Foliated groupoids and their infinitesimal data - Mathematisches ...

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA

M. JOTZ AND C. ORTIZ

Abstract. In this work, we study Lie **groupoids** equipped with multiplicative foliations

**and** the corresponding **infinitesimal** **data**. We determine the **infinitesimal** counterpart

of a multiplicative foliation in terms of its core **and** sides together with a partial connection

satisfying special properties, giving rise to the concept of IM-foliation on a Lie

algebroid. The main result of this paper shows that if G is a source simply connected Lie

groupoid with Lie algebroid A, then there exists a one-to-one correspondence between

multiplicative foliations on G **and** IM-foliations on the Lie algebroid A.

Contents

1. Introduction 2

2. Background 6

2.1. Basics on Lie Theory 6

2.2. The tangent algebroid 7

2.3. Flat partial connections 11

3. **Foliated** **groupoids** 12

3.1. Definition **and** properties 12

3.2. The connection associated to a foliated groupoid 13

3.3. Involutivity of a multiplicative subbundle of T G 16

3.4. Examples 18

4. **Foliated** algebroids 19

4.1. Definition **and** properties 19

4.2. The Lie algebroid of a multiplicative foliation 21

4.3. Integration of foliated algebroids 24

4.4. The connection associated to a foliated algebroid 25

5. The connection as the **infinitesimal** **data** of a multiplicative foliation 28

5.1. Infinitesimal description of a multiplicative foliation 28

5.2. Reconstruction of the foliated Lie algebroid from the connection 28

6. The leaf space of a foliated algebroid 32

7. Further remarks **and** applications 39

7.1. Regular Dirac structures **and** the kernel of the associated presymplectic

**groupoids** 39

7.2. **Foliated** algebroids in the sense of Vaisman 40

Appendix A. Invariance of bundles under flows 40

References 42

2010 Mathematics Subject Classification. Primary 22A22, 53D17, 53C12; Secondary 53C05.

The first author was partially supported by Swiss NSF grant 200021-121512 **and** by the Dorothea Schlözer

Programme of the University of Göttingen.

1

2 M. JOTZ AND C. ORTIZ

1. Introduction

This paper is the first part of a series of two articles devoted to the study of the **infinitesimal**

**data** of Dirac **groupoids**, i.e. Lie **groupoids** equipped with Dirac structures suitably

compatible with the groupoid multiplication.

Since Dirac structures generalize Poisson bivectors, closed 2-forms **and** regular foliations,

multiplicative Dirac structures [24, 11] provide a unified framework for the study of a variety

of geometric structures compatible with a group(oid) multiplication, including Poisson

**groupoids** **and** multiplicative closed 2-forms. It is well known that Poisson **groupoids** are

**infinitesimal**ly described by Lie bialgebroids [21]. Similarly, the **infinitesimal** counterpart

of multiplicative closed 2-forms on a Lie groupoid are IM-2-forms on the corresponding Lie

algebroid [6, 4]. Here we are concerned with Lie **groupoids** equipped with a multiplicative foliation.

In particular, our main goal is to describe multiplicative foliations at the **infinitesimal**

level.

Consider a Lie group G with Lie algebra g **and** multiplication map m : G × G → G. Then

the tangent space T G of G is also a Lie group with multiplication map T m : T G × T G →

T G. It was observed in [13, 12, 23] that a multiplicative distribution F G ⊆ T G, that is,

a distribution on G that is a subgroup of T G, is the bi-invariant image of an ideal f ⊆ g.

Hence, it is automatically an involutive subbundle of T G which is completely determined by

its fiber f over the unit of the Lie group.

If G ⇒ M is a Lie groupoid, then the application of the tangent functor to all the groupoid

maps gives rise to a Lie groupoid T G ⇒ T M, called the tangent groupoid. A distribution

F G ⊆ T G on G is said to be multiplicative if F G is a subgroupoid of T G ⇒ T M over

F M ⊆ T M. In this more general situation it is no longer true that F G is automatically

involutive **and** has constant rank. Indeed, every smooth manifold can be viewed as a Lie

groupoid over itself, **and**, in this case, any distribution F ⊆ T M is multiplicative.

We study in this paper how the easy properties of multiplicative distributions on Lie

groups can be extended to the more general framework of Lie **groupoids**. A multiplicative

foliation on a Lie groupoid G ⇒ M is a multiplicative subbundle F G ⊆ T G which is also

involutive with respect to the Lie bracket of vector fields. We say that the pair (G ⇒ M, F G )

is a foliated groupoid. The main goal of this paper is to describe foliated **groupoids**

**infinitesimal**ly, that is, in terms of Lie algebroid **data**.

First, we observe that given a Lie groupoid G ⇒ M with Lie algebroid A, then the

space of units F M ⊆ T M of a multiplicative foliation F G ⊆ T G is necessarily an involutive

subbundle, **and** similarly the core F c := F G ∩ TM s G of F G is a Lie subalgebroid of the Lie

algebroid A = TM s G. In particular, a multiplicative foliation F G ⊆ T G can be thought of

as a LA-subgroupoid of T G ⇒ T M. As a consequence, following [18, 24] we prove that

the Lie algebroid of F G is, modulo the canonical identification A(T G) ≃ T A, a subalgebroid

F A −→ F M of the tangent Lie algebroid T A −→ T M which at the same time is an involutive

subbundle F A ⊆ T A. In this case we say that (A, F A ) is a foliated algebroid **and** F A is

referred to as a morphic foliation on A. We show that if G ⇒ M is a source simply

connected Lie groupoid with Lie algebroid A, then there is a one-to-one correspondence

between multiplicative foliations on G **and** morphic foliations on A.

We prove that if G is a Lie groupoid with Lie algebroid A endowed with a multiplicative

foliation F G with core F c ⊆ A **and** units F M ⊆ T M, then there is a well-defined partial

F M -connection ∇ on A/F c , which is naturally induced by the Bott connection associated

to the foliation F G on G. We show that this connection is flat **and** allows us to compute

explicitly the Lie algebroid of F G ⇒ F M in terms of restrictions to F M ⊆ T M of special

sections of the Lie algebroid A(T G) −→ T M of the tangent groupoid. Furthermore, we

show that the involutivity of the multiplicative subbundle F G ⊆ T G is completely encoded

in some **data** involving only F M , F c **and** the flatness of the connection ∇.

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 3

Given a foliated algebroid (A, F A ), we show that the core F c **and** side F M of F A are Lie

subalgebroids of A **and** T M, respectively. We prove that, here also, there is a flat partial

F M -connection on A/F c which is naturally induced by the Bott connection defined by the

foliation F A on A. In addition, we show that if A integrates to a Lie groupoid G **and**

F A ⊆ T A integrates to a multiplicative foliation F G ⊆ T G, then the partial connections

induced by F A **and** F G coincide. We say that the **data** (A, F M , F c , ∇) is an IM-foliation

on A.

The main result of this work states that the **data** (A, F M , F c , ∇) determines completely the

foliated algebroid (A, F A ) in the sense that F A can be reconstructed out of (A, F M , F c , ∇).

In particular, if G ⇒ M is a source simply connected Lie groupoid with Lie algebroid A, then

there is a one-to-one correspondence between multiplicative foliations on G **and** IM-foliations

(A, F M , F c , ∇) on A.

The partial connection induced by a multiplicative foliation already appeared in [13],

where the existence of its parallel sections is used to show that under some completeness

conditions, there is a groupoid structure on the leaf space G/F G ⇒ M/F M of the multiplicative

foliation. Analogously, we study the leaf space of a foliated algebroid (A, F A ). We prove

that, whenever the leaf space M/F M is a smooth manifold **and** the partial connection has

trivial holonomy, then there is a natural Lie algebroid structure on A/F A −→ M/F M . We

also show that if a foliated groupoid (G ⇒ M, F G ) is such that the completeness conditions of

[13] are fulfilled, then the Lie groupoid G/F G ⇒ M/F M has Lie algebroid A/F A −→ M/F M .

Several examples of foliated **groupoids** **and** algebroids are discussed, including actions by

groupoid **and** algebroid automorphisms, complex Lie **groupoids** **and** complex Lie algebroids

**and** integrable regular Dirac structures versus regular presymplectic **groupoids** 1 .

Given a Lie groupoid G ⇒ M with Lie algebroid A, it was proved in [7] that the cotangent

bundle T ∗ G has a Lie groupoid structure over A ∗ . In particular, the direct sum vector bundle

T G⊕T ∗ G inherits a Lie groupoid structure over T M ⊕A ∗ . It is easy to see that if F G ⊆ T G

is a multiplicative foliation on G, then its annihilator F ◦ G ⊆ T ∗ G is a Lie subgroupoid. Hence,

the subbundle F G ⊕ F ◦ G ⊆ T G ⊕ T ∗ G is an example of a multiplicative Dirac structure on

G (see [10, 24, 11]). Every multiplicative Dirac structure induces Lie algebroid structures

on the core **and** the base [10]. Furthermore there is a Courant algebroid that is canonically

associated to the Dirac groupoid [10], which generalizes the Lie bialgebroid of a Poisson

groupoid **and** is closely related to the partial connections studied in this paper.

The underst**and**ing of multiplicative Dirac structures at the **infinitesimal** level in the spirit

of this paper is a very interesting goal, because this generalizes simultaneously the correspondences

between Poisson **groupoids** **and** Lie bialgebroids [21], multiplicative closed 2-forms

**and** IM-2-forms [6, 4], **and** the results in this paper. This **infinitesimal** description of Dirac

**groupoids** can be made with similar techniques as here, by using the core **and** side algebroids

of a multiplicative Dirac structure D G on G together with the Courant algebroid associated

to D G . We will treat this problem in a separate paper [8].

Structure of the paper **and** main results. In Section 2, we start by recalling background

on Lie groupoid theory, especially the tangent **and** cotangent **groupoids** associated to a Lie

groupoid, as well as the Lie algebroid structures on A(T G) → T M **and** on T A → T M in

terms of linear **and** core sections. We also discuss briefly the definition **and** some properties

of flat partial connections.

1 Here, regular means that the characteristic distributions have constant rank.

4 M. JOTZ AND C. ORTIZ

In Section 3, we define foliated Lie **groupoids**, together with the associated subalgebroids

F M ⊆ T M **and** F c ⊆ A. Our first main results are summarized in the following theorem.

Theorem 1 Let G ⇒ M be a Lie groupoid endowed with a multiplicative subbundle F G ⊆

T G. Then there is a map

˜∇ : Γ(F M ) × Γ (A) → Γ (A/F c )

such that the following holds: F G is involutive if **and** only if

a) F M ⊆ T M is involutive,

b) for any ¯X ∈ Γ(F M ), ˜∇( ¯X, ·) vanishes on sections of F c **and**

c) the induced map

∇ : Γ(F M ) × Γ (A/F c ) → Γ (A/F c ) .

is then a flat partial F M -connection on A/F c .

The connection ∇ has in this case the following properties:

(1) If a ∈ Γ(A) is ∇-parallel, i.e., ∇ ¯Xā = 0 for all ¯X ∈ Γ(FM ), where ā is the class of

a in Γ(A/F c ), then [a, b] ∈ Γ(F c ) for all b ∈ Γ(F c ).

(2) If a, b ∈ Γ(A) are ∇-parallel, then [a, b] is also ∇-parallel.

(3) If a ∈ Γ(A) is ∇-parallel, then [ρ(a), ¯X] ∈ Γ(F M ) for all ¯X ∈ Γ(FM ). That is, ρ(a)

is ∇ F M

-parallel, where ∇ F M

denotes the Bott connection defined by F M .

Furthermore, a ∈ Γ(A) is ∇-parallel if **and** only if the right invariant vector field a r on G is

parallel with respect to the Bott connection ∇ F G

defined by F G . That is

[a r , Γ(F G )] ⊆ Γ(F G ).

In Section 4, we introduce **and** study the concept of foliated Lie algebroids. We compute

explicitly the Lie algebroid of a multiplicative foliation in terms of parallel sections of the

flat partial connection of Theorem 1, **and** we show the following integration result.

Theorem 2 Let (G ⇒ M, F G ) be a foliated groupoid. Then (A, F A = j −1

G (A(F G))) is a

foliated algebroid. 2

Conversely, let (A, F A ) be a foliated Lie algebroid. Assume that A integrates to a source

simply connected Lie groupoid G ⇒ M. Then there is a unique multiplicative foliation F G

on G such that F A = j −1

G (A(F G)).

A foliated algebroid (A, F A ) induces similar **data** (A, F M , F c , ∇) as does a foliated groupoid.

Theorem 3 Let (A, F A ) be a foliated algebroid. Then F M ⊆ T M **and** F c ⊆ A are subalgebroids

**and** there is a natural partial F M -connection ∇ A on A/F c induced by F A . This

connection has the same properties as the connection found in Theorem 1.

2 Here, jG : T A → A(T G) is the canonical identification.

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 5

The parallel sections of ∇ A are exactly the sections a ∈ Γ(A) such that [a ↑ , Γ A (F A )] ⊆

Γ A (F A ), that is the core vector field a ↑ on A is parallel with respect to the Bott connection

defined by F A .

The **data** induced by a foliated groupoid **and** its foliated algebroid are the same, according

to the next result.

Theorem 4 Assume that A integrates to a Lie groupoid G ⇒ M **and** that j G (F A ) integrates

to F G ⊆ T G. Then ∇ = ∇ A .

The next theorem, which is proved in Section 5, is the main result of this work. We

prove that the **data** (A, F M , F c , ∇), which is called an IM-foliation on A, is what should

be considered as the **infinitesimal** **data** of a multiplicative foliation.

Theorem 5 Let (A → M, ρ, [· , ·]) be a Lie algebroid **and** (A, F M , F c , ∇) an IM-foliation on

A. Then there exists a unique morphic foliation F A on A with side F M **and** core F c such

that the induced connection ∇ A is equal to ∇.

If A integrates to a source simply connected Lie groupoid G ⇒ M, there is a unique

multiplicative foliation F G on G with associated **infinitesimal** **data** (A, F M , F c , ∇).

In Section 6 we study the leaf space of a foliated algebroid, **and** relate it in the integrable

case to the leaf space of the corresponding foliated groupoid [13].

Theorem 6 Let (A, F M , F c , ∇) be an IM-foliation on A **and** assume that the leaf space

M/F M is a smooth manifold **and** ∇ has trivial holonomy. Then there is natural Lie algebroid

structure on A/F A over M/F M such that the projection (π, π M ) is a Lie algebroid morphism.

A

π

A/F A

q A

M

[q A ]

πM

M/F M

If (A, F A ) integrates to (G ⇒ M, F G ) **and** G/F G ⇒ M/F M is a Lie groupoid, then A/F A

with the structure above is the Lie algebroid of G/F G ⇒ M/F M .

We show in the last section that regular Dirac manifolds provide an interesting example

of IM-foliations. The corresponding multiplicative foliation is the kernel of the presymplectic

groupoid integrating the Dirac structure. Finally, we discuss shortly the relation of this work

with the foliated algebroids in the sense of Vaisman [31].

Notation. Let M be a smooth manifold. We will denote by X(M) **and** Ω 1 (M) the spaces

of (local) smooth sections of the tangent **and** the cotangent bundle, respectively. For an

arbitrary vector bundle E → M, the space of (local) sections of E will be written Γ(E). We

will write Dom(σ) for the open subset of the smooth manifold M where the local section

σ ∈ Γ(E) is defined.

The flow of a vector field X will be written φ X· , unless specified otherwise.

Let f : M → N be a smooth map between two smooth manifolds M **and** N. Then two

vector fields X ∈ X(M) **and** Y ∈ X(N) are said to be f-related if T f ◦ X = Y ◦ f on

Dom(X) ∩ f −1 (Dom(Y )). We write then X ∼ f Y .

6 M. JOTZ AND C. ORTIZ

The pullback or restriction of a vector bundle E → M to an embedded submanifold N of

M will be written E| N . In the special case of the tangent **and** cotangent spaces of M, we

will write T N M **and** T ∗ N M. If f : M → N is a smooth surjective submersion, we write T f M

for the kernel of T f : T M → T N.

Acknowledgements. We would like to thank Henrique Bursztyn, Thiago Drummond, Maria

Amelia Salazar, Mathieu Stiénon **and** Ping Xu for interesting discussions. The first author

wishes to thank Penn State University, IMPA **and** Universidade Federal do Paraná, Curitiba,

where parts of this work have been done. The second author thanks IMPA, EPFL **and** ESI

for the hospitality during several stages of this project.

2.1. Basics on Lie Theory.

2. Background

2.1.1. Lie **groupoids**. A groupoid G over the units M will be written G ⇒ M. The

source **and** target maps are denoted by s, t : G −→ M respectively, the unit section

ɛ : M −→ G, the inversion map i : G −→ G **and** the multiplication m : G (2) −→ G, where

G (2) = {(g, h) ∈ G × G | t(h) = s(g)} is the set of composable groupoid pairs.

A groupoid G over M is called a Lie groupoid if both G **and** M are smooth Hausdorff

manifolds, the source **and** target maps s, t : G −→ M are surjective submersions, **and** all the

other structural maps are smooth. Throughout this work we only consider Lie **groupoids**.

