Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

A Lie groupoid, called \textit{material Lie groupoid}, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called \textit{material algebroid}, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of $G-$structures is discussed in detail. An illustrative example is presented as an application of the theory.


Introduction
The goal of this paper is to show how a Lie groupoid (called material groupoid ) can be naturally associated to any uniform material. This association is indeed one of the most intuitives in Continuum Mechanics.
In fact, given two different points X and Y of the material body B, a material uniformity is just a linear isomorphism P XY : T X B → T Y B such that the mechanical response at X and Y is the same; more precisely, if W is the response functional depending on the gradient deformation F at any point X of B, then, for any F . This notion can be expressed in terms of 1−jets in a more geometric way where P = j 1 X,Y φ for all infinitesimal deformation j 1 y,κ(y) κ. The collection of all material uniformities for all pair of points of B is just a groupoid called the material groupoid, and denoted by Ω (B).
The material groupoid Ω (B) encodes the information about the material symmetries of the body B and let us allow to have a more general framework in terms of G−structures.
The other crucial point about this material groupoid is just the associated Lie algebroid, AΩ (B), which is the infinitesimal version of Ω (B).
Since the material groupoid encodes the mechanical geometric information of B, then its homogeneity can be characterized through the properties of AΩ (B). This is indeed accomplished, and related with the earlier approach developed in [15] in the framework of G−structures.
Let us metion some particularities about this paper. Our intention is that it is selfcontained as far as possible. In consequence, we have included material on Lie groupoids and Lie algebroids which one can find in a few specialized books (for instance, K. MacKenzie [31]).
However, the reader should be disposed to spend a lot of time reading and learning all those concepts. Under our scheme, the reader could have the most relevant information on this subject in a few minutes.
Of course, we always refer to other sources in case the reader wants to have more details and additional information.
Groupoids are an specific class of categories and it were studied by H. Brandt in 1926 [5]. In that paper, the author deals with the composition of quadratic forms in four variables, where this kind of mathematical objects were named for the first time. The notion of Lie groupoid was introduced by Ch. Ehresmann in a series of articles [11,12,13,14] in which topological and differential structures were added to the groupoids in order to use them as tools in differential geometry, in particular in fiber bundle theory and connections. J. Pradines redefined the concept of Lie groupoid in [41].
Although groupoids (and, in particular Lie groupoids) have a long history, their use has been only recently introduced in mechanics. In 1991, Alan Weinstein [42] (see also [48]) suggested that Lie groupoids and Lie algebroids have an important role in Hamiltonian mechanics. Indeed, the use of the so-called banal or pair groupoid is in the base of the construction of geometric integrators (see for example [21]).
On the other hand, as we can see in the present paper, their use in continuum mechanics is crucial. However, there is some reluctance to introduce new concepts in a particularly well-established field like continuum mechanics. But we are confident that the present paper could contribute to a better understanding of the role of Lie groupoids in the discipline.
The paper can be divided in two parts. The firt one can be seen as an introduction to the fundamental mathematical objects which we are going to use trying to be much selfcontained as possible.
The first section is about principal bundles and here, we will introduce the notion of G−structure and, as a particular case, integrable G−structures. In the next section we will introduce Lie groupoids. A particular groupoid, the 1-jets groupoid, is going to be closely related with the frame bundle. This groupoid has been studied in [31]. However, due its importance for continuum mechanics it deserves a detailed study in this paper. So, its properties and, in particular, its subgroupoids are carefully studied. The third section is devoted to study the Lie algebroids which are infinitesimal versions of Lie groupoids. Next, we will generalize the construction of the Lie algebra of a Lie group. Thus, we will be able to construct an associated Lie algebroid for each Lie groupoid. As an important case of this construction is just the 1-jets algebroid which will be studied in detail because of its relation with the 1−jets groupoid. Using this construction, in the next section we can define the exponential map of a section of an associated Lie algebroid to a Lie groupoid. This open us a way to construct a Lie algebroid isomorphism from the 1−jets Lie algebroid to the Lie algebroid of derivations which is the focus of section 6. Finally, in section 7 we will present briefly the Lie theory for Lie groupoids and Lie algebroids which will be important for the rest of this paper.
The second part of this paper is devoted to study the uniformity and homogeneity of a simple material. First, we will introduce some definitions using just the material groupoid. Then, we will present a notion of integrability in the 1−jets Lie groupoid which is related with the notion of integrability of G−structures. We will extend this notion to the 1−jets Lie algebroid. We will finish using the Lie algebroid of derivations to characterize this notion. In section 10 we will present the notion of homogeneity introduced in [15] (see also [16] or [47]) where the authors have used G−structures to characterize this property. Finally, we will prove that both definitions are equivalent.

Principal bundles
Firstly, we are going to present the classical notion of principal bundle and, as an example, we will introduce the concepts of frame bundle and G−structure (we remit to [2], [7], [22], [24], [43] and [8] for a detailed exposition about principal bundles, frame bundles and G−structures). Finally, we will introduce the notion of integrability.
Definition 2.1. Let P be an n−manifold and G be a Lie group which acts over P by the right satisfying: (i) The action of G is free, i.e., where e ∈ G is the identity of G.
Φ is called a trivialization on U.
A principal bundle will be denoted by P (M, G), or simply π : P → M if there is no ambiguity as to the structure group G. P is called the total space, M is the base space, G is the structure group and π is the projection. The closed submanifold π −1 (x), x ∈ M will be called the fibre over x. For each point u ∈ P , we have π −1 (x) uG, where π (u) = x, and uG will be called the fibre through u. Every fibre is diffeomorphic to G, but this diffeomorphism depends on the choice of the trivialization. Now, we want to define the morphism of this category.
Definition 2.2. Given P (M, G) and P ′ (M ′ , G ′ ) principal bundles, a principal bundle morphism from P (M, G) to P ′ (M ′ , G ′ ) consists of a differentiable map Φ : P → P ′ and a Lie group homomorphism ϕ : G → G ′ such that Notice that, in this case, Φ maps fibres into fibres and it induces a differentiable map φ : M → M ′ by the equality φ (x) = π (Φ (u)), where u ∈ π −1 (x).
If these maps are embeddings, the principal bundle morphism will be called embedding. In such a case, we can identify P with Φ (P ), G with ϕ (G) and M with φ (M) and P (M, G) is said to be a subbundle of P ′ (M ′ , G ′ ). Furthermore, if M = M ′ and ϕ = Id M , P (M, G) is called a reduced subbundle and we also say that G ′ reduces to the subgroup G.
As usual, a principal bundle morphism is called isomorphism if it can be inverted by another principal bundle morphism. Example 2.3. Let M be an n−dimensional manifold and G be a Lie group, then we can consider M × G as a principal bundle over M with projection pr 1 : M × G → M and structure group G. The action considered here is given by, This principal bundle is called a trivial principal bundle. Now, we will introduce an important example of principal bundle, the frame bundle of a manifold. In order to do that, we will start presenting the following definition. Remark 2.5. Alternatively, a linear frame at x can be viewed as a linear isomorphism z : R n → T x M identifying a basis on T x M as the image of the canonical basis of R n by z.
There is a third way to interpret a linear frame by using the theory of jets. Indeed, a linear frame z at x ∈ M may be considered as the 1-jet j 1 0,x φ of a local diffeomorphism φ from an open neighbourhood of 0 in R n onto an open neighbourhood of x in M such that φ (0) = x. So, z = T 0 φ. Thus, we denote by F M the set of all linear frames at all the points of M. We can view F M as a principal bundle over M with the structure group Gl (n, R) and projection π M : F M → M given by This principal bundle is called the linear frame bundle of M. Let (x i ) be a local coordinate system on an open set U ⊆ M. Then we can introduce local coordinates (2.1) , is a reduced subbundle of F M with structure group G, a Lie subgroup of Gl (n, R).
Now, we shall introduce the notion of integrability of a G−structure.
Note that there exists a principal bundle isomorphism l : F R n → R n × Gl (n, R) over the identity map such that Indeed, we can consider the global section, x τ x where τ x denote the translation on R n by the vector x. So, an 1−jet j 1 0,x φ can be written in a unique way as a composition of s (x) and a matrix of Gl (n, R).
We have thus obtained a principal isomorphism F R n ∼ = R n ×Gl (n, R) over the identity map on R n . Then, if G is a Lie subgroup of Gl (n, R), we can transport R n × G by this isomorphism to obtain a G−structure on R n . These kind of G−structures will be called standard flat on R n . Definition 2.7. Let ω G (M) be a G−structure over M. ω G (M) is said to be integrable if it is locally isomorphic to the trivial principal bundle R n × G.
Any {e}−structure on M, with e the identity of G, will be called parallelism of M. It is easy to show that any trivial structure is, indeed, a global section of F M. So, we will speak about integrable sections and, using Eq. (2.1) we have that (locally) any integrable sections can be written as follows These results about integrability of G−structures can be found in [22].

