Contact isotropic realisations of Jacobi manifolds
Mar\'ia Amelia Salazar and Daniele Sepe
This paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are contact realisations of minimal dimension. These arise in the study of integrable Hamiltonian systems on contact manifolds (cf. \cite{Banyaga,jovanovic), while also extending the Boothby-Wang construction of regular contact manifolds (cf. \cite{boothby_wang,geiges_book). The approach to the problem conjugates ideas from \cite{dd,dui with the recently developed tools of \cite{Maria-Crainic (in the context of the integrability of Jacobi structures). The main results of the paper are local smooth and contact normal forms for contact isotropic realisations, which, amongst other things, provide an intrinsic proof of the existence of local action-angle coordinates for integrable Hamiltonian systems, as well as a cohomological criterion to construct such realisations. Moreover, one of the smooth invariants of such realisations is interpreted as providing a type of transversal projective structure on the foliation of the underlying Jacobi structures.
c_i_r_line_bundle_1.tex
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\newcommand{\JM}{(P,L_P,\{\cdot,\cdot\})} % Jacobi manifold
\newcommand{\CM}{(M,H)} % Contact manifold
\newcommand{\SM}{(S,\omega)} % Symplectic manifold 
\newcommand{\PM}{(P,\Lambda_P)} % Poisson manifold
\newcommand{\ann}{\mathrm{Ann}} %Annihilator
\newcommand{\cotan}{\mathrm{T}^*} %cotangent
\newcommand{\HH}{\mathrm{H}} 
\newcommand{\JJ}{\mathrm{J}^1} %jet bundle
\newcommand{\PJM}{(P\times \RR, \bar{\pi})} % Poissonisation of \JM
\newcommand{\PCM}{(M \times \RR, \de (e^s \theta))} %Poissonisation of \CM 
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\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem*{cm}{Contact model}
\newtheorem*{coh}{Cohomological criterion}

\title{Contact isotropic realisations of Jacobi manifolds}
\author{Mar\'ia Amelia Salazar and Daniele Sepe}

%\email{dsepe@math.ist.utl.pt}
%\subjclass{37J35, 53A15, 57R17, 70H06}
\begin{document}
\maketitle
\begin{abstract}
  This paper investigates the local and global theory of contact
  isotropic realisations of Jacobi manifolds, which are contact
  realisations of minimal dimension. These
  arise in the study of integrable Hamiltonian systems on contact
  manifolds (cf. \cite{Banyaga,jovanovic}), while also extending the Boothby-Wang
  construction of regular contact manifolds
  (cf. \cite{boothby_wang,geiges_book}). The approach to the problem
  conjugates ideas from \cite{dd,dui} 
  with the recently developed tools of \cite{Maria-Crainic} (in the
  context of the integrability of Jacobi structures). The main results
  of the paper are local smooth and contact normal forms for contact
  isotropic realisations, which, amongst other things, provide an intrinsic proof of the
  existence of local action-angle coordinates for integrable
  Hamiltonian systems, as well as a cohomological criterion to
  construct such realisations. Moreover, one of the smooth invariants
  of such realisations is interpreted as providing a type of
  transversal projective structure on the foliation of the underlying
  Jacobi structures.
\end{abstract}
%\setcounter{tocdepth}{1}
%\tableofcontents

\section{Introduction}\label{sec:introduction}
This paper studies a certain family of contact realisations of Jacobi
manifolds. A {\bf contact structure} on a smooth manifold $M$ is a
maximally non-integrable smooth hyperplane distribution $H \subset TM$
(cf. Definition \ref{defn:contact_structure}); these can be thought of
as odd-dimensional analogues of symplectic structures (which are closed, non-degenerate 2-forms). This analogy is
far-reaching. For instance, as a symplectic manifold $(S,\omega)$
gives rise to a Poisson bracket on $C^{\infty}(S)$ (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 {\bf Jacobi structure} on a smooth manifold $P$ is a
local Lie bracket $\{\cdot,\cdot\}$ on the space of sections of some
fixed real
line bundle $L \to P$ (cf. Definition \ref{defn:jacobi_structure} and
\cite{Kirillov,Lich78,Lich83,Marle}). A contact manifold $(M,H)$ naturally
comes equipped with a Jacobi bracket on the space of sections of the
line bundle $T M
/H \to M$, which is not necessarily trivial, or even trivialisable,
{\it e.g.} the projectivisation of the cotangent bundle with canonical
contact structure
(cf. \cite[Appendix 4]{Arnold} for more details). Continuing with the analogy
\begin{equation*}
  \text{symplectic }: \text{ Poisson } = \text{ contact }: \text{ Jacobi}, 
\end{equation*}
\noindent
just as symplectic realisations of Poisson manifolds play an important
role in understanding the geometry of Poisson structures
(cf. \cite{cdw,CrainicFernandes:jdd,marius_ionut,karasev}), so do {\bf
  contact realisations} of Jacobi manifolds, {\it i.e.} surjective submersions $\phi :
\CM \to (P,L,\{\cdot,\cdot\})$ which are Jacobi morphisms
(cf. Remark \ref{rk:cir}). 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. \cite[Section 7]{CrainicFernandes:jdd} and \cite[Section 3.2]{Zambon}).\\

The types of contact realisations considered in this paper are those
of minimal dimension, {\it i.e.} the dimension of the total space $\CM$ 
is as small as possible. These are called {\bf contact isotropic
  realisations} and are contact realisations $\phi : \CM
\to (P, L,\{\cdot,\cdot\})$ with the extra condition that the fibres
are `isotropic' in the following sense. For each $m \in M$, maximal
non-integrability of $H$ induces a non-degenerate bilinear form on
$H_m$, {\it i.e.} a symplectic form. Then the fibres of $\phi$ are
isotropic if, for each $m \in M$, $\ker D_m \phi \cap H_m $ is an
isotropic subspace (cf. Definition \ref{defn:icr}). The simplest such example is given by the Boothby-Wang
construction of regular contact manifolds
(cf. \cite{boothby_wang,geiges_book}), which can be seen as contact
isotropic realisations of symplectic manifolds viewed as Jacobi
structures (cf. Example \ref{exm:rcm}). 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. \cite{dd,dui});
moreover, recently they have been studied in relation to Poisson
manifolds of compact type (cf. the forthcoming
\cite{PMCT}). The work in \cite{dd} 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. \cite{jovanovic} and
Section \ref{sec:integr-syst-cont} for further details). \\


% In the symplectic context, these are symplectic
% realisations $\phi : (S,\omega) \to (P,\pi)$ of 
% Poisson manifolds whose fibres are isotropic submanifolds. 

% A (non-commutatively)
% integrable Hamiltonian system on a $2n$-dimensional symplectic
% manifold $(S,\omega)$ is a map
% $(f_1,\ldots,f_k,f_{k+1},\ldots,f_{2n-k}) : (S,\omega) \to \RR^{2n-k}$
% with $\{f_i,f_j\} = 0$ for all $i=1,\ldots, k$, $j=1,\ldots, 2n-k$
% which is a submersion almost everywhere. The
% global geometry of (the regular part of) an integrable Hamiltonian
% system can be studied using symplectic isotropic realisations
% (cf. \cite{dd,dui}). This approach motivates the study of {\bf contact
%   isotropic realisations}, {\it i.e.} contact realisations $\phi : \CM
% \to (P, L,\{\cdot,\cdot\})$ with the extra condition that the fibres
% are `isotropic' in a sense that is the appropriate analogue of this
% condition in the symplectic setting (cf. Definition
% \ref{defn:icr}). Intuitively, this condition can be explained as
% follows. For each $m \in M$, maximal
% non-integrability of $H$ induces a non-degenerate bilinear form on
% $H_m$, {\it i.e.} a symplectic form. Then the fibres of $\phi$ are
% isotropic if, for each $m \in M$, $\ker D_m \phi \cap H_m $ is an
% isotropic subspace. There already exist examples of contact isotropic
% realisations in the literature, such as the Boothby-Wang
% construction of regular contact manifolds
% (cf. \cite{boothby_wang,geiges_book}) or the completely integrable
% contact forms of toric type (without singularities) considered in
% \cite{Banyaga}. As shown in this paper, contact isotropic
% realisations can be used to study the global geometry of a recently introduced notion of integrable
% Hamiltonian systems on contact manifolds (cf. \cite{jovanovic} and
% Section \ref{sec:integr-syst-cont} for further details). \\

The aim of this paper is to classify contact isotropic realisations of
Jacobi manifolds. The approach taken combines the framework of
\cite{dd,dui} developed to study symplectic isotropic realisations, with new techniques coming
from recent work on the integrability of Jacobi structures
(cf. \cite{Maria-Crainic} and also \cite{Maria} for even more
general methods applicable to Lie groupoids and
algebroids). The main difficulty to apply the ideas of \cite{dd,dui}
directly is the presence of {\em non-trivial line bundles}, which
complicate matters both conceptually and technically. This is
precisely why the methods of \cite{Maria-Crainic,Maria} are employed
in this paper; the upshot of this approach is to provide
{\em 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. \cite{Banyaga,boothby_wang,jovanovic}). \\

The isotropy condition
imposes a restriction on the type of Jacobi manifolds admitting a
contact isotropic realisation. Associated to a Jacobi
structure $(P,L,\{\cdot,\cdot\})$ is a {\bf Lie algebroid} structure
on the first jet bundle $J^1L$ of $L \to P$, which consists of a Lie
bracket on $\Gamma(J^1L)$ and a compatible map $\rho : J^1 L \to TP$
(cf. Proposition \ref{prop:alg} and \cite{Maria-Crainic}). This plays
a central role in this paper. A necessary
condition for $(P,L,\{\cdot,\cdot\})$ to admit a contact isotropic
realisation is that the structure be
regular with even dimensional
leaves, {\it i.e.} $\rho$ is of constant even rank (cf. Lemma
\ref{lemma:regular}). Examples of such Jacobi manifolds are
regular Poisson structures, zero Jacobi structures (defined on any
line bundle), and locally conformal symplectic manifolds (cf. Example
\ref{exm:lcs}). Once this necessary condition is established, the first main
result is a {\em smooth} classification of contact isotropic
realisations of a fixed Jacobi manifold $(P,L,\{\cdot,\cdot\})$; as in
the case of symplectic isotropic realisations, there are two
invariants:

\begin{itemize}
\item a {\bf period lattice} $\Sigma \subset \ker \rho \subset J^1 L$ (cf. Definition
  \ref{obs:per_lattice}), all of whose sections are {\it holonomic},
  {\it i.e.} of the form $j^1 u$ for some $u \in \Gamma(L)$
  (cf. Corollary \ref{cor:sections_period});
\item the {\bf Chern class} $c \in \HH^2(P;\Sigma)$ (cf. Definition \ref{defn:cc}).
\end{itemize}
\noindent
This follows from Theorem \ref{thm:local_model}, whose proof is
entirely analogous to that of \cite[Theorem 8.15]{Vaisman}. In fact,
the content of Theorem \ref{thm:local_model} is that a smooth local
model for a contact isotropic realisation with period lattice $\Sigma$
is given by $\ker \rho /\Sigma \to P$. \\

\noindent
The second main result is a {\em contact} local normal form (in a
neighbourhood of a fibre) for contact isotropic realisations. Recall
that, given a line bundle $L \to P$, its first jet bundle $J^1 L$
comes equipped with a canonical contact structure $H$, defined as the
kernel of the Cartan 1-form $\theta_{\mathrm{can}}\in
\Omega^1(J^1L;L)$ (cf. Example \ref{ex:jet_bundle} for further
details).

\begin{cm}
  A contact isotropic realisation $\phi :\CM \to
  (P,L,\{\cdot,\cdot\})$ is locally isomorphic to
  $$ \pi : (\ker \rho/\Sigma, \ker(\theta_0 + \pi^*\beta)) \to
  (P,L,\{\cdot,\cdot\}), $$
  \noindent
  where $\theta_0 \in \Omega^1(\ker \rho/\Sigma;L)$ is induced by
  $\theta_{\mathrm{can}}$ and $\beta \in \Omega^1(P;L)$ is a locally
  defined 1-form determined (not uniquely!) by the Jacobi bracket.
\end{cm}
\noindent
This is the content of Theorem \ref{thm:aa}, 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
\cite{Banyaga,jovanovic}), 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. \\

\noindent
Given a regular Jacobi manifold $(P,L,\{\cdot,\cdot\})$ whose
foliation $\mathcal{F}$ 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 {\em once} a suitable $\Sigma \subset \ker \rho$
has been fixed, by providing a cohomological criterion which is
analogous to that of \cite[Theorems 4.2 and 4.3]{dd}.

\begin{coh}
  A cohomology class $c \in \HH^2(P;\Sigma)$ is the Chern class of a
  contact isotropic realisation of $(P,L,\{\cdot,\cdot\})$ if and only
  if $\mathcal{D}c = [\omega]$, where $\mathcal{D}: \HH^2(P;\Sigma)
  \to \HH^2(\mathcal{F};L)$ is a homomorphism induced by the classical Spencer operator of
  $J^1L$, and $[\omega]$ is a the cohomology class of a foliated 2-form with values in $L$
  determined by the Jacobi bracket.
\end{coh}

\noindent
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. \cite[Theorems 7.2.4 and 7.2.5]{geiges_book}), and the cohomological
criterion for the existence of completely integrable contact forms of
toric type (cf. \cite{Banyaga}). At the heart of the above result lies
the {\bf classical Spencer operator} $\mathrm{D} : \Gamma(J^1L) \to \Omega^1(P;L)$,
which recently has been discovered to play a central role in Jacobi geometry
(cf. Observation
\ref{ex:Spencer_operator} for a definition of $\mathrm{D}$ and
\cite{Maria-Crainic} for further results). \\

\noindent
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 $\mathcal{F}$ of a Jacobi manifold $(P,L,\{\cdot,\cdot\})$
admitting a contact isotropic realisation. The idea is to use give an
intrinsic characterisation of the period lattice $\Sigma \subset
J^1L$, {\it i.e.} which does not rely on the given realisation
(cf. Definition \ref{defn:tilp}). As
expected from the case $L = P \times \RR$, the ensuing geometric structure on $\mathcal{F}$ is
{\bf transversally integral projective}, {\it i.e.} it gives rise to a
foliated atlas with values in real projective spaces along with an
integrality condition (cf. Definition \ref{defn:tipa}); what is
surprising is that this result holds even when $L$ is non-trivial:
this is the content of Theorem \ref{thm:tilp_tilpa}. \\

The structure of this paper is as follows. Section
\ref{sec:gener-cont-jacobi} 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 \cite{Maria-Crainic}
(cf. Section \ref{sec:lie-algebr-assoc}). Contact isotropic
realisations are introduced and studied in Section
\ref{sec:cont-isotr-real}. After establishing a simple necessary
condition on the Jacobi structure for the existence of such a realisation (cf. Section
\ref{sec:defin-basic-prop}), 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 \ref{sec:prop-regul-jacobi}. The smooth and contact local
normal forms of contact isotropic realisations are constructed in
Sections \ref{sec:smooth-class} and \ref{sec:cont-class} respectively. The question
of constructing contact isotropic realisations, both locally and
globally, is dealt with in Section
\ref{sec:constr-cont-isotr}. Any regular Jacobi manifold all of whose
leaves are even dimensional admits {\em locally} a contact isotropic
realisation (cf. Section \ref{sec:local-construction}); the
cohomological problem of glueing these local constructions together is
solved in Section \ref{sec:cohom-crit}. The geometric structure
induced by a contact isotropic realisation on the foliation of a
Jacobi manifold is introduced in Section
\ref{sec:transv-integr-proj}. Some interesting examples and
applications of the theory developed are discussed in Section
\ref{sec:exampl-cont-isotr}. 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 \ref{sec:integr-syst-cont}),
and contact isotropic realisations of Poisson manifolds and of locally
conformal symplectic manifolds (viewed as Jacobi
structures on the trivial line bundle) in Sections
\ref{sec:case-poiss-manif} and \ref{sec:locally-conf-sympl}
respectively. Throughout this note, there are two types of comments,
labelled {\bf Observation} and {\it 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.


% Given a
% contact manifold $\CM$, the projection $TM \to TM/H$ can be seen as a
% 1-form $\theta \in \Omega^1(M;TM/H)$; the correspondence given by $H =
% \ker \theta$ allows to identify contact distributions with a class of
% 1-forms (with values in some line bundle). 


% \begin{itemize}
% \item connection to integrable systems on contact manifolds (when the
%   base is a Poisson manifold). Description of their global geometry
%   \'a la Dazord-Delzant \cite{dd} (inf fact, we should consider an
%   example and show what invariants crop up); 
% \item connection to Poisson manifolds of Jacobi compact type. Idea is
%   to investigate what kind of geometric structures are associated to
%   Jacobi manifolds of compact type; 
% \item remember to set up notation (sharp, flat, etc.), and to make it
%   uniform throughout;
% \end{itemize}

\subsection*{Notation and conventions}\label{sec:notation}
Throughout the paper, whenever a collection of open sets $\{U_i\}$ is
considered, $U_{i_1i_2\ldots i_N} = U_{i_1} \cap \ldots \cap U_{i_N}$
for any finite set of indices $i_1,\ldots,i_N$. All line bundles
considered in this paper are real unless otherwise stated.

\subsection*{Acknowledgements}
We would like to thank Camilo Arias Abad for interesting
conversations. M.A.S. would like to thank IMPA, CRM and MPIM Bonn for
hospitality at various stages of the project. M.A.S. was partly
supported by the DevMath programme of the Centre de Recerca
Matem\`atica and by the Max Planck Institute for Mathematics in Bonn. D.S. was partly supported by ERC starting grant 279729 and by the NWO
Veni grant 639.031.345.

\section{Generalities on contact and Jacobi manifolds}\label{sec:gener-cont-jacobi}
This section recalls basic definitions, properties and examples of
contact and Jacobi manifolds that are used throughout this paper
(cf. \cite{contact,Maria-Crainic,Dazord-Lich-Marle91,geiges,Marle}
amongst others for further details and proofs).

%%%%%%%%%%%%%%%%%%%%
%%% Contact manifolds %%%
%%%%%%%%%%%%%%%%%%%%

\subsection{Contact manifolds and their properties}\label{sec:cont-manif-their}

\begin{defn}[Contact manifolds]\label{defn:contact_structure} A {\bf contact manifold} is
  a pair $(M,H)$ consisting of a manifold $M$ and a smooth hyperplane
  distribution $H\subset TM$ with the property that the {\bf
    curvature} map defined on sections by
  \begin{eqnarray}\label{curvature}
    c:\Gamma(H)\times \Gamma(H)\to T M/H,\ c(X,Y):=[X,Y] \text{ mod } H,
  \end{eqnarray}
  \noindent
  is fibre-wise non-degenerate. The distribution $H$ is said to be a
  {\bf contact distribution}.
\end{defn}

\begin{rk}\label{rk:ct_man_trivial}
  If $H$ is co-orientable, {\it i.e.} the
  line bundle $T M/H \to M$ is trivialisable, then a choice of
  trivialisation for $T M/H \to M$ corresponds to a globally defined
  1-form $\theta \in \Omega^1(M)$ satisfying $H = \ker \theta$. In
  this case, the condition of equation
  \eqref{curvature} can be rephrased as non-degeneracy of $\de \theta|_H$.
\end{rk}

\begin{obs}\label{obs:ct_mfs}
  Most works in the literature deal with contact manifolds $(M,H)$ whose
  associated line bundle $L := T M/H \to M$ is co-oriented (cf. Remark
  \ref{rk:ct_man_trivial}). However, there are natural examples of
  contact manifolds which do not satisfy this hypothesis, {\it e.g.} the
  canonical contact structure on the projectivisation of any cotangent
  bundle (cf. Example \ref{ex:jet_bundle} for other important examples
  which are central to this work). In general, given a contact manifold $\CM$ as in Definition
  \ref{defn:contact_structure}, the contact structure $H$ can be
  described tautologically as the kernel of $\theta \in \Omega^1(M,L)$,
  where $$\theta : T M \to L = T M/H$$ is the projection. The
  condition of equation \eqref{curvature} can be rephrased as
  fibre-wise non-degeneracy of $(X,Y) \mapsto \theta([X,Y])$, for $X,Y
  \in H$. In fact, contact manifolds can be equivalently described as
  pairs $(M,\theta)$ consisting of a smooth manifold $M$ and a
  point-wise surjective 1-form $\theta \in \Omega^1(M,L)$, such
  that the curvature defined on sections by 
  \begin{equation}
    \label{eq:3}
    \Gamma(\ker \theta) \times \Gamma(\ker \theta) \to \Gamma(L),\ (X,Y) \mapsto
    c_{\theta}(X,Y):=\theta([X,Y]) 
  \end{equation}
  \noindent
  is fibre-wise non-degenerate. Throughout this paper, both points of
  views are used for different purposes and the 1-form
  satisfying $\theta = \ker H$ is referred to as a {\bf generalised
  contact form}.
\end{obs}

\begin{exm}\label{ex:jet_bundle}
Let $\pi:L\to P$ a line bundle and denote by $pr :J^1L\to P$ the {\bf first jet bundle}
\[J^1 L|_x=\{j^1_xu\mid u\in\Gamma(L)\}.\]
Any (local) section $u \in \Gamma(L)$ induces a (local) section
$j^1u \in \Gamma(J^1L)$ which is defined by $p\mapsto j^1_pu$; these are called
{\bf holonomic} sections. The Cartan contact form   
$$ \theta_{\mathrm{can}}\in\Omega^1(J^1L,pr^*L)$$
detects holonomic section in the sense that a section $\xi$ of
$J^1L$ is holonomic if and only if $\xi^*\theta_\text{can}=0$. The
Cartan contact form is defined as follows. By abuse of notation denote
by $pr:J^1L\to L$ the canonical
projection and let $X \in T_{j^1_p u} J^1 L$. Then  
$$\theta_{\mathrm{can}}(X):=D pr(X)-D u(D\pi(X)),$$ 
lies in $\ker D\pi \cong pr^*L$. As
its name suggests, the Cartan contact form $\theta_{\mathrm{can}}$ is
a generalised contact form.
\end{exm}

Throughout this subsection, fix a contact manifold $(M,H)$ unless
otherwise stated, and denote by $$L=TM/H$$ the induced line bundle and by $\theta:TM\to L$ the projection. 

%\begin{obs}\label{obs:lb}
  %Any contact distribution $H \subset T M$ can be written locally
  %as the kernel of a 1-form, which also defines a local trivialisation
  %of $L$. Cover $M$ by open sets $U_i$ and choose local forms
  %$\theta_i \in \Omega^1(U_i)$ with $H|_{U_i} = \ker \theta_i$. For
  %any indices $i,j$ with $U_{ij} \neq \emptyset$, let $G_{ij}
  %\in C^{\infty}(U_{ij})$ be the nowhere vanishing function
  %such that $\theta_i = G_{ij} \theta_j$ on $U_{ij}$. For each such
  %pair $i,j$, the function $G_{ij}$ can be thought of as a smooth map
  %$G_{ij} : U_{ij} \to \mathrm{GL}(\RR)$. The collection $\{G_{ij}\}$ satisfies the cocycle condition and,
  %therefore, defines a line bundle over $M$ which is
  %isomorphic to $L$.
%\end{obs}

\begin{defn}(Reeb vector fields)\label{defn:Reeb_vector_field} 
  A {\bf Reeb vector field} of the contact manifold $(M,H)$ is any vector field $R$ satisfying
  \[[R,\Gamma(H)]\subset\Gamma(H).\]
  \noindent
  The vector space of Reeb vector fields is henceforth denoted by $\X_{\mathrm{Reeb}}(M,H)$.
\end{defn}

The following lemma relates $\X_{\mathrm{Reeb}}(M,H)$ with $\Gamma(L)$
(cf. \cite{Maria-Crainic} for its proof).

\begin{lemma} \label{lemma:reeb_sfns}
  The map
  \begin{equation*}
    \begin{split}
      \X_{\mathrm{Reeb}}(M,H) &\to \Gamma(L) \\
      R &\mapsto \theta(R)
    \end{split}
  \end{equation*}
  \noindent
  is a vector space isomorphism.
\end{lemma}

\begin{obs}\label{obs:rvf}
  The vector field $R_u$ associated to $u \in \Gamma(L)$
  is called the {\it Reeb vector field} of $u$ and is uniquely defined
  by $\theta(R_u) = u$ and $\theta([R_u,H])=0$, where $\theta \in
  \Omega^1(M,L)$ is a generalised contact form. If $f \in
  C^{\infty}(M)$, it can be checked that
  $$ R_{fu} = fR_u + c^{\sharp}_{\theta}(- \de f|_H \otimes u), $$
  \noindent
  where $\theta ([c^{\sharp}_{\theta}(\eta), \cdot]) = -
  \eta$ for any section $\eta \in \Gamma(\mathrm{Hom}(H,L))$ (and the
  convention followed is the same as in \cite{Marle}).
\end{obs}

\begin{rk}\label{rk:reeb_trivial_case}
  The above notion of Reeb vector fields generalises the commonly found
  notion of \emph{the} Reeb vector field associated to a co-oriented
  contact structure, which, in the above description, is nothing but $R_1$.
\end{rk}

The isomorphism of Lemma \ref{lemma:reeb_sfns} allows
to define another geometric structure associated to a
contact manifold $\CM$, namely the {\bf Reeb bracket} on $\Gamma(L)$,
given by
\begin{equation}
  \label{eq:2}
  \{u,v\} := \theta([R_u,R_v]),
\end{equation}
\noindent
for $u,v \in \Gamma(L)$. The triple $(M, L, \{\cdot, \cdot\})$ gives
rise to a Jacobi structure, as explained in the next subsection.

