# Cosymplectic groupoids

Rui Loja Fernandes  Department of Mathematics, University of Illinois at
Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA
<EMAIL_ADDRESS>and  David Iglesias Ponte  ULL-CSIC Geometría Diferencial
y Mecánica Geométrica, Departamento de Matemáticas, Estadística e
Investigación Operativa and Instituto de Matemáticas y Aplicaciones (IMAULL),
University of La Laguna, San Cristóbal de La Laguna, Spain
<EMAIL_ADDRESS>
###### Abstract.

A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic
structure. We provide several structural results for cosymplectic groupoids
and we discuss the relationship between cosymplectic groupoids, Poisson
groupoids of corank 1, and oversymplectic groupoids of corank 1.

RLF was partially supported by NSF grant DMS-2003223.

##  1\. Introduction

A _cosymplectic groupoid_ is a Lie groupoid $\mathcal{G}\rightrightarrows M$
equipped with a multiplicative cosymplectic structure $(\omega,\alpha)$. This
means that $\omega\in\Omega^{2}(\mathcal{G})$,
$\alpha\in\Omega^{1}(\mathcal{G})$ are closed multiplicative forms and
$\alpha\wedge\omega^{m}$ is nowhere vanishing, so it defines a volume form in
$\mathcal{G}$. As we will see later, one must have $\dim\mathcal{G}=2m+1$
where $m=\dim M$ (we assume $M$ connected). The notion of cosymplectic
groupoid was first studied in [5].

Cosymplectic groupoids lie at the intersection of two well-known, interesting,
classes of Lie groupoids:

  * •

_Poisson groupoids_ , i.e., Lie groupoids with a multiplicative Poisson
structure (see, e.g., [14, 17]). For a cosymplectic groupoid
$(\mathcal{G},\omega,\alpha)$ the associated multiplicative Poisson structure
$\pi_{\mathcal{G}}\in\mathfrak{X}^{2}(\mathcal{G})$ has symplectic foliation
$\ker\alpha$ and leafwise symplectic form the restriction of $\omega$.

  * •

_Oversymplectic groupoids_ , i.e., Lie groupoids with a closed multiplicative
2-form $\omega$ satisfying $\operatorname{rank}\omega_{1_{x}}=2\dim M$, for
all $x\in M$ (see, e.g., [2]). We will see that the 2-form of a cosymplectic
groupoid satisfies this condition.

For both of these classes of Lie groupoids the base $M$ inherits a Poisson
structure. For a cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ the
Poisson structures obtained from $\pi_{\mathcal{G}}$ and from $\omega$
coincide, and will be denoted by $\pi_{M}\in\mathfrak{X}^{2}(M)$. Our aim in
this paper is to give structural results for cosymplectic groupoids and to
establish precise relationships with these two classes of Lie groupoids.

To describe our main results we observe that, as a consequence of the
multiplicativity condition, the standard data associated with a cosymplectic
structure satisfies:

  1. (i)

$\ker\alpha\subset T\mathcal{G}$ defines an integrable _multiplicative_
distribution in $\mathcal{G}$;

  2. (ii)

$\omega$ restricts to a symplectic form on the leaves of $\ker\alpha$,
yielding a _multiplicative_ Poisson structure
$\pi_{\mathcal{G}}\in\mathfrak{X}^{2}(\mathcal{G})$;

  3. (iii)

The Reeb vector field $E\in\mathfrak{X}(G)$, characterized by $i_{E}\omega=0$
and $i_{E}\alpha=1$, is _bi-invariant_ (i.e., it is both left and right
invariant). In particular, it is a complete vector field.

These basic facts, to be proved below, give a rich structure to a cosymplectic
groupoid. For example, the collection of all orbits of the Reeb vector field
intersecting the identity section is a bundle of Lie groups
$\mathcal{K}\subset\mathcal{G}$ and one has:

######  Theorem 1.1.

For any cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ there is a short
exact sequence of topological groupoids over the same base:

(1)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $\Sigma$ is the orbit space of the Reeb vector field. When this space is
smooth, this is a short exact sequence of Lie groupoids and $(\Sigma,\Omega)$
is a symplectic groupoid for a unique symplectic structure making the
projection $\mathcal{G}\to\Sigma$ a Poisson map.

For an arbitrary Poisson groupoid or oversymplectic groupoid, the base Poisson
structure $(M,\pi_{M})$ may fail to be integrable. However, for a cosymplectic
groupoid it is always integrable. Indeed, the identity section of a
cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ is contained in a single
symplectic leaf of the Poisson structure $\pi_{\mathcal{G}}$ and we have:

######  Proposition 1.2.

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid. The symplectic
leaf of $\pi_{\mathcal{G}}$ containing $M$ is a Lie subgroupoid
$\Sigma^{0}\subset\mathcal{G}$, and it yields a symplectic groupoid
$(\Sigma^{0},\omega|_{\Sigma^{0}})\rightrightarrows M$ integrating
$(M,\pi_{M})$.

For a proper cosymplectic groupoid where $\Sigma^{0}$ is an embedded
submanifold, one obtains a picture somewhat dual to Theorem 1.1. Namely, the
flow of the Reeb vector field for some fixed time $t_{0}$ gives a
symplectomorphism of $\Sigma^{0}$ and one finds that the cosymplectic groupoid
is a symplectic mapping torus.

######  Theorem 1.3.

Let $(\mathcal{G},\omega,\alpha)$ be a proper, source connected, cosymplectic
groupoid and assume that the symplectic leaf $\Sigma^{0}\subset\mathcal{G}$
containing the identity is an embedded submanifold. Then there is a
symplectomorphism $\varphi:\Sigma^{0}\to\Sigma^{0}$ such that $\mathcal{G}$ is
isomorphic to the symplectic mapping torus
$\Sigma^{0}\times_{\varphi}\mathbb{S}^{1}$. Moreover, the resulting submersion

$q:\mathcal{G}\to\mathbb{S}^{1},$

is a fibration of Lie groupoids.

The short exact sequence (1) may fail to be smooth and, if smooth, it may fail
to split. However, at the infinitesimal level it always splits. In fact, the
Lie algebroid $A\to M$ of a cosymplectic groupoid
$(\mathcal{G},\omega,\alpha)$ carries a closed IM 2-form $\mu:A\to T^{*}M$ and
a closed IM 1-form $\nu:A\to M\times\mathbb{R}$, corresponding to $\omega$ and
$\alpha$, respectively. We then obtain the following:

######  Theorem 1.4.

If $(\mathcal{G},\omega,\alpha)$ is a cosymplectic groupoid, its Lie algebroid
$(A,\mu,\nu)$ is canonically isomorphic to the trivial central extension of
the cotangent algebroid associated with the base Poisson manifold
$(M,\pi_{M})$:

$(A,\mu,\nu)\simeq(T^{*}M\oplus\mathbb{R},\operatorname{pr}_{T^{*}M},\operatorname{pr}_{M\times\mathbb{R}}).$

Notice that given a source connected cosymplectic groupoid
$(\mathcal{G},\omega,\alpha)$, its source 1-connected cover is a cosymplectic
groupoid. Applying the previous result, the latter are easy to describe:

######  Corollary 1.5.

If $(\mathcal{G},\omega,\alpha)$ is a source 1-connected cosymplectic groupoid
then there is a canonical isomorphism

$(\mathcal{G},\omega,\alpha)\cong(\Sigma(M)\times\mathbb{R},\operatorname{pr}_{\Sigma}^{*}\Omega,\operatorname{pr}_{\mathbb{R}}^{*}\mathrm{d}t)$

where $(\Sigma(M),\Omega)$ is the source 1-connected symplectic integration of
$(M,\pi_{M})$.

In the last part of this note we discuss how far Poisson groupoids and
oversymplectic groupoids are from being cosymplectic groupoids.

We say that a Poisson groupoid $(\mathcal{G},\pi_{\mathcal{G}})$ is of corank
1 if its Poisson structure has constant rank equal to $\dim\mathcal{G}-1$,
i.e., if its symplectic foliation is regular of codimension 1. We have the
following simple criteria:

######  Proposition 1.6.

Let $(\mathcal{G},\pi_{\mathcal{G}})$ be a Poisson groupoid of corank 1. Then
$\pi_{\mathcal{G}}$ is associated with a cosymplectic structure if and only if
there exists a Poisson vector field $E\in\mathfrak{X}(\mathcal{G})$ transverse
the symplectic foliation of $\pi_{\mathcal{G}}$ which is bi-invariant.

One the other hand, when $(\mathcal{G}\rightrightarrows M,\pi_{\mathcal{G}})$
is _proper_ of corank 1 with $\dim\mathcal{G}=2\dim M+1$ we will see that if
its leafwise symplectic form admits a multiplicative extension (closed or not)
then, up to a cover, it is homotopic to a cosymplectic groupoid through a
homotopy that does not change the base Poisson structure:

######  Theorem 1.7.

Let $(\mathcal{G},\pi_{\mathcal{G}})\rightrightarrows(M,\pi_{M})$ be an
orientable proper Poisson groupoid of corank 1 with $\dim\mathcal{G}=2\dim
M+1$ and assume that there exists a multiplicative 2-form extending its
leafwise symplectic form. If
$(\widetilde{\mathcal{G}},\widetilde{\pi_{\mathcal{G}}})$ is its universal
covering groupoid, then there is a path of Poisson structures
$\widetilde{\pi}_{\mathcal{G}}^{\,t}\in\mathfrak{X}^{2}(\widetilde{\mathcal{G}})$,
starting at
$\widetilde{\pi}_{\mathcal{G}}^{\,0}=\widetilde{\pi_{\mathcal{G}}}$, with the
following properties:

  1. (i)

each $\widetilde{\pi}_{\mathcal{G}}^{\,t}$ is multiplicative of corank 1;

  2. (ii)

the Poisson structure on $M$ induced by $\widetilde{\pi}_{\mathcal{G}}^{\,t}$
is $\pi_{M}$;

  3. (iii)

$\widetilde{\pi}_{\mathcal{G}}^{\,1}$ is associated with a multiplicative
cosymplectic structure.

Let us turn now to oversymplectic groupoids. Given such a groupoid
$(\mathcal{G},\omega)$, if the foliation given by $\ker\omega$ is simple then
the leaf space

$\Sigma:=\mathcal{G}/\ker\omega$

is automatically a symplectic groupoid (this is the origin of the term
“oversymplectic”; see [2]). If $(\mathcal{G},\omega)$ is proper and
$\ker\omega$ is an orientable line bundle, the quotient map
$\Phi:\mathcal{G}\to\Sigma$ yields a short exact sequence of Lie groupoids

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It follows from recent results in [6] that associated to such a sequence there
is a well-defined _multiplicative Chern class_ , living in the multiplicative
de Rham cohomology of $\Sigma\rightrightarrows M$,

$c(\mathcal{G})\in H^{2}_{M}(\Sigma).$

We will show that:

######  Theorem 1.8.

Let $(\mathcal{G},\omega)$ be a corank 1, orientable, proper oversymplectic
groupoid. If $\ker\omega$ is a simple foliation, then there exists
$\alpha\in\Omega^{1}(\mathcal{G})$ such that $(\mathcal{G},\omega,\alpha)$ is
a cosymplectic groupoid if and only if the multiplicative Chern class
vanishes.

This paper is organized as follows. In Section 2 we recall some basics about
cosymplectic structures, we introduce cosymplectic groupoids, establish its
basic properties, and we prove Theorems 1.1 and 1.3. In Section 3 we construct
the infinitesimal data associated with a cosymplectic groupoid and we show
that its Lie algebroid fits into a split short exact sequence, proving Theorem
1.4 and Corollary 1.5. In Section 4 we discuss relationships between Poisson
groupoids, oversymplectic groupoids and cosymplectic groupoids, deducing in
particular Proposition 1.6 and Theorems 1.7 and 1.8. We mostly follows the
conventions and notation of the monograph [3], to which we refer for
background on Poisson structures and symplectic groupoids.

_Acknowledgments._ We would like to thank the anonymous referee for pointing
out an error in a first version of the paper.

##  2\. Cosymplectic groupoids

###  2.1. Background on cosymplectic structures

A _cosymplectic structure_ on a manifold $Q$ of dimension $2m+1$ is a pair
$(\omega,\alpha)$, where $\omega$ is a closed 2-form, $\alpha$ is a closed
1-form and $\omega^{m}\wedge\alpha$ is a volume form. These structures were
first introduced by Liberman [11]. We collect here some basic facts about
cosymplectic structures.

Associated with a cosymplectic $(\omega,\alpha)$ on $Q$ there is a non-
vanishing vector field $E\in\mathfrak{X}(Q)$, called the _Reeb vector field_ ,
characterized by

(2) $i_{E}\omega=0,\quad\alpha(E)=1.$

On the other hand, $\ker\alpha\subset TQ$ is an integrable distribution and
the restriction of $\omega$ to its leaves is symplectic. The resulting
symplectic foliation determines a regular _Poisson structure_
$\pi_{Q}\in\mathfrak{X}^{2}(Q)$ of corank 1. Notice that, by construction, the
closed 2-form $\omega$ extends the leafwise symplectic form of $\pi_{Q}$

$\omega(\pi_{Q}^{\sharp}(\beta_{1}),\pi_{Q}^{\sharp}(\beta_{2}))=\langle\beta_{1},\pi_{Q}^{\sharp}(\beta_{2})\rangle,\quad(\beta_{1},\beta_{2}\in
T^{*}Q).$

Moreover, the Reeb vector field is a Poisson vector field everywhere
transverse to the symplectic foliation.