Groupoids
Now, we want to introduce the notion of Lie groupoid which is going to be a fundamental concept for the rest of this paper (a good reference on groupoid is [31]). A particular groupoid, the 1-jets groupoid, is going to be closely related with the frame bundle.
If Γ is a groupoid over M, then M is also denoted by Γ (0) and it is often identified with the set ǫ (M) of identity elements of Γ. Γ is also denoted by Γ (1) . The space of sections of the map (α, β) : Γ → M × M is denoted by Γ (α,β) (Γ). Now, we define the morphisms of the category of groupoids.
Definition 3.2. If Γ 1 ⇒ M 1 and Γ 2 ⇒ M 2 are two groupoids then a morphism from Γ 1 ⇒ M 1 to Γ 2 ⇒ M 2 consists of two maps Φ : Γ 1 → Γ 2 and φ : M 1 → M 2 such that for any g 1 ∈ Γ 1 where α i and β i are the source and the target map of Γ i ⇒ M i respectively, for i = 1, 2, and preserves the composition, i.e., We will denote this morphism as (Φ, φ).
Observe that, as a consequence, Φ preserves the identities, i.e., denoting by ǫ i the section of identities of Γ i ⇒ M i for i = 1, 2, Note that, using equations 3.1, φ is completely determined by Φ.
Using this definition we define a subgroupoid of a groupoid Γ ⇒ M as a groupoid Γ ′ ⇒ M ′ such that M ′ ⊆ M, Γ ′ ⊆ Γ and the inclusion map is a morphism of groupoids.

Remark 3.3.
There is a more abstract way of defining a groupoid. We can say that a groupoid is a "small" category (the class of objects and the class of morphisms are sets) in which each morphism is invertible.
If Γ ⇒ M is the groupoid, then M is the set of objects and Γ is the set of morphisms.
A groupoid morphism is a functor between these categories which is a more natural definition. Now, we present some basic examples of groupoids.
Example 3.4. A group is a groupoid over a point. In fact, let G be a group and e the identity element of G. Then, G ⇒ {e} is a groupoid, where the operation of the groupoid, ·, is the operation in G.
Example 3.5. Any set X may be regarded as a groupoid on itself with α = β = ǫ = i = Id X and the operation on this groupoid is given by Note that, in this case, X (2) =△ X . We call this kind of groupoids as base groupoids and we will denote them as ǫ (X). Example 3.6. For any set A and any map π : A → M, we can consider the pullback space A × π,π A according to the diagram Then the maps, endow A × π,π A with a structure of groupoid over A, called the pair groupoid along π. If π = Id A then this groupoid is called the pair groupoid.
Note that, if Γ ⇒ M is an arbitrary groupoid over M, then the map (α, β) : Γ → M × M, which is sometimes called the anchor of Γ, is a morphism from Γ ⇒ M to the pair groupoid of M.
Example 3.7. Let A be a vector bundle over a manifold M. Let Φ (A) denote the set of all vector space isomorphisms L x,y : A x → A y for x, y ∈ M, where for each z ∈ M, A z is the fibre of A over z. We can consider Φ (A) as a groupoid Φ (A) ⇒ M such that, for all x, y ∈ M and L x,y ∈ Φ (A), This groupoid is called the frame groupoid on A. As a particular case, when A is the tangent bundle over M we have the 1-jets groupoid on M which is denoted by Π 1 (M, M). Now, we are going to generalize the notions of orbit and isotropy group of an action.
is called the isotropy group of Γ at x. The set is called the orbit of x, or the orbit of Γ through x.
If O (x) = {x}, or equivalently, β −1 (x) = α −1 (x) = Γ x then x is called a fixed point. The orbit space of Γ is the space of orbits of Γ on M, i.e., the quotient space of M by the equivalence relation induced by Γ: two points of M are equivalent if and only if they lie on the same orbit.
If O (x) = M for all x ∈ M, or equivalently (α, β) : Γ → M × M is a surjective map, the groupoid Γ ⇒ M is called transitive. If every x ∈ M is fixed point, then the groupoid Γ ⇒ M is called totally intransitive. Furthermore, a subset N of M is called invariant if it is a union of some orbits.
Finally, the preimages of the source map α of a Lie groupoid are called α−fibres. Those of the target map β are called β−fibres.
Note that, (3.2) So, for all g ∈ Γ, the right (left) translation on g, R g (resp. L g ), is a bijective map with inverse R i(g) (resp. L i(g) ), where i : Γ → Γ is the inverse map.
One may impose various topological and geometrical structures on a groupoid, depending on the context. We will be mainly interested in Lie groupoids. Definition 3.10. A Lie groupoid is a groupoid Γ ⇒ M such that Γ is a smooth manifold, M is a smooth manifold and all the structure maps are smooth. Furthermore, the source and the target map are submersions. A Lie groupoid morphism is a groupoid morphism which is differentiable. Observe that, taking into account that α • ǫ = Id M = β • ǫ, then ǫ is an injective immersion.
On the other hand, in the case of a Lie groupoid, R g (resp. L g ) is clearly a diffeomorphism for all g ∈ Γ.
Note also that, for each k ∈ N, Γ (k) is a pullback space given by β and the operation map on Γ (k−1) . Thus, by induction, we may prove that Γ (k) is a smooth manifold for all k ∈ N.  Example 3.14. Let π : A → M be a submersion. It is trivial to prove that the pair groupoid along π is a Lie groupoid. Example 3.16. Let π : P → M be a principal bundle with structure group G. Denote by φ : P × G → P the action of G on P . Now, suppose that Γ ⇒ P is a Lie groupoid, with φ : Γ × G → Γ a free and proper action of G on Γ such that, for each g ∈ G, the pair φ g , φ g is an isomorphism of Lie groupoids. We can construct a Lie groupoid Γ/G ⇒ M such that the source map, α, and the target map, β, are given by β ([g]) = π (β (g)) , α ([g]) = π (α (g)) , for all g ∈ Γ, α and β being the source and the target map on Γ ⇒ P , respectively, and [·] denotes the equivalence class in the quotient space Γ/G. These kind of Lie groupoids are called quotient Lie groupoids by the action of a Lie group.
There is an interesting particular case of the above example.
Example 3.17. Let π : P → M be a principal bundle with structure group G and P × P ⇒ P the pair groupoid. Take φ : (P × P ) × G → P × P the diagonal action of φ, where φ : P × G → P is the action of G on P .
Then it is easy to prove that φ g , φ g is an isomorphism of Lie groupoids and thus, we may construct the groupoid (P × P ) /G ⇒ M. This groupoid is called gauge groupoid and is denoted by Gauge (P ). Example 3.18. Let A be a vector bundle over M then the frame groupoid is a Lie groupoid (see Example 3.7). In fact, let (x i ) and (y j ) be local coordinate systems on open sets U, V ⊆ M and {α p } and {β q } be local basis of sections of A U and A V respectively. The corresponding local coordinates (x i • π, α p ) and (y j • π, β q ) on A U and A V are given by Then, we can consider a local coordinate system Φ (A) • y j (L x,y ) = y j (y).
• y j i (L x,y ) = A Lx,y , where A Lx,y is the induced matrix of the induced map of L x,y by the local coordinates (x i • π, α p ) and (y j • π, β q ).
In particular, if A = T M, then the 1−jets groupoid on M, Π 1 (M, M), is a Lie groupoid and its local coordinates will be denoted as follows where, for each j 1

