Abstract
We prove a version of the local ReebThurston stability theorem for symplectic foliations.
Introduction
A symplectic foliation on a manifold \(M\) is a (regular) foliation \(\mathcal {F}\) endowed with a 2form \(\omega \) on \(T\mathcal {F}\) whose restriction to each leaf \(S\) of \(\mathcal {F}\) is a symplectic form
Equivalently, a symplectic foliation is a Poisson structure of constant rank.
In this paper we prove a normal form theorem for symplectic foliations around embedded leaves. The result uses the cohomological variation of \(\omega \) at the leaf \(S\), which is a linear map (see Sect. 1 for the definition)
where \(\nu \) denotes the normal bundle of \(T\mathcal {F}\), and \(\widetilde{S}_{hol}\) is the holonomy cover of \(S\). The cohomological variation arises in fact from a linear map:
The local model for the foliation around \(S\), which appears in the classical results of Reeb and Thurston, is the flat bundle \((\widetilde{S}_{hol}\times \nu _x){/}\pi _1(S,x)\), where \(\pi _1(S,x)\) acts on the second factor via the linear holonomy
The flat bundle can be endowed with leafwise closed 2forms, which are symplectic in a neighborhood of \(S\); namely, the leaf through \(v\in \nu _x\) carries the closed 2form \(j^1_S(\omega )_v\) whose pullback to \(\widetilde{S}_{hol}\times \{v\}\) is
Our main result is the following:
Theorem 1
Let \(S\) be an embedded leaf of the symplectic foliation \((M,\mathcal {F},\omega )\). If the holonomy group of \(S\) is finite and the cohomological variation (1) at \(S\) is a surjective map, then an open neighborhood of \(S\) is isomorphic as a symplectic foliation to an open neighborhood of \(S\) in the flat bundle \((\widetilde{S}_{hol}\times \nu _x){/}\pi _1(S,x)\) endowed with the family of closed 2forms \(j^1_S(\omega )\) 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 by \(\widetilde{S}_{lin}\). Now, the map (2) is in fact the pullback of a map with values in \(\Omega ^2_{\textit{closed}}(\widetilde{S}_{lin})\). This remark, an extension to noncompact leaves of a result of Thurston (Lemma 2), and Theorem 1 imply the following first order normal form result:
Corollary 1
Under the assumptions that \(S\) is embedded, \(\pi _1(S,x)\) is finitely generated, \(H_{lin}\) is finite, \(H^1(\widetilde{S}_{lin})=0\) and the cohomological variation
is surjective, the conclusion of Theorem 1 holds.
Related phenomena
Our result is clearly related to the normal form theorem for Poisson manifolds around symplectic leaves from [4]. Both results have the same conclusion, yet the conditions of Theorem 1 are substantially weaker. More precisely, for regular Poisson manifolds, the hypotheses of the main result in loc.cit. are (see Corollary 4.1.22 and Lemma 4.1.23 [7]):

the leaf \(S\) is compact,

\(\pi _1(S,x)\) is finite,

the cohomological variation is an isomorphism, when viewed as a map
$$\begin{aligned}{}[\delta _S\omega ]_x:\nu _x^*\longrightarrow H^2(\widetilde{S}_{uni}), \end{aligned}$$where \(\widetilde{S}_{uni}\) is the universal cover of \(S\).
There is yet another essential difference between Theorem 1 and the result from [4], namely, even in the setting of Corollary 1, the results presented here have first order assumptions 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. In fact, the conditions of Theorem 1 allow for Poissondeformations which increase the rank. To explain this phenomenon, let us consider the simplified setting:

\(S\) is compact and simply connected.
Then, by Corollary 2.7 in [8], the local normal form result from [4] holds under the weaker assumptions:

\([\delta _S\omega ]_x\) is surjective, and its kernel is at most onedimensional.
These conditions have an interpretation in the context of stability of symplectic leaves, from [3]. Namely, for \(S\) as above (i.e. compact and simply connected), we have an isomorphism (see the proof of Lemma 2.6 in [8])
where \(H^{\bullet }_{\pi ,S}(M)\) denotes the Poisson cohomology restricted to \(S\) (see [3]); thus the conditions are equivalent to \(H^2_{\pi ,S}(M)=0\). By Theorem 2.2 [3], \(H^2_{\pi ,S}(M)=0\) implies that \(S\) is strongly stable, i.e. any Poisson structure close to \(\pi \) has a leaf symplectomorphic to \((S,\omega _S)\). Now, the hypotheses of Theorem 1 take care of the first component in (4). The second component controls deformations which change the rank of the Poisson structure. Concretely, \(v,w\in \ker ([\delta _S\omega ]_x)\) are linearly independent, and \(\pi _{\mathrm{lin },S}\) is the linearized Poisson structure at \(S\), then we can interpret \(\pi _{\mathrm{lin },S}+\epsilon v\wedge w\), \(\epsilon >0\), as a deformation which increases the rank by two (see the proof of Theorem 2.3 in [3]). Of course, we do not allow such deformations in the context of symplectic foliations.
Motivated by a question of the referee, we included in the end of the paper some remarks on global aspects of symplectic foliations with surjective cohomological variation.
A weaker version of Theorem 1, in which the leaf is assumed to be of finite type, is part of the PhD thesis [7] 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 introduce the cohomological variation of the symplectic structures. In the case of general Poisson manifolds, the local model was first constructed by Vorobjev [12]. The approach presented here is more direct; for the relation between these two constructions see [7].
