$G$-Invariant Deformations of Almost-Coupling Poisson Structures

On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of deformation theory and the method of averaging.


Introduction
In this paper, we develop further the results of [18,19], on the construction of invariant Poisson and Dirac structures via the averaging method on foliated Poisson manifolds with symmetry in the context of deformation theory.
For a Poisson bivector field Π on a foliated manifold (M, F), the F-almost-coupling property [16] means the existence of a normal bundle structure H of F, such that in the H-dependent bigraded decomposition of Π, the mixed term of bidegree (1,1) vanishes. In particular, such class of Poisson structure contains the F-coupling Poisson structures which play important roles in some problems of semi-local Poisson geometry [9,10,16,20]. Moreover, the (almost) coupling constructions can be naturally extended to the Dirac category [5,17,21]. Now, starting with a foliated Poisson manifold (M, F, P ), equipped with a leaf preserving action of a compact connected Lie group G, our point is to study, in the context of the averaging method, some deformations {Π ε } ε∈[0,1] of the leaf-tangent 1 Poisson bivector field P in the class of the F-almost-coupling Poisson structures Π ε . Our approach is based on the averaging technique [18]  For the case of a G-action which is locally Hamiltonian relative to P , we give some results about the realization of the above scheme for a class of F-almost-coupling Poisson deformations of P . Within the framework of perturbation theory, these results can be applied to the study of invariant normal forms for Hamiltonian systems of adiabatic type [1], associated to deformations of Poisson structures [2,4].
In a generalized setting, by a Hamiltonian system of adiabatic type on a Poisson foliation (M, F, P ) with G-symmetry, we mean a Hamiltonian system relative to a deformed Poisson structure Π ε = P + εΛ and a function F ∈ C ∞ (M ). The deformation of P is given by a bivector field Λ ∈ Γ(∧ 2 H), where H ⊂ T M is a normal bundle of the foliation F. In the (adiabatic) limit ε → 0, the unperturbed Hamiltonian system (M, P, F ) is G-invariant and describes the fast dynamics along the leaves of F. The key point is to move the original perturbed system to a Ginvariant one by using a near-identity transformation. The construction of the corresponding G-invariant model is related to the averaging procedure for the deformed Poisson structure Π ε . In a particular setting this problem was studied in [4], here we present a general mechanism.

Preliminaries
Let us recall some basic facts that will be used later on, related to the averaging procedure with respect to the action of compact Lie groups, for Poisson and Dirac structures [18].

Gauge transformations of Dirac manifolds
Let (M, D) be a Dirac manifold, that is, a smooth regular distribution D ⊂ T M ⊕ T * M which is maximally isotropic with respect to the natural symmetric pairing on T M ⊕ T * M , and involutive with respect to the Courant bracket [7,8]. The Dirac manifold (M, D) carries a (singular) presymplectic foliation (S, ω): Its leaves are the maximal integral manifolds of the integrable (singular) distribution C := p T (D) ⊂ T M , and the leafwise presymplectic structure ω is defined by We can modify the leafwise presymplectic structure ω by the pull back of a closed 2-form on the base B ∈ Ω 2 (M ): For each presymplectic leaf (S, ω S ), we define the new presymplectic structure as ω S + ι * S B, where ι S : S → M is the inclusion map. Then, the foliation S equipped with the deformed leafwise presymplectic structure gives rise to the new Dirac structure This transformation τ B preserves the presymplectic foliation of D, and is called the gauge transformation associated to the closed 2-form B [6,14].
In particular, the foliation (S, ω) is symplectic if and only if D is the graph of a Poisson bivector field Π on M , where Π : T * M → T M is the induced vector bundle endomorphism given by α → i α Π. Condition (2.1) can be expressed as follows Notice that, in general, for a given closed 2-form B, the gauge transformation τ B takes Graph Π to another Dirac structure which may not necessarily come from a Poisson bivector field. We have the simple, but useful, criterion (see [14, pp. 5-6]). is invertible, then the Dirac structure τ B (GraphΠ) is the graph of the Poisson tensor τ B (Π) whose induced endomorphism is defined by Here, we denote by B : T M → T * M the vector bundle endomorphism given by X → i X B.