Lie Algebroids
The notion of Lie algebroid was introduced by J. Pradines in 1966 [41] as an infinitesimal version of Lie groupoid and for this reason the author called it infinitesimal groupoid. Now, we want to recall this concept (we also refer to [31]).
A is transitive if ♯ is surjective and totally intransitive if ♯ ≡ 0. Also, A is said to be regular if ♯ has constant rank.
Looking at ♯ as a C ∞ (M)-module morphism from Γ (A) to X (M), for each section α ∈ Γ (A) we are going to denote ♯ (α) by α ♯ . Next, let us prove the following fundamental property: is a Lie algebroid, then the anchor map is a morphism of Lie algebras, i.e.
Proof. Let α, β ∈ Γ (A). By the Jacobi identity, for any section γ ∈ Γ (A) and any function f ∈ C ∞ (M), we have Now, using the Leibniz rule, If we replace these equalities in Eq. (4.3), we have for any γ ∈ Γ (A) and for any f ∈ C ∞ (M). Thus, we conclude that An important remark is that the Lie algebra structure on sections is of local type i.e. [α, β] (x) will depend on β (therefore, on α too) around x only, ∀x ∈ M.
As a consequence, the restriction of a Lie algebroid over M to a open subset of M is again a Lie algebroid. Taking local coordinates (x i ) on M and a local basis of sections of A, {α p }, the corresponding local Such coordinates determine local functions ♯ i p , C r pq on M which contain the local information of the Lie algebroid structure, and accordingly they are called the structure functions of the Lie algebroid. They are given by Imposing Eq. (4.2) and the Jacobi identity over the local basis {α p }, we get the following equations for all i, p, q, where ijk a ijk means the cyclic sum a ijk + a kij + a jki . These equations are usually called structure equations.

Now, we will give some examples of Lie algebroids
Example 4.4. Any Lie algebra is a Lie algebroid over a single point. Indeed, identifying Γ (g) with g, the Lie bracket on sections is simply the Lie algebra bracket and the anchor map is the trivial one.
This kind of Lie algebroid is a particular case of the following example.
Then, the Lie bracket on Γ (A) is a point-wise Lie bracket, that is, the restriction of [·, ·] to the fibres induces a Lie algebra structure on each of them. More precisely, using that ♯ ≡ 0, the Leibniz rule is just Consider x ∈ M and β, β ∈ Γ (A) such that Let {α 1 , . . . , α k } be a basis of local sections. Then, around x, we have . Finally, skew-symmetry allows us to prove that the value of [α, β] in a point x ∈ M depends only on α (x) and β (x). These kind of Lie algebroids (with ♯ ≡ 0) are called Lie algebra bundles. Note that the Lie algebra structures on the fibres are not necessary isomorphic to each other.  (ii) Lie algebra structure over the space of sections is given by: This Lie algebroid is called the Trivial Lie algebroid on M with structure algebra g. Note that, since F is regular, T F is a subbundle of T M, and its sections are vector fields tangent to F. Moreover, T F being regular and integrable, implies that it is involutive and, as a consequence, the Lie bracket of two vector fields tangent to F is again a vector field tangent to F. Example 4.9. Let τ : P → M be a principal bundle with structure group G. Denote by φ : G×P → P the action of G on P . Now, suppose that (A → P, ♯, [·, ·]) is a Lie algebroid, with vector bundle projection π : A → P and that φ : G × A → A is an action of G on A such that π is a vector bundle action under the action φ where for each g ∈ G, the pair φ g , φ g satisfies that . This fact will be equivalent to the fact of that φ g , φ g is a Lie algebroid isomorphism. Let π : A/G → M be the quotient vector bundle of π by the action of G. Then, we are going to construct a Lie algebroid structure on π.
Denote by τ : A → A/G the quotient projection. Then, we may define the anchor map ♯ : and therefore i.e., ♯ is well defined. Furthermore, by construction So, using that τ is a submersion, the anchor is a smooth map. Finally, it is trivial that ♯ is a vector bundle morphism. On the other hand, for each α, β ∈ Γ (A) G and for each g ∈ G i.e. [α, β] ∈ Γ (A) G . As a consequence, the Lie bracket on Γ (A) restricts to Γ (A) G ∼ = Γ (A/G) and then, this structure induces a Lie algebra structure on Γ (A/G). Finally, it is easy to prove that the Leibniz identity is satisfied. This kind of Lie algebroids are called quotient Lie algebroids by the action of a Lie group.
A particular but interesting example of this construction is obtained when we consider the tangent lift of a free and proper action of a Lie group on a manifold. Example 4.10. Let π : P → M be a principal bundle with structure group G. Denote by φ the action of G on P . Let (T P → P, Id T P , [·, ·]) be the tangent algebroid and φ T : G × T P → T P be the tangent lift of φ.
Then, φ T satisfies the conditions of Example 4.9. Thus, one may consider the quotient Lie algebroid T P/G → M, ♯, [·, ·] by the action of G. This algebroid is called the Atiyah algebroid associated with the principal bundle π : P → M.
Note that, as we have seen, the space of sections can be considered as the space of invariant vector field by the action φ over M.
Next, we introduce the definition of a morphism in the category of Lie algebroids. The main problem is that a morphism between vector bundles does not, in general, induce a map between the modules of sections, so it is not immediately clear what should be meant by bracket relation. We will give a direct definition in terms of (Φ, φ) −decompositons of sections which is easy to use, and is amenable to categorical methods.
Let Φ : A ′ → A, φ : M ′ → M be a vector bundle morphism between π : A → M and π ′ : Thus, we are ready to give the definiton of Lie algebroid morphism.
and such that for arbitrary (4. 6) In fact, the right-hand side of Eq. (4.6) is independent of the choice of the (Φ, φ) −decompositions of α ′ and β ′ .
It is easy to prove that the composition preserves Lie agebroid morphisms and, hence, we can define the category of Lie algebroids.
On the other hand, if M = M ′ and φ = Id M then Eq. (4.6) reduces Next, we are going to introduce the notion of Lie subalgebroid.
equipped with a Lie algebroid structure such that the inclusion is a morphism of Lie algebroids. A reduced subalgebroid of A is a transitive Lie subalgebroid with M as the base manifold.
Remark 4.14. Suppose that M ′ ⊆ M is a closed submanifold then, using the (i A ′ , i M ′ ) − decomposition and extending functions, it satisfies that for all α ′ ∈ Γ (A ′ ) there exists α ∈ Γ (A) such that So, Eq. (4.6) reduces to

