Reeb-Thurston stability for symplectic foliations

We prove a version the local Reeb-Thurston stability theorem for symplectic foliations.


Introduction
A symplectic foliation on a manifold M is a (regular) foliation F , endowed with a 2-form ω on T F whose restriction to each leaf S of F is a symplectic form ω S ∈ Ω 2 (S).
Equivalently, a symplectic foliation is a Poisson structure of constant rank.
In this paper we prove a normal form result for symplectic foliations around a leaves. The result uses the cohomological variation of ω at the leaf S, which is a linear map (see section 1 for the definition) (1) [δ S ω] x : ν * x −→ H 2 ( S hol ), x ∈ S, where ν denotes the normal bundle of T F , and S hol is the holonomy cover of S. The cohomological variation arises in fact from a linear map: (2) δ S ω x : ν * x −→ Ω 2 closed ( S hol ). The local model for the foliation around S, which appears in the classical results of Reeb and Thurston, is the flat bundle ( S hol × ν x )/π 1 (S, x), where π 1 (S, x) acts on the second factor via the linear holonomy (3) dh : π 1 (S, x) −→ Gl(ν x ).
For a symplectic foliations the flat bundle can be endowed with leafwise closed 2-forms, which are symplectic in a neighborhood of S; namely, the leaf through v ∈ ν x carries the closed 2-form j 1 S (ω) v whose pull-back to S hol × {v} is p * (j 1 S (ω) v ) = p * (ω S ) + δ S ω x (v). Our main result is the following: Theorem 1. Let S be an embedded leaf of the symplectic foliation (M, F , ω). If the holonomy group of S is finite and the cohomological variation (1) at S is a surjective map, then some open around S is isomorphic as a symplectic foliation to an open around S in the flat bundle ( S hol × ν x )/π 1 (S, x) endowed with the family of closed 2-forms j 1 S (ω) by a diffeomorphism which fixes S. This result is not a first order normal form theorem, since the holonomy group and the holonomy cover depend on the germ of the foliation around the leaf. The first order jet of the foliation at S sees only the linear holonomy group H lin (i.e. the image of dh) and the corresponding linear holonomy cover denoted S lin . Now, the map (2) is in fact the pull-back of a map with values in Ω 2 closed ( S lin ). Using this remark, and an extension to noncompact leaves of a result of Thurston (Lemma 2), we obtain the following consequence of Theorem 1.

Corollary 1.
Under the assumptions that S is embedded, π 1 (S, x) is finitely generated, H lin is finite, H 1 ( S lin ) = 0 and the cohomological variation is surjective, the conclusion of Theorem 1 holds.
Our result is clearly related to the normal form theorem for Poisson manifolds around symplectic leaves from [3]. Both results have the same conclusion, yet the conditions of Theorem 1 are substantially weaker. More precisely, for regular Poisson manifolds, the hypothesis of the main result in loc.cit. are (see Corollary 4.1.22 and Lemma 4.1.23 [5]): • the leaf S is compact, • the cohomological variation is an isomorphism, when viewed as a map There is yet another essential difference between Theorem 1 and the result from [3], namely, even in the setting of Corollary 1, the result presented here is a first order result only in the world of symplectic foliations, and not in that of Poisson structures. The information that a Poisson bivector has constant rank is not detectable from its first jet.
A weaker version of Theorem 1 is part of the PhD thesis [5] of the second author.