Let \((M,\mathcal {F})\) be a foliated manifold, and denote its normal bundle by
Then \(\nu \) carries a flat \(T\mathcal {F}\) connection, called the Bott connection, given by
where, for a vector field \(Z\), we denote by \(\overline{Z}\) its class in \(\Gamma (\nu )\). For a path \(\gamma \) inside a leaf \(S\), parallel transport with respect to \(\nabla \) gives the linear holonomy transformations:
This map depends only on \(\gamma \) 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 \(\widetilde{S}_{lin,x}\) is the covering space corresponding to the kernel of \(dh\); thus it is a principal \(H_{lin,x}\) bundle over \(S\). Also, \(\widetilde{S}_{lin,x}\) 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 \(\mathcal {F}_{\nu }\) on \(\nu \) whose leaves are the orbits of \(dh\); i.e. the leaf of \(\mathcal {F}_{\nu }\) through \(v\in \nu _x\) covers the leaf \(S\) through \(x\), and is given by
Therefore, \(\widetilde{S}_{lin,x}\) covers of the leaves of the foliation \(\mathcal {F}_{\nu }\) above \(S\) via the maps
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 \(\omega \) on the foliation \(\mathcal {F}\), i.e. a 2form on \(T\mathcal {F}\)
whose restriction to each leaf is symplectic. We first construct a closed foliated 2form \(\delta \omega \) on \((\nu ,\mathcal {F}_{\nu })\), which represents the derivative of \(\omega \) in the transversal direction. For this, choose an extension \(\widetilde{\omega }\in \Omega ^2(M)\) of \(\omega \) and let
Since \(\omega \) is closed along the leaves of \(\mathcal {F}\), \(\Omega (X,Y)\in \nu ^*\), thus \(\Omega \in \Omega ^2(T\mathcal {F};\nu ^*)\).
Now, the dual of the Bott connection on \(\nu ^*\) induces a differential \(d_{\nabla }\) on the space of foliated forms with values in the conormal bundle \(\Omega ^{\bullet }(T\mathcal {F};\nu ^*)\); this can be given explicitly by the classical Koszul formula
for \(\eta \in \Omega ^{p}(T\mathcal {F};\nu ^*)\), \(X_i\in \Gamma (T\mathcal {F})\). Denote the resulting cohomology by \(H^{\bullet }(\mathcal {F};\nu ^*)\).
It is easy to see that \(\Omega \) is \(d_{\nabla }\)closed. In fact, this construction can be preformed in all degrees, and it produces a canonical map (see e.g. [2])
which maps \([\omega ]\) to \([\Omega ]\). Also, if \(\widetilde{\omega }+\alpha \) is a second extension of \(\omega \) (where \(\alpha \) vanishes along \(\mathcal {F}\)), then \(\Omega \) changes by \(d_{\nabla }\lambda \), where \(\lambda \in \Omega ^1(T\mathcal {F};\nu ^*)\), is given by
Note that there is a natural embedding
where \(p:\nu \rightarrow M\) is the projection. It is easy to see that under \(\mathcal {J}\) the differential \(d_{\nabla }\) corresponds to the leafwise de Rham differential \(d_{\mathcal {F}_{\nu }}\) on the leaves of \(\mathcal {F}_{\nu }\). In particular, we obtain a closed foliated 2form
which we call the vertical derivative of \(\omega \). Since \(\delta \omega \) vanishes on \(M\) (viewed as the zero section), it follows that \(p^*(\omega )+\delta \omega \) is nondegenerate on the leaves in an open neighborhood of \(M\); thus
is a symplectic foliation around \(M\).
Consider now a symplectic leaf \(S\). Restricting \(p^*(\omega )+\delta \omega \) to the leaves above \(S\), we obtain closed foliated 2forms along the leaves of the \(\mathcal {F}_{\nu _S}\), denoted by
where \(\omega _S:=\omega _{S}\) and \(\delta _S\omega :=\delta \omega _{\nu _S}\). Any open neighborhood of \(S\) in
on which \(j^1_S(\omega )\) 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 \(\omega \), consider first the linear map
where the map \(p_v\) is the covering map defined by (5). By the discussion above, choosing a different extension of \(\omega \) changes \(p_v^*(\delta _S\omega )\) by an exact 2form; hence the cohomology class \([p^*_v(\delta _S\omega )]\) is independent of the 2form \(\Omega \) used to construct \(\delta _{S}\omega \). The induced linear map to the cohomology of \(\widetilde{S}_{lin,x}\), will be called the cohomological variation of \(\omega \) at \(S\)
In the Introduction we denoted the lifts of \([\delta _S\omega _x]\) to the holonomy cover \(\widetilde{S}_{hol}\), respectively to the universal cover \(\widetilde{S}_{uni}\) of \(S\), by the same symbol.
Up to isomorphism, the local model is independent of the choices involved.
Proposition 1
Different choices of \(\Omega \in \Omega ^2(T\mathcal {F},\nu ^*)\) satisfying \(d_{\nu }[\omega ]=[\Omega ]\) produce local models that are isomorphic around \(S\) by a diffeomorphism that fixes \(S\).
The proof of this result is deferred until the end of next section, because it uses Lemma 5, which is a foliated version of the Moser Lemma.
Five lemmas
In this section we prove some auxiliary results used in the proof of Theorem 1.
