Generalized CRF-structures

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 d ψ and the γ-Levi-Civita covariant derivatives ∇ 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.