# Complete Lie algebroid actions and the integrability of Lie algebroids

Daniel Álvarez  Departamento de Matemáticas, Facultad de Ciencias, Universidad
Nacional Autónoma de México, CP 04510, México<EMAIL_ADDRESS>
###### Abstract.

We give a new proof of the equivalence between the existence of a complete
action of a Lie algebroid on a surjective submersion and its integrability.
The main tools in our approach are double Lie groupoids and multiplicative
foliations, our proof relies on a simple characterization of those vacant
double Lie groupoids which induce a Lie groupoid structure on their orbit
spaces.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Lie groupoids and Lie algebroids
      1. 2.1.1 Double Lie groupoids and LA-algebroids
  3. 3 Multiplicative foliations and vacant double Lie groupoids
    1. 3.1 Vacant double Lie groupoids and their orbit spaces
    2. 3.2 Monodromy groupoids of vacant multiplicative foliations
  4. 4 Complete Lie algebroid actions and vacant double Lie groupoids

##  1\. Introduction

Lie algebroids are basic objects in differential geometry, generalizing Lie
algebras on one hand and tangent bundles on the other. Among many other
geometric structures, Lie algebroids allow us to study Poisson manifolds,
foliations and connections on principal bundles in a unified way [13].
Generalizing the correspondence between Lie groups and Lie algebras, Lie
algebroids can be seen as the infinitesimal objects corresponding to global
structures which are known as Lie groupoids. Unlike finite-dimensional Lie
algebras, not every Lie algebroid corresponds to a Lie groupoid. Those Lie
algebroids which can be seen as the infinitesimal counterpart of a Lie
groupoid are called integrable. It is of fundamental importance to recognize
when a Lie algebroid is integrable [5], one of the main results in this
respect is the equivalence between the existence of a complete action of a Lie
algebroid on a surjective submersion and its integrability, see Theorem 4.1
below which originally appeared formulated in terms of Poisson geometry [6]
since it settled a question posed in [3] about the existence of complete
symplectic realizations. In this note we give a self-contained proof of this
result and also we use a definition of completeness which seems to be weaker
than the one that is used in [6]. Our arguments do not depend on the
construction of infinite-dimensional manifolds as in [5, 6] and are instead
purely finite-dimensional, the key concept behind our proof is that of double
Lie groupoid [12]. Double Lie groupoids can be described as objects which
encode global symmetries of Lie groupoids and so they are a useful tool for
the study of quotients of Lie groupoids. In fact, the proof of Theorem 4.1 can
be roughly summarized as follows: a complete Lie algebroid action induces a
multiplicative foliation on a submersion groupoid and this multiplicative
foliation can be integrated to a double Lie groupoid. It turns out that the
orbit space of this double Lie groupoid (with respect to one of its sides) is
the desired integration of the Lie algebroid under consideration. The
technical result that allows us to reach this conclusion describes the
conditions that ensure the existence of a Lie groupoid structure on the orbit
space of a vacant double Lie groupoid and it may be of independent interest,
see Theorem 3.2 below.

### Acknowledgments.

The author thanks CNPq for the financial support and H. Bursztyn for his
continuous advice and support. The author also thanks the referee whose
suggestions improved the readability of this work.

##  2\. Preliminaries

###  2.1. Lie groupoids and Lie algebroids

A smooth groupoid $G$ over a manifold $M$, denoted $G\rightrightarrows M$, is
a groupoid object in the category of smooth manifolds (not necessarily
Hausdorff) such that its source map is a submersion, see [14, 7]. In this
situation, $M$ is called the base of $G\rightrightarrows M$. The structure
maps of a groupoid are its source, target, multiplication, unit map and
inversion, denoted respectively
$\mathtt{s},\mathtt{t},\mathtt{m},\mathtt{u},\mathtt{i}$, we also denote
$\mathtt{m}(a,b)$ by $ab$. A Lie groupoid is a smooth groupoid if its base and
source-fibers are Hausdorff. When several groupoids are involved, we use a
subindex $\mathtt{s}=\mathtt{s}_{G}$, $\mathtt{t}_{G}$, $\mathtt{m}_{G}$ to
specify the groupoid under consideration.

