1-forms and vector fields^{1}

M. Zhitomirskii

Department of Mathematics, Technion, 32000 Haifa, Israel mzhi@techunix.technion.ac.il

Abstract. I show that under a mild condition on a differential 1-form ω_{0} on a compact n-manifold M, the classification of global cooriented hyperplane fields (or global Pfaff equations) on M that are described by 1-forms close to ω_{0} reduces to the global orbital classification of vector fields on M.

1. Introduction and main results.

Given a differential 1-form on a orientable compact n-manifold M = M^{n} denote by (ω) one of the following: if ω vanishes at no points of M then (ω) is the (cooriented) hyperplane field in T M which is the field of kernels of ω; if ω vanishes at least at one point of M then (ω) is the Pfaff equation (1-generated module of 1-forms over the ring of functions) generated by ω. In the both cases the equivalence of (ω_{1}) and (ω_{0}) with respect to the natural action of the group of global diffeomorphisms of M is the equivalence of ω_{1} and ω_{0} with respect to the group of diffeomorphisms and multiplication by nonvanishing functions.

Fix ω_{0} and assume that ω_{1} is sufficiently close to ω_{0} in the C^{r}-topology. If n is odd, r ≥ 1, and ω_{0} is a contact 1-form then (ω_{1}) is equivalent to (ω_{0}). This follows from the classical Gray theorem [Gra].

If n is even or if n is odd and ω_{0} is not a contact 1-form then there is an obstacle for the equivalence of (ω_{1}) and (ω_{0}). This obstacle is related to a characteristic vector field of (ω).

A characteristic vector field X of (ω) is not unique, but any two characteristic vector fields differ by multiplication by a nonvanishing function. The construction of a single characteristic vector field requires fixing a 1-form ω generating the Pfaff equation (ω) or describing the hyperplane field (ω) (two 1-forms making this job differ by multiplication by a nonvanishing function) and fixing a volume form Ω on M^{n}.

If n = 2k then X is a vector field on M^{n} defined by the relation

X ⌋ Ω = ω ∧ (dω)^{k−1}.

If n = 2k +1 then X is defined on the set S consisting of points at which ω is not contact:

S = {H = 0}, H = (ω ∧ (dω)^{k})/Ω,
X(p) ⌋ Ω(p) = (ω ∧ (dω)^{k−1} ∧ dH) (p), p ∈ S.

It is easy to see that X(p) ∈ T_{p}S for any point p ∈ S at which S is a smooth codimension one submanifold of M^{n} and that the set of points at which X vanishes includes all other points of S. The set S is called the Martinet hypersurface.

One can easily check that for any parity of n the multiplication of ω and Ω by nonvanishing functions leads to multiplication of X by a nonvanishing function. Therefore the equivalence of (ω_{1}) and (ω_{0}) implies the equivalence of corresponding characteristic vector fields X_{1} and X_{0} with respect to the group of diffeomorphisms of M and multiplication by nonvanishing functions. This equivalence of vector fields is called orbital equivalence. The condition of orbital equivalence of X_{1} and X_{0} is the obstacle for the equivalence of (ω_{1}) and (ω_{0}).

1Date: November 18, 1999. The work on this paper was supported by the Max-Planck-Institut fur Mathematik

In the present paper I show that under a mild condition on the singular points of X_{0} this is the only obstacle: if ω_{1} is sufficiently close to ω_{0} and X_{1} is orbitally equivalent to X_{0} via a sufficiently close to the identity diffeomorphism of M, then (ω_{1}) is equivalent to (ω_{0}).

The results of this paper are related to the question that J. Martinet asked me in 1989 and to conjectures in my book [Zh].

Convention. In what follows M = M^{n} is a compact orientable manifold of dimension n ≥ 3. All objects are assumed to be of the class C^{∞} and all results hold in the same category.

To formulate the main results I will use the following definition.

Definition, cf. [Mou]. A vector field X on a manifold Q^{d} of dimension d has s-division property if for any i = 1, … s and any differential form (d − i)-form θ_{d−i} on Q^{d} such that X ⌋ θ_{d−i} = 0 there exists a differential (d − i + 1)-form θ_{d−i+1} on Q^{d} such that θ_{d−i} = X ⌋ θ_{d−i+1}.

In the C^{∞} category, due to the division of unity, this property is equivalent to local s-division property (the same property for germs at any fixed point of Q^{d}). If X_{1}, …, X_{d} are the coefficients of the germ of X at a point p ∈ Q^{d} in some coordinate system then the s-division property of the germ of X at p is the property of the ideal I = (X_{1}, … X_{d}) generated by the functions X_{1}, … X_{d}. The simplest cases where the s-division property holds for the germ of X at p ∈ Q^{d} are as follows: (a) X(p) ≠ 0; (b) X(p) = 0 and the ideal (X_{1}, … X_{d}) contains (s + 1) differentially independent functions. The simplest case where the s-division property does not hold is the case where X vanishes on a submanifold of Q^{d} of codimension s or less. The 1-division property can be easily expressed without using exterior forms: X has the 1-division property if and only if for any vector field Y on Q^{d} such that Y ∧ X = 0 there exists a function F such that Y = F X. Here Y ∧ X is a field of exterior 2-vectors, and the condition Y ∧ X = 0 means that at any point p ∈ Q^{d} either the vectors X(p) and Y (p) are proportional or one of these vectors is equal to zero.

In this paper I need the 2-division property of a characteristic vector field X_{0} of (ω_{0}) on M^{2k} (then Q^{d} = M^{2k}) and the 1-division property of a characteristic vector field X_{0} of (ω_{0}) on M^{2k+1}, k ≥ 2. In the latter case X_{0} is defined on the Martinet hypersurface S ⊂ M^{2k+1}, and I assume that S is a smooth codimension one submanifold of M^{2k+1}, then d = 2k and Q^{d} = S. In the 3-dimensional case the division property is not required.

Theorem 1. Let ω_{0} be a 1-form on M = M^{2k}, X_{0} - a characteristic vector field of (ω_{0}). Assume that the following conditions hold:

(a) X_{0} has the 2-division property;

(b) there exists an integer r ≥ 0 such that j_{p}^{r}X_{0} ≠ 0, p ∈ M.

Set q = 2r + 2. Then (ω_{0}) is globally equivalent to (ω_{1}) provided that the 1-form ω_{1} is sufficiently C^{q}-close to ω_{0} and a characteristic vector field X_{1} of (ω_{1}) is orbitally equivalent to X_{0} via a sufficiently C^{q+1}-close to the identity diffeomorphism of M.

Note that the assumptions here do not depend on the choice of the characteristic vector fields X_{0} and X_{1} - I recitate that a characteristic vector field is unique up to multiplication by a nonvanishing function. It is clear that the latter is irrelevant for the orbital equivalence and for the division property.

Note also that Theorem 1 contains no assumptions on points at which ω_{0} or ω_{1} vanish. The assumption (b) holds for some r if and only if the vector field X_{0} is flat (has zero formal series) at no points of M.

The analysis of the condition (a) is not so simple since the vector field X_{0} is not generic in the space of all vector fields on M. In particular, the degeneartion ω_{0} ∧ (dω_{0})^{k−1}(p) = 0 has codimension 3, see [Mar], therefore there exists an open set of 1-forms ω_{0} on M such that X_{0} vanishes on a codimension 3 submanifold of M. Nevertheless, conjecturally the condition (a) is violated only for infinite codimension degenerations of 1-forms. In this paper I prove the simplest (and the main) result in this direction.

Proposition 1. If M is a manifold of dimension 4,6, or 8, then in the space of all 1-forms on M there is an open and dense set in the C^{2}-topology such that any 1-form ω_{0} of this set satisfies the assumptions (a) and (b) of Theorem 1.

