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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006

 

Integrability of f-structures on generalized flag manifolds

Sofía Pinzón

Financiado parcialmente por COLCIENCIAS, contract No. 138-2004.

Abstract: Here we consider a generalized flag manifold 𝔽 = U ∕K, and a differential structure F which satisfy F 3 + F = 0; these structures are called f-structures. Such structure F determines in the tangent bundle of 𝔽 some ad(K) - invariant distributions. Since flag manifolds are homogeneous reductive spaces, they certainly have combinatorial properties that allow us to make some easy calculations about integrability conditions for F itself and the distributions that it determines on 𝔽. An special case corresponds to the case U = U (n) , the unitary group, this is the geometrical classical flag manifold and in fact tools coming from graph theory are very useful.

 1. Introduction

A tensor field F of type (1,1) on a Riemannian manifold is called an f -structure if  3 F + F = , and almost complex if  2 F = - I. Obviously, an almost complex structure is also an f -structure. Integrability of almost complex structures is equivalent to the associate Nijenhuis tensor being null. In [7] Ishihara and Yano present an analogous theorem in the case of f -structures. We use their results to study integrability conditions when one generalized flags manifold are consider.

Let G be a semisimple Lie group. A generalized flag manifold with Lie group G is a reductive homogeneous space 𝔽 = G ∕C(S) where C(S) is a centralizer of a torus. This manifold can be expressed as a 𝔽 = U∕K, where U is the compact connected form of G and K = C(S) ∩ U. This manifold and its tangent space Tb(𝔽) = 𝔮, have a characterization in terms of the corresponding root system terms. We will consider along this paper a generalized flag manifold together with an invariant metric Λ and an U -invariant f -structure F , meaning that F commute with the adjoint action of U. Ww will denote by F the complexification of this f -structure, which is diagonalizable with eigenvalues i, 0,- i and eigenspaces  + - 𝔮 Θ,𝔮0Θ,𝔮 Θ. In analogy with the almost complex case we will distinguish between vectors of types (1,0,0) , (0,1,0) or (0,0,1) corresponding to the eigenvalues of 0,i,- i respectively.

The integrability of F , the distributions that it determines in 𝔮 and the properties of that distributions are our central topic. We characterize in root terms and in graph theoretical terms the integrability conditions.

In particular, we get that the only integrable f -structure, different from the null structure, in the maximal classical flag manifold is the structure which corresponds to the integrable almost complex structure, that is, in graph theoretic terms which corresponds to the canonical tournament.

Theorem A necessary and sufficient condition for F to be integrable is that N ≡ 0. Therefore F is integrable if in 𝔮 there are not exist triples of type {0, 3,0},{2,1,0}, {1,1,1} or {1,2, 0}. In 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraphs (2), (3), (4), (5) or (6) in figure 2, that is, the digraph associated to F must be isomorphic to the null digraph or the canonical tournament.

The Theorem above will appear like Theorem 8.8 and with this result we generalized a Theorem from Burstall [4] given in the context of almost complex structures where he shows that one almost complex structure is integrable if and only if its associated tournament is isomorphic to the canonical tournament, that is, the tournament which does not have three-cycles in root terms it avoids {0,3,0}-triples.

2. Preliminaries

In this section we shall briefly review some general concepts involving generalized flag manifold and some operators and structures which we will use in all this paper. First we need to present a survey about some operators in differential geometry, then we will calculate them specifically on generalized flag manifolds, to this topic we used, specially Props I.3.2, I.3.4, I.3.5 in [8].