Construction of the Lie algebroid
Now, we are going to generalize the construction of the Lie algebra of a Lie group. As an important case of this construction is just the 1-jets algebroid.
In order to do that we should generalize the notion of right-invariant vector fields.
Denote the space of smooth right-invariant vector fields on Γ by X R (Γ).
Similarly to the case of Lie groups, it is clear that the Lie bracket of two right-invariant vector fields is again a right-invariant vector field, say On the other hand, T α has constant rank. Thus, we may define the vector subbundle of T Γ given by Let ǫ : M → Γ be the section of identities. We define the pullback vector bundle on M, where τ Γ : T Γ → Γ is the tangent bundle projection on Γ and M × ǫ,τ Γ Ker (T α) is the pullback space according to the following diagram This vector bundle will be denoted by π ǫ : AΓ → M.
Note that the sections of AΓ are determined by smooth maps Λ : The set of these maps will be denoted by Γ ǫ (Γ).
Thus, for each map Λ ∈ Γ ǫ (Γ) we can define the right-invariant vector field on Γ given by i.e., X Λ is determined by the following equality This construction is a natural extension of the Lie structure in the associated Lie algebra of a Lie group. In that case, Let G be a Lie group and ξ be an element of T e G. Then, we constructed the associated rightinvariant vector field by the equality X ξ (e) = ξ.
Using this equality T e G is endowed with a Lie algebra structure.
Finally, an anchor map can be defined as follows: identify C ∞ (M) with the space C ∞ R (Γ) of right-invariant functions on Γ using the map given by Φ : . In this way, we are going to construct the anchor map as follows: let Λ be a section of Γ (AΓ); then for each f ∈ C ∞ (M) we define . Furthermore, its inherits the Leibniz rule from X α and so, ♯ is well-defined.
It is easy to prove that for all Λ ∈ Γ ǫ (Γ) and x ∈ M. In fact, Therefore, ♯ is a vector bundle morphism and it satisfies the Leibniz rule. So, (AΓ → M, ♯, [·, ·]) is a Lie algebroid, called the Lie algebroid of the Lie groupoid Γ ⇒ M, and sometimes denoted by AΓ.
Remark 5.2. Let Γ ⇒ M be a Lie groupoid. For any x ∈ M, the associated Lie algebra to the isotropy Lie group Γ x , A (Γ x ) is isomorphic to the isotropy Lie algebra through x, i.e., There is a natural functor from the category of Lie groupoids to the category of Lie algebroids.
Proof. We will give an sketch of the proof (a detailed proof can be found in [23]).
We already have given the definition of the correspondence between objects (Γ ⇒ M → AΓ) and we will obtain the correspondence between morphisms. Let This morphism induced by a morphism (Φ, φ) of Lie groupoids over the associated Lie algebroids will be denoted by AΦ. Now, we are going to give some examples of the above general construction.  Example 5.6. Let π : P → M be a principal bundle with structure group G. Denote by φ : P × G → P the action of G on P . Now, suppose that Γ ⇒ P is a Lie groupoid, with φ : Γ × G → Γ a free and proper action of G on Γ such that, for each g ∈ G, the pair φ g , φ g is an isomorphism of Lie groupoids. So, we may construct the quotient Lie groupoid by the action of a Lie group, Γ/G ⇒ M (see Example 3.16).
Then, by construction, we may identify A (Γ/G) with the quotient Lie algebroids by the action of a Lie group, AΓ/G (see Example 4.9).
As a particular case, we may give the following interesting example.
Example 5.7. Let π : P → M be a principal bundle with structure group G and Gauge (P ) be the Gauge groupoid (see Example 3.17). Then, the associated Lie algebroid to Gauge (P ) is the Atiyah algebroid associated with the principal bundle π : P → M (see Example 4.10).  Note that, if U ⊆ Γ is an open reduced Lie subgroupoid of Γ ⇒ M, then it is clear that AU and AΓ are isomorphic.

The exponential map
In this section, we will extend the notion of exponential map for Lie groups to this general context of Lie groupoids and algebroids.
As a first step, we have to introduce the notion of bisection (and local bisection) of a groupoid. A local bisection of Γ ⇒ M is a local section Λ : U → Γ of the source map α, defined on an open subset U of M, such that β • Λ is an embedding. We say that Λ is a local bisection of Γ ⇒ M through g ∈ Γ if g ∈ Λ (U).
The set of bisections of Γ ⇒ M is denoted Ω (Γ) and the set of local bisections on U is denoted by Ω (U). It is easy to see that there exist local bisections through any arrow of Γ ⇒ M.
We may define the following operation in the set of bisections: Let be Λ, Θ ∈ Ω (Γ), then Thus, (Ω (Γ) , * ) is a group with the section of identities ǫ : M → Γ as identity. Furthermore, for each Λ ∈ Ω (Γ) the inverse element is given by Now, given Λ : U → Γ a local bisection of a Lie groupoid Γ ⇒ M, we may define the diffeomorphism L Λ : So, the map Λ → L Λ is an isomorphism of groups. Analogously, we may define the diffeomorphism Indeed, the inverse map is given by . The diffeomorphisms L Λ and R Λ are called local left translation induced by Λ and local right translation induced by Λ, respectively.
Finally, we can define the diffeomorphism, The following result will be used to define the exponential map.
(iii) For all (ξ, ν) ∈ Γ (2) such that ξ ∈ U and ξ · ν ∈ U, Then, Φ is the restriction to U of a unique local left translation L Λ associated to the local bisection of Γ ⇒ M α given by for each x ∈ U and ξ ∈ U ∩ β −1 (x).
Let Γ ⇒ M be a Lie groupoid and Λ ∈ Γ (AΓ) be a section of AΓ. We consider the right-invariant vector field associated to Λ, X Λ ∈ X R (Γ), {ϕ t : U t → U −t } the local flow of X Λ and {ψ t : U t → U −t } the local flow of Λ ♯ . Then for all t we have So, taking into account that X Λ is a right-invariant vector field, ϕ t : U t → U −t and ψ t : U t → U −t satisfy the conditions of proposition 6.2 and then, each ϕ t is the restriction of the unique local left translation Observe that, Therefore, we have the following result.
where Exp t Λ is a local bisection on U t , such that (iii) For all |t|, |s|, |t + s| < δ, We recall the following result by Kumpera and Spencer [29].
Lemma 6.5. Let Γ ⇒ M be any Lie groupoid, and let AΓ be its Lie algebroid with anchor map ♯. For Λ ∈ Γ (AΓ), let X Λ be the right invariant vector field corresponding to Λ. Then X Λ is complete if and only if Λ ♯ is complete. In fact, ϕ t (g) is defined whenever ψ t (β (g)) is defined, where {ϕ t } and {ψ t } are the flows generated by X Λ and Λ ♯ , respectively.
Let Λ ∈ Γ (AΓ) be any section which has a compact support, then Λ ♯ has a compact support. In fact, using that ♯ is a vector bundle morphism, if x ∈ M satisfies that Λ (x) = 0, then ♯ (Λ (x)) = 0.
So, as a corollary, we get that X Λ has a compact support and therefore, Exp (Λ) : R × M → Γ, i.e., we have proved the following collolary.
Corollary 6.6. Let Γ ⇒ M be any Lie groupoid, and let AΓ be its Lie algebroid. Any Λ ∈ Γ (AΓ) with compact support is complete.
Finally, we are ready to define the exponential map. Note that, given that in this case U 1 = Γ and U 1 = M, for all x ∈ M we have i.e., another equivalent definition of the exponential map could be Thus, it satisfies the following properties.
The differentiability means that exp transforms differentiable families of sections of AΓ into differentiable families of bisections of Γ. The uniqueness property is a consequence of local uniqueness property of solutions of ordinary differential equations.

