K̈ahler-Nijenhuis manifolds

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.