Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.


Introduction
This paper studies a certain family of contact realisations of Jacobi manifolds. A contact structure on a smooth manifold M is a maximally non-integrable smooth hyperplane distribution H ⊂ T M (cf. Definition 1); these can be thought of as odddimensional analogues of symplectic structures (which are closed, non-degenerate 2-forms). This analogy is far-reaching. For instance, as a symplectic manifold (S, ω) gives rise to a Poisson bracket on C ∞ (S) (a Lie bracket which satisfies a Leibniz-type rule) which is transitive, so can contact structures be seen as (odd dimensional) transitive examples of Jacobi manifolds. A Jacobi structure on a smooth manifold P is a local Lie bracket {·, ·} on the space of sections of some fixed real line bundle L → P (cf. Definition 4 and [24,26,27,28]). A contact manifold (M, H) naturally comes equipped with a Jacobi bracket on the space of sections of the line bundle T M/H → M , which is not necessarily trivial, or even trivialisable, e.g. the projectivisation of the cotangent bundle with canonical contact structure (cf. [1,Appendix 4] for more details). Continuing with the analogy symplectic : Poisson = contact : Jacobi, just as symplectic realisations of Poisson manifolds play an important role in understanding the geometry of Poisson structures (cf. [5,6,9,22]), so do contact realisations of Jacobi manifolds, i.e. surjective submersions φ : (M, H) → (P, L, {·, ·}) which are Jacobi morphisms (cf. Remark 4). Intuitively, symplectic/contact realisations contain information regarding the 'integration' of the Lie algebraic structure encoded in a Poisson/Jacobi manifold; as such they should be thought of 'desingularisations' of the underlying Poisson/Jacobi structures (cf. [6,Section 7] and [33,Section 3.2]).
The types of contact realisations considered in this paper are those of minimal dimension, i.e. the dimension of the total space (M, H) is as small as possible. These are called contact isotropic realisations and are contact realisations φ : (M, H) → (P, L, {·, ·}) with the extra condition that the fibres are 'isotropic' in the following sense. For each m ∈ M , maximal non-integrability of H induces a non-degenerate bilinear form on H m , i.e. a symplectic form. Then the fibres of φ are isotropic if, for each m ∈ M , ker D m φ ∩ H m is an isotropic subspace (cf. Definition 12). The simplest such example is given by the Boothby-Wang construction of regular contact manifolds (cf. [4,19]), which can be seen as contact isotropic realisations of symplectic manifolds viewed as Jacobi structures (cf. Example 15). Intuitively, contact isotropic realisations exhibit the most rigid behaviour amongst contact realisations: this observation stems from the results concerning symplectic isotropic realisations of Poisson manifolds. These realisations can be used to study the geometry of integrable Hamiltonian systems on symplectic manifolds (cf. [14,17]); moreover, recently they have been studied in relation to Poisson manifolds of compact type (cf. the forthcoming [8]). The work in [14] provides local and global classifications of symplectic isotropic realisations; this begs the question of whether analogous results hold for contact isotropic realisations, especially in light of the recently introduced notion of integrable Hamiltonian systems on contact manifolds (cf. [21] and Section 6.1 for further details).
The aim of this paper is to classify contact isotropic realisations of Jacobi manifolds. The approach taken combines the framework of [14,17] developed to study symplectic isotropic realisations, with new techniques coming from recent work on the integrability of Jacobi structures (cf. [10] and also [11] for even more general methods applicable to Lie groupoids and algebroids). The main difficulty to apply the ideas of [14,17] directly is the presence of non-trivial line bundles, which complicate matters both conceptually and technically. This is precisely why the methods of [10,11] are employed in this paper; the upshot of this approach is to provide intrinsic results on contact/Jacobi manifolds whose underlying line bundles are non-trivial, of which there are not many in the existing literature, while also extending results on the specific types of realisations under consideration in this paper (cf. [2,4,21]).
The isotropy condition imposes a restriction on the type of Jacobi manifolds admitting a contact isotropic realisation. Associated to a Jacobi structure (P, L, {·, ·}) is a Lie algebroid structure on the first jet bundle J 1 L of L → P , which consists of a Lie bracket on Γ(J 1 L) and a compatible map ρ : J 1 L → T P (cf. Proposition 2 and [10]). This plays a central role in this paper. A necessary condition for (P, L, {·, ·}) to admit a contact isotropic realisation is that the structure be regular with even dimensional leaves, i.e. ρ is of constant even rank (cf. Lemma 3). Examples of such Jacobi manifolds are regular Poisson structures, zero Jacobi structures (defined on any line bundle), and locally conformal symplectic manifolds (cf. Example 7). Once this necessary condition is established, the first main result is a smooth classification of contact isotropic realisations of a fixed Jacobi manifold (P, L, {·, ·}); as in the case of symplectic isotropic realisations, there are two invariants: • a period lattice Σ ⊂ ker ρ ⊂ J 1 L (cf. Definition 11), all of whose sections are holonomic, i.e. of the form j 1 u for some u ∈ Γ(L) (cf. Corollary 12); • the Chern class c ∈ H 2 (P ; Σ) (cf. Definition 14).
This follows from Theorem 9, whose proof is entirely analogous to that of [31,Theorem 8.15]. In fact, the content of Theorem 9 is that a smooth local model for a contact isotropic realisation with period lattice Σ is given by ker ρ/Σ → P .
The second main result is a contact local normal form (in a neighbourhood of a fibre) for contact isotropic realisations. Recall that, given a line bundle L → P , its first jet bundle J 1 L comes equipped with a canonical contact structure H, defined as the kernel of the Cartan 1-form θ can ∈ Ω 1 (J 1 L; L) (cf. Example 2 for further details). where θ 0 ∈ Ω 1 (ker ρ/Σ; L) is induced by θ can and β ∈ Ω 1 (P ; L) is a locally defined 1

-form determined (not uniquely!) by the Jacobi bracket.
This is the content of Theorem 13, which provides a more precise statement. While there exist similar results in the literature regarding integrable Hamiltonian systems on contact manifolds (cf. the theorems on existence of action-angle coordinates in [2,21]), the above statement and the proof presented in this paper are more intrinsic, as the various geometric objects related to Jacobi structures are used in a natural fashion to obtain the local normal form.
Given a regular Jacobi manifold (P, L, {·, ·}) whose foliation F consists solely of even dimensional leaves, a natural question to ask is to construct all its contact isotropic realisations. The third main result of the paper answers this question once a suitable Σ ⊂ ker ρ has been fixed, by providing a cohomological criterion which is analogous to that of [14,Theorems 4.2 and 4.3]. The above criterion provides both a common framework and an extension of various existing results in the literature, such as the criterion for constructing regular contact manifolds (cf. [19,Theorems 7.2.4 and 7.2.5]), and the cohomological criterion for the existence of completely integrable contact forms of toric type (cf. [2]). At the heart of the above result lies the classical Spencer operator D : Γ(J 1 L) → Ω 1 (P ; L), which recently has been discovered to play a central role in Jacobi geometry (cf. Observation 6 for a definition of D and [10] for further results).
In analogy with a symplectic isotropic realisation of a Poisson manifold, associated to which is an integral affine structure transversal to the symplectic foliation, the last main result of this paper characterises the underlying geometric structure on the foliation F of a Jacobi manifold (P, L, {·, ·}) admitting a contact isotropic realisation. The idea is to use give an intrinsic characterisation of the period lattice Σ ⊂ J 1 L, i.e. which does not rely on the given realisation (cf. Definition 18). As expected from the case L = P × R, the ensuing geometric structure on F is transversally integral projective, i.e. it gives rise to a foliated atlas with values in real projective spaces along with an integrality condition (cf. Definition 21); what is surprising is that this result holds even when L is non-trivial: this is the content of Theorem 19.