Conversely, assume that $(Q,\pi_{Q})$ is a regular Poisson structure of corank
1. If $E$ is a vector field transverse to the symplectic foliation, then one
obtains

  1. (i)

a 2-form $\omega$ extending the leafwise symplectic form such that
$\ker\omega=\langle E\rangle$ and

  2. (ii)

a 1-form $\alpha$ such that $\alpha(E)=1$ and
$\ker\alpha=\operatorname{Im}\pi_{Q}^{\sharp}$.

It is not hard to check that $\omega$ and $\alpha$ are closed iff $E$ is a
Poisson vector field, so we have ([7, Proposition 18]):

######  Proposition 2.1.

A regular Poisson structure $\pi_{Q}\in\mathfrak{X}^{2}(Q)$ of corank 1 is
defined by a cosymplectic structure if and only if there a Poisson vector
field transverse to the symplectic foliation.

Two cosymplectic structures $(\omega,\alpha)$ and
$(\widetilde{\omega},\widetilde{\alpha})$ define the same Poisson structure
$\pi_{Q}$ if and only $\widetilde{\alpha}=f\alpha$ for a nowhere vanishing
Casimir function $f\in C^{\infty}(Q)$, and $\widetilde{\omega}-\omega$ is a
closed two form vanishing on $\ker\alpha=\ker\widetilde{\alpha}$. In this
case, the corresponding Reeb vector fields are related by
$\widetilde{E}=\frac{1}{f}E$.

The following examples give some basic constructions of cosymplectic manifolds
related with symplectic manifolds:

######  Example 2.2.

If $(S,\omega_{S})$ is a symplectic manifold then $Q=S\times\mathbb{R}$ admits
the cosymplectic structure
$(\operatorname{pr}_{S}^{*}\omega_{S},\operatorname{pr}_{\mathbb{R}}^{*}\mathrm{d}t)$,
where $t$ denotes the coordinate on the second factor. In this case, the
vector field $E$ is just $\frac{\partial}{\partial t}$, while the Poisson
structure is $\pi_{Q}=\omega^{-1}\oplus 0$.

Obviously, one can replace $\mathbb{R}$ by $\mathbb{S}^{1}$ and $\mathrm{d}t$
by $\mathrm{d}\theta$, obtaining a cosymplectic structure on
$S\times\mathbb{S}^{1}$. More generally, one can consider a principal
$\mathbb{S}^{1}$-bundle $p:Q\to S$ over a symplectic manifold $(S,\omega_{S})$
that admits a flat connection 1-form $\alpha\in\Omega^{1}(Q)$. Then
$(p^{*}\omega_{S},\alpha)$ is a cosymplectic structure whose underlying
Poisson structure has symplectic foliation the horizontal foliation of
$\alpha$. The Reeb vector field is the infinitesimal generator of the
$\mathbb{S}^{1}$-action.

######  Example 2.3.

Let $(S,\omega_{S})$ be a symplectic manifold and $\varphi\colon S\to S$ a
symplectomorphism. Recall that the corresponding _symplectic mapping torus_ is
the fiber bundle

$q\colon S_{\varphi}\to\mathbb{S}^{1},$

where $S_{\varphi}=(S\times\mathbb{R})/\mathbb{Z}$ is the orbit space of the
free and proper action

$\mathbb{Z}\curvearrowright S\times\mathbb{R},\quad
n\cdot(x,t)=(\varphi^{n}(x),t+n).$

The manifold $S_{\varphi}$ is equipped with the cosymplectic structure
$(\omega,\alpha)$, where $\omega$ is the 2-form obtained from the basic form
$\operatorname{pr}_{S}^{*}\omega_{S}\in\Omega^{2}(S\times\mathbb{R})$ and
$\alpha=q^{\ast}(\mathrm{d}\theta)$, with $\theta$ the angle coordinate on
$S^{1}$. The Reeb vector field $E\in\mathfrak{X}(S_{\varphi})$ is obtained by
projecting the vector field $\frac{\partial}{\partial
t}\in\mathfrak{X}(S\times\mathbb{R})$.

######  Remark 2.4.

Tischler’s theorem [16] shows that given a nowhere vanishing closed 1-form
$\alpha$ on a compact manifold $Q$, there exists a fibration
$q:Q\to\mathbb{S}^{1}$ with the property that one can choose $c>0$ such that
$c\alpha$ and $\widetilde{\alpha}:=q^{*}\mathrm{d}\theta$ are
$C^{\infty}$-close. Therefore, if $(Q,\omega,\alpha)$ is a compact
cosymplectic manifold, one finds:

  1. (i)

The Poisson structure defined by $(\omega,\alpha)$ is homotopic to the Poisson
structure defined by $(\omega,q^{*}\mathrm{d}\theta)$;

  2. (ii)

$q:Q\to\mathbb{S}^{1}$ can be realized as a symplectic mapping torus with
associated cosymplectic structure $(\widetilde{\omega},q^{*}\mathrm{d}\theta)$
for a modified closed 2-form (see, e.g., [10]).

In this sense, a compact cosymplectic structure $(Q,\omega,\alpha)$ is,
essentially, a symplectic mapping torus.

######  Example 2.5.

Let $(S,\Omega)$ be a symplectic manifold and $\iota:Q\hookrightarrow S$ a
submanifold. If $X$ is a symplectic vector field everywhere transverse to $Q$,
then $(\iota^{*}\Omega,\iota^{*}(i_{X}\Omega))$ defines a cosymplectic
structure on $Q$. If $f$ is a function locally defining $Q$ and such that
$X(f)=1$, then the Reeb vector field is given by $E=X_{f}|_{Q}$, where $X_{f}$
is the hamiltonian vector field of $f$. The associated Poisson structure
$\pi_{Q}\in\mathfrak{X}^{2}(Q)$ is

$\pi_{Q}(\beta_{1},\beta_{2}):=\omega^{-1}(\widetilde{\beta}_{1},\widetilde{\beta}_{2}),\quad(\beta_{1},\beta_{2}\in
T^{*}Q),$

where $\widetilde{\beta}\in T^{*}S$ denotes the unique extension of $\beta\in
T^{*}Q$ satisfying $\beta(X)=1$.

Every cosymplectic manifold $(Q,\omega,\alpha)$ can be realized as a
submanifold of a symplectic manifold $(S,\Omega)$: one lets
$S=Q\times\mathbb{R}$ with symplectic form
$\Omega=\omega+\alpha\wedge\mathrm{d}t$.

###  2.2. Cosymplectic groupoids

Let $\mathcal{G}$ be a Lie groupoid over $M$. We denote by $\mathbf{s}$ and
$\mathbf{t}$ the source and target maps, by
$\mathbf{m}:\mathcal{G}^{(2)}\to\mathcal{G}$ the multiplication (defined on
the space $\mathcal{G}^{(2)}$ of pairs of composable arrows), by
$\textbf{i}:\mathcal{G}\to\mathcal{G}$ the inverse map, and by
$\varepsilon:M\to\mathcal{G}$ the identity section. Our convention for the
groupoid multiplication is such that, given two arrows $x,y\in\mathcal{G}$,
the product $x\cdot y:=\mathbf{m}(x,y)$ is defined provided
$\mathbf{s}(x)=\mathbf{t}(y)$. Also, if $m\in M$ we write
$1_{m}:=\varepsilon(m)$ for the unit arrow over $m$, and if $x\in\mathcal{G}$
we write $x^{-1}:=\textbf{i}(x)$ for the inverse arrow. We denote the groupoid
by $\mathcal{G}\rightrightarrows M$.

Recall that a form $\omega\in\Omega^{k}(\mathcal{G})$ is said to be
_multiplicative_ if

(3) $\mathbf{m}^{*}\omega=\pi_{1}^{*}\omega+\pi_{2}^{*}\omega,$

where $\pi_{i}:\mathcal{G}^{(2)}\to\mathcal{G}$ are the projections on each
factor.

######  Definition 2.6.

A _cosymplectic groupoid_ is a triple $(\mathcal{G},\omega,\alpha)$ where
$\mathcal{G}$ is a Lie groupoid and $(\omega,\alpha)$ is a cosymplectic
structure on $\mathcal{G}$ with $\omega$ and $\alpha$ multiplicative forms.

The following proposition gives some basic properties of a cosymplectic
groupoid. It maybe useful to recall that a multiplicative distribution in a
groupoid $\mathcal{G}\rightrightarrows M$ is a distribution $D\subset
T\mathcal{G}$ with $\mathrm{d}\mathbf{s}(D)=\mathrm{d}\mathbf{t}(D)=D_{0}$ and
such that $D\rightrightarrows D_{0}$ is a subgroupoid of the tangent groupoid
$T\mathcal{G}\rightrightarrows TM$. We refer to [8] for basic facts about
multiplicative distributions.

######  Proposition 2.7.

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid over $M$. Then
$\dim\mathcal{G}=2\dim M+1$ and one has:

  1. (i)

$\ker\alpha\subset T\mathcal{G}$ is a multiplicative distribution;

  2. (ii)

the induced Poisson structure
$\pi_{\mathcal{G}}\in\mathfrak{X}^{2}(\mathcal{G})$ is multiplicative;

  3. (iii)

the Reeb vector field $E\in\mathfrak{X}(\mathcal{G})$ is a bi-invariant
Poisson vector field.

######  Remark 2.8.

One can define multiplicative $k$-vector fields for any natural number $k$
[1]. We will only be interested in the case of a bi-vector field
$\pi_{\mathcal{G}}\in\mathfrak{X}^{2}(\mathcal{G})$ for which multiplicativity
amounts to requiring the map $\pi_{\mathcal{G}}^{\sharp}:T^{*}\mathcal{G}\to
T\mathcal{G}$ to be a Lie groupoid morphism [13]. This is the analogue of the
fact that a 2-form $\omega\in\Omega^{2}(\mathcal{G})$ is multiplicative if and
only if the map $\omega^{\flat}:T\mathcal{G}\to T^{*}\mathcal{G}$ is a Lie
groupoid morphism.

###### Proof.

It follows easily from the multiplicativity condition that for a cosymplectic
groupoid $(\mathcal{G},\omega,\alpha)$ one has

$\textbf{i}^{*}\omega=-\omega,\quad\textbf{i}^{*}\alpha=-\alpha,\quad\textbf{i}_{*}E=-E,$

where $E$ denotes the Reeb vector field. In particular, the pull backs of
$\omega$ and $\alpha$ along the identity section vanish and the Reeb vector
field is transverse to it. It follows that the subspace generated by the
tangent space to the identity section and the Reeb vector field is maximally
isotropic for $\omega$. Since $\omega$ has corank 1, we conclude that for a
cosymplectic groupoid $\mathcal{G}\rightrightarrows M$ one must have
$\dim\mathcal{G}=2\dim M+1$.

The kernel of a multiplicative form is a multiplicative distribution (see
[4]). Hence (i) follows. Since
$\operatorname{Im}(\pi_{\mathcal{G}})^{\sharp}=\ker\alpha$ and the symplectic
forms on the leaves of $\pi_{\mathcal{G}}$ are obtained by restricting the
multiplicative form $\omega$, we must have $\pi_{\mathcal{G}}$ multiplicative
and (ii) also follows.

From the multiplicativity of $\omega$ we also have that

$D:=\langle E\rangle=\ker\omega,$

is a multiplicative distribution, i.e., $D$ is a subgroupoid of
$T\mathcal{G}$. Since $E$ is transverse to the identity section, we find that
$D_{0}=D\cap TM=0_{M}$, i.e., we have
$D\subset\ker\mathrm{d}\mathbf{s}\cap\ker\mathrm{d}\mathbf{t}$. This means
that $D$ is a distribution which is both left and right invariant. In order to
conclude that $E$ itself is both left and right invariant, notice that the
right invariant vector field

$\overrightarrow{e}:\mathcal{G}\to T\mathcal{G},\quad g\mapsto
d_{1_{\mathbf{s}(g)}}R_{g}(E_{1_{\mathbf{s}(g)}}),$

takes values in $D$, so $i_{\overrightarrow{e}}\omega=0$. On the other hand,
using the multiplicativity of $\alpha$, we obtain

$\displaystyle\alpha_{g}(\overrightarrow{e})$
$\displaystyle=\alpha_{g}(\mathrm{d}\mathbf{m}(E_{1_{\mathbf{s}(g)}},0_{g}))$
$\displaystyle=\alpha_{1_{x}}(E_{1_{\mathbf{s}(g)}})+\alpha_{g}(0_{g})=1.$

Hence, by uniqueness, we must have $E=\overrightarrow{e}$. Similarly, we find
that $E=\overleftarrow{e}$, so $E$ is both left and right invariant and (iii)
follows. ∎

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid with underlying
multiplicative foliation $\mathcal{F}_{\mathcal{G}}=\ker\alpha$. From (3) one
can prove that $\epsilon^{*}\alpha=0$, which means that the identity section
is tangent to this foliation, i.e.,

$TM\subset\ker\alpha.$

Since we assume that $M$ is connected, it follows that it is contained in a
single symplectic leaf $\Sigma^{0}$ of $\mathcal{F}_{\mathcal{G}}$. In
general, this leaf is only an immersed submanifold. Still, denoting its
symplectic form by $\omega_{\Sigma^{0}}=\omega|_{\Sigma^{0}}$, we have:

######  Proposition 2.9.

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid. Then
$\Sigma^{0}$ is a Lie subgroupoid of $\mathcal{G}\rightrightarrows M$ and
$(\Sigma^{0},\omega_{\Sigma^{0}})\rightrightarrows M$ is a symplectic groupoid
integrating $(M,\pi_{M})$. In particular, $(M,\pi_{M})$ is an integrable
Poisson manifold.