G-averaging procedure
Let G be a connected, compact Lie group, and g its Lie algebra. Suppose we are given a smooth (left) action Φ : G × M → M on a manifold M . For every a ∈ g, the corresponding infinitesimal generator is denoted by a M ∈ X(M ), For every tensor field T on M , we denote by T G its G-average, which is defined as where dg is the normalized Haar measure on G. This averaging procedure can be also applied to Dirac structures [18]: Let D ⊂ T M ⊕ T * M be a Dirac structure on M , and let (S, ω) be its associated presymplectic foliation, carrying the leafwise presymplectic form ω.
Definition 2.2. The G-action on M is compatible with the Dirac structure D if each leaf S, of S, is invariant under the action of G, and there exists a R-linear mapping ρ ∈ Hom(g, Ω 1 (M )) such that for every a ∈ g (here, ω S is a presymplectic structure on S, and ι S : S → M is the canonical injection).
Equivalently, condition (2.2) can be rewritten as follows (a M , ρ a ) ∈ Γ(D) for all a ∈ g.
Then, as a consequence of the fact that G is compact connected (hence the exponential map is surjective) we have the following fact [18]. (a) The leafwise presymplectic form of D is defined as where Θ ∈ Ω 1 (M ) is the 1-form determined in terms of the G-action and ρ by (c) The Dirac structure D is is related to D by an exact gauge transformation: The Dirac structure D will be called the G-average of D relative to the compatible G-action. As we have mentioned above, for the case in which the Dirac structure is the graph of a Poisson tensor Π on M , D = Graph Π, its G-average D = τ B (D), where B = −dΘ, does not necessarily come from a Poisson structure. In other words: In general, the averaging of Poisson structures via compatible G-actions only leads to G-invariant Dirac structures; but, by Lemma 2.1, in the particular case of an invertible endomorphism we can say something more, namely, that the G-average D is the graph of the G-invariant Poisson tensor Π = τ B (Π).

G-invariant connections
Recall that a (generalized Ehresmann) connection on a manifold M [13], is a vector-valued 1-form γ ∈ Ω 1 (M ; T M ) satisfying the following conditions: γ 2 = γ, and the rank of the distribution Im γ ⊂ T M is constant on M (this last condition is a consequence of the first for M connected). Assume that the action Φ : G × M → M , of the compact, connected Lie group G on the connected manifold M , preserves the image of γ, d q Φ g •γ q = γ Φg(q) , for all q ∈ M , g ∈ G. Then, the G-average of γ is the vector-valued 1-form γ G ∈ Ω 1 (M, T M ) defined by the formula where X ∈ X(M ) is any vector field, and this is a G-invariant connection on M .

