# Isometric Lie 2-group actions on Riemannian groupoids

Juan Sebastián Herrera-Carmona and Fabricio Valencia  J. S. Herrera-Carmona,
F. Valencia - Instituto de Matemática e Estadística, Universidade de São
Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo - Brasil.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study isometric actions of Lie $2$-groups on Riemannian groupoids by
exhibiting some of their immediate properties and implications. Firstly, we
prove an existence result which allows both to obtain 2-equivariant versions
of the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to
construct bi-invariant groupoid metrics on compact Lie $2$-groups. We provide
natural examples, transfer some classical constructions and explain how this
notion of isometric $2$-action yields a way to develop a 2-equivariant Morse
theory on Lie groupoids. Secondly, we give an infinitesimal description of an
isometric Lie $2$-group action. We define an algebra of transversal
infinitesimal isometries associated to any Riemannian $n$-metric on a Lie
groupoid which in turn gives rise to a notion of geometric Killing vector
field on a quotient Riemannian stack. If our Riemannian stack is separated
then we prove that the algebra formed by such geometric Killing vector fields
is always finite dimensional.

###### 2020 Mathematics Subject Classification:

22A22, 58H05, 58D19

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Riemannian groupoids
    2. 2.2 Lie 2-groups
  3. 3 Isometric Lie 2-group actions
    1. 3.1 Orthogonal Lie 2-groups
    2. 3.2 Some examples and constructions
  4. 4 Isometries and geometric Killing vector fields
    1. 4.1 Transversal isometries
    2. 4.2 Morita invariance of transversal Killing vector fields

##  1\. Introduction

Isometric actions of Lie groups on Riemannian manifolds have been widely
studied in the literature, as they constitute a powerful tool applied to deal
with several interesting problems in both mathematics and physics, yielding
important geometric-topological consequences and describing certain physical
phenomena in nature. They appear in all branches of science where symmetries
preserving length and angle measures play a role. In this paper we propose a
generalization of the classical setting of isometric actions by studying a
notion of isometric action of a categorified version of a Lie group [5, 9] on
a categorified version of a Riemannian manifold [18, 19, 21, 38]. More
precisely, we are interested in studying isometric actions of (strict) Lie
$2$-groups on Riemannian groupoids. Our purpose here is twofold. On the one
hand, we will be mainly concerned in studying their existence as well as their
immediate implications, exhibiting examples and transferring some classical
constructions into such a new framework. As an interesting application we will
point out how this notion of isometric $2$-action yields a way to develop a
2-equivariant Morse theory on Lie groupoids. On the other hand, we will bring
an infinitesimal description of an isometric Lie $2$-group action. Our
approach will lead us to the study of an algebra of transversal infinitesimal
isometries associated to any Riemannian $n$-metric on a Lie groupoid, which in
turn will give rise to a notion of geometric Killing vector field on a
quotient Riemannian stack.

Lie groupoids provide a general framework suitable for working with several
classical geometries, as they generalize manifolds, Lie groups, submersions,
Lie group actions, foliations, pseudo-groups, vector and principal bundles,
among others, giving lead to a new perspective on classical geometric
questions and results. Besides, these objects can be seen as an intermediate
step in defining differentiable stacks, some geometric objects admitting
singularities and generalizing both manifolds and orbifolds [15, 33]. The
detailed treatment of the interaction between these two theories provides us
with a clean way to perform differential geometry on singular spaces.

As we mentioned earlier, we want to study some features of a natural notion of
isometric action of a Lie $2$-group on a Riemannian groupoid. On the one side,
as it was explicitly mentioned by Baez and Lauda in [5], the notion of Lie
2-group goes back to Brown and Spencer in [9] where it became clear that
classical group theory is just the beginning of a larger subject that
sometimes is called higher-dimensional group theory. In many contexts where we
are tempted in using groups to tackle certain symmetries-involved problems, it
turns out actually to be more natural to use a richer kind of structure where,
in addition to group elements describing symmetries, we also have isomorphisms
between these, thus describing symmetries between symmetries. On the other
side, Riemannian groupoids recently appear into the picture as a
differentiable model allowing to carry out Riemannian geometry techniques over
more general singular spaces than orbifolds or leaf spaces of singular
foliations. Since the seminal works [18, 19], the notion of Riemannian
groupoid (stack) defined and studied therein has been applied to
satisfactorily extend several theories where Riemannian manifolds have played
an important role. For instance, some of the recent contributions in which
Riemannian groupoid metrics have been used as a tool to describe topological
and geometric features of Lie groupoids as well as their differentiable stacks
come in order.

  * •

Resolutions of proper Riemannian groupoids were introduced in [39] in order to
obtain a desingularization of their underlying differentiable stacks via a
successive blow-up construction.

  * •

A theory of stacky geodesics on Riemannian stacks was developed in [16]
allowing to establish a stacky version of the Hopf–Rinow Theorem.

  * •

The problem of understanding invariant linearization of proper Lie groupoids
was addressed in [17] where the authors fixed and extended previous results in
the literature as well as provided a sufficient criterion that uses compatible
complete metrics and covers the case of proper group actions.

  * •

A notion of Morse Lie groupoid morphism was introduced in [37], allowing to
extend the main results of classical Morse theory to the context of Lie
groupoids and differentiable stacks.

The paper is organized as follows. In Section 2 we briefly introduce the
necessary terminology and notions about Riemannian groupoids and Lie
$2$-groups we will be dealing with along the paper. In Section 3 we define our
notion of isometric Lie $2$-group action on a Riemannian groupoid:

######  Definition.

A $2$-action $\theta=(\theta^{1},\theta^{0}):(G_{1}\times
X_{1}\rightrightarrows G_{0}\times X_{0})\to(X_{1}\rightrightarrows X_{0})$ of
a Lie 2-group $G_{1}\rightrightarrows G_{0}$ on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$ is said to be _isometric_ if $G_{2}$
acts by isometries on $(X_{2},\eta^{(2)})$.

For the sake of simplicity at several stages of our work will be enough to
consider only $1$-metrics $\eta^{(1)}$ on $X_{1}$ so that in those cases we
proceed by supposing that $G_{1}\rightrightarrows G_{0}$ acts isometrically on
$(X_{1}\rightrightarrows X_{0},\eta)$ if $G_{1}$ acts by isometries on
$(X_{1},\eta^{(1)})$. It is important to mention that most of our results
below are also valid when working with general $n$-metrics on $X_{n}$ for
$n\geq 3$, see Remark 3.3. We show that we can pass to the quotient groupoid
metrics by free and proper isometric Lie $2$-group actions and, motivated by
the proof of the groupoid linearization theorem provided in [18], we explain
how to construct $2$-equivariant weak linearizations around $G_{0}$-invariant
saturated submanifolds in $X_{0}$. This is the content of Proposition 3.4 and
Proposition 3.5, respectively. Also, by a dual averaging procedure we obtain
the following result:

######  Theorem (3.7).

Let $\theta$ be a $2$-action of a Lie $2$-group $G_{1}\rightrightarrows G_{0}$
on a proper groupoid $X_{1}\rightrightarrows X_{0}$. If $G_{1}$ is compact
then there always exists a Riemannian groupoid metric on
$X_{1}\rightrightarrows X_{0}$ for which $\theta$ becomes an isometric
$2$-action.

Two interesting applications can be derived from the previous existence
result. Firstly, we can prove analogous statements to those of the Slice
Theorem and the Equivariant Tubular Neighborhood Theorem in our setting. More
precisely, we get that:

######  Theorem (3.15).

Suppose that $\theta$ is a proper $2$-action of a Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. Then, for every $x_{0}\in X_{0}$ there
exist $\epsilon>0$, another groupoid metric $\tilde{\eta}$ and a
$2$-equivariant Lie groupoid isomorphism

$\Psi:(G_{1}\times_{\textnormal{Iso}_{G_{1}}(1_{x_{0}})}S_{1_{x_{0}}}\rightrightarrows
G_{0}\times_{\textnormal{Iso}_{G_{0}}(x_{0})}S_{x_{0}})\xrightarrow[]{\cong}(\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})\rightrightarrows\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})).$

Here $S_{1_{x_{0}}}\rightrightarrows S_{x_{0}}$ denotes a Lie subgroupoid
slice at $x_{0}\in X_{0}$ (see Definition 3.10), which always exists by
Proposition 3.12 and $\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})\rightrightarrows\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})$ is the natural Lie groupoid between the tubular neighborhoods of the
orbits $G_{1}\cdot 1_{x_{0}}$ in $X_{1}$ and $G_{0}\cdot x_{0}$ in $X_{0}$.
Secondly, in Subsection 3.1 we define orthogonal Lie $2$-groups (i.e. Lie
$2$-group endowed with bi-invariant Riemannian groupoid metrics), thus proving
that:

######  Proposition (3.19).

Every Lie $2$-group $G_{1}\rightrightarrows G_{0}$ with $G_{1}$ compact can be
equipped with a bi-invariant Riemannian groupoid metric.

We also describe the infinitesimal counterpart of an orthogonal Lie $2$-group,
comparing it with an existing notion of $\mathcal{L}$-orthogonal crossed
module of Lie algebras already known in the literature [4, 20]. In Subsection
3.2 we present several natural examples and transfer two classical
constructions called principal connection warpings and Cheeger deformations
into our new framework. We also explain how our notion of isometric $2$-action
can be used to develop a $2$-equivariant analogous of the results proved in
[37] which are aimed at extending classical Morse theory to the context of Lie
groupoids and their differentiable stacks, see Example 3.30. More precisely,
let $f:X_{0}\to\mathbb{R}$ be a $G_{0}$-invariant basic function which
satisfies a Morse–Bott (normal) nondegenerate condition along its critical
submanifolds. Thus, based on the results proved in [37] we indicate how to get
a $2$-equivariant version of the Morse lemma around a $G_{0}$-invariant
nondegenerate critical saturated submanifold of $f$, how to describe the
topological behavior of the level subgroupoids of $f$ in a $2$-equivariant
way, and also how to construct an equivariant Morse–Bott double cochain
complex which computes the equivariant cohomology of a 2-action as defined in
[7].

In Section 4 we give an infinitesimal description of an isometric Lie
$2$-group action. In order to do so we define the Lie 2-group of strong
isometries of $(X_{1}\rightrightarrows X_{0},\eta)$ which is associated to a
natural crossed module structure
$(\mathrm{Iso}(X,\eta),\textnormal{Bis}_{\eta}(X),I,\alpha)$ introduced in
Proposition 4.7. The infinitesimal counterpart of such a Lie $2$-group of
isometries is the Lie $2$-algebra of strong multiplicative Killing vector
fields of $(X_{1}\rightrightarrows X_{0},\eta)$ which is the defined by means
of the crossed module structure
$(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D)$ from Proposition 4.9. By
using those crossed module structures we show that:

######  Theorem (4.10).

Let $\theta$ be an isometric right 2-action of a Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. Denote by $(G,H,\rho,\alpha)$ to the
crossed module of Lie groups associated to $G_{1}\rightrightarrows G_{0}$ and
by $(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ to its corresponding
crossed module of Lie algebras. Then:

  * •

there is a morphism of crossed modules of Lie groups

$(\sigma,\Sigma):(G,H,\rho,\alpha)\to(\textnormal{Iso}(X,\eta),\textnormal{Bis}_{\eta}(X),I,\alpha),$

that is defined as $\sigma_{h}(x)=1_{x}h$ and $\Sigma_{g}(p)=p1_{g}$, and

  * •

there is a morphism of crossed modules of Lie algebras

$(j_{-1},j_{0}):(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})\to(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D),$

which is defined as $j_{-1}=\left.\tilde{\xi}\right|_{X_{0}}$ and
$j_{0}(\zeta)=(\tilde{1_{\zeta}},\tilde{\zeta})$ for all $\xi\in\mathfrak{h}$
and $\zeta\in\mathfrak{g}$.

Here the symbol $\tilde{}$ stands to denote the fundamental vector field
associated to an element in a Lie algebra with respect to a given Lie group
action. It is important to mention that our infinitesimal description is
motivated by those results developed in [36] in order to describe the Lie
2-algebra structure that the set of multiplicative vector fields on a Lie
groupoid has. In Subsection 4.1 we weaken the condition for a diffeomorphism
to be an isometry by imposing instead a transversal isometric condition along
groupoid orbits. This leads us to define the Lie $2$-algebra of weak Killing
multiplicative vector fields associated to the crossed module
$(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$ which is a
well defined object for any Riemannian groupoid $n$-metric. By adapting some
of the results proved in [36] to our context, in Subsection 4.2 we get that:

######  Theorem 1.1 (4.16).

If $(X_{1}\rightrightarrows X_{0},\eta^{X})$ and $(Y_{1}\rightrightarrows
Y_{0},\eta^{Y})$ are Morita equivalent Riemannian groupoids then the crossed
modules $(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$ and
$(\mathfrak{o}_{m}^{\textnormal{w}}(Y),\Gamma(A_{Y}),\delta,D)$ are isomorphic
in the derived category of crossed modules. In consequence, the following
quotient spaces are isomorphic as Lie algebras:

$\mathfrak{o}_{m}^{\textnormal{w}}(X)/\textnormal{im}(\delta)\cong\mathfrak{o}_{m}^{\textnormal{w}}(Y)/\textnormal{im}(\delta).$

The elements of the Lie algebra
$\mathfrak{o}_{m}^{\textnormal{w}}(X)/\textnormal{im}(\delta)$ are called
geometric Killing vector fields. It is worth mentioning that such a notion of
Killing vector field recovers the classical notions of Killing vector field on
both a Riemannian manifold and a Riemannian orbifold as defined for instance
in [6] as well as the notion of transverse Killing vector field on a regular
Riemannian foliation as defined in [34, p. 84]. If $([X_{0}/X_{1}],[\eta])$
denotes the quotient Riemannian stack presented by $(X_{1}\rightrightarrows
X_{0},\eta)$ then by using the desingularization theorem for proper Riemannian
groupoids proved in [39] we show:

######  Theorem 1.2 (4.19).

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a proper Riemannian groupoid with
compact orbit space $X_{0}/X_{1}$. Then the algebra of geometric Killing
vector fields on $([X_{0}/X_{1}],[\eta])$ has finite dimension.

Finally, by averaging with respect to a proper Haar measure system for
$(X_{1}\rightrightarrows X_{0},\eta)$ we prove that there exists a projection
from what we call the space of projectable vector fields of Killing type to
the space of weak multiplicative Killing vector fields, see Theorem 4.20.
Hence, motivated by some results proved in [13] we give another face to the
space of geometric Killing vectors fields by using Killing invariant sections,
thus obtaining that Theorem 4.20 provides us with a method to construct
geometric Killing vector fields over the quotient Riemannian stack
$([X_{0}/X_{1}],[\eta])$.

Acknowledgments: We would like to thank Mateus de Melo, Cristian Ortiz and
Luca Vitagliano for several enlightened discussions, comments, and suggestions
which allowed us to improve this work. Valencia is grateful to Matias del Hoyo
for inviting and supporting him to visit IMPA in Rio de Janeiro in December,
2022. Their mathematical discussions during the academic visit encouraged the
authors to start enhancing the presentation and the results stated in the
first version of the present work. We are also thankful to the anonymous
referees who provided many suggestions and corrections that improved the
quality of our results. Herrera-Carmona was supported by the Brazilian Federal
Foundation CAPES - Finance Code 001. Valencia was supported by Grant
2020/07704-7 Sao Paulo Research Foundation - FAPESP.

##  2\. Preliminaries

In this short section we briefly introduce the basics on Lie groupoids we
shall be using throughout. For specific details the reader is recommended to
visit for instance [5, 9, 15, 18, 28].

A _Lie groupoid_ $X_{1}\rightrightarrows X_{0}$ consists of a manifold $X_{0}$
of objects and a manifold $X_{1}$ of arrows, two surjective submersions
$s_{X},t_{X}:X_{1}\to X_{0}$ respectively indicating the source and the target
of the arrows, and a smooth associative composition $m_{X}:X_{2}\to X_{1}$
over the set of composable arrows $X_{2}=X_{1}\times_{X_{0}}X_{1}$, admitting
unit $u_{X}:X_{0}\to X_{1}$ and inverse $i_{X}:X_{1}\to X_{1}$, subject to the
usual groupoid axioms. The collection of maps mentioned above are called
_structural maps_ of the Lie groupoid $X_{1}\rightrightarrows X_{0}$. We shall
drop up the sub-index notation for the structural maps only if there is no
risk of confusion. Special instances of Lie groupoids are given by manifolds,
Lie groups, Lie group actions, surjective submersions, foliations,
pseudogroups, principal bundles, vector bundles, among others [15, 28].

Let $X_{1}\rightrightarrows X_{0}$ be a Lie groupoid. For each $x\in X_{0}$,
its _isotropy group_ $X_{x}:=s_{X}^{-1}(x)\cap t_{X}^{-1}(x)$ is a Lie group
and an embedded submanifold in $X_{1}$. There is an equivalence relation in
$X_{0}$ defined by $x\sim y$ if there exists $p\in X_{1}$ with $s_{X}(p)=x$
and $t_{X}(p)=y$. The corresponding equivalence class of $x\in X_{0}$ is
denoted by $\mathcal{O}_{x}\subseteq X_{0}$ and called the _orbit_ of $x$. The
previous equivalence relation defines a quotient space $X_{0}/X_{1}$ called
the _orbit space_ of $X_{1}\rightrightarrows X_{0}$. This space equipped with
the quotient topology is in general a _singular space_ , that is, it does not
carry a differentiable structure making the quotient projection $X_{0}\to
X_{0}/X_{1}$ a surjective submersion.

A _Lie groupoid morphism_ between two Lie groupoids $X_{1}\rightrightarrows
X_{0}$ and $X_{1}^{\prime}\rightrightarrows X_{0}^{\prime}$ is a pair
$\phi:=(\phi^{1},\phi^{0})$ where $\phi^{1}:X_{1}\to X_{1}^{\prime}$ and
$\phi^{0}:X_{0}\to X_{0}^{\prime}$ are smooth maps commuting with both source
and target maps and preserving the composition maps. The _tangent groupoid_
$TX_{1}\rightrightarrows TX_{0}$ is obtained by applying the tangent functor
at both manifolds of arrows and objects and all of its structural maps. If
$S\subset X_{0}$ is a saturated manifold (only an orbit for instance) then we
can restrict the groupoid structure to $X_{S}=s_{X}^{-1}(S)=t_{X}^{-1}(S)$,
thus obtaining a Lie subgroupoid $X_{S}\rightrightarrows S$ of
$X_{1}\rightrightarrows X_{0}$. Furthermore, the Lie groupoid structure of
$TX_{1}\rightrightarrows TX_{0}$ can be restricted to define a new Lie
groupoid $\nu(X_{S})\rightrightarrows\nu(S)$ having the property that all of
its structural maps are fiberwise isomorphisms. The _Lie algebroid_ associated
to the Lie groupoid $X_{1}\rightrightarrows X_{0}$ is defined to be the vector
bundle $A_{X}:=\textnormal{ker}(ds_{X})|_{X_{0}}\subset TX_{1}$ with anchor
map $\rho:A_{X}\to TX_{0}$ obtained by restricting $dt_{X}:TX_{1}\to TX_{0}$
to $A_{X}$.

