Abstract
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.
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1 Introduction
Since the work of Weinstein [17], in which he proved his famous local splitting theorem for Poisson manifolds, many works appeared concerning different viewpoints on the proof and even giving more general statements, namely normal form theorems. Frejlich and Marcut proved a normal form theorem around Poisson (cosymplectic) transversals of Poisson manifolds in [7]. In [6] they used the techniques of Dual Pairs to prove a similar statement for Dirac structures. Finally, there is a unified approach by Bursztyn, Lima and Meinrenken in [3] to prove normal forms for Poisson related structures.
Jacobi geometry was introduced by Kirillov in [10] as local Lie algebras and independently by Lichnerowicz [11]. They have a deep connection to Poisson geometry, since every Poisson structure defines a Jacobi bracket. Moreover, every Jacobi structure induces a Poisson structure on a manifold of one dimension higher, this is known as the symplectization or homogenization, see [2] and its references for a detailed discussion. In Jacobi geometry there is also a local splitting theorem available, which was proven by Dazord, Lichnerowicz and Marle in [5]. Nevertheless, after this work, the parallels in the work of Poisson and Jacobi geometry stopped, at least in the context of local structure. The aim of this paper is to fill these gaps, prove normal form theorems for Jacobi bundles and give a more intrinsic proof of the splitting theorems. To do so, we will choose the approach of [3] and start with socalled Dirac–Jacobi bundles which generalize the notion of Jacobi structures.
Dirac–Jacobi bundles were introduced in [14] by Vitagliano and are a slight generalization of Wade’s \(\mathcal {E}^1(M)\)Dirac structures (see [16]). Moreover, these bundles are a Dirac theoretic generalizations of Jacobi bundles, as usual Dirac structures are for Poisson manifolds.
We want to stress that the methods used in this note are also suitable for proving splittings for involutive fat anchored vector bundles \((E,L\rightarrow M,\rho )\), i.e. a vector bundle \(E\rightarrow M\), a line bundle \(L\rightarrow M\) and a bundle map \(\rho :E\rightarrow DL\) where DL is the Atiyah algebroid of L, such that \(\Gamma ^{\infty }(\rho (E))\) is closed with respect to the bracket, as well as Jacobialgebroids (see [13]). We do not want to treat that in detail since every involutive fat anchored vector bundle is in particular, by composing \(\rho \) with the anchor of DL, an involutive anchored vector bundle and can be treated with the methods in [3]. The same holds true for Jacobialgebroids.
This note is organized as follows: we recall the necessary structures in order to define the setting for Dirac–Jacobi structures, the omniLie algebroid of a line bundle (see [4]) in Sect. 2. Afterwards, we introduce the notion of Eulerlike derivations, which are the crucial ingredient for the proofs of the main theorems. After this we are able to provide a normal form theorem for Dirac–Jacobi bundles, which is the main part of Sect. 4. In the following section, we want to apply this normal form theorem to the special case of Jacobi bundles, which allows us to state and prove two normal form theorems for Jacobi bundles, which allow us to give a different proof of the splitting theorems of Jacobi pairs, first provided in [5]. Moreover, we can apply these theorems to provide a splitting theorem for homogeneous Poisson structures around points where the homogeneity does not vanish, which was also done in [5]. Note that in [5] the proof works exactly the other way around: they prove a local splitting of homogeneous Poisson structures and use it to prove the splitting of Jacobi structures.
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 Dercomplex with applications to contact and Jacobi geometry. Afterwards, we introduce the arena for the socalled Dirac–Jacobi bundles, the omniLie algebroids, and give a quick reminder of Dirac–Jacobi bundles together with the properties we will need afterwards.
2.1 Notation and a brief reminder on Jacobi geometry
The notions of Atiyah algebroid of a vector bundle and the associated Dercomplex 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 [14] and its references. Nevertheless, the notion of omniLie algebroids was first defined in [4], in order to study Lie algebroids and local Lie algebra structures on vector bundles.
For a vector bundle \(E\rightarrow M\), we denote its gauge or Atiyah algebroid by \(DE\rightarrow M\) and by \(\sigma :DE\rightarrow TM\) 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, for a regular \(\Phi :E\rightarrow E'\), we denote by
the corresponding Lie algebroid morphism. We are mostly dealing with line bundles \(L\rightarrow M\) for which we have the identity \(DL=(J^1L)^{*}\mathbin {\otimes _{\scriptscriptstyle {}}}L\), where \(J^1L\) is the first jet bundle and sections of DL are canonically identified with the first order differential operators \(\text {DiffOp}^{1}(L,L)\) and are called derivations of L. The philosophy is that DL is supposed to replace the tangent bundle in the category of line bundles, and hence sections of it are playing the role of vector fields. The (local) flow of a derivation \(\Delta \in \Gamma ^{\infty }(DL)\) is defined as a oneparameter group of automorphisms \(\Phi _t\in {\text {Aut}}(L)\) covering \(\phi _t\in \text {Diffeo}(M)\) given as the unique solution of the ODE
for all \(\lambda \in \Gamma ^{\infty }(L)\), where \(\Phi _t^*\Delta =D\Phi _t^{1} (\Delta )\circ \phi _t\) and \(\Phi _t^*\lambda (p)=\Phi _t^{1}\lambda (\phi _t(p))\).
The gauge algebroid \(DL\rightarrow M\) has a (tautological) Lie algebroid representation on L. The corresponding Lie algebroid complex with values in L is denoted by
Elements of \(\Omega _L^\bullet (M)\) are referred to as Atiyah forms. Since there is an insertion
we can also define a Lie derivative in the direction of \(\Delta \in \Gamma ^{\infty }(DL)\) by
Note that \(\mathscr {L}_\mathbb {1}={\text {id}}_{\Omega _L^\bullet (M)}\), which can be computed directly. This means nothing else but \(\iota _\mathbb {1}\) is a contracting homotopy of the differential \(\mathop {}\!\textrm{d}_L\).
We briefly discuss Jacobi brackets in this setting. A Jacobi bracket is a local Lie algebra structure on the smooth sections of a line bundle \(L\rightarrow M\), i.e. a Lie bracket \(\{,\}:\Gamma ^{\infty }(L)\times \Gamma ^{\infty }(L)\rightarrow \Gamma ^{\infty }(L)\), such that
for all \(\lambda \in \Gamma ^{\infty }(L)\).
Remark 2.1
Let \(\{,\}\) be a Jacobi bracket on a line bundle \(L\rightarrow M\). Then there is a unique tensor, called the Jacobi tensor, \(J\in \Gamma ^{\infty }(\Lambda ^2(J^1\,L)^*\otimes L )\), such that
for \(\lambda ,\mu \in \Gamma ^{\infty }(L)\). Conversely, every Lvalued 2form J on \(J^1 L\) defines a skewsymmetric 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 GerstenhaberJacobi bracket
such that the Jacobi identity of \(\{,\}\) is equivalent to \([J,J]=0\) see [13, Chapter 1.3] for a detailed discussion. Finally, a Jacobi tensor defines a map \(J^\sharp :J^1\,L\rightarrow (J^1\,L)^*\otimes L=DL\).
When L is the trivial line bundle, than the notion of Jacobi bracket boils down to that of a Jacobi pair.
Remark 2.2
(Trivial Line bundle) Let \(\mathbb {R}_M\rightarrow M\) be the trivial line bundle and let J be a Jacobi tensor on it. Let us denote by \(1_M\in \Gamma ^{\infty }(\mathbb {R}_M)\) the canonical global section. Using the canonical connection
we can see that \(DL\cong TM\oplus \mathbb {R}_M\) and hence
With this splitting, we see that
for some \((\Lambda ,E)\in \Gamma ^{\infty }(\Lambda ^2 TM\oplus TM)\). The Jacobi identity is equivalent to \(\llbracket {\Lambda ,\Lambda }\rrbracket _\text {s}+2E\wedge \Lambda =0\) and \(\mathscr {L}_E\Lambda =0\), where \(\llbracket {\,\cdot \,,\,\cdot \,}\rrbracket _\text{ s }\) is the Schouten bracket. The pair \((\Lambda ,E)\) is often referred to as Jacobi pair and in fact, Jacobi structures have been introduced in [11] as Jacobi pairs and the splitting theorem in [5] is proven for Jacobi pairs. Moreover, if we denote by \( \mathbb {1}^*\in \Gamma ^{\infty }(J^1\mathbb {R}_M)\) the canonical section then we can write any \(\psi \in J^1\mathbb {R}_M\) as \(\psi =\alpha +r\mathbb {1}^*\in \Gamma ^{\infty }(J^1\mathbb {R}_M) \), for some \(\alpha \in T^*M\) and \(r\in \mathbb {R}\). We obtain
A more detailed discussion about Jacobi structures on trivial line bundles can be found in [13, Chapter 2]. In a similar way, we can see that \(\Omega _L(M)^\bullet =\Gamma ^{\infty }(\Lambda ^\bullet (T^*M\oplus \mathbb {R}_M))=\Gamma ^{\infty }(\Lambda ^\bullet T^*M \oplus \mathbb {1}^*\wedge \Lambda ^{\bullet 1}T^*M)\). Here \(\mathbb {1}^*\) is the canonical section of \(\mathbb {R}_M\), moreover the differential \(\mathop {}\!\textrm{d}_{\mathbb {R}_M}\) is defined by the relations
2.2 The OmniLie algebroid of a line bundle and its automorphisms
The omniLie algebroid plays the same role in Dirac–Jacobi geometry as the generalized tangent bundle does in Dirac geometry. In fact, the parallels are evidently enormous. The following definitions and Lemmas are obvious adaptations of the case of Dirac structures, this is why we omit proofs. The following definitions and results in Dirac–Jacobi geometry can be found in [14].
Definition 2.3
Let \(L\rightarrow M\) be a line bundle. The vector bundle \(\mathbb {D} L:=DL\oplus J^1\,L\) together with