F -almost-coupling structures
In this section, we present some basic definitions and facts concerning the coupling procedure on foliated Poisson and Dirac manifolds (for more details, see [16,17,20]). The term 'coupling' comes from Sternberg's coupling method on symplectic fiber bundles (see, for example, [12]). Suppose we are given a manifold with a (regular) foliation (M, F). We will denote by V := T F the tangent bundle of F, and by V 0 = Ann(T F) ⊂ T * M its annihilator. By a normal bundle of the foliation F we mean a sub-bundle H ⊂ T M which is complementary to the tangent bundle V, To a normal bundle H, there is associated a vector-valued 1-form γ ∈ Ω 1 (M ; T M ), defined as the canonical projection along H, γ := pr H : T M → V. This form satisfies γ 2 = γ and Im γ = V, and hence defines an Ehresmann connection on the foliated manifold (M, F) (see [13]). The curvature of the connection γ is the vector-valued 2-form R γ ∈ Ω 2 (M ; V) given by R γ = 1 2 [γ, γ] FN , where [·, ·] FN denotes the Frölicher-Nijenhuis bracket for vector valued forms on M [13]. The curvature controls the integrability of the normal sub-bundle, in the sense that H is integrable if and only if γ is flat, R γ = 0.
According to (3.1), we have the dual splitting Then, (3.1) and (3.2) induce an H-dependent bigrading of multivector fields and differential forms on M . For any A ∈ Γ(∧ k T M ) and α ∈ Ω k (M ), we have A = s+l=k A s,l and α = s+l=k α s,l , where the elements A s,l and α s,l , belonging to the subspaces Γ(∧ s H) ⊗ Γ(∧ l V) and , respectively, are said to be multivector fields and forms of bidegree (s, l). Moreover, the exterior differential d on M inherits the H-bigrading decomposition (see [16]) in the form d = d 1,0 + d 2,−1 + d 0,1 .
We will need the following definition [16,20].
This condition means that, in the bigraded decomposition of Π associated to (3.1), the mixed term Π 1,1 is zero and Π = Π 2,0 Remark 3.2. Notice that conditions (3.1), (3.3) hold whenever the following transversality and regularity conditions are satisfied (in this case we say that the bivector field Π is compatible with the foliation F): The F-coupling situation occurs when, along with (3.4), we have and hence the normal bundle H of F in (3.3), associated to Π, is unique and given by In this case, Π is said to be an F-coupling Poisson structure. The factorization of the Jacobi identity for Π (see [18, equations (6.4)-(6.6)]) implies that the intrinsic connection γ, associated with the normal bundle (3.6), possesses the following properties [16,20]: The connection γ is Poisson on the Poisson bundle (M, F, Π 0,2 ), that is, L X Π 0,2 = 0 for any projectable section X ∈ Γ pr (H) (recall that the projectability property for X on the foliated manifold (M, F) is expressed as [X, Γ(V)] ⊂ Γ(V)). Moreover, the curvature of γ takes values in the space of Hamiltonian vector fields of Π 0,2 : Here, the 2-form σ ∈ Γ(∧ 2 V 0 ), called the coupling form, is uniquely determined by Π, and satisfies the γ-covariant constancy condition d 1,0 σ = 0. Indeed, we can write the following explicit expression in terms of the horizontal part of Π and the horizontal lifting induced by γ [20]: for any u 1 , u 2 ∈ X (M ) vector fields on M .
The notion of F-almost-coupling structures can be naturally generalized to the Dirac setting [17]. Given a Dirac structure D ⊂ T M ⊕ T * M on the foliated manifold (M, F), we define the tangent distribution On the other hand, fixing a normal bundle H of F, we consider the distributions In the general case, these are singular distributions, but the almost-coupling hypothesis implies that their respective ranks are constant, rank D H = rank H and rank D V = rank V. One can show that condition (3.8) is equivalent to the following one: Also, one says that D is an F-coupling Dirac structure [17,21] if H(D, F) is a normal bundle of F, Notice that (3.8) implies the following property: It follows from here that a given an F-almost coupling Dirac structure D via H induces a leaf-tangent Poisson bivector field P ∈ Γ(∧ 2 V) such that The relation between geometric data and F-coupling Dirac structures [17,18,21], carries over to the almost-coupling setting, with some particularities: given a connection γ, the almostcoupling Dirac structure directly induces a leaf-tangent Poisson structure P , and no coupling form σ (in the terminology of [18]) appears. We can use that correspondence to characterize (in terms of P ) a particular class of gauge transformations which preserve 2 the class of F-almost coupling Dirac structures on a given foliated manifold (M, F) (this class naturally appears in the context of the averaging method). To this end, let us pick an arbitrary 1-form Q ∈ Γ(V 0 ) and consider the exact gauge transformation Here, P ∈ Γ( 2 V) is the Poisson bivector field in (3.9).
Notice also that the exact gauge transformation (3.10) leaves invariant the set of all Fcoupling Dirac structures on (M, F) [18].

The averaging theorems for deformations
Let (M, F, P ) be a Poisson foliation, consisting of a regular foliation F on M , and a leaf-tangent Poisson bivector field P ∈ Γ(∧ 2 V). Recall that we denote by V = T F ⊂ T M the tangent bundle of F and by V 0 ⊂ T * M its annihilator. The Poisson tensor P is characterized by the property that the symplectic leaf of P through each point q ∈ M , is contained into the leaf F q of the regular foliation. Another characterization, given in [17], states that the leaves of F are Poisson submanifolds.
Consider a smooth (left) action on M , Φ : G × M → M , of a connected and compact Lie group G, which is compatible with the Poisson structure P ∈ Γ(∧ 2 V), in the sense of Definition 2.2: For every a ∈ g, the infinitesimal generator a M has the form a M = P µ a for all a ∈ g, (4.1) for a certain R-linear mapping µ ∈ Hom(g, Ω 1 (M )). In particular, this property means that the G-action preserves the leaves of the symplectic foliation of P , and condition (2.2) holds, that is, the G-action is compatible with the associated Dirac structure D = Graph P .
A G-action on the Poisson foliation is said to be locally Hamiltonian, if each infinitesimal generator a M is a locally Hamiltonian vector field relative to P , in other words, one can choose the 1-form µ a in (4.1) to be closed on M , µ a ∈ Ω 1 cl (M ) for all a ∈ g.