###  2.1. Riemannian groupoids

The notion of Riemannian metric on a Lie groupoid that we will be dealing with
along the paper was introduced in [18] (see also [21, 38]). Such a notion is
compatible with the groupoid composition so that it plays an important role in
several parts of our work. We start by recalling that a submersion
$\pi:(E,\eta^{E})\to B$ with $(E,\eta^{E})$ a Riemannian manifold is said to
be _Riemannian_ if the fibers of it are equidistant (transverse condition). In
this case the base $B$ gets an induced metric $\eta^{B}:=\pi_{\ast}\eta^{E}$
for which the linear map $d\pi(e):(\textnormal{ker}(d\pi(e)))^{\perp}\to
T_{\pi(e)}B$ is an isometry for all $e\in E$. If $(\eta^{E})^{\ast}$ denotes
the dual metric associated to $\eta^{E}$ then the condition for a Riemannian
submersion can be rephrased as follows. For all $e\in E$ the map
$d\pi(e)^{\ast}:T_{\pi(e)}^{\ast}B\to\textnormal{ker}(d\pi(e))^{\circ}$ is an
isometry, where $\textnormal{ker}(d\pi(e))^{\circ}$ denotes the annihilator of
the vectors tangent to the fiber. If $\pi:E\to B$ is a surjective submersion
then a Riemannian metric $\eta^{E}$ on $E$ is said to be _transverse_ to $\pi$
if for all $x\in B$ and all $e_{1},e_{2}\in\pi^{-1}(x)$ we have that the map

$d\pi(e_{1})^{\ast}\circ(d\pi(e_{2})^{\ast})^{-1}:\textnormal{ker}(d\pi(e_{2}))^{\circ}\to
T_{x}^{\ast}B\to\textnormal{ker}(d\pi(e_{1}))^{\circ},$

is a linear isometry. In this case, there exists a unique metric $\eta^{B}$ on
$B$ such that $\pi$ becomes a Riemannian submersion. Such a metric is defined
by the expression $\eta^{B}(d\pi(v),d\pi(w)):=\eta^{E}(v,w)$ for
$v,w\in\textnormal{ker}(d\pi)^{\perp}$ and is called the _push-forward
metric_. The notation we shall be using for the previous Riemannian metric is
$\eta^{B}:=\pi_{\ast}\eta^{E}$.

It is well known that given a Lie groupoid $X_{1}\rightrightarrows X_{0}$
every pair of composable arrows in $X_{2}$ may be identified with an element
in the space of commutative triangles so that it admits an action of $S_{3}$
determined by permuting the vertices of such triangles. In these terms, a
_Riemannian groupoid_ is a pair $(X_{1}\rightrightarrows X_{0},\eta)$ where
$X_{1}\rightrightarrows X_{0}$ is a Lie groupoid and $\eta=\eta^{(2)}$ is a
Riemannian metric on $X_{2}$ that is invariant by the $S_{3}$-action and
transverse to the composition map $m_{X}:X_{2}\to X_{1}$. The metric
$\eta^{(2)}$ induces metrics
$\eta^{(1)}=((\pi_{2})_{X})_{\ast}\eta^{(2)}=(m_{X})_{\ast}\eta^{(2)}=((\pi_{1})_{X})_{\ast}\eta^{(2)}$
on $X_{1}$ and $\eta^{(0)}=(s_{X})_{\ast}\eta^{(1)}=(t_{X})_{\ast}\eta^{(1)}$
on $X_{0}$ such that $(\pi_{2})_{X},m_{X},(\pi_{1})_{X}:X_{2}\to X_{1}$ and
$s_{X},t_{X}:X_{1}\to X_{0}$ are Riemannian submersions and $i_{X}:X_{1}\to
X_{1}$ is an isometry. This is because the $S_{3}$-action permutes these face
maps.

The metric $\eta^{(j)}$, for $j=2,1,0$, is called a $j$-_metric_. It is
important to mention that every proper groupoid can be endowed with a
$2$-metric (more generally, an $n$-metric as defined below) and if a Lie
groupoid admits a $2$-metric then it is weakly linearizable around any
saturated submanifold. For more details visit [18].

######  Remark 2.1.

The notion of _$n$ -metric_ on Lie groupoids for $n\geq 3$ was introduced in
[18]. This is just a Riemannian metric on the set of $n$-composable arrows
$X_{n}$ that is invariant by the canonical $S_{n+1}$-action on $X_{n}$ and
transverse to one (hence to all) face map $X_{n}\to X_{n-1}$. We can push this
$n$-metric forward with the different face maps $X_{n}\to X_{n-1}$ to define
an $(n-1)$-metric on $X_{n-1}$ in such a way these face maps become Riemannian
submersions. One can use this process to obtain $r$-metrics $\eta^{(r)}$ on
$X_{r}$ for all $0\leq r\leq n-1$ so that we get Riemannian submersions
$(X_{r},\eta^{(r)})\to(X_{r-1},\eta^{(r-1)})$.

###  2.2. Lie 2-groups

A _Lie 2-group_ is a group internal to the category of Lie groupoids. In other
words, a Lie 2-group is a Lie groupoid $G_{1}\rightrightarrows G_{0}$ where
both $G_{1}$ and $G_{0}$ are Lie groups and the structural maps of
$G_{1}\rightrightarrows G_{0}$ are Lie group homomorphisms [5, 9]. We will
denote by $\ast$ the composition of arrows in $G_{1}\rightrightarrows G_{0}$
and by $\cdot$ the product of arrows in $G_{1}$. For instance, the fact that
the multiplication is a morphism of groups amounts to the identity:

$(g_{1}\ast g_{2})\cdot(g_{1}^{\prime}\ast g_{2}^{\prime})=(g_{1}\cdot
g_{1}^{\prime})\ast(g_{2}\cdot
g_{2}^{\prime}),\quad\forall(g_{1},g_{2}),(g_{1}^{\prime},g_{2}^{\prime})\in
G_{2},$ (1)

where $G_{2}$ is the space of pairs of composable arrows of
$G_{1}\rightrightarrows G_{0}$.

It is well known that there are several alternative ways to think of Lie
2-groups. One of them is described in terms of crossed modules. Recall that a
_crossed module of Lie groups_ is a quadruple $(G,H,\rho,\alpha)$ consisting
of a Lie group homomorphism $\rho:H\to G$ together with an action $\alpha$ of
$G$ on $H$, $(g,h)\mapsto gh$, by Lie group automorphisms such that:

$\rho(gh)=g\rho(h)g^{-1},\quad\rho(h)h^{\prime}=hh^{\prime}h^{-1},\quad g\in
G,\ h,h^{\prime}\in H.$

To such a crossed module one associates a Lie $2$-group with objects the Lie
group $G_{0}:=G$, arrows the semi-direct product $G_{1}:=H\rtimes G$ so that:

$(h_{1},g_{1})\cdot(h_{2},g_{2}):=(h_{1}(g_{1}h_{2}),g_{1}g_{2}),$

and structure maps:

$s_{G}(h,g)=g,\quad
t_{G}(h,g)=\rho(h)g,\quad(h_{1},\rho(h_{2})g_{2})\ast(h_{2},g_{2})=(h_{1}h_{2},g_{2}).$

Conversely, any Lie 2-group $G_{1}\rightrightarrows G_{0}$ has an associated
crossed module of Lie groups $(G,H,\rho,\alpha)$, where $G=G_{0}$,
$H=\textnormal{ker}(s_{G})$, $\rho:=t_{G}|_{H}:H\to G$ and $G$ acts on $H$ by
conjugation via the identity bisection: $gh:=1_{g}\cdot h\cdot 1_{g^{-1}}$.
This defines an equivalence of categories between the category of Lie 2-groups
and the category of crossed modules of Lie groups [5].

A _Lie 2-algebra_ is a Lie algebra internal to the category of Lie groupoids.
More concretely, a Lie 2-algebra is a Lie groupoid
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ where both
$\mathfrak{g}_{1}$ and $\mathfrak{g}_{0}$ are Lie algebras and the structural
maps of $\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ are Lie algebra
homomorphisms. As expected, the Lie functor provides a correspondence between
Lie 2-groups and Lie 2-algebras.

The infinitesimal object associated to a crossed module of Lie groups is the
so-called _crossed module of Lie algebras_. This is a quadruple
$(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ where $\mathfrak{g}$ and
$\mathfrak{h}$ are Lie algebras and $\partial:\mathfrak{h}\to\mathfrak{g}$ and
$\mathcal{L}:\mathfrak{g}\to\textnormal{Der}(\mathfrak{h})$ are Lie algebra
homomorphisms verifying

$\partial(\mathcal{L}_{x}y)=[x,\partial(y)]_{\mathfrak{g}}\quad\textnormal{and}\quad\mathcal{L}_{\partial(x)}y=[x,y]_{\mathfrak{h}}.$
(2)

A Lie $2$-algebra $\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ has an
associated crossed module given by the data
$\mathfrak{h}=\textnormal{ker}(s)$, $\mathfrak{g}=\mathfrak{g}_{0}$,
$\partial=t|_{\mathfrak{h}}$ and $\mathcal{L}_{x}=\textnormal{ad}^{1}_{u(x)}$
for all $x\in\mathfrak{g}$. Conversely, a crossed module of Lie algebras
$(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ has an associated Lie
$2$-algebra which in turn is given by the data
$\mathfrak{g}_{1}=\mathfrak{h}\rtimes\mathfrak{g}$ with Lie algebra structure
provided by the semi-direct product with respect to $\mathcal{L}$,
$\mathfrak{g}_{1}=\mathfrak{g}$, and structural maps

$s(x,y)=y,\quad t(x,y)=\partial(x)+y,\quad u(y)=(0,y),\quad
i(x,y)=(-x,y+\partial(x)),$
$m((x^{\prime},y+\partial(x)),(x,y))=(x+x^{\prime},y).$

A _morphism_ of crossed modules
$f:(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})\to(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\partial^{\prime},\mathcal{L}^{\prime})$
consists of two Lie algebra homomorphisms
$f_{0}:\mathfrak{g}\to\mathfrak{g}^{\prime}$ and
$f_{1}:\mathfrak{h}\to\mathfrak{h}^{\prime}$ such that
$f_{0}\circ\partial=\partial^{\prime}\circ f_{1}$ and
$f_{1}(\mathcal{L}_{x}y)=\mathcal{L}^{\prime}_{f_{0}(x)}(f_{1}(y))$. This
induces a pair of Lie algebra homomorphisms
$\textnormal{ker}(\partial)\to\textnormal{ker}(\partial^{\prime})$ and
$\textnormal{coker}(\partial)\to\textnormal{coker}(\partial^{\prime})$. A
morphism of crossed modules is a _quasi-isomorphism_ if both of these Lie
algebra morphisms are isomorphisms. The _derived category_ of crossed modules
of Lie algebras is defined to be the localization of the category of crossed
modules of Lie algebras obtained by inverting all quasi-isomorphisms.

##  3\. Isometric Lie 2-group actions

In the sequel we shall denote by $G_{1}\rightrightarrows G_{0}$ a Lie 2-group
and by $(X_{1}\rightrightarrows X_{0},\eta)$ a Riemannian groupoid. A _left
2-action_ of $G_{1}\rightrightarrows G_{0}$ on $X_{1}\rightrightarrows X_{0}$
is defined to be a Lie groupoid morphism
$\theta=(\theta^{1},\theta^{0}):(G_{1}\times X_{1}\rightrightarrows
G_{0}\times X_{0})\to(X_{1}\rightrightarrows X_{0})$ such that both maps
$\theta^{1}$ and $\theta^{0}$ are usual left Lie group actions. Here
$G_{1}\times X_{1}\rightrightarrows G_{0}\times X_{0}$ denotes the product Lie
groupoid. We shall say that $\theta$ is a free (resp. proper) $2$-action if
both $\theta^{1}$ and $\theta^{0}$ are free (resp. proper) actions. Note that
given a left 2-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$ we immediately get that the structural maps of
$X_{1}\rightrightarrows X_{0}$ are equivariant with respect the structural
maps of $G_{1}\rightrightarrows G_{0}$. More precisely, if $p\in X_{1},x\in
X_{0}$ and $g\in G_{1},g_{0}\in G_{0}$ then we obtain that

$s_{X}(gp)=s_{G}(g)s_{X}(p),\quad t_{X}(gp)=t_{G}(g)t_{X}(p),\quad
u_{X}(g_{0}x)=u_{G}(g_{0})u_{X}(x).$ (3)

Moreover, we have that the action on arrows is _multiplicative_ , meaning that
for pairs of composable arrows $(p,q)\in X_{2}$ and $(g,h)\in G_{2}$ the
following formula holds true

$(p*q)(g*h)=(pg)*(qh).$ (4)

Here we are denoting by $m_{X}(p,q)=p*q$ and $m_{G}(g,h)=g*h$. It is clear
that $G_{2}$ is also a Lie group with the structure induced from the direct
product $G_{1}\times G_{1}$ and that the $2$-action $\theta$ indices a
canonical left action $\theta^{2}$ of $G_{2}$ on $X_{2}$.

######  Definition 3.1.

A $2$-action of $G_{1}\rightrightarrows G_{0}$ on $(X_{1}\rightrightarrows
X_{0},\eta)$ is said to be _isometric_ if $G_{2}$ acts by isometries on
$(X_{2},\eta^{(2)})$.

An immediate consequence of the previous definition is the following.

######  Lemma 3.2.

The action of $G_{1}$ on $(X_{1},\eta^{(1)})$ is by isometries. Consequently,
the action of $G_{0}$ on $(X_{0},\eta^{(0)})$ is also by isometries.

###### Proof.

For simplicity we shall verify that if $G_{1}$ acts on $(X_{1},\eta^{(1)})$ by
isometries then $G_{0}$ acts on $(X_{0},\eta^{(0)})$ by isometries as well.
Let $v,w\in T_{x}X_{0}$ and $g_{0}\in G_{0}$. It is clear that there are
$\tilde{v},\tilde{w}\in\textnormal{ker}(d(s_{X})_{p})^{\perp_{\eta^{(1)}}}$
with $s_{X}(p)=x$ such that $d(s_{X})_{p}(\tilde{v})=v$ and
$d(s_{X})_{p}(\tilde{w})=w$ and there is $g\in G_{1}$ such that
$s_{G}(g)=g_{0}$. Therefore,

$\displaystyle(\theta^{0}_{g_{0}})_{x}^{\ast}\eta^{(0)}(v,w)$ $\displaystyle=$
$\displaystyle\eta^{(0)}_{g_{0}x}(d(\theta^{0}_{g_{0}})_{x}(v),d(\theta^{0}_{g_{0}})_{x}(w))=\eta^{(0)}_{g_{0}x}(d(\theta^{0}_{g_{0}}\circ
s_{X})_{p}(\tilde{v}),d(\theta^{0}_{g_{0}}\circ s_{X})_{p}(\tilde{w}))$
$\displaystyle\star$ $\displaystyle=$
$\displaystyle\eta^{(0)}_{g_{0}x}(d(s_{X}\circ\theta^{1}_{g})_{p}(\tilde{v}),d(s_{X}\circ\theta^{1}_{g})_{p}(\tilde{w}))=\eta^{(1)}_{gp}(d(\theta^{1}_{g})_{p}(\tilde{v}),d(\theta^{1}_{g})_{p}(\tilde{w}))$
$\displaystyle=$
$\displaystyle\eta^{(1)}_{p}(\tilde{v},\tilde{w})=\eta^{(0)}_{x}(v,w).$

In the equality $\star$ above we used the fact that $s_{X}$ is a Riemannian
submersion and that the action $\theta^{1}$ preserves the horizontal
distribution $\textnormal{ker}(ds_{X})^{\perp_{\eta^{(1)}}}$. Note that this
computation does not depend on the choice of $\tilde{v},\tilde{w},p$ and $g$.
Furthermore, we may obtain the same conclusion by choosing
$\eta^{(0)}=(t_{X})_{\ast}\eta^{(1)}$ instead of
$\eta^{(0)}=(s_{X})_{\ast}\eta^{(1)}$. The fact that $G_{1}$ acts on
$(X_{1},\eta^{(1)})$ by isometries provided that $G_{2}$ acts on
$(X_{2},\eta^{(2)})$ by isometries can be similarly shown by using that
$\eta^{(1)}=(\pi_{2,X})_{\ast}\eta^{(2)}=(m_{X})_{\ast}\eta^{(2)}=(\pi_{1,X})_{\ast}\eta^{(2)}$
and that $\pi_{2,X},m_{X},\pi_{1,X}:X_{2}\to X_{1}$ are Riemannian submersions
verifying analogous equivariant relations as those in Equation (3) with
respect to the maps $\pi_{2,G},m_{G},\pi_{1,G}:G_{2}\to G_{1}$. ∎

######  Remark 3.3.

More generally, by using exactly the same arguments as in Lemma 3.2 it is
simple to check that if $\eta^{(n)}$ is an $n$-metric on $X_{n}$ and the
induced left action $\theta^{n}$ of $G_{n}$ on $(X_{n},\eta^{(n)})$ is by
isometries then the action $\theta^{k}$ of $G_{k}$ on $(X_{k},\eta^{(k)})$
will be by isometries for all $0\leq k\leq n-1$.

We warn the reader that for the sake of simplicity at several stages of our
study will be enough to consider only $1$-metrics $\eta^{(1)}$ on $X_{1}$ so
that in those cases we proceed by supposing that $G_{1}\rightrightarrows
G_{0}$ acts isometrically on $(X_{1}\rightrightarrows X_{0},\eta)$ if $G_{1}$
acts by isometries on $(X_{1},\eta^{(1)})$. This is because the computations
as well as the arguments will be both canonical and analogous if we consider
$2$-metrics or even $n$-metrics in general. Indeed, most of the notions and
results we will introduce below shall have an analogous statement if the
simplicial approach from Remark 3.3 is considered.

Two interesting consequences that come up from our definition of isometric Lie
$2$-group action are the following. Firstly, we can pass to the quotient
groupoid metrics by free and proper isometric $2$-actions in a natural
fashion, thus obtaining examples of Riemannian groupoid submersions as defined
in [19, Def. 3.2.1].

######  Proposition 3.4.

If $\theta$ is a free and proper isometric $2$-action of
$G_{1}\rightrightarrows G_{0}$ on $(X_{1}\rightrightarrows X_{0},\eta)$ then
there is a structure of Riemannian groupoid $(X_{1}/G_{1}\rightrightarrows
X_{0}/G_{0},\overline{\eta})$ so that the canonical projection
$\pi=(\pi_{1},\pi_{0}):(X_{1}\rightrightarrows
X_{0})\to(X_{1}/G_{1}\rightrightarrows X_{0}/G_{0})$ becomes a Riemannian
groupoid submersion.

###### Proof.

We carry out the proof for $1$-metrics on $X_{1}$ since the computations are
analogous if we consider instead $2$-metrics on $X_{2}$. We start by
exhibiting the Lie groupoid structure on $X_{1}/G_{1}\rightrightarrows
X_{0}/G_{0}$. This fact was stated for instance in [22, Prop. 3.6] but without
proof so that we exhibit a proof of it by sake of completeness. It is well
known that $X_{j}/G_{j}$ admits a unique manifold structure so that $\pi_{j}$
(for $j=0,1$) is a surjective submersion. We define source and target maps
respectively as $\overline{s}([p])=[s_{X}(p)]$ and
$\overline{t}([p])=[t_{X}(p)]$ for all $p\in X_{1}$. As consequence of
Identities (3) these maps are well defined and, moreover, both of them are
surjective submersions since $\pi_{j}$, $s$, and $t$ are so. We have that
$([p],[q])\in(X_{1}/G_{1})_{2}$ if and only if
$\overline{s}([p])=\overline{t}([q])$ which in turn holds true if and only if
$s_{X}(p)=g_{0}t_{X}(q)$ for some $g_{0}\in G_{0}$. Thus, we define
$\overline{m}([p],[q])=[m_{X}(p,gq)]$ for some $g\in G_{1}$ such that
$t_{G}(g)=g_{0}$. It is simple to check that $\overline{m}$ does not depend on
the choice of $g$ and it is well defined because of Property (4). This is also
clearly smooth and associative since $m_{X}$ is associative and Identity (4)
is satisfied. The unit map and the inversion are respectively defined by
$\overline{u}([x])=[u_{X}(x)]$ and $\overline{i}([p])=[i_{X}(p)]$ for all
$x\in X_{0}$ and $p\in X_{1}$. It is also easy to verify that these maps are
well defined, smooth, and they satisfy the required groupoid conditions.

