0) e ∧ ( ω ∧ ν − α ∧ ρ ) (cid:1) ∈ Λ h ∗ since e is perpendicular to h ∗ by assumption. Consequently, τ = ν ∈ Λ h ∗ and α = 0.Summarizing, we have arrived at Theorem 3.4.
Let g be a seven-dimensional Lie algebra, ϕ ∈ Λ g ∗ be a G -structure on g , h be acodimension-one unimodular ideal in g , e ∈ h ⊥ gϕ of norm one, e ∈ Ann( h ) with e ( e ) = 1 and f := ad( e ) | h . Write ϕ = ω ∧ e + ρ with ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ . Then ϕ is a closed G -eigenform with ∆ ϕ ϕ = ϕ if and only if ( ω, ρ ) is half-flat and there exists a primitive (1 , -form ν ∈ [Λ , h ∗ ] on h suchthat ρ = d h ν and f.ν = ω, (3.3) ω ∧ f.ω − d h ˆ ρ = ω ∧ ν. (3.4)If we only look for exact G -structures ϕ ∈ Λ g ∗ , the same calculations as above show: Theorem 3.5.
Let g be a seven-dimensional Lie algebra, ϕ ∈ Λ g ∗ be a G -structure on g , h be acodimension-one unimodular ideal in g , e ∈ h ⊥ gϕ of norm one, e ∈ Ann( h ) with e ( e ) = 1 and f := ad( e ) | h . Write ϕ = ω ∧ e + ρ with ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ . Then ϕ is an exact G -structure if andonly if there exists a two-form ν ∈ Λ h ∗ on h and a one-form α ∈ h ∗ with ρ = d h ν and (3.5) f.ν − d h α = ω. Results in dimension six
From now on, we restrict to ourselves to a special class of Lie algebras:
Definition 4.1.
A Lie algebra g is called almost nilpotent if it admits a codimension-one nilpotent ideal h . Note that then g ∼ = h ⋊ f R for a derivation f ∈ Der( h ), where h ⋊ f R denotes the semi-direct productof R with h and Lie algebra representation ρ : R → Der( h ) of R on h given by ρ ( t ) = tf for all t ∈ R .In order to investigate the existence of exact G -structures and closed G -eigenforms on seven-dimensionalalmost nilpotent Lie algebras g , we first have to determine which six-dimensional nilpotent Lie algebrasadmit exact SL(3 , C )-structures or half-flat SU(3)-structures ( ω, ρ ) for which there exists a primitive (1 , ν with ρ = dν . Exact
SL(3 , C ) -structures on nilpotent Lie algebras. We start by determining the six-dimensionalnilpotent Lie algebras h admitting an exact SL(3 , C )-structure ρ ∈ Λ h ∗ . To this aim, we rephrase thecondition of being exact in the following way: Proposition 4.2.
Let h be a six-dimensional nilpotent Lie algebra. Then h admits an exact SL(3 , C ) -structure ρ ∈ Λ h ∗ if and only if there exist linear independent one-forms α , α ∈ h ∗ and two-forms ω , ω ∈ Λ h ∗ with ω i ∧ ω j = δ ij ω for i, j ∈ { , } such that ker( ω ) = ker( ω ) is a complement of ker( α ) ∩ ker( α ) in h and such that either (a) dim( z ( h )) = 1 , dim( h ) = 2 , ker( ω ) = ker( ω ) = h and there exists a closed non-zero one-form γ ∈ h ∗ \ { } with γ ( h ) = { } such that dα = ω , dα = ω + γ ∧ α (b) or dim( z ( h )) = 2 , ker( ω ) = ker( ω ) = z ( h ) and dα = ω , dα = ω . In the first case, h is J -invariant and in the second case z ( h ) for the almost complex structure J inducedby ρ .Proof. The forward implication. Assume that h admits an exact SL(3 , C )-structure ρ , i.e. ρ = dν for some ν ∈ Λ h ∗ . Let X, Y ∈ z ( h ). Then ρ ( X, Y, Z ) = dν ( X, Y, Z ) = 0for any Z ∈ h , which implies that X and Y are C -linearly dependent by Lemma 2.5 (a). So dim( z ( h )) ∈{ , } .If dim( z ( h )) = 2, z ( h ) is J -invariant, so we may choose a basis X, JX of z ( h ). By Lemma 2.5 (b), ω := ρ ( X, · , · ) and ω := ρ ( JX, · , · ) have two-dimensional common kernel z ( h ) and fulfill ω i ∧ ω j = δ ij ω for i, j ∈ { , } .Moreover, setting α := − ν ( X, · ) ∈ h ∗ and α := − ν ( JX, · ) ∈ h ∗ , we have dα ( Y, Z ) = − α ([ Y, Z ]) = − ν ([ Y, Z ] , X ) = dν ( X, Y, Z ) = ρ ( X, Y, Z ) = ω ( Y, Z )for all
Y, Z ∈ h , i.e. dα = ω . In the same way, one obtains dα = ω , which then also shows that α and α are linearly independent. Now choose Y ∈ h linearly independent of X and JX . By Lemma 2.5 (a),there exists Z ∈ h with ρ ( X, Y, Z ) = 0 and so0 = ρ ( X, Y, Z ) = ν ( X, [ Y, Z ]) . Since [
Y, Z ] ∈ z ( h ) = span( X, JX ), this shows α ( JX ) = − α ( X ) = ν ( X, JX ) = 0. Thus ker( α ) ∩ ker( α )is complementary to ker( ω ) = ker( ω ) = z ( h ).Next, consider the case dim( z ( h )) = 1 and choose X ∈ z ( h ) and Y ∈ h linearly independent. Then wehave ρ ( X, Y, Z ) = dν ( X, Y, Z ) = ν ( X, [ Y, Z ]) = 0for any Z ∈ h , i.e. X and Y are C -linearly dependent by Lemma 2.5 (a). Hence dim( h ) = 2 and h is J -invariant.Choose a basis X, JX of h such that X ∈ z ( h ) and set again ω := ρ ( X, · , · ), ω := ρ ( JX, · , · ), α := − ν ( X, · ) and α := − ν ( JX, · ). As in the case dim( z ( h )) = 2, we get ker( ω ) = ker( ω ) = span( X, JX ) = h , ω i ∧ ω j = δ ij ω for i, j = 1 , dα = ω .Next, let Y ∈ h linearly independent of X and JX . By Lemma 2.5 (a), we again have some Z ∈ h with 0 = ρ ( X, Y, Z ) = ν ( X, [ Y, Z ]) and from [
Y, Z ] ∈ h = span( X, JX ) we get again that α ( JX ) = − α ( X ) = ν ( X, JX ) = 0, i.e. that ker( α ) ∩ ker( α ) is complementary to ker( ω ) = ker( ω ) = z ( h ). So wefinally have to prove the equation for dα in this case. Thereto, let γ ∈ h ∗ \ { } be the one-form uniquelydefined by [ JX, Y ] = − γ ( Y ) X for all Z ∈ h . Obviously, γ ( X ) = γ ( JX ) = 0, i.e. γ ( h ) = { } . Moreover, dγ = 0 as dγ ( Z, W ) X = − γ ([ Z, W ]) X = [ JX, [ Z, W ]] = [ Z, [ W, JX ]] + [ W, [ JX, Z ]]= − γ ( W )[ Z, X ] + γ ( Z )[ W, X ]= 0
LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 11 for all Z, W ∈ h . Furthermore, dα ( Y, Z ) = − α ([ Y, Z ]) = − ν ([ Y, Z ] , JX )= dν ( JX, Y, Z ) + ν ([ JX, Y ] , Z ) − ν ([ JX, Z ] , Y )= ρ ( JX, Y, Z ) − ν ( γ ( Y ) X, Z ) + ν ( γ ( Z ) X, Y )= ω ( Y, Z ) + γ ( Y ) α ( Z ) − γ ( Z ) α ( Y )= ω ( Y, Z ) + ( γ ∧ α )( Y, Z )as claimed.
The backwards implication.
