THE METHOD OF AVERAGING FOR POISSON CONNECTIONS ON FOLIATIONS AND ITS APPLICATIONS

. On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson ﬁber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.

1. Introduction. In this paper we discuss some aspects of the averaging method for Poisson connections on foliated manifolds with symmetry by generalizing the previous results on Hannay-Berry connections on fibrations due to [12,13]. Poisson connections of such type play an important role in the normal form theory for Hamiltonian systems of adiabatic type (see, for example, [1,2]). One of our main motivations is related to the further development of the averaging procedure for Dirac structures with singular presymplectic foliations [20].
First of all, we describe the averaging procedure for Poisson connections on foliations relative to a wide class of canonical (not necessarilly Hamiltonian) actions which appears in [8,9] in the context of invariant Poisson cohomology. Then, these results are applied in the construction of invariant Dirac structures.

Ehresmann connections on foliated manifolds.
Let (M, F) be a regular foliated manifold and V := T F ⊂ T M the tangent bundle called the vertical distribution. A vector-valued 1-form γ ∈ Ω 1 (M ; V) is said to be a connection on (M, F) if the vector bundle morphism γ : T M → V satisfies the conditions: γ • γ = γ and Im γ = V. (1) In fact, these conditions are equivalent to the following: Then, H = H γ := ker γ is a subbundle of T M normal to F, called the horizontal subbundle. It is clear that id T M −γ is just the projection to H along V. Conversely, given a normal bundle H of F, one can define the associated connection as the projection γ = γ H := pr 2 to V according to the decomposition Then, the cotangent bundle splits as follows where V 0 and H 0 are the annihilators of V and H, respectively. These decompositions give rise to a γ-dependent bigrading of differential forms and tensor fields on M . In particular, for any X ∈ X(M ) and α ∈ Ω 1 (M ), we have X = X 1,0 + X 0,1 and α = α 1,0 + α 0,1 , where X 1,0 = (id T M −γ)(X) ∈ Γ(H) and α 1,0 = (id T * M −γ * )(α) ∈ Γ(V 0 ) are the horizontal components while X 0,1 and α 0,1 are the vertical ones, respectively. Here γ * : T * M → T * M is the adjoint of γ. Moreover, the exterior differential of forms on M has the following bigraded decompostion d = d γ 1,0 +d γ 2,−1 +d γ 0,1 associated with (2) (see, [17,18]). The operator d γ 1,0 is called the covariant exterior derivative and is defined by d γ 1,0 α(X 0 , X 1 , . . . , X k ) := dα((id −γ)X 0 , . . . , (id −γ)X k ), for all α ∈ Ω k (M ) and X 0 , X 1 , . . . , X k ∈ Γ(T M ). In general, the covariant exterior derivative is not a coboundary operator.
The curvature of a connection γ on (M, F) is a vector-valued 2-form Curv γ ∈ Ω 2 (M ; V) on M given by here [·, ·] FN denotes the Frölicher-Nijenhuis bracket [10]. Denote the space of all projectable vector fields on (M, F) by The space of all (local) projectable vector fields which are tangent to the horizontal subbundle of a connection γ will be denoted by Γ pr (H γ ). It follows from (4) that for any Z 1 , Z 2 ∈ Γ pr (H γ ). It is well-known that the set of all connections on a foliated manifold is an affine space. Indeed, fixing a connection γ on (M, F), it is easy to see that any other connectionγ is of the formγ = γ − Ξ, where the vector bundle morphism Ξ : T M → T M is called the connection difference form and satisfies the conditions im Ξ ⊆ V ⊆ ker Ξ. The horizontal subbundle associated withγ is given by (4), one can deduce the transition rule for the curvature: 3. The averaging procedure. First, we recall the averaging procedure for connections on a regular foliated manifold (M, F).
Let G be a compact and connected Lie group and g its Lie algebra. Suppose that we are given an action Φ : For every ξ ∈ g, the corresponding infinitesimal generator ξ M ∈ Γ(T M ) of the G-action is defined by Condition (7) implies that each infinitesimal generator is a projectable vector field, ξ M ∈ X pr (M, F) ∀ξ ∈ g.
