# Generalized CRF-structures

Research paper by Izu Vaisman

Indexed on: 19 Feb '08Published on: 19 Feb '08Published in: Geometriae Dedicata

#### Abstract

A generalized F-structure is a complex, isotropic subbundle E of $${T_cM \oplus T^*_cM}$$ ($$T_cM = TM \otimes_{{\mathbb{R}}} {\mathbb{C}}$$ and the metric is defined by pairing) such that $$E \cap \bar{E}^{\perp} = 0$$. 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 $$TM \oplus T^*M$$ 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 $$({\mathcal{V}}, \sigma)$$ where $${\mathcal{V}}$$ 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 $$M \subseteq \tilde{M}$$ . 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 dψ and the γ-Levi-Civita covariant derivatives ∇ F± hold. If dψ =  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.