# Poisson geometry around Poisson submanifolds

Rui Loja Fernandes  Department of Mathematics, University of Illinois at
Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801 USA
<EMAIL_ADDRESS>and  Ioan Mărcut  Radboud University Nijmegen, IMAPP,
6500 GL, Nijmegen, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

We construct a first order local model for Poisson manifolds around a large
class of Poisson submanifolds and we give conditions under which this model is
a local normal form. The resulting linearization theorem includes as special
cases all the known linearization theorems for fixed points and symplectic
leaves. The symplectic groupoid version of these results gives a solution to
the groupoid coisotropic embedding problem.

RLF was partially supported by NSF grants DMS-1710884, DMS-2003223 and
DMS-2303586, a Simons Fellowship in Mathematics and FCT/Portugal. IM was
partially supported by the NWO VENI grant 613.009.031.

###### Contents

  1. 1 Introduction
  2. 2 Jets of Poisson structures
  3. 3 Over-symplectic groupoids
  4. 4 Examples of over-symplectic groupoids
  5. 5 Groupoid coisotropic embeddings
  6. 6 Examples of groupoid local models
  7. 7 Local models of Poisson structures
  8. 8 Linearization and normal form theorems
  9. 9 Examples of first order jets and applications
  10. A The infinitesimal multiplicative Moser method
  11. B Linearization in Dirac geometry
  12. C Linearization around coregular submanifolds

##  1\. Introduction

A fundamental problem in Poisson geometry is to understand the behavior of a
Poisson manifold around a given symplectic leaf [26, 54, 56]. There is a well-
known first order local model for a Poisson structure around a symplectic leaf
due to Vorobjev [53]. This first order local model is the cornerstone for the
study of the Poisson geometry around the leaf. Using it one can study normal
forms around symplectic leaves [17], stability of symplectic leaves [14], etc.

Symplectic leaves is only one class among several interesting classes of
submanifolds in Poisson geometry. A local normal form around Poisson
transversals has also been found in [29], and one hopes to understand the
geometry around other important classes of submanifolds, such as Poisson
submanifolds, coisotropic submanifolds or Poisson-Dirac submanifolds. Unlike
the special case of symplectic leaves, in general, there is no first order
local model around a Poisson submanifold. In this paper we identify conditions
for the existence of such a model, extending the known model for symplectic
leaves, and we prove a normal formal theorem generalizing the known (smooth)
linearization results for fixed points [12, 26] and symplectic leaves [17].

In order to state our results, we introduce some notation and terminology. Let
us denote by $\mathfrak{X}^{\bullet}(M)$ the space of multivector fields on a
manifold $M$. We have the space of Poisson structures on $M$, denoted by

$\Pi(M):=\\{\pi\in\mathfrak{X}^{2}(M):[\pi,\pi]=0\\}.$

Given a submanifold $S\subset M$, which will always assume to be closed and
embedded, we denote by $\mathfrak{X}^{\bullet}_{S}(M)$ the subspace of
multivector fields $\vartheta\in\mathfrak{X}^{\bullet}(M)$ tangent to $S$,
i.e., such that $\vartheta|_{S}\in\mathfrak{X}^{\bullet}(S)$. Then we have the
space of Poisson structures in $M$ for which $S$ is a Poisson submanifold,
denoted by

$\Pi(M,S):=\\{\pi\in\mathfrak{X}^{2}_{S}(M):[\pi,\pi]=0\\}.$

Let $I_{S}\subset C^{\infty}(M)$ denote the vanishing ideal of $S$. The space
of first order jets of multivector fields tangent to $S$ can be identified
with (see Section 2) the quotient

$J^{1}_{S}\mathfrak{X}_{S}^{\bullet}(M):=\mathfrak{X}^{\bullet}_{S}(M)/I_{S}^{2}\cdot\mathfrak{X}^{\bullet}(M).$

The Schouten bracket descends to $J^{1}_{S}\mathfrak{X}_{S}^{\bullet}(M)$ and
we have the space of _first order jets of Poisson structures at $S$_, defined
as follows

$J^{1}_{S}\Pi(M,S):=\\{\tau\in
J^{1}_{S}\mathfrak{X}_{S}^{2}(M):[\tau,\tau]=0\\}.$

We will see that to specify an element $\tau\in J^{1}_{S}\Pi(M,S)$ amounts to
giving a Lie algebroid structure on the restricted cotangent bundle
$T^{*}_{S}M$ for which the natural projection $\mu_{S}:T^{*}_{S}M\to T^{*}S$
is an infinitesimal multiplicative (IM) closed 2-form. We do not distinguished
between these two descriptions, so we will often denote an element $\tau\in
J^{1}_{S}\Pi(M,S)$ as a pair $\tau=(T^{*}_{S}M,\mu_{S})$.

We have a map which to a Poisson structure tangent to $S$ associates its first
order jet along $S$, denoted by

$J^{1}_{S}:\Pi(M,S)\to J^{1}_{S}\Pi(M,S).$

This map, in general, is not surjective (see Example 2.2).

######  Definition 1.

Given a class $\mathscr{C}\subset J^{1}_{S}\Pi(M,S)$, we call a splitting
$\sigma:\mathscr{C}\to\Pi(M,S)$ of the map $J^{1}_{S}$ a first order local
model for the class $\mathscr{C}$.

We state now a simplified version of our first main result (see Section 7 for
more precise statements).

######  Theorem 2 (Existence of local models).

The class $\mathscr{C}\subset J^{1}_{S}\Pi(M,S)$ of first order jets of
Poisson structures satisfying the partially split condition admits a first
order local model $\sigma:\mathscr{C}\to\Pi(M,S)$.

The “partially split” condition in the statement of the theorem is a condition
on a jet $\tau\in J^{1}_{S}\Pi(M,S)$ expressed in terms of the IM
(infinitesimally multiplicative) geometry of the pair
$\tau=(T^{*}_{S}M,\mu_{S})$. It amounts to the existence of an IM Ehresmann
connection [27], and we will give the precise definition later. We use the IM
Ehresmann connection to construct the local model $\sigma(\tau)$ from Theorem
2 explicitly. Moreover, we show that, up to isomorphism, the result is
independent of this choice. For now, we point out that the class of jets
satisfying this condition is a large class that includes many examples of
interest. For example, we will see that it contains the following three
classes of first order jets $\tau=(T^{*}_{S}M,\mu_{S})$:

  * •

$T^{*}_{S}M$ is transitive;

  * •

$T^{*}_{S}M$ integrates to a proper groupoid;

  * •

$\ker\mu_{S}\subset T^{*}_{S}M$ is a bundle of semi-simple Lie algebras.

######  Definition 3.

Consider the map $\sigma:\mathscr{C}\to\Pi(M,S)$ from Theorem 2. Given a
Poisson structure $\pi\in\Pi(M,S)$ with $J^{1}_{S}\pi\in\mathscr{C}$, we will
call $\sigma(J^{1}_{S}\pi)\in\Pi(M,S)$ the linear approximation for $\pi$
around $S$. If $\pi$ and $\sigma(J^{1}_{S}\pi)$ are isomorphic around $S$ via
a Poisson diffeomorphism fixing $S$, then we will say that
$\sigma(J^{1}_{S}\pi)$ is a normal form for $\pi$ around $S$.

Our second main result is the following linearization theorem (see Theorem
8.7).

######  Theorem 4 (Normal form).

Let $\mathscr{C}_{0}\subset J^{1}_{S}\Pi(M,S)$ consist of first order jets
$\tau=(T^{*}_{S}M,\mu_{S})$ such that $T^{*}_{S}M$ is integrable by a compact,
Hausdorff Lie groupoid whose target fibers have trivial second de Rham
cohomology. Then there exists a first order local model
$\sigma:\mathscr{C}_{0}\to\Pi(M,S)$ and $\sigma(J^{1}_{S}\pi)$ is a normal
form for $\pi\in\Pi(M,S)$ around $S$ whenever
$J^{1}_{S}\pi\in\mathscr{C}_{0}$.

This result includes as special cases the most important smooth linearization
theorems around fixed points and symplectic leaves from the literature.

  1. -

For a fixed point $S=\\{x_{0}\\}$, the first order jet $(T^{*}_{S}M,\mu_{S})$
becomes just a Lie algebra $\mathfrak{g}$. The local model always exists and
is just the linear Poisson structure on $\mathfrak{g}^{*}$. If $\mathfrak{g}$
is compact, semi-simple, then any of its integrations is a compact Lie group
$G$ with $H^{2}(G)=\\{0\\}$, so satisfies the assumptions of the theorem and
one recovers Conn’s Linearization Theorem [12]. Actually, one can allow
$\mathfrak{g}$ to be compact with 1-dimensional center, resulting in the
slightly more general result due to Monnier and Zung [51].

  2. -

For a symplectic leaf $S$, the first order jet $(T^{*}_{S}M,\mu_{S})$ is a
transitive algebroid over a symplectic manifold. The local model always exists
and it coincides with the one constructed by Vorobjev [53]. Theorem 4 includes
the linearization theorem due to Crainic and Mărcut [17] and its
generalization due to Mărcut [46].

  3. -

For a general Poisson submanifold, the local model can be regarded as a way of
gluing together Vorobjev-type local models above the symplectic leaves of
$(S,\pi|_{S})$ (see Proposition 7.20). For Poisson submanifolds, the main
result to date was the Rigidity Theorem due to Mărcut [46]. The following
improved version of this result is a direct consequence of Theorem 4 (see
Corollary 8.10).

######  Corollary 5 (Neighborhood equivalence).

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. If
$T^{*}_{S}M$ is integrable by a compact Hausdorff Lie groupoid whose target
fibers have trivial second de Rham cohomology, then any Poisson structure
$\pi^{\prime}$ on $M$ with $J^{1}_{S}\pi=J^{1}_{S}\pi^{\prime}$ is locally
isomorphic to $\pi$ around $S$.

Our proof of Theorem 4, and hence of Corollary 5, will make use of the
Rigidity Theorem of [46]. However, Corollary 5 is an improvement, since the
Rigidity Theorem assumes that $(M,\pi)$ is _a priori_ integrable, while our
result assumes that only the restriction $T^{*}_{S}M$ is integrable. This is a
consequence of the existence of our local model. For example, we obtain the
following statement (see Corollary 8.9).

######  Corollary 6.

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. If
$T^{*}_{S}M$ is integrable by a compact Hausdorff Lie groupoid whose target
fibers have trivial second de Rham cohomology, then there is a neighborhood
$U\subset M$ of $S$ consisting of compact symplectic leaves and such that
$\pi|_{U}$ is integrable.

The proof of the Rigidity Theorem in [46] uses a version of the Nash-Moser
fast convergence method, similar to Conn’s proof [12]. We will also give a
direct proof of Theorem 4 under the additional assumption that the integration
is target 1-connected, by using techniques developed by Crainic and Fernandes
in [15]. This then leads to a geometric proof of Theorem 5 under this
additional assumption, which is independent from the results in [46].

The results above can be generalized in several directions. For example, we
will consider extensions to Dirac geometry, constructing first order local
models and proving normal forms results for Dirac manifolds around invariant
submanifolds (i.e., manifolds saturated by presymplectic leaves). Another
possible direction, is to construct local models and normal forms around other
types of submanifolds. We will introduce and discuss one such class, that we
call _coregular submanifolds_. These include as special cases transversals
submanifolds and invariant submanifolds. By passing to a saturation of a
coregular submanifold we are able to extend our results to such submanifolds.

Another goal of the paper is to give a symplectic groupoid version of the
previous results. Besides being interesting on its own right, it provides a
guide for both the construction of local models and the proofs of various
linearization results for Poisson structures. It can be understood by applying
a dictionary _global_ $\leftrightarrow$ _infinitesimal_ , whose first entries
are as follows:

_Global side_ | _Infinitesimal side_
---|---
symplectic groupoid $(\mathcal{G},\omega)\rightrightarrows M$ |  Poisson manifold $(M,\pi)$
saturated submanifold $S\subset M$ |  Poisson submanifold $S\subset M$
over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})\rightrightarrows S$ |  first order local data $\tau\in J^{1}_{S}\Pi(S,M)$
coisotropic embedding $(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$ |  A Poisson structure $\pi\in\Pi(M,S)$ with $\tau=J^{1}_{S}\pi$

The first two entries in this dictionary are well-known (see, e.g., [57, 13]),
so let us discuss briefly the other entries. Given a symplectic groupoid
$(\mathcal{G},\omega)\rightrightarrows M$ integrating a Poisson manifold
$(M,\pi)$, its restriction to a closed, embedded Poisson submanifold $S\subset
M$ yields a coisotropic groupoid embedding

$(\mathcal{G}|_{S},\omega|_{S})\hookrightarrow(\mathcal{G},\omega).$

The restriction $(\mathcal{G}|_{S},\omega|_{S})$ is an example of an _over-
symplectic groupoid_ in the sense of [7] (see Definition 3.2). The groupoid
coisotropic embedding problem asks:

  * •

Given an over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$, is there a
groupoid coisotropic embedding
$i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$ into some
symplectic groupoid?

  * •

Given two groupoid coisotropic embeddings
$i_{k}:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G}_{k},\omega_{k})$,
$k=1,2$, are they locally isomorphic around $\mathcal{G}_{S}$, i.e., is there
a local symplectic groupoid isomorphism
$\Phi:(\mathcal{G}_{1},\omega_{1})\to(\mathcal{G}_{2},\omega_{2})$ such that
$\Phi\circ i_{1}=i_{2}$?

We will obtain answers to these questions by applying a multiplicative version
of the classical Gotay coisotropic embedding theorem [36]. For that, given an
over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$, the kernel of
$\omega_{S}$ yields a vector bundle (VB) subgroupoid (in the sense of [43] or
[6]) of the tangent groupoid

$\ker\omega_{S}\subset T\mathcal{G}_{S}.$

######  Definition 7.

An over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ is called partially
split if there exists a VB subgroupoid $E\subset T\mathcal{G}_{S}$
complementary to the kernel of $\omega_{S}$,

$T\mathcal{G}_{S}=\ker\omega_{S}\oplus E.$

The distribution $E$ in this definition is an example of a _multiplicative
Ehresmann connection_ , introduced in [27]. The results of [27] imply, for
example, that over-symplectic groupoids that are either transitive or proper
are partially split. The partially split condition for first order jets of
Poisson structures, mentioned before, is the infinitesimal version of this
condition under the aforementioned dictionary.

Given a partially split over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ we construct a local model as follows. On the
one hand, the groupoid $\mathcal{G}_{S}\rightrightarrows S$ acts on the bundle
$\mathfrak{k}=(\ker\omega_{S})|_{S}\to S$, and on its dual, and builds the
action groupoid

$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}.$

On the other hand, we will see that a partial splitting can be encoded by a
_multiplicative connection 1-form_
$\alpha\in\Omega^{1}(\mathcal{G}_{S};\mathfrak{k})$ (see Section 5). This
allows us to define on the action groupoid the closed, multiplicative, 2-form

$\omega_{0}:=\operatorname{pr}_{\mathcal{G}_{S}}^{*}\omega_{S}+\mathrm{d}\langle\alpha,\cdot\rangle.$

Then $\omega_{0}$ is non-degenerate on an open subgroupoid
$\mathcal{G}_{0}\subset\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ which contains
$\mathcal{G}_{S}\simeq\mathcal{G}_{S}\ltimes 0_{S}$. So
$(\mathcal{G}_{0},\omega_{0})$ is a symplectic groupoid, and we obtain the
next result (see Proposition 5.7).

######  Theorem 8 (Existence of groupoid coisotropic embeddings).

Given a partially split over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ the zero section
$i_{0}:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G}_{0},\omega_{0})$
is a coisotropic embedding.

We will show that the partially split condition is in fact equivalent to the
existence of a multiplicative symplectic structure $\omega_{0}$ on a
neighborhood of $\mathcal{G}_{S}$ in the groupoid local model
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$ for
which $i_{0}$ is a coisotropic embedding. Moreover, we will show that any two
such symplectic structures are related by a groupoid isomorphism, defined
around $\mathcal{G}_{S}$.

The specific Lie groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$
appearing in the local model coincides with the local model around invariant
submanifolds for general Lie groupoids [22]. Namely, if
$(\mathcal{G}_{S},\omega_{S})$ is embedded as a coisotropic subgroupoid of the
symplectic groupoid $(\mathcal{G},\omega)$, then we have a canonical
isomorphism of groupoids

$\nu(\mathcal{G}_{S})\rightrightarrows\nu(S)\quad\simeq\quad\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*},$

where $\nu(\mathcal{G}_{S})\rightrightarrows\nu(S)$ is the normal bundle of
$\mathcal{G}_{S}$ in $\mathcal{G}$, which plays the role of the linear
approximation of $\mathcal{G}$ around $\mathcal{G}_{S}$.

In the proper case, we obtain a normal form result for groupoid coisotropic
embeddings, which is the global counterpart of Theorem 4 (see Corollary 5.21
and the comments following it).

######  Theorem 9 (Normal form).

Let $(\mathcal{G}_{S},\omega_{S})$ be a target-proper, Hausdorff, over-
symplectic groupoid with connected target fibers. Any groupoid coisotropic
embedding into a Hausdorff symplectic groupoid
$i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$ is
isomorphic around $\mathcal{G}_{S}$ to the coisotropic groupoid embedding
$i_{0}:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G}_{0},\omega_{0})$
from Theorem 8.

Finally, as a consequence of this result, we have a groupoid coisotropic
neighborhood equivalence theorem, which is the global counterpart of Corollary
5.