Let G **and** G ′ be Lie **groupoids** over M **and** M ′ , respectively. A morphism of Lie

**groupoids** is a smooth map Φ : G ′ −→ G over φ : M ′ → M that is compatible with all the

structural maps. When Φ is injective we say that G ′ is a Lie subgroupoid of G. See [19]

for more details.

2.1.2. Lie algebroids. A Lie algebroid is a vector bundle q A : A −→ M equipped with a Lie

bracket [·, ·] A on the space of smooth sections Γ M (A) **and** a vector bundle map ρ A : A −→

T M called the anchor, such that

[a, fb] A = f[a, b] A + (£ ρA (a)f)b,

for every a, b ∈ Γ M (A) **and** f ∈ C ∞ (M).

Let A **and** A ′ be Lie algebroids over M **and** M ′ , respectively. A Lie algebroid morphism

is a bundle map φ : A ′ −→ A over φ : M ′ → M which is compatible with the anchor **and**

bracket, see [19] for more details. If φ : A ′ −→ A is injective, we say that A ′ is a Lie

subalgebroid of A.

2.1.3. The Lie functor. Let G ⇒ M be a Lie groupoid. The Lie algebroid of G is defined

in this paper to be AG = T s M G, with anchor ρ AG := T t| AG **and** bracket [· , ·] AG defined by

using right invariant vector fields.

We will write A(·) for the functor that sends Lie **groupoids** to Lie algebroids **and** Lie

groupoid morphisms to Lie algebroid morphisms. For simplicity, (AG, ρ AG , [· , ·] AG ) will be

written (A, ρ, [· , ·]) in the following.

Note that if a ∈ Γ M (A), the vector field a r satisfies a r ∼ t ρ(a) ∈ X(M) since we have

T g ta r (g) = T g t(T t(g) r g a(t(g))) = T t(g) t a(t(g)) for all g ∈ G.

2.1.4. Tangent **and** cotangent **groupoids**. Let G be a Lie groupoid over M with Lie algebroid

A. The tangent bundle T G has a natural Lie groupoid structure over T M. This structure is

obtained by applying the tangent functor to each of the structure maps defining G (source,

target, multiplication, inversion **and** identity section). We refer to T G with the groupoid

structure over T M as the tangent groupoid of G [19]. The set of composable pairs (T G) (2)

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 7

of this groupoid is equal to T (G (2) ). For (g, h) ∈ G (2) **and** a pair (v g , w h ) ∈ (T G) (2) , the

multiplication is

v g ⋆ w h := T m(v g , w h ).

As in [18], we define star vector fields on G or star sections of T G to be vector fields

X ∈ X(G) such that there exists ¯X ∈ X(M) with X ∼ s

¯X **and** ¯X ∼ɛ X, i.e., X **and** ¯X are

s-related **and** ¯X **and** X are ɛ-related, or X restricts to ¯X on M. We write then X ∼ ⋆ s

¯X. In

the same manner, we can define t-star sections, X ∼ ⋆ t

¯X with ¯X ∈ X(M) **and** X ∈ X(G).

It is easy to see that the tangent space T G is spanned by star vector fields at each point in

G \ M. Not also that the Lie bracket of two star sections of T G is easily seen to be a star

section.

Consider now the cotangent bundle T ∗ G. It was shown in [7], that T ∗ G is a Lie groupoid

over A ∗ . The source **and** target of α g ∈ T ∗ g G are defined by

**and**

˜s(α g ) ∈ A ∗ s(g) , ˜s(α g)(a) = α g (T l g (a − T t(a))) for all a ∈ A s(g)

˜t(α g ) ∈ A ∗ t(g) , ˜t(α g )(b) = α g (T r g (b)) for all b ∈ A t(g) .

The multiplication on T ∗ G will also be denoted by ⋆ **and** is defined by

(α g ⋆ β h )(v g ⋆ w h ) = α g (v g ) + β h (w h )

for (v g , w h ) ∈ T (g,h) G (2) .

We refer to T ∗ G with the groupoid structure over A ∗ as the cotangent groupoid of G.

2.2. The tangent algebroid. Let M be a smooth manifold. The tangent bundle of M is

denoted by p M : T M −→ M. Consider now q A : A −→ M a vector bundle over M. The

tangent bundle T A has a natural structure of vector bundle over T M, defined by applying

the tangent functor to each of the structure maps that define the vector bundle q A : A → M.

This yields a commutative diagram

(2.1)

T A T q A

T M

p A

A qA

M

where all the structure maps of T A → T M are vector bundle morphisms over the corresponding

structure maps of A → M. In the terminology of [19, 25] this defines a double

vector bundle. Note finally that the zero element in the fiber of T A over v ∈ T M is T 0 A v,

i.e., the zero section is just T 0 A ∈ Γ T M (T A).

2.2.1. The tangent algebroid T A → T M. Assume now that q A : A −→ M has a Lie algebroid

structure with anchor map ρ : A −→ T M **and** Lie bracket [·, ·] on Γ M (A). Then there is a

Lie algebroid structure on T A over T M referred to as the tangent Lie algebroid.

In order to describe explicitly the algebroid structure on T q A : T A −→ T M, we recall

first that there exists a canonical involution

p M

(2.2)

T T M J M

T T M

p T M

T M IdT M

T M

T p M

8 M. JOTZ AND C. ORTIZ

which is given as follows [19, 30]. Elements (ξ; v, x; m) ∈ T T M, that is, with p T M (ξ) = v ∈

T m M **and** T p M (ξ) = x ∈ T m M, are considered as second derivatives

ξ = ∂2 σ

(0, 0),

∂t∂u

where σ : R 2 → M is a smooth square of elements of M. The notation means that σ is first

differentiated with respect to u, yielding a curve v(t) = ∂σ

∂u (t, 0) in T M with d ⏐

dt t=0

v(t) = ξ.

Thus, v = ∂σ

∂u (0, 0) = p T M (ξ) **and** x = ∂σ

∂t (0, 0) = T p M (ξ). The canonical involution

J M : T T M → T T M is defined by

J M (ξ) := ∂2 σ

(0, 0).

∂u∂t

Now we can apply the tangent functor to the anchor map ρ : A −→ T M, **and** then compose

with the canonical involution to obtain a bundle map ρ T A : T A −→ T T M defined by

ρ T A = J M ◦ T ρ.

This defines the tangent anchor map. In order to define the tangent Lie bracket, we observe

that every section a ∈ Γ M (A) induces two types of sections of T A −→ T M. The first

type of section is T a : T M −→ T A, called linear section 3 , which is given by applying the

tangent functor to the section a : M −→ A. The second type of section is the core section

â : T M −→ T A, which is defined by

(2.3) â(v m ) = T m 0 A (v m ) + pA a(m),

where 0 A : M −→ A denotes the zero section, **and** a(m) = d ⏐

dt t=0

ta(m) ∈ T 0 A m

A. As

observed in [20], sections of the form T a **and** â generate the module of sections Γ T M (T A).

Therefore, the tangent Lie bracket [· , ·] T A is completely determined by

[

[T a, T b] T A

= T [a, b], T a, ˆb

]

= [a, ̂ [

b], â, ˆb

]

= 0

for all a, b ∈ Γ(A), the extension to general sections is done using the Leibniz rule with

respect to the tangent anchor ρ T A .

T A

T A

2.2.2. The Lie algebroid of the tangent groupoid. If A is now the Lie algebroid of a Lie

groupoid G ⇒ M, then we can consider the Lie algebroid q A(T G) : A(T G) → T M of the

tangent Lie groupoid T G ⇒ T M. Since the projection p G : T G → G is a Lie groupoid

morphism, we have a Lie algebroid morphism A(p G ) : A(T G) → A over p M : T M → M **and**

the following diagram commutes.

(2.4)

A(T G) A(pG) A

q A(T G)

T M pM

M

q A

3 Note that not all linear sections are of this type: if φ : T M → A is a vector bundle homomorphism over

Id M , **and** a ∈ Γ(A), then the section a φ of T A → T M defined by

a φ (v m) = T a(v m) + d dt ⏐ a m + t · φ(v m)

t=0

for all v m ∈ T M is also a linear section, i.e., a φ : T M → T A is a vector bundle homomorphism over

a : M → A.

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 9

Let a be a section of the Lie algebroid A, choose v ∈ T M **and** consider the curve γ :

(−ε, ε) → T G defined by

γ(t) = T Exp(ta)v

for ε small enough. Then we have γ(0) = v **and** T s(γ(t)) = v for all t ∈ (−ε, ε). Hence,

˙γ(0) ∈ A v (T G) **and** we can define a linear section β a : T M → A(T G) by

(2.5) β a (v) = d dt⏐ T Exp(ta)v

t=0

for all v ∈ T M. It is easy to check that β r a ∈ X(T G) r is the complete lift of a r (see [19]). In

particular the flow of β r a is T L Exp(·a) , **and** (β a , a) is a morphism of vector bundles.

In the same manner, we can consider v ∈ T M, a ∈ A pM (v) **and** the curve γ : R → T G

defined by

γ(t) = v + ta,

where T M **and** A are seen as subsets of T G, T M G = T M ⊕ A. We have again γ(0) = v

**and** T s(γ(t)) = v for all t, which yields ˙γ(0) ∈ A v (T G). Given a ∈ Γ M (A), we define a core

section ã of A(T G) by

(2.6) ã(v) = d dt⏐ v + ta(p M (v))

t=0

for all v ∈ T M. We have for v g ∈ T g G with T g t(v g ) = v m :

ã r (v g ) = ã(v m ) ⋆ 0 vg = d dt⏐ v g + ta r (g).

t=0

The vector bundle A(T G) is spanned by the two types of sections β a **and** ã, for a ∈ Γ M (A),

**and**, using the flows of βa r **and** ˜b r ∈ X r (T G), it is easy to check that the equalities

[

[β a , β b ] A(T G)

= β [a,b] , β a , ˜b

]

= [a, ˜ [

b], ã, ˜b

]

= 0

hold for all a, b ∈ Γ M (A).

There exists a natural injective bundle map

A(T G)

(2.7) ι A : A −→ T G

A(T G)

over ɛ : M → G. The canonical involution J G : T T G −→ T T G restricts to an isomorphism

of Lie algebroids j G : T A −→ A(T G). More precisely, there exists a commutative diagram

(2.8)

T A

j G

A(T G)

T ι A

We get easily the following identities

where â is defined as in (2.3). The equality

T T G

JG

T T G

ι A(T G)

j G ◦ T a = β a **and** j G ◦ â = ã,

ρ A(T G) ◦ j G = J M ◦ T ρ A = ρ T A

is verified on linear **and** core sections. This shows that the Lie algebroid A(T G) → T M of

the tangent groupoid is canonically isomorphic to the tangent Lie algebroid T A → T M of

A.

10 M. JOTZ AND C. ORTIZ

2.2.3. The st**and**ard Lie algebroid T A → A. We study now the Lie algebroid structure T A →

A in terms of special sections. Consider a linear vector field on A, i.e., a section X of T A → A

such that the map

A

X

T A

q A

M

¯X

T M

T q A

is a vector bundle morphism over ¯X ∈ X(M). For any b ∈ Γ M (A), the core section b ↑ of

T A → A associated to b is defined by

(2.9) b ↑ (a m ) = d dt⏐ a m + tb(m)

t=0

for all a m ∈ A.

The flow φ X of X is then a vector bundle morphism over the flow φ ¯X of ¯X

A

q A

M

φ X t

φ ¯X

t

A

M

q A

for t ∈ R such that this makes sense. The flow of b ↑ is given by φ b↑

t (a m ) = a m + tb(m) for all

a m ∈ A **and** all t ∈ R. In particular, for every t ∈ R, φ b↑

t is a vector bundle automorphism

of A, covering the identity.

A

φ b↑

t

A

q A

M IdM

M

Lemma 2.1. Let A be a Lie algebroid. Choose linear sections (X, ¯X), (Y, Ȳ ) of T A → A

**and** sections a, b ∈ Γ M (A). Then

(

[X, Y ],

[ ¯X, Ȳ ])

q A

is a linear section of T A → A,

**and**

[

a ↑ , b ↑] = 0

[

X, a

↑ ] = (D X a) ↑ ,

where

for all m ∈ M.

D X a ∈ Γ(A),

(D X a)(m) = d (

) dt⏐ φ X −t ◦ a ◦ φ ¯X

t (m)

t=0 } {{ }

∈A m

Proof. This is easy to show using the flows of linear **and** core sections, see also [17].

□

Recall that if G ⇒ M is a Lie groupoid, then the tangent space T G is spanned outside of

M by star sections X ⋆ ∼ s

¯X, i.e. with X ∈ X(G), ¯X ∈ X(M) such that X ∼s ¯X **and** ¯X ∼ɛ X

(see Subsection 2.1.4). The bracket of two star sections is a star section **and** for any a m ∈ A,

the properties of the star vector field X ∈ X(G) over ¯X ∈ X(M) imply that (T X)(a m )

has value in A ¯X(m) (T G). Hence, we can mimic the construction of the Lie algebroid map

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 11

associated to a Lie groupoid morphism **and** we can consider the map A(X) : A → A(T G),

A(X)(a m ) = T m X(a m ) over ¯X : M → T M. In the same manner, if a ∈ Γ M (A), then we

can define ā : A → A(T G) by ā(b m ) = β b (0 T m

M ) + qA(T G) ã(0 T m

M ) for any section b ∈ Γ M (A)

such that b(m) = b m . We have j −1

G ◦ ā = a↑ : A → T A **and** for a star section X ∼ ⋆ s

¯X of

T G → T M, the map ˜X := j −1

G ◦ A(X) : A → T A is a linear section ( ˜X, ¯X) of T A → A.

There is a unique Lie algebroid structure on A(T G) over A making j G : T A → A(T G) into

a Lie algebroid isomorphism over the identity on A [17, 20, 18].

2.3. Flat partial connections. The following definition will be crucial in this paper.

Definition 2.2. ([2]) Let M be a smooth manifold **and** F ⊆ T M a smooth involutive vector

subbundle of the tangent bundle. Let E → M be a vector bundle over M. A F -partial

connection is a map ∇ : Γ(F ) × Γ(E) → Γ(E), written ∇(X, e) =: ∇ X e for X ∈ Γ(F ) **and**

e ∈ Γ(E), such that:

(1) ∇ is tensorial in the F -argument,

(2) ∇ is R-linear in the E-argument,

(3) ∇ satisfies the Leibniz rule

∇ X (fe) = X(f)e + f∇ X e

for all X ∈ Γ(F ), e ∈ Γ(E), f ∈ C ∞ (M).

The connection is flat if

for all X, Y ∈ Γ(F ) **and** e ∈ Γ(E).

∇ [X,Y ] e = (∇ X ∇ Y − ∇ Y ∇ X ) e

Example 2.3 (The Bott connection). Let M be a smooth manifold **and** F ⊆ T M an

involutive subbundle. The Bott connection

defined by

∇ F : Γ(F ) × Γ(T M/F ) → Γ(T M/F )

∇ X Ȳ = [X, Y ],

where Ȳ ∈ Γ(T M/F ) is the projection of Y ∈ X(M), is a flat F -partial connection on

T M/F → M.

The class Ȳ ∈ Γ(T M/F ) of a vector field is ∇F -parallel if **and** only if [Y, Γ(F )] ⊆ Γ(F ).

Since F is involutive, this does not depend on the representative of Ȳ . We say by abuse of

notation that Y is ∇ F -parallel.

The following proposition can be easily shown by using the fact that the parallel transport

defined by a flat connection does not depend on the chosen path in simply connected sets

(see [13], [14] for similar statements).

Proposition 2.4. Let E → M be a smooth vector bundle of rank k, F ⊆ T M an involutive

subbundle **and** ∇ a flat partial F -connection on E. Then there exists for each point m ∈ M

a frame of local ∇-parallel sections e 1 , . . . , e k ∈ Γ(E) defined on an open neighborhood U of

m in M.

We have also the following lemma. The first statement is a straightforward consequence

of the Leibniz identity **and** the proof of the second statement can be checked easily in

coordinates adapted to the foliation.

Lemma 2.5. Let E → M be a smooth vector bundle of rank k, F ⊆ T M an involutive

subbundle **and** ∇ a partial F -connection on E.

(1) Assume that f ∈ C ∞ (M) is F -invariant, i.e., X(f) = 0 for all X ∈ Γ(F ). Then

f · e is ∇-parallel for any ∇-parallel section e ∈ Γ(E).

12 M. JOTZ AND C. ORTIZ

(2) Assume that the foliation defined by F on M is simple, i.e. the leaf space has

a smooth manifold structure such that the quotient π : M → M/F is a smooth

surjective submersion. Then X ∈ X(M) is ∇ F -parallel if **and** only if there exists

¯X ∈ X(M/F ) such that X ∼ π ¯X.

3.1. Definition **and** properties.

3. **Foliated** **groupoids**

Definition 3.1. Let G ⇒ M be a Lie groupoid. A subbundle F G ⊆ T G is multiplicative if

it is a subgroupoid of T G ⇒ T M over F G ∩T M =: F M . We say that F G is a multiplicative

foliation on G if it is involutive **and** multiplicative. The pair (G ⇒ M, F G ) is then called a

foliated groupoid.

