ON EXISTENCE OF REGULAR JACOBI STRUCTURES

MAHUYA DATTA AND SAUVIK MUKHERJEE

Abstract. We prove h-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.

Key words: Contact foliations, locally conformal symplectic foliations, regular Jacobi structures. Mathematics Subject Classification 2010: 53D99, 57R17, 57R30.

1. Introduction

Symplectic foliations on a manifold M are closely related to regular Poisson structures on it. Recall that a Poisson structure π is a bivector field satisfying the condition [π, π] = 0, where the bracket denotes the Schouten bracket of multivector fields [11]. The bivector field π induces a vector bundle morphism π^{#} : T^{∗}M → T M by π^{#}(α)(β) = π(α, β) for all α, β ∈ T_{x}^{∗}M, x ∈ M. The characteristic distribution D = Image π^{#} is, in general, a singular distribution which, however, integrates to a foliation. The restriction of the Poisson structure to a leaf of the foliation has the maximum rank and so we obtain a symplectic form on the leaf by dualizing π. Thus, the characteristic foliation is a (singular) symplectic foliation. A Poisson bivector field π is said to be regular if the rank of π^{#} is constant. In this case the characteristic foliation is a regular symplectic foliation on M. On the other hand, given a regular symplectic foliation F on M one can associate a Poisson bivector field π having F as its characteristic foliation.

In a recent article, Fernandes and Frejlich [4] have shown that a regular bivector field π_{0} whose characteristic distribution is integrable can be homotoped to a Poisson bivector field, provided the manifold is open. In general, an arbitrary distribution need not be integrable. Taking into account Haefliger’s result [8] Fernandes and Frejlich have proved the following.

Theorem 1.1. [4] Every regular bivector field π_{0} on an open manifold can be homotoped to a regular Poisson bivector provided the characteristic distribution of π_{0} is homotopic to an integrable one.

Since a symplectic form on a manifold corresponds to a non-degenerate Poisson structure, the above result may be seen as a generalisation of a classical result due to Gromov which says that an open manifold admitting a non-degenerate 2-form must also support a symplectic form. In this connection we also recall a result of Bertelson ([1]).

Theorem 1.2. Any 2-form on a foliated manifold (M, F) which is non-degenerate on the leaves is homotopic through the space of such 2-forms to a leafwise symplectic form, provided the foliation F satisfies some ‘openness’ condition.

The openness condition on the foliation is a technical one and we refer to [1] for the definition. The theorem above follows as a direct consequence of a general h-principle type result proved in [1]. The openness condition on foliation is crucial as there are examples to show that the theorem breaks down without this condition. Theroem 1.2 can be stated in terms of bivector fields as follows:

Let π_{0} be a regular bivector field (on a manifold M) for which the distribution D = Im π_{0}^{#} integrates to a foliation satisfying some open-ness condition. Then π_{0} can be homotoped through regular bivector fields π_{t} to a Poisson bivector field π_{1} having D as its characteristic distribution. Moreover, the homotopy of underlying distributions Im π_{t}^{#} remains constant.

In fact, to keep the underlying foliation constant during homotopy, Bertelson had to impose the openness condition on F.

The present article is inspired by a comment in [4], where the authors mention that there should be analogues of Theorem 1.1 for foliated locally conformal symplectic manifolds, foliated contact manifolds or more generally for Jacobi manifolds. The main results of this article are stated below in Theorems 1.3 and 1.4.

Theorem 1.3. Let M^{2n+q} be an open manifold with a codimension q foliation F_{0} and a 2-form ω_{0} which is non-degenerate on the leaves of F_{0}. Fix a de Rham cohomology class ξ ∈ H_{deR}^{1}(M). Then there exists a homotopy (F_{t}, ω_{t}) and a closed 1-form θ representing ξ such that

(1) ω_{t} is F_{t}-leafwise non-degenerate

(2) ω_{1} satisfies the relation dω_{1} − θ ∧ ω_{1} = 0. In particular, ω_{1} is F_{1}-leafwise locally conformal symplectic form with the foliated Lee class ξ_{F_{1}} which is the image of ξ under the canonical projection H_{deR}^{1}(M) → H^{1}(M, F_{1}), where H^{1}(M, F_{1}) denotes the foliated de Rham cohomology group of (M, F_{1}).

As an immediate corollary we can deduce that any open manifold M admits a locally conformal symplectic form ω with a given Lee class ξ (see Section 2) provided that there exists a nondegenerate 2-form on M. Further, when ξ = 0 this gives a globally conformal symplectic form and in particular, a symplectic form on the manifold.