One of the steps in the proof of Theorem 1 is the reduction to the following deformation theorem. This theorem requires the s-division property with parameter t ∈ [0, 1]. The definition is the same except that the forms θ_{d−i} and θ_{d−i+1} must be replaced by families θ_{d−i,t} and θ_{d−i+1,t}, t ∈ [0, 1]. Here and in what follows all families depend C^{∞} on the parameter. The simplest cases where a vector field has the s-division property with parameter t ∈ [0, 1] are the cases (a) and (b) given after the definition of the s-division property without parameter.

Theorem 1.A. Let ω_{t} be a family of 1-forms on M = M^{2k}, t ∈ [0, 1]. Assume that a single vector field X_{0} is a characteristic vector field for all (ω_{t}). If X_{0} has the 2-division property with parameter t ∈ [0, 1] then there exists a family Φ_{t} of global diffeomorphsims of M such that Φ_{t}^{∗}(ω_{t}) = (ω_{0}).

Remark. If ω_{t} depends on t polynomially then the 2-division property with parameter t ∈ [0, 1] can be replaced by the same division property without parameter.

The results in the odd-dimensional case, M = M^{2k+1}, require a bit more assumptions since they reduce the classification of hyperplane fields or Pfaff equations on M^{n} to the orbital classification of vector fields on a hypersurface of M. Within this paper I assume that the Martinet hypersurface S_{0} of a Pfaff equation or hyperplane field (ω_{0}) is a smooth codimension one submanifold of M^{2k+1} and that the same remains true under a small deformation of ω_{0}. This condition (that can be called the structural smoothness of the Martinet hypersurface, see [JaZh1]) can be expressed as follows: j_{p}^{1} (ω_{0} ∧ (dω_{0})^{k}) ≠ 0, p ∈ M. In fact, S_{0} is the zero level of a function H_{0} = (ω_{0} ∧ (dω_{0})^{k}) /Ω, where Ω is a volume form, and the condition above means that the 1-form dH_{0} does not vanish at any point of S_{0}.

Like in the even-dimensional case, the theorem below for the odd-dimensional case admits vanishing of ω_{0} on a subset of M, but at any point of this subset a certain transversality condition is required.

Theorem 2. Let ω_{0} be a 1-form on M = M^{2k+1}, X_{0} a characteristic vector field of (ω_{0}) defined on the Martinet hypersurface S_{0}. Assume that the following conditions hold:

(a) j_{p}^{1}(ω_{0} ∧ (dω_{0})^{k}) ≠ 0, p ∈ S_{0};

(b) if ω_{0}(p) = 0 at a point p ∈ S_{0} then the kernel of the 2-form dω_{0}(p) is transversal to S_{0};

(c) If k ≥ 2 then X_{0} has the 1-division property. (This assumption is not required if k = 1);

(d) there exists an integer r ≥ 0 such that j_{p}^{r}X_{0} ≠ 0, p ∈ S_{0}.

Set q = r + 2 if k = 1 and q = max(2r + 2, r + 3) if k ≥ 2. Then (ω_{0}) is globally equivalent to (ω_{1}) provided that the 1-form ω_{1} is sufficiently C^{q}-close to ω_{0} and a characteristic vector field X_{1} of (ω_{1}), defined on the Martinet hypersurface S_{1} of (ω_{1}), is orbitally equivalent to X_{0} via a sufficiently C^{q+1}-close to the identity diffeomorphism of M sending S_{1} to S_{0}.

I explain the condition (b). If ω_{0}(p) = 0 then the space ker dω_{0}(p) consisting of tangent vectors a ∈ T_{p}M^{2k+1} such that a ⌋ dω_{0}(p) = 0 is invariantly related to the Pfaff equation (ω_{0}) (if ω_{0}(p) ≠ 0 then this is, of course, not so). The structural smoothness of S_{0} at p, i.e. the condition j_{p}^{1}(ω_{0} ∧ (dω_{0})^{k}) ≠ 0, implies that if ω_{0}(p) = 0 then (dω_{0})^{k}(p) ≠ 0, and consequently the space ker dω_{0}(p) is a line in T_{p}M. The condition (b) excludes degenerations when this line is tangent to S_{0}.

One can expect that the condition (a) (the structural smoothness of S_{0}) holds for a generic 1-form ω_{0}, but this is so only if M is a 3 or 5-dimensional manifold. The thing is that S_{0} fails to be a smooth hypersurface near points p such that ω_{0} ∧ (dω_{0})^{k−1}(p) = 0. This degeneration has codimension 6, see [Mar], and therefore it is typical if dimM ≥ 7. This explains the following proposition.

Proposition 2. If M is a 3-dimensional or a 5-dimensional manifold then in the space of all 1-forms on M there is an open and dense set in the C^{3}-topology such that any 1-form ω_{0} of this set satisfies the assumptions (a-d) of Theorem 2.

The deformation theorem, corresponding to Theorem 2, is as follows.

Theorem 2.A. Let ω_{t} be a family of 1-forms on M = M^{2k+1} such that j_{p}^{1} (ω_{t} ∧ (dω_{t})^{k−1}) ≠ 0, p ∈ M, t ∈ [0, 1]. Assume that a single hypersurface S_{0} and a single vector field X_{0} on S_{0} are the Martinet hypersurface and a characteristic vector field respectively for all (ω_{t}). Assume that for any point (p, t) ∈ S_{0} × [0, 1] such that ω_{t}(p) = 0 the line ker dω_{t}(p) ⊂ T_{p}M is transversal to S_{0}. If k ≥ 2 (and only in this case) assume that X_{0} has 1-division property with parameter t ∈ [0, 1]. Then there exists a family Φ_{t} of global diffeomorphsims of M such that Φ_{t}^{∗}(ω_{t}) = (ω_{0}).

Remark. If ω_{t} depends on t polynomially then the 1-division property with t ∈ [0, 1] can be replaced by the same division property without parameter.

The proofs of Theorem 1 and 1.A are contained in sections 2 and 3. The proof of Theorems 2 and 2.A for the 3-dimensional case are contained in section 4. The proofs of Theorems 2 and 2.A for the case k ≥ 2 are the most complicated since they combine the techniques of of all previous proofs. They are contained in sections 5 and 6. The Propositions 1 and 2 are proved in section 7.

The reduction of Theorem 1 to Theorem 1.A is based on the techniques in [MoZh, Appendix A], where Theorems 1 and 1.A are proved for the simplest case where X_{0} vanishes at no points of the manifold, i.e. (ω_{0}) is a quasi-contact structure. Some steps in the proof of Theorems 2 and 2.A involve the techniques of the paper [JaZh2], where the significance of the division properties of a characteristic vector field for reduction theorems was realized.

Acknowledgements. I am thankful to the administration of the Max-Planck-Institut fur Mathematik, where this work was done, for the hospitality and excellent working atmosphere during my stay. I am grateful to A. Bolibruch, B.Jakubczyk, and R.Montgomery for useful discussions related to this paper.