As a consequence, the G-action preserves the space of all projectable vector fields, Φ * g (X pr (M, F)) = X pr (M, F).
Let Ω k (M ; T M ) the space of vector-valued k-forms. For any K ∈ Ω k (M ; T M ), the G-average of K is the vector-valued form K G ∈ Ω k (M ; T M ) defined by the standard formula: Here, the pull-back Φ * g K of K is given by and the integral is taken with respect to the normalized Haar measure dg on G, G dg = 1.
Recall that a vector-valued k-form K is said to be G-invariant if Φ * g K = K for all g ∈ G. Since the Lie group G is connected, this invariance condition can be expressed in infinitesimal terms: L ξ M K = 0 for all ξ ∈ g. It is clear that the Gaverage K G is G-invariant for any K. We have the following invariance criterion: Property (7) implies that the averaging operator preserves the set of all connections. In other words, for any connection γ on (M, F), its G-averageγ := γ G is a G-invariant vector-valued 1-form which again satisfies the conditions in (1). From the property that the Frölicher-Nijenhuis bracket is a natural operation with respect to the pull-back, it follows that the curvature form ofγ is also G-invariant, Lemma 3.1. We have the following representation Remark 3.1. The integral over G in the right hand side of (9) is well-defined because of the properties of the exponential map of a compact and connected Lie group (for details, see [20]).
Proof of Lemma 3.1. By the fundamental theorem of calculus, we obtain Integrating the equality (10) with respect to the Haar measure, we get From (8) and the identity L ξ M γ = −[γ, ξ M ] F N , we obtain formula (9).
For connections on foliated manifolds, we also have the following invariance criteria.
Proof. The equivalence between the items (i) and (ii) follows by straightforward computations. The implication (i) ⇒ (iii) follows from the relations: Conversely, condition (iii) together with (11) and the connectedness of G imply the invariance condition (i). Here we use the fact [23]: every element of a connected Lie group is the product of exp(ξ 1 ) and exp(ξ 2 ) for some ξ 1 , ξ 2 ∈ g. The equivalence between the items (i) and (iv) follows from the fact that, the G-invariance of γ is equivalent to the equation Finally, the equivalence between the items (ii) and (v) follows directly from (8).
Next, we formulate some key properties of the averaged connection in the case of a leaf-tangent G-action.
Then, for every connection γ on (M, F) the following assertions hold: (b) The space of horizontal projectable vector fields associated with the averaged connectionγ = γ G is described as (c) For every Z ∈ Γ pr (H γ ), Proof. (a) First, we suppose that the G-action is arbitrary. For each Z ∈ Γ pr (H γ ), we have By item (iii) of Proposition 3.2, the G-invariance of γ is equivalent to the condition that [Z, ξ M ] is a horizontal vector field. If the action is leaf-tangent, then the vector field [Z, ξ M ] is always vertical and hence it must be equals to zero. Conversely, if condition (13) holds, then [γ, . Moreover Z is a G-invariant vector field by the item (a). From here and the fact that the average of Ξ G is zero, we get that Z = Z G = Z G . This proves (14). (c) For every Z ∈ Γ pr (H γ ), and the G-invariant vector field Z := Z + Ξ G (Z) ∈ Γ pr (Hγ) it follows from formula (9) that Then, formula (15) follows from (16) and the equality:

Hannay-Berry connections on foliations.
We start by recalling some general facts and notions for Lie group actions on Poisson manifolds due to [8,9,11]. Canonical and cotangential Actions. Let (M, P ) be a Poisson manifold. Suppose that we are given an action Φ : G × M → M of a Lie group G with Lie algebra g. Consider the infinitesimal action (anti-homomorphism) g ξ → ξ M ∈ X(M ). Recall that the G-action is said to be canonical or Poisson if the Lie group acts by Poisson diffeomorphisms, Φ * g P = P for all g ∈ G. If the Lie group G is connected, then this condition can be written in infinitesimal terms: L ξ M P = 0 for all ξ ∈ g. The property for the action Φ to be tangential means that all infinitesimal generators are tangent to the symplectic foliation of P , that is, ξ M (m) ∈ P (T * m M ) for any m ∈ M and ξ ∈ g. Follow [8,9], we say that the G-action on the Poisson manifold is cotangential if there exists a linear map µ : g → Ω 1 (M ) such that where µ ξ := µ(ξ). It is clear that a cotangential action is tangential, but the converse is not true, in general [8]. According to [8,9] the map µ is called a cotangent lift or, a pre-momentum mapping as in [11] .
