On the structure of co-Kähler manifolds

Bazzoni, Giovanni; Oprea, John

doi:10.1007/s10711-013-9869-7

Cited by 13 publications

(54 citation statements)

References 15 publications

(4 reference statements)

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“…However, in higher dimension, K-cosymplectic structures strictly generalize coKähler structures. To see this, recall that compact coKähler manifolds satisfy very strong topological properties (see [5,12]). We collect them: Proposition 2.10.…”

Section: First Of All We Define a New Riemannian Metricg Bỹmentioning

confidence: 99%

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K-Cosymplectic manifolds

Bazzoni

Goertsches

2014

Ann Glob Anal Geom

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In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coKähler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost Kähler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.

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Section: First Of All We Define a New Riemannian Metricg Bỹmentioning

confidence: 99%

“…In fact, until very recently, the word cosymplectic indicated what we call here a coKähler structure (see [7,8,12,17]). The terminology used in this paper was introduced in [33] and seems to have been adopted since (see [4,5,10,26]).…”

Section: Introductionmentioning

confidence: 99%

K-Cosymplectic manifolds

Bazzoni

Goertsches

2014

Ann Glob Anal Geom

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“…The coKähler structure on an odd-dimensional manifold M can be associated to a Kähler structure on an S 1 -bundle (using a symplectomorphism) [21]. Further results on coKähler structures were given in [3].…”

Section: Generalised Cokähler Geometrymentioning

confidence: 99%

Generalised contact geometry as reduced generalised Complex geometry

Wright

2018

Journal of Geometry and Physics

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Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the reduction procedure, generalised coKähler structures are defined in a way which extends the Kähler/coKähler correspondence.1 where H ∈ Ω 3 cl (M ), is given by(for details see [5]). The Leibniz identity for • H (2.5a), gives the Maurer-Cartan identity dH = 0. The Dorfman product, • H , is natural in the sense that it is the derived bracket of d H := d + H∧ (with d the de Rham differential), acting on Γ(Equivalent exact Courant algebroids are classified by [H] ∈ H 3 (M, R), a point first noted by Severa [25]. The equivalence of the exact Courant algebroid under isotropic splittings corresponding to some B ∈ Ω 2 (M ) is closely related to the concepts of symmetries in generalised geometry structures. 2.1. Courant algebroid symmetries. Perhaps the most interesting aspect of generalised geometry is the enhanced symmetry group. The symmetry group of the Lie algebroid, given by the commutator of vector fields on T M , is Diff(M ). Exact Courant algebroids have a symmetry group given by Diff(M ) ⋉ Ω 2 cl (M ) if H = 0. Definition 2.3. A generalised almost complex structure on TM is given by J ∈ End(TM ), satisfying J * = −J and J 2 = −id.A generalised almost complex structure can equivalently be described by a maximal isotropic complex subbundle L ⊂ TM ⊗ C, satisfying L ∩L = {0}, forThere is a local one-to-one correspondence between generalised almost complex structures and conformal classes of complex pure spinors, where a complex pure spinor satisfies the non-degeneracy condition (φ, ϕ) M = 0.A generalised almost complex structure, J , is H-involutive if all sections X of the +i-eigenbundle L J are involutive with respect to • H : X 1 , X 2 ∈ L J ⇒ X 1 • H X 2 ∈ L J . Definition 2.4. A generalised complex structure is an H-involutive generalised almost complex structure.

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“…Two crucial facts about co-Kähler manifolds are contained in the following lemma. For a direct proof of these facts, see [1]. Lemma 1.3.…”

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confidence: 93%

Parallel Forms, Co-Kähler Manifolds and their Models

Bazzoni¹,

Lupton²,

Oprea³

2018

Bull. Belg. Math. Soc. Simon Stevin

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We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context.

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On the structure of co-Kähler manifolds

Cited by 13 publications

References 15 publications

K-Cosymplectic manifolds

K-Cosymplectic manifolds

Generalised contact geometry as reduced generalised Complex geometry

Parallel Forms, Co-Kähler Manifolds and their Models

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