2.1. Operators on a general differential manifold.

  • Lie derivative. This is the resulting derivative when a tensor field or a differential form is differentiated with respect to a vector field.
    1) Lie derivative on tensor fields: LieX Y = [X, Y] .
    2) Lie derivative on tensors: On tensors K of type (1,r) we get
     ∑r (LieX K)(Y1, ...,Yr) = [X, K(Y1, ...,Yr)] - K(Y1, ...,[X, Yi]...,Yr). i=1
    (1)

    3) Lie derivative on forms: If ω is an r -form, then

     r ∑ (LieXω)(Y1, ...,Yr) = X ω(Y1, ...,Yr) - w(Y1, ...,[X, Yi]...,Yr). i=1
    (2)
  • The riemannian invariant connection. Each Riemannian manifold admits a unique metric connection with vanishing torsion, called the Riemannian connection or Levi-Civita connection, and it satisfies T(X, Y ) = [X, Y ] - ∇X Y + ∇Y X = 0 and Xg(Y, Z) = g(∇X Y,Z) + g(Y,∇X Z), where g is the metric on the manifold and T is the torsion tensor [8].

    1) The covariant derivative on tensors. Given a tensor field K of type (r,s), the covariant differential ∇K of K is a tensor field of type (r,s + 1) defined as follows. (∇K)(X1, ...,Xs; Y ) = (∇Y K)(X1, ...,Xs).

    2) The covariant derivative on forms. If ω is an r -form, then

     ∑r (∇ ω)((X1, ...,Xr), Y) = Y ω(X1, ...,Xs) - ω(X1, ...,∇Y Xi,...,Xr). i=1

  • Exterior derivative on forms. Exterior differentiation d can be characterize as follow:
    • d is a degree-increasing ℝ -linear mapping, that is if ω is a p form, dω is a p + 1 -form;
    • signed derivation w.r.t. the wedge product, that is, if ω is a p -form and θ is a q -form, then d(ω ∧ θ) = dω ∧ θ + (- 1)pω ∧ d θ ;
    • d2 = 0.
    • For 0-forms d is defined by ⟨df,X ⟩ = Xf.

    d extends to r -forms coefficient-wise, using basis expansions, resulting in

     ∑ r (r + 1)dω(X0, X1, ...,Xr) = (- 1)iXi(ω(X0, X1, ...,Xˆi, ...,Xr)) + i=0 ∑ i+j (- 1) ω([Xi,Xj], X0,...,Xˆi, ...,Xˆj, ...,Xr). 0≤i<j≤r

    Here ˆXi means that you not consider that component. On functions the 1-form df is defined by df (Y) = Y f. Otherwise we get

     ∑ dω(X)--= - ±(∇ ω)(Xˆ, Xi).
    (3)

    On 1-forms we get

    2dω(X, Y) = X( ω(Y )) - Y ω(X) - ω([X, Y]).
    (4)

    On 2-forms,

    3dω(X, Y, Z) = Cyclic {X ω(Y,Z) - ω([X, Y ],Z)}
    (5)

    where Cyclic is the cyclic symmetrization operator w.r.t. the vector fields involved. See [8] Prop. I.3.11. By duality we get ⟨∇f, ∇g ⟩ = ⟨df,dg ⟩.

3. Generalized Flag Manifold

A generalized flag manifold is and homogeneous space of the form G ∕K = G ∕C(S), where G is a semisimple compact Lie group and C(S) is the centralizer of some torus S in G. If the torus S is maximal, say T, then 𝔽 = G ∕T is called a maximal (full) flag manifold, if S is not maximal, G∕C(S) is called a partial flag manifold. Lets us describe generalized flag manifolds associated with the Lie group G in terms of the root systems associated with the corresponding semisimple Lie algebra 𝔤.

Assume 𝔤 complex and let 𝔥 be a Cartan subalgebra of 𝔤; denote by Π and Π+ the root system and the positive root system , respectively, (of 𝔤 with respect to 𝔥 ) and

 ∑ 𝔤 = 𝔥 ⊕ α∈Π

its root decomposition, where 𝔤α = {X ∈ 𝔤 : [H, X] = α(H)X, ∀H ∈ 𝔥} , α ∈ Π , is the one-dimensional complex root space corresponding to α . As Π is the generator of 𝔥* (the dual of 𝔥), we have the elements H α, defined by α(⋅) = ⟨H α,⋅⟩. Denote by 𝔥ℝ the subspace of 𝔥 generated over ℝ by H α, α ∈ Π. Choose now  + Σ = {α1,...,αl} ⊂ Π a simple root system, take Θ ⊂ Σ and denote by ⟨Θ ⟩ the set of roots generated by Θ. Each subset ⟨Θ ⟩± = ⟨Θ ⟩ ∩ Π ± splits 𝔤 as follows