%%%%%%%%%%%%%%%%%%%
%%% Jacobi structures %%%
%%%%%%%%%%%%%%%%%%%

\subsection{Jacobi structures}\label{sec:jacobi-struct-defin}
As symplectic forms are special examples of Poisson structures, so are
contact distributions instances of a more general geometric structure, first
introduced in \cite{Kirillov,Lich78}, and further explained in \cite{Marle}.

\begin{defn}[Jacobi structure]\label{defn:jacobi_structure} A {\bf
    Jacobi structure} on a manifold $P$ is a pair
  $(L,\{\cdot,\cdot\})$ consisting of a line bundle $L\to P$, and a
  Lie bracket $\{\cdot,\cdot\}$ on $\Gamma(L)$ with the property that
  it is {\it local}, in the sense that 
  \[
  \mathrm{supp}(\{u,v\})\subset \mathrm{supp}(u)\cap \mathrm{supp} (v)\
  \ \ \forall u,v\in\Gamma(L),
  \]
  \noindent
  where $\mathrm{supp}(u)$ denotes the support of $u$. The triple
  $(P,L,\{\cdot,\cdot\})$ consisting of a manifold $P$ and a Jacobi
  structure $(L,\{\cdot,\cdot\})$ is henceforth referred to as a
  {\it Jacobi manifold}.
\end{defn}

% \begin{rk}\label{rk:local}
%   The local condition in Definition \ref{defn:jacobi_structure}
%   is proved to be equivalent to $\{\cdot,\cdot\}$ being a differential
%   operator of order at most one in each argument
%   (cf. \cite{Kirillov}). In fact, the objects defined above first
%   appear in \cite{Kirillov} under the name {\em local Lie algebras}.
% \end{rk}

% Before providing a list of examples, the following observation
% reformulates Definition \ref{defn:jacobi_structure} in the case when
% $L = P \times \RR$.

\begin{obs}[cf. \cite{Lich78}]
\label{obs:kir}
If $L=P \times \RR$, a Jacobi structure $\{\cdot, \cdot\}$ is
completely described by a pair
$(\Lambda, R) \in \X^2(P) \times \X(P)$, satisfying 
\begin{equation}
  \label{eq:10}
  \llbracket \Lambda,\Lambda \rrbracket =2R\wedge\Lambda,\ \
  \llbracket \Lambda,R \rrbracket=0,
\end{equation}
\noindent
where $\llbracket \cdot, \cdot \rrbracket$ is the Schouten
bracket. The Lie bracket on $\Gamma(L) = C^{\infty}(P)$ is
given by 
$$ \{f,g\} := \Lambda(\de f, \de g) + f (Rg) - g(Rf), $$
\noindent
for $f,g \in C^{\infty}(P)$.
\end{obs}

\begin{exm}\label{exm:ctc_jacobi}
  A {\bf contact manifold} $\CM$ comes equipped with a natural Jacobi
  structure $(L, \{\cdot,\cdot\})$, where $L = T M/H \to P$ and
  $\{\cdot,\cdot\}$ is the Reeb bracket
  \eqref{eq:2}.
\end{exm}

\begin{exm}\label{exm:po_as_ja}
  A {\bf Poisson manifold} $(P,\Lambda)$ is a manifold $P$ along
  with a bivector field $\Lambda \in \X^2(P)$, which satisfies $\llbracket
  \Lambda,\Lambda \rrbracket=0$. In light of Observation \ref{obs:kir},
  Poisson manifolds $(P,\Lambda)$ are precisely Jacobi structures on $P
  \times \RR \to P$ with $\Lambda_P = \Lambda$ and $R_P
  \equiv 0$.
\end{exm}

\begin{exm}\label{exm:lcs}
  A {\bf locally conformal symplectic manifold} is a triple
  $(P,\sigma, \tau)$, where $P$ is a manifold, $(\sigma, \tau)
  \in \Omega^2(P) \times \Omega^1(P)$, $\sigma$ is non-degenerate, $\de \tau = 0$,
  and $\de \sigma = - \tau \wedge \sigma$. Following \cite{Marle},
  given $(P,\sigma,\tau)$ 
  define a bivector $\Lambda \in \X^2(P)$ 
  and a vector field $R \in \X(P)$ uniquely by
  \begin{equation*}
    \iota(R) \sigma = - \tau \qquad \text{and} \qquad
    \iota(\Lambda^{\sharp}(\eta)) \sigma = - \eta
  \end{equation*}
  \noindent
  for all $\eta \in \Omega^{1}(P)$, where $\iota$ denotes the
  interior product. Then $(P,\Lambda_P,R_P)$ is a Jacobi manifold
  (cf. \cite{Marle}).
\end{exm}

% \begin{exm}\label{exm:marle}
%   The work in \cite{Marle} provides examples of Jacobi structures
%   defined on not necessarily trivial line bundles which generalise
%   Examples \ref{exm:po_as_ja} and \ref{exm:lcs}.
% \end{exm}

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 \ref{defn:jacobi_structure}.

\begin{defn}[Jacobi maps]\label{defn:jacobi_maps}
  Let $(N, L_N,\{\cdot,\cdot\}_N)$ and $(P, L_P, \{\cdot,\cdot\}_P)$ be Jacobi
  manifolds such that there exists an isomorphism $F: \phi^*L_P \to
  L_N$. A map $\phi : N \to P$ is said to be {\bf Jacobi} with 
  {\bf bundle component} $F$ if for all $u,v \in \Gamma(L_P)$ 
  $$ \{F \circ \phi^*u, F \circ \phi^*v\}_N = F \circ \phi^*\{u,v\}_P. $$
\end{defn}

\begin{obs}\label{obs:Jac_struc_locally}
  Fix a Jacobi manifold $(P, L, \{\cdot,\cdot\})$. Let $\{U_i\}$ be an
  open cover of $P$ with the property that, for each $i$, there exists
  a nowhere vanishing section $v_i : U_i \to L$, which 
  trivialises $L|_{U_i} \cong U_i \times \RR$. Then, for each $i$,
  there is a unique Jacobi structure $(U_i \times \RR,
  \{\cdot,\cdot\}_i:=\{\cdot,\cdot\}|_{U_i})$. If $i,j$ are such that $U_{ij} \neq \emptyset$, then there exists a nowhere vanishing function
  $g_{ij} : U_{ij} \to \RR$ such that $v_j = g_{ij} v_i$ on
  $U_{ij}$. The Jacobi manifolds $(U_{ij},U_{ij} \times \RR , \{
  \cdot, \cdot\}_j|_{U_{ij}})$ and $(U_{ij},U_{ij} \times \RR , \{
  \cdot, \cdot\}_i|_{U_{ij}})$ are Jacobi isomorphic with bundle
  component $g_{ij}$ (viewed as a bundle isomorphism $U_{ij} \times
  \RR \to U_{ij} \times \RR$). Conversely, given an open cover
  $\{U_i\}$ of $P$ such that
  \begin{itemize}
  \item for each $i$, there is a Jacobi structure on $U_i \times \RR$;
  \item for each pair $i,j$ with $U_{ij} \neq \emptyset$, the Jacobi
    structures on $U_{ij} \times \RR$ are isomorphic with bundle
    component $g_{ij} : U_{ij} \to \mathrm{GL}(\RR)$;
  \item the collection of nowhere vanishing functions $\{g_{ij}\}$
    satisfy the cocycle condition, {\it i.e.} for all $i,j,l$ for
    which $U_{ijl}\neq \emptyset$, $g_{il} = g_{jl} g_{ij}$; 
  \end{itemize}
  then there exists a unique Jacobi structure on the line bundle $L
  \to P$ whose (isomorphism class) is uniquely defined by the
  collection of $\{g_{ij}\}$ (cf. \cite[Section
  1.4]{Dazord-Lich-Marle91}).
\end{obs}

\begin{exm}\label{obs:jac_trivial}
  If, in the above definition, $L_N$ and $L_P$ are trivial, and the
  bundle component is the identity, the condition that $\phi :
  (N,L_N,\{\cdot,\cdot\}_N) \to (P, L_P, \{\cdot,\cdot\}_P)$ is a
  Jacobi map is equivalent to asking that $\Lambda_N$ and $R_N$ are
  $\phi$-related to $\Lambda_P$ and $R_P$ respectively, where
  $(\Lambda_N,R_N)$ and $(\Lambda_P,R_P)$ are the bivector and vector
  fields which uniquely determine the Jacobi structures on $L_N$ and
  $L_P$ respectively (cf. Observation \ref{obs:kir}).
\end{exm}

\subsection{The Lie algebroid associated to a Jacobi
  manifold}\label{sec:lie-algebr-assoc} 
Given a Jacobi manifold $(P,L,\{\cdot,\cdot\})$, the first jet bundle
$J^1 L \to P$ can be endowed with the structure of a {\bf Lie algebroid}, which
codifies the geometry of $(P,L,\{\cdot,\cdot\})$ --
cf. \cite{CrainicFernandes:lecture} for a definition and examples of
Lie algebroids. This
Lie algebroid is described in \cite{Kerbrat} for the trivial line
bundle, in \cite{Dazord} in the general case, and more recently from a
different point of view in \cite{Maria-Crainic}. Its defining properties are given in the proposition
below, stated without proof (cf. \cite{Maria-Crainic}).

\begin{prop}\label{prop:alg}
  Given a  Jacobi manifold $(P, L,\{\cdot,\cdot\})$, there exists a
  Lie algebroid structure on $J^1 L$ which is 
  uniquely characterised by the following properties:
  \begin{enumerate}[label=(\Roman*), ref=(\Roman*)]
  \item\label{item:A} The anchor map $\rho: J^1 L\to T P$ satisfies 
    \[\{u,fv\}=f\{u,v\}+L_{\rho(j^1u)}(f)v,\]
    for all $u,v\in\Gamma(L),$ $f\in C^\infty(M)$;
  \item\label{item:B} The Lie bracket on $\Gamma(J^1 L)$ satisfies
    \[[j^1u,j^1v]=j^1\{u,v\},\ \ \ \forall u,v\in\Gamma(L).\]
  \end{enumerate}
  This Lie algebroid is henceforth referred to as {\it the associated}
  Lie algebroid to the Jacobi manifold $(P,L,\{\cdot,\cdot\})$.
\end{prop}

\begin{exm}[Trivial line bundle, cf. \cite{jacobi}] \label{exm:trivial}
  For Jacobi structures on the trivial line bundle $L=P \times
  \mathbb{R}$, uniquely defined by a pair $(\Lambda,R)$ as in
  Observation \ref{obs:kir}, the Lie algebroid structure on $J^1 L =
  J^1 P = T^*P\oplus \mathbb{R}$ has anchor $\rho=\rho_{\Lambda,R}$ given by
  \begin{equation*} \rho(\eta, \lambda)=
    \Lambda_P^{\sharp}(\eta)+ \lambda R_P,
  \end{equation*}
  \noindent
  for any element $(\eta,\lambda) \in \cotan P \oplus \RR$, and Lie
  bracket defined by
  \begin{equation*}
    [(\eta, f), (\xi, g)]= ([\eta, \xi]_{\Lambda}- i_{R}(\eta\wedge
    \xi)+ fL_R(\xi)- gL_{R}(\eta), \Lambda(\eta,\xi)+ L_{\rho(\eta,
      f)}(g)- L_{\rho(\xi, g)}(f)), 
  \end{equation*}
  \noindent
  for any two sections $(\eta,f), (\xi,g) \in \Gamma(J^1 P)$.
  % (extended to general elements using bilinearity and the Leibniz identity).
\end{exm}


\begin{obs}\label{obs:contact_algebroid} 
  For a contact manifold $\CM$, where $H = \ker \theta$, with $\theta
  \in \Omega^1(M,L)$, the Lie algebroid structure of $J^1L$ associated to
  the underlying Jacobi structure defined in Example
  \ref{exm:ctc_jacobi} can be expressed in terms of the
  Reeb vector fields, and the curvature map $c_{\theta}$
  (cf. equation \eqref{eq:3}). Using the defining property
  \ref{item:A} of the anchor and the definition of the Reeb bracket of
  equation \eqref{eq:2}, it follows that $\rho(j^1u) = R_u$. This
  implies that, given $(u,\eta) \in
  \Gamma(J^1 L)$, 
  \begin{equation*}
    \rho(u, \eta) = R_u + c^{\sharp}_{\theta}(\eta|_H) = R_u + c^{\sharp}_{\theta}(\mathrm{D}(u,\eta)|_H).
  \end{equation*}
\end{obs}

\begin{obs}[The Spencer operator of a Jacobi manifold, cf. \cite{Maria-Crainic}]\label{ex:Spencer_operator}
The first jet bundle $J^1 L \to P$ fits into a short exact sequence of
vector bundles
\begin{equation}
  \label{eq:11}
  0\rmap \cotan P \otimes L \stackrel{i}{\to} J^1L \stackrel{pr}{\to} L\rmap 0,
\end{equation}
\noindent
where $pr: J^1 L \to L$ is the standard projection, and $ i(df\otimes
u)= f j^1(u)- j^1(fu)$ for $f \in C^{\infty}(P)$ and $u \in \Gamma(L)$. While the above
sequence is {\em 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 {\bf
  Spencer decomposition}, is used
extensively in this paper; it is defined by $u \mapsto j^1u$,
for $u \in \Gamma(L)$. This induces the Spencer decomposition 
\[ \Gamma(J^1L)\cong \Gamma(L)\oplus\Gamma( \textrm{Hom}(TP, L))  ,\]
\noindent
where the $C^{\infty}(P)$-module structure on the right hand
is given by 
\[ f\cdot (u, \phi)= (f u, \phi+ df\otimes u).\]
The projection to the second component 
$$\mathrm{D}:\Gamma(J^1L)\to\Omega(P;L), \ \ (u,\eta)\mapsto\eta$$ 
\noindent
is the so-called {\bf classical Spencer
  operator}; $\mathrm{D}$ is compatible with
the Lie algebroid of a Jacobi manifold $(P,L,\{\cdot,\cdot\})$ in the
following sense. First, $L$ is a representation of $J^1L$ with flat connection uniquely defined by 
\begin{eqnarray}\label{eq:nabla}
  \nabla:\Gamma(J^1L)\times\Gamma(L)\to\Gamma(L),\
  \ \nabla_{j^1u}(v)= \{u,v\}
\end{eqnarray} 
\noindent
(and extended using the Leibniz rule, cf. \cite{Maria-Crainic} for details). Secondly, $\mathrm{D}$ satisfies the
following conditions (of Spencer operators on Lie algebroids \cite{Maria})
\begin{align}
  \label{horizontal}\mathrm{D}_{\rho(\alpha)}(\alpha')&
  =\nabla_{\alpha'}(pr(\alpha))+ pr([\alpha, \alpha']) \\
  \label{vertical}
  \mathrm{D}_X[\alpha,\alpha']=\nabla_\alpha(\mathrm{D}_X\alpha')-&\mathrm{D}_{[\rho(\alpha),X]}\alpha'-\nabla_{\alpha'}(\mathrm{D}_X\alpha)+\mathrm{D}_{[\rho(\alpha'),X]}\alpha,
\end{align}
\noindent
for any $\alpha,\alpha'\in\Gamma(J^1L)$ and $X \in \X(P)$. Henceforth, the above Spencer
decomposition is going to be used tacitly unless otherwise stated.
\end{obs}





Jacobi manifolds whose underlying characteristic distribution $\Im\rho\subset TP$ has
constant rank play an important role in this paper (cf. Lemma \ref{lemma:regular}).

\begin{defn}\label{defn:rj}
  A Jacobi manifold $\JM$ is said to be {\bf regular} of corank $k$ if,
  for all $p \in P$, $\dim \ker \rho_{P,p} = k$.
\end{defn}

% \begin{obs}\label{obs:corank}
%   It is important to remark that the characteristic distribution $T
%   \mathcal{F}$ of a
%   regular Jacobi manifold $\JM$ of corank $k$ satisfies
%   $$ \dim T_p \mathcal{F} = \dim T_p P - (k-1) $$
%   \noindent
%   for any $p \in P$.
% \end{obs}

\begin{rk}\label{rk:notion_regular_jacobi}
   There exist other notions of `regular' Jacobi manifolds in the
   literature which are different to the one of Definition
   \ref{defn:rj} (cf. \cite{dlmp}). % The adjective `regular'
   % above refers to regularity of the associated Lie algebroid and, as
   % such, it generalises the notion of regularity for Poisson manifolds
   % to their Jacobi counterparts naturally. 
 \end{rk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Contact isotropic realisations %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Contact isotropic realisations: definition and invariants}\label{sec:cont-isotr-real}

 

\subsection{Definition and basic
  properties}\label{sec:defin-basic-prop}
The following definition introduces the main object studied in this
paper. 
\begin{defn}[Contact isotropic realisations]\label{defn:icr}
A {\bf contact isotropic realisation} of a Jacobi manifold
$\JM$ is a contact manifold $(M,H)$ together with
a surjective submersion $\phi:(M,H) \to (P,L_P,\{\cdot,\cdot\})$ with
connected, compact fibers satisfying the following properties
\begin{enumerate}[label=(IR\arabic*),ref=(IR\arabic*)]
\item \label{item:IR1}$\phi$ is a Jacobi map with bundle component $F:\phi^*(L_P)\to L_M$ (cf. Definition \ref{defn:jacobi_maps});
\item \label{item:IR2} $H = \ker \theta$ is transversal to $\phi$, {\it i.e.} 
$$H+\Ker D\phi=\mathrm{T}M;$$
\item \label{item:IR3} $\Ker D\phi\subset (\Ker D\phi)^{\perp} $,
  where 
  $$ (\Ker D\phi)^{\perp} := \rho_M(\phi^*J^1L_P)$$ 
  \noindent
  is the {\em pseudo-orthogonal distribution} of $\ker D\phi$, where
  $\phi^*J^1L_P$ is identified with its image in $J^1L_M$ via the map $F$.
\end{enumerate}
\end{defn}

Henceforth, whenever referring to an isotropic contact realisation
$\phi: (M,H) \to (P,L_P,\{\cdot,\cdot\})$, the dimensions of $M$ and of $P$ are fixed to be $2n+1$ and $2n+1-k$ respectively, unless otherwise stated.
\begin{rk}\label{rk:cir}
  \mbox{}
  \begin{itemize}
  \item The assumption on compactness of the fibres of a contact
    isotropic realisation can be weakened for what follows, but it is
    imposed at this stage to simplify the exposition;
  \item A surjective submersion $\phi : (M,H) \to
    (P,L_P,\{\cdot,\cdot\})$ with connected fibres satisfying only
    properties \ref{item:IR1} and \ref{item:IR2} is called a {\it
      contact realisation}. 
  \end{itemize}
\end{rk}

\begin{obs}\label{obs:icr}
Each of the properties in Definition \ref{defn:icr} have important implications.
\begin{enumerate}[label=(\arabic*),ref=\arabic*]
\item\label{item:comm} Property \ref{item:IR1} implies commutativity of the following diagram
$$ \xymatrix{J^1L_M \ar[r]^-{\rho_M} & TM \ar[d]^-{D\phi} \\
\phi^*J^1L_P \ar[u]^-{\phi^*} \ar[r]_-{\rho_M} & \phi^*TP,}$$
\noindent
where $\rho_M,\rho_P$ are the anchor maps for the Jacobi structures on
$P$ and $M$ respectively, and $\phi^*:\phi^*J^1L_P\to J^1L_M$  is
the injective vector bundle morphism given by $j^1u\mapsto
j^1(F\phi^*u)$. This can be seen as follows. Consider $\{u,gv\}_P$ for
any sections $u,v\in\Gamma(L)$ and any function $g \in C^\infty (P)$;
using the defining property \ref{item:A} of the anchor of a Lie
algebroid associated to a Jacobi structure together with property
\ref{item:IR1}, it can be shown that  
$$ \phi^*(\rho_P(j^1u)g) = \rho_M(j^1(F\phi^*u)\phi^*g),$$ 
\noindent
which, in turn, implies the claimed result;
\item \label{item:trans}Property \ref{item:IR2} can be used to show
  that for all $m \in M$, $\dim (H_m \cap \ker D_m\phi) = k-1$, since 
$$ \dim (H_m + \ker D_m\phi) - \dim H_m - \dim \ker D_m\phi = \dim
(H_m \cap \ker D_m\phi) $$ 
\noindent
and by property \ref{item:IR2}, for all $m \in M$, $H_m + \ker D_m\phi =T_mM$;
\item \label{item:po} When $\JM$ is Poisson, 
  condition \ref{item:IR3} is equivalent to requiring that, for all $m
  \in M$, the vector space $ \ker D_m\phi \cap H_m$ is isotropic with
  respect to the symplectic structure on $H_m$ induced by a suitable
  choice of contact form (cf. \cite{lib_leg} for details). This explains the nomenclature used in the
  definition. 
  % the pseudo-orthogonal $(\ker D\phi)^{\perp}$ of
  % $\ker D\phi$ can be written as ${\mathbb R}\langle R_M\rangle \oplus
  % \Lambda_M^\sharp(\ann(\Ker D\phi))\subset \mathrm{T}M$, since, by
  % Example \ref{exm:trivial}, $J^1P =T^*P \oplus \mathbb{R}$, which
  % implies that $\phi^*J^1P = \phi^* T^*P \oplus \mathbb{R}$. The fact
  % that $\phi$ is a surjective submersion implies that $\ann(\ker
  % D\phi) = \phi^* T^*P$, which in turn gives that ${\mathbb R}\langle R_M\rangle \oplus
  % \Lambda_M^\sharp(\ann(\Ker D\phi))=\rho_M(\phi^*J^1P)$ as claimed. Thus property
  % \ref{item:IR3} can be rephrased as saying that $$\ker D \phi \subset
  % {\mathbb R}\langle R_M\rangle \oplus
  % \Lambda_M^\sharp(\ann(\Ker D\phi)).$$
\end{enumerate}
\end{obs}

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 \cite[Proposition 8.10]{Vaisman}, which
proves an analogous statement for isotropic realisations of Poisson manifolds.

\begin{lemma}\label{lemma:regular}
For a contact isotropic realisation $\phi: (M,H) \to (P,L_P,\{\cdot,\cdot\})$ the Jacobi
structure on $P$ is regular with corank equal to $k$, and all its
leaves are even dimensional.
\end{lemma}

\begin{proof}
Fix points $p \in P$ and $m \in M$ such that $m \in
\phi^{-1}(p)$. Using item \eqref{item:comm} of Observation \ref{obs:icr},
\begin{equation}
\label{contando_rangos}
\rank \rho_{P,p}(J_p^1(L_P)))=\rank
(\rho_{M,m} \circ \phi^*(J^1L_P)) -\rank\ker (D\phi|_{\rho_{M,m}\circ\phi^*(J^1L_P)}).
\end{equation}
\noindent
On the one hand, property \ref{item:IR2} implies that
$\rho_M\circ\phi^* : \phi^*J^1L_P \to T M$ is injective: indeed,
Observation \ref{obs:contact_algebroid} gives that for
$(u,\eta)\in\Gamma(L_P)\oplus \Omega^1(P,L_P)\simeq \Gamma(J^1L_P)$,  
$$\rho_M(\phi^*(u,\eta))=R_{F(\phi^*u)}-c^\sharp(F\circ(\phi^*\eta|_H)).$$
If $\rho_{M,m}(\phi^*(u,\eta))=0$, then
$R_{F(\phi^*u)}(m)=c^\sharp(F\circ(\phi^*\eta|_H))(m)\in H$. By
definition of the Reeb vector field, this means
that $$F(\phi^*u)(m)=R_{F(\phi^*u)}(m)\mod H=0.$$ 
Hence, $F\circ(\phi^*\eta_m|_H)=0$ which in turns implies that
$\phi^*\eta_m|_H=0$. Condition \ref{item:IR2} implies that
$D\phi|_{H_m}:H_m\to T_{\phi(m)}P$ is surjective (as $\phi$ is a
surjective submersion); thus $\phi^*\eta_m|_H=0$ holds if and only if
$\eta_{\phi(m)}=0$, thus proving injectivity of $\rho_M \circ
\phi^*$. \\

Condition \ref{item:IR3} gives that $\rank\ker
(D\phi|_{\rho_M\circ\phi^*J^1L_P})=\rank \ker D\phi=k$. 
By definition $\rank J^1L_P=2n-k+2$, while injectivity of $\rho_M \circ
\phi^*$ implies that $\rank \rho_{M,m}(\phi^* J^1L_P) = 2n-k+2$;
equation \eqref{contando_rangos} yields $\rank
\rho_{P,p}(J_p^1(L_P)))=2n+2-k - k = 2n+2-2k$. Since $p \in P$ is
arbitrary, the proof of the lemma is completed. 
\end{proof} 


\subsection{Properties of regular Jacobi manifolds all of whose
  leaves are even dimensional}\label{sec:prop-regul-jacobi}
In light of Lemma \ref{lemma:regular} 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 $\JM$ be a regular Jacobi manifold all of
whose leaves are even dimensional, unless otherwise stated. The
notation employed below follows from that of Section
\ref{sec:defin-basic-prop}. % The following proposition
% shows that the vector bundle $\pi : \ker \rho_P \to P$ is naturally a
% bundle of abelian Lie algebras.
 
\begin{prop}\label{prop:bla}
  Let $(P,L_P,\{\cdot,\cdot\})$ be a regular Jacobi manifold with even dimensional
  leaves. Then $\pi: \ker \rho_P \to P$ is a
  bundle of abelian Lie algebras, so that the inclusion $\ker \rho_P
  \hookrightarrow J^1L_P$ is a morphism of Lie algebroids. 
\end{prop}


\begin{proof}
  Since $\ker \rho_P$ is the kernel of the anchor $\rho : J^1L_P \to
  TP$ and the restriction of $\pi : J^1L_P \to P $ to $\ker
  \rho_P$ is a vector bundle, it follows that $\pi: \ker
  \rho_P \to P$ is a bundle of Lie
  algebras. It remains to show that each such Lie algebra is
  abelian. \\ 
  