Remark 3.2. Multiplicative subbundles were introduced in [29] as follows. A subbundle

F G ⊆ T G is multiplicative if for all composable g, h ∈ G **and** u ∈ F G (g ⋆ h), there exist

v ∈ F G (g), w ∈ F G (h) such that u = v ⋆ w. It is easy to check that a multiplicative foliation

in the sense of Definition 3.1 is multiplicative in the sense of [29], but the converse is not

necessarily true, unless for instance if the Lie groupoid is a Lie group (see [12]). The case of

involutive wide sub**groupoids** of T G ⇒ T M has also been studied in [1].

The following result about multiplicative subbundles can be found in [13].

Lemma 3.3. Let G ⇒ M be a Lie groupoid **and** F G ⊆ T G a multiplicative subbundle. Then

the intersection F M := F G ∩ T M has constant rank on M. Since it is the set of units of F G

seen as a subgroupoid of T G, the pair F G ⇒ F M is a Lie groupoid.

The bundle F G | M splits as F G | M = F M ⊕ F c , where F c := F G ∩ A. We have

(F G ∩ T s G)(g) = F c (t(g)) ⋆ 0 g = T t(g) r g (F c (t(g)))

for all g ∈ G.

In the same manner, if F t := (F G ∩ T t G)| M , we have (F G ∩ T t G)(g) = 0 g ⋆ F t (s(g)) for

all g ∈ G.

As a consequence, the intersections F G ∩ T t G **and** F G ∩ T s G have constant rank on G.

The previous lemma says that a multiplicative subbundle F G ⊆ T G determines a VBgroupoid

p G

F G

G

T t

T s

F M pM

M

with core F c . As a corollary, one gets that the source **and** target maps of F G ⇒ F M are

fiberwise surjective.

Corollary 3.4. Let G ⇒ M be a Lie groupoid **and** F G a multiplicative subbundle of T G.

The induced maps T g s : F G (g) → F G (s(g)) ∩ T s(g) M **and** T g t : F G (g) → F G (t(g)) ∩ T t(g) M

are surjective for each g ∈ G.

Proposition 3.5. Let F G be a multiplicative foliation on a Lie groupoid G ⇒ M. Then F c

is a subalgebroid of A **and** F M is an involutive subbundle of T M.

Proof. Choose first two sections a, b ∈ Γ(F c ). Then we have a r , b r ∈ Γ(F G ) r by Lemma 3.3

**and** hence [a, b] r = [a r , b r ] ∈ Γ(F G ) since F G is involutive. Again by Lemma 3.3, we find

thus that [a, b] ∈ Γ(F c ).

t

s

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 13

Choose now two sections ¯X, Ȳ ∈ Γ(F M ) **and** X, Y ∈ Γ(F G ) that are s-related to ¯X, Ȳ .

This is possible by Corollary 3.4. We find then that [X, Y ] ∈ Γ(F G ) **and** [X, Y ] ∼ s [ ¯X, Ȳ ].

Hence, [ ¯X, Ȳ ] ∈ Γ(F M ).

□

As explained in [10, 24], a multiplicative foliation is a LA-groupoid in the sense of [18],

T s

F G

F M

ι T t

FG

T G

p G

s

G

M

p G

t

ι FM

T s

T t

p

M

where **and** ι FM , ι FG are the inclusions of F M in T M **and** of F G in T G, respectively. That is,

(F M → M, ι FM , [· , ·]) is a Lie algebroid over M **and** the quadruple (F G ; G, F M ; M) is such

that F G has both a Lie groupoid structure over F M **and** a Lie algebroid structure over G

such that the two structures on F G are compatible in the sense that the maps defining the

groupoid structure are all Lie algebroid morphisms. Furthermore, the double source map

T M

(p G , T s) : F G → G × M F M = {(g, v) ∈ G × F M | v ∈ F M (s(g))}

is a surjective submersion by Corollary 3.4.

3.2. The connection associated to a foliated groupoid. Recall that if G ⇒ M is a Lie

groupoid **and** F G is a multiplicative foliation on G, then F M := F G ∩ T M **and** F c = F G ∩ A

are subalgebroids of T M → M **and** A → M, respectively (Proposition 3.5).

In the main theorem of this subsection, we show that if F G is a multiplicative foliation on

the Lie groupoid G ⇒ M, then the Bott F G -connection on T G/F G induces a well-defined

partial F M -connection on A/F c . We write ā for the class in A/F c of a ∈ Γ(A).

Theorem 3.6. Let (G ⇒ M, F G ) be a Lie groupoid endowed with a multiplicative foliation.

Then there is a partial F M -connection on A/F c

(3.10)

∇ : Γ(F M ) × Γ (A/F c ) → Γ (A/F c ) .

with the following properties:

(1) ∇ is flat.

(2) If a ∈ Γ(A) is ∇-parallel, i.e., ∇ ¯Xā = 0 for all ¯X ∈ Γ(FM ), then [a, b] ∈ Γ(F c ) for

all b ∈ Γ(F c ).

(3) If a, b ∈ Γ(A) are ∇-parallel, then [a, b] is also ∇-parallel.

(4) If a ∈ Γ(A) is ∇-parallel, then [ρ(a), ¯X] ∈ Γ(F M ) for all ¯X ∈ Γ(FM ). That is,

ρ(a) ∈ X(M) is ∇ F M

-parallel.

Furthermore, a ∈ Γ(A) is ∇-parallel if **and** only if the right invariant vector field a r ∈ X(G)

is parallel with respect to the Bott connection defined by F G , that is

[a r , Γ(F G )] ⊆ Γ(F G ).

In the following, we call a vector field X ∈ X(G) a t-section if there exists ¯X ∈ X(M)

such that X ∼ t

¯X. Similarly, a one-form η ∈ Ω 1 (G) is a t-section of T ∗ G if ˜t ◦ η = ¯η ◦ t for

some ¯η ∈ Γ(A ∗ ). Analogously, we can define s-sections of T G **and** T ∗ G.

It is easy to see that F G **and** its annihilator F ◦ G ⊆ T ∗ G are spanned by **their** t-sections.

Lemma 3.7. Let (G ⇒ M, F G ) be a Lie groupoid endowed with a multiplicative subbundle

F G ⊆ T G. Choose ¯X ∈ Γ(F M ), ¯η ∈ Γ(FG ◦ ∩ A∗ ) **and** t-sections X ∼ t

¯X, η ∼˜t ¯η of F G **and**

FG ◦ , respectively. Then the identity

(3.11) η(£ a rX) = t ∗ ɛ ∗(¯η(£ a rX) )

14 M. JOTZ AND C. ORTIZ

holds for any section a ∈ Γ(A).

Proof. Choose g ∈ G **and** set p = t(g). For all t ∈ (−ε, ε) for a small ε, we have

The vector

X(Exp(ta)(p) ⋆ g) = (X(Exp(ta)(p))) ⋆ ( X(Exp(ta)(p)) ) −1

⋆ X(Exp(ta)(p) ⋆ g).

(

X(Exp(ta)(p))

) −1

⋆ X(Exp(ta)(p) ⋆ g)

is an element of F G (g) for all t ∈ (−ε, ε) **and** will be written v t (g) to simplify the notation.

We compute

( )

d (

η(£ a rX)(g) = η(g)

dt⏐

⏐ TLExp(ta) (g)L Exp(−ta) X(Exp(ta)(p) ⋆ g)

t=0

= d (TExp(ta) dt⏐ (¯η(p) ⋆ η(g))(

L Exp(−ta) X(Exp(ta)(p)) ) )

⋆ v t (g)

t=0

= d dt⏐ ¯η(p) ( T Exp(ta) L Exp(−ta) X(Exp(ta)(p)) ) + d t=0

dt⏐ η(g)(v t (g))

( )

t=0

d = ¯η(£ a rX)(p) +

dt⏐

⏐ 0 = ¯η(£ a rX)(t(g)).

t=0

Let G ⇒ M be a Lie groupoid **and** F G a multiplicative subbundle of T G. Recall that the

restriction to M of F G splits as

(3.12) F G | M = F M ⊕ F c ,

where F c = F G ∩ A **and** F M = F G ∩ T M. Recall also the notation F t = F G ∩ T t M G.

Theorem 3.8. Let (G ⇒ M, F G ) be a Lie groupoid endowed with a multiplicative subbundle

F G ⊆ T G, X a t-section of F G , i.e., t-related to some ¯X ∈ Γ(F M ), **and** consider a ∈ Γ(A).

Then the derivative £ a rX can be written as a sum

£ a rX = Z a,X + b r a,X

with b a,X ∈ Γ(A), **and** Z a,X a t-section of F G . In addition, if X ∼ t 0, then £ a rX ∈

Γ(F G ∩ T t G). In particular, its restriction to M is a section of F t **and** b a,X is a section of

F c .

Proof. Set

for all m ∈ M **and**

b a,X (m) = £ a rX(m) − T m s (£ a rX(m))

Z a,X := £ a rX − b r a,X.

In particular, Z a,X (m) = T m s (£ a rX(m)) for all m ∈ M. First, we see that Z a,X is a

t-section of T G since

Z a,X = £ a rX − b r a,X ∼ t

[

ρ(a), ¯X] − ρ(ba,X ).

Recall that F G ◦ ⊆ T ∗ G is spanned by its t-sections. Hence, to show that Z a,X ∈ Γ(F G ),

it suffices to show that η(Z a,X ) = 0 for all t-sections η of F G ◦ .

Setting t(g) = p, we have immediately

η(Z a,X )(g) = η(£ a rX)(g) − ¯η(b a,X )(p)

(3.11)

= ¯η(£ a rX)(p) − ¯η(b a,X )(p)

= ¯η(T p s(£ a rX(p))) = 0

since ¯η vanishes on T M. This shows that Z a,X is a section of F G .

□

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 15

Assume now that X ∼ t 0. This means that X ∈ Γ(F G ∩ T t G) **and** X can be written

X = ∑ r

i=1 f i · b l i with f 1, . . . , f r ∈ C ∞ (G) **and** b 1 , . . . , b r ∈ Γ(F t ). We get then

( r∑

)

r∑

r∑

£ a rX = £ a r f i · b l i = a r (f i )b l i + f i · [a r , b l ]

r∑

i = a r (f i )b l i,

i=1

i=1

which is again a section of F G ∩ T t G.

□

i=1

i=1

by

Assume now that F G is involutive **and** define

∇ : Γ(F M ) × Γ(A/F c ) → Γ(A/F c )

∇ ¯Xā = −b a,X ,

with b a,X as in Theorem 3.8, for any choice of t-section X ∈ Γ(F G ) such that X ∼ t

¯X **and**

any choice of representative a ∈ Γ(A) for ā. We will show that this is a well-defined partial

F M -connection **and** complete the proof of Theorem 3.6.

Proof of Theorem 3.6. Choose X, X ′ ∈ Γ(F G ) such that X ∼ t

¯X **and** X

′

∼ t

¯X. Then

Y := X − X ′ ∼ t 0 **and**, by Theorem 3.8, we find b a,Y ∈ Γ(F c ) for any a ∈ Γ(A), i.e.,

b a,X = b a,X ′.

Choose now a ∈ Γ(F c ) **and** X ∈ Γ(F G ), X ∼ t

¯X ∈ Γ(FM ). Then we have a r ∈ Γ(F G ) **and**

since F G is involutive, £ a rX ∈ Γ(F G ). Again, since Z a,X ∈ Γ(F G ), we find b a,X ∈ Γ(F c ).

This shows that ∇ is well-defined.

By definition, if a ∈ Γ(A) is such that ∇ ¯Xā = 0 for all ¯X ∈ Γ(FM ), then we have

£ a rX = Z a,X + b r a,X ∈ Γ(F G) for all t-descending sections X ∈ Γ(F G ). Since Γ(F G ) is

spanned as a C ∞ (G)-module by its t-descending sections, we get

[a r , Γ(F G )] ⊆ Γ(F G ).

Conversely, [a r , Γ(F G )] ⊆ Γ(F G ) implies immediately ∇ ¯Xā = 0 for all ¯X ∈ Γ(FM ). This

proves the second claim of the theorem.

We check that ∇ is a flat partial F M -connection. Choose a ∈ Γ(A), ¯X ∈ Γ(FM ), X ∈

Γ(F G ) such that X ∼ t

¯X **and** f ∈ C ∞ (M). Then we have t ∗ f · X ∼ t f ¯X **and**

In particular, we find

£ a r(t ∗ f · X) = t ∗ (ρ(a)(f)) · X + t ∗ f · £ a rX.

b a,t ∗ f·X = (1 − T s) (t ∗ (ρ(a)(f)) · X + t ∗ f · £ a rX) | M

= ρ(a)(f) · (1 − T s)X| M + f · (1 − T s)(£ a rX)| M .

Since (T s − 1)X| M ∈ Γ(F c ), this leads to b a,t ∗ f·X = f · b a,X **and** hence ∇ f ¯Xā = −f · b a,X =

f · ∇ ¯Xā.

Since (fa) r = t ∗ f · a r , we have in the same manner

£ (fa) rX = −£ X (t ∗ f · a r ) = −t ∗ ( ¯X(f)) · a r + t ∗ f · £ a rX,

which leads to ∇ ¯X(f · ā) = ¯X(f) · ā + f · ∇ ¯Xā.

Choose ¯X, Ȳ ∈ Γ(F M ) **and** X, Y ∈ Γ(F G ) such that X ∼ t

¯X **and** Y ∼t Ȳ . Then we have

[X, Y ] ∼ t [ ¯X, Ȳ ] **and** [X, Y ] ∈ Γ(F G) since F G is involutive. For any a ∈ Γ(A), we have by

16 M. JOTZ AND C. ORTIZ

the Jacobi-identity:

£ a r[X, Y ] = [£ a rX, Y ] − [£ a rY, X]

= [ Z a,X + b r a,X, Y ] − [ Z a,Y + b r a,Y , X ]

= [Z a,X , Y ] − [Z a,Y , X] + £ b r

a,X

Y − £ b r

a,Y

X

= [Z a,X , Y ] − [Z a,Y , X] + Z ba,X ,Y + b ba,X ,Y r − Z ba,Y ,X − b ba,Y ,X r .

Since [Z a,X , Y ] − [Z a,Y , X] + Z ba,X ,Y − Z ba,Y ,X is a t-section of F G , we find that

∇ [ ¯X, Ȳ ]ā = b ba,Y ,X − b ba,X ,Y = ∇ ¯X∇ Ȳ ā − ∇ Ȳ ∇ ¯Xā

which shows the flatness of ∇.

Choose now a ∈ Γ(A) such that ∇ ¯Xā = 0 ∈ Γ(A/F c ) for all ¯X ∈ Γ(FM ). If b ∈ Γ(F c ),

then b r ∈ Γ(F G ), ρ(b) ∈ Γ(F M ) **and** b r ∼ t ρ(b). This leads to

[b, a] = ∇ ρ(b) ā = 0 ∈ Γ(A/F c )

**and** hence [a, b] ∈ Γ(F c ). This shows 2. For each ¯X ∈ Γ(F M ), there exists X ∈ Γ(F G ) such

that X ∼ t

¯X. Since [a r , X] ∈ Γ(F G ), a r ∼ t ρ(a) **and** T t(F G ) = F M , we find [ρ(a), ¯X] ∈

Γ(F M ), which proves 4.

To show 3., choose two sections a, b ∈ Γ(A) such that ā **and** ¯b are ∇-parallel. We have

then for any t-section X ∼ t

¯X of FG :

£ [a,b] rX = £ a r(Z b,X + b r b,X) − £ b r(Z a,X + b r a,X)

= £ a r(Z b,X ) + [a, b b,X ] r − £ b r(Z a,X ) − [b, b a,X ] r .

Since ā **and** ¯b are ∇-parallel, this yields ∇ ¯X[a, b] = −[a, b b,X ] + [b, b a,X ]. Since ā **and** ¯b are

parallel, we have b b,X , b a,X ∈ Γ(F c ) **and** 3. follows using 2.

□

Remark 3.9. Let (G ⇒ M, F G ) be a foliated Lie groupoid **and** consider the multiplicative

Dirac structure F G ⊕F G ◦ on G. The “tangent” part of the Courant algebroid B(F G ⊕F G ◦ ) in

[10] is in this case just (F M ⊕A)/F t , which is isomorphic as a vector bundle to F M ⊕(A/F c ).

The induced bracket on sections of F M ⊕(A/F c ) defines a map Γ(F M )×Γ(A/F c ) → Γ(A/F c ),

( ¯X, ā

)

↦→ prA/Fc

[( ¯X, 0

)

, (0, ā)

]

,

which can be checked to be exactly the connection ∇.

3.3. Involutivity of a multiplicative subbundle of T G. It is natural to ask here how

the involutivity of F G is encoded in the **data** (F M , F c , ∇). For an arbitrary (not necessarily

involutive) multiplicative subbundle F G ⊆ T G, we can consider the map

˜∇ : Γ(F M ) × Γ(A) → Γ(A/F c ),

˜∇ ¯Xa = −b a,X

which is well-defined by the proof of Theorem 3.6.

Theorem 3.10. Let (G, F G ) be a source-connected Lie groupoid endowed with a multiplicative

subbundle. Then F G is involutive if **and** only if the following holds:

(1) F M ⊆ T M is involutive,

(2) ˜∇ vanishes on sections of F c ,

(3) the induced map ∇ : Γ(F M ) × Γ(A/F c ) → Γ(A/F c ) is a flat partial F M -connection

on A/F c .