Assume that there exist linear independent one-forms α , α ∈ h ∗ and two-forms ω , ω ∈ Λ h ∗ as in the statement. Note that dim(ker( α ) ∩ ker( α )) = 4 since α , α are linearlyindependent. Consequently, dim(ker( ω )) = dim(ker( ω )) = 2 and so ω , ω are non-degenerate two-formson V := ker ( α ) ∩ ker( α ) satisfying ω i ∧ ω j = δ ij ω for i, j = 1 , ω = 0. Then it is well-known thatthere is a basis ( v , . . . , v ) such with respect to the dual basis ( v , . . . , v ) we have ω = v + v , ω = v − v , cf., e.g., the proof of Lemma 2.2 in [FY]. Consider ( v , . . . , v ) as one-forms on h by identifying V ∗ withthe annihilator of ker( ω ) = ker( ω ) and set ρ := α ∧ ω − α ∧ ω = − α ∧ v + α ∧ v + α ∧ v + α ∧ v ρ is an SL(3 , C )-structure on h ∗ since an adapted basis is given by ( α , α , v , v , v , − v ). Moreover, ρ isexact since ν := α ∈ Λ h ∗ satisfies dν = dα ∧ α − α ∧ dα = α ∧ ω − α ∧ ω = ρ in both cases. (cid:3) There are 34 (isomorphism classes of) real six-dimensional nilpotent Lie algebras. Of these, exactlythose five admits an exact SL(3 , C )-structures which are listed in Table 1. The notation for these Liealgebras is obtained by numbering the 34 six-dimensional nilpotent Lie algebras from n to n in theorder in which they occur in Table A.1 in [Sa]. Corollary 4.3.
Let h be a six-dimensional nilpotent Lie algebra. Then h admits an exact SL(3 , C ) -structure if and only if h is one of the five Lie algebras listed in Table 1. g differentials n (0 , , , ,
14 + 23 , − n (0 , , , ,
14 + 23 ,
24 + 15) n (0 , , , , − ,
15 + 34) n (0 , , , ,
13 + 42 ,
14 + 23) n (0 , , , ,
13 + 42 ,
14 + 23)
Table 1.
Six-dimensional nilpotent Lie algebras admitting an exact SL(3 , C )-structure Proof.
By Proposition 4.2, we either have dim( z ( h )) = 2 or dim( z ( h )) = 1 and dim( h ) = 2.Let us first assume that dim( z ( h )) = 2. By Proposition 4.2, there are closed two-forms ω , ω ∈ Λ h ∗ with common kernel ker( ω ) = ker( ω ) = z ( h ) such that ω i ∧ ω j = δ ij ω . Moreover, there are linearlyindependent one-forms α , α ∈ h ∗ such that dα i = ω i for i = 1 , α ) ∩ ker( α ) iscomplementary to z ( h ). Now h / z ( h ) is a four-dimensional nilpotent Lie algebra and ω , ω descend toclosed two-forms on h / z ( h ), again called ω , ω . It is well known, see e.g. [Ov], that there are exactly threefour-dimensional nilpotent Lie algebras, namely (0 , , , , , ,
0) and (0 , , , , , ,
0) and (0 , , ,
0) admit closed two-forms ω , ω with ω i ∧ ω j = δ ij ω .If h / z ( h ) ∼ = (0 , , , e , . . . , e of thedual space of (0 , , ,
0) such that ω = e − e and ω = e + e . We may extend this basis to a basis e , . . . , e of h ∗ by e := α and e := α and so h ∼ = (0 , , , ,
13 + 42 ,
14 + 23) = n . If h / z ( h ) ∼ = (0 , , , ω is a symplectic form on (0 , , ,
0) and so symplectomorphic to e + e by [Ov], i.e. we may assume that ω = e + e . Then, since ω is closed, ω ∧ ω = 0 and ω = ω , onechecks that ω = a ( e − e ) + b e − b e + ce for certain a, b , b , c ∈ R with b b − a = 1. But so f := ab − ab − cb − acb b is an automorphism of (0 , , ,
0) with f ∗ ω = ω and f ∗ ω = b e − b e . Next, g := diag (cid:0) b − , b , b − , b (cid:1) fulfills g ∗ f ∗ ω = ω and g ∗ f ∗ ω = e − e . Extending e , . . . , e to a basis e , . . . , e by setting e := α , e := α , we do get h ∼ = (0 , , , ,
14 + 23 ,
13 + 42) ∼ = (0 , , , ,
13 + 42 ,
14 + 23) = n , where the latterisomorphism F is, e.g., given by the one with by F ( e ) = − e , F ( e ) = e , F ( e ) = e , F ( e ) = − e , F ( e ) = e and F ( e ) = e .Next, let dim( z ( h )) = 1 and dim( h ) = 2. By Proposition 4.2, there are two-forms ω , ω ∈ Λ h ∗ withcommon kernel ker( ω ) = ker( ω ) = h such that ω i ∧ ω j = δ ij ω . Moreover, there are linearly independentone-forms α , α ∈ h ∗ and γ ∈ h ∗ \ { } closed with γ ( h ) = { } such that dα = ω , dα = ω + γ ∧ α and such that ker( α ) ∩ ker( α ) is complementary to z ( h ). Note that then ω is closed and dω = γ ∧ ω . Hence, h y dω = 0 and so ω , ω , γ descend to forms on a := h / h with dω = 0 and dω = γ ∧ ω . Since dω = 0 on a , a cannot be Abelian and we must either have a ∼ = (0 , , ,
0) or a ∼ = (0 , , , a ∼ = (0 , , , a are symplec-tomorphic to each other. Hence, we may assume that ω = e − e . Then ω = a e + a e + b ( e + e ) + c e + c e for certain a , a , b, c , c ∈ R with a a + c c − b = 1. We must have a = 0 asotherwise dω = 0, a contradiction. But so the automorphism − a a c a − ba − a − ba c a a of (0 , , ,
0) is well-defined. That automorphism fixes ω and transforms ω into − e − e . Hence,we may assume that ω = e − e and that ω = − e − e . Then dω = − e = e ∧ ω , i.e. γ = e . Thus, extending e , . . . , e to a basis e , . . . , e of h ∗ by e := α and e := α + e , we have h = (0 , , , , − , − ∼ = (0 , , , , − ,
15 + 34) = n , where the latter isomorphism fixes e i for i / ∈ { , } and interchanges e and e .Next, let us consider the case a ∼ = (0 , , , a are symplecto-morphic. Thus, we may assume that ω = e + e . Then ω = a e + a e + b e − b e + c ( e − e )for certain a , a , b , b , c ∈ R with a a + b b − c = 1.Let us first assume that a = 0. Then − b a − a c + b a b a a b + b ca − ca b a is an automorphism of (0 , , ,
13) which fixes ω and maps ω to a e + a e , i.e. may assume that ω = a e + a e . Then dω = a e = − a e ∧ ω , i.e. γ = − a e . Hence, extending e , . . . , e to abasis e , . . . , e of h ∗ by e := α and e := a ( α − a e ), we have h = (0 , , , ,
14 + 23 , −
25) = n . LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 13 Finally, we consider the case a = 0. Then b b − c = 1 and so, in particular, b = 0. Thus, cb − cb − a b + cb c b − cb is a well-defined automorphism of (0 , , ,
13) which fixes ω and maps ω to b e − b e . Hence,we may assume that ω = b e − b e and then γ = − b e as dω = − b e = − b e ∧ ω . Thus,extending e , . . . , e to a basis e , . . . , e of h ∗ by e := α and e := − b (cid:16) α − b e (cid:17) , we have h =(0 , , , ,
14 + 23 ,
24 + 15) = n .Conversely, the existence of forms as in Proposition 4.2 follows from the discussion above on any of theLie algebras n , n , n , n and n . (cid:3) Half-flat
SU(3) -structures ( ω, ρ ) with exact ρ . Here, we determine the six-dimensional nilpotentLie algebras which admit a half-flat SU(3)-structure ( ω, ρ ) for which ρ = dν for a primitive (1 , ν .In fact, we will determine all nilpotent Lie algebras which admit a half-flat SU(3)-structure ( ω, ρ ) withexact ρ and show that these are the same for which ρ = dν with a primitive (1 , ν .For this, note that by Corollary 4.3, only n , n , n , n , n may admit a half-flat SU(3)-structure( ω, ρ ) with exact ρ . Now Conti determined the six-dimensional nilpotent Lie algebras admitting a half-flatSU(3)-structures in [C] and his results reduce the possible cases to n , n , n . We will show that n cannotadmit a half-flat SU(3)-structure ( ω, ρ ) with exact ρ and for the proof we use the following obstructionby Schulte-Hengesbach and the first author [FrSH] adapted to our setting. Note that this obstruction is arefinement of one used by Conti in [C]: Lemma 4.4.
Let h be a six-dimensional Lie algebra and ν ∈ Λ h ∗ \ { } . If there is a non-zero one-form α ∈ h ∗ satisfying (4.1) α ∧ ˜ J ∗ τ α ∧ σ = 0 for all exact three-forms τ ∈ Λ h ∗ and all closed four-forms σ ∈ Λ h ∗ , where ˜ J ∗ τ α is defined for X ∈ h ∗ by ˜ J ∗ τ α ( X ) ν = α ∧ ( X y τ ) ∧ τ, (4.2) then g does not admit a half-flat SU(3) -structure ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ with exact ρ . This allows us now to prove:
Theorem 4.5.