The following theorem may be viewed as a foliated version of Gromov’s result on the existence of a contact form on an open manifold ([5]).

Theorem 1.4. Let M^{(2n+1)+q} be an open manifold and F_{0} a codimension q foliation on M. Let (θ_{0}, ω_{0}) ∈ Ω^{1}(M) × Ω^{2}(M) be such that its restrictions to the leaves of F_{0} are almost contact structures. Then there exists a homotopy (F_{t}, θ_{t}, ω_{t}) such that (θ_{t}, ω_{t}) is a F_{t}-foliated almost contact structure and ω_{1} = dθ_{1}. In particular, θ_{1} is a leafwise contact form on (M, F_{1}).

Both the theorems, Theorem 1.3 and Theorem 1.4, can be interpreted in terms of regular Jacobi structures, which are more general geometric structures than Poisson structures (see Section 2). Locally conformal symplectic foliations and contact foliations arise naturally in connection with regular Jacobi structures.

We organise the paper as follows. In Section 2 we recall basic theory of Jacobi manifolds. We briefly discuss the Holonomic Approximation Theorem in Section 3 as it plays an important role in the proof. The proofs of Theorems 1.3, 1.4 are given in Sections 4 and 5 respectively. In section 6 we interpret Theorems 1.3 and 1.4 in terms of regular Jacobi structure on a manifold.

2. Preliminaries of Jacobi manifolds

A Jacobi structure on a smooth manifold M is given by a pair (Λ, E), where Λ is a bivector field, that is, a section of ∧^{2}T M and E is a vector field on M, satisfying

(1)  [Λ, Λ] = 2E ∧ Λ,  [E, Λ] = 0,

where [ , ] is the Schouten bracket. If E = 0 then Λ is a Poisson structure on M.

Example 2.1. Let M be a 2n-dimensional manifold with a symplectic form ω (i.e. a closed, non-degenerate 2-form). The non-degeneracy condition implies that b : T M → T^{∗}M, given by b(X) = i_{x}ω is a vector bundle isomorphism, where i_{X} denotes the interior multiplication by X ∈ T M. Then M has a Poisson structure defined by

π(α, β) = ω(b^{−1}(α), b^{−1}(β)), for all α, β ∈ T_{x}^{∗}M, x ∈ M.

Example 2.2. A locally conformal symplectic manifold (in short, an l.c.s manifold) is an even dimensional manifold M with a non-degenerate 2-form ω and a closed 1-form θ such that

dω = θ ∧ ω.

This equation can be expressed as d_{θ}(ω) = 0, where d_{θ} = d − θ∧ . It can be seen easily that d_{θ}^{2} = 0; hence d_{θ} is a coboundary operator. The 1-form θ is called the Lee form of the l.c.s structure ω and the de Rham cohomology class of θ is called the Lee class of ω. For an l.c.s manifold (M, ω, θ) the Jacobi pairing is given by

Λ(α, β) = ω(b^{−1}(α), b^{−1}(β)) and E = b^{−1}(θ),

where b : T M → T^{∗}M is defined as in Example 2.1.

Example 2.3. Contact manifolds are the odd dimensional counterpart of symplectic manifolds. A 1-form η on a (2n + 1)-dimensional manifold is said to be a contact form on M if η ∧ (dη)^{n} is nowhere vanishing. For contact manifold (M, η) the Jacobi pairing is given by

Λ(α, β) = dη(b^{−1}(α), b^{−1}(β)), and E = b^{−1}(η),

where b : T M → T^{∗}M is the isomorphism given by b(X) = i_{X}dη + η(X)η for all X ∈ T M.

Let (M, Λ, E) be a Jacobi manifold. The bivector field Λ defines a bundle homomorphism Λ^{#} : T^{∗}M → T M by

Λ^{#}(α)(β) = Λ(α, β),

where α, β ∈ T_{x}^{∗}M, x ∈ M. The Jacobi pair (Λ, E) defines a distribution D as follows:

D(x) = Λ^{#}(T_{x}^{∗}M) + ⟨E_{x}⟩, x ∈ M,

where ⟨E_{x}⟩ denotes the subspace of T_{x}M spanned by the vector E_{x}. In general, D is a singular distribution; however, it integrates to a (singular) foliation F on M. This foliation is referred as the characteristic foliation of the Jacobi manifold. If D = T M then the manifold is locally conformally symplectic or contact according as the dimension of M is even or odd. More generally, the leaves of the characteristic foliation are either locally conformally symplectic or contact with the induced Jacobi structures on them. In particular, when E = 0, Λ is a Poisson bivector field and the induced Poisson structure on the leaves are non-degenerate. Thus the leaves are symplectic manifolds.

