aa r X i v : . [ m a t h . DG ] F e b THE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS
ROHIN BERICHON
Abstract.
We show that noncompact homogeneous spaces not diffeomorphicto Euclidean space of dimension 9 or 10 admit no homogeneous Einstein met-rics of negative Ricci curvature, with only three potential exceptions. Themain ingredient in the proof is to show, via a cohomogeneity-one approach,that noncompact homogeneous spaces admitting an ideal isomorphic to sl ( R )admit no homogeneous Einstein metrics of negative Ricci curvature. Introduction
A Riemannian manifold ( M , g ) is called Einstein if the Ricci tensor satisfiesRic( g ) = λ · g for some λ ∈ R . In general, it is far too optimistic to provideexistence criteria for solutions to the Einstein equation. Instead, a widespreadmethod is to impose some sort of symmetry assumption, or to restrict to metricswith special holonomy. In this article, we study the existence of Einstein metricson noncompact homogeneous spaces.The study of homogeneous Einstein metrics is roughly grouped into the caseswhen the Ricci curvature is positive, zero, or negative. If the Ricci curvature ispositive, the manifold is compact, and has finite fundamental group by the Myerstheorem [Mye41]. It was proven in [AK75] that homogeneous Ricci flat manifoldsare flat. Finally, it is known that homogeneous Einstein manifolds of negative Riccicurvature are noncompact, by [Boc48]. At present, all known examples of thesespaces are isometric to simply-connected solvmanifolds. We have the following Conjecture 1.1 (Alekseevskii, 1975, [Bes08]) . Any connected homogeneous Ein-stein manifold of negative scalar curvature is diffeomorphic to a Euclidean space.
Current results verify the Alekseevskii conjecture in dimension at most 8 with3 possible exceptions (see [AL16] and references therein). In 2 and 3 dimensions,homogeneous Einstein manifolds have constant sectional curvature, so are diffeo-morphic to R n . [Jen69] classified the 4 dimensional simply-connected homogeneousEinstein spaces of negative curvature, implying the conjecture. [Nik05] verifiedthe conjecture in 5 dimensions with the possible exception of spaces admitting atransitive SL ( C ) action (see Proposition 4.2). In the 6 dimensional case, the con-jecture was verified with the exception of SL ( C ) and the universal cover of SL ( R ) by [AL15, JP17], the latter of which being resolved by [BL19, Corollary 6.4]. Fi-nally, [AL16, Theorem B, C] verified the conjecture in dimensions 7 and 8 with thepossible exception of SU (2 ,
1) and SL ( R ).The primary goal of this paper is to prove the following Theorem A.
Let ( M n , g ) be a simply-connected homogeneous Einstein manifoldwith λ < of dimension at most 10, which is de Rham irreducible. If ( M , g ) is notan invariant metric on the universal covers of Sp (1 , / ∆ U (1) , Sp (1 , , Sp (2 , R ) , SL ( R ) , SU (2 , , or SL ( C ) , then M n is diffeomorphic to Euclidean space. Regarding the simply-connected assumption, suppose ( M n , g ) is a homogeneousEinstein metric of negative Ricci curvature. If the universal cover f M , which is also homogeneous and Einstein, is diffeomorphic to Euclidean space, then by[BL19], f M is isometric to a simply-connected Einstein solvmanifold. Therefore, by[Jab15, Theorem 1.1], f M does not admit any non-trivial quotient, thus in order toverify the Alekseevskii conjecture, it is sufficient to check only the simply-connectedhomogeneous spaces.It is also important to note the de Rham irreducible assumption in the statementof the theorem. Assuming the Alekseevskii conjecture holds in dimensions at most8, Theorem A verifies the conjecture in dimensions 9, and 10, with the possibleexceptions detailed therein, since on de Rham reducible manifolds, the Einsteinequation splits into the Einstein equation on each of the irreducible factors.Unfortunately, our methods do not allow us to extend these results to higherdimensions. In 11 dimensions for example, the Einstein equation on the homoge-neous space ( SL ( C ) · R ) ⋉ C , with SL ( C ) acting with the standard representationon C reduces to an equation for left-invariant metrics on SL ( C ) [AL16].Our main tool for proving Theorem A is the following result, which allows us toignore homogeneous spaces G / H with an ideal in g isomorphic to sl ( R ). Theorem B. If ( M n , g ) is a simply-connected homogeneous Einstein manifold withminimal presentation G / H , with G semisimple, with Ric( g ) = − g , then no ideal in g = Lie( G ) is isomorphic to sl ( R ) . We may also restrict ourselves to studying only the semisimple homogeneousspaces by [Dot88, Theorem 2], since the case when G is nonunimodular was al-ready verified in [AL16, Theorem D]. Moreover, irreducible symmetric spaces areall diffeomorphic to solvmanifolds [Hel78].The proof of Theorem A is broken into two parts. The first is a classification,which provides a complete account for the remaining AU spaces (see Definition 4.1)and their corresponding compact dual AU* spaces. In this part, we provide thedecompositions of the isotropy representations of each AU space into irreduciblesubmodules, indicating isomorphisms between them. In the second part, we verifythe non-existence of invariant Einstein metrics of negative curvature in the re-maining AU spaces from the first part which are not covered by other results (see[AL16, Nik00]).The proof of Theorem B proceeds by contradiction, assuming g admits an idealisomorphic to sl ( R ). We show that such a homogeneous space admits an effectivecohomogeneity-one action of a closed subgroup of G satisfying the conditions of[BL19, Theorem D]. Using these results, we show that ( M n , g ) is locally isometricto a Riemannian product of Einstein metrics with one factor isometric to PSL ( R ),giving us our contradiction.The existence of homogeneous Einstein metrics on compact homogeneousmanifolds is extensively investigated in low dimensions in [BK06], with severalother existence results in [DK08, Wan82, WZ86, YCD19]. In the case of non-compact homogeneous manifolds, structural results are known for solvmanifolds[Heb98, Lau09, Lau10] and more generally [AL16, BL19, JP17].In §
2, we provide an overview of the necessary theory of homogeneous manifoldsrequired for the proof of Theorems A and B. In §
3, we prove Theorem B, and in §
4, we prove Theorem A. 2.