 ∑ ∑ ∑ ∑ 𝔤 = 𝔥 ⊕ 𝔤α ⊕ 𝔤-α ⊕ 𝔤β ⊕ 𝔤-β. α∈ ⟨Θ⟩+ α∈⟨Θ ⟩+ β∈Π+ \⟨Θ⟩+ β∈Π+\⟨Θ ⟩+
(6)

Fix a Weyl basis of 𝔤, that is, a set of vectors {X α ∈ 𝔤 ∣ α ∈ Π} which satisfies [X ,X ] = H α -α α or equivalently ⟨X ,X ⟩ = 1, α -α since [X ,X ] = ⟨X ,X ⟩H α -α α -α α 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 ∈∕ Π .

Let now

 ∑ ∑ ∑ 𝔭 Θ = 𝔥 ⊕ 𝔤α ⊕ 𝔤 -α ⊕ 𝔤β. α∈⟨Θ⟩+ α∈⟨Θ⟩+ β∈Π+\⟨Θ⟩+
(7)

𝔭Θ is the parabolic subalgebra determined by Θ in 𝔤. Then equation ((6)) becomes

 ∑ 𝔤 = 𝔭Θ ⊕ 𝔤 -β. + + β∈Π \⟨Θ⟩
(8)

Then the generalized flag manifold 𝔽 Θ associated to {𝔤,Θ} corresponds to the homogeneous space 𝔽Θ = G∕P Θ , where PΘ is the normalizer of 𝔭 Θ in G .

Denote by 𝔲 a real compact form of 𝔤 , and by U ⊂ G the connected subgroup associated to 𝔲 . Assume 𝔲 the real subspace generated by i𝔥ℝ,A α,Sα, with α ∈ Π \ Θ , where Aα = Xα - X -α and S α = i(X α + X -α). Let K Θ = PΘ ∩ U, which, by construction, is the centralizer of a torus. U acts in a transitively way on 𝔽Θ, and we can write 𝔽Θ = U∕K Θ . If Θ = ∅ , 𝔽Θ = 𝔽 correspondes to the maximal flag manifold 𝔽, otherwise it corresponds to a partial flag manifold.

The generalized flag manifold 𝔽Θ = U∕K Θ is a reductive homogeneous space. In fact let 𝔲β = 𝔲 ∩ (𝔤β ⊕ 𝔤- β), β ∈ Π \ ⟨Θ ⟩ and
 ∑ 𝔮Θ = β∈Π\⟨Θ ⟩𝔲β. Then,

  • 𝔲 = 𝔱Θ ⊕ 𝔮Θ, 𝔱Θ ∩ 𝔮 Θ = ∅;
  • Ad(K )𝔮 ⊂ 𝔮 , Θ Θ Θ that is, [𝔱 ,𝔮 ] ⊂ 𝔮 , Θ Θ Θ

and 𝔽 Θ satisfies the condition to be a a reductive homogeneous space (see [8]).

Denote by b the origin of 𝔽 = U ∕K . Θ Θ We identify with 𝔮 = T (𝔽 ). Θ b Θ This identification is given by X ∈ 𝔮Θ → Xb ∈ Tb(𝔽 Θ), that is, by evaluation of X ∈ 𝔮 Θ in b like a vector field in Tb(𝔽 Θ). The tangent space to 𝔽 Θ in b is, naturally, identify with  ∑ 𝔮Θ = 𝔲 ⊖ 𝔱 = 𝔲 β β∈ Π\⟨Θ⟩ generated by A α,Sα, α ∈ Π \ ⟨Θ⟩, where A α = X α - X -α and S α = i(X α + X -α). In the same way, the complexified tangent space of 𝔽Θ is identified with 𝔮ℂΘ = 𝔤 ⊖ 𝔥 = ⊕ α∈Π\⟨Θ⟩𝔤α.