The proof of this theorem is a simplified version of the proof of the general criterion for

the closedness of multiplicative Dirac structures (see [11]).

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 17

Proof. We have already shown in Proposition 3.5 **and** Theorem 3.6 that the involutivity of

F G implies (1), (2) **and** (3).

For the converse implication, recall that the t-star sections of F G span F G outside of

the set of units M. Hence, it is sufficient to show involutivity on t-star sections **and** rightinvariant

sections of F G . Choose first two right-invariant sections a r , b r of F G , i.e. with

a, b ∈ Γ(F c ). Then we have ρ(b) ∈ Γ(F M ), b r ∼ t ρ(b) **and**, since ˜∇ ρ(b) a = 0 by (2), we

find that [a r , b r ] ∈ Γ(F G ). In the same manner, by the definition of ˜∇ **and** Theorem 3.8,

Condition (2) implies that the bracket of a right-invariant section of F G **and** a t-star section

is always a section of F G .

We have thus only to show that the bracket of two t-star sections of F G is again a section

of F G . Let F ◦ c be the annihilator of F c in A ∗ **and** consider the dual F M -connection on

(A/F c ) ∗ ≃ F c ◦ ⊆ A ∗ , i.e. the (by (3)) flat connection

given by

∇ ∗ : Γ(F M ) × Γ(F c ◦ ) → Γ(F c ◦ )

(

∇

∗¯Xα ) (ā) = ¯X(α(ā)) − α (∇ ¯Xā)

for all ¯X ∈ Γ(FM ), α ∈ Γ(F ◦ c ) **and** a ∈ Γ(A).

Choose a t-star section X ∈ Γ(F G ), X ∼ t

¯X, X|M = ¯X **and** a ˜t-section η ∈ Γ(FG ◦), η ∼˜t ¯η.

Then, for any section a of A, we have

(£ X η)(a r ) = X (η(a r )) + η (£ a rX)

= t ∗ ( ) ( ¯X(¯η(ā)) + η Za,X + b r )

a,X

by Theorem 3.8

= t ∗ ( ¯X(¯η(ā)) − ¯η (∇ ¯Xā) ) since η ∈ Γ(F ◦ G) **and** Z a,X ∈ Γ(F G )

= t ∗ (∇ ∗¯X ¯η(ā)).

This shows that £ X η ∼˜t ∇ ∗¯X ¯η ∈ Γ(Fc ◦ ). Note that we have not shown yet that £ X η is a

section of FG ◦. Choose a second t-star section Y of F G, Y ∼ t Ȳ **and** Y | M = Ȳ . An easy

computation using η(X) = η(Y ) = 0 yields

Hence, we get for a ∈ Γ(A):

−2d (η([X, Y ])) = £ X £ Y η − £ Y £ X η − £ [X,Y ] η.

−2 · a r (η([X, Y ])) = ( £ X £ Y η − £ Y £ X η − £ [X,Y ] η ) (a r )

=X(£ Y η(a r )) − £ Y η([X, a r ]) − Y (£ X η(a r )) − £ X η([Y, a r ])

− [X, Y ](η(a r )) + η([[X, Y ], a r ])

=t ∗ ¯X(∇

∗Ȳ ¯η(ā)) + (£ Y η)(Z a,X + b r a,X)

− t ∗ Ȳ (∇ ∗¯X ¯η(ā)) − (£ X η)(Z a,Y + b r a,Y )

− t ∗ [ ¯X, Ȳ ](¯η(ā)) − η ([£ a rX, Y ] + [X, £ arY ])

=t ∗ ¯X(∇

∗Ȳ ¯η(ā)) + £ Y η(Z a,X ) − t ∗ (∇ ∗ Ȳ ¯η)(∇ ¯Xā)

− t ∗ Ȳ (∇ ∗¯X ¯η(ā)) − £ X η(Z a,Y ) + t ∗ (∇ ∗¯X ¯η)(∇ Ȳ ā)

− t ∗ [ ¯X, Ȳ ](¯η(ā)) − η ( [Z a,X + b r a,X, Y ] + [X, Z a,Y + b r a,Y ] )

=t ∗ (( ∇ ∗¯X∇ ∗ Ȳ ¯η − ∇∗ Ȳ ∇∗¯X ¯η ) (ā) − [ ¯X, Ȳ ](¯η(ā)))

(

)

+ Y (η(Z a,X )) − X(η(Z a,Y )) − η Z ba,X ,Y + b r b a,X ,Y − Z ba,Y ,X − b r b a,Y ,X

(3)

=t ∗ (¯η(−∇ [ ¯X, Ȳ ]ā − ∇ Ȳ ∇ ¯Xā + ∇ ¯X∇ Ȳ ā) ) (3)

= 0.

Hence, a r (η([X, Y ])) = 0 for all a ∈ Γ(A) **and** since G is source-connected, this implies that

η([X, Y ])(g) = η([X, Y ])(s(g)) for all g ∈ G. But since for m ∈ M, we have

[X, Y ](m) = [ ¯X, Ȳ ](m)

18 M. JOTZ AND C. ORTIZ

**and** F M is a subalgebroid of T M by (1), we find that [X, Y ](s(g)) ∈ F G (s(g)) for all g ∈ G

**and** hence η([X, Y ])(g) = η([X, Y ])(s(g)) = 0. Since η was a ˜t-section of FG ◦ **and** ˜t-sections

of FG ◦ span F G ◦ on G, we have shown that [X, Y ] ∈ Γ(F G) **and** the proof is complete. □

Remark 3.11. (1) We have seen in this proof that Condition (2) implies the fact that

F c is a subalgebroid of A.

(2) The same result has been shown independently by M. A. Salazar **and** M. Crainic,

using Lie groupoid **and** Lie algebroid cocycles, in the special case where F M = T M,

i.e. where F G is a wide subgroupoid of T G.

3.4. Examples.

Example 3.12. Assume that G is a Lie group (hence with M = {e}) with Lie algebra g.

Let F G be a multiplicative distribution. In this case, the core F c =: f is the fiber of F G over

the identity **and** F M = 0. As a consequence, any partial F M -connection on g/f is trivial. We

check that all the conditions in Theorem 3.6 are automatically satisfied.

First of all, any element ξ of g is ∇-parallel. This implies that

[ξ r , Γ(F G )] ⊆ Γ(F G ) for all ξ ∈ g,

i.e., F G is left-invariant, in agreement with [23, 12, 13].

1), 3) **and** 4) are trivially satisfied **and** 2) is exactly the fact that f is an ideal in g. This

recovers the results proved in [23, 12, 13].

Example 3.13. Let G ⇒ M be a Lie groupoid with a smooth, free **and** proper action of

a Lie group H by Lie groupoid automorphisms. Let V G be the vertical space of the action,

i.e., the smooth subbundle of T G that is generated by the **infinitesimal** vector fields ξ G , for

all ξ ∈ h, where h is the Lie algebra of H. The involutive subbundle V G is easily seen to be

multiplicative (see for instance [13]).

The action restricts to a free **and** proper action of H on M, **and** it is easy to check

that V G ∩ T M = V M is the vertical vector space of the action of H on M. Furthermore,

V G ∩ T s G = V G ∩ T t G = 0 T G **and** we get V c = 0 A .

The **infinitesimal** vector fields (ξ G , ξ M ), ξ ∈ h, are multiplicative (in the sense of [20] for

instance). We get hence from [20] that the Lie bracket [a r , ξ G ] is right-invariant for any ξ ∈ h

**and** a ∈ Γ(A). We obtain a map (see also [20])

**and** we recover the connection

h × Γ(A) → Γ(A)

(ξ, a) ↦→ [ξ G , a r ]| M

,

∇ : Γ(V M ) × Γ(A) → Γ(A)

defined by ∇ ξM a = [ξ G , a r ]| M for all ξ ∈ h **and** a ∈ Γ(A). This connection is obviously flat

**and** satisfies all the conditions in Theorem 3.6.

Example 3.14. Let (G ⇒ M, J G ) be a complex Lie groupoid, i.e., a Lie groupoid endowed

with a complex structure J G that is multiplicative in the sense that the map

T G J G

T G

T t

T s

T t

T M

JM

T M

is a Lie groupoid morphism over some map J M . Since J 2 G = −Id T G , we conclude that

J 2 M = −Id T M **and** the Nijenhuis condition for J M is easy to prove using s-related vector

fields. The map J G restricts also to a map j A on the core A, i.e., a fiberwise complex

T s

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 19

structure that satisfies also a Nijenhuis condition. (This can be seen by noting that the

Nijenhuis tensor of J G restricts to right-invariant vector fields.)

The subbundles T 1,0 G = E i **and** T 0,1 G = E −i of T G ⊗ C are multiplicative **and** involutive

with bases T 1,0 M **and** T 0,1 M **and** cores A 1,0 **and** A 0,1 . The quotient (A ⊗ C)/A 1,0 is

isomorphic as a vector bundle to A 0,1 **and** a straightforward computation shows that the connection

that we get from the multiplicative complex foliation T 1,0 G is exactly the connection

∇ : Γ(T 1,0 M) × Γ(A 0,1 ) → Γ(A 0,1 ) as in Lemma 4.7 of [15].

Since the parallel sections of the connection in this Lemma are exactly the holomorphic

sections of A 0,1 , one can reconstruct the map J A : T A → T A defined by J A = σ −1 ◦A(J G )◦σ

4 as in [16] by requiring that J A (T a) = T a ◦ J M for all parallel sections a ∈ Γ(A), **and**

J A (ˆb) = ĵA(b) ◦ J M for all sections b ∈ Γ(A).

By Lemma 4.7 in [15] **and** the integration results in [16], the complex structure J G is hence

equivalent to the **data** J M , j A **and** this connection with its properties. This is in agreement

with the results that we will prove in section 5.

4. **Foliated** algebroids

In this section we study Lie algebroids equipped with foliations compatible with both Lie

algebroid structures T A → T M **and** T A → A on T A. This is the first step towards an

**infinitesimal** description of multiplicative foliations.

4.1. Definition **and** properties.

Definition 4.1. Let A → M be a Lie algebroid. A subbundle F A ⊆ T A is called morphic

if it is a Lie subalgebroid of T A → T M over some subbundle F M ⊆ T M.

If F A is involutive **and** morphic, then the pair (A, F A ) is referred to as a foliated Lie

algebroid.

Consider a foliated Lie algebroid (A, F A ). We have the VB-algebroid

F A

T q A | FA

F M

p M | FM

**and** the Lie algebroid morphisms

p A | FA

A qA

M

F A

ι FA

T A

F A

ι FA

T A

.

T q A | FA

F M ιFM

T M

T q A

p A | FA

A IdA

A

p A

We have

(

(4.13) F A 0 A (m) ) = T m 0 A (F M (m)) ⊕ ker(T q A | FA (0 A (m)))

for all m ∈ M. If v 0 A (m) ∈ F A (0 A m) is such that T q A (v 0 A (m)) = 0 m ∈ F M (m), then

v 0 A (m) = d dt⏐ 0 A m + ta m

t=0

4 To avoid confusions, we write in this example σ : T A → A(T G) for the canonical flip map.

20 M. JOTZ AND C. ORTIZ

for some a m ∈ A m . Set

{ ∣ ∣∣∣ d

F c (m) := a m ∈ A m dt⏐ 0 A ( ) }

m + ta m ∈ F A 0

A

m

t=0

for all m ∈ M. Equation (4.13) shows that this defines a vector subbundle of A.

Furthermore, using the fact that F A is closed under the addition in T A → T M, one can

check that

(4.14) ker ( { ∣ }

) d ∣∣∣

T q A | FA (b m) = dt⏐ b m + ta m a m ∈ F c (m)

t=0

for all b m ∈ A m . The vector bundle F c ≃ ker(T q A ) ∩ ker(p A ) is the core of F A .

Recall that the core sections of T A → A are the sections a ↑ as in (2.9) for a ∈ Γ(A), **and**

the core sections of T A → T M are the sections â as in (2.3). The core sections of T A → A

that have image in F A are hence exactly the core sections a ↑ for a ∈ Γ(F c ). Now if

â(v m ) = T m 0 A d

v m + pA

dt⏐ ta(m) ∈ F A

t=0

then we have v m = T q A (â(v m )) ∈ F M (m) **and** since F A → F M is a vector subbundle of

T A → T M, the zero element T m 0 A v m belongs to the fiber of F A over v m . This leads to

d ⏐

dt t=0

ta(m) ∈ F A , using the addition in the fiber of F A over v m , **and** hence a(m) ∈ F c (m).

This shows that the core sections â of T A → T M that restrict to core sections of F A → F M

are the restrictions â| FM for a ∈ Γ(F c ).

The vector bundles F A → A **and** F A → F M are spanned by **their** linear **and** core sections.

Lemma 4.2. Let (A, F A ) be a foliated Lie algebroid.

T a| FM ∈ Γ FM (F A ) if **and** only if [ a ↑ , Γ(F A ) ] ⊆ Γ(F A ).

A section a ∈ Γ(A) is such that

Proof. Choose any linear section X of F A → A over a non-vanishing section ¯X ∈ Γ(F M ).

Since ¯X(m) ≠ 0 for all m, we get that for each c m ∈ A m , the vector X(c m ) can be written

X(c m ) = T m c( ¯X(m)) for some section c ∈ Γ(A) with c(m) = c m . We write for simplicity φ a·

for the flow of a ↑ , i.e., φ a t (c ′ ) = c ′ + ta(q A (c ′ )) for all c ′ ∈ A **and** t ∈ R, **and** ¯φ· for the flow

of ¯X. We have, writing vm = ¯X(m),

(4.15)

T cm φ a t X(c m ) = T m (φ a t ◦ c)( ¯X(m))

= d ds⏐ c ( ¯φs (m) ) + ta ( ¯φs (m) )

s=0

= T m c(v m ) + tT m a(v m )

= X(c m ) + tT m a(v m ).

If [ a ↑ , Γ(F A ) ] ⊆ Γ(F A ), the flow φ a of a ↑ preserves F A . Hence, T cm φ a t X(c m ) ∈ F A (c m +

ta(m)) is a vector of F A → F M over ¯X(m) **and** since X(c m ) is also a vector of F A → F M

over ¯X(m), we find that tT m a(v m ) ∈ F A (ta(m)) over ¯X(m) for all t ∈ R. In particular we

get T a| FM ∈ Γ FM (F A ).

Conversely, if T a| FM ∈ Γ FM (F A ), then tT m a(v m ) ∈ F A (ta(m)) for all t ∈ R **and** v m ∈ F M .

Since X(c m ) is an element of F A for all c m ∈ A m , we find hence T cm φ a t X(c m ) ∈ F A (c m +

ta(m)) **and** [a ↑ , X] ∈ Γ(F A ). This implies the inclusion [ a ↑ , Γ(F A ) ] ⊆ Γ(F A ) since the bracket

of two core sections is 0 **and** F A → A is spanned by its linear **and** its core sections. □

Proposition 4.3. Let (A, F A ) be a foliated Lie algebroid. Then the core F c → M of F A is

a subalgebroid of A **and** the base F M → M is an involutive subbundle of T M.

Proof. Choose a, b ∈ Γ(F c ). Since F A → A is involutive **and** a ↑ ∈ Γ(F A ), we know that

[

a ↑ , Γ(F A ) ] ⊆ Γ(F A ). By the preceding lemma, this implies that T a| FM is a section of

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 21

F A → F M . Since ˆb| FM is also a section of the subalgebroid F A → F M of T A → T M, we find

that [a, ̂ b]| FM =

[T a| FM , ˆb|

]

FM ∈ Γ FM (F A ) **and** hence [a, b] ∈ Γ(F c ).

The second statement is immediate by Lemma 2.1 **and** the fact that q A is a surjective

submersion.

□

The following lemma will also be useful later.

Lemma 4.4. Let (A, F A ) be a foliated Lie algebroid. Let ( X, 0 T M ) be a linear section of

T A covering the zero section. Then we have

X ∈ Γ(F A ) ⇔ D X a ∈ Γ(F c ) for any a ∈ Γ(A).

Proof. Choose a m ∈ A. Since T am q A (X(a m )) = 0 T m M , we can write X(a m ) = (c(a m )) ↑ (a m )

for some c(a m ) ∈ Γ(A). We have then, for any section ϕ ∈ Γ((F c ) ◦ ), where (F c ) ◦ is the

annihilator of F c in A ∗ , **and** any section a ∈ Γ(A) such that a(m) = a m :

〈ϕ(m), D X a(m)〉 = d dt⏐ 〈ϕ(m), φ X −t(a m )〉 = −X(a m )(l ϕ )

t=0

= d dt⏐ 〈ϕ(m), a m − tc(a m )(m)〉 = −〈ϕ(m), c(a m )(m)〉.

t=0

Hence, (D X a)(m) ∈ F c (m) if **and** only if c(a m )(m) ∈ F c (m). If D X a ∈ Γ(F c ) for all a ∈ Γ(A),

we find hence X ∈ Γ(F A ) by (4.14), **and** conversely, if X ∈ Γ(F A ), we get D X a ∈ Γ(F c ) for

all a ∈ Γ(A).

□

4.2. The Lie algebroid of a multiplicative foliation. The following construction can

be found in [24] in the more general setting of multiplicative Dirac structures.