Preliminaries
In what follows, we will cover some of the well-known theory of homogeneousEinstein manifolds necessary for the proof of the main theorems. Throughout,we assume that all manifolds are connected, and all presentations of homogeneousspaces are effective with connected transitive group and isotropy.
HE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS 3
Let G / H be a homogeneous space with H compact and fix a reductive decompo-sition g = h ⊕ m for G / H . Then, it is well known that G / H admits G -invariant Rie-mannian metrics. Moreover, we have the isomorphism [KN69, Chapter X, Propo-sition 3.1] M G := (cid:26) G -invariant metricson G / H (cid:27) ! (cid:26) Ad( H )-invariant innerproducts on m (cid:27) . On the right is the set of positive-definite, symmetric, non-degenerate, Ad ( H )-endomorphisms on m . Decompose m into Ad ( H )-irreducible submodules m = m ⊕ m ⊕ · · · ⊕ m k , and let Q be an Ad( H )-homomorphism defining an invari-ant inner product on m . Denote by Q ij the restriction of Q to m i projected onto m j with respect to the decomposition above. Then, Q ij is an Ad ( H )-homomorphismbetween irreducible modules. If m i m j , then by Schur’s Lemma, Q ij = 0. Onthe other hand, if m i ≃ m j , then Q ij is invertible. Moreover, by the FrobeniusTheorem, End Ad( H ) ( m i ) is isomorphic to one of R , C , or H . We say that m i is of real , complex , or quaternionic type, respectively.Without loss of generality, assume that m is decomposed into isotypical sum-mands m n ⊕ · · · ⊕ m n ℓ ℓ . Then, summarising the above, the space of G -invariantmetrics on G / H is given by [Lau19, Proposition 3.3.10] M G = h + n ( F ) × · · · × h + n ℓ ( F ℓ ) , n + · · · + n ℓ = k, where each h + n i ( F i ) is the subspace of gl n i ( F i ) consisting of symmetric , positivedefinite, non-degenerate matrices with entries in F i := End H ( m i ).In this notation, recall the following result. Suppose G is a noncompact semisim-ple Lie group, and H is a compact subgroup contained strictly within a maximalcompact subgroup K of G . Writing g = k ⊕ p = h ⊕ q ⊕ p , with the first equalitygiving the Cartan decomposition of g and where q is the orthogonal complement of h in k with respect to the Killing form, we obtain a decomposition of the reductivecomplement of h into Ad( H )-submodules m = q ⊕ p . Suppose q = q ( n )1 ⊕ · · · ⊕ q ( n k ) k , p = p ( m )1 ⊕ · · · ⊕ p ( m ℓ ) ℓ is a decomposition of q and p into Ad( H )-irreducible submodules q ( n i ) i and p ( m j ) j ,where n i = dim q ( n i ) i and m j = dim p ( m j ) j . Then, Theorem 2.1 ([Nik00, Theorem 1]) . Under the above assumptions, suppose g is a G -invariant metric on G / H such that g ( q , p ) = 0 . Then, g is not Einstein. By Schur’s Lemma, if q ( n i ) i p ( m j ) j for any i, j , then p and q are orthogonal forevery G -invariant metric, and hence none of them can be Einstein. On the otherhand, if there are q ( n i ) i ≃ p ( m j ) j , then all we can gather from the decomposition ofthe isotropy representation into irreducible summands is a parameterisation of thespace of invariant metrics in the sense of the prequel. Definition 2.2.