The structure of this paper is as follows. Section 2 recalls basic notions and properties of contact and Jacobi manifolds which are used throughout; particular importance is given the Lie algebroid associated to a Jacobi manifold, described from the point of view developed in [10] (cf. Section 2.3). Contact isotropic realisations are introduced and studied in Section 3. After establishing a simple necessary condition on the Jacobi structure for the existence of such a realisation (cf. Section 3.1), there is an intermezzo dealing with properties of the family of Jacobi manifolds that occur: these share many properties with regular Poisson manifolds. This is the content of Section 3.2. The smooth and contact local normal forms of contact isotropic realisations are constructed in Sections 3.3 and 3.4 respectively. The question of constructing contact isotropic realisations, both locally and globally, is dealt with in Section 4. Any regular Jacobi manifold all of whose leaves are even dimensional admits locally a contact isotropic realisation (cf. Section 4.1); the cohomological problem of glueing these local constructions together is solved in Section 4.2. The geometric structure induced by a contact isotropic realisation on the foliation of a Jacobi manifold is introduced in Section 5. Some interesting examples and applications of the theory developed are discussed in Section 6. These include integrable Hamiltonian systems on contact manifolds, which are shown to be equivalent to contact isotropic realisations of a special class of Jacobi manifolds (cf. Section 6.1), and contact isotropic realisations of Poisson manifolds and of locally conformal symplectic manifolds (viewed as Jacobi structures on the trivial line bundle) in Sections 6.2 and 6.3 respectively. Throughout this note, there are two types of comments, labelled Observation and Remark respectively; those with the former label are central to the problems studied in this paper, while those with the latter may be skipped at a first reading.
Notation and conventions. Throughout the paper, whenever a collection of open sets {U i } is considered, U i1i2...iN = U i1 ∩ . . . ∩ U iN for any finite set of indices i 1 , . . . , i N . All line bundles considered in this paper are real unless otherwise stated.

Generalities on contact and Jacobi manifolds
This section recalls basic definitions, properties and examples of contact and Jacobi manifolds that are used throughout this paper (cf. [3,10,15,18,28] amongst others for further details and proofs).

Contact manifolds and their properties.
Definition 1 (Contact manifolds). A contact manifold is a pair (M, H) consisting of a manifold M and a smooth hyperplane distribution H ⊂ T M with the property that the curvature map defined on sections by is fibre-wise non-degenerate. The distribution H is said to be a contact distribution.
Remark 1. If H is co-orientable, i.e. the line bundle T M/H → M is trivialisable, then a choice of trivialisation for T M/H → M corresponds to a globally defined 1-form θ ∈ Ω 1 (M ) satisfying H = ker θ. In this case, the condition of equation (1) can be rephrased as non-degeneracy of dθ| H .  (1) can be rephrased as fibre-wise non-degeneracy of (X, Y ) → θ([X, Y ]), for X, Y ∈ H. In fact, contact manifolds can be equivalently described as pairs (M, θ) consisting of a smooth manifold M and a point-wise surjective 1-form θ ∈ Ω 1 (M, L), such that the curvature defined on sections by is fibre-wise non-degenerate. Throughout this paper, both points of views are used for different purposes and the 1-form satisfying θ = ker H is referred to as a generalised contact form.
Example 2. Let π : L → P a line bundle and denote by pr : p u; these are called holonomic sections. The Cartan contact form θ can ∈ Ω 1 (J 1 L, pr * L) detects holonomic section in the sense that a section ξ of J 1 L is holonomic if and only if ξ * θ can = 0. The Cartan contact form is defined as follows. By abuse of notation denote by pr : J 1 L → L the canonical projection and let X ∈ T j 1 p u J 1 L. Then θ can (X) := Dpr(X) − Du(Dπ(X)), lies in ker Dπ ∼ = pr * L. As its name suggests, the Cartan contact form θ can is a generalised contact form.  The vector space of Reeb vector fields is henceforth denoted by X Reeb (M, H).
is a vector space isomorphism.
Observation 2. The vector field R u associated to u ∈ Γ(L) is called the Reeb vector field of u and is uniquely defined by θ(R u ) = u and θ([R u , H]) = 0, where θ ∈ Ω 1 (M, L) is a generalised contact form. If f ∈ C ∞ (M ), it can be checked that , ·]) = −η for any section η ∈ Γ(Hom(H, L)) (and the convention followed is the same as in [28]).
Remark 2. The above notion of Reeb vector fields generalises the commonly found notion of the Reeb vector field associated to a co-oriented contact structure, which, in the above description, is nothing but R 1 .
The isomorphism of Lemma 1 allows to define another geometric structure associated to a contact manifold (M, H), namely the Reeb bracket on Γ(L), given by for u, v ∈ Γ(L). The triple (M, L, {·, ·}) gives rise to a Jacobi structure, as explained in the next subsection.
2.2. Jacobi structures. As symplectic forms are special examples of Poisson structures, so are contact distributions instances of a more general geometric structure, first introduced in [24,26], and further explained in [28].
Definition 4 (Jacobi structure). A Jacobi structure on a manifold P is a pair (L, {·, ·}) consisting of a line bundle L → P , and a Lie bracket {·, ·} on Γ(L) with the property that it is local, in the sense that where supp(u) denotes the support of u. The triple (P, L, {·, ·}) consisting of a manifold P and a Jacobi structure (L, {·, ·}) is henceforth referred to as a Jacobi manifold.
In the literature on Jacobi manifolds two types of maps between Jacobi manifolds are considered, namely those which are, in some sense, 'strictly' Jacobi and those which are up to some rescaling (conformal). The following definition provides a common framework for both of the above types of maps in the more general setting of Jacobi manifolds in the sense of Definition 4.
Definition 8 (Jacobi maps). Let (N, L N , {·, ·} N ) and (P, L P , {·, ·} P ) be Jacobi manifolds such that there exists an isomorphism F : φ * L P → L N . A map φ : N → • the collection of nowhere vanishing functions {g ij } satisfy the cocycle condition, i.e. for all i, j, l for which U ijl = ∅, g il = g jl g ij ; then there exists a unique Jacobi structure on the line bundle L → P whose (isomorphism class) is uniquely defined by the collection of {g ij } (cf. [15,Section 1.4]).
Example 9. If, in the above definition, L N and L P are trivial, and the bundle component is the identity, the condition that φ : (N, L N , {·, ·} N ) → (P, L P , {·, ·} P ) is a Jacobi map is equivalent to asking that Λ N and R N are φ-related to Λ P and R P respectively, where (Λ N , R N ) and (Λ P , R P ) are the bivector and vector fields which uniquely determine the Jacobi structures on L N and L P respectively (cf. Observation 3).
2.3. The Lie algebroid associated to a Jacobi manifold. Given a Jacobi manifold (P, L, {·, ·}), the first jet bundle J 1 L → P can be endowed with the structure of a Lie algebroid, which codifies the geometry of (P, L, {·, ·}) -cf. [7] for a definition and examples of Lie algebroids. This Lie algebroid is described in [23] for the trivial line bundle, in [13] in the general case, and more recently from a different point of view in [10]. Its defining properties are given in the proposition below, stated without proof (cf. [10]). This Lie algebroid is henceforth referred to as the associated Lie algebroid to the Jacobi manifold (P, L, {·, ·}).
Observation 5. For a contact manifold (M, H), where H = ker θ, with θ ∈ Ω 1 (M, L), the Lie algebroid structure of J 1 L associated to the underlying Jacobi structure defined in Example 5 can be expressed in terms of the Reeb vector fields, and the curvature map c θ (cf. equation (2)). Using the defining property (I) of the anchor and the definition of the Reeb bracket of equation (3), it follows that ρ(j 1 u) = R u . This implies that, given (u, η) ∈ Γ(J 1 L), Observation 6 (The Spencer operator of a Jacobi manifold, cf. [10]). The first jet bundle J 1 L → P fits into a short exact sequence of vector bundles where pr : J 1 L → L is the standard projection, and i(df ⊗ u) = f j 1 (u) − j 1 (f u) for f ∈ C ∞ (P ) and u ∈ Γ(L). While the above sequence is not canonically split, there is a canonical splitting at the level of sections, which is very useful when studying geometric structures on J 1 L. This splitting, known as the Spencer decomposition, is used extensively in this paper; it is defined by u → j 1 u, for u ∈ Γ(L). This induces the Spencer decomposition where the C ∞ (P )-module structure on the right hand is given by The projection to the second component is the so-called classical Spencer operator; D is compatible with the Lie algebroid of a Jacobi manifold (P, L, {·, ·}) in the following sense. First, L is a representation of J 1 L with flat connection uniquely defined by (and extended using the Leibniz rule, cf. [10] for details). Secondly, D satisfies the following conditions (of Spencer operators on Lie algebroids [11]) for any α, α ′ ∈ Γ(J 1 L) and X ∈ X(P ). Henceforth, the above Spencer decomposition is going to be used tacitly unless otherwise stated.