  Let $\nu^*\subset \cotan P$ denote the  {\em conormal bundle} of the
  regular foliation $\mathcal{F}$ whose distribution is
  $T\mathcal{F}:=\Im (\rho:J^1L_P\to TP)$. By assumption, the corank
  of the Jacobi structure is $k$, which implies that the rank of
  $\nu^*\to P$ is $k-1$. The claim is that $\ker
  \rho_P$ fits in a short exact sequence of vector bundles
  $$0\to\nu^*\otimes L_P\to \ker\rho_P\to L_P\to 0,$$
  \noindent
  where the projection $\ker \rho_P \to L_P$ is nothing but the
  restriction of $pr: J^1 L_P \to L_P$ (cf. Observation
  \ref{ex:Spencer_operator}). To this end, it suffices to prove that 
  \begin{equation}
    \label{eq:1}
    \nu^* \otimes L_P = \ker \rho_P \cap (\cotan P \otimes L_P) =
    \ker pr|_{\ker \rho_P}.
  \end{equation}
  \noindent
  For, if
  equation \eqref{eq:1} holds, then dimension counting implies that
  the restriction $pr : \ker \rho_P \to L_P$ is surjective. First, it
  is shown that $\nu^* \otimes L_P \subset \ker \rho_P$. Suppose that
  $\de f\otimes u \in\Gamma( \nu^*\otimes L_P)$, and fix $g\in C^\infty(P)$ and
  $v\in \Gamma(L_P)$. Using the $C^\infty(M)$-module
  structure of the Spencer decomposition of $\Gamma(J^1 L_P)$
  (cf. Observation \ref{ex:Spencer_operator}), have that $i(\de f\otimes
  u)=fj^1u-j^1(fu),$ and therefore $\rho_P(i(\de f\otimes
  u))=f\rho_P(j^1u)-\rho_P(j^1(fu))$. The defining property \ref{item:A}
  of the anchor implies that
  \begin{equation}
    \label{antisimetric}
    \begin{split}
      \mathcal{L}_{\rho(\de f\otimes u)}(g)v&=f\mathcal{L}_{\rho(j^1u)}(g)v-\mathcal{L}_{\rho(j^1(fu))}(g)v\\
      &=f(\{u,gv\}-g\{u,v\})-(\{fu,gv\}-g\{fu,v\})\\
      &=-\mathcal{L}_{\rho(\de g\otimes v)}(f)u\ =\ 0.
    \end{split}
  \end{equation}
  \noindent
  This shows that $\nu^*\otimes L_P \subset \ker\rho_P \cap
  (T^*P\otimes L_P)$. In order to show that equality holds, it suffices
  to check that, for each $p \in P$, $\dim \ker \rho_{P,p}
  \cap (\cotan P \otimes L_P)_p = k-1$. To this end, note that $\ker\rho_P \cap
  (T^*P\otimes L_P) = \ker \rho_P|_{\cotan P \otimes L_P}$. The map
  $\rho_P:T^*P\otimes L_P\to TP$ is, in fact, 
  antisymmetric, {\it i.e.} for any $\eta, \xi \in \cotan P \otimes
  L_P$, $\eta(\rho(\xi))=-\xi(\rho(\eta))$. This can be checked
  directly from the defining property of the anchor $\rho_P$ (in the
  case of trivial coefficients, this map is given by the sharp of a
  bivector). Thus, for each $p \in P$, the vector space
  $\rho_{P,p}(\cotan_p P \otimes L_{P,p}) \subset T_p \mathcal{F}$
  is even dimensional. On the other hand, since $k \geq \dim \ker \rho_{P,p}|_{\cotan_p
    P \otimes L_{P,p}} \geq k-1$, it follows that 
  $$ 2n - 1 \leq \dim \rho_{P,p}(\cotan_p P \otimes L_{P,p}) \leq
  2n, $$
  \noindent
  with $\dim \rho_{P,p}(\cotan_p P \otimes L_{P,p}) = 2n $ if
  and only if $\dim \ker \rho_{P,p}|_{\cotan_p P \otimes L_{P,p}} =
  k-1$. Since $\dim \rho_{P,p}(\cotan_p P \otimes L_{P,p})$ is even
  dimensional, it follows that $\dim \ker \rho_{P,p}|_{\cotan_p P \otimes L_{P,p}} =
  k-1$; as $p \in P$ is arbitrary, this completes the proof that
  equation \eqref{eq:1} holds. \\

  Using the above description of $\ker \rho_P$, it is possible to prove
  that $\ker\rho_P\to P$ is a bundle of abelian Lie algebras. Let
  $\alpha,\alpha'\in \Gamma(\ker\rho_P)$; the Spencer decomposition of
  Observation \ref{ex:Spencer_operator} gives that 
  $[\alpha,\alpha']=(pr[\alpha,\alpha'],\mathrm{D}[\alpha,\alpha'])$,
  where $\mathrm{D}:\Gamma(J^1 L_P)\to \Omega^1(M;L_P)$ is the classical
  Spencer operator. Writing
  $\alpha=(u,\eta)$ and $\alpha'=(v,\xi)$, where $u,v \in \Gamma(L_P)$
  and $\eta, \xi \in \Omega^1(P;L_P)$, the compatibility
  conditions of equations \eqref{horizontal} and \eqref{vertical}
  yield the following formula for the Lie bracket restricted to
  sections of $\ker \rho_P$
  \begin{equation}
    \label{eq:backet}
    [\alpha,\alpha']=(-\nabla_{\alpha'}u,\nabla_{\alpha}(\xi(\cdot))-\nabla_{\alpha'}(\eta(\cdot))),
  \end{equation}
  \noindent
  where $\nabla$ is the $J^1 L_P$-connection defined by equation
  \eqref{eq:nabla}. There are two cases to consider: the
  first case is when $u =0 = v$, and the second is when $ u = 0$ and
  $v \neq 0$. Note that, for dimensional
  reasons, $\Gamma(\ker\rho_P)$ is (locally) generated by elements of
  this form, and therefore, as $\ker\rho_P$ 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 $\nabla$ and of $\rho_P$ (cf. equation
  \eqref{eq:nabla} and property
  \ref{item:A} respectively), the following equality can be proved 
  \begin{equation}\label{segundo}
    \nabla_{\xi}s=-\xi(\rho_P(j^1s)),
  \end{equation}
  where $s\in\Gamma(L_P)$ is arbitrary (and analogously for $\nabla_\eta
  s$). Since $\xi, \eta \in \Gamma(\nu^* \otimes L_P)$, equation \eqref{segundo}
  implies that $[\eta,\xi] = 0$. Thus it only remains to consider the
  case $u=0$ and $v \neq 0$. Write $\alpha' = j^1 z + i(\gamma)$, for
  $\gamma \in \Omega^1(P,L_P)$; without loss of generality, the
  section $z \in \Gamma(L_P)$ can be taken to be nowhere vanishing at
  the cost of restricting its domain of definition, say to $U \subset
  P$. This does not matter to the purpose at hand, as the aim is to
  check that a {\em fibre-wise} Lie algebra structure is abelian and,
  to do so, there is a freedom in choosing the sections $\alpha,
  \alpha'$ under consideration. Since $\rho_P(\alpha') =0$, the
  defining property \ref{item:A} of $\rho_P$ implies that the operator 
  $\nabla_{\alpha'}(\cdot):\Gamma(L_P|_U)\to\Gamma(L_P|_U)$
  is $C^{\infty}(U)$-linear and hence it defines a vector bundle
  automorphism $L_P|_U\to L_P|_U$. 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 $G\in C^\infty(U)$ such that
  $$\nabla_{\alpha'} (s)(p)=G(p) s(p),$$
  for all $s\in\Gamma(L_P|_U)$ and $p\in U$. Hence,
  \begin{equation*}
  Gz=\nabla_{\alpha'}(z)=\{z,z\}-\gamma(\rho_P(j^1z))=-\gamma(\rho_P(\gamma))=0,
  \end{equation*}
  \noindent
  where antisymmetry of $\rho_P : \cotan P \otimes L_P \to L_P$ is
  used to obtain the last equality. As $z\in\Gamma(L_P|_U)$ does not
  vanish at any point, it follows that $G \equiv 0$. This, in turn,
  implies that $[\alpha,\alpha']=0$ in this case as well, thus
  completing the proof. 
\end{proof}

\begin{obs}\label{rk:nabla} 
  The previous proof shows that under the
  hypotheses of Proposition \ref{prop:bla}, the connection
  $\nabla:\Gamma(J^1L_P)\times
  \Gamma(L_P)\to\Gamma(L_P)$ (cf. equation \eqref{eq:nabla}) 
  restricted to $\Gamma(\ker\rho)\subset \Gamma(J^1L)$ is zero. 
\end{obs}

The next aim is to construct a cohomology class associated to $\JM$
which generalises the foliated cohomology class given by the leafwise
symplectic form induced by a regular Poisson manifold. As in the proof
of Proposition \ref{prop:bla}, denote the
characteristic foliation of $\JM$ by $\mathcal{F}$. 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 $T \mathcal{F}$-connection on $L_P$ associated to the Jacobi structure $(L_P,
\{\cdot,\cdot\}_P)$,
where the foliation $T \mathcal{F} \subset T P$ is viewed as a Lie
subalgebroid. Let $\nabla$ be the $J^1 L_P$-connection
on $L_P$ defined by equation \eqref{eq:nabla}. 



\begin{prop}\label{prop:well-defined}
  There is a unique $T \mathcal{F}$-connection $\bar{\nabla} : \Gamma(T \mathcal{F}) \times
  \Gamma(L_P) \to \Gamma(L_P)$ on $L_P$ uniquely
  defined by
  \begin{equation}
    \label{eq:26}
    \bar{\nabla}_{\rho_P(j^1u)} v := \nabla_{j^1 u}v = \{u,v\}.
  \end{equation}
\end{prop}

\begin{proof} 
  Observation \ref{rk:nabla} implies that $\bar{\nabla}$ is well-defined
  as a connection. Flatness of $\bar{\nabla}$ follows from that of
  $\nabla$, thus completing the proof.

  % To this end, recall that there is a short exact
  % sequence of Lie algebroids over $P$
  %$$ 0 \to \ker \rho_P \to J^1 L_P \xrightarrow{\rho_P} T
  %\mathcal{F} \to 0; $$
  %\noindent
  %restrict $\nabla$ to a $\ker \rho_P$-connection. Then it suffices to
  %prove that this restriction is trivial, {\it i.e.} for all $(u,\eta)
  %\in \Gamma(\ker \rho_P)$ and all $v \in \Gamma(L_P)$,
  %$\nabla_{(u,\eta)} v = 0$. \\
  
  %First, observe that since $\ker \rho_P \to P$ is a bundle of Lie
  %algebras, for any fixed $(u,\eta) \in \Gamma(\ker \rho_P)$, the map
  %$\nabla_{(u,\eta)} : \Gamma(L_P) \to \Gamma(L_P)$ is
  %$C^{\infty}(P)$-linear. For, if $f \in
  %C^{\infty}(P)$ and $v \in \Gamma(L_P)$,
  %$$ \nabla_{(u,\eta)} (fv) = f  \nabla_{(u,\eta)} v +
  %\mathcal{L}_{\rho_P(u,\eta)}f v = f  \nabla_{(u,\eta)} v, $$
  %\noindent
  %since $\rho_P(u,\eta) = 0$. Therefore it determines uniquely a
  %vector bundle morphism $L_P \to L_P$ or, in other words, a section
  %of $\mathrm{End}(L_P) \cong P \times \RR$, where triviality follows
  %from the fact that $L_P \to P$ is a line bundle (cf. the proof of
  %Proposition \ref{prop:bla}). However, the induced map $\nabla :
  %\Gamma(\ker \rho_P) \to \Gamma(\mathrm{End}(L_P))$ is
  %$C^{\infty}(P)$-linear by definition of $\nabla$. Thus it
  %determines a unique vector bundle morphism $\nabla : \ker \rho_P \to
  %\mathrm{End}(L_P)$, which is defined as follows. Let $(u_p,\eta_p)
  %\in \ker \rho_{P,p}$ and let $v_p \in L_p$ for a fixed point $p \in
  %P$. Then 
  %$$ \nabla(u_p,\eta_p)(v_p) := \nabla_{(u,\eta)} v, $$
  %\noindent
  %where $(u,\eta) \in \Gamma(\ker \rho_P)$ and $v \in \Gamma(L_P)$ are
  %(possibly locally defined) sections which satisfy $(u,\eta)(p) =
  %(u_p,\eta_p)$ and $v(p) = v_p$ respectively. To prove that $\nabla:
  %\Gamma(\ker \rho_P) \times \Gamma(L_P) \to \Gamma(L_P)$ is zero, it
  %suffices to show that the vector bundle map $\nabla : \ker \rho_P
  %\to \mathrm{End}(L_P)$ is zero. Fix $0 \neq (u_p,\eta_p) \in \ker
  %\rho_{P,p}$. Then either $u_p =0$ and $\eta_p \in \nu^*_p \otimes
  %L_{P,p}$, or $u_p \neq 0$ (cf. the proof of Proposition
  %\ref{prop:bla} for the notation). In the first case, pick a section
  %$\eta \in \Gamma(\nu^* \otimes L_P)$ with $\eta(p) = \eta_p$. Then
  %for any section $v \in \Gamma(L_P)$, 
  %$$ \nabla_{(0,\eta)} v = \eta(\rho_P(j^1v)) = 0$$
  %\noindent
  %by definition of $\nabla$ and since $\eta \in \Gamma(\nu^* \otimes
  %L_P)$. In particular, $\nabla(0,\eta_p) = 0$. In the second case,
  %since $\mathrm{End}(L_P) \cong P \times \RR$, associated to any
  %section $(u,\eta) \in \Gamma(\ker \rho_P)$, there exists a unique
  %smooth function $f_{(u,\eta)} \in C^*(P)$ satisfying
  %$$ \nabla_{(u,\eta)} v = f_{(u,\eta)} v $$
  %\noindent
  %for any $v \in \Gamma(L_P)$. The aim is to pick a suitable choice of section
  %$(u,\eta) \in \Gamma(\ker \rho_P)$ with $(u,\eta)(p) = (u_p,\eta_p)$
  %to prove that $f_{(u,\eta)} \equiv 0$, which implies that
  %$\nabla(u_p,\eta_p) = 0$. The claim is that there exists a (possibly
  %locally defined) section $(u,\eta) \in \Gamma(L_P)$ with
  %$(u,\eta)(p) = (u_p,\eta_p)$ and $u$ never vanishing. This can be
  %constructed as follows. First, extend $u_p \in L_p$ to a (possibly
  %locally defined) nowhere
  %vanishing section $u \in \Gamma(L_P)$. Since the map $\rho_P :
  %\cotan P \otimes L_P \to T \mathcal{F} = \rho_P(J^1 L_P)$ is
  %surjective (cf. the proof of Proposition \ref{prop:bla}), there
  %exists a section $\xi \in \Gamma(\cotan P \otimes L_P)$ with
  %$(u,\xi) \in \Gamma(\ker \rho_P)$, {\it i.e.} $\rho_P(u) = -
  %\rho_P(\xi)$. Consider the element 
  %$$ (u_p,\eta_p) - (u,\xi)_p = (0_p, \eta_p - \xi(p)) \in \nu^*_p
  %\otimes L_{P,p}; $$
  %\noindent
  %then there exists a section $\tau \in \Gamma(\nu^* \otimes L_P)$
  %with $\tau(p) = \eta_p - \xi(p)$. The sum $(u,\eta) := (u,\xi) + (0,\tau) \in
  %\Gamma(\ker \rho_P)$ satisfies $(u,\eta)(p) = (u_p,\eta_p)$ and $u$
  %never vanishes, as claimed. Then
  %\begin{equation}
    %\label{eq:28}
    %f_{(u,\eta)} u = \nabla_{(u,\eta)} u = \eta(\rho_P(j^1 u)) = -\eta
    %(\rho_P(\eta)) = 0, 
  %\end{equation}
  %\noindent
  %where the second equality follows from the definition of $\nabla$,
  %the third from the fact that $(u,\eta) \in \Gamma(\ker \rho_P)$, and
  %the last by antisymmetry of $\rho_P : \cotan P \otimes L_P \to T
  %\mathcal{F}$ (cf. the proof of Proposition \ref{prop:bla}). Since
  %$u$ never vanishes, equation \eqref{eq:28} implies that
  %$f_{(u,\eta)} \equiv 0$, which implies that $\nabla(u_p,\eta_p) =
  %0$. This, in turn, shows that the vector bundle map $\nabla : \ker
  %\rho_P \to \mathrm{End}(L_P)$ is equal to zero, which completes the proof.
\end{proof}

In light of Proposition \ref{prop:bla}, it makes sense to consider the
Lie algebroid cohomology $\HH^*(\mathcal{F};L_P)$, which is defined as
the cohomology of the complex of smooth foliated forms with values in $L_P$
with the following (standard) differential $\de_{\mathcal{F}}$
\begin{equation}
  \label{eq:17}
  \begin{split}
    \de_{\mathcal{F}}\omega(X_1,\ldots,X_{l+1}) := &\sum\limits_{i<j}
    (-1)^{i+j} \omega([X_i,X_j],X_1,\ldots, \hat{X}_i,\ldots, \hat{X}_j,
    \ldots,X_{l+1})  \\
     + &\sum\limits_i (-1)^{i+1} \bar{\nabla}_{X_i}(\omega(X_1,\ldots,
    \hat{X}_i,\ldots,X_{l+1})), 
  \end{split}
\end{equation}
\noindent
for $X_1,\ldots,X_{l+1} \in \Gamma(T \mathcal{F})$ and $\omega \in
\Omega^l(\mathcal{F};L_P) := \Gamma(\wedge^l \cotan \mathcal{F}
\otimes L_P)$, and where hat
denotes omission. The next lemma associates a cohomology class in
$\HH^2(\mathcal{F};L_P)$ to $\JM$.

\begin{lemma}\label{lemma:coho_class}
  The foliated 2-form with values in $L_P$ uniquely defined by
  \begin{equation}
    \label{eq:29}
    \omega(\rho_P(j^1 u), \rho_P(j^1 v)) : = \{u,v\}_P,
  \end{equation}
  \noindent
  for any $u,v \in \Gamma(L_P)$, is a 2-cocycle, {\it i.e.} $\de_{\mathcal{F}} \omega = 0$. 
\end{lemma}
\begin{proof}
  Since sections of the form $j^1 u$, for $u \in \Gamma(L_P)$ form a
  $C^{\infty}(P)$-basis of $\Gamma(J^1 L_P)$ and $T
  \mathcal{F} = \rho_P(J^1 L_P)$, equation
  \eqref{eq:29} defines a unique map $\Gamma(T \mathcal{F}) \times
  \Gamma(T \mathcal{F}) \to \Gamma(L_P)$ which is manifestly
  antisymmetric. Let $f \in C^{\infty}(P)$ and fix $u,v \in
  \Gamma(L_P)$; then
  \begin{equation*}
    \begin{split}
      \omega(f\rho_P(j^1 u)&, \rho_P(j^1 v)) = \omega(\rho_P(fu,\de f
      \otimes u), \rho_P(j^1v)) = \{fu,v\}_P + \de f \otimes
      u(\rho_P(j^1v)) \\
      &=f\{u,v\}_P - \mathcal{L}_{\rho_P(j^1v)}f u +
      \mathcal{L}_{\rho_P(j^1v)}f u = f \omega(\rho_P(j^1 u), \rho_P(j^1 v)),
    \end{split}
  \end{equation*}
  \noindent
  where the first equality follows from the
  $C^{\infty}(P)$-structure on $\Gamma(J^1 L_P)$ arising from
  the Spencer decomposition (cf. Observation
  \ref{ex:Spencer_operator}), the second by definition of $\omega$,
  and the third by the characterising property \ref{item:A} of the anchor $\rho_P$
  (cf. Proposition \ref{prop:alg}). The above calculation shows that
  equation \eqref{eq:29} indeed defines a foliated 2-form with values
  in $L_P$ (also denoted by $\omega$) as it is $C^{\infty}(P)$-bilinear. \\

  To check that $\de_{\mathcal{F}}\omega =0$ it suffices to check that
  $\de_{\mathcal{F}}\omega(\rho_P(j^1u),\rho_P(j^1v),\rho_P(j^1w)) =0
  $ for any $u,v,w \in \Gamma(L_P)$. This follows from the fact that
  $\de_{\mathcal{F}}\omega$ is $C^{\infty}(P)$-linear in each
  entry and that sections of the form $\rho_P(j^1u)$ form a
  $C^{\infty}(P)$-basis of $\Gamma(T \mathcal{F})$. Then
  \begin{equation*}
    \begin{split}
      &\de_{\mathcal{F}}\omega(\rho_P(j^1u),\rho_P(j^1v),\rho_P(j^1w))
      = \bar{\nabla}_{\rho_P(j^1u)}(\omega(\rho_P(j^1v),\rho_P(j^1w)))
      + \mathrm{c.p.}  \\
      & -
      (\omega([\rho_P(j^1u),\rho_P(j^1v)],\rho_P(j^1w)) +
      \mathrm{c.p.}) = \{u,\{v,w\}_P\}_P + \mathrm{c.p.} \\
      & - (\omega(\rho_P([j^1u,j^1v]), \rho_P(j^1w)) + \mathrm{c.p.}) = -(\omega(\rho_P(j^1\{u,v\}_P),\rho_P(j^1w)) + \mathrm{c.p.}) \\
      & =-(\{\{u,v\}_P,w\}_P + \mathrm{c.p.}) = 0,
    \end{split}
  \end{equation*}
  \noindent
  where c.p. stands for cyclic permutation, the second equality
  follows from the fact that $\rho_P$ is a map of Lie algebroids, the
  third by the defining property \ref{item:B} of the Lie bracket on
  $J^1 L_P$ (cf. Proposition \ref{prop:alg}) and by the Jacobi
  identity for $\{\cdot,\cdot\}$, which also implies the last
  equality. This shows that $\de_{\mathcal{F}}\omega = 0$, as required.
\end{proof}

\begin{obs}\label{obs:unravelling_omega}
  The following is an explicit formula for
  $\omega$
  $$ \omega(\rho_P(u,\eta),\rho_P(v,\zeta)) : = \{u,v\}_P +
  \eta(\rho_P(v,\zeta)) - \zeta(\rho_P(j^1u)), $$
  \noindent
  for $(u,\eta), (v,\zeta) \in \Gamma(J^1 P)$.
\end{obs}


%%%%%%%%%%%%%%%%%%%%%%
%%% Smooth classification %%%
%%%%%%%%%%%%%%%%%%%%%%


\subsection{Smooth classification}\label{sec:smooth-class}
Any contact realisation $\phi:\CM\to\JM$ (cf. Remark \ref{rk:cir})
comes equipped with an {\bf action} of the Lie algebroid $J^1L_P$, in the
sense that there exists a vector bundle map $\psi: \Gamma(J^1L_P) \to
\X(M)$ satisfying
\begin{enumerate}[label=(A\arabic*), ref=(A\arabic*)]
\item \label{item:3}  it induces a Lie algebra homomorphism $\Gamma(J^1L_P)\to \X(M)$;
\item \label{item:4} for all $m \in M$, $D_m\phi\circ
  \psi_m=\rho_{\phi(m)}$.
\end{enumerate}

\begin{lemma}\label{lemma:action} Let $\phi:\CM\to\JM$ be a contact realisation. The map 
$\psi:\phi^*J^1L_P\to T M$ given at the level of sections
by $$\phi^*j^1 u\mapsto R_{F \circ \phi^*u},\ \ \ \forall u\in \Gamma(L_P)$$ 
defines a faithful action of $J^1L_P$ on $\phi : M \to P$.
\end{lemma}

\begin{obs}\label{rk:action} 
Note that $\psi$ can be alternatively described as the composite
$\rho_M \circ \phi^*: \phi^*J^1L_P \to T M$, where the notation is
as in item \ref{item:comm} of Observation \ref{obs:icr}.
\end{obs}

\begin{proof}[Proof of Lemma \ref{lemma:action}]
  Since $\phi: \CM \to \JM$ is a Jacobi map, item \ref{item:comm} in Observation
  \ref{obs:icr} implies that  
  \begin{equation}\label{action}
    D\phi\circ\rho_M\circ\phi^*=\phi^*(\rho_P).
  \end{equation}
  \noindent
  In light of Observation \ref{rk:action}, it follows that property
  \ref{item:4} holds. It remains to prove that $\psi$ induces a Lie
  algebra morphism, {\it i.e.}  
  \begin{equation}\label{respect.bracket}
    \psi([\alpha,\alpha'])-[\psi(\alpha),\psi(\alpha')]=0,
  \end{equation}
  \noindent
  for all $\alpha,\alpha' \in \Gamma(J^1L_P)$. This equation holds when
  $\alpha,\alpha'\in\Gamma(J^1P)$ are holonomic sections, {\it i.e.} of the
  form $j^1u,j^1v$ for smooth sections $u,v \in C^{\infty}(P)$. This is
  because $\phi:M\to P$ is a Jacobi map with bundle component $F$, which implies that    
  \begin{equation}\label{foliation}
    R_{F \circ \phi^*\{u,v\}}=R_{\{F\circ\phi^*u,F \circ \phi^*v\}}=[R_{F
      \circ \phi^*u},R_{F \circ \phi^*v}].
  \end{equation}
  \noindent
  for $u,v\in \Gamma(L_P)$, where in the last equality property
  \eqref{eq:2} of the Jacobi bracket of $\CM$ is used. In
  general, notice that the left hand side of equation
  \eqref{respect.bracket} satisfies   
  $$ \psi([f\alpha,\alpha'])-[\psi(f\alpha),\psi(\alpha')] =
  (\phi^*f)(\psi([\alpha,\alpha'])-[\psi(\alpha),\psi(\alpha')]); $$ 
  \noindent
  for $f \in C^{\infty}(P)$ and $\alpha,\alpha' \in \Gamma(J^1L_P)$. This again
  follows from the fact that $\phi :\CM \to \JM$ is a Jacobi map, so
  that equation \eqref{action} holds. Thus, since equation
  \eqref{respect.bracket} holds for holonomic sections and these
  generate $\Gamma(J^1L_P)$ as a $C^{\infty}(P)$-module, it follows that
  equation \eqref{respect.bracket} holds for all sections of $J^1P$,
  which proves that $\psi$ satisfies property \ref{item:3}. The proof
  of Lemma \ref{lemma:regular} shows that $\psi$ is injective (this
  follows from property \ref{item:IR2}). 
  % To show that
  % $\psi$ is injective, let $(u,)\in J^1 L_P$ be so that  
  % \[0=\psi(\eta,\lambda)=\lambda R_M+\Lambda_M^\sharp(\phi^*\eta).\]
  % As $\Lambda^\sharp(\phi^*\eta)=c_{\theta}^\sharp(-\phi^*\eta|_H)$,
  % where $c_{\theta}^\sharp$ is the dual of the curvature map (see
  % equation \eqref{curvature} and Observation \ref{rk:contact bracket}),
  % and $c_{\theta}^\sharp$ takes values in $H=\ker\theta$, then $\lambda
  % R_M\in H$ and therefore $0=\theta(\lambda R_M)=\lambda$. This implies
  % that $\phi^*\eta|_H=0$ ($c_{\theta}^\sharp$ is injective because
  % $c_{\theta}$ is non-degenerate) which means that
  % $\eta|_{D\phi(H)}=0$. But $D\phi(H)=TP$ as $H$ is transversal to
  % $\phi$, and therefore $\eta=0$, which concludes the proof.     
\end{proof}

The following proposition shows that, for a contact isotropic realisation $\phi : \CM \to \JM$, the restriction of the action of
Lemma \ref{lemma:action} to the Lie subalgebroid $\ker \rho_P \to P$
yields an action.
  
\begin{prop}\label{obs:trans_inf_action}
Let $\phi : \CM \to \JM$ be a contact isotropic realisation of a Jacobi manifold. The action $\psi$ restricted to the bundle of abelian Lie algebras $\ker\rho_P$,
is a vector bundle isomorphism over $M$,
$$\psi:\phi^*\ker\rho_P\to \ker D\phi,$$
which at the level of sections $\psi:\Gamma(\ker\rho_P)\to \Gamma(\ker D\phi)$ is a Lie algebra map.
\end{prop}

\begin{proof}
  %Let $\phi : \CM \to \JM$ be a contact isotropic realisation of a
  %Jacobi manifold. By Proposition \ref{prop:bla} $\ker \rho_P \subset
  %J^1 L_P \to P$ is a bundle of abelian Lie algebras. Restricting the action defined in the statement of Lemma
  %\ref{lemma:action} to this Lie subalgebroid, obtain an action of
  %$\pi: \ker \rho_P \to P$ along $\phi : \CM \to \JM$, also
  %denoted by $\psi : \phi^*\ker \rho_P \to T M$. Note that, 
  Property \ref{item:4} implies that the image of $\psi|_{\ker\rho_P}$
  consists of {\it vertical} vector fields, {\it i.e.} tangent to the
  fibres of $\phi$.  On the other hand, property \ref{item:3} implies that at the level of sections $\psi:\Gamma(\ker\rho_P)\to \Gamma(\ker D\phi)$ is a Lie algebra map. Now, since by Lemma \ref{lemma:action}, the action
  $\psi : \phi^* \ker \rho_P \to T M$ is injective and for all $p
  \in P$, $\dim \ker \rho_{P,p} = k = \dim \phi^{-1}(p)$, dimension
  counting implies that for all $m \in M$, 
  $$\psi((\phi^* \ker \rho_P)_m) = T_m \phi^{-1}(\phi(m)).$$ 
\end{proof}

The action $\psi : \phi^* \ker \rho_P \to T M$ should be thought of
as being infinitesimal; since the fibres of $\phi$ are compact, $\psi$
can be integrated to an action of $\pi : \ker \rho_P \to P$ considered
as a bundle of abelian Lie groups.

\begin{rk}\label{rk:complete}
  In fact, the action $\psi$ can be integrated under a weaker
  assumption on the fibres, which is generally known as {\em
    completeness} (cf. \cite{CrainicFernandes:jdd}). All following
  results hold in this more general setting with the appropriate modifications.
\end{rk}

% , it is natural to ask whether it is
% possible to `integrate' $\psi$. In general, this is not necessarily
% true unless a notion of completeness of the infinitesimal action is
% introduced.  