We call a homogeneous space G / H minimally presented if H iscompact, and dim G is minimal amongst all presentations of G / H with compactisotropy.In what follows, G / H is a homogeneous space in minimal presentation. In[Heb98], Heber attained structural results on so-called standard Einstein solvman-ifolds, providing criteria for uniqueness results, which were then developed on by[Lau10] by showing all Einstein solvmanifolds are standard. Motivated by these with respect to the ground field F i . That is, if F i = R , C , or H , then h + n i ( F i ) consists ofsymmetric, hermitian, or quaternionic hermitian matrices respectively. ROHIN BERICHON results, [BL19] generalises the standardness condition to arbitrary homogeneousspaces in the following
Definition 2.3 ([BL19]) . Let ( G / H , g ) be an homogeneous space with compactisotropy and reductive decomposition g = h ⊕ m , and denote by ˆ g an Ad( H )-invariant extension of g e H to g such that h ⊥ m . Then, we say that ( G / H , g ) is standard if the ˆ g -orthogonal complement of the nilradical of g is a Lie subalgebraof g .We have the following result, relating the existence of cohomogeneity-one actionson ( G / H , g ) with the structure underlying g . Theorem 2.4 ([BL19, Theorem D]) . Suppose ( G / H , g ) is a homogeneous Einsteinmanifold admitting an effective, cohomogeneity-one action of the closed subgroup G of G . If G \ G / H = S and consists of integrally minimal orbits, then the G -orbitsare standard homogeneous spaces. Let ( G / H , g ) is as in the theorem above, with G an effective, cohomogeneity-oneaction on G / H . Suppose γ : S → G / H is a unit speed normal geodesic to the G -orbits, and denote by tr L t the mean curvature of the hypersurfaces Σ t = G · γ ( t ).Then, we have the following Definition 2.5.
In the notation above, we say that the orbits are integrally minimal (in the sense of [BL19]) if Z S tr L t d t = 0 . If Σ t are all minimal hypersurfaces, then the orbits are integrally minimal. Wesay that (cid:0) G / H , g, G (cid:1) is orbit-Einstein with negative Einstein constant λ < g ( X, X ) = λ · g ( X, X ) for all vectors X tangent to the orbits of G . If ( G / H , g )is Einstein with negative Einstein constant, then ( G / H , g, G ) is orbit-Einstein.3. Proof of Theorem B
The purpose of this section is to prove Theorem B. We will argue by contra-diction, using a series of lemmas to show that if g contains an ideal isomorphic to sl ( R ), then ( G / H , g ) is locally isometric to a Riemannian product of negativelycurved Einstein spaces with one factor isometric to PSL ( R ).Suppose ( M n , g ) is a simply-connected, noncompact homogeneous manifold pre-sented as G / H , with G semisimple. Let g be a simple ideal of g and g the semisim-ple ideal complementary to g , and denote by h ⊆ g the projection of h onto thesimple factor g with respect to the splitting g = g ⊕ g . Lemma 3.1.
Under the above assumptions, if h is a maximal compactly embeddedsubalgebra, then G / H is not a minimal presentation.Proof. Suppose q ⊆ g is a subalgebra such that g = h ⊕ q . Then, define g = q ⊕ g , and denote by Q and G the connected Lie subgroups of G with Liealgebras q and g respectively. We claim that the action of G on G / H is transitive.It is sufficient to look at the infinitesimal generators of the action by [Aud04],and show that the homomorphism ι : g → ι ( g ) ⊆ Kill( G / H ) defined pointwise by X ι ( X ) p := dd t (cid:12)(cid:12)(cid:12) t =0 exp( tX ) · p, restricts to an epimorphism ι ( · ) p : g → T p G / H on every tangent space. Indeed,since T p G / H ≃ g / h , and ι ( g ) p ≃ g / ker ι ( · ) p by the first isomorphism theorem, ι ( · ) p is an epimorphism if and only if dim ker ι ( · ) p = dim h − (dim g − dim g ). Since HE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS 5 G p := (cid:8) g ∈ G : g · p = p (cid:9) = (cid:8) g ∈ G : g · p = p and g ∈ G (cid:9) = H ∩ G , by [Aud04, Theorem 1.1.1],ker ι ( · ) p = g p = Lie( G p ) = g ∩ h . Therefore, by the second isomorphism theorem, and the fact that h + g = g , wehave dim ker ι ( · ) p = dim( g ∩ h ) = dim h − (dim g − dim g ) , and hence the action of G on G / H is transitive, so G / H is not a minimal presentation. (cid:3) Suppose now that G / H is a minimal presentation for ( M n , g ) in the above, with g = sl ( R ). Then, by Lemma 3.1, h is trivial, since otherwise it is a maximalcompactly embedded subalgebra.Denote by G and G the connected Lie subgroups of G corresponding to the Liealgebras g = sl ( R ) and g respectively. Then, since g = g ⊕ g is a decompositioninto ideals, the intersection I = G ∩ G is a normal and discrete subgroup of G ,and so I is also central in G [OV88, Proposition 4.6]. Therefore, Λ = Z ( G ) I actsproperly discontinuously on ( M , g ) by isometries, so G / H is locally isometric to Λ \ G / H ≃ PSL ( R ) × ( G / H ′ ), where H ′ = H ∩ Λ . Since the Einstein condition islocal, it suffices to prove Theorem B in the case where M is presented minimallyas PSL ( R ) × ( G / H ). Lemma 3.2.