Derivation algebroid
Now, we are going to use the exponential map in order to get another way of interpreting the associated Lie algebroid of Π 1 (M, M). A more detailed construction of this algebroid can be found in [31] (see also [28]).
To do this, first we have to introduce some notions.
. is a zeroth-order differential operator on A. Equivalently, for all f, g ∈ C ∞ (M) and Λ ∈ Γ (A), a vector field X ∈ X (M) such that for each f ∈ C ∞ (M) and Λ ∈ Γ (A), We call X the base vector field of D. So, a derivation on A is characterized by two geometrical objects, D and X.
Notice that the space of zeroth-order differential operators is contained in the space of derivations on A (X = 0) and this space is contained in the space of first-order differential operators. , Then, any vector field X ∈ X (M) generates a derivation on A, ∇ X , (with base vector field X) fixing the first coordinate again, i.e., Now, associated to any first-order differential operator, D, there is a map from 1−forms on M to zeroth-order differential operators on A, called symbol of D, which is determined by With this, the symbol of D evaluated at any 1−form ξ at a point x ∈ M is a scalar multiple of the identity map of the fibre A x over x, σ (D) (ξ) = ξ (x) (X (x)) Id Ax . We have thus obtained that a firstorder differential operator is a derivation if and only if it has scalar symbol. Furthermore, it is obvious that σ (D) = 0 if and only if D is a zeroth-order differential operator. Now, the space of first-order differential operators on A can be con- which is going to be called the symbol of A.
It turns out that σ is a surjective submersion and its kernel the zeroth-order differential operators. Thus, σ induces a short exact sequence of vector bundles over M End (A) ֒→ Dif f 1 (A) → Hom (T * M, End (A)) . Now, we can define D (A) to be the pullback vector bundle defined by the symbol map and the injective map Furthermore, taking into account that the left-hand vertical arrow is an injective inmersion we can consider D (A) as a subbundle of Dif f 1 (A). We will denote the top arrow by a and, clearly, as we have noticed before, the kernel of a is End (A). So, using a, we can consider another exact sequence So, we have defined a vector bundle in which the sections are the derivations. Now we will endow this vector bundle of a Lie algebroid structure.
• Let D 1 , D 2 be derivations on A, we can define [D 1 , D 2 ] as the commutator, i.e., A simple computation shows that the commutator of two derivations is again a derivation, indeed, the base vector field of [D 1 , D 2 ] is given by where X 1 and X 2 are the base vector fields of D 1 and D 2 respectively. • Let D be a derivation on A, then D ♯ is its base vector field. Thus, with this structure D (A) is a transitive Lie algebroid called the Lie algebroid of derivations on A. The space of sections of D (A), the derivations on A, will be denoted by Der (A).
Note that in this Lie algebroid the linear sections of ♯ are C ∞ (M) −linear maps from X (M) to Der (A). So, the space of linear sections of ♯ is, indeed, the space of covariant derivatives on M. Finally, it is easy to see that a covariant derivative ∇ is a section (in the category of Lie algebroids) of ♯ if and only if ∇ is flat, i.e., for all X, Y ∈ X (M) Now, it is turn to relate this algebroid with the frame algebroid. In order to do that, given the frame groupoid Φ (A) ⇒ M and a bisection θ ∈ Ω (Φ (A)) we can define a vector bundle automorphism over β • θ, Conversely, given a vector bundle automorphism L : A → A we can find a bisection on A, θ, such that θ = L. We will generalize this using its pullback. Thus, we can consider θ * : Γ (A) → Γ (A) satisfying for each Λ ∈ Γ (A) and x ∈ M. Thus, we can give the following result, More detail, for each X ∈ Γ (A) and x ∈ M we have This theorem gives us another way of interpreting the 1−jets Lie algebroid. In fact, the 1−jets Lie algebroid AΠ 1 (M, M) is isomorphic to the algebroid of derivations on T M.
Notice that using this isomorphism, we can consider a one-to-one map from linear sections of ♯ in AΠ 1 (M, M) to covariant derivatives over M. Thus, having a section Λ of ♯ in AΠ 1 (M, M) we will denote its associated covariant derivative by ∇ Λ . Furthermore, if Λ is a morphism of Lie algebroids, ∇ Λ is flat.

Lie theory for groupoids
As we have seen Lie groupoids and Lie algebroids can be seen as a generalization of the Lie groups and the Lie algebras. In this sense, using foliations and connections on principal Lie groupoid bundles we can extend Lie's fundamental theorems (see [10]).
Every finite dimensional Lie algebra is the Lie algebra of a unique connected simply connected Lie group. Something similar holds for integrable Lie algebroids.
It is said to be source-simply connected if each α −1 (x) is connected and simply connected.
So, the Lie's first fundamental theorem can be written as follows. This has been proved in [36] (see also [31]). The same methods involved in the construction of the source-simply connected groupoid can be used to prove the following integrability result.

Proposition 8.3. Any Lie subalgebroid of an integrable Lie algebroid is integrable.
Next, the Lie's second fundamental theorem shows that any morphism of integrable Lie algebroids can be integrated to a unique morphism of the integral Lie groupoids, provided that the domain groupoid is source-simply connected. This result has been proved by Mackenzie and Xu [32] (see also [36]). Finally the Lie's third fundamental theorem cannot been generalized, i.e., not every Lie algebroid can be integrated by a Lie groupoid. Almeida and Molino showed in [1] that there exist non integrable Lie algebroids. This result is of great importance because it contrasts with the one for Lie groups and Lie algebras. In [9] the author give necessary and sufficient conditions for the integrability of a Lie algebroid.