###### Proof.

The condition that $\mathcal{F}_{\mathcal{G}}$ is multiplicative amounts to
the identities

(4)
$\textbf{i}_{*}\mathcal{F}_{\mathcal{G}}=\mathcal{F}_{\mathcal{G}},\quad\mathbf{m}_{*}(\mathcal{F}_{\mathcal{G}}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{F}_{\mathcal{G}})=\mathcal{F}_{\mathcal{G}},$

where
$\mathcal{F}_{\mathcal{G}}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{F}_{\mathcal{G}}=T\mathcal{G}^{(2)}\cap(\mathcal{F}\times\mathcal{F})$
(note that since
$\mathcal{F}+\ker\mathrm{d}\mathbf{s}=\mathcal{F}+\ker\mathrm{d}\mathbf{t}$
this intersection is transverse). Since $M$ is contained in the leaf
$\Sigma^{0}$, it follows that the restriction of $\mathbf{s}$ and $\mathbf{t}$
to $\Sigma^{0}$ are surjective submersions.

Now observe that inversion fixes the identity section, so the first condition
in (4) implies that inversion maps leaves of $\mathcal{F}_{\mathcal{G}}$ to
leaves of $\mathcal{F}_{\mathcal{G}}$. Since inversion fixes the identity
section, it follows that it maps $\Sigma^{0}$ into itself.

Similarly, the second condition in (4) implies that multiplication maps leaves
of
$\mathcal{F}_{\mathcal{G}}^{(2)}:=\mathcal{F}_{\mathcal{G}}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{F}_{\mathcal{G}}$
to leaves of $\mathcal{F}_{\mathcal{G}}$. Since
$\Sigma^{0}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\Sigma^{0}$ is a
leaf of $\mathcal{F}_{\mathcal{G}}^{(2)}$ containing all the pairs
$(1_{x},1_{x})$, it follows that $\Sigma^{0}$ is closed under multiplication.

Therefore we have smooth maps $\textbf{i}:\Sigma^{0}\to\mathcal{G}$ and
$\mathbf{m}:\Sigma^{0}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\Sigma^{0}\to\mathcal{G}$
with image lying in $\Sigma^{0}$. The fact that these are also smooth as maps
into $\Sigma^{0}$ follows from the general fact that leaves of foliations are
regularly immersed submanifolds.

The inclusion $\Sigma^{0}\hookrightarrow\mathcal{G}$ is a groupoid morphism,
so obviously $\omega_{\Sigma^{0}}:=\omega|_{\Sigma^{0}}$ is a multiplicative
symplectic form, so $(\Sigma^{0},\omega_{\Sigma^{0}})\rightrightarrows M$ is a
symplectic groupoid. The composition of this inclusion with the target of
$\mathcal{G}$ gives a Poisson map

$\mathbf{t}:(\Sigma^{0},\omega_{\Sigma^{0}})\to(M,\pi_{M}),$

so this symplectic groupoid integrates $(M,\pi_{M})$. ∎

###  2.3. Examples of cosymplectic groupoids

The basic examples of cosymplectic structures mentioned in Section 2.1 all
have multiplicative versions, yielding examples of cosymplectic groupoids.

######  Example 2.10.

Let $(\Sigma,\Omega_{\Sigma})$ be a symplectic groupoid over $M$. Then we can
form the trivial abelian extension

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

and equip $\mathcal{G}=\Sigma\times\mathbb{R}$ with the cosymplectic structure
$(\operatorname{pr}_{\Sigma}^{*}\Omega_{\Sigma},\operatorname{pr}_{\mathbb{R}}^{*}\mathrm{d}t)$,
where $t$ denotes the coordinate in the second factor. This gives a
cosymplectic groupoid, called the _trivial central extension_ of the
symplectic groupoid $(\Sigma,\Omega_{\Sigma})$ by $\mathbb{R}$.

A similar construction holds with $\mathbb{R}$ replaced by $\mathbb{S}^{1}$
and $\mathrm{d}\theta$ instead of $\mathrm{d}t$. More generally, one can
consider a central extension of Lie groupoids with trivial kernel

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\times\mathbb{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $(\Sigma,\Omega_{\Sigma})$ a symplectic groupoid. A multiplicative
Ehresmann connection for this extension is specified by a multiplicative
1-form $\alpha\in\Omega^{1}(\mathcal{G},\mathbb{R})$ (see [6, 9]). This
connection is flat if and only if the form is closed, and will see in Section
4.4 that in this case we obtain a multiplicative cosymplectic structure
$(\operatorname{pr}_{\Sigma}^{*}\Omega_{\Sigma},\alpha)$ in $\mathcal{G}$.

######  Example 2.11.

Let $(\Sigma\rightrightarrows M,\omega)$ be a symplectic groupoid and
$\varphi:\Sigma\to\Sigma$ a symplectomorphism satisfying

$\displaystyle\mathbf{s}\circ\varphi$
$\displaystyle=\mathbf{s},\qquad\mathbf{t}\circ\varphi=\mathbf{t},$
$\displaystyle\varphi(gh)$
$\displaystyle=g\varphi(h)=\varphi(g)h,\qquad(g,h)\in\Sigma^{(2)}.$

for all $(g,h)\in\Sigma^{(2)}$. Notice that these properties are satisfied by
the time-one map of any bi-invariant vector field on a Lie groupoid (e.g., the
Reeb vector field of a cosymplectic groupoid). These properties ensure that
the map

$M\times\mathbb{Z}\hookrightarrow\Sigma\times\mathbb{R},\quad(x,n)\mapsto(\varphi^{n}(1_{x}),n),$

make the trivial bundle of groups $M\times\mathbb{Z}\rightrightarrows M$ a
closed, normal, subgroupoid inside the isotropy of the direct product groupoid
$\Sigma\times\mathbb{R}\rightrightarrows M$. It follows that the mapping torus

$\Sigma\times_{\varphi}\mathbb{S}^{1}:=(\Sigma\times\mathbb{R})/\mathbb{Z},$

has a unique Lie groupoid structure making the following sequence of Lie
groupoids exact

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Since $\Sigma\times_{\varphi}\mathbb{S}^{1}$ is a symplectic mapping torus, it
has a cosymplectic structure which one checks is multiplicative. Hence, it is
a cosymplectic groupoid. Notice that the map

$\Sigma\times_{\varphi}\mathbb{S}^{1}\to\mathbb{S}^{1},\quad\left[g,t\right]\mapsto
e^{2\pi it},$

is a fibration of Lie groupoids.

######  Example 2.12.

Let $(\Sigma,\Omega)$ be a symplectic groupoid over $M$ and let
$\iota:\mathcal{G}\hookrightarrow\Sigma$ be a Lie subgroupoid. Assume that
there exists a multiplicative symplectic vector field
$X\in\mathfrak{X}(\Sigma)$ transverse to $\mathcal{G}$. Then
$\omega:=\iota^{*}\Omega$ and $\alpha:=\iota^{*}(i_{X}\Omega)$ define a
multiplicative cosymplectic structure on the groupoid $\mathcal{G}$.

Conversely, given a cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ we can
form a symplectic groupoid $(\Sigma,\Omega)$ so that
$(\mathcal{G},\omega,\alpha)$ is obtained from $(\Sigma,\Omega)$. We let
$\Sigma$ be the product of the groupoid $\mathcal{G}\rightrightarrows M$ with
the identity groupoid $\mathbb{R}\rightrightarrows\mathbb{R}$, so $\Sigma$ is
a groupoid with space of arrows $\mathcal{G}\times\mathbb{R}$ and space of
objects $M\times\mathbb{R}$. The symplectic form on $\Sigma$ is given by
$\Omega=\omega+\alpha\wedge\mathrm{d}t$. One checks easily that $\Omega$ is
multiplicative and that $\frac{\partial}{\partial t}$ is a multiplicative
symplectic vector field transverse to
$\mathcal{G}\times\\{0\\}\cong\mathcal{G}$.

######  Example 2.13.

For a concrete example of a cosymplectic groupoid which is not a central
extension, let
$\Sigma=\mathbb{T}^{n}\times\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ be
the trivial bundle of Lie groups with fiber the torus $\mathbb{T}^{n}$.
Denoting by $(\theta^{1},\dots,\theta^{n})$ angle coordinates on the torus and
$(x^{1},\dots,x^{n})$ linear coordinates on $\mathbb{R}^{n}$, we let
$\Omega:=\sum_{i=1}^{n}\mathrm{d}\theta^{i}\wedge\mathrm{d}x^{i}$. This is a
multiplicative form so $(\Sigma,\Omega)$ is a symplectic groupoid. Now fix
some $a=(a_{1},\dots,a_{n})\in\mathbb{R}^{n}$ with $||a||=1$. The vector field

$X:=\sum_{i=1}^{n}a_{i}\frac{\partial}{\partial x^{i}},$

is a symplectic vector field transverse to the subgroupoid
$\mathcal{G}=\mathbb{T}^{n}\times M\rightrightarrows M$ where $M$ is the
hyperplane

$M=\\{(x^{1},\dots,x^{n})\in\mathbb{R}^{n}:\sum_{i=1}^{n}a_{i}x^{i}=0\\}.$

Moreover, $X$ is multiplicative since its flow is a 1-parameter group of
automorphisms of $\Sigma$. Hence, we are in the situation of the previous
example, so we obtain a cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$.
In particular, we find that $\alpha=-\sum_{i=1}^{n}a_{i}\mathrm{d}\theta^{i}$.
The hamiltonian vector field $X_{f}$ associated with the function
$f(x,\theta)=\sum_{i=1}^{n}a_{i}x^{i}$ satisfies
$i_{X_{f}}\omega=\mathrm{d}f|_{T\mathcal{G}}=0$ and
$\alpha(X_{f})=-\sum_{i=1}^{n}a_{i}^{2}=-1$. Hence, we conclude that the Reeb
vector field is

$E=-X_{f}|_{\mathcal{G}}=-\sum_{i=1}^{n}a_{i}\frac{\partial}{\partial\theta^{i}}.$

It follows that the orbit space of $E$ is smooth if and only if $\mathbb{Z}a$
defines a discrete subgroup of $\mathbb{T}^{n}$, i.e., if and only if the
ratios $a_{i}:a_{j}$ are all rational. Therefore this yields examples of
cosymplectic groupoids which are not central extensions.

###  2.4. Central extensions and cosymplectic groupoids

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid over $M$. Let us
denote by $\mathcal{K}\subset\mathcal{G}$ the collection of all orbits of the
Reeb vector field $E$ which intersect the identity section of $\mathcal{G}$.
We call $\mathcal{K}$ the _kernel of the cosymplectic groupoid_
$(\mathcal{G},\omega,\alpha)$.

######  Theorem 2.14.

The kernel $\mathcal{K}$ of a cosymplectic groupoid
$(\mathcal{G},\omega,\alpha)$ is a bundle of abelian groups which fits into a
short exact sequence of topological groupoids over the same base

(5)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $\Sigma$ is the orbit space of the Reeb vector field. When this space is
smooth, this is a short exact sequence of Lie groupoids and $(\Sigma,\Omega)$
is a symplectic groupoid for a unique symplectic structure making the
projection $\mathcal{G}\to\Sigma$ a Poisson map.

###### Proof.

Since $E$ is non-vanishing and transverse to the identity section, it follows
that $\mathcal{K}$ is a submanifold of $\mathcal{G}$. By Proposition 2.7,
since $E$ is bi-invariant it follows that $\mathcal{K}$ is a Lie subgroupoid
of $\mathcal{G}$, which is actually a bundle of Lie groups contained in the
isotropy of $\mathcal{G}$. One can form the quotient groupoid
$\Sigma=\mathcal{G}/\mathcal{K}$, which is a topological groupoid, giving the
short exact sequence (5).

Notice that the quotient groupoid $\Sigma=\mathcal{G}/\mathcal{K}$ can be
identified with the space of orbits of the $\mathbb{R}$-action defined by the
Reeb vector field. When this orbit space is a smooth manifold, the form
$\omega$ is basic for the $\mathbb{R}$-action so there is a unique symplectic
2-form $\Omega$ in $\Sigma$ such $p^{*}\Omega=\omega$, where
$p:\mathcal{G}\to\Sigma$ is the projection. Since $\omega$ is multiplicative,
it follows that $\Omega$ is also multiplicative, so $(\Sigma,\Omega)$ is a
symplectic groupoid. One checks easily that
$p:(\mathcal{G},\pi_{\mathcal{G}})\to(\Sigma,\Omega)$ is a Poisson map. Since
$p$ is a submersion, it follows that $\Omega$ is the unique symplectic form
with this property. ∎

We will call a cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ a _central
extension Lie groupoid_ whenever the orbit space of the Reeb vector field is
smooth, so $\mathcal{G}$ fits into a short exact sequence of Lie groupoids
with abelian one-dimensional kernel.