A Jacobi manifold will be called regular if its characteristic foliation is regular. Thus a regular Jacobi manifold has a regular l.c.s foliation or a regular contact foliation depending on the dimension of the foliation. Recall that a foliated k-form on a foliated manifold (M, F) is a section of the exterior bundle ∧^{k}(T^{∗}F). The exterior differential operator d on the de Rham complex defines in a canonical way a coboundary operator d̄ on the foliated de Rham complex Γ(∧^{k}T^{∗}F), k ≥ 0. The resulting cohomology is called the foliated de Rham cohomology of the pair (M, F). By a foliated l.c.s. structure on a regular foliation F, we mean a pair (ω, θ), where θ is a foliated closed 1-form and ω is a non-degenerate foliated 2-form such that d̄_{θ}ω = 0, where d̄_{θ} = d̄+θ∧ω. The cohomology class of θ in the foliated de Rham cohomology group H^{1}(M, F) is called the Lee class of the foliated l.c.s. structure ω. On the other hand given a regular foliation with leafwise l.c.s or contact form there is a Jacobi structure on M whose characteristic foliation is the given one.

For general theory of symplectic and contact manifolds and Poisson, Jacobi manifolds we refer to [2], [10] and [11].

3. Holonomic Approximation Theorem

In this section we review the Holonomic Approximation Theorem from the theory of h-principle as it is going to play an important role in the proofs. Let p : E → M be a smooth fibration and let E^{(r)} → M denote the r-jet bundle of local sections of E. A section σ of E^{(r)} is said to be holonomic if it is the r-jet of some section f of E.

Let A be a polyhedron (possibly non-compact) in M of positive codimension. Let σ be any section of the r-jet bundle E^{(r)} over Op A. Here Op A denotes an unspecified open neighbourhood of A which may change in course of an argument. The Holonomic Approximation Theorem (see [3]) says that given any positive functions ε and δ on M there exist a (small) diffeotopy δ_{t} and a holonomic section σ′ : Op δ_{1}(A) → E^{(r)} such that

(1) dist(x, δ_{t}(x)) < δ(x) for all x ∈ M and t ∈ [0, 1] and

(2) dist(σ(x), σ′(x)) < ε(x) for all x ∈ Op (δ_{1}(A)).

This means that under the given conditions any σ can be approximated (in the fine C^{0} topology) by a holonomic section j^{r}f in a neighbourhood of δ_{1}(A), where δ_{t} is an arbitrary small diffeotopy of M. By a small diffeotopy we mean that δ_{t}(A) remains within the domain of σ for all t

We can summarise the main argument used in the proofs of Theorems 1.3 and 1.4 in the following result.

Theorem 3.1. Let M be an open manifold and p : E → M a smooth vector bundle. Let R be an open subset of the jet space E^{(r)}. Then given any section σ of R there exist a core K of M and a holonomic section σ′ : Op K → E such that the linear homotopy between σ and σ′ lies completely within Γ(R) over Op K.

Proof. Since R is an open subset of E^{(r)}, the space of sections of R is an open subset of Γ(E^{(r)}) in the fine C^{0}-topology. Therefore, given a section σ of R, there exists a positive function ε satisfying the following condition:

σ′ ∈ Γ(E^{(r)}) and dist (σ(x), σ′(x)) < ε(x) ⇒ σ′ is a section of R

Consider a core A of M and a δ-tubular neighbourhood of A for some positive δ. By the Holonomic Approximation Theorem there exist a diffeotopy δ_{t} and a holonomic section σ′ satisfying (1) and (2). Observe that K = δ_{1}(A) is also a core. Now, suppose that σ and σ′ are ε-close on the ρ-neighbourhood U_{ρ} of δ_{1}(A) in the δ-tubular neighbourhood, for some positive number ρ > 0. Now take a smooth map χ : M → [0, 1] satisfying the following conditions:

χ ≡ 1, on U_{ρ/2} and supp χ ⊂ U_{ρ}

Define a homotopy σ_{t}as follows:

σ_{t} = σ + tχ(σ′ − σ), t ∈ [0, 1].