Let F G be a multiplicative subbundle of T G with space of units F M ⊆ T M. Since

F G ⊆ T G is a Lie subgroupoid, we can apply the Lie functor, leading to a Lie subalgebroid

A(F G ) ⊆ A(T G) over F M ⊆ T M.

As we have seen in Subsection 2.2, the canonical involution J G : T T G −→ T T G restricts

to an isomorphism of double vector bundles j G : T A −→ A(T G) inducing the identity map

on both the side bundles **and** the core. Since j G : T A −→ A(T G) is an isomorphism of Lie

algebroids over T M, we conclude that

is a Lie algebroid over F M ⊆ T M. Since

F A := j −1

G (A(F G)) ⊆ T A

p G

F G

G

T t

T s

F M pM

M

t

s

is a VB-subgroupoid of

T G p G

G

,

T t

T s t

T M pM

M

s

22 M. JOTZ AND C. ORTIZ

the Lie algebroid

A(F G ) A(p G)

A

F M pM

M

is a VB-subalgebroid of

A(T G) A(p G)

A

T M pM

M

([3]), **and** F A → A is also a subbundle of T A → A.

Before we study the sections of A(F G ) → F M , we show that the vector subbundle F A ⊆

T A is involutive. The space of sections of F A → A is spanned by the core sections a ↑ for

a ∈ Γ(F c ) **and** linear sections ( ˜X, ¯X) = (j −1

G (A(X)), ¯X) coming from star sections X ∼ ⋆ s

¯X

of F G → F M . Hence, the involutivity of F A follows immediately from the involutivity of

F G ⊆ T G **and** the considerations following Lemma 2.1.

The Lie algebroid A(F G ) → F M is a subalgebroid of A(T G) → T M. We want to find the

linear sections of A(T G) that restrict to linear sections of A(F G ). If a ∈ Γ(F c ), then ã r is

easily seen to be tangent to F G on F M . Choose an arbitrary section a ∈ Γ(A). We want to

find a condition for β a | FM 5 to be a section of A(F G ). We have to find

β a (v) ∈ T v F G

for all v ∈ F M . A sufficient condition is [a r , Γ(F G )] ⊆ Γ(F G ). We will show in the remainder

of this subsection that this condition is also necessary.

Now we define a more natural, partial F M -connection ∇ s on A/F c , that will help us to

underst**and** which linear sections of A(T G) → T M restrict to linear sections of A(F G ) → F M .

Choose a ∈ Γ(A), ¯X ∈ Γ(FM ) **and** set

(4.16) ∇ s¯Xā := [X, a r ]| M

for any section X ∈ Γ(F G ) such that X ⋆ ∼ s

¯X. We show the following proposition.

Proposition 4.5. Let (G ⇒ M, F G ) be a foliated Lie groupoid. Then ∇ s as in (4.16) is a

well-defined flat partial F M -connection on A/F c .

Proof. For any section a ∈ Γ(A), we have a r ∼ s 0. Since X ∼ ⋆ s

¯X implies in particular

X ∼ s

¯X, we find [X, a r ] ∼ s 0 **and** [X, a r ]| M is thus a section of A.

We show that [X, a r ]| M ∈ Γ(A/F c ) does not depend on the choice of X ∈ Γ(F G ). If

X ′ ∈ Γ(F G ), X ′ ⋆

∼s ¯X, then we have X − X ′ ∈ Γ(F G ) **and** X − X ′ ⋆

∼s 0. Hence, if a 1 , . . . , a r

is a local frame for F c on an open set U ⊆ M, we can write X − X ′ = ∑ r

i=1 f ia r i on

Ũ := t −1 (U), with f 1 , . . . , f r ∈ C ∞ (Ũ) such that f i| U = 0 for i = 1, . . . , r. We have then

r∑

r∑

[X − X ′ , a r ] = − a r (f i )a r i + f i [a i , a] r .

i=1

But since the second term of this sum vanishes on U, we conclude that [X−X ′ , a r ]| M ∈ Γ(F c ).

This proves that ∇ s is well-defined.

5 Recall the definitions of ã **and** βa from (2.6) **and** (2.5).

i=1

Assume now that X ⋆ ∼ s

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 23

¯X **and** choose f ∈ C ∞ (M). Then we have (s ∗ f) · X ⋆ ∼ s f · ¯X **and**

[s ∗ f · X, a r ] = s ∗ f · [X, a r ] − a r (s ∗ f)X = s ∗ f · [X, a r ],

which shows

∇ s f· ¯Xā

= f · ∇s¯Xā.

Analogously, (f · a) r = t ∗ f · a r **and**, since X| M = ¯X ∈ X(M),

[X, t ∗ f · a r ] = t ∗ f · [X, a r ] + X(t ∗ f) · a r

restricts to

f · [X, a r ]| M + ¯X(f) · a

on M.

We show finally that ∇ s is flat. Choose X ∼ ⋆ ⋆

s

¯X **and** Y ∼s Ȳ . Then, since

[X, a r ] − ([X, a r ]| M ) r

vanishes on M **and** Y is tangent to M on M, we find that

[

Y, [X, a r ] − ([X, a r ]| M ) r]

vanishes on M. Thus, we have

[

Y, [X, a r ] ] | M = [ Y, ([X, a r ]| M ) r] | M

**and** the flatness of ∇ s follows easily.

Recall that we say that a ∈ Γ(A) is ∇- (or ∇ s -)parallel if ā is ∇-parallel, since this doesn’t

depend on the chosen representative.

Theorem 4.6. Let (G ⇒ M, F G ) be a foliated Lie groupoid **and** choose a ∈ Γ(A). Then the

following are equivalent:

(1)

[a r , Γ(F G )] ⊆ Γ(F G ),

(2) a is ∇-parallel, where ∇ is the connection in Theorem 3.6,

(3) a is ∇ s -parallel,

(4) β a | FM is a section of A(F G ),

(5) [X, a r ]| M ∈ Γ(F c ) for any section X ∈ Γ(F G ) such that X ⋆ ∼ s

¯X.

This theorem shows in particular that the ∇- (or equivalently ∇ s -) parallel sections of A

are exactly the sections of A such that β a | FM is a section of A(F G ).

Since the two connections are flat **and** **their** parallel sections coincide, we get the following

corollary.

Corollary 4.7. Let (G ⇒ M, F G ) be a foliated Lie groupoid. Then ∇ = ∇ s .

Remark 4.8. Note that in Example 3.13, the two connections can immediatly be seen to

be the same by definition, since the multiplicative vector fields ξ G , ξ ∈ h are t-sections **and**

star sections at the same time.

Proof of Theorem 4.6. We already know by Theorem 3.6 that (1) **and** (2) are equivalent. By

definition of ∇ s , (3) **and** (5) are equivalent.

We show that a is ∇-parallel if **and** only if a is ∇ s -parallel. If a is ∇-parallel, then

[a r , Γ(F G )] ⊆ Γ(F G ). This yields immediately [X, a r ]| M ∈ Γ(F c ) for any star-section X ∼ ⋆ s

¯X

of F G . Conversely, assume that a is ∇ s -parallel. Since ∇ is flat, a can be written locally as

a sum a = ∑ n

i=1 f ia i with f i ∈ C ∞ (M), i = 1, . . . , n **and** ∇-parallel sections a 1 , . . . , a n . We

have then for any section ¯X of F M :

n∑

∇ ¯Xā = ¯X(f i )ā i .

i=1

□

24 M. JOTZ AND C. ORTIZ

But since the sections a i , i = 1, . . . , n are then also ∇ s -parallel by the considerations above,

we have also

n∑

¯X(f i )ā i = ∇ s¯Xā = 0,

i=1

which shows that a is ∇-parallel.

We show that (1) implies (4). If [a r , Γ(F G )] ⊆ Γ(F G ), then the flow L Exp(·a) of a r leaves

F G invariant by Corollary A.2. Hence, we have T m Exp(ta)v m ∈ F G (Exp(ta)(m)) for all

v m ∈ F M (m). This yields β a (v m ) = d ⏐

dt t=0

T m Exp(ta)v m ∈ Tv T s

m

F G = A vm (F G ).

We show that (4) implies (5). Recall that the flow of β r a is equal to T L Exp(·a) . Hence, if

β a | FM ∈ A(F G ), we have that

T m L Exp(sa) (v m ) ∈ F G (Exp(sa)(m))

for all |s| ∈ R small enough **and** v m ∈ F M . We have also

for all u m ∈ F t . This yields

T m L Exp(sa) (u m ) = 0 Exp(sa)(m) ⋆ u m ∈ F G (Exp(sa)(m))

T m L Exp(sa) F G (m) ⊆ F G (Exp(sa)(m))

for all m ∈ M. Since both sides are vector spaces of the same dimension, we get an equality

**and** in particular

F G (m) = T Exp(sa)(m) L Exp(−sa) F G (Exp(sa)(m)).

Choose now any star section X ∼ ⋆ s

¯X. We have for any m ∈ M:

[a r , X](m) = d ds⏐ T Exp(sa)(m) L Exp(−sa) X(Exp(sa)(m)).

s=0

Since T Exp(sa)(m) L Exp(−sa) X(Exp(sa)(m)) ∈ F G (m) for all s small enough, we find that

[a r , X](m) ∈ F c (m). □

4.3. Integration of foliated algebroids. The main theorem of this subsection is the following.

Theorem 4.9. Let (G ⇒ M, F G ) be a foliated groupoid. Then (A, F A = j −1

G (A(F G))) is a

foliated algebroid.

Conversely, let (A, F A ) be a foliated Lie algebroid. Assume that A integrates to a source

simply connected Lie groupoid G ⇒ M. Then there is a unique multiplicative foliation F G

on G such that F A = j −1

G (A(F G)).

We will use a result of [3], which states that a VB-algebroid

integrates to a VB-groupoid

q v E

E qh E

A

B qB

M

q A

G(E) q G(E)

G(A) .

B q B

M

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 25

Furthermore, if E ′ ↩→ E, B ′ ↩→ B is a VB-subalgebroid with the same horizontal base A,

E ′ q h E

A

,

q v E

B ′ q B ′

M

q A

then E ′ → B ′ integrates to an embedded VB-subgroupoid G(E ′ ) ↩→ G(E) over B ′ ↩→ B,

G(E ′ ) q G(E)

G(A) .

B ′ q M

M

This is done in [3] using the characterization of vector bundles via homogeneous structures

(see [9]).

Proof of Theorem 4.9. The first part has been shown in the previous subsection.

Let (A, F A ) be a foliated Lie algebroid. Then the core F c ⊆ A **and** the base F M ⊆ T M

are subalgebroids by Proposition 4.3. The VB-subgroupoid of (T G, G; T M, M) integrating

the subalgebroid j G (F A ) → F M of A(T G) → T M is a multiplicative subbundle F G ⇒ F M

of T G ⇒ T M with core F c . By Theorem 3.10, F G is involutive.

□

4.4. The connection associated to a foliated algebroid. Let A be a Lie algebroid

endowed with an involutive subbundle F A ⊆ T A. We show that if F A → F M is also morphic,

that is, a subalgebroid of the tangent algebroid T A → T M, then the Bott connection

∇ F A

: Γ(F A ) × Γ(A/F A ) → Γ(A/F A )

induces a natural partial F M -connection on A/F c with the same properties as the connection

in Theorem 3.6.

For ¯X in Γ(F M ), we choose any linear section X ∈ Γ A (F A ) over ¯X. Then, for any core

section a ↑ of T A → A, we have

[

X, a

↑ ] = D X a ↑

for some section D X a of A, see Lemma 2.1. Set

(4.17) ∇ Ā Xā = D Xa ∈ Γ(A/F c ).

We show the following proposition.

Proposition 4.10. Let (A, F A ) be a foliated Lie algebroid. The map

∇ A : Γ(F M ) × Γ(A/F c ) → Γ(A/F c )

as defined in (4.17) is a well-defined flat partial F M -connection on A/F c .

Proof. Choose ¯X ∈ Γ(F M ) **and** two linear sections X, Y ∈ Γ A (F A ) covering ¯X. Then

(X − Y, 0) is a vector bundle morphism **and** Z := X − Y ∈ Γ A (F A ). By Lemma 4.4, we get

D Z a ∈ Γ(F c ) for any a ∈ Γ(A) **and** since

(D Z a) ↑ = [X − Y, a ↑ ] = (D X a − D Y a) ↑ ,

this shows that ∇ A is well-defined. The properties of a connection can be checked using the

equality (f · a) ↑ = qA ∗ f · a↑ for a ∈ Γ(A) **and** f ∈ C ∞ (M).

The flatness of ∇ A follows immediately from the flatness of the Bott connection ∇ F A

. □

26 M. JOTZ AND C. ORTIZ

Example 4.11. Let H be a connected Lie group with Lie algebra h. Assume that H acts

on a Lie algebroid A −→ M in a free **and** proper manner, by Lie algebroid automorphisms.

That is, for all h ∈ H, the diffeomorphism Φ h is a Lie algebroid morphism over φ h : M → M.

Consider the vertical spaces V A , V M defined as follows

V A (a) = {ξ A (a) | ξ ∈ h},

V M (m) = {ξ M (m) | ξ ∈ h},

for a ∈ A **and** m ∈ M. We check that V A inherits a Lie algebroid structure over V M making

the pair (A, V A ) into a foliated algebroid with core zero. Choose ξ A (a m ) ∈ V A (a m ) for some

ξ ∈ g, then T am q A ξ A (a m ) = ξ M (m) ∈ V M (m). Hence, if T am q A ξ A (a m ) = 0, then ξ = 0 since

the action is supposed to be free **and** we find F c = 0. Notice that, since the action is by

algebroid automorphisms, the **infinitesimal** generators ξ A are in fact morphic vector fields

covering ξ M .

We have, writing ρ(a m ) = ċ(0),

( )

d ρ T A (ξ A (a m )) = J M (T ρ(ξ A (a m )) = J M dt⏐ (ρ ◦ Φ exp(tξ) )(a m )

( t=0

)

d = J M dt⏐ (T φ exp(tξ) ◦ ρ)(a m ) = d t=0

ds⏐ ξ M (c(s)) ∈ T ξM (m)V M .

s=0

If a ∈ Γ(A) is such that T m a(ξ M (m)) ∈ V A (a m ) for some ξ ∈ g **and** m ∈ M, then there

exists η ∈ g such that T m a(ξ M (m)) = η A (a(m)). But applying T am q A to both sides of this

equality yields then ξ M (m) = η M (m), which leads to ξ = η, since the action is free, **and**

hence T m a(ξ M (m)) = ξ A (a(m)).

Here, the induced partial V M -connection on A is given by ∇ A ξ M

a ∈ Γ(A) where

[ξ A , a ↑ ] = (∇ A ξ M

a) ↑

for any a ∈ Γ(A). If a ∈ Γ(A) is ∇-parallel, then we find [ξ A , a ↑ ] = 0 for all ξ ∈ g **and** hence

0 = d d

dt⏐ t=0

ds⏐ Φ exp(sξ) (b m + ta(m)) − ta(φ exp(sξ) )

s=0

= d d

(

dt⏐ ξ A (b m ) + t ·

t=0

ds⏐ Φexp(sξ) (a(m)) − a(φ exp(sξ) ) )

s=0

for all b m ∈ A. This leads to ξ A (a(m)) = T m aξ M (m) **and** hence

(4.18) T a(V M ) ⊆ V A .

The parallel sections of ∇ A are hence exactly the sections of A satisfying (4.18). This will

be shown in the general situation in the next lemma.

If a ∈ Γ(A) is ∇ A -parallel, we find easily that a(φ h (m)) = Φ h (a(m)) for all h ∈ H (recall

that H is assumed to be connected) **and** m ∈ M. Hence, if a, b ∈ Γ(A) are such that

T a(V M ) ⊆ V A **and** T b(V M ) ⊆ V A , we have a ◦ φ h = Φ h ◦ a **and** b ◦ φ h = Φ h ◦ b for all h ∈ H.

Since the action is by Lie algebroid morphisms, we get then [a, b] ◦ φ h = Φ h ◦ [a, b], which

implies T [a, b](V M ) ⊆ V A . Since the core sections of V A are all trivial, this shows that the

Lie bracket of T A → T M restricts to V A → V M .

Lemma 4.12. Let (A, F A ) be a foliated Lie algebroid **and** ∇ A the connection defined in(4.17).

The following are equivalent for a ∈ Γ(A):

(1) the class ā is ∇ A -parallel,

(2) T a| FM is a section of F A ,

(3) the core section a ↑ of F A → A is parallel with respect to the Bott connection ∇ F A

,

that is [ a ↑ , Γ(F A ) ] ⊆ Γ(F A ).

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 27

Proof. The equivalence (2) ⇔ (3) has been shown in Lemma 4.2 **and** the implication (3) ⇒

(1) is obvious.

If ∇ Ā Xā = 0 for all ¯X ∈ Γ(FM ), we get that

[

X, a

↑ ] ∈ Γ(F A )

for all linear sections X ∈ Γ(F A ). Since

[

a ↑ , b ↑] = 0 ∈ Γ(F A )

for all b ∈ Γ(F c ) **and** the linear sections of F A together with the core sections of F A span

F A , we get

[

a ↑ , Γ(F A ) ] ⊆ Γ(F A )

**and** the flow of a ↑ leaves F A invariant.

Hence, we have shown (1) ⇒ (3).

