Generalized CRF-structures

Abstract

A generalized F-structure is a complex, isotropic subbundle $E$ of $T_{c}M\oplus T_c^{\ast }M$ ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ 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 $\Phi$ of $TM\oplus T^{\ast }M$ that satisfies the condition ${\Phi }^3+\Phi =0$ and we express the CRF-condition by means of the Courant-Nijenhuis torsion of $\Phi$. 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 $\sigma$ 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 $(\gamma ,F_{+},F_{-},\psi )$, where $(\gamma ,F_{\pm })$ are classical, metric CRF-structures, $\psi$ is a 2-form and some conditions expressible in terms of the exterior differential $d\psi$ and the $\gamma$-Levi-Civita covariant derivative $\nabla F_{\pm }$ hold. If $d\psi =0$, the conditions reduce to the existence of two partially Kähler reductions of the metric $\gamma$. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.

Appendix on generalized Sasakian structures added