1.
the (Dorfmanlike) bracket
$$\begin{aligned}{}[\![(\Delta _1,\psi _1) ,(\Delta _2,\psi _2 )]\!] =([\Delta _1,\Delta _2],\mathscr {L}_{\Delta _1} \psi _2 \iota _{\Delta _2}\mathop {}\!\textrm{d}_L\psi _1) \end{aligned}$$ 
2.
the nondegenerate Lvalued pairing
$$\begin{aligned} \langle \langle (\Delta _1,\psi _1) ,(\Delta _2,\psi _2 )\rangle \rangle := \psi _1(\Delta _2)+\psi _2(\Delta _1) \end{aligned}$$ 
3.
the canonical projection \({\text {pr}}_D:\mathbb {D}L\rightarrow DL\)
is called the omniLie algebroid of \(L\rightarrow M\).
Remark 2.4
In principle, one can consider the omniLie algebroid together with a bracket twisted by an Atiyah 3form, as it is done in Dirac geometry. But in the case of line bundles the cohomology of the Dercomplex is trivial and hence we prefer not to include it since anyway, we can find an isomorphism of the two brackets.
We shall now introduce automorphisms of the omniLie algebroid, which mirrors the definition of automorphisms of the generalized tangent bundle.
Definition 2.5
Let \(L\rightarrow M\) be a line bundle. A pair \((F,\Phi )\in {\text {Aut}}(\mathbb {D}L)\times {\text {Aut}}(L)\) is called CourantJacobi automorphism, if

1.
\(D\Phi \circ {\text {pr}}_D={\text {pr}}_D\circ F\),

2.
\( \Phi ^*\langle \langle ,\rangle \rangle =\langle \langle F,F\rangle \rangle \) and

3.
\(F^* [\![,]\!]=[\![F^*,F^* ]\!]\).
The group of CourantJacobi automorphisms is denoted by \({\text {Aut}}_{CJ}(L)\).
For a line bundle \(L\rightarrow M\) and \(\Phi \in {\text {Aut}}(L)\), we define
which gives canonically an automorphism \(\mathbb {D}\Phi \in {\text {Aut}}(\mathbb {D}L)\), moreover \((\mathbb {D}\Phi ,\Phi )\) is a CourantJacobi automorphism. For a closed 2form \(B\in \Omega _L^2(M)\), we define
and see that \((\exp (B),{\text {id}})\in {\text {Aut}}_{CJ}(L)\). We can combine these two special kinds of morphisms together with the action of \({\text {Aut}}(L)\) on \(\mathbb {D}L\) and find the following
Lemma 2.6
Let \(L\rightarrow M\) be a line bundle. If we denote by \(Z^2_L(M)\) the closed 2forms, then
is an ismorphism of groups.
The group structure of the semidirect product \(Z^2_L(M)\rtimes {\text {Aut}}(L)\) is given by
for \((\Omega _i,\Phi _i)\in Z^2_L(M)\rtimes {\text {Aut}}(L)\) for \(i=1,2\). In a similar way, we can define infinitesimal automorphisms of the omniLie algebroid.
Definition 2.7
Let \(L\rightarrow M\) be line bundle. A pair \((D,\Delta )\in \Gamma ^{\infty }(D\mathbb {D}L)\times \Gamma ^{\infty }(DL)\) is called infinitesimal CourantJacobi automorphism, if

1.
\([\Delta ,{\text {pr}}_D(\varepsilon )]={\text {pr}}_D(D(\varepsilon ))\),

2.
\( \Delta \langle \langle \varepsilon ,\chi \rangle \rangle =\langle \langle D(\varepsilon ),\xi \rangle \rangle + \langle \langle \epsilon ,D(\chi )\rangle \rangle \) and

3.
\(D ([\![\varepsilon ,\chi ]\!]_H)=[\![D(\epsilon ),\chi ]\!]+[\![\epsilon ,D(\chi )]\!]\)
for all \(\varepsilon ,\chi \in \Gamma ^{\infty }(\mathbb {D}L)\). The Lie algebra of infinitesimal CourantJacobi automorphisms is denoted by \(\mathfrak {aut}_{CJ}(L)\).
Note that the flow of an infinitesimal CourantJacobi automorphism gives a (local) CourantJacobi automorphism, in this sense, we can see \(\mathfrak {aut}_{CJ}(L)\) as the Lie algebra of \({\text {Aut}}_{CJ}(L)\). Similarly to the autmorphism case, we have
Lemma 2.8
Let \(L\rightarrow M\) be a line bundle. Then
is an isomorphism of Lie algebras.
The Lie algebra structure on the semidirect product is given by
for \((\Omega _i,\Delta _i)\in Z^2_L(M)\rtimes \Gamma ^{\infty }(DL)\) and \(i=1,2\).
For every section \((\Delta ,\alpha )\in \Gamma ^{\infty }(\mathbb {D}L)\) the map \([\![(\Delta ,\alpha ),]\!]\) is an infinitesimal CourantJacobi automorphism, in fact it is realized in \(Z^2_L(M)\rtimes \Gamma ^{\infty }(DL)\) by
For later use, we want to talk about the flow of infinitesimal CourantJacobi automorphisms and want to compute them as explicit as possible.
Lemma 2.9
Let \(L\rightarrow M\) be line bundle. Let additionally \((B,\Delta )\in Z^2_L(M)\rtimes \Gamma ^{\infty }(DL)\). The flow of \(\mathfrak {i}(B,\Delta )\) is given by
Corollary 2.10
Let \(L\rightarrow M\) be a line bundle. For every \((\Delta ,\alpha )\in \Gamma ^{\infty }(\mathbb {D}L)\) the flow of \([\![(\Delta ,\alpha ),]\!]\) is given by
2.3 Dirac–Jacobi bundles
After having discussed the arena, we want to introduce the subbundles of interest: socalled Dirac–Jacobi Bundles. As the name suggests, they are the analogue of Dirac structures in the generalized tangent bundle. In fact, the definition is (up to some obvious replacements) the same.
Definition 2.11
Let \(L\rightarrow M\) be a line bundle. A subbundle \(\mathcal {L}\subseteq \mathbb {D}L\) is called a Dirac–Jacobi structure if

1.
\(\mathcal L\) is involutive with respect to \([\![,]\!]\),

2.
\(\mathcal L\) is maximally isotropic with respect to \(\langle \langle ,\rangle \rangle \).
Remark 2.12
Let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi structure on a line bundle \(L\rightarrow M\) and let \((F,\Phi )\) be a CourantJacobi automorphism, then
is a Dirac–Jacobi structure on \(L\rightarrow M\). Moreover, we denote a transformation with a closed 2form \(B\in \Omega _L^2(M)\) by
Proposition 2.13
Let \(L\rightarrow M \) be a line bundle and let \(J\in \Gamma ^{\infty }(\Lambda ^2 (J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi structure, then
is a Dirac–Jacobi structure. If a Dirac–Jacobi structure \(\mathcal {L}\subseteq \mathbb {D}L\) fulfills
then there is a unique Jacobi structure \(J\in \Gamma ^{\infty }(\Lambda ^2 (J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\), such that \(\mathcal {L}_J=\mathcal {L}\)
Proof
The result follows the same lines as the wellknown fact in Poisson geometry. \(\square \)
Another interesting example of Dirac–Jacobi bundles, which also plops up in Jacobi geometry, is
Definition 2.14
Let \(L\rightarrow M\) be a line bundle. A Dirac–Jacobi structure \(\mathcal {L}\subseteq \mathbb {D}L\) is called of homogeneous Poisson type, if
The name of these objects is justified by the following
Lemma 2.15
Let \(L\rightarrow M\) be a line bundle and let \(\mathcal {L}\subseteq \mathbb {D}L\) a Dirac–Jacobi structure of homogeneous Poisson type, then for every point \(p\in M\) there exists a local trivialization \(L_U=U\times \mathbb {R}\), a flat connection \(\nabla :TU\rightarrow DL_U\cong TU\oplus \mathbb {R}_U\) and a homogeneous Poisson structure \(\pi \in \Gamma ^{\infty }(\Lambda ^2TU)\) with homogeneity \(Z\in \Gamma ^{\infty }(TM)\), i.e. \(\mathscr {L}_Z\pi =\pi \), such that
where we use the inclusion \(T^{*}M\rightarrow J^1L\) by \(\alpha (\nabla _X)=\alpha (X)\) and \(\alpha (\mathbb {1})=0\).
Proof
Let \(p\in M\) and \(U\subseteq M\) be an open subset containing p, such that \(L_U\cong U\times \mathbb {R}\) with corresponding trivialization of the gauge algebroid \(DL_U=TU\oplus \mathbb {R}_U\) together with the trivialization \(J^1L_U=T^*U\oplus \mathbb {R}_U\). Let us denote by \(\nabla ^{\textrm{can}}:TU\ni X\mapsto (X,0) \in TU\oplus \mathbb {R}_U\) the canonical flat connection. In a possibly smaller neighborhood, denoted also by U, we find a nonvanishing section \(\Delta =(X,f)\in \Gamma ^{\infty }(\mathcal {L}\cap DL)\). We can distinguish two cases: the first is that \(f(p)\ne 0\), then we find a (possibly smaller) neighbourhood of p, such that f is nonvanishing, hence \((\frac{X}{f},1)=:(Z,1)\) spans \(\mathcal {L}\cap DL\) in that neighbourhood and hence \(DL_U=\langle (Z,1)\rangle \oplus \langle (X,0)\rangle _{X\in TU}\). Moreover, we have the short exact sequence
and hence \({\text {rank}}({\text {pr}}_{J^1L}\mathcal {L})=\dim (M)\), with the canonical inclusion \(T^*U\ni \alpha \rightarrow (\alpha ,\alpha (Z))\in {\text {pr}}_{J^1\,L}\mathcal {L} \). Summarizing, for all \(\alpha \in T^*U\) there exists a unique \(X\in TU\), such that
Because of the maximal isotropy of \(\mathcal {L}\), this assignment comes from a bivector field \(\pi \in \Gamma ^{\infty }(\Lambda ^2TU)\), i.e. \(X=\pi ^\sharp (\alpha )\). Finally, we can write
which is nothing else but
The claim follows by using the flatness of \(\nabla ^\textrm{can}\) and the involutivity of \(\mathcal {L}\).
Now we have to treat the case \(f(p)=0\). Since \(\Delta =(X,f)\) is nonvanishing, we conclude that \(X(p)\ne 0\), hence there is a closed one form \(\beta \in \Gamma ^{\infty }(T^* U)\) such that \(\beta (X)=1\) around p. We define the flat connection
With this connection we see that \(\Delta =(f1)\mathbb {1}\nabla _X\) and since \(f(p)=0\), we have that \(f1\ne 0\) in a whole neighbourhood of p and hence we choose \(\Delta '=\frac{1}{f1}\Delta \) as a generating section of \(\mathcal {L}\cap DL\) around p. We can now repeat the same argument as for the case \(f(p)\ne 0\) by using the connection \(\nabla \) instead of \(\nabla ^\textrm{can}\), since \(\Delta '= \mathbb {1}\nabla _Z\) for \(Z=\frac{1}{f1} X\). \(\square \)
In the category of Dirac–Jacobi bundles there are not just automorphisms of the omniLie algebroid as morphisms, one of the possibilities is to include socalled backwards transformations as in the Dirac geometry case.
Definition 2.16
Let \(L_i\rightarrow M_i\) for \(i=1,2\) be two line bundles and let \(\Phi :L_1\rightarrow L_2\) be a regular line bundle morphism covering \(\phi :M_1\rightarrow M_2\). Let \(\mathcal {L}\subseteq \mathbb {D}L_2\) be a Dirac–Jacobi bundle. The (not necessarily smooth) family of vector spaces
is called backwards transformation of \(\mathcal {L}\).
Remark 2.17
Let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle on a line bundle \(L\rightarrow M\) and let \(\Phi \in {\text {Aut}}(L)\) an automorphism, then
The backwards transformation of a Dirac–Jacobi bundle need not to be Dirac–Jacobi any more, but there are sufficient conditions on the subbundle \(\mathcal {L}\) and the line bundle morphism \(\Phi \) which can be seen, e.g. in [14]:
Theorem 2.18
Let \(\Phi :L_1\rightarrow L_2\) be a regular line bundle morphism over \(\phi :M_1\rightarrow M_2\) and let \(\mathcal {L}\in \mathbb {D}L_2 \) be a Dirac–Jacobi bundle. If \(\ker D\Phi ^*\cap \phi ^*\mathcal {L}\) has constant rank, then \(\mathfrak {B}_\Phi (\mathcal {L})\) is a Dirac–Jacobi bundle, where \(\phi ^*\mathcal {L}\) defines the pullback bundle of \(\mathcal {L}\rightarrow M_2\) seen as a vector bundle.
Proof
The proof can be found in [14, Proposition 8.4]. \(\square \)
A Dirac–Jacobi bundle \(\mathcal {L}\subseteq \mathbb {D}L\) has, as Dirac structures, a canonical involutive and integrable (singular) distribution obtained by
This distribution generally has two different kind of leaves, whereas in Dirac geometry all of the leaves are presymplectic. This mirrors the fact that Jacobi bundles may have two types of leaves: contact and locally conformal symplectic.
The details about this facts can be found in [14], we just want to briefly recall some properties.
Lemma 2.19
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(S\hookrightarrow M\) be a leaf of the characteristic distribution \(\sigma ({\text {pr}}_D (\mathcal {L}))\subseteq TM\). Then the two cases