Let h be a six-dimensional nilpotent Lie algebra. Then h admits a half-flat SU(3) -structure ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ with exact ρ if and only if h is isomorphic to n or n . In these cases, h also admitsa half-flat SU(3) -structure (˜ ω, ˜ ρ ) ∈ Λ h ∗ × Λ h ∗ with ˜ ρ = dν for some primitive (1 , -form ν ∈ [Λ , h ∗ ] .Proof. As explained above, by the results of [C] and Corollary 4.3, only n , n or n may admit a half-flat( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ with exact ρ Now a direct computation, efficiently carried out with a computer algebra system like MAPLE, showsthat one may the obstruction in Lemma 4.4 with, e.g. α = e or α = e , to exclude the existence of ahalf-flat SU(3)-structure ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ with exact ρ on n .For the other two cases, we provide a half-flat SU(3)-structure (˜ ω, ˜ ρ ) ∈ Λ h ∗ × Λ h ∗ and some ν ∈ [Λ , h ∗ ]with ˜ ρ = dν : • n : Here, we may take the SU(3)-structure defined by the adapted basis ( e , e , e , e , e , − e ), i.e. ω = e + e − e , ρ = e + e − e − e . Then one checks that d ( ω ) = 0. Moreover, set ν := e + e + e + e . Then dν = ρ and ρ is a (1 , ν ∧ ω = 0, ν is primitive as well, i.e. ν ∈ [Λ , h ∗ ]. • n : Take the SU(3)-structure defined by the adapted basis ( e , e , e , e , e , e ), i.e. ω = e + e − e , ρ = e − e − e − e . Then d ( ω ) = 0. Setting now ν := e + e , we get dν = ρ and that ρ is a (1 , ν ∧ ω = 0 and so ν is primitive, i.e. ν ∈ [Λ , h ∗ ]. (cid:3) Remark . Fino and Raffero determined in [FR] all six-dimensional nilpotent Lie algebras admitting aso-called coupled half-flat SU(3)-structure, i.e. a half-flat SU(3)-structure ( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ with dω = ρ .Interestingly, the six-dimensional nilpotent Lie algebras admitting a coupled half-flat SU(3)-structure arealso n and n .Our proof of Theorem 4.5 is independent of the coupled approach, and in some sense more direct. Finoand Raffero compute with a computer algebra system for all 24 six-dimensional nilpotent Lie algebrasadmitting a half-flat SU(3)-structure the most general exact three-form ρ and check if the quartic invariant λ of ρ can be negative. This way they obtain that, of the six-dimensional nilpotent Lie algebras admittinga half-flat SU(3)-structure, those which admit a maybe non half-flat SU(3)-structure with exact three-formpart are precisely n , n and n . Then they show by different methods that n cannot admit a coupledSU(3)-structure.5. Exact G -structures on compact almost nilpotent solvmanifolds Here, we prove that a compact almost nilpotent solvmanifold cannot admit an invariant exact G -structure.For this, note first that Corollary 4.3 implies the following: Corollary 5.1.
Let g be a seven-dimensional almost nilpotent Lie algebra with codimension-one nilpotentideal h . If g admits an exact G -structure, then h is isomorphic to n , n , n , n or n . We show now that four of the five cases of a codimension-one nilpotent ideal, namely n , n , n and n , may occur in Corollary 5.1 leaving open if there is an almost nilpotent Lie algebra with codimensionone ideal n which admits an exact G -structure.For this, note that Theorem 6.9 below even classifies all the almost nilpotent Lie algebra with codimension-one nilpotent ideal isomorphic to n which admit an exact G -structure. For h ∈ { n , n n } , we providenow one example of an exact G -structure on a seven-dimensional almost nilpotent Lie algebra withcodimension-one nilpotent ideal h : Example 5.2.
For the six-dimensional nilpotent Lie algebras h ∈ { n , n , n } , we give an SU(3)-structure( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ , a two-form ν ∈ Λ h ∗ , a one-form α ∈ h ∗ and a derivation f ∈ Der( h ) such that ρ = dν and such that (3.5) is valid. • n : Take the SU(3)-structure ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ defined by the adapted basis (cid:0) − e + ae , e + ae ,e , e , e , e (cid:1) with a := √ , i.e. ω = − e − a e + e − a e + (cid:0) a (cid:1) e , ρ = − e + e + e − e . Setting ν := (cid:0) − a (cid:1) e − e + (cid:0) − a (cid:1) e − e + e , one gets dν = ρ . Moreover, f := − a − a − a − a − a − − a . is a derivation of n and one computes f.ν = ω + e . Hence, choosing α := e , we have dα = e and so f.ν − dα = ω , i.e. (3.5) is fulfilled. • n : In this case, we choose the SU(3)-structure ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ defined by the adapted basis( e , e , e , e , e , − e ), i.e. ω = e + e − e , ρ = e + e − e − e . Setting ν := − e + e + e + e + e , one obtains dν = ρ . Moreover, f := diag (cid:0) , − , , − , , (cid:1) is a derivation of n and f.ν = e + e − e = ω . Thus, for α := 0, (3.5) is satisfied. LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 15 • n : Here, we look at the SU(3)-structure ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ defined by the adapted basis( e , e , e , e , e , e ), i.e. ω = e + e − e , ρ = e − e − e − e . Taking ν := e − e + e , we get dν = ρ . Now one checks that f := −
00 0 0 0 0 . is a derivation of n and that f.ν = e − e . Thus, for α := − e , we have dα = − e and so f.ν − dα = e + e − e = ω , i.e. (3.5) is valid for our choices.Next, we look at compact almost nilpotent solvmanifolds, i.e. manifolds of the form Γ \ G , where G isa simply-connected almost nilpotent Lie group and Γ a cocompact lattice in G . A necessary condition forthe existence of such a lattice is that the associated Lie algebra g is strongly unimodular , cf. [G]: Definition 5.3.
Let g be a solvable Lie algebra g , n be its nilradical and n , n , . . . be the descendingcentral series of n . One checks that ad X preserves n i for all X ∈ g and all i ∈ N . g is called stronglyunimodular if tr(ad X | n i / n i +1 ) = 0 for all i ∈ N and all X ∈ g . Remark . Since the commutator ideal [ g , g ] of a solvable Lie algera g is nilpotent, the nilradical n contains the commutator ideal [ g , g ]. Hence, if g is strongly unimodular, one has tr(ad X ) = 0 for all X ∈ g , i.e. g is unimodular. Theorem 5.5.
Let g be a seven-dimensional strongly unimodular almost nilpotent Lie algebra. Then g does not admit an exact G -structure.Proof. Assume the contrary. By Corollary 5.1 the Lie algebra g then admits a codimension-one nilpotentideal h which is isomorphic to n , n , n , n or n . Moreover, h is the nilradical as the entire Lie algebracannot be nilpotent according to Proposition 2.17. Furthermore, we have an induced SU(3)-structure( ω, ρ ) ∈ Λ h ∗ × Λ h ∗ on h with exact ρ , i.e. there is some ν ∈ Λ h ∗ with dν = ρ which has to fulfill f.ν − d h α = ω for some one-form α ∈ h ∗ by Theorem 3.5. Now we know that in the cases n , n and n , we havedim( z ( h )) = 1 and dim( h ) = 2 with h being J -invariant by Proposition 4.2 for the almost complexstructure J induced by ρ . Moreover, in all theses cases, one checks that h is the sum of quotient spacesof the form h i / h i +1 , i.e. the trace of each ad X , X ∈ g , has to be trace-free on h . In these cases, we set a := h .In the cases n and n , we have dim( z ( h )) = 2 and z ( h ) is J -invariant by Proposition 4.2. Moreover, z ( h ) equals in both cases the last non-zero h i , so is of the form h i / h i +1 . Hence, each ad X , X ∈ g , has tobe trace-free when restricted to z ( h ). Here, we set a := z ( h ).Now coming back to genereal case, choose some 0 = X ∈ z ( h ) ⊆ a . Then we get0 = − tr( f | a ) ν ( X, JX ) = ( f.ν − d h α )( X, JX ) = ω ( X, JX ) = − k X k = 0 , since f has to preserve a = span( X, JX ). This yields the desired contradiction and so g cannot admit anexact G -structure. (cid:3) In general, if G is a simply-connected solvable Lie group which admits a cocompact lattice Γ, then anyleft-invariant differential form β induces a differential form ˜ β on the compact quotient Γ \ G . We then call˜ β invariant . By [OTr, Theorem 3.2.10], the assignment β ˜ β induces an injection H ∗ ( g ) → H ∗ dR (Γ \ G ).Hence, Theorem 5.5 implies that no compact almost nilpotent solvmanifold can admit an invariant exact G -structure. If G is completely solvable , i.e. ad X has only real eigenvalues for all X ∈ g , then H ∗ ( g ) → H ∗ dR (Γ \ G ) is an isomorphism by [H] and so one may skip the word ‘invariant’ in the statement: Corollary 5.6.