Jacobi manifolds whose underlying characteristic distribution Im ρ ⊂ T P has constant rank play an important role in this paper (cf. Lemma 3).
Definition 11. A Jacobi manifold (P, L P , {·, ·}) is said to be regular of corank k if, for all p ∈ P , dim ker ρ P,p = k.
Remark 3. There exist other notions of 'regular' Jacobi manifolds in the literature which are different to the one of Definition 11 (cf. [16]).

Contact isotropic realisations: definition and invariants
3.1. Definition and basic properties. The following definition introduces the main object studied in this paper.  (1) Property (IR1) implies commutativity of the following diagram where ρ M , ρ P are the anchor maps for the Jacobi structures on P and M respectively, and φ * : φ * J 1 L P → J 1 L M is the injective vector bundle morphism given by j 1 u → j 1 (F φ * u). This can be seen as follows. Consider {u, gv} P for any sections u, v ∈ Γ(L) and any function g ∈ C ∞ (P ); using the defining property (I) of the anchor of a Lie algebroid associated to a Jacobi structure together with property (IR1), it can be shown that which, in turn, implies the claimed result; and by property (IR2), for all m ∈ M , H m + ker D m φ = T m M ; (3) When (P, L P , {·, ·}) is Poisson, condition (IR3) is equivalent to requiring that, for all m ∈ M , the vector space ker D m φ∩H m is isotropic with respect to the symplectic structure on H m induced by a suitable choice of contact form (cf. [25] for details). This explains the nomenclature used in the definition.
Not all Jacobi manifolds admit isotropic contact realisations; in fact, a necessary condition is that the structure be regular and all its leaves be even dimensional. This is the content of the next lemma, which should be compared to [31,Proposition 8.10], which proves an analogous statement for isotropic realisations of Poisson manifolds.
Lemma 3. For a contact isotropic realisation φ : (M, H) → (P, L P , {·, ·}) the Jacobi structure on P is regular with corank equal to k, and all its leaves are even dimensional.

3.2.
Properties of regular Jacobi manifolds all of whose leaves are even dimensional. In light of Lemma 3 a necessary condition for a Jacobi manifold to admit a contact isotropic realisation is that it is regular and all its leaves are even dimensional. The aim of this subsection is to prove some properties of Jacobi manifolds of this kind, which generalise those enjoyed by regular Poisson manifolds (which are an example of this family of Jacobi structures). Throughout this section, let (P, L P , {·, ·}) be a regular Jacobi manifold all of whose leaves are even dimensional, unless otherwise stated. The notation employed below follows from that of Section 3.1.
Proposition 4. Let (P, L P , {·, ·}) be a regular Jacobi manifold with even dimensional leaves. Then π : ker ρ P → P is a bundle of abelian Lie algebras, so that the inclusion ker ρ P ֒→ J 1 L P is a morphism of Lie algebroids.
Proof. Since ker ρ P is the kernel of the anchor ρ : J 1 L P → T P and the restriction of π : J 1 L P → P to ker ρ P is a vector bundle, it follows that π : ker ρ P → P is a bundle of Lie algebras. It remains to show that each such Lie algebra is abelian.
Let ν * ⊂ T * P denote the conormal bundle of the regular foliation F whose distribution is T F := Im (ρ : J 1 L P → T P ). By assumption, the corank of the Jacobi structure is k, which implies that the rank of ν * → P is k − 1. The claim is that ker ρ P fits in a short exact sequence of vector bundles where the projection ker ρ P → L P is nothing but the restriction of pr : J 1 L P → L P (cf. Observation 6). To this end, it suffices to prove that (10) ν * ⊗ L P = ker ρ P ∩ (T * P ⊗ L P ) = ker pr| ker ρP .
For, if equation (10) holds, then dimension counting implies that the restriction pr : ker ρ P → L P is surjective. First, it is shown that ν * ⊗ L P ⊂ ker ρ P . Suppose that df ⊗u ∈ Γ(ν * ⊗L P ), and fix g ∈ C ∞ (P ) and v ∈ Γ(L P ). Using the C ∞ (M )-module structure of the Spencer decomposition of Γ(J 1 L P ) (cf. Observation 6), have that The defining property (I) of the anchor implies that This shows that ν * ⊗ L P ⊂ ker ρ P ∩ (T * P ⊗ L P ). In order to show that equality holds, it suffices to check that, for each p ∈ P , dim ker ρ P,p ∩ (T * P ⊗ L P ) p = k − 1. To this end, note that ker ρ P ∩ (T * P ⊗ L P ) = ker ρ P | T * P ⊗LP . The map ρ P : T * P ⊗ L P → T P is, in fact, antisymmetric, i.e. for any η, ξ ∈ T * P ⊗ L P , η(ρ(ξ)) = −ξ(ρ(η)). This can be checked directly from the defining property of the anchor ρ P (in the case of trivial coefficients, this map is given by the sharp of a bivector). Thus, for each p ∈ P , the vector space ρ P,p (T * p P ⊗ L P,p ) ⊂ T p F is even dimensional. On the other hand, since k ≥ dim ker ρ P,p | T * p P ⊗LP,p ≥ k − 1, it follows that 2n − 1 ≤ dim ρ P,p (T * p P ⊗ L P,p ) ≤ 2n, with dim ρ P,p (T * p P ⊗ L P,p ) = 2n if and only if dim ker ρ P,p | T * p P ⊗LP,p = k − 1. Since dim ρ P,p (T * p P ⊗ L P,p ) is even dimensional, it follows that dim ker ρ P,p | T * p P ⊗LP,p = k − 1; as p ∈ P is arbitrary, this completes the proof that equation (10) holds.
Using the above description of ker ρ P , it is possible to prove that ker ρ P → P is a bundle of abelian Lie algebras. Let α, α ′ ∈ Γ(ker ρ P ); the Spencer decomposition of Observation 6 gives that [α, where u, v ∈ Γ(L P ) and η, ξ ∈ Ω 1 (P ; L P ), the compatibility conditions of equations (7) and (8) yield the following formula for the Lie bracket restricted to sections of ker ρ P (12) [α, where ∇ is the J 1 L P -connection defined by equation (6). There are two cases to consider: the first case is when u = 0 = v, and the second is when u = 0 and v = 0. Note that, for dimensional reasons, Γ(ker ρ P ) is (locally) generated by elements of this form, and therefore, as ker ρ P is a bundle of Lie algebras it is enough to check that the bracket is zero on these elements. Suppose first that u = 0 = v; using the defining properties of ∇ and of ρ P (cf. equation (6) and property (I) respectively), the following equality can be proved where s ∈ Γ(L P ) is arbitrary (and analogously for ∇ η s). Since ξ, η ∈ Γ(ν * ⊗ L P ), equation (13) implies that [η, ξ] = 0. Thus it only remains to consider the case u = 0 and v = 0. Write α ′ = j 1 z + i(γ), for γ ∈ Ω 1 (P, L P ); without loss of generality, the section z ∈ Γ(L P ) can be taken to be nowhere vanishing at the cost of restricting its domain of definition, say to U ⊂ P . This does not matter to the purpose at hand, as the aim is to check that a fibre-wise Lie algebra structure is abelian and, to do so, there is a freedom in choosing the sections α, α ′ under consideration. Since ρ P (α ′ ) = 0, the defining property (I) of ρ P implies that the -linear and hence it defines a vector bundle automorphism L P | U → L P | U . As Hom(L P , L P ) is trivial and of rank 1 (because L P is of rank 1), then there exists a smooth function G ∈ C ∞ (U ) such that for all s ∈ Γ(L P | U ) and p ∈ U . Hence, where antisymmetry of ρ P : T * P ⊗ L P → L P is used to obtain the last equality. As z ∈ Γ(L P | U ) does not vanish at any point, it follows that G ≡ 0. This, in turn, implies that [α, α ′ ] = 0 in this case as well, thus completing the proof.