Lemma 4.1. A cotangential action Φ of a connected Lie group G on a Poisson manifold (M, P ) is canonical if and only if for every ξ ∈ g, the pull-back of the 1-form µ ξ to each symplectic leaf of P is closed, Proof. By definition of the morphism P and the standard properties of the Lie derivative, we have (P dµ ξ )(α, β) = dµ ξ (P α, P β) and for any α, β ∈ Ω 1 (M ). This proves the assertion.
Notice that a pre-momentum mapping µ is determined uniquely modulo the transformation µ → µ + ν for arbitrary ν ∈ Hom(g, Ω 1 (M )) such that P ν ξ = 0 for all ξ ∈ g. It is easy to see that the leafwise closedness condition (17) is independent of this freedom.
Not every cotangential and canonical action is Hamiltonian, as we show in the following example.
y \{(y 1 = y 2 = 0} be the complement of the y 3 -axis. Consider the S 1 -action on U with infinitesimal generator where g 0 (y) = − 1 2 ln(y 2 1 + y 2 2 ). Now, consider the Poisson structure on U where h(y) := Then, it follows that P 0 µ ξ = ξ U and the S 1 -action is cotangential, and canonical on (U, P 0 ) but not Hamiltonian. The last assertion follows from the fact that there exists a symplectic leaf i : S → U of P 0 such that i * µ ξ is not exact.
In the 3-dimensional case, to construct a new Poisson G-space with premomentum map from a given one, we can try to use the conformal invariance property of Poisson structures. dβ ∧ β = 0. Suppose that there exists a canonical and cotangential G-action on M with premomentum map µ : g → Ω 1 (M ). Then, for any nowhere vanishing function k ∈ C ∞ (M ), the Poisson structures P and kP have one and the same symplectic foliation and hence, the G-action is also cotangential relative to kP . Moreover, this action is canonical on (M, kP ) if and only if dk ∧ µ ∧ β = 0.
In general, the previous example gives rise to the question on the study of deformation of Poisson G-spaces with pre-momentum. The Case of Poisson Foliations. Recall that a Poisson foliation is a triple (M, F, P ) consisting of a regular foliated manifold (M, F) equipped with a vertical Poisson bivector field P ∈ Γ(∧ 2 V), [P, P ] SCH = 0. Thus, the Poisson structure P is characterized by the property: every symplectic leaf of P is contained in a leaf of F.
A connection γ on the foliated manifold (M, F, P ) is said to be Poisson if every The class of Poisson connections on foliated manifolds also appears in the context of semilocal Poisson geometry [18,19,21]. Lemma 4.6. Let (M, F, P ) be a Poisson foliation. Suppose that the G-action is leaf-tangent (condition (12)) and canonical relative to P , Then, the G-averageγ of every Poisson connection γ on (M, F, P ) is again Poisson. Moreover, the curvature ofγ has the following property: if Proof. Taking into account that the action is canonical and γ is a Poisson connection, by standard properties of the averaging operator we obtain for all Z ∈ Γ pr (H γ ), that is, the average of a γ-horizontal projectable vector field is Poisson. Under the assumption that the action is leaf tangent, the item (b) of Lemma 3.3 implies that theγ-horizontal projectable vector fields are Poisson and hence,γ is a Poisson connection. The last assertion of the lemma follows directly from (5). Now, for the class of canonical and cotangential actions on a Poisson foliated manifold, we get the following result.