Consider, for $j=0,1$, the induced Riemannian metric
$\overline{\eta}^{(j)}=(\pi_{j})_{\ast}\eta^{(j)}$ on $X_{j}/G_{j}$ making of
$\pi_{j}$ a Riemannian submersion. Moreover, note that

$\overline{i}_{\ast}\overline{\eta}^{(1)}=(\overline{i}\circ\pi_{1})_{\ast}\eta^{(1)}=(\pi_{1}\circ
i_{X})_{\ast}\eta^{(1)}=(\pi_{1})_{\ast}\eta^{(1)}=\overline{\eta}^{(1)},$

since $i$ is an isometry, and

$\overline{s}_{\ast}\overline{\eta}^{(1)}=(\overline{s}\circ\pi_{1})_{\ast}\eta^{(1)}=(\pi_{0}\circ
s_{X})_{\ast}\eta^{(1)}=(\pi_{0})_{\ast}\eta^{(0)}=(\pi_{0}\circ
t_{X})_{\ast}\eta^{(1)}=(\overline{t}\circ\pi_{1})_{\ast}\eta^{(1)}=\overline{t}_{\ast}\overline{\eta}^{(1)},$

so that
$\overline{s}_{\ast}\overline{\eta}^{(1)}=\overline{t}_{\ast}\overline{\eta}^{(1)}=\overline{\eta}^{(0)}$.
Therefore, $(X_{1}/G_{1}\rightrightarrows X_{0}/G_{0},\overline{\eta})$ with
$\overline{\eta}=(\overline{\eta}^{(1)},\overline{\eta}^{(0)})$ is a
Riemannian groupoid and $\pi=(\pi_{1},\pi_{0})$ is a Riemannian submersion of
groupoids, as desired. ∎

Secondly, we can obtain $2$-equivariant weak linearizations around
$G_{0}$-invariant saturated submanifolds in $X_{0}$. Namely, let $\theta$ be a
$2$-action of $G_{1}\rightrightarrows G_{0}$ on $X_{1}\rightrightarrows X_{0}$
and let $S\subset X_{0}$ be a $G_{0}$-invariant saturated submanifold. We say
that $X_{1}\rightrightarrows X_{0}$ is _$2$ -equivariant weakly linearizable_
at $S$ if there are $G$-invariant Lie groupoid neighborhoods
$\widetilde{V}\rightrightarrows V$ of $X_{S}\rightrightarrows S$ in
$\nu(X_{S})\rightrightarrows\nu(S)$ (seen as the zero section) and
$\widetilde{U}\rightrightarrows U$ of $X_{S}\rightrightarrows S$ in
$X_{1}\rightrightarrows X_{0}$, and a $2$-equivariant Lie groupoid isomorphism
$\phi:(\widetilde{V}\rightrightarrows
V)\xrightarrow[]{\cong}(\widetilde{U}\rightrightarrows U)$ which is the
identity on $X_{S}\rightrightarrows S$.

The $G$-invariant property of the Lie groupoid neighborhood in
$\nu(X_{S})\rightrightarrows\nu(S)$ used above makes sense because of the
following facts. Let us assume this time that we are given with a $2$-metric
$\eta^{(2)}$ on $X_{2}$ and that $G_{2}$ acts on $(X_{2},\eta^{(2)})$ by
isometries. It is simple to check that every $2$-action
$\theta=(\theta^{1},\theta^{0})$ of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$ induces a $2$-action
$T\theta=(T\theta^{1},T\theta^{0})$ of $G_{1}\rightrightarrows G_{0}$ on
$TX_{1}\rightrightarrows TX_{0}$ by differentiating the actions $\theta^{1}$
and $\theta^{0}$. Let us pick a $G_{0}$-invariant saturated submanifold $S$ in
$X_{0}$. This implies that $X_{S}$ is $G_{1}$-invariant and that $(X_{S})_{2}$
is $G_{2}$-invariant. If we respectively use $\eta^{(2)}$, $\eta^{(1)}$, and
$\eta^{(0)}$ to identify $\nu(X_{S})_{2}\cong\nu((X_{S})_{2})$ with
$(T(X_{S})_{2})^{\perp}$, $\nu(X_{S})$ with $TX_{S}^{\perp}$, and $\nu(S)$
with $TS^{\perp}$ then it follows that the $2$-action $T\theta$ restrict to a
well defined $2$-action $\overline{T\theta}$ of $G_{1}\rightrightarrows G_{0}$
on $\nu(X_{S})\rightrightarrows\nu(S)$ since $\theta$ is isometric.
Furthermore, the latter fact also implies that the exponential maps
$\textnormal{exp}^{(2)}$, $\textnormal{exp}^{(1)}$, and
$\textnormal{exp}^{(0)}$ are equivariant local diffeomorphisms. Therefore, as
isometries preserve (horizontal) geodesics then by following the classical
proofs of the equivariant tubular neighborhood theorem in [8, VI. Thm. 2.2]
and [24, 25, Thm. 4.4] with the proof of the weak linearization theorem given
in [18, Thm. 5.11] we easily obtain:

######  Proposition 3.5 (2-equivariant weak groupoid linearization).

Let $G_{1}\rightrightarrows G_{0}$ be a Lie $2$-group acting by isometries on
a Riemannian groupoid $(X_{1}\rightrightarrows X_{0},\eta)$. Then there exists
a $2$-equivariant weak linearization of $X_{1}\rightrightarrows X_{0}$ around
any $G_{0}$-invariant saturated submanifold $S$ in $X_{0}$.

Let us now deal with the classical approach used to ensure the existence of
invariant Riemannian metrics on $G$-manifolds to provide a similar
construction in our context.

######  Remark 3.6.

Let us denote by $\eta^{\ast}$ the dual metric associated to a Riemannian
metric $\eta$. From [18, Prop. 2.2] we know that if $\pi:E\to B$ is a
submersion and $\\{\eta_{1},\cdots,\eta_{k}\\}$ is a collection of
$\pi$-transverse metrics then its tangent average
$\frac{1}{k}\sum_{l=1}^{k}\eta_{l}$ fails to be $\pi$-transverse again in
general. Nevertheless, its cotangent average
$\frac{1}{k}\left(\sum_{l=1}^{k}\eta_{l}^{\ast}\right)^{\ast}$ is always
$\pi$-transverse which sometimes makes more advantageous to take a cotangent
space point of view in the study of Riemannian submersions.

Assume for a moment that $G_{1}$ is compact so that $G_{0}$ is also compact.
If $\mu_{1}$ is the normalized Haar measure on $G_{1}$ then, by uniqueness,
the pushforward measure $s_{G\ast}\mu_{1}$ agrees with the normalized Haar
measure $\mu_{0}$ on $G_{0}$ since $s_{G}$ a surjective Lie group homomorphism
and

$\int_{G_{0}}fd(s_{G\ast}\mu_{1})=\int_{G_{1}}f\circ s_{G}d\mu_{1},$ (5)

for each continuous function $f:G_{0}\to\mathbb{R}$. Analogously,
$t_{G\ast}\mu_{1}=\mu_{0}$ and $i_{G\ast}\mu_{1}=\mu_{1}$. Note also that the
fact that $i_{G\ast}\mu_{1}=\mu_{1}$ and $s_{G}\circ i_{G}=t_{G}$ immediately
implies that $t_{G\ast}\mu_{1}=s_{G\ast}\mu_{1}$. Thus, we are in conditions
to state:

######  Theorem 3.7.

Suppose that $G_{1}\rightrightarrows G_{0}$ is a Lie $2$-group acting on a
Riemannian groupoid $(X_{1}\rightrightarrows X_{0},\eta)$. If $G_{1}$ compact
then there exists a groupoid metric $\overline{\eta}$ on
$X_{1}\rightrightarrows X_{0}$ for which the 2-action $\theta$ becomes
isometric.

###### Proof.

To simplify the computations we shall suppose that $\eta=\eta^{(1)}$ is a
$1$-metric on $X_{1}$. By dual averaging we define:

$(\overline{\eta}^{(1)})^{\ast}:=\int_{G_{1}}(\theta^{1})^{\ast}_{g}(\eta^{(1)})^{\ast}d\mu_{1}(g)\quad\textnormal{and}\quad(\overline{\eta}^{(0)})^{\ast}:=\int_{G_{0}}(\theta^{0})^{\ast}_{g_{0}}(\eta^{(0)})^{\ast}d\mu_{0}(g_{0}).$

The result will follow from the fact that $\theta$ is a $2$-action, $\eta$ is
already a $1$-metric, and Identity (5) holds true. Indeed, on the one hand, a
straightforward computation using the fact that $i$ is an isometry of
$(X_{1},\eta^{(1)})$ verifying both Identity (3) and
$i_{G\ast}\mu_{1}=\mu_{1}$ implies that $i_{X}$ is an isometry of
$(X_{1},\overline{\eta}^{(1)})$ as well. The fact that $s_{X}:X_{1}\to X_{0}$
is a Riemannian submersion tells us that
$ds_{X}(p)^{\ast}:T_{s_{X}(p)}^{\ast}X_{0}\to\textnormal{ker}(ds_{X}(p))^{\circ}$
is an isometry for all $p\in X_{1}$ where
$\textnormal{ker}(ds_{X}(p))^{\circ}$ denotes the annihilator of the vectors
tangent to the fiber. Given $p\in X_{1}$, covectors $\alpha,\beta\in
T_{s_{X}(p)}^{\ast}X_{0}$, and $g\in G_{1}$ such that $s_{G}(g)=g_{0}$ we get
the following chain of equalities:

$\displaystyle(\overline{\eta}^{(0)}_{s_{X}(p)})^{\ast}(\alpha,\beta)$
$\displaystyle=$
$\displaystyle\int_{G_{0}}(\eta^{(0)}_{g_{0}s_{X}(p)})^{\ast}((\theta^{0}_{g_{0}})^{\ast}(\alpha),(\theta^{0}_{g_{0}})^{\ast}(\beta))d\mu_{0}$
$\displaystyle\eqref{PushHaar}$ $\displaystyle=$
$\displaystyle\int_{G_{1}}(\eta^{(0)}_{s_{G}(g)s_{X}(p)})^{\ast}((\theta^{0}_{s_{G}(g)})^{\ast}(\alpha),(\theta^{0}_{s_{G}(g)})^{\ast}(\beta))d\mu_{1}$
$\displaystyle=$
$\displaystyle\int_{G_{1}}(\eta^{(1)}_{gp})^{\ast}(ds_{X}(gp)^{\ast}((\theta^{0}_{s_{G}(g)})^{\ast}(\alpha)),ds_{X}(gp)^{\ast}((\theta^{0}_{s_{G}(g)})^{\ast}(\beta)))d\mu_{1}$
$\displaystyle=$
$\displaystyle\int_{G_{1}}(\eta^{(1)}_{gp})^{\ast}((\theta^{1}_{g})^{\ast}(ds_{X}(p)^{\ast}(\alpha)),(\theta^{1}_{g})^{\ast}(ds_{X}(p)^{\ast}(\beta)))d\mu_{1}$
$\displaystyle=$
$\displaystyle(\overline{\eta}^{(1)}_{p})^{\ast}(ds_{X}(p)^{\ast}(\alpha),ds_{X}(p)^{\ast}(\beta)),$

from which we conclude that $s_{X}$ is also Riemannian for the averaged
metrics. With analogous computations we get a similar conclusion for
$t_{X}:X_{1}\to X_{0}$ so that the result follows as claimed. ∎

######  Remark 3.8.

It is worth noticing that if we consider an $n$-metric on $X_{n}$ for $n\geq
2$ and take into account the simplicial approach described in Remark 3.3 then
by applying similar averaging arguments as in the proof of Theorem 3.7 it is
possible to show the existence of another $n$-metric on $X_{n}$ that is
invariant by the action of $G_{n}$ on $X_{n}$. For instance, let $\eta^{(2)}$
be a $2$-metric on $X_{2}$ and let $\mu_{2}$ denote the normalized Haar
measure on $G_{2}$. On the one hand, by dual averaging we define
$\displaystyle(\overline{\eta}^{(2)})^{\ast}:=\int_{G_{2}}(\theta^{2})^{\ast}_{(g,h)}(\eta^{(2)})^{\ast}d\mu_{2}(g,h)$.
On the other hand, it is simple to see that we can obtain analogous formulas
as that in Identity (5) by using instead $\pi_{2,G},m_{G},\pi_{1,G}:G_{2}\to
G_{1}$ as well as the Lie group isomorphisms $G_{2}\to G_{2}$ determined by
the canonical action of $S_{3}$ on $G_{2}$. Hence, by similar computations as
those in the main part of Theorem 3.7 it follows that $\overline{\eta}^{(2)}$
is a $2$-metric on $X_{2}$ for which the action of $G_{2}$ on
$(X_{2},\overline{\eta}^{(2)})$ becomes isometric.

It is known that every proper Lie groupoid can be equipped with a groupoid
metric, see [18, Thm. 4.13]. Hence, we get that:

######  Corollary 3.9.

Let $\theta$ be a $2$-action of a Lie $2$-group $G_{1}\rightrightarrows G_{0}$
on a proper groupoid $X_{1}\rightrightarrows X_{0}$. If $G_{1}$ is compact
then there always exists a Riemannian groupoid metric on
$X_{1}\rightrightarrows X_{0}$ for which $\theta$ becomes an isometric
$2$-action.

As an interesting application of Theorem 3.7 and Proposition 3.4 we can prove
analogous statements to those of the Slice Theorem and the Equivariant Tubular
Neighborhood Theorem in our setting. Let us start by introducing the following
terminology. If $\theta^{0}:G\times M\to M$ is a smooth action of a Lie group
$G$ on a smooth manifold $M$ then a _standard slice_ at $x_{0}\in M$ is an
embedded submanifold $S_{x_{0}}$ containing $x_{0}$ and satisfying the
following properties:

  1. a.

$T_{x_{0}}M=T_{x_{0}}(G\cdot x_{0})\oplus T_{x_{0}}S_{x_{0}}$ and
$T_{x}M=T_{x}(G\cdot x)+T_{x}S_{x_{0}}$ for all $x\in S_{x_{0}}$,

  2. b.

$S_{x_{0}}$ is $\textnormal{Iso}_{G}(x_{0})$-invariant, and

  3. c.

if $x\in S_{x_{0}}$ and $g\in G$ are such that $\theta_{g}(x)\in S_{x_{0}}$
then $g\in\textnormal{Iso}_{G}(x_{0})$.

Here $G\cdot x_{0}$ and $\textnormal{Iso}_{G}(x_{0})$ respectively denote the
$G$-orbit and the $G$-isotropy group at $x_{0}$ with respect to $\theta$.

######  Definition 3.10.

Let $\theta$ be a $2$-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$. A _groupoid slice_ at $x_{0}\in X_{0}$ is
defined to be a Lie subgroupoid $S_{1_{x_{0}}}\rightrightarrows S_{x_{0}}$ of
$X_{1}\rightrightarrows X_{0}$ such that $S_{x_{0}}$ and $S_{1_{x_{0}}}$ are
standard slices at $x_{0}$ and $1_{x_{0}}$, respectively.

The following is a straightforward result.

######  Lemma 3.11.

Take any $x_{0}\in X_{0}$. There are natural structures of:

  * •

Lie 2-subgroup between the $G$-isotropy groups
$\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows\textnormal{Iso}_{G_{0}}(x_{0})$,
and

  * •

Lie subgroupoid between the $G$-orbits $G_{1}\cdot 1_{x_{0}}\rightrightarrows
G_{0}\cdot x_{0}$.

Our version of the Slice Theorem is as follows.

######  Proposition 3.12 (Groupoid slice).

Let $\theta$ be a proper $2$-action of a Lie $2$-group $G_{1}\rightrightarrows
G_{0}$ on a Riemannian groupoid $(X_{1}\rightrightarrows X_{0},\eta)$. Then
there exists a groupoid slice $S_{1_{x_{0}}}\rightrightarrows S_{x_{0}}$ at
each $x_{0}\in X_{0}$.

###### Proof.

Consider the induced $2$-action of
$\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows\textnormal{Iso}_{G_{0}}(x_{0})$
on $X_{1}\rightrightarrows X_{0}$. By applying Theorem 3.7 together with
Remark 3.8 we may use the $2$-metric $\eta$ on $X_{2}$ and use it to construct
another $2$-metric $\tilde{\eta}$ on $X_{2}$ in such a way
$\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows\textnormal{Iso}_{G_{0}}(x_{0})$
acts isometrically on $(X_{1}\rightrightarrows X_{0},\tilde{\eta})$. As in the
classical case [3, Thm. 3.49], we define $S_{x_{0}}$ by setting
$S_{x_{0}}=\tilde{\exp}^{(0)}_{x_{0}}(B_{\epsilon}(0))$ where
$B_{\epsilon}(0)$ is an open ball of radius $\epsilon>0$ around the origin in
the normal space $\nu_{x_{0}}(G_{0}\cdot x_{0})$ to the $G_{0}$-orbit through
$x_{0}$ (normal domain).

As $G_{1}\cdot 1_{x_{0}}\rightrightarrows G_{0}\cdot x_{0}$ is a Lie
subgroupoid of $X_{1}\rightrightarrows X_{0}$ we have a well defined Lie
subgroupoid $\nu(G_{1}\cdot 1_{x_{0}})\rightrightarrows\nu(G_{0}\cdot x_{0})$
of $TX_{1}\rightrightarrows TX_{0}$ so that we may also consider the Lie
groupoid $V_{B_{\epsilon}(0)}\rightrightarrows B_{\epsilon}(0)$ where
$V_{B_{\epsilon}(0)}=\overline{ds}_{1_{x_{0}}}^{-1}(B_{\epsilon}(0))\cap\overline{dt}_{1_{x_{0}}}^{-1}(B_{\epsilon}(0))$.
Let us use the groupoid metric $\tilde{\eta}$ to identify $\nu(G_{1}\cdot
1_{x_{0}})\cong T(G_{1}\cdot 1_{x_{0}})^{\perp}$ and $\nu(G_{0}\cdot
x_{0})\cong T(G_{0}\cdot x_{0})^{\perp}$. By shrinking $B_{\epsilon}(0)$ if
necessary we may assume that $V_{B_{\epsilon}(0)}$ is an open ball around the
origin in the normal space $T_{1_{x_{0}}}(G_{1}\cdot 1_{x_{0}})^{\perp}$ to
the $G_{1}$-orbit through $1_{x_{0}}$ on which
$\tilde{\exp}^{(1)}_{1_{x_{0}}}$ is well defined. Therefore, we now set
$S_{1_{x_{0}}}=\tilde{\exp}^{(1)}_{1_{x_{0}}}(V_{B_{\epsilon}(0)})$. Hence, by
arguing with similar arguments as those used to prove the multiplicative
property of the exponential maps associated to a Riemannian $2$-metric in [18,
Thm. 5.11] together with the equivariant property these exponential maps have,
we conclude that $S_{1_{x_{0}}}\rightrightarrows S_{x_{0}}$ is the Lie
subgroupoid of $X_{1}\rightrightarrows X_{0}$ we are looking for. ∎

The previous proposition will be used in order to obtain a $2$-equivariant
Tubular Neighborhood Theorem for the $G$-orbit groupoid $G_{1}\cdot
1_{x_{0}}\rightrightarrows G_{0}\cdot x_{0}$. In addition, we need to consider
the construction of the associated groupoid bundle of a groupoid principal
$2$-bundle. A particular case of such a construction can be found in [23, Lem.
9.1.2].

