Generalized CRF-structures

Abstract

ABSTRACT. A generalized F-structure is a complex, isotropic subbundle E of T c M ⊕ T * c M (T c M = T M ⊗ R C and the metric is defined by pairing) such that E ∩Ē ⊥ = 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 T M ⊕ T * M that satisfies the condition Φ 3 + Φ = 0 and we express the CRF-condition by means of the CourantNijenhuis 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 (V, σ) where V is an integrable subbundle of T M 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 manifoldM induces generalized CRFstructures into some submanifolds M ⊆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 dψ and the γ-Levi-Civita covariant derivative ∇F ± 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.* 2000 Mathematics Subject Classification: 53C15 .

AN ABSTRACT MORIMOTO THEOREM FOR GENERALIZEDF-STRUCTURES

“…Split generalized F -structures are a particular case of the generalized F -structures introduced in [17]. In particular, the following two characterizations of SGF(E) can be easily deduced from the results of [17]. Extending J ∈ SGF(E) to TM by 0 provides a bijection between SGF(E) and the set of all orthogonal endomorphisms Φ of TM such that Φ 3 + Φ = 0 and ker(Φ) = E. On the other hand, assigning to J the subbundle…”

Geometric quantization of weak-Hamiltonian functions

“…The notion of an integrable big-isotropic structure may be complexified and all the previous result hold if complex values are assumed overall. Important examples are offered by the generalized CRF-structures, where E appears as the i-eigenbundle of a skew-symmetric endomorphism F of T big M such that F 3 + F = 0 [13] and, in the Dirac case, by generalized complex structures, i.e., integrable i-eigenbundles of a skew-symmetric endomorphism I of T big M such that…”

“…This example extends the classical notion of a Kähler polarization to generalized geometry. We briefly recall the framework following [2,13]. A classical metric F-structure on a manifold M is a pair (F, γ) where γ is a Riemannian metric, F ∈ End T M , F 3 + F = 0 and…”

Generalized contact bundles

“…Several approaches to odd-dimensional analogues of generalized complex structures can be found in the literature [13,22,18,19,1]. They are often named generalized contact structures and all of them include contact structures globally defined by a contact 1-form.…”