Stability of holomorphic-horizontal maps and Einstein metrics on flag manifolds

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

Stability of holomorphic-horizontal maps and Einstein metrics on flag manifolds

Caio J. C. Negreiros

Partialy supported by CNPq grant 303695/2005-6 and Fapesp grant 02/10246-2

Abstract: In this note we announce several results concerning the stability of certain families of harmonic maps that we call holomorphic-horizontal frames, with respect to families of invariant Hermitian structures on flag manifolds. Special emphasis is given to the Einstein case. See [23] for additional detail and the proofs of the results mentioned in this survey.

1. Introduction

Let be 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 . Thus,

∑ 𝔤 = 𝔭Θ ⊕ 𝔤- β. β∈Π+-⟨Θ⟩

(3)

The flag manifold is defined as 𝔽 Θ = G ∕P , where has Lie algebra and is the normalizer of in .

for Θ = ∅ , is called the full flag manifold and is denoted by . This case is nowadays well understood. Starting with the work of Borel (cf. [3]), the classification of all invariant Hermitian structures is known and it was described in [25].

The main purpose for this note is to announce some results discussing the stability phenomenon for the energy functional for to a special class of maps ψ : (M 2,J,g) → (𝔽Θ,J,ds2Λ) , called holomorphic-horizontal frames. The energy functional is taken with respect to several families of invariant Hermitian structures on 𝔽 Θ . These maps are deeply connected with the study of harmonic/minimal surfaces in n 𝕊 , n ℂℙ , n 𝔾k(ℂ ), n ℍℙ , Twistor Theory and so on (cf. [27], [8], [4], [15]).

The layout of the paper is as follows. In the first two sections, we state general results on the invariant Hermitian geometry of , and state the holomorphic and harmonic map equations. We give in this note, examples of families holomorphic-horizontal frames only in the (A ) l case, but in [23] we also discuss the cases (B ) l , (C ) l and (Dl ) .

We generalize and give additional results to the approach initiated by Black in [2] and the author in [21]. As a reference for harmonic maps theory we suggest the Eells-Lemaire [14] article.

In Section 3 we compute the second variation of energy for an arbitrary harmonic map on 𝔽 Θ . We state a basic perturbation lemma and a result for holomorphic-horizontal frames on 𝔽 Θ .

According to the classification results in [25], among all Hermitian invariant structures there are two main classes. Thus, in the last section we state results concerning the stability of holomorphic-horizontal frames regarding metrics in such classes and in particular, the case of metrics that are Einstein and non Kähler on the geometrical flag manifold .

2. Generalities on invariant Hermitian geometry of flag manifolds

Let be a root system and a simple root system for a simple Lie algebra . If is a subset of , ⟨Θ ⟩ denotes the roots generated by . We have the root decomposition :

∑ ∑ 𝔤 = 𝔥 ⊕ 𝔤α ⊕ 𝔤 -α ⊕ (4 ) α∈⟨Θ⟩ α∈ ⟨Θ⟩ ∑ ∑ 𝔤β ⊕ 𝔤-β, (5 ) β∈Π+ -⟨Θ⟩ β∈Π+- ⟨Θ ⟩

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⟩

(6)

The space 𝔽Θ = G ∕PΘ is called a flag manifold, where and are the Lie algebras of and P Θ , respectively.

Each manifold 𝔽 Θ has families of complex geometries, i.e., families of invariant Hermitian structures denoted by 2 (𝔽 Θ,J,dsΛ ) .

We denote by ⟨⋅,⋅⟩ the Cartan-Killing form of , and fix once and for all a Weyl basis of , which amounts to take X α ∈ 𝔤α such that ⟨X α,X -α ⟩ = 1 , and [X α,X β] = m α,βX α+β with m α,β ∈ ℝ , m -α,-β = - m α,β , and m α,β = 0 if α + β is not a root.