% \begin{defn}\label{defn:complete}
%   a contact isotropic realisation $\phi: \CM \to \JM$ is {\bf
%     complete} if $\psi(\phi^*\alpha)$ is a complete vector field
%   whenever $\alpha \in \Gamma(\ker \rho_P)$ is compactly supported. 
% \end{defn}

% \begin{rk}\label{rk:complete_ir}
%   \mbox{}
%   \begin{enumerate}[label=\arabic*), ref=\arabic*]
%   \item \label{item:5} The notion introduced in Definition
%     \ref{defn:complete} is a special case of that of complete Lie
%     algebroid actions (cf. \cite{CrainicFernandes:jdd}); 
%   \item \label{item:6} Complete contact realisations of Jacobi
%     manifolds play an analogous role in Jacobi geometry to their
%     symplectic counterpart in the Poisson realm. In particular,
%     existence of such a realisation is equivalent to integrability of
%     the underlying Jacobi manifolds. 
%   \end{enumerate}
% \end{rk}

% \begin{exm}\label{exm:compact_fibres}
%   If the fibres of a contact isotropic realisation $\phi : \CM \to
%   \JM$ are all compact, then the realisation is complete. 
% \end{exm}

% Henceforth, any isotropic contact realisation $\phi: \CM \to \JM$ is
% assumed to have compact fibres (hence, by Example
% \ref{exm:compact_fibres} it is complete), unless otherwise
% stated. Many of the results in the following sections hold more
% generally for complete isotropic contact realisations; crucial
% differences are pointed out in several {\it Remarks} below. However,
% the case with compact fibres is the most relevant both from the point
% of view of integrable Hamiltonian systems on contact manifolds and
% from the point of view of `compactness' in Jacobi geometry. \\

Fix a contact isotropic realisation $\phi : \CM \to \JM$ and let
$\psi: \phi^* \ker \rho_P \to T M$ be the associated infinitesimal
$\pi : \ker \rho_P \to P$ action. The integrated action is given by  
\begin{equation}
  \label{eq:6}
  \begin{split}
    \Psi : \ker \rho_P \tensor[_\pi]{\times}{_{\phi}} M &\to M \\
    (\alpha,m) &\mapsto \varphi^1_{\alpha}(m),
  \end{split}
\end{equation}
\noindent
where
$$  \ker \rho_P \tensor[_\pi]{\times}{_{\phi}} M :=\{
(\alpha,m) \in \ker \rho_P \times M \mid
\pi(\alpha) = \phi(m) \} $$ 
\noindent
is a smooth manifold, and $\varphi^1_{\alpha}$ is the time-1
flow of $\psi(\phi^*\alpha)$. 

\begin{obs}\label{obs:per_lattice}
  For each $p \in P$, the action of equation \eqref{eq:6} restricts to
  an action of the abelian Lie group $\ker \rho_{P,p} \cong \RR^k$ on
  $\phi^{-1}(p)$. Connectivity of $\phi^{-1}(p)$ and Observation
  \ref{obs:trans_inf_action} imply that the action of $\ker
  \rho_{P,p}$ is transitive. Moreover, since $\ker \rho_{P,p}$ is an
  abelian Lie group, if $j^1_pu \in \ker \rho_{P,p}$ is
  such that $\Psi(j^1_pu,m) = m$, then
  $\Psi(j^1_pu,m') = m'$ for {\em any} other $m' \in
  \phi^{-1}(p)$. This is because the isotropy of the action at any two
  points on $\phi^{-1}(p)$ are equal. Therefore, if 
$$ \Sigma_p := \{ j^1_pu \in \ker \rho_{P,p} \mid
\varphi^1_{j^1_pu} = \mathrm{id} \}, $$ 
\noindent
then $\phi^{-1}(p) \cong \ker \rho_{P,p}/\Sigma_p$; since
$\phi^{-1}(p)$ is compact by assumption, it follows that
$\phi^{-1}(p)$ is diffeomorphic to $\mathbb{T}^k$ and that $\Sigma_p$
is a cocompact lattice in $\ker \rho_{P,p}$ and, therefore, isomorphic
to $\ZZ^k$. 
\end{obs}

% \begin{rk}\label{rmk:complete_not_compact}
%   If the contact isotropic realisation is only assumed to be complete,
%   the arguments of Observation \ref{obs:per_lattice} imply that all
%   the fibres are isomorphic to quotients of $\RR^k$ by discrete
%   subgroups (whose rank may vary!), and that, for each $p \in P$,
%   $\Sigma_p$ is isomorphic to $\ZZ^l$ for some $0 \leq l \leq k$. 
% \end{rk}

Just as in the theory of symplectic isotropic realisations of Poisson manifolds,
the isotropy of the action of equation \eqref{eq:6} plays an important
role in the classification of contact isotropic realisations of Jacobi
manifolds. 

\begin{defn}[Period lattice]\label{defn:period_lattice}
  The subset
  $$ \Sigma := \coprod\limits_{p \in P} \Sigma_p \subset \ker \rho_{P,p}$$
  \noindent
  is called the {\bf period lattice} of the isotropic contact realisation $\phi : \CM \to \JM$.
\end{defn}

\begin{rk}\label{rk:legendre}
  The above notion of period lattice extends that of a {\em Legendre
    lattice} introduced in \cite{Banyaga}.
\end{rk}

The following theorem provides a local model for isotropic
realisations $\phi : \CM \to \JM$; its proof is
omitted as it is entirely analogous to that of \cite[Theorem
2.1]{dd}. 

\begin{thm}\label{thm:local_model} Let $\phi:\CM\to\JM$ be an isotropic realisation with associated period lattice $\Sigma$. Then
\begin{enumerate}[label=(\Roman*), ref=(\Roman*)]
\item \label{item:1} $\Sigma$ is a closed submanifold of $\ker\rho_P$;
\item \label{item:2} the quotient $\ker\rho_P/\Sigma$ is a smooth manifold and the
  projection $\pi : \ker\rho_P/\Sigma \to P$ is a fibre bundle
  with fibre $\mathbb{T}^k$;
\item \label{item:7} upon a choice of a local section $\sigma:U\subset M\to \phi^{-1}(U)$, the map
\begin{eqnarray}\label{eqn:local_isomorphism}
\Psi_\sigma:\ker \rho_P/\Sigma|_U\to \phi^{-1}(U), \ \ [\alpha]\mapsto \varphi^1_{\alpha}(\sigma(\pi(\alpha)))
\end{eqnarray}
is an isomorphism of fibres bundles, making the following diagram
\begin{eqnarray*}
\xymatrix{
  \ker \rho_P/\Sigma|_U \ar[rr]^{\Psi_\sigma} \ar[rd]_{\pi} &  & \phi^{-1}(U) \ar[ld]^{\phi}\\
  & U & 
}
\end{eqnarray*}
commute.
\end{enumerate}
\end{thm}

% \begin{rk}\label{rk:theorem_complete_not_compact}
%   Theorem \ref{thm:local_model} holds also almost {\it verbatim} for complete isotropic
%   realisations, the only difference being, in this case, that the
%   fibres of the projection $\pi: \ker\rho_P/\Sigma \to P$ are
%   not necessarily compact (or all diffeomorphic to one another).
% \end{rk}

Fix a good open cover $\mathcal{U} := \{U_i\}$ of $P$, {\it i.e.} all
finite intersections of the elements of $\mathcal{U}$ are
contractible, and, for each $i$, fix a section $\sigma_i : U_i \to
M$. Suppose that $U_{ij} \neq \emptyset$; then the
composite $\Psi_{\sigma_j}^{-1} \circ \Psi_{\sigma_i} : \ker
\rho_P /\Sigma |_{U_{ij}} \to \ker \rho_P /\Sigma |_{U_{ij}}$ is given
by translation along the fibres by a section $t_{ij} : U_{ij} \to \ker
\rho_P/\Sigma$, {\it i.e.}
$$\Psi_{\sigma_j}^{-1} \circ \Psi_{\sigma_i} ([\alpha])
= [\alpha] + t_{ij}(\pi(\alpha)), $$
\noindent
where $t_{ij}$ is uniquely defined by $\sigma_j(p) =
\varphi^1_{t_{ij}(p)}(\sigma_i(p))$ for all $p \in U_{ij}$. The
collection $\{t_{ij}\}$ defines a \v Cech cohomology class $t \in
\HH^1(\mathcal{U};\mathcal{C}^{\infty}(\ker \rho_P /\Sigma)) \cong
\HH^1(P;\mathcal{C}^{\infty}(\ker \rho_P /\Sigma))$, where
$\mathcal{C}^{\infty}(\ker \rho_P /\Sigma)$ is the sheaf of smooth sections
of $\ker \rho_P /\Sigma \to P$, and the isomorphism follows from the
fact that open cover $\mathcal{U}$ is good. \\

The short exact sequence of sheaves

$$  1 \to \Sigma \hookrightarrow \mathcal{C}^{\infty} (\ker \rho_P)
\to \mathcal{C}^{\infty}(\ker \rho_P/\Sigma) \to 1, $$
\noindent
where, by abuse of notation, $\Sigma$ denotes the sheaf of sections of
$\Sigma \to P$, induces a long exact sequence in cohomology whose
connecting morphisms 
$$ \delta: \HH^{k}(P; \mathcal{C}^{\infty}(\ker \rho_P/\Sigma)) \to
\HH^{k+1}(P;\Sigma) $$
\noindent
are isomorphisms for all $k \geq 1$, since $\mathcal{C}^{\infty}(\ker \rho_P)$ is fine.

\begin{defn}\label{defn:cc}
  The cohomology class $c = \delta(t) \in \HH^2(P;\Sigma)$ is called
  the {\bf Chern class} of the contact isotropic realisation $\phi :
  \CM \to \JM$.
\end{defn}

% \begin{rk}\label{rk:pb}
%   The above discussion proves that $\phi : \CM \to \JM$ is a $\ker
%   \rho_P/\Sigma \to P$-principal bundle in the sense of
%   \cite[Definition 2.1]{rossi}, whose isomorphism class is classified
%   by its Chern class $c$.
% \end{rk}

In conclusion, this section proves that a contact isotropic
realisation $\phi : \CM \to \JM$ is classified smoothly by 
\begin{itemize}
\item its period lattice $\Sigma \subset \ker \rho_P \subset J^1 L_P$;
\item its Chern class $c \in \HH^2(P;\Sigma)$;
\end{itemize}
\noindent
{\it i.e.} two contact isotropic realisations $\phi : \CM \to \JM$ and
$\phi': (M',H') \to \JM$ are smoothly equivalent if and only if their
period lattices and Chern classes coincide, where smooth equivalence
means the existence of a diffeomorphism $B: \CM \to (M', H')$
satisfying $ B \circ \phi = \phi'$.


%%%%%%%%%%%%%%%%%%%%%%
%%% Contact classification %%%
%%%%%%%%%%%%%%%%%%%%%%



\subsection{Contact classification}\label{sec:cont-class}
This section studies the problem of finding a (local) {\em contact}
classification of contact isotropic realisations, which complements
the results of Section \ref{sec:smooth-class}. The main result in this
section is Theorem \ref{thm:aa} which provides a local normal form for
generalised contact forms arising in contact isotropic
realisations. Before proving Theorem \ref{thm:aa}, some
preparatory results are required. Throughout, fix a contact isotropic realisation $\phi:
\CM \to \JM$ with compact fibres, generalised contact form $\theta \in
\Omega^1(M,L_M)$, and bundle component denoted by $F:\phi^*L_P\to
TM/H$. Denote by $\bar\theta\in\Omega^1(M;\phi^*L_P)$ the one form
given by the composition $F^{-1}\circ\theta.$\\ 

Recall that for any section 
$\alpha\in\Gamma(\ker\rho_P)$, $\psi( \phi^*\alpha) \in
\Gamma(\ker D\phi)$, where $\psi : \phi^* \ker \rho_P \to TM$ is defined as in Lemma \ref{lemma:action}. Therefore, the flow $\varphi^t_{\alpha}:M\to
M$ of $\psi(\phi^*\alpha)$ preserves the fibres, {\it i.e.}
$\phi\circ \varphi^t_{\alpha}=\phi$ for all $t \in \RR$. Hence,
$(\varphi^t_{\alpha})^* \phi^* L_P \cong \phi^* L_P$ canonically. This allows to define a `Lie
derivative' type operator on $\Omega^*(M;\phi^*L_P)$ by 
\begin{equation}\label{eq:lie_derivative}(\mathcal{L}_{\alpha}\omega)_m=\frac{d}{dt}\Big\rvert_{t=0}((\varphi^t_{\alpha})^*\omega)_m\end{equation}
\noindent
for any $m\in M$ and $\omega \in \Omega^l(M,\phi^*L_P)$. It obeys rules which are analogous to that of the
standard Lie derivative, for instance,
\begin{equation}
  \label{eq:8}
  \frac{d}{dt}\Big\rvert_{t=s}((\varphi^t_{\alpha})^*\omega)_m
  = ((\varphi^s_{\alpha})^*(\mathcal{L}_{\alpha} \omega))_m. 
\end{equation}
\noindent
The following lemma, stated below without proof,
shows that the above Lie derivative also satisfies a familiar identity involving the
interior product (cf. \cite[Lemma 3.8]{Maria}). 

\begin{lemma}\label{lemma:imt}
  For any $\omega\in\Omega^l(M;\phi^*L_P)$ and $X\in\X(M),$
  $$[i_X,\mathcal{L}_{\alpha}]\omega=
  (i_{[X,\psi(\phi^*\alpha)]}\omega).$$
\end{lemma}

These differential geometric tools allow to prove an important
property of the flows $\varphi^t_{\alpha}$, which
generalises \cite[Lemma 2.1]{dd}. 

% Let $X \in \X(P)$ and choose any
% $\phi$-projectable $\tilde{X} \in \Gamma(H)$; the existence of one
% such follows from property \ref{item:IR2} and the fact that $\phi$ is
% a surjective submersion.

\begin{lemma}\label{lemma:fundamental}
  For any section $\alpha = (u,\eta) \in \Gamma(\ker \rho_P)$, 
  $$ (\varphi^1_{(u,\eta)})^*\bar\theta - \bar\theta =
  \phi^*\eta. $$
\end{lemma}
\begin{proof}
  Fix points $p \in P$ and $m \in \phi^{-1}(p)$, a tangent vector $X_m \in
  T_m M$, and a section $(u,\eta) \in \Gamma(\ker \rho_P)$ defined
  near $p$. Extend $X_m$ to a (locally defined)
  vector field $\tilde{X}$. Since $\phi \circ \varphi^1_{(u,\eta)} =
  \phi$, the following expression makes sense
  \begin{equation}
    \label{eq:14}
    ((\varphi^1_{(u,\eta)})^*\bar\theta - \bar\theta)(\tilde{X}) =
     \int\limits_0^1 \Bigg( \frac{\de}{\de t} (\varphi^t_{(u,\eta)})^*
     \bar\theta\Bigg)(\tilde{X}) \de t. 
  \end{equation}
  \noindent
  Consider the integrand
  \begin{equation}
    \label{eq:9}
    \begin{split}
       &\Bigg( \frac{\de}{\de t} (\varphi^t_{(u,\eta)})^*
       \bar\theta\Bigg)(\tilde{X})
       =((\varphi^t_{(u,\eta)})^*(\mathcal{L}_{(u,\eta)} \bar\theta))(\tilde{X}) \\
       & =
       (\varphi^t_{(u,\eta)})^*((\mathcal{L}_{(u,\eta)}\bar\theta)(D
       \varphi^t_{(u,\eta)}(\tilde{X}))) \\
       & =  (\varphi^t_{(u,\eta)})^*(\mathcal{L}_{(u,\eta)}(\bar\theta(D \varphi^t_{(u,\eta)}(\tilde{X}))) +
       \bar\theta([D \varphi^t_{(u,\eta)}(\tilde{X}),\psi(\phi^* u, \phi^*\eta)])),
     \end{split}
  \end{equation}
  \noindent
  where the first equality follows from equation \eqref{eq:8} and the
  last by Lemma \ref{lemma:imt}; write 
  $$ D\varphi^t_{(u,\eta)}(\tilde{X}) = \sum\limits_i f_{t,i} A_{t,i} + \tilde{X}_t, $$
  \noindent
  where $f_{t,i} \in C^{\infty}(M)$,
  $A_{t,i} := \psi(\phi^*u_{t,i},\phi^*\eta_{t,i}) \in \Gamma(\ker D\phi)$, and $\tilde{X}_t \in
  \Gamma(H)$ are all time-dependent, with $D\phi(\tilde{X}_t) = D\phi(\tilde{X})$. This decomposition follows from property
  \ref{item:IR2} and the fact that the action $\psi : \Gamma(\phi^*
  \ker \rho_P) \to \Gamma(\ker D\phi)$ is an isomorphism of Lie
  algebras. On the one hand, 
  $$ \bar\theta(D\varphi^t_{(u,\eta)}(\tilde{X})) = \bar\theta (\sum\limits_i
  f_{t,i}\psi(\phi^*u_{t,i},\phi^* \eta_{t,i})) = \sum\limits_i
  f_{t,i} \phi^*u_{t,i}, $$
   \noindent
   since $\psi(\phi^*u_{t,i},\phi^* \eta_{t,i})) = \rho_M(F\circ
   (\phi^*u_{t,i}, \phi^* \eta_{t,i})) = R_{F \circ \phi^*u_{t,i}} +
   c^{\sharp}_{\theta}(F\circ \phi^* \eta_{t,i}\rvert_H)$, $c^{\sharp}_{\theta}(F\circ \phi^* \eta_{t,i}\rvert_H) \in \Gamma(H)$ for
   all $t,i$, and by definition of Reeb vector fields (cf. Observation
   \ref{obs:rvf}). Thus 
   \begin{equation}
     \label{eq:12}
     \mathcal{L}_{(u,\eta)}(\bar \theta(D \varphi^t_{(u,\eta)}(\tilde{X})))  =
     \mathcal{L}_{(u,\eta)}(\sum\limits_i f_{t,i}\phi^*u_{t,i}) =
     \sum\limits_i(\mathcal{L}_{\psi(u,\eta)}f_{t,i}) \phi^*u_{t,i}
 \end{equation}
 \noindent
 since $\psi(\phi^*u,\phi^*\eta) \in \Gamma(\ker D\phi)$. On the
 other hand, 
 \begin{equation}
   \label{eq:15}
   \begin{split}
     \bar\theta([D\varphi^t_{(u,\eta)}(\tilde{X}),&
     \psi(\phi^*u,\phi^*\eta)])=\bar\theta([\tilde{X}_t,\psi(\phi^*u,\phi^*\eta)])
     + \sum\limits_i \bar\theta([f_{t,i}A_{t,i},\psi(\phi^*u,\phi^*\eta)])
     \\
     & = \bar\theta([\tilde{X}_t,\psi(\phi^*u,\phi^*\eta)]) - \sum\limits_i
     \mathcal{L}_{\psi(\phi^*u,\phi^*\eta)}f_{t,i}\bar\theta(A_{t,i}) \\
     & = \bar\theta([\tilde{X}_t,\psi(\phi^*u,\phi^*\eta)]) - \sum\limits_i
     \mathcal{L}_{\psi(\phi^*u,\phi^*\eta)}f_{t,i} \phi^* u_{t,i},
   \end{split}
 \end{equation}
 \noindent
 where the second equality follows from the fact that $A_{t,i},
 \psi(\phi^*u,\phi^*\eta) \in \Gamma(\ker D\phi)$ and this is a bundle
 of abelian Lie algebras (since $\ker \rho_P \to P$ is and $\ker D\phi
 \cong \phi^*\ker \rho_P$ as bundles of Lie algebras), and the third by
 definition of $A_{t,i}$. Plugging equations \eqref{eq:12} and
 \eqref{eq:15} into equation \eqref{eq:9}, obtain that 
 \begin{equation}
   \label{eq:27}
   %\begin{split}
     \Bigg( \frac{\de}{\de t} (\varphi^t_{(u,\eta)})^*
       \bar\theta\Bigg)(\tilde{X})= ((\varphi^t_{(u,\eta)})^*(\mathcal{L}_{(u,\eta)} \bar\theta))(\tilde{X}) \\
        = (\varphi^t_{(u,\eta)})^*(\bar\theta([\tilde{X}_t,\psi(\phi^*u,\phi^*\eta)]));
   %\end{split}
 \end{equation}
 \noindent
 the right hand side of equation \eqref{eq:27} can be computed to be
 $$ \bar\theta([\tilde{X}_t,c^{\sharp}_{\theta}( \phi^* \eta
 \rvert_H)]) = \phi^*\eta \rvert_H(\tilde{X}_t) =
 \phi^*(\eta(D\phi(\tilde{X}_t)))) = \phi^*(\eta(D\phi(\tilde{X})), $$
 \noindent
 where the second equality follows from property \ref{item:IR2},
 and the last from $D\phi(\tilde{X}_t) = D\phi(\tilde{X})$. Therefore,
 equation \eqref{eq:27} becomes 
 \begin{equation}\label{eq:expresion}
  \Bigg( \frac{\de}{\de t} (\varphi^t_{(u,\eta)})^*
 \bar\theta\Bigg)(\tilde{X}) = (\varphi^t_{(u,\eta)})^* \phi^*(\eta
 D\phi(\tilde{X})) = \phi^*(\eta (D\phi\tilde{X})) = (\phi^*\eta)(\tilde{X}), \end{equation}
 \noindent
 where the second equality follows from $\phi \circ
 \varphi^t_{(u,\eta)} = \phi$. The above equation implies that the
 integrand of equation \eqref{eq:14} is independent of $t$, thus
 yielding that 
 $$ ((\varphi^1_{(u,\eta)})^*\bar\theta - \bar\theta)(\tilde{X}) = (\phi^*\eta)(\tilde{X}). $$
 \noindent
 This implies that at the point $m \in M$ the 1-forms
 $(\varphi^1_{(u,\eta)})^*\bar\theta - \bar\theta$ and
 $\phi^*\eta$ are equal; since $m$ is arbitrary, this completes the proof.
\end{proof}


As in the case of symplectic isotropic realisations of Poisson
manifolds, Lemma \ref{lemma:imt} provides a local description of the
sections of the period lattice $\Sigma \to P$.

\begin{cor}\label{cor:sections_period}
  Any local section of $\Sigma \to P$ is of the form $j^1 u$, for some
  locally defined section $u \in \Gamma(L_P)$.
\end{cor}

\begin{proof}
  Suppose that $(u,\eta) \in \Gamma(\Sigma)$ (possibly locally
  defined). By definition, $\varphi^1_{(u,\eta)} =
  \mathrm{id}$. Fix $p \in P$, $X_p \in T_p P$, $m \in \phi^{-1}(p)$
  and $\tilde{X}_m \in H_m$ with $D_m \phi(\tilde{X}_m) =
  X_p$. Applying Lemma 
  \ref{lemma:fundamental} to the section $(u,\eta)$, obtain that
  $\eta(X_p) = 0$. Since $p \in P$ and $X_p \in T_p P$ are
  arbitrary, it follows that $\eta = 0$. This means precisely that
  $(u,\eta)$ is holonomic, {\it i.e.} of the form $j^1 u$ as required.
\end{proof}

\begin{obs}\label{obs:contact form} Consider the canonical contact
  form $\theta_{\mathrm{can}}\in \Omega^1(J^1 L_P)$ described in
  Example \ref{ex:jet_bundle}. By Corollary \ref{cor:sections_period},
  translations by elements of $\Sigma$ preserve $\theta_{\mathrm{can}}$ (as
  $(u,\eta)^*\theta_{\mathrm{can}}=\eta$, for $u\in\Gamma(L_P),\eta\in\Omega^1(P,L_P)$, and sections of $\Sigma$ are holonomic), and
  therefore its restriction to $\ker\rho_P\subset J^1L_P$ descends to
  a 1-form
  $$\theta_0\in \Omega^1(\ker\rho_P/\Sigma;\pi^*L_P),$$ 
  \noindent
  which does not necessarily define a contact distribution (unless
  $\ker\rho_P=J^1 L_P$).
\end{obs}

The following theorem gives a local normal form for sufficiently small
open $\phi$-saturated neighbourhoods of the fibres of $\phi : \CM \to
\JM$; as such, it can be thought of as providing local {\bf action-angle}
variables in analogy with \cite[Corollary 2]{dd} in the case of
complete isotropic realisations of Poisson manifolds, and
\cite[Theorem 1]{Banyaga} and \cite[Theorems 4 and 5]{jovanovic} in the case
of non-commutative integrable systems on contact manifolds (the former
with elliptic-type singularities) {\em once} a suitable local contact form
has been fixed (cf. Section \ref{sec:integr-syst-cont} for the
relation between integrable Hamiltonian systems on contact manifolds
and contact isotropic realisations). % It is important to remark that the methods used below
% are different to those employed in \cite{Banyaga,jovanovic}, and are
% closer to those used in \cite{dd}.

