Einstein metrics on flag manifolds

dos Santos, Evandro C. F.; Negreiros, Caio J. C.

Servicios Personalizados

Revista

Articulo

Indicadores

Citado por SciELO

Links relacionados

Similares en SciELO

Otros
Otros

Permalink

Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dic. 2006

Einstein metrics on flag manifolds

Evandro C. F. dos Santos^* and Caio J. C. Negreiros^**

^*Partially supported by FUNCAP
^**Partialy supported by CNPq grant 303695/2005-6 and Fapesp grant 02/10246-2

Abstract: In this survey we describe new invariant Einstein metrics on flag manifolds. Following closely San Martin-Negreiros's paper [26] we state results relating Kähler, (1,2)-symplectic and Einstein structures on flags. For the proofs see [11] and [10].

2000 Mathematics Subject Classification. 53C55; 58D17; 53C25; 22F30.

Key words and phrases. Flag manifolds, Einstein metrics, Semi-simple Lie groups.

Introduction

We recall that a Riemannian metric on a manifold is called Einstein if Ric (g) = cg for some constant . As we know Einstein metrics form a special class of metrics on a given manifold (see [4]). In this note we announce properties of these metrics and new examples of Einstein metrics on flag manifolds as described in [11] and [10].

With this purpose in mind, we consider as being a complex semi-simple Lie algebra and a simple root system for . If is an arbitrary subset of , ⟨Θ ⟩ denotes the roots spanned by . We have

∑ ∑ ∑ ∑ 𝔤 = 𝔥 ⊕ 𝔤α ⊕ 𝔤 -α ⊕ 𝔤 β ⊕ 𝔤-β, α∈⟨Θ⟩ α∈ ⟨Θ⟩ β∈Π+-⟨Θ⟩ β∈Π+ -⟨Θ⟩

(1)

where is a Cartan subalgebra of and is the root space associated to the root . Let

∑ ∑ ∑ 𝔭Θ := 𝔥 ⊕ 𝔤α ⊕ 𝔤-α ⊕ 𝔤β, α∈⟨Θ ⟩ α∈⟨Θ⟩ β∈Π+ -⟨Θ⟩

(2)

the canonical parabolic subalgebra determined by . Hence

∑ 𝔤 = 𝔭Θ ⊕ 𝔤- β. β∈Π+-⟨Θ⟩

(3)

𝔽Θ = G ∕P is called a flag manifold, where has the Lie algebra and is the normalizer of in . Each manifold 𝔽 Θ has a very rich complex geometry, containing families of invariant Hermitian structures denoted by (𝔽Θ, J,ds2) Λ .

The case 𝔽 = 𝔽 Θ for Θ = ∅ , i.e., the full flag manifold is nowadays well understood. Starting with the work of Borel (cf. [7]), the classification of all invariant Hermitian structures is known and it was derived in [26].

On the other hand, the case for Θ ⁄= ∅ is much less known so far. Some partial results are derived in [27] and [28].

We now describe the contents of this survey. In the first two sections we discuss all the invariant Hermitian structures on 𝔽 Θ and the associated Einstein system of equations. In Section 3 we present new invariant Einstein metrics on generalized flag manifolds of type Al. We suggest Besse's book [4] as a reference for Einstein manifolds.

In Section 4 we state the classification of all invariant Einstein metrics on 𝔽 (4) and state some partial results relating Kähler, (1,2)-symplectic and Einstein structures on 𝔽 (n) .

For a very stimulating article see [1].

All manifolds and maps between them will be assumed to be ∞ C in this survey.

We would like to thank to FAEP-UNICAMP and FAPESP (grant 02/10246-2) for the financial support.

1. General results on the invariant Hermitian geometry of flag manifolds

We denote by ⟨⋅,⋅⟩ the Cartan-Killing form of , and we fix a Weyl basis {X α} α∈Π for . We define the compact real form of , as the real subalgebra

𝔲 = span ℝ{i𝔥ℝ, Aα,iSα : α ∈ Π},

where Aα = Xα - X -α and S α = X α + X - α .