We define the compact real form of , as the real subalgebra

𝔲 = spanℝ {i𝔥 ℝ,Aα, iS α : 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

$TxΘ𝔽Θ ≈ ηΘ =∑ span ℝ{A α,iS α : α ∕∈ ⟨Θ ⟩} = = 𝔲α, α∈Π \⟨Θ⟩=Π Θ$

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

$𝔮 = ∑ 𝔤 . Θ β β∈Π\⟨Θ⟩$

We denote the irreducible components of as , where is the set of roots with 𝔤α ⊂ 𝔤σ , thus 𝔤σ = Σα∈ σ𝔤α .

Let Π(Θ ) be the collection of sets originating the irreducible components. We write

𝔮Θ = ⊕ σ∈Π(Θ )𝔤 σ = ⊕α ∈ΠΘ𝔤α.

Each σ ∈ Π (Θ ) defines a field of complex subspaces (Eσ)σ∈Π (Θ) such that ℂ ∑ T x 𝔽 Θ = σ∈Π (Θ )Eσ (x) for each x ∈ 𝔽Θ .

A -invariant almost complex structure on is completely determined by a linear map J : ηΘ → ηΘ . The map satisfies J2 = - 1 and commutes with the adjoint action of K Θ on . We denote also by its complexification to .

The invariance of entails that J(𝔤σ) = 𝔤σ for all σ ∈ Π (Θ) . The eigenvalues of are √ --- ± - 1 , and the eigenvectors in are X α , α ∈ Π Θ . Hence, in each irreducible component , √ --- J = - 1εσid with εσ = ±1 satisfying ε- σ = - εσ . A -invariant almost complex structure on is completely determined by the numbers ɛσ = ±1 , σ ∈ Π (Θ ) .

A -invariant Riemannian metric ds2 Λ on 𝔽 Θ is completely determined by the following inner product ⟨⋅,⋅⟩ on

⟨X, Y ⟩ := - ⟨ΛX, Y ⟩ Λ

(7)

with Λ : ηΘ → ηΘ definite-positive with respect to the Cartan-Killing form. On each irreducible component of , Λ = λ σid with λ-σ = λ σ > 0 .

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 .

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

√ --- Ω (X α,X β) = - - 1λ αεβ⟨Xα, Xβ⟩.

(8)

We recall that an 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.

3. Maps on

From now on, for abuse of notation we will denote a map φ : M → 𝔽Θ by (φσ)σ∈Π(Θ) where φ σ : M 2 → E σ , despite and (φ σ)σ∈Π(Θ) being completely different objects.

Black for the cases , and (see [2]) and the author in the case (see ([21]), obtained the Cauchy-Riemann equations in our situation:

Proposition 3.1. A map φ : M 2 → (𝔽Θ, J ) is -holomorphic on p ∈ M if and only if for every σ ∈ Π (Θ ) , φ (p ) ⁄= 0 σ implies φ (p) = 0 -σ .

We define the energy of as:

1 ∫ ( ⟨⟨∂φ, ∂φ⟩⟩ + ) E (φ) = -- ⟨∂⟨z∂φ ∂z∂φ⟩Λ⟩ υg 2 M ∂¯z, ∂¯z Λ 1 ∑ ∫ ( ⟨⟨φ εσ(p),φ εσ(p)⟩⟩ + ) = -- ⟨⟨ σ σ Λ⟩⟩ υg 2 σ∈Π(Θ)+ M φ ε-σσ(p),φ ε-σσ(p) Λ

To deduce the harmonic map equations, a basic remark is that

∫ ⟨ ⟨ ∂q ⟩⟩ M2 ( --1-, q2 + ∂z ⟨⟨ ∂q ⟩ ⟩ q1 ,---2 ) ∂z υg = 0 , for any perturbations

2 q1,q2 : M → g . In fact, every map : 2 M → ℂ satisfies

df (p) ∂f(p) ∂x dx + ∂f-(p)dy = ∂y

√ --- = 1(∂f-(p)dz + - 1 2 ∂z √ --- ∂f-(p)d( - 1z ) ∂z ) =

1∂f = 2---(p)dz+ ∂z √--1√ ---∂f (- -2--) - 1---(p) = ∂z ∂f ---(p)dz ∂z .