###  2.5. Proper cosymplectic groupoids

Recall that a Lie groupoid $\mathcal{G}\rightrightarrows M$ is called _proper_
if its space of arrows is Hausdorff and the map
$(\mathbf{s},\mathbf{t}):\mathcal{G}\to M\times M$ is proper. In this sections
we restrict our attention to proper cosymplectic groupoids.

######  Example 2.15.

The cosymplectic groupoids arising as symplectic mapping torus, as in Example
2.11, are proper whenever one starts with a proper symplectic groupoid
$(\Sigma\rightrightarrows M,\omega)$. In this case, the symplectic leaves of
the resulting cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ are the
fibers of $q:\mathcal{G}\to\mathbb{S}^{1}$, hence are embedded submanifolds.

The cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ constructed in Example
2.13, which is not a central extension, is also proper, being a bundle of
compact Lie groups. However, in this example the symplectic leaf through the
identity section $\Sigma^{0}$ is not embedded.

It turns out that a proper cosymplectic groupoid is a symplectic mapping
torus, as in Example 2.11, if and only if the symplectic leaf $\Sigma^{0}$
containing the identity section is embedded.

######  Theorem 2.16.

Let $(\mathcal{G},\omega,\alpha)$ be a proper, source connected, cosymplectic
groupoid and assume that the symplectic leaf $\Sigma^{0}\subset\mathcal{G}$ is
embedded. Then:

  1. (i)

there is a time $t_{0}$ such that the flow of the Reeb vector field at time
$t_{0}$ maps $\Sigma^{0}$ to itself yielding a symplectomorphism

$\varphi:=\varphi^{t_{0}}_{E}:\Sigma^{0}\to\Sigma^{0}.$

  2. (ii)

$\mathcal{G}$ is isomorphic to the symplectic mapping torus
$\Sigma^{0}\times_{\varphi}\mathbb{S}^{1}$ and the resulting submersion

$q:\mathcal{G}\to\mathbb{S}^{1},$

is a fibration of Lie groupoids.

###### Proof.

The Reeb vector field is a complete Poisson vector field transverse to the
sympletic leaves of $(\mathcal{G},\pi_{\mathcal{G}})$. Hence, for each fix
$t$, its flow $\varphi^{t}_{E}:\mathcal{G}\to\mathcal{G}$ maps leaves to
leaves. We claim that there exists some smallest $t_{0}>0$ such that
$\varphi^{t_{0}}_{E}(\Sigma^{0})=\Sigma^{0}$.

Since the Reeb vector field $E$ is both left and right invariant, it satisfies

$\displaystyle\mathbf{s}\circ\varphi^{t}_{E}$
$\displaystyle=\mathbf{s},\qquad\mathbf{t}\circ\varphi^{t}_{E}=\mathbf{t},$
$\displaystyle\varphi^{t}_{E}(gh)$
$\displaystyle=g\varphi^{t}_{E}(h)=\varphi^{t}_{E}(g)h,\qquad((g,h)\in\Sigma^{(2)}).$

It follows that the map

(6)
$\Phi:\Sigma^{0}\times\mathbb{R}\to\mathcal{G},\quad(g,t)\mapsto\varphi^{t}_{E}(g).$

is a Lie groupoid morphism. Since $\Sigma^{0}$ is embedded, this map is also a
local diffeomorphism and the image of $\Phi$ is open and closed in
$\mathcal{G}$. Since $M$ is connected and $\mathcal{G}$ is source connected,
we have that $\mathcal{G}$ is connected, so $\Phi$ is surjective.

Now observe that, for each $x\in M$, the map $\Phi$ restricts to a Lie group
map

$\Phi_{x}:\mathbb{R}\to\mathcal{G}_{x},\quad t\mapsto\Phi(1_{x},t),$

whose image is closed. Since $\mathcal{G}$ is proper, isotropy groups are
compact, so the image of this map is compact. Hence, there exists a first time
$t_{0}>0$ such that

$\Phi(1_{x},t_{0})\in\Sigma^{0}\cap\mathcal{G}_{x}.$

Since, for each $t$, $\varphi^{t}_{E}:\mathcal{G}\to\mathcal{G}$ maps leaves
to leaves we conclude that

$\varphi^{t_{0}}_{E}(\Sigma^{0})=\Sigma^{0},$

and $t_{0}$ is the smallest positive real satisfying this property, proving
our claim.

######  Lemma 2.17.

The morphism (6) yields a short exact sequence of Lie groupoids:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\times\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{0}\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where the first map is $(x,n)\mapsto(\varphi^{nt_{0}}_{E}(1_{x}),-nt_{0})$. In
particular, the groupoid $\mathcal{G}$ is isomorphic to a mapping torus:

$\mathcal{G}\simeq(\Sigma^{0}\times\mathbb{R})/\mathbb{Z},$

where the $\mathbb{Z}$-action is generated by
$(g,t)\mapsto(\varphi^{t_{0}}_{E}(g),t-t_{0})$.

Assuming this lemma, it remains to prove is that $\Phi$ pulls back the
cosymplectic structure $(\omega,\alpha)$ to the cosymplectic structure
$(\operatorname{pr}^{*}_{\Sigma^{0}}\omega_{\Sigma^{0}},\operatorname{pr}^{*}_{\mathbb{R}}\mathrm{d}t)$.
This follows because:

  1. (a)

$\Phi$ is a map of the underlying foliations;

  2. (b)

The Reeb vector fields $\partial_{t}$ and $E$ are $\Phi$-related

$\mathrm{d}_{(g,t)}\Phi(\partial_{t})=\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=t}\varphi^{s}_{E}(g)=E|_{\Phi(g,t)}.$

In fact, from (b), we find that

$i_{\partial_{t}}\Phi^{*}\omega=\Phi^{*}i_{E}\omega=0.$

Since $\omega$ is closed, it follows that $\Phi^{*}\omega$ is basic relative
to $\operatorname{pr}_{\Sigma^{0}}:\Sigma^{0}\times\mathbb{R}\to\Sigma^{0}$.
On the other hand, for the section
$s:\Sigma^{0}\to\Sigma^{0}\times\mathbb{R}$, $g\mapsto(g,0)$, we have

$s^{*}\Phi^{*}\omega=(\Phi\circ s)^{*}\omega=\omega_{\Sigma^{0}},$

so we conclude that

$\Phi^{*}\omega=\operatorname{pr}^{*}_{\Sigma^{0}}\omega_{\Sigma^{0}}.$

Similarly, from (a), we find that for any tangent vector $(v,0)\in
T(\Sigma^{0}\times\mathbb{R})$

$i_{(v,0)}\Phi^{*}\alpha=\Phi^{*}(i_{\mathrm{d}\Phi(v,0)}\alpha)=0.$

Since $\alpha$ is closed, it follows that $\Phi^{*}\alpha$ is basic relative
to
$\operatorname{pr}^{*}_{\mathbb{R}}:\Sigma^{0}\times\mathbb{R}\to\mathbb{R}$.
But using (b) again we find

$i_{\partial_{t}}\Phi^{*}\alpha=\Phi^{*}i_{E}\alpha=1,$

so we conclude that

$\Phi^{*}\alpha=\operatorname{pr}^{*}_{\mathbb{R}}\mathrm{d}t.$

∎

###### Proof of Lemma 2.17.

Observe that for each $x$ there is a smallest positive integer $n_{0}$ such
that

$\varphi^{n_{0}t_{0}}_{E}(1_{x})=1_{x}.$

Note that $n_{0}$ is independent of $x$. This follows, e.g., because $n_{0}$
is the order of the group

$\Sigma^{0}\cap\Phi_{x}(\mathbb{R})=\\{1_{x},\varphi^{t_{0}}_{E}(1_{x}),\dots,\varphi^{(n_{0}-1)t_{0}}_{E}(1_{x})\\},$

and these groups form a Lie group bundle when $x$ vary in $M$. From this it
follows also that

$g\in\Sigma^{0},\
\varphi^{t}_{E}(g)=1_{x}\quad\Leftrightarrow\quad\left\\{\begin{array}[]{l}g=\varphi^{nt_{0}}_{E}(1_{x})\\\
\hskip 93.95122pt\text{for some }n,k\in\mathbb{Z},\\\
t=-nt_{0}+k(n_{0}t_{0}).\end{array}\right.$

Since $\varphi^{nt_{0}}_{E}(1_{x})=\varphi^{(n-kn_{0})t_{0}}_{E}(1_{x})$ we
conclude that the kernel of the morphism (6) is

$\operatorname{Ker}\Phi=\\{(\varphi^{mt_{0}}_{E}(1_{x}),-mt_{0}):m\in\mathbb{Z}\\}.$

This proves the lemma and completes also the proof of the theorem. ∎

##  3\. The Infinitesimal picture

###  3.1. Infinitesimal data of a cosymplectic groupoid

Let $(\mathcal{G},\omega,\alpha)$ be a cosymplectic groupoid, with Reeb vector
field $E$ and associated Poisson structure $\pi_{\mathcal{G}}$. All these
geometric structures have infinitesimal versions, as we now explain.

In general, will denote by $A$ a Lie algebroid with bundle projection $p:A\to
M$, anchor $\rho_{A}:A\to TM$, and Lie bracket $[~{},~{}]_{A}$ on its space of
sections $\Gamma(A)$. Our conventions are such that if
$\mathcal{G}\rightrightarrows M$ is a Lie groupoid, then its Lie algebroid
$A=A(\mathcal{G})$ has fiber
$A_{x}:=\operatorname{Ker}\mathrm{d}_{1_{x}}\mathbf{s}$ and anchor
$\rho_{A}|_{x}:=\mathrm{d}_{1_{x}}\mathbf{t}$. Moreover, its space of sections
$\Gamma(A)$ is identified with the space $\mathfrak{X}_{r}(\mathcal{G})$ of
right invariant vector fields on $\mathcal{G}$ and we will denote by
$\overrightarrow{X}\in\mathfrak{X}_{r}(\mathcal{G})$ the right invariant
vector field corresponding to $X\in\Gamma(A)$. A multiplicative form
$\omega\in\Omega^{k}(\mathcal{G})$ induces a pair of bundle maps
$\mu:A\to\wedge^{k-1}T^{*}M$, $\tilde{\mu}:A\to\wedge^{k}T^{*}M$, defined by

$\displaystyle\mu(a)(v_{1},\dots,v_{k-1})$
$\displaystyle=\omega(a,\mathrm{d}\varepsilon(v_{1}),\dots,\mathrm{d}\varepsilon(v_{k-1})),$
$\displaystyle\tilde{\mu}(a)(v_{1},\dots,v_{k})$
$\displaystyle=\mathrm{d}\omega(a,\mathrm{d}\varepsilon(v_{1}),\dots,\mathrm{d}\varepsilon(v_{k})).$

These maps satisfy the following conditions that characterize infinitesimal
multiplicative (IM) forms (see, e.g., [1, 2]):

$\displaystyle i_{\rho_{A}(b)}\mu(a)$ $\displaystyle=-i_{\rho_{A}(a)}\mu(b),$
(7) $\displaystyle\mu([a,b]_{A})$
$\displaystyle=\mathscr{L}_{\rho_{A}(a)}\mu(b)-i_{\rho_{A}(b)}(\mathrm{d}\mu(a)+\tilde{\mu}(a)),$
$\displaystyle\tilde{\mu}([a,b]_{A})$
$\displaystyle=\mathscr{L}_{\rho_{A}(a)}\tilde{\mu}(b)-i_{\rho_{A}(b)}\mathrm{d}\tilde{\mu}(a),$

for all sections $a,b\in\Gamma(A)$. For a closed IM form the component
$\tilde{\mu}$ vanishes.

After these preliminaries we can now list the infinitesimal data corresponding
to a cosymplectic groupoid. Let $(\mathcal{G},\omega,\alpha)$ be a
cosymplectic groupoid and denote by $A\to M$ its Lie algebroid. Then:

  1. (i)

The multiplicative closed 1-form $\alpha$ induces a closed IM 1-form
$\nu:A\to\mathbb{R}$;

  2. (ii)

The multiplicative closed 2-form $\omega$ induces a closed IM 2-form $\mu:A\to
T^{*}M$;

  3. (iii)

The Reeb vector field $E$ induces a central section $e\in\Gamma(A)$, i.e.,
$E=\overrightarrow{e}=\overleftarrow{e}$ where $\rho(e)=0$ and $[e,a]=0$, for
all $a\in\Gamma(A)$;

  4. (iv)

The multiplicative Poisson structure $\pi_{\mathcal{G}}$ induces a unique
Poisson structure $\pi_{M}\in\mathfrak{X}^{2}(M)$, for which the target is a
Poisson map, and the source is an anti-Poisson map.