□

Now we can complete the proof of the main theorem of this subsection.

Theorem 4.13. Let (A, F A ) be a foliated Lie algebroid. Then the partial F M -connection

∇ A on A/F c as in (4.17) has the same properties as the connection ∇ in Theorem 3.6, **and**

such that

ā is ∇ A -parallel ⇔ T a(F M ) ⊆ F A .

Proof. By Proposition 4.10, we know that ∇ A as in (4.17) is well-defined **and** flat **and** we

have shown in Lemma 4.12 that ā is ∇ A -parallel if **and** only if T a(F M ) ⊆ F A .

If c ∈ Γ(F c ), then ĉ| FM is a section of F A → F M , **and** [a, ̂ c]| FM = [T a| FM , ĉ| FM ] ∈ Γ FM (F A ).

Hence, [a, c] ∈ Γ(F c ).

If b ∈ Γ(A) is also such that ¯b is ∇ A -parallel, **and** T a| FM , T b| FM ∈ Γ FM (F A ), then, since

F A → F M is a subalgebroid of T A → T M, we find that [T a| FM , T b| FM ] ∈ Γ FM (F A ). But

since [T a| FM , T b| FM ] = T [a, b]| FM , this shows that [a, b] is ∇ A -parallel by Lemma 4.12.

It remains to show that ρ(a) ∈ X(M) is ∇ F M

-parallel if a is ∇ A -parallel. Since T a| FM ∈

Γ FM (F A ) **and** F A → F M is a subalgebroid of T A → T M, we find that

ρ T A (T m a(v m )) = J M (T m (ρ A (a))v m )

is an element of T vm F M for any v m ∈ F M . Choose X ∈ Γ(F M ) **and** α ∈ Γ(FM ◦ ). Then, if

l α ∈ C ∞ (T M) is the linear function defined by α, we have

But the identity

d X(m) l α (ρ T A (T m a(X(m)))) = 0.

J M (T m (ρ A (a))X(m)) = d dt⏐ T m φ t X(m),

t=0

where φ· is the flow of ρ A (a), yields then immediately

(

£ρA (a)α ) (X(m)) = 0

for all m, **and** the equality

0 = £ ρA (a)(α(X)) = (£ ρA (α)α)(X) + α([ρ A (a), X])

leads thus to α([ρ A (a), X]) = 0. Since α **and** X were arbitrary sections of FM ◦ **and** F M ,

respectively, we have shown that [ρ A (a), Γ(F M )] ⊆ Γ(F M ).

□

Theorem 4.14. Let (A, F A ) be a foliated algebroid. Assume that A integrates to a Lie

groupoid G ⇒ M **and** that j G (F A ) integrates to F G ⊆ T G. Then

∇ = ∇ s = ∇ A .

28 M. JOTZ AND C. ORTIZ

Proof. It is clear that if j G (F A ) integrates to F G , then the core **and** the base of F G **and** of

F A coincide. Hence, the three connections are flat partial F M -connections on A/F c . Since

∇ = ∇ s by Corollary 4.7, we only need to show that ∇ **and** ∇ A have the same parallel

sections. For that we use Theorem 4.6. The class ā of a ∈ Γ(A) is ∇-parallel if **and** only

if β a | FM is a section of A(F G ). Since F A = j −1

G (A(F G)) **and** j −1

G ◦ β a| FM = T a| FM , this is

equivalent to T a| FM ∈ Γ FM (F A ). But by the preceding theorem, this is true if **and** only if ā

is ∇ A -parallel.

□

5. The connection as the **infinitesimal** **data** of a multiplicative foliation

5.1. Infinitesimal description of a multiplicative foliation. We have seen that both

foliated **groupoids** **and** foliated algebroids induce partial connections which have important

properties. This motivates the following concept.

Definition 5.1. Let (A → M, ρ, [· , ·]) be a Lie algebroid, F M ⊆ T M an involutive subbundle,

F c ⊆ A a subalgebroid over M such that ρ(F c ) ⊆ F M **and** ∇ a partial F M -connection on

A/F c with the following properties:

(1) ∇ is flat.

(2) If a ∈ Γ(A) is ∇-parallel, then [a, b] ∈ Γ(F c ) for all b ∈ Γ(F c ).

(3) If a, b ∈ Γ(A) are ∇-parallel, then [a, b] is also parallel.

(4) If a ∈ Γ(A) is ∇-parallel, then [ρ(a), ¯X] ∈ Γ(F M ) for all ¯X ∈ Γ(FM ). That is, ρ(a)

is ∇ F M

-parallel.

The quadruple (A, F M , F c , ∇) will be referred to as an IM-foliation 6 on A, or simply an

IM-foliation.

Remark 5.2. Note that this is an **infinitesimal** version of the ideal systems in [19].

We have shown in Theorem 3.6 that each foliated Lie groupoid (G ⇒ M, F G ) induces an

IM-foliation on A. In the same manner, we have seen in Theorem 4.13 that each foliated

Lie algebroid (A, F A ) induces an IM-foliation on A. Moreover, a foliated Lie groupoid (G ⇒

M, F G ) **and** its foliated Lie algebroid (A, j −1

G (A(F G))) induce the same IM-foliation on A.

We will show here that an IM-foliation on a Lie algebroid is equivalent to a morphic foliation

on this Lie algebroid. This implies that IM-foliations on integrable Lie algebroids are in

one-to-one correspondence with multiplicative foliations on the corresponding Lie **groupoids**.

5.2. Reconstruction of the foliated Lie algebroid from the connection. Let now

(q A : A → M, ρ, [· , ·]) be a Lie algebroid **and** (A, F M , F c , ∇) an IM-foliation on A.

The Lie algebroid T A → T M is spanned by the sections T a **and** â for all a ∈ Γ(A). Recall

that â(v m ) = T m 0 A d

v m + pA

⏐

t=0

ta(m) for v m ∈ T m M **and** that the bracket of the Lie

dt

algebroid T A → T M is given by

[

[T a, T b] T A

= T [a, b], T a, ˆb

]

= [a, ̂ b],

T A

for all a, b ∈ Γ(A).

For each v m ∈ F M , define the subset F A (v m ) of (T A) vm as follows:

⎧

⎨ T m a(v m ) ∈ T a(m) A,

(5.19) F A (v m ) = span

⎩

∣

ˆb(vm )

[

â, ˆb

]

= 0

T A

a ∈ Γ(A) is ∇-parallel,

b ∈ Γ(F c )

That is, F A is spanned by the restrictions to F M of the linear sections T a of T A defined by

a ∈ Γ(A) such that ā ∈ Γ(A/F c ) is ∇-parallel **and** the core sections ˆb defined by sections

b ∈ Γ(F c ). We will show that F A is a well-defined subalgebroid of T A → T M.

6 Here, “IM” st**and**s for “**infinitesimal** multiplicative”, as for the IM-2-forms in [5].

⎫

⎬

⎭ .

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 29

First, we check that F A is a double vector bundle

F A

A .

F M

By construction, F A has core F c **and** there will be an injective double vector bundle morphism

to

p A

T A A

T q A

M

T M pM

M

which has core A.

By definition, we have T am q A (T m a(v ) m )) = T m (q A ◦ a)v m = v m ∈ F M (m) for all a ∈ Γ(A)

**and** v m ∈ F M (m) **and** T vm q A

(ˆb(vm ) = v m ∈ F M (m) for all b ∈ Γ(F c ).

Choose m ∈ M. Then there exists a neighborhood U of m in M **and** sections a 1 , . . . , a n ∈

Γ(A) such that a r+1 , . . . , a n are ∇-parallel **and** form a basis for A/F c on U, **and** a 1 , . . . , a r

form a basis for F c on U. Note that a 1 , . . . , a r are in a trivial manner ∇-parallel. Choose any

∇-parallel section a ∈ Γ(A). Then, on U, we have a = ∑ n

i=1 f ia i with f 1 , . . . , f r ∈ C ∞ (U)

**and** F M -invariant functions f r+1 , . . . , f n ∈ C ∞ (U). For any v m ∈ F M (m), the equality

T m a(v m ) =

r∑

v m (f i )â i (v m ) +

i=1

q A

n∑

f i (m)T m a i (v m )

can be checked in coordinates, using v m (f i ) = 0 for i = r + 1, . . . , n. If

n∑

r∑

α i T m a i (v m ) + β i â i (v m ) = 0 T v A

m

i=1

for some v m ∈ F m **and** α 1 , . . . , α n , β 1 , . . . , β r ∈ R, we find

(

n∑

n

)

∑

r∑

α i a i (m) = p A α i T m a i (v m ) + β i â i (v m ) = 0 A m.

i=1

i=1

i=1

Since a 1 , . . . , a n is a local frame for A, this yields α 1 = . . . = α n = 0 **and** the equality

β 1 = . . . = β r = 0 follows easily. The set of sections T a 1 , . . . , T a n **and** â 1 , . . . , â r is thus a

local frame for F A as a vector bundle over F M .

We next show that F A is also a vector bundle over A (a subbundle of T A → A). Choose

a m ∈ A m . Then there exists a ∇-parallel section a ∈ Γ(A) defined on a neighborhood U of

m such that a(m) = a m . Let r be the rank of F c **and** l the rank of F M . Choose a basis of

vector fields V 1 , . . . , V l for F M on U **and** a basis of sections b 1 , . . . , b r for F c on U. Choose

a local frame a 1 , . . . , a n for A on U by ∇-parallel sections **and** define for i = 1, . . . , l the

sections Ṽi : q −1

A (U) → F A by

Ṽ i (a m ) = T

i=1

i=1

(

φ a↑ 1

α 1

◦ . . . ◦ φ a↑ l

α l

◦ 0 A )

(V i (m))

where α 1 , . . . , α n ∈ R are such that a m = ∑ n

i=1 α i · a i (m). By construction, the vector fields

Ṽ i are sections of F A → A (recall Lemma 4.12), **and**, using the fact that core vector fields

commute, it is easy to show that Ṽi is linear over V i for each i = 1, . . . , l. The sections

Ṽ 1 , . . . , Ṽl, b ↑ 1 , . . . , b↑ r form a basis for F A (seen as vector bundle over A) on q −1

A (U).

Now we check that (F A , ρ T A , [· , ·] T A ) is a Lie algebroid over F M (a subalgebroid of T A →

T M). Choose two linear sections T a| FM **and** T b| FM of F A , i.e., with ā, ¯b ∈ Γ(A/F c ) that

30 M. JOTZ AND C. ORTIZ

are ∇-parallel. We have then [T a, T b] T A = T [a, b]. By the properties of the connection,

[a, b] is ∇-parallel **and** T [a, b]| FM is a section of F A . Choose now a core section ˆb| FM **and** a

linear section T a| FM of F A , i.e., with b ∈ Γ(F c ) **and** a ∈ Γ(A) a ∇-parallel section. We get

[T a, ˆb] T A = [a, ̂ b], whose restriction to F M is a section of F A since, by the properties of the

connection, [a, b] ∈ Γ(F c ). Since the bracket of two core sections vanishes, we have shown

that [· , ·] T A restricts to F A .

Next, we show that the anchor map of T A restricts to a map F A → T F M . Again, it

is sufficient to show that the spanning sections of F A are sent by ρ T A to vector fields on

F M . Recall that the anchor map is given by ρ T A = J M ◦ T ρ. Choose a ∇-parallel section

a ∈ Γ(A), set X := ρ(a) ∈ X(M) **and** compute for any v m = ċ(0) ∈ F M :

ρ T A (T m a(v m )) = J M (T m (ρ(a))v m ) =

d ds⏐ T m φ X s (v m ).

s=0

By the properties of the connection, the vector field X is such that [X, Γ(F M )] ⊆ Γ(F M ). By

Corollary A.2, this implies T m φ X s F M (m) = F M (φ X s (m)) for all m ∈ M **and** s ∈ R where this

makes sense. This implies that the curve s ↦→ T m φ X s (v m ) has image in F M **and** consequently,

ρ T A (T m a(v m )) ∈ T vm F M .

Similarly, choose b ∈ Γ(F c ), v m = ċ(0) ∈ F M (m), set Y = ρ(b) ∈ Γ(F M ) **and** compute:

(

)

ρ T A T m 0 A d

(v m ) + pA

dt⏐ tb(m)

(

t=0 ))

= J M

(T 0m ρ T m 0 A d

(v m ) + pA

dt⏐ tb(m)

( t=0 )

d = J M dt⏐ 0 T M d

(c(t)) + pT M

t=0

dt⏐ tY (m)

( t=0 )

d = J M d

d

dt⏐ t=0

ds⏐ c(t) + pT d

M

s=0

dt⏐ t=0

ds⏐ φ Y st(m)

s=0

= d d

d

ds⏐ s=0

dt⏐ c(t) + T pM

d

t=0

ds⏐ s=0

dt⏐ φ Y st(m)

t=0

= d ds⏐ v m + T M sY (m) ∈ T vm F M .

s=0

Note also that the induced map

T q A

p A

F A

A

F M pM

M

is a surjective Lie algebroid morphism by construction.

We have proved the following proposition.

Proposition 5.3. Let (A → M, F M , F c , ∇) be an IM-foliation on A. Consider F A ⊆ T A

constructed as in (5.19). Then F A → F M is a subalgebroid of T A → T M.

We show now that F A → A is also a subalgebroid of T A → A.

Proposition 5.4. Let (A → M, F M , F c , ∇) be an IM-foliation on A. Consider F A → F M

the Lie subalgebroid of T A → T M as above. The vector subbundle F A → A of T A → A is

involutive.

q A

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 31

Proof. By definition, the linear sections of T A → T M that restrict to sections of F A → F M

are the sections T a for all a ∈ Γ(A) such that ā is a ∇-parallel section of A/F c . Note also

that, by definition, every section of F c is ∇-parallel in this sense.

As in the proof of Lemma 4.2, this implies [a ↑ , Γ(F A )] ⊆ Γ(F A ) if a ∈ Γ(A) is a ∇-parallel

section. As a consequence, for any linear section X of F A over ¯X, we have [X, a ↑ ] = (D X a) ↑

with D X a ∈ Γ(F c ). We find thus the trivial equality ∇ ¯Xā = D X a for ∇-parallel sections

a ∈ Γ(A).

Since ∇ is flat, there exist local frames for A of ∇-parallel sections. Choose a ∈ Γ(A) **and**

write a = ∑ n

i=1 f ia i with a 1 , . . . , a n ∈ Γ(A) ∇-parallel sections **and** f 1 , . . . , f n ∈ C ∞ (M).

Then ∇ ¯Xā = ∑ n ¯X(f i=1 i )ā i . But, on the other h**and**, if X ∈ Γ(F A ) is a linear section over

¯X, we have

which leads to

⎡

(D X a) ↑ = ⎣X,

∇ ¯Xā =

( n

∑

n∑

i=1

i=1

f i a i

) ↑

⎤

⎦ =

¯X(f i )ā i =

n∑

i=1

n∑

i=1

q ∗ A

¯X(f i )ā i +

( ¯X(fi ) ) n a ↑ i + ∑

qAf ∗ i (D X a i ) ↑ ,

i=1

n∑

f i · D X a i = D X a,

since D X a i ∈ Γ(F c ) for i = 1, . . . , n.

Choose now two linear sections (X, ¯X) **and** (Y, Ȳ ) of F A → A. Since ∇ is flat **and** F M is

involutive , we have ∇ [ ¯X, Ȳ ]ā = ∇ ¯X(∇ Ȳ ā)−∇ Ȳ (∇ ¯Xā) for all a ∈ Γ(A). Hence, if ∇ [ ¯X, Ȳ ]ā = ¯k

for some k ∈ Γ(A), then

(k + b) ↑ = [ X, [ Y, a ↑]] − [ Y, [ X, a ↑]] = [ [X, Y ], a ↑]

for some b ∈ Γ(F c ) **and**, if Z ∼ qA [ ¯X, Ȳ ] is a linear section of F A covering [ ¯X, Ȳ ], then

(k + b ′ ) ↑ = [ Z, a ↑]

for some b ′ ∈ Γ(F c ). The linear section W := Z − [X, Y ] covers hence 0 T M **and** satisfies

[

W, a

↑ ] = (b − b ′ ) ↑ .

Since b − b ′ ∈ Γ(F c ), we have found D W a ∈ Γ(F c ) for any a ∈ Γ(A), **and**, combining this

with Lemma 4.4, one concludes that W ∈ Γ(F A ) **and** hence [X, Y ] ∈ Γ(F A ). Since F A → A

is spanned by its linear **and** core sections, we have shown that F A is involutive. □

This completes the proof of our main theorem.

Theorem 5.5. Let (q A : A → M, ρ A , [· , ·] A ) be a Lie algebroid **and** (A, F M , F c , ∇) an IMfoliation

on A. Then F A defined as in (5.19) is a VB-algebroid with sides F M **and** A **and**

core F c , such that the inclusion is a VB-algebroid morphism:

F A

i=1

A

F M

T A

M

A

T M

Hence, (A, F A ) is a foliated Lie algebroid **and** the connection ∇ A induced by F A is equal to

∇.

M

32 M. JOTZ AND C. ORTIZ

Example 5.6. Assume that H acts on a Lie groupoid G over M by groupoid automorphisms.

