Mathematics

In this note we study, for n = 5, 6, 7, the geometry of the full flag . By using tournaments we characterize all of the (1,2)-symplectic invariant metrics on F (n), for n = 5, 6, 7, corresponding to different classes of non-integrable invariant almost complex structure.

We characterize the f -structures F on the classical maximal flag manifold F(n) which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of F-harmonic maps from any cosymplectic Riemannian manifold onto F(n).

We give a new proof of a classification theorem of (1,2)symplectic metrics on maximal flag manifolds proved by Cohen, Negreiros and San Martin. We use locally transitive tournaments in order to simplify the demonstration of this theorem. Finally, using a result due to Brouwer we count the number of invariant almost complex structures which admit (1,2)-symplectic metrics on a maximal flag manifold.

Here we consider the general flag manifold F Θ as a naturally reductive homogeneous space endowed with an U -invariant metric Λ Θ and an invariant almost-complex structure J Θ . The main objective of this work is to explore the riemannian connection associated with the metric Λ Θ in order to calculate some classes of curvatures which should allow us to confirm, in a simple way, that flag manifolds are either not biholomorfically equivalent nor holomorphically isometric to any complex projective space.