A Lie groupoid $G\rightrightarrows M$ is _proper_ if the map
$(\mathtt{t},\mathtt{s}):G\rightarrow M\times M$ is proper. A Lie groupoid is
_source-simply-connected_ if its source fibres are 1-connected. _A left
(right) Lie groupoid action_ of a Lie groupoid $G\rightrightarrows M$ on a map
$J:S\rightarrow M$ is a smooth map $a:G_{\mathtt{s}}\times_{J}S\rightarrow S$
(respectively, $a:S_{J}\times_{\mathtt{t}}G\rightarrow S$) which satisfies (1)
$a(\mathtt{m}(g,h),x)=a(g,a(h,x))$ (respectively,
$a(x,\mathtt{m}(g,h))=a(a(x,g),h)$) for all $g,h\in G$ and for all $x\in S$
for which $a$ and $\mathtt{m}$ are defined and (2) $a(\mathtt{u}(J(x)),x)=x$
for all $x\in M$. For the sake of brevity, we denote the fiber product
$G_{\mathtt{t}}\times_{J}S$ by $G\times_{M}S$. In this situation,
$G\times_{M}S$ becomes a Lie groupoid over $S$ with the projection being the
source map, $a$ being the target map and the multiplication is defined by
$(g,a(h,p))(h,p)=(gh,p)$. The Lie groupoid $G\times_{M}S\rightrightarrows S$
thus defined is called an action groupoid. Another construction which is
relevant to us is the following: let $q:S\rightarrow M$ be a surjective
submersion. Then the fiber product $S\times_{M}S$ is a Lie groupoid over $S$
with the source and target maps being the projections to $S$ and the
multiplication given by $\mathtt{m}((x,y),(y,z))=(x,z)$. The Lie groupoid thus
obtained is called the submersion groupoid associated to $q$.

A Lie algebroid consists of the following data: (1) a vector bundle $A$ over a
manifold $M$, (2) a bundle map $\mathtt{a}:A\rightarrow TM$ called the anchor
and (3) a Lie algebra structure $[\,,\,]$ on $\Gamma(A)$ such that the Leibniz
rule holds:

$[u,fv]=f[u,v]+\left(\mathcal{L}_{\mathtt{a}(u)}f\right)v,$

for all $u,v\in\Gamma(A)$ and $f\in C^{\infty}(M)$. The definition of Lie
algebroid morphism is more involved, see [9, 17]. The tangent Lie algebroid
$\text{Lie}(G)$ of a Lie groupoid $G\rightrightarrows M$ is the vector bundle
$\ker T\mathtt{s}|_{M}$ endowed with the restriction of $T\mathtt{t}$ to $A$
as the anchor and with the bracket determined by $[u,v]^{r}=[u^{r},v^{r}]$ for
all $u,v\in\Gamma(A)$, where $w^{r}\in\Gamma(G)$ is defined by
$w^{r}(g)=T\mathtt{m}(w_{\mathtt{t}(g)},0_{g})$ for all $w\in\Gamma(\ker
T\mathtt{s}|_{M})$ and all $g\in G$. Also, we have that a Lie groupoid
morphism induces a Lie algebroid morphism, thereby producing a functor from
the category of Lie groupoids to the category of Lie algebroids which is
called the Lie functor [14].

A family of Lie algebroids which is relevant to us is given by the following
construction. A Lie algebroid $A$ over $M$ acts on a map $J:S\rightarrow M$ if
there is a Lie algebra morphism $\rho:\Gamma(A)\rightarrow\mathfrak{X}(S)$
such that $u_{S}:=\rho(u)$ is $J$-related to $\mathtt{a}(u)$ for all
$u\in\Gamma(A)$. In this situation, the pullback vector bundle $J^{*}A$
inherits a Lie algebroid structure over $S$ called the action Lie algebroid
structure [13]. Since $\Gamma(J^{*}A)\cong
C^{\infty}(S)\otimes_{C^{\infty}(M)}\Gamma(A)$, the Lie bracket on
$\Gamma(J^{*}A)$ is determined by extending the bracket on
$\Gamma(A)\hookrightarrow\Gamma(J^{*}A)$ using the Leibniz rule and the anchor
$\widehat{\rho}:J^{*}A\rightarrow TS$ is defined by extending $\rho$ in a
$C^{\infty}(S)$-linear fashion to $\Gamma(J^{*}A)$. The algebroid action
$\rho$ of $A$ on $J$ is complete if $\rho(u)\in\mathfrak{X}(S)$ is a complete
vector field whenever $\mathtt{a}(u)\in\mathfrak{X}(M)$ is complete.