Reeb Stability around noncompact leaves
Consider a foliated manifold \((M,\mathcal {F})\) and let \(S\) be an embedded leaf. The classical Reeb Stability Theorem (see e.g. [9]) 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 \((M,\mathcal {F})\) is isomorphic as a foliated manifold with the flat bundle from the previous section
Generalizations to noncompact leaves of the Reeb stability theorem were studied before (see [1, 6] and the references therein). The goal in these works is to produce saturated neighborhoods of embedded leaves (also called proper leaves in foliation literature) which are isomorphic to the flat bundle (7). For this, finiteness of the holonomy group is not sufficient [6]. Results are known to hold only in low dimensions [1], for compact ambient manifolds, and with extra restrictions on the topology of the leaf. Below, we show that the direct generalization of the Reeb Stability Theorem for noncompact leaves does work, but at the expense of giving up on the conclusion that the neighborhood be saturated. The proof is a careful adaptation to the noncompact case of the proof of the classical result from [9].
Lemma 1
Let \((M,\mathcal {F})\) be a foliation and let \(S\subset M\) be an embedded leaf. If \(S\) has finite holonomy, then an open neighborhood of \(S\) in \(M\) is isomorphic as a foliated space to an open neighborhood of \(S\) in the local model (7), by a diffeomorphism which fixes \(S\).
Proof
Since the holonomy is finite, it equals the linear holonomy, and we denote \(H:=H_{hol}=H_{lin}\) and \(\widetilde{S}:=\widetilde{S}_{hol}=\widetilde{S}_{lin}\).
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 \(p:E\rightarrow S\) (with \(E\cong \nu _{S}\)), 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 \(U\subset E\) satisfying

\(S\cap U\) is 1connected,

for \(x\in S\cap U\), \(E_x\cap U\) is a connected neighborhood of \(x\),

for every \(x,y\in S\cap U\), the holonomy along any path in \(S\cap U\) connecting them is defined as a diffeomorphism between the spaces
$$\begin{aligned} h_{x}^y:E_x\cap U\mathop \rightarrow \limits ^{\sim }E_y\cap U. \end{aligned}$$
Let \(\mathfrak {U}\) be locally finite cover of \(S\) by open sets \(U\subset E\) of the type just described, such that for all \(U,U^{\prime }\in \mathfrak {U}\), \(U\cap U^{\prime }\cap S\) is connected (or empty), and such that each \(U\in \mathfrak {U}\) is relatively compact.
We fix \(x_0\in S\), \(U_0\in \mathfrak {U}\) an open set containing \(x_0\), and denote by
Consider a path \(\gamma \) in \(S\) starting at \(x_0\) and with endpoint \(x\). Cover the path by a chain of open sets in \(\mathfrak {U}\)
such that there is a partition
with \(\gamma ([t_{j1},t_{j}])\subset U_{j}\). Since the holonomy transformations inside \(U_{j}\) are all trivial, and all the intersections \(U_i\cap U_j\cap S\) are connected, it follows that the holonomy of \(\gamma \) only depends on the chain \(\xi \) and is defined as an embedding
where \(O(\xi )\subset V\) is an open neighborhood of \(x_0\), which is independent of \(x\in U_{k(\xi )}\). Denote by \(\mathcal {Z}\) the space of all chains in \(\mathfrak {U}\)
Denote by \(K\) the kernel of \(\pi _1(S,x_0)\rightarrow H\). The holonomy cover \(\widetilde{S}\rightarrow S\) can be described as the space of all paths \(\gamma \) in \(S\) starting at \(x_0\), and two such paths \(\gamma _1\) and \(\gamma _2\) are equivalent if they have the same endpoint, and the homotopy class of \(\gamma _2^{1}\circ \gamma _1\) lies in \(K\). The projection is then given by \([\gamma ]\mapsto \gamma (1)\). Denote by \(\widetilde{x}_0\) the point in \(\widetilde{S}\) corresponding to the constant path at \(x_0\). So, we can represent each point in \(\widetilde{S}\) (not uniquely!) by a pair \((\xi ,x)\) with \(\xi \in \mathcal {Z}\) and endpoint \(x\in U_{k(\xi )}\cap S\).
The group \(H\) acts freely on \(\widetilde{S}\) by precomposing paths. For every \(g\in H\) fix a chain \(\xi _g\in \mathcal {Z}\), such that \((\xi _g,x_0)\) represents \(\widetilde{x}_0g\). Consider the open set
on which all holonomies \(h_{x_0}^{x_0}(\xi _{g})\) are defined, and a smaller open set \(\widetilde{O}_1\subset \widetilde{O}_0\) around \(x_0\), such that \(h_{x_0}^{x_0}(\xi _{g})\) maps \(\widetilde{O}_1\) into \(\widetilde{O}_0\). Hence the composition
is well defined. Since the germs of \(h_{x_0}^{x_0}(\xi _g)\circ h_{x_0}^{x_0}(\xi _h)\) and \(h_{x_0}^{x_0}(\xi _{gh})\) are the same, by shrinking \(\widetilde{O}_1\) if necessary, we may assume that
Consider the following open set
Then \(O\subset \widetilde{O}_{1}\), and for \(h\in H\), we have that
So \(h_{x_0}^{x_0}(\xi _h)\) maps \(O\) to \(O\), and by (8) it follows that the holonomy transport along \(\xi _g\) defines an action of \(H\) on \(O\), which we further denote by
Since \(H\) is a finite group acting on \(O\) with a fixed point \(x_0\), by Bochner’s Linearization Theorem, we can linearize the action around \(x_0\). So, by shrinking \(O\) if necessary, the action is isomorphic to the linear holonomy action of \(H\) on \(V\). In particular, this implies that \(O\) contains arbitrarily small \(H\)invariant open neighborhoods of \(x_0\).