Theorem 4.7. Suppose we have a G-action on the Poisson foliation (M, F, P ) which is canonical and cotangential with pre-momentum µ. Let γ be an arbitrary Poisson connection andγ := γ G its G-average. Then, the connection difference form Ξ G = γ −γ takes values on Hamiltonian vector fields of the leaf-tangent Poisson structure P , that is where Moreover, the curvature of the averaged connectionγ is given by Proof. Observe that the mapμ : g → Ω 1 (M ) given bȳ is also a pre-momentum for the G-action. Indeed, since P •γ * = P , we have Now, pick Z ∈ Γ pr (H γ ). By item (c) of Lemma 3.3, taking into account that Z ∈ Γ(H γ ) is Poisson, and the fact that the G-action is canonical, we obtain Combining the above results, we get Therefore, the first term in (23) vanishes. This proves (19) and (20). Formula (21) follows from (6).
As consequences of the proof of Theorem 4.7, we have the following results.
Corollary 4.8. The following assertions hold.
(c) Let Z ∈ Γ pr (Hγ). Since Z is a G-invariant Poisson vector field, it follows that 0 = [Z, ξ M ] = P (L Z µ ξ ), for all ξ ∈ g. From this fact and condition (25), we have for every α ∈ Ω 1 (M ). This implies that i Zμ ξ is a Casimir function.
By Lemma 3.1, we derive the following consequence of Theorem 4.7.
Corollary 4.9. The horizontal distribution of the averaged connectionγ is generated by the G-invariant Poisson vector fields of the form where Z runs over Γ pr (H γ ).
Remark 4.1. In the context of the Poisson cohomology of (M, P ), one can derive from Corollary 4.9 the following fact [2]: for every γ-horizontal k-cocycle A ∈ Γ(∧ k H γ ), [P, A] SCH = 0, its Poisson cohomology class is represented by a G-invariant k-tensor. This partially recovers a result on the Poisson invariant cohomology due to [8].
To end this section, let us consider some special cases. It is clear that the hypotheses of Theorem 4.7 hold in the case when the G-action is locally Hamiltonian on (M ; P ), that is, the pre-momentum map µ is closed- In particular, in the standard case [12] of a Hamiltonian G-action with momentum map J : M → g * , ξ M = P dJ ξ , formula (26) for the horizontal 1-form Q reads Theorem 4.7 presents a generalized version of the results on Hannay-Berry connections obtained in [12] for the case of a Poisson fiber bundle equipped with Hamiltonian G-action with momentum map. Thus, in the case of a canonical and cotangential G-action with pre-momentum map µ on a Poisson foliation (M, F, P ), the averaged Poisson connectionγ = γ G can be also called a Hannay-Berry connection on (M, F, P ).
Here is an example of a cotangential and canonical G-action on a Poisson fibration which is not Hamiltonian.
, one can verify without serious difficulty that γ is a Poisson connection on (M → B, P ). Consider the leaf-preserving S 1 -action on M given by the infinitesimal generator ξ U = y 3 y 1 ∂ ∂y1 + y 2 ∂ ∂y2 + g(x, y) ∂ ∂y3 . It follows from Example 4.2 that this action is canonical and contangential with premomentum map µ ξ = − y2 y 2 1 +y 2 2 d y 1 + y1 y 2 1 +y 2 2 d y 2 , but not Hamiltonian. By Theorem 4.7, the difference connection form Ξ S 1 takes values in Hamiltonian vector fields. By (26) and straightforward computations, we get that and the Hannay-Berry connetion of γ is

Poisson connections with Hamiltonian curvature.
Starting with a Poisson foliation (M, F, P ), let us denote by Conn H (M, F, P ) the set of all Poisson connections γ whose curvature form takes values in the space of Hamiltonian vector fields of the vertical Poisson structure P . More precisely, for a certain horizontal 2-from σ γ ∈ Γ(∧ 2 V 0 ) which is called a Hamiltonian form of the curvature. Denote by C k := C k (M, F, P ) the space of all horizontal k-forms β ∈ Γ(∧ k V 0 ) which take values in the space Casim(M, P ) of Casimir functions of P , β(X 1 , ..., X k ) ∈ Casim(M, P ) ∀X i ∈ X pr (M, F).

Then, it is clear that a Hamiltonian form σ γ of the curvature in (27) is unique up to the transformations
In particular, if σ γ ∈ C 2 , then the connection is flat and the covariant exterior derivative d γ 1,0 is a coboundary operator.
