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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)
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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