2. Proof of Theorem 1.A.

Theorems like Theorem 1.A are usually proved by the homotopy method. The required family Φ_{t} of diffeomorphisms will be constructed as the solution of the ordinary differential equation with a parameter p ∈ M = M^{2k} of the form

\frac{dΦ_{t}(p)}{dt} = Z_{t} (Φ_{t}(p)) , Φ_{0}(p) = p, p ∈ M,  (2.1)

where Z_{t} is a family of vector fields on M. The requirement Φ_{t}^{∗}(ω_{t}) = (ω_{0}) holds if

L_{Z_{t}}ω_{t} + h_{t}ω_{t} + \frac{d}{dt}ω_{t} = 0,  (2.2)

where h_{t} is any family of functions on M, and L_{Z_{t}} is the Lie derivative along the vector field Z_{t}. In fact, due to the compactness of M the equation (2.1) has a solution for all t, and the basic uniquiness theorems imply that for any t the mapping p → Φ^{t}(p) is a diffeomorphism of M. The function A(t) = Φ_{t}^{∗}ω_{t} with values in the space of 1-forms on M satisfies the equation

A′(t) = Φ_{t}^{∗} (L_{Z_{t}}ω_{t} + \frac{d}{dt}ω_{t}) = −h_{t}(Φ_{t})A(t),

and it follows that A(t) = H_{t}A(0) = H_{t}ω_{0}, where H_{t} is a family of nonvanishing functions.

The first step in solving the equation (2.2) with respect to Z_{t} and h_{t} is to exclude h_{t} using the following division property of ω_{t}.

Lemma 2.1. Let μ_{t} be any family of 1-forms on M such that μ_{t} ∧ ω_{t} = 0, t ∈ [0, 1]. Then there exists a family of functions h_{t} such that μ_{t} = h_{t}ω_{t}.

This property is not trivial since ω_{t} might vanish at some points (p, t) ∈ M × [0, 1]. The proof uses the division property of a characteristic vector field. Lemma 2.1 allows to reduce the equation (2.2) to the equation

(L_{Z_{t}}ω_{t} + \frac{d}{dt}ω_{t}) ∧ ω_{t} = 0  (2.3)

with respect to the family Z_{t} of vector fields. The solution of this equation will be constructed as follows.

Lemma 2.2. There exists a family Z_{t} of vector fields on M such that

Z_{t} ⌋ (ω_{t} ∧ (dω_{t})^{k−1}) = (1 − k)(\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−2}.  (2.4)

The proof of this lemma is based on the assumption that (ω_{t}) have the same characteristic vector field X_{0}. I will show that the family of vector field Z_{t} satisfying (2.4) is a solution of the equation (2.3).

At first note that (2.4) implies that the form (Z_{t} ⌋ ω_{t}) · (dω_{t})^{k−1} is the external product of ω_{t} with a certain (2k − 3)-form. It follows that

(Z_{t} ⌋ ω_{t}) · ω_{t} ∧ (dω_{t})^{k−1} = 0.  (2.5)

Notation. Denote by R the subset of M consisting of points at which the vector field X_{0} does not vanish.

Since X_{0} is a characteristic vector field of (ω_{t}) for all t then (ω_{t} ∧ (dω_{t})^{k−1}) (p) ≠ 0 for any p ∈ R and any t, and (2.5) implies that the function Z_{t} ⌋ ω_{t} vanishes at any point p ∈ R, for any t. The set R is everywhere dense in M - it is easy to see that otherwice the field X_{0} would not have the 2-division property. Therefore Z_{t} ⌋ ω_{t} ≡ 0. Then L_{Z_{t}}ω_{t} = Z_{t} ⌋ dω_{t} and the relation (2.4) easily implies

(L_{Z_{t}}ω_{t} + \frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−2} = 0.  (2.6)

Now the following simple algebraic statement will be used. Assume that ω is a quasicontact form at a point p ∈ M^{2k}, i.e. (ω ∧ (dω)^{k−1})(p) ≠ 0. Let μ be a 1-form such that (μ ∧ ω ∧ (dω)^{k−2}) (p) = 0. Then (μ ∧ ω)(p) = 0. In fact, if (μ ∧ ω)(p) ≠ 0 then (dω)^{k−2}(p) vanishes when being restricted to the space kerω(p) ∩ kerμ(p) which is a hyperplane in the space kerω(p). This contradicts to the assumption (ω ∧ (dω)^{k−1})(p) ≠ 0.

Using the given algebraic statement and the relation (2.6) I conclude that (2.3) holds for all points at which ω_{t} is quasi-contact, i.e. for all points (p, t) such that p ∈ R. Since R is everywhere dense in M then (2.3) holds for all t and all points of M. Theorem 1.A is proved modulo Lemmas 2.1 and 2.2.

Proof of Lemma 2.2. Since X_{0} is a characteristic vector field of (ω_{t}) then

X_{0} ⌋ (ω_{t} ∧ (dω_{t})^{k−1}) = 0.  (2.7)

Let p ∈ R. Then (ω_{t} ∧ (dω_{t})^{k−1}) (p) ≠ 0 and it is easy to see that near p the relation (2.7) implies that X_{0} ⌋ ω_{t} = 0 and X_{0} ⌋ dω_{t} = h_{t}ω_{t} for some function h_{t}. It follows that near p we have, for any t:

X_{0} ⌋ ((\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−2}) = 0.  (2.8)

Since the set R is everywhere dense in M then (2.8) holds for all t and all points of the manifold. Using the 2-division property of X_{0} with parameter t ∈ [0, 1] I conclude that (2.8) implies the existence of a family θ_{t,2k−1} of (2k − 1)-form on M such that

(\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−2} = X_{0} ⌋ θ_{t,2k−1}.  (2.9)

The relation (2.7) and the 1-division property of X_{0} (following from the 2-division property) imply the existence of a family Ω_{t} of 2k-forms such that

X_{0} ⌋ Ω_{t} = ω_{t} ∧ (dω_{t})^{k−1}.  (2.10)

The condition that X_{0} is one of the characteristic vector fields of (ω_{t}) implies that Ω_{t} vanishes at no point (p, t) ∈ M × [0, 1]. Therefore one can define a family Z_{t} of vector fields by the relation

Z_{t} ⌋ Ω_{t} = (k − 1)θ_{t,2k−1},

and then (2.9) and (2.10) imply (2.4). ■

Proof of Lemma 2.1. Let μ_{t} ∧ ω_{t} = 0. Take any point p ∈ R. It was shown that near p the vector field X_{0} is annihilated by ω_{t} and X_{0} ⌋ dω_{t} = h_{t}ω_{t} for some function h_{t} (see the proof of Lemma 2.2). Then X_{t} ⌋ μ_{t} = 0 and X_{0} ⌋ (μ_{t} ∧ (dω_{t})^{k−1}) = 0 for all t. Since the set R is dense, these relations hold at all points p ∈ M and for all t. The latter relation implies, together with the 1-division property of X_{0}, the existence of a family Ω′_{t} of (2k)-forms on M such that μ_{t} ∧ (dω_{t})^{k−1} = X_{0} ⌋ Ω′_{t}. Let Ω_{t} be the family of nondegenerate volume forms in (2.10), and let Ω′_{t} = h_{t}Ω_{t}. Then (μ_{t} − h_{t}ω_{t}) ∧ (dω_{t})^{k−1} = 0. Let γ_{t} = μ_{t} − h_{t}ω_{t}. Note that γ_{t} ∧ ω_{t} = 0. This relation and the relation γ_{t} ∧ (dω_{t})^{k−1} = 0 easily imply that γ_{t}(p) = 0 for all points (p, t) such that ω_{t} is quasi-contact at p, i.e. for all points (p, t) such that p ∈ R. Using again the density of the set R I conclude that γ_{t}(p) = 0 for all p and t, i.e. μ_{t} = h_{t}ω_{t}. ■

Proof of remark after Theorem 1.A. It is clear that if X_{0} has the 1-division property without parameter then it has the same property with parameter t provided that all families depend on t polynomially. If ω_{t} depends on t polynomially then all objects in the proof of Theorem 1.A depend on t polynomially, and the proof remains the same.

3. Proof of Theorem 1.

The proof is based on the following lemma.