Only the last item needs some justification. One can show directly from the
condition that $\pi_{\mathcal{G}}$ is multiplicative that the Poisson bracket
of functions locally constant on the $\mathbf{t}$-fibers is a function locally
constant on the $\mathbf{t}$-fibers (see, e.g., [17]), so that there is a
unique Poisson structure on $M$ for which the submersion
$\mathbf{t}:\mathcal{G}\to M$ is Poisson.

The infinitesimal data above has various relationships between themselves,
which can be stated in a concise form as follows:

######  Proposition 3.1.

The Lie algebroid $A\to M$ of a cosymplectic groupoid
$(\mathcal{G},\omega,\alpha)$ is a central extension

(8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\nu}}$$\scriptstyle{\mu}$$\textstyle{T^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $T^{*}M$ is equipped with the cotangent Lie algebroid structure
associated with the Poisson manifold $(M,\pi_{M})$ and $\mathfrak{k}$ is the
trivial line bundle generated by the central section $e\in\Gamma(A)$. This
extension has a natural splitting given by:

$\underline{\nu}:A\to\mathfrak{k},\quad a\mapsto\nu(a)e.$

###### Proof.

(i) _$\mu:A\to T^{*}M$ is a surjective Lie algebroid map:_ The definition of
$\mu$ shows that we have a commutative diagram:

$\textstyle{T\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\flat}}$$\textstyle{T^{*}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A^{*}}$

where $\mu^{*}$ is the transpose of $\mu$. By Proposition 2.7, we know that
$E\in\ker\omega^{\flat}$ is transverse to the identity section, so we conclude
that $\mu^{*}$ is injective.

Now observe that since
$\mathbf{t}:(\mathcal{G},\pi_{\mathcal{G}})\to(M,\pi_{M})$ is a Poisson map,
using the definition of $\pi_{\mathcal{G}}$ and $\mu$, we obtain

$\pi_{M}^{\sharp}(\mu(a))=\mathrm{d}_{1_{x}}\mathbf{t}\cdot\pi_{\mathcal{G}}^{\\#}\cdot(\mathrm{d}_{1_{x}}\mathbf{t})^{*}(\mu(a))=\rho_{A}(a)$

where $a\in A_{x}$. On the other hand, since $\mu$ is a closed IM form,
relations (7) give

$\displaystyle\mu([a,b]_{A})$
$\displaystyle=\mathscr{L}_{\rho_{A}(a)}\mu(b)-i_{\rho_{A}(b)}\mathrm{d}\mu(a)$
$\displaystyle=\mathscr{L}_{\pi_{M}^{\sharp}(\mu(a))}\mu(b)-i_{\pi_{M}^{\sharp}(\mu(b))}\mathrm{d}\mu(a)=[\mu(a),\mu(b)]_{\pi_{M}},$

so $\mu:A\to T^{*}M$ is a Lie algebroid morphism.

(ii) $\ker(\mu)=\langle e\rangle$: Since $\mu$ is surjective, its kernel is a
rank 1 vector sub-bundle. Since $e\in\Gamma(A)$ is a non-vanishing section,
all we have to check is that $\mu(e)=0$. This is clear from the definition of
$\mu$ since $\mu(e)=(i_{E}\omega)|_{TM}=0$.

(iii) _$\underline{\nu}:A\to\langle e\rangle$ , $a\mapsto\nu(a)e$, splits the
short exact sequence (8):_ Notice that we have $\nu(e)=1$, since

$\nu(e)(x)=\alpha_{1_{x}}(E_{1_{x}})=1.$

This shows that $\underline{\nu}$ is a splitting as a short exact sequence of
vector bundles. Associated with this splitting there is a $T^{*}M$-connection
on the bundle $\mathfrak{k}$. Because $e$ commutes with any section of $A$, we
have that $e$ is a flat section

$\nabla_{\beta}e=0.$

On the other hand, the curvature 2-form of this splitting is given by

$c(\beta_{1},\beta_{2})=\nu([a_{1},a_{2}]_{A}),$

where $a_{i}\in\operatorname{Ker}\nu$ is the unique element such that
$\mu(a_{i})=\beta_{i}$. But this curvature 2-form vanishes since $\nu$ is a
closed IM form and from (7) find

$\nu([a_{1},a_{2}]_{A})=\mathscr{L}_{\rho_{A}(a_{1})}\nu(a_{2})-i_{\rho_{A}(a_{2})}\mathrm{d}\nu(a_{1})=0,$

whenever $a_{1},a_{2}\in\Gamma(\operatorname{Ker}\nu)$. ∎

The previous proposition shows that the space of objects of a cosymplectic
groupoid is a Poisson manifold, and that we have a canonical isomorphism:

(9) $A\cong T^{*}M\oplus\mathbb{R},\quad a\mapsto(\mu(a),\nu(a)).$

Under this isomorphism, the anchor becomes

$\rho_{A}:T^{*}M\oplus\mathbb{R}\to
TM,\quad(\beta,\lambda)\mapsto\pi^{\sharp}_{M}(\beta),$

while the bracket on sections $(\beta_{i},f_{i})\in\Omega^{1}(M)\times
C^{\infty}(M)$ can be written as:

$[(\beta_{1},f_{1}),(\beta_{2},f_{2})]_{A}=([\beta_{1},\beta_{2}]_{\pi_{M}},\pi_{M}^{\sharp}(\beta_{1})(f_{2})-\pi_{M}^{\sharp}(\beta_{2})(f_{1})).$

In other words, we have:

######  Corollary 3.2.

If $(\mathcal{G},\omega,\alpha)$ is a cosymplectic groupoid, its Lie algebroid
$(A,\mu,\nu)$ is canonically isomorphic via (9) to the trivial central
extension of the cotangent algebroid associated with the base Poisson manifold
$(M,\pi_{M})$:

$(A,\mu,\nu)\simeq(T^{*}M\oplus\mathbb{R},\operatorname{pr}_{T^{*}M},\operatorname{pr}_{M\times\mathbb{R}}).$

The only missing piece on the infinitesimal side is what corresponds to the
multiplicative Poisson structure $\pi_{\mathcal{G}}$. At the infinitesimal
level such a structure corresponds to a _Lie bialgebroid_. We recall that a
Lie bialgebroid is a pair of Lie algebroid structures $(A,A^{*})$, on a vector
bundle $A\to M$ and on its dual bundle $A^{*}\to M$, such that for any
$X,Y\in\Gamma(A)$,

$\mathrm{d}_{*}[X,Y]_{A}=[X,\mathrm{d}_{*}Y]_{A}-[Y,\mathrm{d}_{*}X]_{A}.$

where $\mathrm{d}_{*}$ denotes the $A^{*}$-differential ([14]). Given a
multiplicative Poisson structure $\pi_{\mathcal{G}}$ on a groupoid
$\mathcal{G}$ with Lie algebroid $A$ one obtains a Lie algebroid structure on
$A^{*}$, so that $(A,A^{*})$ is a Lie bialgebroid. Namely, the
$A^{*}$-differential is characterized by

$\overrightarrow{\mathrm{d}_{*}X}=-[\overrightarrow{X},\pi_{\mathcal{G}}],\qquad(X\in\Gamma(A)).$

The Lie bialgebroid of a cosymplectic groupoid is again rather special.

######  Proposition 3.3.

If $(\mathcal{G},\omega,\alpha)$ is a cosymplectic groupoid, (9) gives an
isomorphism of Lie bialgeboids

$(A,A^{*})\simeq(T^{*}M\oplus\mathbb{R},TM\oplus\mathbb{R}),$

where $T^{*}M$ denotes the cotangent Lie algebroid of the base Poisson
manifold $(M,\pi_{M})$.

###### Proof.

By Corollary 3.2, we already know that the IM forms corresponding to
$(\omega,\alpha)$ give a Lie algebroid isomorphism:

$A\cong T^{*}M\oplus\mathbb{R},\quad a\mapsto(\mu(a),\nu(a)).$

On the other hand, the central section $e\in\Gamma(A)$ satisfies:

$\mathrm{d}_{*}e=0,$

since the Reeb vector field $E=\overrightarrow{e}$ is a Poisson vector field
on $\mathcal{G}$ (see, [13, Thm 11.4.7]). This implies that the transpose of
the map $(\mu,\nu)$ is also a Lie algebroid isomorphism:

$TM\oplus\mathbb{R}\cong
A^{*},\quad({u},\lambda)\mapsto(\mu^{*}({u}),\nu^{*}(\lambda)).$

∎

###  3.2. Source 1-connected cosymplectic groupoids

By the results in the previous section, source 1-connected cosymplectic
groupoids are very easy to describe:

######  Theorem 3.4.

The base of any cosymplectic groupoid $(\mathcal{G},\omega,\alpha)$ is an
integrable Poisson manifold $(M,\pi_{M})$. If $\mathcal{G}$ is source
1-connected then there is a canonical isomorphism

$(\mathcal{G},\omega,\alpha)\cong(\Sigma(M)\times\mathbb{R},\operatorname{pr}_{\Sigma}^{*}\Omega,\operatorname{pr}_{\mathbb{R}}^{*}\mathrm{d}t)$

where $(\Sigma(M),\Omega)$ the source 1-connected symplectic integration of
$(M,\pi_{M})$.

###### Proof.

Since $A\cong T^{*}M\oplus\mathbb{R}$, $a\mapsto(\mu(a),\nu(a))$, is a Lie
algebroid isomorphism, it follows that $A$ is integrable iff and only if
$T^{*}M$ is integrable. When $\mathcal{G}$ is source 1-connected, the
integration of this isomorphism gives the desired groupoid isomorphism. This
groupoid isomorphism maps $\operatorname{pr}_{\Sigma}^{*}\Omega$ to $\omega$
and $\operatorname{pr}_{\mathbb{R}}^{*}\mathrm{d}t$ to $\alpha$. ∎

In general, if $(\mathcal{G},\omega,\alpha)$ is only source connected, we have

$(\Sigma(M)\times\mathbb{R})/\Lambda,$

where $i:\Lambda\hookrightarrow\Sigma(M)\times\mathbb{R}$ is an embedded
bundle of discrete Lie groups such that $i^{*}\Omega=0$ and
$i^{*}\mathrm{d}t=0$. If we assume that the orbit space of the Reeb vector
field $E\in\mathfrak{X}(\mathcal{G})$ is smooth, we obtain that $\mathcal{G}$
is a central extension of some symplectic integration $\Sigma$ of
$(M,\pi_{M})$:

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In general, this sequence fails to split. Moreover, it may have a groupoid
splitting, so that $\mathcal{G}\cong\Sigma\times\mathcal{K}$, while the
cosymplectic structure may not be the trivial one (as in Example 2.10). This
is illustrated in the next example.

######  Example 3.5.

Let
$\mathcal{G}=\mathbb{R}^{2}\times\mathbb{S}^{1}\rightrightarrows\mathbb{R}$ be
the trivial bundle of Lie groups with projection $(x,y,\theta)\mapsto x$ and
fiber $\mathbb{R}\times\mathbb{S}^{1}$. The forms

$\omega:=\mathrm{d}x\wedge\mathrm{d}y+\mathrm{d}x\wedge\mathrm{d}\theta,\quad\alpha:=\frac{1}{2}\left(\mathrm{d}y-\mathrm{d}\theta\right),$

define a multiplicative cosymplectic structure on $\mathcal{G}$. The
corresponding Reeb vector field is

$E=\partial_{y}-\partial_{\theta}.$

It follows that the kernel of this cosymplectic groupoid is a trivial bundle
$\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ and the leaf space of this vector
field can be identified with $\mathbb{R}\times\mathbb{S}^{1}$, giving rise to
the central extension:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}^{2}\times\mathbb{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}\times\mathbb{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

Here the first map is given by $(x,y)\mapsto(x,y,-y)$ while the second map is
given by $(x,y,\theta)\mapsto(x,y+\theta)$. This sequence has the splitting
$(x,\theta)\mapsto(x,0,\theta)$.

We claim that although $\mathcal{G}\cong\Sigma\times\mathcal{K}$ as a
groupoid, the cosymplectic structure structure is not isomorphic to
$(p_{\Sigma}^{*}\Omega,p_{\mathcal{K}}^{*}\mathrm{d}\theta)$. In fact, the
symplectic leaves of $\mathcal{G}$ are the leaves of the distribution
$\mathrm{d}y-\mathrm{d}\theta=0$, so admit the parametrization
$(x,y)\mapsto(x,y,y+c)$, with $c\in\mathbb{S}^{1}$. Hence, the symplectic
leaves are diffeomorphic to $\mathbb{R}^{2}$, while a trivial extension
$\Sigma\times\mathcal{K}$ has symplectic leaves diffeomorphic to
$\mathbb{R}\times\mathbb{S}^{1}$.

##  4\. Poisson groupoids

###  4.1. Poisson groupoids of corank 1

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid equipped with a regular
multiplicative Poisson structure $\pi_{\mathcal{G}}$ of corank 1. Recall that
a Poisson structure $\pi_{\mathcal{G}}$ is multiplicative if and only if
$\pi_{\mathcal{G}}^{\sharp}\colon T^{*}\mathcal{G}\to T\mathcal{G}$ is a
morphism of Lie groupoids

$\textstyle{T^{*}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{G}}^{\sharp}}$$\textstyle{T\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{A^{*}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
TM}$

In this diagram $A^{*}$ is identified with the annihilator subbundle
$(TM)^{0}\subset T^{*}_{M}\mathcal{G}$ and the anchor $\rho_{A^{*}}$ is the
restriction of $\pi_{\mathcal{G}}^{\sharp}$ to this subbundle. The condition
that $\pi_{\mathcal{G}}$ is regular of corank 1 places restrictions on the
codimension of $M$ in $\mathcal{G}$.