Since \(\mathfrak {U}\) is a locally finite cover by relatively compact open sets, there are only finitely many chains in \(\mathcal {Z}\) of a certain length. Denote by \(\mathcal {Z}_n\) the set of chains of length at most \(n\). Let \(c\ge 1\) be such that \(\xi _g\in \mathcal {Z}_c\) for all \(g\in H\).
By the above, and by the basic properties of holonomy, there exist open neighborhoods \(\{O_n\}_{n\ge 1}\) of \(x_0\) in \(O\):
satisfying the following:

(1)
for every chain \(\xi \in \mathcal {Z}_n\), \(O_n\subset O({\xi })\),

(2)
for every two chains \(\xi ,\xi ^{\prime }\in \mathcal {Z}_n\) and \(x\in U_{k(\xi )}\cap U_{k(\xi ^{\prime })}\cap S\), such that the pairs \((\xi ,x)\) and \((\xi ^{\prime },x)\) represent the same element in \(\widetilde{S}\), we have that
$$\begin{aligned} h_{x_0}^x(\xi )=h_{x_0}^x(\xi ^{\prime }):O_n\hookrightarrow E_x, \end{aligned}$$ 
(3)
\(O_n\) is \(H\)invariant,

(4)
for every \(g\in H\), \(\xi \in \mathcal {Z}_{n}\) and \(x\in U_{k(\xi )}\cap S\), we have that
$$\begin{aligned} h_{x_0}^x(\xi _g\cup \xi )=h_{x_0}^x(\xi )\circ h(g): O_{n+c}\hookrightarrow E_{x}. \end{aligned}$$
Denote by \(\widetilde{S}_n\) the set of points in \(\widetilde{x}\in \widetilde{S}\) for which every element in the orbit \(\widetilde{x}H\) can be represented by a pair \((\xi ,x)\) with \(\xi \in \mathcal {Z}_n\). Note that for \(n\ge c\), \(\widetilde{S}_n\) is nonempty, \(H\)invariant, open, and connected. Consider the following \(H\)invariant open neighborhood of \(\widetilde{S}\times \{x_0\}\):
On \(\mathcal {V}\) we define the map
for \((\widetilde{x},v)\in \widetilde{S}_n\times O_{n+c}\), where \((\xi ,x)\) is pair representing \(\widetilde{x}\) with \(\xi \in \mathcal {Z}_{n}\) and \(x\in U_{k(\xi )}\). By the properties of the open sets \(O_n\), \(\widetilde{\mathcal {H}}\) is well defined. Since the holonomy transport is by germs of diffeomorphisms and preserves the foliation, it follows that \(\widetilde{\mathcal {H}}\) is a foliated local diffeomorphism, which sends the trivial foliation on \(\mathcal {V}\) with leaves \(\mathcal {V}\cap \big (\widetilde{S}\times \{v\}\big )\) to \(\mathcal {F}_{E}\).
We prove now that \(\widetilde{\mathcal {H}}\) is \(H\)invariant. Let \((\widetilde{x},v)\in \widetilde{S}_n\times O_{n+c}\) and \(g\in H\). Consider chains \(\xi \) and \(\xi ^{\prime }\) in \(\mathcal {Z}_{n}\) representing \(\widetilde{x}\) and \(\widetilde{x}g\) respectively, with \(x\in U_{k(\xi )}\cap U_{k(\xi ^{\prime })}\cap S\). Then \(\xi ^{\prime }\) and \(\xi _g\cup \xi \) both belong to \(\mathcal {Z}_{n+c}\) and \((\xi ^{\prime },x)\), \((\xi _g\cup \xi ,x)\) both represent \(\widetilde{x}g\in \widetilde{S}\). Using properties 2) and 4) of the open sets \(O_n\), we obtain \(H\)invariance:
Since the action of \(H\) on \(\mathcal {V}\) is free and preserves the foliation on \(\mathcal {V}\), we obtain an induced local diffeomorphism of foliated manifolds:
We prove now that \(\mathcal {H}\) is injective. Let \((\widetilde{x},v), (\widetilde{x}^{\prime },v^{\prime })\in \mathcal {V}\) be such that
Denoting by \(x=p(\widetilde{\mathcal {H}}(\widetilde{x},v))=p(\widetilde{\mathcal {H}}(\widetilde{x}^{\prime },v^{\prime }))\), we have that \(\widetilde{\mathcal {H}}(\widetilde{x},v)\), \(\widetilde{\mathcal {H}}(\widetilde{x}^{\prime },v^{\prime })\in E_x\). Hence \(\widetilde{x}\) and \(\widetilde{x}^{\prime }\), both lie in the fiber of \(\widetilde{S}\rightarrow S\) over \(x\), thus there is a unique \(g\in H\) with \(\widetilde{x}^{\prime }=\widetilde{x}g\). Let \(n,m\ge c\) be such that \((\widetilde{x},v)\in \widetilde{S}_n\times O_{n+c}\) and \((\widetilde{x}^{\prime },v^{\prime })\in \widetilde{S}_m\times O_{m+c}\), and assume also that \(n\le m\). Consider \(\xi \in \mathcal {Z}_n\) and \(\xi ^{\prime }\in \mathcal {Z}_m\) such that \((\xi ,x)\) represents \(\widetilde{x}\) and \((\xi ^{\prime },x)\) represents \(\widetilde{x}^{\prime }\). Then we have that
Since both \((\xi ^{\prime },x)\) and \((\xi _g\cup \xi ,x)\) represent \(\widetilde{x}^{\prime }\in \widetilde{S}\), and both have length \(\le m+c\), again by the properties 2) and 4) we obtain
Since \(h_{x_0}^x(\xi )\) is injective, (9) implies that \(v=h(g)(v^{\prime })\). So, we obtain
which proves injectivity of \(\mathcal {H}\). \(\square \)
Thurston Stability around noncompact leaves
To obtain the first order normal form result (Corollary 1), we will use the following extension to noncompact leaves of a result of Thurston [11].