Uniformity and Homogeneity
A body B is a three-dimensional differentiable manifold which can be covered with just one chart. An embedding φ : B → R 3 is called a configuration of B and its 1−jet j 1 x,φ(x) φ at x ∈ B is called an infinitesimal configuration at x. We usually identify the body with any one of its configurations, say φ 0 , called reference configuration. Given any arbitrary configuration φ, the change of configurations κ = φ • φ −1 0 is called a deformation, and its 1−jet j 1 φ 0 (x),φ(x) κ is called an infinitesimal deformation at φ 0 (x).
From now on we make the following identification: B ∼ = φ 0 (B). For elastic bodies, the mechanical response of a material is completely characterized by one function W which depends, at each point x ∈ B, on the gradient of the deformations evaluated at the point. Thus, W is defined (see [15]) as a differentiable map denoted by the same letter W : Gl (3, R) × B → V, where V is a real vector space. Another equivalent way of considering W is as a differentiable map which doesn't depend on the final point, i.e., for all x, y, z ∈ B where τ v is the translation map on R 3 by the vector v. This map will be called response functional. Notice that, using Eq. (9.1), we can define W over Π 1 (B, R 3 ), which is the open subset of Π 1 (R 3 , R 3 ) given by (α, β) −1 (B × R 3 ) (see Example 3.18). Now, suppose that an infinitesimal neighbourhood of the material around the point y can be grafted so perfecly into a neighbourhood of x, that the graft cannot be detected by any mechanical experiment. If this condition is satisfied with every point x of B, the body is said uniform. We can express this physical property in a geometric way as follows. y,κ(y) κ · j 1 x,y ψ = W j 1 y,κ(y) κ , (9.2) for all infinitesimal deformation j 1 y,κ(y) κ.
These kind of maps are going to be important and we will endow these maps of a groupoid structure. For each two points we will denote by G (x, y) the collection of all 1−jets j 1 x,y ψ which satisfy Eq. (9.2). So, the set Ω (B) = ∪ x,y∈B G (x, y) can be considered as a groupoid over B which is, indeed, a subgroupoid of the 1−jets groupoid Π 1 (B, B). We will denote α −1 (x) by Ω x (B).
Definition 9.2. Given a point x ∈ B a material symmetry at x is an We denote by G (x) the set of all material symmetries which is, indeed, the isotropy group of Ω (B) at x.
So, the following result is obvious. Notice that, at general, we cannot ensure that Ω (B) ⊆ Π 1 (B, B) is a Lie subgroupoid. Our assumption is that Ω (B) is in fact a Lie subgroupoid and, in this case, Ω (B) is said to be the material groupoid of B.
Remark 9.4. Suppose that V = R and W is multiplicative, i.e., W j 1 z,y φ · j 1 x,z ψ = W j 1 z,y φ W j 1 x,z ψ , for all j 1 x,z ψ, j 1 z,y φ ∈ Π 1 (B, B). Then, it is easy to prove that Hence, if W is a submersion, Ω (B) is, indeed, a Lie subgroupoid of Π 1 (B, B).
As we have seen, a body is uniform if the function W does not depend on the point x. In addition, a body is said to be homogeneous if we can choose a global section of the material groupoid which is constant on the body, more precisely: Definition 9.5. A body B is said to be homogeneous if it admits a global deformation κ which induces a global section of (α, β) in Ω (B), P , i.e., for each x, y ∈ B where τ κ(y)−κ(x) : R 3 → R 3 denotes the translation on R 3 by the vector κ (y) − κ (x). B is said to be locally homogeneous if there exists a covering of B by homogeneous open sets.
As we will prove later (see Proposition 10.2), B is locally homogeneous if and only if for each x 0 , y 0 ∈ B there exist two open sets U, V ⊆ B with x 0 ∈ U and y 0 ∈ V and two local deformations κ andκ over x 0 and y 0 respectively such that the map P : Now, suppose that B is homogeneous. Then, if we take global coordinates (x i ) given by the induced diffeomorphism κ, we deduce that P is locally expressed by If B is locally homogeneous we can cover B by local sections of (α, β) in Ω (B) which satisfy Eq. (9.3).
Next, we want to give some equivalent definitions and relate this definition with the classical one, which is given for G−structures.