One special case of flag manifold corresponds to the geometrical or classical flag manifold, in this case U = U(n) the unitary group and C(S) has to be conjugate to some subgroup S(Un × Un × ...× Un ), 1 2 k with n1,n2,...,nk positive integers and n1 + n2 + ...+ nk = n . If mi = n1 + ...+ ni , the homogeneous space SUn ∕S(Un1 × ...× Unk ) can be identified with the set of "partial flags" F (m1, ...,mk), that is, the manifold of the flags {0}= E0 ⊂ Em ⊂ ...⊂ Em ⊂ Em = ℂn, 1 k-1 k where Ei is an i -dimensional subspace of  n ℂ . The flag manifold corresponding to the case nr = 1 for all 1 ≤ r ≤ k will be denoted by 𝔽(n), the space of "full flags" or maximal flags in ℂn. Each flag consists of the sequence  n {0}= E0 ⊂ E1 ⊂ ... ⊂ En -1 ⊂ En = ℂ . In particular, the vectors Ejk, j ⁄= k y Ejj - Ekk, j < k, is a Weyl basis for 𝔰𝔩(n,ℂ), and 𝔥 is the subalgebra of diagonal matrices in 𝔰𝔩(n,ℂ) , Θ = ∅, then Tb𝔽 = 𝔮 ⊂ 𝔰𝔲(n) is spanned by Ajk = Ejk - Ekj and Sjk = i(Ejk + Ekj), where Ejk is the usual canonical basis in 𝔤𝔩(n,ℂ).

Example 1.1: Consider

𝔽(4) = U (4)∕(U (1) × U (1) × U(1) × U (1)) = U(4)∕T,

 ( ( ) ) ||| 0 a b c ||| { || - ¯a 0 d e || } 𝔮 = T (𝔽(4))(b) = | |( ¯ ¯ |) : a,b,c,d,e,f ∈ ℂ | , ||( - b - d 0 f ||) - ¯c - ¯e -f¯ 0

here,

 ( ) ( ) 0 0 0 0 0 0 0 0 || 0 0 0 0 || || 0 0 0 0 || √ --- A34 = |( 0 0 0 1 |) , S34 = |( 0 0 0 i |) : i = - 1. 0 0 - 1 0 0 0 i 0

For an interesting and complete review about homogeneous spaces we recommend [1].

4. Invariant Metrics

A U -invariant Riemannian metric  2 ds Λ in 𝔽Θ is completely determined by its values in b , that is, by an inner product (⋅,⋅) in 𝔮Θ, invariant under the associated action of K Θ ([3]). Any inner product in 𝔮Θ , invariant under the associated action of KΘ, has the form (X, Y )ΛΘ = - ⟨ΛΘ ∘ X, Y ⟩, with Λ Θ : 𝔮Θ → 𝔮Θ definite with respect to the Cartan-Killing form and ∘ is the Hadamard product or term by term product [3]. The inner product (⋅,⋅)ΛΘ admits a natural extension to a bilinear symmetric form on  ℂ 𝔮 Θ . We use the same notation (⋅,⋅)ΛΘ for this form, as well as for the correspondent complexified form  Θ Λ . K Θ -invariance of (⋅,⋅)ΛΘ amounts to the Weyl basis being a complex basis of eigenvectors for the action of Λ Θ , in other words in 𝔮ℂΘ we have

Λ ΘX α = λΘ αX α,
(9)

with λ Θα = λΘ -α > 0. for α ∈ Π \ ⟨Θ⟩

for the real space 𝔮 , Θ the elements of the canonical base A , S , α α with α ∈ Π \ ⟨Θ ⟩ , are eigenvectors for the same eigenvalue λΘα. We denote by ds2ΛΘ the U -invariant metric associated with Λ Θ. In what follows we will use Λ Θ as synonymous of  2 dsΛ Θ.