####  2.1.1. Double Lie groupoids and LA-algebroids

######  Definition 2.1 ([2, 12]).

A double topological groupoid is a groupoid object in the category of
topological groupoids and is represented as follows:

$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S.}$

In the previous diagram, each of the sides denotes a groupoid structure and
the structure maps of $\mathcal{G}$ over $K$ are groupoid morphisms with
respect to the groupoids $\mathcal{G}\rightrightarrows H$ and
$K\rightrightarrows S$. A double Lie groupoid is a double topological groupoid
such that: (1) each of the side groupoids is a smooth groupoid, (2) $H$ and
$K$ are Lie groupoids over $S$, and (3) the double source map
$(\mathtt{s}^{H},\mathtt{s}^{K}):\mathcal{G}\rightarrow{H}\times_{S}{K}:=H_{\mathtt{s}_{H}}\times_{\mathtt{s}_{K}}K$
is a surjective submersion (the superindices $\quad{}^{H},\quad^{K}$ denote
the groupoid structures $\mathcal{G}\rightrightarrows H$,
$\mathcal{G}\rightrightarrows K$ respectively).

The infinitesimal counterpart of a double Lie groupoid is provided by the
following concept.

######  Definition 2.2 ([12]).

An LA-groupoid is a Lie groupoid $\mathcal{B}\rightrightarrows B$ such that:
(1) $\mathcal{B}$ and $B$ are Lie algebroids and all the structure maps are
Lie algebroid morphisms over the structure maps of a base groupoid
$G\rightrightarrows M$, (2) the map
$\mathtt{s}^{\prime}_{\mathcal{B}}:\mathcal{B}\rightarrow\mathtt{s}_{G}^{*}B$
induced by $\mathtt{s}_{\mathcal{B}}$ is surjective. An LA-groupoid is
_vacant_ if $\mathtt{s}^{\prime}_{\mathcal{B}}$ is an isomorphism.

The most fundamental example of an LA-groupoid is the tangent bundle of a Lie
groupoid. A _multiplicative foliation_ is an LA-subgroupoid of the tangent
groupoid of a Lie groupoid [8, 16].

##  3\. Multiplicative foliations and vacant double Lie groupoids

A Lie groupoid action on another Lie groupoid [9, Definition 3.1] induces a
Lie groupoid structure on the quotient if the action is principal on the base,
see [15, Lemma 2.1]. In this section we generalize this observation thanks to
the notion of vacant double Lie groupoid, see Theorem 3.2. Theorem 3.2 will be
the key ingredient in our proof that a Lie algebroid acting on a surjective
submersion is integrable if such an action is complete, see Theorem 4.1.

###  3.1. Vacant double Lie groupoids and their orbit spaces

The global counterpart of a vacant LA-groupoid is given by the following
notion.

######  Definition 3.1 ([12]).

A double Lie groupoid in which the double source map is a diffeomorphism is
called vacant.

In the proof of the following theorem, we shall use the following equivalent
description of a vacant double Lie groupoid. If $\mathcal{G}$ is a Lie
groupoid over $S$ and $H,K$ are Lie subgroupoids such that the multiplication
map ${H}\times_{S}K\rightarrow\mathcal{G}$ is a diffeomorphism, then
$\mathcal{G}$ is a vacant double Lie groupoid with sides ${H}$ and $K$. In
fact, each element in $\mathcal{G}$ can be written as
$\mathtt{m}(h,k)=\mathtt{m}(\overline{k},\overline{h})$ for some
$h,\overline{h}\in{H}$ $k,\overline{k}\in K$. So we get two action groupoids
$H\times_{S}K\rightrightarrows{K}$ and ${H}\times_{S}K\rightrightarrows H$ for
the actions $(h,k)\mapsto{}^{h}k:=\overline{k}$, $(h,k)\mapsto
h^{k}:=\overline{h}$ respectively. These structures make $\mathcal{G}$ with
sides $H,K$ into a vacant double Lie groupoid over $S$ and all vacant double
Lie groupoids are of this form, see [12, Thm. 2.15]. In one of the simplest
situations, if $G$ and $G^{*}$ are 1-connected complete Poisson groups in
duality, then we get a vacant double Lie groupoid on the double $G\times
G^{*}$ with sides $G$ and $G^{*}$ over a point: the action groupoid structures
over $G$ and over $G^{*}$ being the dressing actions and the group structure
on $G\times G^{*}$ is the one that integrates the double
$\mathfrak{g}\bowtie\mathfrak{g}^{*}$ of the tangent Lie bialgebra
$(\mathfrak{g},\mathfrak{g}^{*})$, see [11, 12].

