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

Using moving frames we obtain a formula to calculate the codifferential of the Kahler 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. | Turk J Math 34 (2010) , 305 – 315. ¨ ITAK ˙ c TUB doi:10.3906/mat-0810-19 On the codifferential of the K¨ ahler form and cosymplectic metrics on maximal flag manifolds Marlio Paredes and Sof´ıa Pinz´ on Abstract Using moving frames we obtain a formula to calculate the codifferential of the K¨ ahler 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, K¨ ahler form, flag manifolds, differential forms. 1. Introduction In this note we study the K¨ ahler 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 K¨ ahler 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 K¨ ahler 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 .