Lemma 2
Let \(S\) be an embedded leaf of a foliation such that \(K_{lin}\), the kernel of \(dh:\pi _1(S,x)\rightarrow H_{lin}\), is finitely generated and \(H^1(\widetilde{S}_{lin})=0\). Then the holonomy group \(H_{hol}\) of \(S\) coincides with the linear holonomy group \(H_{lin}\) of \(S\).
Proof
Denote by \(V:=\nu _x\), the normal space at some \(x\in S\). The linear holonomy gives an identification of the normal bundle of \(S\) in \(M\) with the vector bundle \((\widetilde{S}_{lin}\times V){/}H_{lin}\). Passing to a tubular neighborhood, we may assume that the foliation \(\mathcal {F}\) is on \((\widetilde{S}_{lin}\times V){/}H_{lin}\), and that its linear holonomy coincides with the holonomy of the flat bundle, i.e. the first order jet along \(S\) of \(\mathcal {F}\) equals the first order jet along \(S\) of flat bundle foliation. Consider the covering map
The leaf \(\widetilde{S}_{0}:=\widetilde{S}_{lin}\times \{0\}\) of the pullback foliation \(p^*(\mathcal {F})\) on \(\widetilde{S}_{lin}\times V\) satisfies:

(1)
\(\widetilde{S}_{0}\) has trivial linear holonomy;

(2)
\(H^1(\widetilde{S}_{0})=0\);

(3)
\(\pi _1(\widetilde{S}_{0})\cong K_{lin}\) is finitely generated.
Thurston shows in [11] that, under the assumption that \(\widetilde{S}_0\) is compact, the first two conditions imply that the holonomy group of \(\widetilde{S}_0\) 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 \(\widetilde{S}_0\) of \(p^*(\mathcal {F})\) vanishes.
Now consider a loop \(\gamma \) in \(S\) based at \(x\) such that \([\gamma ]\in K_{lin}\). This is equivalent to saying that \(\gamma \) lifts to a loop in \(\widetilde{S}_{lin}\), hence to a loop \(\widetilde{\gamma }\) in \(\widetilde{S}_0\). The holonomy transport along \(\widetilde{\gamma }\) induced by \(p^*(\mathcal {F})\) projects to the holonomy transport of \(\gamma \) induced by \(\mathcal {F}\), and since the first is trivial, so is the latter. This proves that \(K_{lin}\) is included in the kernel of \(\pi _1(S,x)\rightarrow H_{hol}\), 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 \(TM\times N\) on \(M\times N\), with leaves
We denote the complex computing the corresponding foliated cohomology by
The elements of \(\Omega ^{\bullet }(TM\times N)\) 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 \(M\times \{x\}\), for a fixed \(x\in N\). Denote the subcomplex of foliated forms vanishing on \(M\times \{x\}\) by
and the associated cohomology by
Finally, consider the complex of germs at \(M\times \{x\}\) of foliated forms
This space is the quotient of \(\Omega ^\bullet (TM\times N)\) by the space of foliated forms that vanish on some open neighborhood of \(M\times \{x\}\) in \(M\times N\). The leafwise de Rham differential induces a differential on \(\Omega ^\bullet _{\mathrm {g}_{x}}(TM\times N)\). Denote the resulting cohomology by
Let \(C^{\infty }_{x}(N)\) denote the space of smooth functions on \(N\) vanishing at \(x\), and let \(C^{\infty }_{\mathrm {g}_{x}}(N)\) denote the space of germs of smooth functions on \(N\) around \(x\).
These three versions of foliated cohomology come with natural pairings with the homology of \(M\), which yield maps:
We explain the third map; the first two are constructed similarly. Consider an element \([\eta ]\in H^q_{\mathrm {g}_{x}}(TM\times N)\), which is represented by a foliated \(q\)form \(\eta \) which is closed on some open set containing \(M\times \{x\}\). The corresponding linear map:
is constructed as follows. Represent an element \([c]\in H_{q}(M)\) as \(c=\sum _i a_i \sigma _i\), where \(\sigma _i:\Delta _q\rightarrow M\) are smooth \(q\)simplices. Define
The germ at \(x\) of the function \(\langle \eta , c\rangle \) is independent of the choice of the representatives, yielding a welldefined element \(\Psi _{\mathrm {g}_{x}}([\eta ])([c]):=\langle [\eta ],[c]\rangle \in C^{\infty }_{\mathrm {g}_x}(N)\).
Lemma 3
The maps from (10) are linear isomorphisms.
Proof
Denote the constant sheaves on \(M\) associated to the groups \(C^{\infty }(N)\), \(C^{\infty }_x(N)\) and \(C^{\infty }_{\mathrm {g}_x}(N)\) by \(\mathcal {S}_1\), \(\mathcal {S}_2\) and \(\mathcal {S}_3\), respectively. By standard arguments, the de Rham differential along \(M\) induces resolutions \(\mathcal {S}_i\rightarrow \mathcal {C}_i^{\bullet }\) by fine sheaves on \(M\):
Hence, the foliated cohomologies from (10) are isomorphic to the sheaf cohomologies with coefficients in \(\mathcal {S}_1\), \(\mathcal {S}_2\) and \(\mathcal {S}_3\) respectively. On the other hand, for any vector space \(V\), denoting by \(\underline{V}\) 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 (10). For this we will exploit the naturality of the maps in (11).