######  Remark 3.13.

Let $\pi:(P_{1}\rightrightarrows P_{0})\to(X_{1}\rightrightarrows X_{0})$ be a
groupoid principal $2$-bundle with structural Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ [10]. Assume that there exists a left
$2$-action of $G_{1}\rightrightarrows G_{0}$ over another Lie groupoid
$F_{1}\rightrightarrows F_{0}$. Given this data we can construct two
associated fiber bundles $E_{j}:=P_{j}\times_{G_{j}}F_{j}$ over $X_{j}$ for
$j=0,1$. These are defined as the quotient spaces $(P_{j}\times F_{j})/G_{j}$
with respect to the actions
$g_{j}\cdot(p_{j},f_{j})=(p_{j}g_{j}^{-1},g_{j}f_{j})$, for all $g_{j}\in
G_{j}$, $p_{j}\in P_{j}$ and $f_{j}\in F_{j}$, together with projections
$\overline{\pi}_{j}([p_{j},f_{j}])=\pi_{j}(p_{j})$ onto $X_{j}$. It is simple
to check that there exists a natural Lie groupoid structure
$E_{1}\rightrightarrows E_{0}$ for which the projection
$\overline{\pi}:(E_{1}\rightrightarrows E_{0})\to(X_{1}\rightrightarrows
X_{0})$ becomes a Lie groupoid fibration. The source and target maps are the
obvious ones, namely:

$s_{E}([p,f])=[s_{P}(p),s_{F}(f)]\qquad\textnormal{and}\qquad
t_{E}([p,f])=[t_{P}(p),t_{F}(f)],$

and the groupoid composition is defined as follows. If
$s_{E}([p,f])=t_{E}([q,l])$ then there is $g_{0}\in G_{0}$ such that
$(s_{P}(p)g_{0}^{-1},g_{0}s_{F}(f))=(t_{P}(q),t_{F}(l))$. So, we set

$m_{E}([p,f],[q,l])=[(pg^{-1})\ast q,(gf)\ast l],$

for some $g$ inside the $s_{G}$-fiber at $g_{0}$. From Formula (4) it follows
that the latter equality is well defined and that it does not depend on the
choice of $g\in s_{G}^{-1}(g_{0})$. Furthermore, the same formula and the
associativity of $m_{X}$ together imply the associativity of the composition
$m_{E}$. As expected, the inversion $i_{E}$ and the unit map $u_{E}$ are
defined in the obvious way.

Let $\theta$ be a $2$-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$. From Lemma 3.11, Proposition 3.12, the
quotient construction described in Proposition 3.4, and Remark 3.13 we easily
deduce that:

######  Lemma 3.14.

There exists a principal groupoid $2$-bundle $\pi:(G_{1}\rightrightarrows
G_{0})\to(G_{1}/\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows
G_{0}/\textnormal{Iso}_{G_{0}}(x_{0}))$ with structural Lie $2$-group
$\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows\textnormal{Iso}_{G_{0}}(x_{0})$,
yielding an associated groupoid fibration

$(G_{1}\times_{\textnormal{Iso}_{G_{1}}(1_{x_{0}})}S_{1_{x_{0}}}\rightrightarrows
G_{0}\times_{\textnormal{Iso}_{G_{0}}(x_{0})}S_{x_{0}})\to(G_{1}/\textnormal{Iso}_{G_{1}}(1_{x_{0}})\rightrightarrows
G_{0}/\textnormal{Iso}_{G_{0}}(x_{0})),$

with groupoid fiber $S_{1_{x_{0}}}\rightrightarrows S_{x_{0}}$.

Consider the classical $G$-invariant tubular neighborhoods
$\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})=\theta^{1}(G_{1},S_{1_{x_{0}}})$ and
$\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})=\theta^{0}(G_{0},S_{x_{0}})$ of $G_{1}\cdot 1_{x_{0}}$ and $G_{0}\cdot
x_{0}$, respectively. Note that we have emphasized the inclusion of
$\tilde{\eta}$ and $\epsilon$, from the proof of Proposition 3.12, at the
definition of the above tubular neighborhoods since the existence of our
groupoid slice relies on them. It is clear that
$\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})\rightrightarrows\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})$ is a Lie subgroupoid of $X_{1}\rightrightarrows X_{0}$ and that the
$2$-action $\theta$ restricts well over it. Furthermore,
$\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})\rightrightarrows\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})$ determines an open Lie groupoid neighborhood of the $G$-orbit groupoid
$G_{1}\cdot 1_{x_{0}}\rightrightarrows G_{0}\cdot x_{0}$.

We are now in conditions to state our version of the Equivariant Tubular
Neighborhood Theorem. Namely:

######  Theorem 3.15 (Equivariant groupoid tubular neighborhood).

Suppose that $\theta$ is a proper $2$-action of a Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. Then, for every $x_{0}\in X_{0}$ there
exists a $2$-equivariant Lie groupoid isomorphism

$\Psi:(G_{1}\times_{\textnormal{Iso}_{G_{1}}(1_{x_{0}})}S_{1_{x_{0}}}\rightrightarrows
G_{0}\times_{\textnormal{Iso}_{G_{0}}(x_{0})}S_{x_{0}})\xrightarrow[]{\cong}(\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{1}\cdot
1_{x_{0}})\rightrightarrows\textnormal{Tub}_{\epsilon}^{\tilde{\eta}}(G_{0}\cdot
x_{0})).$

###### Proof.

The left $2$-action of $G_{1}\rightrightarrows G_{0}$ on
$G_{1}\times_{\textnormal{Iso}_{G_{1}}(1_{x_{0}})}S_{1_{x_{0}}}\rightrightarrows
G_{0}\times_{\textnormal{Iso}_{G_{0}}(x_{0})}S_{x_{0}}$ we will consider here
is the one given by
$\overline{g_{j}}\cdot[g_{j},f_{j}]=[\overline{g_{j}}g_{j},f_{j}]$ for all
$\overline{g_{j}},g_{j}\in G_{j}$ and $f_{j}\in S_{j}$. Here
$S_{1}=S_{1_{x_{0}}}$ and $S_{0}=S_{x_{0}}$. The Lie groupoid isomorphism
$\Psi$ is defined as

$\Psi^{j}([g_{j},f_{j}])=\theta^{j}(g_{j},f_{j}).$

From the proof of Theorem 3.57 in [3] we already know that $\Psi^{j}$ is a
$G_{j}$-equivariant diffeomorphism. Thus, we only have to check that $\Psi$
defines indeed a Lie groupoid morphism. By using the structural maps defined
in Remark 3.13 we have that

$(\Psi^{0}\circ
s_{E})([g,f])=\Psi^{0}([s_{G}(g),s_{X}(f)])=s_{G}(g)s_{X}(f)=s_{X}(gf)=(s_{X}\circ\Psi^{1})([g,f]).$

We can similarly obtain that $\Psi^{0}\circ t_{E}=t_{X}\circ\Psi^{1}$.
Moreover, by applying Formula (4) we get

$\displaystyle\Psi^{1}([g,f]\ast[h,l])$ $\displaystyle=$
$\displaystyle\Psi^{1}([(g\overline{g}^{-1})\ast h,(\overline{g}f)\ast
l])=((g\overline{g}^{-1})\ast h)\cdot((\overline{g}f)\ast l)$ $\displaystyle=$
$\displaystyle(gf)\ast(hl)=\Psi^{1}([g,f])\ast\Psi^{1}([h,l]).$

Hence, the result follows as desired. ∎

We finish the discussion of this subsection by briefly commenting that it is
possible to define a groupoid $G$-orbit type topological stratification for
$X_{1}\rightrightarrows X_{0}$. We say that $x_{0}$ and $y_{0}$ in $X_{0}$
have the same $G$-_orbit type_ if there exists a Lie groupoid isomorphism
$\Phi$ between $G_{1}\cdot 1_{x_{0}}\rightrightarrows G_{0}\cdot x_{0}$ and
$G_{1}\cdot 1_{y_{0}}\rightrightarrows G_{0}\cdot y_{0}$ such that $\Phi^{1}$
is $G_{1}$-equivariant. This automatically implies that $\Phi^{0}$ is
$G_{0}$-equivariant. It is clear that this $G$-orbit type requirement defines
an equivalent relation $\sim$ on $X_{0}$ for which we denote by
$M_{x_{0}}^{\sim}$ the equivalent class at $x_{0}\in X_{0}$ associated to such
a relation. We claim that $M_{x_{0}}^{\sim}$ is saturated in $X_{0}$, that is,
$s_{X}^{-1}(M_{x_{0}}^{\sim})=t_{X}^{-1}(M_{x_{0}}^{\sim})$. If $p\in
s_{X}^{-1}(M_{x_{0}}^{\sim})$ then $s_{X}(p)\sim x_{0}$ so that there is a
$(G_{1}\rightrightarrows G_{0})$-equivariant isomorphism between $G_{1}\cdot
1_{s_{X}(p)}\rightrightarrows G_{0}\cdot s_{X}(p)$ and $G_{1}\cdot
1_{x_{0}}\rightrightarrows G_{0}\cdot x_{0}$. It is simple to check that
$\Phi_{p}$ defined by $\Phi_{p}^{1}(g1_{t_{X}(p)})=g1_{s_{X}(p)}$ and
$\Phi_{p}^{0}(g_{0}t_{X}(p))=g_{0}s_{X}(p)$ defines another
$(G_{1}\rightrightarrows G_{0})$-equivariant isomorphism between $G_{1}\cdot
1_{t_{X}(p)}\rightrightarrows G_{0}\cdot t_{X}(p)$ and $G_{1}\cdot
1_{s_{X}(p)}\rightrightarrows G_{0}\cdot s_{X}(p)$ so that by taking
$\Phi\circ\Phi_{p}$ we conclude that $t_{X}(p)\sim x_{0}$. The other inclusion
may be similarly verified. Thus, by setting
$M_{1_{x_{0}}}^{\sim}=s_{X}^{-1}(M_{x_{0}}^{\sim})=t_{X}^{-1}(M_{x_{0}}^{\sim})$
we obtain a collection of topological groupoids
$\\{M_{1_{x_{0}}}^{\sim}\rightrightarrows M_{x_{0}}^{\sim}\\}_{x_{0}\in
X_{0}}$ which somehow stratifies the Lie groupoid $X_{1}\rightrightarrows
X_{0}$. The previous observation suggests that by combining classical ideas
from [3, s. 3.5] with some of the results obtained in this section it could be
possible to show that each of the $M_{1_{x_{0}}}^{\sim}\rightrightarrows
M_{x_{0}}^{\sim}$ is a honest Lie subgroupoid. Nevertheless, the local
$G$-orbit type notion from the classical case does not extended directly in
our case. _We conjecture that our guess is true_. That is,
$M_{1_{x_{0}}}^{\sim}\rightrightarrows M_{x_{0}}^{\sim}$ is Lie subgroupoid of
$X_{1}\rightrightarrows X_{0}$ for all $x_{0}\in X_{0}$.

###  3.1. Orthogonal Lie 2-groups

In this subsection we derive another application of Theorem 3.7 which has to
do with the construction of groupoid bi-invariant Riemannian metrics on
compact Lie $2$-groups. To simplify computations in this subsection will be
enough to consider only $1$-metrics. Let $G_{1}\rightrightarrows G_{0}$ be a
Lie 2-group and consider the pairs $L=(L^{1},L^{0})$ and $R=(R^{1},R^{0})$
where $L^{j}$ and $R^{j}$ for $j=0,1$ are respectively the actions of $G_{j}$
on itself determined by left and right multiplications. It is simple to check
that as consequence of Identity (1) and the fact that the structural maps of
$G_{1}\rightrightarrows G_{0}$ are Lie group homomorphisms it follows that $L$
and $R$ determine left $2$-actions of $G_{1}\rightrightarrows G_{0}$ on
itself.

######  Definition 3.16.

A Lie $2$-group $G_{1}\rightrightarrows G_{0}$ is said to be _orthogonal_ if
it may be equipped with a $1$-metric for which both $L$ and $R$ are isometric
$2$-actions. Such a $1$-metric will be called _bi-invariant_.

We will think of $1$-metrics on a Lie $2$-algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ as pairs of inner products
$\langle\cdot,\cdot\rangle=(\langle\cdot,\cdot\rangle^{(1)},\langle\cdot,\cdot\rangle^{(0)})$
verifying the required conditions of $1$-metric. Let $G_{1}\rightrightarrows
G_{0}$ be a Lie 2-group with respective Lie 2-algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$. We denote by
$\textnormal{Ad}=(\textnormal{Ad}^{1},\textnormal{Ad}^{0})$ the $2$-action of
$G_{1}\rightrightarrows G_{0}$ on
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ determined by the adjoint
actions $\textnormal{Ad}^{j}$ of $G_{j}$ on $\mathfrak{g}_{j}$ for $j=0,1$.
This $2$-action will be called _adjoint $2$-action_ of $G_{1}\rightrightarrows
G_{0}$.

######  Proposition 3.17.

A Lie $2$-group $G_{1}\rightrightarrows G_{0}$ is orthogonal if and only if
there exists a $1$-metric $\langle\cdot,\cdot\rangle$ on its Lie $2$-algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ for which the adjoint
$2$-action is by linear isometries.

###### Proof.

It is clear that if $\eta$ is a bi-invariant $1$-metric then
$\eta_{e}=(\eta^{1}_{e_{1}},\eta^{0}_{e_{0}})$, where $e_{j}$ is the identity
element in $G_{j}$, defines a $1$-metric on
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ for which the adjoint
$2$-action is by linear isometries. Conversely, given such a
$\langle\cdot,\cdot\rangle$, we define $\eta$ on $G_{1}\rightrightarrows
G_{0}$ by setting

$\eta^{(1)}_{g}(v,w)=\langle
d(L^{1}_{g^{-1}})_{g}(v),d(L^{1}_{g^{-1}})_{g}(w)\rangle^{(1)}.$

The metric $\eta^{(0)}$ has a similar defining formula but using $L^{0}$ and
$\langle\cdot,\cdot\rangle^{(0)}$ instead of $L^{1}$ and
$\langle\cdot,\cdot\rangle^{(1)}$. It is clear that $\eta^{(1)}$ is bi-
invariant. Therefore, it remains to prove that $\eta$ defines indeed a
$1$-metric. Firstly, for $g\in G_{1}$ and $v,w\in T_{g}G_{1}$ we have

$\displaystyle i^{\ast}_{g}\eta^{(1)}(v,w)$ $\displaystyle=$
$\displaystyle\langle
d(L^{1}_{i(g)^{-1}})_{i(g)}(di_{g}(v)),d(L^{1}_{i(g)^{-1}})_{i(g)}(di_{g}(w))\rangle^{(1)}$
$\displaystyle=$ $\displaystyle\langle d(L^{1}_{i(g^{-1})}\circ
i)_{g}(v),d(L^{1}_{i(g^{-1})}\circ i)_{g}(w)\rangle^{(1)}$ $\displaystyle=$
$\displaystyle\langle d(i\circ L^{1}_{g^{-1}})_{g}(v),d(i\circ
L^{1}_{g^{-1}})_{g}(w)\rangle^{(1)}$ $\displaystyle=$ $\displaystyle\langle
di_{e_{1}}(d(L^{1}_{g^{-1}})_{g}(v)),di_{e_{1}}(d(L^{1}_{g^{-1}})_{g}(v))\rangle^{(1)}=\eta^{(1)}_{g}(v,w).$

Secondly, let $g_{0}\in G_{0}$ and $v,w\in T_{g_{0}}G_{0}$. It is clear that
there are $\tilde{v},\tilde{w}\in\textnormal{ker}(ds(g))^{\perp_{\eta^{(1)}}}$
with $s(g)=g_{0}$ such that $ds_{g}(\tilde{v})=v$ and $ds_{g}(\tilde{w})=w$.
Thus

$\displaystyle(s_{\ast}\eta^{(1)})_{g_{0}}(v,w)$ $\displaystyle=$
$\displaystyle\eta^{(1)}_{g}(\tilde{v},\tilde{w})=\langle
d(L^{1}_{g^{-1}})_{g}(\tilde{v}),d(L^{1}_{g^{-1}})_{g}(\tilde{w})\rangle^{(1)}$
$\displaystyle=$ $\displaystyle\langle
ds_{e_{1}}(d(L^{1}_{g^{-1}})_{g}(\tilde{v})),ds_{e_{1}}(d(L^{1}_{g^{-1}})_{g}(\tilde{w}))\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\langle d(s\circ
L^{1}_{g^{-1}})_{g}(\tilde{v}),d(i\circ
L^{1}_{g^{-1}})_{g}(\tilde{w})\rangle^{(0)}$ $\displaystyle=$
$\displaystyle\langle d(L^{0}_{g_{0}^{-1}}\circ
s)_{g}(\tilde{v}),d(L^{0}_{g_{0}^{-1}}\circ s)_{g}(\tilde{w})\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\langle
d(L^{0}_{g_{0}^{-1}})_{g_{0}}(v),d(L^{0}_{g_{0}^{-1}})_{g_{0}}(v)\rangle^{(0)}=\eta^{(0)}_{g_{0}}(v,w).$

Analogously, $t_{\ast}\eta^{(1)}=\eta^{(0)}$. So, the result follows. ∎

From now on we assume that the Lie groups we are working with are connected.
It is well known that bi-invariant metrics on a Lie group are in one-to-one
correspondence with inner products on its Lie algebra for which the adjoint
representation determines infinitesimal isometries [29, 31]. A Lie $2$-algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ is said to be _orthogonal_
if it admits a $1$-metric
$\langle\cdot,\cdot\rangle=(\langle\cdot,\cdot\rangle^{(1)},\langle\cdot,\cdot\rangle^{(0)})$
for which the adjoint representation
$\textnormal{ad}^{1}:\mathfrak{g}_{1}\to\textnormal{Der}(\mathfrak{g}_{1})$
acts by infinitesimal isometries on
$(\mathfrak{g}_{1},\langle\cdot,\cdot\rangle^{(1)})$. Therefore, as
consequence of Proposition 3.17 and [31, Lem. 7.2] we get:

######  Corollary 3.18.

A Lie $2$-group is orthogonal if and only if its Lie algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ is orthogonal.

From Theorem 3.7 we get that:

######  Corollary 3.19.

Every Lie $2$-group $G_{1}\rightrightarrows G_{0}$ with $G_{1}$ compact can be
endowed with a bi-invariant $1$-metric.

Given an orthogonal Lie $2$-algebra
$\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ we may split
$\mathfrak{g}_{1}$ as a direct sum of ideals
$\mathfrak{g}_{1}=\mathfrak{h}\oplus\mathfrak{h}^{\perp_{1}}$ where
$\mathfrak{h}=\textnormal{ker}(s)$. As the unit map $u$ is a canonical
bisection we would expect that $u(x)\in\mathfrak{h}^{\perp_{1}}$ for all
$x\in\mathfrak{g}_{0}$ but, however, this is not true in general unless we
assume “non-canonical” identifications. We say that an orthogonal Lie
$2$-algebra is _trivial_ if $\mathfrak{h}^{\perp_{1}}=u(\mathfrak{g}_{0})$.
This notion comes up by the following simple result.

######  Proposition 3.20.

If $\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ is a trivial orthogonal
Lie $2$-algebra then $\textnormal{ad}^{1}_{u(x)}|_{\mathfrak{h}}=0$ for all
$x\in\mathfrak{g}_{0}$. In consequence, $\mathfrak{h}$ is abelian and
$\textnormal{im}(t|_{\mathfrak{h}})\subseteq\mathfrak{z}(\mathfrak{g}_{0})$.

