Normal forms for Dirac–Jacobi bundles and splitting theorems for Jacobi structures

The aim of this paper is to prove a normal form Theorem for Dirac–Jacobi bundles using a recent techniques of Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As another application we provide an alternative proof of the splitting theorem of homogeneous Poisson structures.


Introduction 2 Preliminaries and Notation
This 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 the so-called Dirac-Jacobi bundles in odd dimensions, the omni-Lie algebroids, and give a quick reminder of Dirac-Jacobi bundles together with the properties we will need afterwards.

Notation and a brief reminder on Jacobi Geometry
The 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 [12] and its references. Nevertheless, the notion of Omni-Lie algebroids was first defined in [4], in order to study Lie algebroids and local Lie algebra structures on vector bundles.
For a vector bundle E → M , we denote its gauge or Atiyah algebroid by DE → M and by σ : DE → T M its anchor. Note that D is a functor from the category of vector bundles with regular, i.e. fiberwise invertible, vector bundle morphisms to Lie algebroids. Hence, we denote for a regular Φ : E → E ′ by for λ, µ ∈ Γ ∞ (L). Conversely, every L-valued 2-form J on J 1 L defines a skew-symmetric bilinear bracket {−, −}, but the latter needs not to be a Jacobi bracket. Specifically, it does not need to fulfill the Jacobi identity. However, there is the notion of a Gerstenhaber-Jacobi bracket When L is the trivial line bundle, than the notion of Jacobi bracket boils down to that of Jacobi pair.

Remark 2.2 (Trivial Line bundle)
Let R M → M be the trivial line bundle and let J be a Jacobi tensor on it. Let us denote by 1 M ∈ Γ ∞ (R M ) the canonical global section. Using the canonical connection we can see that DL ∼ = T M ⊕ R M and hence With this splitting, we see that The Jacobi identity is equivalent to [Λ, Λ] + E ∧ Λ = 0 and L E Λ = 0. The pair (Λ, E) is often referred to as Jacobi pair. Moreover, if we denote by ½ * ∈ Γ ∞ (J 1 R M ) the canonical section then we can write any ψ ∈ J 1 R M as ψ = α+r½ * ∈ Γ ∞ (J 1 R M ), for some α ∈ T * M and r ∈ R. We obtain A more detailed discussion about Jacobi structures on trivial line bundles can be found in [11,Chapter 2]. In a similar way, we can see that Here ½ * is the canonical section of R M , moreover the differential d R M is defined by the relations d R M (½ * ) = 0 and d R M = d dR + ½ * ∧ .

The Omni-Lie Algebroid of a line bundle and its automorphisms
The omni-Lie algebroid plays the same role as the generalized tangent bundle does in Dirac geometry. In fact, the parallels are evidently enormous. Moreover, since the canonical inner product of it will be line-bundle valued, one can easily drop the word local Courant algebroid. Note that the following definitions and Lemmas are obvious adaptions of the case of H-twisted Dirac structure, this is why we omit proofs. The non-twisted versions of the following definitions and resulats in Dirac-Jacobi geometry can be found in [12].
ii.) the non-degenerate L-valued pairing iii.) the canonical projection pr D : DL → DL is called the H-twisted Omni-Lie algebroid of L → M . We shall now introduce automorphisms of the omni-Lie algebroid, which mirrors the definition of automorphisms of the generalized tangent bundle.
The group of H-twisted Courant-Jacobi automorphisms is denoted by Aut H CJ (L).
For a line bundle L → M and Φ ∈ Aut(L), we define which gives canonically an automorphism DΦ ∈ Aut(DL). Moreover, the pair (DΦ, Φ) fulfills conditions i.) and ii.) in Definition 2.5, nevertheless it is not an (H-twisted) a Courant-Jacobi automorphism for an arbitrary H. For a 2-form B ∈ Ω 2 L (M ), we define which also fulfills conditions i.) and ii.) in Definition 2.5, seen as pair (exp(B), id). We can combine this two special kinds of morphisms together with an H-dependent action on DL and find the following Lemma 2.6 Let L → M be a line bundle and let H ∈ Ω 3 L (M ) be closed. If we denote by Z 2 L (M ) the closed 2-forms, then is an ismorphism of groups.
In a similar way, we can define infinitesimal automorphisms of the Omni lie algebroid is an isomorphism of Lie algebras.

