Generalized CRF-structures

Details Generalized CRF-structures Generalized CRF-structures

2007-05-27

arxiv.org/abs/0705.3934v2

Mathematics

Differential Geometry

Appendix on generalized Sasakian structures added

Scientific paper

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 $\Phi$ of $TM\oplus T^*M$ 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 K\"ahler 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 K\"ahler reductions of the metric $\gamma$. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.

Affiliated with

Also associated with

No associations

Rating

Generalized CRF-structures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized CRF-structures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized CRF-structures will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-99064