###### Proof.

On the one hand, note that for any $y\in\mathfrak{h}$ and $z\in\mathfrak{g}$
one gets

$\langle\textnormal{ad}^{1}_{u(x)}(y),z\rangle^{(1)}=\langle
u(x),\textnormal{ad}^{1}_{y}(z)\rangle^{(1)}=0,$

since $\mathfrak{h}$ is an ideal. Thus, as $\langle\cdot,\cdot\rangle^{(1)}$
is nondegenerate we get that $\textnormal{ad}^{1}_{u(x)}|_{\mathfrak{h}}=0$
for all $x\in\mathfrak{g}_{0}$. On the other hand, for all
$y,y^{\prime}\in\mathfrak{h}$ and $x\in\mathfrak{g}$ it follows that
$[y,y^{\prime}]=\textnormal{ad}^{1}_{u(t(y))}(y^{\prime})=0$ and
$0=t([u(x),y])=[x,t(y)]$ since $t$ is a Lie algebra homomorphism. ∎

This in particular implies that the crossed module of Lie algebras associated
to a trivial orthogonal Lie $2$-algebra is of the form
$(\mathfrak{g},\mathfrak{h},\partial,0)$ with $\mathfrak{h}$ an abelian Lie
algebra.

######  Remark 3.21.

Suppose that we have a crossed module of Lie algebras
$(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ which is
$\mathcal{L}$-orthogonal in the sense of [4, 20]. That is, there exist inner
products $\langle\cdot,\cdot\rangle_{\mathfrak{g}}$ and
$\langle\cdot,\cdot\rangle_{\mathfrak{h}}$ such that the Lie algebra
representation $\mathcal{L}:\mathfrak{g}\to\textnormal{Der}(\mathfrak{h})$
acts by infinitesimal isometries on
$(\mathfrak{h},\langle\cdot,\cdot\rangle_{\mathfrak{h}})$ and the adjoint
representation of $\mathfrak{g}$ acts by infinitesimal isometries on
$(\mathfrak{g},\langle\cdot,\cdot\rangle_{\mathfrak{g}})$. Note that this
directly implies that the adjoint representation of $\mathfrak{h}$ acts by
infinitesimal isometries on
$(\mathfrak{h},\langle\cdot,\cdot\rangle_{\mathfrak{h}})$. On the one hand,
recall that the associated Lie $2$-algebra constructed with the crossed module
data has $\mathfrak{g}_{1}=\mathfrak{h}\rtimes\mathfrak{g}$ with Lie algebra
structure provided by the semi-direct product with respect to $\mathcal{L}$.
Therefore, a straightforward computation allows us to conclude that the
adjoint representation of $\mathfrak{g}_{1}$ acts by infinitesimal isometries
with respect to
$\langle\cdot,\cdot\rangle_{\mathfrak{h}}+\langle\cdot,\cdot\rangle_{\mathfrak{g}}$
if and only if $\mathcal{L}=0$. On the other hand, since the inversion
$i:\mathfrak{g}_{1}\to\mathfrak{g}_{1}$ is given by
$i(x,y)=(-x,y+\partial(x))$ for all $x,x^{\prime}\in\mathfrak{h}$ and
$y,y^{\prime}\in\mathfrak{g}$ then it follows that $i$ is an isometry with
respect to
$\langle\cdot,\cdot\rangle_{\mathfrak{h}}+\langle\cdot,\cdot\rangle_{\mathfrak{g}}$
if and only if $\partial=0$. As consequence, there is no canonical
correspondence between our notion of orthogonal Lie $2$-algebras and the
notion of $\mathcal{L}$-orthogonal crossed module of Lie algebras which is
known in the literature.

###  3.2. Some examples and constructions

It this short subsection we exhibit some examples and interesting
constructions in which isometric Lie $2$-group actions naturally appear.

######  Example 3.22.

Classical isometric actions of Lie groups $G$ on Riemannian manifolds $M$ are
recovered from isometric $2$-actions of unit Lie $2$-groups
$G\rightrightarrows G$ acting upon unit Riemannian groupoids
$M\rightrightarrows M$.

######  Example 3.23.

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a Riemannian groupoid and
$(\xi,v)$ be a complete multiplicative Killing vector field on
$X_{1}\rightrightarrows X_{0}$. Here we consider $\xi$ a Killing vector field
on $(X_{1},\eta^{(1)})$ and $v$ a Killing vector field on
$(X_{0},\eta^{(0)})$. From [27, Prop. 3.5] we know that the pair of flows
defined by $(\xi,v)$ determine global automorphisms on $X_{1}\rightrightarrows
X_{0}$ so that we get a well defined isometric $2$-action of
$\mathbb{R}\rightrightarrows\mathbb{R}$ on $(X_{1}\rightrightarrows
X_{0},\eta)$. In particular, the flow of the multiplicative vector field
$((1_{\xi})_{X_{1}},\xi_{X_{0}})$ formed by the fundamental vector fields of
an isometric $2$-action determines another isometric $2$-action.

######  Example 3.24.

Let $G$ be an orthogonal Lie group and $H\leq G$ be a normal Lie subgroup. It
is clear that $H$ acts on $G$ by left multiplication, leading to the action
Lie groupoid $H\times G\rightrightarrows G$. Note that its space of arrows has
a group structure, namely the semi-direct product by the conjugation action
$C_{g}(h)=ghg^{-1}$ of $G$ on $H$ so that we get a well defined Lie $2$-group.
More importantly, by applying the gauge trick construction behind Proposition
4.7 and Example 4.9 in [18] we conclude that it is possible to cook up a
$1$-metric on $H\rtimes G\rightrightarrows G$ made out from the initial bi-
invariant metric on $G$ in such a way it becomes an orthogonal Lie $2$-group.

######  Example 3.25.

Let $(M,\mathcal{F})$ be a regular Riemannian foliation and consider a free
and proper isometric foliated action
$G\times(M,\mathcal{F})\to(M,\mathcal{F})$ of a Lie group $G$. From [22, Thm.
3.7] it is known that the Lie $2$-group $G\rtimes G\rightrightarrows G$, as
defined in Example 3.24, determines a canonical $2$-action on the holonomy
groupoid $\textnormal{Hol}(M,\mathcal{F})\rightrightarrows M$ which extends
the given action of $G$ on $M$. The Riemannian metric on $M$ completely
determines a $0$-metric on $\textnormal{Hol}(M,\mathcal{F})\rightrightarrows
M$ which can be extended to a $1$-metric [18, Ex. 3.12]. Thus, the fact that
$G$ acts on $M$ isometrically implies that the extended $2$-action of
$G\rtimes G\rightrightarrows G$ on
$\textnormal{Hol}(M,\mathcal{F})\rightrightarrows M$ is by isometries.

######  Example 3.26.

If $(M,\eta^{(0)})$ is a Riemannian manifold then
$\eta^{(1)}=\eta^{(0)}\oplus\eta^{(0)}$ defines a $1$-metric on the pair
groupoid $M\times M\rightrightarrows M$. Thus, if $G$ is a Lie group acting
freely, properly and isometrically on $(M,\eta^{(0)})$ then the unit Lie
$2$-group $G\rightrightarrows G$ acts isometrically on $M\times
M\rightrightarrows M$ and in light of Proposition 3.4 it follows that the
quotient groupoid $(M\times M)/G\rightrightarrows M/G$ canonically inherits a
$1$-metric. If $G$ admits a bi-invariant metric then the previous procedure
yields an easy way to construct $1$-metrics on the gauge groupoid associated
to a principal $G$-bundle over a Riemannian manifold (see Example 3.27 below).

In the next construction we will consider the notion of multiplicative
$2$-connection $\omega=(\omega^{1},\omega^{0})$ on a groupoid principal
$2$-bundle $\pi:(P_{1}\rightrightarrows P_{0})\to(X_{1}\rightrightarrows
X_{0})$ with structural Lie $2$-group $G_{1}\rightrightarrows G_{0}$ as
defined for instance in [10, s. 5].

######  Example 3.27 (Principal groupoid warping).

Suppose that $X_{1}\rightrightarrows X_{0}$ is a Lie groupoid equipped with a
$1$-metric and that $G_{1}\rightrightarrows G_{0}$ is an orthogonal Lie
$2$-group. Let us prove that there exists a $1$-metric $\overline{\eta}$ on
$P_{1}\rightrightarrows P_{0}$ for which the $2$-action of
$G_{1}\rightrightarrows G_{0}$ is isometric and such that
$\pi=(\pi_{1},\pi_{0}):(P_{1}\rightrightarrows
P_{0})\to(X_{1}\rightrightarrows X_{0})$ is a Riemannian groupoid submersion.
Consider the associated $1$-metric $\langle\cdot,\cdot\rangle$ on the Lie
$2$-algebra $\mathfrak{g}_{1}\rightrightarrows\mathfrak{g}_{0}$ of
$G_{1}\rightrightarrows G_{0}$ and define

$\overline{\eta}^{(1)}(v,w)=\eta^{(1)}(d\pi_{1}(v),d\pi_{1}(w))+\langle\omega^{1}(v),\omega^{1}(w)\rangle^{(1)}.$

The metric $\overline{\eta}^{(0)}$ is similarly defined by using instead
$\eta^{(0)}$, $\pi_{0}$, $\langle\cdot,\cdot\rangle^{(0)}$, and $\omega^{0}$.
It is simple to check that this expression yields a well defined right
$G_{j}$-invariant metric on $P_{j}$ for which $\pi_{j}:P_{j}\to X_{j}$ becomes
a Riemannian submersion ($j=0,1$). This is because $\pi_{j}$ is constant along
the action orbits, $\omega^{j}$ is of $\textnormal{Ad}^{j}$-invariant type,
and
$\textnormal{ker}(d\pi_{j})^{\perp_{\overline{\eta}^{(j)}}}=\textnormal{ker}(\omega^{j})$
(compare [35, p. 467]). Let us verify that
$\overline{\eta}=(\overline{\eta}^{(1)},\overline{\eta}^{(0)})$ determines a
$1$-metric on $P_{1}\rightrightarrows P_{0}$. Note that, by abusing a little
on the notation, we may rewrite $\overline{\eta}^{(j)}$ in a simpler way as
$\overline{\eta}^{(j)}=(\pi_{j})^{\ast}\eta^{(j)}+(\omega^{j})^{\ast}\langle\cdot,\cdot\rangle^{(j)}$.
Recall that $\pi$ and $\omega:(TP_{1}\rightrightarrows
TP_{0})\to(\mathfrak{g}_{1}\times P_{1}\rightrightarrows\mathfrak{g}_{0}\times
P_{0})$ are Lie groupoid morphisms. So, on the one hand we get

$\displaystyle(i_{P})^{\ast}\overline{\eta}^{(1)}$ $\displaystyle=$
$\displaystyle(\pi_{1}\circ i_{P})^{\ast}\eta^{(1)}+(\omega^{1}\circ
i_{P})^{\ast}\langle\cdot,\cdot\rangle^{(1)}=(i_{X}\circ\pi_{1})^{\ast}\eta^{(1)}+(d(i_{G})_{e_{1}}\circ\omega^{1})^{\ast}\langle\cdot,\cdot\rangle^{(1)}$
$\displaystyle=$
$\displaystyle(\pi_{1})^{\ast}\eta^{(1)}+(\omega^{1})^{\ast}\langle\cdot,\cdot\rangle^{(1)}=\overline{\eta}^{(1)}.$

On the other hand, if
$v\in\textnormal{ker}(d(s_{P})_{p})^{\perp_{\overline{\eta}^{(1)}}}$ then it
is simple to verify that the identities $\pi_{0}\circ s_{P}=s_{X}\circ\pi_{1}$
and $\omega^{0}\circ s_{P}=d(s_{G})_{e_{1}}\circ\omega^{1}$ imply that
$d(\pi_{1})_{p}(v)\in\textnormal{ker}(d(s_{X})_{\pi_{1}(p)})^{\perp_{\eta^{(1)}}}$
and
$\omega^{1}(v)\in\textnormal{ker}(d(s_{G})_{e_{1}})^{\perp_{\langle\cdot,\cdot\rangle^{(1)}}}$.
Let us pick
$v_{1},v_{2}\in\textnormal{ker}(d(s_{P})_{p})^{\perp_{\overline{\eta}^{(1)}}}$.
Thus,

$\displaystyle\overline{\eta}^{(0)}_{s_{P}(p)}(ds_{P}(p)(v_{1}),ds_{P}(p)(v_{2}))$
$\displaystyle=$
$\displaystyle\eta^{(0)}_{\pi_{0}(s_{P}(p))}(d(\pi_{0})_{s_{P}(p)}(ds_{P}(p)(v_{1})),d(\pi_{0})_{s_{P}(p)}(ds_{P}(p)(v_{2})))$
$\displaystyle+$
$\displaystyle\langle\omega^{0}(ds_{P}(p)(v_{1})),\omega^{0}(ds_{P}(p)(v_{2}))\rangle^{(0)}$
$\displaystyle=$
$\displaystyle\eta^{(0)}_{s_{X}(\pi_{1}(p))}(d(s_{X})_{\pi_{1}(p)}(d\pi_{1}(p)(v_{1})),d(s_{X})_{\pi_{1}(p)}(d\pi_{1}(p)(v_{1})))$
$\displaystyle+$ $\displaystyle\langle
d(s_{G})_{e_{1}}(\omega^{1}(v_{1})),d(s_{G})_{e_{1}}(\omega^{1}(v_{2}))\rangle^{(0)}$
$\displaystyle=$
$\displaystyle\eta^{(1)}_{\pi_{1}(p)}(d\pi_{1}(p)(v_{1}),d\pi_{1}(p)(v_{1}))+\langle\omega^{1}(v_{1}),\omega^{1}(v_{2})\rangle^{(0)}$
$\displaystyle=$ $\displaystyle\overline{\eta}^{(1)}(v_{1},v_{2}).$

Analogously, it follows that
$(t_{P})_{\ast}\overline{\eta}^{(1)}=\overline{\eta}^{(0)}$. Hence, we have
shown that $(P_{1}\rightrightarrows P_{0},\overline{\eta})$ is a Lie groupoid
equipped with a $1$-metric $\overline{\eta}$ for which $G_{1}$ acts
isometrically on $(P_{1},\overline{\eta})$ and such that $\pi$ is a Riemannian
groupoid submersion, as claimed.

After performing similar computations it is simple to verify an analogous
result for $2$-metrics instead. Namely, let us now suppose that $X_{2}$ can be
endowed with a $2$-metric $\eta^{(2)}$ and that $G_{2}$ admits a bi-invariant
$2$-metric with associated $2$-metric $\langle\cdot,\cdot\rangle^{(2)}$ on
$\mathfrak{g}_{2}$. If $\omega^{2}:TP_{2}\to\mathfrak{g}_{2}\times P_{2}$
denotes the induced connection $1$-form by $\omega$ on the principal bundle
$\pi_{2}:P_{2}\to X_{2}$ with structural group $G_{2}$ then the formula
$\overline{\eta}^{(2)}((v,w),(v^{\prime},w^{\prime}))=\eta^{(2)}(d\pi_{2}(v,w),d\pi_{2}(v^{\prime},w^{\prime}))+\langle\omega^{2}(v,w),\omega^{2}(v^{\prime},w^{\prime})\rangle^{(2)}$
defines a $2$-metric on $P_{2}$ for which the action of $G_{2}$ on
$(P_{2},\overline{\eta}^{(2)})$ is by isometries and $\pi_{2}$ becomes a
Riemannian submersion.

######  Example 3.28.

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a Riemannian étale groupoid with
$n=\dim X_{0}=\dim X_{1}$. We denote by $\pi_{1}:O(X_{1})\to X_{1}$ and
$\pi_{0}:O(X_{0})\to X_{0}$ to the principal $O(n,\mathbb{R})$-bundles of
orthonormal frames over $(X_{1},\eta^{(1)})$ and $(X_{0},\eta^{(0)})$,
respectively. There exists a canonical Lie groupoid structure
$O(X_{1})\rightrightarrows O(X_{0})$ with the property that
$\pi:(O(X_{1})\rightrightarrows O(X_{0}))\to(X_{1}\rightrightarrows X_{0})$
becomes a principal $2$-bundle with structural Lie $2$-group
$O(n,\mathbb{R})\rightrightarrows O(n,\mathbb{R})$. Moreover, it admits a
canonical multiplicative $2$-connection $\omega=(\omega^{1},\omega^{0})$,
where $\omega^{1}$ and $\omega^{0}$ are respectively determined by the Levi-
Civita connections on $(X_{1},\eta^{(1)})$ and $(X_{0},\eta^{(0)})$. The
source and target maps of $O(X_{1})\rightrightarrows O(X_{0})$ are given by

$\tilde{s}(v_{1},\cdots,v_{n})=(ds(v_{1}),\cdots,ds(v_{n}))\quad\textnormal{and}\quad\tilde{t}(v_{1},\cdots,v_{n})=(dt(v_{1}),\cdots,dt(v_{n})),$

and the composition is

$\tilde{m}((v_{1},\cdots,v_{n}),(v_{1}^{\prime},\cdots,v_{n}^{\prime}))=(dm(v_{1},v_{1}^{\prime}),\cdots,dm(v_{n},v_{n}^{\prime})).$

The other structural maps can be defined in a similar fashion. Note that the
principal warping construction from Example 3.27 applies in this case.

Next construction comes motivated by a beautiful notion known in the
literature as _Cheeger deformation_. The classical construction can be found
for instance in [3, s. 6.1].

######  Example 3.29 (Cheeger groupoid deformation).

Suppose that $(G_{1}\rightrightarrows G_{0},Q)$ is an orthogonal Lie 2-group,
with $G_{1}$ compact, acting isometrically on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. On the Lie groupoid product
$X_{1}\times G_{1}\rightrightarrows X_{0}\times G_{0}$ we can consider the
1-metric $\eta\oplus\frac{1}{\tau}Q$.

There is a natural free $2$-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\times G_{1}\rightrightarrows X_{0}\times G_{0}$ where the
$G_{j}$-actions, $j=1,0$, on $X_{j}\times G_{j}$ are given by

$h_{j}\cdot(p_{j},g_{j})=(h_{j}p_{j},h_{j}g_{j}),\qquad p_{j}\in X_{j},\
h_{j},g_{j}\in G_{j}.$ (6)

We claim that the quotient groupoid $\frac{X_{1}\times
G_{1}}{G_{1}}\rightrightarrows\frac{X_{0}\times G_{0}}{G_{0}}$ determined by
the previous actions is isomorphic to $X_{1}\rightrightarrows X_{0}$. Let us
consider the groupoid principal $2$-bundle $G_{1}\rightrightarrows G_{0}$ over
the point groupoid $\\{e_{1}\\}\rightrightarrows\\{e_{0}\\}$ with structural
Lie $2$-group $G_{1}\rightrightarrows G_{0}$. By applying Remark 3.13 we know
that by using the $2$-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$ we can construct the associated Lie groupoid
bundle $(X_{1}\times_{G_{1}}G_{1}\rightrightarrows
X_{0}\times_{G_{0}}G_{0})\to(\\{e_{1}\\}\rightrightarrows\\{e_{0}\\})$. It is
important to notice that $X_{1}\times_{G_{1}}G_{1}\rightrightarrows
X_{0}\times_{G_{0}}G_{0}$ is precisely the quotient groupoid
$\frac{X_{1}\times G_{1}}{G_{1}}\rightrightarrows\frac{X_{0}\times
G_{0}}{G_{0}}$. Therefore, as the new Lie groupoid bundle has groupoid fiber
$X_{1}\rightrightarrows X_{0}$ and base groupoid
$\\{e_{1}\\}\rightrightarrows\\{e_{0}\\}$ then we have the desired
isomorphism. Under this identification the canonical groupoid projection
$\pi=(X_{1}\times G_{1}\rightrightarrows X_{0}\times
G_{0})\to(X_{1}\rightrightarrows X_{0})$ is formed by the maps
$\pi_{j}(p_{j},g_{j})=g_{j}^{-1}p_{j}$.

Observe that the $2$-action (6) is also isometric. Thus, as consequence of
Proposition 3.4 there is a unique 1-metric $\eta_{\tau}$ on
$X_{1}\rightrightarrows X_{0}$ making of the projection $\pi=(X_{1}\times
G_{1}\rightrightarrows X_{0}\times
G_{0},\eta\oplus\frac{1}{\tau}Q)\to(X_{1}\rightrightarrows X_{0},\eta_{\tau})$
a Riemannian groupoid submersion.