The first term of each summand vanishes due to the Poisson property of γ. Taking into account (27), the fact d γ 1,0 σ ∈ C 3 follows from the Bianchi identity: On the other hand, one can also show from (27) that Observe that d γ 1,0 (C k ) ⊂ C k+1 . Hence, one can define the coboundary operator d γ : C k → C k+1 just byd γ := d γ 1,0 C k . Thus, one can associate to the setup (M, F, P, γ) the cochain complex (⊕ ∞ k=0 C k ,d γ ) called the de Rham-Casimir complex [14,21]. In terms of this complex, (30) and (31) say that d γ 1,0 σ is a 3-cocycle. Taking into account that the freedom in the choice of σ γ is given by the transformation (28), we derive the following fact: the connection γ is admissible if and only if the cohomology class of d γ 1,0 σ relative tod γ is trivial. Indeed, under the triviality property, we have d γ 1,0 σ = d γ 1,0 c for some c ∈ C 2 . From here, we have that σ − c is a Hamiltonian form for Curv γ satisfying (29).
We remark that if we have γ,γ ∈ Conn H (M, F, P ) such that the connection difference form Ξ := γ −γ takes values in Hamiltonian vector fields, thendγ =d γ . By Theorem 4.7, for a contangential and canocial G-action on (M, F, P, γ) the de Rham-Casimir complex associated to the complex (⊕ ∞ k=0 C k ,d γ ) remains the same after averaging the connection form γ.
Example 5.2. Let π : N → B be a principal G-bundle and (F, P F ) a Poisson G-space with pre-momentum map µ F : g → Ω 1 (F ). Consider the associated bundle M = N × G F which is a locally tivial fiber bundle (M, P ) over B with typical fiber bundle (F, P F ). Here, P is a vertical Poisson tensor on M defined by P = (π N ×F ) * P F , where π N ×F = N × F → N × G F is the canonical projection. A given connection θ ∈ Ω 1 (N ; g) on the principal bundle N induces an Ehresmann connection γ on M with horizontal lift Then, γ is a Poisson connection on (M, P ) [22] . Moreover, the curvature form of γ is given by Curv γ (hor γ (u), hor γ (v)) = P (π N ×F ) * µ F ((Curv θ )(hor θ (u), hor θ (v))) for u, v ∈ X(B). According to [22], the connection γ is admissible in the sense of Definition 5.1, if (F, P F ) is a Hamiltonian G-space with a momentum map. Now, suppose that we are given an action Φ on M of a connected and compact Lie group G which is canonical and cotangential with a pre-momentum map µ. Since all infinitesimal generators ξ M of the G-action are tangent to the symplectic foliation of P , we have Hence, any horizontal 2-form C ∈ C 2 is G-invariant, L ξ M C = 0. It follows that the G-invariance of a Hamiltonian form σ γ is preserved under transformation (28). Furthermore, the G-invariance of Poisson connection γ implies the G-invariance of any Hamiltonian form σ. Proof. First, we shall prove that L ξ M σ is G-invariant for all ξ ∈ g. Since the Frölicher-Nijenhuis bracket is natural with respect to the pull-back, the curvature Curv γ is G-invariant. So, for every ξ ∈ g and Z 0 , Z 1 ∈ Γ pr (H γ ), we have Thus, L ξ M σ ∈ C 2 . It follows from here that L ξ M (L ξ M σ) = 0 for all ξ ∈ g and hence L ξ M σ is G-invariant. Due to the compactness of G [15, Theorem 5.18] g = g ⊕ z(g), where g , z(g) denote the derived algebra and the center of g, respectively. The G-invariance of L ξ M σ for each ξ ∈ g implies that L ξ M σ = 0 for all ξ ∈ g . On the other hand, z(g) is generated by elements ξ such that exp tξ is a closed curve. Then, the flow of ξ M is periodic with period, say T ξ > 0. The G-invariance of L ξ M σ implies that d dt (Fl t ξ M ) * σ = L ξ M σ. By integrating in t, we obtain Since T ξ > 0, we conclude that L ξ M σ = 0 for all ξ ∈ z(g). Therefore, σ is Ginvariant.