Let M = Γ \ G be an almost nilpotent solvmanifold, i.e. G is a simply-connected almostnilpotent Lie group and Γ a cocompact lattice in G . Then M does not admit an invariant exact G -structure. If G is completely solvable, then M does not admit any exact G -structure at all. Closed G -eigenforms on almost nilpotent Lie algebras In this section, we establish:
Theorem 6.1.
Let g be a seven-dimensional almost nilpotent Lie algebra. Then g does not admit a closed G -eigenform. To start the proof, note that by Theorem 3.4 and Theorem 4.5, the codimension-one nilpotent ideal h of an almost nilpotent Lie algebra admitting a closed G -eigenform has to be isomorphic to n or to n .In Subsection 6.1, we will show in Theorem 6.6 that no almost nilpotent Lie algebra with codimension-one nilpotent ideal isomorphic to n can admit a closed G -eigenform and in Subsection 6.2, we will showin Theorem 6.12 that no almost nilpotent Lie algebra with codimension-one nilpotent ideal isomorphic to n can admit a closed G -eigenform. This work completes the proof of Theorem 6.1.In Subsection 6.2, we also give a classification of all almost nilpotent Lie algebras with codimension-onenilpotent ideal isomorphic to n that admit an exact G -structure, and we distinguish those with specialtorsion of positive type or of negative type, respectively.6.1. The case n . Note first that the Lie algebra Der( n ) of all derivations of n is given byDer( n ) = f , − f , f , − f , +2 f , f , − f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , − f , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f i,j ∈ R . (6.1)with respect to the basis ( e , . . . , e ) of n . This can be checked by a lengthy but straightforward calculationdone efficiently with a computer algebra system like MAPLE. Exponentials of these derivations are then(inner) automorphisms of the Lie algebra n . Using these automorphisms, one obtains: Lemma 6.2.
Let ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ be an SU(3) -structure on n with exact ρ , i.e. there exists some ν ∈ Λ n ∗ with dν = ρ . Then ω, ρ and ν are given, up to automorphism, by (6.2) ρ = ǫ ( e + e − e − e ) =: ǫρ ,ω = a e + a e + a e + a ( e + e ) + a (cid:0) e − e (cid:1) + a (cid:0) e − e (cid:1) + a (cid:0) e + e (cid:1) ,ν = b e + b e + b e + b (cid:0) e + e (cid:1) + b (cid:0) e + e (cid:1) + b e + b e + ǫ (cid:0) e + e (cid:1) + ǫ e , for certain ǫ ∈ { , − } , a , . . . , a ∈ R with a a > , a a < and certain b , . . . , b ∈ R . If ( ω, ρ ) ishalf-flat and ν of type (1 , , then we may assume that a = a = 0 , b = b = 0 and b = b in (6.2) .Proof. First of all, observe that the most general exact three-form ρ is given by ρ = c e + c e + c e + c e + c e + c e + c e + c e + c e − c e − c e for certain c i ∈ R , i = 1 , . . . ,
8. The quartic invariant λ ( ρ ) of ρ computes to be equal to λ ( ρ ) = − c ( c c − c ) (cid:0) e (cid:1) ⊗ . As λ ( ρ ) has to be negative, we surely must have c = 0. Now exponentials of matrices as in (6.1) giveautomorphisms of n . We consider first the automorphism F := exp( A ) for the matrix A as in (6.1)with f , = 0, f , = 0, f , = 0, f , = 0, f , := 0, f , = 0 and f , = − f , f , . For notationalsimplicity, we set a := f , , b := f , , s := f , and so have A := a b − ab s a b s b The exponential of this matrix is easily computed to be F := a b S a b
A B C S b LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 17 with A = as − ab , B = bs + b , C = ab and S = s + b . Then( F ∗ ρ )( e , e , e ) = c + bc , ( F ∗ ρ )( e , e , e ) = c + ac , ( F ∗ ρ )( e , e , e ) = c − sc So, setting a := − c c and b := − c c and s := c c , we get that ( F ∗ ρ )( e , e , e ) = ( F ∗ ρ )( e , e , e ) = 0,( F ∗ ρ )( e , e , e ) = 0 and ( F ∗ ρ )( e , e , e ) = 0, i.e. we have F ∗ ρ = ˜ c e + ˜ c e + ˜ c e + ˜ c e + ˜ c (cid:0) e − e − e (cid:1) , for certain ˜ c i ∈ R , i = 1 , . . . ,
5, now with ˜ c = 0 (in fact, ˜ c = c ). Next, we consider the automorphism F := exp( A ) with the matrix A as in (6.1) with f , := 0, f , = 0, f , = 0, f , = 0, f , = 0, f , = 0and f , = 0, i.e. we have A := p q r , where we set p := f , , q := f , and r := f , . With F := exp( A ) = I + A , we obtain( F ∗ F ∗ ρ )( e , e , e ) = ˜ c − ˜ c p, ( F ∗ F ∗ ρ )( e , e , e ) = ˜ c − ˜ c q, ( F ∗ F ∗ ρ )( e , e , e ) = ˜ c − ˜ c r. Thus, setting p := ˜ c ˜ c , q := ˜ c ˜ c and r := ˜ c ˜ c , we get that F ∗ F ∗ ρ = Ae + B (cid:0) e − e − e (cid:1) . Then λ ( F ∗ F ∗ ρ ) = − AB , i.e. we must have A · B >
0. It is now fairly easy to see that the exponential F := exp( A ) of a diagonal matrix A as in (6.1) allows to normalise A = B ∈ {− , } , i.e. we have( F ∗ F ∗ F ∗ ρ ) = ǫ ( e + e − e − e ) = ǫρ for some ǫ ∈ {− , } .From now on, we will use again ρ for F ∗ F ∗ F ∗ ρ , so that ρ = ǫρ . Note that ǫ ( e , − e , e , − e , e , e ) or ǫ ( e , e , e , e , e , − e ) is an oriented adapted basis for ρ = ǫρ , depending on the orientation induced by ω . Hence, the induced almost complex structure J = J ρ is either given by J e = − e , J e = − e and J e = e or by − J . A straightforward computation shows that a two-form ν ∈ Λ n ∗ with dν = ρ has tobe as claimed and ν is of type (1 ,
1) precisely when b = b = 0 and b = b .Next, we are interested in bringing ω into a canonical form. To this aim, note first that ω has to be a(1 , J and so ω = a e + a e + a e + a (cid:0) e + e (cid:1) + a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1) + a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1) + a (cid:0) e + e (cid:1) for certain a , . . . , a ∈ R . Observe that a = ω ( e , e ) = ω ( e , ∓ Je ) = ± g ( e , e ), a = ω ( e , e ) = ± g ( e , e ) and a = ω ( e , e ) = ∓ g ( e , e ), and so a a > a a < ω into a form with less parameters without changing ρ , we need to look at thosematrices A in (6.1) that are in the Lie algebra a of the stabiliser group of ρ . Such an A has to commutewith J , which is the case if and only if f , = f , , f , = 0, f , = 0, f , = 0, f , = − f , , f , = 0, f , = 0. Moreover, the complex ± J -trace of A must be equal to zero, which additionally gives us f , = 0.So A is given by A = x x y − y for x := f , and y := f , . Then F := exp( A ) = I + A and( F ∗ ω )( e , e ) = a + xa − ya , ( F ∗ ω )( e , e ) = a + xa + ya . Now g ( e , e ) = ± a , g ( e , e ) = ∓ a , as we observed above, and g ( e , e ) = ω ( e , Je ) = ± ω ( e , e ) = ± a . Since any minor of g has to be non-zero, we get 0 = − ( g ( e , e ) g ( e , e ) − g ( e , e ) ) = a a + a .Thus, setting x := − a a + a a a a + a , y := − a a − a a a a + a yields ( F ∗ ω )( e , e ) = ( F ∗ ω )( e , e ) = 0 and ( F ∗ ω )( e , e ) = ( F ∗ ω )( e , e ) = 0. Hence, renaming F ∗ ω by ω and using again coefficients labeled a , . . . , a , we have ω = a e + a e + a e + a ( e + e ) + a (cid:0) e − e (cid:1) + a (cid:0) e − e (cid:1) + a (cid:0) e + e (cid:1) . Then d (cid:0) ω (cid:1) = ( a a + a a + ( a a − a a )) e + ( a a − a a ) e , i.e. ( ω, ρ ) is half-flat if and only if a a + a a = 0 , a a − a a = 0 . The first equation gives us a = − a a a and inserting this into the second equation yields0 = a a − a a = − a a a − a a a a = − a ( a a + a ) a Since a plays the role of the former a , we showed above that a a + a = 0. Thus, a = 0 and so a = 0.The equations a a + a a = 0, a a − a a = 0 are now fulfilled, so this finishes the proof. (cid:3) For the rest of this subsection, we assume that ( ω, ρ, ν ) is as in (6.2) with ( ω, ρ ) being half-flat and ν being a primitive form of type (1 , ω ∈ R + · e , i.e. ω induces the orientation in which the ordered basis( e , . . . , e ) is oriented. This follows from the observation that with ( ω, ρ, ν, f ) also ( − ω, ρ, ν, − f )satisfies (3.3) and (3.4).(ii) Moreover, we may assume that ǫ = 1 as with ( ω, ρ, ν, f ) also ( ω, − ρ, − ν, − f ) fulfills (3.3) and (3.4)Using these simplifications, we obtain: Lemma 6.3.