######  Proposition 4.1.

Let $(\mathcal{G}\rightrightarrows M,\pi_{\mathcal{G}})$ be a regular Poisson
groupoid of corank 1. Then either:

$\mathrm{(C1)}\quad\dim\mathcal{G}=2\dim
M+1\qquad\textrm{or}\quad\mathrm{(C2)}\quad\dim\mathcal{G}=2\dim M-1.$

###### Proof.

Since $\pi_{\mathcal{G}}$ has constant rank, we can choose an extension
$\omega\in\Omega^{2}(\mathcal{G})$ of the leafwise symplectic form, i.e.,

$\omega(\pi_{\mathcal{G}}^{\sharp}(\alpha),\pi_{\mathcal{G}}^{\sharp}(\beta))=\langle\alpha,\pi_{\mathcal{G}}^{\sharp}(\beta)\rangle.$

Note that $\omega$ is not assumed to be multiplicative. We claim that
$-\textbf{i}^{*}\omega$ is also an extension of the leafwise symplectic form.
Indeed, since i is anti-Poisson, we find

$\displaystyle-\textbf{i}^{*}\omega(\pi_{\mathcal{G}}^{\sharp}(\alpha),\pi_{\mathcal{G}}^{\sharp}(\beta))$
$\displaystyle=-\omega(\mathrm{d}\textbf{i}\mbox{\,\tiny{$\circ$}\,}\pi_{\mathcal{G}}^{\sharp}(\alpha),\mathrm{d}\textbf{i}\mbox{\,\tiny{$\circ$}\,}\pi_{\mathcal{G}}^{\sharp}(\beta))$
$\displaystyle=-\omega(\pi_{\mathcal{G}}^{\sharp}(\textbf{i}^{*}(\alpha)),\pi_{\mathcal{G}}^{\sharp}(\textbf{i}^{*}(\beta)))$
$\displaystyle=-\langle\textbf{i}^{*}(\alpha),\pi_{\mathcal{G}}^{\sharp}(\textbf{i}^{*}(\beta)))=\langle\alpha,\pi_{\mathcal{G}}^{\sharp}(\beta)\rangle.$

It follows that
$\tilde{\omega}=\frac{1}{2}\left(\omega-\textbf{i}^{*}\omega\right)$ is also
an extension satisfying additionally:

$\textbf{i}^{*}\tilde{\omega}=-\tilde{\omega}.$

Since $\textbf{i}\circ\epsilon=\epsilon$, we deduce that
$\epsilon^{*}\tilde{\omega}=0$. In addition,
$\operatorname{rank}\tilde{\omega}=\dim\mathcal{G}-1$ implies that

$2\dim M\leq\dim\mathcal{G}+1.$

Now observe that since $\pi_{\mathcal{G}}$ is multiplicative, the identity
section is coisotropic, i.e., we have
$\pi^{\sharp}_{\mathcal{G}}(TM)^{0}\subset TM$. So we also have

$2\dim M\geq\dim\mathcal{G}-1.$

Since $\dim\mathcal{G}$ is odd, it follows from these two estimates that
$\dim\mathcal{G}=2\dim M\pm 1$. ∎

Note that both cases listed in this proposition exist. For example, let
$(\Sigma,\Omega)$ be a symplectic groupoid then:

  1. (C1)

the product of $\Sigma$ with a 1-dimensional Lie group, e.g.
$\mathbb{S}^{1}\rightrightarrows\\{*\\}$, is a Poisson groupoid
$(\mathcal{G}\rightrightarrows M,\pi_{\mathcal{G}})$ of corank 1 with
$\dim\mathcal{G}=2\dim M+1$;

  2. (C2)

the product of $\Sigma$ with a 1-dimensional identity groupoid, e.g.
$\mathbb{S}^{1}\rightrightarrows\mathbb{S}^{1}$, is a Poisson groupoid
$(\mathcal{G}\rightrightarrows M,\pi_{\mathcal{G}})$ of corank 1 with
$\dim\mathcal{G}=2\dim M-1$.

We will be interested in Poisson groupoids of type (C1) since these are the
ones that arise in connection with cosymplectic groupoids.

######  Proposition 4.2.

If $(\mathcal{G}\rightrightarrows M,\pi_{\mathcal{G}})$ is a Poisson groupoid
of corank 1 then the following conditions are equivalent:

  1. (i)

$\dim\mathcal{G}=2\dim M+1$;

  2. (ii)

$\operatorname{rank}(\pi_{\mathcal{G}})=2\dim M$;

  3. (iii)

$\rho_{A^{*}}:A^{*}\to TM$ surjective;

  4. (iv)

the identity section $M$ is contained in a symplectic leaf of
$\pi_{\mathcal{G}}$.

###### Proof.

(i) $\Rightarrow$ (ii) Obvious.

(ii) $\Rightarrow$ (iii) Since $\dim\mathcal{G}=2\dim M+1$, we have
$\operatorname{rank}(TM)^{0}=\dim M+1$. Hence, the map
$\rho_{A^{*}}:=(\pi_{\mathcal{G}}^{\sharp})|_{(TM)^{0}}:(TM)^{0}\to TM$ must
be surjective.

(iii) $\Rightarrow$ (iv) If (iii) holds, then
$TM=\operatorname{Im}(\rho_{A^{*}})\subset\operatorname{Im}(\pi_{\mathcal{G}}^{\sharp})$.

(iv) $\Rightarrow$ (i) For any Poisson groupoid $(\mathcal{G}\rightrightarrows
M,\pi_{\mathcal{G}})$ one has:

$\pi_{\mathcal{G}}^{\sharp}(TM)^{0}\subset
TM,\quad\pi_{\mathcal{G}}^{\sharp}(\ker\mathrm{d}\mathbf{s})^{0}\subset\ker\mathrm{d}\mathbf{t}.$

If (iv) holds, then we must have
$\operatorname{rank}(TM)^{0}\geq\operatorname{rank}(TM)$, so case (C2) is
excluded. ∎

In the sequel, by a “Poisson groupoid of type (C1)” we mean a Poisson groupoid
of corank 1 satisfying any of the equivalent conditions of Proposition 4.2.

######  Corollary 4.3.

Let $(\mathcal{G},\pi_{\mathcal{G}})$ be a Poisson groupoid of type (C1). Then
the symplectic leaf $\Sigma^{0}$ of $\pi_{\mathcal{G}}$ containing the
identity section is a (symplectic) subgroupoid of
$(\mathcal{G},\pi_{\mathcal{G}})$ integrating $(M,\pi_{M})$. In particular,
$(M,\pi_{M})$ is an integrable Poisson manifold.

###### Proof.

The transpose $\rho_{A^{*}}^{*}:T^{*}M\to A$ is a Lie algebroid morphism,
where $T^{*}M$ is the cotangent bundle algebroid associated with the base
Poisson manifold $(M,\pi_{M})$ (see [13]). By Proposition 4.2 (iii), this
morphism is injective, so $T^{*}M$ is a Lie subalgebroid of the integrable
algebroid $A$, and the result follows. Alternatively, one can also apply the
argument in the proof of Proposition 2.9. ∎

###  4.2. Poisson groupoids vs cosymplectic groupoids

When is a Poisson groupoid of corank 1 a cosymplectic groupoid? By Proposition
3.3, its Lie bialgebroid must be a central extension. A necessary and
sufficient condition is the following multiplicative version of the criteria
for a Poisson manifold of corank 1 to be cosymplectic:

######  Proposition 4.4.

A Poisson groupoid $(\mathcal{G},\pi_{\mathcal{G}})$ is cosymplectic if and
only if $\pi_{\mathcal{G}}$ is regular of corank 1 and there exists a non-
vanishing, bi-invariant, Poisson vector field $E\in\mathfrak{X}(\mathcal{G})$
transverse to the symplectic foliation.

###### Proof.

In one direction, we already know that the Reeb vector field of a cosymplectic
groupoid is a Poisson vector field transverse to the symplectic foliation,
which is both left and right invariant.

To prove the reverse direction, assume that $(\mathcal{G},\pi_{\mathcal{G}})$
is a Poisson groupoid of corank 1 that admits a non-vanishing Poisson vector
field $E\in\mathfrak{X}(\mathcal{G})$ transverse to the symplectic foliation,
which is both left and right invariant. We extend the symplectic forms on the
leaves to a 2-form $\omega$ by requiring $i_{E}\omega=0$. Also, we define a
1-form $\alpha$ by requiring $\alpha(E)=1$ and
$\ker\alpha=\operatorname{Im}\pi^{\sharp}_{\mathcal{G}}$. Since $E$ is a
Poisson vector field, one checks easily that $\omega$ and $\alpha$ are closed.
We claim that $\omega$ and $\alpha$ are multiplicative.

Take $(X_{1},X_{2})\in T\mathcal{G}^{(2)}$. Since $E$ is tranverse to the
symplectic foliation
$X_{i}=\lambda_{i}E+\pi_{\mathcal{G}}^{\sharp}(\gamma^{i})$, $i=1,2$.
Moreover, the fact that $E$ is left and right invariant implies that
$\mathrm{d}\mathbf{s}(\pi_{\mathcal{G}}^{\sharp}(\gamma_{1}))=\mathrm{d}\mathbf{t}(\pi_{\mathcal{G}}^{\sharp}(\gamma_{2}))$.
Since $\pi_{\mathcal{G}}$ is multiplicative,
$\operatorname{Im}\pi_{\mathcal{G}}^{\sharp}\subseteq T\mathcal{G}$ is a
multiplicative distribution, so we find

$\mathrm{d}\mathbf{m}(X_{1},X_{2})=(\lambda_{1}+\lambda_{2})E+\pi_{\mathcal{G}}^{\sharp}(\gamma_{1}\cdot\gamma_{2}).$

Using that $\alpha(E)=1$,
$\operatorname{Ker}\alpha=\operatorname{Im}\pi_{\mathcal{G}}^{\sharp}$, and
that $\omega$ is an extension of $\pi_{\mathcal{G}}$ with $i_{E}\omega=0$, we
can conclude that $\omega$ and $\alpha$ are multiplicative. For instance, for
$\alpha$,

$\mathbf{m}^{*}\alpha(X_{1},X_{2})=\alpha(\mathrm{d}\mathbf{m}(X_{1},X_{2}))=\alpha((\lambda_{1}+\lambda_{2})E+\pi_{\mathcal{G}}^{\sharp}(\gamma_{1}\cdot\gamma_{2}))=\lambda_{1}+\lambda_{2}.$

On the other hand,

$(\pi_{1}^{*}\alpha+\pi_{2}^{*}\alpha)(X_{1},X_{2})=\alpha(\lambda_{1}E+\pi_{\mathcal{G}}^{\sharp}(\gamma^{1}))+\alpha(\lambda_{2}E+\pi_{\mathcal{G}}^{\sharp}(\gamma^{2}))=\lambda_{1}+\lambda_{2}.$

Thus, $\alpha$ is multiplicative. ∎

###  4.3. Proper Poisson groupoids of corank 1

Proposition 4.4 shows that given a Poisson groupoid
$(\mathcal{G},\pi_{\mathcal{G}})$ of corank 1, in order to have a compatible
multiplicative cosymplectic structure one needs a vector field
$E\in\mathfrak{X}(\mathcal{G})$ transverse to symplectic foliation satisfying
two properties:

  1. (a)

$E$ is bi-invariant;

  2. (b)

$E$ is a Poisson vector field.

We now analyze these properties for the important special case of _proper_
Poisson groupoids.

First, as a general remark, note that the existence of a vector field
transverse to the symplectic foliation means that this foliation is co-
orientable. Since the leaves are oriented (being symplectic), this is
equivalent to $\mathcal{G}$ being orientable. Hence, we will assume this
condition throughout this discussion.

We start by looking into condition (a).

######  Proposition 4.5.

Let $(\mathcal{G},\pi_{\mathcal{G}})$ be an orientable Poisson groupoid of
corank 1 with symplectic foliation $\mathcal{F}_{\pi_{\mathcal{G}}}$. The
following two conditions are equivalent:

  1. (i)

There exists a bi-invariant vector field transverse to
$\mathcal{F}_{\pi_{\mathcal{G}}}$;

  2. (ii)

There exists a multiplicative 1-form whose kernel is
$\mathcal{F}_{\pi_{\mathcal{G}}}$;

and they imply that

  1. (iii)

There exists a multiplicative 2-form extending the leafwise symplectic form.

If $\mathcal{G}$ is proper and $\dim\mathcal{G}=2\dim M+1$ then the 3
conditions are equivalent.