Assume also that the action is free **and** proper. Starting from the **data** (A, V M , 0, ∇) where

∇ is the partial V M -connection on A determined by

[ξ G , a r ] = (∇ ξM a) r

for all ξ ∈ g **and** a ∈ Γ(A), the last theorem states that we recover exactly the Lie foliated

algebroid V A −→ V M obtained by applying the Lie functor to the foliated groupoid V G ⇒

V M .

Example 5.7. Assume that g is a Lie algebra, i.e. a Lie algebroid over a point. In this

case, the tangent Lie algebroid T g is also a Lie algebroid over a point, that is, T g is a Lie

algebra. It is easy to see that the Lie algebra structure on T g = g × g is the semi-direct

product Lie algebra g ⋉ g with respect to the adjoint representation of g on itself. Note also

that the fact that a quadruple (g, 0, f, ∇ = 0) is an IM-foliation on g is equivalent to saying

that f ⊆ g is an ideal.

The morphic foliation F g associated to the IM-foliation (g, 0, f, ∇ = 0) is given by F g =

g × f. This follows immediately from the fact that every element a ∈ g can be viewed, in a

trivial way, as a ∇-parallel section of g → {0}. The property that F g = g × f is a morphic

foliation is equivalent to saying that g × f is a Lie subalgebra of g ⋉ g. In particular, if G is

the connected **and** simply connected Lie group integrating g, we conclude that the foliated

algebroid F g = g × f integrates to a Lie subgroup G × f of the semi-direct Lie group G ⋉ g

determined by the adjoint action of G on its Lie algebra g. Using right (or left) translations,

we get a subbundle F G ⊆ T G which is involutive **and** multiplicative. We conclude that F G

is the multiplicative foliation on G integrating the IM-foliation (g, 0, f, ∇ = 0) on g. Thus,

in the case of Lie groups **and** Lie algebras, this recovers the results in [23, 12, 13].

6. The leaf space of a foliated algebroid

Assume that (A, F M , F c , ∇) is an IM-foliation on A. Then there is an induced involutive

subbundle F A ⊆ T A as in Theorem 5.5. We will show that if the leaf space M/F M is a

smooth manifold, such that the quotient map π M : M → M/F M is a surjective submersion,

then there is an induced Lie algebroid structure ([q A ] : A/F A → M/F M , [ρ], [· , ·] A/FA ) such

that the projection

A

π

A/F A

q A

M

[q A ]

πM

M/F M

is a Lie algebroid morphism. Furthermore, if A → M integrates to a Lie groupoid G ⇒ M

**and** the completeness **and** regularity conditions for the leaf space G/F G ⇒ M/F M to be a

Lie groupoid are satisfied (see [13]), then A/F A → M/F M is the Lie algebroid of G/F G ⇒

M/F M . We will see in the proofs that the important **data** for this reduction process is the

IM-foliation (A, F M , F c , ∇).

The class of a m ∈ A m is written [a m ] ∈ A/F A , **and** in the same manner, the class of

m ∈ M is denoted by [m] ∈ M/F M . The class of a m ∈ A m in A/F c will be written ā m .

Proposition 6.1. Let (A → M, F M , F c , ∇) be an IM-foliation on A **and** F A

corresponding morphic foliation as in Theorem 5.5.

⊆ T A the

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 33

(1) The map π : A → A/F A factors as a composition

A

π

A/F c π c

A/F A

That is, we have π(a m + b m ) = π(a m ) for all a m ∈ A **and** b m ∈ F c (m).

(2) The equivalence relation ∼:=∼ FA on A can be described as follows.

(6.20) a m ∼ a n ⇔

There exist linear sections (X 1 , ¯X 1 ), . . . , (X r , ¯X r ) of F A → A

with flows φ 1 , . . . , φ r such that

a m ∈ φ 1 t 1

◦ . . . ◦ φ r t r

(a n ) + F c (m)

for some t 1 , . . . , t r ∈ R.

(3) The map q A induces a map [q A ] : A/F A → M/F M such that

A

q A

M

π

A/F A

[q A ]

πM

M/F M

commutes.

(4) Let a ∈ Γ(A) be such that ā ∈ Γ(A/F c ) is ∇-parallel. Let U := Dom(a) ⊆ M. Then

there is an induced map [a] : π M (U) → A/F A such that

π M

U

π M (U)

a

[a]

A

π

A/F A

commutes. The map [a] is a section of [q A ] in the sense that

[q A ] ◦ [a] = Id πM (U).

Proof. (1) Recall that all the core sections b ↑ ∈ X(A) with b ∈ Γ(F c ) are sections of

F A . Choose a m ∈ A **and** b m ∈ F c (m). Then there exists a section b ∈ Γ(F c ) with

b(m) = b m . The flow φ b↑ of b ↑ is given by φ b↑

t (a) = a + tb(q A (a)) for all a ∈ A **and**

t ∈ R. Hence, we have a m ∼ a m + tb(m) = a m + tb m for all t ∈ R, **and** in particular

a m ∼ a m + b m . The map π c : A/F c → A/F A , π c (a m + F c (m)) = [a m ] is hence

well-defined **and** the diagram commutes.

(2) Since the family of linear sections of F A **and** the family of core sections of F A span

together F A , its leaves are the accessible sets of these two families of vector fields

([26, 27, 28], see [22] for a review of these results). Hence, two points a m **and** a n in

A are in the same leaf of F A if they can be joined by finitely many curves along flow

lines of core sections b ↑ for b ∈ Γ(F c ) **and** linear vector fields X ∈ Γ(F A ). By the

involutivity of F A , we have D X b ∈ Γ(F c ) for all b ∈ Γ(F c ) **and** linear vector fields

X ∈ Γ(F A ). Hence, by Lemma A.3, we get that F c is invariant under the flow lines

of linear vector fields with values in F A . That is, using the fact that φ X t is a vector

bundle homomorphism, we have

) ( )

(φ X t ◦ φ b↑

s (a m ) ∈ φ X t (a m + F c (m)) = φ X t (a m ) + F c φ ¯X

t (m)

for all a m ∈ A, t ∈ R where this makes sense **and** s ∈ R. Since

)

φ b↑

s ◦ φ X t (a m ) ∈ φ X t (a m ) + F c

(φ ¯X

t (m) ,

34 M. JOTZ AND C. ORTIZ

the proof is finished.

(3) Assume that a m ∼ a n for some elements a m , a n ∈ A. Then there exists, without

loss of generality, one linear vector field X ∈ Γ(F A ) over ¯X ∈ Γ(F M ), an element

b m ∈ F c (m) **and** t ∈ R such that a m = φ X t (a n ) + b m . We have then immediately

( ) (

m = q A (a m ) = q A φ

X

t (a n ) + b m = qA ◦ φ X )

t (an ) = φ ¯X

t (n),

which shows m ∼ FM n.

(4) Assume first that a does not vanish on its domain of definition. Since a is ∇-parallel,

we have D X a ∈ Γ(F c ) for all linear vector fields X ∈ Γ(F A ), X ∼ qA

¯X ∈ Γ(FM ). By

Lemma A.3, this yields

( ) ( )

(6.21) φ X t (a(m)) ∈ a φ ¯X

t (m) + F c φ ¯X

t (m)

for all t ∈ R where this makes sense **and** consequently

( ( ))

π (a(m)) = π a φ ¯X

t (m) .

Since F M is spanned by projections ¯X of linear vector fields X ∈ Γ(F A ), this shows

that a projects to the map [a].

In general, we have a = ∑ n

i=1 f ia i on an open set U with non-vanishing ∇-parallel

sections a 1 , . . . , a n of A such that a 1 , . . . , a r ∈ Γ(F c ) **and** functions f 1 , . . . , f n ∈

C ∞ (U) such that f r+1 , . . . , f n are F M -invariant. This yields using (6.21):

( n

)

∑

n∑

φ X t (a(m)) = φ X t f i (m)a i (m) = f i (m)φ X t (a i (m))

∈

n∑

i=r+1

i=1

i=1

) ( )

f i

(φ ¯X

t (m) φ X t (a i (m)) + F c φ ¯X

t (m)

**and** we get the statement in the same manner as above.

We have

( ) ( )

= a φ ¯X

t (m) + F c φ ¯X

t (m)

([q A ] ◦ [a]) ◦ π M = [q A ] ◦ π ◦ a = π M ◦ q A ◦ a = π M ◦ Id M = π M ,

which shows the last claim since π M is surjective.

Corollary 6.2. Let (A → M, F M , F c , ∇) be an IM-foliation on a Lie algebroid A. Choose

ā m **and** ā n in A/F c .

(1) Then π c (ā m ) = π c (ā n ) if **and** only if a m ∈ A/F c is the ∇-parallel transport of a n

over a piecewise smooth path along the foliation defined by F M on M.

(2) If ∇ has trivial holonomy, then π c (a m ) = π c (a ′ m) if **and** only if a m = a ′ m.

Proof.

(1) Assume first that π c (ā m ) = π c (ā n ). Then there exists without loss of generality

one linear vector field X ∈ Γ(F A ) over ¯X ∈ Γ(F M ) **and** t ∈ R such that

ā m = φ X t (a n ).

Consider the curve a : [0, t] → A/F c over c := φ ¯X· (n) defined by

a(τ) = φ X τ (a n )

for τ ∈ [0, t]. For each τ ∈ [0, t], we find ε τ > 0 **and** a parallel section a τ of A

such that a τ (c(τ)) = a(τ). As in the proof of Proposition 6.1, 4), we get then that

a τ (c(s)) = a(s) for s ∈ [τ − ε τ , τ + ε τ ]. This yields ∇ ¯X(c(τ)) a = 0 for all τ.

Conversely, assume that a m ∈ A/F c is the ∇-parallel transport of a n over a

piecewise smooth path along a path lying in the leaf of F M through n. Without loss

□

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 35

of generality, this path is a segment of a flow curve of a vector field ¯X ∈ Γ(F M ),

m = φ ¯X

t (n) for some t ∈ R, **and** there exists a ∇-parallel section a of A such that

a(m) = a m **and** a(n) = a n . Choose any linear vector field X ∈ Γ(F A ) over ¯X. Then

we get as in the proof of Proposition 6.1, 4) that

( )

a(m) = a φ ¯X

t (n) = φ X t (a(n))

**and** hence a m ∼ a n by Proposition 6.1, 2).

(2) This is immediate since here, parallel transport does not depend on the path along

the leaf of F M through m.

□

Using Proposition 6.1, we show that if M/F M is a smooth manifold **and** the projection is

a submersion, then the quotient A/F A is a vector bundle over M/F M .

Proposition 6.3. Let (A → M, F M , F c , ∇) be an IM-foliation on a Lie algebroid A. Assume

that M/F M is a smooth manifold such that the projection is a submersion, **and** that the

connection ∇ has no holonomy. The quotient space A/F A inherits a vector bundle structure

over ¯M such that the projection (π, π M ) is a vector bundle homomorphism.

A

π

A/F A

q A

M

[q A ]

πM

M/F M

Proof. Choose a local frame for A/F c of ∇-parallel sections ā 1 , . . . , ā k defined on U ⊆ M,

k = rank(A/F c ). Write q A/Fc : A/F c → M for the vector bundle projection **and** consider the

local trivialization of A/F c :

Φ :

q −1

A/F c

(U) → U × R k

b m ↦→ (m, ξ 1 (b m ), . . . , ξ k (b m )),

where b m = ∑ k

i=1 ξ i(b m )a i (m) + c m with some c m ∈ F c (m).

By Corollary 6.2, we find that for ¯b m , ¯b n ∈ q −1

A/F c

(U), the equality

π c

(

bm

)

= πc

(

bn

)

implies

ξ i (b m ) = ξ i (b n ) for i = 1, . . . , k,

since ¯b m is the parallel transport of ¯b n along any path in the leaf of F M through m **and** n,

**and** so in particular along a path in U. That is, the map Φ factors to a well-defined map

such that

−1

[Φ] : [q A ] (Ū) → Ū × Rk

[Φ] ◦ π c = (π M , Id R k) ◦ Φ.

The map [Φ] is the projection to A/F A of the “well chosen” local trivialization q A/Fc ×ξ 1 ×

. . . × ξ k of A/F c . Since by Proposition 2.4, we can cover A/F c by this type of F A -invariant

trivializations, we find that we can construct trivializations for A/F A , which is hence shown

to be a vector bundle over ¯M.

□

Remark 6.4. Corollary 6.2 implies that the quotient space π c : A/F c → A/F A is the

quotient by the equivalence relation given by parallel transport. Constructions like this were

made in [32], see also [14]. This idea will be used (implicitly) in the proofs of the following

statements.

36 M. JOTZ AND C. ORTIZ

Note that this shows also that the **data** (A, F M , F c , ∇) is an **infinitesimal** version of the

ideal systems as in [19], **and** the methods of construction of the quotient algebroid are similar

in this sense.

By construction, we have also the following vector bundle homomorphism

q A/Fc

π c

A/F c

A/F A

[q A ]

M πM

M/F M

which is an isomorphism in every fiber, **and** we find that for each local section [a] of A/F A

defined on Ū ⊆ ¯M, there exists a ∇-parallel section a of A defined on π −1

M

(Ū) such that

π ◦ a = [a] ◦ π M .

Proposition 6.5. Let (A, F M , F c , ∇) be an IM-foliation **and** assume that the quotient space

¯M = M/F M is a smooth manifold **and** ∇ has trivial holonomy. Then there is an induced

map [ρ] : A/F A → T ¯M such that

A

π

ρ

T M

A/F A

[ρ]

T ¯M

commutes.

If a ∈ Γ(A) is ∇-parallel, then ρ(a) ∈ X(M) is ∇ F M

-parallel **and** π M -related to [ρ][a] ∈

X( ¯M).

Proof. Define [ρ] : A/F A → T (M/F M ) by

T π M

[ρ]([a m ]) = T m π M (ρ(a m )) ∈ T [m] (M/F M ) ≃ T m M/F M (m).

To see that [ρ] is well-defined, recall first that ρ(F c ) ⊆ F M . If [a m ] = [a n ], then a m =

b m + φ X t (a n ) for some linear section X ∈ Γ(F A ) over ¯X ∈ Γ(F M ), t ∈ R **and** b m ∈ F c (m).

Consider the curve a : [0, t] → A/F c over c := φ ¯X· (n) defined by

a(τ) = φ X τ (a n )

for τ ∈ [0, t]. Then ∇ ¯X(c(τ)) a = 0 for all τ. For each τ ∈ [0, t], we find ε τ > 0 **and** a parallel

section a τ of A such that a τ (c(s)) = a(s) for s ∈ [τ −ε τ , τ +ε τ ]. Then, ρ◦a τ is ∇ F M

-parallel,

**and** we get by Lemma 2.5 that (ρ ◦ a τ ) ∼ πM Y τ for some Y τ ∈ X( ¯M).

Since [c(τ − ε τ )] = [c(s)] for all s ∈ [τ − ε τ , τ + ε τ ], we have then

T c(τ−ετ )π M (ρ(a(τ − ε τ ))) = T c(τ−ετ )π M (ρ(a τ (c(τ − ε τ ))))

= Y τ ([c(τ − ε τ )]) = Y τ ([c(s)]) = T c(s) π M (ρ(a(s)))

for all s ∈ (τ −ε τ , τ +ε τ ). Since [0, t] is covered by intervals like this, we get T m π M (ρ(a m )) =

T m π M (ρ(a(0))) = T n π M (ρ(a n )), which shows that [ρ] is well-defined.

□

Now we will define a Lie bracket on the space of sections of A/F A . For [a], [b] ∈ Γ(A/F A ),

choose ∇-parallel sections a, b ∈ Γ(A) such that [a] ◦ π M = π ◦ a **and** [b] ◦ π M = π ◦ b. Then

[a, b] is ∇-parallel by the properties of ∇ **and** we can define

[ ]

[a], [b] ∈ Γ(A/F A )

A/F A

by [ ]

[a], [b] ◦ π M = π ◦ [a, b].

A/F A

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 37

By the properties of ∇, this definition does not depend on the choice of the ∇-parallel

sections (which can be made up to sections of F c ), **and** by definition **and** with Proposition

6.5, we get

( [ ) [ ]

[ρ] [a], [b] = [ρ][a], [ρ][b] ,

]A/F A T ¯M

where the bracket on the right-h**and** side is the Lie bracket on vector fields on ¯M.

We can now complete the proof of the following theorem.

Theorem 6.6. Let (A, F M , F c , ∇) be an IM-foliation on a Lie algebroid A. Assume that

¯M = M/F M is a smooth manifold **and** ∇ has trivial holonomy. Then the triple

(A/F A , [ρ], [· , ·] A/FA ) is a Lie algebroid over ¯M such that the projection (π, π M ) is a Lie

algebroid morphism.

A

q A

M

π

A/F A

[q A ]

π

M/F M

Proof. The Jacobi identity follows immediately from the properties of the Lie bracket [· , ·]

on Γ M (A) **and** the definition of [· , ·] A/FA . For the Leibniz identity choose [a], [b] ∈ Γ(A/F A )

corresponding to ∇-parallel sections a, b ∈ Γ M (A), **and** f ∈ C ∞ ( ¯M). We have then πM ∗ f ∈

C ∞ (M) F M

**and** f · [b] corresponds to the ∇-parallel section (πM ∗ f) · b of A (see Lemma 2.5).

We have hence

[[a], f · [b]] A/FA ◦ π M = π ◦ [a, (π ∗ M f) · b]

= π ◦ (π ∗ M f · [a, b] + ρ(a)(π ∗ M f) · b)

= π ◦ (π ∗ M f · [a, b] + π ∗ M ([ρ][a](f)) · b)

= ( f · [[a], [b]] A/FA + [ρ][a](f) · [b] ) ◦ π M ,

where we have used Proposition 6.5 in the third equality.