We now consider the map f = ⟨⟨q , q ⟩⟩ 1 2 : M 2 → ℂ . ∂f- = -∂- ⟨⟨q , q ⟩⟩ = ∂z ∂z 1 2

⟨ ⟩ ∂--⟨q1 , q2⟩ = ∂q1-, q2 + ∂z ∂z ⟨ ⟩ q1 ,-∂-(q2) = ∂z ⟨ ⟨ ⟩⟩ ∂q1-, q2 + ∂z ⟨ -----⟩ q1 ,(∂q2) = ∂z

⟨ ⟨ ⟩ ⟩ ∂q1-, q2 + ∂z ⟨ ⟨ ⟩ ⟩ q1 , ∂q2 ∂ z . According to Stokes' Theorem we have

∫ ∫ M2 df(p)υg = ∂(M2)f (p)υg = 0 . Thus,

⟨ ⟨ ⟩⟩ ∫ ∂q1- M2 ( ∂z , q2 + ⟨⟨ ⟩ ⟩ ∂q2- q1 , ∂z- ) υg = 0 .

We now perturb the map in the following natural way

t tq(p) φ (p ) := e ∘ φ(p) - ε < t < ε

We are considering here the natural action of Gl (n,C ) on and taking an arbitrary ∞ C map 2 q : M → 𝔤𝔩(𝔫, ℂ) .

In [23] we deduce the following Euler-Lagrange equations for our variational problem

Proposition 3.2. A map φ : (M 2,g) → (𝔽 ,ds2) Θ Λ is harmonic if and only if -d| E (φ ) = 0 dt t=0 t if and only if

( ∑ ) Re λ ∇ -φ (p) = 0, for every p ∈ M α z α α∈Π

(9)

We will use a generalization of to an -structure following Yano ([29]). An -structure on is a section of End (T (𝔽Θ)) such that F 3 + F = 0 .

An invariant -structure is given by the matrix ε(F) = (fα)α∈Π Θ with fα = 1 , or , according to the eigenvalues of .

We now state the Cauchy-Riemann equations in the case of -structures

Proposition 3.3. A map φ : (M 2,J) → (FΘ, F ) is -holomorphic if and only if it is subordinate to ε(F ) .

Definition 3.1. Consider an invariant -structure on 𝔽 Θ . Let

∑ F+ := fα α∈ΠΘ fα=1

and

∑ F- := fα. α∈ΠΘ fα=- 1

is said horizontal if [F ,F ] ⊂ 𝔭 + - Θ .

The following theorem due to Black ([2]) is essential in our study

Theorem 3.4. Let φ = (φ α) : (M 2,J,g) → (𝔽Θ,F ,ds2 ) Λ be subordinate to a horizontal -structure . Then is equi-harmonic.

Consider now ∂φ ∑ ---(p) = α∈ΠΘ xα(p)X α ∂z . We can prove that φ H := { α ∈ Π Θ, x α(p) = 0} is a horizontal -structure, and we will call it by -structure associated to .

We will now exhibit families of equi-harmonic and holomorphic maps 2 2 ψ : (M ,J,g) → (𝔽Θ,J,ds Λ) subordinate to an horizontal -structure, thus all of them are equi-harmonic according to Theorem 3.4. Any map in these families, is called a holomorphic and horizontal frame.

Let h : M 2 → ℂP n-1 be a holomorphic and non-degenerate map. We consider its associate curve θ : M 2 → G (ℂn) k k , where θ (p) := h(z) ∧ ∂h(p) ∧ ⋅⋅⋅ ∧ ∂(k-1)h(p) k ∂z ∂z and ⊥ Πk := θk ∩ θk-1 .