Lemma 3.1. Let ω_{0} be a 1-form on M = M^{2k} satisfying the conditions (a) and (b) of Theorem 1. Let X_{0} be a characteristic vector field of (ω_{0}). Let ω_{1} be a 1-form on M that is sufficiently C^{r+1}-close to ω_{0} and such that

ω_{1} ∧ (dω_{1})^{k−1} = ω_{0} ∧ (dω_{0})^{k−1}.  (3.1)

Then X_{0} is a characteristic vector field of (ω_{t}), where ω_{t} = ω_{0} + t(ω_{1} − ω_{0}).

At first I will show that Theorem 1 follows from this lemma and Theorem 1.A. Let ω_{0} and ω_{1} be 1-forms satisfying the conditions of Theorem 1. Let Φ be a diffeomorphism realizing the orbital equivalence of a characteristic vector field X_{1} of (ω_{1}) and the characteristic vector field X_{0} of (ω_{0}). Set ω_{2} = Φ^{∗}ω_{1}. Then

ω_{2} ∧ (dω_{2})^{k−1} = F ω_{0} ∧ (dω_{0})^{k−1}

for some function F on M. Since Φ is C^{q+1}-close to the identity and ω_{1} is C^{q}-close to ω_{0} then ω_{2} is C^{q}-close to ω_{0} and consequently the form ω_{2} ∧(dω_{2})^{k−1} is C^{q−1}-close to ω_{0} ∧(dω_{0})^{k−1}. By the condition (b) of Theorem 1 the r-jet of the form ω_{0} ∧ (dω_{0})^{k−1} vanishes at no points of M, therefore the function F is C^{q−r−1}-close to 1. Note that q − r − 1 = r + 1. Set ω_{3} = F^{1/k}ω_{2}. Then ω_{3} is C^{r+1}-close to ω_{0}, and ω_{3} satisfies the relation

ω_{3} ∧ (dω_{3})^{k−1} = ω_{0} ∧ (dω_{0})^{k−1}.

By Lemma 3.1 the vector field X_{0} is a characteristic vector field of (ω̂_{t}), where ω̂_{t} = ω_{0} + t(ω_{3} − ω_{0}). By Theorem 1.A (ω_{t}) is equivalent to (ω_{0}). In particular, (ω_{3}) is equivalent to (ω_{0}) and consequently (ω_{1}) is equivalent to (ω_{0}).

Proof of Lemma 3.1. I use again the set R ⊂ M consisting of points at which the field X_{0} does not vanish. The fact that X_{0} is a characteristic vector field of (ω_{0}) and of (ω_{1}) implies that near any point p ∈ R the following holds:

X_{0} ⌋ ω_{0} = X_{0} ⌋ ω_{1} = 0,  (3.2)

X_{0} ⌋ dω_{0} = h_{0}ω_{0}, X_{0} ⌋ dω_{1} = h_{1}ω_{1}  (3.3)

for some functions h_{0} and h_{1} defined near p. I will show that the relation (3.1) implies that

h_{0} = h_{1}.  (3.4)

In order to prove this I take the Lie derivative of the relation (3.1) along the vector field X_{0}. Using (3.2) and (3.3) one obtains the following:

L_{X_{0}} (ω_{0} ∧ (dω_{0})^{k−1}) = (k − 1)h_{0}ω_{0} ∧ (dω_{0})^{k−1},

L_{X_{0}} (ω_{1} ∧ (dω_{1})^{k−1}) = (k − 1)h_{1}ω_{1} ∧ (dω_{1})^{k−1}.

These relations and (3.1) imply (3.4). The relation (3.4) easily implies

X_{0} ⌋dω_{t} = h_{0}ω_{t}.

Using this and (3.2) one obtains that near p ∈ R the field X_{0} is annihilated by the form ω_{t} ∧ (dω_{t})^{k−1}:

X_{0} ⌋ (ω_{t} ∧ (dω_{t})^{k−1}) = 0.

The set R is dense, therefore the latter relation holds for all points p ∈ M and for all t. This relation and the 2-division property of X_{0} allow to conclude that

ω_{t} ∧ (dω_{t})^{k−1} = Q_{t} · ω_{0} ∧ (dω_{0})^{k−1},  (3.5)

where Q_{t} is a family of functions on M. To show that X_{0} is a characteristic vector field of (ω_{t}) one has to prove that the function Q_{t} vanishes at no points of M. This follows from the C^{r+1}-closeness of ω_{1} to ω_{0}. In fact, the latter implies that the form ω_{t} ∧ (dω_{t})^{k−1} is C^{r}-close to ω_{0} ∧ (dω_{0})^{k−1} (the fact that t varies on the compact segment [0, 1] is used here).

By the condition (b) of Theorem 1 the r-jet of the form ω_{0} ∧ (dω_{0})^{k−1} vanishes at no points of M, therefore the function Q_{t} is C^{0}-close to 1. ■

4. Proof of Theorems 2 and 2.A, k = 1.

At first I prove Theorem 2.A. Fix the function

H_{0} = (ω_{0} ∧ dω_{0})/Ω,

where Ω is a fixed volume form on M = M^{3}. The assumption that (ω_{t}) have the same characteristic vector field X_{0} on the Martinet surface S_{0} = {H_{0} = 0} implies

(ω_{t} ∧ dH_{0})(p) = (Q_{t}ω_{0} ∧ dH_{0})(p), p ∈ S_{0},

where Q_{t} is a family of functions vanishing at no points of M. There is no loss of generality to assume that Q_{t} ≡ 1 since the family ω_{t} can be replaced by ω̂_{t} = Q_{t}ω_{t}. Then the following relation holds

((ω_{t} − ω_{0}) ∧ dH_{0}) (p) = 0, p ∈ S_{0}.

Since the 1-jet of the form ω_{0} ∧ dω_{0} does not vanish at points of S_{0} then dH_{0}(p) ≠ 0, p ∈ S_{0}. It follows that the latter display relation implies the existence of a family F_{t} of functions on M and a family α_{t} of 1-forms on M such that

ω_{t} = ω_{0} + F_{t}dH_{0} + H_{0}α_{t}.  (4.1)

Like in the even-dimensional case, the homotopy method is used: I will show that (4.1) allows to solve the equation

L_{Z_{t}}ω_{t} + h_{t}ω_{t} + \frac{d}{dt}ω_{t} = 0  (4.2)

with respect to the unknown family Z_{t} of vector fields on M and the unknown family h_{t} of functions on M. Here L_{Z_{t}} is the Lie derivative along the vector field Z_{t}.

The equation (4.2) will be solved in two steps. At first I will show that the equation

L_{Y_{t}}ω_{t} + \frac{d}{dt}ω_{t} = 0 mod (H_{0})  (4.3)

is solvable with respect to the unknown family Y_{t} of vector fields on M. Here and below ν_{t} = 0 mod (H_{0}) for an s-form ν_{t} means that ν_{t} = H_{0}μ_{t} for some s-form μ_{t} (s is any integer). The solvability of the equation (4.3) reduces the equation (4.2) to the equation

L_{Z_{t}}ω_{t} + h_{t}ω_{t} = H_{0}γ_{t},  (4.4)

where γ_{t} is a certain family of 1-forms on M. The second step is to prove that (4.4) is solvable for any family γ_{t}.

So, to prove Theorem 2.A in the 3-dimensional case, it suffices to prove the solvability of the equations (4.3) and (4.4).

