K̈ahler-Nijenhuis manifolds

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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 languageEnglish
Pages (from-to)125-131
Number of pages7
JournalBalkan Journal of Geometry and its Applications
Issue number1
StatePublished - 2003


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

ASJC Scopus subject areas

  • Geometry and Topology


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