Then (1) σ_{0} = σ and each σ_{t} is globally defined;

(2) σ_{t} = σ outside U_{ρ} for each t;

(3) σ_{1} is holonomic on U_{ρ/2}. Moreover, since the above homotopy between σ and σ′ is linear, σ_{t} lies in the ε-neighbourhood of σ for each t. Hence the homotopy σ_{t} lies completely within R by the choice of ε. This completes the proof of the proposition. □

Remark 3.2. In the language of h-principle, R is called an open differential relation of order r. If σ = j_{f}^{r} is a holonomic section of R then f is called a solution of R. The above theorem is a an instance of local h-principle near a core K of M. Note that the core K can not be fixed a priori in the statement of the proposition. The theorem is, in fact, true in the general set up where E is a smooth fibration [6]. In this case, however, the linearity condition on the homotopy σ_{t} has to be dropped for obvious reason.

4. The foliated l.c.s case

In this section we prove that an open manifold together with a regular foliation and a leafwise non-degenerate 2-form can be homotoped through such pairs to a regular foliation with a leafwise locally conformal symplectic form having prescribed Lee class.

Lemma 4.1. Let M^{n} be a given manifold equipped with a 1-form θ. Then there exists an epimorphism D_{θ} : E^{(1)} = (∧^{1}M)^{(1)} → ∧^{2}M satisfying D_{θ} ∘ j^{1}α = d_{θ}α so that the following diagram is commutative:

E^{(1)} ⟶^{D_{θ}} ∧^{2}(M)
↓  ↓
M ⟶^{id_{M}} M

In particular, given any 2-form ω there exists a section F_{ω} : E^{(1)} → M such that D_{θ} ∘ F_{ω} = ω.

Proof. Let θ be as in the hypothesis. Define D_{θ}(j^{1}α(x_{0})) = d_{θ}α(x_{0}) for any local 1-form α on M. To prove that the right hand side is independent of the choice of a representative α, choose a local coordinate system (x^{1}, ..., x^{n}) around x_{0} ∈ M. We may then express α and θ in the following forms:

α = Σ_{i=1}^{n}α_{i}dx^{i}, θ = Σ_{i=1}^{n}θ_{i}dx^{i}

where α_{i} and θ_{i} are smooth (local) functions defined in a neighbourhood of x_{0}. The 1-jet j^{1}α(x_{0}) is completely determined by the ordered tuple (a_{i}, a_{ij}) ∈ ℝ^{(n+n^{2})}, where

a_{i} = α_{i}(x_{0}), a_{ij} = \frac{∂α_{i}}{∂x^{j}} (x_{0}), i, j = 1, 2, …, n.

Now,

d_{θ}α(x_{0}) = dα(x_{0}) − θ(x_{0}) ∧ α(x_{0}) = Σ_{i<j}[(a_{ji} − a_{ij}) + (θ_{i}(x_{0})a_{j} − a_{i}θ_{j}(x_{0}))]dx^{i} ∧ dx^{j}

This shows that d_{θ}α(x_{0}) depends only on the 1-jet of α at x_{0} and the value of θ(x_{0}). Since θ is fixed, D_{θ} is well-defined. Hence, D_{θ} ∘ j^{1}α = d_{θ}(α)_{x_{0}} for any 1-form α.

It is easy to check that D_{θ} is a vector bundle morphism. In fact, D_{θ} is a vector bundle epimorphism. Indeed, given b_{ij}, 1 ≤ i < j ≤ n the following system of linear equations

(a_{ij} − a_{ji}) + (a_{i}θ_{j}(x_{0}) − a_{j}θ_{i}(x_{0})) = b_{ij}

is clearly solvable by a_{i} = 0, a_{ij} = −a_{ji} = \frac{b_{ij}}{2} .

Since D_{θ} is an epimorphism, every section ω : M → ∧^{2}M can be lifted up to a section F_{ω} : M → (T^{∗}M)^{(1)} such that D_{θ}F_{ω} = ω. Moreover, any two such lifts of a given section ω : M → ∧^{2}T^{∗}M are linearly homotopic. □

Proposition 4.2. Let M be an open manifold and F_{0} be a regular foliation on M. Let θ be a closed 1-form on M. Then any F_{0}-leafwise non-degenerate 2-form ω_{0} on M can be homotoped through such forms to a 2-form ω_{1} which is d_{θ}-exact on a neighbourhood U of some core K of M.