1.
\(\mathbb {1}_p\in {\text {pr}}_D\mathcal {L}\) for some point \(p\in S\) and

2.
\(\mathbb {1}_p\notin {\text {pr}}_D\mathcal {L}\) for some point \(p\in S\)
are mutually exclusive.
This motivates the following definition.
Definition 2.20
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(\iota :S\hookrightarrow M\) be a leaf. Then

1.
S is called precontact, if \(\mathbb {1}\in \Gamma ^{\infty }({\text {pr}}_D\mathcal {L}\big \vert _{S})\) and

2.
S is called locally conformal presymplectic, if it is not precontact.
To justify this we have the following
Corollary 2.21
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq DL\) be a Dirac–Jacobi bundle and let \(\iota :S\hookrightarrow M\) be a leaf of its characteristic foliation and denote by \(I:L_S\rightarrow L\) the embedding of L restricted to S. If S is a

1.
precontact leaf, then there exists a \(\omega \in \Omega ^2_{L_S}(S)\), such that
$$\begin{aligned} \mathfrak {B}_I(\mathcal {L})=\{(\Delta ,\iota _\Delta \omega )\in \mathbb {D}L_S \  \ \Delta \in DL_S\} \end{aligned}$$and \(\mathop {}\!\textrm{d}_{L_S}\omega =0\).

2.
locally conformal presymplectic leaf, then there exists a flat connection \(\nabla :TS\rightarrow DL_S\) and an \(L_S\)valued 2form \(\omega \in \Gamma ^{\infty }(\Lambda ^2 T^*S\mathbin {\otimes _{\scriptscriptstyle {}}}L_S)\), such that
$$\begin{aligned} \mathfrak {B}_I(\mathcal {L})= \{(\nabla _X,\sigma ^*(\iota _X\omega )+\alpha ) \in \mathbb {D}L_S\  \ X\in TS\text { and } \alpha \in \textrm{Ann}({\text {im}}(\nabla ))\} \end{aligned}$$and \(\mathop {}\!\textrm{d}^\nabla \omega =0\).
3 Submanifolds and Eulerlike vector fields
In this subsection we want to discuss Eulerlike 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, to the manifold being the 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 Eulerlike 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 maps, which can be found in [3] and describe afterwards the notion of Eulerlike vector fields and extend this notion to derivations of a line bundle. As a final remark, we want to stress that all of the used submanifolds are actually embedded submanifolds.
3.1 Normal bundles and tubular neighborhoods
For a pair of manifolds (M, N), i.e. a submanifold \(N\hookrightarrow M\), we denote
the normal bundle. If the ambient space is clear, we will just write \(\nu _N\) instead. Given a map of pairs
i.e. a map \(\Phi :M\rightarrow M\), such that \(\Phi (N)\subseteq N'\), we denote by
the induced map on the normal bundle. For a vector field X on M tangent to N, we have that the flow \(\Phi ^X_t\) is a map of pairs from (M, N) to itself. Hence we define
Moreover, for a vector bundle \(E\rightarrow M\) and \(\sigma \in \Gamma ^{\infty }(E)\), such that \(\sigma \big \vert _{N}=0\) for a submanifold \(N\hookrightarrow M\), we denote by
the map which is \(\nu (\sigma )\), for \(\sigma \) seen as a map \(\sigma :(M,N)\rightarrow (E,M)\), followed by the canonical identification \(\nu (E,M)=E\), given by
Before we prove the next results, we want to find a useful description of \(C_E^{1}\). Let us therefore consider a curve \(\gamma :I\rightarrow E\) for an open interval I containing 0, such that \(\gamma (0)=0_p\) for \(p\in M\), then one can prove in local coordinates
Proposition 3.1
Let \(E_i\rightarrow M_i\) be vector bundles for \(i=1,2\) and let \(\Phi :E_1\rightarrow E_2\) be a vector bundle morphism. Then, for \(\Phi :(E_1,M_1)\rightarrow (E_2,M_2)\),
Proof
Let \(v_p\in E_1\), then
\(\square \)
Proposition 3.2
Let \(E_i\rightarrow M\) be vector bundles for \(i=1,2\) and let \(\Phi :E_1\rightarrow E_2\) be a vector bundle morphism covering the identity. Then, for every section \(\sigma \in \Gamma ^{\infty }(E_1)\), such that \(\sigma \big \vert _{N}=0\) for some submanifold \(N\hookrightarrow M\),
holds.
Proof
We consider the map \(\Phi (\sigma ):(M,N)\rightarrow (E_2,M)\), then we have
and the claim follows if we restrict these maps. \(\square \)
Proposition 3.3
Let (M, N) be a pair of manifolds and let \(X\in \Gamma ^{\infty }(TM)\) be such that \(X\big \vert _{N}=0\). Then
for a unique \(D_X\in \Gamma ^{\infty }({\text {End}}(TM\big \vert _{N}))\), moreover \(TN \subseteq \ker (D_X)\) and
commutes.
Proof
Since \(X\big \vert _{N}=0\) its flow fixes all elements of N. This means that
for all \(t\in \mathbb {R}\) and \(p\in N\). Moreover, it fulfills the property,
and \(T\Phi ^X_0={\text {id}}\) and hence the claim follows. \(\square \)
Definition 3.4
Let (M, N) be a pair of manifolds. A tubular neighbourhood of N is an open subset \(U\subseteq M\) containing N together with a diffeomorphism
such that \(\psi \big \vert _{N}:N\rightarrow N\) is the identity and for \(\psi :(\nu _N,N)\rightarrow (M,N)\) the map
is inverse of \(C_{\nu _N}:\nu _N\rightarrow \nu (\nu _N,N)\).
3.2 Eulerlike vector fields and derivations
In this part, we basically recall just the notion of Eulerlike vector fields from [3] and extend this notion to derivations of a line bundle.
Definition 3.5
Let (M, N) be a pair of manifolds. A vector field \(X\in \Gamma ^{\infty }(TM)\) is called Eulerlike, if

1.
\(X\big \vert _{N}=0\),

2.
X has complete flow,

3.
\(T\nu (X)=\mathcal {E}\),
where \(\mathcal {E}\) is the Euler vector field on \(\nu _N\rightarrow N\).
Proposition 3.6
Let (M, N) be a pair of manifolds, then there exists an Eulerlike vector field.
Proof
The proof can be found in [3]. \(\square \)
Lemma 3.7
Let M be a manifold, \(N\hookrightarrow M\) a submanifold and \(X\in \Gamma ^{\infty }(TM)\) be a Eulerlike vector field. Then there exists a unique tubular neighbourhood embedding
such that \(\psi ^* X=\mathcal {E}\).
Proof
The proof can be found in [3]. \(\square \)
Proposition 3.8
Let (M, N) be a pair of manifolds and let \(X\in \Gamma ^{\infty }(TM)\) be a complete vector field, such that \(X\big \vert _{N}=0\). Then X is Eulerlike if and only if \(\mathop {}\!\textrm{d}^N X\) followed by the projection \(TM\big \vert _{N}\rightarrow \nu _N\) is identity.
Proof
The proof can be found in [3]. \(\square \)
Note that for a pair of manifolds (M, N) and an Euler like vector field \(X\in \Gamma ^{\infty }(TM)\), the set
is an open subset in M containing N, such that that the action of \(\Phi _t^X\) restricts to this set. Moreover, for a tubular neighbourhood \(\psi :\nu _N \rightarrow U\), such that \(\psi ^*X=\mathcal {E}\), we have that
The proof of this statement and a more detailed discussion can be found in [3]. Let us denote by \(\lambda _s=\Phi ^X_{\log (s)}\big \vert _{U}\). We obtain, that \(\lambda _s\) is smooth for all \(s\in \mathbb {R}_0^+\). Moreover, we have that
where we denote by \(\kappa _s:\nu _N\rightarrow \nu _N\) the map \([X_p]\mapsto [s X_p]\). Note that \(\kappa _0:\nu _N\rightarrow N\) coincides with the bundle projection, to be more precise \(\kappa _0=j\circ {\text {pr}}_\nu \), where \({\text {pr}}_\nu \) is the bundle projection and \(j:N\rightarrow \nu _N\) the canonical inclusion.
Let us add now the line bundle case
Definition 3.9
Let \(L\rightarrow M\) be a line bundle and \(N\hookrightarrow M\) be a submanifold. A derivation \(\Delta \in \Gamma ^{\infty }(DL)\) is called Eulerlike, if