For every section
For later use, we want to talk about the flow of infintesimal (H-twisted ) Courant-Jacobi automorphisms and want to compute them as explicit as possible.

Dirac-Jacobi bundles
After having discussed the arena, we want to introduce the subbundles of interest: so-called Dirac-Jacobi Bundles. As the name suggest, they are the analogue of Dirac structures on the generalized tangent bundle. In fact, the definition is (up to some obvious replacements) the same. Moreover, if H = 0, we will call L simply Dirac-Jacobi structure.
Example 2.12 Let L → M be a line bundle and let J ∈ Γ ∞ (Λ 2 (J 1 L) * ⊗ L) be a Jacobi structure, then is a Dirac-Jacobi structure. Then there is a unique Jacobi structure J ∈ Γ ∞ (Λ 2 (J 1 L) * ⊗ L), such that L J = L Proof: The result follows the same lines as the well-known fact in Poisson geometry.
Another interesting example of Dirac-Jacobi bundles, which also plops up in Jacobi geometry, is Definition 2.14 Let L → M be a line bundle. A Dirac-Jacobi structure L ⊆ DL is called of homogeneous Poisson type, if rank(L ∩ DL) = 1.
The name of these objects is justified by the following Lemma 2.15 Let L → M be a line bundle and let L ⊆ DL a Dirac-Jacobi structure of homogeneous Poisson type, then for every point p ∈ M there exists a local trivialization L U = U × R, a flat connection ∇ : T U → DL U ∼ = T U ⊕ R U and a homogeneous Poisson structure π ∈ Γ ∞ (Λ 2 T U ) with homogeneity Z ∈ Γ ∞ (T M ), such that where we use the inclusion T * M → J 1 L by α(∇ X ) = α(X) and α(½) = 0.
Proof: Let p ∈ M and U ⊆ M be an open subset containing p, such that L U ∼ = U × R with corresponding trivialization of the gauge algebroid DL U = T U ⊕ R U , and hence we are using the canonical flat connection ∇ can : T U → T U ⊕ R U . In a possibly smaller neighbourhood, notated also by U , we find a non-vanishing section ∆ = (−X, f ) ∈ Γ ∞ (L ∩ DL). We can distinguish two cases: the first is that f (p) = 0, the we find a (possibly smaller) neighbourhood of p, such that f is nonvanishing, hence (− X f , 1) =: (−Z, 1) spans L ∩ DL in that neighbourhood. Exploting the isotropy, we see that L U is of the form and not further specified Y ∈ T U , since the J 1 L U part has to vanish at sections of the form r(½−∇ can Z ). We can write this as Note that, because of the isotropy, hZ + Y is completely determined by α, hence there is a bi-vector π ∈ Γ ∞ (Λ 2 T U ) such that π ♯ (α) = hZ + Y and we can write The claim follows by using the flatness of ∇ can and the involutivity of L. Now we have to treat the case f (p) = 0. Since ∆ = (−X, f ) is non-vanishing, we conclude that X(p) = 0, hence there is a closed two form β ∈ Γ ∞ (T * U ) such that β(X) = −1 around p. We define the flat connection With this connection we see that ∆ = (f − 1)½ − ∇ X and since f (p) = 0, we have that f − 1 = 0 in a whole neighbourhood of p and hence we choose ∆ ′ = 1 f −1 ∆ as a generating section of L ∩ DL around p. We can now repeat the same argument as for the case f (p) = 0 by using the connection ∇ In the category of Dirac-Jacobi bundles there are not just automorphism of the omni-Lie algebroid as morphisms, one of the possibilities is to include so-called backwards transformations as in the Dirac geometry case.
is called Backwards transformation of L.
The backwards transform of a Dirac-Jacobi bundle need not to be Dirac-Jacobi anymore, but there are sufficient conditions on the subbundle L and the line bundle morphism Φ which can be seen, i.e. in [12]: Theorem 2.17 Let Φ : L 1 → L 2 be a regular line bundle morphism over φ : M 1 → M 2 and let L ∈ DL 2 be a Dirac-Jacobi bundle. If ker DΦ * ∩ φ * L has constant rank, then B Φ (L) is a Dirac-Jacobi bundle.
Proof: The proof can be found in [12,Proposition 8.4].