Proof. Let S denote the set of all elements ω_{x} in ∧^{2}(T_{x}^{∗}M), x ∈ M, such that the restriction of ω_{x} is non-degenerate on D = T_{x}F. Since non-degeneracy is an open condition, S is an open subset of ∧^{2}(T^{∗}M). Given a closed 1-form θ ∈ ξ, let

R_{θ} = D_{θ}^{−1}(S) ⊂ E^{(1)},

where D_{θ} is defined as in Lemma 4.1. Then R_{θ} is an open relation. Let σ_{0} be a section of E^{(1)} such that D_{θ}(σ_{0}) = ω_{0}. By Theorem 3.1, there exists a homotopy of sections σ_{t} of σ_{0} lying in R_{θ} such that σ_{1} = j^{1}α (that is σ_{1} is holonomic) on an open neighbourhood U of A.

Then the 2-forms ω_{t} = D_{θ} ∘ σ_{t}, t ∈ [0, 1], take values in S. Hence,

(1) ω_{t} is F_{0}-leafwise non-degenerate for all t ∈ [0, 1].

(2) ω_{1} = d_{θ}α is d_{θ}-exact on U.

□

Proof of Theorem 1.3. To prove the result we proceed as in [4]. Consider the canonical Grassmann bundle G_{2n}(T M) ⟶^{π} M for which the fibres π^{−1}(x) over a point x ∈ M is the Grassmannian of 2n-planes in T_{x}M. The space Dist_{q}(M) of codimension q distributions on M can be identified with the section space Γ(G_{2n}(M)). We topologize Dist_{q}(M) and Γ(∧^{2}T^{∗}M) with the compact open topology. Let Δ_{q} denote the subset of Dist_{q}(M) × Γ(∧^{2}T^{∗}M) defined by

\begin{aligned}Δ_{q} & := & \{(D, ω) : (ι_{D}^{∗}ω)^{n} ≠ 0\} and \\ Φ_{q} & := & F ol_{q}(M) × Γ(∧^{2}T^{∗}M) ∩ Δ_{q} \end{aligned}

where F ol_{q}(M) ⊂ Dist_{q}(M) is the space of all codimension q foliations (that is, integrable distributions) on M.

Let ω_{t} be a homotopy of 2-forms as obtained in Proposition 4.2. Then (F_{0}, ω_{t}) ∈ Φ_{q} for 0 ≤ t ≤ 1. Since M is an open manifold there exists an isotopy g_{t}, 0 ≤ t ≤ 1, with g_{0} = id_{M} such that g_{1} takes M into U. Note that U is a neighborhood of the core K defined by Proposition 4.2. Now we define (F′_{t}, ω′_{t}) ∈ Φ_{q} for t ∈ [0, 1] by setting

F′_{t} = g_{t}^{−1}F_{0}, ω′_{t} = g_{t}^{∗}ω_{1}.

Then, ω′_{t} is F′_{t}-leafwise non-degenerate. Further, it is easy to see that ω′_{1} is d_{g_{1}^{∗}θ} closed.

d_{g_{1}^{∗}θ}ω′_{1} = d_{g_{1}^{∗}θ}(g_{1}^{∗}ω_{1})
= dg_{1}^{∗}ω_{1} − g_{1}^{∗}θ ∧ g_{1}^{∗}ω_{1}
= g_{1}^{∗}[dω_{1} − θ ∧ ω_{1}]
= g_{1}^{∗}d_{θ}ω_{1} = 0

since ω_{1} is d_{θ}-closed on U and g_{1} maps M into U. Since g_{1} is homotopic to the identity map the de Rham cohomology class [g_{1}^{∗}θ] = [θ] = ξ. Note that ω_{1} defines a F_{1}-leafwise locally conformal symplectic structure. Let H^{1}(F_{1}) denote the first foliated de Rham cohomology group of the foliated manifold (M, F). Denote by ξ_{F_{1}} the image of ξ under the canonical map H^{1}(M) → H^{1}(F_{1}). Then it follows that the foliated Lee class of the foliated 2-form ω′_{1} is ξ_{F_{1}}. The desired homotopy is obtained by the concatenation of the two homotopies namely (F_{0}, ω_{t}) and (F′_{t}, ω′_{t}). □

Remark If we take θ to be the zero form we get back the result of Fernandes and Frejlich in[4].

