NDLI: Generalized CRF-structures

Content Provider	Springer Nature Link
Author	Vaisman, Izu
Copyright Year	2008
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.
Ending Page	154
Page Count	26
Starting Page	129
File Format	PDF
ISSN	00465755
e-ISSN	15729168
Journal	Geometriae Dedicata
Issue Number	1
Volume Number	133
Language	English
Publisher	Springer Netherlands
Publisher Date	2008-02-19
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry F-structure Courant bracket CRFK-structure CRF-structure Sasakian structure General geometric structures on manifolds (almost complex, almost product structures, etc.)
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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