\begin{thm}\label{thm:aa}[Action-angle coordinates] Given a local section
  $\sigma:U\subset P\to \CM$, then  
$$\Psi_\sigma^*\bar{\theta}=\theta_0+\pi^*\sigma^*\bar{\theta},$$ 
where $\Psi_\sigma:\ker \rho_P/\Sigma|_U\to \phi^{-1}(U)$ is defined
by equation \eqref{eqn:local_isomorphism}. 
\end{thm}

It is important to remark that all forms in the above theorem take values in $\pi^* L_P$.

\begin{proof}
First, it is shown that the difference
$\Psi_\sigma^*\bar{\theta}-\theta_0\in\Omega^1(\ker\rho_P/\Sigma|_U;\pi^*L_P)$
is basic, {\it i.e.} 
\begin{eqnarray}\label{eqn:contact
    form}\Psi_\sigma^*\bar{\theta}-\theta_0=\pi^*\beta
\end{eqnarray}  
for some 1-form $\beta\in\Omega^1(U;L_P)$. This is the case if
$i_Z(\Psi_\sigma^*\bar{\theta}-\theta_0)=0$ and
$\mathcal{L}_Z(\Psi_\sigma^*\bar{\theta}-\theta_0)=0$ for any $Z$
tangent to $\ker(\pi:\ker\rho_P/\Sigma|_U\to U)$, (where
$\mathcal{L}_Z\omega\in\Omega^k(\ker\rho_P/\Sigma;\pi^*L_P)$ is
defined as in \eqref{eq:lie_derivative} for any
$\omega\in\Omega^k(\ker\rho_P/\Sigma;\pi^*L_P)$). In order to prove
this, identify (canonically)
$T^\pi_{[j^1_pv]}(\ker_{P,p}/\Sigma_p)$ with $\ker\rho_{P,p}$ via the isomorphism
$T^\pi(\ker\rho_P/\Sigma)\simeq \pi^*\ker\rho_P$. Observe that for
sections $(u',\eta'),(u,\eta)\in \Gamma(\ker\rho_P)$,
$\Psi_\sigma((u',\eta')+t(u,\eta))=\phi^t_{(u,\eta)}(\Psi_\sigma(u',\eta'))$
for any $t\in\RR$, where $\varphi^t_{(u,\eta)} : M\to M$ is the flow of
$\psi(\phi^*(u,\eta))$. Equivalently, 
\begin{equation}\label{eq:conmuta}\Psi_\sigma\circ \varphi^t_{(u,v)}=\varphi^t_{(u,v)}\circ\Psi_\sigma,\end{equation}
where on the left hand side
$\varphi^t_{(u,v)}:\ker\rho_P/\Sigma\to\ker\rho_P/\Sigma$ stands for
the flow of $\pi^*(u,\eta)$, given by $j^1_pv\mapsto
j^1_pv+t(u,\eta)(p)$. Differentiating equation \eqref{eq:conmuta},
obtain that 
$$D_{[j^1_pv]}\Psi_\sigma(\pi^*(u,\eta))=\psi(\phi^*u,\phi^*\eta)_{\Psi_\sigma(j^1_pv)}.$$
\noindent
Consider $i_Z(\Psi_\sigma^*\bar{\theta}-\theta_0) $; the above yields
$$i_Z\Psi^*_\sigma\bar{\theta}(j^1_pv)=\bar{\theta}_{\Psi_\sigma(j^1_v)}(\psi(\phi^*u,\phi^*\eta))=\bar\theta_{\Psi_\sigma(j^1_v)}(R_{F\phi^*u}+c(F\phi^*\eta|_H))=u_p.$$
\noindent
for any $p \in P$. On the other hand, $\theta_0$ restricted to $\pi^*\ker\rho_P$ is equal
to the projection $pr:\ker\rho_P\to L$ which implies that
$i_Z\theta_0=\theta_{0,j^1_pv}(u,\eta)=u_p$. Thus
$i_Z(\Psi_\sigma^*\bar{\theta}-\theta_0)=0$ follows. To compute the Lie
derivative along $Z$, observe that  
$$\mathcal{L}_Z\Psi^*_\sigma\bar\theta =
\frac{d}{dt}(\varphi^t_{(u,\eta)})^*(\Psi^*_\sigma\bar\theta)|_{t=0} =\frac{d}{dt}\Psi^*_\sigma\circ
(\varphi^t_{(u,\eta)})^*\bar\theta|_{t=0}$$ 
\noindent
where the second equality uses equation \eqref{eq:conmuta}. By
equation \eqref{eq:expresion}, have that 
$(\varphi^t_{(u,\eta)})^*\bar\theta|_{t=0}=\phi^*\eta$, which implies that
$\mathcal{L}_Z\Psi^*_\sigma\bar\theta=\pi^*\eta$. On the other hand,  
$$\mathcal{L}_Z\theta_0=\frac{d}{dt}(\varphi^t_{(u,\eta)})^*\theta_0|_{t=0}=\frac{d}{dt}(id^*+t\pi^*\circ(u,\eta)^*)\theta_0|_{t=0}=\frac{d}{dt}(\theta_0+t\pi^*\eta)|_{t=0}=\pi^*\eta,$$
where the third equality uses that for a section $(u,\eta):P\to
J^1L$, 
$(u,\eta)^*\theta_{\mathrm{can}}=\eta$. With this,
$\mathcal{L}_Z(\Psi_\sigma^*\bar{\theta}-\theta_0)=0$. As vector fields
of the form $\pi^*(u,v)$ generate $\pi^*\ker\rho_P\simeq
T^\pi(\ker\rho_P/\Sigma)$ as a $C^\infty(\ker\rho_P/\Sigma)$-module,
this implies that equation \eqref{eqn:contact
  form} holds. \\

To show that $\beta=\sigma^*\bar{\theta}$, consider the section $z:P\to
\ker\rho_P/\Sigma, p\mapsto [\Sigma_p]=0$ of $\pi:\ker/\Sigma\to
P$. Then $z^*\mathrm{pr}^*\beta=\beta$,
$z^*\Psi_\sigma^*\bar{\theta}=\sigma^*\bar\theta$ as $\Psi_\sigma\circ
z=\sigma$, and $z^*\theta_0=0$ as for any $s\in\Gamma(\Sigma)$,
$s^*\theta_{\mathrm{can}}=0$ (see Observation \ref{obs:contact
  form}). Therefore, $\beta=\sigma^*\bar\theta$. 
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Constructing contact isotropic realisations %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Constructing contact isotropic
  realisations} \label{sec:constr-cont-isotr}

% Sections \ref{sec:smooth-class} and \ref{sec:cont-class} imply that the smooth invariants of a
% contact isotropic realisation $\phi : \CM \to \JM$ are 
% \begin{itemize}
% \item its period lattice $\Sigma \subset \ker \rho_P$ which is a
%   $\ZZ^k$-bundle all of whose are holonomic (cf. Corollary
%   \ref{cor:sections_period});
% \item its Chern class $c \in \HH^2(P;\Sigma)$.
% \end{itemize}

In analogy with the classification of symplectic isotropic
realisations of Poisson manifolds carried out in \cite{dd}, the classification of
contact isotropic realisations over a given regular Jacobi manifold
$\JM$ of corank $k$, all of whose leaves are even dimensional, 
consists of the following two parts:

\begin{enumerate}[label={\bf Part \arabic*:}, ref=Part \arabic*]
\item \label{item:16} Classify closed submanifolds $\Sigma \subset \ker
  \rho_P$  for which the
  composite $\Sigma \hookrightarrow \ker \rho_P \to P$ is a
  $\ZZ^k$-bundle and all of its sections are holonomic. Such a
  submanifold is henceforth called {\bf period lattice} even though it
  may not necessarily arise from a contact isotropic realisation;
\item \label{item:17} Having fixed a period lattice $\Sigma \subset
  \ker \rho_P$, determine which cohomology classes in
  $\HH^2(P;\Sigma)$ give rise to contact isotropic realisations, {\it
    i.e.} whether the total space of the (isomorphism class of the) principal $\ker \rho_P/\Sigma$-bundle associated
  to $c$ admits a contact structure which makes the projection to $P$
  into a contact isotropic realisation.
\end{enumerate}

\ref{item:16}, while very interesting in its own right, is beyond the
scope of this paper; Section \ref{sec:transv-integr-proj} gives an
alternative geometric interpretation of a period lattice. This section concentrates on solving the problem
outlined in \ref{item:17}. To this end, it is important to remark that there
already exist partial solutions to this question, as the following
examples show.

\begin{exm}[Regular contact manifolds]\label{exm:rcm}
  Suppose that $\JM$ is a closed symplectic manifold, so that $L_P = P \times
  \RR$ and the Jacobi bracket is nothing but the Poisson bracket on
  $C^{\infty}(P)$ induced by a symplectic form $\omega$. Then
  $\ker \rho_P = P \times \RR \subset J^1 L_P$, which implies that any period lattice
  $\Sigma$ is trivial. Let $\Sigma \subset \ker \rho_P$ be a period
  lattice and suppose that $\alpha \in \Gamma(\Sigma)$ is a nowhere
  vanishing section. Corollary \ref{cor:sections_period} gives that
  $\alpha = j^1 f$ for some smooth function $f$, which implies that
  $f$ is a non-zero constant function. Without loss of generality, fix
  $f \equiv 1$. A contact isotropic realisation $\phi : (M,\theta) \to
  (P,\omega)$ 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!) $\theta \in
  \Omega^1(M)$ (cf. Remark \ref{rk:reeb_trivial_case}). A co-oriented
  contact manifold $(M,\theta)$ as above is known in the literature as
  a {\bf regular contact manifold}
  (cf. \cite{boothby_wang,geiges_book}). Such a bundle exists if and only if the
  cohomology class $[\omega] \in \HH^2(P;\RR)$ is integral and the
  Chern class of the bundle must then be equal to
  $[\omega]$ (cf. \cite[Theorems 7.2.4 and 7.2.5]{geiges_book}). Reformulating
  this result in this paper's language, $(P,\omega)$ endowed with the
  period lattice $\Sigma \to P$ admits a contact isotropic
  realisation if and only if $[\omega]$ is integral and the
  corresponding Chern class equals $[\omega]$, where
  $\HH^2(P;\Sigma)$ is identified with $\HH^2(P;\ZZ)$. In
  particular, not all elements of $\HH^2(P;\Sigma)$ give rise to a
  contact isotropic realisation (in fact, there may be none that does).
\end{exm}

\begin{exm}[Zero Jacobi structure]\label{exm:zero_structure}
  Suppose that $\JM$ is endowed with the zero Jacobi structure, so
  that $L_P = P \times \RR$ and the Jacobi bracket is identically
  zero. In this case, $\ker \rho_P = J^1 L_P := J^1 P$. Given any
  period lattice $\Sigma \subset J^1 P$, it can be shown that {\em every} cohomology class
  in $\HH^2(P;\Sigma)$ gives rise to a contact isotropic realisation,
  as proved in \cite{Banyaga}.
\end{exm}


The main result of this section provides a cohomological criterion
that solves \ref{item:17} in
the spirit of \cite[Theorems 4.2 and 4.3]{dd} which unifies
and extends Examples \ref{exm:rcm} and \ref{exm:zero_structure}
above (cf. Theorem \ref{thm:main}). The rest of the section is
structured as follows. First, it is shown that {\em locally} any
regular Jacobi manifold $\JM$ all of whose leaves are even dimensional with
a period lattice $\Sigma \subset \ker \rho_P$ admits a contact
isotropic realisation (cf. Section \ref{sec:local-construction}). The
global obstruction to constructing these objects is investigated in
Section \ref{sec:cohom-crit}.

\subsection{Local construction}\label{sec:local-construction}
Fix notation as in Theorem \ref{thm:aa} and its proof. Let $\phi : \CM
\to \JM$ be a contact isotropic realisation with characteristic
foliation denoted by $\mathcal{F}$, and let $\sigma : U \to M$
be a local section. Theorem \ref{thm:aa} proves that
$\Psi_\sigma^*\bar{\theta}=\theta_0+\pi^*\sigma^*\bar{\theta}$, where
$\theta_0 \in \Omega^1(\ker \rho_P/\Sigma;\pi^* L_P)$ is the 1-form
constructed in Observation \ref{obs:contact form}. The next lemma
shows that $\sigma^*\bar{\theta} \in
\Omega^1(P;\sigma^*\phi^*L_P) = \Omega^1(P;L_P)$ is related to the foliated 2-form uniquely defined by
$\omega(\rho_P(j^1u), \rho_P(j^1v)) = \{u,v\}_P$ (cf. Proposition \ref{lemma:coho_class}).

\begin{lemma}\label{lemma:relation}
  The restriction of the 1-form $\sigma^*\bar{\theta}$ to the
  foliation $\mathcal{F}$ satisfies
  $\de_{\mathcal{F}}\sigma^*\bar{\theta} = \omega$, {\it i.e.} it is a
  local primitive for $\omega$.
\end{lemma}
\begin{proof}
  As in the proof of Proposition \ref{lemma:coho_class}, it suffices
  to check that
  $$\de_{\mathcal{F}}\sigma^*\bar{\theta}(\rho_P(j^1u),\rho_P(j^1v)) =
  \omega(\rho_P(j^1u),\rho_P(j^1v)).$$
  \noindent
  By definition,
  \begin{equation}
    \label{eq:30}
    \begin{split}
      \de_{\mathcal{F}}\sigma^*\bar{\theta}(\rho_P(j^1u),\rho_P(j^1v))
      = -\sigma^*\bar{\theta}([\rho_P(j^1u), \rho_P(j^1v)]) &+
      \bar{\nabla}_{\rho_P(j^1u)}(\sigma^*\bar{\theta}(\rho_P(j^1v))) \\
      &- \bar{\nabla}_{\rho_P(j^1v)}(\sigma^*\bar{\theta}(\rho_P(j^1u))).
    \end{split}
  \end{equation}
  \noindent
  First, consider 
  $$ \bar{\nabla}_{\rho_P(j^1u)}(\sigma^*\bar{\theta}(\rho_P(j^1v))) =
  \bar{\nabla}_{\rho_P(j^1u)}(\sigma^*(\bar{\theta}(D\sigma(\rho_P(j^1v))))); $$
  \noindent
  since $\phi \circ \sigma = id$ and $\phi : \CM \to \JM$ is a Jacobi
  map with bundle morphism $F$, it follows that 
  $$ D\sigma(\rho_P(j^1v)) = \rho_M(j^1(F\circ \phi^*v)) + X_v, $$
  \noindent
  where $X_v \in \Gamma(\ker D\phi|_{\sigma(U)})$ (and similarly for $u$). Note that since $\ker D\phi =\rho_M\circ F( \phi^*
  \ker \rho_P)$ and the vector fields $X_u,X_v$ are only defined on
  $\sigma(U)$, it follows that there exist local sections $(w,\eta),
  (z,\zeta) \in \Gamma(\ker \rho_P)$ with 
  $$X_u =
  \rho_M(F(\phi^*w,\phi^*\eta)) \text{ and } X_v =
  \rho_M(F(\phi^*z,\phi^*\zeta)).$$
  \noindent
  Thus by definition of $\rho_M$ (cf. Observation \ref{obs:contact_algebroid})
  $$\bar{\theta}(D\sigma(\rho_P(j^1v))) =
  \bar{\theta}(\rho_M(j^1(F\circ \phi^*v)) +
  \rho_M(F(\phi^*z,\phi^*\zeta))) = \phi^*(v + z), $$
  \noindent
  and, similarly, $\bar{\theta}(D\sigma(\rho_P(j^1u))) =
  \phi^*(u+w)$. Therefore, by definition of $\bar{\nabla}$,
  \begin{equation}
    \label{eq:31}
     \bar{\nabla}_{\rho_P(j^1u)}(\sigma^*\bar{\theta}(\rho_P(j^1v)))
    - \bar{\nabla}_{\rho_P(j^1v)}(\sigma^*\bar{\theta}(\rho_P(j^1u)))
    = \{u,v+z\}_P - \{v, u + w\}_P.
  \end{equation}
  \noindent
  On the other hand, 
  \begin{equation}
    \label{eq:32}
    \begin{split}
      &(\sigma^*\bar{\theta})([\rho_P(j^1u), \rho_P(j^1v)]) =
      \sigma^*(\bar{\theta}(D\sigma ([\rho_P(j^1u), \rho_P(j^1v)]) )) \\
      & = \sigma^*(\bar{\theta}(([D\sigma (\rho_P(j^1u)),
      D\sigma(\rho_P(j^1v))]))) \\
      & =
      \sigma^*(\bar{\theta}([R_{F(\phi^*(u+w))} +
      c^{\sharp}_{\theta}(F\circ \phi^*\eta|_H), R_{F(\phi^*(v+z))} +
      c^{\sharp}_{\theta}(F\circ \phi^*\zeta|_H)])) \\
      & = \sigma^*(\bar{\theta}(R_{\{F(\phi^*(u+w)),
        F(\phi^*(v+z))\}_M} + [c^{\sharp}_{\theta}(F\circ \phi^*\eta|_H),
      c^{\sharp}_{\theta}(F\circ \phi^*\zeta|_H)])) \\
      & = \sigma^*(\phi^*(\{u+w,v+z\}_P) - \phi^*\eta(\rho_M(F\circ
      \phi^*\zeta))) \\
      & = \{u+w,v+z\}_P - \eta(D\phi (\rho_M(F\circ
      \phi^*\zeta))) = \{u+w,v+z\}_P - \eta(\rho_P(\zeta)), 
    \end{split}
  \end{equation}
  \noindent
  where the third equality follows from the definition of $\rho_M$ and
  the above conventions, the fourth from the defining property of
  Reeb vector fields (cf. Definition \ref{defn:Reeb_vector_field}),
  the fifth by definition of Reeb vector fields and the curvature map,
  and the last from the fact that $\phi$ is a Jacobi map with bundle
  component $F$. Observe that since $(w,\eta),(z,\zeta) \in \Gamma(\ker
  \rho_P)$,
  $$ \{w,z\}_P - \eta (\rho_P(\zeta)) = \{w,z\}_P + \eta(\rho_P(j^1z))
  = \omega (\rho_P(w,\eta),\rho_P(z,\zeta)) = 0, $$
  \noindent
  where the general definition of $\omega$ is used (cf. Observation
  \ref{obs:unravelling_omega}). Thus equation \eqref{eq:32} yields
  that 
  $$ (\sigma^*\bar{\theta})([\rho_P(j^1u), \rho_P(j^1v)]) = \{u,v\}_P
  + \{u,z\}_P + \{w,v\}_P. $$
  \noindent
  Using this identity together with equation \eqref{eq:31} in equation
  \eqref{eq:30} yields the required result.
\end{proof}

The condition of Lemma \ref{lemma:relation} 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 $\JM$
and a period lattice $\Sigma \subset \ker \rho_P$.

\begin{prop}\label{prop:local_cir}
Let $U \subset P$ be an open set and fix $\beta\in\Omega^1(U,L)$ with the property that $\de_{\mathcal{F}}\beta=\omega$.
Then 
$$\pi:(\ker\rho/\Sigma|_U,\theta_0+\pi^*\beta)\to (U,L_U,\{\cdot,\cdot\})$$
is a contact isotropic realisation.
\end{prop}  


%%%%%%%%%%%%%%%%%%%%%%%%%
%%% A cohomological criterion %%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A cohomological criterion}\label{sec:cohom-crit}
Fix a regular Jacobi manifold $\JM$ all of whose leaves
are even dimensional, whose underlying foliation is denoted by
$\mathcal{F}$, and a period lattice $\Sigma \subset J^1 L_P$. By assumption, $\Gamma(\Sigma)$ consists
solely of holonomic sections (cf. Corollary
\ref{cor:sections_period}); thus the restriction of the Spencer
operator $\mathrm{D} : \Gamma(\ker \rho_P) \to \Omega^1(P;L_P)$
(cf. Observation \ref{ex:Spencer_operator}) descends to a homomorphism
$\hat{\mathrm{D}} : \Gamma(\ker \rho_P/\Sigma) \to \Omega^1(P;L_P)$. 

The next
lemma shows that the image $\hat{\mathrm{D}}(\Gamma(\ker \rho_P/\Sigma))$
consists solely of $\de_{\mathcal{F}}$-closed 1-forms, where
$\de_{\mathcal{F}}$ is defined as in equation \eqref{eq:17}.

\begin{lemma}\label{lemma:sod}
  For any $\alpha \in \Gamma(\ker \rho_P)$,
  $\de_{\mathcal{F}}\mathrm{D}(\alpha) = 0$.
\end{lemma}

\begin{proof}
  It suffices to check that
  $\de_{\mathcal{F}}\mathrm{D}(\alpha)(\rho_P(j^1u),\rho_P(j^1v)) = 0$ for any
  $u,v \in \Gamma(L_P)$ (cf. the
  proof of Lemma \ref{lemma:coho_class}). Fix two such sections of
  $L_P$; by definition of
  $\de_{\mathcal{F}}$, 
  \begin{equation}
    \label{eq:33}
    \begin{split}
      & \de_{\mathcal{F}}\mathrm{D}(\alpha)(\rho_P(j^1u),\rho_P(j^1v))
      = -
      \mathrm{D}_{[\rho_P(j^1u), \rho_P(j^1v)] }(\alpha) \\
      & + \bar{\nabla}_{\rho_P(j^1u)}(\mathrm{D}_{\rho_P(j^1v)}(\alpha))
      - \bar{\nabla}_{\rho_P(j^1v)}(\mathrm{D}_{\rho_P(j^1u)}(\alpha)) \\
      & = -\mathrm{D}_{\rho_P(j^1\{u,v\}_P)}(\alpha) +
      \{u,\mathrm{D}_{\rho_P(j^1v)}(\alpha)\}_P - \{v,\mathrm{D}_{\rho_P(j^1u)}(\alpha)\}_P,
    \end{split}
  \end{equation}
  \noindent
  where the second equality uses properties of the anchor map
  $\rho_P$, of the Lie bracket on $\Gamma(J^1 L_P)$, and the
  definition of $\bar{\nabla}$ (cf. equation \eqref{eq:26}). Let $z
  \in \Gamma(L_P)$ be any section; then
  \begin{equation*}
    \begin{split}
      \mathrm{D}_{\rho_P(j^1z)}(\alpha) & = \nabla_{\alpha} z +
      pr([j^1z,\alpha])  = pr([j^1z,\alpha]) = - pr([\alpha,j^1z]) \\
      & = \nabla_{j^1z}(pr(\alpha)) - \mathrm{D}_{\rho(\alpha) }(j^1z) = \{z,pr(\alpha)\}_P,
    \end{split}
  \end{equation*}
  \noindent
  where the first and fourth equalities use the compatibility
  condition for the Spencer operator of equation \eqref{horizontal},
  the second uses that $\alpha \in \Gamma(\ker \rho_P)$, the third
  exploits anti-symmetry of the Lie bracket, and the last follows by
  definition of the $J^1 L_P$-connection $\nabla$ (cf. equation
  \eqref{eq:nabla}). Applying the above equality to the right hand
  side of equation \eqref{eq:33}, it follows that
  $$ \de_{\mathcal{F}}\mathrm{D}(\alpha)(\rho_P(j^1u),\rho_P(j^1v)) =
  -(\{\{u,v\}_P,pr(\alpha)\}_P + \mathrm{c.p.}) = 0, $$
  \noindent
  where c.p. stands for cyclic permutation and the last equality
  follows from the Jacobi identity. This proves that
  $\de_{\mathcal{F}}D(\alpha) = 0$, thus completing the proof.
\end{proof}

% \begin{obs}\label{obs:so_pullback}
%   For any $\alpha + \Sigma \in \Gamma(\ker \rho_P/\Sigma)$,
%   $\hat{\mathrm{D}}(\alpha + \Sigma) = \alpha^* \theta_{\mathrm{can}}$, where $\theta_{\mathrm{can}} \in \Omega^1(J^1
%   L_P;\pi^* L_P)$ is the Cartan contact form. 
% \end{obs}

Let $\mathcal{C}^{\infty}(\ker \rho_P/\Sigma)$ and
$\mathcal{Z}^{1}(\mathcal{F};L_P)$ denote the sheaves of smooth
sections of $\ker \rho_P/\Sigma \to P$ and of foliated closed 1-forms
with values in $L_P$ respectively. Lemma \ref{lemma:sod} implies that
there is a morphism of sheaves 
\begin{equation}
  \label{eq:20}
  \hat{\mathrm{D}} : \mathcal{C}^{\infty}(\ker \rho_P/\Sigma) \to
  \mathcal{Z}^{1}(\mathcal{F};L_P),
\end{equation}
\noindent
which induces a homomorphism at the level of cohomology with values in
the above sheaves, 
\begin{equation}
  \label{eq:18}
  \mathcal{D} : \HH^1(P;\mathcal{C}^{\infty}(\ker \rho_P/\Sigma)) \to
  \HH^1(P;\mathcal{Z}^{1}(\mathcal{F};L_P)).
\end{equation}
\noindent
Recall that $\HH^1(P;\mathcal{C}^{\infty}(\ker \rho_P/\Sigma)) \cong
\HH^2(P;\Sigma)$ (cf. the discussion preceding Definition
\ref{defn:cc}), while a standard double \v Cech-Lie algebroid
differential complex argument implies that
$\HH^1(P;\mathcal{Z}^{1}(\mathcal{F};L_P)) \cong
\HH^2(\mathcal{F};L_P)$. Using these identifications tacitly, the
above yields a homomorphism $ \mathcal{D} : \HH^2(P;\Sigma) \to
\HH^2(\mathcal{F};L_P)$ which plays a central role in the following
theorem, which is the main result of this section.