######  Theorem 3.2.

Let $\mathcal{G}$ be a vacant double Lie groupoid with sides ${H},K$ over $S$.
If $H\rightrightarrows S$ is proper with trivial isotropy, then the orbit
space $K/\mathcal{G}$ inherits a unique Lie groupoid structure over $S/H$ such
that the projection $K\rightarrow K/\mathcal{G}$ is a Lie groupoid morphism.

###### Proof.

Thanks to the description in the previous paragraph, we get that
$\mathcal{G}\rightrightarrows K$ is isomorphic to the action groupoid
$H\times_{S}K\rightrightarrows K$ associated to the action
$(h,k)\mapsto{}^{h}k$. The fact that $H\rightrightarrows S$ is proper with
trivial isotropy implies that the orbit space $S/H$ inherits a unique manifold
structure such that the projection $S\rightarrow S/H$ is a submersion, see
[14, Example 5.33]111Notice that $H$ is automatically Hausdorff since the map
$(\mathtt{t}_{H},\mathtt{s}_{H}):H\rightarrow S\times S$ is injective and its
image is Hausdorff.. We shall see that the isotropy groups of
$H\times_{S}K\rightrightarrows K$ are also trivial. Suppose that $(h,k)\in
H\times_{S}K$ is such that ${}^{h}k=k$, this means that
$hk={}^{h}kh^{k}=kh^{k}$ and so $\mathtt{s}(h)=\mathtt{t}(k)=\mathtt{t}(h)$.
Then $h$ is a unit as desired. Since the $H$-action on $K$ is free, we get
that $K/\mathcal{G}=K/H$ inherits a unique manifold structure such that the
projection $K\rightarrow K/H$ is a submersion. In fact, if we take the
pullback with $\mathtt{s}_{K}$ of a slice for the natural $H$-action on $S$,
then we get a slice for the $H$-action on $K$, see the proof of [15, Lemma
2.1].

We just have to check now that the groupoid structure on $K$ descends to
$K/H$. For instance, let us check that $\mathtt{s}_{K}$ induces a well defined
map $\overline{\mathtt{s}}:K/H\rightarrow S/H$. Take $(h,k)\in H\times_{S}K$,
then $hk={}^{h}kh^{k}$ which means that
$\mathtt{s}_{K}({}^{h}k)=\mathtt{t}(h^{k})$. Since $h^{k}\in H$ satisfies that
$\mathtt{s}_{K}(h^{k})=\mathtt{s}_{K}(k)$, we get that $\mathtt{s}_{K}(k)$ and
$\mathtt{s}_{K}({}^{h}k)$ lie in the same $H$-orbit as desired. It is somewhat
less clear that the multiplication in $K/H$ is well defined. Take
$k,k^{\prime}\in K$ such that there exists $h\in H$ with the property that
$\mathtt{s}_{K}(k)=\mathtt{t}_{H}(h)$ and
$\mathtt{t}_{K}(k^{\prime})=\mathtt{s}_{H}(h)$. So we define
$\overline{\mathtt{m}}(H\cdot k,H\cdot
k^{\prime})=H\cdot\mathtt{m}_{K}(k,{}^{h}k^{\prime})$. If $h^{\prime}\in H$ is
such that ${}^{h^{\prime}}k$ is defined, let us notice that
$\mathtt{s}_{K}({}^{h^{\prime}}k)=\mathtt{t}_{K}({}^{(h^{\prime
k})h}k^{\prime})$. We have to check that $\mathtt{m}_{K}(k,{}^{h}k^{\prime})$
and $\mathtt{m}_{K}({}^{h^{\prime}}k,{}^{(h^{\prime k})h}k^{\prime})$ lie in
the same $H$-orbit. Indeed:

${}^{h^{\prime}}(k({}^{h}k^{\prime}))h^{\prime
k({}^{h}k^{\prime})}=h^{\prime}k({}^{h}k^{\prime})={}^{h^{\prime}}k(h^{\prime
k})({}^{h}k^{\prime})={}^{h^{\prime}}k({}^{(h^{\prime
k})h}k^{\prime})h^{\prime k({}^{h}k^{\prime})}$

and hence we have that $({}^{h^{\prime}}k)({}^{(h^{\prime
k})h}k^{\prime})={}^{h^{\prime}}(k({}^{h}k^{\prime}))$ as we wanted to show.
Therefore, the groupoid structure on the quotient is well defined. ∎

######  Remark 3.3.

In the previous theorem, if $\mathcal{G}\rightrightarrows K$ and
$H\rightrightarrows S$ are the action groupoids associated to a Lie groupoid
action of $G\rightrightarrows M$ on $K\rightrightarrows S$ as in [9,
Definition 3.1], then we recover [15, Lemma 2.1].