1.
\(\Delta \big \vert _{N}=0\),

2.
\(\sigma (\Delta )\) is an Eulerlike vector field.
This definition turns out to be the correct one for our purposes, since we can prove basically all results, which are available for Eulerlike vector fields. Let us start collecting them.
Proposition 3.10
Let \(L\rightarrow M\) be a line bundle and let \(\Delta \in \Gamma ^{\infty }(DL)\) be an Eulerlike derivation with respect to \(N\hookrightarrow M\), then the (complete) flow \(\Phi ^\Delta _t\in {\text {Aut}}(L)\) of \(\Delta \) induces the map
which, restricted to \(U=\{p\in M\  \ \lim _{t\rightarrow \infty } \Phi _t^{\sigma (X)}(p) \text { exists and lies in } N\}\), can be 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 \(\psi :\nu _N\rightarrow U\), such that \(\psi ^*\sigma (X)=\mathcal {E}\). \(\square \)
Definition 3.11
Let \(L\rightarrow M\) be a line bundle and \(N\hookrightarrow M\) be a submanifold. A fat tubular neighbourhood is a regular line bundle morphism
where the line bundle \(L_\nu \) is given by the pullback
covering a tubular neighborhood \(\psi :\nu _N\rightarrow U\), such that \(\Psi \big \vert _{N}:L_N\rightarrow L_N\) is the identity.
Lemma 3.12
Let \(L\rightarrow M\) be a line bundle, let \(N\hookrightarrow M\) be a submanifold and let \(\psi :\nu _N\rightarrow U\) be a tubular neighborhood. Then there exists a fat tubular neighbourhood covering \(\psi \).
Proof
The proof can be found in [13, Chapter 3]. \(\square \)
For a line bundle \(L\rightarrow N\) and a vector bundle \(E\rightarrow N\) there is always a canonical Derivation \(\Delta _{\mathcal {E}}\in \Gamma ^{\infty }(DL_E)\), such that \(\sigma (\Delta _{\mathcal {E}})=\mathcal {E}\) constructed as follows: Consider the pullback \(L_E\) of L along \(p:E\rightarrow M\) as in the diagram
and the corresponding map \(DP:L_E\rightarrow L_N\). We have that canonically \(\ker (DP)\cong \textrm{Ver}(E):=\ker (Tp)\subseteq TE\), which induces a flat (partial) connection \(\nabla :\textrm{Ver}(E)\rightarrow DL_E\). Since the Euler vector field is canonically vertical, we can define \(\Delta _\mathcal {E}=\nabla _\mathcal {E}\).
Proposition 3.13
Let \(L\rightarrow N\) be a line bundle and let \(E\rightarrow N\) be a vector bundle. Then the flow \(\Phi _t\) of \(\Delta _\mathcal {E}\in \Gamma ^{\infty }(DL_E)\) is given by
for all \((v_p,l_p)\in L_E\).
Proof
Note that since \(\nabla _\mathcal {E}\) is in the kernel of DP, it is related to the 0 derivation on \(L\rightarrow M\) and hence we have for its flow
Since \(L_E=E\times _M L\), we have that
where \(\phi _t\) is the flow of the symbol of \(\nabla _\mathcal {E}\), which is by construction the Euler vector field and hence the claim follows. \(\square \)
Note that for the flow \(\Phi _t\) of the canonical Eulerlike derivation \(\Delta _\mathcal {E}\in \Gamma ^{\infty }(DL_E)\), 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 \(P:L_E\rightarrow L\) followed by the canonical inclusion \(J:L\rightarrow L_E\).
Lemma 3.14
Let \(L\rightarrow M\) be a line bundle, let \(N\hookrightarrow M\) be a submanifold and let \(\Delta \in \Gamma ^{\infty }(DL)\) be an Eulerlike derivation. Then there is a unique fat tubular neighbourhood \(\Psi :L_\nu \rightarrow L_U\) such that \(\Psi ^*\Delta = \Delta _{\mathcal {E}}\).
Proof
First, we want to prove existence. It is clear that any such \(\Psi \) has to cover the unique tubular neighbourhood \(\psi :\nu _N\rightarrow U\), such that \(\psi ^*\sigma (\Delta )=\mathcal {E}\). So let us choose a fat tubular neighbourhood \(\tilde{\Psi }:L_\nu \rightarrow L_U\) covering \(\psi \). We consider now \(\tilde{\Psi }^*\Delta \in \Gamma ^{\infty }(DL_\nu )\). We have \(\sigma (\tilde{\Psi }^*\Delta )=\psi ^*\sigma (\Delta )=\mathcal {E}\). Thus \(\sigma (\Delta _{\mathcal {E}})= \sigma (\tilde{\Psi }^*\Delta )\). Consider now the derivation \(\Box =\Delta _\mathcal {E}\tilde{\Psi }^*\Delta \) and
where \(\Phi _t \) is the flow of \(\Delta _\mathcal {E}\) and extend it smoothly to \(t=0\). Let us denote the flow of \(\Box _t\) by \(\phi _t\). Note that it is complete, since \(\sigma (\Box _t)=0\), indeed there is even a explicit formula for it, which we do not need. Note however, that \(\phi _t\in {\text {Aut}}(L_\nu )\) and it fixes every base point for all \(t\in \mathbb {R}\), since \(\sigma (\Box _t)=0\). Let us compute
Hence we see that \(\Delta _\mathcal {E}=\phi _0^*(\Delta _\mathcal {E})=\phi _1^*(\Delta _\mathcal {E}+\Box _1)= \phi _1^*(\tilde{\Psi }^*\Delta )\). Therefore, we have that the map \(\Psi =\tilde{\Psi }\circ \phi _1\) will do the job, since \(\phi _1\big \vert _{N}={\text {id}}\), because \(\Box \big \vert _{N}\) is trivial and hence also \(\Box _t\).
Let us now assume that we have \(\Psi _1,\Psi _2:L_\nu \rightarrow L_U\), such that \(\Psi _1^*\Delta =\Psi _2^*\Delta =\Delta _\mathcal {E}\). Note that since both have to cover the unique \(\psi :\nu _N\rightarrow U\), the target \(L_U\) is the same for both. Let us consider \(\Xi :=\Psi _1^{1}\circ \Psi _2 :L_\nu \rightarrow L_\nu \), which covers the identity, which implies that there is a nowhere vanishing function \(f\in \mathscr {C}^{\infty }(\nu _N)\), such that \(\Xi (l_p)=f(p)l_p\) for all \(l_p\in L_\nu \). Moreover, we have that \(\Xi \big \vert _{N}= {\text {id}}_{L_\nu }\big \vert _{N}\), hence \(f(0_n)=1\) for all \(n\in N\), and \(\Xi ^*\Delta _{\mathcal {E}}=\Delta _\mathcal {E}\). We consider now an arbitrary section \(\lambda \in \Gamma ^{\infty }(L_\nu )\) and compute
Hence \(\mathcal {E}(f)=0\), which means that \(f={\text {pr}}_\nu ^*g\) for some function \(g\in \mathscr {C}^{\infty }(N)\), but since \(1=f(0_n)=g(n)\) for all \(n\in N\), we have that \(\Xi ={\text {id}}_{L_\nu }\). \(\square \)
For a line bundle \(L\rightarrow M\), a submanifold N and an Eulerlike derivation \(\Delta \in \Gamma ^{\infty }(DL)\) with respect to N, 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 \(\Psi :L_\nu \rightarrow L_U\), such that \(\Psi ^*\Delta =\Delta _\mathcal {E}\). Moreover, we have that
for all \(s\ge 0\). Note that if we project this equation to the manifold level, this simply gives Eq. 3.2.
4 Normal forms of Dirac–Jacobi bundles
Using Eulerlike 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 Eulerlike derivations and afterwards, we are able to prove a normal form theorem and deduce some corolloraries from it.
Definition 4.1
Let \(L\rightarrow M\) be a line bundle and let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle. A submanifold \(N\hookrightarrow M\) is called transversal, if
Proposition 4.2
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(N\hookrightarrow M\) be a transversal. Then
is a Dirac–Jacobi bundle, where \(I:L_N\rightarrow L\) is the canonical inclusion.
Proof
This is an easy consequence of Theorem 2.18. \(\square \)
Lemma 4.3
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(\iota :N\hookrightarrow M\) be a transversal. The backwards transformation \(\mathfrak {B}_{I}(\mathfrak {L})\) is canonically isomorphic (as vector bundles) to the fibred product \(I^!\mathcal {L}\), which is defined by the diagram
Proof
We consider the linear map
which is welldefined since \(DI(\Delta _p)=\Box _{\iota (p)}\). We claim now that this map is injective, let us therefore consider \((\Delta _p,(\Box _{\iota (p)},\alpha _{\iota (p)}))\in \ker (\Xi )\). It follows immediately, that \(\Delta _p=0\) and hence \(\Box _{\iota (p)}=0\). If \((0,\alpha _{\iota (p)})\in \mathcal {L}\) then \(\alpha _{\iota _{p}}\in \textrm{Ann}({\text {pr}}_D L )\). Since \(DI^*\alpha _{\iota (p)} =0\), we have that \(\alpha _{\iota (p)}\in \textrm{Ann}(DL_N)\), hence \(\alpha _{\iota (p)}=0\) and the claim follows. For dimensional reasons we have that \(\Xi \) is an isomorphism. \(\square \)
Proposition 4.4
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(N\hookrightarrow M\) be a transversal. Then there exists \(\varepsilon \in \Gamma ^{\infty }(\mathcal {L})\), such that \(\varepsilon \big \vert _{N}=0\) and \({\text {pr}}_D(\varepsilon )\) is Eulerlike.
Proof
We consider the exact sequence
where the first arrow is defined by the identification \(\mathfrak {B}_I(\mathcal {L})\cong I^!\mathcal {L}\) from Lemma 4.3 followed by the canonical map \(I^!\mathcal {L}\rightarrow \mathcal {L}\). The second arrow is the projection \({\text {pr}}_D:\mathcal {L}\big \vert _{N}\rightarrow DL \big \vert _{N}\) followed by the symbol map \( \sigma :DL\big \vert _{N}\rightarrow TM\big \vert _{N}\)and finally followed by the the projection to the normal bundle \({\text {pr}}_{\nu _N}:TM\big \vert _{N}\rightarrow \nu _N\). This map is surjective, because N is a transversal. Let us choose a section \(\varepsilon \in \Gamma ^{\infty }(\mathcal {L})\) with \(\varepsilon \big \vert _{N}=0\), such that \(\mathop {}\!\textrm{d}^N\varepsilon :\nu _N\rightarrow \mathcal {L}\big \vert _{N}\) defines a splitting of the sequence. We consider now
and see that if \(\mathop {}\!\textrm{d}^N\varepsilon \) splits the above sequence then \((\sigma \circ {\text {pr}}_D)\mathop {}\!\textrm{d}^N\varepsilon \) splits the lower sequence. Using Proposition 3.2, we see that \((\sigma \circ {\text {pr}}_D)\mathop {}\!\textrm{d}^N\varepsilon =\mathop {}\!\textrm{d}^N((\sigma \circ {\text {pr}}_D)(\varepsilon ))\) and by Proposition 3.8, we see that \(T\nu (\sigma \circ {\text {pr}}_D)(\varepsilon )=\mathcal {E}\). Multiplying \(\varepsilon \) by a suitable bump function we may arrange that \((\sigma \circ {\text {pr}}_D)(\varepsilon )\) is complete and hence an Eulerlike vector field. By definition \({\text {pr}}_D(\varepsilon )\) is hence an Eulerlike derivation. \(\square \)