Remark 2.18
Note that for a line bundle automorphism Φ ∈ Aut L, we have that DΦ(L) = B Φ −1 (L). but not every backwards transform needs to be of this form.

Submanifolds and Euler-like Vector Fields
In this subsection we want to discuss Euler-like vector fields. These vector fields, in particular, induce a homogeneity structure on the manifold, which is equivalent, under some additional conditions which are in our case always fulfilled, that the manifold is total space of a vector bundle, see e.g. [8]. This total space turns out to be the normal bundle for some submanifold, which is an input datum for an Euler-like vector field. Nevertheless, we will not go more in details with these features, since we work directly with tubular neighbourhoods. We will begin collecting facts about tubular neighbourhoods, submanifolds and corresponding mappings and describe afterwards the notion of Euler-like vector fields and extend this notion the derivations of a line bundle.

Normal Bundles and tubular neighbourhoods
the induced map on the normal bundle. For a vector field X on M tangent to N , we have that the flow Φ X t is a map of pairs from (M, N ) to itself. Hence we define Moreover, for a vector bundle E → M and σ ∈ Γ ∞ (E), such that σ N = 0 for a submanifold N ֒→ M , we denote by the map which is ν(σ), for σ seen as a map σ : (M, N ) → (E, M ), followed by the canonical identification ν(E, M ) = E, given by Before we prove the next results, we want to find a useful description of C −1 E . Let us therefore consider a curve γ : I → E for an open interval I containing 0, such that γ(0) = 0 p for p ∈ M , then one can prove in local coordinates Proposition 3.2 Let E i → M be vector bundles for i = 1, 2 and let Φ : E 1 → E 2 be a vector bundle morphism covering the identity. Then, for every section σ ∈ Γ ∞ (E 1 ), such that σ N = 0 for some submanifold N ֒→ M , holds.
and the claim follows if we restrict this maps.
for a unique D X ∈ Γ ∞ (End(T M N )), moreover T N ⊆ ker(D X ) and such that ψ N : N → N is the identity and for ψ :

