On the codifferential of the Kahler form and cosymplectic metrics on maximal flag manifolds

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Turk J Math 34 (2010) , 305 – 315. c TÜBİTAK  doi:10.3906/mat-0810-19 On the codifferential of the Kähler form and cosymplectic metrics on maximal flag manifolds Marlio Paredes and Sofı́a Pinzón Abstract Using moving frames we obtain a formula to calculate the codifferential of the Kähler form on a maximal flag manifold. We use this formula to obtain some differential type conditions so that a metric on the classical maximal flag manifold be cosymplectic. Key Words: Codifferential, Kähler form, flag manifolds, differential forms. 1. Introduction In this note we study the Kähler form on the classical maximal flag manifold F(n) = U (n)/(U (1) × · · · U (1)). The geometry of this manifold has been studied in several papers. Burstall and Salamon [2] showed the existence of a bijective relation between almost complex structures on F(n) and tournaments with n vertices. This correspondence has been very important to study the geometry of the maximal complex manifold, see for example [5], [6], [9], [11], [12] and [13]. In [6], was showed the existence of a one to-one correspondence between (1, 2)-symplectic metrics and locally transitive tournaments. In [4], this result was generalized for (1, 2)-symplectic metrics defined using f -structures. Mo and Negreiros [9], by using moving frames and tournaments, showed explicitly the existence of an ndimensional family of invariant (1, 2)-symplectic metrics on F(n). In order to do this, they obtained a formula to calculate the differential of the Kähler form by using the moving frames technique. In the present work we use a similar method in order to obtain a formula to calculate the codifferential of the Kähler form. An important reference to our calculations is the book by Griffiths and Harris [8]; we use definitions, results and notations contained in this book to differential forms of type (p, q). Finally, we use such formula to find some differential type conditions in order for a metric on a maximal flag manifold be cosymplectic. We show that a metric on the classical flag manifold is cosymplectic if and only if the complex functions fkij in the Kähler form (see (13)) satisfy different types of partial differential equations. 2000 AMS Mathematics Subject Classification: 14M15, 32M10, 58A10. 305 PAREDES, PINZÓN 2. Flag manifolds The usual manifold of full flags of subspaces of Cn is defined by F(n) = {(V1 , . . . , Vn ) : Vi ⊂ Vi+1 , dim Vi = i}. (1) The unitary group U (n) acts transitively on F(n) turning this manifold into the homogeneous space F(n) = U (n) U (n) = , U (1) × U (1) × · · · × U (1) M (2) where M = U (1) × U (1) × · · · × U (1) is any maximal torus of U (n). Let p be the tangent space of F(n) at the point (M ). It is known that u(n), the Lie algebra of skewhermitian matrices, decomposes as u(n) = p ⊕ u(1) ⊕ · · · ⊕ u(1) , where p ⊂ u(n) is the subspace of zero-diagonal matrices. In order to define any tensor on F(n) it is sufficient to give it on p , because the action of U (n) on F(n) is transitive. An invariant almost complex structure on F(n) is determined by a linear map J : p → p such that J 2 = −I and commutes with the adjoint representation of the torus M on p . For each almost complex structure we assign a tournament, a special class of directed graph. A tournament or n-tournament T , consists of a finite set T = {p1 , . . . , pn } of n players together with a dominance relation, → , which assigns to every pair of players a winner, that is, pi → pj or pj → pi . A tournament T can be represented by a directed graph in which T is the set of vertices and any two vertices are joined by an oriented edge. If the dominance relation is transitive, then the tournament is called transitive. For a complete reference on tournaments see [10]. Given an invariant complex structure J , we define the associated tournament T (J) in the following way: if J(aij ) = (aij ), then T (J) is such that for i < j   √ i → j ⇔ aij = −1 aij  or  √ i ← j ⇔ aij = − −1 aij ; see [9]. We consider Cn equipped with the standard Hermitian inner product, that is, for V = (v1 , . . . , vn ) and n  W = (w1 , . . . , wn) in Cn , we have V, W = vi wi . We use the convention vı̄ = vi and fı̄j = fij̄ . i=1 A frame consists of an ordered set of n vectors (Z1 , . . . , Zn ) such that Z1 ∧ . . . ∧ Zn = 0 , and it is called unitary if Zi , Zj = δij̄ . The set of unitary frames can be identified with the unitary group U (n).  If we write dZi = j ωij̄ Zj , the coefficients ωij̄ are the Maurer-Cartan forms of the unitary group U (n). They are skew-Hermitian, this is, ωij̄ + ωj̄i = 0 . For more details see [3]. We may define all left-invariant metrics on (F(n), J) by (see [1]) ds2Λ =  i,j 306 λij ωij̄ ⊗ ωı̄j , (3) PAREDES, PINZÓN where Λ = (λij ) is a simetric real matrix such that  λij > 0, λij = 0, if i = j, if i = j, (4) and the Maurer-Cartan forms ωij̄ are such that ωij̄ ∈ C1,0 (forms of type (1,0)) ⇐⇒ T (J) i −→ j. (5) The metrics (3) are called of Borel type and they are almost Hermitian for every invariant almost complex structure J , that is, ds2Λ (JX, JY ) = ds2Λ (X, Y ) for all tangent vectors X, Y . When J is integrable, ds2Λ is said to be Hermitian. Given J an invariant almost complex structure on F(n) and ds2Λ an invariant metric, the Kähler form with respect to J and ds2Λ is defined by Ω(X, Y ) = ds2Λ (X, JY ), (6) for any tangent vectors X, Y . For each permutation σ of n elements, this Kähler form can be written as (see [9])  √ μσ(i)σ(j) ωσ(i)σ(j) ∧ ωσ(i)σ(j) , Ω = −2 −1 (7) i
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