It is important to point out that by construction $\eta_{\tau}$ goes to $\eta$
as $\tau$ goes to $\infty$ and the $2$-action of $G_{1}\rightrightarrows
G_{0}$ on $(X_{1}\rightrightarrows X_{0},\eta_{\tau})$ is still isometric when
$\tau>0$. More importantly, the 1-parameter family of $1$-metrics
$\eta_{\tau}$ on $X_{1}\rightrightarrows X_{0}$ varies smoothly with $\tau$
and extends smoothly to $\tau=0$ with $\eta_{0}=\eta$. Hence, $\eta_{\tau}$
with $\tau\geq 0$ is a deformation of $\eta$ by other $(G_{1}\rightrightarrows
G_{0})$-invariant metrics on $X_{1}\rightrightarrows X_{0}$ which we call
_Cheeger groupoid deformation_ of $\eta$. The reader is recommended to visit
[3, s. 6,1] for specific details about the last assertions in the classical
case.

We end this section by commenting that our notion of isometric $2$-action can
be used to develop a $2$-equivariant analogous of the results proved in [37]
which are aimed at extending classical Morse theory to the context of Lie
groupoids and their differentiable stacks.

######  Example 3.30 (2-Equivariant Morse theory on groupoids).

Let $\theta$ be an isometric $2$-action of a Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ on a proper Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. Assume that $G_{1}$ is compact so that
we can ensure that such a $2$-action always exists. Suppose that there is a
$G_{0}$-invariant basic function $f:X_{0}\to\mathbb{R}$, where by basic we
mean that $s_{X}^{\ast}f=t_{X}^{\ast}f$. Note that the critical points of $f$
come in families of $G_{0}$-invariant saturated submanifolds in $X_{0}$ so we
may impose a Morse–Bott (normal) nondegenerate condition along those critical
submanifolds. As it was proven in [37], examples of these kinds of functions
are provided by the components of a moment map which is associated to a
Hamiltonian 2-action of a Lie $2$-torus on a $0$-symplectic groupoid in the
sense of [23].

Hence, without so many changes along the proofs in [37] it is possible to
prove the following results.

  * •

A $2$-equivariant version of the Morse lemma around a $G_{0}$-invariant
nondegenerate critical saturated submanifold of $f$. This can be done by
following two possible approaches. The first one is by using the
$2$-equivariant linearization from Proposition 3.5 together with the classical
equivariant Morse Lemma proved in [41]. The second one is by applying the
ideas of the proof of the classical Morse–Bott lemma provided in [30, s. 4.2 &
Thm. 4.5] and then by constructing an Euler–like multiplicative vector field
which must be made $G$-invariant by averaging with respect to the $2$-action
$\theta$.

  * •

To describe the topological behavior of the level subgroupoids of $f$ in a
$2$-equivariant way. Firstly, the $2$-action $\theta$ induces $2$-actions of
$G_{1}\rightrightarrows G_{0}$ on the level subgroupoids of $f$ since this is
$G_{0}$-invariant. Secondly, the gradient vector field of $f$ with respect to
$\eta$ is a $G$-invariant multiplicative vector field so that its flow
determines a 1-parametric family of $2$-equivariant (local) Lie groupoid
automorphisms of $X_{1}\rightrightarrows X_{0}$. As expected, such a flow
allows to study how are the topology changes of our Lie groupoid whether or
not we cross by a critical subgroupoid level of $f$. Namely, if $f^{-1}[a,b]$
does not contain critical points of $f$ then the subgroupoid levels
$X_{1}^{a}$ and $X_{1}^{b}$ are $2$-equivariant isomorphic. Otherwise, if
$f^{-1}[a,b]$ contains no critical points besides a $G_{0}$-invariant
nondegenerate critical saturated submanifold $S$ then the subgroupoid level
$X_{1}^{b}$ is $2$-equivariant homotopy equivalent to $X_{1}^{a}\cup_{\partial
D_{-}(G_{S})}D_{-}(G_{S})$. Here $D_{-}(G_{S})$ and $\partial D_{-}(G_{S})$
are the groupoids of disks and spheres of $S$ which are defined with respect
to $\eta$ and also depend on $f$, see [37]. In our case, the $2$-action
$\theta$ also induces canonical $2$-actions of $G_{1}\rightrightarrows G_{0}$
on those groupoids. It is important to mention that these results strongly
depend on the fact that our groupoid metric $\eta$ is invariant by the
$2$-action of $G_{1}\rightrightarrows G_{0}$.

  * •

To construct an equivariant Morse–Bott double cochain complex which computes
the equivariant cohomology of a 2-action as defined in [7]. Such a double
complex is an extension of the equivariant Austin–Braam’ complex introduced in
[3, s. 4] to the 2-equivariant setting. Our approach mainly depends on the
fact that the gradient vector field of $f$ with respect to $\eta$ is a
$G$-invariant multiplicative vector field which implies that the stable and
unstable groupoids of any $G_{0}$-invariant nondegenerate critical saturated
submanifold of $f$ inherit $2$-actions of $G_{1}\rightrightarrows G_{0}$
induced by $\theta$. The details needed to construct such an equivariant
Morse–Bott double cochain complex were already completed in [37, s. 9.3].

##  4\. Isometries and geometric Killing vector fields

Let $G$ be a Lie group, with Lie algebra $\mathfrak{g}$, acting isometrically
on a Riemannian manifold $(M,\eta^{(0)})$. It is well known that the
fundamental vector field of each element in $\mathfrak{g}$ determines a
Killing vector field on $M$ so that we get a Lie algebra homomorphism from
$\mathfrak{g}$ to the finite dimensional Lie subalgebra
$\mathfrak{o}(M,\eta^{(0)})\leq\mathfrak{X}(M)$ of Killing vector fields on
$M$. The aim of this section is to bring an infinitesimal description of an
isometric Lie $2$-action. Our approach will lead us to the study an algebra of
transversal infinitesimal isometries associated to any Riemannian $n$-metric
on a Lie groupoid. These transversal isometries will give rise to a notion of
geometric Killing vector field on a quotient Riemannian stack.

We start by describing what would be our attempt to set the _diffeomorphism
group_ of a Lie groupoid. Some of the references we shall be following
throughout are [28, 36] and [2, App. D]. A _bisection_ of a Lie groupoid
$X_{1}\rightrightarrows X_{0}$ is a smooth map $\sigma:X_{0}\to X_{1}$ such
that $s_{X}\circ\sigma=\textnormal{id}_{X_{0}}$ and $\iota_{\sigma}:X_{0}\to
X_{0}$ defined by $\iota_{\sigma}(x):=t_{X}(\sigma(x))$ is a diffeomorphism.
The set of all bisections of $X_{1}\rightrightarrows X_{0}$ will be denoted by
$\textnormal{Bis}(X)$. This has the structure of an infinite-dimensional Lie
group where the multiplication of two bisections $\sigma$ and
$\sigma^{\prime}$ is given by
$\sigma\bullet\sigma^{\prime}(x):=\sigma((t_{X}\circ\sigma^{\prime})(x))*\sigma^{\prime}(x)$
for all $x\in X_{0}$; see for instance [40] and [28, s. 1.4]. Let us denote by
$\textnormal{Aut}(X)$ the group of Lie groupoid automorphisms of
$X_{1}\rightrightarrows X_{0}$. Given a bisection
$\sigma\in\textnormal{Bis}(X)$ one has an inner automorphism
$I_{\sigma}:X_{1}\to X_{1}$ defined by

$I_{\sigma}(p):=\sigma(t_{X}(p))*p*i_{X}(\sigma(s_{X}(p))).$ (7)

Clearly, $I_{\sigma}$ covers the map $\iota_{\sigma}$. This inner automorphism
allows us to define what we call the _crossed module of automorphisms of a Lie
groupoid_ $(\textnormal{Aut}(X),\textnormal{Bis}(X),I,\alpha)$ where the map
$\alpha$ is defined as $\alpha_{\Phi}(\sigma):=\Phi\circ\sigma\circ\phi^{-1}$
for all $\Phi\in\textnormal{Aut}(X)$ covering $\phi:X_{0}\to X_{0}$ and
$\sigma\in\textnormal{Bis}(X)$. Accordingly, we have a 2-group
$\textnormal{Bis}(X)\ltimes\textnormal{Aut}(X)\rightrightarrows\textnormal{Aut}(X)$
called the _2-group of Lie groupoid automorphisms_. The first property we
obtain of such a $2$-group is provided below.

######  Proposition 4.1.

The orbit space of the 2-group of Lie groupoid automorphisms equals the set of
Lie groupoid automorphisms up to smooth natural equivalences. Namely:

$\textnormal{Aut}(X)/\textnormal{Bis}(X)=\left\\{[\Phi]\,|\,\Phi\in\textnormal{Aut}(X),\quad\Psi\sim\Phi\Leftrightarrow\exists_{\alpha}\left(\Psi\stackrel{{\scriptstyle\alpha}}{{\Rightarrow}}\Phi\right)\right\\}.$

###### Proof.

Let us describe the orbit of an element in $\Phi\in\textnormal{Aut}(X)$
covering $\phi$. If we pick $\sigma\in\textnormal{Bis}(X)$ and
$\Psi\in\textnormal{Aut}(X)$ covering $\psi$ such that $I_{\sigma}\Phi=\Psi$
then it follows that

$\Psi(p)=I_{\sigma}(\Phi(p))=\sigma(\phi(t_{X}(p)))*\Phi(p)*i_{X}(\sigma(\phi(s_{X}(p)))),$

for some $p\in X_{1}$ so that
$\Psi(p)*\sigma(\phi(s_{X}(p)))=\sigma(\phi(t_{X}(p)))*\Phi(p)$. Therefore, by
setting $\alpha:=\sigma\circ\phi$ we get a smooth natural transformation
$\Phi\stackrel{{\scriptstyle\alpha}}{{\Rightarrow}}\Psi$. Conversely, note
that if $\Phi\stackrel{{\scriptstyle\alpha}}{{\Rightarrow}}\Psi$ a smooth
natural transformation then for some $p\in X$ it holds
$\alpha(t_{X}(p))*\Phi(p)=\Psi(p)*\alpha(s_{X}(p))$, thus obtaining that
$t_{X}(\Phi(p))=s_{X}(\alpha(t_{X}(p)))$ and
$s_{X}(\Psi(p))=t_{X}(\alpha(s_{X}(p)))$. On the one hand, by setting
$\sigma:=\alpha\circ\phi^{-1}$ it follows that
$s_{X}\circ\sigma=\textnormal{id}_{X_{0}}$. On the other hand, observe that

$\psi(s_{X}(p))=t_{X}(\alpha(s_{X}(p)))=t_{X}(\sigma(\phi(s_{X})))(p)=\iota_{\sigma}(\phi(s_{X}(p))).$

Hence, we have obtained that $i_{\sigma}=\psi\circ\phi^{-1}$ so that it is a
diffeomorphism. ∎

Some simple examples which illustrate the naturality of the previous result
are the following.

######  Example 4.2.

If $G\rightrightarrows*$ is a Lie group then $\textnormal{Bis}(G)\simeq G$. We
may think of $\textnormal{Bis}(G)$ as the subset in $\textnormal{Aut}(G)$
determined by conjugations with respect to the elements in $G$. This implies
that the crossed module of automorphisms of $G$ is
$(\textnormal{Aut}(G),G,j,c)$ where $j$ is the inclusion and $c$ is the
identity representation. Thus, the orbit space $\textnormal{Aut}(G)/G$
corresponds to the set of automorphisms of $G$ up to conjugations. That is,
the outer automorphisms of $G$.

######  Example 4.3.

Let $\pi:M\to N$ be a surjective submersion and let
$M\times_{N}M\rightrightarrows M$ denote its corresponding submersion
groupoid. A straightforward computation shows that
$\textnormal{Aut}(M\times_{N}M)$ is in one-to-one correspondence with
$\textnormal{Aut}(\pi)$ which stands for the set of pairs
$(\widetilde{f},f)\in\textnormal{Diff}(M)\times\textnormal{Diff}(N)$ commuting
with $\pi$ and that $\textnormal{Bis}(M\times_{N}M)$ corresponds to
$\textnormal{Gau}(\pi)$ that is the set of pairs
$(\widetilde{f},\textnormal{id})\in\textnormal{Aut}(\pi)$. In this case the
crossed module of automorphisms of $M\times_{N}M$ is
$(\textnormal{Aut}(\pi),\textnormal{Gau}(\pi),j,c)$ where $j$ is the inclusion
and $c$ is the representation by conjugations. Therefore, the orbit space

$\textnormal{Aut}(M\times_{N}M)/\textnormal{Bis}(M\times_{N}M)\simeq\textnormal{Aut}(\pi)/\textnormal{Gau}(\pi)\simeq\textnormal{Diff}(N).$

Let $G_{1}\rightrightarrows G_{0}$ be a Lie $2$-group acting on
$X_{1}\rightrightarrows X_{0}$ by the left.

######  Lemma 4.4.

The normal subgroup $H=\textnormal{ker}(s_{G})$ acts on
$X_{1}\rightrightarrows X_{0}$ by bisections and $G_{0}$ acts by Lie groupoid
automorphisms. Moreover, the right multiplication map defined on
$s_{X}$-fibers for each arrow is $H$-equivariant.

###### Proof.

Consider $h\in H$ and define the map $\sigma_{h}:X_{0}\to X_{1}$ as
$\sigma_{h}(x):=h1_{x}.$ It is clear that $s_{X}(\sigma_{h}(x))=x$ and
$t(\sigma_{h}(x))=\rho(h)x$ so that $\sigma_{h}$ is a well defined bisection.
Let us now take $g\in G_{0}$ and define the map $\Sigma_{g}:X_{1}\to X_{1}$ as
$\Sigma_{g}(x):=1_{g}x$. Equation (4) implies that

$1_{g}(x*y)=(1_{g}*1_{g})(p*q)=(1_{g}p)*(1_{g}q),$

for all $g\in G_{0}$ and $(p,q)\in X_{2}$, thus obtaining that $\Sigma_{g}$ is
a Lie groupoid morphism which clearly satisfies
$(\Sigma_{g})^{-1}=\Sigma_{g^{-1}}$. Note that $s_{X}(hp)=s_{X}(p)$ for all
$p\in X_{1}$ and $h\in H$ so that the left action of $H$ on $X_{1}$ preserves
the $s_{X}$-fibers. Therefore, for each $y\xleftarrow[]{\it p}x$ the right
action $R_{p}:s_{X}^{-1}(y)\to s_{X}^{-1}(x)$ satisfies that

$R_{p}(hq)=(hq)*(1_{e}p)=(h*1_{e})(q*p)=hR_{p}(q).$

In consequence, $R_{p}$ is $H$-equivariant as claimed. ∎

It is clear that the same result can be obtained if we consider right
2-actions instead of left ones. Let $(G,H,\rho,\alpha)$ denote the crossed
module of Lie groups associated to $G_{1}\rightrightarrows G_{0}$. Then:

######  Lemma 4.5.

There is a natural morphism of crossed modules of Lie groups
$(\sigma,\Sigma):(G,H,\rho,\alpha)\to(\textnormal{Aut}(X),\textnormal{Bis}(X),I,\alpha)$
where $\sigma_{h}$ and $\Sigma_{g}$ are defined as in Lemma 4.4.

###### Proof.

Let us check that $\Sigma\circ\rho=I\circ\sigma$ and
$\sigma_{\alpha_{g}(h)}=\alpha_{\Sigma_{g}}(\sigma_{h})$ for all $g\in G$ and
$h\in H$. Firstly, for $p\in X_{1}$ and $h\in H$ we obtain

$\displaystyle I_{\sigma_{h}}(p)$ $\displaystyle=$
$\displaystyle\sigma_{h}(t_{X}(p))*p*i(\sigma_{h}(s_{X}(p)))=(h1_{t_{X}(p)})*p*i(h1_{s_{X}(p)})$
$\displaystyle=$
$\displaystyle(h*e)(1_{t_{X}(p)}*p)*i(h1_{s_{X}(p)})=hp*i_{G}(h)1_{s(p)}=1_{\rho(h)}p=\Sigma_{\rho(h)}(p).$

Secondly, for $x\in X_{0}$, $g\in G_{0}$ and $h\in H$ we get

$\alpha_{\Sigma_{g}}(\sigma_{h})(x)=\Sigma_{g}(h1_{(g^{-1}x)})=1_{g}h1_{g^{-1}}1_{x}=\alpha_{g}(h)1_{x}=\sigma_{\alpha_{g}(h)}(x).$

∎

We are now in conditions to define an _infinitesimal 2-action_ associated to a
right Lie 2-group action. In order to do so we need to introduce the following
structure. Let $A_{X}$ be the Lie algebroid of $X_{1}\rightrightarrows X_{0}$
and for each $\xi\in\mathfrak{g}_{0}$ we denote its fundamental vector fields
as $\tilde{\xi}$. We also denote by $\mathfrak{X}_{m}(X)$ the set of
multiplicative vector fields on $X_{1}\rightrightarrows X_{0}$ and by
$(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D)$ its associated crossed module.
Here we have that $\delta(\alpha)=\alpha^{r}-\alpha^{l}$ and
$D_{(\xi,v)}\alpha=[\xi,\alpha^{r}]|_{X_{0}}$ for all $\alpha\in\Gamma(A_{X})$
and $(\xi,v)\in\mathfrak{X}_{m}(X)$. The reader is recommended to visit [36,
s. 7.1] for getting specific details.

######  Theorem 4.6.

Suppose that we have a right Lie 2-action of $G_{1}\rightrightarrows G_{0}$ on
$X_{1}\rightrightarrows X_{0}$. Let
$(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ be the crossed module of
Lie algebras associated to the Lie $2$-algebra of $G_{1}\rightrightarrows
G_{0}$. Then there is a canonical homomorphism of Lie 2-algebra
$j=(j_{-1},j_{0}):(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})\to(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D)$
defined by $j_{-1}(\xi)=\left.\tilde{\xi}\right|_{X_{0}}$ and
$j_{0}(\zeta)=(\tilde{1_{\zeta}},\tilde{\zeta})$ for all $\xi\in\mathfrak{h}$
and $\zeta\in\mathfrak{g}$.

###### Proof.

To see that $j$ is a well defined morphism we have to check that for any
$\xi\in\mathfrak{h}$ its fundamental vector field $\tilde{\xi}$ belongs to
$\mathfrak{X}_{inv}^{s}(X)$ and that for any $\zeta\in\mathfrak{g}$ it holds
that $(\tilde{1_{\zeta}},\tilde{\zeta})$ is in $\mathfrak{X}_{m}(X)$. The
latter assertion is clear since if $\exp{t\zeta}\in G_{0}$ then Lemma 4.4
implies that the pair of flows
$(\varphi_{t}^{\tilde{1_{\zeta}}},\varphi_{t}^{\tilde{\zeta}})$ determines a
Lie groupoid morphism. Now, if $\xi\in\mathfrak{h}$ then again from Lemma 4.4
it follows that the flow of its fundamental vector field lies inside the
$s$-fibers since
$s_{X}(\varphi_{t}^{\tilde{\xi}}(p))=s_{X}(p\exp(t\xi))=s_{X}(p)$. Therefore,
$\tilde{\xi}\in\mathfrak{X}^{s}(X)$. Furthermore, for $y\xleftarrow[]{\it p}x$
and $q\in s_{X}^{-1}(y)$ one has that

$R_{p}(\varphi^{\tilde{\xi}}_{t}(q))=(q\exp(t\xi))*p=(q*p)(\exp{t\xi}*1_{e})=\varphi_{t}^{\tilde{\xi}}(q*p)=\varphi_{t}^{\tilde{\xi}}(R_{p}(q)),$

thus obtaining that $d(R_{p})_{q}(\tilde{\xi}_{q})=\tilde{\xi}_{q*p}$ so that
$\tilde{\xi}\in\mathfrak{X}_{inv}^{s}(X)$ and
$\left.\tilde{\xi}\right|_{X_{0}}\in\Gamma(A_{X})$.