We have f , = f , = f , = 0 , f , = − f , , f , = 2 f , ,a = f , − f , , a = f , , a = − f , + f , and b = b = 0 or f , = 0 ,Proof. First of all,0 = ( ω − f.ν )( e , e ) = f , , ω − f.ν )( e , e ) = b f , − f , , ω − f.ν )( e , e ) = b f , − f , + f , , i.e. f , = f , = 0 and b f , = 0. Moreover, we get0 = ( ω − f.ν )( e , e ) = f.ν = − f , + f , + b (3 f , − f , ) , ω − f.ν )( e , e ) = f.ν = f , + f , + b (3 f , − f , ) , which yields f , = − f , . Furthermore,0 = ( ω − f.ν )( e , e ) = a + f , + f , f , , ω − f.ν )( e , e ) = a + f , + f , − f , , which gives f , = 0 as well as a = − f , + f , . Next, we have0 = ( ω − f.ν )( e , e ) = a − f , + f , , ω − f.ν )( e , e ) = − a + f , , i.e. f , = 2 f , and a = f , . Moreover, we get0 = ( ω − f.ν )( e , e ) = a − ( f , − f , ) , ω − f.ν )( e , e ) = f , ( b + 2 b ) , i.e. a = f , − f , , and, since also b f , = 0, f , = 0 or b = b = 0. (cid:3) Lemma 6.4.
In Lemma 6.3, we must have f , = 0 .Proof. Assume that f , = 0. Then b = b = 0 by Lemma 6.3 and so0 = ( ω − f.ν )( e , e ) = f , + f , , ω − f.ν )( e , e ) = a + 3 b ( f , − f , ) , ω − f.ν )( e , e ) = a + b (3 f , − f , ) , i.e. a = 3 b ( f , − f , ), a = b ( f , − f , ) and f , = − f , . Imposing these identities, we get0 = ( ω − f.ν )( e , e ) = f , (1 + 4 b )4 , LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 19 and so either f , = 0 or b = − holds.We show that f , = 0 and argue by contradiction, i.e. we assume that f , = 0 and so b = − . Then(3.4) gives us 0 = ( f.ω ∧ ω − ω ∧ ν − d ˆ ρ )( e , e , e , e ) = − f , (( f , − f , ) + f , ) , so that f , = f , and f , = 0 due to f , = 0. But then one checks that ω = 0, a contradiction. Hence,we must have f , = 0.Assuming f , = 0, one computes0 = ( f.ω ∧ ω − ω ∧ ν − d ˆ ρ )( e , e , e , e ) = − ǫb f , (cid:0) f , − f , f , + 24 f , − (cid:1) f.ω ∧ ω − ω ∧ ν − d ˆ ρ )( e , e , e , e ) = 2 b (3 f , − f , f , + 9 f , − f , − f , )(6.3)One checks that b = 0 implies ω = 0, and so we must have b = 0. Since f , = 0 by assumption, (6.3)yields 0 = 12 f , − f , f , + 24 f , − , f , − f , f , + 9 f , − f , − f , )So either 3 f , − f , f , + 9 f , − f , − f , = 0. However, both cases give us a contradiction:Namely, if 3 f , − f , f , + 9 f , − f , − f , f , + 9 f , = 1 = 12 f , − f , f , + 24 f , , and so 0 = 9 f , − f , f , + 15 f , = (3 f , − f , ) − f , , i.e. 3 f , − f , = ± f , . So either f , − f , = 0 or f , = f , . However, in the first case, one checks that ω = 0, a contradiction.Thus, we must have f , = f , . But then1 = 3 f , − f , f , + 9 f , = 3 f , − f , + f , = − f , ≤ , again a contradiction.Consider now the case 2 f , − f , = 0. Then f , = f , and so1 = 12 f , − f , f , + 24 f , = − f , ≤ , which is a also a contradiction.Thus, we must have f , = 0. (cid:3) To simplify the notation, we set from now on x := f , , y := f , , z := f , . With this new notation, one gets:
Lemma 6.5.
We have a = ( y − x ) b , f , = (2 x − y ) b − z , x + y = 0 , b + 1 = 0 , b = 0 and a = b x + y ) .Proof. Firstly0 = ( ω − f.ν )( e , e ) = a + b (3 x − y ) , ω − f.ν )( e , e ) = (3 y − x ) b + z + f , , i.e. a = ( y − x ) b , f , = (2 x − y ) b − z . Next, set A := (12 b − x − (40 b + 2) xy + (12 b − y . Then 0 = ω = a Ae , which implies a = 0 and A = 0. Since0 = ν ∧ ω = (cid:0) Ab + a (4 b + 1)( x + y ) (cid:1) e , we either have b = 0, and then (4 b + 1)( x + y ) = 0 as well, or b = 0, and so x + y = 0, 4 b + 1 = 0, and A = − a (4 b +1)( x + y ) b . Moreover, we have(6.4) 0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = ( x + y )( A +4 b +1)2 Assume now first that b = 0. Then we must have x + y = 0, as otherwise 4 b + 1 = 0 and so ( x + y ) A = 0,a contradiction to x + y = 0 and A = 0. But then0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − a , f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = a (40 x − , from which we obtain a = 2 and x = δ q for some δ ∈ {− , } . But then0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = b − , i.e. b = . Hence, 0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = z − δ √ b , f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − δ (3 √ b − δz ) , and so b = δ √ z , b = δ √ z . However,0 = ( ω − f.ν )( e , e ) = z , i.e. z = 0, and so 0 = ( ω − f.ν )( e , e ) = 2 , a contradiction.Hence, we must have b = 0, x + y = 0, 4 b + 1 = 0 and A = − a (4 b +1)( x + y ) b . But then (6.4) gives us A = − (4 b + 1). Thus, a (4 b +1)( x + y ) b = − A = 4 b + 1and so, since 4 b + 1 = 0, a = b x + y ) . (cid:3) This allows us now to prove:
Theorem 6.6.
