## Abstract

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.

Original language | English |
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Pages (from-to) | 125-131 |

Number of pages | 7 |

Journal | Balkan Journal of Geometry and its Applications |

Volume | 8 |

Issue number | 1 |

State | Published - 2003 |

## Keywords

- Kähler metric
- Kähler-Nijenhuis manifold
- Poisson-Nijenhuis structure

## ASJC Scopus subject areas

- Geometry and Topology