The next aim is to construct a cohomology class associated to (P, L P , {·, ·}) which generalises the foliated cohomology class given by the leafwise symplectic form induced by a regular Poisson manifold. As in the proof of Proposition 4, denote the characteristic foliation of (P, L P , {·, ·}) by F . Intuitively, the cohomology class defined below is a foliated 2-form with values in L P . However, in order to formalise this intuition, it is necessary to prove that there is a T F -connection on L P associated to the Jacobi structure (L P , {·, ·} P ), where the foliation T F ⊂ T P is viewed as a Lie subalgebroid. Let ∇ be the J 1 L P -connection on L P defined by equation (6).
Proof. Observation 8 implies that∇ is well-defined as a connection. Flatness of∇ follows from that of ∇, thus completing the proof.
In light of Proposition 4, it makes sense to consider the Lie algebroid cohomology H * (F ; L P ), which is defined as the cohomology of the complex of smooth foliated forms with values in L P with the following (standard) differential d F for X 1 , . . . , X l+1 ∈ Γ(T F ) and ω ∈ Ω l (F ; L P ) := Γ(∧ l T * F ⊗ L P ), and where hat denotes omission. The next lemma associates a cohomology class in H 2 (F ; L P ) to (P, L P , {·, ·}).
Lemma 6. The foliated 2-form with values in L P uniquely defined by where the first equality follows from the C ∞ (P )-structure on Γ(J 1 L P ) arising from the Spencer decomposition (cf. Observation 6), the second by definition of ω, and the third by the characterising property (I) of the anchor ρ P (cf. Proposition 2). The above calculation shows that equation (16) indeed defines a foliated 2-form with values in L P (also denoted by ω) as it is C ∞ (P )-bilinear.
To check that d F ω = 0 it suffices to check that d F ω(ρ P (j 1 u), ρ P (j 1 v), ρ P (j 1 w)) = 0 for any u, v, w ∈ Γ(L P ). This follows from the fact that d F ω is C ∞ (P )-linear in each entry and that sections of the form ρ P (j 1 u) form a C ∞ (P )-basis of Γ(T F ). Then where c.p. stands for cyclic permutation, the second equality follows from the fact that ρ P is a map of Lie algebroids, the third by the defining property (II) of the Lie bracket on J 1 L P (cf. Proposition 2) and by the Jacobi identity for {·, ·}, which also implies the last equality. This shows that d F ω = 0, as required.
Proof of Lemma 7. Since φ : (M, H) → (P, L P , {·, ·}) is a Jacobi map, item 1 in Observation 7 implies that In light of Observation 10, it follows that property (A2) holds. It remains to prove that ψ induces a Lie algebra morphism, i.e.
. This equation holds when α, α ′ ∈ Γ(J 1 P ) are holonomic sections, i.e. of the form j 1 u, j 1 v for smooth sections u, v ∈ C ∞ (P ). This is because φ : M → P is a Jacobi map with bundle component F , which implies that for u, v ∈ Γ(L P ), where in the last equality property (3) of the Jacobi bracket of (M, H) is used. In general, notice that the left hand side of equation (18) satisfies for f ∈ C ∞ (P ) and α, α ′ ∈ Γ(J 1 L P ). This again follows from the fact that φ : (M, H) → (P, L P , {·, ·}) is a Jacobi map, so that equation (17) holds. Thus, since equation (18) holds for holonomic sections and these generate Γ(J 1 L P ) as a C ∞ (P )module, it follows that equation (18) holds for all sections of J 1 P , which proves that ψ satisfies property (A1). The proof of Lemma 3 shows that ψ is injective (this follows from property (IR2)).
The following proposition shows that, for a contact isotropic realisation φ : (M, H) → (P, L P , {·, ·}), the restriction of the action of Lemma 7 to the Lie subalgebroid ker ρ P → P yields an action. Proof. Property (A2) implies that the image of ψ| ker ρP consists of vertical vector fields, i.e. tangent to the fibres of φ. On the other hand, property (A1) implies that at the level of sections ψ : Γ(ker ρ P ) → Γ(ker Dφ) is a Lie algebra map. Now, since by Lemma 7, the action ψ : φ * ker ρ P → T M is injective and for all p ∈ P , dim ker ρ P,p = k = dim φ −1 (p), dimension counting implies that for all m ∈ M , The action ψ : φ * ker ρ P → T M should be thought of as being infinitesimal; since the fibres of φ are compact, ψ can be integrated to an action of π : ker ρ P → P considered as a bundle of abelian Lie groups.
Remark 5. In fact, the action ψ can be integrated under a weaker assumption on the fibres, which is generally known as completeness (cf. [6]). All following results hold in this more general setting with the appropriate modifications.
Fix a contact isotropic realisation φ : (M, H) → (P, L P , {·, ·}) and let ψ : φ * ker ρ P → T M be the associated infinitesimal π : ker ρ P → P action. The integrated action is given by where } is a smooth manifold, and ϕ 1 α is the time-1 flow of ψ(φ * α). Observation 11. For each p ∈ P , the action of equation (20) restricts to an action of the abelian Lie group ker ρ P,p ∼ = R k on φ −1 (p). Connectivity of φ −1 (p) and Observation 8 imply that the action of ker ρ P,p is transitive. Moreover, since ker ρ P,p is an abelian Lie group, if j 1 p u ∈ ker ρ P,p is such that Ψ(j 1 p u, m) = m, then Ψ(j 1 p u, m ′ ) = m ′ for any other m ′ ∈ φ −1 (p). This is because the isotropy of the action at any two points on φ −1 (p) are equal. Therefore, if Σ p := {j 1 p u ∈ ker ρ P,p | ϕ 1 j 1 p u = id}, then φ −1 (p) ∼ = ker ρ P,p /Σ p ; since φ −1 (p) is compact by assumption, it follows that φ −1 (p) is diffeomorphic to T k and that Σ p is a cocompact lattice in ker ρ P,p and, therefore, isomorphic to Z k .
Just as in the theory of symplectic isotropic realisations of Poisson manifolds, the isotropy of the action of equation (20) plays an important role in the classification of contact isotropic realisations of Jacobi manifolds. (II) the quotient ker ρ P /Σ is a smooth manifold and the projection π : ker ρ P /Σ → P is a fibre bundle with fibre T k ; (III) upon a choice of a local section σ : is an isomorphism of fibres bundles, making the following diagram Fix a good open cover U := {U i } of P , i.e. all finite intersections of the elements of U are contractible, and, for each i, fix a section σ i : U i → M . Suppose that U ij = ∅; then the composite Ψ −1 σj • Ψ σi : ker ρ P /Σ| Uij → ker ρ P /Σ| Uij is given by translation along the fibres by a section t ij : U ij → ker ρ P /Σ, i.e.
As in the case of symplectic isotropic realisations of Poisson manifolds, Lemma 10 provides a local description of the sections of the period lattice Σ → P .
Corollary 12. Any local section of Σ → P is of the form j 1 u, for some locally defined section u ∈ Γ(L P ).
The following theorem gives a local normal form for sufficiently small open φsaturated neighbourhoods of the fibres of φ : (M, H) → (P, L P , {·, ·}); as such, it can be thought of as providing local action-angle variables in analogy with [14,Corollary 2] in the case of complete isotropic realisations of Poisson manifolds, and [2, Theorem 1] and [21,Theorems 4 and 5] in the case of non-commutative integrable systems on contact manifolds (the former with elliptic-type singularities) once a suitable local contact form has been fixed (cf. Section 6.1 for the relation between integrable Hamiltonian systems on contact manifolds and contact isotropic realisations).