The local model and the cohomological variation
In this section we describe the local model of a symplectic foliation around a leaf, and define the cohomological variation of the symplectic structure on the leaves. In the case of general Poisson manifolds, the local model was first constructed by Vorobjev [9]. The approach presented here is more direct; for the relation between these two constructions see [5].
Let (M, F ) be a foliated manifold, and denote its normal bundle by Then ν carries a flat T F connection, called the Bott connection, given by where, for a vector field Z, we denote by Z its class in Γ(ν). For a path γ inside a leaf S, parallel transport with respect to ∇ gives the linear holonomy transformations: This map depends only on γ modulo homotopies inside S with fixed endpoints. Applying dh to closed loops at x, we obtain the linear holonomy group The linear holonomy cover of a leaf S at x, denoted by S lin,x is the covering space corresponding to the kernel of dh; thus it is a principal H lin,x bundle over S. Also, S lin,x can be defined as the space of classes of paths in S starting at x, where we identify two such paths if they have the same endpoint and they induce the same holonomy transport.
The Bott connection induces a foliation F ν on ν whose leaves are the orbits of dh; i.e. the leaf of F ν through v ∈ ν x covers the leaf S through x, and is given by Therefore, S lin,x covers of the leaves of the foliation F ν above S via the maps (4) p The local model of the foliation around the leaf S is the foliated manifold The linear holonomy induces an isomorphism between the local model and the flat bundle from the Introduction Consider now a symplectic structure ω on the foliation F , i.e. a 2-form on T F ω ∈ Ω 2 (T F ) whose restriction to each leaf is symplectic. We first construct a closed foliated 2form δω on (ν, F ν ), which represents the derivative of ω in the transversal direction. For this, choose an extension ω ∈ Ω 2 (M ) of ω and let Since ω is closed along the leaves of F , Ω(X, Y ) ∈ ν * , thus Ω ∈ Ω 2 (T F ; ν * ). Now, the dual of the Bott connection on ν * induces a differential d ∇ on the space of foliated forms with values in the conormal bundle Ω • (T F ; ν * ); this can be given explicitly by the classical Koszul formula , X 0 , . . . , X i , . . . , X j , . . . , X p ), for η ∈ Ω p (T F ; ν * ), X i ∈ Γ(T F ). Denote the resulting cohomology by H • (F ; ν * ).
It is easy to see that Ω is d ∇ -closed. In fact, this construction can be preformed in all degrees, and it produces a canonical map (see e.g. [2]) which maps [ω] to [Ω]. Also, if ω + α is a second extension of ω (where α vanishes along F ), then Ω changes by d ∇ λ, where λ ∈ Ω 1 (T F ; ν * ), is given by Note that there is a natural embedding where p : ν → M is the projection. It is easy to see that under J the differential d ∇ corresponds to the leafwise de Rham differential d Fν on the leaves of F ν . In particular, we obtain a closed foliated 2-form δω := J (Ω) ∈ Ω 2 (T F ν ), which we call the vertical derivative of ω. Since δω vanishes on M (viewed as the zero section), it follows that p * (ω) + δω is nondegenerate on the leaves in an open around M ; thus (ν, F ν , p * (ω) + δω) is a symplectic foliation around M .
Consider now a symplectic leaf S. Restricting p * (ω) + δω to the leaves above S, we obtain closed foliated 2-forms along the leaves of the F νS , denoted by is symplectic will be regarded as the local model of the symplectic foliation around S; i.e. we think about the local model as a germ of a symplectic foliation around S.
In order to define the cohomological variation of ω, consider first the linear map , where the map p v is the covering map defined by (4). By the discussion above, choosing a different extension of ω changes p * v (δ S ω) by an exact 2-form; hence the cohomology class [p * v (δ S ω)] is independent of the 2-form Ω used to construct δ S ω. The induced linear map to the cohomology of S lin,x , will be called the cohomological variation of ω at S In the Introduction we denoted the lifts of [δ S ω x ] to the holonomy cover S hol , respectively to the universal cover S uni of S, by the same symbol.
We finish this section by proving that, up to isomorphism, the local model is independent of the choices involved. The proof uses a version of the Moser Lemma for symplectic foliations (Lemma 5 from the next section).
produce local models that are isomorphic around S by a diffeomorphism that fixes S.

Proof. A second 2-form is of the form
for some λ ∈ Ω 1 (T F ; ν * ). We apply the Lemma 5 to the symplectic foliation (ν, F ν , p * (ω) + δω), and the foliated 1-form α := J (λ) which vanishes along M . The resulting diffeomorphism is foliated. In particular, above any leaf S of F it sends the local model corresponding to Ω to the local model corresponding to Ω ′ .

