广义CRF结构
Generalized CRF-structures
摘要：
广义F-结构是$T_cM\oplus T^*_cM$的复各向同性子丛$E$（$T_cM=TM\otimes_{\mathds{R}}\mathds{C}$，度量由配对定义），使得$E\cap\bar E^{\perp}=0$。如果$E$也被Courant括号封闭，则$E$是广义CRF结构。证明了广义F-结构等价于满足条件$Φ^3+Φ=0$的$TM\oplus T^*M$的反对称自同态$Φ$，并利用$Φ$的Courant-Nijenhuis挠率表示CRF-条件.我们考虑的结构是Yano定义的F-结构和CR（Cauchy-Riemann）结构的推广。我们构造了广义CRF-结构：一个经典的F-结构，一个对$(\mathcal{V},σ)$，其中$\mathcal{V}$是$TM$的可积子丛，$σ$是$M$上的2-形式，一个余维$h$的广义正规几乎切触结构.证明了流形$\tilde M$上的广义复结构将广义CRF-结构诱导到子流形$M\subseteq\tilde M$中。最后，我们考虑相容的，广义的，黎曼度量，我们定义了广义CRFK结构，它扩展了广义Kähler结构，并与四重结构$(γ,F_+,F_-,ψ)$等价，其中$(γ,F_\pm)$是经典的度量CRF结构，$ψ$是2-形式，并且一些条件可以用外微分$dψ$和$γ$ -Levi-Civita协变导数$\nabla F_\pm$来表示。如果$dψ=0$，条件减少到度量$γ$的两个部分Kähler约化的存在。本文最后的附录中，我们定义和广义Sasakian结构的特点。
英文摘要：
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.
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