Euler-like Vector fields and Derivations
In this part, we recall basically just the notion of Euler-like vector fields from [3] and extend this notion to derivations of a line bundle.
where E is the Euler vector field on ν N → N . Proof: Let us choose a tubular neighbourhood For the vector field X = ψ * E multiplied by a suitable bump function which is 1 in a neighbourhood of N , we have where we used Proposition 3.1 and the fact that ν(ψ) = C −1 ν N .
Lemma 3.7 Let M be a manifold, N ֒→ M a submanifold and X ∈ Γ ∞ (T M ) be a Euler-like vector field. Then there exists a tubular unique neighbourhood embedding such that ψ * X = E.
Proof: The proof can be found in [3].
Proposition 3.8 Let (M, N ) be a pair of manifolds and let X ∈ Γ ∞ (T M ) be a vector field, such that X N = 0 and is complete. Then X is Euler-like, if and only if d N X followed by the projection T M N → ν N is identity.
Proof: Let X ∈ Γ ∞ (T M ) be given as in the proposition. According to Proposition 3.3, there exists This is just equal to the flow of the Euler vector field, if pr ν N • D X = id ν N . Using Proposition 3.3, we have d N X = D X and hence the claim.
Note that for a pair of manifolds (M, N ) and a Euler like vector field X ∈ Γ ∞ (T M ), the set is an open subset in M containing N , such that that the action of Φ X t shrinks to this set. Moreover, for a tubular neighbourhood ψ : ν N → U , such that ψ * X = E, we have that Let us denote by λ s = Φ X log(s) U . We obtain, that λ s is smooth for all s ∈ R + 0 . Moreover, we have that where we denote by κ s : Note that κ 0 : ν N → N coincides with the bundle projection, to be more precise k 0 = pr ν • j, where pr ν is the bundle projection and j : N → ν N the canonical inclusion. Let us add now the line bundle case This definition turns out to be the correct one for our purposes, since we can prove basically all results, which are available for Euler-like vector fields. Let us start collecting them.
(p) exists and lies in N }, extended smoothly to s = 0. Moreover, the map is a regular line bundle morphism.
Proof: The proof is an easy verification using a tubular neighbourhood ψ : ν N → U , such that ψ * σ(X) = E. Definition 3.11 Let L → M be a line bundle and N ֒→ M be a submanifold. A fat tubular neighbourhood is a regular line bundle morphism where the line bundle L ν is given by the pull-back Proof: The proof can be found in [11,Chapter 3].
For a line bundle L → N and a vector bundle E → N there is always a canonical Derivation Proof: This proof is an easy verification using the fact that Φ t covers the flow of the Euler vector field.
Note that for the flow Φ t of the canonical Euler-like derivation ∆ E ∈ Γ ∞ (DL E ), we have that is defined for all s > 0 and can be extended smoothly to s = 0, moreover P 0 coincides with the canonical projection P : L E → L followed by the canonical inclusion J : L → L E .
Lemma 3.14 Let L → M be a line bundle, let N ֒→ M be a submanifold and let ∆ ∈ Γ ∞ (DL) be an Euler-like derivation. Then there is a unique fat tubular neighbourhood Ψ : Proof: First, we want to proof existence. It is clear that any such Ψ has to cover the unique tubular neighbourhood ψ : ν N → U , such that ψ * σ(∆) = E. So let us choose a fat tubular neighbourhood Ψ : Hence σ(∆ E ) = σ(Ψ * ∆). Consider now the derivation = ∆ E −Ψ * ∆ and where Φ t is the flow of ∆ E . Let us denote the flow of t by φ t . Note that it is complete, since σ( t ) = 0, indeed there is even a explicit formula for it, which we do not use. Note however, that φ t ∈ Gau(L ν ) for all t ∈ R. Let us compute . Therefore, we have that the map Ψ =Ψ • φ 1 will do the job, since obviously φ 1 N = id.
Let us now assume that we have Ψ 1 , Note that since both have to cover the unique ψ : ν N → U , the target L U is for both the same. Let us consider Ξ := Ψ −1 1 • Ψ 2 : L ν → L ν , which covers the identity, which implies that there is a nowhere vanishing function f ∈ C ∞ (ν N ), such that Ξ(l p ) = f (p)l p for all l p ∈ L ν . Moreover, we have that Ξ N = id Lν N , hence f (0 n ) = 1 for all n ∈ N , and Ξ * ∆ E = ∆ E . We consider now an arbitrary section λ ∈ Γ ∞ (L ν ) and compute .
Hence E(f ) = 0, which means that f = pr * ν g for some function g ∈ C ∞ (N ), but since 1 = f (0 n ) = g(n) for all n ∈ N , we have that Ξ = id Lν .
For a line bundle L → M , a submanifold N and an Euler-like derivation ∆ ∈ Γ ∞ (DL), we have that is well defined for s > 0 and can be extended smoothly to s = 0, where L U is the target of the unique fat tubular neighbourhood Ψ : L ν → L U , such that Ψ * ∆ = ∆ E . Moreover, we have that for all s ≥ 0. Note that if we project this equation to the manifold level, this simply gives Eq. 3.2.