Corollary 4.3. Let M be an open manifold and ω be a non-degenerate 2-form on M. Given any de Rham cohomology class ξ ∈ H^{1}(M), ω can be homotoped through non-degenerate 2-forms to a locally conformal symplectic form d_{θ}α, where the deRham cohomology class of θ is ξ.

Proof. This is a direct consequence of Theorem 1.3. □

5. The foliated contact case

In this section we prove Theorem 1.4. The proof is very similar to the l.c.s case.

Definition 5.1. Let M be a smooth manifold. A pair (α, β) consisting of a 1-form α and a 2-form β on M is said to be an almost contact structure on M if α ∧ β^{n} is nowhere vanishing on M; equivalently, the restriction of β to the vector bundle ker α is a symplectic structure on the bundle.

Lemma 5.2. Let M^{n} be a smooth manifold and E = ∧^{1}M. Then there exists a vector bundle epimorphism D̄

E^{(1)} ⟶^{D̄} ∧^{1}M ⊕ ∧^{2}M
↓  ↓
M ⟶^{id_{M}} M

such that D̄ ∘ (j^{1}α) = (α, dα) for any 1-form α on M.

Proof. Define D̄ by D̄(j^{1}α(x_{0})) = (α(x_{0}), dα(x_{0})) for any local 1-form α defined near a point x_{0}. It follows from the proof of Lemma 4.1 that this map is well defined. Hence D̄ ∘ j^{1}α = dα for any 1-form α. Let (x^{1}, ..., x^{n}) be a local coordinate system around x_{0} ∈ M and α = Σ_{i=1}^{n}α_{i}dx^{i} be the representation of α with respect to these coordinates. Then j_{x_{0}}^{1}(α) is uniquely determined by the ordered tuple (a_{i}, a_{ij}) ∈ ℝ^{n+n^{2}} as in Lemma 4.1 and

D̄(j_{x_{0}}^{1}α) = (α(x_{0}), dα(x_{0})) = (Σ_{i=1}^{n}a_{i}dx^{i}, Σ_{i<j}(a_{ij} − a_{ji})dx^{i} ∧ dx^{j})

Then D̄ is an epimorphism since the following system of equations

a_{i} = b_{i} and a_{ij} − a_{ji} = b_{ij} for all i ≠ j, i, j = 1, ..., n.

is clearly solvable in a_{i} and a_{ij}. Hence any section (θ, ω) : M → ∧^{1}M ⊕ ∧^{2}M can be lifted to a section F_{(θ,ω)} : M → E^{(1)} such that D̄ ∘ F_{(θ,ω)} = (θ, ω) and any two such lifts of a given (θ, ω) are homotopic. □

Proposition 5.3. Let M be an open manifold and F_{0} be a regular foliation on M. Let (θ, ω) be a pair consisting of a 1-form θ and a 2-form ω on M such that the restriction of (θ, ω) to the leaves are almost contact structures. Then (θ, ω) can be homotoped through such pairs to a pair (θ′, ω′), where ω′ = dθ′ on a neighbourhood U of some core K of M.

Proof. Let C denote the set of all pairs (θ_{x}, ω_{x}) ∈ T_{x}^{∗}M × ∧^{2}(T_{x}^{∗}M), x ∈ M, such that ι_{D}^{∗}θ_{x} ∧ ι_{D}^{∗}ω_{x} ≠ 0, where D is the tangent space to the leaf of F through x. Then C is an open subset of T^{∗}M ⊕ ∧^{2}(T^{∗}M). Let

R = D̄^{−1}(C) ⊂ E^{(1)},

where E = T^{∗}M and D̄ is as in Lemma 5.2. Then R is an open relation. Let σ be such that D̄(σ_{0}) = (θ_{0}, ω_{0}). By Theorem 3.1, there exists a homotopy of sections σ_{t} lying in R such that σ_{1} = j^{1}θ_{1} for some 1-form θ_{1} (that is σ_{1} is holonomic) on an open neighbourhood U of K.

Then the pairs (θ_{t}, ω_{t}) = D ∘ σ_{t}, t ∈ [0, 1], take values in C. It may be seen easily that

(1) (θ_{t}, ω_{t}) is a F_{0}-leafwise almost contact structure.