Integrability
As a first step we will introduce the notion of integrability of reduced subgroupoids of the 1-jets groupoid which is going to be closely related with the notion of integrability of G−structures.
Note that there exists a Lie groupoids isomorphism L : Π 1 (R n , R n ) → R n × R n × Gl (n, R) over the identity map defined by L j 1 x,y φ = x, y, dφ |x , ∀j 1 x,y φ ∈ Π 1 (R n , R n ) . Another way of expressing this isomorphism is identifying Gl (n, R) with the fibre of F R n at 0. Then, the isomorphism is given by where τ z denote the translation on R n by the vector z ∈ R n . So, the inverse map satisfies We have thus obtained a Lie groupoid isomorphism Π 1 (R n , R n ) ∼ = R n × R n ×Gl (n, R) over the identity map on R n . Then, if G is a Lie subgroup of Gl (n, R), we can transport R n × R n × G by this isomorphism to obtain a reduced Lie subgroupoid of Π 1 (R n , R n ). These kind of reduced subgroupoids will be called standard flat on Π 1 (R n , R n ).
Let U, V ⊆ M be two open subsets of M. We denote by Π 1 (U, V ) the open subset of Π 1 (M, M) defined by (α, β) −1 (U × V ). Note that if U = V , then, Π 1 (U, U) is in fact the 1-jets groupoid of U and, in this way, our notation is consistent. Furthermore, we are going to think about Π 1 (U, V ) as the restriction of the Lie groupoid Π 1 (M, M) equipped with the restriction of the structure maps (this could not be a Lie groupoid). We will also use this notation for subgroupoids of Π 1 (M, M).
Before continuing, we need to explain what we understand by "locally diffeomorphic" in this case. So, Π 1 G (M, M) is locally diffeomorphic to R n × R n × G ⇒ R n if for all x, y ∈ M there exist two open sets U, V ⊆ M with x ∈ U, y ∈ V and two local charts, ψ U : U → U and . Notice that, Π 1 G (U, V ) and U × V × G are Lie groupoids if and only if U = V and U = V . Suppose that U = V and U = V , then, for all x ∈ U Ψ U,U j 1 x,x Id ∈ G. However, Ψ U,U j 1 x,x Id is not necessarily the identity map and, hence, Ψ U,U is not an isomorphism of Lie groupoids. Proof. On the one hand, suppose that Π 1 (M, M) is integrable. Let x 0 ∈ M be a point in M and ψ U : U → U and ψ V : V → V be local charts through x 0 which induced diffeomorphism Therefore, denoting U ∩ V by W , the map . On the other hand, suppose that for each x ∈ M there exists a local chart (ψ U , U) through x which induces a Lie groupoid isomorphism over ψ U , namely . Take open sets U, V ⊆ M such that there exist ψ U and ψ V satisfy Eq. (10.2). Suppose that U ∩ V = ∅. Then, for all x, y ∈ U ∩ V , we have So, we define the diffeomorphism ψ V A · ψ V : V → A · V . Then, using Eq. (10.3) for all y ∈ U ∩ V , we deduce that (10.4) In this way, we consider We will check that Ψ U,V is well-defined. We fix j 1 x,y φ ∈ Π 1 G (U, V ). Then, we can consider two cases: (i) y ∈ U ∩ V . Then, using Eq. (10.4) (ii) y / ∈ U ∩ V . Then, Thus, it is immediate to prove that Ψ U,V is a diffeomorphism which commutes with the restrictions of the structure maps.
Finally, if U ∩ V = ∅ we can find a finite family of local neighbour- Thus, we can find Ψ U,V following a similar procedure than above.  (n, R), i.e., locally diffeomorphic to R n × R n × G. Suppose that there exists another subgroup of Gl (n, R),G, such that Π 1 G (M, M) is locally diffeomorphic to R n × R n ×G. Then, using the above result, it is easy to see that G andG are conjugated subgroups of Gl (n, R). Conversely, if G andG are conjugated subgroups of Gl (n, R) then, Π 1 G (M, M) is locally diffeomorphic to R n × R n × G if and only if Π 1 G (M, M) locally diffeomorphic to R n × R n ×G. It is easy to prove that P is, indeed, a global section of (α, β). Conversely, every global section of (α, β) (understanding "section" as section in the category of Lie groupoids, i.e., Lie groupoid morphism from the pair groupoid M × M to Π 1 (M, M) which is a section of the morphism (α, β)) can be seen as a parallelism of Π 1 (M, M). Using this, we can also speak about integrable sections of (α, β). Now, using the induced coordinates given in Eq. (3.3) an integrable section can be written locally as follows, for some two local charts (ϕ, U) , (ψ, V ) on M.
Notice that, using Proposition 10.2, P is an integrable section if and only if we can cover M by local charts (ϕ, U) such that Next, analogously to the case of G−structures, we can characterize the integrable subgroupoids using (local) integrable sections (see Proposition 2.8). However, in this case it is not so easy because having a reduced subgroupoid we don't know anything about the group G. So, firstly, we will have to solve this problem. Let Π 1 G (M, M) be a reduced subgroupoid of Π 1 (M, M) and Z 0 ∈ F M be a frame at z 0 ∈ M. Then, we define where Π 1 G (z 0 ) is the isotropy group of Π 1 (M, M) at z 0 . Therefore, G is a Lie subgroup of Gl (n, R). This Lie group will be called associated Lie group to Π 1 G (M). Note that, as a difference with G−structures, we don't have a unique Lie group G. In fact, letZ 0 be a frame atz 0 andG be the associated Lie group, then, if we take is integrable if and only if for each two points x, y ∈ M there exist coordinate systems (x i ) and (y j ) over U, V ⊆ M, respectively with x ∈ U and y ∈ V such that the local section, Proof. First, it is obvious that if Π 1 G (M, M) is integrable then, we can restrict the maps Ψ −1 U,V to U × V × {e} to get (local) integrable sections of (α, β) which takes values on Π 1 G (M, M). Conversely, in a similar way to Proposition 10.2 we can claim that for each x ∈ M there exists an open set U ⊆ M with x ∈ U and P : U × U → Π 1 G (U, U) an integrable sections of (α, β) given by Then, we can build the map , defined in the obvious way. Now, let z 0 ∈ U be a point at U, Z 0 j 1 0,z 0 ψ −1 U • τ ψ U (z 0 ) ∈ F U be a frame at z 0 and G be the Lie subgroup satisfying Eq. (10.8). Then, we can define is an isomorphism of Lie groupoids induced by ψ U .
To end the proof, we only have to use Proposition 10.2.
Let B be a body. Taking into account the definition of homogeneity (see Definition 9.5) and the above result we can give the following proposition: Proposition 10.6. Let B be a uniform body. If B is homogeneous then Ω (B) is integrable. Conversely, Ω (B) is integrable implies that B is locally homogeneous. Now, we want to work with the notion of integrability in the associated Lie algebroid of the 1-jets groupoid. So, we will introduce this notion and relate it with the integrability of reduced subgroupoids of Π 1 (M, M). Note that the induced map of the Lie groupoid isomorphism L : Π 1 (R n , R n ) → R n × R n × Gl (n, R) is given by a Lie algebroid isomorphism where T R n ⊕ (R n × gl (n, R)) is the trivial Lie algebroid on R n with structure algebra gl (n, R). Now, if g is a Lie subalgebra of gl (n, R), we can transport T R n ⊕ (R n × g) by this isomorphism to obtain a reduced Lie subalgebroid of AΠ 1 (R n , R n ). These kind of reduced subalgebroids will be called standard flat on AΠ 1 (R n , R n ). Let such that AΨ U,U is the induced map of the following isomorphism of Lie groupoids Ψ U,U : 11) for all j 1 x,y φ ∈ Π 1 G (U, U).
So, for each open U ⊆ M, AΠ 1 G (U, U) is integrable by a Lie subgroupoid Π 1 G (U, U) of Π 1 (U, U). Using the uniqueness of integrating immersed (source-connected) subgroupoids (see for example [37]), Analogously to the case of 1−jets groupoid, a parallelism of AΠ 1 (M, M) is an associated Lie algebroid of a parallelism of Π 1 (M, M). Hence, using the Lie's second fundamental theorem, a parallelism is a section of ♯ (understanding "section" as section in the category of Lie algebroids, i.e., Lie algebroid morphism from the tangent algebroid T M to AΠ 1 (M, M) which is a section of the morhism ♯) and reciprocally. In this way, we will also speak about integrable sections of ♯.
Let (x i ) be a local coordinate system defined on some open subset U ⊆ M. Then, we will use the local coordinate system defined in Eq. (5.9), which are, indeed, induced coordinates by the functor A from local coordinates on Π 1 (U, U).
Notice that each integrable section of (α, β) in Π 1 (M, M), P , is a Lie groupoid morphism. Hence, P induces a Lie algebroid morphism AP : T M → AΠ 1 (M, M) which is a section of ♯ and is given by So, taking into account that, locally, P x i , y j = x i , y j , δ j i , we have that each integrable section can be written locally as follows Now, using Proposition 10.5, we have the following analogous proposition. Equivalently, for each point x ∈ M there exists a local coordinate system (x i ) over an open set U ⊆ M with x ∈ U such that the local sections Let B be a body. The induced subalgebroid of the material groupoid, AΩ (B), will be called material algebroid of B.
Thus, taking into account that Θ ∈ Γ (AΩ (B)) . So, Eq. (10.15) gives us a way to characterize the material Lie algebroid without using the material Lie groupoid. Now, using the above results we can give the following results.
Finally, we will use the algebroid of derivations on T B. Thus, using Theorem 7.4, the map D : Γ (AΠ 1 (B, B)) → Der (T B) given by and the base vector field of D Λ is given locally by (x i , Λ j ).
Proof. Let Λ ∈ Γ (AΠ 1 (M, M)) be a section of the 1−jets algebroid and X Λ its associated right-invariant vector field over Π 1 (M, M). Considering the flow of X Λ , {ϕ t : U t → U −t } we have by construction that where ξ ∈ U t ∩ β −1 (x). Now, take local coordinate systems (x i ) and (y j ) and its induced local coordinates over Λ Then, the associated right-invariant vector field is (locally) as follows (ϕ t (y j )) = Λ j i .
So, with these coordinates, we obtain Then, Therefore, i.e., the matrix −Λ j i is (locally) the associated matrix to D Λ .
Let Λ be a linear section of ♯ in AΠ 1 (M, M). Then, D induces a covariant derivative on M, ∇ Λ . Thus, for each (x i ) local coordinate system on M where Λ j i depends on ∂ ∂x j . Taking into account that Λ j i is linear in the second coordinate we will change the notation as follows Therefore, locally Λ will be written in the following way and thus where Λ ∂ ∂x j is the (local) section of AΠ 1 (M, M) given by So, −Λ k i,j end up being the Christoffel symbols of ∇ Λ .
With this fact in mind, we can give another characterization of the integrability over the 1−jets algebroid. Remark 10.14. There is still another interesting way of interpreting the 1−jets Lie groupoid on a body B (and, hence, of interpreting the integrability of a reduced subgroupoid Π 1 G (B, B) of Π 1 (B, B)). As we know, there exists another structure of Lie groupoid related with F B, the Gauge groupoid of the principal bundle F B.
We only have to take into account that Furthermore, translating points we can construct an isomorphism of Lie groupoid from B × F B to Π 1 (B, B) (notice that this isomorphism depends on the reference configuration φ 0 ). Thus, the 1−jets Lie groupoid can be seen as the gauge groupoid of the principal bundle F B and, therefore, the 1−jets Lie algebroid can be seen as the Atiyah algebroid associated with F B. Let (x i ) be a local coordinate system on B and D be a derivation on B with base vector field X. We denote Then, D is in D (B) if and only if over any (x i ) local coordinate system on B it is satisfied that for all material symmetry g ∈ G (x) which is locally written as follows