######  Corollary 10 (Neighborhood equivalence).

Let $(\mathcal{G}_{S},\omega_{S})$ be a target-proper, Hausdorff, over-
symplectic groupoid with connected target fibers. Any two groupoid coisotropic
embeddings into Hausdorff symplectic groupoids
$i_{k}:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G}_{k},\omega_{k})$,
$k=0,1$, are locally isomorphic around $\mathcal{G}_{S}$.

Organization of the paper. We start by introducing, in Section 2, the
appropriate first order data of a Poisson structure along a Poisson
submanifold using the language of jets. In Section 3, we recall some results
about over-symplectic groupoids and explain the relation to first order jets.
In Section 5, we work on the global side and consider the groupoid coisotropic
embedding problem, proving Theorems 8 and 9. We turn to the infinitesimal side
in Section 7, where we introduce the first order local model and prove Theorem
2. In Section 8 we discuss linearization and we prove the local normal form
(Theorem 4). Sections 4, 6 and 9 present many examples, are independent of the
rest of the text. In Appendix A, we discuss the Cartan calculus of
multiplicative forms and infinitesimal multiplicative forms, and we prove the
multiplicative and IM versions of Moser’s theorem. Appendices B and C contain
extensions of our results to the Dirac setting and to coregular submanifolds,
respectively.

_Acknowledgments_. We would like to thank Olivier Brahic, Alejandro Cabrera,
Marius Crainic, Matias del Hoyo, Pedro Frejlich, Stephane Geudens and Florian
Zeiser, for various comments and suggestions while this work was completed.

##  2\. Jets of Poisson structures

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. We
will always assume that $S$ is embedded and closed in $M$. Since we are
interested only in the local picture around $S$, the second assumption is not
very restrictive, as it always holds in a tubular neighborhood of $S$.

In describing the Poisson geometry of $(M,\pi)$ around $S$ the first order jet
of $\pi$ _along_ $S$ plays a crucial role. The space of multivector fields
tangent to $S$ is closed under the Schouten bracket, and will be denoted by

$\mathfrak{X}^{\bullet}_{S}(M):=\\{\vartheta\in\mathfrak{X}^{\bullet}(M):\vartheta|_{S}\in\Gamma(\wedge^{\bullet}TS)\\}.$

The space of Poisson structures tangent to $S$, denoted by

$\Pi(M,S):=\\{\pi\in\mathfrak{X}^{2}_{S}(M):[\pi,\pi]=0\\},$

consists precisely of the Poisson structures on $M$ for which $S$ is a Poisson
submanifold. We have the obvious restriction map to Poisson structures on $S$

$\Pi(M,S)\to\Pi(S),\quad\pi\mapsto\pi_{S}:=\pi|_{S}.$

On the other hand, let $I_{S}\subset C^{\infty}(M)$ denote the vanishing ideal
of $S$. For each $k\geq 1$, $I^{k}_{S}\cdot\mathfrak{X}^{\bullet}(M)$ is also
a Lie ideal in $\mathfrak{X}^{\bullet}_{S}(M)$. In particular the quotient

$J^{1}_{S}\mathfrak{X}^{\bullet}_{S}(M):=\mathfrak{X}^{\bullet}_{S}(M)/(I^{2}_{S}\cdot\mathfrak{X}^{\bullet}(M))$

inherits the structure of a graded Lie algebra.

######  Definition 2.1.

An element of the space

$J^{1}_{S}\Pi(M,S):=\big{\\{}\tau\in
J^{1}_{S}\mathfrak{X}^{2}_{S}(M)\,:\,[\tau,\tau]=0\big{\\}},$

is called a first order jet of a Poisson structure at $S$.

We will give in later a more practical definition (see Remark 2.4). For now, a
first order jet of a Poisson structure at $S$ is a class of a bivector field
tangent to $S$, modulo bivector fields that vanish to second order along $S$,
and which satisfies the equation of being a Poisson structure up to second
order

$J^{1}_{S}\Pi(M,S)=\frac{\\{\pi\in\mathfrak{X}^{2}_{S}(M):[\pi,\pi]\in
I^{2}_{S}\cdot\mathfrak{X}^{3}(M)\\}}{I^{2}_{S}\cdot\mathfrak{X}^{2}(M)}.$

Notice that we have a commutative diagram

$\textstyle{\Pi(M,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J^{1}_{S}}$$\textstyle{J^{1}_{S}\Pi(M,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi(S)}$
(2.1)

Although one can express any first order jet $\tau\in J^{1}_{S}\Pi(M,S)$ in
the form $\tau=J^{1}_{S}\pi$ for some bivector field
$\pi\in\mathfrak{X}^{2}_{S}(M)$, in general, one may not be able to choose
$\pi$ to be a Poisson structure. In other words, the map $J^{1}_{S}$ in the
previous diagram, in general, is not surjective. This is illustrated in the
next example.

######  Example 2.2 ([45]).

Let $M=\mathbb{R}^{3}$ and
$S=\\{(x,y,0):x,y\in\mathbb{R}\\}\subset\mathbb{R}^{3}$. The bivector field
$\pi\in\mathfrak{X}_{S}^{2}(M)$ given by

$\pi=z\partial_{x}\wedge\partial_{y}+xz\partial_{x}\wedge\partial_{z}$

satisfies

$[\pi,\pi]=2z^{2}\partial_{x}\wedge\partial_{y}\wedge\partial_{z}\in
I_{S}^{2}\cdot\mathfrak{X}^{3}(M).$

It follows that $\tau=J^{1}_{S}(\pi)\in J^{1}_{S}\Pi(M,S)$, with induced
Poisson structure $\pi_{S}=0$.

Any bivector field $\tilde{\pi}\in\mathfrak{X}^{2}_{S}(M)$ such that
$J^{1}_{S}(\tilde{\pi})=\tau$ takes the form

$\tilde{\pi}=(z+z^{2}f_{1})\partial_{x}\wedge\partial_{y}+(xz+z^{2}f_{2})\partial_{x}\wedge\partial_{z}+z^{2}f_{3}\partial_{y}\wedge\partial_{z},$

for some smooth functions $f_{1},f_{2},f_{3}\in C^{\infty}(\mathbb{R}^{3})$.
Next, one finds that

$[\tilde{\pi},\tilde{\pi}]=2z^{2}(1+O(1))\partial_{x}\wedge\partial_{y}\wedge\partial_{z}.$

Therefore, $[\tilde{\pi},\tilde{\pi}]\neq 0$, and so $\tau$ does not belong to
the image of $J^{1}_{S}$.

In the rest of this section we give an alternative description of first order
jets in terms of infinitesimal multiplicative (IM) closed 2-forms (see
Appendix A for a recollection of definitions and basic facts about IM forms).

In order to motivate this description, recall that a Poisson manifold
$(M,\pi)$ has an associated cotangent Lie algebroid $T^{*}M\Rightarrow M$ with
anchor $\pi^{\sharp}$ and Lie bracket

$\displaystyle[\alpha,\beta]_{\pi}:$
$\displaystyle=\mathscr{L}_{\pi^{\sharp}(\alpha)}\beta-\mathscr{L}_{\pi^{\sharp}(\beta)}\alpha-\mathrm{d}\pi(\alpha,\beta).$
(2.2)

It is a classical result, first observed in [21], that one can actually
characterize Poisson structures by such Lie algebroid structures. More
precisely, it is proved in [21] that there is a 1-to-1 correspondence between

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-8.98553pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{}\\\\\hbox{{\ignorespaces
Poisson
structures}\\\\\hbox{{\ignorespaces$\pi\in\mathfrak{X}^{2}(M)$}\\\\\hbox{{\ignorespaces\vspace*{0.1cm}
\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-15.9272pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroids $(T^{*}M,[\cdot,\cdot],\rho)$
with}\\\\\hbox{{\ignorespaces$\rho:T^{*}M\to TM$ skew-
symmetric}\\\\\hbox{{\ignorespaces$[\Omega^{1}_{\mathrm{cl}}(M),\Omega^{1}_{\mathrm{cl}}(M)]\subset\Omega^{1}_{\mathrm{cl}}(M)$\,}\crcr}}}}\right\\}}}}}}}}$

Using the language of IM forms, this classical result can be reformulated as
stating that there is a 1-to-1 correspondence (see [7] and also Corollary 2.9
below)

$\Pi(M)\equiv\left\\{\hbox{\hbox{\kern
0.0pt\raise-8.98553pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Poisson structures}\\\\\hbox{{\ignorespaces$\pi\in\mathfrak{X}^{2}(M)$
\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-9.3861pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroids $(T^{*}M,[\cdot,\cdot],\rho)$ with
}\\\\\hbox{{\ignorespaces$\mathrm{Id}:T^{*}M\to T^{*}M$ a closed IM
2-form\,}\crcr}}}}\right\\}}}}}$

Now the main result of this section is the following submanifold analogue of
this classical result. We denote by $\mu_{S}:T^{*}_{S}M\to T^{*}S$ the
restriction map $\alpha\mapsto\alpha|_{TS}$.

######  Proposition 2.3.

Given a submanifold $S\subset M$, there is a 1-to-1 correspondence

$J^{1}_{S}\Pi(M,S)=\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.56888pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode$\tau\in
J^{1}_{S}\mathfrak{X}^{2}_{S}(M)$ with
}\\\\\hbox{{\ignorespaces$[\tau,\tau]=0$
}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-13.83054pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroids $(T^{*}_{S}M,[\cdot,\cdot],\rho)$}\\\\\hbox{{\ignorespaces with
$\mu_{S}:T^{*}_{S}M\to T^{*}S$}\\\\\hbox{{\ignorespaces a closed IM
2-form\,}\crcr}}}}\right\\}.}}}}}}$

######  Remark 2.4 (Working definition of first order jet of a Poisson
structure).

This description of first order jets turns out to be much more practical and
we will adopt it as our working definition. Hence, we will often denote an
element $\tau\in J^{1}_{S}\Pi(M,S)$ as a pair $\tau=(T^{*}_{S}M,\mu_{S})$.

######  Example 2.5.

In Example 2.2, the first order jet $\tau\in J^{1}_{S}\Pi(M,S)$ corresponds to
the pair $(T^{*}_{S}M,\mu_{S})$, where $T^{*}_{S}M$ is the Lie algebra bundle
(i.e., $\rho=0$)

$\mathbb{R}^{2}\times\mathbb{R}^{3}\to\mathbb{R}^{2}$

with Lie bracket given on the basis $e_{1}=(\mathrm{d}x)|_{S}$,
$e_{2}=(\mathrm{d}y)|_{S}$, $e_{3}=(\mathrm{d}z)|_{S}$ by:

$[e_{1},e_{2}]=e_{3},\quad[e_{1},e_{3}]=xe_{3},\quad[e_{2},e_{3}]=0,$

and closed IM 2-form $\mu_{S}:T^{*}_{S}M\to T^{*}S$ given by:

$\mu_{S}(e_{1})=\mathrm{d}x,\quad\mu_{S}(e_{2})=\mathrm{d}y,\quad\mu_{S}(e_{3})=0.$
(2.3)

The rest of this section is dedicated to proving the correspondence above as
well as relating it with the usual theory of jet bundles. The reader who is
willing to accept this identification can skip the remainder of the section
altogether.

######  Remark 2.6.

There is yet another approach to first order jets along a submanifold. It
starts with the observation that if $S$ is a Poisson submanifold of $(M,\pi)$
then $I_{S}$ is a Poisson ideal of $(C^{\infty}(M),\\{\cdot,\cdot\\})$. The
quotient $C^{\infty}(M)/I_{S}$ is a Poisson algebra canonically isomorphic to
$(C^{\infty}(S),\\{\cdot,\cdot\\}_{S})$, and the higher order quotients
$C^{\infty}(M)/I_{S}^{k+1}$ can be interpreted as $k$th-order jets of $\pi$ at
$S$. In fact, given any manifold $M$ with a closed embedded submanifold
$S\subset M$, it is not hard to see that there is 1-to-1 correspondence
between Poisson algebra structures on $C^{\infty}(M)/I_{S}^{2}$ such that
$I_{S}/I_{S}^{2}$ is a Lie ideal and first order jets of a Poisson structure
at $S$, as defined above (see [44] for a proof, and [33] for further
applications).

###  2.1. Jets of multivector fields and derivations

We denote by $J^{n}E\to M$ the $n$-th order jet bundle of a fiber bundle $E\to
M$. A section $s\in\Gamma(J^{n}E)$ is holonomic if it is the $n$th-order jet
of a section $f\in\Gamma(E)$, i.e., if $s=J^{n}f$. We will be interested in
first order jets of the bundle $E=\wedge^{2}TM$. This is motivated by the
following observation. The Lie bracket (2.2) of the cotangent algebroid of a
Poisson manifold $(M,\pi)$ depends pointwise on the values and the first order
partial derivatives of $\pi$, i.e., on the values of the section
$J^{1}\pi\in\Gamma(J^{1}(\wedge^{2}TM))$. Moreover, we will see that the first
jet encodes precisely the data of the cotangent Lie algebroid. More generally,
we will see which (possibly non-holonomic) sections
$\tau\in\Gamma(J^{1}(\wedge^{2}TM))$ give rise to Lie algebroids on $T^{*}M$.
For this, we recall briefly the approach to Lie algebroid structures using
multiderivations of a vector bundle developed in [18].

A derivation of degree $d$ of a vector bundle $E\to M$ is an
$\mathbb{R}$-multilinear, alternating map

$D:\underbrace{\Gamma(E)\times\dots\times\Gamma(E)}_{d+1\text{-times}}\to\Gamma(E),$

for which there exists a bundle map $\sigma_{D}:\wedge^{d}E\to TM$, called the
symbol, such that

$D(s_{0},\dots,fs_{d})=fD(s_{0},\dots,s_{d})+(\mathscr{L}_{\sigma_{D}(s_{0},\dots,s_{d-1})}f)s_{d},$

for any $s_{0},\dots,s_{d}\in\Gamma(E)$ and $f\in C^{\infty}(M)$. The space of
derivations of degree $d$ is denoted by $\operatorname{Der}^{d}(E)$. It is the
space of sections of a vector bundle $\mathcal{D}^{d}E\to M$, which fits into
a short exact sequence of vector bundles

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\wedge^{d+1}E^{*}\otimes
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{d}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{D}}$$\textstyle{\wedge^{d}E^{*}\otimes
TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

There is a graded Lie bracket

$[~{},~{}]:\operatorname{Der}^{k}(E)\times\operatorname{Der}^{l}(E)\to\operatorname{Der}^{k+l}(E),$

and it is proved in [18] that specifying a Lie algebroid structure on $E\to M$
amounts to giving a degree 1 derivation $D\in\operatorname{Der}^{1}(E)$ such
that $[D,D]=0$. The derivation $D$ defines the Lie bracket by
$[\alpha,\beta]:=D(\alpha,\beta)$, while the anchor coincides with its symbol:
$\rho:=\sigma_{D}$.

In our case of interest $E=T^{*}M$, we can generalize formula (2.2) and obtain
a map from jets of multivector fields to multiderivations.

######  Proposition 2.7.

  1.   2. (a)

We have a map of graded Lie algebras

$i:(\mathfrak{X}^{d+1}(M),[~{},~{}])\to(\operatorname{Der}^{d}(T^{*}M),[~{},~{}]),$
$i(\vartheta)(\alpha_{0},\ldots,\alpha_{d})=\mathrm{d}(\vartheta(\alpha_{0},\dots,\alpha_{d}))+\sum_{i=0}^{d}(-1)^{d-i}i_{\vartheta^{\sharp}(\alpha_{0},\dots,\widehat{\alpha_{i}},\dots,\alpha_{d})}\mathrm{d}\alpha_{i},$

  3. (b)

The map in (a) induces an injective map of vector bundles

$\tilde{i}:J^{1}(\wedge^{d+1}TM)\hookrightarrow\mathcal{D}^{d}(T^{*}M),$

whose image consists of elements $D\in\mathcal{D}^{d}(T^{*}M)$ with skew-
symmetric symbol $\sigma_{D}\in\wedge^{d+1}TM\subset\wedge^{d}TM\otimes TM$.

###### Proof.

Note that $i(\vartheta)$ is determined by the relations

$i(\vartheta)(\mathrm{d}f_{0},\ldots,\mathrm{d}f_{d})=\mathrm{d}(\vartheta(f_{0},\ldots,f_{d})),\quad\sigma_{i(\vartheta)}(\mathrm{d}f_{1},\ldots,\mathrm{d}f_{d})=\vartheta^{\sharp}(\mathrm{d}f_{1},\ldots,\mathrm{d}f_{d}),$

for all $f_{0},\ldots,f_{d}\in C^{\infty}(M)$. Both the Schouten bracket and
the bracket on derivations can be described as graded commutators. Using
these, one obtains the relation $i([\vartheta,\tau])=[i(\vartheta),i(\tau)]$
on exact 1-forms. Using the formula for the symbol of the commutator of
derivations given in the proof of [18, Proposition 1], one obtains that
$\sigma_{[i(\vartheta),i(\tau)]}=\sigma_{i([\vartheta,\tau])}$. This implies
item $(a)$. Item $(b)$ is straightforward. ∎

######  Remark 2.8.

The derivations with skew-symmetric symbol do not form a subalgebra of all
derivations. Therefore, one cannot use the map from item (b) of the
proposition to obtain a bracket on first jets.