5. Invariant f -estructures

K. Yano [20] in 1961 introduced f -structure for general manifolds; here we shall be interested in invariant structures. An U -invariant f -structure in 𝔽 Θ is completely determined by an endomorphism  Θ F : 𝔮Θ → 𝔮Θ , satisfying  Θ 3 Θ (F ) + F = 0, which commutes with the adjoint action of KΘ. We also denote by F Θ its complexification F Θ : 𝔮ℂ → 𝔮 ℂ Θ Θ which is diagonalizable with eigenvalues i,0,- i , and denote by 𝔮+,𝔮0 ,𝔮- Θ Θ Θ the corresponding eigenspaces. Then we have  ℂ + 0 - 𝔮Θ = 𝔮Θ + 𝔮Θ + 𝔮Θ with -+- - 𝔮Θ = 𝔮 Θ. The U -invariance of F Θ guarantees that F Θ(𝔤α) ⊆ 𝔤 α for all α ∈ Π \ ⟨Θ⟩, with equality when  Θ F is an invariant almost complex structure (see [17]). Thus  Θ F is determined uniquely by the values ɛΘα ∈ {0,±1}, α ∈ Π \ ⟨Θ ⟩, defined by F Θ(X α) = iɛΘX α α . These values satisfy ɛΘ = - ɛΘ , - α α therefore F Θ is defined by its values in  + + Π \ ⟨Θ⟩ . In the sequel we allow some abuse of notation and identify the invariant f -structure F Θ on 𝔽Θ with {ɛΘα : α ∈ Π \ ⟨Θ ⟩}. In our invariant context if the f -structure is an invariant almost complex structure this amounts to  Θ ɛα ⁄= 0 for all α ∈ Π \ ⟨Θ ⟩.

In what follows we shall simplify notation by suppressing the subscript Θ in the context of partial flag manifolds.

Denote by p and l the complementary projections onto the spaces 𝔮0 and 𝔮+ + 𝔮- denoted as p and l , respectively and defined as follow

 2 2 I = - F , p = F + I.

Since F is an f -structure we have

 2 2 l + p = 1, l = l, p = p, lp = pl = 0,
(11)

where I denote the identity. In other words l and p are complementary projection operators in 𝔮.

Consider an f -estructure F and l , p like below. Then:

 2 F l = lF = F , F p = pF = 0, F l = - l,

that is, F act in 𝔮+ + 𝔮- like an almost complex structure and in 𝔮0 like the null operator.

We are interested in studying integrability conditions for F , the distributions 𝔮0 and 𝔮+ + 𝔮 - . For this purpose, we need the Nijenhuis tensor which describes the torsion of F . It is given by

N (X, Y ) = 2([F(X), F (Y )] - F ([F (X), Y ]) - F ([X, F (Y )]) - L([X, Y ]).

Using Weyl basis properties we get

1∕2N (X ,X ) = [F (X ),F (X )] - F ([F(X ),X ])+ α β α β α β - F ([X α,F (X β)]) - L([X α,X β] = (- ɛαɛβ + ɛαɛα+ β + ɛβ ɛα+β + ɛ2 )[Xα, Xβ]. α+β

Because l + p = 1,

N (X α,X β) = lN (lX α,lX β) + pN (lX α,lX β)+ +N (lXα, pX β) + N (pX α,lX β) + N (pX α, pX β).