We define the map Ψ = (Π1, Π2, ...,Πn ) : M 2 → F (n ) . We can prove that ∑ ∂Ψ∂z = α∈Π xα (p )X α , with xα ≡ 0 if ∑ α ∈ Π - , where ∑ = {α12,α23,...,α (n-1)n} denotes a simple root system for sl(n + 1,ℂ ) .

We can prove that any such map is holomorphic and subordinate to H Ψ , thus, again according to Theorem 3.4, it is an equi-harmonic map.

More generally, we will now construct families of holomorphic and equi-harmonic maps Ψ : M 2 → 𝔽(n;n1, n2,...,nk) .

Let be the geometric flag manifold 𝔽 = -------U(n)--------= 𝔽(n; n1,n2,..., U(n1)×U(n2)×...×U (nk) nk) , where ni > 0 , k ≥ 3 and n1 + n2 + ...+ nk = n .

A root system of height one with respect to , is given by:

( ) ||| α1(n1+1),...,α1(n1+n2),...,αn1(n1+1),..., ||| ||{ α ,...,α ,...,α(n1+1)(n1+n2+n3 ), ||} ∑ n1(n1+1) (n1+1)(n1+n2+1) (Θ) = || ,...,α(n1+n2)(n1+n2+1),...,α (n1+n2)(n1+n2+n3),..., || ||| ...,α(n1+n2+...+n )(n1+...+n +1),..., ||| ( ...,α k-1 k-1 ) (n1+n2+...+nk-1)n

Let h : M 2 → CP n-1 any holomorphic and nondegenerate map, and Ψ = (Π ,...,Π ) : M 2 → F(n ) 0 1 n as we have defined above.

We define the map 2 Ψ = (Ψ1, ...,Ψk ) : M → F (n; n1,...,nk) by: Ψ1 = Π1 + ...+ Πn1,...,Ψk = Πnk- 1+1 + ...+ Πnk .

We prove in [23] that any such is a holomorphic-horizontal frame.

4. The second variation of energy and stability on

We compute now the second variation of the energy in our situation.

Theorem 4.1. Consider a harmonic map φ : (M 2,g) → (𝔽 Θ,ds2) Λ .

Thus,

where the map q : M → 𝔤 𝔩(𝔫,ℭ ) is defined by ∂q ∑ ∂z(p ) = α∈Π qα(p)X α .

Definition 4.1. A harmonic map φ : (M 2,g) → (𝔽 ,ds2) Θ Λ is said stable if φ IΛ(q) ≥ 0 , for any variation 2 q : M → g . Otherwise, is said unstable.

The following Theorem due to Lichnerowicz ([19]) is fundamental in our study of stability on flags.

Theorem 4.2. Let φ : (M 2,J,g ) → (𝔽 ,J,ds2) Θ Λ be a -holomorphic map and 2 (𝔽 Θ,J,dsΛ ) a Kähler structure. Then is stable.

Definition 4.2. We say that ′ Λ ′P = (λσ )σ∈Π (Θ ) is a -perturbation of Λ = (λσ)σ subordinate to ψ = (ψ σ) : M 2 → 𝔽Θ σ∈Π(Θ) if

;
if ;
, if ;
if .

Regarding the families of the holomorphic and horizontal frames we have just defined, we can simply consider P =Σ (Θ ) .

Using the above definition of perturbation we derive the following basic lemma.

Lemma 4.3. Let ψ  (ψσ)σ : (M 2,g) → (𝔽Θ,ds2Λ ) Θ a holomorphic and horizontal frame. Then,

Iψ′(q) = IψΛ(q) + (15 ) ΛP ( ∫ ) ∑ M λσ(⟨qσ(p),qσ(p)⟩+ + ξσ ⟨q-σ(p),q-σ(p)⟩)υg . (16 ) σ∈ Π(Θ )- P

According to Gray-Hervella ([16]) the almost Hermitian structures can be decomposed into four irreducible components. For instance, {0} corresponds to Kähler metrics, W1 ⊕ W2 to the (1,2) -symplectic ones and, so on. See [25] and [26].