###### Proof.

(i) $\Leftrightarrow$ (ii) A vector field $E\in\mathfrak{X}(\mathcal{G})$
transverse to $\mathcal{F}_{\pi_{\mathcal{G}}}$ determines a unique 1-form
$\alpha\in\Omega^{1}(\mathcal{G})$ by

$i_{E}\alpha=1,\quad\ker\alpha=T\mathcal{F}_{\pi_{\mathcal{G}}}.$

Conversely, given $\alpha$ these conditions determine $E$. By an argument
entirely similar to the last part of the proof of Proposition 4.4, one checks
that $\alpha$ is multiplicative if and only if $E$ is both left and right
invariant.

(i) $\Rightarrow$ (iii) Given a vector field $E\in\mathfrak{X}(\mathcal{G})$
transverse to $\mathcal{F}_{\pi_{\mathcal{G}}}$ which is both right and left
invariant, we define an extension $\omega\in\Omega^{2}(\mathcal{G})$ of the
leafwise symplectic form $\omega_{\mathcal{F}_{\pi_{\mathcal{G}}}}$ by
requiring

$i_{E}\omega=0.$

Again, one checks easily that $\omega$ is multiplicative.

(iii) $\Rightarrow$ (i) We assume now that $\mathcal{G}$ is proper and
$\dim\mathcal{G}=2\dim M+1$. We let $\omega\in\Omega^{2}(\mathcal{G})$ be a
multiplicative 2-form extending the leafwise symplectic form. By [2, Lemma
3.3], the multiplicativity of $\omega$ implies that at points $x\in M$ one has

$\displaystyle\ker\omega_{x}$
$\displaystyle=(\ker\omega_{x}\cap\ker\mathrm{d}_{x}\mathbf{s})\oplus(\ker\omega_{x}\cap
T_{x}M)$
$\displaystyle=(\ker\omega_{x}\cap\ker\mathrm{d}_{x}\mathbf{t})\oplus(\ker\omega_{x}\cap
T_{x}M).$

By Proposition 4.2 (iv), we have $\ker\omega_{x}\cap T_{x}M=\\{0\\}$, so we
conclude that:

$\mathfrak{k}:=(\ker\omega)|_{M}\subset\ker\mathrm{d}_{M}\mathbf{t}\cap\ker\mathrm{d}_{M}\mathbf{s}=\ker\rho_{A}.$

Since $\ker\omega$ is multiplicative, it is a distribution which is both right
and left invariant and, as a consequence, $\mathfrak{k}$ is invariant under
the adjoint action of $\mathcal{G}$. In addition, since
$\mathcal{F}_{\pi_{\mathcal{G}}}$ is co-orientable, there exists a non-
vanishing section $\tilde{e}\in{\Gamma}(\mathfrak{k})$. The corresponding
right-invariant vector field $\tilde{E}=\overrightarrow{\tilde{e}}$ may fail
to be left-invariant. To correct this we use properness of $\mathcal{G}$.
Since $\mathfrak{k}$ is $\operatorname{Ad}$-invariant, the function
$c\colon\mathcal{G}\to\mathbb{R}^{+}$ defined by

$\mathrm{d}L_{g}(\tilde{E}_{\mathbf{s}(g)})=c(g)\tilde{E}_{g},$

is multiplicative:

$c(gh)=c(g)c(h),\quad((g,h)\in\mathcal{G}^{(2)}).$

Since $\mathcal{G}$ is proper, there is function $f\colon M\to\mathbb{R}^{+}$
such that

$c(g)=f(\mathbf{t}(g))/f(\mathbf{s}(g)).$

The section $e:=f\tilde{e}\in{\Gamma}(\mathfrak{k})$ gives the desired vector
field $E:=\overrightarrow{e}=\overleftarrow{e}$. ∎

We now turn to condition (b) assuming than condition (a) holds. We show that,
up to a cover, a proper Poisson groupoid of type (C1) satisfying (a) is
homotopic to one satisfying also (b) through a homotopy that does not change
the Poisson structure on the base:

######  Theorem 4.6.

Let $(\mathcal{G},\pi_{\mathcal{G}})\rightrightarrows(M,\pi_{M})$ be an
orientable proper Poisson groupoid of type (C1) and assume that there exists a
multiplicative 2-form extending its leafwise symplectic form. If
$(\widetilde{\mathcal{G}},\widetilde{\pi_{\mathcal{G}}})$ is its universal
covering groupoid, then there is a path of
$\widetilde{\pi}_{\mathcal{G}}^{\,t}\in\mathfrak{X}^{2}(\widetilde{\mathcal{G}})$
of Poisson structures starting at
$\widetilde{\pi}_{\mathcal{G}}^{\,0}=\widetilde{\pi_{\mathcal{G}}}$ with the
following properties:

  1. (i)

each $\widetilde{\pi}_{\mathcal{G}}^{\,t}$ is multiplicative of corank 1;

  2. (ii)

the Poisson structure on $M$ induced by $\widetilde{\pi}_{\mathcal{G}}^{\,t}$
is $\pi_{M}$;

  3. (iii)

$\widetilde{\pi}_{\mathcal{G}}^{\,1}$ is associated with a multiplicative
cosymplectic structure.

The rest of this section will be dedicated to the proof of this theorem.

Since $\pi_{\mathcal{G}}$ is a Poisson groupoid of type (C1), $\rho_{A^{*}}$
is surjective and there is a short exact sequence

(10) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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90.69933pt\raise 5.37639pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.6375pt\hbox{$\scriptstyle{\rho_{A^{*}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 107.8778pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
107.8778pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
155.9021pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
155.9021pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

and a dual sequence

(11) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T^{*}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
59.18369pt\raise 5.84167pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.10278pt\hbox{$\scriptstyle{\rho^{*}_{A^{*}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 80.32431pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.32431pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
117.82431pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
117.82431pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
153.10211pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
153.10211pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Let $\omega\in\Omega^{2}(\mathcal{G})$ be the multiplicative 2-form extending
the leafwise symplectic form, and $\alpha\in\Omega^{1}(\mathcal{G})$ and
$E\in\mathfrak{X}(\mathcal{G})$ be the corresponding multiplicative 1-form and
bi-invariant vector field, given by Proposition 4.5. At the infinitesimal
level these give:

  1. (i)

an IM 2-form $(\mu,\tilde{\mu}):A\to T^{*}M\oplus\wedge^{2}T^{*}M$;

  2. (ii)

an IM 1-form $(\nu,\tilde{\nu}):A\to\mathbb{R}\oplus T^{*}M$;

  3. (iii)

a section $e\in{\Gamma}(\ker\rho_{A})$ which is central: $[e,X]_{A}=0$, for
any $X\in{\Gamma}(A)$.

This data make the second short exact sequence (11) a split exact sequence of
Lie algebroids. Indeed, one has $\mathfrak{k}=\mathbb{R}e$, $\mu(e)=0$ and
(11) becomes

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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50.75655pt\raise-3.82864pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
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where $i(x,\lambda)=\lambda e$. Because $e$ is central and
$\rho^{*}_{A^{*}}:T^{*}M\to A$ is a Lie algebroid map, we conclude that we
have a Lie algebroid isomorphism

$(\mu,\nu):A\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}M\oplus\mathbb{R},$

where the right-hand side has anchor and bracket

(12) $\displaystyle\rho_{A}(\alpha,f)$
$\displaystyle=\pi_{M}^{\sharp}(\alpha),$ (13)
$\displaystyle[(\alpha,f),(\beta,g)]_{A}$
$\displaystyle=([\alpha,\beta]_{\pi_{M}},\pi_{M}^{\sharp}(\alpha)(g)-\pi_{M}^{\sharp}(\beta)(f)).,$

Now let us look at the exact sequence (10). If $e^{*}$ is the section of
$\mathfrak{k}^{*}$ defined by $\langle e^{*},e\rangle=1$, then
$\mathfrak{k}^{*}=\mathbb{R}e^{*}$ and we have a vector bundle splitting

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\times\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{*}}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{A^{*}}}$$\scriptstyle{i^{*}}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}}$$\textstyle{0}$

The Lie algebroid structure of $A^{*}$ can be described in terms of the
central section $e$.

######  Lemma 4.7.

There is a Lie algebroid isomorphism

$(\rho_{A^{*}},i^{*}):A^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}TM\oplus\mathbb{R},$

where the right-hand side has anchor and Lie bracket given by

(14) $\displaystyle\rho_{A^{*}}(X,a)$ $\displaystyle=X,$ (15)
$\displaystyle[(X,a),(Y,b)]_{A^{*}}$
$\displaystyle=([X,Y],\nabla_{X}b-\nabla_{Y}a+\Omega(X,Y)),$

with $\nabla$ the flat connection on the trivial line bundle
$M\times\mathbb{R}\to M$ given by

$\nabla_{X}a=X(a)+a\gamma(X),\qquad\gamma(X):=\langle d_{\ast}e,X\wedge
e^{*}\rangle,$

and $\Omega\in\Omega^{2}(M)$ given by

$\Omega(X,Y):=\langle d_{\ast}e,X\wedge Y\rangle.$

Moreover, for all $\alpha\in\Omega^{1}(M)$ one has

$i_{\pi_{M}^{\sharp}(\alpha)}\gamma=0,\quad
i_{\pi_{M}^{\sharp}(\alpha)}\Omega=0.$

######  Remark 4.8.

Note that Jacobi identity for a Lie bracket of the form (15) is equivalent to
the connection $\nabla$ being flat, i.e., to $\gamma$ being a closed 1-form,
and the 2-form $\Omega$ being $\mathrm{d}^{\nabla}$-closed

$\mathrm{d}\gamma=0,\qquad\mathrm{d}^{\nabla}\Omega=0.$

One can also expressed $\nabla$ and $\Omega$ in terms of the 2nd components of
the IM forms associated with $(\omega,\alpha)$ as follows

$\gamma=\tilde{\nu}(e),\quad\Omega=\tilde{\mu}(e).$

Hence, $A^{*}$ becomes the trivial extension of $TM$ precisely when
$(\omega,\alpha)$ is cosymplectic, in agreement with Proposition 3.3.

###### Proof of the Lemma 4.7.

Under the identification
$(\rho_{A^{*}},i^{*}):A^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}TM\oplus\mathbb{R}$,
one has $\nabla_{X}e^{*}=[X,e^{*}]_{A^{*}}$ and
$\Omega(X,Y)e^{*}=[X,Y]_{A^{*}}$. Hence, using the definition of
$\mathrm{d}_{*}$, we find

$\displaystyle\gamma(X)$ $\displaystyle=\langle e,[X,e^{*}]_{A^{*}}\rangle$
$\displaystyle=\langle\mathrm{d}_{*}e,X\wedge e^{*}\rangle+X(\langle
e,e^{*}\rangle)-\rho_{A^{*}}(e^{*})(\langle
e,X\rangle)=\langle\mathrm{d}_{*}e,X\wedge e^{*}\rangle.$
$\displaystyle\Omega(X,Y)$ $\displaystyle=\langle e^{*},[X,Y]_{A^{*}}\rangle$
$\displaystyle=\langle\mathrm{d}_{*}e,X\wedge Y\rangle+X(\langle
e,Y\rangle)-Y(\langle e,X\rangle)=\langle\mathrm{d}_{*}e,X\wedge Y\rangle.$

On the other hand, by [14, Cor. 3.9], we have

${i_{\rho_{A}^{*}(\alpha)}\mathrm{d}_{*}e=[e,{\rho_{A^{*}}^{*}(\alpha)}]_{A}-\mathrm{d}_{*}(\langle\alpha,\rho_{A}(e)\rangle)-\rho^{*}_{A^{*}}(i_{\rho_{A}(e)}\mathrm{d}\alpha).}$

Observing that $\rho^{*}_{A}(\alpha)=-\pi_{M}^{\sharp}(\alpha)$ and
$\rho_{A}(e)=0$, the result follows. ∎

Next, we will see that $(A^{*},A)$ is a triangular Lie bialgebroid in the
sense of Mackenzie and Xu [14], i.e., there exists an element
$\Lambda\in\Gamma(\wedge^{2}A^{*})$ satisfying

$[\Lambda,\Lambda]_{A^{*}}=0,$

such that the anchor and Lie bracket on $A=(A^{*})^{*}$ are given by

(16) $\displaystyle\rho_{A}(\xi)$ $\displaystyle=\Lambda^{\sharp}(\xi),$ (17)
$\displaystyle[\xi_{1},\xi_{2}]_{A}$
$\displaystyle=[\xi_{1},\xi_{2}]_{\Lambda}:=\mathscr{L}_{\Lambda^{\sharp}(\xi_{1})}\xi_{2}-\mathscr{L}_{\Lambda^{\sharp}(\xi_{2})}\xi_{1}-\mathrm{d}_{A}(\Lambda(\xi_{1},\xi_{2})).$

In fact, we have the following general result which is an analogue for central
extensions of the fact that for any Poisson structure $(M,\pi_{M})$ the pair
$(TM,T^{*}M)$ is a triangular Lie bialgebroid.

######  Proposition 4.9.

Let $\gamma\in\Omega^{1}(M)$, $\Omega\in\Omega^{2}(M)$ and
$\pi_{M}\in\mathfrak{X}^{2}(M)$, and denote by $\nabla$ the connection on the
trivial line bundle given by $\gamma$. Assume that:

  1. (i)

$\gamma$ is closed: $\mathrm{d}\gamma=0$;

  2. (ii)

$\Omega$ is $\mathrm{d}^{\nabla}$-closed: $\mathrm{d}^{\nabla}\Omega=0$;

  3. (iii)

$\pi_{M}$ is Poisson: $[\pi_{M},\pi_{M}]=0$.