A solution Y_{t} of the equation (4.3) will be found in the form Y_{t} = H_{0}X_{t}, where X_{t} is a family of vector fields on M. Then L_{Y_{t}}ω_{t} = (X_{t} ⌋ ω_{t}) · dH_{0}. Since \frac{d}{dt}ω_{t} = G_{t}dH_{0} with G_{t} = \frac{d}{dt}F_{t} (by the relation (4.1)), then to solve (4.3) it suffices to solve the equation

X_{t} ⌋ ω_{t} + G_{t} = 0.  (4.5)

The solvability of (4.5) is not trivial since ω_{t} might vanish at some points. Differentiating the relation ω_{t} ∧ dω_{t} = 0 mod (H_{0}) one obtains \frac{d}{dt}ω_{t} ∧ dω_{t} = δ_{t} ∧ ω_{t} mod (H_{0}), where δ_{t} is a certain family of 2-forms. Then (4.1) implies

G_{t}dH_{0} ∧ dω_{t} = δ_{t} ∧ ω_{t} mod (H_{0}).  (4.6)

Let (p_{0}, t_{0}) be a fixed point of S_{0} × [0, 1] such that ω_{t}(p) = 0. The transversality condition in Theorem 2.A means that dH_{0} ∧ dω_{t} ≠ 0 near (p_{0}, t_{0}). Therefore the relation (4.6) implies that the equation (4.5) is solvable locall near (p_{0}, t_{0}). Consequently, (4.5) is solvable near any point (p, t) ∈ M × [0, 1]. By the division of unity (4.5) has a global solution.

It remains to solve the equation (4.4). Since S_{0} is the Martinet surface of (ω_{t}) for all t then

ω_{t} ∧ dω_{t} = H_{0}Ω_{t}  (4.7)

for some family Ω_{t} of volume forms on M. By the condition j_{p}^{1}(ω_{t} ∧ dω_{t}) ≠ 0, p ∈ S_{0} the volume forms Ω_{t} are not degenerated. Define families Z_{t} and V_{t} of vector fields fields on M by the relations

Z_{t} ⌋ Ω_{t} = γ_{t} ∧ ω_{t},  (4.8)
V_{t} ⌋ Ω_{t} = dω_{t}  (4.9)

and a family h_{t} of functions by the relation

h_{t} = V_{t} ⌋ γ_{t}.

I will show that (Z_{t}, h_{t}) is a solution of (4.4). In fact, (4.8) implies that Z_{t}⌋ω_{t} = 0 and then by (4.8) and (4.9)

L_{Z_{t}}ω_{t} = Z_{t} ⌋ dω_{t} = Z_{t} ⌋ (V_{t}⌋ Ω_{t}) =
= −V_{t} ⌋ (Z_{t} ⌋ Ω_{t}) = −V_{t} ⌋ (γ_{t} ∧ ω_{t}) = −h_{t}ω_{t} + (V_{t} ⌋ ω_{t}) · γ_{t}.  (4.10)

Note now that (4.9) imples that V_{t} ⌋dω_{t} = 0. Using this, (4.7) and (4.9) one obtains that (V_{t} ⌋ ω_{t}) · dω_{t} = H_{0}dω_{t}. Then the function V_{t} ⌋ ω_{t} − H_{0} vanishes at points at which the 2-form dω_{t} does not vanish. In particular, this function vanishes at points away from S_{0} and consequently V_{t} ⌋ ω_{t} ≡ H_{0}. This and (4.10) imply the required relation (4.4). Theorem 2.A is proved.

Theorem 2 can be reduced to Theorem 2.A by the folowing arguments. Let ω_{0} and ω_{1} be 1-forms satisfying the conditions of Theorem 2. Arguing like in the proof of Theorem 1, it is easy to see that there is no loss of generality to assume that ω_{1} is C^{r+2}-close to ω_{0}, that these forms have the same Martinet surface S_{0} = {H_{0} = 0}, where H_{0} = (ω_{0} ∧ dω_{0})/Ω, Ω is a fixed volume form, and that (ω_{0}) and (ω_{1}) have the same characteristic vector field X_{0}. The latter means that

ω_{1} ∧ dH_{0} = Qω_{0} ∧ dH_{0} mod (H_{0}),

where Q is a nonvanishing function on M. The condition that j_{p}^{r}X_{0} ≠ 0, p ∈ S_{0}, implies that j_{p}^{r}(ω_{0} ∧ dH_{0}) ≠ 0, p ∈ S_{0}. Therefore the C^{r+2}-closeness of ω_{1} to ω_{0} implies the C^{2}-closeness of the function Q to 1. Set ω_{2} = Qω_{1}. Then the form ω_{2} is C^{2}-close to ω_{0} and these forms satisfy the relation

ω_{2} ∧ dH_{0} = ω_{0} ∧ dH_{0} mod (H_{0}).  (4.11)

It follows that ω_{2} − ω_{0} = F dH_{0} mod (H_{0}) for some function F (the condition dH_{0}(p) ≠ 0, p ∈ S_{0} is used). Consider the family ω_{t} = ω_{0} + t(ω_{2} − ω_{0}), t ∈ [0, 1]. Note that at any point p ∈ S_{0} the 3-form ω_{t} ∧ dω_{t} depends on t linearly: (ω_{t} ∧ dω_{t})(p) = (ω_{0} ∧ dω_{0})(p) + tΩ_{1}, where Ω_{1} is a certain 3-form. Since S_{0} is the Martinet surface of ω_{0} and of ω_{2} then the forms ω_{0} ∧ dω_{0} and Ω_{1} vanish on S_{0} and therefore ω_{t} ∧ dω_{t} vanishes on S_{0} for all t.

The C^{2}-closemess of ω_{2} to ω_{0} implies the C^{1}-closeness of ω_{t} ∧ dω_{t} to ω_{0} ∧ dω_{0} (the compactness of the segment [0, 1] is used). Since dH_{0} does not vanish on S_{0}, it follows that the 1-jet of ω_{t} ∧ dω_{t} does not vanish on S_{0} and that S_{0} is the Martinet syrface of (ω_{t}) for all t. Note also that the 3-form dω_{t} ∧ dH_{0} does not vanish at points of S_{0} at which ω_{t} vanishes; this follows from the assumption (b) of Theorem 2 and the C^{2}-closeness of ω_{2} to ω_{0}. Therefore the family ω_{t} satisfies the assimptions of Theorem 2.A and by this theorem (ω_{t}) is equaivalent to (ω_{0}). In particular, (ω_{2}) is equivalent to (ω_{0}) and consequently (ω_{1}) is equivalent to (ω_{0}).

5. Proof of Theorem 2.A, k ≥ 2.

Like in previous sections the homotopy method is used, and to prove Theorem 2.A it suffices to solve the equation

L_{Z_{t}}ω_{t} + h_{t}ω_{t} + \frac{d}{dt}ω_{t} = 0.

The solvability of this equation with respect to the unknown family Z_{t} of vector fields and the unknown family h_{t} of functions on M = M^{2k+1} follows from the Propositions 5.1 and 5.2 below. The proof of these propositions uses the techniques involved in the both even-dimensional and 3-dimensional cases.

Fix a volume form Ω on M and a function

H_{0} = ω_{0} ∧ (dω_{0})^{k}/Ω.

The Martinet hypersurface S_{0} of (ω_{0}) is the zero level of H_{0}. I will use the following notations. If A_{t} and B_{t} are families of s-forms on M then A_{t} = B_{t} mod (H_{0}) means that A_{t} − B_{t} = H_{0}C_{t} for some family C_{t} of 1-forms on M. Here s is any integer.