Normal Forms of Dirac-Jacobi bundles
Using the techniques of Euler-like derivations, we want to prove a normal form theorem for Dirac-Jacobi bundles. In fact, if the submanifold N is a transversal, then we can find special Euler like derivations which are, in some sense, controlling the behaviour of the Dirac-Jacobi bundles near N . The aim is now to prove the existence of this special kind of Euler-like derivations and afterwards, we are able to prove a normal form theorem. and conclude some corolloraries from it. Proof: This is an easy consequence of Theorem 2.17. Proof: We consider the linear map which is well-defined since DI(∆ p ) = ι(p) . We claim now that this map is injective, let us therefore consider (∆ p , ( ι(p) , α ι(p) )) ∈ ker(Ξ). It follows immediately, that ∆ p = 0 and hence ι(p) = 0. If (0, α ι(p) ) ∈ L then α ιp ∈ Ann(pr D L). Since DI * α ι(p) 0 = 0, we have that α ι(p) ∈ Ann(DL N ), hence α ι(p) = 0 and the claim follows.
For dimenional reasons we have that Ξ is an isomorphism. Proof: We consider the exact sequence where the first arrow is defined by the identifiaction B I (L) ∼ = I ! L from Lemma 4.3 followed by the canonical map I ! L → L. The second arrow is the projection pr D : L N → DL N followed by the symbol map σ : DL N → T M N and finally followed by the the projection to the normal bundle pr ν N : T M N → ν N . Let us choose a section ε ∈ Γ ∞ (L) with ε N = 0, such that d N ε : ν N → L N defines a splitting of the sequence. We consider now and see that if d N ε splits the above sequence then (σ • pr D ) d N ǫ splits the lower sequence. Using Proposition 3.2, we see that (σ • pr D ) d N ǫ = d N ((σ • pr D )(ε)) and by Proposition 3.8, we see that T ν(σ • pr D )(ε) = E. Multiplying ε by a suitable bump function we may arrange that (σ • pr D )(ε) is complete and hence an Euler-like vector field. By definition pr D (ε) is hence an Euler-like derivation.
Let us fix now a H-twisted Dirac-Jacobi structure L ⊆ DL for a line bundle L → M . Let us also consider a transversal ι : N ֒→ M and a section ε = (∆, α) ∈ Γ ∞ (L), such that ε N = 0 and ∆ is an Euler-like derivation. Due to the Lemma 3.14, we find a unique fat tubular neighbourhood For sure we have that the action of (γ t , Φ ∆ t ) preserves L, which is explicitly This leads us directly to the following theorem Proof: According to Proposition 4.4, we can find (∆, α) ∈ Γ ∞ (L), such that ∆ is Euler-like. Then there is a unique fat tubular neighbourhood Ψ : L ν → L U , such that Ψ * ∆ = ∆ E , due to Lemma 3.14. Let us denote by preserves L for all t ∈ R and so will (γ − log(s) , Φ ∆ − log(s) ) for all s > 0. Let us take a closer look to and we obtain that it is smoothly extendable to s = 0 and let us denote its limit s → 0 by ω ′ and ω = Ψ * ω ′ . We have which holds for all s ≥ 0. Hence we have for s = 0, using that for the canonical inclusion J : L N → L ν we have that P 0 = J • P and Ψ • J = I, that Note that this Theorem says, that up to a B-field, the Dirac-Jacobi structure is fully encoded in a given transversal, and hence the term "normal form" is justified by this fact. Moreover, it is possible to distinguish two different kind of leaves in Dirac-Jacobi geometry, see [12], so it is also possible to distinguih two kinds of transversals, which are more interesting in the Jacobi setting, since in the general Dirac-Jacobi setting the normal forms will be the same. Nevertheless, we will introduce them here and use them more excessively in the next section.  This transversals naturally appear as minimal transversal to locally conformal pre-symplectic leaves, see [12] for a more detailed discussion.
So a corollary of this normal form theorem using the new notion of cosymplectic transversals   and hence, for dimensional reasons, DL p 0 = (DL N + B I (L)) p 0 ⊕ α p 0 . Therefore this equality holds in a whole neighbourhood of p 0 , so rank(DL N + B I (L)) = 2n + 1 in this neighbourhood, which implies rank(DL N ∩ B I (L)) = 1 around p 0 .

Remark 4.11
Note that a cocontact transversal does not inhert a Jacobi structure, but nevertheless the Dirac-Jacobi structure is of homogeneous Poisson type.

Remark 4.13
The definition of a homogeneous cocontact transversal seems a bit strange, since it includes a connection. This fact can be explained quite easily using the homogenezation described in [12], which turns a Dirac-Jacobi structure on a line bundle L → M into a Dirac structure on L × := L * \{0 M } which is homogeneous (in the sense of [10]) with respect to the shrinked Euler vector field E on L * . The pre-symplectic leaves of this Dirac structure have the additional property that E is either tangential to it or transversal. If E is tangential, then the leaf corresponds to a pre-contact leaf on the base M . Hence a minimal transversal N to it is transversal to the Euler vector field and defines therefore a horizontal bundle on L * pr(N ) and hence a connection.
where (π N , Z N ) is the homogeneous Poisson structure on the transversal from Lemma 2.15.
This last two corollaries can be seen as the Jacobi-geometric analogue of the results obtained by Blohmann in [1].

