  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.