In the first case, consider the evaluation map \(ev_y:C^{\infty }(N)\rightarrow \mathbb {R}\), for \(y\in N\). This induces a sheaf map \(ev_y^M:\mathcal {S}_1\rightarrow \underline{\mathbb {R}}\) into the constant sheaf over \(M\), which is covered by a map \(ev_y^M:\mathcal {C}_1^{\bullet }\rightarrow \Omega _M^{\bullet }\) into the standard de Rham resolution of \(\underline{\mathbb {R}}\). Hence the map \(H^{\bullet }(M;\mathcal {S}_1)\rightarrow H^{\bullet }(M;\underline{\mathbb {R}})\) induced by \(ev_y\) becomes
By naturality of (11), it follows that the following square commutes:
Since \(\Phi _{\mathbb {R}}\) is the usual isomorphism given by integration, and by the explicit description of the map \(\Psi \), this implies that \(\Psi =\Phi \).
For the second map in (10) and (12) we proceed similarly, but using the inclusion \(i:C^{\infty }_x(N)\rightarrow C^{\infty }(N)\) instead of \(ev_y\). This gives rise to a sheaf map \(\mathcal {S}_2\rightarrow \mathcal {S}_1\) which lifts to their resolutions, and then we obtain a commutative square
Using also that \(\Psi =\Phi \), this implies the equality \(\Psi _x=\Phi _x\).
Similarly, for the third map in (10) and (12), but using the projection map \(p:C^{\infty }(N)\rightarrow C^{\infty }_{\mathrm {g}_x}(N)\) (instead of the inclusion), we obtain a commutative square
Again, since \(\Psi =\Phi \), we obtain that \(\Psi _{\mathrm {g}_x}=\Phi _{\mathrm {g}_x}\). This concludes the proof. \(\square \)
We will use the following consequences of Lemma 3 (the first appeared in [5]).
Corollary 2
Let \(\eta \in \Omega ^{q}(TM\times N)\) be a foliated \(q\)form such that \(\eta _y\in \Omega ^{q}(M)\) is exact for all \(y\in N\). Then there exists \(\theta \in \Omega ^{q1}(TM\times N)\) such that \(d\theta =\eta \). Moreover, if \(\eta _{x}=0\) for some \(x\in N\), then one can choose \(\theta \) such that \(\theta _{x}=0\).
Proof
In the first case, we need that \([\eta ]=0\) in \(H^{\bullet }(TM\times N)\), and in the second, that \([\eta ]=0\) in \(H^{\bullet }_x(TM\times N)\). Since \(\langle [\eta _y],[c]\rangle =0\), for all \([c]\in H_{q}(M)\) and all \(y\in N\), the description of the maps \(\Psi \) and \(\Psi _x\) and Lemma 3 imply the result. \(\square \)
Corollary 3
Let \(\eta \) be a closed foliated \(q\)form defined on some open neighborhood \(\mathcal {U}\subset M\times N\) of \(M\times \{x\}\). Then there exists a closed foliated \(q\)form \(\widetilde{\eta }\) on \(M\times N\), such that \(\eta _{\widetilde{\mathcal {U}}}=\widetilde{\eta }_{\widetilde{\mathcal {U}}}\), for some open set \(\widetilde{\mathcal {U}}\subset \mathcal {U}\) containing \(M\times \{x\}\).
Proof
First, we claim that the projection \(p:\Omega ^{\bullet }(TM\times N)\rightarrow \Omega ^{\bullet }_{\mathrm {g}_x}(TM\times N)\) induces a surjective map in cohomology. By the description of the maps \(\Psi \) and \(\Psi _{\mathrm {g}_x}\), 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.
Consider a foliated \(q\)form \(\eta ^{\prime }\in \Omega ^q(TM\times N)\), such that \(\eta ^{\prime }_{\mathcal {U}^{\prime }}=\eta _{\mathcal {U}^{\prime }}\) for some open neighborhood \(\mathcal {U}^{\prime }\subset \mathcal {U}\) of \(M\times \{x\}\). Then \(\eta ^{\prime }\) defines a class \([\eta ^{\prime }]\in H^q_{\mathrm {g}_{x}}(TM\times N)\). By the above, there is a closed foliated \(q\)form \(\eta ^{\prime \prime }\in \Omega ^q(TM\times N)\), such that \([\eta ^{\prime \prime }]=[\eta ^{\prime }]\in H^{q}_{\mathrm {g}_x}(TM\times N)\). Thus, there is some foliated \(q1\)form \(\theta \) and an open neighborhood \(\widetilde{\mathcal {U}}\subset \mathcal {U}^{\prime }\) of \(M\times \{x\}\) such that \(\eta ^{\prime }_{\widetilde{\mathcal {U}}}=(\eta ^{\prime \prime }+d\theta )_{\widetilde{\mathcal {U}}}\). The closed foliated \(q\)form \(\widetilde{\eta }:=\eta ^{\prime \prime }+d\theta \) satisfies the conclusion: \(\widetilde{\eta }_{\widetilde{\mathcal {U}}}=\eta _{\widetilde{\mathcal {U}}}\). \(\square \)
Equivariant submersions
We prove that submersions can be equivariantly linearized.
