Generalized CRF-structures

Research output: Contribution to journalArticlepeer-review


A generalized F-structure is a complex, isotropic subbundle E of T cM ⊕ T*cM ( TcM = TM ⊗ ℝ ℂ and the metric is defined by pairing) such that E ∩ E = 0. If E is also closed by the Courant bracket, E is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism Φ of TM ⊕ T*M that satisfies the condition Φ3 + Φ = 0 and we express the CRF-condition by means of the Courant-Nijenhuis torsion of Φ. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair (V σ) where V is an integrable subbundle of TM and σ is a 2-form on M, a generalized, normal, almost contact structure of codimension h. We show that a generalized complex structure on a manifold M̃ induces generalized CRF-structures into some submanifolds M ̃M. Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kähler structures and are equivalent with quadruples (γ, F +, F -, ψ), where (γ, F ±) are classical, metric CRF-structures, ψ is a 2-form and some conditions expressible in terms of the exterior differential d ψ and the γ-Levi-Civita covariant derivatives δ F ± hold. If d ψ = 0, the conditions reduce to the existence of two partially Kähler reductions of the metric γ. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.

Original languageEnglish
Pages (from-to)129-154
Number of pages26
JournalGeometriae Dedicata
Issue number1
StatePublished - Apr 2008


  • CRF-structure
  • CRFK-structure
  • Courant bracket
  • F-structure
  • Sasakian structure

ASJC Scopus subject areas

  • Geometry and Topology


Dive into the research topics of 'Generalized CRF-structures'. Together they form a unique fingerprint.

Cite this