Proposition 5.1. There exists a family Y_{t} of vector fields on M such that

(L_{Y_{t}}ω_{t} + \frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−1} = 0 mod (H_{0}).  (5.1)

Proposition 5.2. Assume that γ_{t} is a family of 1-forms on M such that

γ_{t} ∧ ω_{t} ∧ (dω_{t})^{k−1} = 0 mod (H_{0}).  (5.2)

Then there exist a family Z_{t} of vector fields and a family h_{t} of functions such that

L_{Z_{t}}ω_{t} + h_{t}ω_{t} = γ_{t}.  (5.3)

To prove Proposition 5.1 fix a characteristic vector field X_{0} of (ω_{0}). Since X_{0} is a characteristic vector field of (ω_{t}) for all t then

X_{0} ⌋ (ω_{t} ∧ (dω_{t})^{k−1} ∧ dH_{0}) = 0 mod (H_{0}).  (5.4)

Lemma 5.1. The relation (5.4) implies that there exists a family F_{t} of functions on M such that

(\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−1} = F_{t} · ω_{t} ∧ (dω_{t})^{k−1} ∧ dH_{0} mod (H_{0}).  (5.5)

Lemma 5.1 allows to find a solution Y_{t} of the equation (5.1) in the form

Y_{t} = H_{0}X_{t},

where X_{t} is a family of vector fields on M. Then

L_{Y_{t}}ω_{t} = (X_{t} ⌋ ω_{t}) · dH_{0} mod (H_{0})

and by Lemma 5.1 to solve (5.1) it suffices to solve the equation

X_{t} ⌋ ω_{t} − F_{t} = 0 mod (H_{0}).  (5.6)

Note that the form ω_{t} might vanish at some points, therefore the solvability of (5.6) is not trivial. By the division of unity, it suffices to solve (5.6) locally.

Lemma 5.2. The equation (5.6) has a local solution, near any point (p, t) ∈ M × [0, 1].

Now I will prove Proposition 5.2. A solution (Z_{t}, h_{t}) of the equation (5.3) will be constructed explicitly.

Since S_{0} is the Martinet hypersurface of (ω_{t}) then

ω_{t} ∧ (dω_{t})^{k} = H_{0}Ω_{t},  (5.7)

where Ω_{t} is a family of nondegenerated volume forms on M. Define a family V_{t} of vector fields on M by the relation

V_{t} ⌋ Ω_{t} = (dω_{t})^{k}.  (5.8)

The relations (5.7), (5.8) and the fact that the form (dω_{t})^{k} does not vanish at points of a dense set in M (at points away from S_{0}) imply

V_{t} ⌋ dω_{t} = 0, V_{t} ⌋ ω_{t} = H_{0}.  (5.9)

Using (5.2) and (5.9) one obtains

(V_{t} ⌋ γ_{t}) · ω_{t} ∧ (dω_{t})^{k−1} = 0 mod (H_{0}).

Notation. Denote by R the set of points of S_{0} at which the field X_{0} does not vanish.

Since X_{0} is a characteristic vector field of (ω_{t}) then the form ω_{t} ∧ (dω_{t})^{k−1} does not vanish at points of the set R. The set R is dense in S_{0} (it is easy to see that otherwice X_{0} cannot have the 1-division property). Therefore the latter relation implies that the function V_{t} ⌋ γ_{t} vanishes at any point of S_{0}. Since dH_{0}(p) ≠≠ 0, p ∈ S_{0} then there exists a family h_{t} of functions on M such that

V_{t} ⌋ γ_{t} = h_{t}H_{0}.  (5.10)

I will show that the pair (h_{t}, Z_{t}), where h_{t} is defined by (5.10) and Z_{t} by the relation

Z_{t} ⌋ Ω_{t} = k \frac{γ_{t} ∧ ω_{t} ∧ (dω_{t})^{k−1}}{H_{0}}  (5.11)

is a solution of the equation (5.3).

Note that (5.11) implies that Z_{t} ⌋ ω_{t} = 0 and therefore L_{Z_{t}}ω_{t} = Z_{t} ⌋ dω_{t}. Using this and (5.8) evaluate

(L_{Z_{t}}ω_{t}) ∧ (dω_{t})^{k−1} = \frac{Z_{t} ⌋ (dω_{t})^{k}}{k} = \frac{Z_{t} ⌋ (V_{t} ⌋ Ω_{t})}{k} =

= −\frac{V_{t} ⌋ (Z_{t} ⌋ Ω_{t})}{k} = −V_{t} ⌋ (\frac{γ_{t} ∧ ω_{t} ∧ (dω_{t})^{k−1}}{H_{0}} ) .

Using again (5.9) and (5.10), I obtain

(L_{Z_{t}}ω_{t} + h_{t}ω_{t} − γ_{t}) ∧ (dω_{t})^{k−1} = 0.

This relation implies the required relation (5.3) at points (p, t) ∈ M × [0, 1] such that (dω_{t})^{k−1}(p) ≠ 0. Therefore (5.3) holds at points (p, t) such that p ∉ S_{0}. Consequently (5.3) holds for all points (p, t) ∈ M × [0, 1]. Theorem 2.A is proved modulo Lemmas 5.1 and 5.2.

Proof of Lemma 5.1. I use the set R ⊂ S_{0} defined above. The relation (5.4) easily implies:

X_{0}(p) ⌋ ω_{t}(p) = 0, X_{0}(p) ⌋ dH_{0}(p) = 0,
X_{0}(p) ⌋ dω_{t}(p) = a(t, p)ω_{t}(p) + b(t, p)dH_{0}(p), p ∈ R,

where a(t, p) and b(t, p) are some numbers depending on t and p. Since S_{0} is the Martinet hypersurface of (ω_{t}) for all t then (ω_{t} ∧ (dω_{t})^{k}) (p) = 0, p ∈ S_{0}, and it follows that b(t, p) = 0, p ∈ R, t ∈ [0, 1]. This and the density of the set R in S_{0} allow to conclude that

X_{0}(p) ⌋ ((\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−1}) (p) = 0, p ∈ R.  (5.12)

Define a family A_{t} of vector fields on M by the relation

A_{t}(p) ⌋ Ω(p) = ((\frac{d}{dt}ω_{t}) ∧ ω_{t} ∧ (dω_{t})^{k−1}) (p), p ∈ S_{0}.

Then by (5.12) X_{0}(p) ∧ A_{t}(p) = 0, p ∈ S_{0}. It follows that the field A_{t} is tangent to S_{0} at any point of the set R and consequently at any point of S_{0}. By the 1-division property of X_{0} there exists a family F_{t} of functions on S_{0} such that A_{t} = F_{t}X_{0}. This relation and the condition dH_{0}(p) ≠ 0, p ∈ S_{0} imply (5.5). ■

Proof of Lemma 5.2. I will use the following notations. The relation

A_{t} = B_{t} mod (H_{0}, ω_{t}),

where A_{t} and B_{t} are s-forms, means that A_{t} − B_{t} = ω_{t} ∧ ν_{t} + H_{0}μ_{t} for some (s − 1)-form ν_{t} and s-form μ_{t}. Write the relation (5.5) in the form

\frac{d}{dt}ω_{t} ∧ ω_{t} ∧ (dω_{t})^{k−1} = F_{t} · ω_{t} ∧ (dω_{t})^{k−1} ∧ dH_{0} + H_{0}α_{t},  (5.13)

where α_{t} is a family of (2k)-forms. Taking the differential of this relation gives

(\frac{d}{dt}ω_{t}) ∧ (dω_{t})^{k} = F_{t} · (dω_{t})^{k} ∧ dH_{0} + dH_{0} ∧ α_{t} mod (H_{0}, ω_{t}).  (5.14)

To prove the solvability of the equation (5.6) I will show that the following relations hold:

(\frac{d}{dt}ω_{t}) ∧ (dω_{t})^{k} = 0 mod (H_{0}, ω_{t}),  (5.15)

α_{t} = 0 mod (H_{0}, ω_{t}).  (5.16)

These relations and (5.14) imply

F_{t} · (dω_{t})^{k} ∧ dH_{0} = 0 mod (H_{0}, ω_{t}).  (5.17)

The assumption of Theorem 2.A that the kernel of dω_{t} is transversal to S_{0} at points at which ω_{t} vanishes implies that at such points the (2k + 1)-form (dω_{t})^{k} ∧ dH_{0} does not vanish. Therefore (5.17) implies that locally (near any fixed point (p, t)) the equation (5.6) can be solved locally.