It is important to remark that all forms in the above theorem take values in π * L P .

Constructing contact isotropic realisations
In analogy with the classification of symplectic isotropic realisations of Poisson manifolds carried out in [14], the classification of contact isotropic realisations over a given regular Jacobi manifold (P, L P , {·, ·}) of corank k, all of whose leaves are even dimensional, consists of the following two parts: Part 1: Classify closed submanifolds Σ ⊂ ker ρ P for which the composite Σ ֒→ ker ρ P → P is a Z k -bundle and all of its sections are holonomic. Such a submanifold is henceforth called period lattice even though it may not necessarily arise from a contact isotropic realisation; Part 2: Having fixed a period lattice Σ ⊂ ker ρ P , determine which cohomology classes in H 2 (P ; Σ) give rise to contact isotropic realisations, i.e. whether the total space of the (isomorphism class of the) principal ker ρ P /Σ-bundle associated to c admits a contact structure which makes the projection to P into a contact isotropic realisation.
Part 1, while very interesting in its own right, is beyond the scope of this paper; Section 5 gives an alternative geometric interpretation of a period lattice. This section concentrates on solving the problem outlined in Part 2. To this end, it is important to remark that there already exist partial solutions to this question, as the following examples show.
Example 15 (Regular contact manifolds). Suppose that (P, L P , {·, ·}) is a closed symplectic manifold, so that L P = P × R and the Jacobi bracket is nothing but the Poisson bracket on C ∞ (P ) induced by a symplectic form ω. Then ker ρ P = P × R ⊂ J 1 L P , which implies that any period lattice Σ is trivial. Let Σ ⊂ ker ρ P be a period lattice and suppose that α ∈ Γ(Σ) is a nowhere vanishing section. Corollary 12 gives that α = j 1 f for some smooth function f , which implies that f is a non-zero constant function. Without loss of generality, fix f ≡ 1. A contact isotropic realisation φ : (M, θ) → (P, ω) is nothing but a principal S 1 -bundle whose infinitesimal action is generated by the Reeb vector field R M associated to the connection 1-form (which is a contact form!) θ ∈ Ω 1 (M ) (cf. Remark 2). A cooriented contact manifold (M, θ) as above is known in the literature as a regular contact manifold (cf. [4,19] , where H 2 (P ; Σ) is identified with H 2 (P ; Z). In particular, not all elements of H 2 (P ; Σ) give rise to a contact isotropic realisation (in fact, there may be none that does).
Example 16 (Zero Jacobi structure). Suppose that (P, L P , {·, ·}) is endowed with the zero Jacobi structure, so that L P = P × R and the Jacobi bracket is identically zero. In this case, ker ρ P = J 1 L P := J 1 P . Given any period lattice Σ ⊂ J 1 P , it can be shown that every cohomology class in H 2 (P ; Σ) gives rise to a contact isotropic realisation, as proved in [2].
The main result of this section provides a cohomological criterion that solves Part 2 in the spirit of [14, Theorems 4.2 and 4.3] which unifies and extends Examples 15 and 16 above (cf. Theorem 17). The rest of the section is structured as follows. First, it is shown that locally any regular Jacobi manifold (P, L P , {·, ·}) all of whose leaves are even dimensional with a period lattice Σ ⊂ ker ρ P admits a contact isotropic realisation (cf. Section 4.1). The global obstruction to constructing these objects is investigated in Section 4.2.

Lemma 14.
The restriction of the 1-form σ * θ to the foliation F satisfies d F σ * θ = ω, i.e. it is a local primitive for ω.
The condition of Lemma 14 is, in fact, sufficient to construct a local contact isotropic realisation of a regular Jacobi manifold all of whose leaves are even dimensional; this is the content of the next proposition, stated below without proof (it can be checked using local coordinates, for instance). Fix one such (P, L P , {·, ·}) and a period lattice Σ ⊂ ker ρ P .
Proposition 15. Let U ⊂ P be an open set and fix β ∈ Ω 1 (U, L) with the property that d F β = ω. Then is a contact isotropic realisation.
The next lemma shows that the imageD(Γ(ker ρ P /Σ)) consists solely of d Fclosed 1-forms, where d F is defined as in equation (15).
Proof. It suffices to check that d F D(α)(ρ P (j 1 u), ρ P (j 1 v)) = 0 for any u, v ∈ Γ(L P ) (cf. the proof of Lemma 6). Fix two such sections of L P ; by definition of d F , where the second equality uses properties of the anchor map ρ P , of the Lie bracket on Γ(J 1 L P ), and the definition of∇ (cf. equation (14)). Let z ∈ Γ(L P ) be any section; then where the first and fourth equalities use the compatibility condition for the Spencer operator of equation (7), the second uses that α ∈ Γ(ker ρ P ), the third exploits anti-symmetry of the Lie bracket, and the last follows by definition of the J 1 L Pconnection ∇ (cf. equation (6)). Applying the above equality to the right hand side of equation (35), it follows that where c.p. stands for cyclic permutation and the last equality follows from the Jacobi identity. This proves that d F D(α) = 0, thus completing the proof.
Let C ∞ (ker ρ P /Σ) and Z 1 (F ; L P ) denote the sheaves of smooth sections of ker ρ P /Σ → P and of foliated closed 1-forms with values in L P respectively. Lemma 16 implies that there is a morphism of sheaves (36)D : C ∞ (ker ρ P /Σ) → Z 1 (F ; L P ), which induces a homomorphism at the level of cohomology with values in the above sheaves, (37) D : H 1 (P ; C ∞ (ker ρ P /Σ)) → H 1 (P ; Z 1 (F ; L P )).
Recall that H 1 (P ; C ∞ (ker ρ P /Σ)) ∼ = H 2 (P ; Σ) (cf. the discussion preceding Definition 14), while a standard doubleČech-Lie algebroid differential complex argument implies that H 1 (P ; Z 1 (F ; L P )) ∼ = H 2 (F ; L P ). Using these identifications tacitly, the above yields a homomorphism D : H 2 (P ; Σ) → H 2 (F ; L P ) which plays a central role in the following theorem, which is the main result of this section.
Conversely, suppose that Dc = [ω]. Choose the good open cover U so that, for each i, there exists a primitive for ω| Ui , i.e. there exists β i ∈ Ω 1 (F | Ui ; L P ) with d F β i = ω| Ui . Since F is regular, the foliated 1-forms β i can be extended to honest 1-forms (i.e. elements of Ω 1 (U i ; L P )), which are henceforth also denoted by β i by abuse of notation. Since Dc = [ω], there exists aČech cocycle {t ij } representing c (subordinate to the above good open cover) which satisfies t * ij θ 0 = β j − β i on U ij . For each i, consider the contact isotropic realisation π : (ker ρ P /Σ| Ui , θ 0 + π * β i ) → (U i , L P | Ui , {·, ·} P | Ui ) (cf. Proposition 15). The principal ker ρ P /Σ-bundle over P with Chern class c is constructed (up to isomorphism) by glueing the above local models using the translations t ij . The condition t * ij θ 0 = β j − β i ensures that the locally defined generalised contact forms θ 0 + π * β i patch together to give a globally defined generalised contact form θ ∈ Ω 1 (M ; φ * L P ), where φ : M → P is the principal ker ρ P /Σ-bundle over P with Chern class c constructed above. Setting H := ker θ, have that φ : (M, H) → (P, L P , {·, ·}) is a contact isotropic realisation as it suffices to check that the properties (IR1), (IR2) and (IR3) hold locally and they do by construction, since the bundle is obtained by glueing contact isotropic realisations. This completes the proof of the theorem.