Lemma 4.4. A necessary and sufficient condition for 2 (𝔽,J, dsΛ) to be in W1 ⊕ W3 is: λ α = λβ = λγ if {α, β,γ} is a (0,3) -triple.

As an immediate consequence of this lemma we notice that the Cartan-Killing structure is in W1 ⊕ W3 . We will now consider perturbations of the Cartan-Killing structure.

We consider a J = (εα) and denote by C (J) the subset of roots such that there exists a (0,3) -triple { α,β,γ } .

Let ds2 Λ0=(λ0α) given by λ0 = k > 0 α for each α ∈ Σ ∪ C (J) , and 0 < λ0 ≤ k α otherwise. According to Lemma 4.4, 2 (𝔽,J, dsΛ0) ∈ W1 ⊕ W3 . We can prove the following theorem.

Theorem 4.5. Let ψ = (ψ ) : M 2 → 𝔽 α be an arbitrary holomorphic-horizontal frame. Then, 2 2 ψ : (M ,g ) → (𝔽,dsΛ0) is unstable.

5. Stability results on

According to the results obtained in [25], among all the invariant Hermitian structures, the main cases are W1 ⊕ W2 and W1 ⊕ W3 . We will now discuss the stability phenomenon of holomorphic-horizontal frames in these two main classes.

Based on a crucial result derived in [25] we present the following definition.

Definition 5.1. Let ′ ′ ε = (εα)α∈Π , ′ εα = ±1 and ′ ′ ε-α = - εα . We fix a Kähler structure (𝔽,(εα)α∈Π,(λ α)α∈Π) . The metric ′ ′ Λ = (λα )α ∈Π is said a perturbation of type (1,2)-symplectic of Λ = (λ α)α∈Π if

for each with and, we have .
for each with and if , where is the highest root, and each , then .

We now are ready to discuss the W1 ⊕ W2 case. We consider ′ 2 (𝔽, (εα),dsΛ′) equipped with an invariant Hermitian structure that comes from a perturbation of type (1,2)-symplectic of a Kähler structure (𝔽,(εα),ds2Λ) . Thus, in [23] we prove

Theorem 5.1. Let ψ : (M 2,g) → (𝔽,ds2 ′) Λ be a holomorphic-horizontal frame. Then is stable.

We will now concentrate our attention on the family of invariant Hermitian structures on that are in W1 ⊕ W3 . We begin our discussion mentioning the classification of Einstein metrics on 𝔽 (3 ) and 𝔽(4) derived in [17] and [22], exploiting these results and obtaining (see [23])

Theorem 5.2. Let ψ : (M 2,g) → (𝔽(n),ds2 ) Λ= (λij) a holomorphic and horizontal frame with n = 3 or and 2 dsΛ=(λij) a Einstein and non-Kähler metric. Then, the map is unstable.

A basic result due to Arvanitoyeorgos [1] and Kimura [17] is the following one.

Theorem 5.3. The space 𝔽 (n ) for n = 3 admits as Einstein metrics only the normal and the Kähler-Einstein metrics. If n ≥ 4 it admits at least n2!+ n + 1 Einstein metrics. The n2! metrics are the already mentioned Kähler-Einstein metrics described by Borel, one is the usual normal metric and the remaining n are given explicitly as follows:

More generally, in his Ph.D. thesis ([12]), dos Santos has found new families of Einstein non-Kähler metrics on arbitrary 𝔽 (n) . See also [9] and [13] for additional details.

We notice that any known invariant Einstein metric on 𝔽(n ) has a common feature: either it is Kähler or is in W ⊕ W 1 3 . In fact, we believe that this fact is true for any Einstein metric on 𝔽 (n) .

Using an appropriate Cartan-Killing perturbation (as in Theorem 4.5) we can prove.