(4.2)
Suppose now that we start with a smooth deformation {Π ε } ε∈[0,1] of the leaf-tangent Poisson structure P , so each Π ε is a Poisson tensor on M , the whole family is smoothly dependent in ε, and is such that Π 0 = P . Moreover, we assume that, for each ε ∈ [0, 1], the bivector field Π ε is an F-almost-coupling Poisson structure via an ε-independent normal bundle H of F: Such a family {Π ε }, will be called an F-almost-coupling Poisson deformation of P via H. It follows from (4.3) that each Π ε admits the following bigraded decomposition relative to (3.1): We will also assume that the leaf-tangent component of Π ε of bidegree (0, 2) is independent of ε and hence Then, (Π ε ) 2,0 = εΛ ε for a certain bivector field Λ ε of bidegree (2, 0), smoothly varying in ε, and therefore, the deformation of P can be parameterized as For a fixed R-linear mapping µ ∈ Hom(g, Ω 1 (M )) in (4.1), we can decompose µ = µ 1,0 + µ 0,1 relative to splitting (3.2), associated to a fixed normal bundle H in (4.3).
Finally, let us associate to the family {Π ε } (as in (4.4)) a smooth ε-dependent family of Dirac structures D ε := Graph Π ε = Graph P ⊕ Graph εΛ ε , (4.5) which is to be viewed as a deformation of the 'limiting' Dirac structure D 0 = Graph Π 0 = Graph P . Then, for each ε, the Dirac structure D ε also satisfies the F-almost-coupling condition (3.8). Under the above hypotheses, we are in position to state our main results regarding the existence of the G-average for these deformations of Dirac and Poisson structures. The first one is a consequence of Lemma 2.3.
Theorem 4.2. If the G-action is locally Hamiltonian (condition (4.2)), then, for any Ginvariant open connected set with compact closure, N ⊂ M , and for sufficiently small ε ∈ ]0, 1[, the restriction of the Dirac structure (4.6), D ε | N , is the graph of a G-invariant Poisson structure Π ε on N defined by with Π 0 = P . Here Q ∈ Γ(V 0 ) is expressed as Moreover, the Poisson structure Π ε is F-almost-coupling via the following G-invariant normal bundle of F: Proof . Under the hypothesis that the G-action is locally Hamiltonian, let us fix a G-invariant relatively compact domain N ⊆ M . Let B = −dΘ, with Θ constructed as in (2.3). By Lemma 2.1 it suffices to show that there exists a 0 < δ < 1 such that In terms of the bigraded components of µ, the closedness condition dµ = 0 splits into By using these relations, we see from (2.3) that B = −dΘ = −dQ, where Q ∈ Γ(V 0 ) is given by (4.8). It follows that B = B 2,0 + B 1,1 , where B 2,0 = −d 1,0 Q and B 1,1 = −d 0,1 Q. From here, taking into account (4.4), for an arbitrary α = α 1,0 + α 0,1 ∈ Ω 1 (M ), we get This shows that the matrix of the morphism Id −B • Π ε : T * M → T * M , in a local basis compatible with the splitting (3.2), has the form This fact, together with a standard compactness argument, proves (4.10). Furthermore, it follows that and P • (dQ) • P = 0. (4.12) From here and (4.7), we conclude that Π 0 = P . Finally, to show that the Poisson structure Π ε is F-almost coupling, we consider the connection γ associated to the normal bundle H and its G-average γ G . Then, the normal bundle ker γ G just coincides with H in (4.9) (see [18]). Applying Lemma 3.5 ends the proof.
Consider now the case of Π ε a coupling Poisson structure for ε = 0, that is, H = Π ε (V 0 ) is a normal bundle of F. Then, the corresponding Dirac structure is represented as where the 2-form σ ε ∈ Γ(∧ 2 V 0 ) smoothly depends in ε, satisfies the H-covariant constancy condition d 1,0 σ ε = 0, and has the following expansion around ε = 0: Here, the 2-form c ∈ Γ(∧ 2 V 0 ) takes values in the Casimir functions of P , c(X, Y ) ∈ Casim(M, P ) for all X, Y ∈ Γ pr (H), and the 2-form σ ∈ Γ(∧ 2 V 0 ) satisfies the curvature identity (3.7) with Π 0,2 = P (for more details, see [16,18]). From these remarks and Theorem 4.2, we deduce the following consequence. Corollary 4.3. In the coupling case, for a locally Hamiltonian G-action, the G-average D ε is an F-coupling Dirac structure via the invariant normal bundleH given by In particular, D 0 = Graph(P ).