Example 17. Theorem 17 can be used to construct examples of contact isotropic realisations all of whose invariants are non-trivial, i.e. with non-trivial line bundle, with period lattice Σ defining a non-trivial local system of coefficients and with non-zero Chern class. To this end, consider RP 2 with the tautological line bundle τ → RP 2 and the zero Jacobi bracket. Example 20 below shows how to construct a period lattice Σ RP 2 ⊂ J 1 τ . It can be checked that H 2 (RP 2 ; Σ RP 2 ) ∼ = Z 3 (cf. [30] for the explicit calculation). Since the Jacobi structure under consideration is trivial, the induced characteristic foliation is by points; in other words, H 2 (F ; τ ) = 0. This implies that the homomorphism D : H 2 (RP 2 ; Σ RP 2 ) → H 2 (F ; τ ) is identically zero; Theorem 17 therefore gives that all c ∈ H 2 (RP 2 ; Σ RP 2 ) give rise to contact isotropic realisations. Taking c = 0, obtain an example of contact isotropic realisation defined on a non-trivial line bundle with non-trivial invariants.

Transversal integral L-projective structures
The period lattice Σ associated to a contact isotropic realisation φ : (M, H) → (P, L P , {·, ·}) induces a geometric structure on the characteristic foliation of the Jacobi manifold (P, L P , {·, ·}). The aim of this section is to introduce this structure, which, in the case of the zero Jacobi structure on the trivial line bundle, has already been noticed (cf. [2]). Intuitively, this is the analogue of the geometric structure induced on the symplectic (regular!) foliation of on the base of a symplectic isotropic realisation of a Poisson manifold.
Before giving some examples of the above structure, it is useful to reformulate the existence of a flat T F -connection and property (T1) in Definition 18 in equivalent terms; this is done in the following lemma.
where ν * ⊂ T * N is the conormal bundle to the foliation, and Σ R → L is the restriction of the canonical projection J 1 L → L.
Proof. Suppose that (N, F ) admits a transversal integral L-projective structure, so that there exist ∇ and Σ as in Definition 18. The aim is to prove that the short exact sequence of equation (39) holds. First, it is shown that the map Σ R → L is pointwise surjective. Fix a point p ∈ P , an open neighbourhood p ∈ U and a trivialisation L| U ∼ = U ×R. By shrinking U if needed, it may be assumed that Σ| U is also trivialisable; properties (T1) and (T2) imply that there exist smooth functions u 1 , . . . , u l+1 ∈ C ∞ (U ) with the property that j 1 u 1 , . . . , j 1 u l+1 are a frame of Σ| U . It suffices to prove that there exists i such that u i (p) = 0. Suppose the contrary; since ∇ is a flat connection, it can be locally written by where X ∈ Γ(T F | U ), u ∈ C ∞ (U ) and β ∈ Ω 1 (F ) is a closed foliated 1-form. Property (T3) implies that, for all i, and for all However, dim ν * p = l, which contradicts the independence of j 1 u 1 , . . . , j 1 u l+1 on U . Since p ∈ P is arbitrary, this argument shows that Σ R → L is pointwise surjective. To complete the proof that equation (39) holds, it suffices to show that ν * ⊗ L = (T * P ⊗ L) ∩ Σ R . To this end, fix p ∈ P , let U be an open neighbourhood over which both L and J 1 L can be trivialised as above. Given any element of α p ∈ (T * p P ⊗ L) ∩ Σ R , (by possibly shrinking U if needed) there exists a smooth section α : U → (T * P ⊗ L)∩Σ R with α(p) = α p , since (T * P ⊗ L)∩Σ R = ker(Σ R → L) → P is a vector bundle of rank l. As above, fix a frame j 1 u 1 , . . . , j 1 u l+1 of Σ| U ; then for all X ∈ Γ(T F | U ), where D is the Spencer operator. This implies that D(α) ∈ Γ(ν * ⊗ L| U ), which, in turn, gives that α ∈ Γ(ν * ⊗ L| U ), since α ∈ Γ(ker(J 1 L → L)| U ). Thus α p ∈ ν * p ⊗ L p ; since both p and α p are arbitrary, this proves that (T * P ⊗L)∩Σ R ⊂ ν * ⊗L. Equality follows by dimension counting; this shows that Σ R fits in the short exact sequence of equation (39).
Conversely, suppose that Σ ⊂ J 1 L satisfies properties (T1) and (T2), and that the short exact sequence of equation (39) holds. Let v ∈ Γ(L) and p ∈ P ; since Σ R → L is surjective, there exists an open neighbourhood U of p and a section j 1 u ∈ Γ(Σ| U ) with u(q) = 0 for all q ∈ U . Then, locally, there exists a smooth The above is well-defined, for if v = f ′ u ′ for some other smooth nowhere vanishing section u ′ ∈ Γ(L| U ) with j 1 u ′ ∈ Γ(Σ| U ), then, by definition, f j 1 u − f ′ j 1 u ′ ∈ Γ(Σ R ), and Thus f j 1 u − f ′ j 1 u ′ ∈ ν * ⊗ L, which implies that, for all X ∈ Γ(T F | U ), L X (f )u = L X (f ′ )u ′ , giving that the formula (40) is independent of the choices made. It can be checked that equation (40) defines a flat T F -connection and that Σ satisfies property (T3) as required. This completes the proof of the lemma.
Example 19 (Radiant transversal integral affine structures). Let Ξ ⊂ ν * be a transversal integral affine structure on the foliated manifold (N, F ), i.e. Ξ is an embedded submanifold which is a full rank lattice of ν * and such that any (locally defined) smooth section σ ∈ Γ(Ξ) is closed. Such structures arise naturally in the study of isotropic realisations of Poisson manifolds (cf. [14]). The existence of a Ξ as above is equivalent to the existence of an atlas {(U i , κ i )} of submersions κ i : U i → R l locally defining the foliation F such that for all i, j with U ij = ∅, there exists h ij := (A ij , c ij ) ∈ GL(l; Z) ⋉ R l with κ j = h ij • κ i on U ij (cf. [32] and Theorem 19 below for an analogous statement for TIP-structures). Suppose further that Ξ is radiant, i.e. for all indices i, j as above, c ij = 0 (cf. [20,32] for a more intrinsic definition). Then Ξ gives rise to a TIP-structure on the trivial line bundle The following example is, in some sense, 'universal' (cf. Definition 21 and Theorem 19).
Example 20 (The tautological line bundle over RP l ). Let RP l be the real projective space and denote by τ → RP l the tautological line bundle, whose fibre at a point [l] ∈ RP l is given by τ [l] = l. The aim is to construct a transversal integral τ -projective structure on RP l endowed with the trivial foliation. Thus it suffices to construct a full-rank lattice Σ RP l ⊂ J 1 τ all of whose sections are holonomic, i.e. so that it satisfies properties (T1) and (T2).
The universal cover of RP l can be identified with R l+1 \ 0 and the pull-back of τ to R l+1 \ 0, denoted byτ is trivialisable. In fact, it can be described as follows: it is the line bundle whose fibre over r ∈ R l+1 \ 0 is given by R r . A nowhere vanishing section of τ is given by z : r → (r, r). The bundle J 1τ admits a full-rank lattice given byΣ := Z j 1 u 1 , . . . , j 1 u l+1 , where u i := ri r z, and r = (r 1 , . . . , r l+1 ). It can be checked thatΣ defines a transversal integralτ -projective structure on R l+1 \ 0. The quotient of R l+1 \ 0 by the R * -action by homotheties can be identified with RP l ; the quotient ofτ by the obvious lifted action yields τ . By construction, the lifted R * -action on J 1τ preservesΣ, which implies thatΣ induces a full-rank lattice Σ RP l ⊂ J 1 τ which satisfies property (T2), asΣ does. Thus Σ RP l defines a transversal integral τprojective structure.
Another way of constructing TIP-structures is through the following special class of foliated atlases.

Definition 21.
A transversal integral projective atlas of (N, F ) (TIP-atlas for short), is an atlas A := {(U i , χ i )} of submersions χ i : U i → RP l , locally defining F , and smooth maps A ij : U ij → GL(l + 1; Z) for all i, j with U ij = 0 satisfying Remark 7. There is a natural notion of two TIP-atlases being compatible (essentially inducing the same structure) and, thus, of maximal TIP-atlases. Henceforth, whenever two TIP-atlases are said to be equal, it means that they are compatible (and hence contained in the same maximal one).