Let us fix now a Dirac–Jacobi structure \(\mathcal {L}\subseteq \mathbb {D}L\) for a line bundle \(L\rightarrow M\). Let us also consider a transversal \(\iota :N\hookrightarrow M \) and a section \(\varepsilon =(\Delta ,\alpha ) \in \Gamma ^{\infty }(\mathcal {L})\), such that \(\varepsilon \big \vert _{N}=0\) and \(\Delta \) is an Eulerlike derivation. Due to the Lemma 3.14, we find a unique fat tubular neighbourhood
such that \(\Psi ^*\Delta =\Delta _\mathcal {E}\). With this we have now two ways to construct a Dirac–Jacobi bundle on \(L_\nu \rightarrow \nu _N\), namely we can take the backwards transformation \(\mathfrak {B}_\Psi (\mathcal L_U)\) and, if we consider
we can take the backwards transformation \(\mathfrak {B}_{I\circ P}(\mathcal {L})=\mathfrak {B}_{P}(\mathfrak {B}_I(\mathcal {L}))\). The aim is now to compare these two structures. Let us therefore consider the flow of \([\![(\Delta ,\alpha ),]\!]\), which is given by
where \(\Phi ^\Delta _t\) is the flow of \(\Delta \) and \(\gamma _t=\int _0^t (\Phi _{\tau }^\Delta )^* \mathop {}\!\textrm{d}_L\alpha \mathop {}\!\textrm{d}\tau \) by Corollary 2.10. For sure we have that the action of \(( \gamma _t,\Phi ^\Delta _t)\) preserves \(\mathcal {L}\), which is explicitly
This leads us directly to the following theorem.
Theorem 4.5
(Normal form for Dirac–Jacobi bundles) Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi bundle and let \(N\hookrightarrow M\) be a transversal. Then there exists an open neighbourhood \(U\subseteq M\) of N and a fat tubular neighbourhood \(\Psi :L_\nu \rightarrow L_U\), such that
for an \(\omega \in \Omega ^2_{L_\nu }(\nu _N)\).
Proof
According to Proposition 4.4, we can find \((\Delta ,\alpha )\in \Gamma ^{\infty }(\mathcal {L})\), such that \(\Delta \) is Eulerlike. Then there is a unique fat tubular neighbourhood \(\Psi :L_\nu \rightarrow L_U\), such that \(\Psi ^*\Delta =\Delta _\mathcal {E}\), due to Lemma 3.14. Let us denote by \( ( \gamma _t,\Phi ^\Delta _t)\in Z^2_L(M)\rtimes {\text {Aut}}(L)\) the flow of \([\![(\Delta ,\alpha ),]\!]\). We know that \(( \gamma _t,\Phi ^\Delta _t)\) preserves \(\mathcal {L}\) for all \(t\in \mathbb {R}\) and so will \((\gamma _{\log (s)},\Phi ^\Delta _{\log (s)})\) for all \(s>0\). Let us take a closer look at
and we obtain that it is smoothly extendable to \(s=0\). Let us denote its limit \(s\rightarrow 0\) by \(\omega '\) and \(\omega =\Psi ^*\omega '\). We have
which holds for all \(s\ge 0\). Here we used Remark 2.17 to identify CourantJacobi automorphisms with backwards transformations as well as Eq. 3.3. Hence, using that for the canonical inclusion \(J:L_N\rightarrow L_\nu \) we have that \(P_0=J\circ P\) and \(\Psi \circ J=I\), we get
for \(s=0\). \(\square \)
Note that this theorem says that, up to a Bfield, 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 kinds of leaves in Dirac–Jacobi geometry, see [14], so it is also possible to distinguish 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 extensively in the next section.
Definition 4.6
(Cosymplectic Transversal) Let \(L\rightarrow M\) be a line bundle and let \(\mathcal {L}\in \mathbb {D}L\) be a Dirac–Jacobi structure. A transversal \(\iota :N\hookrightarrow M\) is called cosymplectic, if
Remark 4.7
The term cosymplectic is already occupied in the literature: it is corank one Poisson manifold with some properties. Since there is no possible risk of confusion in this note, we use this term as short for transversal to a locally conformal presymplectic leaf. Note that a cosymplectic transversal always inherts a Dirac–Jacobi bundle coming from a Jacobi tensor by Proposition 2.13. So let us denote \(\mathcal {L}_{J_N}=\mathfrak {B}_I(\mathcal {L}_J)\subseteq \mathbb {D}L_N\).
These transversals naturally appear as minimal transversals to locally conformal presymplectic leaves, i.e. submanifolds of minimal dimension intersecting the leaf transversally, see [14] for a more detailed discussion.
Using the normal form theorem, we get in the case of cosymplectic transversals:
Corollary 4.8
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi structure and let \(\iota :N\hookrightarrow M\) be a minimal transversal to \(\mathcal {L}\) at a point \(p_0\) in a locally conformal presymplectic leaf, i.e. \(\sigma ({\text {pr}}_D(\mathcal {L}))\big \vert _{p_0}\oplus T_{p_0}N=T_{p_0}M\) and let \(\nu _N=V\times N\) be trivializable and trivialized. Then locally around \(p_0\):
where \(J_N\) is the Jacobi structure on the transversal and the canonical identification \(DL_{\nu _N}= TV\oplus DL_N\).
Proof
First, we note that a minimal transversal to a leaf is always a transversal as in Definition 4.1 and that for a minimal transversal N at a locally conformal presymplectic point \(p_0\), i.e. a point in a locally conformal presymplectic leaf, we have the equation
at \(p_0\) and hence in a whole neighborhood. The rest is an application of Theorem 4.5 and the usage of the splitting \(DL_{\nu _N}= TV\oplus DL_N\), since \(TV=\ker (Tp)\) using the discussion in front of Proposition 3.13. \(\square \)
The other kind of leaves of a Dirac–Jacobi structure are the socalled precontact leaves. Their minimal transversals possess the following structure:
Definition 4.9
(Cocontact Transversal) Let \(L\rightarrow M\) be a line bundle and let \(\mathcal {L}\in \mathbb {D}L\) be a Dirac–Jacobi structure. A transversal \(\iota :N\hookrightarrow M\) is called cocontact, if
Lemma 4.10
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi structure and let \(\iota :N\hookrightarrow M\) be a minimal transversal to \(\mathcal {L}\) at a precontact point \(p_0\). Then
holds in a neighbourhood of \(p_0\).
Proof
Recall that a minimal transversal at \(p_0\) is a transversal of minimal dimension, which in particular implies that
It is easy to see that
which follows because N is minimal and \(p_0\) is a precontact point, i.e. \(\mathbb {1}_{p_0}\in {\text {pr}}_D\mathcal {L}\). To be more precise, by using the precontact property of \(p_0\) and the minimality of N, we see \(({\text {pr}}_D\mathcal {L}\cap DL_N) \big \vert _{p_0}=\langle \mathbb {1}_{p_0}\rangle \) and hence there is \(\alpha \in J^1_{p_0}L\), such that \((\mathbb {1}_{p_0},\alpha ) \in \mathcal {L}\). Let us define \(\beta \in J^1_{p_0}L\) by
and
Then \(\beta \) is welldefined, since \({\text {pr}}_D\mathcal {L}\cap DL_N=\mathbb {1}\) and \(\alpha (\mathbb {1})=0\) and moreover \((0, \beta )\in \mathcal {L}\), since \(\langle \langle (0,\beta ),\mathcal {L}\rangle \rangle =0\) and \(\mathcal {L}\) is maximal isotropic. We consider now the element \((\mathbb {1}_{p_0},\alpha \beta )\in \mathcal {L}\), thus \((\mathbb {1}_{p_0},DI^*(\alpha \beta ))= (\mathbb {1}_{p_0},0)\in \mathfrak {B}_I(\mathcal {L})\). Moreover, since \({\text {pr}}_D\mathcal {L}\cap DL_N=\mathbb {1}\), we conclude \(DL_N\cap \mathfrak {B}_I(\mathcal {L})=\langle \mathbb {1}_{p_0}\rangle \) and hence \({\text {rank}}(DL_N\cap \mathfrak {B}_I(\mathcal {L}))\big \vert _{p_0}=1\).
Now we want to argue why this holds in a whole neighbourhood. Let us therefore consider the sum \(DL_N+ \mathfrak {B}_I(\mathcal {L})\subseteq \mathbb {D}L\) and a (local) section \(\alpha \in \Omega ^1_L(M)\) such that \(\alpha (\mathbb {1})\big \vert _{p_0} \ne 0\). Let \((0,\beta )\in \big (DL_N+ \mathfrak {B}_I(\mathcal {L})\big )\big \vert _{p_0}\cap \langle \alpha \rangle \big \vert _{p_0}\), then there exists \(\Delta \in D_{p_0}L\) such that \((\Delta ,\beta )\in \mathfrak {B}_I(\mathcal {L})\), but since \((\mathbb {1},0)\in \mathfrak {B}_I(\mathcal {L})\), using the isotropy of \(\mathfrak {B}_I(\mathcal {L})\), we have
but \(\beta =k\alpha \) for \(k\in \mathbb {R}\), we conclude \(k=0\) and thus \(\beta =0\) and therefore \(\big (DL_N+ \mathfrak {B}_I(\mathcal {L})\big )\big \vert _{p_0}\cap \langle \alpha \rangle \big \vert _{p_0}=\{0\}\). For dimensional reasons we conclude \(\mathbb {D}L\big \vert _{p_0}=(DL_N+ \mathfrak {B}_I(\mathcal {L}))\big \vert _{p_0}\oplus \langle \alpha \rangle \big \vert _{p_0}\). Therefore this equality holds in a whole neighbourhood of \(p_0\), so \({\text {rank}}(DL_N+ \mathfrak {B}_I(\mathcal {L}))=2n+1\) in this neighbourhood, which implies \({\text {rank}}(DL_N\cap \mathfrak {B}_I(\mathcal {L}))=1\) around \(p_0\). \(\square \)
Remark 4.11
Note that a cocontact transversal does not inherit a Jacobi structure, but the pulledback Dirac–Jacobi is of homogeneous Poisson type (see Definition 2.14).
Definition 4.12
Let \(L\rightarrow M\) be a line bundle and let \(\mathcal {L}\in \mathbb {D}L\) be a Dirac–Jacobi structure. A homogeneous cocontact transversal \(\iota :N\hookrightarrow M\) is a cocontact transversal together with a flat connection \(\nabla :TN\rightarrow DL_N\), such that
Remark 4.13