Let ( M n , g ) be a homogeneous Riemannian manifold with minimalpresentation PSL ( R ) × G / H as in the notation above. Let q = a ⊕ n ⊆ sl ( R ) be a Borel subalgbera with matching Iwasawa decomposition sl ( R ) = k ⊕ a ⊕ n ,and denote g = q ⊕ g as in the above. Then (1) G is closed in G ; (2) the action of G on M is effective, and of cohomogeneity-one; (3) the orbits are integrally minimal; (4) G \M = S .Proof. Since Q and G are both closed, G = Q × G is closed. Moreover, theeffective action of G on M decends to an effective action of G on M .We claim that G \M = S . Indeed, since the center of PSL ( R ) is finite, K iscompact by [Hel78, Chapter 6, Theorem 1.1], and so K ≃ S . Writing G = ( K Q ) × G , we have that every element in M is written uniquely as k · ( q, g ) H = ( kq, g ) H for some k ∈ K , q ∈ Q , and g ∈ G . That is, M ≃ K ( G / H ), so the orbits of the G action on M are parametrised by the action of K on G / H , so M\ G = K = S .It remains to show integral minimality in the sense of [BL19]. We will showmore generally that the G -orbits are minimal hypersurfaces. Let γ : S → M be aunit speed normal geodesic to the orbits of the G -action, and set Σ t = G · γ ( t ). Let N denote the unit normal vector field to each Σ t , satisfying γ ′ ( t ) = N γ ( t ) for all t ∈ S . Following the same construction as [BL19, § t ,denoted L t ∈ End( T Σ t ) satisfies g t ( L t X, Y ) = g ( ∇ X N, Y ) γ ( t ) on Killing fields, where g t is the induced metric on the submanifold Σ t of M . TheKoszul formula for Killing fields [Bes08, (7.27)] implies that for all Killing fields X on Σ t , g t ( L t X, X ) = g ( ∇ X N, X ) γ ( t ) = (cid:0) g ([ X, N ] , X ) γ ( t ) + g ([ X, X ] , N ) γ ( t ) + g ([ X, N ] , X ) γ ( t ) (cid:1) ROHIN BERICHON = − g ([ N, X ] , X ) γ ( t ) . Since sl ( R ) is unimodular, tr L t = − tr ad N = 0. Hence, Σ t is a minimal hyper-surface. (cid:3) Suppose now that ( M , g ) is in addition Einstein with negative Einstein constant.Then, by Theorem 2.4, the cohomogeneity-one G -orbits are standard homogeneousspaces. Lemma 3.3.
In the notation above and of Definition 2.3, if the G -orbits are stan-dard, then g is ˆ g -orthogonal to the nilradical of q .Proof. It is sufficient to prove that g ⊆ u := n ⊥ , the orthogonal complement ofnilrad( g ) = n under the Ad( H )-invariant extension of the inner product g to g .Since the G -orbits are standard, g = u ⊕ n is a Lie algebra direct sum. Moreover, u ≃ g / n is reductive by [Var84, Theorem 3.16.3], and so u (1) = [ u , u ] is semisimple,and(1) g (1) = [ g , g ] ⊆ n ⊕ [ u , u ] = n ⊕ u (1) . Now, the bracket (cid:2) u (1) , n (cid:3) defines n as a trivial 1-dimensional u (1) -representation, since u (1) is semisimple. Hence, n ⊕ u (1) is a Lie algebra directsum. Also, since g = q ⊕ g is a decomposition into ideals and q is at most 2-stepsolvable, g = g (2) (1) ⊆ h n ⊕ u (1) , n ⊕ u (1) i ⊆ u (2) ⊆ u , as required. (cid:3) Proof of Theorem B.
As in the above notation, let ( M , g ) be a simply-connectedhomogeneous Einstein manifold of negative Ricci curvature, with minimal presenta-tion PSL ( R ) × ( G / H ) such that G is semisimple. We seek three Borel subalgebrasof g = sl ( R ) whose nilradicals form a basis. In the basis n ˆ h, ˆ e, ˆ f o of sl ( R ), withcanonical commutation relations[ e, f ] = h, [ h, e ] = 2 e, [ h, f ] = − f, three such Borel subalgebras take the form q = span { h, e } , q = span { h, f } , q = span { e + f, e − f + h } . Applying Lemma 3.3 on each q i , we see that sl ( R ) ⊥ g . Therefore, G / H = PSL ( R ) × G / H splits as a Riemannian product, and hence the Einstein equationssplits into the Einstein equation on the PSL ( R ) and G / H factors. But it is wellknown (for example [Mil76]) that PSL ( R ) admits no left invariant Einstein metricsof negative Einstein constant. (cid:3) Proof of Theorem A
