Unconjugated Contact Forms

In the acts of Colloquium of Brussels in 1958, Libermann [1] addressed the study of the automorphisms of the contact structures on a differentiable manifold M. She has proved that these automorphisms corresponds bijectively to functions on this manifold. This allows to transport the Lie algebra structure on the vector space F(M) of the functions on M: We obtain, for given functions f, g ∈ F(M), a Poisson bracket [f, g] that depends of the contact form ω: The study of the infinite dimensional Lie algebras obtained is far from be advanced. Thus, in 1973 Lichnerowicz [2] who hopes distinguish the contact structures by their Lie algebras, has given series of results that are all however of general character. Some works that have appeared after have emphasis on the similarities of these algebras. In 1979, Lutz [3] has proved the existence of infinitely many non-isomorphic contact structures on the sphere S3. In 1989, as reported by Lutz [3] himself, we have opened in our thesis [4] new perspectives in the other direction by studying the sub-algebras of finite dimension of these algebras. We know that if two contact structures [ω1] and [ω2] are isomorphic then their Lie algebras (of infinite dimension of course) A([ω1]) and A([ω2]) are also isomorphic.


Introduction
In the acts of Colloquium of Brussels in 1958, Libermann [1] addressed the study of the automorphisms of the contact structures on a differentiable manifold M. She has proved that these automorphisms corresponds bijectively to functions on this manifold. This allows to transport the Lie algebra structure on the vector space F(M) of the functions on M: We obtain, for given functions f, g ∈ F(M), a Poisson bracket [f, g] that depends of the contact form ω: The study of the infinite dimensional Lie algebras obtained is far from be advanced. Thus, in 1973 Lichnerowicz [2] who hopes distinguish the contact structures by their Lie algebras, has given series of results that are all however of general character. Some works that have appeared after have emphasis on the similarities of these algebras. In 1979, Lutz [3] has proved the existence of infinitely many non-isomorphic contact structures on the sphere S 3 . In 1989, as reported by Lutz [3] himself, we have opened in our thesis [4] new perspectives in the other direction by studying the sub-algebras of finite dimension of these algebras. We know that if two contact structures [ω 1 ] and [ω 2 ] are isomorphic then their Lie algebras (of infinite dimension of course) A([ω 1 ]) and A([ω 2 ]) are also isomorphic.
Given an n-dimensional smooth manifold M, and a point p 2 M, a contact element of M with contact point p is an (n-1)-dimensional linear subspace of the tangent space to M at p: A contact contact element can be given by the zeros of a 1-form on the tangent space to M at p: However, if a contact element is given by the zeros of a1-form ω, then it will also be given by the zeros of λω where λ ≠ 0: thus {λω: λ ≠ 0} all give the same contact element. It follows that the space of all contact elements of M can be identified with a quotient of the cotangent bundle PT*M, where PT*M=T*M/R; where, for ω i ∈2 T* p M, ω 1 R ω 2 if there exists λ ≠ 0: ω 1 =λω 2 .
A contact structure on an odd dimensional manifold M, of dimension 2k + 1, is a smooth distribution of contact elements, denoted by ξ, which is generic at each point. The genericity condition is that ξ is non-integrable.
Assume that we have a smooth distribution of contact elements ξ given locally by a differential 1-form; i.e. a smooth section of the cotangent bundle. The non-integrability condition can be given explicitly as Notice that if ξ is given by the differential 1-form, then the same distribution is given locally by β=fα, where f is a non-zero smooth function. If ξ is co-orientable then is defined globally.
If is a contact form for a given contact structure, the Reeb vector field R can be defined as the unique element of the kernel of dα such that α(R)=1.

The Main Result
The main result is contained in the following theorem:

Theorem 1
On the torus T 3 the contact structures defined by the contact forms To establish this result, we need the following lemma.

Lemma 2
Let f a C ∞ -function on the torus T 3 and R n the Reeb field of ω n defined by If R n (f)=0, then f depend only on θ 3 .
We return now to the eqns. (1) and (2); we obtain:  This completes the proof of the theorem.

Conclusion
The technics used in this work to find nonisomporphic contact structures can be extended to the sphere S 3 in a first step and may be to other manifolds suitably choosen. It is also interesting to find the group of diffeomorphisms that leaves the contact structure invariante.