Infinitesimally, the situation described by Theorem 3.2 corresponds to a
vacant multiplicative foliation given by the LA-groupoid
$\text{Lie}(\mathcal{G})\rightrightarrows\text{Lie}(H)$ over
$K\rightrightarrows S$. In general, the leaf space of a (vacant)
multiplicative foliation does not inherit any groupoid structure compatible
with the quotient map, see [10] for a general description of the necessary and
sufficient conditions for that conclusion to hold.

###  3.2. Monodromy groupoids of vacant multiplicative foliations

Now we are going to give a simple criterion that allows us to recognize when
the monodromy groupoids of a vacant multiplicative foliation constitute a
vacant double Lie groupoid. Let $K\rightrightarrows S$ be a Lie groupoid and
let $\mathcal{B}\rightrightarrows B$ be multiplicative foliation on
$H\rightrightarrows S$ which is vacant as an LA-groupoid. In this situation,
we have the following result.

######  Proposition 3.4.

Let $\mathcal{G}$ and $H$ be the monodromy groupoids of $\mathcal{B}$ and $B$
respectively. If the source map of $K$ restricts to a covering map between
every compatible pair of $\mathcal{B}$ and $B$-orbits, then $\mathcal{G}$ is a
vacant double Lie groupoid with sides $H$ and $K$.

###### Proof.

Suppose that $\mathtt{s}_{K}$ restricted to every $\mathcal{B}$-orbit is a
covering map and let $\mathtt{s}^{H}$, $\mathtt{t}^{H}$ be the Lie groupoid
morphisms $\mathcal{G}\rightarrow H$ which integrate the source and target
maps of $\mathcal{B}\rightrightarrows B$ respectively. Take
$[a],[b]\in\mathcal{G}$ such that $\mathtt{s}^{H}(a)$ and $\mathtt{t}^{H}(b)$
are homotopic relative to their endpoints by means of $h$. Then we can lift
$h$ to a homotopy from $a$ to $a^{\prime}$ where $a^{\prime}$ is pointwise
composable with $b$, so we can define $\mathtt{m}^{H}([a],[b])$ as the class
of the path defined by $t\mapsto\mathtt{m}_{K}(a^{\prime}(t),b(t))$. This
multiplication over $H$ satisfies the interchange law with respect to
$\mathcal{G}\rightrightarrows K$. It remains to check that the double Lie
groupoid thus obtained is vacant. We claim that the double source map
$(\mathtt{s}^{H},\mathtt{s}^{K}):\mathcal{G}\rightarrow
H\times_{S}K:=H_{\mathtt{s}_{H}}\times_{\mathtt{s}_{K}}K$ is a local
diffeomorphism. In fact, since $\mathtt{s}^{H}$ integrates the source map of
$\mathcal{B}\rightrightarrows B$ which is fiberwise injective by the vacant
assumption, we have that $T\mathtt{s}^{H}$ restricted to $\ker
T\mathtt{s}^{K}$ is injective and then $T(\mathtt{s}^{H},\mathtt{s}^{K})$ has
trivial kernel. On the other hand, $\mathcal{G}$ and $H\times_{S}K$ have the
same dimension so the claim follows. Finally, the homotopy lifiting property
of the leaves implies that for every $k\in K$ and every $[a]\in H$ such that
$a(0)=\mathtt{s}_{K}(k)$, there exists a unique path $\widetilde{a}$ tangent
to $\mathcal{B}$ which starts at $k$ and satisfies
$\mathtt{s}^{H}([\widetilde{a}])=[a]$. As a consequence, the double source map
$(\mathtt{s}^{H},\mathtt{s}^{K}):\mathcal{G}\rightarrow H\times_{S}K$ is a
bijection and, therefore, it is a diffeomorphism. ∎

The covering map condition in Proposition 3.4 cannot be dropped: [1, Example
2.4] shows a vacant multiplicative foliation in which the associated monodromy
groupoids do not admit any double Lie groupoid structure.