Our main purpose of this section is to prove Theorem A. Assume ( M , g ) is asimply-connected homogeneous Einstein manifold of negative Ricci curvature ofdimension at most 10. Then, by [AL16, BL19, JP17, Jen69, LL14, Nik05] and refer-ences therein, we may assume dim M is 9 or 10. Moreover, we may assume withoutloss of generality that for the minimal presentation G / H for M , G is semisimple,and has no compact simple factors. By Theorem B, g has no ideal isomorphic to sl ( R ). We may also assume G / H is not an irreducible symmetric spaces, sincethese are all diffeomorphic to simply-connected solvmanifolds by [Hel78, Chapter6, Theorem 5.1]. HE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS 7
We will conduct a case-by-case analysis of the remaining homogeneous spacesnot covered by literature, showing that each of these spaces do not admit invari-ant Einstein metrics of negative Einstein constant. By using classifications of lowdimensional compact semisimple homogeneous spaces (see, for example [BK06]),we obtain a classification of these spaces. We do this by using the partial dual-ity between compact semisimple homogeneous spaces and semisimple homogeneousspaces without compact simple factors, in the sense of [Ale12, Nik05] to construct alist of all semisimple homogeneous spaces without compact simple factors. We willdetail this procedure below:Let e G / e H be a compact semisimple homogeneous space, with e g = Lie( e G ) = e g = e g ⊕· · ·⊕ e g r for e g i simple, and denote by e h i the projection of the isotropy subalgebraonto e g i in the decomposition. Suppose ( e g , k ) is a symmetric pair with k containing e h . Denote k = k ⊕ · · · ⊕ k r with each k i the projection of k onto e g i as before.Then, each ( e g i , k i ) is an irreducible symmetric space of the compact type. For each i , denote by g i the corresponding noncompact symmetric dual of e g i , and h i thecorresponding subalgebra isomorphic to e h i contained in g i . Then, the Lie algebra g = g ⊕ · · · ⊕ g r gives rise to a simply-connected Lie group G and a connectedsubgroup H such that h = Lie( H ). The corresponding homogeneous space G / H hasno compact simple factors. By considering all possible symmetric pairs ( e g , k ), weattain a complete classification of semisimple homogeneous spaces without compactsimple factors [Nik05]. Definition 4.1.
We call a minimally presented homogeneous space G / H Alek-seevskii Unresolved (AU) if it is simply-connected, noncompact, non-symmetric,non-product, with G semisimple without compact simple factors, and without sl ( R )ideals in g . Analogously, if G / H is simply-connected, compact, non-symmetric, non-product, semisimple without su (2) ideals in g , then we say that G / H is AU*.In order to prove Theorem A, we may argue in the following way:(1) Use classifications of AU* spaces [BK06] and the partial duality detailedabove [Nik05] to construct a list of AU spaces (see Table 1).(2) Decompose the isotropy representation for each AU space into irreduciblesummands (in the sense of [Nik00]), and ignore the spaces with q i p j forevery i, j .(3) Analyse the space of invariant metrics for each remaining AU space, andshow that none them are Einstein.In Table 1, we give a classification of AU spaces with non-trivial isotropy byproviding the transitive group G , isotropy subgroup H giving rise to the AU space G / H , along with its corresponding maximal compact subgroup K containing H ,and corresonding compact dual e G used in the dualisation procedure detailed above.All embeddings of K into G are taken from [Hel78], and the embeddings of H into K are taken to be the canonical ones, unless otherwise specified in the notesbelow. It should also be understood that for each space in the table, we are actu-ally considering the universal covers of these spaces, since they are assumed to besimply-connected. For each AU space, we provide the decomposition of the isotropyrepresentation into irreducible summands, following the notational conventions of[AL16] also used throughout. ROHIN BERICHON dim e G H K G