With some simple calculations and using, again, Weyl basis properties we obtain this other identities:

 lN (lX α,lX β) = ɛαɛβɛα+β(ɛα + ɛβ - ɛα+β - ɛαɛ βɛα+β)[Xα, Xβ]; (12) 2 pN (lX α,lX β) = p[F X α,F X β] = ɛαɛβ(ɛα+β - 1)[Xα, Xβ]; (13) N (lX α,lXβ) = F {l(LiepXβF )lX α}; (14) N (pX α,pX β) = lN (pX α,pX β) = (ɛαɛ2 - ɛ2 - ɛ2 + 1)ɛ2 [X α,X β]. (15) β α β α+β

6. Forms on 𝔽

∙ Riemannian Connection. Since 𝔽Θ is a naturally reductive homogeneous space its Riemannian connection is given by

2 ∇X Y = [X, Y ]𝔮Θ + 2U (X, Y),
(16)

where U is a symmetric bilinear application U : 𝔮Θ × 𝔮Θ → 𝔮Θ defined by

2 ΛΘ(U (X, Y ),Z) = Λ Θ(X, [Y, Z]𝔮Θ) + ΛΘ([Z, X] 𝔮Θ, Y),

for all X, Y, Z ∈ 𝔮Θ.

Then the concrete action of the Riemannian connection on the elements of the Weyl basis is given by:

Proposition 6.1. For (𝔽Θ, ΛΘ) let α,β, α + β ∈ Π \ ⟨Θ⟩, and X α,X β,Xα+ β ∈ 𝔮Θ, elements in the Weyl base. Then

 λΘα+ β + λ Θβ - λΘα ∇X αX β = m α,β-------Θ--------X α+β. 2λα+ β
(17)

∙ Kähler form. σ(X α,X β) = (X α,F (X β))Λ = - ⟨ Λ(X α),F (X β) ⟩,

 { iɛαλ α β = - α, ɛα ⁄= 0, σ(X α,X β) = 0 otherwise.

for all α, β ∈ Π ′ .

∙ Derivative in the connection.

(d∇F )(X ,X ) = ∇ F X - ∇ F X - F [X ,X ] α β (Xλαα-λβ)(βɛα-ɛβ)+Xβλα+β(ɛαα+ɛβ-2ɛαα+β) β = i------------2(λα+β)------------[X α,X β].

∙ Lie derivative:

(Lie F )X = F [X ,X ] - [F X ,X ] Xβ α α β α β = i(ɛα+β - ɛβ)[X α,X β].

7. Graph theoretic description of (𝔽(n),F ,Λ)

On 𝔽(n) invariant f -structures are in 1:1 correspondence with digraphs G = (V,E). The correspondence is given by associating with the f -structure F (Ejk) = iɛjkEjk a digraph G whose vertices are {1, ⋅⋅⋅ ,n} and whose arrows are given by the following rules: For j < k

j → k ⇐ ⇒ ɛjk = 1,

j ← k ⇐ ⇒ ɛjk = - 1,

j ⁄↔ k ⇐ ⇒ ɛjk = 0.

Similarly, through the matrix Λ = {λjk} we may identify an invariant metric ds2 on (𝔽(n), F) with a positive weighting on the edge set E of the digraph.

Example 7.1. Again in 𝔽(4) let the f -structure

( ) ( ) 0 a b c 0 ia - ib - ic || - ¯a 0 d e || || - i¯a 0 0 ie || | ¯ ¯ | ↦- → | ¯ | ( - b -d 0 f ) ( ib 0 0 0 ) - ¯c -e¯ - f¯ 0 i¯c - i¯e 0 0

PICT
Figure 1: Digraph associated to the f-structure in Example 7.1.

There exists a complete classification for invariant f -structures on 𝔽(n) (see [5]). Here we present, up to isomorphism, the invariant f -structures in the case 𝔽(3), this graphs are the most relevant in our present work.

PICT
Figure 2: Isomorphism classes for n = 3

We are now ready to present our characterization of integrability in graph theoretical terms (classical case) or in root terms (general case).

8. Integrability via projections

In this section we will use all the formulas given before to establish integrability conditions for F and for the associated distributions 𝔮 0 and 𝔮 + 𝔮 + - .

First, we will call the root triple {α,β, γ} with α, β,γ ∈ Π whenever α + β + γ = 0, {α,β,γ} is called a zero-sum triple. Given an invariant f -structure F , each root assumes a sign in {0,±1} .