This leads to the following interpretation of jets as Lie algebroids:

######  Corollary 2.9.

There is a 1-to-1 correspondence

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.56888pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode$\tau\in\Gamma(J^{1}(\wedge^{2}TM))$
with }\\\\\hbox{{\ignorespaces$[\tilde{i}(\tau),\tilde{i}(\tau)]=0$
}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-14.52498pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroid structures on $T^{*}M$}\\\\\hbox{{\ignorespaces with skew-
symmetric anchor}\\\\\hbox{{\ignorespaces$\rho:T^{*}M\to
TM$}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}}}}}}}}$

Under this correspondence, holonomic sections (i.e., Poisson structures)
corresponds to Lie algebroid structures for which the identity map
$\mathrm{Id}:T^{*}M\to T^{*}M$ is a closed IM 2-form.

###### Proof.

The first part of the corollary follows from the proposition and the results
in [18]. For the second part, consider a jet $\tau$ satisfying
$[\tilde{i}(\tau),\tilde{i}(\tau)]=0$. The anchor of the corresponding Lie
algebroid bracket $[~{},~{}]_{\tau}$ is given by $\rho=\pi^{\sharp}$, where
$\pi\in\Gamma(\wedge^{2}TM)$ is the bivector field covered by $\tau$. By (A.1)
the identity map $\mathrm{Id}:T^{*}M\to T^{*}M$ is a closed IM form if and
only if

$[\alpha,\beta]_{\tau}=\mathscr{L}_{\pi^{\sharp}(\alpha)}\mu(\beta)-i_{\pi^{\sharp}(\beta)}\mathrm{d}\mu(\alpha),$

which is equivalent to $\tilde{i}(\tau)=i(\pi)=\tilde{i}(J^{1}\pi)$. Since
$\tilde{i}$ is injective, the conclusion follows. ∎

###  2.2. Jets along a submanifold

We can describe first order jets of Poisson structures as sections of a
certain jet bundle satisfying a PDE. Namely, the restriction
$J^{1}(\wedge^{\bullet}TM)|_{S}\to S$ has the subbundle of jets tangent to
$S$, denoted by

$J^{1}(\wedge^{\bullet}TM:\wedge^{\bullet}TS):=\\{\tau\in
J^{1}(\wedge^{\bullet}TM)|_{S}:\operatorname{pr}_{\wedge^{\bullet}T_{S}M}(\tau)\in\wedge^{\bullet}TS\\}.$

This subbundle has an induced projection

$\operatorname{pr}_{J^{1}(\wedge^{\bullet}TS)}:J^{1}(\wedge^{\bullet}TM:\wedge^{\bullet}TS)\longrightarrow
J^{1}(\wedge^{\bullet}TS).$

and we call a section
$\tau\in\Gamma(J^{1}(\wedge^{\bullet}TM:\wedge^{\bullet}TS))$ _holonomic_ if
its image under this projection is holonomic, i.e., if there
$\gamma\in\mathfrak{X}^{\bullet}(S)$ such that

$\operatorname{pr}_{J^{1}(\wedge^{\bullet}TS)}(\tau)=J^{1}\gamma.$

Since $S$ is closed and embedded, this is equivalent to the existence of a
multivector field $\vartheta\in\mathfrak{X}^{\bullet}_{S}(M)$ such that
$J^{1}\vartheta|_{S}=\tau$ (then $\vartheta$ is automatically tangent to $S$,
with $\vartheta|_{S}=\gamma$). Note that the section $\vartheta$ is determined
only up elements in $I^{2}(S)\cdot\mathfrak{X}^{\bullet}(M)$. Therefore we
have a canonical identification between the set of holonomic sections of
$J^{1}(\wedge^{\bullet}TM:\wedge^{\bullet}TS)$ and the quotient
$J^{1}_{S}\mathfrak{X}^{\bullet}_{S}(M)=\mathfrak{X}^{\bullet}_{S}(M)/(I^{2}_{S}\cdot\mathfrak{X}^{\bullet}(M))$,
which we will use with no further notice.

In general, a Poisson submanifold $S\subset(M,\pi)$ gives rise to a Lie
subalgebroid $T^{*}_{S}M\subset(T^{*}M,[~{},~{}]_{\pi},\pi^{\sharp})$ of the
tangent bundle. This Lie subalgebroid encodes precisely the first jet data of
$J^{1}_{S}\pi\in J^{1}_{S}\Pi(M,S)$. To make this precise, we consider the
following. First consider multiderivations of $T^{*}M$ with symbol tangent to
$S$,

$\operatorname{Der}^{d}_{S}(T^{*}M):=\\{D\in\operatorname{Der}^{d}(T^{*}M):\sigma_{D}(\alpha_{1},\ldots,\alpha_{d})\in
TS,\ \forall\,\alpha_{i}\in T^{*}_{S}M\\}.$

We have the following relative version of Proposition 2.7.

######  Proposition 2.10.

  1. (a)

$\operatorname{Der}^{\bullet}_{S}(T^{*}M)$ is closed under the commutator
bracket, and we have a commutative diagram of graded Lie algebra homomorphisms

$\textstyle{\mathfrak{X}^{\bullet+1}_{S}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{J^{1}_{S}}$$\textstyle{\operatorname{Der}^{\bullet}_{S}(T^{*}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{S}}$$\textstyle{J^{1}_{S}\mathfrak{X}^{\bullet+1}_{S}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{S}}$$\textstyle{\operatorname{Der}^{\bullet}(T^{*}_{S}M),}$

where $i$ is the map from Proposition 2.7 and where $r_{S}$ is the operation
of restricting a derivation of $T^{*}M$ tangent to $S$ to a derivation of
$T^{*}_{S}M$.

  2. (b)

The map $i_{S}$ induces an injective vector bundle map

$\tilde{i}_{S}:J^{1}(\wedge^{d+1}TM:\wedge^{d+1}TS)\hookrightarrow\mathcal{D}^{d}(T^{*}_{S}M),$

whose image consists of elements $D\in\mathcal{D}^{d}(T^{*}_{S}M)$whose symbol
is a multivector $\sigma_{D}\in\wedge^{d}TS\subset\wedge^{d-1}T_{S}M\otimes
TS$.

We have the following analogue of Corollary 2.9. The proof is entirely
similar, where one uses instead Proposition 2.10. In particular, this yields
the main result of this section.

######  Corollary 2.11.

There is a 1-to-1 correspondence

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.56888pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode$\tau\in\Gamma(J^{1}(\wedge^{2}TM:\wedge^{2}TS))$
}\\\\\hbox{{\ignorespaces with $[\tilde{i}_{S}(\tau),\tilde{i}_{S}(\tau)]=0$
}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-15.08054pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroid structures on $T^{*}_{S}M$}\\\\\hbox{{\ignorespaces with anchor
$\rho:T^{*}_{S}M\to TS$
s.t.:}\\\\\hbox{{\ignorespaces$\langle\mu_{S}(\alpha),\rho(\beta)\rangle=-\langle\mu_{S}(\beta),\rho(\alpha)\rangle$}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}.}}}}}}}$

Under this correspondence, holonomic sections (i.e., first order jets of
Poisson structures) correspond to Lie algebroids for which $\mu_{S}$ is a
closed IM 2-form

$J^{1}_{S}\Pi(M,S)=\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.56888pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode$\tau\in
J^{1}_{S}\mathfrak{X}^{2}_{S}(M)$ with
}\\\\\hbox{{\ignorespaces$[\tau,\tau]=0$
}\\\\\hbox{{\ignorespaces\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-13.83054pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
Lie algebroids $(T^{*}_{S}M,[\cdot,\cdot],\rho)$}\\\\\hbox{{\ignorespaces with
$\mu_{S}:T^{*}_{S}M\to T^{*}S$}\\\\\hbox{{\ignorespaces a closed IM
2-form\,}\crcr}}}}\right\\}.}}}}}}$

##  3\. Over-symplectic groupoids

In this section we start by recalling the notion of over-symplectic groupoid
and its properties. This is the type of groupoid one obtains by restricting a
symplectic groupoid to an invariant submanifold, i.e., a (complete) Poisson
submanifold of the underlying Poisson manifold. This is also the type of
groupoid for which one can hope to find coisotropic embeddings. We show in
this section that the infinitesimal data corresponding to an over-symplectic
groupoid $(\mathcal{G}_{S},\omega_{S})$ is precisely a first order jet of a
Poisson structure at $S$. Our conventions for Lie groupoids and Lie algebroids
are the same as in the companion paper [27] and in the monograph [16].

###  3.1. Over-symplectic groupoids

Unless otherwise stated, all groupoids are assumed to be Hausdorff and target
connected. The latter implies that the orbits of a groupoid coincide with the
orbits of its Lie algebroid. We will denote the restriction of a Lie groupoid
$\mathcal{G}\rightrightarrows M$ to a submanifold $S\subset M$ by

$\mathcal{G}|_{S}:=\mathbf{s}^{-1}(S)\cap\mathbf{t}^{-1}(S).$

This is always a groupoid $\mathcal{G}|_{S}\rightrightarrows S$, but it may
fail to be smooth, i.e., to be a Lie subgroupoid. However, if $S$ is saturated
by orbits of $\mathcal{G}$ then

$\mathcal{G}|_{S}=\mathbf{s}^{-1}(S)=\mathbf{t}^{-1}(S),$

and $\mathcal{G}|_{S}\rightrightarrows S$ is a target connected Lie
subgroupoid of $\mathcal{G}\rightrightarrows M$.

######  Proposition 3.1.

Let $(\mathcal{G},\omega)\rightrightarrows(M,\pi)$ be a symplectic groupoid
and $S\subset M$ a closed embedded Poisson submanifold. Then
$\mathcal{G}|_{S}\rightrightarrows S$ is a closed embedded coisotropic Lie
subgroupoid of $\mathcal{G}\rightrightarrows M$. The restriction of the
symplectic form $\omega_{S}:=\omega|_{\mathcal{G}|_{S}}$ is a closed
multiplicative form on $\mathcal{G}|_{S}$ satisfying

$\ker\omega_{S}\subset\ker\mathrm{d}\mathbf{s}_{S}\cap\ker\mathrm{d}\mathbf{t}_{S},$

where are $\mathbf{s}_{S}$ and $\mathbf{t}_{S}$ are the source and target maps
of $\mathcal{G}|_{S}\rightrightarrows S$.

###### Proof.

Since $S$ is a closed Poisson submanifold, it is a union of symplectic leaves,
and since $\mathcal{G}$ is target connected, it follows that $S$ is a
saturated submanifold for $\mathcal{G}$. Therefore,
$\mathcal{G}|_{S}\rightrightarrows S$ is a closed embedded Lie subgroupoid of
$\mathcal{G}\rightrightarrows M$. Moreover,
$\ker\mathrm{d}\mathbf{s}=\ker\mathrm{d}\mathbf{s}_{S}\subset
T\mathcal{G}|_{S}$ and
$\ker\mathrm{d}\mathbf{t}=\ker\mathrm{d}\mathbf{t}_{S}\subset
T\mathcal{G}|_{S}$. Since $\omega$ is multiplicative and closed, the
restriction $\omega_{S}$ is also multiplicative and closed.

Now, recall that if a Poisson map is transverse to a coisotropic submanifold,
then it pulls it back to a coisotropic submanifold. Since a Poisson
submanifold is also coisotropic, we can apply this fact to the target map
$\mathbf{t}:(\mathcal{G},\omega)\to(M,\pi)$ to conclude that
$\mathcal{G}|_{S}=\mathbf{t}^{-1}(S)\subset\mathcal{G}$ is a coisotropic
submanifold.

In order to prove the statement concerning the kernel of $\omega_{S}$ observe
that, since $\ker\mathrm{d}\mathbf{s},\ker\mathrm{d}\mathbf{t}\subset
T(\mathcal{G}|_{S})$, for $v\in\ker\omega_{S}$ we have that

$\omega_{S}(v,w)=\omega(v,w)=0,\quad\forall
w\in\ker\mathrm{d}\mathbf{s}+\ker\mathrm{d}\mathbf{t}.$

Hence, we obtain that

$\ker\omega_{S}\subset(\ker\mathrm{d}\mathbf{t})^{\perp_{\omega}}\cap(\ker\mathrm{d}\mathbf{s})^{\perp_{\omega}}=\ker\mathrm{d}\mathbf{s}\cap\ker\mathrm{d}\mathbf{t}=\ker\mathrm{d}\mathbf{s}_{S}\cap\ker\mathrm{d}\mathbf{t}_{S}.\qed$

The previous proposition shows that the restriction of a symplectic groupoid
to a Poisson submanifold is an _over-symplectic groupoid_ in the sense of [7].

######  Definition 3.2.

An over-symplectic groupoid is a groupoid $\mathcal{G}_{S}\rightrightarrows S$
together with a closed multiplicative 2-form
$\omega_{S}\in\Omega^{2}(\mathcal{G}_{S})$ such that

$\ker\omega_{S}\subset\ker\mathrm{d}\mathbf{s}_{S}\cap\ker\mathrm{d}\mathbf{t}_{S}.$

We will need the following properties of over-symplectic groupoids.

######  Proposition 3.3 (Proposition 4.5 from [7]).

Let $\mathcal{G}_{S}\rightrightarrows S$ be a Lie groupoid and
$\omega_{S}\in\Omega^{2}(\mathcal{G}_{S})$ a closed multiplicative 2-form. The
following are equivalent

  1. (a)

$(\mathcal{G}_{S},\omega_{S})$ is over-symplectic;

  2. (b)

$(\ker\mathrm{d}\mathbf{t}_{S})^{\perp_{\omega_{S}}}=\ker\mathrm{d}\mathbf{s}_{S}$;

  3. (c)

$\mathrm{rank}(\omega_{S}|_{x})=2\dim S$, for all $x\in S$;

  4. (d)

$\ker\omega_{S}|_{x}\subset\ker\mathrm{d}_{x}\mathbf{s}_{S}\cap\ker\mathrm{d}_{x}\mathbf{t}_{S}$,
for all $x\in S$;

  5. (e)

There exists a Poisson structure on $S$, such that
$\mathbf{t}_{S}:(\mathcal{G}_{S},\omega_{S})\to(S,\pi_{S})$ is a forward Dirac
map.

Moreover, if these hold, $\omega_{S}$ has constant rank and the orbits of
$\mathcal{G}_{S}$ coincide with the symplectic leaves of $(S,\pi_{S})$.

We have the following converse to Proposition 3.1.

######  Proposition 3.4.

Let $i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$ be a
coisotropic embedding of an over-symplectic groupoid in a symplectic groupoid.
The corresponding embedding of units $(S,\pi_{S})\hookrightarrow(M,\pi)$ is a
Poisson map. If the embedding is closed then $S$ is a saturated submanifold of
$M$ and $\mathcal{G}_{S}=\mathcal{G}|_{S}$.

###### Proof.

Since $\mathcal{G}_{S}$ is an embedded subgroupoid of $\mathcal{G}$, the
source/target fibers of $\mathcal{G}_{S}$ are embedded submanifolds of the
source/target fibers of $\mathcal{G}$

$\mathbf{s}_{S}^{-1}(x)\subset\mathbf{s}^{-1}(x),\quad\mathbf{t}_{S}^{-1}(x)\subset\mathbf{t}^{-1}(x),\quad\forall\,x\in
S.$

We claim that they have the same dimension, so these are open inclusions.
Indeed,

$\displaystyle\dim\mathbf{s}_{S}^{-1}(x)$
$\displaystyle=\dim\mathbf{t}_{S}^{-1}(x)=\dim\mathcal{G}_{S}-\dim S,$
$\displaystyle\dim\mathbf{s}^{-1}(x)$
$\displaystyle=\dim\mathbf{t}^{-1}(x)=\dim\mathcal{G}-\dim
M=\frac{1}{2}\dim\mathcal{G},$ $\displaystyle\dim\mathcal{G}$
$\displaystyle=\dim(\ker\omega_{S})+\dim\mathcal{G}_{S}=2(\dim\mathcal{G}_{S}-\dim
S),$

where in the second equation we used that $\mathcal{G}$ is a symplectic
groupoid and in the last equation we used Proposition 3.3 and the fact that
the embedding is coisotropic. These equations imply our claim

$\dim\mathbf{s}_{S}^{-1}(x)=\dim\mathbf{t}_{S}^{-1}(x)=\dim\mathbf{s}^{-1}(x)=\dim\mathbf{t}^{-1}(x).$

It follows that the orbits of $\mathcal{G}_{S}\rightrightarrows S$ are open
subsets of the orbits of $\mathcal{G}\rightrightarrows M$. Therefore, $S$ is a
Poisson submanifold with Poisson structure $\pi_{S}$. Thus we obtain the
following commutative diagram

$\textstyle{(\mathcal{G}_{S},\omega_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\mathbf{t}_{S}}$$\textstyle{(\mathcal{G},\omega)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{t}}$$\textstyle{(S,\pi_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{(M,\pi)}$

where the horizontal arrows are backward Dirac and $\mathbf{t}$ forward Dirac.
It follows by [30, Lemma 3] applied to $\mathcal{G}|_{S}$ that also
$\mathbf{t}_{S}$ is forward Dirac, i.e., the induced Poisson structure on the
base of the over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ coincides
with $\pi_{S}$.