(2) dθ_{1} = ω_{1} on U. Hence (θ_{t}, ω_{t}) is the desired homotopy. □

Proof of Theorem 1.4. Let D_{0} denote the distribution defined by the foliation F_{0}. Define Δ_{q}, Φ_{q} as subsets of Dist_{q}(M) × Ω^{1}(M) × Ω^{2}(M) as follows:

Δ_{q} := {(D, θ, ω) : (ι_{D}^{∗}θ) ∧ (ι_{D}^{∗}ω)^{n} ≠ 0 at each point of M}
Φ_{q} := F ol_{q}(M) × Γ(∧^{1}M ⊕ ∧^{2}M) ∩ Δ_{q},

where i_{D} is the inclusion of D in T M. By the given hypothesis, (F_{0}, θ_{0}, ω_{0}) is in Φ_{q}. Let (θ_{t}, ω_{t}) be a homotopy of (θ_{0}, ω_{0}) as obtained in Proposition 5.3. Then (F_{0}, θ_{t}, ω_{t}) belongs to Φ_{q} for 0 ≤ t ≤ 1. Choose an isotopy g_{t} : M → M such that g_{0} = id_{M} and g_{1}(M) ⊂ U, where U is a neighborhood of the core K defined by Proposition 5.3. Such an isotopy exists because M is an open manifold and U is a neighbourhood of a core of M. Now we define (F′_{t}, θ′_{t}, ω′_{t}) ∈ Φ_{q}, t ∈ [0, 1] by setting

F′_{t} = g_{t}^{−1}(F_{0}), θ′_{t} = g_{t}^{∗}θ_{1}, ω′_{t} = g_{t}^{∗}ω_{1}.

Finally observe that

dθ′_{1} = d[g_{1}^{∗}θ_{1}] = g_{1}^{∗}[dθ_{1}] = g_{1}^{∗}[ω_{1}] = ω′_{1},

since g_{1}(M) ⊂ U and dθ_{1} = ω_{1} on U. Therefore, θ′_{1} is a F′_{1}-leafwise contact form. Concatenating the homotopies (F_{0}, θ_{t}, ω_{t}) and (F′_{t}, θ′_{t}, ω′_{t}) we obtain the desired homotopy. □

Remark 5.4. An analogue of Theorem 1.2 holds true in the contact case. Suppose that M is a smooth manifold with a foliation F of dimension 2n + 1. Let (α, β) be a pair consisting of a foliated 1-form α and a foliated 2-form β on (M, F) such that α ∧ β^{n} is nowhere vanishing on the leaves of F. The leafwise non-vanishing condition on (α, β) is an open condition and hence defines an open subset R in the 1-jet space E^{(1)}, where E → M is the vector bundle whose total space is T^{∗}F ⊕Λ^{2}(T^{∗}F). Since the condition is invariant under the action of foliated diffeotopies Bertelson’s theorem applies to this relation provided we assume that the foliation (M, F) satisfies certain openness condition as defined in [1]. Hence, the given pair (α, β) can be homotoped through such pairs to (η, d_{F}η) for some foliated 1-form η on (M, F), so that η is a foliated contact form. (Here d_{F} denotes the coboundary map of the foliated de Rham complex).

6. Regular Jacobi structures on open manifolds

We can reformulate Theorems 1.3 and 1.4 in terms of the Jacobi structure as follows. Let ν^{k}(M) denote the space of sections of the alternating bundle Λ^{k}(T M). We shall refer to these sections as k-multivector fields on M. A pair (Λ, E) ∈ ν^{2}(M) × ν^{1}(M) will be called a regular pair if D = Λ^{#}(T^{∗}M) + ⟨E⟩ is a regular distribution on M.

Let (Λ, E) be a regular pair such that the distribution D = Im Λ^{#} + ⟨E⟩ is even dimensional. Define ϕ : D^{∗} → D by ϕ(i^{∗}α) = Λ^{#}α for all α ∈ T^{∗}M. Using the skew symmetry property of Λ we can show that ker Λ^{#} = ker α. Hence ϕ is well defined. Clearly ϕ is an isomorphism and therefore, it defines a section ω of ∧^{2}D^{∗} by ω(X, Y) = ϕ^{−1}(X)(Y) which is non-degenerate at each point. Conversely given any section ω of ∧^{2}D^{∗}, define Λ by the following commutative diagram:

(2)  \begin{matrix}T^{∗}M & ⟶^{Λ^{#}} & T M \\ i^{∗}↓ & & ↑i \\ D^{∗} & ⟶^{ω̃^{−1}} & D \end{matrix}

where ω̃ is defined by ω̃(X) = i_{X}ω for all X ∈ ΓD.

In view of the above correspondence, we can interpret Theorem 1.3 as follows.