It remains to prove the relations (5.15) and (5.16). The relation (5.15) can be obtained by differiantiating by t of the relation ω_{t} ∧ (dω_{t})^{k} = 0 mod (H_{0}). The proof of (5.16) is as follows. The relation (5.13) can be written in the form

H_{0}α_{t} = ω_{t} ∧ η_{t}  (5.17)

for some (2k − 1)-form η_{t}. I work near a point (p, t) ∈ S_{0} × [0, 1] such that ω_{t}(p) = 0, otherwice (5.16) is trivial. The transversality of kerdω_{t} to S_{0} allows to fix a vector field B_{t} which is transversal to S_{0} and such that B_{t} ⌋ dω_{t} = 0. Note that (B_{t} ⌋ ω_{t})(p) = 0, p ∈ S_{0} since the form ω_{t} ∧ (dω_{t})^{k} vanishes on S_{0}. Then by (5.17) the function B_{t} ⌋ ν_{t} = 0 vanishes at points of S_{0} at which ω_{t} does not vanish and consequently it vanishes at any point of S_{0}. Take the Lie derivative of the relation (5.17) along B_{t} at a point p ∈ S_{0}. Using the relations

(B_{t} ⌋ ω_{t})(p) = (B_{t} ⌋ dω_{t})(p) = (B_{t} ⌋ ν_{t})(p) = 0, p ∈ S_{0}

one obtains

(B_{t} ⌋ dH_{0})(p) · α_{t}(p) = ω_{t}(p) ∧ β_{t}(p), p ∈ S_{0},

where β_{t} is a family of 1-forms on S_{0}. The transversality of B_{t} to S_{0} means that (B_{t} ⌋ dH_{0})(p) ≠ 0, therefore the latter relation implies (5.16). ■

6. Proof of Theorem 2, k ≥ 2

Fix a volume form Ω on M = M^{2k+1}, the function H_{0} = ω_{0} ∧ (dω_{0})^{k}/Ω and a characteristic vector field X_{0} of (ω_{0}) defined by the relation

X_{0}(p) ⌋ Ω_{0}(p) = (ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0}) (p), p ∈ S_{0} = {H_{0} = 0}.

There is no loss of generality to assume that ω_{1} is C^{q}-close to ω_{0} (recitate that q = max(2r+ 2, r + 3)), that (ω_{1}) and (ω_{0}) have the same Martinet hypersurface S_{0} = {H_{0} = 0} and that X_{0} is a characteristic vector field of (ω_{1}). Then

(ω_{1} ∧ (dω_{1})^{k−1} ∧ dH_{0}) (p) = (Qω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0}) (p), p ∈ S_{0}

for some function Q on S_{0}. The (2k)-forms in this relation are C^{q−1}-close. Since the r-jet of X_{0} does not vanish on S_{0} then the r-jet of the form ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0} does not vanish on S_{0} and one can conclude that the function Q is C^{q−r−1}-close to 1. Define ω_{2} = Q^{1/k}ω_{1}. Then ω_{2} is C^{q−r−1}-close to ω_{0} and these forms satisfy the relation

(ω_{2} ∧ (dω_{2})^{k−1} ∧ dH_{0}) (p) = (ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0}) (p), p ∈ S_{0}.  (6.1)

Let R be the set of points of S_{0} at which the characteristic vector field X_{0} does not vanish. In section 5 it was shown that X_{0}(p) ⌋ ω_{0}(p) = 0, X_{0}(p) ⌋ dω_{0}(p) = h_{0}(p)ω_{0}(p), p ∈ R for some function h_{0} defined in a neighbourhood of p of S_{0}. Using this relations and taking the Lie derivative of the form ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0} along X_{0}, one obtains

(L_{X_{0}} (ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0})) (p) = k (h_{0}ω_{0} ∧ (dω_{0})^{k−1} ∧ dH_{0}) (p), p ∈ R.  (6.2)

Since X_{0} is also a characteristic vector field of (ω_{2}), one obtains:

X_{0}(p) ⌋ ω_{2}(p) = 0, X_{0}(p) ⌋ dω_{2}(p) = h_{2}(p)ω_{0}(p), p ∈ R,
(L_{X_{0}} (ω_{2} ∧ (dω_{2})^{k−1} ∧ dH_{0})) (p) = k (h_{2}ω_{2} ∧ (dω_{2})^{k−1} ∧ dH_{0}) (p), p ∈ R  (6.3)

for some function h_{2} on S_{0}. The relations (6.1)-(6.3) imply

h_{0}(p) = h_{2}(p).

This allows to conclude that

X_{0}(p) ⌋ ω_{t}(p) = 0, X_{0}(p) ⌋ dω_{t}(p) = h_{0}(p)ω_{t}(p), p ∈ R,

where ω_{t} = ω_{0} + t(ω_{2} − ω_{0}) is the segment joining ω_{0} to ω_{2}. It follows

(ω_{t} ∧ (dω_{t})^{k}) (p) = 0, p ∈ R,  (6.4)
X_{0}(p) ⌋ (ω_{t} ∧ (dω_{t})^{k−1} ∧ dH_{0}) (p) = 0, p ∈ R.  (6.5)

Since the set R is dense in S_{0} (otherwice the condition (d) of Theorem 2 is violated) then the relations (6.4) and (6.5) hold at all points p ∈ S_{0} (not only at points of R).

Using these relations and the C^{q−r−1}-closeness of ω_{t} to ω_{0} (following from the C^{q−r−1}- closeness of ω_{2} to ω_{0}) I will show that S_{0} is the Martinet hypersurface of (ω_{t}) for all t and X_{0} is a characteristic vector field of (ω_{t}) for all t. If this is so then the family (ω_{t}) satisfies the assumptions of Theorem 2.A. This follows from the C^{q−r−1}-closeness of ω_{t} to ω_{0}, the observation that q − r − 1 ≥ 2 and the condition (b) of Theorem 2. By Theorem 2.A (ω_{t}) is equivalent to (ω_{0}). In particular (ω_{2}) is equivalent to ω_{0} and consequently (ω_{1}) is equivalent to (ω_{0}).

The proof that S_{0} is the Martinet hypersurface of (ω_{t}) for all t is as follows. The relation (6.4) (which is valid for all p ∈ S_{0}) and the condition dH_{0}(p) ≠ 0, p ∈ S_{0} imply

ω_{t} ∧ (dω_{t})^{k} = E_{t}H_{0}Ω = E_{t}ω_{0} ∧ (dω_{0})^{k}  (6.6)

for some family E_{t} of functions on M. Since the form ω_{t} ∧ (dω_{t})^{k} is C^{q−r−2}-close to ω_{0} ∧ (dω_{0})^{k} and the 1-jet of the function H_{0} does not vanish then E_{t} is C^{q−r−3}-close to 1. Since q ≥ r + 3 then E_{t} vanishes at no points of M for any t ∈ [0, 1]. Therefore S_{0} is the Martinet hypersurface of (ω_{t}) for all t.