Then $A^{*}=TM\oplus\mathbb{R}$, with anchor (14) and Lie bracket (15), and
$A=T^{*}M\oplus\mathbb{R}$ with anchor (12) and Lie bracket (13) are both Lie
algebroids. If, additionally, one has

$i_{\pi_{M}^{\sharp}(\alpha)}\gamma=0,\quad
i_{\pi_{M}^{\sharp}(\alpha)}\Omega=0,\quad(\alpha\in\Omega^{1}(M)),$

then $(A^{*},A)$ is a triangular Lie bialgebroid for the section
$\Lambda\in\Gamma(\wedge^{2}A^{*})$ given by

$\Lambda((\alpha,f),(\beta,g)):=\pi_{M}(\alpha,\beta).$

In particular, in this case one has

$[\Lambda,\Lambda]_{A^{*}}=0.$

###### Proof of Proposition 4.9.

The fact that both $A$ and $A^{*}$ are Lie algebroids is standard. To check
that under the additional assumptions on $\gamma$ and $\Omega$ the pair
$(A^{*},A)$ is a triangular bialgebroid, notice that

$\rho_{A}(\alpha,f)=\pi_{M}^{\sharp}(\alpha)=\Lambda^{\sharp}(\alpha,f),\qquad((\alpha,f)\in\Gamma(A)),$

so (16) holds. On the other hand, we find

$\displaystyle\langle\mathscr{L}_{\Lambda^{\sharp}(\alpha,f)}(\beta,g),(Y,b)\rangle$
$\displaystyle=\langle\mathscr{L}_{(\pi_{M}^{\sharp}(\alpha),0)}(\beta,g),(Y,b)\rangle$
$\displaystyle=\pi_{M}^{\sharp}(\alpha)(\langle(\beta,g),(Y,b)\rangle)-\langle(\beta,g),[\pi_{M}^{\sharp}(\alpha,0),(Y,b)]_{A}\rangle$
$\displaystyle=\pi_{M}^{\sharp}(\alpha)(\langle\beta,Y\rangle)+\pi_{M}^{\sharp}(\alpha)(gb)+$
$\displaystyle\qquad\qquad-\langle\beta,[\pi_{M}^{\sharp}(\alpha),Y]\rangle-g\nabla_{\sharp(\alpha)}b-g\Omega(\pi_{M}^{\sharp}(\alpha),Y)$
$\displaystyle=\langle\mathscr{L}_{\pi_{M}^{\sharp}(\alpha)}\beta-g\,i_{\pi_{M}^{\sharp}(\alpha)}\Omega,Y\rangle+b\pi_{M}^{\sharp}(\alpha)(g)-gb(i_{\pi_{M}^{\sharp}(\alpha)}\gamma)$
$\displaystyle=\langle\mathscr{L}_{\pi_{M}^{\sharp}(\alpha)}\beta,Y\rangle+b\pi_{M}^{\sharp}(\alpha)(g)\rangle,$

where in the last line we have used the extra assumptions on on $\gamma$ and
$\Omega$. Using this we find that the Lie bracket on $A$ is indeed given by
(17), namely

$\displaystyle[(\alpha,f),(\beta,g)]_{\Lambda}$
$\displaystyle=\mathscr{L}_{\Lambda^{\sharp}(\alpha,f)}(\beta,g)-\mathscr{L}_{\Lambda^{\sharp}(\beta,g)}(\alpha,f)-\mathrm{d}_{A}(\Lambda((\alpha,f)(\beta,g)))$
$\displaystyle=(\mathscr{L}_{\pi_{M}^{\sharp}(\alpha)}\beta,\pi_{M}^{\sharp}(\alpha)(g))-(\mathscr{L}_{\pi_{M}^{\sharp}(\beta)}\alpha,\pi_{M}^{\sharp}(\beta)(f))-(\mathrm{d}(\pi_{M}(\alpha,\beta)),0)$
$\displaystyle=([\alpha,\beta]_{\pi_{M}},\pi_{M}^{\sharp}(\alpha)(g)-\pi_{M}^{\sharp}(\beta)(f))$
$\displaystyle=[(\alpha,f),(\beta,g)]_{A}.$

To complete the proof we show that $[\Lambda,\Lambda]_{A^{*}}=0$. For this we
observe that by the computation above we have

$\displaystyle[\Lambda,\Lambda]_{A^{*}}^{\sharp}((\alpha,f),(\beta,g))$
$\displaystyle=\Lambda^{\sharp}\big{(}[(\alpha,f),(\beta,g)]_{\Lambda}\big{)}-[\Lambda^{\sharp}(\alpha,f),\Lambda^{\sharp}(\beta,g)]$
$\displaystyle=\Lambda^{\sharp}\big{(}[(\alpha,f),(\beta,g)]_{A}\big{)}-[\Lambda^{\sharp}(\alpha,f),\Lambda^{\sharp}(\beta,g)]$
$\displaystyle=\pi^{\sharp}_{M}([\alpha,\beta]_{\pi_{M}})-[\pi^{\sharp}_{M}(\alpha),\pi^{\sharp}_{M}(\beta)]$
$\displaystyle=[\pi_{M},\pi_{M}]^{\sharp}(\alpha,\beta)=0,$

where the first identity is Lemma 2.2 in [12]. ∎

We can now complete the proof of Theorem 4.6. We perform two consecutive
homotopies of Lie bialgebroids as follows:

  1. (1)

Starting with the original Poisson groupoid, its Lie bialgebroid is a
triangular Lie bialgebroid $(A^{*},A)$ as in Proposition 4.9 with associated
data $(\gamma,\Omega,\pi_{M})$. We can rescale the 2-form $\Omega$, obtaining
a family of triangular Lie bialgebroids $(A^{*}_{t},A)$ with data
$(\gamma,(1-t)\Omega,\pi_{M})$, $t\in[0,1]$ (note that this triple still
satisfies for each $t$ all the conditions in the proposition).

  2. (2)

The previous homotopy gives at $t=1$ a Lie bialgebroid $(A^{*}_{1},A)$with
corresponding triple $(\gamma,\Omega=0,\pi_{M})$. Now we can rescale the
connection 1-form $\gamma$, obtaining a family of triangular Lie bialgebroids
$(A^{*}_{t},A)$ with data $((2-t)\gamma,0,\pi_{M})$, $t\in[1,2]$ (notice again
that this triple still satisfies for each $t$ all the conditions in the
proposition).

The result of these two consecutive deformations is a Lie bialgebroid
$(A_{2}^{*},A)$ whose associated triple has both $\gamma$ and $\Omega$ equal
to zero, i..e, it is of cosymplectic type (cf. Proposition 3.3).

Finally, we observe that in these deformations $(A^{*}_{t},A)$ the Lie bracket
of $A$ and the anchors of both Lie algebroids are fixed, and so is the
underling Poisson structure. Using the Mackenzie-Xu correspondence between Lie
bialgebroids and source 1-connected Lie groupoids [15], we conclude that the
Lie groupoid $\widetilde{\mathcal{G}}$ is fixed (since $A$ is fixed) and we
have a path of multiplicative Poisson structures
$\widetilde{\pi}_{\mathcal{G}}^{\,t}\in\mathfrak{X}^{2}(\widetilde{\mathcal{G}})$
as in the statement of Theorem 4.6. ∎

######  Remark 4.10.

A geometric way of thinking about the two deformations in the proof is as
follows. We start with a Poisson groupoid $(\mathcal{G},\pi_{\mathcal{G}})$
which can be described by a pair $(\omega,\alpha)$ consisting of a
multiplicative 2-form and a multiplicative 1-form, which fail to be closed
but, nonetheless, $\ker\alpha$ is an integrable distribuition and the
restriction of $\omega$ to the leaves of $\ker\alpha$ is symplectic. After
replacing $\mathcal{G}$ by $\widetilde{\mathcal{G}}$, we are able to construct
homotopies as follows:

  1. (1)

the first homotopy consists of a deformation $(\omega_{t},\alpha)$ where the
1-form $\alpha$ is fixed, the 2-form $\omega_{t}$ is multiplicative, at $t=0$
equals $\omega$ and at $t=1$ is closed;

  2. (2)

the second homotopy consists of a deformation $(\omega_{1},\alpha_{t})$ where
the 2-form $\omega_{1}$ is fixed, the 1-form $\alpha_{t}$ is multiplicative,
at $t=0$ equals $\alpha$ and at $t=1$ is closed;

Moreover, through out these deformations the 1-form always defines an integral
distribution and the restriction of the 2-form to its leaves is symplectic, so
they define a multiplicative Poisson structure $\pi_{t}$ on $\mathcal{G}$.

###  4.4. Proper over-symplectic groupoids of corank 1

Consider an oversymplectic groupoid $(\mathcal{G},\omega)$ of corank 1 for
which $\ker\omega$ is a simple foliation. Then we obtain an extension

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{G},\omega)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{(\Sigma,\Omega_{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $(\Sigma,\Omega_{\Sigma})$ is a symplectic groupoid and
$\omega=q^{*}\Omega_{\Sigma}$. If $\mathcal{G}$ is proper and orientable, then
$\mathcal{K}$ is the trivial $\mathbb{S}^{1}$-bundle of groups and we have a
$\mathbb{S}^{1}$-central extension

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M\times\mathbb{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
88.64719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
88.64719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
109.65015pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 126.49442pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
126.49442pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
163.71664pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
163.71664pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Notice that this also makes $\mathcal{G}$ into a $\mathbb{S}^{1}$-principal
bundle and we denote the generator of the $\mathbb{S}^{1}$-action by
$\partial_{\theta}\in\mathfrak{X}(\mathcal{G})$. A _multiplicative Ehresmann
connection_ for such an extension is given by a multiplicative 1-form
$\alpha\in\Omega^{1}(\mathcal{G})$ with the property that:

(18) $i_{\partial_{\theta}}\alpha=1.$

We refer the reader to [6, 9] for the theory of such connections and its
relation to ordinary principal bundle connections.

It is proved in [6] that a $\mathbb{S}^{1}$-central extension of a proper
groupoid always admits a multiplicative Ehresmann connection $\alpha$. Its
curvature 2-form is the multiplicative 2-form

$\Omega:=\mathrm{d}\alpha\in\Omega^{2}(\mathcal{G}).$

This form is closed and so by (18) it is basic. Hence, we have a
multiplicative, closed, 2-form $\underline{\Omega}\in\Omega^{2}(\Sigma)$ such
that:

$\Omega=q^{*}\underline{\Omega}.$

Denoting by $H^{\bullet}_{M}(\Sigma)$ the _multiplicative de Rham cohomology_
of $\Sigma\rightrightarrows M$, we have:

######  Proposition 4.11.

Given a $\mathbb{S}^{1}$-central extension of a proper Lie groupoid $\Sigma$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M\times\mathbb{S}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
88.64719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
88.64719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
109.65015pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 126.49442pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
126.49442pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
163.71664pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
163.71664pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

the class of the basic curvature of a multiplicative Ehresmann connection

$[\underline{\Omega}]\in H^{2}_{M}(\Sigma)$

is independent of the choice of connection.

###### Proof.

If $\alpha_{1}$ and $\alpha_{2}$ are two multiplicative Ehresmann connections
then their difference $\alpha_{1}-\alpha_{2}$ is a basic multiplicative
1-form, i.e., we have

$\alpha_{1}-\alpha_{2}=q^{*}\beta,\quad\text{with $\beta\in\Omega^{1}(\Sigma)$
multiplicative}.$

It follows that their basic curvature 2-forms differ by an exact
multiplicative form:

$\underline{\Omega}_{1}-\underline{\Omega}_{2}=q^{*}\mathrm{d}\beta.$

∎

We call the class $[\underline{\Omega}]\in H^{2}_{M}(\Sigma)$ the
_multiplicative Chern class_ of the extension. This class vanishes if and only
if the extension admits a flat multiplicative Ehresmann connection.

######  Theorem 4.12.

Let $(\mathcal{G},\omega)$ be a corank 1, orientable, proper oversymplectic
groupoid. If $\ker\omega$ is a simple foliation, then there exists
$\alpha\in\Omega^{1}(\mathcal{G})$ such that $(\mathcal{G},\omega,\alpha)$ is
a cosymplectic groupoid if and only if the corresponding
$\mathbb{S}^{1}$-central extension has vanishing multiplicative Chern class.

###### Proof.

If we can complete $\omega$ to a multiplcative cosymplectic stucture
$(\omega,\alpha)$ then obviously $\alpha$ is a flat multiplicative Ehresmann
connection.

For the reverse direction, assume that the multiplicative Chern class vanishes
so the extension admits a multiplicative Ehresmann connection $\alpha$. Let
$\dim\mathcal{G}=2n+1$, where $2n=\dim\Sigma$. From (18) and the fact that
$\omega=q^{*}\Omega_{\Sigma}$, with $\Omega_{\Sigma}$ non-degenerate, it
follows that $\alpha\wedge\omega^{n}$ is nowhere vanishing. Hence,
$(\omega,\alpha)$ is a multiplicative cosymplectic structure. ∎

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