Theorem 6.1. Let M be an open manifold and (Λ_{0}, E_{0}) be a regular pair such that the distribution D = Im Λ^{#} + ⟨E⟩ is even dimensional and integrable. Fix a de Rham cohomology class ξ in H^{2}(M). Then (Λ_{0}, E_{0}) can be homotoped through regular pairs to a Jacobi pair (Λ_{1}, E_{1}) such that ϕ_{1}^{−1}(E_{1}) is a closed 1-form in the foliated de Rham cohomology of (M, F_{1}) and the cohomology class of ϕ_{1}^{−1}(E_{1}) in H^{1}(M, F) is the same as ξ_{F_{1}}, where F_{1} is the characteristic foliation of the Jacobi pair (Λ_{1}, E_{1}) and ϕ_{1} : T^{∗}F_{1} → T F_{1} is defined as above.

Theorem 6.2. Let (Λ_{0}, E_{0}) ∈ ν^{2}(M) × ν^{1}(M) be a regular pair an open manifold M. Suppose that the distribution D_{0} := Λ_{0}^{#}(T^{∗}M)+⟨E_{0}⟩ is odd dimensional. Then (Λ_{0}, E_{0}) can be homotoped through regular pairs to a Jacobi pair (Λ_{1}, E_{1}) provided D_{0} is an integrable distribution.

Proof. Let (Λ, E) be a regular pair and the distribution D = Λ^{#}(T^{∗}M)+⟨E⟩ is odd dimensional. Define a section α of D^{∗} by the relations

(3)  α(Im Λ^{#}) = 0 and α(E) = 1.

Define β as a section of ∧^{2}(D^{∗}) by

(4)  β(Λ^{#}η, Λ^{#}ξ) = Λ(η, ξ), i_{E}β = 0,

where i_{E} denotes the interior multiplication by E. It can be shown easily that β is non-degenerate on Im Λ^{#}. Hence α ∧ β^{n} is nowhere vanishing. Further, note that Λ^{#} satisfies the relation i_{Λ^{#}ξ}β = −ξ|_{ker α}.

On the other hand given a pair (α, β) as above we can write D = ker α ⊕ ker β and define a vector field E on M satisfying

(5)  i_{E}β = 0, and α(E) = 1

Since β is non-degenerate on ker α by our hypothesis, β̃ : ker α → (ker α)^{∗} is an isomorphism. For any ξ ∈ T^{∗}M define Λ^{#}(ξ) as the unique element of ker α such that

(6)  β̃(Λ^{#}ξ) = −ξ|_{ker α}.

Then (Λ, E) is a regular pair. Thus there is a one to one correspondence between regular pairs (Λ, E) and the triples (D, α, β) such that α∧β^{n} is nowhere vanishing. Further, the regular contact foliations correspond to regular Jacobi pairs with odd-dimensional characteristic distributions under this correspondence [9].

The result now follows directly from Theorem 1.4. Let (Λ_{0}, E_{0}) be as in the hypothesis and F_{0} be the foliation associated with D_{0}. Then we can define (α_{0}, β_{0}) by the equations (3) and (4) so that α_{0} ∧ β_{0}^{n} is non-vanishing on D_{0}. The foliated forms α_{0} and β_{0} can be extended to differential forms on M. Let α̃_{0} and β̃_{0} be any two such extensions of α_{0} and β_{0} respectively. By Theorem 1.4, we obtain a homotopy (F_{t}, α̃_{t}, β̃_{t}) of (F_{0}, α̃_{0}, β̃_{0}) such that α̃_{t} ∧ β̃_{t}^{n} is a nowhere vanishing form on T F_{t} and β̃_{1} = dα̃_{1} on F_{1}, so that α̃_{1} is a F_{1}-leafwise contact form. Let (α_{t}, β_{t}) be a pair of foliated forms obtained by restricting (α̃_{t}, β̃_{t}) to the tangent distribution of the foliation F_{t}. The desired homotopy (Λ_{t}, E_{t}) is then obtained from (α_{t}, β_{t}) by (5) and (6). □

We conclude with the following remark.

Remark 6.3. The integrability condition on the initial distribution in Theorems 1.3 and 1.4 can be relaxed to the extent that it is enough to have the distribution homotopic to the distribution of a foliation. This can be seen by taking into account the classification of foliations due to Haefliger[8]. We refer to [4] for a detailed argument.

References

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Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India, e-mail: mahuya@isical.ac.in; Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India, e-mail: mukherjeesauvik@yahoo.com