\begin{thm}\label{thm:main}
  A Jacobi manifold $\JM$ admits a contact isotropic realisation
  with period lattice $\Sigma$ and Chern class $c \in \HH^2(P;\Sigma)$
  if and only if $\mathcal{D}c = [\omega]$, where $\omega \in
  \Omega^2(\mathcal{F};L_P)$ is the closed foliated 2-form constructed
  in Lemma \ref{lemma:coho_class}.
\end{thm}
\begin{proof}
  The strategy is similar to that of \cite[Theorems 4.2 and
  4.3]{dd}. Throughout the proof, let $\mathcal{U} := \{U_i\}$ denote
  a good open cover of $P$ unless otherwise stated. Suppose first that
  $\phi : \CM \to \JM$ is a contact isotropic realisation with Chern
  class $c \in \HH^2(P;\Sigma)$, and denote by $\bar{\theta} \in
  \Omega^1(M;\phi^* L_P)$ the induced generalised contact form. For each $i$, let $\sigma_i : U_i \to
  M$ be a local smooth section of $\phi$; as in Theorem
  \ref{thm:local_model}, the section $\sigma_i$ induces a local
  trivialisation $\Psi_i : \ker \rho_P/\Sigma|_{U_i} \to
  \phi^{-1}(U_i)$. By Definition \ref{defn:cc}, a \v Cech cocycle
  representing $c$ is given by the smooth maps $t_{ij} : U_{ij} \to \ker \rho_P /\Sigma$ defined by 
  $$ \Psi_j^{-1} \circ \Psi_i([\alpha]) = [\alpha] +
  t_{ij}(\pi(\alpha)),$$
  \noindent
  where $\pi: \ker \rho_P/\Sigma \to P$ is the projection (cf. the
  discussion following Theorem \ref{thm:local_model}). Thus a \v
  Cech cocycle representing $\mathcal{D}c$ is given by
  $\{t_{ij}^*\theta_0 = \hat{\mathrm{D}}(t_{ij})\}$, where the
  equality follows by definition of $\theta_0$ (cf. Observation
  \ref{obs:contact form}). By definition of $t_{ij}$, $\Psi \circ
  t_{ij} = \sigma_j$ on $U_{ij}$; thus
  \begin{equation}
    \label{eq:13}
      \sigma_j^* \bar{\theta} =t_{ij}^* \circ \Psi_i^*\bar{\theta} =
      t_{ij}^*(\theta_0 + \pi^* \sigma_i^*\bar{\theta}) = t_{ij}^*\theta_0 + \sigma_i^* \bar{\theta},
  \end{equation}
  \noindent
  where the second equality uses Theorem
  \ref{thm:aa}, and the last follows by noticing that $\pi \circ
  t_{ij} = id$. Equation \eqref{eq:13} gives that $t_{ij}^*\theta_0 =
  \sigma_j^*\bar{\theta} - \sigma_i^*\bar{\theta}$. By Lemma
  \ref{lemma:relation}, $\de_{\mathcal{F}}(\sigma_j^* \bar{\theta}) = 
  \omega = \de_{\mathcal{F}}(\sigma_i^*\bar{\theta})$; this implies
  that the cohomology class corresponding to the \v Cech cocycle
  $\{\sigma_j^*\bar{\theta} - \sigma_i^* \bar{\theta}\}$ is precisely
  $[\omega]$, as it is the difference of two primitives of
  $\omega$. \\

  Conversely, suppose that $\mathcal{D}c =  [\omega]$. Choose the good
  open cover $\mathcal{U}$ so that, for each $i$, there exists a
  primitive for $\omega|_{U_i}$, {\it i.e.} there exists $\beta_i \in
  \Omega^1(\mathcal{F}|_{U_i};L_P)$ with $\de_{\mathcal{F}}\beta_i =
  \omega|_{U_i}$. Since $\mathcal{F}$ is regular, the foliated 1-forms
  $\beta_i$ can be extended to honest 1-forms ({\it i.e.} elements of
  $\Omega^1(U_i;L_P)$), which are henceforth also denoted by $\beta_i$
  by abuse of notation. Since $\mathcal{D}c = [\omega]$, there exists
  a \v Cech cocycle $\{t_{ij}\}$ representing $c$ (subordinate to the above good open
  cover) which satisfies $t_{ij}^* \theta_0 = \beta_j - \beta_i$ on
  $U_{ij}$. For each $i$, consider the contact isotropic realisation
  $\pi : (\ker \rho_P/\Sigma|_{U_i},\theta_0 + \pi^*\beta_i) \to
  (U_i,L_P|_{U_i}, \{\cdot,\cdot\}_P|_{U_i})$ (cf. Proposition
  \ref{prop:local_cir}). The principal $\ker \rho_P/\Sigma$-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
  $t_{ij}^* \theta_0 = \beta_j - \beta_i$ ensures that the locally
  defined generalised contact forms $\theta_0 + \pi^*\beta_i$ patch
  together to give a globally defined generalised contact form $\theta
  \in \Omega^1(M;\phi^* L_P)$, where $\phi : M \to P$ is the principal $\ker \rho_P/\Sigma$-bundle
  over $P$ with Chern class $c$ constructed above. Setting $H := \ker
  \theta$, have that $\phi: \CM \to \JM$ is a contact isotropic
  realisation as it suffices to check that the properties \ref{item:IR1}, \ref{item:IR2} and
  \ref{item: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.
\end{proof}

\begin{exm}\label{exm:non-trivial}
  Theorem \ref{thm:main} can be used to construct examples of contact
  isotropic realisations all of whose invariants are non-trivial, {\it
    i.e.} with non-trivial line bundle, with period lattice $\Sigma$ defining a non-trivial local
  system of coefficients and with non-zero Chern class. To this end,
  consider $\RR \mathrm{P}^2$ with the tautological line bundle $\tau
  \to \RR \mathrm{P}^2$ and the zero Jacobi bracket. Example
  \ref{ex:projective_plane} below shows how to construct a period
  lattice $\Sigma_{\RR \mathrm{P}^2} \subset J^1 \tau$. It can be
  checked that $\HH^2(\RR \mathrm{P}^2;\Sigma_{\RR \mathrm{P}^2})
  \cong \ZZ^3$ (cf. \cite{sepe} for the explicit calculation). Since the
  Jacobi structure under consideration is trivial, the induced
  characteristic foliation is by points; in other words,
  $\HH^2(\mathcal{F};\tau) = 0$. This implies that the homomorphism
  $\mathcal{D} : \HH^2(\RR \mathrm{P}^2;\Sigma_{\RR \mathrm{P}^2}) \to
  \HH^2(\mathcal{F};\tau)$ is identically zero; Theorem \ref{thm:main}
  therefore gives that all $c \in \HH^2(\RR \mathrm{P}^2;\Sigma_{\RR
    \mathrm{P}^2})$ give rise to contact isotropic
  realisations. Taking $c \neq 0$, obtain an example of contact
  isotropic realisation defined on a non-trivial line bundle with
  non-trivial invariants.
\end{exm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Transversal integral projective structures %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Transversal integral $L$-projective structures} \label{sec:transv-integr-proj}
The period lattice $\Sigma$ associated to a contact isotropic realisation
$\phi : \CM \to \JM$ induces a geometric structure on the
characteristic foliation of the Jacobi manifold $\JM$. 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. \cite{Banyaga}). 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. \\

Throughout this section, let $(N,\mathcal{F})$ denote a foliated
manifold whose foliation has codimension $l$, and let $\pi : L \to N$
be a line bundle, unless otherwise stated.

\begin{defn}\label{defn:tilp}
  A {\bf transversal integral
    $L$-projective structure} (TIP-structure for short) on
  $(N,\mathcal{F})$ is a choice of
  \begin{itemize}
  \item flat $T\mathcal{F}$-connection on $\pi : L \to N$, 
    $\nabla:\Gamma(T\mathcal{F})\times\Gamma(L)\to\Gamma(L)$;
  \item an embedded smooth submanifold $\Sigma\subset J^1L$ satisfying
  \end{itemize}
  \begin{enumerate}[label=(T\arabic*), ref=(T\arabic*)]
    \item\label{item:8} the composite $\Sigma\hookrightarrow J^1L\to N$ is
      a fibre bundle with fibre $\ZZ^{l+1}$ and structure group
      $\mathrm{GL}(l+1;\ZZ)$ (henceforth referred to as a {\em $\ZZ^{l+1}$-bundle});
      % \item\label{item:1'} the composite $\Sigma\hookrightarrow J^1L\to L$ is different from zero on each fibre of $\pi$;
    \item\label{item:9} any locally defined section $\xi:U\subset N\to
      \Sigma$ is holonomic, i.e. $\xi=j^1u$ for some $u\in\Gamma(L)$; 
    \item\label{item:10} any locally defined section $j^1u:U\subset N\to
      \Sigma$ is flat, i.e. $\nabla u=0$.
    \end{enumerate}
\end{defn}

Before giving some examples of the above structure, it is useful to
reformulate the existence of a flat $T\mathcal{F}$-connection and
property \ref{item:8} in Definition \ref{defn:tilp} in equivalent
terms; this is done in the following lemma.

\begin{lemma}\label{lemma:tilp}
  The foliated manifold $(N,\mathcal{F})$ admits a transversal
  integral $L$-projective structure if and only if there exists an
  embedded submanifold $\Sigma \subset J^1L$ satisfying properties
  \ref{item:8}, \ref{item:9}, and 
  \begin{enumerate}[label=\emph{(T\arabic*)}, ref=(T\arabic*),start=4]
  \item \label{item:11} $\Sigma^{\RR} := \Sigma
    \otimes \RR$ fits in a short exact sequence of vector bundles 
    \begin{equation}
      \label{eq:4}
      0 \to \nu^* \otimes L \to \Sigma^{\RR} \to L
      \to 0,
    \end{equation}
  \end{enumerate}
  \noindent
  where $\nu^* \subset \cotan N$ is the conormal bundle to the
  foliation, and $\Sigma^{\RR} \to L$ is the restriction of the
  canonical projection $J^1 L \to L$.
\end{lemma}

\begin{proof}
  Suppose that $(N,\mathcal{F})$ admits a transversal integral
  $L$-projective structure, so that there exist $\nabla$ and $\Sigma$
  as in Definition \ref{defn:tilp}. The aim is to prove that the short
  exact sequence of equation \eqref{eq:4} holds. First, it is shown
  that the map $\Sigma^{\RR} \to L$ is pointwise surjective. Fix a
  point $p \in P$, an open neighbourhood $p \in U$ and a
  trivialisation $L|_U \cong U \times \RR$. By shrinking $U$ if
  needed, it may be assumed that $\Sigma|_U$ is also trivialisable;
  properties \ref{item:8} and \ref{item:9} imply that there exist
  smooth functions $u_1,\ldots, u_{l+1} \in C^{\infty}(U)$ with the
  property that $j^1u_1,\ldots,j^1u_{l+1}$ are a frame of
  $\Sigma|_U$. It suffices to prove that there exists $i$ such that
  $u_i(p) \neq 0$. Suppose the contrary; since $\nabla$ is a flat
  connection, it can be locally written by 
  $$ \nabla_X u := \de u(X) + \beta(X)u, $$
  \noindent
  where $X \in \Gamma(T \mathcal{F}|_U)$, $u \in C^{\infty}(U)$ and
  $\beta \in \Omega^1(\mathcal{F})$ is a closed foliated
  1-form. Property \ref{item:10} implies that, for all $i$, and for
  all $X \in T_p\mathcal{F}$, $\de_p u_i(X_p) = \beta_p(X_p)u_i(p) =
  0$. Thus, for all $i$, $\de_p u_i \in \nu_p^*$. Using the
  identification $J^1 U \cong \cotan U \oplus \RR$, have that
  $j^1_pu_1,\ldots,j^1_pu_{l+1} \in \nu^*_p \oplus 0 \subset J^1
  U$. However, $\dim \nu^*_p = l$, which contradicts the independence
  of $j^1u_1,\ldots,j^1u_{l+1}$ on $U$. Since $p \in P$ is arbitrary,
  this argument shows that $\Sigma^{\RR} \to L$ is pointwise
  surjective. \\
  \noindent
  To complete the proof that equation \eqref{eq:4} holds, it suffices
  to show that $\nu^* \otimes L = (\cotan P \otimes L) \cap \Sigma^{\RR}$. To this end,
  fix $p \in P$, let $U$ be an open neighbourhood over which both
  $L$ and $J^1 L$ can be trivialised as above. Given any element of
  $\alpha_p \in (\cotan_p
  P \otimes L) \cap \Sigma^{\RR}$, (by possibly shrinking $U$ if
  needed) there exists a smooth section
  $\alpha : U \to (\cotan P \otimes L) \cap \Sigma^{\RR}$ with
  $\alpha(p) = \alpha_p$, since $(\cotan P \otimes L) \cap
  \Sigma^{\RR}=\ker(\Sigma^{\RR} \to L) \to P$ is a vector bundle of
  rank $l$. As above, fix a frame $j^1u_1,\ldots,j^1u_{l+1}$ of $\Sigma|_U$;
  then there exist smooth functions $f_1,\ldots f_{l+1} \in
  C^{\infty}(U)$ with $\alpha = \sum\limits_{i=1}^{l+1} f_i
  j^1u_i$. Since $\alpha \in \Gamma(\ker( J^1L\to L)|_U)$, it follows
  that $\sum\limits_{i=1}^{l+1}f_i u_i = 0$ on $U$. Using property
  \ref{item:10}, for all $X \in \Gamma(T\mathcal{F}|_U)$, 
  $$ 0 = \nabla_X(\sum\limits_{i=1}^{l+1}f_i u_i) =
  \sum\limits_{i=1}^{l+1}\de f_i(X) u_i = \mathrm{D}(\alpha)(X), $$
  \noindent
  where $\mathrm{D}$ is the Spencer operator. This implies that
  $\mathrm{D}(\alpha) \in \Gamma(\nu^* \otimes L|_U)$, which, in turn, gives that
  $\alpha \in \Gamma(\nu^* \otimes L|_U)$, since $\alpha \in \Gamma(
  \ker(J^1L \to L)|_U)$. Thus $\alpha_p \in \nu^*_p \otimes L_p$;
  since both $p$ and $\alpha_p$ are arbitrary, this proves that $
  (\cotan P \otimes L) \cap \Sigma^{\RR} \subset \nu^* \otimes
  L$. Equality follows by dimension counting; this shows that
  $\Sigma^{\RR}$ fits in the short exact sequence of equation
  \eqref{eq:4}.\\

  Conversely, suppose that $\Sigma \subset J^1 L$ satisfies properties
  \ref{item:8} and \ref{item:9}, and that the short exact sequence of
  equation \eqref{eq:4} holds. Let $v \in \Gamma(L)$ and $p\in P$;
  since $\Sigma^{\RR} \to L$ is surjective, there exists an open
  neighbourhood $U$ of $p$ and a section $j^1u \in \Gamma(\Sigma|_U)$
  with $u(q) \neq 0$ for all $q \in U$. Then, locally, there exists a
  smooth function $f \in C^{\infty}(U)$ with $v =f u$. For $X \in
  \Gamma(T\mathcal{F}|_U)$, define 
  \begin{equation}
    \label{eq:16}
    \nabla_X v := \mathcal{L}_X(f) u. 
  \end{equation}
  \noindent
  The above is well-defined, for if $v = f'u'$ for some other smooth
  nowhere vanishing section $u' \in \Gamma(L|_U)$ with $j^1 u' \in
  \Gamma(\Sigma|_U)$, then, by definition, $fj^1u-f'j^1u'\in
  \Gamma(\Sigma^{\RR})$, and
  \begin{equation*}
    \begin{split}
      fj^1u-f'j^1u'& =(j^1(fu),\mathrm{D}(fj^1u)) - (j^1(f'u'),
      \mathrm{D}(f'j^1 u')) \\
      &= (0,\mathrm{D}(fj^1u - f'j^1u')) \in
      \Gamma(\cotan P \otimes L).
    \end{split}
  \end{equation*}
  \noindent
  Thus $fj^1u-f'j^1u' \in \nu^*\otimes L$, which implies that, for all
  $X \in  \Gamma(T\mathcal{F}|_U)$, $\mathcal{L}_X(f) u =
  \mathcal{L}_X(f') u'$, giving that the formula \eqref{eq:16} is
  independent of the choices made. It can be checked that equation
  \eqref{eq:16} defines a flat $T \mathcal{F}$-connection and that
  $\Sigma$ satisfies property \ref{item:10} as required. This
  completes the proof of the lemma. 
\end{proof}

\begin{ex}[Radiant transversal integral affine structures]\label{exm:tias}
  Let $\Xi \subset \nu^*$ be a transversal integral affine
  structure on the foliated manifold $(N,\mathcal{F})$, {\it i.e.}
  $\Xi$ is an embedded submanifold which is a full rank lattice of
  $\nu^*$ and such that any (locally defined) smooth section $\sigma
  \in \Gamma(\Xi)$ is closed. Such structures arise naturally in the
  study of isotropic realisations of Poisson manifolds
  (cf. \cite{dd}). The existence of a $\Xi$ as above is equivalent to
  the existence of an atlas $\{(U_i,\varkappa_i)\}$ of submersions
  $\varkappa_i : U_i \to \RR^l$ locally defining the foliation
  $\mathcal{F}$ such that for all $i,j$ with $U_{ij} \neq \emptyset$,
  there exists $h_{ij}:=(A_{ij},c_{ij}) \in \mathrm{GL}(l;\ZZ) \ltimes \RR^l$
  with $\varkappa_j = h_{ij} \circ \varkappa_i$ on $U_{ij}$
  (cf. \cite{wolak} and Theorem \ref{thm:tilp_tilpa} below for an
  analogous statement for TIP-structures). Suppose further that $\Xi$
  is {\bf radiant}, {\it i.e.} for all indices $i,j$ as above, $c_{ij}
  = 0$ (cf. \cite{ghl,wolak} for a more intrinsic definition). Then
  $\Xi$ gives rise to a TIP-structure on the trivial line bundle $L:=N
  \times \RR \to \RR$ with flat connection
  $\nabla:=\de_{\mathcal{F}}$, where $\de_{\mathcal{F}}$ denotes the
  restriction of the exterior derivative to $T \mathcal{F}$, as
  follows. Let $\{(U_i,\varkappa_i)\}$ be an atlas of submersions as
  above; for each $i$, set $\varkappa_i:= (f^1_i,\ldots,f^l_i)$. The
  1-forms $\de f^1_i,\ldots, \de f^l_i$ are linearly independent over
  $U_i$ and $\Xi|_{U_i} = \ZZ \langle \de f^1_i,\ldots, \de f^l_i
  \rangle$. Locally $\Xi$ can be extended to a lattice of $J^1 L|_{U_i}$ by considering 
  $$ \ZZ\langle j^1f^1_i,\ldots j^1f^l_i \rangle; $$
  \noindent
  since $\Xi$ is radiant, these locally defined lattices patch
  together to yield a globally defined lattice $\bar{\Xi} \subset
  J^1L$. The lattice
  $$ \Sigma:= \bar{\Xi} \oplus \ZZ\langle j^1 1\rangle $$
  \noindent
  satisfies properties \ref{item:8}, \ref{item:9} and \ref{item:11} by
  construction. In light of Lemma \ref{lemma:tilp}, it defines a TIP-structure.
\end{ex}

The following example is, in some sense, `universal' (cf. Definition \ref{defn:tipa} and Theorem
\ref{thm:tilp_tilpa}).

\begin{ex}[The tautological line bundle over
    $\mathbb{R}\mathrm{P}^l$]\label{ex:projective_plane} 
  Let $\mathbb{R}\mathrm{P}^l$ be the real projective space and denote
  by $\tau\to \mathbb{R}\mathrm{P}^l$ the {\bf tautological line bundle}, whose fibre at a point
  $[\mathfrak{l}]\in\mathbb{R}\mathrm{P}^l$ is given by $\tau_{[\mathfrak{l}]}=\mathfrak{l}$. The aim is to
  construct a transversal integral $\tau$-projective structure on
  $\mathbb{R}\mathrm{P}^l$ endowed with the trivial foliation. Thus it suffices to construct a full-rank lattice
  $\Sigma_{\RR\mathrm{P}^l} \subset J^1 \tau$ all of whose sections
  are holonomic, {\it i.e.} so that it satisfies properties
  \ref{item:8} and \ref{item:9}. \\

  The universal cover of $\mathbb{R}\mathrm{P}^l$ can be identified
  with $\RR^{l+1}\setminus 0$ and the pull-back of $\tau$ to
  $\RR^{l+1}\setminus 0$, denoted by $\tilde{\tau}$ is trivialisable. In fact, it can be
  described as follows: it is the line bundle whose fibre over $r\in
  \RR^{l+1}\setminus 0$ is given by $\RR\langle r \rangle$. A nowhere
  vanishing section of $\tau$ is given by $z:r\mapsto (r,r)$. The
  bundle $J^1 \tilde{\tau}$ admits a full-rank lattice given by 
  $$ \tilde{\Sigma} : = \ZZ \langle j^1u_1,\ldots,j^1u_{l+1}
  \rangle, $$
  \noindent
  where $u_i := \frac{r_i}{\lVert r \rVert} z$, and $r
  =(r_1,\ldots,r_{l+1})$. It can be checked that $\tilde{\Sigma}$
  defines a transversal integral $\tilde{\tau}$-projective structure
  on $\RR^{l+1}\setminus 0$. The quotient of $\RR^{l+1}\setminus 0$ by
  the $\RR^*$-action by homotheties can be identified with $\RR
  \mathrm{P}^l$; the quotient of $\tilde{\tau}$ by the obvious lifted
  action yields $\tau$. By construction, the lifted $\RR^*$-action on
  $J^1\tilde{\tau}$ preserves $\tilde{\Sigma}$, which implies that
  $\tilde{\Sigma}$ induces a full-rank lattice $\Sigma_{\RR \mathrm{P}^l} \subset J^1
  \tau$ which satisfies property \ref{item:9}, as $\tilde{\Sigma}$
  does. Thus $\Sigma_{\RR \mathrm{P}^l}$ defines a
  transversal integral $\tau$-projective structure.
  
  % To show that $J^1\tau$ has a canonical lattice
  % $\Sigma_{\RR\mathrm{P}^k}\subset J^1\tau$ of rank $k+1$, whose
  % sections are holonomic, we do it as follows: consider
  % $\RR^{k+1}\setminus 0$. Consider the line bundle
  % $\tau_{\RR^{k+1}\setminus 0}$ over $\RR^{k+1}\setminus 0$, with
  % fibre at $r\in \RR^{k+1}\setminus 0$ given by the line passing
  % through $r$ and the origin. Of course, $\tau_{\RR^{k+1}\setminus 0}$
  % is trivializable with non-vanishing section $z:r\mapsto
  % r\in\tau_{\RR^{k+1}\setminus 0,r}$. With this, we consider the
  % $k+1$-canonical sections $x_1z,\ldots x_{k+1}z$ where
  % $x_i:\RR^{k+1}\setminus 0\to \RR$ is the $i$-th component of
  % $r/||r||$. Of course we have a full lattice
  % $\Sigma_{\RR^{k+1}\setminus 0}\subset J^1(\tau_{\RR^{k+1}\setminus
  %   0})$ as the $\ZZ$-span of the $k+1$-canonical linearly independent
  % holonomic sections $j^1(x_1z),\ldots j^{1}(x_{k+1}z)\in
  % J^1(\tau_{\RR^{k+1}\setminus 0})$.  

  
  % On the other hand, $\RR\setminus 0$ acts on $\RR^{k+1}\setminus 0$ by
  % multiplication $m_t:\RR^{k+1}\setminus 0\to \RR^{k+1}\setminus 0$ by
  % $t$, and this action lifts to the vector bundle isomorphism $m_t\times
  % id:\tau_{\RR^{k+1}\setminus 0}\to \tau_{\RR^{k+1}\setminus 0},
  % (r,v)\mapsto (m_t(r),v)$ (covering $m_t$). Note that the quotients of
  % $\RR^{k+1}\setminus 0$ and $\tau_{\RR^{k+1}\setminus 0}$ by the action
  % are precisely $\RR\mathrm{P}^{k}$ and $\tau$, and that
  % $\Sigma_{\RR^{k+1}\setminus 0}$ is preserved by the action. Therefore,
  % $\Sigma_{\RR^{k+1}\setminus 0}/(\RR\setminus 0)$ defines the desired
  % lattice $\Sigma_{\RR\mathrm{P}^k}$, which locally is given
  % by $$\Sigma_{\RR\mathrm{P}^k}=\ZZ\langle
  % j^1[x_1],\ldots,j^1[x_{k+1}]\rangle\subset J^1\tau.$$ 
\end{ex}

Another way of constructing TIP-structures is through the
following special class of foliated atlases. 

\begin{defn}\label{defn:tipa}
  A {\bf transversal integral projective atlas} of
  $(N,\mathcal{F})$ (TIP-atlas for short), is an atlas $\mathcal{A}
  :=\{(U_i, \chi_i)\}$ of 
  submersions $\chi_i : U_i \to \RR\mathrm{P}^l$, locally defining
  $\mathcal{F}$, and smooth maps $A_{ij}: U_{ij} \to
  \mathrm{GL}(l+1;\ZZ)$ for all $i,j$ with $U_{ij} \neq 0$ satisfying
  \begin{itemize}
  \item $\chi_j = [A_{ij}]\circ \chi_i$ on $U_{ij}$, where $[A_{\ij}]
    \in \mathrm{PGL}(l+1;\RR)$ is the equivalence class of $A_{ij}$;
  \item $\{A_{ij}\}$ satisfies the cocycle condition, {\it i.e.} for
    all $i,j,k$ with $U_{ijk} \neq \emptyset$, $A_{ik} = A_{jk}A_{ij}$.
  \end{itemize}
\end{defn}

\begin{rk}\label{rmk:maximal_atlas}
  There is a natural notion of two TIP-atlases being compatible
  (essentially inducing the same structure) and, thus, of {\it
    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).
\end{rk}

\begin{obs}\label{obs:from_atlas}
A TIP-atlas $\{(U_i, \chi_i)\}$ on $(N,\mathcal{F})$ gives rise to a
TIP-structure as follows. The line bundle $L\to N$ is defined by 
\begin{equation}\label{eq:line_bundle}
  L=\chi^*(\tau):=\coprod\limits_{i} \chi_i^*(\tau)/\sim,
\end{equation}
where $(m,v_{[\mathfrak{l}]})\in\chi_i^*(\tau)$ is related to
$(m',v_{[\mathfrak{l}']})\in\chi_j^*(\tau)$ if $m=m'\in N$ (hence $A_{ij}$ maps
the line $\mathfrak{l}$ to $\mathfrak{l}'\subset\RR^{l+1}$) and
$v_{[\mathfrak{l}']}=A_{ij}(v_{[\mathfrak{l}']})\in \RR^{l+1}$. The reason why equation
\eqref{eq:line_bundle} 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  
$$\chi^*: \chi^*(J^1\tau)\hookrightarrow J^1L,$$
where $\chi^*(J^1\tau):=\coprod\limits_i \chi_i^*(J^1\tau)/\sim$, and
$\gamma:T_{\chi_i(m)}\RR\mathrm{P}^l\to
T_{v_{\chi_i(m)}}\tau\in\chi_i^*(J^1\tau)_m$ is related to
$\gamma':T_{\chi_j(m')}\RR\mathrm{P}^l\to
T_{v_{\chi_j(m')}}\tau\in\chi_j^*(J^1\tau)_{m'}$ if $m=m'$ and
$DA_{ij}\circ\gamma=\gamma'\circ DA_{ij}$. Explicitly, $\chi$ is
defined by sending the class of $\gamma\in\chi_i^*(J^1\tau)_m$ to the
linear map 
$$\chi^*(\gamma):T_mN\to T_{[m,v_{\chi_i(m)}]}L, \ \ X\mapsto
[X,\gamma(D\chi_i(X))].$$  
As $A_{ij}\in\mathrm{GL}(l+1;\ZZ),$ then 
$$\Sigma:=\chi^*(\Sigma_{\RR\mathrm{P}^l})=\coprod\limits_i
\chi_i^*(\Sigma_{\RR\mathrm{P}^l})/\sim\;\subset J^1L$$  
is a well defined lattice sitting inside $J^1L$ via the map
$\chi$. Note that $\rank(\Sigma)=\rank(\Sigma_{\RR\mathrm{P}^l})=l+1$,
and that $\Sigma$ has properties \ref{item:8} and
\ref{item:9}. Finally, Lemma \ref{lemma:tilp} implies that $\Sigma\subset J^1L$ defines a
TIP-structure on $(N,\mathcal{F})$, since 
$\Sigma^{\RR}$ fits into the exact sequence \eqref{eq:4} by
construction.