Since the G-action is canonical relative to P and preserves the vertical distribution V, it is easy to see that the group G naturally acts on the set of Poisson connections Conn H (M, F, P ), γ → Φ * g γ. For every β ∈ Γ(∧ q V 0 ) and Q ∈ Γ(∧ 1 V 0 ), denote by {Q ∧ β} P the element of Γ( q+1 V 0 ) given by where {·, ·} is the Poisson bracket associated to P .
As a consequence of Theorem 4.7, we get the following fact.
Theorem 5.4. Suppose that we are given an action Φ on M of a connected and compact Lie group G which is canonical and cotangential with a pre-momentum map µ.The averaging procedure preserves the set Conn H (M, F, P ), that is, Moreover, if σ is a Hamiltonian of the curvature of γ, then a Hamiltonian form of the curvature of γ is given bȳ where the horizontal 1-form Q ∈ Γ(V 0 ) is defined in terms of µ by formula (20). Furthermore, the admissibility of γ, implies that γ is also admisible, Proof. The first part of the theorem is a direct consequence of Theorem 4.7. In particular, the formula for the Hamiltonian form of γ follows from equation (21). So, it remains to prove that the averaging procedure preserves the admissibility property. Assume that d γ 1,0 σ = 0. Since σ ∈ Γ(∧ 2 V 0 ), the relation (8) implies the equality Recall that the exterior differential has the following bigraded decomposition d = d γ 1,0 + d γ 0,1 + d γ 2,−1 depending on the connection γ. Taking account that d 2 = 0, we obtain the following identity (d γ In particular, for Q ∈ Ω 1,0 (M ), the equation (27) implies that On the other hand, since γ is a Poisson connection, by straightforward computation, we obtain that From relations (32)-(35), it follows that dγ 1,0σ = 0.
6. Cotangential actions on Dirac manifolds. First, we recall some facts from the theory of Dirac structures (for more details, see [5,6,7] provided that D is smooth. Suppose we are given an action Φ : G × M → M of a connected Lie group G on the Dirac manifold (M, D). The G-action is said to be Dirac or canonical if for all (X, α) ∈ Γ(D) and g ∈ G. In other words, this condition is just the invariance of the distribution D with respect to the G-action. Infinitesimally, (36) reads Conversely, if the Lie group G is connected, condition (37) means that the G-action is Dirac.
Definition 6.1. A G-action on the Dirac manifold (M, D) is said to be cotangential if there exists a R-linear mapping µ ∈ Hom(g, Ω 1 (M )) such that In this case, we say that µ is a pre-momentum map for the cotangential G-action on (M, D).
By ( In terms of the presymplectic form, condition (38) can be rewritten as follows i ξ M ω S = −i * S µ ξ , for any presymplectic leaf S and ξ ∈ g.
Denote by } the set of all 1-forms α on M vanishing along the characteristic foliation S, i * S α = 0 for each leaf S of S. Then, µ is uniquely determined by (38) modulo elements κ ∈ Hom(g, I 1 T S (M )), µ −→ µ + κ. Since i * S dκ ξ = 0, the pull-back i * S dµ ξ is independent of the freedom in the choice of µ ξ .
Our next result states necessary and sufficient conditions under which a cotangential G-action of (M, D) is canonical.
that is, µ ξ is closed along every presymplectic leaf S.
Proof. Computing the Courant bracket between (ξ M , µ ξ ) and a section (X, α) ∈ Γ(D) and using the isotropy property, we get It follows that condition (37) is equivalent to (39).
We remark that Lemma 6.2 can be reformulated in the following way: the Dirac structure D is canonical with respect to a cotangential G-action if and only if the pull-back of µ to each presymplectic leaf is closed.
In particular, if µ in (38) is exact-valued, µ = d • J, for some R-linear function J : g → C ∞ (M ), then one says that we have a Hamiltonian G-action on the Dirac manifold with momentum map J (see, also [3]).
Finally, we observe that if a cotangential G-action on (M, D) is locally Hamiltonian, dµ ξ = 0 for all ξ ∈ g, then this action is canonical.

Averaging the coupling Dirac structures.