The definition of a homogeneous cocontact transversal seems a bit strange, since it includes a flat connection. This fact can be explained quite easily using the homogenization described in [14], which turns a Dirac–Jacobi structure on a line bundle \(L\rightarrow M\) into a Dirac structure on \(L^\times :=L^*\backslash \{0_M\}\) which is homogeneous (in the sense of [14]) with respect to the restricted Euler vector field \(\mathcal {E}\) on \(L^*\). The presymplectic leaves of this Dirac structure have the additional property that \(\mathcal {E}\) is either tangential to it at very point or nowhere tangential to it. If \(\mathcal {E}\) is tangential, then the leaf corresponds to a precontact leaf on the base M. Hence a minimal transversal N to it is nowhere tangential to the Euler vector field and defines therefore a horizontal bundle on \(L_{{\text {pr}}(N)}^*\) and hence a flat connection. On the other hand, given a minimal transversal to the projected leaf on M, we need a flat connection to lift it to a minimal transversal to the leaf on \(L^\times \). So a homogeneous cocontact transversal to a leaf of a Dirac–Jacobi structure is equivalent to a transversal of a leaf of its homogenization. Moreover, if we (locally) choose the horizontal bundle to be integrable, we have a flat connection. But we stress that in both cases, the homogenization and the line bundle point of view, this is an additional property one has to impose on the transversal.
Proposition 4.14
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi structure and let \(\iota :N\hookrightarrow M\) be a minimal transversal to \(\mathcal {L}\) at a precontact point \(p_0\). Then every flat connection \(\nabla \) locally gives to N the structure of a homogeneous cocontact transversal.
Proof
In the proof of Lemma 4.10, we have seen that
and hence for every flat connection \(\nabla \), we have that \({\text {im}}(\nabla )\big \vert _{p_0}\oplus (DL_N\cap \mathfrak {B}_I(\mathcal {L}))\big \vert _{p_0}=DL_N\) and hence this decomposition holds in a whole neighborhood of \(p_0\). \(\square \)
An immediate consequence is:
Corollary 4.15
Let \(L\rightarrow M\) be a line bundle, let \(\mathcal {L}\subseteq \mathbb {D}L\) be a Dirac–Jacobi structure and let \(\iota :N\hookrightarrow M\) be a homogeneous cocontact transversal with connection \(\nabla \). Then there exists a local trivialization of \(\nu \) such that
where we used the trivializations \(DL_\nu =T\nu \oplus \mathbb {R}_M\) and \(J^1L=T^*M\oplus \mathbb {R}_M\) corresponding to \(\nabla \) as well as \(L\big \vert _{U}\cong L_\nu \) and the homogeneous Poisson \((\pi _N,Z_N)\) structure on the transversal from Lemma 2.15.
This last two corollaries can be seen as the Jacobigeometric analogue of the results obtained by Blohmann in [1, Theorem 3.2, Corollary 3.6].
5 Normal forms and splitting Theorems of Jacobi bundles
As explained in Proposition 2.13, Jacobi bundles are a special kind of Dirac–Jacobi structures. In addition, we have that Jacobi isomorphism induces an isomorphism of the corresponding Dirac structures (this holds even for morphisms if one considers forward maps of Dirac–Jacobi structures which we will not explain here, see [14]). 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 Bfields. Nevertheless, we can keep track of them, if we make further assumptions on the transversals, namely cosymplectic and cocontact transversals.
5.1 Cosymplectic transversals of Jacobi structures
In this part, we are using the notion of cosymplectic transversals as explained in the previous section. The difference is now that in Jacobi geometry 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 \(L\rightarrow M\) be a line bundle, \(J\in \Gamma ^{\infty }(\Lambda ^2(J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi tensor with corresponding Dirac–Jacobi structure \(\mathcal {L}_J\in \mathbb {D}L\) and let \(\iota :N\hookrightarrow M\) be a cosymplectic transversal. Then
Proof
First we prove that \(J^\sharp \big \vert _{\textrm{Ann}(DL_N)}\) is injective. Let therefore \(\alpha \in \textrm{Ann}(DL_N)\) be such that \(J^\sharp (\alpha )=0\). Hence, for an arbitrary \(\beta \in J^1L\) we have that
Hence, \(\alpha =\textrm{Ann}(DL_N)\cap \textrm{Ann}({\text {im}}(J^\sharp ))=\textrm{Ann}(DL_N)\cap \textrm{Ann}({\text {pr}}_D\mathcal {L}_J)= \textrm{Ann}(DL_N+{\text {pr}}_D\mathcal {L}_J)=\{0\}\), and \(J^\sharp \big \vert _{\textrm{Ann}(DL_N)}\) is injective. Let \(\Delta \in DL_N\cap J^\sharp (\textrm{Ann}(DL_N))\), then there exists an \(\alpha \in \textrm{Ann}(DL_N)\), such that \(J^\sharp (\alpha )=\Delta \). Thus, we have that \((\Delta ,\alpha )\in \mathcal {L}_J\) and moreover \((\Delta ,DI^*\alpha )\in \mathfrak {B}_I(\mathcal {L}_J)\), but since \(\alpha \in \textrm{Ann}(DL_N)\), we have that \(DI^*\alpha =0\) and hence \(\Delta =0\), since N is cosymplectic. The claim follows counting dimensions. \(\square \)
Suppose that \(\iota :N\hookrightarrow M\) is a cosymplectic transversal, then we have that
is an isomorphism. Let us chose \(\alpha \in \Gamma ^{\infty }( J^1L)\), such that \(\alpha \big \vert _{N}=0\) and such that \(\mathop {}\!\textrm{d}^N\alpha :\nu _N\rightarrow \textrm{Ann}(DL_N)\subseteq J^1\,L\big \vert _{N}\) is a rightinverse to \({\text {pr}}_\nu \circ \sigma \circ J^\sharp \). We then have
and hence we have that \(T\nu (\sigma (J^\sharp (\alpha )))=\mathcal {E}\). Multiplying \(\alpha \) by a bumpfunction, which is 1 near N, we may arrange that \(\sigma (J^\sharp (\alpha ))\) is complete and hence \(J^\sharp (\alpha )\) is an Eulerlike derivation. By Theorem 4.5 and the definition of the Jacobi structure \(J_N\) on the transversal (see Remark 4.7), we have that
where \(\omega = \Psi ^*\int _{0}^{1}\frac{1}{t}(\Phi _{\log (t)}^\Delta )^*(\mathop {}\!\textrm{d}_L\alpha )\mathop {}\!\textrm{d}t\) and \(\Psi :L_\nu \rightarrow L_U\) is the unique tubular neighborhood, such that \(\Psi ^*(J^\sharp (\alpha ))=\Delta _\mathcal {E}\).
Proposition 5.2
The 2form \(\omega \in \Omega _{L_\nu }^2(\nu _N)\) evaluated along N has kernel \(DL_N\).
Proof
One can show, in local coordinates, that \(\mathop {}\!\textrm{d}^N\alpha ([\sigma (\Box )\big \vert _{N}])=\mathscr {L}_\Box \alpha \big \vert _{N}\) for all \(\Box \in \Gamma ^{\infty }(DL)\). Hence we have trivially \(\mathscr {L}_\Delta \alpha \big \vert _{N}=0\) for \(\Delta \in \Gamma ^{\infty }(DL)\), such that \(\Delta \big \vert _{N}\in \Gamma ^{\infty }(DL_N)\). Let now \(\Delta ,\Box \in \Gamma ^{\infty }(DL)\) be such that \(\Delta \big \vert _{N}\in \Gamma ^{\infty }(DL_N)\), then
where the last equality follows since \(\mathop {}\!\textrm{d}^N\alpha :\nu _N\rightarrow \textrm{Ann}(DL_N)\). Hence we have that \(\ker ((\mathop {}\!\textrm{d}_L\alpha )^\flat )\supseteq DL_N\), in particular this is true for \(\frac{1}{t}(\Phi _{\log (t)}^\Delta )^*(\mathop {}\!\textrm{d}_L\alpha )\), since \(\Phi _{\log (s)}\big \vert _{N}\) is a gauge transformation fixing \(DL_N\). Thus it is true also for \(\omega \), since \(D\Psi \big \vert _{DL_N}={\text {id}}\). The equality follows from the fact that \(\mathop {}\!\textrm{d}^N\alpha \) is chosen to be injective, since it has a leftinverse. \(\square \)
We want to describe the structure of \(\omega \) at N. Note that, for a cosymplectic transversal N, the normal bundle always comes together with a canonical symplectic (i.e. nondegenerate) \(L_N\)valued 2form \(\Theta \in \Gamma ^{\infty }(\Lambda ^2 \nu _N^*\mathbin {\otimes _{\scriptscriptstyle {}}}L_N)\) defined by
Lemma 5.3
The 2form \(\omega \in \Omega _{L_{\nu }}^2(\nu _N)\) coincides, restricted to \(\nu _N\subseteq DL_{\nu _N}\), with \(\Theta \).
Proof
Note that for a cosymplectic transversal, we have
with the canonical identification
where the last equality is the identification via the symbol \(\sigma :DL\rightarrow TM\). Moreover, we have
where we include \(\nu _N\) by the following map:
It is clear that \(D\Psi \) fixes \(DL_N\), since \(\Psi \big \vert _{N}:L_N\rightarrow L_N\) is the identity. We want to show that \(D\Psi (\nu _N)\subseteq J^\sharp (\textrm{Ann}(DL_N))\). Let \(v_n\in \nu _N\) and \(s\in \Gamma ^{\infty }(L_U)\), then
where we used Eqs. 3.2 and 3.3 as well as the definition of \(\Lambda _t\) and its relation to the flow of \(\Delta \), see Proposition 3.10. The equality
follows from the fact that \(\Lambda _0\big \vert _{N}={\text {id}}\).
But, by definition, we have that
hence \(D\Psi \circ \chi =\mathop {}\!\textrm{d}^N\Delta = J^\sharp \circ \mathop {}\!\textrm{d}^N\alpha \), but \(\alpha \) was chosen in such a way that \(\mathop {}\!\textrm{d}^N\alpha \) takes values in \(\textrm{Ann}(DL_N)\). Thus \(D\Psi \big \vert _{N}\) respects the splitting. Using this,
\(\ker (\omega ^\flat )\big \vert _{N}=DL_N\) and the definition of \(\Theta \), we see that they have to coincide at N. \(\square \)
This leads us to the normal form theorem for Jacobi manifolds.
Theorem 5.4
(Normal Form for Jacobi bundles I) Let \(L\rightarrow M\) be a line bundle, let J be a Jacobi structure and let \(N\rightarrow M\) be a cosymplectic transversal. For a closed 2form \(\omega \in \Omega _{L_\nu }^2(\nu _N)\), such that \(\ker (\omega ^\flat )\big \vert _{N}=DL_N\) and \(\omega \) coincides with \(\Theta \) at \(\nu _N\subseteq DL_\nu \), the Dirac–Jacobi structure
is the graph of a Jacobi structure near the zero section and there exists a fat tubular neighbourhood \(\Psi :L_\nu \rightarrow L_U\) which is a Jacobi map near the zero section.
Proof
The theorem is true for
due to Theorem 4.5 with \(\alpha \in \Gamma ^{\infty }(J^1\,L)\) chosen as in the discussion before, which ensures that \(\ker (\omega ^\flat )\big \vert _{N}=DL_N\) and \(\omega \) coincides with \(\Theta \) at \(\nu _N\subseteq DL_\nu \). Let \(\omega '\) be a second 2form fulfilling these requirements, then