% , because
% $\RR\langle\Sigma_{\RR\mathrm{P}^k}\rangle=J^1\tau$ (which implies
% that $\RR\langle\Sigma\rangle\to L$ is surjective), and for any $m\in
% N$, $m\in U_i$, $D_m\chi_i$ descends to an isomorphism
% $D_m\chi_i:T_mN/\mathcal{F}_m\to T_{\chi_i(m)}\RR\mathrm{P}^k$ (which
% implies that $\nu^*\otimes L\subset \RR\langle\Sigma\rangle$).   
\end{obs}


In fact, Observation \ref{obs:from_atlas} can be strengthened, since
every TIP-structure also gives rise to a TIP-atlas. This is the
content of the following theorem.

\begin{thm}\label{thm:tilp_tilpa}
  Then there is a 1-1 correspondence between 
  \begin{itemize}
  \item a transversal integral $L$-projective structure $\Sigma\subset J^1L$ of $(N,\mathcal{F})$;
  \item a transversal integral projective atlas $\{(U_i,\chi_i)\}$ of $(N,\mathcal{F})$.
  \end{itemize}
  The correspondence is given by 
  $$L=\chi^*(\tau),\ \ \ \Sigma=\chi^*(\Sigma_{\RR\mathrm{P}^l}),$$
  \noindent
  where the notation is as in Observation \ref{obs:from_atlas}.
\end{thm}

\begin{proof}
  Observation \ref{obs:from_atlas} 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 $\Sigma\subset J^1L$ of $(N,\mathcal{F})$.
  Choose an open cover $\{U_i\}$ of $N$ with the property that over each $U_i$ the line
  bundle $\pi:L\to N$ trivialises via the nowhere vanishing section $z_i:U_i\times
  \RR\to\pi^{-1}(U_i)$, and denote the transition functions by
  $c_{ij}:U_{ij} \to \mathrm{GL}(\RR)$. Without loss of generality,
  assume that all $U_i$ are small enough so
  that $\Sigma|_{U_i}$  can be trivialised with local frame
  $\alpha^i_1,\ldots \alpha^i_{l+1}\in \Gamma(\Sigma|_{U_i})$ and transition functions
  denoted by $A_{ij}:U_{ij} \to \mathrm{GL}(l+1;\ZZ)$. Property
  \ref{item:9} implies that for all $i$ and each $r = 1,\ldots, l+1$,
  there exist smooth functions $g^i_r:U_i\to \RR$ with $\alpha^i_r =
  j^1(g_r^iz_i)$. 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.
  
  \begin{claim}\label{claim:trans}
    For each $i$, the smooth map $\bar\chi_i:U_i\to
    \RR^{l+1},\bar\chi_i:=(g^i_1,\ldots,g^i_{l+1})$ takes values on
    $\RR^{l+1}\setminus 0$ and is transversal to the Euler vector field
    $E=\sum\limits_rx_r\frac{\partial}{\partial x_r}$. 
  \end{claim}
  \begin{proof}[Proof of Claim \ref{claim:trans}]
    Indeed, as $j^1g^i_1,\ldots,j^1g^i_{l+1}$ are a frame of 
    $\Sigma|_{U_i}$ and 
    $\Sigma^{\RR}\to L$ is pointwise onto by property \ref{item:11}, it
    follows that $\bar\chi_i \neq 0$. Secondly, to prove that
    $\bar\chi_i$ is transversal to $E$, {\it i.e.} $\Im
    D\bar\chi_i + E=T\RR^{l+1}$, it suffices to show that
    $\bar\chi_i^*:\Ann(E)\to \nu^*$ is injective. Using the same
    argument of Lemma \ref{lemma:tilp}, have that
    $dg^i_r\mid_{\mathcal{F}}=g^i_r\beta\mid_{\mathcal{F}}$, for some
    fixed closed, foliated 1-form $\beta$; hence  
    \begin{equation}\label{eq:euler}
      D\bar\chi_i|_{\mathcal{F}}=\beta \otimes E,
    \end{equation}
    which implies that $\chi_i^*(\Ann(E))\subset \nu^*$. To show
    that $\bar\chi_i^*:\Ann(E)_{\bar\chi_i(p)}\to \nu^*_p$ is injective
    for any $p\in U_i$, it suffices to show that it is surjective (which is, of
    course, equivalent by dimension counting). Let
    $\gamma_p\in\nu^*_p$ and extend it to a locally defined smooth
    section $\gamma \in \Gamma(\nu^*|_{U_i})$. By property \ref{item:11} there exist
    smooth functions $f^1,\ldots,f^{l+1}\in C^\infty(U_i)$ with   
    $$\gamma=\sum_rf^rj^1g^i_r\in \nu^*\subset J^1L\mid_{U_i}.$$
    Using the Spencer decomposition, it follows that  $\sum_rf^rg^i_r=0$
    and $\sum_rf^rdg^i_r=\gamma$. Therefore,
    $\chi_i^*(\gamma')=\gamma$ for 
    $$\gamma'=\sum\limits_rf^r(p)\de x_r\in \Gamma(Ann(E)|_{\bar\chi_i(U_i)},$$
    and hence surjectivity follows. 
  \end{proof}
  
  \begin{claim}\label{claim:commutative}
    The diagram
    \begin{eqnarray*}\xymatrix{
        & U_{ij} \ar[dl]_{\bar\chi_i} \ar[dr]^{\bar\chi_j} & \\
        \RR^{l+1}\setminus 0\ar[rr]^{c_{ji}A_{ij}} &  & \RR^{l+1}\setminus 0
      } 
    \end{eqnarray*}
    commutes.
  \end{claim}
  \begin{proof}[Proof of Claim \ref{claim:commutative}]
    This follows by definition of the maps $\bar{\chi}_i$.
  \end{proof}

  % Indeed, using the trivializations of $L$ and $\Sigma$ that we fixed
  % from the beginning, we have that on $U_i\cap U_j$, $c_{ij}z_i=z_j$,
  % and $A_{ij}(\alpha^i)=(\alpha^j)$ where $(\alpha^i)$ is the matrix
  % whose columns are the vectors $\alpha^i_1,\ldots,\alpha^{i}_{k+1}$,
  % and respectively for $(\alpha^j)$. Since $\alpha^i_r=j^1(g^i_rz_i)$
  % $(0\leq r\leq k+1)$, then $A_{ij}(g^iz_i)=(g^jz_j)$, which implies
  % that $A_{ij}(g^iz_i)=(g^jc_{ij}z_i)$ (where $(g^iz_i)$ is the vector
  % whose $r$-th entry is $g^i_rz_i$, respct. for $(g^jc_{ij}z_i)$), or in
  % other words, $c_{ji}A_{ij}(\bar\chi_i)=\bar\chi_j$. 

  Using Claims \ref{claim:trans} and \ref{claim:commutative}, a
  TIP-atlas can be defined as follows. Set $\chi_i:U_i\to
  \RR\mathrm{P}^l$ to be the composite
  $$\chi_i:U_i\overset{\bar\chi_i}{\to} \RR^{l+1}\setminus
  0\overset{q}{\to}\RR\mathrm{P}^l,$$ 
  where $q:\RR^{l+1}\setminus 0\to \RR\mathrm{P}^l$ is the quotient
  map. By Claim 1, $\chi_i$ is a submersion, and Claim 2 implies that
  $\chi_j = [A_{ij}] \circ \chi_i $ on $U_{ij}$.
  % send the the line passing through $\bar\chi_i(p)$ and the origin, to
  % the line passing through $\bar\chi_i(p)$ and the origin, hence
  % $A_{ij}(\chi_i(p))=\chi_j(p)$. 
  Moreover, since $Dq(E)=0$, equation
  \eqref{eq:euler} implies that $D\chi_i|_{\mathcal{F}}=0$, hence
  $\mathcal{F}$ is tangent to the fibres of $\chi_i$ on the one hand,
  and on the other, dimension counting shows that for $p\in U_i$,
  $T\mathcal{F}_p=\ker D_p\chi_i$. This shows that the atlas $\{(U_i,\chi_i)\}$
  defines an TIP-atlas for $(N,\mathcal{F})$. \\
  
  It remains to show that these constructions are inverse to each
  other. Start with a TIP-structure $\Sigma\subset J^1L$, and construct
  the TIP-atlas $\{(U_i,\chi_i)\}$ as above. First, it is shown that $L$ is
  isomorphic to $\chi^*(\tau)$, as constructed in Observation
  \ref{obs:from_atlas}. Observe that, for each $i$, $\chi^*_i\tau$ is
  trivialisable; a nowhere vanishing section is given by $p \mapsto
  (p,\chi_i(p))$; this follows from the fact that $\chi_i = q \circ
  \bar{\chi}_i$ and $q^* \tau $ is trivialisable. For each $i$, define a local
  trivialisation of $\chi^*(\tau)|_{U_i}$ by 
  $$\varphi_i:U_i\times \RR\to \chi^*(\tau)|_{U_i},\ \ (p, t)\mapsto(p,
  t \bar\chi_i(p))\in\chi_i^*(\tau).$$
  \noindent
  The transition functions with respect to these trivialisations are
  given by 
  \begin{equation*}
    \begin{split}
      \varphi_i(p,t)&=(p, t \bar\chi_i(p))\sim
      (p,A_{ij}(t\bar\chi_i(p))) \\
      &=(m,t\frac{c_{ji}}{c_{ji}}
      A_{ij}(\bar\chi_i(p)))=(p,tc_{ij}\bar\chi_j(p))=\varphi_j(p,tc_{ij}),
    \end{split}
  \end{equation*}
  \noindent 
  for $p \in U_{ij}$. Since the cocycle of transition functions of
  $\chi^* (\tau)$ 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
  TIP-atlas $\{(U_i,\chi_i)\}$ as outlined in Observation
  \ref{obs:from_atlas} is the same as the original
  TIP-structure. Moreover, it is straightforward to check that given a
  TIP-structure obtained from a TIP-atlas as in Observation
  \ref{obs:from_atlas}, the induced TIP-atlas is compatible with the
  original one. In both cases, details are left to the reader to check. 
\end{proof}

\section{Some examples}\label{sec:exampl-cont-isotr}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Integrable systems on contact manifolds %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Integrable systems on contact manifolds}\label{sec:integr-syst-cont}
% One of the main reasons behind the work in this paper is to provide a
% geometric framework to study the global geometry of non-commutative
% integrable Hamiltonian systems on contact manifolds as 
% defined in \cite{Banyaga,jovanovic}. The aim of this section is to
% prove that a non-commutative integrable Hamiltonian system on a
% contact manifold First, the main definitions
% and results in \cite{jovanovic} are recalled below without proof to
% illustrate the connection with contact isotropic realisations (which
% are stated and proved). \\

Let $(M,H)$ be a $2n+1$-dimensional contact manifold. A vector field
$X \in \X(M)$ is said to be
an {\bf infinitesimal automorphism} of $(M,H)$ if its flow preserves
$H$, or, equivalently, if $[X,Y] \in H$ for any $Y \in
\Gamma(H)$. Suppose that $\phi : (M, H) \to P$ is a proper surjective
submersion onto a $2n+1-k$-dimensional smooth manifold $P$ with
connected fibres ({\it i.e.} a fibre bundle). % The following definition
% encapsulates that of (the regular part of) a non-commutative integrable
% Hamiltonian system on a contact manifold (cf. \cite[Definition 2]{jovanovic}).

\begin{defn}\label{defn:cpics}
  The quadruple $(M, H, \phi, \mathcal{X})$, where $\mathcal{X}$ is an
  Abelian Lie algebra of infinitesimal automorphisms of
  $(M,H)$ of dimension $k$, is a {\bf complete
    pre-isotropic contact structure} if
  \begin{itemize}
  \item $\ker D \phi + H = T M$;
  \item the subbundle $\ker D\phi \cap H$ is isotropic or,
    equivalently, it is involutive (cf. \cite[Lemma 2]{jovanovic});
  \item the orbits of $\mathcal{X}$ coincide with the fibres of
    $\phi$, {\it i.e.} the orbits of the action of the simply
    connected abelian Lie group integrating $\mathcal{X}$ are the
    fibres of $\phi$.
  \end{itemize}
\end{defn}

\begin{rk}\label{rk:connectedness}
  While connectedness of the fibres of $\phi$ is not required in
  \cite{jovanovic}, it is assumed here for simplicity. Many of the
  arguments below carry over to the case of disconnected fibres with
  small modifications.
\end{rk}

The aim of this section is to prove the following result, which
relates the notion of Definition \ref{defn:cpics} to contact isotropic
realisations.

\begin{prop}\label{prop:gcpics}
  Given a complete pre-isotropic contact structure
  $(M,H,\phi,\mathcal{X})$, there exists a unique line bundle
  $L_P \to P$ (up to isomorphism), a bundle isomorphism $F: \phi^*L_P \to L$, and a unique
  Jacobi structure $(L_P,\{\cdot,\cdot\})$ so that $\phi : \CM \to
  \JM$ is a contact isotropic realisation with bundle component $F$.
  % \begin{itemize}
  % \item $\phi : \CM \to \JM$ is a Jacobi map with bundle component
  %   $F$;
  % \item $\ker D\phi \subset \rho_P(\phi^* J^1 P)$, where $\rho_P : J^1
  %   L_P \to P$ is the anchor of the Lie algebroid structure on $J^1 P
  %   \to P$ induced by the Jacobi structure $(L_P,\{\cdot,\cdot\})$
  %   (cf. property \ref{item:A} of Proposition \ref{prop:alg}).
  % \end{itemize}
\end{prop}

The strategy of the proof is, first, to obtain the result locally
(with the trivial line bundle), and then globalise. In order to prove
the local statement, recall the following theorem, stated below
without proof (cf. \cite[Theorem 3]{jovanovic}).

\begin{thm}\label{thm:complete}
  Given a complete pre-isotropic contact structure
  $(M,H,\phi,\mathcal{X})$, for any $m \in M$ there exists an open
  $\phi$-saturated set $U$ with $m \in U$, a contact form $\theta_U$
  such that $\mathcal{X}$ preserves $\theta_U$, and the foliation $\ker
  D\phi|_{U}$ is {\bf $\theta_U$-complete}, {\it i.e.} first integrals
  of $\ker D\phi|_{U}$ are closed under the Jacobi bracket induced by $\theta_U$.
\end{thm}

The key idea behind the proof of Theorem \ref{thm:complete} is to
construct $\theta_U$ by fixing an element of $\ker D\phi$ to be its
Reeb vector field. This can be achieved as follows. Fix $m \in M$ and
consider a small enough neighbourhood of $m$ such that
\begin{itemize}
\item there exists a local
section $X$ of $\ker D\phi$ which is transversal to $H$;
\item the contact distribution is locally defined as $\ker \theta$,
  for $\theta$ a locally defined contact form.
\end{itemize}

The 1-form $\theta_U := \theta/\theta(X)$ 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. \cite[proof of Theorem
3]{jovanovic}). Theorem \ref{thm:complete} implies the local form of
Proposition \ref{prop:gcpics}, as stated in the following corollary,
which, in fact, proves a slightly stronger statement:

\begin{cor}\label{cor:cpics_cir}
  There exists a
  unique Poisson structure $\Lambda_W$ on $W = \phi(U) \subset P$
  which makes $\phi : (U,\theta_U) \to (W,\Lambda_W, W \times \RR)$
  into a contact isotropic realisation with the identity as bundle component.
\end{cor}

\begin{proof}
  Since $\phi$ is a submersion, it is an open map, which implies that
  $W = \phi(U) \subset P$ is an open set. Let $f,g \in
  C^{\infty}(W)$ be smooth functions. 
  % \begin{itemize}
  % \item $\phi : (U,\theta_U) \to (W,\Lambda_W)$ is a Jacobi map with the
  %   identity map as bundle component;
  % \item $\ker D\phi \subset \RR \langle X \rangle \oplus
  %   \Lambda_U^{\sharp}(\mathrm{Ann} (\ker D\phi))$, where $(\Lambda_U,X) \in
  %   \X^2(U) \times \X(U)$ are the bivector and vector fields which
  %   determine the Jacobi structure on the trivial line bundle $U
  %   \times \RR \to U$ induced by the (co-oriented!) contact manifold
  %   $(U,\theta_U)$ (cf. Observation \ref{obs:kir} and Example \ref{exm:ctc_jacobi}).
  % \end{itemize}
  Then, 
  \begin{equation}
    \label{eq:25}
    \{\phi^*f, \phi^*g\} :=\Lambda_U^{\sharp}(\phi^* \de f, \phi^*
    \de g) + (\phi^*f) X(\phi^*g) - (\phi^*g) X(\phi^*f) = \Lambda_U^{\sharp}(\phi^* \de f, \phi^*
    \de g), 
  \end{equation}
  \noindent
  where $\Lambda_U$ is the bivector defined by $\theta_U$, and the
  equality follows from the fact that $X =R_U\in \Gamma(\ker
  D\phi)$. Since first integrals of $\ker D\phi|_U$ are closed under
  the above Jacobi bracket (cf. Theorem \ref{thm:complete} above),
  equation \eqref{eq:25} implies that there 
  exists a unique smooth bivector $\Lambda_W$ on $W$ which is defined by the following equation
  \begin{equation}
    \label{eq:5}
    \phi^*(\Lambda_W(\de f, \de g)) = \Lambda_U^{\sharp}(\phi^* \de f, \phi^*
    \de g).
  \end{equation}
  \noindent
  Note that equation \eqref{eq:5} implies that $D\phi(\Lambda_U) =
  \Lambda_W$.  The bivector $\Lambda_W$ is Poisson since 
  $$ \llbracket \Lambda_W, \Lambda_W \rrbracket = \llbracket D\phi (\Lambda_U),
  D\phi (\Lambda_U) \rrbracket = D\phi (\llbracket \Lambda_U, \Lambda_U
  \rrbracket) = D \phi(2X \wedge \Lambda_U) = 0, $$
  \noindent
  since $D\phi (X) =0$ by Theorem \ref{thm:complete}. As the Jacobi structure defined by
  $\Lambda_W$ is completely determined by the pair $(\Lambda_W,0) \in
  \X^2(W) \times \X(W)$, and $D\phi(\Lambda_U) = \Lambda_W$ and
  $D\phi(X) = 0$, this implies that $
  \phi : (U,\theta_U) \to (W,\Lambda_W)$ is Jacobi with the identity
  as bundle component (cf. Example \ref{obs:jac_trivial}); this proves property \ref{item:IR1}. Property
  \ref{item:IR2} follows from the fact that $X \in \Gamma(\ker
  D\phi)$. The distribution $\ker D\phi$ contains the Reeb vector
  field of $\theta_U$ and is {\em pre-isotropic} in the sense of
  \cite[Definition 1]{jovanovic}; by \cite[part (i) of Theorem
  2]{jovanovic}, it follows that $\ker D\phi \subset (\ker
  D\phi)^{\perp}$, so that property \ref{item:IR3} holds. This completes the proof of the corollary.
\end{proof}

% Corollary \ref{cor:cpics_cir} In particular, this implies
% that the Poisson structure $(W,\Lambda_W)$ is regular (cf. Lemma
% \ref{lemma:regular}); this is entirely analogous to the case of
% non-commutative integrable Hamiltonian systems on symplectic manifolds
% (cf. \cite{fasso}). 

% In fact, a similar result to Corollary \ref{cor:cpics_cir} holds {\em
%   globally} and does not depend on the choices made to obtain local
% contact forms as in Theorem \ref{thm:complete}; this is the content of the next proposition.


\begin{proof}[Proof of Proposition \ref{prop:gcpics}]
  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
  \ref{thm:complete}, so that, for each $i$, there exists a
  $\mathcal{X}$-invariant contact form $\theta_{U_i}$. For each $i$, Corollary \ref{cor:cpics_cir}
  gives a Poisson (and hence Jacobi) structure on $W_i :=
  \phi(U_i)$. Suppose that $i,j$ are such that $U_{ij} \neq
  \emptyset$; since both $\theta_{U_i}$ and $\theta_{U_j}$ are contact
  forms on $U_{ij}$, there exists a nowhere vanishing function $G_{ji}
  : U_{ij} \to \RR$ with $\theta_j = G_{ji} \theta_i$. As both
  $\theta_{U_i}$ and $\theta_{U_j}$ are $\mathcal{X}$-invariant, it
  follows that for all $X \in \mathcal{X}$, $\mathcal{L}_X G_{ji} =
  0$. Given that the orbits of $\mathcal{X}$ coincide with the fibres
  of $\phi$, this implies that $G_{ji}$ is {\em basic}, {\it i.e.}
  there exists a unique nowhere vanishing function $g_{ji} : W_{ij}
  \to \RR$, where $W_i = \phi(W_i)$ and $W_{ij} = W_i \cap W_j =
  \phi(U_{ij})$, with $G_{ji} = \phi^* g_{ji}$. If $\Lambda_i$ denotes
  the Poisson bivector induced on $W_i$ as in Corollary
  \ref{cor:cpics_cir} (and similarly for $j$), it follows from the
  construction that $\Lambda_j = g_{ji}^{-1} \Lambda_i = g_{ij}
  \Lambda_i$ on $W_{ij}$, where the last equality follows from the fact that
  $G_{ij} = G_{ji}^{-1}$. Thus the Poisson bivectors $\Lambda_i$ and
  $\Lambda_j$ 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 $\Lambda_i$ and $\Lambda_j$ 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
  \ref{obs:Jac_struc_locally}, there exists a line bundle $L_p \to P$
  (determined up to isomorphism), and a unique Jacobi structure
  $(L_P,\{\cdot,\cdot\}_P)$ with the property that there exists a
  bundle isomorphism $F : \phi^* L_P \to L_M := T M/H$ (the latter
  depending on the explicit isomorphism used to identify $L_P\to P$
  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 \ref{cor:cpics_cir}, thus completing the proof.
\end{proof}

% Proposition \ref{prop:gcpics} motivates the main definition in this
% paper, that of a contact isotropic realisation of a Jacobi manifold 
%  (cf. Definition \ref{defn:icr} below), so that the map $\phi :
%  \CM \to \JM $  (and its restriction to any $U_i$ as in the above
%  proof) is one such.

% Therefore, non-commutative integrable Hamiltonian systems locally
% define contact isotropic realisations of the Reeb type
% (cf. Observation \ref{obs:reeb}); conversely, a contact isotropic
% realisation of the Reeb type locally gives rise to a (non-singular)
% non-commutative integrable Hamiltonian system on a contact
% manifold. For, if $\phi : \CM \to \JM$ is such a realisation, then
% $\JM$ is in fact a regular Poisson manifold and $\ker \rho_P = \nu^* \oplus
% \RR \langle (0,1) \rangle \subset \cotan P \oplus \RR$. Pick a small
% enough open subset $W \subset P$ over which $\nu^*$ can be
% trivialised, say by fixing coordinates $a^1,\ldots a^{k-1}$
% transversal to the symplectic foliation $\mathcal{F}$. Then the
% abelian Lie algebra $\mathcal{X} :=\RR \langle \rho_M(\phi^* \de a^1), \ldots,
% \rho_M(\phi^* \de a^{k-1}), \rho_M(1) \rangle$ makes the quadruple
% $(\phi^{-1}(W) =U,\ker \theta|_U, \phi|_U, \mathcal{X})$ into a
% complete pre-isotropic contact structure and thus, into a
% non-commutative integrable Hamiltonian system. 

\begin{rk}\label{rk:marginally_stronger}
  A closer look at the proof of Proposition \ref{prop:gcpics} yields
  further restrictions on (the isomorphism class of) the line bundle
  $L_P \to P$. Using the notation of the above proof, let $\Lambda_i$,
  $\Lambda_j$ be Poisson bivectors on $W_i, W_j$ respectively; then
  there exists a nowhere vanishing function $g_{ij} : W_{ij} \to \RR$
  with $\Lambda_j = g_{ij} \Lambda_i$ on $W_{ij}$. Since the
  self-commutator of both
  $\Lambda_i, \Lambda_j$ under the Schouten bracket vanishes, the
  above equality implies that 
  \begin{equation}
    \label{eq:7}
    -2g_{ij} \Lambda_j^{\sharp}(\de g_{ij}) \Lambda_j
    = 0
  \end{equation}
  \noindent
  on $W_{ij}$, where the sign convention used is that of
  \cite{marle_schouten}. Since $\phi:(U_j,\theta_{U_j}) \to
  (W_j,\Lambda_j)$ is a complete isotropic realisation, it follows
  that $\Lambda_j$ is a regular Poisson bivector on $W_j$ (and, thus,
  on $W_{ij}$, which is an open subset) -- cf. Lemma \ref{lemma:regular}. Therefore, it can be checked
  that equation \eqref{eq:7} gives that $\Lambda^{\sharp}_j(\de
  g_{ij}) = 0$, {\it i.e.} the function $g_{ij}$ is a local Casimir for
  $\Lambda_j$. However, since $\Lambda_i = g^{-1}_{ij} \Lambda_j$, it
  follows that $g_{ij}$ is also a local Casimir for $\Lambda_i$. In
  other words, the transition functions of the line bundle $L_P \to P$
  can be chosen to be {\it basic} functions with respect to the
  foliation $\mathcal{F}$ which is locally defined as the symplectic
  foliation of the Poisson bivectors $\Lambda_i$.
\end{rk}

Complete pre-isotropic contact structures lie at the heart of non-commutative
Hamiltonian integrability in the contact setting (cf. \cite[Definition
3]{jovanovic}).

\begin{defn}\label{defn:is}
  Given an infinitesimal automorphism $X$ of $(M,H)$, the associated
  differential equation 
  $$ \dot{x} = X $$
  \noindent
  is said to be {\bf non-commutatively Hamiltonian integrable} if
  there exists an open dense subset $M_{\mathrm{reg}} \subset M$, a
  proper surjective submersion $\phi : M_{\mathrm{reg}} \to P$ onto a
  smooth manifold $P$, and an Abelian Lie algebra $\mathcal{X}$ of
  infinitesimal automorphisms of $(M,H)$ such that
  $(M,H,\phi,\mathcal{X})$ is a complete pre-isotropic contact
  structure. 
\end{defn}

In light of Proposition \ref{prop:gcpics}, (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
\ref{sec:cont-isotr-real} and \ref{sec:constr-cont-isotr} can be
applied to find invariants of the former. For instance, Theorem
\ref{thm:aa} provides local action-angle coordinates for
non-commutatively integrable Hamiltonian systems on contact manifolds
(cf. \cite{Banyaga,jovanovic} for alternative proofs).

