J an 2 00 3 Kähler - Nijenhuis Manifolds by Izu Vaisman

A Kähler-Nijenhuis manifold is a Kähler manifold M , with metric g, complex structure J and Kähler form Ω, endowed with a Nijenhuis tensor field A that is compatible with the Poisson structure defined by Ω in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if AJ = ±JA, M is foliated by im A into non degenerate Kähler-Nijenhuis submanifolds. If A is a non degenerate (1, 1)-tensor field on M , (M, g, J, A) is a Kähler-Nijenhuis manifold iff one of the following two properties holds: 1) A is associated with a symplectic structure of M that defines a Poisson structure compatible with the Poisson structure defined by Ω; 2) A and A −1 are associated with closed 2-forms. On a Kähler-Nijenhuis manifold, if A is non degenerate and AJ = −JA, A must be a parallel tensor field. A Kähler manifold is a particular case of a symplectic 2n-dimensional mani-fold (M, Ω) with a symplectic form defined as where Γ denotes the space of global cross sections, J is a complex structure on M, and g is a Hermitian metric on (M, J) [4]. Accordingly, on M one has the Poisson bivector field Π defined by the Poisson brackets computed with * 2000 Mathematics Subject Classification: 53C55, 53D17 .