is a (timedependent) 2form such that \(\sigma _0=0\) and moreover \(\sigma _t\big \vert _{N}=0\). Thus,
is a Jacobi structure near N, since \(\sigma _t\big \vert _{N}=0\) and hence the condition \(DL_\nu \cap \mathfrak {B}_P(\mathcal {L}_{J_N})^{\omega +\sigma _t} =\{0\}\) is fulfilled along N and thus in a open neighbourhood of N. This is equivalent to \(\mathfrak {B}_P(\mathcal {L}_{J_N})^{\omega +\sigma _t}\) being a Jacobi structure by Proposition 2.13. Now we can apply Appendix A to get the result. \(\square \)
An immediate consequence of this theorem is the Splitting Theorem for Jacobi manifolds around a locally conformal symplectic leaf, proven by Dazord, Lichnerowicz and Marle in [5].
Theorem 5.5
Let \(L\rightarrow M \) be a line bundle, let \(J\in \Gamma ^{\infty }(\Lambda ^2 (J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi tensor and let \(p_0\in M\) be a locally conformal symplectic point. Then there are a line bundle trivialization \(L_U\cong U\times \mathbb {R}\) around \(p_0\) and a minimal cosymplectic transversal \(N \hookrightarrow U\), such that \(U\cong U_{2q}\times N\) for an open subset \(0\in U_{2q}\subseteq \mathbb {R}^{2q}\) and the corresponding Jacobi pair \((\Lambda ,E)\) is transformed (via this isomorphism) to
where \((\Lambda _N, E_N)\) is the induced Jacobi structure on the transversal N and the canonical stuctures on the fiber are given by \((\pi _{\textrm{can}}, Z_{\textrm{can}})= (\frac{\partial }{\partial p_i}\wedge \frac{\partial }{\partial q^i},p_i\frac{\partial }{\partial p_i})\).
Proof
We can assume from the beginning that the line bundle is trivial, since otherwise we can trivialize around \(p_0\) and and restrict the line bundle to this open neighbourhood. Let us choose an minimal transversal N to the leaf S at \(p_0\) (in the sense, that \(S\times N = M\) holds locally around \(p_0\)). It is easy to see that
and hence we can restrict to an open neighburhood of \(p_0\), where this equality holds. This means that every minimal 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 \(\nu _N\cong T_{s_0}S \times N \cong \mathbb {R}^{2k}\times N\). Since the line bundle is trivial, we can identify \(\nu _N\) together with \(\Theta \) as a symplectic vector bundle, hence we find a possible smaller N and a vector bundle automorphism of \(\nu _N\), such that \(\Theta \) is the constant symplectic form. We can now choose
where (q, p) are the symplectic coordinates on \(\nu _N\rightarrow N\). This 2from is \(\mathop {}\!\textrm{d}_L\)closed and coincides with \(\Theta \) on N, moreover \(\ker (\omega ^\flat )\big \vert _{N}=DL_N\). Hence the requirements of Theorem 5.4 are fulfilled and the claim follows by an easy computation. \(\square \)
5.2 Cocontact transversals of Jacobi structures
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.
Lemma 5.6
Let \(L\rightarrow M\) be a line bundle, \(J\in \Gamma ^{\infty }(\Lambda ^2(J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi tensor with corresponding Dirac–Jacobi structure \(\mathcal {L}_J\in \mathbb {D}L\) and let \(\iota :N\hookrightarrow M\) be a homogeneous cocontact transversal with connection \(\nabla :TN\rightarrow DL_N\). Then
Proof
The proof follows the same lines as Lemma 5.1. \(\square \)
Now, as in the cosymplectic case, we pick an \(\alpha \in \Gamma ^{\infty }(J^1\,L)\), such that \(\alpha \big \vert _{N}=0\) and
defines a splitting of \(I^!\mathcal {L}\rightarrow \mathcal {L}\big \vert _{N}\rightarrow \nu _N\), i.e. \({\text {pr}}_\nu \circ \sigma \circ J^\sharp \circ \mathop {}\!\textrm{d}^N\alpha ={\text {id}}_{\nu _N}\). Hence we have that \(J^\sharp (\alpha )\), multiplied by a suitable bump function which is 1 close to N, is an Eulerlike derivation. By Theorem 4.5, we have that
where \(\omega = \Psi ^*\int _{0}^{1}\frac{1}{t}(\Phi _{\log (t)}^\Delta )^*(\mathop {}\!\textrm{d}_L\alpha )\mathop {}\!\textrm{d}t\) and \(\Psi :L_\nu \rightarrow L_U\) is the unique tubular neighbourhood, such that \(\Psi ^*(J^\sharp (\alpha ))=\Delta _\mathcal {E}\). We can prove, as before, the following
Proposition 5.7
The 2form \(\omega \in \Omega _{L_\nu }^2(\nu _N)\) restricted to N has kernel \({\text {im}}(\nabla )\).
Proof
This proof follows the same lines as the proof of Proposition 5.2. \(\square \)
As in the cosymplectic transversal case, we can define a skew symmetric 2form
by
It is easy to see that \(\Theta \) is nondegenerate. Moreover, we have
Lemma 5.8
The 2form \(\omega \in \Omega _{L_{\nu }}^2(\nu _N)\) coincides, restricted to \(\nu _N\oplus K \subseteq DL_{\nu _N}\), with \(\Theta \), where we denote \(K:=(DL_N\cap \mathfrak {B}_I(\mathcal {L}_J))\).
Proof
Using the ideas of the proof of Lemma 5.3, we can show that the fat tubular neighbourhood transports \(J^\sharp (\textrm{Ann}({\text {im}}(\nabla ))\) to \(\nu _N\oplus K\), hence the proof is copy and paste of this Lemma. \(\square \)
Theorem 5.9
(Normal Form for Jacobi bundles II) Let \(L\rightarrow M\) be a line bundle, let J be a Jacobi structure and let \(N\rightarrow M\) be a homogenous cocontact transversal with connection \(\nabla :TN\rightarrow DL_N\). For a closed 2form \(\omega \in \Omega _{L_\nu }^2(\nu _N)\), such that \(\ker (\omega ^\flat )\big \vert _{N}={\text {im}}(\nabla )\) and \(\omega \) coincides with \(\Theta \) at \(\nu _N\oplus (\mathfrak {B}_I (\mathcal {L}_J)\cap DL_N)\subseteq DL_\nu \), the Dirac–Jacobi structure
is the graph of a Jacobi structure near the zero section and there exists a fat tubular neighbourhood \(\Psi :L_\nu \rightarrow L_U\) which is a Jacobi map near the zero section.
Proof
The proof follows the lines of Theorem 5.4 with the obvious adaptions. \(\square \)
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 \(L\rightarrow M \) be a line bundle, let \(J\in \Gamma ^{\infty }(\Lambda ^2 (J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi tensor and let \(p_0\in M\) be a contact point. Then there are a line bundle trivialization \(L_U\cong U\times \mathbb {R}\) around \(p_0\) and a minimal homogeneous cocontact transversal \(N\hookrightarrow U\), such that \(U\cong U_{2q+1}\times N\) for an open subset \(0\in U_{2q+1}\subseteq \mathbb {R}^{2q+1}\) and the corresponding Jacobi pair \((\Lambda ,E)\) is transformed (via this isomorphism) to
where \((\pi _N, Z_N)\) is the induced homogeneous Poisson structure on the transversal N and the contact structure on the fiber is given by \((\Lambda _{\textrm{can}}, E_\textrm{can}) =((p_i\frac{\partial }{\partial u}+\frac{\partial }{\partial q^i}) \wedge \frac{\partial }{\partial q_i},\frac{\partial }{\partial u})\).
Proof
Let \(p_0\in M\) be a contact point and let \(N\subseteq M\) be a minimal transversal, such that
Note using Proposition 4.14,this means in particular, that N is a homogeneous cocontact transversal for a any flat connection, so we assume that the line bundle \(L\rightarrow M\) is trivial and choose the flat connection \(\nabla \) induced by this trivialization. In a possibly smaller neighbourhood, we can assume that also the normal bundle \(\nu _N=V\times N\rightarrow N\) is trivial. We want to show that there is a trivialization of \(\nu _N\), such that \(\Theta \) looks trivial, where we specialize on the way through the proof what we mean by trivial. Let us therefore denote by \(\lambda \) the local trivializing section of \(L_N\), thus we can write
for \(\Delta ,\Box \in \nu _N\oplus K\) with \(K=DL_N\cap \mathfrak {B}_I(\mathcal {L}_J)\subseteq DL_N\). Hence, we can find a (local) nowhere vanishing section of K of the form \(\mathbb {1} Z\) for a unique Z. Let us now restrict
since \(\nu _N\) is odd dimensional and \(\Theta \) is a skewsymmetric pairing, we can find a local nonvanishing \(X\in \Gamma ^{\infty }(\nu _N)\), such that \(\Theta (X,\cdot )=0\), moreover, since \(\Theta \) is nondegenerate, we can modify X by multiplying by a nonvanishing section in such a way that
It is now easy to see that the symplectic complement \(S:=\langle \mathbb {1}Z, X\rangle ^\perp \subseteq \nu _N\). Finally, we find a trivialization of S such that \(\Omega \big \vert _{S}\) is the trivial symplectic form with Darboux frame \(\{e_2,e_{k+2},\dots \}\). Hence, by extending this trivialization to \(\nu _N=V\times N\) by using the coordinate X as \(e_0\), we find that \(\{e_0,\mathbb {1}Z, e_1,e_{k+1},e_2,e_{k+2}, \dots \}\) is a Darboux frame of \(\Omega \) in this trivialization. with the decomposition \(DL_\nu =TV\oplus TN \oplus \mathbb {R}_{\nu _N}\) we can choose
which coincides with \(\Theta \) on \(\nu _N\oplus K\) and is \(\mathop {}\!\textrm{d}_L\)closed. By applying Theorem 5.9, since N together with \(\nabla \) is a homogeneous cocontact transversal, we find a Jacobi morphism
An easy computation shows that \(\mathfrak {B}_{P}(\mathcal {L}_N)^\omega \) is the graph of the Jacobi structure of the form in the theorem. \(\square \)
6 Application: splitting theorem for homogeneous Poisson structures
Using the homogenization 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 \((\pi ,Z)\) be a homogeneous Poisson structure on a manifold M and let \(p_0\in M\) be a point such that \(Z_{p_0}\ne 0\) such that \({\text {rank}}(\pi ^\sharp )=2k\). Then there exist an open neighbourhood U of \(p_0\), an open neighbourhood \(U_{2k}\) of \(0\in \mathbb {R}^{2k}\), a manifold N with a homogeneous Poisson structure \((\pi _N,Z_N)\) and a diffeomorphism \(\psi :U\rightarrow U_{2k}\times N\), such that
Additionally,

