An h-Principle for Symplectic Foliations

Codimension-one foliations calibrated by nondegenerate closed 2-forms

“…Determining which codimension-one foliations are the symplectic foliations of a Poisson structure is very complicated; there exist partial results which use h-principles and only apply to open manifolds [Bertelson 2001;Bertelson 2002;Fernandes and Frejlich 2012]. The existence of a closed lift of a foliated 2-form ω Ᏺ is controlled by three obstructions associated to the spectral sequence which relates basic cohomology, leafwise cohomology and the cohomology of the total space [El Kacimi-Alaoui 1983] (see [Alcalde-Cuesta and Hector 1993] for a treatment in the setting of Poisson geometry); if the foliation is defined by a closed 1-form, then the obstruction to the existence of a closed lift admits a simpler description [Guillemin et al 2011, Section 2.2].…”

Universal models via embedding and reduction for locally conformal symplectic structures

“…structure [13], on reduction [14], on Moser stability type results [1] and on existence of l.c.s. structures on open manifolds via a h-principle [7] have been obtained. Another very active line of research in the subject is centered in locally conformal Kähler (l.c.K.)…”

“…is a morphism. Therefore according to (7), e −f π * β is closed whenever β ∈ Ω k (M ) is d ω -closed, where f is a scaling function. The smallest exact covering space of (M, ω) is the one with fundamental group the kernel of the additive character ω.…”

Contact Structures & Codimension-one Symplectic Foliations

“…Using holonomic approximation it follows almost immediately that the h-principle for contact structures holds on open manifolds. The same techniques have been used for symplectic foliations on open manifolds [12,49].…”

“…Remark 3.2. 49. If n " dimpY q ´1 and r " 1, the structure we just constructed is precisely the space of contact elements.…”