Let be the origin of 𝔽 Θ . TxΘ𝔽 Θ is identified with

$TxΘ 𝔽Θ ≈ 𝔪 Θ =∑ span ℝ{A α,iSα : α ∈∕⟨Θ ⟩} = = 𝔲 , α α∈Π\⟨Θ⟩=ΠΘ$

where 𝔲 = (𝔤 ⊕ 𝔤 ) ∩ 𝔲 = span {A ,iS } α α -α ℝ α α . Complexifying 𝔪 Θ we obtain Tℂ 𝔽 xΘ Θ , which can be identified with

$ℂ ∑ 𝔪 Θ = 𝔤β. β∈Π \⟨Θ⟩$

(4)

A -invariant almost complex structure on , is completely determined by a collection of numbers ɛσ = ±1 , σ ∈ ΠΘ .

A -invariant Riemannian metric ds2Λ on is completely characterized by the following inner product ⟨⋅,⋅⟩ on 𝔪 Θ

⟨X, Y ⟩Λ := - ⟨ΛX, Y ⟩,

(5)

where Λ : 𝔪 Θ → 𝔪 Θ is definite-positive with respect to the Cartan-Killing form. On each irreducible component of 𝔪 Θ , Λ = λ id σ with λ = λ > 0 -σ σ .

Consider the conjugation of relatively to . Hence, ⟨⟨X, Y ⟩⟩Λ = ⟨X, τY ⟩Λ is a Hermitian form on , that originates a -invariant Hermitian form on 𝔽 Θ .

If Ω = ΩJ,Λ denotes the corresponding Kähler form then

--- Ω (X ,X ) = - √ - 1λ ε ⟨X ,X ⟩. α β α β α β

(6)

We recall that a almost-Hermitian manifold is said (1, 2) -symplectic if dΩ (X,Y, Z ) = 0 when one of the vectors , , is of type (1,0) , and the other two are of type (0,1) . If is integrable and d Ω ≡ 0 , we say (𝔽 Θ,J,ds2Λ) is a Kähler manifold.

2. Ricci tensor and the Einstein system of equations

We now consider {eα} a -orthogonal basis adapted to a decomposition of l 𝔪 = ⊕ 𝔪 . k=1 k In other words, e ∈ 𝔪 α i for some i ∈ {1,⋅⋅⋅ ,l}, and α < β if i < j with eα ∈ 𝔪i, eβ ∈ 𝔪j. Define, as in [29],

γ A αβ = ([eα,eβ],eγ),

(7)

that is,

[ k ] ∑ γ 2 i j (A αβ) =

(8)

where in the second equation we take all indices α,β,γ with e ∈ 𝔪 ,e ∈ 𝔪 ,e ∈ 𝔪 . α i β j γ k Notice that [ ] k i j is independent of orthonormal frame chosen for 𝔪i, 𝔪j,𝔪k and ∑ [eα,eβ ] = γ A γαβ

[ k ] i j = [ k ] [ j ] j i = k i

Furthermore, if is an element of Weyl´s group then

[ w (γ) ] [ γ ] w(α) w (β) = α β

(9)

The following result is due to Wang-Ziller [29] (see also [2]):

Lemma 2.1. The components of the Ricci tensor of an -invariant metric on M = U∕K are given by:

∑l [ k ] ∑l [ j ] r = -1--+ -1-- -λk-- i j - --1- -λk-- k i (k = 1,⋅⋅⋅ ,l), k 2λk 4dk λiλj 2dk λiλj i,j=1 i,j=1

(10)

where ⊕l 𝔪 = k=1 𝔪k, dk = dim 𝔪k .

