We characterize the invariant 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). In the special case of almost complex structures, our analysis extends and unifies two previous approaches: a paper of Brouwer in 1980 on locally transitive digraphs, involving unpublished work by Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs. We also discuss the construction of (1,2)-symplectic metrics and calculate their dimension. Our approach is graph theoretic.
The (1, 1)-symplectic property for f -structures on a complex Riemannian manifold M is a natural extension of the (1, 2)-symplectic property for almost-complex structures on M , and arises in the analysis of complex harmonic maps with values in M .
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.
For a complex semisimple Lie group G and a real form G 0 we define a Poisson structure on the variety of Borel subgroups of G with the property that all G 0-orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a G 0-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.
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