Isotropy Representation Notes5 SU (3) SU (2) U (2) SU (2 , q (1)0 ⊕ p (4)1 SU (2) × SU (2) ∆ p,q U (1) ∆ SU (2) SL ( C ) q (2)1 ⊕ p (1)0 ⊕ p (2)1 , q (2)1 ≃ p (2)1 (b)6 SU (3) T U (2) SU (2 , q (2)1 ⊕ p (2)1 ⊕ p (2)2 Sp (2) U (1) Sp (1) Sp (1) × Sp (1) Sp (1 , q (2)1 ⊕ p (4)1 SU (3) ∆ p,q U (1) SO (3) SL ( R ) q (2)1 ⊕ p (1)0 ⊕ p (2)1 ⊕ p (2)2 , q (2)1 ≃ p (2)1 (c) U (2) SU (2 , q (1)0 ⊕ q (2)1 ⊕ p (2)1 ⊕ p (2)2 , p (2)1 ≃ p (2)2 ⇔ p = q = 1 , q (2)1 ≃ p (2)1 ⇔ p = 0 , q = 1 (a) SU (4) SU (3) U (3) SU (3 , q (1)0 ⊕ p (6)1 Sp (2) Sp (1) Sp (1) × Sp (1) Sp (1 , q (3)1 ⊕ p (4)1 (e) SO (5) SO (3) SO (4) SO (4 , q (3)1 ⊕ p (1)0 ⊕ p (3)1 , p (3)1 ≃ q (3)1 SO (3) × SO (2) SO (3 , q (1)0 ⊕ p (3)1 ⊕ p (3)2 , p (3)1 ≃ p (3)2 Sp (2) T Sp (1) × Sp (1) Sp (1 , q (2)1 ⊕ q (2)2 ⊕ p (2)1 ⊕ p (2)2 U (2) Sp (2 , R ) q (2)1 ⊕ p (2)1 ⊕ p (2)2 ⊕ p (2)3 SO (6) SO (4) SO (4) × SO (2) SO (4 , q (1)0 ⊕ p (4)1 ⊕ p (4)2 , p (4)1 ≃ p (4)2 SU (4) Sp (1) × Sp (1) Sp (2) SL ( H ) q (4)1 ⊕ p (1)0 ⊕ p (4)1 , q (4)1 ≃ p (4)1 SU (5) SU (4) U (4) SU (4 , q (1)0 ⊕ p (8)1 Sp (2) ∆ p,q U (1) Sp (1) × Sp (1) Sp (1 , q (1)0 ⊕ q (2)1 ⊕ q (2)2 ⊕ p (2)1 ⊕ p (2)2 , p (2)1 ≃ p (2)2 ⇔ p = 0 , q = 1 , q (2)1 ≃ q (2)2 ≃ p (2)2 ⇔ p = 0 , q = 1 (a) U (2) Sp (2 , R ) q (1)0 ⊕ q (2)1 ⊕ p (2)1 ⊕ p (2)2 ⊕ p (2)3 , p (2)1 ≃ p (2)2 ≃ p (2)3 ⇔ p = q = 1 , p (2)2 ≃ p (2)3 ⇔ p = 0 , q = 1 (a) SU (2) ∆(∆ SU (2)) (∆ SU (2)) SL ( C ) q (3)1 ⊕ p (3)1 ⊕ p (3)2 q (3)1 ≃ p (3)1 ≃ p (3)2 SU (3) SU (2) ∆ p,q U (1) U (2) SU (2 , q (1)0 ⊕ p (4)1 ⊕ p (4)2 p (4)1 ≃ p (4)2 (a)10 SU (4) S ( U (1) U (1) U (2)) U (3) SU (3 , q (4)1 ⊕ p (2)1 ⊕ p (4)2 S ( U (2) ) SU (2 , q (2)1 ⊕ p (4)1 ⊕ p (4)2 Sp (3) U (1) Sp (2) Sp (1) Sp (2) Sp (1 , q (2)1 ⊕ p (8)1 G U (2) SO (4) G q (2)1 ⊕ p (8)1 (d) G U (2) SO (4) G q (2)1 ⊕ p (4)1 ⊕ p (4)2 (d) SU (3) SU (2) U (2) SU (2 , q (2)0 ⊕ p (4)1 ⊕ p (4)2 SU (2) ∆ p,q U (1)∆ r,s U (1) (∆ SU (2)) SL ( C ) × SL ( C ) q (2)1 ⊕ q (2)2 ⊕ p (1)0 ⊕ p (2)1 ⊕ p (2)2 , q (2)1 ≃ p (2)1 , q (2)2 ≃ p (2)2 (a, b) SU (3) × SU (2) ∆ p,q U (1)( SU (2) × { e } ) U (2)∆ SU (2) SU (2 , × SL ( C ) q (1)0 ⊕ q (2)1 ⊕ p (1)0 ⊕ p (4)1 ⊕ p (2)2 , p (2)2 ≃ q (2)1 (a) SU (3) × SU (2) SU (2) × ∆ p,q U (1) U (2)∆ SU (2) SU (2 , × SL ( C ) q (1)0 ⊕ q (2)1 ⊕ p (1)0 ⊕ p (4)1 ⊕ p (2)2 , p (2)2 ≃ q (2)1 (a) SU (3) × SU (2) U (1)∆ SU (2) U (2) SU (2) SU (2 , × SL ( C ) q (3)1 ⊕ p (4)1 ⊕ p (3)2 q (3)1 ≃ p (3)2 Sp (2) × SU (2) Sp (1)∆ SU (2) Sp (1) × SU (2) Sp (1 , × SL ( C ) q (3)1 ⊕ p (4)1 ⊕ p (3)2 q (3)1 ≃ p (3)2 Table 1.
Non-symmetric, non-product, noncompact, homoge-neous spaces with semisimple transitive group without compactsimple factors, without sl ( R ) ideals in g = Lie( G ), with non-trivialisotropy, and their corresponding compact duals, in dimensions lessthan or equal to 10. HE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS 9
Notes on Table 1:(a) p, q ∈ Z , 0 ≤ p ≤ q , gcd( p, q ) = 1, see [Wan82].(b) The embedding of ∆ p,q U (1) into SL ( C ) gives rise to the homogeneous space SL ( C ) / U (1).(c) p = q .(d) Precise meanings of the embeddings of U (2) and notations may be foundin [DK08].(e) The diagonal embedding ∆ Sp (1) and embeddings onto each of the factorsof Sp (1) into Sp (1) × Sp (1) both give equivalent isotropy representations. Proposition 4.2.
Suppose G / H is a homogeneous space with G semisimple with an sl ( C ) ideal of g . Then, if the projection of h onto any sl ( C ) factor is isomorphicto su (2) , there exists a closed Lie subgroup G of smaller dimension than G actingtransitively on M .Proof. Take g = sl ( C ) and let q be any Borel subalgebra complementary to h and apply Lemma 3.1. (cid:3) The upshot of this is that the following spaces are not minimally presented: SL ( C ) / ∆ SU (2) , SU (2 , × SL ( C ) / U (1)∆ SU (2) , Sp (1 , × SL ( C ) / Sp (1)∆ SU (2) . We will now prove Theorem A by conducting a case-by-case analysis on theremaining AU spaces.