Roots triples may then be classified by their sign characteristic, which is a triple (p,q,r) (p + q + r = 3) where p corresponds to the quantity of roots in triple who has 0 like its eigenvalue, q corresponds to the quantity of roots in triple who has i like its eigenvalue and r corresponds to the quantity of roots in triple who has - i like its eigenvalue. There are six possible sign characteristics: (3,0,0), {2,1,0}= {(2,1, 0), (2,0, 1)}, {1,2,0}= {(1,2, 0),(1, 0,2)}, (1,1,1), {0, 2,1}= {(0,2,1),(0,1,2)}, {0, 3,0}= {(0,3,0),(0,0,3)}.

By the Frobenious Theorem we know that a distribution is integrable if and only if it is involutive. Thus 𝔮0 is integrable if and only if l[pX α,pX β] = 0, that is,

l[pX α,pX β] = (ɛ2- 1)(ɛ2 - 1)ɛ2 [X α,X β] = 0. α β α+ β
(18)

Theorem 8.1. A necessary and sufficient condition for the distribution 𝔮0 to be integrable is that in 𝔮 does not admit triples of type {2, 1,0}. In case of 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraph (2) in Figure 2.

PROOF. By equation (18) 𝔮0 is not integrable in case where (ɛ2α - 1)(ɛ2β - 1)ɛ2α+β ⁄= 0. This can only occur when the triple α,β, α + β is a {2,1,0} -triple. In the classical case, this corresponds the configuration (2) in Figure 2. □

Now 𝔮+ + 𝔮- is integrable if and only if p[lX α,lX β] = 0 but p[lXα, lX β] = (F 2 + 1)[- F 2X α,- F 2Xβ] (19) 2 2 2 = ɛα ɛβ(F + 1)[X α,X β] (20) 2 2 2 = ɛα ɛβ(1 - ɛα+β)[X α,X β]. (21)

Thus we have the following.

Theorem 8.2. A necessary and sufficient condition for distribution 𝔮+ + 𝔮- to be integrable is that in 𝔮 does not amit triples of type {1,2,0} and {1,1,1}. In case of 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraph (3),(4) and (5) in Figure 2.

PROOF. By equation (21) 𝔮+ + 𝔮- is not integrable in case  2 2 2 ɛαɛβ(1 - ɛα+β) ⁄= 0. This can only occur when the triple α,β, α + β is a {1,2,0} -triple or a {1, 1,1} -triple. In the classical case, this corresponds to configurations (3), (4) and (5) in Figure 2. □

When 𝔮0 and 𝔮+ + 𝔮- are integrable, the structure of the submanifolds defined by these distributions and its properties is of interest. We hope to be able to report on this structure in a future communication.

Looking for the integrability of F we need another definition from [7].

Definition 8.3. Assume 𝔮+ + 𝔮- integrable and let X ′ be an arbitrary vector field which is tangent to an integrable manifold of 𝔮+ + 𝔮- . It is defined F ′X ′ = F X ′. Then  ′ F is an almost-complex structure on each integral manifold of 𝔮+ + 𝔮- . F is called partially integrable if both, 𝔮+ + 𝔮- and F ′ are integrable.

Theorem 8.4. A necessary and sufficient condition for F be partially integrable is that one of the following equivalent conditions be satisfied:

N (lX α,lX β) = 0, or N (F X α,F X β) = 0.

Using Weyl basis properties the Theorem 8.4 is equivalent to the following.

Theorem 8.5. A necessary and sufficient condition for F to be partially integrable is that in 𝔮 does not admit triples of type {0,3,0}, {1,2,0} and {1,1,1}. In the case of 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraph (3), (4), (5) and (6) in Figure 2.

PROOF. By Theorem 8.4 is enough to see the conditions N (LX α, LX β) = 0. Doing the respective calculations we have

N (LX α,LX β) = ɛαɛβ(- 1 - ɛβ ɛα+β - ɛαɛα+β + ɛαɛβɛ2α+ β)[X α,X β].