Now if the embedding is closed, the source/target fibers of $\mathcal{G}_{S}$
are closed submanifolds of the source/target fibers of $\mathcal{G}$. Since
they are connected, they coincide

$\mathbf{s}_{S}^{-1}(x)=\mathbf{s}^{-1}(x),\quad\mathbf{t}_{S}^{-1}(x)=\mathbf{t}^{-1}(x),\quad\forall
x\in S.$

This implies that $S$ is saturated and that
$\mathcal{G}_{S}=\mathcal{G}|_{S}$. ∎

Note that for an over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ the
distribution $\ker\omega_{S}$ is integrable. If $\ker\omega_{S}$ is a simple
foliation then $\omega_{S}$ descends to a symplectic form
$\underline{\omega_{S}}$ on the leaf space $\mathcal{G}_{S}/\ker\omega_{S}$,
which inherits the structure of a symplectic groupoid

$(\mathcal{G}_{S}/\ker\omega_{S},\underline{\omega_{S}})\rightrightarrows S$

integrating the Poisson manifold $(S,\pi_{S})$. This explains the term “over-
symplectic”.

We can summarize this discussion by the following diagram

}$\textstyle{\left\\{\hbox{\hbox{\kern
0.0pt\raise-7.55556pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
over-symplectic
groupoids}\\\\\hbox{{\ignorespaces$(\mathcal{G}_{S},\omega_{S})\rightrightarrows
S$}\\\\}}}}\right\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}$
!V{ symplectic groupoids
$(\mathcal{G}_{S}/\ker\omega_{S},\underline{\omega_{S}})\rightrightarrows S$
}}$\textstyle{\left\\{\hbox{\hbox{\kern
0.0pt\raise-7.55556pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
symplectic
groupoids}\\\\\hbox{{\ignorespaces$(\mathcal{G}_{S}/\ker\omega_{S},\underline{\omega_{S}})\rightrightarrows
S$}\\\\}}}}\right\\}}}}$ !V We will see in Subsection 4.4 an example of an
over-symplectic groupoid which does not admit a coisotropic embedding into a
symplectic groupoid.

###  3.2. Over-symplectic groupoids and first order jets

We will show now that over-symplectic groupoids are the global objects
corresponding to first order jets of Poisson structures along Poisson
submanifolds. In other words, the previous diagram is the groupoid version of
the diagram (2.1). Let $(\mathcal{G}_{S},\omega_{S})\rightrightarrows S$ be an
over-symplectic groupoid. If $A_{S}\to S$ denotes its Lie algebroid, then
$\omega_{S}$ induces a closed IM 2-form $\mu:A_{S}\to T^{*}S$ (see Appendix
A). By Proposition 3.3, this bundle map is surjective. In fact, the following
holds.

######  Proposition 3.5.

If $\mathcal{G}_{S}\rightrightarrows S$ is a target 1-connected Lie groupoid
with Lie algebroid $A_{S}\to S$, there is a 1-to-1 correspondence

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.01332pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
over-symplectic
structures}\\\\\hbox{{\ignorespaces${\omega_{S}}\in\Omega^{2}(\mathcal{G}_{S})$
\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-9.48471pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
surjective IM closed 2-forms}\\\\\hbox{{\ignorespaces$\mu:A_{S}\to T^{*}S$
\,}\crcr}}}}\right\\}}}}}$

###### Proof.

According to our conventions, $A_{S}=\ker\mathrm{d}\mathbf{t}|_{S}$. Then, as
explained in Section A, the 1-to-1 correspondence between multiplicative forms
and IM forms associates to a _closed_ multiplicative form
${\omega_{S}}\in\Omega^{2}_{\mathrm{M}}(\mathcal{G}_{S})$ the _closed_ IM form
$\mu:A_{S}\to T^{*}S$ given by composing the maps

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.67001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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where $\operatorname{pr}$ is the pullback along the unit map
$S\hookrightarrow\mathcal{G}_{S}$. By Proposition 3.3,
${\omega_{S}}\in\Omega^{2}_{\mathrm{M}}(\mathcal{G}_{S})$ is an over-
symplectic structure if and only if

$(\ker\mathrm{d}\mathbf{t}|_{S})^{\perp_{{\omega_{S}}}}=\ker\mathrm{d}\mathbf{s}|_{S}.$
(3.1)

The inclusion
$\ker\mathrm{d}\mathbf{s}\subset(\ker\mathrm{d}\mathbf{t})^{\perp_{{\omega_{S}}}}$
holds for any multiplicative 2-form, and using the decomposition
$T_{S}\mathcal{G}_{S}=TS\oplus\ker\mathrm{d}\mathbf{s}|_{S}$ we see that (3.1)
is equivalent to

$(\ker\mathrm{d}\mathbf{t}|_{S})^{\perp_{{\omega_{S}}}}\cap TS=\\{0\\}.$

This condition is equivalent to $\mu$ being surjective. ∎

Given a pair $(A_{S},\mu)$, where $\mu:A_{S}\to T^{*}S$ a surjective closed IM
form, we can choose a manifold and an isomorphism of vector bundles
$A_{S}\simeq T^{*}_{S}M$ such that $\mu$ becomes the canonical projection
$\mu_{S}:T^{*}_{S}M\to T^{*}S$: for example, we can choose $M:=(\ker\mu)^{*}$.
So by Corollary 2.11, we see that the infinitesimal data $(A_{S},\mu)$
codifying an over-symplectic groupoid is the same thing as a first order jet
of a Poisson structure $(T^{*}_{S}M,\mu_{S})$. This justifies one of the
entries in the dictionary from the Introduction. Now if
$(\mathcal{G}_{S},\omega_{S})\rightrightarrows S$ is an over-symplectic
groupoid integrating the pair $(A_{S},\mu)$ and it admits a coisotropic
embedding $i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$
into a symplectic groupoid $(\mathcal{G},\omega)\rightrightarrows(M,\pi)$,
then we obtain

  * •

a Poisson embedding $(S,\pi_{S})\hookrightarrow(M,\pi)$;

  * •

an isomorphism $(A_{S},\mu)\simeq(T^{*}_{S}M,\mu_{S})$;

  * •

$(M,\pi)$ is a solution of the realization problem for
$\tau=(T^{*}_{S}M,\mu_{S})$:

$J^{1}_{S}\pi=\tau.$

Next we will look at some examples, before we discuss the problem of existence
of coisotropic groupoid embeddings.

##  4\. Examples of over-symplectic groupoids

We give a few examples of over-symplectic groupoids which will be useful
later.

###  4.1. Products

Let $(\Sigma,\omega_{\Sigma})\rightrightarrows S$ be a symplectic groupoid and
$G$ be a Lie group. We obtain an over-symplectic groupoid by forming the
product

$\mathcal{G}_{S}:=\Sigma\times
G,\quad\omega_{S}:=\operatorname{pr}_{\Sigma}^{*}\omega_{\Sigma}.$

###  4.2. Transitive over-symplectic groupoids

By Proposition 3.3, an over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})\rightrightarrows S$ is transitive if and only if
the Poisson structure induced on its base is non-degenerate:
$\pi_{S}=\omega^{-1}$, for a symplectic form $\omega\in\Omega^{2}(S)$.
Assuming this to be the case, and denoting the isotropy group at $x\in S$ by
$G=\mathcal{G}_{S,x}$, we obtain a principal $G$-bundle
$\mathbf{t}:\mathbf{s}^{-1}(x)\to S$ with a symplectic form $\omega$ on the
base. Applying again Proposition 3.3 (e), one sees that
$\mathbf{t}^{*}\omega=\omega_{S}|_{\mathbf{s}^{-1}(x)}$. Conversely, if we are
given a principal $G$-bundle $p:P\to S$ with a symplectic form
$\omega\in\Omega^{2}(S)$, then the gauge construction produces a transitive
groupoid

$\mathcal{G}_{S}=P\times_{G}P\rightrightarrows S,$

together with the closed, multiplicative 2-form

$\omega_{S}=(p\circ\operatorname{pr}_{2})^{*}\omega-(p\circ\operatorname{pr}_{1})^{*}\omega\in\Omega^{2}(\mathcal{G}_{S}).$

This makes $(\mathcal{G}_{S},\omega_{S})$ into a transitive over-symplectic
groupoid. These constructions are inverse to each other, up to isomorphism. So
transitive over-symplectic groupoids correspond to principal bundles with a
symplectic form on the base. At the infinitesimal level, if $A_{S}$ is the Lie
algebroid of $\mathcal{G}_{S}$, the corresponding closed IM-2 form is given by
$\mu=\omega^{\flat}\circ\rho$.

###  4.3. Over-symplectic bundles of Lie groups

At the other extreme, Proposition 3.3 implies that an over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})\rightrightarrows S$ is a bundle of Lie groups if
and only if the Poisson structure on its base vanishes: $\pi_{S}=0$. These
already provide non-trivial examples even in the case of a trivial group
bundle, i.e., if $\mathcal{G}_{S}$ is the product

$\mathcal{G}_{S}\simeq G\times S\to S,$

for a fixed Lie group $G$. Since any multiplicative 2-form on a Lie group is
trivial, a multiplicative 2-form $\omega_{S}\in\Omega^{2}(G\times S)$ has the
form

$\omega_{S}=\eta+c$

where (see [7, Section 6.4])

  * •

the mixed component $\eta_{(g,x)}\in T^{*}_{g}G\otimes T^{*}_{x}S$ satisfies
that, for any $w\in T_{x}S$, $i_{w}\eta$ is a multiplicative 1-form on $G$,
which is the same as a bi-invariant 1-form on $G$, or a $G$-invariant element
of $\mathfrak{g}^{*}$;

  * •

the horizontal component is a group homomorphism

$G\ni g\mapsto c(g)\in\Omega^{2}(S).$

Then, the condition that $\omega_{S}$ is closed translates to:

$\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}c(\exp(tv))=\mathrm{d}(i_{v}\eta)|_{\\{e\\}\times
S}\in\Omega^{2}(S),\quad v\in\mathfrak{g}.$

At the infinitesimal level we have the trivial Lie algebra bundle
$A_{S}=\mathfrak{g}\times S\to S$, and the induced closed IM 2-form is

$\mu:\mathfrak{g}\times S\to T^{*}S,\ \langle
w,\mu(v)\rangle=i_{w}i_{v}\eta|_{g=e},\quad v\in\mathfrak{g},\,w\in TS.$

The IM-condition says that $\mu:\mathfrak{g}\to\Omega^{1}(S)$ satisfies

$\mu([v,w])=0.$ (4.1)

Conversely, the reconstruction of $\omega_{S}$ from $\mu$ works as follows.
The component $\eta$ can be always defined

$\eta(v,w)=i_{w}\mu(\mathrm{d}L_{g^{-1}}v),\ \quad v\in T_{g}G,\,w\in T_{x}S.$

The component $c:G\to\Omega^{2}(S)$ is a group homomorphism integrating the
Lie algebra homomorphism $\mathrm{d}\circ\mu:\mathfrak{g}\to\Omega^{2}(S)$,
which exists e.g. if $G$ is simply connected. Moreover, if $\mu$ itself
integrates to a group homomorphism $\lambda:G\to\Omega^{1}(S)$, then

$\omega_{S}=\mathrm{d}\tilde{\lambda},\quad\textrm{where}\quad\tilde{\lambda}_{(g,x)}=\operatorname{pr}_{S}^{*}(\lambda(g)).$

When $G$ is a compact Lie group, the group homomorphism $c:G\to\Omega^{2}(S)$
must be trivial, and so its Lie algebra version
$\mathrm{d}\circ\mu:\mathfrak{g}\to\Omega^{2}(S)$ vanishes. The IM condition
(4.1) then means we can view $\mu:\mathfrak{g}\to\Omega^{1}(S)$ as an 1-form
$\mu\in\Omega^{1}(S,(\mathfrak{g}^{*})^{G})$. When this form is exact (e.g.,
if $H^{1}(S)=0$), there is $f\in C^{\infty}(S)\otimes(\mathfrak{g}^{*})^{G}$
such that

$\mu(v)=\langle\mathrm{d}f,v\rangle,\quad v\in\mathfrak{g}.$

The nondegeneracy of $\mu$ is equivalent to $f$ being an immersion, and we
obtain the following expression for $\omega_{S}$ in terms of $f$

$\omega_{S}((w_{1},v_{1}),(w_{2},v_{2}))=\langle\mathrm{d}f(v_{1}),\mathrm{d}L_{g}^{-1}(w_{2})\rangle-\langle\mathrm{d}f(v_{2}),\mathrm{d}L_{g}^{-1}(w_{1})\rangle,$

for all $(w_{i},v_{i})\in T_{g}G\oplus T_{x}S$.

###  4.4. Non-existence of a coisotropic embedding

The jet $\tau\in J^{1}_{S}\Pi(M,S)$ in Example 2.2 satisfies $\pi_{S}\equiv
0$. The corresponding pair $(T_{S}^{*}M,\mu_{S})$ has zero anchor, i.e., it is
a bundle of Lie algebras. This Lie algebroid integrates to a bundle of
1-connected Lie groups $\mathcal{G}_{S}\rightrightarrows S$ which is
Hausdorff. An explicit description is the following. We use coordinates
$(x,y)$ in $S=\mathbb{R}^{2}$ and $\mathbf{s}=\mathbf{t}:\mathcal{G}_{S}\to S$
is the trivial bundle
$\operatorname{pr}:\mathbb{R}^{5}=\mathbb{R}^{2}\times\mathbb{R}^{3}\to\mathbb{R}^{2}$,
where on the fiber we use coordinates $(u,v,w)$. The multiplication
$m:\mathcal{G}_{S}\times_{S}\mathcal{G}_{S}\to\mathcal{G}_{S}$ of the group
over $(x,y)\in S$ is given by

$(u_{1},v_{1},w_{1})\cdot(u_{2},v_{2},w_{2})=(u_{1}+u_{2},v_{1}+e^{xu_{1}}v_{2},w_{1}+w_{2}+\frac{e^{xu_{1}}-1}{x}v_{2}).$

Note that this is not a trivial group bundle. By Proposition 3.5, there is a
multiplicative closed 2-form $\omega_{S}$ on $\mathcal{G}_{S}$. The resulting
over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ does not admit a
coisotropic embedding, for otherwise $\tau$ would be in the image of
$J^{1}_{S}$, contradicting Example 2.2. It is not hard to show that the over-
symplectic structure is

$\omega_{S}=\mathrm{d}x\wedge\mathrm{d}u+\mathrm{d}y\wedge\mathrm{d}v+w\,\mathrm{d}x\wedge\mathrm{d}y-x\,\mathrm{d}y\wedge\mathrm{d}w,$

by proving that this form is closed, multiplicative and induces the IM form
(2.3).

###  4.5. Principal type

Let $(\Sigma,\omega_{\Sigma})\rightrightarrows S$ be a symplectic groupoid,
and let $\mathcal{H}\rightrightarrows S$ be a transitive Lie groupoid.
Consider their product over $S$

$\mathcal{G}_{S}:=\\{(\sigma,h)\in\Sigma\times\mathcal{H}\,:\,\mathbf{s}_{\Sigma}(\sigma)=\mathbf{s}_{\mathcal{H}}(h),\,\mathbf{t}_{\Sigma}(\sigma)=\mathbf{t}_{\mathcal{H}}(h)\\}\rightrightarrows
S,$

with groupoid structure such that the inclusion
$\mathcal{G}_{S}\hookrightarrow\Sigma\times\mathcal{H}$ is a groupoid map.
Smoothness of $\mathcal{G}_{S}$ follows because, as for any transitive Lie
groupoid, the anchor map
$(\mathbf{t}_{\mathcal{H}},\mathbf{s}_{\mathcal{H}}):\mathcal{H}\to S\times S$
is a submersion. Then $\mathcal{G}_{S}$ is an over-symplectic groupoid with
multiplicative 2-form

$\omega_{S}:=\operatorname{pr}_{\Sigma}^{*}(\omega_{\Sigma})\in\Omega^{2}(\mathcal{G}_{S}),$

which we will call the over-symplectic groupoid of principal type associated
to $(\Sigma,\omega)$ and $\mathcal{H}$. The terminology comes from the fact
that, if one fixes a base point $x_{0}\in S$, the transitive groupoid
$\mathcal{H}$ can be identified with the gauge groupoid $\mathcal{H}\simeq
P\times_{G}P$ associated to the principal bundle
$P:=\mathbf{s}_{\mathcal{H}}^{-1}(x_{0})$, with projection
$\mathbf{t}_{\mathcal{H}}:P\to S$ and structure group
$G:=\mathbf{s}_{\mathcal{H}}^{-1}(x_{0})\cap\mathbf{t}_{\mathcal{H}}^{-1}(x_{0})$.
The Poisson structure $\pi_{S}$ induced on $S$ by the over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ is the same as the one induced by the
symplectic groupoid $(\Sigma,\omega_{\Sigma})$. Some of the previous examples
fit into this setting

  * •

When $P$ is the trivial principal bundle, $P\simeq G\times S$, or equivalently
$\mathcal{H}=(S\times S)\times G$, then $\mathcal{G}_{S}$ is the product
$\mathcal{G}_{S}=\Sigma\times G$ of Example 4.1.

  * •

For a symplectic pair groupoid
$(\Sigma,\omega_{\Sigma})=(S,\omega_{S})\times(S,-\omega_{S})$, we have
$\mathcal{G}_{S}\simeq\mathcal{H}$ and we obtain the transitive over-
symplectic groupoid of Example 4.2.

  * •

When $\Sigma$ is the symplectic groupoid $(T^{*}S,\omega_{\mathrm{can}})$,
then we obtain the bundle of Lie groups
$\mathcal{G}_{S}=T^{*}S\times_{S}\mathcal{K}$, with
$\omega_{S}=\operatorname{pr}_{T^{*}S}^{*}\omega_{\mathrm{can}}$, where
$\mathcal{K}$ is the bundle of isotropy groups of $\mathcal{H}$, or
equivalently, the associated bundle $P\times_{G}G$, where $G$ acts on $G$ by
conjugation.