Proof of Theorem A.
As explained above, we may assume G is semisimple. In lieu ofthe discussion above, it is sufficient to verify the non-existence of invariant Einsteinmetrics with negative Einstein constant on the following cases: SL ( H ) / Sp (1) , Sp (1 , / ∆ p,q U (1) , SL ( C ) × SL ( C ) / U (1) , SU (2 , × SL ( C ) / ∆ p,q U (1)( SU (2) × { e } ) , SU (2 , × SL ( C ) / SU (2) × ∆ p,q U (1) . SL ( H ) / Sp (1) . As in Table 1 we have the following decomposition of p ⊕ q into ad sp (1) ⊕ sp (1) -irreducible submodules; q (4)1 = (cid:26)(cid:20) h − h (cid:21) : h ∈ H (cid:27) , p (1)0 = (cid:26)(cid:20) z − z (cid:21) : z ∈ R (cid:27) , p (4)1 = (cid:26)(cid:20) h h (cid:21) : h ∈ H (cid:27) , with a 1-parameter family of isomorphisms ψ λ : q (4)1 → p (4)1 given by ψ λ (cid:18)(cid:20) h − h (cid:21)(cid:19) = (cid:20) λh λh (cid:21) , λ ∈ R . Hence, by Schur’s Lemma, every ad h -homomorphism φ : q → p is given by φ = ψ λ for some λ ∈ R , and so Hom( q , p ) = R . Unfortunately, [ q , q ] h and [ p , p ] h , so SL ( H ) / Sp (1) admits SL ( H )-invariant metrics with no orthogonal Cartandecomposition.Writing sl ( H ) = sl ( R ) ⊕ i sl ( R ) ⊕ j sl ( R ) ⊕ k sl ( R ) and using the standardordered basis n ˆ h, ˆ e, ˆ f o for sl ( R ), consider the following ordered bases for the sub-modules of q ⊕ p , which form an ordered basis B q ⊕ p = { e i } i =1 = B p (1)0 ∪B p (4)1 ∪B q (4)1 ; B p (1)0 = n ˆ h o , B p (4)1 = n ˆ e + ˆ f , i ( ˆ f − ˆ e ) , j ( ˆ f − ˆ e ) , k ( ˆ f − ˆ e ) o , B q (4)1 = n ˆ e − ˆ f , i ( ˆ f + ˆ e ) , j ( ˆ f + ˆ e ) , k ( ˆ f + ˆ e ) o , and fix an Ad( Sp (1) )-invariant inner product h· , ·i on p ⊕ q making B p ⊕ q or-thonormal. For example, one choice of inner product is, up to scaling, the Killingform of sl ( H ) restricted to the isotropy submodules p ⊕ q . We may then parametrisethe space of SL ( H )-invariant metrics on SL ( H ) / Sp (1) by using the fact that q (4)1 and p (4)1 are both of real type in the following Lemma 4.3.
Up to isometry, the space of SL ( H ) -invariant metrics on SL ( H ) / Sp (1) is parametrised by the 4-parameter family of Ad( Sp (1) ) -invariantinner products on p ⊕ q of the form h· , ·i Q = h Q · , ·i , where Q ∈ GL ( R ) is given by Q = a b I d I d I c I , a, b, c > , d < bc. Suppose h· , ·i Q induces an SL ( H )-invariant Einstein metric on SL ( H ) / Sp (1) with negative Einstein constant, and denote the orthonormal basis given by runningthe Gram-Schmidt algorithm on B p ⊕ q by B p ⊕ q = { e i } i =1 . Then, the Ricci curvaturesatisfies Ric Q ( e , e ) = 2(7 a − b + 2 c ) dab √ bc − d . Since h· , ·i Q is an Einstein metric, either d = 0 or b = (7 a + 2 c ) /
2. In the formercase, the Cartan decomposition is orthogonal, giving us a contradiction by Theorem2.1. In the latter case, all of the off diagonal terms of the Ricci curvature vanish,and Ric Q ( e , e ) = 45 a bc − d ) , which is positive, a contradiction to the negativity of the Einstein constant.4.2. Sp (1 , / ∆ p,q U (1) . As long as p = 1 and q = 1, every Sp (1 , p = q = 1, since the space of Sp (1 , SL ( C ) × SL ( C ) / U (1) . Writing sl ( C ) = sl ( R ) ⊕ i sl ( R ) and using the stan-dard ordered basis n ˆ h, ˆ e, ˆ f o for sl ( R ), let us fix an Ad( U (1) )-invariant innerproduct h· , ·i on the reductive complement of u (1) ⊕ u (1) such that the orderedbasis B p ⊕ q = { e i } i =1 = B p (2)0 ∪ B q (2)1 ∪ B p (2)1 ∪ B q (2)2 ∪ B p (2)2 for p ⊕ q in terms of thedecomposition sl ( C ) ⊕ sl ( C ) given by B p (2)0 = n ˆ h ⊕ , ⊕ ˆ h, o , B q (2)1 = n (ˆ e − ˆ f ) ⊕ , i (ˆ e + ˆ f ) ⊕ o , B q (2)2 = n ⊕ (ˆ e − ˆ f ) , ⊕ i (ˆ e + ˆ f ) o , B p (2)1 = n (ˆ e + ˆ f ) ⊕ , i (ˆ e − ˆ f ) ⊕ o , B p (2)2 = n ⊕ (ˆ e + ˆ f ) , ⊕ i (ˆ e − ˆ f ) o is orthonormal. As before, this is just the rescaled Killing form restricted to theisotropy submodules. Again, we may parametrise the space of SL ( C ) -invariantmetrics on SL ( C ) / U (1) as the space of Ad( U (1) )-invariant inner products on p ⊕ q in the following HE ALEKSEEVSKII CONJECTURE IN 9 AND 10 DIMENSIONS 11
Lemma 4.4.
