Generalized para-Kähler manifolds

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We define a generalized almost para-Hermitian structure to be a commuting pair (F,J) of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähler structure. This class of structures contains both the classical para-Kähler structure and the classical Kähler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple (γ, ψ, F), where γ is a (pseudo) Riemannian metric, ψ is a 2-form and F is a complex (1, 1)-tensor field such that F2=Id, γ(FX, Y)+γ(X, FY)=0. We deduce integrability conditions similar to those of the generalized Kähler structures and give several examples of generalized para-Kähler manifolds. We discuss submanifolds that bear induced para-Kähler structures and, on the other hand, we define a reduction process of para-Kähler structures.

Original languageEnglish
Pages (from-to)84-103
Number of pages20
JournalDifferential Geometry and its Application
StatePublished - 1 Oct 2015

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V.


  • 53C15
  • Generalized geometry
  • Generalized para-Hermitian structure
  • Generalized para-Kähler structure

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Computational Theory and Mathematics


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