Our point is to construct G-invariant Dirac structures on (M, F, P ) by combining the averaging procedure for Poisson connections in Conn H (M, F, P ) with the so-called coupling method (see also [7,19,20]). Dirac structures from admissible Poisson connections. Now, pick an admissible Poisson connection γ ∈ Conn H (M, F, P ), and fix a Hamiltonian 2-form of the curvature σ = σ γ in (27). Then, one can introduce the distribution D γ,σ given by the following subbundle of T M ⊕ T * M : It is clear that D γ,σ is a regular distribution whose rank is just equal to dim M . By straightforward computations, one can show that D γ,σ is an isotropic distribution relative to the symmetric form. Moreover, we have the following fact. Proof. We only need to prove that D γ,σ is closed under the Courant bracket. Taking into account that D γ,σ = Graph(P )| H 0 ⊕ Graph(σ)| H , we fix a set of (local) generators of D, defined by the elements of the form e α = (P (α), α) and e X = (X, −i X σ), with X ∈ Γ pr (H γ ) and α ∈ Γ((H γ ) 0 ). Since the bivector P and the connection γ are Poisson, we have e α , e β = P L P (α) β − i P (β) dα , L P (α) β − i P (β) dα ∈ D γ,σ .
Finally, the admissibility of γ and the curvature identity imply that Remark 7.1. In fact, D γ,σ is a coupling Dirac structure on the foliated manifold (M, F) associated with the geometric data (P, γ, σ), [19,20].
Invariant Dirac Structures. Now we suppose that on the Poisson foliation (M, F, P ), we are given a canonical action of a connected and compact Lie group G which is contangential respect to P with pre-momentum map µ, that is, ξ M = P µ ξ for all ξ ∈ g. Lemma 7.2. Under the hypothesis above, the G-action is cotangential on the Dirac structure D γ,σ .
Evaluatingω S on the generating elements, we conclude thatω S = ω S − dQ| S . Since the G-action admits a pre-momentum map, the average of ω S can be written as ω S G = ω S − i * S dQ, where i S : S → M is the canonical injection, (see [20]). Hence, ω S = ω S G and the G-invariance ofσ follows from here and the G-invariance of τ . The G-invariance of Dγ ,σ+C is a consequence of Lemma 7.3. Therefore, the G-action on Dγ ,σ+C is canonical.
Corollary 7.5. The Dirac structures D γ,σ and Dγ ,σ are related by gauge transformation associated with the exact 2-form dQ.
Remark 7.2. An alternative way to prove the G-invariance of the Dirac structure Dγ ,σ+C in Theorem 7.4 is to apply Proposition 5.3 with Lemma 7.3. Combining Theorem 7.4 with the fact that a Dirac structure D on M around an embedded pre-symplectic leaf S is realized as a coupling Dirac structure [19], one can apply the averaging method relative to a cotangential and canonical G-action on (M, D) in order to construct a G-invariant Dirac structure in a neighborhood of S.
Example 7.6. Let (M, F) be a foliated manifold equipped with a leaf-tangent G-action. Let D be a Dirac structure on M and S an embedded pre-symplectic leaf of D such that T S M = T S ⊕ T S F. Assume that the G-action is canonical and cotangential relative to D with pre-momentum map µ : g → Ω 1 (M ). If S is invariant with respect to the G-action then there exists a G-invariant neighborhood U of S in M with the following properties: (a) the restriction D U = D| U is a coupling Dirac structure on (U, F U ), D U = D γ,σ , where γ, σ and the vertical Poisson tensor P in (40) are uniquely defined by D U [19]; (b) the G-action on U is canonical and cotangential relative to P , ξ M | U = P ( µ ξ U ). By Theorem 7.4, Dγ ,σ is a Ginvariant Dirac structure on the neighborhood U which is gauge equivalent to the original one.
The Poisson Case. Under the above assumptions, suppose that the Lie group G acts in Hamiltonian fashion with momentum map J : g → C ∞ (M ), µ ξ = dJ ξ . Suppose that the Dirac structure D is the graph of a bivector fieldΠ. Then,Π is a Poisson tensor on M which is G-invariant. If the cohomology class of thed γ -cocycle d γ 1,0 J G is trivial, then the G-action is canonical with momentum map J relative toΠ (see, also [1]).