Theorem 5.4. Let (𝔽(n),ds2 ) Λ= (λij) equipped with any of the known Einstein non-Kähler metrics above described, and ψ : (M 2,g ) → (𝔽(n ),ds2Λ=(λij)) be any arbitrary holomorphic-horizontal frame. Then, is unstable.

References

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

[2] M. Black, Harmonic Maps into Homogeneous Spaces, Pitman Res. Notes in Math. vol. 255, Longman, Harlow (1991). [ Links ]

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

[4] R. Bryant, Lie groups and twistor spaces, Duke Math. J. 52 (1985), 223-261. [ Links ]

[5] F.E. Burstall and S. Salamon, Tournaments, flags and harmonics maps, Math. Ann. 277 (1987), 249-265. [ Links ]

[6] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom. 1 (1967), 111-125. [ Links ]

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

[8] S. S. Chern and J. G. Wolfson, Minimal surfaces by moving frames, Am. J. Math. 105 (1983), 59-83. [ Links ]

[9] Nir Cohen, E. C. F. dos Santos, Caio J.C. Negreiros, Properties of Einstein metrics on flag manifolds, IMECC-UNICAMP, preliminary version (2006). [ Links ]

[10] Nir Cohen, Caio J.C. Negreiros, Marlio Paredes, Sofia Pinzón and Luiz A. B. San Martin, F - structures on the classical flag manifold wich admit (1,2)-simplectic metrics, Tohoku Math. J., 57 (2005), 261-271. [ Links ]

[11] N. Cohen, C. J. C. Negreiros and L. A. B. San Martin, A rank-three condition for invariant (1,2)-symplectic metrics on flag manifolds and tournaments, Bull. London Math. Soc. 34 (2002), 641-649. [ Links ]

[12] E. C. F. dos Santos, Métricas de Einstein em variedades bandeira, Ph.D. thesis, State University of Campinas, 2005. [ Links ]

[13] E. C. F. dos Santos and C. J. C. Negreiros , Einstein metrics on flag manifolds [ Links ]

[14] J. Eells and L. Lemaire, Another Report on Harmonic Maps, Bull. London Mat. Soc. 20 (1988), 385-524. [ Links ]

[15] J. Eells and J. C. Wood, Harmonic maps from surfaces to complex projective spaces, Adv. Math. 49 (1983), 217-263. [ Links ]

[16] 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 ]

[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 II., (1969). [ Links ]

[19] A. Lichnerowicz, Applications harmoniques et variétés Kählériennes, Symposia Mathematica 3, Bologna (1970), 341-402. [ Links ]

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

[21] C. J. C. Negreiros, Some remarks about harmonic maps into flag manifolds, Indiana Univ. Math. J. 37 (1988), 617-636. [ Links ]

[22] Y. Sakane, Homogeneous Einstein metrics on flag manifolds, Lobatchevskii J. of Math., vol. 4, (1999), 71-87. [ Links ]

[23] C. J. C. Negreiros and L. A. B. San Martin, Stability properties of holomorphic maps on flag manifolds, preprint, State University of Campinas (2005). [ Links ]

[24] J. Rawnsley, f-structures, f-twistor spaces and harmonic maps, in Geometry Seminar "Luigi Bianchi", II, 1984, Lecture Notes in Math., vol. 1164 (1985); Ed.: E. Vesentini. [ Links ]

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

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

[27] K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model), J. Diff. Geom. 30 (1989), 1-50. [ Links ]

[28] K. Yang, Horizontal holomorphic curves in Sp(n) -flag manifolds, Proc. A. M. S. 103, # 1 (1988), 265-273. [ Links ]

[29] K. Yano, On a structure defined by a tensor field of type (1,1) satisfying 3 F + F = 0 , Tensor 14 (1963), 99-109. [ Links ]

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

Caio J. C. Negreiros
Department of Mathematics.
Universidade Estadual de Campinas.
Cx. Postal 6065 13081-970, Campinas-SP, Brasil

Recibido: 5 de octubre de 2005
Aceptado: 19 de septiembre de 2006