More generally, Arvanitoyeorgos proved in [3] the following result

Proposition 2.2. The Ricci tensor of an invariant metric (Λα) = {λ α > 0, α ∈ ΠM } on a flag manifold is given by

Ric (X α,X β) = 0, if α∑,β ∈ ΠM , α + β∑ ∕∈ ΠM 2 2 2 Ric (X ,X ) = (α, α) + m2 + 1- m-α,β(λ-α --(λ-α+β---λβ)-) α -α α,φ 4 λ α+βλβ αφ∈+φΠ∈ΘΠ αβ+∈βΠ∈MΠM

We have the following non-homogeneous version of this equation

1 ∑ λ2 - (λik - λjk)2 λij = 2 + -- --ij--------------- 2 k⁄=i,j λijλjk

With each solution we associate the Einstein constant, which is defined as the value of the Ricci tensor rij when is re-normalized to have unit volume.

3. New Einstein metrics

Using the Einstein system of equations described above, we describe now the known and new Einstein metrics on 𝔽(n ) as in [11] and [10].

a) The normal metric. We notice this metric is not Kähler.

b) Kähler-Einstein metrics

On the flag manifold 𝔽(n) ( n ≥ 3) , up to permutation there is a unique integrable structure , and associated with it a unique (up to scaling) Kähler-Einstein metric (which corresponds to the choice 1 ∑ c = δ = 2 β β according to Matsushima [19] or [4]):

( ) 0 -1 -1 ... n-1- | 1 2n n1 . 2n. | || 2n- 0 2n .. .. || | 1 -1 0 ... 1 | Λ = || n. 2.n . . n || . || .. .. .. .. 12n- || ( n-1- ... -1 1- 0 ) 2n n 2n

Thus, counting in the symmetry of this metric, we have n! 2 Kähler-Einstein metrics on 𝔽 (n) .

c) The Arvanitoyeorgos metrics

Arvanitoyeorgos ([3]) considers for all s ∈ [1,n] metrics in 𝔽 (n) ( n ≥ 4 ) satisfying

λij = A ( s ∈ {i,j} ), λij = B otherwise

The Einstein system is reduced to the equations in whose solution is A = n - 1 and B = n + 1 . Counting permutations, we get Arvanitoyeorgos metrics whose Einstein constant is seen to be

2 ∘n -------2-------n-2 c = (n----n-+-2)---(n --1)-(n +-1)---. Arv. 4n (n - 1)2

d) The Sakane-Senda metrics

Sakane and Senda in [25] consider metrics in 𝔽(2m ) ( m ≥ 3 ) satisfying

λij = A ( i,j ≤ m or i,j > m ), λij = B otherwise

Again, the the Einstein system is reduced to two equations in whose solution is A = m + 2 and B = 3m - 2 .

e) A new family

If m ≥ 6 we find another solution in 𝔽 (2m ) , for A = m + 5 and B = 3m - 5 .

f) Two new families

On 𝔽(2m + 1) ( m ≥ 6 ) we consider

λij = A ( i,j ≤ m + 1 or i,j > m + 1 ), λij = B otherwise

There are two families as solution of the Einstein system. The Einstein constants for these two families are, respectively,

∘ ----------------------- 1 1 2n-2 2 4n n+3 ± 2√ (n-5)(n-13) n-1 c± = (2 + 4n((n+3) ±√2(n-1)2--4n+16-)) ( --------4--------) . 2

g) A new metric

Still assuming the same pattern, with m = 2 , we find on 𝔽 (5) the invariant Einstein metric with A = 1 and B = 2 . The Einstein constant of this metric is √5- 1140-4 .

We define the class EN K = {ds2Λ; ds2Λ is a Einstein non -K¨ahler metric in a) or c) or d) or e) or f) or g). } .

A complete classification of the Einstein metrics for 𝔽 (n) (n ⁄= 3,4) is completely unknown. It is not even know if the number of such metrics is finite (the Bohn-Wang-Ziller conjecture).

In [11] and [10] we use the procedure described above in order to obtain new Einstein metrics on non-maximal A-type manifolds. Our notation will be 𝔽(n;n1, ...,nk ) where (n ,...,n ) 1 k represents block-matrices of size n = ∑k n i=1 i . All the entries in each block are equal, so that the metric is completely expressed by a reduced matrix, which we denote by .

