Kaehler-Nijenhuis Manifolds

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A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is compatible with the Poisson stucture defined by F in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M is foliated by im A into non degenerate Kaehler-Nijenhuis submanifolds. If A is a non degenerate (1,1)-tensor field on M, (M,g,J,A) is a Kaehler-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 F; 2) A and its inverse are associated with closed 2-forms. On a Kaehler-Nijenhuis manifold, if A is non degenerate and AJ=-JA, A must be a parallel tensor field.