Normal forms and Splitting Theorems of Jacobi bundles
As explained in Example 2.12, Jacobi bundles are a special kind of Dirac-Jacobi structures. In addition, we have that Jacobi isomorphism induces an isomorphism of the corrsponding Dirac structures (this holds even for morphisms if one considers forward maps of Dirac-Jacobi structures which we will not explain here, see [12]). The converse is unfortunately not true: if the Dirac-Jacobi structures of two Jacobi structures are isomorphic, it does not follow in general that the Jacobi structures are isomorphic. The parts which are not "allowed" in Jacobi geometry are the B-fields. Nevertheless, we can keep track of them, if we make further assumptions on the transversals, namely cosymplectic and cocontact transversals.

Cosymplectic Transversals
In this part, we are using the notion of cosymplectic transversals as explained in the previous section. The difference is now that in Jacobi geoemtry this transversal gives us more than on arbitrary Dirac-Jacobi manifolds. In fact, the Jacobi structure induces a line bundle valued symplectic structure on the normal bundle, to be seen in the following Lemma 5.1 Let L → M be a line bundle, J ∈ Γ ∞ (Λ 2 (J 1 L) * ⊗ L) be a Jacobi tensor with corresponding Dirac-Jacobi structure L J ∈ DL and let ι : N ֒→ M be a cosymplectic transversal. Then Proof: First we prove that J ♯ Ann(DL N ) is injective. Let therefore α ∈ Ann(DL N ), such that J ♯ (α) = 0. Hence we have for an arbitrary β ∈ J 1 L, that α(J ♯ (β)) = −β(J ♯ (α)) = 0.
where the last equality follows since d N α : ν N → Ann(DL N ). Hence we have that ker((d L α) ♭ ) ⊇ DL N , in particular this is true for 1 t (Φ ∆ log(t) ) * (d L α), since Φ log(s) N is a gauge transformation fixing DL N . Thus it is true also for ω, since DΨ DL N = id.
We want to describe the structure of ω at N . Note that for a cosymplectic transversal N , the normal bundle always comes together with a canonical symplectic (i.e. non-degenerate) Proof: Note that for a cosymplectic transversal, we have

Moreover, we have
where we include ν N by the following map: It is clear that DΨ fixes DL N , since Ψ N : L N → L N is identity. We want to show that DΨ(ν N ) ⊆ J ♯ (Ann(DL N )). One can show that by an elementary calculation, that DΨ(χ(v n )) = lim t→0 ∆ λt(ψ(vn)) t using Equation 3.3. But by defintion, we have that  is the graph of a Jacobi structure near the zero section and there exists a fat tubular neighbourhood Ψ : L ν → L U which is a Jacobi map near the zero section.
Proof: We have proven this theorem for the special ω given by Let ω ′ be a second 2-form fulfilling the requirements of the theorem, then is a (time-dependent) 2-form such that σ 0 = 0 and moreover σ t N = 0. Thus, is a Jacobi structure near N . Now we can apply Appendix A to get the result.
An immediaty consequence of this theorem is the Splitting for Jacobi manifolds around a locally conformal symplectic leaf, proven by Dazord, Lichnerowicz and Marle in [5].
Theorem 5.5 Let L → M be a line bundle, let J ∈ Γ ∞ (Λ 2 (J 1 L) * ⊗ L) be a Jacobi tensor and let p 0 ∈ M be a locally conformal symplectic point. Then there are a line bundle trivialization L U ∼ = U ×R around p 0 and a cosymplectic transversal N ֒→ U , such that U ∼ = U 2q × N for an open subset 0 ∈ U 2q ⊆ R 2q and the corresponding Jacobi pair (Λ, E) is transformed (via this isomorphism) to where (Λ N , E N ) is the induced Jacobi structure on the transversal N and the canonical stuctures on the fiber are given by (π can , Z can ) = ( Proof: We can assume from the beginning that the line bundle is trivial, since otherwise we can trivialize around p 0 and and shrink the line bundle to this open neighbourhood. Let us choose an arbitrary transversal N to the leaf S at p 0 (in the sense, that S × N = M ). It is easy to see that and hence we can shrink to an open neighbourhood of p 0 , where this equality holds. This means every transversal to a leaf is a cosymplectic transversal near the intersection point. Let us from now on denote p 0 = (s 0 , n 0 ), hence ν N ∼ = T s 0 S × N ∼ = R 2k × N . Since the line bundle is trivial, we can identify ν N together with Θ as a symplectic vector bundle, hence we find a possible smaller N and a vector bundle automorphism of ν N , such that Θ is the constant symplectic form. We can now choose where (q, p) are the symplectic coordinates on ν N → N . This 2-from is d L -closed and coincides with Θ on N , moreover ker(ω ♭ ) N = DL N . Hence the requirements of Theorem 5.4 are fulfilled and the claim follows by an easy computation.