Homogeneity with G-structures
In this last section we will prove that our definition of homogeneity (which is given using the material groupoid) is, indeed, equivalent to that used in [15] (see [16] or [47]; see also [4] and [35]) where the authors use G−structures to characterize this property.
Let B be a uniform body. First, we will introduce the definition of homogeneity in terms of G−structures.
Fix z 0 ∈ B and Z 0 = j 1 0,z 0 φ ∈ F B a frame at z 0 . Define a map h : Then, h is differentiable and h (G (z 0 )) G 0 is a Lie subgroup of Gl (3, R). G 0 is called the isotropy group of B. Now, we can consider a family of (local) sections of the principal bundle Ω z 0 (B), {σ a : U a → Ω z 0 (B)}, such that ∪ a U a = B. Furthermore, we can assume that for some a 0 , σ a 0 (z 0 ) is the identity of G (z 0 ) (in another case we could change σ a by σ a = σ a · σ −1 a 0 (z 0 ) for all a).
where g ab (x) ∈ G (z 0 ). On the other hand, this family of (local) sections induces a family of (local) section of F B {S a : U a → F B}, Therefore this family defines a G 0 −structure ω G 0 (B) on B with transition functions {h (g ab (x))}. A local section of ω G 0 (B) will be called local uniform reference. A global section of ω G 0 (B) will be called global uniform reference. We call reference crystal to any frame Z 0 ∈ F B at z 0 .
(1) If we change the point z 0 to another point z 1 then we obtain an isomorphic G 0 −structure. We only have to take a frame Z 1 as the composition of Z 0 with a j 1 z 0 ,z 1 ψ ∈ G (z 0 , z 1 ).
(2) We have fixed a configuration φ 0 . Suppose that φ 1 is another reference configuration such that the change of configuration is given by Ψ = φ −1 1 • φ 0 . Transporting the reference crystal Z 0 via F Ψ we get another reference crystal such that the G 0 −structures are isomorphic.
(3) Finally suppose that we have another crystal reference Z ′ 0 at z 0 . Hence, S ′ a (x) = S a (x) · A, with A ∈ Gl (3, R). Therefore the new G ′ 0 −structure, ω G ′ 0 (B), is conjugate of ω G 0 (B), namely, In this way, the definition of homogeneity in terms of G−structures is the following, Definition 11.2. A body B is said to be homogeneous with respect to a given frame Z 0 if it admits a global deformation κ such that κ −1 induces a uniform reference P , i.e., for each x ∈ B , where τ κ(x) : R 3 → R 3 denotes the translation on R 3 by the vector κ (x). B is said to be locally homogeneous if every x ∈ B has a neighbourhood which is homogeneous.
It is easy to prove the following result: Proposition 11.3. If B is homogeneous then ω G 0 (B) is integrable. Conversely, ω G 0 (B) is integrable implies that B is locally homogeneous.
Notice that, using Remark 11.1, this result shows us that the homogeneity doesn't depend on the point, the reference configuration and the frame Z 0 .
Finally we will prove that both definitions are equivalents. In order to do this, we are going to construct the following map: such that, GP (x, y) = P (y) · [P (x) −1 ]. It is obvious that G is well-defined.
Before starting to work with integrable sections we are interested in dilucidating when an element of Γ (α,β) (Π 1 (M, M)) can be inverted by G. First, we consider P ∈ Γ (F M); then for all x, y, z ∈ M, we have GP (y, z) · GP (x, y) = GP (x, z) , (11.1) i.e., GP is a morphism of Lie groupoids over the identity map on M from the pair groupoid M × M to Π 1 (M, M). Therefore, not every element of Γ (α,β) (Π 1 (M, M)) can be inverted by G but we can prove the following result. Proof. We have proved the right implication. Conversely, if Eq. (11.1) is satisfied we can define P ∈ Γ (F M) as follows P (x) = P (z, x) · j 1 0,z ψ, where j 1 0,z ψ ∈ F M is fixed. Then, using Eq. (11.1), we have GP = P.
However, there is not a unique P such that GP = P. We will study this problem in Remark 11.5. Notice that the relevant sections of (α, β) are going to be the parallelisms which are, indeed, the morphisms of Lie groupoids over the identity map from the pair groupoid M × M to Π 1 (M, M).
Next, suppose that P is an integrable section of F M. Then, for each point x ∈ M there exists a local coordinate system (x i ) on M such that P x i = x i , δ i j , or equivalently, P (x) = j 1 0,x ϕ −1 • τ ϕ(x) , (11.2) where ϕ is the local chart over x and τ ϕ(x) denote the translation on R n by the vector ϕ (x).
Conversely, for each 1−jet Z 0 = j 1 0,0 φ ∈ F R n 0 , where F R n 0 is the fibre of F R n over 0, and each section of F M, P : M → F M, the section of F M given by Q (x) = P (x) · Z 0 , (11.4) satisfies that GP = GQ. Thus, we have shown that for each section on F M P : M → F M G −1 (GP ) = {P · Z 0 / Z 0 ∈ F R n 0 }, i.e., the map G can be considered as an injective map over the quotient space by Eq. (11.4).
Using this, it is obvious that, if GP is integrable, then, P is integrable too, i.e., the map G restricted to the integrable sections can be considered as a one-to-one map over the quotient space by Eq. (11.4).
Finally, we can generalize the map G into a map which takes G−structures on M into reduced subgroupoids of Π 1 (M, M). Let ω G (M) be a G−structure on M, then we consider the following set, x ] / L x , L y ∈ ω G (M)}. It is straightforward to prove that G (ω G (M)) is a reduced subgroupoid of Π 1 (M, M). In fact, taking a local section of ω G (M), the map given by is a diffeomorphism which satisfies that F U (Π 1 G (U, U)) = ω G (U).
Analogously to parallelisms, we can prove that every reduced subgroupoid can be inverted by G into a G−structure on M, where G is defined by Eq. (10.8) with Z 0 ∈ F M fixed.
We consider z 0 = π M (Z 0 ). Then, we can generate a G−structure over M in the following way ω G (M) := {L z 0 ,x · Z 0 · g / g ∈ G, L z 0 ,x ∈ Π 1 G (M, M) z 0 }. Notice that the fibre of ω G (M) at x ∈ M is given by the set {L z 0 ,x · Z 0 · g / g ∈ G}, for any fixed L z 0 ,x ∈ Π 1 G (M, M) z 0 . In fact, for two L z 0 ,x , G z 0 ,x ∈ Π 1 G (M, M) z 0 , [L z 0 ,x · Z 0 ] −1 · G z 0 ,x · Z 0 ∈ G. Notice that the map L z 0 ,x → L z 0 ,x · Z 0 defines an isomorphism of principal bundles from Π 1 G (M, M) z 0 to ω G (M).
Finally, to prove the converse we only have to construct ω G (M) using Eq. (11.5) and repeat the above construction of a G−structure which inverts Π 1 G (M, M).
Using this, if G (ω G (M)) is integrable, then ω G (M) is integrable too.
Let B be a uniform body. Suppose that {σ a : U a → Ω z 0 (B)} is the family of (local) sections of the principal bundle Ω z 0 (B) defined at the beginning of the section and ω G 0 (B) the associated G 0 −structure on B. Using the above results, the G 0 −structure ω G 0 (B) is integrable if and only if G (ω G 0 (B)) is integrable.

Conclusions and future work
In this paper we have used the theory of Lie groupoids and Lie algebroids to characterize the uniformity and homogeneity of simple materials. In this sense, we have shown that the uniformity is equivalent to the transitivity of the material groupoid. We have also characterized the (local) homogeneity of a simple material B usinng the associated Lie algebroid of the material groupoid (the material algebroid).
These results have been related with those in [33,15] where the homogeneity is studied in terms of G−structures.
In future research we would like to get similar results for Cosserat media and time-dependet materials ( see [17], [6] and [19]). We will also discuss the dynamics, in the context of Lie groupoids, following the lines developed in [34].