Lemma 4
Let \(G\) be compact Lie group acting linearly on the vector spaces \(V\) and \(W\). Consider \(f:V\rightarrow W\) a smooth \(G\)equivariant map, such that \(f(0)=0\). If \(f\) is a submersion at \(0\), then there exists a \(G\)equivariant embedding \(\chi :U\hookrightarrow V\), where \(U\) is an invariant open around \(0\) in \(V\), such that \(\chi (0)=0\) and
Proof
Since \(G\) is compact, we can find a \(G\)equivariant projection \(p_K:V\rightarrow K\), where \(K:=\ker (df_0)\). The differential at \(0\) of the \(G\)equivariant map
is \((df_{0},p_K)\). So \((f,p_K)\) is a diffeomorphism when restricted to some open \(U_0\) in \(V\) around \(0\), which we may assume to be \(G\)invariant. Define the embedding as follows
where \(U:=(df_{0},p_K)^{1}(U_0)\). Clearly \(U\) is \(G\)invariant, \(\chi \) is \(G\)equivariant and \(\chi (0)=0\). Since
we also have that \(f(\chi (v))=df_{0}(v)\). \(\square \)
The Moser Lemma for symplectic foliations
The following is a version for symplectic foliations of the Moser Lemma.
Lemma 5
Let \((M,\mathcal {F},\omega )\) be a symplectic foliation. Consider a foliated 1form
that vanishes on an embedded saturated submanifold \(Z\) of \(M\). Then \(\omega +d_{\mathcal {F}}\alpha \) is nondegenerate in a neighborhood \(U\) of \(Z\), and the resulting symplectic foliation
is isomorphic around \(Z\) to \((M,\mathcal {F},\omega )\) by a foliated diffeomorphism that fixes \(Z\).
Proof
Since \(\alpha \) vanishes on \(Z\) and \(Z\) is saturated, it follows that also \(d_{\mathcal {F}}\alpha \) vanishes on \(Z\). Thus, there is an open \(V\) around \(Z\) such that \(\omega +d_{\mathcal {F}}\alpha \) is nondegenerate along the leaves of \(\mathcal {F}_{V}\). Moreover, by the classical tube lemma from topology, we may choose \(V\) such that
is nondegenerate along the leaves of \(\mathcal {F}_{V}\), for all \(t\in [0,1]\). Consider the time dependent vector field \(X_t\) on \(V\), tangent to \(\mathcal {F}\), determined by
Since \(X_t\) vanishes along \(Z\), again by the tube lemma, there is an open \(O\subset V\) around \(Z\), such that the flow \(\Phi ^t_{X}\) of \(X_t\) is defined up to time 1 on \(O\). We claim that \(\Phi ^1_X\) gives the desired isomorphism. Clearly \(\Phi ^1_X\) preserves the foliation and is the identity on \(Z\). On each leaf \(S\), we have that
Thus \(\Phi _X^{t*}(\omega _{t}_S)\) is constant, and since \(\Phi _X^0=\mathrm{Id }\), we have that
So, \(\Phi ^1_X\) is an isomorphism between the symplectic foliations
where \(U:=\Phi ^1_X(O)\). \(\square \)
We conclude this section with:
Proof of Proposition 1
A second 2form is of the form
for some \(\lambda \in \Omega ^1(T\mathcal {F};\nu ^*)\). We apply the Lemma 5 to the symplectic foliation
and the foliated 1form \(\alpha :=\mathcal {J}(\lambda )\) which vanishes along \(M\). The resulting diffeomorphism is foliated. In particular, above any leaf \(S\) of \(\mathcal {F}\), the diffeomorphism sends the local model corresponding to \(\Omega \) to the local model corresponding to \(\Omega ^{\prime }\). \(\square \)
Proof of Theorem 1
Since the holonomy of \(S\) is finite, it coincides with the linear holonomy. Consider \(x\in S\) and denote by \(V:=\nu _{x}\), by \(H:=H_{hol}=H_{lin}\), and by \(\widetilde{S}:=\widetilde{S}_{hol}=\widetilde{S}_{lin}\). Applying Lemma 1, we obtain that some neighborhood of \(S\) in \(M\) is diffeomorphic as a foliated manifold to an open \(\mathcal {U}\) around \(S\) in the flat bundle
The symplectic leaves correspond to the connected components of \(S_v\cap \mathcal {U}\), where
We claim that there exists \(\omega _1\), a closed foliated 2form on \((\widetilde{S}\times V){/}H\) that extends \(\omega _{\mathcal {U}_1}\), for some open \(\mathcal {U}_1\subset \mathcal {U}\) around \(S\). For this, consider the projection
and denote by \(\widetilde{\mathcal {U}}:=p^{1}(\mathcal {U})\) and by \(\widetilde{\omega }:=p^*(\omega )\), which is a closed foliated 2form on the product foliation restricted to \(\widetilde{\mathcal {U}}\). By Corollary 3, there is a closed extension \(\widetilde{\omega }_0\) of \(\widetilde{\omega }_{\widetilde{\mathcal {U}}_0}\), where \(\widetilde{\mathcal {U}}_0\subset \widetilde{\mathcal {U}}\) is an open around \(\widetilde{S}\times \{0\}\). Define \(\widetilde{\omega }_1\) by averaging over \(H\)
Since \(\widetilde{\omega }\) is invariant, it follows that \(\widetilde{\omega }_1\) coincides with \(\widetilde{\omega }\) on \(\widetilde{\mathcal {U}}_1:=\bigcap _{g\in H} g\widetilde{\mathcal {U}}_0\). Since \(\widetilde{\omega }_1\) is invariant, it is of the form \(\widetilde{\omega }_1=p^*(\omega _1)\), where \(\omega _1\) is a closed foliated 2form on \((\widetilde{S}\times V){/}H\), which extends the restriction to \(\mathcal {U}_1:=p(\widetilde{\mathcal {U}}_1)\) of \(\omega \).
