Tohoku Mathematical Journal
2005
June
SECOND SERIES VOL. 57, NO. 2

Tohoku Math. J.
57 (2005), 261-271

Title $f$-STRUCTURES ON THE CLASSICAL FLAG MANIFOLD WHICH ADMIT $(1, 2)$-SYMPLECTIC METRICS

Author Nir Cohen, Caio J. C. Negreiros, Marlio Paredes, Sofia Pinzón and Luiz A. B. San Martin

(Received August 22, 2003, revised April 16, 2004)
Abstract. We characterize the invariant $f$-structures $\mathcal{F}$ on the classical maximal flag manifold $\boldsymbol{F}(n)$ which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of $\mathcal{F}$-harmonic maps from any cosymplectic Riemannian manifold onto $\boldsymbol{F}(n)$. 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extends and unifies two previous approaches: a paper of Brouwer in 1980 on locally transitive digraphs, involving unpublished work by Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs.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.

2000 Mathematics Subject Classification. Primary 53C55; Secondary 22F30, 17B45, 05C20.

Key words and phrases. Flag manifolds, (1,2)-symplectic structures, directed graphs.

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