It remains to show that X_{0} is a characteristic vector field of (ω_{t}) for all t. Define a family C_{t} of vector fields on S_{0} by the relation C_{t}(p) ⌋ Ω(p) = (ω_{t} ∧ (dω_{t})^{k−1} ∧ dH_{0}) (p), p ∈ S_{0}. It is easy to see that C_{t} is tangent to S_{0}. The relation (6.5) implies C_{t} ∧ X_{0} = 0. Using the 1-division property of X_{0} gives C_{t} = K_{t}X_{0} for some family K_{t} of functions on S_{0}. The field C_{t} is a characteristic vector field of (ω_{t}), and to prove that X_{0} is also a characteristic vector field of (ω_{t}) one has show that the function K_{t} vanishes at no points of S_{0}. The C^{q−r−1}-closeness of ω_{t} to ω_{0} implies the C^{q−r−2}-closeness of C_{t} to X_{0}. The r-jet of X_{0} vanishes at no points of S_{0}, therefore the relation C_{t} = K_{t}X_{0} implies that the function K_{t} is C^{q−2r−2}-close to 1. Since q ≥ 2r + 2 then K_{t} vanishes at no points of M. Theorem 2 is proved.

7. Proof of Propositions 1 and 2.

The proof is based on the analysis of singularities of 1-forms in [Mar] and [Zh] and the following simple facts:

(a) The global s-division property reduces, in the C^{∞} category, to the local s-division property (due to the division of unity);

(b) If X is a germ at 0 of a vector field vanishing at 0 ∈ ℝ^{n} then the simplest sufficient condition guarantying the s-division property of X is the existence of at least (s + 1) differentially independent germs in the ideal I_{X} generated by the function germs f_{1}, …, f_{n} that are the coefficients of X in a coordinate system near 0 (the ideal does not depend on the choice of a coordinate system).

Lemma 7.1, [Mar]. Let k ≥ 2, 0 ≤ p ≤ k − 2. The set of 1-jets 0 ∈ ℝ^{2k} of 1-forms ω on ℝ^{2k} such that ω(0) ≠ 0 and

(ω ∧ (dω)^{k−1−p}) (0) = 0  (7.1)

is a stratified submanifold of codimension (p + 1)(2p + 3) of the manifold consisting of all 1-jets at 0 of 1-forms on ℝ^{2k}.

By this lemma the set of 1-jets at 0 of 1-forms such that

(ω ∧ (dω)^{k−1}) (0) = 0  (7.1)

is a stratified submanifold of codimension 3 in the space of all 1-jets at 0. A characteristic vector field of (ω) vanishes at 0 if and only if (7.1) holds. This explains the following statement.

Lemma 7.2, [Zh, sec.25,13]. In the space of all 2-jets at 0 of 1-forms on ℝ^{2k}, k ≥ 2, there is a stratified submanifold E of codimension (2k + 1) such that any 1-form ω whose 2-jet at 0 is away from E satisfies the following property. Let I be the ideal generated by the coefficients of the germ at 0 of a characteristic vector field X of (ω). If X(0) = 0 and ω(0) ≠ 0 then I is generated by 3 differentially independent at 0 functions provided that

(ω ∧ (dω)^{k−2}) (0) ≠ 0.  (7.2)

If ω(0) = 0 (and consequently X(0) = 0) then I is generated by 2k differentially independent at 0 functions.

By Lemma 7.1 the degeneration (7.2) has codimension 10. Therefore Proposition 1 follows from Lemmas 7.1, 7.2, the transversality theorem and the mentioned above sufficient condition for the s-division property, s = 2.

In the odd-dimensional case, the main lemma used in the proof of Proposition 2 is as follows.

Lemma 7.3, [Mar]. Let 0 ≤ p ≤ k − 1. The set J_{p} consisting of 1-jets at a nonfixed point x ∈ M^{2k+1} of 1-forms ω on M^{2k+1} such that ω(x) ≠ 0 and

(ω ∧ (dω)^{k−p}) (x) = 0  (7.4)

is a stratified submanifold of codimension (p + 1)(2p + 1) in the space J of all 1-jets (at a nonfixed point of the manifold). The set J_{0} is not a smooth hypersurface near a point j_{x}^{1}ω if and only if

(ω ∧ (dω)^{k−1}) (x) = 0  (7.4)

Consequently, the set of points at which J_{0} is not smooth is a stratified codimension 6 submanifold of J.

The condition (a) of Theorem 2 (the structural smoothness of the Martinet hypersurface) means that the set J_{0} is a smooth hypersurface near the point j_{p}^{1}ω_{0} and the mapping x → j_{x}^{1}ω_{0} is transversal to J_{0}. Therefore by the transversality theorem and Lemma 7.3 the condition (a) of Theorem 2 holds for an open and dense subset of the space of all nonvanishing 1-forms on M = M^{2k+1} in the C^{2}-topology provided that dimM ≤ 5.

The genericity of the conditions (c) and (d) of Theorem 2 for the case where dimM ≤ 5 and ω_{0} vanishes at no points of the manifold follows from the genericity of the condition (a) and the following lemma.

Lemma 7.4, [Zh, sec.17]. In the space of all 3-jets at 0 of 1-forms on ℝ^{2k+1} there is a set E of codimension 2k + 2 such that any 1-form ω whose 3-jet at 0 is away from E satisfies the following property. If ω(0) ≠ 0 and the Martinet hypersurface S is structurally smooth at 0 (i.e. j_{0}^{1} (ω ∧ (dω)^{k}) ≠ 0) then the ideal generated by the coefficients of the germ at 0 of a characteristic vector field X of (ω) is generated by 2 differentially independent at 0 function germs.

Lemmas 7.3 and 7.4 prove Proposition 2 for the case where (ω_{0}) is a hyperplane field, i.e. ω_{0} vanishes at no points of the manifold. To complete the proof I note that the conditions (a)-(d) of Theorem 2 can be easily checked for the example where the germ of at p of ω_{0} has the normal form

(x_{1}dy_{1} − y_{1}dx_{1}) + ⋯ + (x_{k}dy_{k} − y_{k}dx_{k}) + dϕ(x, y) ± zdz

in some coordinate system centered at p, where ϕ(x, y) is a generic quadratic function, and use a result in [Zh, sec. 13] stating that this normal form serves for all germs at 0 of 1forms ω on ℝ^{2k+1} such that ω(0) = 0 and j_{0}^{2}ω does not belong to a certain (fixed) stratified submanifold of codimension (2k + 2) of the space of all 2-jets at 0.

References.

[Gra] J.W. Gray, Some global properties of contact structures, Ann. of Math., No.2, Vol. 69 (1959), 421 - 450

[JaZh1] B. Jakubczyk, M. Zhitomirskii, Odd-dimensional Pfaffian equations; reduction to the hypersurface of singular points, Comptes Rendus Acad. Sci. Paris, t. 325, Série I (1997), 423-428.

[JaZh2] B. Jakubczyk, M. Zhitomirskii, Local reduction theorems and invariants for singular contact structures, preprint, Banach Center, Warsaw

[Mar] J. Martinet, Sur les singularites des formes differentielles, Ann. Inst. Fourier, Vol. 20, No.1 (1970), 95-178.

[Mou] R. Moussu, Sur l’existence d’intégrales prèmieres pour un germe de forme de Pfaff, Ann. Inst. Fourier, Vol. 26, No. 2 (1976), 171-220.

[MoZh] R. Montgomery, M. Zhitomirskii, Geometric approach to Goursat flags, preprint, University of California at Santa Cruz, see the home-page of R. Montgomery

[Zh] M. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations, Translations of Math. Monographs, Vol. 113, AMS, Providence, 1992.