###  4.6. Over-symplectic groupoid inducing a non-integrable Poisson structure

Let $H$ be a compact semi-simple Lie group, and consider the symplectic
groupoid $(T^{*}H,\omega_{\mathrm{can}})\rightrightarrows\mathfrak{h}^{*}$
integrating the linear Poisson structure on $\mathfrak{h}^{*}$. Let
$S\subset\mathfrak{h}^{*}$ be the unit sphere around the origin with respect
to an $H$-invariant inner product. Then $S$ is a Poisson submanifold, and
therefore we obtain the over-symplectic groupoid

$\mathcal{G}_{S}:=T^{*}H|_{S}\simeq H\ltimes S\rightrightarrows
S,\quad\textrm{with}\quad\omega_{S}:=\omega_{\mathrm{can}}|_{\mathcal{G}_{S}}$

For $\mathfrak{h}\not\simeq\mathfrak{so}(3,\mathbb{R})$, the Poisson manifold
$(S,\pi_{S})$ is not integrable, and so $\mathcal{G}_{S}$ is not of principal
type.

##  5\. Groupoid coisotropic embeddings

In this section, given a over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ admitting a _multiplicative Ehresmann
connection_ we construct a local model: it is a symplectic groupoid
$(\mathcal{G}_{0},\omega_{0})$ where $(\mathcal{G}_{S},\omega_{S})$ sits as a
coisotropic subgroupoid. We will show that the local model is essentially
unique and we discuss a normal form theorem. We will see that the local model
can be thought of as a multiplicative (or groupoid) version of the classical
Gotay coisotropic embedding theorem.

###  5.1. The groupoid local model

Let $(\mathcal{G}_{S},\omega_{S})$ be an over-symplectic groupoid. We start by
observing that $\mathcal{G}_{S}$ acts on the vector bundle

$\mathfrak{k}:=(\ker\omega_{S})|_{S}\subset A_{S},$

where $A_{S}$ is the Lie algebroid of $\mathcal{G}_{S}$. By Proposition 3.3,
$\mathfrak{k}\subset\ker\rho_{S}$. Then, an arrow $g\in\mathcal{G}_{S}$ from
$x:=\mathbf{s}_{S}(g)$ to $y:=\mathbf{t}_{S}(g)$ acts by conjugation on the
isotropies:

$g:\ker\rho_{S}|_{x}\to\ker\rho_{S}|_{y},\quad
g\cdot\alpha=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}g\,\exp_{x}(t\alpha)\,g^{-1},$
(5.1)

where $\exp_{x}:\ker\rho_{S}|_{x}\to(\mathcal{G}_{S})_{x}$ is the Lie group
exponential. Since $\omega_{S}$ is multiplicative, this action preserves
$\mathfrak{k}\subset\ker\rho_{S}$. So $\mathfrak{k}$ is a
$\mathcal{G}_{S}$-representation. The dual representation allows us to define
the action groupoid

$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}.$

We would like to construct a multiplicative symplectic structure on this
groupoid (which is not always possible; see Subsection 4.4). It will be
convenient to introduce the following notion, which originates from the theory
of multiplicative connections developed in [27].

######  Definition 5.1.

A multiplicative connection 1-form on the over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ is a $\mathfrak{k}$-valued, multiplicative,
1-form $\alpha\in\Omega^{1}(\mathcal{G}_{S};\mathfrak{k})$ satisfying

$\alpha(\xi^{L})=\xi,$

where $\xi^{L}\in\mathfrak{X}(\mathcal{G}_{S})$ is the left-invariant vector
field determined by $\xi\in\Gamma(\mathfrak{k})$. An over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ which admits a multiplicative connection 1-form
is called partially split (in the next subsection, we will motivate the choice
of this terminology).

Now, given a multiplicative connection 1-form
$\alpha\in\Omega^{1}_{\mathrm{M}}(\mathcal{G}_{S},\mathfrak{k})$ we have an
associated (ordinary) 1-form
$\langle\alpha,\cdot\rangle\in\Omega^{1}(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})$,
defined by

$(v,z)\mapsto\langle\alpha_{g}(v),\eta\rangle,\quad\textrm{for}\quad(v,z)\in
T_{g}\mathcal{G}_{S}\times_{TS}T_{\eta}\mathfrak{k}^{*}.$

######  Proposition 5.2.

Given an over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ and a
multiplicative connection 1-form
$\alpha\in\Omega^{1}_{\mathrm{M}}(\mathcal{G}_{S},\mathfrak{k})$ the closed
2-form

$\omega_{0}=\operatorname{pr}^{*}_{\mathcal{G}_{S}}\omega_{S}+\mathrm{d}\langle\alpha,\cdot\rangle\in\Omega^{2}(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})$
(5.2)

is multiplicative. Moreover, there is an open groupoid neighborhood

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}_{0}\rightrightarrows
M_{0})\subset(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*})$

on which $\omega_{0}$ is non-degenerate and the first map is a coisotropic
embedding

$i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G}_{0},\omega_{0}).$

###### Proof.

Since $\alpha$ is multiplicative, a direct computation shows that
$\langle\alpha,\cdot\rangle$ is a multiplicative 1-form on
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$, and so is its differential. Since
the projection
$\operatorname{pr}_{\mathcal{G}_{S}}:\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\to\mathcal{G}_{S}$
is a groupoid morphism, it pulls back multiplicative forms to multiplicative
forms, and we conclude that $\omega_{0}$ is multiplicative. That $\omega_{0}$
is non-degenerate along the zero-section
$\mathcal{G}_{S}\hookrightarrow\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ and
that $i$ is a coisotropic embedding can be checked directly using the
condition $\alpha(\xi)=\xi^{L}$, and that $(\mathcal{G}_{S},\omega_{S})$ is
over-symplectic. Alternatively, this follow from Gotay’s coisotropic embedding
theorem – see the next section. Finally, we need to find an open set
$\mathcal{G}_{0}$ where $\omega_{0}$ is non-degenerate and which is a
subgroupoid. Consider the open subset $M_{0}\subset\mathfrak{k}^{*}$
consisting of points $x$ such that $\omega_{0}|_{x}$ is non-degenerate. By
Proposition 3.3 (c),
$\mathcal{G}:=(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})|_{M_{0}}$ is over-
symplectic, and by the last part of that proposition, $\omega_{0}$ is non-
degenerate on $\mathcal{G}$. Thus $(\mathcal{G},\omega_{0})$ is indeed a
symplectic groupoid. In order to comply with our convention that Lie groupoids
are target-connected, we let $\mathcal{G}_{0}$ be the open subgroupoid of
$\mathcal{G}$ consisting of the connected components of the identities. ∎

######  Definition 5.3.

Given a partially split over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$, with a multiplicative connection 1-form
$\alpha$, the symplectic groupoid $(\mathcal{G}_{0},\omega_{0})$ constructed
in the previous proposition is called a groupoid local model for
$(\mathcal{G}_{S},\omega_{S})$.

We will see in the next subsection alternative expressions for $\omega_{0}$
and we will show that for any two multiplicative connection 1-forms the
corresponding groupoid local models are isomorphic around $\mathcal{G}_{S}$.

######  Remark 5.4.

In general, the subgroupoid $\mathcal{G}_{0}\rightrightarrows M_{0}$ of
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$
cannot be taken to be a full groupoid neighborhood, i.e., of the form
$\mathcal{G}_{S}\ltimes M_{0}\rightrightarrows M_{0}$, for some
$\mathcal{G}_{S}$-invariant open set $M_{0}\subset\mathfrak{k}^{*}$ containing
the zero section. However, this is always possible if
$\mathcal{G}_{S}\rightrightarrows S$ is a proper groupoid (see [22, Lemma
5.3]).

###  5.2. The partially split condition

Over-symplectic groupoids are not always partially split. The natural setup
for this notion is the theory of _multiplicative Ehresmann connections_
developed in [27]. We now recall this notion briefly and give alternative
descriptions. In the case of over-symplectic groupoids, this will lead to
alternative descriptions of the symplectic form on the local model. Given a
Lie groupoid $\mathcal{G}_{S}\rightrightarrows S$ with Lie algebroid $A_{S}$,
we call a vector subbundle $\mathfrak{k}\subset A_{S}$ a bundle of ideals of
$\mathcal{G}_{S}$ if $\mathfrak{k}\subset\ker\rho_{S}$ and $\mathfrak{k}$ is
invariant under the $\mathcal{G}_{S}$-action by conjugation (5.1). The fact
that $\mathfrak{k}$ is invariant under conjugation implies that

$\alpha\in\Gamma(A_{S}),\
\xi\in\Gamma(\mathfrak{k})\quad\Longrightarrow\quad[\alpha,\xi]\in\Gamma(\mathfrak{k}).$

so this justifies the use of the term “bundle of ideals”.

We have the following equivalent descriptions of the notion of “connection”
for a bundle of ideals $\mathfrak{k}\subset A_{S}$ for $\mathcal{G}_{S}$ (for
details and terminology see [27, Section 2.3]).

  1. (i)

A multiplicative connection 1-form for $\mathfrak{k}$ is a multiplicative
1-form $\alpha\in\Omega^{1}_{\mathrm{M}}(\mathcal{G}_{S},\mathfrak{k})$
satisfying

$\alpha(\xi^{L})=\xi\quad(\xi\in\Gamma(\mathfrak{k})).$

By associating to a multiplicative connection 1-form $\alpha$ the distribution

$E:=\ker\alpha\subset T\mathcal{G}_{S},$

we obtain the following equivalent notion.

  2. (ii)

A multiplicative Ehresmann connection for $\mathfrak{k}$ is a wide subgroupoid
$E$ of $T\mathcal{G}_{S}\rightrightarrows TS$ such that

$T\mathcal{G}_{S}=E\oplus K,$

where $K\subset T\mathcal{G}_{S}$ is obtained by spreading $\mathfrak{k}$
using (left or right) translations

$K_{g}:=\mathrm{d}L_{g}(\mathfrak{k}_{\mathbf{s}(g)})=\mathrm{d}R_{g}(\mathfrak{k}_{\mathbf{t}(g)})\subset\ker\mathrm{d}\mathbf{s}\cap\ker\mathrm{d}\mathbf{t}.$

Note that $K\rightrightarrows 0_{S}$ is a subgroupoid of the tangent groupoid
$T\mathcal{G}_{S}\rightrightarrows TS$, which is canonically isomorphic to the
semi-direct product $\mathcal{G}_{S}\times_{S}\mathfrak{k}\rightrightarrows
S$, via

$\mathcal{G}_{S}\times_{S}\mathfrak{k}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}K,\quad(g,v)\mapsto\mathrm{d}L_{g}(v).$

Then a multiplicative Ehresmann connection gives a splitting of the inclusion
of VB groupoids

---
$\textstyle{{}\ignorespaces
K\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\mathcal{G}_{S}\times_{S}\mathfrak{k}\simeq$
---
$\textstyle{T\mathcal{G}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{S}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

Under the duality operation in the category of VB groupoids [6], we obtain the
VB groupoids $T^{*}\mathcal{G}_{S}\rightrightarrows A_{S}^{*}$ and
$K^{*}\rightrightarrows\mathfrak{k}^{*}$, and the dual VB groupoid map

---
$\textstyle{{}\ignorespaces
K^{*}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\simeq$
---
$\textstyle{T^{*}\mathcal{G}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

The groupoid $K^{*}\rightrightarrows\mathfrak{k}^{*}$ is isomorphic to the
action groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$ of
the dual action of $\mathcal{G}_{S}$ on $\mathfrak{k}^{*}$. This leads to the
following equivalent notion.

  3. (iii)

A partial splitting is a VB groupoid morphism $\Theta:K^{*}\to
T^{*}\mathcal{G}_{S}$ that splits the projection $p:T^{*}\mathcal{G}_{S}\to
K^{*}$

$p\circ\,\Theta=\mathrm{Id}.$

The corresponding Ehresmann connection is given by

$E=(\operatorname{Im}\Theta)^{0}.$

Another equivalent notion is the following.

  4. (iv)

A linear, closed, multiplicative, 2-form
$\omega^{\mathrm{lin}}\in\Omega^{2}_{\mathrm{M}}(K^{*})$ that restricts to the
canonical symplectic form on $\mathfrak{k}\times_{S}\mathfrak{k}^{*}\subset
T_{S}K^{*}$

$\omega^{\mathrm{lin}}((v_{1},\xi_{1}),(v_{2},\xi_{2}))=\xi_{2}(v_{1})-\xi_{1}(v_{2}),\quad\text{if}\quad(v_{k},\xi_{k})\in\mathfrak{k}\times_{S}\mathfrak{k}^{*},$

where linear means that

$m_{t}^{*}(\omega^{\mathrm{lin}})=t\omega^{\mathrm{lin}},\quad(t>0),$

where $m_{t}:K^{*}\to K^{*}$ is fiberwise multiplication by $t>0$. Using the
partial splitting or the connection 1-form, the linear form is given by

$\omega^{\mathrm{lin}}=\Theta^{*}\omega_{\mathrm{can}}=\mathrm{d}\langle\alpha,\cdot\rangle.$
(5.3)

A bundle of ideals admitting such a structure is called partially split.

######  Remark 5.5.

There are several reasons for our choice of the terminology “partial
splitting”. The main reason, as explained in the companion paper [27], is the
following. Multiplicative Ehresmann connections appeared first as Ehresmann
connections for a surjective, submersive, groupoid morphism
$\Phi:\mathcal{G}\to\mathcal{H}$ covering the identity. Such a morphism
determines a short exact sequence of groupoids:

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3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The existence of an Ehresmann connection for $\Phi$, which is equivalent to
the bundle of ideals $\mathfrak{k}=\textrm{Lie}(\mathcal{K})$ being partially
split, is _weaker_ than the existence of a splitting for this short exact
sequence. Moreover, a partial splitting can be thought of as an infinitesimal
version of a splitting of this sequence. We refer to [27] for more details.
The designation “partial splitting” also allows us to distinguish these from
other splittings that appear in the sequel.

Let us return to our discussion of over-symplectic groupoids and their local
models. For an over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$, the
relevant bundle of ideals is $\mathfrak{k}=(\ker\omega_{S})|_{S}$ and we have
$K=\ker\omega_{S}$. In particular, note that

$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\simeq(\ker\omega_{S})^{*}.$

The symplectic form $\omega_{0}$ on the local model $\mathcal{G}_{0}$ can be
written as follows:

  * •

using a multiplicative connection 1-form
$\alpha\in\Omega^{1}_{\mathrm{M}}(\mathcal{G}_{S},\mathfrak{k})$ as in the
definition

$\omega_{0}=\operatorname{pr}_{\mathcal{G}_{S}}^{*}\omega_{S}+\mathrm{d}\langle\alpha,\cdot\rangle;$

  * •

using a partial splitting $\Theta:\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\to
T^{*}\mathcal{G}_{S}$

$\omega_{0}=\operatorname{pr}_{\mathcal{G}_{S}}^{*}\omega_{S}+\Theta^{*}\omega_{\mathrm{can}};$

  * •

using a linear, closed, multiplicative, 2-forms
$\omega^{\mathrm{lin}}\in\Omega^{2}_{\mathrm{M}}(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})$

$\omega_{0}=\operatorname{pr}_{\mathcal{G}_{S}}^{*}\omega_{S}+\omega^{\mathrm{lin}}.$

The last expression shows that $\omega_{0}$ is the sum of a constant
multiplicative form $\operatorname{pr}^{*}\omega_{S}$ and a linear
multiplicative form $\omega^{\mathrm{lin}}$. We can also explain the origins
of the local model. It is a multiplicative version of the well-known Gotay’s
coisotropic embedding [36]. In the classical case, starting with a pair
$(C,\omega_{C})$, where $\omega_{C}$ is a closed 2-form of constant rank, one
constructs a symplectic manifold $(X_{0},\omega_{0})$ and a coisotropic
embedding $(C,\omega_{C})\hookrightarrow(X_{0},\omega_{0})$ as follows. A
choice of complementary subbundle

$TC=\ker\omega_{C}\oplus E,$

determines an embedding

$\Theta:(\ker\omega_{C})^{*}\hookrightarrow T^{*}C.$

The form

$\omega_{0}:=\operatorname{pr}_{C}^{*}\omega_{C}+\Theta^{*}\omega_{\mathrm{can}}\in\Omega^{2}((\ker\omega_{C})^{*}),$

is non-degenerate along the zero section $C\simeq 0_{C}$ and hence defines a
symplectic form in a neighborhood $X_{0}\subset(\ker\omega_{C})^{*}$ of $C$.
Moreover, the zero section $C$ becomes a coisotropic submanifold of
$(X_{0},\omega_{0})$ with induced 2-form $\omega_{C}$. Gotay’s result says
that any coisotropic embedding of $(C,\omega_{C})$ in a symplectic manifold is
locally isomorphic to the local model $(X_{0},\omega_{0})$. The local model we
constructed is the multiplicative version of Gotay’s model. The construction
in Proposition 5.2 can be viewed in a slightly different way. For any over-
symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$, we have an associated
symplectic groupoid

$(T^{*}\mathcal{G}_{S},\omega_{\mathrm{can}}+\operatorname{pr}^{*}\omega_{S})\rightrightarrows
A_{S}^{*}.$

######  Corollary 5.6.

For any splitting $\Theta$, the local model is a symplectic subgroupoid

$\Theta:(\mathcal{G}_{0},\omega_{0})\hookrightarrow(T^{*}\mathcal{G}_{S},\omega_{\mathrm{can}}+\operatorname{pr}^{*}\omega_{S}).$

In Subsection 7.4, we will discuss this construction at the level of Lie
algebroids, and then we will describe explicitly the corresponding Poisson
structure on $A_{S}^{*}$.