Observation 13. A TIP-atlas {(U i , χ i )} on (N, F ) gives rise to a TIP-structure as follows. The line bundle L → N is defined by The reason why equation (41) yields a well-defined smooth line bundle is that {A ij } satisfies the cocycle condition. With this, there is a canonical inclusion of vector bundles Explicitly, χ is defined by sending the class of γ ∈ χ * i (J 1 τ ) m to the linear map As A ij ∈ GL(l + 1; Z), then is a well defined lattice sitting inside J 1 L via the map χ. Note that rk (Σ) = rk (Σ RP l ) = l + 1, and that Σ has properties (T1) and (T2). Finally, Lemma 18 implies that Σ ⊂ J 1 L defines a TIP-structure on (N, F ), since Σ R fits into the exact sequence (39) by construction.
In fact, Observation 13 can be strengthened, since every TIP-structure also gives rise to a TIP-atlas. This is the content of the following theorem.

Theorem 19. Then there is a 1-1 correspondence between
• a transversal integral L-projective structure Σ ⊂ J 1 L of (N, F ); • a transversal integral projective atlas {(U i , χ i )} of (N, F ). The correspondence is given by where the notation is as in Observation 13. Proof. Observation 13 provides a way to construct a TIP-structure starting from a TIP-atlas. It remains to prove that that a TIP-structure gives rise to a TIP-atlas and that the two procedures are inverse to one another.
Fix a TIP-structure Σ ⊂ J 1 L of (N, F ). Choose an open cover {U i } of N with the property that over each U i the line bundle π : L → N trivialises via the nowhere vanishing section z i : U i × R → π −1 (U i ), and denote the transition functions by c ij : U ij → GL(R). Without loss of generality, assume that all U i are small enough so that Σ| Ui can be trivialised with local frame α i 1 , . . . α i l+1 ∈ Γ(Σ| Ui ) and transition functions denoted by A ij : U ij → GL(l+1; Z). Property (T2) implies that for all i and each r = 1, . . . , l+1, there exist smooth functions g i r : U i → R with α i r = j 1 (g i r z i ). The TIP-atlas is going to be constructed using these functions g i r ; before proceeding to the construction, two preparatory claims need to be proved. Proof of Claim 1. Indeed, as j 1 g i 1 , . . . , j 1 g i l+1 are a frame of Σ| Ui and Σ R → L is pointwise onto by property (T4), it follows thatχ i = 0. Secondly, to prove thatχ i is transversal to E, i.e. Im Dχ i + E = T R l+1 , it suffices to show that χ * i : Ann(E) → ν * is injective. Using the same argument of Lemma 18, have that dg i r | F = g i r β | F , for some fixed closed, foliated 1-form β; hence (42) which implies that χ * i (Ann(E)) ⊂ ν * . To show thatχ * i : Ann(E)χ i (p) → ν * p is injective for any p ∈ U i , it suffices to show that it is surjective (which is, of course, equivalent by dimension counting). Let γ p ∈ ν * p and extend it to a locally defined smooth section γ ∈ Γ(ν * | Ui ). By property (T4) there exist smooth functions Using the Spencer decomposition, it follows that r f r g i r = 0 and r f r dg i r = γ. Therefore, and hence surjectivity follows.
Proof of Claim 2. This follows by definition of the mapsχ i .
Using Claims 1 and 2, a TIP-atlas can be defined as follows. Set χ i : U i → RP l to be the composite where q : R l+1 \ 0 → RP l is the quotient map. By Claim 1, χ i is a submersion, and Claim 2 implies that χ j = [A ij ] • χ i on U ij . Moreover, since Dq(E) = 0, equation (42) implies that Dχ i | F = 0, hence F is tangent to the fibres of χ i on the one hand, and on the other, dimension counting shows that for p ∈ U i , T F p = ker D p χ i . This shows that the atlas {(U i , χ i )} defines an TIP-atlas for (N, F ).
It remains to show that these constructions are inverse to each other. Start with a TIP-structure Σ ⊂ J 1 L, and construct the TIP-atlas {(U i , χ i )} as above. First, it is shown that L is isomorphic to χ * (τ ), as constructed in Observation 13. Observe that, for each i, χ * i τ is trivialisable; a nowhere vanishing section is given by p → (p, χ i (p)); this follows from the fact that χ i = q •χ i and q * τ is trivialisable. For each i, define a local trivialisation of χ * (τ )| Ui by . The transition functions with respect to these trivialisations are given by for p ∈ U ij . Since the cocycle of transition functions of χ * (τ ) and L are equal over the same open cover, it follows that they are canonically isomorphic. This identification can be used to show that the TIP-structure obtained from the TIPatlas {(U i , χ i )} as outlined in Observation 13 is the same as the original TIPstructure. Moreover, it is straightforward to check that given a TIP-structure obtained from a TIP-atlas as in Observation 13, the induced TIP-atlas is compatible with the original one. In both cases, details are left to the reader to check. Remark 8. While connectedness of the fibres of φ is not required in [21], it is assumed here for simplicity. Many of the arguments below carry over to the case of disconnected fibres with small modifications.
The aim of this section is to prove the following result, which relates the notion of Definition 22 to contact isotropic realisations. The key idea behind the proof of Theorem 21 is to construct θ U by fixing an element of ker Dφ to be its Reeb vector field. This can be achieved as follows. Fix m ∈ M and consider a small enough neighbourhood of m such that • there exists a local section X of ker Dφ which is transversal to H; • the contact distribution is locally defined as ker θ, for θ a locally defined contact form.
The 1-form θ U := θ/θ(X) is again a contact form which locally defines the contact distribution H and it has the property that X is its Reeb vector field (cf. [21, proof of Theorem 3]). Theorem 21 implies the local form of Proposition 20, as stated in the following corollary, which, in fact, proves a slightly stronger statement: There exists a unique Poisson structure Λ W on W = φ(U ) ⊂ P which makes φ : (U, θ U ) → (W, Λ W , W × R) into a contact isotropic realisation with the identity as bundle component.
Proof. Since φ is a submersion, it is an open map, which implies that W = φ(U ) ⊂ P is an open set. Let f, g ∈ C ∞ (W ) be smooth functions. Then, , where Λ U is the bivector defined by θ U , and the equality follows from the fact that X = R U ∈ Γ(ker Dφ). Since first integrals of ker Dφ| U are closed under the above Jacobi bracket (cf. Theorem 21 above), equation (43) implies that there exists a unique smooth bivector Λ W on W which is defined by the following equation . Note that equation (44) implies that Dφ(Λ U ) = Λ W . The bivector Λ W is Poisson since since Dφ(X) = 0 by Theorem 21. As the Jacobi structure defined by Λ W is completely determined by the pair (Λ W , 0) ∈ X 2 (W ) × X(W ), and Dφ(Λ U ) = Λ W and Dφ(X) = 0, this implies that φ : (U, θ U ) → (W, Λ W ) is Jacobi with the identity as bundle component (cf. Example 9); this proves property (IR1). Property (IR2) follows from the fact that X ∈ Γ(ker Dφ). The distribution ker Dφ contains the Reeb vector field of θ U and is pre-isotropic in the sense of [21,Definition 1]; by [21, part (i) of Theorem 2], it follows that ker Dφ ⊂ (ker Dφ) ⊥ , so that property (IR3) holds. This completes the proof of the corollary.