Cocontact transversals
The second kind of transversals we want to discuss in the context of Jacobi geometry are cocontact transversals, which were also introduced before in Definition 4.9. In fact this notion is not enough for our purposes and we need to assume more information on the structure of the transversal, which is precisely the notion of homogeneous cocontact transversal from Definition 4.9.
Proof: The proof follows the same lines as Lemma 5.1.
Proof: This proof follows the same lines as the proof of Proposition 5.2.
As in the cosymplectic transversal case, we can define a skew symmetric 2-form It is easy to see that Θ is non-degenerate. Moreover, we have Proof: Using the ideas of the proof of Lemma 5.3, we can show that the fat tubular neighbourhood transports J ♯ (Ann(im(∇)) to ν N ⊕ K, hence the proof is copy and paste of this Lemma. The next step is to prove the second splitting Theorem of Dazord and Lichnerowicz and Marle in [5], namely the splitting of Jacobi manifolds around contact leaves.
Theorem 5.10 Let L → M be a line bundle, let J ∈ Γ ∞ (Λ 2 (J 1 L) * ⊗ L) be a Jacobi tensor and let p 0 ∈ M be a contact point. Then there are a line bundle trivialization L U ∼ = U × R around p 0 and a homogeneous cocontact transversal N ֒→ U , such that U ∼ = U 2q+1 × N for an open subset 0 ∈ U 2q+1 ⊆ R 2q+1 and the corresponding Jacobi pair (Λ, E) is transformed (via this isomorphism) to where (π N , Z N ) is the induced homogeneous Poisson structure on the transversal N and the contact structure on the fiber is given by Proof: Let p 0 ∈ M be a contact point and let N ⊆ M be a transversal, such that We can again assume that the line bundle L → M is trivial, since we want to prove a local statement.
In a possibly smaller neighbourhood, we can assume that also the normal bundle ν N = V × N → N is trivial. We want to show that there is a trivialization of ν N , such that Θ looks trivial, where we specialize on the way through the proof what we mean by trivial. Let us therefore denote by λ the local trivializing section of L N , thus we can write Θ(∆, ) = Ω(∆, ) · λ for ∆, ∈ ν N ⊕ K. Since L N → N is trivial, we identify DL N = T N ⊕ R N and choose the trivial connection ∇. Hence, we can find a (local) nowhere vanishing section of K of the form ½ − Z for a unique Z. Let us now shrink since ν N is odd dimensional and Θ is a skew-symmetric pairing, we can find a local non-vanishing X ∈ Γ ∞ (ν N ), such that Θ(X, ·) = 0, moreover, since Θ is non-degenerate, we can modify X in such a way that It is now easy to see that symplectic complement S := ½ − Z, X ⊥ ⊆ ν N . Finally, we find a trivialization of S such that Ω S is the trivial symplectic form with Darboux frame {e 2 , e k+2 , . . . }.
Hence, by extending this trivialization to ν N = V × N by using the coordinate X as e 0 , we find that {e 0 , ½−Z, e 1 , e k+1 , e 2 , e k+2 , . . . } is a Darboux frame of Ω in this trivialization. with the decomposition which coincides with Θ on ν N ⊕ K and is d L -closed. By applying Theorem 5.9, since N together with ∇ is a homogeneous cocontact transversal, we find a Jacobi morphism An easy computation shows that B P (L N ) ω is the graph of the Jacobi structure of the form in the theorem.