Five lemmas
In this section we prove some auxiliary results used in the proof of Theorem 1.

Reeb Stability around non-compact leaves
Consider a foliated manifold (M, F ) and let S be an embedded leaf. The classical Reeb Stability Theorem (see e.g. [6]) says that, if the holonomy group H hol is finite and S is compact, then a saturated neighborhood of S in M is isomorphic as a foliated manifold to the flat bundle where T is a small transversal that is invariant under the holonomy action of H hol . Since actions of finite groups can be linearized, it follows that the holonomy of S equals the linear holonomy of S. So, some neighborhood of S in (M, F ) is isomorphic as a foliated manifold with the flat bundle from the previous section Below we show that the proof of the Reeb Stability Theorem from [6] can be adapted to the non-compact case, at the expense of saturation of the open.  (6), by a diffeomorphism that fixes S.
Proof. Since the holonomy is finite, it equals the linear holonomy, and we denote H := H hol = H lin and S := S hol = S lin .
The assumption that S be embedded allows us to restrict to a tubular neighborhood; so we assume that the foliation is on a vector bundle p : E → S (with E ∼ = ν S ), for which S, identified with the zero section, is a leaf. Then the holonomy of paths in S is represented by germs of a diffeomorphism between the fibers of E.
Each point in S has an open neighborhood U ⊂ E satisfying • for every x, y ∈ S ∩ U , the holonomy along any path in S ∩ U connecting them is defined as a diffeomorphism between the spaces Let U be locally finite cover of S by opens U ⊂ E of the type just described, such that for all U, U ′ ∈ U, U ∩ U ′ ∩ S is connected (or empty), and such that each U ∈ U is relatively compact.
We fix x 0 ∈ S, U 0 ∈ U an open containing x 0 , and denote by Consider a path γ in S starting at x 0 and with endpoint x. Cover the path by a chain of opens in U ξ = (U 0 , . . . , U k(ξ) ), such that there is a partition Since the holonomy transformations inside U j are all trivial, and all the intersections U i ∩ U j ∩ S are connected, it follows that the holonomy of γ only depends on the chain ξ and is defined as an embedding Denote by K the kernel of π 1 (S, x 0 ) → H. The holonomy cover S → S can be described as the space of all paths γ in S starting at x 0 , and two such paths γ 1 and γ 2 are equivalent if they have the same endpoint, and the homotopy class of γ −1 2 • γ 1 lies in K. The projection is then given by [γ] → γ(1). Denote by x 0 the point in S corresponding to the constant path at x 0 . So, we can represent each point in S (not uniquely!) by a pair (ξ, x) with ξ ∈ Z and endpoint x ∈ U k(ξ) ∩ S.
The group H acts freely on S by pre-composing paths. For every g ∈ H fix a chain ξ g ∈ Z, such that (ξ g , x 0 ) represents x 0 g. Consider the open on which all holonomies h x0 x0 (ξ g ) are defined, and a smaller open and h x0 x0 (ξ gh ) are the same, by shrinking O 1 if necessary, we may assume that Consider the following open Then O ⊂ O 1 , and for h ∈ H, we have that So h x0 x0 (ξ h ) maps O to O, and by (7) it follows that the holonomy transport along ξ g defines an action of H on O, which we further denote by Since U is a locally finite cover by relatively compact opens, there are only finitely many chains in Z of a certain length. Denote by Z n the set of chains of length at most n. Let c ≥ 1 be such that ξ g ∈ Z c for all g ∈ H.
By the above, and by the basic properties of holonomy, there exist open neighborhoods {O n } n≥1 of x 0 in O: satisfying the following: 1) for every chain ξ ∈ Z n , O n ⊂ O(ξ), 2) for every two chains ξ, ξ ′ ∈ Z n and x ∈ U k(ξ) ∩ U k(ξ ′ ) ∩ S, such that the pairs (ξ, x) and (ξ ′ , x) represent the same element in S, we have that Denote by S n the set of points in x ∈ S for which every element in the orbit xH can be represented by a pair (ξ, x) with ξ ∈ Z n . Note that for n ≥ c, S n is nonempty, H-invariant, open, and connected. Consider the following H-invariant open neighborhood of S × {x 0 }: On V we define the map for ( x, v) ∈ S n ×O n+c , where (ξ, x) is pair representing x with ξ ∈ Z n and x ∈ U k(ξ) . By the properties of the opens O n , H is well defined. Since the holonomy transport is by germs of diffeomorphisms and preserves the foliation, it follows that H is a foliated local diffeomorphism, which sends the trivial foliation on V with leaves V ∩ S × {v} to F | E . We prove now that H is H-invariant. Let ( x, v) ∈ S n ×O n+c and g ∈ H. Consider chains ξ and ξ ′ in Z n representing x and xg respectively, with x ∈ U k(ξ) ∩ U k(ξ ′ ) ∩ S. Then ξ ′ and ξ g ∪ξ both belong to Z n+c and (ξ ′ , x), (ξ g ∪ξ, x) both represent xg ∈ S. Using properties 2) and 4) of the opens O n , we obtain H-invariance: . Since the action of H on V is free and preserves the foliation on V, we obtain an induced local diffeomorphism of foliated manifolds: Hence x and x ′ , both lie in the fiber of S → S over x, thus there is a unique g ∈ H with x ′ = xg. Let n, m ≥ c be such that ( x, v) ∈ S n ×O n+c and ( x ′ , v ′ ) ∈ S m ×O m+c , and assume also that n ≤ m. Consider ξ ∈ Z n and ξ ′ ∈ Z m such that (ξ, x) represents x and (ξ ′ , x) represents x ′ . Then we have that . Since both (ξ ′ , x) and (ξ g ∪ ξ, x) represent x ′ ∈ S, and both have length ≤ m + c, again by the properties 2) and 4) we obtain which proves injectivity of H.