###  5.3. Existence of multiplicative Ehresmann connections

The following result gives some geometric meaning to the existence of a
partially split condition: it shows that a multiplicative Ehresmann connection
exists if and only if the groupoid
$(\ker\omega_{S})^{*}=\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ can be made into
a symplectic groupoid around $\mathcal{G}_{S}$ such that the symplectic form
restricts to $\omega_{S}$. Therefore the partially split condition is optimal
for the existence of a groupoid local model on
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$.

######  Proposition 5.7.

An over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ is partially split
if and only if there exists a groupoid neighborhood

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}_{0}\rightrightarrows
M_{0})\subset(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}),$

together with multiplicative symplectic structures
$\omega_{0}\in\Omega^{2}(\mathcal{G}_{0})$ whose pullback along the zero
section
$i:\mathcal{G}_{S}\hookrightarrow\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ is
$\omega_{S}$,

$i^{*}\omega_{0}=\omega_{S}.$

###### Proof.

Proposition 5.2 gives one implication. For the converse, let
$(\mathcal{G}_{0},\omega_{0})$ be as in the statement. That $\mathcal{G}_{S}$
is coisotropic follows from Proposition 3.1. The restriction of the tangent
groupoid $T(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})\rightrightarrows
T\mathfrak{k}^{*}$ to the invariant submanifold $T_{S}\mathfrak{k}^{*}\simeq
TS\oplus\mathfrak{k}^{*}$ is the action groupoid:
$T\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows
TS\oplus\mathfrak{k}^{*}$. This has the subgroupoids

$K:=(T\mathcal{G}_{S})^{\perp_{\omega_{0}}}\rightrightarrows S,\quad
T\mathcal{G}_{S}\rightrightarrows
TS\quad\textrm{and}\quad\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*},$

where the last is an embedded normal bundle for $\mathcal{G}_{S}$. We obtain
that

$E:=(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})^{\perp_{\omega_{0}}}\cap
T\mathcal{G}_{S}$

is a vector bundle complementary to $K$ in $T\mathcal{G}_{S}$.
Multiplicativity of $\omega_{0}$ implies that $E$ is subgroupoid. Hence
$(\mathcal{G}_{S},\omega_{S})$ is partially split. ∎

######  Example 5.8.

Consider the over-symplectic groupoid $(\mathcal{G}_{S},\omega_{S})$ from
Subsection 4.4. The action of $\mathcal{G}_{S}$ on the line bundle

$\mathfrak{k}=(\ker\omega_{S})|_{S}=\langle
x\partial_{v}+\partial_{w}\rangle|_{S}$

is given on a section $s\in\Gamma(\mathfrak{k})$ by

$(x,y,u,v,w)\cdot s{(x,y)}=e^{xu}s{(x,y)}.$

Therefore, the groupoid
$(\ker\omega_{S})^{*}=\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ has one
dimensional orbits, and so it cannot be a symplectic groupoid. The previous
proposition implies that $(\mathcal{G}_{S},\omega_{S})$ is not partially
split. This also follows from the fact that the corresponding first jet cannot
be realized as a Poisson structure.

Another type of obstructions to the partially split condition, different from
the previous example, derive from the general theory in [27] of multiplicative
connections. In particular, the results from [27, Section 2.3] when applied to
over-symplectic groupoids, give the following

######  Proposition 5.9.

Let $(\mathcal{G}_{S},\omega_{S})$ be partially split over-symplectic groupoid
and let $\mathfrak{k}=(\ker\omega_{S})|_{S}$. A choice of partial splitting
$\Theta:(\ker\omega_{S})^{*}\to T^{*}\mathcal{G}_{S}$ yields

  1. (i)

A linear connection $\nabla$ on $\mathfrak{k}$ which preserves the bracket

$\nabla_{X}[\xi,\eta]_{\mathfrak{k}}=[\nabla_{X}\xi,\eta]_{\mathfrak{k}}+[\xi,\nabla_{X}\eta]_{\mathfrak{k}},\quad
X\in\mathfrak{X}(S),\xi,\eta\in\Gamma(\mathfrak{k});$

In particular, $\mathfrak{k}$ is a locally trivial bundle of Lie algebras.

  2. (ii)

For each $x\in S$, a splitting of the isotropy Lie algebra
$\mathfrak{g}_{x}=\ker\rho_{S}|_{x}$ into a direct sum of ideals

$\mathfrak{g}_{x}=\mathfrak{k}_{x}\oplus\mathfrak{l}_{x}.$

There are also some classes of over-symplectic groupoids that are always
partially split. For example, we have the following fundamental result from
[27].

######  Theorem 5.10.

Every bundle of ideals for a proper Lie groupoid is partially split.

######  Corollary 5.11.

Every proper over-symplectic groupoid is partially split.

Other classes of partially split over-symplectic groupoids are discussed in
the example section.

######  Remark 5.12.

Bundle of ideals arising from locally trivial groupoid fibrations have been
studied in [42] in the context of the theory of non-abelian gerbes. There the
authors have defined a cohomology which contains an obstruction class for the
existence of multiplicative Ehresmann connections.

###  5.4. Uniqueness of the local model

We will extend the Moser method to the groupoid setting. Consider a linear
action groupoid

$\mathcal{G}_{S}\ltimes V\rightrightarrows V,$

where $V\to S$ is a representation of $\mathcal{G}_{S}\rightrightarrows S$.

######  Proposition 5.13.

Let $(\mathcal{G}_{S},\omega_{S})$ be an over-symplectic groupoid. Consider
two open groupoid neighborhoods

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}_{k}\rightrightarrows
M_{k})\subset(\mathcal{G}_{S}\ltimes V\rightrightarrows V),\quad k=0,1,$

together with multiplicative symplectic structures
$\omega_{k}\in\Omega^{2}(\mathcal{G}_{k})$ extending $\omega_{S}$

$i^{*}\omega_{k}=\omega_{S},\quad k=0,1.$

Then there exists a symplectic groupoid isomorphism
$\Phi:(\mathcal{G}_{0}^{\prime},\omega_{0})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(\mathcal{G}_{1}^{\prime},\omega_{1})$,
defined between open groupoid neighborhoods
$\mathcal{G}_{S}\subset\mathcal{G}_{k}^{\prime}\subset\mathcal{G}_{k}$,
satisfying $\Phi|_{\mathcal{G}_{S}}=\mathrm{Id}$.

Applying this result to the representation $V=\mathfrak{k}^{*}$, we obtain
that the local model is well-defined up to isomorphism.

######  Corollary 5.14.

The germs of any two groupoid local models for $(\mathcal{G}_{S},\omega_{S})$
are isomorphic.

We begin by making the following observations about such groupoids.

######  Lemma 5.15.

Any open groupoid neighborhood,

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}\rightrightarrows
M)\subset(\mathcal{G}_{S}\ltimes V\rightrightarrows V),$

has an open subgroupoid

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}^{\prime}\rightrightarrows
M^{\prime})\subset(\mathcal{G}\rightrightarrows M)$

which is invariant under multiplication by $\lambda\in[0,1]$

$(g,e)\in\mathcal{G}^{\prime}\ \Longrightarrow\ m_{\lambda}(g,e)=(g,\lambda
e)\in\mathcal{G}^{\prime}.$

###### Proof.

It is easy to see that the following is a subgroupoid of $\mathcal{G}$

$\mathcal{G}^{\prime}:=\big{\\{}(g,e)\,:\,(g,\lambda
e)\in\mathcal{G},\forall\,\lambda\in[0,1]\big{\\}}\subset\mathcal{G},$

which is open and invariant under multiplication by $\lambda\in[0,1]$. ∎

######  Lemma 5.16.

Consider an open groupoid neighborhood

$(\mathcal{G}_{S}\rightrightarrows
S)\hookrightarrow(\mathcal{G}\rightrightarrows
M)\subset(\mathcal{G}_{S}\ltimes V\rightrightarrows V),$

which is invariant under multiplication by $\lambda\in[0,1]$. Any closed
multiplicative form $\alpha\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ whose
pullback along the zero section $i:\mathcal{G}_{S}\hookrightarrow\mathcal{G}$
vanishes, i.e.

$i^{*}\alpha=0\in\Omega^{k}_{\mathrm{M}}(\mathcal{G}_{S}),$

is exact with primitive a multiplicative form, i.e.

$\alpha=\mathrm{d}\beta,\quad\text{with}\quad\beta\in\Omega^{k-1}_{\mathrm{M}}(\mathcal{G}).$

Moreover, we can take $\beta$ to vanish at points of $\mathcal{G}_{S}$.

###### Proof.

Assume first that $\mathcal{G}=\mathcal{G}_{S}\ltimes V$. We only need to
observe that the usual homotopy operator preserves multiplicative forms.
Indeed, let $i:\mathcal{G}_{S}\hookrightarrow\mathcal{G}_{S}\ltimes V$ be the
zero section, $P:\mathcal{G}_{S}\ltimes V\to\mathcal{G}_{S}$ the projection,
denote by $X$ the Euler vector field of the vector bundle $p:V\to S$ and set
$Y=(0,X)\in\mathfrak{X}(\mathcal{G}_{S}\ltimes V)$. Then

$\mathrm{Id}-(i\circ P)^{*}=\mathrm{d}H+H\mathrm{d},$

where $H:\Omega^{\bullet}(\mathcal{G}_{S}\ltimes
V)\to\Omega^{\bullet-1}(\mathcal{G}_{S}\ltimes V)$, $\bullet\geq 1$, is the
homotopy operator

$H(\gamma):=\int_{0}^{1}\frac{1}{\lambda}m_{\lambda}^{*}(i_{Y}\gamma)\,\mathrm{d}\lambda.$

That $\frac{1}{{\lambda}}m_{\lambda}^{*}(i_{Y}\gamma)$ is smooth at
$\lambda=0$ follows because $m_{0}^{*}(i_{Y}\gamma)=P^{*}\circ
i^{*}(i_{Y}\gamma)=0$. Since $m_{\lambda}:\mathcal{G}_{S}\ltimes
V\to\mathcal{G}_{S}\ltimes V$ is a groupoid morphism and $Y$ is a
multiplicative vector field, $H$ maps multiplicative forms to multiplicative
forms. So if $\alpha$ is a closed multiplicative form whose pullback to
$\mathcal{G}_{S}$ vanishes, we have $\alpha=\mathrm{d}H(\alpha)$. The form
$\beta=H(\alpha)$ is multiplicative and vanishes at points of
$\mathcal{G}_{S}$. Finally, note that if
$\mathcal{G}\subset\mathcal{G}_{S}\ltimes V$ is invariant under multiplication
by ${\lambda}\in[0,1]$, the same formula gives the homotopy operators for
$\mathcal{G}$. ∎

The following shows that the representation $V$ from Proposition 5.13 is
actually isomorphic to $\mathfrak{k}^{*}$ (see Proposition 5.18 for a more
general version of this result).

######  Lemma 5.17.

Consider an open groupoid neighborhood,

$(\mathcal{G}_{S}\rightrightarrows S)\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}(\mathcal{G}\rightrightarrows
M)\subset(\mathcal{G}_{S}\ltimes V\rightrightarrows V),$

together with a multiplicative symplectic form
$\omega\in\Omega^{2}(\mathcal{G})$. Then $\mathcal{G}_{S}$ is a coisotropic
submanifold. Let $\mathfrak{k}:=(T_{S}\mathcal{G}_{S})^{\perp_{\omega}}\subset
T_{S}\mathcal{G}_{S}$. Then the vector bundle map:

$\varphi:V\to\mathfrak{k}^{*},\quad\langle\varphi(u),v\rangle=\omega((0,v),(u,0)),\quad
u\in V_{x},\ v\in\mathfrak{k}_{x},$

is a $\mathcal{G}_{S}$-equivariant isomorphism, where we regard

$(v,0)\in T_{(1_{x},0)}(\mathcal{G}_{S}\ltimes
V),\quad(0,u)=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(1_{x},tu)\in
T_{(1_{x},0)}(\mathcal{G}_{S}\ltimes V).$

###### Proof.

That $\mathcal{G}_{S}$ is coisotropic follows from Proposition 3.1. Note that
$\varphi$ is injective

$\ker\varphi=V\cap\mathfrak{k}^{\perp_{\omega}}=V\cap T_{S}\mathcal{G}_{S}=0.$

That $\mathcal{G}_{S}$ is a coisotropic submanifold implies also that
$\mathfrak{k}$ and $V$ have the same rank, so $\varphi$ is a linear
isomorphism. It remains to show that $\varphi$ is equivariant. Let
$g\in\mathcal{G}_{S}$, $x:=\mathbf{s}(g)$, $y:=\mathbf{t}(g)$, and let
$v\in\mathfrak{k}_{y}$, $u\in V_{x}$. Denote
$\gamma_{t}:=\exp_{y}(tv)\in(\mathcal{G}_{S})_{y}$ and $u_{t}:=tu$. We have
that

$\displaystyle\omega((g^{-1}\cdot v,0),(0,u))$
$\displaystyle=\omega\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1}\gamma_{t}g,0),\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(1_{x},u_{t})\Big{)}$
$\displaystyle=\omega\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1}\gamma_{t},0)(g,0),\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1},g\cdot
u_{t})(g,u_{t})\Big{)}$
$\displaystyle=\omega\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1}\gamma_{t},0),\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1},g\cdot
u_{t})\Big{)}+0$
$\displaystyle=\omega\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1},0)(\gamma_{t},0),\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(g^{-1},g\cdot
u_{t})(1_{y},g\cdot u_{t})\Big{)}$
$\displaystyle=0+\omega\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\gamma_{t},0),\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(1_{y},g\cdot
u_{t})\Big{)}$ $\displaystyle=\omega((v,0),(0,g\cdot u)),$

where we have used twice that $\omega$ is multiplicative. This equality is
equivalent to $\varphi$ being $\mathcal{G}_{S}$ equivariant: $\varphi(g\cdot
u)=g\cdot\varphi(u)$. ∎

We are now ready to prove the multiplicative Weinstein-Moser theorem.

###### Proof of Proposition 5.13.

Let $\mathfrak{k}:=\ker\omega_{S}|_{S}$. Consider the
$\mathcal{G}_{S}$-equivariant isomorphisms from Lemma 5.17,
$\varphi_{k}:V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}^{*}$,
corresponding to $\omega_{k}$. By pushing forward $\omega_{k}$ along the
groupoid isomorphisms $\mathrm{Id}\times\varphi_{k}:\mathcal{G}_{S}\times
V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$,
we may assume that $V=\mathfrak{k}^{*}$ and that $\omega_{k}$ restricted to
$\mathfrak{k}\times_{S}\mathfrak{k}^{*}$ is the canonical pairing. Therefore,
for each $t$, the 2-form $\omega_{t}:=t\omega_{1}+(1-t)\omega_{0}$ is non-
degenerate on $T_{S}(\mathcal{G}_{S}\ltimes\mathfrak{k}^{*})\simeq
T_{S}\mathcal{G}_{S}\oplus\mathfrak{k}^{*}$. We claim that there exists an
open subgroupoid $\mathcal{G}\subset\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$
containing $\mathcal{G}_{S}$ and invariant under multiplication $m_{\lambda}$,
$\lambda\in[0,1]$, such that, for all $t\in[0,1]$, $\omega_{t}$ is a
multiplicative symplectic form on $\mathcal{G}$. The 2-form $\omega_{t}$ is
defined and multiplicative on the open subgroupoid
$\mathcal{G}_{2}:=\mathcal{G}_{0}\cap\mathcal{G}_{1}\rightrightarrows
M_{2}:=M_{0}\cap M_{1}$. Let $M_{3}\subset M_{2}$ be the set consisting of
points $x$ such that $\omega_{t}|_{1_{x}}$ is non-degenerate for all
$t\in[0,1]$. As we have seen, $M_{3}$ contains $S$, and by the Tube Lemma,
$M_{3}$ is open. By Proposition 3.3 (d), for all $t\in[0,1]$, $\omega_{t}$ is
over-symplectic on the open subgroupoid
$\mathcal{G}_{3}:=\mathcal{G}_{2}|_{M_{3}}\rightrightarrows M_{3}$, and by the
conclusion of that proposition, $\omega_{t}$ is non-degenerate on
$\mathcal{G}_{3}$. Finally, we let $(\mathcal{G}\rightrightarrows
M)\subset(\mathcal{G}_{3}\rightrightarrows M_{3})$ be an open subgroupoid
containing $\mathcal{G}_{S}$ which is invariant under multiplication by
$t\in[0,1]$, as in Lemma 5.15. We apply a groupoid version of Moser’s trick to
the path of symplectic, multiplicative forms
$\omega_{t}\in\Omega^{2}_{\mathrm{M}}(\mathcal{G})$, $t\in[0,1]$. If $X_{t}$
is a time-dependent multiplicative vector field generating an isotopy of
groupoid automorphisms $\Phi^{t}:\mathcal{G}\to\mathcal{G}$, starting at the
identity at $t=0$, such that

$(\Phi^{t})^{*}\omega_{t}=\omega_{0},\quad\forall t\in[0,1],$

then, by the usual argument, $X_{t}$ must satisfy the equation

$\mathrm{d}i_{X_{t}}\omega_{t}=\omega_{0}-\omega_{1}.$

Since $\omega_{0}-\omega_{1}$ is a closed, multiplicative 2-form that vanishes
on points of $\mathcal{G}_{S}$ and $\mathcal{G}$ is invariant under
multiplication by $t\in[0,1]$, by Lemma 5.16, we can find a primitive $\beta$
which is multiplicative and vanishes on $\mathcal{G}_{S}$. Consider the vector
field $X_{t}\in\mathfrak{X}(\mathcal{G})$ defined by the equation

$i_{X_{t}}\omega_{t}=\beta,\quad t\in[0,1].$

Then $X_{t}$ is a multiplicative vector field vanishing on $\mathcal{G}_{S}$.
The isotopy $\Phi^{t}$, generated by $X_{t}$, is defined on an open
subgroupoid of $\mathcal{G}$ containing $\mathcal{G}_{S}$, and satisfies
$(\Phi^{t})^{*}(\omega_{t})=\omega_{0}$. Then $\Phi^{1}$ is the desired
symplectic groupoid automorphism. ∎

###  5.5. Linearization of symplectic groupoids

Having established conditions for the existence of a groupoid local model, we
now ask

  * •

To what extent can one expect the groupoid local model of a partially split
$(\mathcal{G}_{S},\omega_{S})$ to be a _normal form_?

