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Kähler-Nijenhuis manifolds. (English) Zbl 1059.53059
The paper defines and studies the notion of a Kähler-Nijenhuis manifold. A Kähler-Nijenhuis manifold is a Kähler manifold together with a Nijenhuis tensor which is compatible with the Poisson structure coming from the symplectic structure of the Kähler manifold (in the sense of Poisson-Nijenhuis manifolds). The author gives a characterization of Kähler-compatible Nijenhuis tensors coming from the known characterization of the Poisson-Nijenhuis tensors. The particular cases of complex-compatible and skew-complex compatible Kähler-Nijenhuis manifolds are studied in more details and several examples are given. Several results on the geometry of Kähler-Nijenhuis manifolds are also obtained.
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53D17 Poisson manifolds; Poisson groupoids and algebroids
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