##  4\. Complete Lie algebroid actions and vacant double Lie groupoids

As an application of the arguments developed in the previous section, we
recover the following classical result.

######  Theorem 4.1 ([6, 4]).

A Lie algebroid $A\rightarrow M$ is integrable if there exists a complete
action of $A$ on a surjective submersion $q:S\rightarrow M$ which induces an
injective anchor $q^{*}A\rightarrow TS$ on the associated action Lie
algebroid. Moreover, if the foliation induced by the $A$-action has no
vanishing cycles, then $A$ is integrable by a Hausdorff Lie groupoid.

######  Remark 4.2.

R. L. Fernandes pointed out to the author that this result was proved in [6]
under an apparently stronger assumption on the action map
$\rho:\Gamma(A)\rightarrow\mathfrak{X}(S)$: $\rho(u_{t})\in\mathfrak{X}(S)$
should be complete for every time-dependent smooth family $u_{t}\in\Gamma(A)$
such that $\mathtt{a}(u_{t})\in\mathfrak{X}(M)$ is complete. We do not know if
this condition is equivalent to the one we use here.

Let us stress that in the following proof we shall only use monodromy
groupoids of foliations so we will not need to consider Banach manifolds as in
[5, 6]. The main steps in our proof are the following:

  1. (1)

first of all, we consider the multiplicative foliation
$\mathcal{B}\rightrightarrows B$ induced by the $A$-action on the submersion
groupoid $S\times_{M}S\rightrightarrows S$ associated to $q$,

  2. (2)

we observe how the completeness assumption on the action implies that the
monodromy groupoids associated to $\mathfrak{B}\rightrightarrows B$ constitute
a vacant double Lie groupoid,

  3. (3)

finally, we apply Theorem 3.2 to the vacant double Lie groupoid thus obtained,
since the submersion groupoid is proper with trivial isotropy.

######  Lemma 4.3.

Let $\mathcal{B}\rightrightarrows B$ be a vacant multiplicative foliation on a
Lie groupoid $K\rightrightarrows S$. Suppose that there is a subspace
$\mathcal{W}\subset\Gamma(\mathcal{B})\subset\mathfrak{X}(K)$ consisting of
complete multiplicative vector fields with the following property: for every
$v\in B_{p}$ and every $p\in S$, there exists $\mathcal{X}\in\mathcal{W}$ such
that $\mathcal{X}_{p}=v$. Then the $\mathcal{B}$-orbits are coverings of the
corresponding $B$-orbits by means of $\mathtt{s}:K\rightarrow S$.

###### Proof.

Take $p\in S$ and an ordered basis $\\{e_{i}\\}_{i=1\dots k}\subset B_{p}$.
Let $\mathcal{X}_{i}\in\mathcal{W}$ be vector fields such that
$\mathcal{X}_{i}(p)=e_{i}$ for all $i$. Define a map
$\Phi:\mathbb{R}^{k}\rightarrow S$ as
$\Phi(t_{1},\dots,t_{k})=\Phi^{t_{k}}_{\mathcal{X}_{k}}\circ\dots\circ\Phi^{t_{1}}_{\mathcal{X}_{1}}(p)$,
where $\Phi^{t}_{\mathcal{X}_{i}}$ is the flow of $\mathcal{X}_{i}$. Then
$\Phi$ is an injective immersion on a neighborhood $V\subset\mathbb{R}^{k}$ of
the origin whose image $U=\Phi(V)$ is embedded in $S$ and is an open set in
the $B$-orbit $\mathcal{O}$ that passes through $p$. For every $x\in K$ such
that $\mathtt{s}(x)=p$ we can define a neighborhood of $x$ in the
$\mathcal{B}$-orbit that passes through $x$ as follows. Take
$\widetilde{U}_{x}=\\{\Phi^{t_{k}}_{\mathcal{X}_{k}}\circ\dots\circ\Phi^{t_{1}}_{\mathcal{X}_{1}}(x):(t_{1},\dots,t_{k})\in
V\\}$. We claim that $\mathtt{s}|:{\widetilde{U}_{x}}\rightarrow U$ is a
diffeomorphism. In fact, since $\mathcal{X}_{i}$ is multiplicative, it is
$\mathtt{s}$-related to $\mathcal{X}_{i}|_{S}$. So we have that
$\mathtt{s}(\Phi^{t}_{\mathcal{X}_{i}}(x))=\Phi^{t}_{\mathcal{X}_{i}}(\mathtt{s}(x))$
for all $x\in K$ and all $t\in\mathbb{R}$ and then it follows that
$\mathtt{s}(\Phi^{t_{k}}_{\mathcal{X}_{k}}\circ\dots\circ\Phi^{t_{1}}_{\mathcal{X}_{1}}(x))=\Phi(t_{1},\dots,t_{k})$
as long as $\mathtt{s}(x)=p$. As a consequence, $\mathtt{s}$ establishes a
bijection between $\widetilde{U}_{x}$ and $U$. On the other hand, since the
foliation $\mathcal{B}$ is vacant and the $\mathcal{X}_{i}$ are
multiplicative, the vector fields $\mathcal{X}_{i}$ are also pointwise
independent over $\widetilde{U}_{x}$ and hence
$\mathtt{s}|:{\widetilde{U}_{x}}\rightarrow U$ is indeed a diffeomorphism. Let
$x,y\in K$ be such that $\mathtt{s}(x)=\mathtt{s}(y)=p$. Since each
$\Phi^{t_{1}}_{\mathcal{X}_{1}}$ is an automorphism of $K$, we have that
$\widetilde{U}_{x}\cap\widetilde{U}_{y}=\emptyset$ if $x\neq y$. Then
$\mathtt{s}^{-1}(U)\cap\widetilde{\mathcal{O}}\cong\coprod_{x\in\mathtt{s}^{-1}(p)\cap\widetilde{\mathcal{O}}}\widetilde{U}_{[a]}$,
where $\widetilde{\mathcal{O}}$ is the $\mathcal{B}$-orbit that passes through
$x$. Therefore, $\mathtt{s}|:\widetilde{\mathcal{O}}\rightarrow\mathcal{O}$ is
a covering map. ∎