Recall that for a Lie groupoid $\mathcal{G}\rightrightarrows M$ with a
_saturated_ submanifold $S\subset M$, one defines the linear approximation to
$\mathcal{G}$ around $S$ as the normal bundle to the restriction
$\mathcal{G}_{S}=\mathcal{G}|_{S}$

$\nu(\mathcal{G}_{S})\rightrightarrows\nu(S),$

where the groupoid structure is induced from the restriction of
$T\mathcal{G}\rightrightarrows TM$ to $T_{S}M$. More concretely, it can also
be identified with the action groupoid

$\nu(\mathcal{G}_{S})=\mathcal{G}_{S}\ltimes\nu(S),$

associated with the normal representation of $\mathcal{G}_{S}\rightrightarrows
S$ on $\nu(S)$. Recall (see [22]) that a Lie groupoid
$\mathcal{G}\rightrightarrows M$ is called linearizable around a saturated
submanifold $S\subset M$ if there are groupoid neighborhoods
$\mathcal{G}_{S}\subset\mathcal{U}\subset\mathcal{G}$ and
$\mathcal{G}_{S}\subset\mathcal{V}\subset\nu(\mathcal{G}_{S})$ and a groupoid
isomorphism

$\Phi:\mathcal{U}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathcal{V},$

which is the identity on $\mathcal{G}_{S}$. It is called invariantly
linearizable around $S$, if there are saturated neighborhoods $S\subset
U\subset M$ and $S\subset V\subset\nu(S)$ such that

$\mathcal{U}=\mathcal{G}|_{U}\quad\text{and}\quad\mathcal{V}=\nu(\mathcal{G}_{S})|_{V}.$

We call the isomorphism $\Phi$ a linearization of $\mathcal{G}$ around $S$.
The linearization problem for Lie groupoids has been intensively studied [20,
22, 58, 59, 62]. In particular, it is well-known that proper (respectively,
$\mathbf{s}$-proper) Hausdorff Lie groupoids can be linearized (respectively,
invariantly linearized) around saturated submanifolds. Returning to the
coisotropic embedding problem of an over-symplectic groupoid, it turns out
that the groupoid from the local model is naturally isomorphic to the linear
approximation. Notice that this property is independent of the existence of a
multiplicative Ehresmann connection.

######  Proposition 5.18.

Let $(\mathcal{G}_{S},\omega_{S})$ be an over-symplectic groupoid. Given a
groupoid coisotropic embedding
$i:(\mathcal{G}_{S},\omega_{S})\hookrightarrow(\mathcal{G},\omega)$, then the
isomorphism given by the symplectic form descends to a groupoid isomorphism

$\textstyle{T_{\mathcal{G}_{S}}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\flat}}$$\textstyle{T^{*}_{\mathcal{G}_{S}}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu(\mathcal{G}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\ker\omega_{S})^{*}}$

whose base map is an isomorphism of $\mathcal{G}_{S}$-representations

$\nu(S)\simeq\mathfrak{k}^{*},\quad\textrm{where}\ \
\mathfrak{k}:=\ker(\omega_{S})|_{S}.$

###### Proof.

Since $i$ is a coisotropic embedding, it follows that it induces an
isomorphism of vector bundles making the diagram commute. Now observe that,
since $\omega$ is multiplicative, all solid arrows in the diagram are groupoid
morphisms, so it follows that the dashed arrow is also a groupoid morphism.
Denote this isomorphism by
$\Phi:\nu(\mathcal{G}_{S})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(\ker\omega_{S})^{*}$
and its base map by
$\varphi:\nu(S)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}^{*}$.
As we have remarked before, these groupoids are isomorphic to action groupoids

$\nu(\mathcal{G}_{S})\simeq\mathcal{G}_{S}\ltimes\nu(S),\quad(\ker\omega_{S})^{*}\simeq\mathcal{G}_{S}\ltimes\mathfrak{k}^{*},$

where the isomorphisms are the pairs
$(\operatorname{pr}_{\mathcal{G}_{S}},\mathbf{s})$, bundle projection and
source map. Since $\Phi$ is a vector bundle map covering the identity, we
obtain a commutative diagram of groupoid isomorphisms

$\textstyle{\nu(\mathcal{G}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{(\ker\omega_{S})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}_{S}\ltimes\nu(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{Id},\varphi)}$$\textstyle{\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}.}$

In particular, the map $\varphi$ is an isomorphism of
$\mathcal{G}_{S}$-representations. ∎

The previous proposition allows to restate Proposition 5.7 as follows.

######  Corollary 5.19.

Given a symplectic groupoid $(\mathcal{G},\omega)\rightrightarrows M$, its
restriction $(\mathcal{G}_{S},\omega_{S})$ to a saturated submanifold $S$ is
partially split if and only if the linear approximation
$\nu(\mathcal{G}_{S})\rightrightarrows\nu(S)$ admits a closed multiplicative
2-form $\omega_{0}$ extending $\omega_{S}$ and which is non-degenerate along
$\mathcal{G}_{S}$.

Proposition 5.18 also shows that if a symplectic groupoid
$(\mathcal{G},\omega)$ is isomorphic to the local model
$(\mathcal{G}_{0},\omega_{0})$ around $\mathcal{G}_{S}$, then the groupoid
$\mathcal{G}$ is linearizable around $S$. Next, we show that Proposition 5.13
implies that this is the only obstruction.

######  Theorem 5.20.

If a symplectic groupoid $(\mathcal{G},\omega)$ is linearizable (as a Lie
groupoid) around a saturated submanifold $S$, then $(\mathcal{G},\omega)$ is
locally isomorphic (as a symplectic groupoid) to the groupoid local model of
$(\mathcal{G}_{S},\omega_{S})$.

###### Proof.

A linearization of $\mathcal{G}$ around $S$, induces a multiplicative
symplectic structure $\omega_{1}$ on a groupoid neighborhood
$\mathcal{G}_{1}\subset\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}$ of
$\mathcal{G}_{S}$ such that $i^{*}(\omega_{1})=\omega_{S}$. By Proposition
5.7, $(\mathcal{G}_{S},\omega_{S})$ is partially split, and so the groupoid
local model $(\mathcal{G}_{0},\omega_{0})$ of Proposition 5.2 exists. Since
$i^{*}(\omega_{0})=\omega_{S}$, Proposition 5.13 implies that
$(\mathcal{G}_{0},\omega_{0})$ and $(\mathcal{G}_{1},\omega_{1})$ are
isomorphic around $\mathcal{G}_{S}$, and so $(\mathcal{G},\omega)$ and the
groupoid local model $(\mathcal{G}_{0},\omega_{0})$ are isomorphic around
$\mathcal{G}_{S}$. ∎

The linearization theorems mentioned before (see, e.g., [22]) imply the
following.

######  Corollary 5.21.

Let $(\mathcal{G},\omega)\rightrightarrows M$ be a proper symplectic groupoid.
For any closed embedded invariant submanifold $S\subset M$,
$(\mathcal{G},\omega)$ is isomorphic around $S$ to the groupoid local model of
$(\mathcal{G}_{S},\omega_{S})$. If, additionally, $(\mathcal{G},\omega)$ is
target-proper, then $\mathcal{G}$ is invariantly linearizable.

Let us now see that Theorem 9 in the Introduction follows from Corollary 5.21.
First, the assumption in Theorem 9 looks weaker than in this corollary, since
there one only assumes the restriction $\mathcal{G}_{S}\rightrightarrows S$ to
be target-proper. However, this is only apparent, as the following general
fact shows.

######  Lemma 5.22.

Let $f:X\to Y$ be a submersion with connected fibers, and let $Y_{c}\subset Y$
be the collection of points $y\in Y$ whose preimage $f^{-1}(y)\subset X$ is
compact. Then $Y_{c}$ is open in $Y$, and $f:f^{-1}(Y_{c})\to Y_{c}$ is a
locally trivial fibration.

###### Proof.

The foliation on $X$ given by the fibers of $f$ has trivial holonomy. By
applying the Local Reeb Stability Theorem (see, e.g., [50]) to a fiber
$f^{-1}(y)$, $y\in Y_{c}$, we obtain a saturated open neighborhood
$f^{-1}(y)\subset U\subset X$ and a diffeomorphism $U\simeq f^{-1}(y)\times
V$, with $y\subset V\subset Y$, under which $f$ becomes $\mathrm{pr}_{V}$.
Since the fibers of $f$ are connected, it follows that $U=f^{-1}(V)$. Hence
$V\subset Y_{c}$, and $f$ is can be trivialized above $V$. ∎

###### Proof of Theorem 9.

The lemma applied to $\mathbf{t}:\mathcal{G}\to M$ yields the saturated open
set $M_{c}$ for which $\mathcal{G}|_{M_{c}}$ is target-proper. The result
follows from Corollary 5.21. ∎

##  6\. Examples of groupoid local models

In [27, Section 3] we give many examples of classes of groupoids with bundles
of ideals that are partially split. These lead to examples of over-symplectic
groupoids which are partially split. We give here some classes where one can
find the symplectic groupoid local models explicitly.

###  6.1. Transitive over-symplectic groupoids

Given a principal $G$-bundle $P\to S$, the gauge groupoid

$\mathcal{G}_{S}:=P\times_{G}P\rightrightarrows S,$

is a transitive groupoid for which the adjoint bundle
$\mathfrak{k}=P\times_{G}\mathfrak{g}$ is a bundle of ideals. It was observed
in [27, Section 3] that this is always partially split and that there is a
1-to-1 correspondence

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.01332pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
multiplicative connection}\\\\\hbox{{\ignorespaces 1-forms
$\alpha\in\Omega^{1}(\mathcal{G}_{S},\mathfrak{k})$
\,}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-10.01332pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
principal bundle}\\\\\hbox{{\ignorespaces connections
$\eta\in\Omega^{1}(P;\mathfrak{g})$\,}\crcr}}}}\right\\}}}}}$

This correspondence is as follows: given a principal bundle connection
$\eta\in\Omega^{1}(P,\mathfrak{g})$, the 1-form
$\operatorname{pr}_{2}^{*}\eta-\operatorname{pr}_{1}^{*}\eta\in\Omega^{1}(P\times
P,\mathfrak{g})$ descends to a unique multiplicative connection 1-form
$\alpha\in\Omega^{1}(\mathcal{G}_{S},\mathfrak{k})$ such that

$q^{*}\alpha=\operatorname{pr}_{2}^{*}\eta-\operatorname{pr}_{1}^{*}\eta,$
(6.1)

where $q:P\times P\to P\times_{G}P$ is the projection. If
$(\mathcal{G}_{S},\omega_{S})$ is a transitive over-symplectic groupoid, then
we saw in the example of Subsection 4.2 that it is isomorphic to the gauge
groupoid of a principal $G$-bundle $p:P\to S$ over a symplectic manifold
$(S,\omega)$,

$\mathcal{G}_{S}\simeq P\times_{G}P\rightrightarrows S.$

Under this isomorphism, the over-symplectic structure is given by

$\omega_{S}=(p\circ\operatorname{pr}_{2})^{*}\omega-(p\circ\operatorname{pr}_{1})^{*}\omega\in\Omega^{2}(\mathcal{G}_{S}).$

It follows that the bundle of ideals is the adjoint bundle

$\mathfrak{k}=(\ker\omega_{S})|_{S}=P\times_{G}\mathfrak{g}.$

Hence, every transitive over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ is partially split. An explicit form for the
local model $(\mathcal{G}_{0},\omega_{0})$ associated with a principal
connection $\eta$ is given by a groupoid neighborhood of the zero section in

$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}=(P\times
P)\times_{G}\mathfrak{g}^{*},$

and the closed two 2-form $\omega_{0}$ is obtained from expression (5.2)

$\omega_{0}=\operatorname{pr}_{\mathcal{G}_{S}}^{*}\omega_{S}+\mathrm{d}\langle\alpha,\cdot\rangle,$

and $\alpha$ is determined by (6.1). This local model is well-known [17, 44]
and is the groupoid version of Vorobjev’s local model around symplectic leaves
[53].

###  6.2. Principal type

Consider an over-symplectic groupoid of principal type,
$\mathcal{G}_{S}:=\Sigma\times_{S\times S}\mathcal{H}$, associated to the
symplectic groupoid $(\Sigma,\omega_{\Sigma})\rightrightarrows S$ and the
transitive groupoid $\mathcal{H}\simeq P\times_{G}P\rightrightarrows S$, as in
the example from Subsection 4.5. Note that

$\ker\omega_{S}=\Sigma\times_{S\times
S}\big{(}\ker\mathrm{d}\mathbf{t}_{\mathcal{H}}\cap\ker\mathrm{d}\mathbf{s}_{\mathcal{H}}\big{)}\simeq\mathcal{G}_{S}\times_{S}\mathfrak{k},$

where, as in the previous example, the bundle $\mathfrak{k}$ coincides with
the adjoint bundle

$\mathfrak{k}=(\ker\omega_{S})|_{S}=P\times_{G}\mathfrak{g}.$

Also as in the previous example, a principal connection
$\eta\in\Omega^{1}(P;\mathfrak{g})$ gives rise to a multiplicative connection
$\alpha\in\Omega^{1}(\mathcal{H},\mathfrak{k})$. The projection
$\operatorname{pr}_{\mathcal{H}}:\mathcal{G}_{S}\to\mathcal{H}$ is a groupoid
map, so it follows that we have a multiplicative connection 1-form on
$\mathcal{G}_{S}$

$\operatorname{pr}_{\mathcal{H}}^{*}\alpha\in\Omega_{\mathrm{M}}^{1}(\mathcal{G}_{S},\mathfrak{k}).$

We conclude that over-symplectic groupoids of principal type are partially
split. The local model has supporting groupoid an open subgroupoid

$\mathcal{G}_{0}\subset(P\times_{S}\Sigma\times_{S}P)\times_{G}\mathfrak{g}^{*}\rightrightarrows
P\times_{G}\mathfrak{g}^{*},$

and multiplicative form

$\omega_{0}=\operatorname{pr}^{*}_{\Sigma}\omega_{\Sigma}+\mathrm{d}\langle\operatorname{pr}^{*}_{\mathcal{H}}\alpha,\cdot\rangle.$

###  6.3. Action groupoids

Let $(\mathcal{G}_{S},\omega_{S})$ be an over-symplectic groupoid, where
$\mathcal{G}_{S}$ is the action groupoid of an action of a Lie group $G$ on a
manifold $S$

$\mathcal{G}_{S}=G\ltimes S\rightrightarrows S.$

The kernel of $\omega_{S}$ defines a bundle of ideals
$\mathfrak{k}\hookrightarrow\mathfrak{g}\times S$. This means that each fiber
is an ideal $\mathfrak{k}_{x}\subset\ker\rho_{x}$ such that

$\operatorname{Ad}_{g}(\mathfrak{k}_{x})=\mathfrak{k}_{gx},$

where $\rho:\mathfrak{g}\to\mathfrak{X}(S)$ denotes the infinitesimal action.
If there exists a $G$-equivariant splitting $l:\mathfrak{g}\times
S\to\mathfrak{k}$ then the over-symplectic groupoid
$(\mathcal{G}_{S},\omega_{S})$ is partially split: this splitting yields a
multiplicative connection 1-form $\alpha\in\Omega^{1}_{M}(G\ltimes
S;\mathfrak{k})$ by setting

$\alpha_{(g,x)}(\mathrm{d}L_{g}(v),w):=l(v,x).$

Consider for example a compact Lie group $G$ with Lie algebra $\mathfrak{g}$
and denote by $(\cdot,\cdot)$ a $G$-invariant inner product on $\mathfrak{g}$.
Given any $G$-invariant submanifold $S\subset\mathfrak{g}$ the action groupoid

$\mathcal{G}_{S}:=G\ltimes S\rightrightarrows S.$

is an over-symplectic groupoid with closed 2-form

$(\omega_{S})_{(g,x)}((\mathrm{d}L_{g}(v_{1}),z_{1}),(\mathrm{d}L_{g}(v_{2}),z_{2}))=(v_{2},z_{1})-(v_{1},z_{2})-([v_{1},v_{2}],x),$
(6.2)

where $(g,x)\in G\times S$, $v_{1},v_{2}\in\mathfrak{g}$ and $z_{1},z_{2}\in
T_{x}S\subset\mathfrak{g}$. The corresponding Lie algebra bundle
$\mathfrak{k}\subset\mathfrak{g}\times S$ is:

$\mathfrak{k}_{x}=\\{(w,x):w\in(T_{x}S)^{\perp}\\}.$ (6.3)