Let g be a seven-dimensional almost nilpotent Lie algebra with codimension-one nilpotentideal isomorphic to n . Then g does not admit a closed G -eigenform.Proof. We assume that the parameters fulfill all the conditions that we derived in all the previous lemmas.Then we first get 0 = ( ω − f.ν )( e , e ) = − (4 x − y ) b − yb − (4 b +1) z , i.e. z = (4 x − y ) b − yb b +1 since 4 b + 1 = 0 by Lemma 6.5. Then one computes0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − b (6 xy +6 y − x + y )4( x + y ) , i.e. b (6 xy + 6 y −
1) = − x + y ). Since x + y = 0, also 6 xy + 6 y − = 0, and so b = − x + y )6 xy +6 y − . Moreover,0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = x + y (cid:0) b (12 x − xy + 12 y + 4) + 1 − ( x + y ) (cid:1) , f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − b xy +6 y − (2 x − xy + 24 y − , that is, 2 x − xy + 24 y − , b (12 x − xy + 12 y + 4) = ( x + y ) − , since b = 0, x + y = 0 by Lemma 6.5.We show now that 12 x − xy + 12 y + 4 = 0: LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 21 If this is not the true, then ( x + y ) = 1, i.e. y = δ − x for some δ ∈ {− , } . But then 0 =12 x − xy + 12 y + 4 = 16 (2 x − δ ) , i.e. x = δ = y and so 2 x − xy + 24 y − = 0, acontradiction.Thus, 12 x − xy + 12 y + 4 = 0 and we have b = ( x + y ) − x − xy +12 y +4 . On then computes 0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = ( b +2 b )( x − y )2 . Hence, b = − b or x = 4 y . However, x = 4 y is impossible since then0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = y − , a contradiction. Thus b = − b and one gets0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − b ( x − y ) ( x +2 y )3 x − xy +3 y +1 and 0 = ω = x − y ) (6 xy +6 y − · (3 x − xy +3 y +1) e . Thus, x − y = 0 and so b ( x + 2 y ) = 0. We show that b = 0 and, consequently, x = − y :If b = 0, then b = 0 as well and we do get0 = ( ω − f.ν )( e , e ) = x − y − xy +6 y − , f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − x +28 xy − y +26 xy +6 y − . One easily checks that all solutions of − x + 12 y + 2 = 0 , − x + 28 xy − y + 2 = 0are given by ( x, y ) = δ √ (cid:0) , (cid:1) , ( x, y ) = δ (5 , − δ , δ ∈ {− , } . However, in the first case, one computes0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − and in the second case one obtains0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − , and so a contradiction in both cases.Hence, b = 0 and so x = − y . But then0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = y − y +1 , i.e. y = δ q for some δ ∈ {− , } and we finally obtain0 = ( f.ω ∧ ω − d ˆ ρ − ν ∧ ω ) ( e , e , e , e ) = − , a contradiction.Hence, g does not admit a closed G -eigenform. (cid:3) The case n . In this subsection, we are considering exact G -structures and closed G -eigenformson seven-dimensional almost nilpotent Lie algebras with codimension-one nilpotent ideal isomorphic to n . We will determine all such Lie algebras which admit an exact G -structure and we will show that nosuch Lie algebra can admit a closed G -eigenform.First of all, note that n is a well-known real six-dimensional nilpotent Lie algebra, namely the oneunderlying the complex three-dimensional Heisenberg Lie algebra, and the Iwasawa manifold, see e.g.[KSa]. Moreover, for this subsection, denote by J the almost complex structure on n uniquely definedby J e i − = e i for i = 1 , J e = − e .The Lie algebra of all derivations of n is given by(6.5) Der( n ) = (cid:26) (cid:18) A B tr C A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) A ∈ C × , B ∈ R × (cid:27) , with respect to the basis ( e , . . . , e ), where we consider A ∈ C × as a real 4 × C A ∈ C as a real 2 × n ) of inner automorphism of n is given by the Lie groupgenerated by the exponentials of elements in Der( n ), and so equals(6.6) Inn( n ) = (cid:26) (cid:18) C D det C ( C ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) C ∈ GL(2 , C ) , D ∈ R × (cid:27) . Now split n = V ⊕ W with V := span( e , e , e , e ) and W := span( e , e ) and do the same for the dualspace n ∗ = V ∗ ⊕ W ∗ . Let an SU(3)-structure ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ with exact ρ be given, i.e. thereexists ν ∈ Λ n ∗ with dν = ρ . Write ω = ae + e ∧ α + e ∧ α + ˜ ω for a ∈ R , α , α ∈ V ∗ and ˜ ω ∈ Λ V ∗ and, similarly, ν = be + e ∧ β + e ∧ β + ˜ ν for b ∈ R , β , β ∈ V ∗ and ˜ ν ∈ Λ V ∗ . Moreover, let σ := de and σ := de , note that σ , σ ∈ [[Λ , V ∗ ]]with respect to the almost complex structure J on V . Set ρ := e ∧ σ − e ∧ σ . Then ρ induces the complex structure J (if one chooses the right orientation on n ) and we have ρ = dν = bρ + σ ∧ β + σ ∧ β . This shows that b = 0 as otherwise ρ ( e , · , · ) = 0, contradicting Lemma 2.5 (a). Moreover, ρ ( e , e , · ) = 0,i.e. e and e are J := J ρ -linearly dependent by Lemma 2.5 (a). But so0 = g ( e , e ) = ω ( Je , e ) = ae ( Je , e ) , implies a = 0 and we may apply an inner automorphism F as in (6.6) with C = I and suitable D ∈ R × to get rid of α and α in ω , i.e. we may assume that ω = ae + ˜ ω with ˜ ω = 0 due to the non-degeneracyof ω . Now we must have0 = ω ∧ ρ = ae ∧ ( σ ∧ β + σ ∧ β ) + be ∧ σ ∧ ˜ ω − be ∧ σ ∧ ˜ ω Thus, σ ∧ β + σ ∧ β = 0, and so ρ = bρ , and σ i ∧ ˜ ω = 0 for i = 2 ,
3. Now σ , σ span [[Λ , V ∗ ]] andso the latter identity shows ˜ ω ∈ [Λ , V ∗ ]. A straightforward computation shows that σ ∧ β = − σ ∧ β implies β = J ∗ β .Next, ( σ , J | V ), σ := e + e , defines an almost Hermitian structure on V and SU(2) preserves σ and acts as SO(3) on [Λ , V ∗ ]. Since matrices of the block-diagonal form diag( A, I ) with A ∈ SU(2)are in Inn( n ) and preserve ρ , we may thus assume that ˜ ω = a e + a e . Moreover, an element ofthe form diag ( b , b , b , b , b b , b b ) is in Inn( n ) and only scales ρ , and so we may even assume that | a | = | a | = | a | , i.e. there are ǫ , ǫ ∈ {− , } such that ˜ ω = ǫ ( aǫ e + e ). Since J = ± J , we do get ǫ a = ω ( e , e ) ω ( e , e ) = ω ( Je , e ) ω ( Je , e ) = g ( e , e ) g ( e , e ) > , and so ǫ = 1, i.e. ω = aǫ ( e + e ) + ae . Moreover, ǫ a = ω ( e , e ) ω ( e , e ) = − ω ( Je , e ) ω ( Je , e ) = − g ( e , e ) g ( e , e ) < , i.e. ǫ = − ω = a ( − e + e ) + e ) =: aω . Noting that for a >
0, the ordered basis( e , − e , e , − e , e , − e ) is oriented and so J = J , and otherwise J = − J , the normalisation conditionreads | a | = b . Hence, we may write a = ǫλ and b = λ for some λ ∈ R \ { } and some ǫ ∈ {− , } .Finally, we may use block-diagonal matrices diag( A, I ) with A ∈ SU(2) to bring ˜ ν into a canonical.For this, note that diag( A, I ) is in Inn( n ) and preserves ( ω, ρ ) and so we may assume that ˜ ν = c e + c e + c σ + c σ for certain c , c , c , c ∈ R . Thus, ν = λ e + e ∧ β + e ∧ J ∗ β + c e + c e + c σ + c σ for β := β ∈ V ∗ . If ν ∈ [Λ , n ∗ ], then c = c = 0 and c = λ − c . Summarizing, we have arrived at LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 23 Lemma 6.7.
Let ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ be an SU(3) -structure for which there exists ν ∈ Λ n ∗ with ρ = dν . Then ( ω, ρ, ν ) , are, up to automorphism, given by (6.7) ω = ǫλ ω = ǫλ (cid:0) − e − e + e (cid:1) ,ρ = λ ρ = λ (cid:0) e − e − e − e (cid:1) ,ν = λ e + e ∧ β + e ∧ J ∗ β + c e + c e + c σ + c σ , for certain c , c , c , c ∈ R , λ ∈ R \ { } , ǫ ∈ {− , } and β ∈ V ∗ .If ν ∈ [Λ , n ∗ ] , then, up to an automorphism, ( ω, ρ ) take the form as in (6.7) and (6.8) ν = λ e + e ∧ β + e ∧ J ∗ β + ce + ( λ − c ) e for some c ∈ R and β ∈ n ∗ . Next, we determine those seven-dimensional almost nilpotent Lie algebras g with codimension-onenilpotent ideal n which admit an exact G -structure. First of all, we get some restriction on f if g admits an exact G -structure, i.e. if (3.5) is valid: Lemma 6.8.
