The Local Structure of Generalized Contact Bundles

Schnitzer, Jonas; Vitagliano, Luca

doi:10.1093/imrn/rnz009

Cited by 9 publications

(16 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will not discuss the automorphisms of DL in detail. A conceptual discussion can be found in [14]. Nevertheless, we will need a kind of action of Atiyah 2-forms on H-generalized contact structures:…”

Section: Lemma 219mentioning

confidence: 99%

“…The remarkable feature of this counterexample is, that it is, as manifolds, a global product of a (locally conformal) symplectic manifold and an Atiyah-complex manifold. Note that in [14] it was proven that every generalized contact bundle is locally isomorphic to a product, however not all products arise in this way. a tiny abuse of notation we will see ω is an element of Γ ∞ (Λ 2 T * S 2 ) ⊆ E (0,0),2 0…”

Section: A Counterexamplementioning

confidence: 99%

“…Generalized contact structures (in the sense of Vitagliano and Wade) encompass non necessarily co-orientable contact structures and integrable complex structures on the Atiyah-algebroid of a line bundle. In [14] it was proven that, locally, generalized contact structures are products of either a contact and a homogeneous generalized complex or a symplectic and a generalized contact structure (of a simpler type). This local normal form theorem can be seen as a manual to build up examples which are not included in the extreme cases by taking products.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weakly regular Jacobi structures and generalized contact bundles

Schnitzer

2019

Ann Glob Anal Geom

Self Cite

View full text Add to dashboard Cite

A Jacobi structure J on a line bundle L → M is weakly regular if the sharp map J ♯ : J 1 L → DL has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure. * jschnitzer@unisa.it Preliminaries and NotationThis introductory section is divided into two parts: first we recall the Atiyah algebroid of a vector bundle and the corresponding Der-complex with applications to contact and Jacobi geometry. Afterwards, we introduce the arena for generalized geometry in odd dimensions, the omni-Lie algebroids, and give a quick reminder of generalized contact bundles together with the properties we will need 2 afterwards. The Atiyah algebroid and the Der-complexThe notions of Atiyah algebroid of a vector bundle and the associated Der-complex are known and are used in many other situations. This section is basically meant to fix notation. A more complete introduction to this can be found in [17], which also discusses the notion of Dirac structures on the omni-Lie algebroid of line bundles in more detail, nevertheless the notion of Omni-Lie algebroids was first defined in [5], in order to study Lie algebroids and local Lie algebra structures on vector bundles.for a necessarily unique X ∈ Γ ∞ (T M ).Remark 2.2 Derivations of a vector bundle form a subspace of the first order differential operators from E to itself. Moreover, if the vector bundle has rank one then each differential operator is a derivation. Lemma 2.3 Let E → M be a vector bundle. The derivations of of E → M are the sections of a Lie algebroid DE → M . The Lie bracket of Γ ∞ (DE) is the commutator. An element in the fiber DThe Atiyah algebroid fits into the following exact sequence of Lie algebroids, which is called the Spencer sequence,where the Lie algebroid structure of End(E) is the (pointwise) commutator and the trivial anchor. The first arrow is given by the inclusion. With this we see immediatly that rank(DE) = rank(E) 2 + dim(M ).We assign a vector bundle E → M to a Lie algebroid DE → M . A natural question is: is this assignment functorial? In fact it is not, unless we restrict the category of vector bundles by just taking regular vector bundle morphisms, i.e. vector bundle morphisms which are fiber-wise invertible. Note that a regula...

show abstract

Section: Lemma 219mentioning

confidence: 99%

Section: A Counterexamplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weakly regular Jacobi structures and generalized contact bundles

Schnitzer

2019

Ann Glob Anal Geom

Self Cite

View full text Add to dashboard Cite

show abstract

“…Generalized contact structures are odd analouge of generalized complex structures and generalization of contact structures [11] (see also [22,26]). Motivated from the notion of B-field transformation of generalized complex structures, in section 7, we define a similar notion for generalized contact structures.…”

Section: Introductionmentioning

confidence: 99%

“…Kirillov's local Lie algebras are same as Jacobi structures when the line bundle L is trivial. We refer [4,22,25,26] for more details on the line bundle approach of contact structures, contact groupoids, Dirac-Jacobi structures and generalized contact structures. Finally, we remark that the contents of the present paper can also be discussed in the line bundle framework.…”

Section: Introductionmentioning

confidence: 99%