A natural question concerns the relationship between the original, ε-dependent, Poisson structure Π ε , and its G-average Π ε , as ε tends to 0.
Theorem 4.4. Let Π ε be the G-average of the F-almost-coupling Poison structure Π ε defined in (4.7). Let N ⊂ M be a G-invariant open subset whose closure is compact. There exists a smooth isotopy φ ε : N → M , such that, for ε sufficiently small, Proof . Following the reasoning in the proof of Theorem 4.2, we can show that there exists a δ > 0 such that the gauge transformation of Π ε , associated to the exact 2-form tB = −tdQ, exists for all ε ∈ [0, δ], t ∈ [0, 1], and gives the 2-parameter family of Poisson structures on N , Π ε,t , characterized by Then, Π ε,0 = Π ε and Π ε,1 = Π ε . Fixing ε ∈ [0, δ], one can verify [11,18] that the time-dependent vector field on N given by
Proof . For the infinitesimal generator Z ε,t in (4.14), and the family of Poisson structures Π ε,t in (4.13), we can evaluate their expansion around ε = 0: Z ε,t = εW t + O(ε 2 ) and Π ε,t = P + εΨ t + O(ε 2 ), where W t and Ψ t are a time-dependent vector field and a bivector field, respectively. Putting these expansions into (4.15), and collecting all first-order terms in ε, leads to the equation [[W t , P ]] = − dΨt dt . Integrating this equation with respect to t, and taking into account that Ψ 0 = Λ 0 , and Remark 5.2. These results can be used to construct normal forms for perturbed dynamics associated to almost-coupling deformations of foliated Poisson manifolds with symmetry. Typically, such perturbed dynamics appear in the context of adiabatic theory [1] on nontrivial phase spaces, particularly those in which no global action-angle coordinates can be introduced [2, 3, 4].
We have a simple example of an F-almost-coupling Poisson deformation when the foliation F is a fibration. Example 5.3. Suppose we start with a Poisson fiber bundle (π : M → S, P ) whose base S carries a Poisson structure ψ ∈ Γ(∧ 2 T S). Assume that there exists a flat Poisson connection γ on M , associated to an integrable distribution H, which is complementary to the vertical one V = ker dπ. Let hor γ (ψ) ∈ Γ(∧ 2 H) be the γ-horizontal lift of the Poisson bivector field ψ. Then, by using the standard properties of the Schouten bracket (see [15]), one can show that the bivector field Π ε = P + ε hor γ (ψ) (5.2) satisfies the Jacobi identity and gives an almost-coupling Poisson structure for every ε ∈ [0, 1]. It is clear that Π ε is a coupling Poisson tensor on M if and only if the bivector field ψ is non-degenerate, and hence induces a symplectic form on S.
Finally, let us say some words about the physical meaning of the preceding constructions. Consider the Hamiltonian vector field X(ε) = i dF P + εi dF hor γ (ψ), relative to a function F ∈ C ∞ (M ) and the Poisson structure (5.2). As stated in the Introduction, the corresponding dynamical systeṁ ξ = εi dF hor γ (ψ),ẋ = i dF P, (5.3) belongs to the class of the so-called slow-fast Hamiltonian systems [1], where the coordinates ξ = (ξ i ), along the base S, and the coordinates x = (x α ), along the fibers of π, are called the slow and fast variables, respectively. We observe that the infinitesimal generator of the Poisson isotopy φ ε in Theorem 4.4, controls the 'geometric' part of an invariant normalization transformation for the system (5.3) as ε tends to zero. As stated in Remark 5.2, these transformations, and the normal forms to which they lead, are crucial for determining asymptotic properties of the system, such as stability, existence of periodic orbits, or the computation of adiabatic invariants, see [2] for some examples.