Let ( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ be as in (6.7) and assume that ν ∈ Λ n ∗ satisfies dν = ρ . Then α ∈ n ∗ and f ∈ Der( n ) fulfill (3.5) if and only if f.ν , = ω, f.ν , = dα, [ f, J ] = 0 , where ν , is the (1 , -part and ν , is the (2 ,
0) + (0 , -part of ν . If this is the case, then no eigenvalueof f is purely imaginary.Proof. We decompose f = f + f into its J -invariant part f and its J -anti-invariant part f . Then f preserves the splitting Λ n ∗ = [Λ , n ∗ ] ⊕ [[Λ , n ∗ ]] while f interchanges the two summands. As dα isof type (2 ,
0) + (0 , f .ν , + f .ν , = ω, f .ν , + f .ν , = dα. if we decompose ν = ν , + ν , as in the statement. Note that f is a strictly lower triangular block matrixwith respect to the splitting n = V ⊕ W , while f is a lower triangular block matrix with respect to thesame splitting. Moreover, ν , , dα ∈ Λ V ∗ and ν , has a non-trivial Λ W ∗ -part. Thus, f .ν , ∈ Λ V ∗ and f .ν , ∈ W ∗ ∧ V ∗ ⊕ Λ V ∗ with non-trivial W ∗ ∧ V ∗ -part if f = 0. Hence, f = 0, i.e. f = f , andso [ f, J ] = 0, and the above equations simplify to f.ν , = ω, f.ν , = dα as stated.Finally, assume that there is an eigenvector X ∈ n \ { } of f with purely imaginary eigenvalue ic , c ∈ R . Then f ( X ) = cJ X , and so f ( J X ) = J f ( X ) = − cX , and we get0 = g ( X, X ) = ω ( J X, X ) = f.ν , ( J X, X ) = − ν , ( f ( J X ) , X ) − ν , ( J X, f ( X ))= cν , ( X, X ) − cν , ( J X, J X ) = 0a contradiction. Hence, no eigenvalue of f can be purely imaginary. (cid:3) We are now in the position to give a classification of those almost nilpotent Lie algebras with codimen-sion-one nilpotent ideal isomorphic to n which admit an exact G -structure: Theorem 6.9.
Let g be a seven-dimensional almost nilpotent Lie algebras g with codimension-one nilpotentideal n , i.e. g ∼ = n ⋊ f R for some f ∈ Der( n ) . Then g admits an exact G -structure if and onlyif f has no purely imaginary eigenvalues. Equivalently, g admits an exact G -structure if and only if g ∼ = n ⋊ f a,b ,b R for certain a ∈ (cid:2) − , ∞ (cid:1) \ { } , b , b ∈ R or g ∼ = n ⋊ h b R for some b ∈ R , where f a,b ,b := a + ib − − a + ib −
12 + i ( b + b ) ! , h b := −
14 + ib −
14 + ib −
12 +2 ib . Proof.
We note that the forward implication in the first statement is incorporated in Lemma 6.8.So let g ∼ = n ⋊ f R for some f ∈ Der( n ) which has no purely imaginary eigenvalues. By (6.5), weknow that f = (cid:18) A B tr C A (cid:19) for some A ∈ C × and B ∈ R × . If we conjugate f with an automorphism F of n as in (6.6) with C = I , we surely get a Lie algebra n ⋊ F fF − R which is isomorphic to g , where F f F − = (cid:18) A B + D ( A − (tr C A ) I ) tr C A (cid:19) . Thus,
F f F − is block-diagonal if A − (tr C A ) I is invertible, i.e. if tr C A is not a complex eigenvalue ofthe complex matrix A . However, if tr C A would be an eigenvalue of the complex matrix A , then the othercomplex eigenvalue would have to be zero and so the real matrix f would have one eigenvalue equal tozero, which is excluded since f has no purely imaginary eigenvalues.Thus, calling F f F − again f , we may assume that f = diag( A, tr C A ). But then we may use anautomorphism of n as in (6.6) to bring A into complex Jordan normal form. Hence, we may assume thateither A = diag( w , w ) for w , w ∈ C with Re( w ) = 0, Re( w ) = 0, Re( w + w ) = tr C A = 0 or A = (cid:18) w w (cid:19) for some w ∈ C with Re( w ) = 0. We provide now in both cases an example of an SU(3)-structure( ω, ρ ) ∈ Λ n ∗ × Λ n ∗ a two-form ν ∈ Λ n ∗ and a one-form α ∈ n ∗ such that ρ = dν and such that (3.5)is fullfilled, where the latter equation is valid by Lemma 6.8 if and only if f.ν , = ω , f.ν , = dα . Wewill always choose α = 0 and a (1 , ν , so that the second equation is automatically fulfilled and weonly have to deal with the first one.In the first case, one checks by a straightforward computation that ω = λ ω , ρ = λ ρ , ν = λ w ) e + λ w ) e + λ e with λ := −
12 Re( w + w ) fulfills all necessary equations, whereas in the second case ω = λ ω , ρ = λ ρ , ν = − λ e − (16 λ + 2 λ ) e − λ · ( e − e ) + λ e with λ := − w ) does the job.The second statement in the assertion follows immediately from the considerations above by notingthat rescaling f by a non-zero scalar gives an isomorphic Lie algebra and by noting that we may order thereal parts of the eigenvalues of A in such a way that the first one is greater or equal to the second one. (cid:3) Remark . Note that f a,b ,b and h b in Theorem 6.9 both fix e and so one easily sees that ( ω, ρ ) asin Lemma 6.7 with λ = 1, i.e. ω = − e − e + e , ρ = e − e − e − e , give rise to an exactG -structure on n ⋊ f a,b ,b R and n ⋊ h b R , respectively. This explains the strange ‘normalisation’ ofthe endomorphisms in Theorem 6.9.Looking for exact G -structures of special torsion, we obtain: Theorem 6.11.
Let g be a seven-dimensional almost nilpotent Lie algebra with codimension-one nilpotentideal isomorphic to n which admits an exact G -structure. Then: (a) g admits an exact G -structure with special torsion of negative type. (b) g admits an exact G -structure with special torsion of positive type if and only if g = n ⋊ f − / ,b,b R for all b ∈ R .Proof. By Theorem 6.9, we may assume that g = n ⋊ f R with either f = f a,b ,b for certain a ∈ (cid:2) − , ∞ (cid:1) , b , b ∈ R or f = h b for some b ∈ R . We divide the proof into four different parts:(I) We first show that g = n ⋊ f a,b ,b R for a, b , b ∈ (cid:0)(cid:2) − , (cid:3) \ { } (cid:1) × R with a = − or b = b and n ⋊ h b R for b ∈ R admits an exact G -structure with special torsion of positive type. LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 25 For this, note that under the assumptions on a, b , b , the Lie algebra n ⋊ f a,b ,b R is isomorphicto n ⋊ g R with g = g a,b ,b ,c = a + ib c − − a + ib −
12 + i ( b + b ) ! If a = − and b = b =: b , then, for any c ∈ R \ { } , we have n ⋊ g − / ,b,b,c R ∼ = n ⋊ h b R .So we are looking for exact G -structures with special torsion of positive type on n ⋊ g R . Forthis, note that a = 0 and 2 a + 1 = 0 by assumption. Thus, ν = 12 a e + 2 c − a ( b − b ) + 1) a (4( b − b ) + 1)(2 a + 1) e + e − b − b ) ca (4( b − b ) + 1) ( e + e )+ ca (4( b − b ) + 1) ( e − e ) ∈ Λ n ∗ is well-defined and one checks that dν = ρ = e − e − e − e and g.ν = ω = − e − e + e .Thus, the pair ( ω, ρ ) gives rise to an exact G -structure ϕ . As ˆ ρ = e − e − e − e , we get d ⋆ ϕ ϕ = d (cid:0) ω + e ∧ ˆ ρ (cid:1) = e ∧ g. (cid:0) ω (cid:1) − e ∧ d n ˆ ρ = e ∧ (cid:0) − e + (2 a − e − (2 + 2 a ) e + c ( e − e ) (cid:1) due to d ( ω ) = 0. Hence, the torsion two-form τ is given by τ = − ⋆ ϕ d ⋆ ϕ ϕ = (2 + 2 a ) e − (2 a − e + c ( e − e ) + 3 e . Now the exact G -structure ϕ has special torsion of positive type if and only if τ = 0, which isequivalent to ((2 + 2 a ) e − (2 a − e + c ( e − e )) = 0, and so to0 = (2 + 2 a )(2 a −
