an:05314043
Zbl 1161.53065
Cohen, N.; Pinzon, S.
An extension of the \((1,2)\)-symplectic property for \( f\)-structures on flag manifolds
EN
Izv. Math. 72, No. 3, 479-496 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 3, 69-88 (2008).
00229549
2008
j
53C55 22F30 17B45 05C20
\(f\)-structures; flag manifolds; \((1,1)\)-symplectic; \((1,2)\)-symplectic; intersection graph; root system
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 K??hler 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.
Anna Fino (Torino)