Theorem 3.1. a) On 𝔽(5;2,1,1, 1). The set of restrictions λ = λ = λ and λ = λ = λ , 12 13 14 23 24 34 produce two invariant non-Kähler Einstein. On the other hand the restrictions λ12 = λ13 = λ23 = λ24 and λ14 = λ34 do not produce any solution.

b) On ---U(n)--- 𝔽 (n;k,q,q,⋅⋅⋅ ,q) = U(k)×U (q)s, i.e. n = k + sq ( √--2--- q( s - 4 + 2 - s) < 2k, ) we look for a (s + 1) (s + 1) reduced matrix with

^ λij = A ( 1 ∈ {i,j} ), otherwise

In this way we can produce two non-Kähler Einstein metrics.

c) On 𝔽 (n; k,k,⋅⋅⋅ ,k) with n = sk the invariant metric represented by the matrix is Einstein if, and only if, the same matrix represents an Einstein metric on 𝔽 (s) .

4. Results on the classification of Einstein metrics on

Gray and Hervella in [13] gave a complete classification of triples (M, g,J ) into sixteen classes for arbitrary almost Hermitian manifolds. San Martin-Negreiros discussed in [26] the case where is a maximal flag manifold. They have proved that the invariant almost Hermitian structures on maximal flag manifolds can be divided only in three classes, namely

where the class contains any invariant almost Hermitian structures.

In [26] it is proved that an invariant pair (J, Λ) ∈ W1 ⊕ W2 if and only if for all {1, 2} -triple of roots {α, β,γ}

ε λ + ε λ + ε λ = 0. α α β β γ γ

(11)

The next lemma characterizes the Hermitian structures belonging to W1 ⊕ W3 (see [26]) for more details.

Lemma 4.1. A necessary and sufficient condition for an invariant pair (J,Λ ) to be in W1 ⊕ W3 ≈ W1 ⊕ W3 ⊕ W4 is λα = λβ = λ γ ∀ {0,3} -triple {α,β, γ} .

In [11] or [10] the following result is proved:

Theorem 4.2. If ds2 ∈ EN K Λ for n ≥ 4 , then this metric belongs to W1 ⊕ W3 .

This result leads us to conjecture that any invariant Einstein non-Kähler metric on 𝔽 (n) is in W ⊕ W 1 3 . One result supporting this conjecture is

Theorem 4.3. The space 𝔽 (4 ) admits (up to scaling) precisely 3 classes of invariant Einstein metrics: The Kähler-Einstein [7], the 4 Arvanitoyeorgos's class [3], and the class of the normal metric [30].

References

[1] V. I. Arnold, Symplectization, Complexification and Mathematical Trinities, Fields Institute Communications, 24 (1999), 23-36. [ Links ]

[2] A. Arvanitoyeorgos, Invariant Einstein metrics on generalized flag manifolds, PhD thesis, Rochester University,1991. [ Links ]

[3] A. Arvanitoyeorgos, New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. math.Soc., 337(1993), 981-995. [ Links ]

[4] A. Besse, Einstein Manifolds, Springer Verlag, 1987 [ Links ]

[5] C. Böhm, M. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geometric and Functional Analysis 4 Vol.14, (2004) 681-733. [ Links ]

[6] M. Bordemann, M. Forger and H. Romer, Homogeneous Kähler manifolds:Paving the way towards supersymmetric sigma-models, Commun. Math. Physics,102, (1986), 605-647. [ Links ]

[7] A. Borel, Kählerian coset spaces of semi-simple Lie groups, Proc. Nat. Acad. Sci. USA 40 (1954), 1147-1151. [ Links ]

[8] É. Cartan, Sur les domaines bornés homogènes de l'espace des n variables complexes, Abhandl. Math. Sem. Hamburg, 11 (1935), 116-162 [ Links ]