Then F is not integrable in case ɛ ɛ (- 1 - ɛ ɛ - ɛ ɛ + ɛ ɛ ɛ2 ) ⁄= 0, α β β α+β α α+ β α β α+β this can only occur when the triples are the type in the Theorem. In the classical case this corresponds to configurations (3), (4), (5) and (6) in Figure 2. □

When 𝔮 0 and 𝔮 + 𝔮 + - are both integrables, it is possible to choose a local coordinates system such that the operators l and p can be supposed to have the components of the form:

 ( ) ( ) Ir 0 0 0 l = , p = , 0 0 0 In-r

where n is the dimension of the manifold, r is the rank of F and 1s means the s -identity matrix. This coordinate system is called "adapted." Since F satisfy F l = lF = F and F p = pF = 0 then in an adapted coordinate system, we can express F in the following way:

 ( ) F = Fr 0 . 0 0

But LiepX αF = 0 means that the components of F are independent of the coordinates, thus in the next theorem we are interested on this condition in root terms.

Theorem 8.6. Suppose that 𝔮0 and 𝔮+ + 𝔮- are both integrable and that an adapted coordinate system has been chosen. A necessary and sufficient condition for the local components of F to be functions independent of the coordinates is that N (lX α,pX β) = 0, in combinatorial terms a necessary and sufficient condition is that in 𝔮 does not admit triples of type {1,1,1}. In case of 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraphs (4) and (5) in Figure 2.

PROOF. By equation (14) we have the first affirmation and with some calculus we have

N (lX ,pX ) = ɛ ɛ (1 - ɛ 2)(ɛ ɛ - 1)[X ,X ]. α β α α+β β α α+β α β

Thus N (lX ,pX ) α β will be different from zero in case ɛ ɛ (1 - ɛ 2)(ɛ ɛ - 1) ⁄= 0. α α+ β β α α+β

This can only occur when the triple is {1,1,1} -triple. In the classical case this corresponds to configurations mentioned in the theorem. □

When F is an almost complex structure integrability is associated with the existence of canonical coordinate systems, which allows us to consider the manifold as a complex manifold as well is known, integrability is equivalent to N ≡ 0. The following definition in the context of general differential manifolds appears in [7]. Here we present it in the case of flag manifolds.

Definition 8.7. The f -structure F , is called integrable if it satisfies the following three conditions:

  • F is partially integrable.
  • 𝔮0 is integrable.
  • The components of F are independent of the coordinates.

With Definition 8.7, we arrive at a general integrability theorem for f -structures.

Theorem 8.8. A necessary and sufficient condition for F to be integrable is that N ≡ 0. Therefore F is integrable if in 𝔮 does not admit triples of type {0, 3,0},{2,1,0}, {1,1,1} and {1,2,0}. In 𝔽(n) this condition is equivalent to the associated digraph avoiding the subdigraphs (2), (3), (4), (5) and (6) in figure 2, that is, the associated digraph to F must be isomorphic to the null digraph or canonical tournament.

PROOF. It is immediate by Definition 8.7 and Theorems 8.1, 8.6 and 8.5. □

Theorem 8.8 is a generalization of the results obtained by Burstall [4] in the case of almost complex structures for classical maximal flag manifolds.

At this point we would like to point out an important connection between integrability and complex structures. It is well known that for a general almost complex structure J on a differential manifold is parallel, namely  ∇ d J = 0 if the manifold with that structure is Kähler. That is, parallelism means that J is integrable and the manifold is Kähler.

Now a manifold with an f -structure F will be called Kähler if  ∇ d F = 0.

Observe that in the case of generalized flag manifolds integrability condition is stronger than Kähler condition, for example, in 𝔽(3) the invariant f -structures to the digraphs (4) and (5) are Kähler but not integrable.

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Sofía Pinzón
Escuela de Matemáticas,
Universidad Industrial de Santander,
A.A. 678, Bucaramanga, Colombia.

Recibido: 30 de septiembre de 2005
Aceptado: 18 de septiembre de 2006