The fact that (π, π M ) is compatible with the Lie algebroid brackets is immediate by

construction **and** the compatibility of the anchor maps is given by the definition of [ρ]. □

Assume now that F G ⊆ T G is a multiplicative foliation on G ⇒ M such that the leaf

space G/F G is a Lie groupoid over the leaf space M/F M (recall that there are topological

conditions for this to be true [13]). The multiplicative foliation F G determines an IM-foliation

(A, F M , F c , ∇) **and** a Lie algebroid (A/F A , [ρ], [·, ·] A/FA ) as in the preceding theorem. We

conclude this subsection with the comparison of this Lie algebroid with the Lie algebroid of

the leaf space G/F G ⇒ M/F M .

Theorem 6.7. Let (A, F M , F c , ∇) be an IM-foliation on A. Assume that A integrates to a

Lie groupoid G ⇒ M, **and** F A to a multiplicative foliation F G on G. If G/F G **and** M/F M

are smooth manifold, ∇ has trivial holonomy **and** F G is such that G/F G ⇒ M/F M is a Lie

groupoid, then we have

A(G/F G ) = A/F A ,

where A/F A is equipped with the Lie algebroid structure in the previous theorem.

Remark 6.8. It would be interesting to study the relation between the trivial holonomy

property of ∇ **and** the condition of F G for G/F G ⇒ M/F M to be a Lie groupoid.

Proof of Theorem 6.7. Let π G : G → G/F G be the projection, **and** [s], [t] the source **and**

target maps of G/F G ⇒ M/F M . Recall from Theorem 3.6 that a section a ∈ Γ(A) is ∇-

parallel if **and** only if [a r , Γ(F G )] ⊆ Γ(F G ) **and** the vector field a r is then π G -related to a

38 M. JOTZ AND C. ORTIZ

vector field a r ∈ X(G/F G ). We have

T [s] ◦ a r ◦ π G = T [s] ◦ T π G ◦ a r = T π G (T s ◦ a r ) = 0,

which shows that a r is tangent to the [s]-fibers. By Lemma 3.18 in [13], we get

a r ([g]) = T g π G (a r (g)) = T g π G (a(t(g)) ⋆ 0 g ) = T t(g) π G (a(t(g))) ⋆ 0 [g] = a r ([t][g]) ⋆ 0 [g] ,

which shows that a r = ã r for ã := a r | M/FM ∈ Γ(A(G/F G )).

Since (π G , π M ) is a Lie groupoid morphism

G

π G

t

M

s

G/F G

[t] [s]

πM

M/F M

the map A(π G ) = T π G | A

A A(πG) A(G/F G )

M πM

M/F M

is a Lie algebroid morphism **and** A(π G )(b m ) = 0 for all b m ∈ F c . For any ∇-parallel section

a ∈ Γ(A) we have a r ∼ πG ã r **and** hence

(6.22) A(π G ) ◦ a = T π G a = ã ◦ π M .

Define the map

A/F Ψ A

A(G/F G )

by

¯M

Id M

M/F M

Ψ([a m ]) = A(π G )(a m )

for all a m ∈ A. To see that this does not depend on the representative, note that F c =

ker(A(π G )) **and** recall that a m ∼ a n if **and** only if ā m is the ∇-parallel transport of ā n along

a path lying in the leaf through m of F M (Corollary 6.2). Without loss of generality, there

exists a ∇-parallel section a ∈ Γ(A) such that a(m) = a m **and** a(n) = a n + b n for some

b n ∈ F c (n). Then, using (6.22), we get

Ψ([a m ]) = A(π G )(a m ) = (A(π G ) ◦ a) (m) = (ã ◦ π M ) (m)

= (ã ◦ π M ) (n) = A(π G )(a n ) = Ψ([a n ]).

Hence, Ψ is a well-defined vector bundle homomorphism over the identity on M/F M . Furthermore,

the considerations above show that for any section [a] of A/F A , **and** corresponding

∇-parallel section a of A, we get

Ψ ◦ [a] = ã.

The compatibility of the Lie algebroid brackets **and** anchors is then immediate by the construction

of A/F A , **and** the fact that A(π G ) is a Lie algebroid morphism.

□

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 39

7. Further remarks **and** applications

7.1. Regular Dirac structures **and** the kernel of the associated presymplectic

**groupoids**. Let (M, D) be a Dirac manifold. Recall that (D → M, pr T M , [· , ·]) is then a Lie

algebroid, where [· , ·] is the Courant Dorfman bracket on sections of T M ⊕ T ∗ M.

Assume that the characteristic distribution F M ⊆ T M, defined by

F M (m) = {v m ∈ T m M | (v m , 0 m ) ∈ D(m)}

for all m ∈ M, is a subbundle of T M. The involutivity of F M follows from the properties of

the Dirac structure. Set F c := F M ⊕ {0} ⊆ D. It is easy to check that F c is a subalgebroid

of D. Define

∇ : Γ(F M ) × Γ (D/F c ) → Γ (D/F c )

∇ X

( ¯d

)

= [(X, 0), d].

This map is easily seen to be a well-defined, flat, partial F M -connection on D/F c , **and** the

verification of the fact that (D, F M , F c , ∇) is an IM-foliation on the Lie algebroid D → M is

straightforward.

We show that if D → M integrates to a presymplectic groupoid (G ⇒ M, ω G ) [6, 4], then

(D, F M , F c , ∇) integrates to the foliation F G = ker ω G ⊆ T G.

The map ρ := pr T M : D → T M is the anchor of the Dirac structure D viewed as a

Lie algebroid over M, **and** the map σ := pr T ∗ M : D → T ∗ M defines an IM-2-form on

the Lie algebroid D (see [6], [4]). Note that F c is the kernel of σ **and** F M is the kernel of

σ t : T M → D ∗ . The two-form Λ := σ ∗ ω can ∈ Ω 2 (D) is morphic in the sense that

T D Λ♯ T ∗ D

T M

−σ

t

D ∗

is a Lie algebroid morphism ([4]). See, for instance [19], for the Lie algebroid structure

on T ∗ D → D ∗ . If D → M integrates to a presymplectic groupoid (G ⇒ M, ω G ), the Lie

algebroid T ∗ D → D ∗ is isomorphic to the Lie algebroid of the cotangent groupoid T ∗ G → D ∗

**and** the map Λ ♯ integrates via the identifications T D ≃ A(T G) **and** T ∗ D ≃ A(T ∗ G) to the

vector bundle map ω ♯ G

, that is a Lie groupoid morphism. See [4] for more details.

We show that the morphic foliation F D ⊆ T D defined by (D, F M , F c , ∇) as in Theorem

5.5 is equal to the kernel of Λ ♯ .

Let n be the dimension of M **and** k the rank of F M . Then D is spanned locally by frames

of n parallel sections, the first k of them spanning F c . If d is a parallel section of D, we have

£ X d ∈ Γ(F c ) for all X ∈ Γ(F M ), that is, £ X (σ(d)) = 0 for all X ∈ Γ(F M ). Since i X σ(d) = 0

for all X ∈ Γ(F M ), this yields i X d(σ(d)) = 0 for all X ∈ Γ(F M ). Hence, using this type

of frames, we find with formulas (4.57) **and** (4.58) in [4], that the kernel of Λ ♯ is spanned

by the restriction to F M of the linear sections defined by parallel sections of D, **and** by the

restrictions to F M of the core sections defined by sections of F c . Hence, by construction, the

foliation F D is the kernel of Λ ♯ .

Since the kernel of ω ♯ G is multiplicative with Lie algebroid equal to the kernel of Λ♯ , this

yields F G = ker ω ♯ G .

Note that if F M ⊆ T M is simple, then the leaf space M/F M has a natural Poisson

structure such that the projection (M, D) → (M/F M , π) is a forward Dirac map. Under

a completeness condition **and** if F G ⊆ T G is also simple, we get a Lie groupoid G/F G ⇒

M/F M , with a natural symplectic structure ω such that the projection π G : G → G/F G

satisfies πG ∗ ω = ω G. It would be interesting to study the relation between the integrability

40 M. JOTZ AND C. ORTIZ

of the Poisson manifold (M/F M , π) **and** the completeness conditions on F G (see [13]) so that

the quotient (G/F G ⇒ M/F M , ω) is a symplectic groupoid.

7.2. **Foliated** algebroids in the sense of Vaisman. In [31], foliated Lie algebroids are

defined as follows. A foliated Lie algebroid is a Lie algebroid A → M together with a

subalgebroid B of A **and** an involutive subbundle F M ⊆ T M such that

(1) ρ(B) ⊆ F M ,

(2) A is locally spanned over C ∞ (M) by B-foliated cross sections, i.e., sections a of A

such that [a, b] ∈ Γ(B) for all b ∈ B.

Recall our definition of IM-foliation on a Lie algebroid (Definition 5.1). Since the F M -

partial connection is flat, we get by Proposition 2.4 the existence of frames of parallel sections

for A. By the properties of the connection, these are F c -foliated cross sections. Since (1)

is also satisfied by hypothesis, our IM-foliations are foliated Lie algebroids in the sense of

Vaisman if we set B := F c .

The object that integrates the foliated algebroid in the sense of Vaisman is the right

invariant image of B, which defines a foliation on G that is tangent to the s-fibers **and**

invariant under left multiplication. This is exactly the intersection of our multiplicative

subbundle F G ⊆ T G integrating (A → M, F M , F c , ∇) with T s G.

Appendix A. Invariance of bundles under flows

We prove here a result that is st**and**ard, but the proof of which is difficult to find in the

literature.

Theorem A.1. Let M be a smooth manifold **and** E be a subbundle of the direct sum vector

bundle TM := T M ⊕ T ∗ M. Let Z ∈ X(M) be a smooth vector field on M **and** denote its

flow by φ t . If

£ Z e ∈ Γ(E) for all e ∈ Γ(E),

then

φ ∗ t e ∈ Γ(E) for all e ∈ Γ(E) **and** t ∈ R where this makes sense.

Corollary A.2. Let F be a subbundle of the tangent bundle T M of a smooth manifold M.

Let Z ∈ X(M) be a smooth vector field on M **and** denote its flow by φ t . If

then

for all m ∈ M **and** t where this makes sense.

[Z, Γ(F )] ⊆ Γ(F ),

T m φ t F (m) = F (φ t (m))

Proof. Choose X ∈ Γ(F ) **and** m ∈ M. Then, by Theorem A.1, we have T φ t ◦ X ◦ φ −t =

φ ∗ −tX ∈ Γ(F ) for all t where this makes sense, **and** hence:

T m φ t X(m) = ( φ ∗ −tX ) (φ t (m)) ∈ F (φ t (m)).

Proof of Theorem A.1. The subbundle E of TM is an embedded submanifold of TM. For

each section σ of TM, the smooth function l σ : TM → R is defined by

l σ (v, α) = 〈σ(p(v, α)), (v, α)〉

for all (v, α) ∈ TM, where p : TM → M is the projection. For all e ∈ E, the tangent space

T e E of the submanifold E of TM is equal to

ker { d e l σ | σ ∈ Γ ( E ⊥)} .

□

FOLIATED GROUPOIDS AND THEIR INFINITESIMAL DATA 41

Consider the complete lift ˜Z to TM of Z, i.e., the vector field ˜Z ∈ X(TM) defined by

˜Z(l σ ) = l £Z σ **and** ˜Z(p ∗ f) = p ∗ (Z(f))

for all σ ∈ Γ(TM) **and** f ∈ C ∞ (M) (see [19]).

Choose e ∈ E **and** σ ∈ Γ ( E ⊥) . Then we have £ Z σ ∈ Γ ( E ⊥) since for all τ ∈ Γ(E):

〈£ Z σ, τ〉 = Z (〈σ, τ〉) − 〈σ, £ Z τ〉 = 0.

This leads to

(d e l σ )( ˜Z(e))

( )

= ˜Z(lσ ) (e) = l £Z σ(e) = 0.

Hence, the vector field ˜Z is tangent to E on E. As a consequence, its flow curves starting

at points of e remain in the submanifold E.

It is easy to check that the flow Φ t of the vector field ˜Z is equal to (T φ t , (φ −t ) ∗ ), i.e.,

Φ t (v m , α m ) = (T m φ t (v m ), α m ◦ T φt(m)φ −t )

for all (v m , α m ) ∈ TM(m). Choose a section (X, α) ∈ Γ(E) **and** a point m ∈ M. We find

(φ ∗ t (X, α))(m) = ( T φt(m)φ −t X(φ t (m)), α φt(m) ◦ T m φ t

)

= Φ−t ((X, α)(φ t (m))) ∈ E(m)

since (X, α)(φ t (m)) ∈ E(φ t (m)). Thus, we have shown that φ ∗ t (X, α) is a section of E.

□

Assume now that q A : A → M is a vector bundle, **and** consider a linear vector field X on

A, i.e., the map X : A → T A is a vector bundle homomorphism over ¯X : M → T M such

that X ∼ qA

¯X. Let φ

X· be the flow of X **and** φ ¯X· the flow of ¯X. Then φ

X

t : A → A is a

vector bundle homomorphism over φ ¯X

t for all t ∈ R where this is defined.

Recall that for any a ∈ Γ(A), the section D X a ∈ Γ(A) is defined by

(D X a)(m) = d dt⏐ φ X −t(a(φ ¯X

t (m)))

t=0

for all m ∈ M. In the same manner, if ϕ ∈ Γ(A ∗ ), we can define

(D X ϕ)(m) = d dt⏐ (φ X t ) ∗ (ϕ(φ ¯X

t (m)))

t=0

for all m ∈ M. We have then ϕ(a) ∈ C ∞ (M), **and**

(1.23) ¯X(m)(ϕ(a)) = ϕ(DX a)(m) + (D X ϕ)(a)(m).

We can hence show the following lemma.

Lemma A.3. Let A be a vector bundle **and** B ⊆ A a subbundle.

(1) If (X, ¯X) is a linear vector field on A such that

D X b ∈ Γ(B)

( )

for all b ∈ Γ(B), then φ X t (b m ) ∈ B φ ¯X

t (m) for all b m ∈ B m .

(2) Assume furthermore that a ∈ Γ(A) is such that a(m) is linearly independent to B(m)

for all m in Dom(a) **and**

D X a ∈ Γ(B).

Then

( ) ( )

φ X t (a(m)) ∈ a φ ¯X

t (m) + B φ ¯X

t (m)

for all m ∈ U **and** t ∈ R where this makes sense.

42 M. JOTZ AND C. ORTIZ

Proof.

(1) We check that the vector field X is tangent to B on points in B. Let ϕ ∈

Γ(A ∗ ) be a section that vanishes on B, i.e., ϕ m (b m ) = 0 for all b m ∈ B. Let l ϕ ∈

C ∞ (A) be the linear function defined by ϕ. By (1.23), we have then D X ϕ ∈ Γ(B ◦ ).

Choose b m ∈ B. We have then

d bm l ϕ (X(b m )) = d dt⏐ l ϕ (φ X t (b m )) = d t=0

dt⏐ ϕ φ ¯Xt (m) (φX t (b m ))

t=0

= (D X ϕ)(b m ) = 0.

Thus, X is tangent to B on B **and** the flow of X preserves B.

(2) Assume now that (b 1 , . . . , b k ) is a local frame for B on an open set U ⊆ M. Complete

this frame to a local frame (b 1 , . . . , b n ) for A defined on an open U such that b k+1 :=

a ∈ Γ(A). Let ϕ 1 , . . . , ϕ n be a frame for A ∗ that is dual to (b 1 , . . . , b n ), i.e., such that

(ϕ k+1 , . . . , ϕ n ) is a frame for B ◦ **and** ϕ k+1 (a) = 1. Then, the closed submanifold C

of A| U defined by C(m) = a(m) + B(m) is the level set with value (1, 0, . . . , 0) of the

function

(l ϕk+1 , . . . , l ϕn ) : A| U → R n−k .

Since D X a ∈ Γ(B) for the linear vector field (X, ¯X) on A, we get

0 = ¯X(ϕ i (a)) = ϕ i (D X a) + D X ϕ i (a) = 0 + D X ϕ i (a)

for i = k + 1, . . . , n **and** this yields as before for all b m ∈ B:

d a(m)+bm l ϕi (X(a(m) + b m )) = d dt⏐ l ϕi (φ X t (a(m) + b m ))

t=0

= (D X ϕ i )(a(m) + b m ) = 0.

Hence, X is tangent to C on points of C. That is, the flow of X preserves C **and**

the proof is finished.

□

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Georg-August-Universität Göttingen, **Mathematisches** Institut, Bunsenstr. 3-5, 37073 Göttingen,

Germany

E-mail address: mjotz@uni-math.gwdg.de

Departamento de Matemática, Universidade Federal do Paraná, Setor de Ciencias Exatas -

Centro Politecnico, 81531-990 Curitiba, Brazil

E-mail address: cristian.ortiz@ufpr.br