The Lie group $G$ acts on the total space of the bundle $\mathfrak{k}$

$g\cdot(w,x)=(\operatorname{Ad}_{g}(w),gx),$

and the groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}\rightrightarrows\mathfrak{k}$ is then
identified with the action groupoid
$G\ltimes\mathfrak{k}\rightrightarrows\mathfrak{k}$. Using the inner product,
we can identify $\mathfrak{g}\simeq\mathfrak{g}^{*}$ and
$\mathfrak{k}\simeq\mathfrak{k}^{*}$. The orthogonal projection gives a
$G$-invariant splitting

$l:\mathfrak{g}\times S\to\mathfrak{k},\quad
l_{x}(v):=\pi_{\mathfrak{k}_{x}}(v).$

Then we obtain the multiplicative connections 1-form
$\alpha\in\Omega^{1}_{M}(G\ltimes S;\mathfrak{k})$ defined by

$\alpha_{(g,x)}(\mathrm{d}L_{g}(v),z):=\pi_{\mathfrak{k}_{x}}(v).$

This gives a closed, multiplicative, 1-form
$\langle\alpha,\cdot\rangle\in\Omega^{1}_{M}(G\ltimes\mathfrak{k})$ (recall
that $\mathfrak{k}\simeq\mathfrak{k}^{*}$)

$\langle\alpha,\cdot\rangle_{(g,w,x)}(\mathrm{d}L_{g}(v),u,z)=(v,w),$ (6.4)

where $g\in G$, $(w,x)\in\mathfrak{k}$, $v\in\mathfrak{g}$ and $(u,z)\in
T_{(w,x)}\mathfrak{k}\subset\mathfrak{g}\times\mathfrak{g}$. The symplectic
form on the local model $\mathcal{G}_{0}\subset G\ltimes\mathfrak{k}$ is
given, as usual, by restricting the closed, multiplicative, 2-form

$\omega_{0}=\operatorname{pr}_{G\ltimes
S}^{*}\omega_{S}+\mathrm{d}\langle\alpha,\cdot\rangle\in\Omega^{2}_{M}(G\ltimes\mathfrak{k}).$

Using (6.2) and (6.4) we find

$\displaystyle(\omega_{0})_{(g,w,x)}$
$\displaystyle((\mathrm{d}L_{g}(v_{1}),u_{1},z_{1}),(\mathrm{d}L_{g}(v_{2}),u_{2},z_{2}))=$
$\displaystyle=(v_{2},z_{1})-(v_{1},z_{2})-([v_{1},v_{2}],x)+(v_{2},u_{1})-(v_{1},u_{2})-([v_{1},v_{2}],w)$
$\displaystyle=(v_{2},z_{1}+u_{1})-(v_{1},z_{2}+u_{2})-([v_{1},v_{2}],x+w).$
(6.5)

Note that $\mathfrak{k}$ is the normal bundle of $S$ in $\mathfrak{g}$ (see
(6.3)) and that the Riemannian exponential map $\mathfrak{k}\to\mathfrak{g}$
is given by $(w,x)\mapsto w+x$. In particular, it is $G$-equivariant, and so
it induces a map of groupoids

$G\ltimes\mathfrak{k}\to G\ltimes\mathfrak{g},$

which is an open embedding around $G\ltimes S$. Formula (6.5) shows that it is
in fact an embedding of symplectic groupoids, where we use the identifications
$T^{*}G\simeq G\times\mathfrak{g}^{*}\simeq G\ltimes\mathfrak{g}$ given by
left-invariant translations and the invariant inner product, and endow
$T^{*}G$ with the canonical symplectic structure. We conclude that for a
compact Lie group $G$ and any $G$-invariant submanifold
$S\subset\mathfrak{g}^{*}$, a choice of $G$-invariant inner product gives an
identification of the local model $\mathcal{G}_{0}$ around $S$ with $T^{*}G$.
In other words, the symplectic groupoid
$T^{*}G\rightrightarrows\mathfrak{g}^{*}$ is linearizable around any invariant
submanifold. This also agrees, of course, with Corollary 5.21, since
$T^{*}G\rightrightarrows\mathfrak{g}^{*}$ is an s-proper groupoid whenever $G$
is compact. The next example shows that the assumption that $G$ is compact is
crucial.

######  Example 6.1.

Consider the non-abelian 2-dimensional Lie group $G$ and the associated
symplectic groupoid
$(T^{*}G,\omega_{\mathrm{can}})\rightrightarrows\mathfrak{g}^{*}$, which
integrates the linear Poisson structure on
$\mathfrak{g}^{*}\simeq\mathbb{R}^{2}$ given by

$\pi=x\,\partial_{x}\wedge\partial_{y}.$

Then $T^{*}G$ is not linearizable around the closed, embedded Poisson
submanifold $S=\\{x=0\\}$. For this, note that the over-symplectic groupoid
$\mathcal{G}_{S}:=(T^{*}G)|_{S}$, is, as groupoid, the trivial bundle of Lie
groups $G\times S\to S$, and that the orbits of $\mathcal{G}_{S}$ on the
normal bundle $\nu(S)=\mathfrak{k}^{*}$ are one dimensional. So, as in Example
5.8, a partial splitting does not exist.

##  7\. Local models of Poisson structures

We now come back to the infinitesimal level with the aim of constructing local
models for elements $\tau\in J^{1}_{S}\Pi(M,S)$. Since any such element can be
represented by a pair $\tau=(T^{*}_{S}M,\mu_{S})$, i.e., as the infinitesimal
counterpart of an over-symplectic groupoid, all we have to do is apply the
dictionary to transfer the construction in the previous section to obtain a
local model for Poisson structures.

###  7.1. The local model

Let us fix a first order jet of a Poisson structure

$(A_{S},[\cdot,\cdot]_{A_{S}},\rho_{A_{S}}),\quad\mu_{S}:A_{S}\to T^{*}S,$

i.e., a Lie algebroid together with a surjective IM 2-form. As usual,
$\pi_{S}$ denotes the induced Poisson structure on $S$. Let

$\mathfrak{k}:=\ker(\mu_{S}).$

The IM conditions imply that

$0\longrightarrow\mathfrak{k}\longrightarrow
A_{S}\stackrel{{\scriptstyle\mu_{S}}}{{\longrightarrow}}T^{*}S\longrightarrow
0,$ (7.1)

is a short exact sequence of Lie algebroids, where $T^{*}S\Rightarrow S$ is
regarded as the cotangent Lie algebroid of $\pi_{S}$. In particular
$\mathfrak{k}$ is a bundle of ideals in $A_{S}$. Therefore, $A_{S}$ has a
canonical representation on $\mathfrak{k}$

$\nabla^{\mathfrak{k}}_{\alpha}\gamma:=[\alpha,\gamma]_{A_{S}},\quad\alpha\in\Gamma(A_{S}),\gamma\in\Gamma(\mathfrak{k}).$
(7.2)

This is the infinitesimal version of the groupoid representation (5.1). It
gives rise to the semi-direct product Lie algebroid
$A_{S}\times_{S}\mathfrak{k}\Rightarrow S$, with Lie bracket

$[(\alpha_{1},\gamma_{1}),(\alpha_{2},\gamma_{2})]:=([\alpha_{1},\alpha_{2}]_{A_{S}},\nabla^{\mathfrak{k}}_{\alpha_{1}}\gamma_{2}-\nabla^{\mathfrak{k}}_{\alpha_{2}}\gamma_{1}),$

and anchor $\rho_{A_{S}}\circ\operatorname{pr}_{A_{S}}$. This is the
infinitesimal version of the groupoid
$\mathcal{G}_{S}\times_{S}\mathfrak{k}\rightrightarrows S$. The infinitesimal
version of the groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$ is,
of course, the action algebroid
$A_{S}\ltimes\mathfrak{k}^{*}\Rightarrow\mathfrak{k}^{*}$ associated with the
dual representation $\nabla^{\mathfrak{k}^{*}}$ of $A_{S}$ on
$\mathfrak{k}^{*}$

$\mathscr{L}_{\rho_{A_{S}}(\alpha)}\langle\beta,X\rangle=\langle\nabla^{\mathfrak{k}}_{\alpha}\beta,X\rangle+\langle\beta,\nabla^{\mathfrak{k}^{*}}_{\alpha}X\rangle.$

To build the local model, we need know whether the Lie algebroid
$A_{S}\ltimes\mathfrak{k}^{*}\Rightarrow\mathfrak{k}^{*}$ admits a closed IM
2-form $\mu_{0}$ which is non-degenerate around $S$ and whose restriction to
$A_{S}$ is $\mu_{S}$. From the global picture, we know that such an IM form
might not exist, and so we need the infinitesimal analogue of multiplicative
connections and the partially split condition. Using the notion of
infinitesimal multiplicative (IM) form with coefficients – see Appendix A –
one obtains the infinitesimal version of Definition 5.1.

######  Definition 7.1.

An IM connection 1-form on a first order jet $(A_{S},\mu_{S})$ with kernel
$\mathfrak{k}=\ker\mu_{S}$ is a IM 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ whose symbol satisfies

$l(\xi)=\xi,\quad\forall\xi\in\Gamma(\mathfrak{k}).$

A first order jet is called partially split if it admits an IM connection
1-form.

The general theory of IM connections 1-forms was developed in [27], and will
be recalled briefly in the next subsection. Given an IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ we have an associated
IM 1-form
$\langle(L,l),\cdot\rangle\in\Omega^{1}_{\mathrm{IM}}(A_{S}\ltimes\mathfrak{k}^{*})$
which has components $\mu:A_{S}\ltimes\mathfrak{k}^{*}\to\mathbb{R}$ and
$\zeta:A_{S}\ltimes\mathfrak{k}^{*}\to T\mathfrak{k}^{*}$ defined on constant
sections $\alpha\in\Gamma(A_{S})$ by

$\displaystyle\mu(\alpha)$ $\displaystyle=\widetilde{l(\alpha)},$ (7.3)
$\displaystyle\zeta(\alpha)$
$\displaystyle=\mathrm{d}\widetilde{l(\alpha)}-\widetilde{L(\alpha)},$ (7.4)

where for a $\mathfrak{k}$-valued form $\eta\in\Omega^{k}(S,\mathfrak{k})$ we
denote by $\widetilde{\eta}\in\Omega^{k}(\mathfrak{k}^{*})$ the $k$-form given
by $\widetilde{\eta}|_{\xi}:=\langle\operatorname{pr}^{*}(\eta),\xi\rangle$,
with $\operatorname{pr}:\mathfrak{k}^{*}\to S$ denoting the bundle projection.
That $\langle(L,l),\cdot\rangle=(\mu,\zeta)$ is indeed a IM form is proven in
[27, Proposition 5.4], and follows directly from the IM conditions. Using the
IM-differential from definition (A.3), one obtains the closed IM 2-form

$\mathrm{d}_{\mathrm{IM}}\langle(L,l),\cdot\rangle=(\zeta,0)\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes\mathfrak{k}^{*}).$

Then we can state the infinitesimal analogue of Proposition 5.2.

######  Proposition 7.2.

Given a first order jet $(A_{S},\mu_{S})$ and an IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ the closed IM 2-form

$\mu_{0}=\operatorname{pr}^{*}\mu_{S}+\mathrm{d}_{\mathrm{IM}}\langle(L,l),\cdot\rangle\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes\mathfrak{k}^{*}),$
(7.5)

is non-degenerate along $S$. Moreover, the zero section
$i_{0}:A_{S}\hookrightarrow A_{S}\ltimes\mathfrak{k}^{*}$ is an algebroid
embedding satisfying $i_{0}^{*}\mu_{0}=\mu_{S}$.

###### Proof.

The pullback of a (closed) IM form by a Lie algebroid morphism is a (closed)
IM form. Since the projection
$\operatorname{pr}:A_{S}\ltimes\mathfrak{k}^{*}\to A_{S}$ is a Lie algebroid
morphism, it follows that (7.5) defines a closed IM 2-form. Using the
canonical isomorphism

$T^{*}_{S}\mathfrak{k}^{*}\simeq\mathfrak{k}\oplus T^{*}S,$

the restriction

$(\mu_{0})|_{0_{S}}:A_{S}\to T^{*}_{S}\mathfrak{k}^{*}$

has components

  * •

$(\operatorname{pr}^{*}\mu_{S})|_{0_{S}}:A_{S}\to\mathfrak{k}\oplus T^{*}S$,
$\alpha\mapsto(0,\mu_{S}(\alpha))$;

  * •

$(\mathrm{d}_{\mathrm{IM}}\langle(L,l),\cdot\rangle)|_{0_{S}}:A_{S}\to\mathfrak{k}\oplus
T^{*}S$, $\alpha\mapsto(l(\alpha),0)$.

Hence, $(\mu_{0})|_{0_{S}}$ is a fiberwise surjective map. Since the two
vector bundles have the same rank, $\mu_{0}$ is a non-degenerate IM 2-form
along $0_{S}$. ∎

In the setting of the previous proposition, let $M_{0}\subset\mathfrak{k}^{*}$
be the open neighborhood of $S$ where $\mu_{0}$ is non-degenerate. Then

$A_{0}:=(A_{S}\ltimes\mathfrak{k}^{*})|_{M_{0}}\simeq A_{S}\ltimes M_{0},$

carries the non-degenerate, closed, IM 2-form

$\mu_{0}|_{A_{0}}:A_{0}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}M_{0}.$

######  Definition 7.3.

Given a partially split first order jet $(A_{S},\mu_{S})$, the pair
$(A_{0},\mu_{0})$ is called the Lie algebroid local model corresponding to the
IM connection 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$.
The corresponding Poisson local model is the Poisson manifold
$(M_{0},\pi_{0})$, where

$\pi_{0}^{\sharp}:=\rho_{0}\circ\mu_{0}^{-1}.$

Note that
$\mu_{0}:A_{0}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}M_{0}$
is an isomorphism of Lie algebroids, giving an identification

$J^{1}_{S}\pi_{0}=(A_{S},\mu_{S}).$

In conclusion, we have constructed a local model, in the sense of Definition
1, for the class $\mathscr{C}\subset J^{1}_{S}\Pi(M,S)$ of partially split
first order jets of Poisson structures. Namely, by the axiom of choice, for
each $\tau\in\mathscr{C}$, we can chose a IM connection 1-form
$(L_{\tau},l_{\tau})$, and we let $\sigma(\tau)$ be the associated local
model, i.e., $\sigma(\tau):=(M_{0},\pi_{0})$. We obtain Theorem 2 from the
Introduction. In the next subsection we will see that different choices of IM
connection 1-forms produce isomorphic local models.

###  7.2. The partially split condition for jets

In order to understand the partially split condition and the existence of IM
connection 1-forms, as in Definition 7.1, we recall briefly the notion of IM
Ehresmann connection introduced in [27]. Consider a Lie algebroid $A_{S}$ and
a bundle of ideals $\mathfrak{k}\subset A_{S}$, i.e.,
$\mathfrak{k}\subset\ker\rho_{S}$ and
$[\alpha,\gamma]_{A_{S}}\in\Gamma(\mathfrak{k})$, for all
$\alpha\in\Gamma(A_{S})$, $\gamma\in\Gamma(\mathfrak{k})$. So $\mathfrak{k}$
is a representation of $A_{S}$

$\nabla^{\mathfrak{k}}_{\alpha}\gamma:=[\alpha,\gamma]_{A_{S}}.$

In [27] we gave the following equivalent descriptions of IM connections for
$\mathfrak{k}$

  1. (i)

An IM Ehresmann connection for $\mathfrak{k}$ is a VB subalgebroid
$E\Rightarrow TS$ of $TA\Rightarrow TS$ such that

$TA=\mathfrak{k}\oplus E.$

  2. (ii)

There is a natural inclusion $A_{S}\times_{S}\mathfrak{k}\subset TA_{S}$

$(\alpha,\gamma)\mapsto\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\alpha+t\gamma)$

which is a map of VB algebroids

---
$\textstyle{A_{S}\times_{S}\mathfrak{k}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TA_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{S}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

An IM Ehresmann connection yields a partial splitting
$\theta:A_{S}\ltimes\mathfrak{k}^{*}\to T^{*}A_{S}$, i.e., a VB-algebroid map
which is a splitting of the dual projection

---
$\textstyle{A_{S}\ltimes\mathfrak{k}^{*}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{T^{*}A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

  3. (iii)

Given a partial splitting $\theta:A_{S}\ltimes\mathfrak{k}^{*}\to T^{*}A_{S}$
for $\mathfrak{k}$, one obtains an IM connection 1-form, i.e., an IM 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ whose symbol satisfies

$l(\xi)=\xi,\quad\forall\xi\in\Gamma(\mathfrak{k}).$

It is defined by setting for any $\alpha\in\Gamma(A)$ and $v\in TM$

$\displaystyle L(\alpha)(v)$
$\displaystyle:=\operatorname{pr}_{E}(\theta^{\vee}(\mathrm{d}\alpha(v))),$
$\displaystyle l(\alpha)$
$\displaystyle:=\operatorname{pr}_{E}\theta^{\vee}(\hat{\alpha})|_{M},$

where $\hat{\alpha}\in\mathfrak{X}(A)$ is the vertical lift of the section
$\alpha$.

  4. (iv)

Finally, given an IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$, one obtains a linear,
closed multiplicative 2-form
$\mu^{\mathrm{lin}}\in\Omega^{2}(A_{S}\ltimes\mathfrak{k}^{*})$ by

$\mu^{\mathrm{lin}}:=\mathrm{d}_{\mathrm{IM}}\langle(L,l),\cdot\rangle.$

We now return to our case of interest, namely first order jets