Application: Splitting theorem for homogeneous Poisson Structures
Using the homogenezation scheme from [2], one can see that Jacobi bundles are nothing else but special kinds of homogeneous Poisson manifolds. Moreover, the two most important examples of Poisson manifolds are of this kind: the cotangent bundle and the dual of a Lie algebra. Using this insight, it is easy to see that proving something for Jacobi structures gives a proof for something in homogeneous Poisson Geometry. We want to apply this philosophy to give a splitting theorem for homogeneous Poisson manifolds. The first appearance of such a theorem was [5,Theorem 5.5] in order to prove the local splitting of Jacobi pairs. Here we want to attack the problem from the other side: we use the splitting of Jacobi manifolds to prove the splitting of homogeneous Poisson structures.
Theorem 6.1 Let (π, Z) be a homogeneous Poisson structure on a manifold M and let p 0 ∈ M be a point such that Z p 0 = 0. Then there exist an open neighbourhood U of p 0 , an open neighbourhood U 2k of 0 ∈ R 2k , a manifold N with a homogeneous Poisson structure (π N , Z N ) and a diffeomorphism ψ : U → U 2k × N , such that Additionally, i.) if Z ∈ im(π ♯ ), then ψ * Z = p i ∂ ∂p i + ∂ ∂p k + Z N .
Proof: Note that since Z p 0 = 0, we find coordinates {u, x 1 , . . . , x q } with p 0 = (1, 0, . . . , 0), such that Z = u ∂ ∂u . In this chart, we have, using L Z π = −π, for unique Λ ∈ Γ ∞ (Λ 2 T M ) and E ∈ Γ ∞ (T M ) which do not depend on u. It is easy to see, that we have [Λ, Λ] = −E ∧ Λ and L E Λ = 0, which means that (Λ, E) is a Jacobi pair. This allows us to use Theorem 5.5 and Theorem 5.10 to prove the result. We will do it just for the case where p 0 is a contact point, which means, translated to Jacobi pairs, that E p 0 is transversal to im(Λ ♯ ) p 0 and thus Z ∈ im(π ♯ ), since the other case is exactly the same. Note that, we can apply Theorem 5.10: there exists coordinates {x, q i , p i , y j } and a local non-vanishing function a( which is basically the line bundle trivialization), such that Λ = 1 a (Λ can + π N + E can ∧ Z N ) and E = 1 a (E can + Λ ♯ (da)), where Λ can and E can are just depending on {x, q i , p i } and (φ N , Z N ) is a homogeneous Poisson structure just depending on y j -coordinates. If we apply the diffeomorphism (u, x 1 , . . . , x q ) → (a · u, x 1 , . . . , x q ), we have π = 1 u (Λ can + π N + E can ∧ Z N + u ∂ ∂u ∧ E can ).
A (quite) long and not very insightful computation shows that the diffeomorphism Φ(u, x 1 , . . . , x q ) = (u, Φ Z N log(u) (Φ Ecan −log(u) (x 1 , . . . , x q ))), where Φ Z N t (resp. Φ Ecan t ) is the flow uf Z N (resp. E can ), gives us and with some obvious variations and renaming coordinates of π we get the result.
This Application shows us that, eventhough we can see Poisson structures as Jacobi manifolds, which suggests that they are more general objects than Poisson structures, the splitting theorems (of Jacobi pairs) are a refinement of the known splitting theorems for Poisson structures.

Generalized Contact bundles
In this last section, we want to drop a word about generalized contact bundles. They were introduced recently in [13] and they are modeled to be the odd dimensional analogue to generalized complex structures. A generalized contact structure can be also seen as an endomorphism of DL of the form where φ ∈ End(DL), J ∈ Γ ∞ ((J 1 L) * ⊗L) and α ∈ Ω 2 L (M ) (see [13] and [10]). This endomorphism has to fulfill certain properties: it has to be almost complex, compatible with the pairing and integrable, which we do not explain what it means here and refer the reader to [13]. The +i-Eigenbundle produces a generalized contact structure in the sense of Definition 7.1. Moreover, we have that among many more conditions that J is a Jacobi structure. Let us now pick a (cosymplectic or cocontact) transversal to J together with an Euler-like derivation ∆ = J ♯ (α), then (∆, iα − φ * (α)) ∈ Γ ∞ (L). With the techniques from Section 4 and Section 5, one can show that 1 t (Φ ∆ log(t) ) * d L φ * (α) dt. This is nothing else but a normal form for generalized contact bundles. This can be pushed more forward to prove a local splitting of generalized bundles, but this has already be done in [10] with similar techniques. and hence we can compute