[9] N. Cohen, C. J. C. Negreiros, M. Paredes, S. Pinzón, L. A. B. San Martin, f-structures on the classical flag manifold which admit (1,2)-symplectic metrics. Tôhoku Math. J. 57(2) (2005), 262-271. [ Links ]

[10] N. Cohen, E. C. F. dos Santos and C. J. C. Negreiros, Properties of Einstein metrics on flag manifolds, Preprint. State University of Campinas, 2005. [ Links ]

[11] Evandro C. F. dos Santos,Métricas de Einstein em varedades bandeira, PhD thesis. State University of Campinas, 2005. [ Links ]

[12] Evandro C. F. dos Santos, Caio J. C. Negreiros and Sofia Pinzòn, (1,2)-symplectic submanifols of a flag manifold, Preprint. State University of Campinas, 2005. [ Links ]

[13] A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35-58. [ Links ]

[14] M.A.Guest, Geometry of maps between generalized flag manifolds, J. Differential Geometry, 25 (1987), 223-247 [ Links ]

[15] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. [ Links ]

[16] D.Hilbert, Die Grundlagen der Physik,Nachr. Akad. Wiss. Gött. 395-407 (1915). Reprinted in Math. Ann., 92 (1924),1-32. [ Links ]

[17] M. Kimura, Homogeneous Einstein Metrics on Certain Kähler C-spaces, Advanced Studies in Pure Mathematics. 18-I, 1990. Recent topics in Differential and analytic Geometry (1986), 303-320. [ Links ]

[18] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Publishers, vol.2 (1969). [ Links ]

[19] Y. Matsushima, Remarks on Kähler-Einstein Manifolds, Nagoya Math. Journal 46, (1972), 161-173. [ Links ]

[20] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. in Mathematics 21, (1976), 293-329. [ Links ]

[21] X. Mo and C. J. C. Negreiros, (1,2)-Symplectic structures on flag manifolds, Tohoku Math. J. 52 (2000), 271-282. [ Links ]

[22] Y. Mutö, On Einstein metrics, Journal of Differential geometry, 9 (1974), 521-530. [ Links ]

[23] M. Paredes, Aspectos da geometria complexa das variedades bandeira, Tese de Doutorado, Universidade Estadual de Campinas,2000. [ Links ]

[24] S. Pinzòn, Variedades bandeira,estruturas-f e mètricas (1,2)-simplèticas, Tese de Doutorado, Universidade Estadual de Campinas, 2003. [ Links ]

[25] Y. Sakane, Homogeneous Einstein metrics on flag manifolds, Lobatchevskii Journal of Mathematics, vol.4, (1999), 71-87. [ Links ]

[26] Luiz A. B. San Martin and Caio J. C. Negreiros, Invariant almost Hermitian structures on flag manifolds, Advances in Math., 178 (2003), 277-310. [ Links ]

[27] Luiz. A. B. San Martin and Rita. C. J. Silva, Invariant nearly-Kähler structures, preprint, State University of Campinas (2003). [ Links ]

[28] Rita de Cássia de J. Silva, Estruturas quase Hermitianas invariantes em espaços homogêneos de grupos semi-simples, Ph.D. thesis, State University of Campinas, 2003. [ Links ]

[29] M. Wang and W. Ziller, Existence and Non-Existence of Homogeneous Einstein Metrics, Invent. math., 84, (1986), 177-194. [ Links ]

[30] M. Wang and W. Ziller, On normal homogeneous Einstein metrics, Ann. Sci. Ecole norm. Sup.18 (1985), 563-633. [ Links ]

Evandro C. F. dos Santos
Department of Mathematics
Universidade Regional do Cariri
Av. Leão Sampaio s/n Km 4
Juazeiro do Norte-Ce
Cep 63040-000 - Brazil
evandrocfsantos@gmail.com

Caio J. C. Negreiros
Departament of Mathematics - IMECC - Unicamp
PO Box 6065 - Campinas - Brazil
caione@ime.unicamp.br

Recibido: 20 de octubre de 2005
Aceptado: 15 de noviembre de 2006