% A list of examples of such integrable Hamiltonian systems can be found
% in \cite[Section 5.3]{jovanovic}. The main goal of \cite{jovanovic} is to
% prove the existence of local action-angle coordinates near regular
% orbits of non-commutatively Hamiltonian integrable systems on
% contact manifolds (cf. \cite[Theorems 4 and 5]{jovanovic}). An
% important step towards achieving this result is the following local property
% of complete pre-isotropic contact structures, stated below without
% proof (cf. \cite[Theorem 3]{jovanovic}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% The case of Poisson manifolds %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The case of Poisson manifolds}\label{sec:case-poiss-manif}
Since a Jacobi manifold $\JM$ admits a contact isotropic realisation
only if it is regular and all of its leaves are even dimensional
(cf. Lemma \ref{lemma:regular}), it is worth rephrasing some of the
results of Sections \ref{sec:cont-isotr-real} and
\ref{sec:constr-cont-isotr} for regular Poisson manifolds, as many of
the above objects simplify in this case. Henceforth,
let $\PM$ be a regular Poisson manifold with symplectic foliation
denoted by $\mathcal{F}$; the induced Jacobi structure
is defined on the trivial line bundle $L_P := P \times \RR$
(cf. Example \ref{exm:po_as_ja}). Following Example \ref{exm:trivial},
the anchor $\rho_P : \cotan P \oplus \RR \to TP$ is given by 
$\rho_P(\eta, f) = \Lambda_P^{\sharp}(\eta),$
\noindent
for any $(\eta,f) \in \cotan P \oplus \RR$. Thus 
$$\ker \rho_P = \nu^*
\oplus \RR \subset \cotan P \oplus \RR,$$
\noindent
while the Spencer operator is given by 
$$ \mathrm{D}(\eta,f) := \de f - \eta. $$ 
\noindent
Moreover, the $T\mathcal{F}$-connection given by Proposition
\ref{prop:well-defined} is
\begin{equation*}
  \bar{\nabla} : \Gamma(T\mathcal{F}) \times C^{\infty}(P) \to
  C^{\infty}(P), \qquad (X,f) \mapsto \de f(X);
\end{equation*}
\noindent
thus the differential
$\de_{\mathcal{F}}$ defined by equation \eqref{eq:17} is the
restriction of the exterior derivative to the foliation
$\mathcal{F}$. The cohomology class $[\omega] \in
\HH^2(\mathcal{F};L_P) = \HH^2(\mathcal{F})$ defined by Lemma
\ref{lemma:coho_class} is that of the foliated symplectic form defined
by $\Lambda_P$.\\

Without loss of generality, consider a contact
isotropic realisation $\phi : (M,\theta) \to \PM$ with the identity as
bundle component, where $\theta \in \Omega^1(M)$ is a contact
form. Let $\Sigma \subset \ker \rho_P$ denote the period lattice
of $\phi$; by Corollary \ref{cor:sections_period}, any local section
of $\Sigma$ is of the form $j^1 f$, for some $f \in
C^{\infty}(P)$. However, since $\Sigma \subset \nu^* \oplus \RR$,
\begin{equation}
  \label{eq:21}
  j^1 f \in \Gamma(\Sigma) \quad \Rightarrow \quad \de f \in
  \Gamma(\nu^*);
\end{equation}
\noindent
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 $\ker \rho_P = \nu^* \oplus \RR$ can be exploited to give
another way to construct the homomorphism of equation \eqref{eq:18}
which underpins the cohomological criterion of Theorem
\ref{thm:main}. Denote the sheaf of {\bf basic} smooth functions on
$(P,\mathcal{F})$ by
$\mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})$, {\it i.e.} it
consists of smooth functions which are locally constant on the leaves
of $\mathcal{F}$. There is a short exact sequence of sheaves 
$$ 1 \to 
\mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F}) \hookrightarrow
\mathcal{C}^{\infty} (P) \xrightarrow{\de_{\mathcal{F}}}
\mathcal{Z}^1(\mathcal{F}) \to 1, $$
\noindent
where $\mathcal{C}^{\infty} (P)$ and $\mathcal{Z}^1(\mathcal{F})$ are
the sheaves of smooth functions on $P$ and of closed foliated 1-forms
respectively, and $\de_{\mathcal{F}}$ is the foliated exterior
derivative. 

\begin{lemma}\label{lemma:poisson}
  The following is a commutative diagram of short exact sequences of
  sheaves
  \begin{equation}
    \label{eq:19}
    \xymatrix{ 1 \ar[r] & \Sigma \ar[d]^-{pr} \ar@{^{(}->}[r] &
      \mathcal{C}^{\infty}(\ker \rho_P) \ar[r] \ar[d]^-{pr} &
      \mathcal{C}^{\infty}(\ker \rho_P/\Sigma)
      \ar[d]^-{\hat{\mathrm{D}}} \ar[r] & 1\\ 
    1 \ar[r] & \mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})
    \ar@{^{(}->}[r] & \mathcal{C}^{\infty} (P)
    \ar[r]_-{\de_{\mathcal{F}}} & \mathcal{Z}^1(\mathcal{F}) \ar[r] & 1,}
  \end{equation}
  \noindent
  where $pr$ denotes the homomorphism of sheaves induced by the
  projection $pr : \cotan P \oplus \RR \to P \times \RR$,
  $\hat{\mathrm{D}}$ is the homomorphism of sheaves defined by
  equation \eqref{eq:20} which is induced by the Spencer operator
  $\mathrm{D}$, and $\Sigma$ denotes the sheaf of smooth sections of
  $\Sigma \to P$.
\end{lemma}
\begin{proof}
  First, observe that equation \eqref{eq:21} implies that the
  image $pr(\Sigma)$ of the sheaf homomorphism $pr :
  \mathcal{C}^{\infty}(\ker \rho_P) \to \mathcal{C}^{\infty}$ lies in
  the sheaf
  $\mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})$. The only
  non-trivial fact that needs checking is commutativity on the right
  hand side of the diagram \eqref{eq:19}. Let $[\alpha] \in
  \Gamma(\ker \rho_P/\Sigma)$ and set $\alpha = (\eta,f) \in
  \Gamma(\ker \rho_P)$ be a lift of $[\alpha]$. Then 
  $$ \hat{\mathrm{D}}([\alpha]) = \mathrm{D}(\eta,f)|_{\mathcal{F}} =
  (\de f - \eta)|_{\mathcal{F}} = \de_{\mathcal{F}} f =
  \de_{\mathcal{F}} \circ pr(\alpha),$$
  \noindent
  where the third equality follows from the fact that $\ker \rho_P =
  \nu^* \oplus \RR$.
\end{proof}

Since both $\mathcal{C}^{\infty}(P)$ and $\mathcal{C}^{\infty}(\ker
\rho_P)$ are fine sheaves, there is a commutative diagram
\begin{equation}
  \label{eq:22}
  \xymatrix{\HH^1(P;\mathcal{C}^{\infty}(\ker\rho_P
    /\Sigma))\ar[d]_-{\mathcal{D}} \ar[r]^-{\cong} & \HH^2(P;\Sigma)
    \ar[d] \\
    \HH^1(P;\mathcal{Z}^{1}(\mathcal{F})) \ar[r]^-{\cong} & \HH^2(\mathcal{F}),}
\end{equation}
\noindent
where the vertical maps are induced by the outer vertical maps of
diagram \eqref{eq:19} (cf. equation \eqref{eq:18} also for the
definition of $\mathcal{D}$), and the horizontal isomorphisms are the
connecting morphisms induced by the short exact sequences in equation
\eqref{eq:19}. Therefore, the following corollary holds.

\begin{cor}\label{cor:po}
  The map in cohomology 
  $$\HH^2(P;\Sigma) \to
  \HH^2(P;\mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})) \cong
  \HH^2(\mathcal{F})$$
  \noindent
  induced by the sheaf homomorphism $pr : \Sigma \to
  \mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})$ equals
  $\mathcal{D}$ via the identifications of equation \eqref{eq:22}.
\end{cor}

Corollary \ref{cor:po} indicates how to calculate the map
$\mathcal{D}$, which is central to understanding whether a regular
Poisson manifold $\PM$ with given period lattice $\Sigma \subset \nu^*
\oplus \RR$ admits a contact isotropic realisation (cf. Theorem
\ref{thm:main}). For instance, suppose that $\Sigma = \ZZ \langle j^1
f_1 ,\ldots, j^1 f_k \rangle$ for some basic functions $f_i$, so that it induces the trivial
$\ZZ^k$-system of coefficients. Then (modulo torsion), an element in
$\HH^2(P;\Sigma)$ can be written as 
$$\sum\limits_{i=1}^k [\omega_i] \otimes j^1 f_i, $$
\noindent
where, for each $i$, $\omega_i \in \Omega^2(P)$ is a 2-form with
integral cohomology class. Since the map $\mathcal{D} :
\HH^2(P;\Sigma) \to \HH^2(\mathcal{F})$ is induced by the projection
$pr : \Sigma \to \mathcal{C}^{\infty}_{\mathrm{basic}}(P;\mathcal{F})$
and $\Sigma$ is a trivial bundle,
$$ \mathcal{D} \Big(\sum\limits_{i=1}^k [\omega_i] \otimes j^1
f_i\Big) = \sum\limits_{i=1}^k [f_i \omega_i], $$
\noindent
where, for each $i$, $f_i\omega_i$ is a closed foliated 2-form since
$f_i$ is basic. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Locally conformal symplectic %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Locally conformal symplectic manifolds}\label{sec:locally-conf-sympl}
As a final example, the question of which locally conformal symplectic
(lcs for short) manifolds admit contact isotropic realisations is
tackled. Let $(P,\sigma, \tau)$ be a lcs manifold; without loss of
generality, it suffices to consider contact isotropic realisations of
the form $\phi : (M,\theta) \to (P,\sigma, \tau)$, where $\theta \in
\Omega^1(P)$ is a contact form. Denote by
$(\Lambda_M,R_M)$ the bivector and vector fields which uniquely
define the Jacobi structure on $(M,\theta)$ (cf. Observation
\ref{obs:kir}). % s and, without loss of
% generality, suppose that is a
% contact isotropic realisation with the identity as bundle component,
% where 

% We now investigate lcs manifolds $(P,\sigma,\eta)$ (as in example \ref{exm:lcs}) that admit contact isotropic realisations. In what it follows we will assume that $P$ is connected. We will explain and proof the following result:

\begin{prop}\label{cor:lcs} A lcs manifold $(P,\sigma,\tau)$ admits a
  contact isotropic realisation if and only if the following conditions hold
\begin{enumerate}
\item \label{item:12} $(P,\sigma,\eta)$ is a globally conformal symplectic manifold, i.e. $\tau=\de f$, for some $f\in C^\infty(P)$;
\item \label{item:13} the cohomology class of the symplectic form $\bar\sigma=e^f\sigma$ is integral, i.e. $[\bar\sigma]\in H^2(P;\ZZ)$.
\end{enumerate}
Moreover, a contact isotropic realisation $\phi:(M,\theta)\to
(P,\sigma,\de f)$ is the same thing as a principal $S^1$-bundle whose
infinitesimal action is generated
by $$e^{-\phi^*f}(R_M-\Lambda^\sharp_M(\phi^*\de f)),$$
\noindent
whose Chern class is $[\bar\sigma]\in H^2(P;\ZZ)$ and satisfying
$\de\theta=\phi^*\sigma-\phi^*\de f\wedge\theta$.
\end{prop}

The proof of Proposition \ref{cor:lcs} relies on the following two results.

\begin{lemma}\label{lemma:gcs}
  A necessary condition for a lcs manifold $(P,\sigma, \tau)$ to
  admit a contact isotropic realisation is that
  $\tau = \de f$ for some smooth function $f \in
  C^{\infty}(P)$. Moreover, $\Sigma = \ZZ \langle B
  e^{-f}(\tau,-1)\rangle$ for some constant $B \in \RR$.
\end{lemma}
\begin{proof}
  Let $\phi : (M,\tau) \to (P,\sigma, \tau)$ 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 \rho_P = \RR \langle (\tau,-1) \rangle $; since
  the fibres of $\phi$ are compact and by smoothness of the period lattice, $\Sigma = \ZZ \langle g(\tau,-1)
  \rangle$ for some positive smooth function $f \in
  C^{\infty}(P)$. Corollary \ref{cor:sections_period} implies
  that there exists a smooth function $\lambda$ such that $j^1 \lambda
  = (g\tau, -g)$; in other words, $\tau = \de (- \log g)$, as required.
\end{proof} 

The following result is stated without proof, as it follows from a
simple calculation. 

\begin{lemma}\label{lemma:lcs}
Let $(P,\sigma,\de f)$ be a globally conformal symplectic
manifold. Then $\phi:(M,\theta)\to(P,\sigma,\de f)$ is a contact isotropic
realisation if and only if $\phi:(M,\bar\theta)\to (P,\bar\sigma)$ is
a contact isotropic realisation, where $\bar\theta\in\Omega^1(M)$ is
the contact form $\bar\theta:=e^{f}\theta$ and
$\bar\sigma\in\Omega^2(P)$ is the symplectic form
$\bar\sigma=e^f\sigma.$ 
\end{lemma}

% \begin{rk}
% The above lemma is in fact more general in the following sense: 

% Let $\phi:\CM\to (P,\Lambda,R)$ be a surjective submersion and $a\in C^\infty(P)$ a nowhere vanishing function. Then $\phi:\CM\to (P,\Lambda,R)$ is a contact isotropic realisation if and only if $\phi:(M,\phi^*(a^{-1})\theta)\to (P,\Lambda^{a},R^{a})$ is a contact isotropic realisation, where $\Lambda^{a},R^{a}$ are the bivector and vector field of the Jacobi structure $\{\cdot,\cdot\}^{a}$ defined on observation \ref{obs:conformal_structures}
% \end{rk}

% \begin{proof}
%   Suppose that $\phi:(M,\bar\theta)\to (P,\bar\sigma)$ is
%   a contact isotropic realisation, where the notation is as in the
%   statement of the lemma. Then  Note that the underlying Jacobi bracket of $(M,\bar\theta)$ and
%   $(P,\bar\sigma)$ are $\phi^*(e^{-f})$ and $e^{-f}$ conformal (as in
%   observation \ref{obs:conformal_structures}) to the underlying Jacobi
%   bracket of $(M,\theta)$ and $(P,\sigma,\de f)$, respectively. With
%   this, by a straightforward computation one can check that
%   $\phi:\CM\to(P,\sigma,\de f)$ satisfies conditions \eqref{item:IR1}
%   and \eqref{item:IR2} if and only if $\phi:(M,\bar\theta)\to
%   (P,\bar\omega)$ does. For condition \ref{item:IR3}, using equation
%   \ref{eq:conformal_anchor} of Observation
%   \ref{obs:conformal_structures} for $a:=\phi^*(e^{-f})$, it follows
%   that $\rho_M(\phi^*J^1 L_P)=\rho_M^a(\phi^*J^1 L_P)$. 
% \end{proof}

\begin{proof}[Proof of Proposition \ref{cor:lcs}]
Lemma \ref{lemma:gcs} proves that property (\ref{item:12}) in the
statement of the proposition is a necessary
condition for $(P,\sigma,\tau)$ to admit a contact isotropic
realisation. Henceforth, without loss of generality, assume that $\tau = \de f$ and that the aim
is to construct a complete isotropic realisation of $(P,\sigma,\de f)$
with period net $\Sigma = \ZZ \langle j^1 e^{-f} \rangle$ (all other
cases follow by multiplying by a suitable constant).
In this case, Lemma \ref{lemma:lcs} gives that
$\phi:(M,\theta)\to(P,\sigma,\de f)$ is a contact isotropic realisation if and
only if $\phi:(M,\bar\theta)\to (P,\bar\sigma)$ is; by Example
\ref{exm:rcm}, such a
realisation exists if and only if $[\bar\sigma]$ is integral, and
$[\bar\sigma]$ is the Chern class of the $S^1$-bundle $\phi:M\to
P$. Moreover, $\de\bar\theta=\phi^*\bar\sigma$, {\it i.e.} 
$$\phi^*(e^f)(\de\theta+\phi^*\de f\wedge\theta)=\de\bar\theta=\phi^*\bar\sigma=\phi^*(e^f)\phi^*\sigma$$
which is equivalent to $\de\theta=\phi^*\sigma-\phi^*\de fa\wedge\theta$, and the infinitesimal generator of the action is given by the Reeb vector field of $\bar\theta$, which, written in terms of the bivector and Reeb vector field of $\theta$ is given by
$e^{-f}(R_M-\Lambda^\sharp_M(\phi^*\de f))$ as required.   
\end{proof}


% Recall from Lemma \ref{lemma:gcs} that a necessary condition for $(P,\sigma,\eta)$ to admit a contact isotropic realisation with compact fibres is that  $$\eta=\de f$$ for some $f\in C^\infty(P)$.  As $\ker\rho_P=\langle(\eta,1)\rangle\subset T^*P\oplus \RR$, then in this case the only period lattices are of the form $\Sigma_{f,k}:=k\ZZ(e^f\eta,e^f)$ for $k\in \ZZ$. Consider the case $k=1$ (the other cases follow similarly).

% Recall from example \ref{exm:rcm}, that a contact isotropic realisation $\phi:(M,\bar\theta)\to (P,\bar\sigma)$ of a symplectic manifold $(P,\bar\sigma)$ corresponds to an $S^1$-bundle $\phi:M\to P$ whose infinitesimal action is generated by the Reeb vector field of $\bar\theta$. Moreover such $S^1$-bundle exists if and only if $[\bar\sigma]$ is integral, and then $[\bar\sigma]$ is the Chern class of the bundle and $\de\bar\theta=\phi^*\bar\sigma$. With this and Lemma \ref{lemma:lcs} we can now prove Corollary \ref{cor:lcs}.




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c_i_r_line_bundle_1.bbl
\begin{thebibliography}{10}

\bibitem{Arnold}
V.~I. Arnol'd.
\newblock {\em Mathematical methods of classical mechanics}, volume~60 of {\em
  Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York, second edition, 1989.
\newblock Translated from the Russian by K. Vogtmann and A. Weinstein.

\bibitem{Banyaga}
A.~Banyaga and P.~Molino.
\newblock G\'eom\'etrie des formes de contact compl\`etement int\'egrables de
  type toriques.
\newblock In {\em S\'eminaire {G}aston {D}arboux de {G}\'eom\'etrie et
  {T}opologie {D}iff\'erentielle, 1991--1992 ({M}ontpellier)}, pages 1--25.
  Univ. Montpellier II, Montpellier, 1993.

\bibitem{contact}
D.~E. Blair.
\newblock {\em Riemannian geometry of contact and symplectic manifolds}, volume
  203 of {\em Progress in Mathematics}.

\bibitem{boothby_wang}
W.~Boothby and H.~Wang.
\newblock On contact manifolds.
\newblock {\em Ann. of Math. (2)}, 68(3):721--734, 1958.

\bibitem{cdw}
A.~Coste, P.~Dazord, and A.~Weinstein.
\newblock Groupo\"\i des symplectiques.
\newblock In {\em Publications du {D}\'epartement de {M}ath\'ematiques.
  {N}ouvelle {S}\'erie. {A}, {V}ol.\ 2}, volume~87 of {\em Publ. D\'ep. Math.
  Nouvelle S\'er. A}, pages i--ii, 1--62. Univ. Claude-Bernard, Lyon, 1987.

\bibitem{CrainicFernandes:jdd}
M.~Crainic and R.~L. Fernandes.
\newblock Integrability of {P}oisson brackets.
\newblock {\em J. Differential Geom.}, 66(1):71--137, 2004.

\bibitem{CrainicFernandes:lecture}
M.~Crainic and R.~L. Fernandes.
\newblock Lectures on integrability of {L}ie brackets.
\newblock {\em Geometry \& Topology Monographs}, 17:1--94, 2010.

\bibitem{PMCT}
M.~Crainic, R.~L. Fernandes, and D.~Mart\'inez~Torres.
\newblock Poisson manifolds of compact type {I}.
\newblock {\em (in preparation)}, 2014.

\bibitem{marius_ionut}
M.~Crainic and I.~M{\u{a}}rcu{\c{t}}.
\newblock On the existence of symplectic realizations.
\newblock {\em J. Symplectic Geom.}, 9(4):435--444, 2011.

\bibitem{Maria-Crainic}
M.~Crainic and M.~A. Salazar.
\newblock Jacobi structures and {S}pencer operators.
\newblock {\em to appear in J. Math. Pures Appl.}, 2014.

\bibitem{Maria}
M.~Crainic, M.~A. Salazar, and I.~Struchiner.
\newblock Multiplicative forms and {S}pencer operators.
\newblock {\em arXiv:1210.2277}, 2012.

\bibitem{jacobi}
M.~Crainic and C.~Zhu.
\newblock Integrability of {J}acobi and {P}oisson structures.
\newblock {\em Ann. Inst. Fourier (Grenoble)}, 57(4):1181--1216, 2007.

\bibitem{Dazord}
P.~Dazord.
\newblock Sur l'int\'egration des alg\`ebres de {L}ie locales et la
  pr\'equantification.
\newblock {\em Bull. Sci. Math.}, 121(6):423--462, 1997.

\bibitem{dd}
P.~Dazord and T.~Delzant.
\newblock Le probl\`eme g\'en\'eral des variables actions-angles.
\newblock {\em J. Differential Geom.}, 26(2):223--251, 1987.

\bibitem{Dazord-Lich-Marle91}
P.~Dazord, A.~Lichnerowicz, and C.-M. Marle.
\newblock Structure locale des vari\'et\'es de {J}acobi.
\newblock {\em J. Math. Pures Appl. (9)}, 70(1):101--152, 1991.

\bibitem{dlmp}
M.~de~Le{\'o}n, J.~Marrero, and E.~Padr{\'o}n.
\newblock On the geometric quantization of {J}acobi manifolds.
\newblock {\em J. Math. Phys.}, 38(12):6185--6213, 1997.

\bibitem{dui}
J.~J. Duistermaat.
\newblock On global action-angle coordinates.
\newblock {\em Comm. Pure Appl. Math.}, 33(6):687--706, 1980.

\bibitem{geiges}
H.~Geiges.
\newblock Contact geometry.
\newblock In {\em Handbook of differential geometry. {V}ol. {II}}, pages
  315--382. Elsevier/North-Holland, Amsterdam, 2006.

\bibitem{geiges_book}
H.~Geiges.
\newblock {\em An introduction to contact topology}, volume 109 of {\em
  Cambridge Studies in Advanced Mathematics}.
\newblock Cambridge University Press, Cambridge, 2008.

\bibitem{ghl}
W.~M. Goldman, M.~W. Hirsch, and G.~Levitt.
\newblock Invariant measures for affine foliations.
\newblock {\em Proc. Amer. Math. Soc.}, 86(3):511--518, 1982.

\bibitem{jovanovic}
B.~Jovanovi{\'c}.
\newblock Noncommutative integrability and action-angle variables in contact
  geometry.
\newblock {\em J. Symplectic Geom.}, 10(4):535--561, 2012.

\bibitem{karasev}
M.~V. Karas{\"e}v.
\newblock Analogues of objects of the theory of {L}ie groups for nonlinear
  {P}oisson brackets.
\newblock {\em Izv. Akad. Nauk SSSR Ser. Mat.}, 50(3):508--538, 638, 1986.

\bibitem{Kerbrat}
Y.~Kerbrat and Z.~Souici-Benhammadi.
\newblock Vari\'et\'es de {J}acobi et groupo\"\i des de contact.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 317(1):81--86, 1993.

\bibitem{Kirillov}
A.~A. Kirillov.
\newblock Local {L}ie algebras.
\newblock {\em Uspehi Mat. Nauk}, 31(4(190)):57--76, 1976.

\bibitem{lib_leg}
P.~Libermann.
\newblock Legendre foliations on contact manifolds.
\newblock {\em Differential Geom. Appl.}, 1(1):57--76, 1991.

\bibitem{Lich78}
A.~Lichnerowicz.
\newblock Les vari\'et\'es de {J}acobi et leurs alg\`ebres de {L}ie
  associ\'ees.
\newblock {\em J. Math. Pures Appl. (9)}, 57(4):453--488, 1978.

\bibitem{Lich83}
A.~Lichnerowicz.
\newblock Sur les alg\`ebres de {L}ie locales de {K}irillov-{S}higa.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 296(22):915--920, 1983.

\bibitem{Marle}
C.-M. Marle.
\newblock On {J}acobi manifolds and {J}acobi bundles.
\newblock In {\em Symplectic geometry, groupoids, and integrable systems
  ({B}erkeley, {CA}, 1989)}, volume~20 of {\em Math. Sci. Res. Inst. Publ.},
  pages 227--246. Springer, New York, 1991.

\bibitem{marle_schouten}
C.-M. Marle.
\newblock The {S}chouten-{N}ijenhuis bracket and interior products.
\newblock {\em J. Geom. Phys.}, 23(3-4):350--359, 1997.

\bibitem{sepe}
D.~Sepe.
\newblock Topological classification of {L}agrangian fibrations.
\newblock {\em J. Geom. Phys.}, 60(2):341--351, 2010.

\bibitem{Vaisman}
I.~Vaisman.
\newblock {\em Lectures on the geometry of {P}oisson manifolds}, volume 118 of
  {\em Progress in Mathematics}.
\newblock Birkh\"auser Verlag, Basel, 1994.

\bibitem{wolak}
R.~A. Wolak.
\newblock Transversely affine foliations compared with affine manifolds.
\newblock {\em Quart. J. Math. Oxford Ser. (2)}, 41(163):369--384, 1990.

\bibitem{Zambon}
M.~Zambon and C.~Zhu.
\newblock Contact reduction and groupoid actions.
\newblock {\em Trans. Amer. Math. Soc.}, 358(3):1365--1401 (electronic), 2006.

\end{thebibliography}