Up to isometry, the space of SL ( C ) -invariant metrics on SL ( C ) / U (1) is parametrised by the 7-parameter family of Ad( U (1) ) -invariantinner products on p ⊕ q of the form h· , ·i Q = h Q · , ·i , where Q ∈ GL ( R ) is given by Q = a c c b d ℓ d − ℓ − ℓ q ℓ q f n f − n
00 0 0 0 0 0 0 − n g
00 0 0 0 0 0 n g , where a, d, q, f, g > , ℓ < dq, n < f g, c < ab .Proof. Since we have the equivalences p (2)1 ≃ q (2)1 and p (2)2 ≃ q (2)2 of complex typeisotropy submodules, we may parametrise the space of Ad( U (1) )-invariant innerproducts Q in the form of the lemma, except with a block of the form h p ℓ − ℓ p i mapping q (2)1 to p (2)1 , and a block of the form [ m n − n m ] mapping q (2)2 to p (2)2 with ℓ + p < dq and m + n < f g . Since p (2)0 is a trivial module, the two-parameterfamily of automorphisms, given by P ( t, s ) = Ad(exp( te + se )) give rise to anisometric family of metrics of the form P T QP . Moreover, in this family, P T QP has the form of the Lemma if and only if there exist t, s ∈ R such that( e s − f + ( e s − g + 2(1 + e s ) m = 0 , ( e t − d + ( e t − q + 2(1 + e t ) p = 0 . Using the relations m < f g and p < dq , it is easy to find t, s such that P T QP satisfies the form of Q in the Lemma. (cid:3) Suppose Q gives rise to an SL ( C ) -invariant Einstein metric on SL ( C ) / U (1) ,and denote the orthonormal basis given by running the Gram-Schmidt algorithm on B p ⊕ q by B p ⊕ q = { e i } i =1 . Then, looking at the Ricci curvature in the off-diagonaldirections, we find thatRic Q ( e , e ) = 2 c ∆ a √ ab − c ∆ = 2 + 2 ℓ d + ab − c f g − n + 2( ab − c ) n ( n − f g ) + ( d + ℓ ) d ( dq − ℓ ) + dq − ℓ d . Since dq − ℓ , f g − n , ab − c , a, b, d, q, f, g >
0, we have that ∆ >
0, and soRic Q ( e , e ) = 0 if and only if c = 0, so that SL ( C ) / U (1) splits as a Riemann-ian product. But SL ( C ) / U (1) admits no invariant Einstein metrics by [AL16], acontradiction.4.4. SU (2 , × SL ( C ) / ∆ p,q U (1)( SU (2) × { e } ) . Again, this space admits invari-ant metrics for which no Cartan decomposition is orthogonal. Looking at thedecomposition of the isotropy representation, we have the following orderedbases B q ⊕ p = { e i } i =1 = B q (1)0 ∪ B p (1)0 ∪ B p (2)2 ∪ B q (2)1 ∪ B p (4)1 and B h = { h i } i =1 : B h = − ip i ( p + q ) 0 0 00 0 − iq i ( p − q ) 00 0 0 0 i ( q − p ) , i − i , − , i i , B q (1)0 = i i − i , B p (1)0 = − B q (2)1 = − , i i , B p (2)2 = , i − i , B p (4)1 = , i − i , , i − i . We have the equivalences q (2)1 ≃ p (2)2 , so any invariant metric would make p (1)0 ⊕ q (1)0 , q (2)1 ⊕ p (2)2 and p (4)1 orthogonal. Moreover, ad( e ) acts trivially on p (1)0 ⊕ q (1)0 and q (2)1 ⊕ p (2)2 , and as ad( h + ( p + q ) h ) on p (4)1 , and so acts by skew symmetricendomorphisms on q ⊕ p for any invariant metric. By [AL16, Lemma 2.10], for anyinvariant metric g , Ric g ( e , e ) = X i,j g ([ e i , e j ] , e ) ≥ . SU (2 , × SL ( C ) / SU (2) × ∆ p,q U (1) . Since the isotropy representations of thisand the previous space are equivalent up to the p (4)1 modules, the computation hereis identical to the above. (cid:3) References [AK75] Dimitri Alekseevskii and Boris Kimel’fe’ld,
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