We will identify foliated \(q\)forms \(\eta \) on \((\widetilde{S}\times V){/}H\), with smooth \(H\)equivariant families \(\{\eta _v\in \Omega ^q(\widetilde{S})\}_{v\in V}\), where \(\eta _v:=p^*(\eta )_{\widetilde{S}\times \{v\}}\).
We compute now the variation of \(\omega \) at \(S\). Since \(\omega \) and \(\omega _1\) coincide around \(S\), they have the same variation at \(S\). Using the extension of \(\omega \) (or equivalently of \(\omega _1\)) that vanishes on vectors tangent to the fibers of the projection to \(S\), we see that the variation \(\delta _S\omega \) is given by the \(H\)equivariant family:
The local model is represented by the \(H\)equivariant family of 2forms:
Consider the \(H\)equivariant map
Smoothness of \(f\) follows from Lemma 3. Clearly, \(f(0)=0\) and its differential at \(0\) is the cohomological variation
By our hypothesis, \(f\) is a submersion at \(0\). So we can apply Lemma 4 to find an \(H\)equivariant embedding
where \(U\) is an \(H\)invariant open neighborhoods of \(0\) in \(V\), such that
By \(H\)equivariance, \(\chi \) induces a foliation preserving embedding
that restricts to a diffeomorphism between the leaf \(S_v\) and the leaf \(S_{\chi (v)}\). The pullback of \(\omega _1\) under \(\widetilde{\chi }\) is the \(H\)equivariant family
We have that
Equivalently, this relation can be rewritten as
where \(\{\alpha _v\}_{v\in U}\) is an \(H\)equivariant family of exact 2forms that vanishes for \(v=0\). By Corollary 2, \(p^*(\alpha )\) is an exact foliated form on \(\widetilde{S}\times U\), and moreover, we can choose a primitive \(\widetilde{\beta }\in \Omega ^1(T\widetilde{S}\times U)\) such that \(\widetilde{\beta }_0=0\). By averaging, we may also assume that \(\widetilde{\beta }\) is \(H\)equivariant, thus it is of the form \(\widetilde{\beta }=p^*(\beta )\) for a foliated 1form on \(\beta \) on \((\widetilde{S}\times U){/}H\) that vanishes along \(S\). We obtain:
Applying Lemma 5, we conclude that, on some open around \(S\), \(j^1_S\omega \) and \(\omega _2\) are related by a foliated diffeomorphism. Now, \(\omega _2\) and \(\omega _1\) are related by \(\widetilde{\chi }\), and \(\omega _1\) and \(\omega \) have the same germ around \(S\). This concludes the proof.
Proof of Corollary 1
Schreier’s Lemma says that a subgroup of finite index of a finitely generated group is also finitely generated (see e.g. Sect. 5.6 in [10]). Hence, \(K_{lin}\) is finitely generated. By Lemma 2, \(H_{hol}=H_{lin}\), in particular \(H_{hol}\) is finite, and so we are in the setting of Theorem 1.
Some global considerations
A natural question, which was proposed by the referee, is that of giving a version of the global Reeb stability for symplectic foliations. In this last section, we discuss an interesting aspect of this problem, namely, that the appropriate compactness conditions are incompatible with the surjectivity of the cohomological variation. The argument used is similar to the one in Sect. 2.1 of [8].
We consider a symplectic foliation \((M,\mathcal {F},\omega )\), and we assume that all leaves of \(\mathcal {F}\) are compact with finite holonomy group; in particular, this setting includes the conclusion of the classical global Reeb stability. This allows us to define a smooth volume map on \(M\), which is constant on the leaves:
where \(H_x\) is the holonomy group of the leaf \((S_x,\omega _{S_x})\) through \(x\). To see that this map is indeed smooth, write locally the foliation as the flat bundle \((\widetilde{S}_x\times V){/}H_x\), and represent \(\omega \) by an \(H_x\)equivariant smooth family of 2forms \(\{\omega _v\in \Omega ^2(\widetilde{S}_x)\}_{v\in V}\). The pullback of the map \(\mathrm{vh }\) to \(\widetilde{S}_x\times V\) is simply the volume map \((y,v)\mapsto \int _{\widetilde{S}_x}\omega _v^k\), which is clearly smooth. Next, we claim that if the cohomological variation is onto, then \(\mathrm {vh}\) is submersion. In the local model, the pullback to \(\widetilde{S}_x\times V\) of the map \(\mathrm{vh }\) is decomposable as
The derivative of the first map is the cohomological variation, which is surjective; at \(\eta :=\omega _0\), since \([\eta ]^k\ne 0\), the derivative in the direction of \([\eta ]\) is nonzero, thus it is surjective; and finally, the last map is an isomorphism. This argument, implies that \(\mathrm{vh }\) is a submersion. In particular, we conclude that \(M\) cannot be compact.
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Acknowledgments
This research was financially supported by the ERC Starting Grant no. 279729.
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Crainic, M., Mărcuţ, I. ReebThurston stability for symplectic foliations. Math. Ann. 363, 217–235 (2015). https://doi.org/10.1007/s0020801411677
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DOI: https://doi.org/10.1007/s0020801411677