1) + c = 4 a + 2 a − c . Here, a is fixed and we are searching for a solution of this equation for c , which is possible if4 a + 2 a − ≤
0, i.e. if a ∈ (cid:2) − , (cid:3) . Note that for a = − , we have c = ± = 0 and so n ⋊ h b R admits an exact G -structure with special torsion of positive type for any b ∈ R .(II) We show now that ( ω , ρ ) defines also an exact G -structure on n ⋊ g − / ,b,b,c R ∼ = n ⋊ h b R withspecial torsion of negative type for a suitable chosen c ∈ R \ { } .The computations in (I) show that τ = ( e + e ) + c ( e − e ) + 3 e . Hence, ϕ has specialtorsion of negative type if and only if | τ | ϕ = (cid:12)(cid:12) τ (cid:12)(cid:12) ϕ , which here is equivalent to ( + 2 c ) = (cid:0) (cid:0) − c (cid:1)(cid:1) ⇐⇒ c (cid:0) c − c + 2187 (cid:1) = 0 , i.e. to c = 0 or c = ± . Thus, for c = , we get an exact G -structure with special torsion ofnegative type on n ⋊ g − / ,b,b, R ∼ = n ⋊ h b R .(III) Next, we show that n ⋊ f a,b ,b R admits an exact G -structure with special torsion of positivetype if ( a, b , b ) ∈ (cid:0) , ∞ (cid:1) × R and that it admits an exact G -structure with special torsion ofnegative type for any possible values of ( a, b , b ), i.e. for any ( a, b , b ) ∈ (cid:16)(cid:2) , ∞ (cid:1) \ { } (cid:17) × R .For this, we note that n ⋊ f a,b ,b R is isomorphic to n ⋊ h R with h := h a,b ,b ,r := a − b b a − − a − b b − − ar − − ( b + b ) − r b + b − for any r ∈ R . The minus sign occuring before one of the r s is due to ( e , − e , e , − e , e , − e )being a complex basis, i.e. due to the shift in the order of e and e .We have dν = ρ and h.ν = ω for ν = a e − ( b +2 b ) +(2 r + a +1)( a +1)(2 a +1)(( a +1) +( b +2 b ) ) e + e + r ( b +2 b )( a +1) +( b +2 b ) (cid:0) e − e (cid:1) + r (1+ a )( a +1) +( b +2 b ) (cid:0) e + e (cid:1) . Hence, the pair ( ω , ρ ) defines an exact G -structure ϕ on n ⋊ h R for any value of r ∈ R .Moreover, we have d ⋆ ϕ ϕ = e ∧ (cid:0) h. (cid:0) ω (cid:1) − d n ˆ ρ (cid:1) = e ∧ (cid:0) − e + (2 a − e − (2 + 2 a ) e + r ( e + e ) (cid:1) and so the torsion two-form τ is given by τ = − ⋆ ϕ d ⋆ ϕ ϕ = (2 + 2 a ) e − (2 a − e − r ( e + e ) + 3 e . Hence, τ = − a )(6 a − − r ) e and τ = 0, i.e. ϕ has special torsion of positive type, if and only if r = 6 a −
3. But we assumed a > and so have 6 a − > ϕ has special torsion of positive type for r = √ a − ∈ R .Moreover, ϕ has special torsion of negative type if and only if ((2 + 2 a ) + (2 a − + 2 r + 9) = | τ | ϕ = (cid:12)(cid:12) τ (cid:12)(cid:12) ϕ = (cid:0) a )(6 a − − r ) (cid:1) . Bringing both terms on one side and factorising gives (cid:0) ( a − + r (cid:1) · (cid:0) a + 22 a + 5 − r (cid:1) = 0 . So one may find some r ∈ R such that ϕ has special torsion of negative type if (2 a + 5)(4 a + 1) =8 a +22 a +5 ≥
0. But this is the case if a ≥ − and so n ⋊ f a,b ,b R admits an exact G -structurewith special torsion of negative type for any possible values of ( a, b , b ).(IV) Finally, we need to show that for any b ∈ R , the Lie algebra n ⋊ f − / ,b,b R does not admit anexact G -structure with special torsion of positive type.For this, let ( ω, ρ ) be a half-flat SU(3)-structure which determines an exact G -structure ϕ on n ⋊ f − / ,b,b R and let ν ∈ Λ n ∗ and α ∈ n be such that (3.5) holds. By Lemma 6.7, we mayassume that ω = ǫλ ω = ǫλ ( − e − e + e ) , ρ = λ ρ = λ ( e − e − e − e )for some λ ∈ R \{ } , up to an automorphism F of n , i.e. ( ω, ρ ) are of this form on n ⋊ F f − / ,b,b F − R . Now one computes that f := F f − / ,b,b F − is of the form f = (cid:18)(cid:0) − + ib (cid:1) I B − + 2 ib (cid:19) for some B ∈ R × . By Lemma 6.8, we have [ f, J ] = 0, which amounts to B being of the form B = (cid:18) a a a a a − a a − a (cid:19) for certain a , a , a , a ∈ R . Moreover, by Lemma 6.8, we have f.ν , = ω and ν , = λ e + e ∧ β + e ∧ J ∗ β + c e + c e for certain β ∈ span( e , e , e , e ) and c , c ∈ R by Lemma 6.7. Thus, ǫλ ( − e − e + e ) = ω = f.ν , = λ e + e ∧ γ + e ∧ J ∗ γ + c e + c e for some γ ∈ span( e , e , e , e ), which implies, in particular, λ = ǫ , i.e. ω = ǫω and ρ = ǫρ .Since for ǫ = −
1, the induced orientation is the opposite of that for ǫ = 1, we always haveˆ ρ = e − e − e − e . Thus, one cpmputes d ⋆ ϕ ϕ = e ∧ (cid:0) f. (cid:0) ω (cid:1) − d n ˆ ρ (cid:1) = e ∧ (cid:0) − e − e − e + a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1) + a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1)(cid:1) independently of ǫ . Hence, τ = − ⋆ ϕ d ⋆ ϕ ϕ = ǫ (cid:0) (cid:0) e + e (cid:1) + 3 e − a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1) − a (cid:0) e + e (cid:1) + a (cid:0) e − e (cid:1)(cid:1) , LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 27 and so τ = 9 ǫ (cid:0) + a + a + a + a (cid:1) e = 0 , i.e. ϕ does not have special torsion of positive type. (cid:3) Finally in this section, we show that a Lie algebra of the form g = n ⋊ f R cannot admit a closedG -eigenform: Theorem 6.12.
Let g be a seven-dimensional almost nilpotent Lie algebra with codimension-one nilpotentideal isomorphic to n . Then g does not admit a closed G -eigenform.Proof. We assume the contrary. Then, by Lemma 6.7, λ ∈ R \ { } , ǫ ∈ {− , } such that ω = ǫλω , ρ = λ ρ and ν ∈ [Λ , n ∗ ] is as in (6.8), i.e. ν = ce + ( λ − c ) e + λ e + e ∧ β + e ∧ J ∗ β for certain c ∈ R and β ∈ V ∗ . Moreover, we may assume that f.ν = ω, ω ∧ ( f.ω − ν ) = d ˆ ρ. for some f ∈ Der( n ). Since then also ( − ω, ρ, ν, − f ) fulfills all necessary equations, and so defines a closedG -eigenform, we may assume that ǫ = 1.But then one computes d ˆ ρ = λ d ˆ ρ = 4 λ e = λ ( − e − e + e ) ∧ − λ ( e + e + e )= ω ∧ − λ ( e + e + e ) . Since wedging with ω is an isomorphism from Λ h ∗ to Λ h ∗ , the latte equation implies f.ω − ν = − λ ( e + e + e )By (6.5), we know that f = (cid:18) A B tr C A (cid:19) for some A = ( a ij ) i,j ∈ C × and some B ∈ R × . Thus, inserting ( e , e ) into the equality f.ω − ν = − λ ( e + e + e ) yields2 Re( a ) λ − c = − a ) ω ( e , e ) − ν ( e , e ) = ( f.ω − ν )( e , e ) = − λ. Similarly, we obtain 2 Re( a ) λ − ( λ − c ) = ( f.ω − ν )( e , e ) = − λ. by inserting ( e , e ). Adding these two equations yields(6.9) 2 Re(tr C A ) λ − λ = − λ. Moreover, inserting ( e , e ), we do get − C A ) λ − λ = ( f.ω − ν )( e , e ) = − λ. Adding (6.9) to this equation, we obtain − λ = − λ , and so, since λ = 0, that λ = 3.However, we also get − C A ) λ = − C A ) ν ( e , e ) = f.ν ( e , e ) = ω ( e , e ) = λ , i.e. 2Re(tr C A ) λ = λ , which, together with (6.9), yields λ − λ = − λ , i.e. λ = 5, a contradiction.Thus, g does not admit a closed G -eigenform. (cid:3) Acknowledgements.
The first author was supported by a
Forschungsstipendium (FR 3473/2-1) fromthe Deutsche Forschungsgemeinschaft (DFG).
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LOSED G -EIGENFORMS AND EXACT G -STRUCTURES 29 Mathematisches Seminar, Christian-Albrechts-Universit¨at zu Kiel, Ludewig-Meyn-Strasse 4, D-24098 Kiel,Germany
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