1.
if \(Z\in {\text {im}}(\pi ^\sharp )\), then \(\psi _*Z=p_i\frac{\partial }{\partial p_i} +\frac{\partial }{\partial p_k}+Z_N\).

2.
if \(Z\notin {\text {im}}(\pi ^\sharp )\), then \(\psi _*Z=p_i\frac{\partial }{\partial p_i}+Z_N\).
Proof
Note that since \(Z_{p_0}\ne 0\), we find coordinates \(\{u,x^1,\dots , x^q\}\) with \(p_0=(1,0,\dots ,0)\), such that \(Z=u\frac{\partial }{\partial u}\). In this chart, using \(\mathscr {L}_Z\pi =\pi \), we have
for unique \(\Lambda \in \Gamma ^{\infty }(\Lambda ^2 TM)\) and \(E\in \Gamma ^{\infty }(TM)\) which do not depend on u. It is easy to see, that we have
which means that \((\Lambda ,E)\) 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 \({\text {im}}(\Lambda ^\sharp )\big \vert _{p_0}\) and thus \(Z\in {\text {im}}(\pi ^\sharp )\), since the other case is exactly the same. Note that, we can apply Theorem 5.10: there exist coordinates \(\{x,q^i,p_i, y^j\}\) and a local nonvanishing function a (which is basically the line bundle trivialization), such that
where \(\Lambda _{\textrm{can}}\) and \(E_{\textrm{can}} \) are just depending on \(\{x,q^i,p_i\}\) and \((\phi _N,Z_N)\) is a homogeneous Poisson structure just depending on \(y^j\)coordinates.
If we apply the diffeomorphism \((u,x^1,\dots ,x^q)\mapsto (a\cdot u,x^1,\dots ,x^q)\), we have
A (quite) long and not very insightful computation shows that the diffeomorphism
where \(\Phi ^{Z_N}_t\) (resp. \(\Phi ^{E_{\textrm{can}}}_t\)) is the flow uf \(Z_N\) (resp. \(E_{\textrm{can}}\)), gives us
and with some obvious variations and renaming the coordinates we get the result. \(\square \)
This Application shows us that, even though 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.
7 Generalized contact bundles
In this last section, we want to drop a word about generalized contact bundles. They were introduced recently in [15] and they are modelled to be the odd dimensional analogue to generalized complex structures. In the same way Dirac–Jacobi bundles are a generalization of Wade’s \(\mathcal {E}^1(M)\)Dirac structures, generalized contact bundles are a generalization of integrable generalized almost contact structure, which were defined in [9].
Definition 7.1
Let \(L\rightarrow M\) be a line bundle. A subbundle \(\mathcal {L}\subseteq \mathbb {D}_\mathbb {C}L:=\mathbb {D}L\mathbin {\otimes _{\scriptscriptstyle {}}}\mathbb {C}\) is called generalized contact structure on L, if

1.
\(\mathcal {L}\) is a (complex) Dirac–Jacobi structure

2.
\(\mathcal {L}\cap \overline{{\mathcal {L}}}=\{0\}\)
A generalized contact structure can be also seen as an endomorphism of \(\mathbb {D}L\) of the form
where \(\phi \in \textrm{End}(DL)\), \(J\in \Gamma ^{\infty }((J^1\,L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) and \(\alpha \in \Omega _L^2(M)\) (see [12] and [15]). 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 [15]. The \(+\textrm{i}\)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 Eulerlike derivation \(\Delta =J^\sharp (\alpha )\), then \((\Delta , \textrm{i}\alpha \phi ^*(\alpha ))=\textrm{i}(0,\alpha )+\mathbb {K}(0,\alpha )\in \Gamma ^{\infty }(\mathcal {L})\). With the techniques from Sect. 4 and Sect. 5, one can show that
where \(\omega =\int _{0}^{1}\frac{1}{t}(\Phi _{\log (t)}^\Delta )^*\mathop {}\!\textrm{d}_L\alpha \mathop {}\!\textrm{d}t\) and \(\beta =\int _{0}^{1}\frac{1}{t} (\Phi _{\log (t)}^\Delta )^*\mathop {}\!\textrm{d}_L\phi ^*(\alpha )\mathop {}\!\textrm{d}t\). 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 been done in [12] with similar techniques.
8 Final remarks
In [14] the Diracization trick is explained, which introduces a onetoone correspondence between Dirac–Jacobi bundles and socalled homogeneous Dirac structures, i.e. Dirac structures \(\mathcal {D}\subseteq \mathbb {T}P\) on the total space of a \(\mathbb {R}^\times \) principal bundle \(P\rightarrow M\), such that
for the fundamental vector field \(E\in \Gamma ^{\infty }(TP)\) of \(1\in \mathbb {R}=\textsf {Lie}(\mathbb {R}^\times )\). Knowing this one may wonder, if one can apply the normal form theorem from [3] to the homogeneous Diracstructure in order to obtain the normal form theorems for Dirac–Jacobi structures. This is not directly possible, at least not to the author’s knowledge. The main problems are the following:

1.
Homogeneous Dirac structures are not invariant with respect to the lifted \(\mathbb {R}^\times \)action on \(\mathbb {T}M\), so the invariant versions of the normal forms from [3] are not applicable.

2.
The homogenization of a submanifold gives always a homogeneous submanifold, i.e. invariant under the principal action, but minimal transversals to contact points (points such that the leaf through the point is tangential to the homogeneity E) are not homogeneous.

3.
Even though there is a notion of product in the category of \(\mathbb {R}^\times \)principal bundles (the cartesian product of the total spaces modulo the diagonal action), the product is not a principal bundle over the cartesian product of the bases of the factors, which makes a splitting in this category very difficult.
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Acknowledgements
I would like to thank my advisor, Luca Vitagliano, who suggested me this project and helped me a lot in turning it in to a paper as well as Chiara Esposito who helped me to improve the presentation. The content of this note was produced almost completely during a stay at IMPA in Rio de Janeiro from April to July in 2018, where I was warmly received in the Poisson Geometry group. In particular, I would like to thank Henrique Bursztyn for discussions and useful suggestions.
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A the Moser trick for Jacobi manifolds
A the Moser trick for Jacobi manifolds
Let \(J\in \Gamma ^{\infty }(\Lambda ^2 (J^1L)^*\mathbin {\otimes _{\scriptscriptstyle {}}}L)\) be a Jacobi structure on a line bundle \(L\rightarrow M\). Moreover, we assume having smooth family of closed 2forms \(\sigma _t\), such that \(\sigma _0=0\) and \(\mathcal {L}_J^{\sigma _t}\) is a Jacobi structure for all t, denoted by \(J_t\). For
the equation
holds. We define the Moserderivation by
and its flow by \(\Phi _t\in {\text {Aut}}(L)\), where we assume it exists for on open subset containing [0, 1]. Let us compute
It is easy to see that
by the equality \(\mathcal {L}_{J_t}=\mathcal {L}_J^{\sigma _t}\). We can compute
and hence \(\tfrac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}t} J_t=J_t^\sharp (\mathop {}\!\textrm{d}_L\alpha _t)\). If we use this equality in Eq. A.1, we find
so we finally have \(J=\Phi ^*_0J_0=\Phi _1^* J_1\) and hence the two Jacobi structures are isomorphic.
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Schnitzer, J. Normal forms for Dirac–Jacobi bundles and splitting theorems for Jacobi structures. Math. Z. 303, 74 (2023). https://doi.org/10.1007/s00209023032229
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DOI: https://doi.org/10.1007/s00209023032229