Proof of Proposition 20. Let {U i } be an open cover of M , where each U i is an open set of the type described in the statement of Theorem 21, so that, for each i, there exists a X -invariant contact form θ Ui . For each i, Corollary 22 gives a Poisson (and hence Jacobi) structure on W i := φ(U i ). Suppose that i, j are such that U ij = ∅; since both θ Ui and θ Uj are contact forms on U ij , there exists a nowhere vanishing function G ji : U ij → R with θ j = G ji θ i . As both θ Ui and θ Uj are X -invariant, it follows that for all X ∈ X , L X G ji = 0. Given that the orbits of X coincide with the fibres of φ, this implies that G ji is basic, i.e. there exists a unique nowhere vanishing function g ji : If Λ i denotes the Poisson bivector induced on W i as in Corollary 22 (and similarly for j), it follows from the construction that Λ j = g −1 ji Λ i = g ij Λ i on W ij , where the last equality follows from the fact that G ij = G −1 ji . Thus the Poisson bivectors Λ i and Λ j are isomorphic on W ij up to a conformal factor; equivalently, the induced Jacobi structures on the trivial line bundle over W ij defined by Λ i and Λ j are isomorphic with bundle component g ij . Moreover, since the collection of functions {G ji } satisfy the cocycle condition, it follows that {g ij } also do. Therefore, by Observation 4, there exists a line bundle L p → P (determined up to isomorphism), and a unique Jacobi structure (L P , {·, ·} P ) with the property that there exists a bundle isomorphism F : φ * L P → L M := T M/H (the latter depending on the explicit isomorphism used to identify L P → P as a representative of the isomorphism class of line bundles determined by the cocycle {g ij }). The properties in the statement of the proposition are local in nature, but these hold locally by Corollary 22, thus completing the proof.
Remark 9. A closer look at the proof of Proposition 20 yields further restrictions on (the isomorphism class of) the line bundle L P → P . Using the notation of the above proof, let Λ i , Λ j be Poisson bivectors on W i , W j respectively; then there exists a nowhere vanishing function g ij : W ij → R with Λ j = g ij Λ i on W ij . Since the self-commutator of both Λ i , Λ j under the Schouten bracket vanishes, the above equality implies that (45) − 2g ij Λ ♯ j (dg ij )Λ j = 0 on W ij , where the sign convention used is that of [29]. Since φ : (U j , θ Uj ) → (W j , Λ j ) is a complete isotropic realisation, it follows that Λ j is a regular Poisson bivector on W j (and, thus, on W ij , which is an open subset) -cf. Lemma 3. Therefore, it can be checked that equation (45) gives that Λ ♯ j (dg ij ) = 0, i.e. the function g ij is a local Casimir for Λ j . However, since Λ i = g −1 ij Λ j , it follows that g ij is also a local Casimir for Λ i . In other words, the transition functions of the line bundle L P → P can be chosen to be basic functions with respect to the foliation F which is locally defined as the symplectic foliation of the Poisson bivectors Λ i .
Complete pre-isotropic contact structures lie at the heart of non-commutative Hamiltonian integrability in the contact setting (cf. [21,Definition 3]).
Definition 23. Given an infinitesimal automorphism X of (M, H), the associated differential equationẋ = X is said to be non-commutatively Hamiltonian integrable if there exists an open dense subset M reg ⊂ M , a proper surjective submersion φ : M reg → P onto a smooth manifold P , and an Abelian Lie algebra X of infinitesimal automorphisms of (M, H) such that (M, H, φ, X ) is a complete pre-isotropic contact structure.
In light of Proposition 20, (the regular part of) a non-commutatively Hamiltonian integrable system on a contact manifold gives rise to a contact isotropic realisation. In particular, the structure theory for the latter developed in Sections 3 and 4 can be applied to find invariants of the former. For instance, Theorem 13 provides local action-angle coordinates for non-commutatively integrable Hamiltonian systems on contact manifolds (cf. [2,21] for alternative proofs).
6.2. The case of Poisson manifolds. Since a Jacobi manifold (P, L P , {·, ·}) admits a contact isotropic realisation only if it is regular and all of its leaves are even dimensional (cf. Lemma 3), it is worth rephrasing some of the results of Sections 3 and 4 for regular Poisson manifolds, as many of the above objects simplify in this case. Henceforth, let (P, Λ P ) be a regular Poisson manifold with symplectic foliation denoted by F ; the induced Jacobi structure is defined on the trivial line bundle L P := P × R (cf. Example 6). Following Example 10, the anchor ρ P : T * P ⊕ R → T P is given by ρ P (η, f ) = Λ ♯ P (η), for any (η, f ) ∈ T * P ⊕ R. Thus ker ρ P = ν * ⊕ R ⊂ T * P ⊕ R, while the Spencer operator is given by D(η, f ) := df − η.
Moreover, the T F -connection given by Proposition 5 is ∇ : Γ(T F ) × C ∞ (P ) → C ∞ (P ), (X, f ) → df (X); thus the differential d F defined by equation (15) is the restriction of the exterior derivative to the foliation F . The cohomology class [ω] ∈ H 2 (F ; L P ) = H 2 (F ) defined by Lemma 6 is that of the foliated symplectic form defined by Λ P .
Without loss of generality, consider a contact isotropic realisation φ : (M, θ) → (P, Λ P ) with the identity as bundle component, where θ ∈ Ω 1 (M ) is a contact form. Let Σ ⊂ ker ρ P denote the period lattice of φ; by Corollary 12, any local section of Σ is of the form j 1 f , for some f ∈ C ∞ (P ). However, since Σ ⊂ ν * ⊕ R, it is important to notice that this does not hold in general, even for regular Jacobi structures defined on the trivial line bundle. The equality ker ρ P = ν * ⊕ R can be exploited to give another way to construct the homomorphism of equation (37) which underpins the cohomological criterion of Theorem 17. Denote the sheaf of basic smooth functions on (P, F ) by C ∞ basic (P ; F ), i.e. it consists of smooth the trivial Z k -system of coefficients. Then (modulo torsion), an element in H 2 (P ; Σ) can be written as where, for each i, ω i ∈ Ω 2 (P ) is a 2-form with integral cohomology class. Since the map D : H 2 (P ; Σ) → H 2 (F ) is induced by the projection pr : Σ → C ∞ basic (P ; F ) and Σ is a trivial bundle, where, for each i, f i ω i is a closed foliated 2-form since f i is basic.
6.3. Locally conformal symplectic manifolds. As a final example, the question of which locally conformal symplectic (lcs for short) manifolds admit contact isotropic realisations is tackled. Let (P, σ, τ ) be a lcs manifold; without loss of generality, it suffices to consider contact isotropic realisations of the form φ : (M, θ) → (P, σ, τ ), where θ ∈ Ω 1 (P ) is a contact form. Denote by (Λ M , R M ) the bivector and vector fields which uniquely define the Jacobi structure on (M, θ) (cf. Observation 3).
The proof of Proposition 25 relies on the following two results.
Lemma 26. A necessary condition for a lcs manifold (P, σ, τ ) to admit a contact isotropic realisation is that τ = df for some smooth function f ∈ C ∞ (P ). Moreover, Σ = Z Be −f (τ, −1) for some constant B ∈ R.
Proof. Let φ : (M, τ ) → (P, σ, τ ) be a contact isotropic realisation. The dimension of its fibres is 1 as it is equal to the corank of the Jacobi structure on P . By definition, ker ρ P = R (τ, −1) ; since the fibres of φ are compact and by smoothness of the period lattice, Σ = Z g(τ, −1) for some positive smooth function f ∈ C ∞ (P ). Corollary 12 implies that there exists a smooth function λ such that j 1 λ = (gτ, −g); in other words, τ = d(− log g), as required.
The following result is stated without proof, as it follows from a simple calculation.
Proof of Proposition 25. Lemma 26 proves that property (1) in the statement of the proposition is a necessary condition for (P, σ, τ ) to admit a contact isotropic realisation. Henceforth, without loss of generality, assume that τ = df and that the aim is to construct a complete isotropic realisation of (P, σ, df ) with period net Σ = Z j 1 e −f (all other cases follow by multiplying by a suitable constant). In this case, Lemma 27 gives that φ : (M, θ) → (P, σ, df ) is a contact isotropic realisation if and only if φ : (M,θ) → (P,σ) is; by Example 15, such a realisation exists if and only if [σ] is integral, and [σ] is the Chern class of the S 1 -bundle φ : M → P . Moreover, dθ = φ * σ , i.e.