Generalized CRF-structures

Abstract

A generalized F-structure is a complex, isotropic subbundle $E$ of $T_cM\oplus T^*_cM$ ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ 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 $Φ^3+Φ=0$ 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 $(\mathcal{V},σ)$ 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 $\tilde M$ 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 $(γ,F_\pm)$ 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 derivative $\nabla F_\pm$ 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.