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An extension of the $$(1,2)$$-symplectic property for $$f$$-structures on flag manifolds. (English. Russian original) Zbl 1161.53065
Izv. Math. 72, No. 3, 479-496 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 3, 69-88 (2008).
An $$f$$-structure on a Riemannian manifold is a tensor field $$F$$ of type $$(1,1)$$ satisfing the condition $$F^3 + F = 0$$. In the present paper the authors study the $$(1,1)$$-symplecticity of an invariant $$f$$-structure $$F$$ on a general flag manifold $$(M, g)$$ endowed with an invariant Riemannian metric $$g$$. This notion is a natural extension of the $$(1,2)$$-symplectic condition for almost complex structures.
An $$f$$- structure $$F$$ is said to be $$(1,1)$$-symplectic if the $$(+, -)$$-part of $$d^{\nabla} F$$ vanishes, where $$\nabla$$ denotes the Riemannian connection of $$g$$ or, equivalently, if the $$(+, -, *)$$-part of $$d \sigma$$ is zero, where $$\sigma(X,Y)= g(X, FY )$$ is the Kähler form associated to $$(F, g)$$. Given a general flag manifold, the authors characterize in combinatorial terms those invariant $$f$$-structures $$F$$ on a flag manifold $$M$$ that are $$(1, 1)$$-symplectic for some invariant Riemannian metric $$g$$ on $$M$$. To this aim, they consider an intersection graph defined in terms of the corresponding root system.
The authors prove that the $$f$$-structure is $$(1, 1)$$-symplectic if the intersection graph is locally transitive.
Reviewer: Anna Fino (Torino)
##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 22F30 Homogeneous spaces 17B45 Lie algebras of linear algebraic groups 05C20 Directed graphs (digraphs), tournaments
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