###### Proof of Theorem 4.1.

Take a complete action $\rho:\Gamma(A)\rightarrow\mathfrak{X}(S)$ of $A$ on
$q$ which induces an injective anchor $\widehat{\rho}:q^{*}A\rightarrow TS$,
where $q:S\rightarrow M$ is a surjective submersion. There is an induced
action of $A$ on $q\circ\mathtt{s}$, where $\mathtt{s}$ is the source map of
the submersion groupoid $K:=S\times_{M}S\rightrightarrows S$. In fact, simply
define the action map $\Gamma(A)\rightarrow\mathfrak{X}(K)$ as follows:
$u\mapsto(\rho(u),\rho(u))$ for all $u\in\Gamma(A)$. This action induces a
vacant multiplicative foliation $\mathcal{B}\rightrightarrows B$ over
$S\times_{M}S\rightrightarrows S$, where $B:=q^{*}A$ is the action Lie
algebroid associated to $\rho$. Let $\mathcal{G}$ be the monodromy groupoid of
$\mathcal{B}$ and let ${H}$ be the monodromy groupoid of $B$. In order to
apply Proposition 3.4 we have to see that the $\mathcal{B}$-orbits cover the
corresponding ${B}$-orbits by means of $\mathtt{s}$. This follows from Lemma
4.3 by putting

$\mathcal{W}=\\{(\rho(u),\rho(u))\in\mathfrak{X}(K)|\text{$\forall
u\in\Gamma(A)$ of compact support}\\}.$

Therefore, Proposition 3.4 implies that $\mathcal{G}$ is a vacant double Lie
groupoid with sides ${H}$ and $K$. Since $S\times_{M}S\rightrightarrows S$ is
proper with trivial isotropy, Theorem 3.2 implies that the orbit space
$H/\mathcal{G}$ of $\mathcal{G}\rightrightarrows H$ inherits a unique Lie
groupoid structure over $M$ such that the projection map $H\rightarrow
H/\mathcal{G}$ is a Lie groupoid morphism. Since we have that
$q^{*}\text{Lie}(H/\mathcal{G})\cong q^{*}A$, it follows that
$\text{Lie}(H/\mathcal{G})\cong A$. If the foliation induced by the $A$-action
has no vanishing cycles, then ${H}$ is Hausdorff [14, Sec. 5.2]. This implies
that the orbit space $H/\mathcal{G}$ is a Hausdorff integration of $A$, since
it coincides with the orbit space $H/(S\times_{M}S)$ for the associated lifted
action of $S\times_{M}S\rightrightarrows S$ on $H$, see the proof of Theorem
3.2. ∎

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