Let us finally verify that for $\xi\in\mathfrak{h}$ and $\zeta\in\mathfrak{g}$
it satisfies that $\delta(j_{-1}(\xi))=j_{0}(\partial\xi)$ and
$j_{-1}(\mathcal{L}_{\zeta}\xi)=D_{j_{0}(\zeta)}(j_{-1}(\xi))$. On the one
hand, by using the flow of vector field
$\delta(j_{0}(\xi))=\left.\tilde{\xi}\right|_{X_{0}}^{r}-\left.\tilde{\xi}\right|_{X_{0}}^{l}$
and Equation (4) we get

$\displaystyle\varphi_{t}^{\delta(j_{-1}(\xi))}(p)$ $\displaystyle=$
$\displaystyle\varphi_{t}^{\tilde{\xi}}(1_{t_{X}(p)})*p*\iota_{X}(\varphi_{t}^{\tilde{\xi}}(1_{s_{X}(p)}))=1_{t_{X}(p)}\exp(t\xi)*p*i_{X}(1_{s_{X}(p)}\exp(t\xi))$
$\displaystyle=$
$\displaystyle(1_{t_{X}(p)}*p)\exp(t\xi)*1_{s_{X}(p)}i_{G}(\exp(t\xi))=p(\exp(t\xi)*i_{G}(\exp(t\xi)))$
$\displaystyle=$ $\displaystyle
p(1_{t_{G}(\exp(t\xi))})=p1_{\exp(t\partial(\xi))}=p\exp(t1_{\partial\xi})=\varphi_{t}^{{j_{0}(\partial\xi)}}(p).$

Hence, $\delta(j_{-1}(\xi))=j_{0}(\partial\xi)$. On the other hand, observe
that

$D_{j_{0}(\zeta)}(j_{-1}(\xi))=\left.\left[\tilde{1_{\zeta}},\tilde{\xi}\right]\right|_{X_{0}}=\left.\tilde{\left[1_{\zeta},\xi\right]}\right|_{X_{0}}=(\left.\tilde{\mathcal{L}_{\zeta}(\xi))}\right|_{X_{0}}=j_{-1}(\mathcal{L}_{\zeta}\xi).$

∎

Let us now put a Riemannian structure into play. Suppose that
$X_{1}\rightrightarrows X_{0}$ can be equipped with a 0-metric $\eta$ and
consider the following sets

$\textnormal{Bis}_{\eta}(X)=\left\\{\sigma\in\textnormal{Bis}(X)\,|\,\iota_{\sigma}^{*}\eta=\eta\right\\}\quad\textnormal{and}\quad\mathrm{Iso}(X,\eta)=\left\\{(\Phi,\phi)\in\mathrm{Aut}(X)\,|\,\phi^{*}\eta=\eta\right\\}.$

######  Proposition 4.7.

The quadruple $(\mathrm{Iso}(X,\eta),\textnormal{Bis}_{\eta}(X),I,\alpha)$
determines a sub-crossed module structure of
$(\textnormal{Aut}(X),\textnormal{Bis}(X),I,\alpha)$.

###### Proof.

It is clear that $I(\textnormal{Bis}_{\eta}(X))\subseteq\mathrm{Iso}(X,\eta)$.
As $I_{\alpha_{\Phi}(\sigma)}=\Phi I_{\sigma}\Phi^{-1}$ for all
$\sigma\in\textnormal{Bis}_{\eta}(X)$ and $\Phi\in\textnormal{Iso}(X,\eta)$
then when restricting to unities we have that
$\iota_{\alpha_{\Phi}(\sigma)}=\phi\circ\iota_{\sigma}\circ\phi^{-1}$, thus
obtaining an isometry. ∎

The Lie 2-group associated to the crossed module
$(\mathrm{Iso}(X,\eta),\textnormal{Bis}_{\eta}(X),I,\alpha)$ will be called
_Lie 2-group of strong isometries_ of $(X_{1}\rightrightarrows X_{0},\eta)$.

######  Example 4.8.

Let $(M,\eta)$ be an $n$-dimensional Riemannian manifold and let $\pi:O(M)\to
M$ denote the corresponding $O(n,\mathbb{R})$-principal bundle of orthonormal
frames. After fixing an Ad-invariant inner product $\langle\cdot,\cdot\rangle$
on the Lie algebra $\mathfrak{o}(n,\mathbb{R})$ and taking the connection
1-form $\omega\in\Omega^{1}(O(M),\mathfrak{o}(n,\mathbb{R}))$ associated to
the Levi–Civita connection on $(M,\eta)$ we can define a Riemannian metric on
$O(M)$ as

$\tilde{\eta}(X,Y)=\eta(d\pi(X),d\pi(Y))+\langle\omega(X),\omega(Y)\rangle.$

It follows that $\pi:O(M)\to M$ becomes a Riemannian submersion and its
corresponding submersion groupoid $O(M)\times_{M}O(M)\rightrightarrows O(M)$
inherits an induced $0$-metric which we also denote by $\eta$ [18, 35].
Therefore,

$\textnormal{Bis}_{\eta}(O(M)\times_{M}O(M))\simeq\textnormal{Gau}(O(M),\theta)\quad\textnormal{and}\quad\textnormal{Iso}(O(M)\times_{M}O(M),\eta)\simeq\textnormal{Aut}(O(M),\theta),$

where $\theta\in\Omega^{1}(O(M),\mathbb{R}^{n})$ is the canonical $1$-form,
$\textnormal{Aut}(O(M),\theta)$ is the group of bundle isomorphisms preserving
$\theta$ and $\textnormal{Gau}(O(M),\theta)$ is its normal subgroup of bundle
isomorphisms covering the identity (compare Example 4.3). Hence, from [26, p.
236] we get that the orbit space

$\textnormal{Iso}(O(M)\times_{M}O(M),\eta)/\textnormal{Bis}_{\eta}(O(M)\times_{M}O(M))\simeq\textnormal{Iso}(M,\eta).$

Let us now consider the sets

$\Gamma_{\eta}(A_{X})=\left\\{\alpha\in\Gamma(A_{X})\,|\,\rho(\alpha)\in\mathfrak{o}(X_{0},\eta)\right\\}\quad\textnormal{and}\quad\mathfrak{o}_{m}(X)=\left\\{(\xi,v)\in\mathfrak{X}_{m}(X)\,|\,v\in\mathfrak{o}(X_{0},\eta)\right\\}$

where $\mathfrak{o}(X_{0},\eta)$ denotes the Lie algebra of Killing vector
fields of $(X_{0},\eta)$. In these terms we may describe the infinitesimal
version of the Lie $2$-group of strong isometries as follows.

######  Proposition 4.9.

The quadruple $(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D)$ defines a
sub-crossed module structure of
$(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D)$.

###### Proof.

Firstly, note that $\mathfrak{o}_{m}(X)$ is a Lie subalgebra of
$\mathfrak{X}_{m}(X)$ since $\mathfrak{o}(X_{0},\eta)$ is a Lie algebra.
Secondly, if $\alpha,\beta\in\Gamma_{\eta}(A_{X})$ then
$\rho([\alpha,\beta])=[\rho(\alpha),\rho(\beta)]\in\mathfrak{o}(X_{0},\eta)$
so that $[\alpha,\beta]\in\Gamma_{\eta}(A)$. If
$(\xi,v)\in\mathfrak{o}_{m}(X)$ and $\alpha\in\Gamma_{\eta}(A)$ then we have
by definition that
$D_{\xi}(\alpha)=\left.[\xi,\alpha^{r}]\right|_{X_{0}}\in\Gamma(A_{X})$.
However, the equivariance identity implies that
$\delta(D_{\xi}(\alpha))=[\xi,\delta(\alpha)]$. Therefore, it holds that
$\left.\delta(D_{\xi}\alpha)\right|_{X_{0}}=\left.[\xi,\delta(\alpha)]\right|_{X_{0}}$
which is the same thing that saying
$\rho(D_{\xi}\alpha)=[v,\rho(\alpha)]\in\mathfrak{o}(X_{0},\eta)$ since $v$ is
also a Killing vector field. ∎

The Lie $2$-algebra associated to the crossed module
$(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D)$ will be called _Lie
2-algebra of strong multiplicative Killing vector fields_ of
$(X_{1}\rightrightarrows X_{0},\eta)$.

Summing up, as consequence of Lemma 3.2 we can now provide an infinitesimal
description of an isometric Lie $2$-group action. Indeed:

######  Corollary 4.10.

Let $\theta$ be an isometric right 2-action of a Lie $2$-group
$G_{1}\rightrightarrows G_{0}$ on a Riemannian groupoid
$(X_{1}\rightrightarrows X_{0},\eta)$. Denote by $(G,H,\rho,\alpha)$ to the
crossed module of Lie groups associated to $G_{1}\rightrightarrows G_{0}$ and
by $(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})$ to its corresponding
crossed module of Lie algebras. Then:

  * •

there is a morphism of crossed modules of Lie groups

$(\sigma,\Sigma):(G,H,\rho,\alpha)\to(\textnormal{Iso}(X,\eta),\textnormal{Bis}_{\eta}(X),I,\alpha),$

that is defined as in Lemma 4.5, and

  * •

there is a morphism of crossed modules of Lie algebras

$(j_{-1},j_{0}):(\mathfrak{g},\mathfrak{h},\partial,\mathcal{L})\to(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D),$

which is defined in Theorem 4.6.

###  4.1. Transversal isometries

Recall that a $0$-metric on a Lie groupoid $X_{1}\rightrightarrows X_{0}$ is a
Riemannian metric $\eta$ on $X_{0}$ which is transversely invariant by the
canonical left action of $X_{1}\rightrightarrows X_{0}$ on $X_{0}$, compare
[18, 38]. Note that this is the same that requiting that $\eta$ is
transversely invariant by the action of the group of bisections
$\textnormal{Bis}(X)\times X_{0}\to X_{0}$ which is defined by $\sigma\cdot
x:=\iota_{\sigma}(x)$. It is clear that this action preserves the orbits so
that it induces a well defined action on the normal space of an orbit. In
consequence, $\eta$ is a 0-metric if and only if for all
$\sigma\in\textnormal{Bis}(X)$ the map
$\overline{d\iota_{\sigma}}:(\nu_{x}(\mathcal{O}),\overline{\eta})\to(\nu_{\iota_{\sigma}(x)}(\mathcal{O}),\overline{\eta})$
is a linear isometry since we may identify $\nu(\mathcal{O})\cong
T\mathcal{O}^{\perp}$. Therefore, motivated by this fact and what we did in
the previous section we now plan to weaken the condition for a diffeomorphism
to be an isometry by imposing instead a transversal isometric condition along
groupoid orbits. This will lead us to define a Lie $2$-algebra of transverse
infinitesimal isometries with respect to any Riemannian groupoid $n$-metric,
which at the end turns out to be Morita invariant.

Let $\Phi:X_{1}\to X_{1}$ be a Lie groupoid automorphism covering
$\phi:X_{0}\to X_{0}$. From now on we assume the identification
$\nu(\mathcal{O})\cong T\mathcal{O}^{\perp}$ for each groupoid orbit
$\mathcal{O}$ in $(X_{0},\eta^{(0)})$ without stating it explicitly unless it
is necessary. The diffeomorphism $\phi:X_{0}\to X_{0}$ is said to be a
_transversal isometry_ of $(X_{0},\eta)$ if
$\overline{d\phi}:\nu(\mathcal{O}_{x})\to\nu(\mathcal{O}_{\phi(x)})$ is a
fiberwise isometry for every groupoid orbit $\mathcal{O}_{x}$ in $X_{0}$.
Consider the set

$\textnormal{Iso}_{\textnormal{w}}(X,\eta)=\left\\{(\Phi,\phi)\in\mathrm{Aut}(X)\,|\,\phi\
\textnormal{transversal isometry of}\ (X_{0},\eta)\right\\}.$

Note that for every $\sigma\in\textnormal{Bis}(X)$ it follows that
$\iota_{\sigma}$ is a transversal isometry of $(X_{0},\eta)$. Thus, by arguing
as in Proposition 4.7 we easily get that:

######  Lemma 4.11.

The quadruple
$(\mathrm{Iso}_{\textnormal{w}}(X,\eta),\textnormal{Bis}(X),I,\alpha)$
determines a sub-crossed module structure of
$(\textnormal{Aut}(X),\textnormal{Bis}(X),I,\alpha)$.

The Lie $2$-group determined by the crossed module
$(\mathrm{Iso}_{\textnormal{w}}(X,\eta),\textnormal{Bis}(X),I,\alpha)$ is
called _Lie 2-group of weak isometries_ of $(X_{1}\rightrightarrows
X_{0},\eta)$. Let us denote by $\mathfrak{o}^{\textnormal{w}}(X_{0},\eta)$ the
set of vector fields on $X_{0}$ which are covered by multiplicative vector
fields on $X_{1}$ whose flow determines a (local) transversal isometry of
$(X_{0},\eta)$ and consider the set
$\mathfrak{o}_{m}^{\textnormal{w}}(X)=\left\\{(\xi,v)\in\mathfrak{X}_{m}(X)\,|\,v\in\mathfrak{o}^{\textnormal{w}}(X_{0},\eta)\right\\}$.
Observe that if $v_{1},v_{2}\in\mathfrak{o}^{\textnormal{w}}(X_{0},\eta)$ then
for $t$ small enough we have that the commutator flow
$\varphi_{-\sqrt{t}}^{v_{2}}\varphi_{-\sqrt{t}}^{v_{1}}\varphi_{\sqrt{t}}^{v_{2}}\varphi_{\sqrt{t}}^{v_{1}}$
is a transversal isometry so that $\mathfrak{o}^{\textnormal{w}}(X_{0},\eta)$
is a Lie subalgebra of $\mathfrak{X}(X_{0})$. From [40, Thm. D] we know that
$\textnormal{Lie}(\textnormal{Bis}(X))$ is identified with $\Gamma(A_{X})$.
Therefore, by using similar arguments as those in Proposition 4.9 we obtain a
description of the Lie $2$-group of weak isometries of
$(X_{1}\rightrightarrows X_{0},\eta)$. Namely:

######  Proposition 4.12.

The quadruple $(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$
defines a sub-crossed module structure of
$(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D)$.

Accordingly, the Lie $2$-algebra associated to the crossed module
$(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$ will be called
_Lie 2-algebra of weak multiplicative Killing vector fields_ of
$(X_{1}\rightrightarrows X_{0},\eta)$.

######  Remark 4.13.

Let us suppose that we are equipped with an $n$-metric $\eta^{(n)}$ on $X_{n}$
and that $(\xi,v)$ is a multiplicative vector field on $X_{1}\rightrightarrows
X_{0}$ with $v\in\mathfrak{o}^{\textnormal{w}}(X_{0},\eta^{(0)})$. It is
simple to see that the fact that $s_{X}:X_{1}\to X_{0}$ (or $t_{X}:X_{1}\to
X_{0}$) is a Riemannian submersion clearly implies that
$\xi\in\mathfrak{o}^{\textnormal{w}}(X_{1},\eta^{(1)})$. More importantly, if
$\xi_{n}$ denotes the vector field on $X_{n}$ induced by $(\xi,v)$ for all
$n\geq 2$ then if follows that
$\xi_{n}\in\mathfrak{o}^{\textnormal{w}}(X_{n},\eta^{(n)})$ since the face
maps $X_{n}\to X_{n-1}$ are Riemannian submersions. As a consequence of this
we have that the notion of weak multiplicative Killing vector field can be
extended to a notion associated to any Riemannian $n$-metric. This immediately
implies that the Lie 2-algebra of weak multiplicative Killing vector fields
$(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$ can be thought
of as an algebraic object associated to any Riemannian $n$-metric on $X_{n}$.

###  4.2. Morita invariance of transversal Killing vector fields

In this subsection we apply some of the results from [36] to our context in
order to define a notion of geometric Killing vector field on a quotient
Riemannian stack. Then we use some of the results from [13] to describe
several interesting features that these kinds of vector fields have.

Let us start by introducing some necessary terminology. A _Morita map_ is a
groupoid morphism $\phi:(X_{1}\rightrightarrows
X_{0})\to(Y_{1}\rightrightarrows Y_{0})$ which is _fully faithful_ and
_essentially surjective_ , in the sense that the source/target maps define a
fibred product of manifolds $X_{1}\cong(X_{0}\times X_{0})\times_{(Y_{0}\times
Y_{0})}Y_{1}$ and that the map $Y_{1}\times_{Y_{0}}X_{0}\to X_{0}$ sending
$(\phi^{0}(x)\to y)\mapsto y$ is a surjective submersion [15, 33]. An
important fact shown in [15, Thm. 4.3.1] is that a Lie groupoid morphism is a
Morita map if and only if it yields an isomorphism between _transversal data_.
That is, the morphism must induce: a homeomorphism between the orbit spaces, a
Lie group isomorphism $X_{x}\cong Y_{\phi^{0}(x)}$ between the isotropies and
isomorphisms between the normal representations
$X_{x}\curvearrowright\nu_{x}\to
Y_{\phi^{0}(x)}\curvearrowright\nu^{\prime}_{\phi^{0}(x)}$. We think of a
quotient stack as a Lie groupoid up to Morita equivalence in the sense that
two Lie groupoids $X$ and $Y$ define the same stack if there is a third
groupoid $Z$ and Morita maps $Z\to X$ and $Z\to Y$ [15]. These two Morita maps
may be assumed to be Morita fibrations (surjective submersion at the level of
objects) [33, p. 131]. The quotient stack associated to the Lie groupoid
$X_{1}\rightrightarrows X_{0}$ will be denoted by $[X_{0}/X_{1}]$.

Two Riemannian metrics $\eta_{1}$ and $\eta_{2}$ on $X_{1}\rightrightarrows
X_{0}$ are said to be _equivalent_ if they induce the same inner product on
the normal vector spaces over the groupoid orbits [19]. More generally, we
define a _Riemannian Morita map_ (_fibration_) $\phi:(Z_{1}\rightrightarrows
Z_{0})\to(X_{1}\rightrightarrows X_{0})$ as a Morita map between Riemannian
groupoids that induces isometries on the normal vector spaces to the groupoid
orbits $\nu_{z}(\mathcal{O}^{Z})\to\nu_{\phi(z)}(\mathcal{O}^{X})$ (Riemannian
submersion at the level of objects). By using this terminology we have that
$\eta_{1}$ and $\eta_{2}$ are equivalent if and only if the identity
$\textnormal{id}:(X_{1}\rightrightarrows
X_{0},\eta_{1})\to(X_{1}\rightrightarrows X_{0},\eta_{2})$ is a Riemannian
Morita map.