Thurston Stability around non-compact leaves
To obtain the first order normal form result (Corollary 1), we will use the following extension to non-compact leaves of a result of Thurston [8].
Lemma 2. Let S be an embedded leaf of a foliation such that K lin , the kernel of dh : π 1 (S, x) → H lin , is finitely generated and H 1 ( S lin ) = 0. Then the holonomy group H hol of S coincides with the linear holonomy group H lin of S.
Proof. Denote by V := ν x , the normal space at some x ∈ S. The linear holonomy gives an identification of the normal bundle of S in M with the vector bundle ( S lin × V )/H lin . Passing to a tubular neighborhood, we may assume that the foliation F is on ( S lin × V )/H lin , and that its linear holonomy coincides with the holonomy of the flat bundle, i.e. the first order jet along S of F equals the first order jet along S of flat bundle foliation. Consider the covering map The leaf S 0 := S lin × {0} of the pull-back foliation p * (F ) on S lin × V satisfies: (1) S 0 has trivial linear holonomy; (2) H 1 ( S 0 ) = 0; (3) π 1 ( S 0 ) ∼ = K lin is finitely generated.
Thurston shows in [8] that, under the assumption that S 0 is compact, the first two conditions imply that the holonomy group of S 0 vanishes. It is straightforward to check that Thurston's argument actually doesn't use the compactness assumption, but it only uses condition (3); and we conclude that also in our case the holonomy at S 0 of p * (F ) vanishes. Now consider a loop γ in S based at x such that [γ] ∈ K lin . This is equivalent to saying that γ lifts to a loop in S lin , hence to a loop γ in S 0 . The holonomy transport along γ induced by p * (F ) projects to the holonomy transport of γ induced by F , and since the first is trivial, so is the latter. This proves that K lin is included in the kernel of π 1 (S, x) → H hol , and since the other inclusion always holds, we obtain that H hol = H lin .