Following [36, s. 6], given a Morita fibration $\phi:(Z_{1}\rightrightarrows
Z_{0})\to(X_{1}\rightrightarrows X_{0})$ we denote the set of projectable
sections by

$\Gamma(A_{Z})^{\phi}=\\{\alpha\in\Gamma(A_{Z}):\textnormal{there exists}\
\alpha^{\prime}\in\Gamma(A_{X})\ \textnormal{such that}\
\phi_{\ast}\alpha=\alpha^{\prime}\phi\\}.$

If $\alpha\in\Gamma(A_{Z})$ then the surjectivity of $\phi$ at the level of
objects implies that there exists at most one section
$\alpha^{\prime}\in\Gamma(A_{X})$ such that
$\phi_{\ast}\alpha=\alpha^{\prime}\phi$, so that it follows that there is a
natural linear map $\phi_{\ast}:\Gamma(A_{Z})^{\phi}\to\Gamma(A_{X})$. We
denote by $\Gamma(A_{Z})^{\phi}\hookrightarrow\Gamma(A_{Z})$ the inclusion
map. It is clear that we can similarly define the set of projectable
multiplicative sections $\mathfrak{X}_{m}(Z)^{\phi}$, a natural map
$\phi_{\ast}:\mathfrak{X}_{m}(Z)^{\phi}\to\mathfrak{X}_{m}(X)$ and an
inclusion $\mathfrak{X}_{m}(Z)^{\phi}\hookrightarrow\mathfrak{X}_{m}(Z)$. From
[36, Prop. 7.4] it follows that
$(\mathfrak{X}_{m}(Z)^{\phi},\Gamma(A_{Z})^{\phi},\delta,D)$ is a sub-crossed
module of $(\mathfrak{X}_{m}(Z),\Gamma(A_{Z}),\delta,D)$ and both maps
$\phi_{\ast}:(\mathfrak{X}_{m}(Z)^{\phi},\Gamma(A_{Z})^{\phi},\delta,D)\to(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D)$
and
$(\mathfrak{X}_{m}(Z)^{\phi},\Gamma(A_{Z})^{\phi},\delta,D)\hookrightarrow(\mathfrak{X}_{m}(Z),\Gamma(A_{Z}),\delta,D)$
are morphisms of crossed-modules. More importantly,

$(\mathfrak{X}_{m}(Z),\Gamma(A_{Z}),\delta,D)\hookleftarrow(\mathfrak{X}_{m}(Z)^{\phi},\Gamma(A_{Z})^{\phi},\delta,D)\xrightarrow[]{\it\phi_{\ast}}(\mathfrak{X}_{m}(X),\Gamma(A_{X}),\delta,D),$

are quasi-isomorphisms of crossed modules. For specific details visit [36].

######  Lemma 4.14.

Let $\phi:(Z_{1}\rightrightarrows Z_{0},\eta^{Z})\to(X_{1}\rightrightarrows
X_{0},\eta^{X})$ be a Morita Riemannian fibration. Then

  * •

$(\mathfrak{o}_{m}(Z)^{\phi},\Gamma_{\eta}(A_{Z})^{\phi},\delta,D)$ is a sub-
crossed module of $(\mathfrak{o}_{m}(Z),\Gamma_{\eta}(A_{Z}),\delta,D)$,

  * •

the inclusion
$(\mathfrak{o}_{m}(Z)^{\phi},\Gamma_{\eta}(A_{Z})^{\phi},\delta,D)\hookrightarrow(\mathfrak{o}_{m}(Z),\Gamma_{\eta}(A_{Z}),\delta,D)$
is a morphism of crossed modules, and

  * •

the projection
$\phi_{\ast}:(\mathfrak{o}_{m}(Z)^{\phi},\Gamma_{\eta}(A_{Z})^{\phi},\delta,D)\to(\mathfrak{o}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D)$
is a morphism of crossed modules.

Moreover,

$(\mathfrak{o}_{m}(Z),\Gamma_{\eta}(A_{Z}),\delta,D)\hookleftarrow(\mathfrak{o}_{m}(Z)^{\phi},\Gamma_{\eta}(A_{Z})^{\phi},\delta,D)\xrightarrow[]{\it\phi_{\ast}}(\mathfrak{0}_{m}(X),\Gamma_{\eta}(A_{X}),\delta,D),$

are quasi-isomorphisms of crossed modules. Same conclusion holds true for the
weak counterpart.

###### Proof.

By using similar arguments as those in Lemma 3.2 if follows that if $v$ is a
(weak) Killing vector field on $(Z_{0},\eta^{Z})$ then $\phi_{\ast}(v)$ is
also a (weak) Killing vector field on $(X_{0},\eta^{X})$. Therefore, the
result follows by applying Proposition 7.4 and Theorem 7.3 from [36] after
restricting the structure. ∎

The following result is clear.

######  Lemma 4.15.

If $\eta_{1}$ and $\eta_{2}$ are equivalent Riemannian metrics on
$X_{1}\rightrightarrows X_{0}$ then the crossed modules
$(\mathfrak{o}_{m}^{\textnormal{w}}(X,\eta_{1}),\Gamma(A_{X}),\delta,D)$ and
$(\mathfrak{o}_{m}^{\textnormal{w}}(X,\eta_{2}),\Gamma(A_{X}),\delta,D)$
agree.

Suppose that $X$ and $Y$ are Morita equivalent Lie groupoids so that there is
a third Lie groupoid $Z$ with Morita fibrations $Z\to X$ and $Z\to Y$. From
[19, Prop. 6.3.1] we know that if $\eta^{X}$ is a Riemannian metric on $X$
then there exists a Riemannian metric $\eta^{Z}$ on $Z$ that makes the
fibration $Z\to X$ Riemannian. We can slightly modify $\eta^{Z}$ by a
cotangent averaging procedure so that we get another Riemannian metric
$\tilde{\eta}^{Z}$ on $Z$ which descends to $Y$ defining a Riemannian metric
$\eta^{Y}$ making of the fibration $Z\to Y$ Riemannian. It turns out that
these pullback and pushforward constructions are well-defined and mutually
inverse modulo equivalence of metrics. This is because $\eta^{Z}$ and
$\tilde{\eta}^{Z}$ turn out to be equivalent, see the proof of Theorem 6.3.3
in [19]. In this case we refer to $(X,\eta^{X})$ and $(Y,\eta^{Y})$ as being
_Morita equivalent Riemannian groupoids_. It suggests a definition for
Riemannian metrics over differentiable stacks. Namely, a stacky metric on the
orbit stack $[X_{0}/X_{1}]$ presented by a Lie groupoid
$X_{1}\rightrightarrows X_{0}$ is defined to be an equivalence class $[\eta]$
of a Riemannian metric $\eta$ on $X$. For further details the reader is
recommended to visit [19].

Summing up, we get that:

######  Theorem 4.16.

If $(X_{1}\rightrightarrows X_{0},\eta^{X})$ and $(Y_{1}\rightrightarrows
Y_{0},\eta^{Y})$ are Morita equivalent Riemannian groupoids then the crossed
modules $(\mathfrak{o}_{m}^{\textnormal{w}}(X),\Gamma(A_{X}),\delta,D)$ and
$(\mathfrak{o}_{m}^{\textnormal{w}}(Y),\Gamma(A_{Y}),\delta,D)$ are isomorphic
in the derived category of crossed modules. In consequence, the following
quotient spaces are isomorphic as Lie algebras:

$\mathfrak{o}_{m}^{\textnormal{w}}(X)/\textnormal{im}(\delta)\cong\mathfrak{o}_{m}^{\textnormal{w}}(Y)/\textnormal{im}(\delta).$

###### Proof.

This result is consequence of Lemmas 4.14 and 4.15 together with Theorem 7.4
and Corollaries 7.1 and 7.2 from [36]. ∎

Motivated by the previous result and [36, Def. 8.1] we set up the following
definition.

######  Definition 4.17.

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a Riemannian groupoid. A
_geometric Killing vector field_ on the quotient Riemannian stack
$([X_{0}/X_{1}],[\eta])$ is defined to be an element of the quotient

$\mathfrak{o}([X_{0}/X_{1}],[\eta]):=\mathfrak{o}_{m}^{\textnormal{w}}(X)/\textnormal{im}(\delta).$

On the one hand, if we consider proper étale Riemannian groupoids then
geometric Killing vector fields recover the classical notions of Killing
vector fields on both Riemannian manifolds and Riemannian orbifolds as defined
for instance in [6]. On the other hand, if we consider the Riemannian groupoid
$\textnormal{Hol}(M,\mathcal{F})\rightrightarrows M$ associated to a regular
Riemannian foliation $(M,\mathcal{F})$ then geometric Killing vector fields
recover the notion of transverse Killing vector fields as defined in [34, p.
84]. Each of these particular cases has the property that the obtained algebra
of geometric Killing vector fields is finite dimensional. We finish this
section by proving that this is always the case if our quotient Riemannian
stack is separated (i.e. it is presented by a proper Riemannian groupoid). Let
us start by analyzing the Riemannian foliation groupoid case. A _foliation
groupoid_ is a Lie groupoid $X_{1}\rightrightarrows X_{0}$ whose space of
objects $X_{0}$ is Hausdorff and whose isotropy groups $X_{x}$ are discrete
for all $x\in X_{0}$. For instance, every étale Lie groupoid with Hausdorff
objects manifold is a foliation groupoid. The converse is not true, however
every foliation groupoid is Morita equivalent to an étale groupoid. As shown
in [12, Thm. 1] (see also [11]), being a foliation groupoid is equivalent to
the associated Lie algebroid anchor map $\rho:A_{X}\to TX_{0}$ being
injective. As a consequence, the manifold $X_{0}$ comes with a regular
foliation $\mathcal{F}$ tangent to the leaves of $\mathrm{im}(\rho)\subseteq
TX_{0}$. Note that if $X_{1}\rightrightarrows X_{0}$ is source-connected then
the leaves of $\mathrm{im}(\rho)\subseteq TX_{0}$ coincide with the groupoid
orbits.

######  Lemma 4.18.

If $(X_{1}\rightrightarrows X_{0},\eta)$ is a Riemannian foliation groupoid
with compact orbit space $X_{0}/X_{1}$ then the algebra of geometric Killing
vector fields on $([X_{0}/X_{1}],[\eta])$ has finite dimension.

###### Proof.

Let $\iota:T\hookrightarrow X_{0}$ be a complete transversal submanifold to
the orbit foliation $\mathcal{F}$ of $X_{1}\rightrightarrows X_{0}$ and
consider its restricted groupoid $X_{T}\rightrightarrows T$. As
$X_{T}\rightrightarrows T$ is étale and Morita equivalent to
$X_{1}\rightrightarrows X_{0}$ (see [33, p. 136]), from Theorem 4.16 it
follows that

$\mathfrak{o}([X_{0}/X_{1}],[\eta])=\mathfrak{o}_{m}^{\textnormal{w}}(X_{T},\iota^{*}\eta)/\textnormal{im}(\delta)\cong\mathfrak{o}(T)^{X}.$

Here $\mathfrak{o}(T)^{X}$ denotes the transversal Killing vector fields that
are invariant by the normal action. As $X_{0}/X_{1}$ is compact we have that
$T$ has a finite number of connected components (see [33, p. 135]), so that
the dimension of $\mathfrak{o}(T)^{X}$ is finite by Theorem 3.3 from [26, p.
238] (see also [34, p. 85]). That is, $\mathfrak{o}([X_{0}/X_{1}],[\eta])$ is
finite dimensional. ∎

It is simple to see that similar arguments to those used in the proof of the
previous Lemma work if we consider regular Riemannian groupoids instead of the
Riemannian foliation ones. Here by regular we mean that the anchor map
$\rho:A_{X}\to TX_{0}$ has locally constant rank. Therefore, as every proper
groupoid is regular over a dense and open subset then one would expect that a
similar result can be proven if we consider proper Riemannian groupoids. We
show below that such a finite dimensional result is true by using the
desingularization theorem for proper Riemannian groupoids proved in [39, s.
6]. Namely:

######  Theorem 4.19.

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a proper Riemannian groupoid with
compact orbit space $X_{0}/X_{1}$. Then the algebra of geometric Killing
vector fields on $([X_{0}/X_{1}],[\eta])$ has finite dimension.

###### Proof.

Let us denote by
$(\tilde{X_{1}}\rightrightarrows\tilde{X_{0}},\tilde{\eta},\pi)$ the
Riemannian desingularization of $(X_{1}\rightrightarrows X_{0},\eta)$, see
Theorem 6.10 in [39]. We have that
$(\tilde{X_{1}}\rightrightarrows\tilde{X_{0}},\tilde{\eta})$ is a proper
regular Riemannian groupoid and $\pi:\tilde{X_{1}}\to X_{1}$ is a proper
Riemannian fibration which is almost-everywhere an isometry. As consequence of
the functoriality properties described in [39, s. 5.2] it follows that any Lie
groupoid automorphism of $X_{1}$ preserves its co-dimensional stratum data
(see [39, s. 3]), so that they can be lifted to Lie groupoid automorphism of
$\tilde{X_{1}}$. That is, we can lift multiplicative vector fields on $X_{1}$
to multiplicative vector fields on $\tilde{X_{1}}$ by lifting their
1-parametric (local) families of Lie groupoid automorphisms determined by
their flows. In particular, by using both Proposition 6.13 and Theorem 6.14
from [39] we get that weak multiplicative Killing vector fields on
$(X_{1},\eta)$ can be lifted to weak multiplicative Killing vector fields on
its desingularization $(\tilde{X_{1}},\tilde{\eta})$, thus obtaining an
surjective algebra homomorphism

$[\pi]:\mathfrak{o}([\tilde{X_{0}}/\tilde{X_{1}}],[\tilde{\eta}])\to\mathfrak{o}([X_{0}/X_{1}],[\eta]).$

By arguing as in Proposition 6.3.2 from [19] and by using the classification
of regular Lie groupoids given in [32], since the groupoid $\tilde{X_{1}}$ is
regular we may assume that it fits into an extension of Riemannian groupoids
$(K,\iota^{\ast}\tilde{\eta})\xrightarrow[]{\it\iota}(\tilde{X_{1}},\tilde{\eta})\xrightarrow[]{\it
q}(E,q_{\ast}\tilde{\eta})$ over $\tilde{X_{0}}$ where $K$ is a bundle of
connected Lie groups (i.e. a Lie groupoid whose source and target maps agree),
$E$ is a foliation groupoid, $\iota$ is a groupoid Riemannian embedding and
$\pi$ is a groupoid Riemannian submersion with connected fibers
($\tilde{\eta}$ possibly needs to be averaged in order to define an equivalent
groupoid metric which descends to the quotient). This in turn induces an
extension of algebras

$0\to\mathfrak{o}(\tilde{X_{0}},\iota^{\ast}\tilde{\eta})\to\mathfrak{o}([\tilde{X_{0}}/\tilde{X_{1}}],[\tilde{\eta}])\to\mathfrak{o}([\tilde{X_{0}}/E],[q_{\ast}\tilde{\eta}])\to
0.$

As $X_{0}/X_{1}$ is compact and $\pi$ is proper and surjective it follows
$\tilde{X_{0}}/\tilde{X_{1}}$ is also compact and from Lemma 4.18 together
with Theorem 3.3 from [26, p. 238] (see also [34, p. 85]) it follows that both
$\mathfrak{o}([\tilde{X_{0}}/E],[q_{\ast}\tilde{\eta}])$ and
$\mathfrak{o}(\tilde{X_{0}},\iota^{\ast}\tilde{\eta})$ have finite dimension
so that $\mathfrak{o}([\tilde{X_{0}}/\tilde{X_{1}}],[\tilde{\eta}])$ has also
finite dimension. That is, $\mathfrak{o}([X_{0}/X_{1}],[\eta])$ is finite
dimensional since $[\pi]$ is surjective. ∎

We end the paper by describing some features about our notion of geometric
Killing vector field which are motivated by some results proved in [13]. Let
$(X_{1}\rightrightarrows X_{0},\eta)$ be a Riemannian groupoid. A vector field
$\xi_{1}$ on $X_{1}$ is said to be _projectable of Killing type_ if there
exists a vector field $v_{1}$ on $(X_{0},\eta^{(0)})$ such that the following
conditions are satisfied:

  1. a.

$\xi_{1}$ and $v_{1}$ are both $s_{X}$-related and $t_{X}$-related, and

  2. b.

if $\varPhi_{\tau}$ and $\varphi_{\tau}$ respectively denote the (local) flows
of $\xi_{1}$ and $v_{1}$ then the induced map
$\overline{d\varphi_{\tau}}:\nu(\mathcal{O}_{x})\to\nu(\mathcal{O}_{\varphi_{\tau}(x)})$
is a linear isometry for each groupoid orbit $\mathcal{O}_{x}$ for which
$\varphi_{\tau}(x)$ is defined.

Compare with [13, Def. 4.6]. We shall also refer to $v_{1}$ as a _weak Killing
vector field_ on $(X_{0},\eta^{(0)})$. The space of vector fields of Killing
type on $X_{1}$ will be denoted by $\Gamma^{\textnormal{proj}}_{\eta}(X_{1})$.
Firstly, note that Condition b. makes sense because the local flows
$\varPhi_{\tau}$ and $\varphi_{\tau}$ commute with both $s_{X}$ and $t_{X}$ as
consequence of Condition a. Secondly, it follows that $\xi_{1}$ is also a weak
Killing vector field on $(X_{1},\eta^{(1)})$ in the sense that
$\overline{d\varPhi_{\tau}}:\nu(G_{\mathcal{O}_{x}})\to\nu(G_{\mathcal{O}_{\varphi_{\tau}(x)}})$
is a linear isometry since $s_{X}$ and $t_{X}$ are Riemannian submersions.
Actually, it is simple to see that a similar property holds true for the
vector field $\xi_{2}(p,q)=(\xi_{1}(p),\xi_{1}(q))$ on $(X_{2},\eta^{(2)})$
induced by $\xi_{1}$.

We explain below how proper Haar measure systems on proper Riemannian
groupoids determine projections from the space of projectable vector fields of
Killing type on $(X_{1},\eta^{(1)})$ to the space of weak multiplicative
Killing vector fields on $(X_{1}\rightrightarrows X_{0},\eta)$. This can be
done by following [14, s. 2.5].

######  Theorem 4.20.

Let $(X_{1}\rightrightarrows X_{0},\eta)$ be a proper Riemannian groupoid.
Then any proper Haar measure system $\\{\mu^{x}\\}$ for
$X_{1}\rightrightarrows X_{0}$ induces a linear map from
$\Gamma^{\textnormal{proj}}_{\eta}(X_{1})$ to
$\mathfrak{o}_{m}^{\textnormal{w}}(X,\eta)$.

###### Proof.

Let us pick a projectable vector field of Killing type $\xi_{1}$ on
$(X_{1},\eta^{(1)})$. From [14, s. 2.5] we know that we can construct a
multiplicative vector field $\xi:X_{1}\to TX_{1}$ by taking the average with
respect to $\\{\mu^{x}\\}$:

$\xi(p)=\int_{a\in
t^{-1}(s(p))}dm_{(pa,a^{-1})}(\xi_{1}(ap),di_{a}(\xi_{1}(a))\mu(a).$

This vector field is $s_{X}$-related with the vector field $v$ on $X_{0}$
defined as

$v(x)=\int_{a\in t^{-1}(x)}dt_{a}(\xi_{1}(a))\mu(a).$

Therefore, our result will follow once we prove that the flow of $v$ is a
local transversal isometry of $(X_{0},\eta)$ since $\xi$ and $v$ are
$s_{X}$-related. However, as $\xi_{1}$ is a weak Killing vector field on
$(X_{1},\eta^{(1)})$, this follows from a straightforward computation after
noting that the flow of $v$ is given by $\varphi^{v}_{r}(x)=\int_{a\in
t^{-1}(x)}(t\circ\varphi^{\xi_{1}}_{r})(a)\mu(a)$ and
$\overline{dt}:\nu(G_{\mathcal{O}})\to\nu(\mathcal{O})$ is a fiberwise
isometry since $t_{X}$ is a Riemannian submersion. Hence, the assignment
$\xi_{1}\mapsto(\xi,v)$ establishes the desired projection. ∎

As consequence of Remark 4.13 we may conclude that the previous result is
actually a fact that can be proven for any $n$-metric $\eta^{(n)}$ on $X_{n}$.

In [13, s. 4.4] it was studied the “normal” bundle
$\vartheta:=TX_{0}/\rho(A_{X})$ where $\rho:A_{X}\to TX_{0}$ is the anchor map
of the Lie algebroid $A_{X}$ of $X_{1}\rightrightarrows X_{0}$. This is a
smooth vector bundle only in the regular case. Its space of sections is
defined to be the quotient