Foliated cohomology of products
Let M and N be two manifolds. Consider the product foliation T M × N on We denote the complex computing the corresponding foliated cohomology by The elements of Ω • (T M × N ) can be regarded as smooth families of forms on M : Denote the corresponding cohomology groups by We need two versions of these groups associated to a leaf M × {x}, for a fixed We explain the third map; the first two are constructed similarly. Consider an element [η] ∈ H q gx (T M × N ), which is represented by a foliated q-form η that is closed on some open containing M × {x}. We define the corresponding linear map: The germ at x of the function η, c is independent of the choice of the representatives, yielding a well-defined element Ψ gx ( Proof. Denote the constant sheaves on M associated to the groups C ∞ (N ), C ∞ x (N ) and C ∞ gx (N ) by S 1 , S 2 and S 3 , respectively. By standard arguments, the de Rham differential along M induces resolutions S i → C • i by fine sheaves on M : N ). Hence, the foliated cohomologies from (9) are isomorphic to the sheaf cohomologies with coefficients in S 1 , S 2 and S 3 respectively. On the other hand, for any vector space V , denoting by V the constant sheaf on M , one has a natural isomorphism: Hence, we obtain isomorphisms: ). We still have to check that these maps coincide with those from (9). For this we will exploit the naturality of the maps in (10).
In the first case, consider the evaluation map ev y : C ∞ (N ) → R, for y ∈ N . This induces a sheaf map ev M y : S 1 → R into the constant sheaf over M , which is covered by a map ev By naturality of (10), it follows that the following square commutes: Since Φ R is the usual isomorphism given by integration, and by the explicit description of the map Ψ, this implies that Ψ = Φ. For the second map in (9) and (11) we proceed similarly, but using the inclusion i : C ∞ x (N ) → C ∞ (N ) instead of ev y . This gives rise to a sheaf map S 2 → S 1 which lifts to their resolutions, and then we obtain a commutative square Using also that Ψ = Φ, this implies the equality Ψ x = Φ x .
Similarly, for the third map in (9) and (11), but using the projection map p : C ∞ (N ) → C ∞ gx (N ) (instead of the inclusion), we obtain a commutative square (N ) . Again, since Ψ = Φ, we obtain that Ψ gx = Φ gx . This concludes the proof.
We will use the following consequences of Lemma 3 (the first appeared in [4]).
is exact for all y ∈ N . Then there exists θ ∈ Ω q−1 (T M × N ) such that dθ = η. Moreover, if η x = 0 for some x ∈ N , then one can choose θ such that θ x = 0.  Proof. First, we claim that the projection p : ×N ) induces a surjective map in cohomology. By the description of the maps Ψ and Ψ gx , we have a commutative diagram . By Lemma 3, the horizontal maps are isomorphisms, and since the vertical map on the right is surjective, so is the vertical map on the left.

Equivariant submersions
We prove now that submersions can be equivariantly linearized.
Since ω is invariant, it follows that ω 1 coincides with ω on U 1 := g∈H g U 0 .
We compute now the variation of ω at S. Since ω and ω 1 coincide around S, they have the same variation at S. Using the extension of ω (or equivalently of ω 1 ) that vanishes on vectors tangent to the fibers of the projection to S, we see that the variation δ S ω is given by the H-equivariant family: The local model is represented by the H-equivariant family of 2-forms: Smoothness of f follows from Lemma 3. Clearly, f (0) = 0 and its differential at 0 is the cohomological variation that restricts to a diffeomorphism between the leaf S v and the leaf S χ(v) . The pullback of ω 1 under χ is the H-equivariant family We have that Equivalently, this relation can be rewritten as where {α v } v∈U is an H-equivariant family of exact 2-forms that vanishes for v = 0. By Corollary 2, p * (α) is an exact foliated form on S × U , and moreover, we can choose a primitive β ∈ Ω 1 (T S × U ) such that β 0 = 0. By averaging, we may also assume that β is H-equivariant, thus it is of the form β = p * (β) for a foliated 1-form on β on ( S × U )/H that vanishes along S. We obtain: