Jacobi Manifolds, Contact Manifolds and Contactomorphism

Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and thus, we characterize the Contact diffeomorphisms.


Jacobi Manifolds
The notion of Jacobi manifolds has been introduced by A. Lichnerowicz (Lichnerowicz, 1978). These manifolds are a generalization of the notion of locally conformally symplectic manifolds and of contact manifolds (which can be used in the representation of energy in thermodynamics). A Jacobi structure on a manifold M is a pair (Λ, E) where Λ is a contravariant skew-symmetric 2-tensor and E is a vector field on M such that with ∧ the Grassmann wedge product and θ E the Lie derivative with respect to the vector field E. The bracket defined in (1) is the Schouten-Nijenhuis bracket on multivectors (Koszul, 1985).
Using the classical Darboux's theorem, around every point of M there exist canonical coordinates (x 1 , ..., x 2n ) and a local smooth function f on M such that Defining a Jacobi structure on (Λ, E) is equivalent to defining an internal composition law {·, ·} on C ∞ (M) such that for all f , g ∈ C ∞ (M) where d is the exterior differentiation operator (Okassa, 2007). The bracket {·, ·} is skew-symmetric and satisfies the Jacobi identity if and only if (1) is fulfilled. Furthermore, one has the locality condition A natural generalization of Poisson bracket is Jacobi bracket. The only difference is that we only replace the Leibniz rule by Proposition 1. The Jacobi bracket {·, ·} defines a Lie algebra structure on C ∞ (M), if and only if (1) is satisfied.
The vector field E is the hamiltonian vector field associated with the constant function 1. We which equivalent to θ E Λ = 0. If the vector field E vanishes, we recover the Poisson bracket.
Remark 2. (Lichnerowicz, 1977) Let (Λ, E) be a Jacobi structure on a manifold M and consider a product manifold M × R, the bivector Λ given by Λ = e −t Λ + ∂ ∂t ∧ E , where t is the usual coordinate on R. Then, Λ defines a Poisson structure on M ×R. The manifold M ×R with the structure Λ is called the poissonization of the Jacobi manifold (M, Λ, E).
Note that a smooth manifold M is a Jacobi manifold when the algebra of numerical functions of class C ∞ on M, C ∞ (M) admits a Jacobi algebra structure (Okassa, 2007).

Contact Manifolds
By a contact manifold in this paper, we mean a C ∞ manifold M 2n+1 together with a 1-form α such that α ∧ (dα) n 0. In particular, α ∧ (dα) n is a volume element on M 2n+1 , so that a contact manifold is orientable. Also (dα) n has rank 2n on the Grassmann algebra ΛT p M at each point p ∈ M, and thus we have a 1-dimensional subspace, on which α 0 and which is complementary to the subspace defined by α = 0. Therefore choosing R p in this subspace normalized by α(R p ) = 1, we have a global vector field R satisfying α(R) = 1 and dα(R, X) = 0. In this case, the C ∞ (M)-module of vector fields X(M) admits a inner sum direct decomposition where, R is called Reeb vector field of the contact structure α. We deduce that θ R α = 0 and θ R dα = 0.
One denotes by D the contact distribution defined by the space Roughly speaking, the meaning of the contact condition, α ∧ (dα) n 0, is that the contact subbundle is as far from being integrable as possible.
By a smooth manifold, we will mean a paracompact connected finite dimensional C ∞ manifold without boundary.
Here one lists a few examples of contact manifolds and Reeb vector field.
The map h is called a morphism of contact manifolds, so long as h * α = α (Gatsé, 2016).
Given a contact diffeomorphism h, h −1 is also contact, and h is called a contactomorphism, where by definition of the pullback of a 1-form α for x element of M and for any u element of T x M.
The main goal of this paper is to show that the contact manifolds are the Jacobi manifolds. We also will give the generalization of the notion of contactomorphism and will characterize the contact diffeomorphisms.
In what follows, M denotes a paracompact and connected smooth manifold, C ∞ (M) the algebra of numerical functions of class C ∞ on M, D(M) the C ∞ (M)-module of first order differential operators on C ∞ (M) and δ (Okassa, 2008) the cohomology operator associated with the identically map The term "differential operator" will mean "first order differential operator".

Link Between Jacobi Manifolds and Contact Manifolds
Let Ω 1 (M) R be the C ∞ (M)-module of differential forms of degree 1 on M which vanish on R. Then the map is an isomorphism of C ∞ (M)-modules. Moreover, for any function f ∈ C ∞ (M), we obtain the following properties: The differential form d f − R( f ) · α vanishes on R and we have easily One notes X f the unique vector field on M belonging to ker α such that We notice that (i) R ∈ ker(i X f dα).
A differential operator on M is a R-linear map such that for any f , g ∈ C ∞ (M). We verify that the map is a differential operator on M. We set we deduce the following results.
Theorem 3. For all f , g and h in C ∞ (M), we get (ii) The proof of this assertion is obvious.
(iii) For any f ∈ C ∞ (M), we get (iv) Since R X f (g) = X f (R(g)) + X g (R( f )), we deduce that (v) We calculate That ends the proof.
Proposition 4. The map defines a real Lie algebra structure on the contact manifold M.
Proof. We write Hense the assertion is proved.
Give the foregoing, it is concluded that a contact manifold is a Jacobi manifold.

Contactomorphism
If D(M) is the set of differential operators on M and is a nondegenerate skew-symmetric 2-form on M, then the pair (M, ω) is a contact manifold (fore more details, see Okassa, 2011). Let be the 1-form on M such that Proposition 5. (Okassa, 2008) If ω is a nondegenerate skew-symmetric 2-form such that δ ω = 0, then ω = δ (i 1 ω).
Proposition 6. (Gatsé, 2019) Suppose that f is a diffeomorphism on M. Let ϕ be a differential operator of C ∞ (M). Then the map is a differential operator of C ∞ (N).
Proposition 7. When ϕ is a differential operator, when f is a diffeomorphism then for any g element of C ∞ (M), we obtain Proof. Let x be an element of M, we write Proposition 8. If f is a diffeomorphism and let ξ be a multilinear p-form on N then for all ψ 1 , ψ 2 , ..., ψ p ∈ D(M), That ends the proof.
Theorem 9. When f is a diffeomorphism then δ and f * commute.
Proof. We have On the other hand, we get That ends the proof.
Proof. Suppose that f is a contactomorphism then for any vector field X on M, we obtain For any ϕ ∈ D(M), ϕ = ϕ(1) + X, and Theorem 11. The map f is a contactomorphism if and only if f * ω = ω.

Conversely let
In particular if ψ = Y , we have That ends the proof.
Let Di f f ω (M) be the group of C ∞ (M)-diffeomorphisms of M, which preserve ω. This group is the automorphism group of the contact geometry of (M, ω). It acts transitively on the contact manifold (provided that is connected). Hence the contact manifold (M, ω) can be viewed as a homogeneous space of Di f f ω (M).
Theorem 12. Let (M, ω) and (M , ω ) be two smooth contact manifolds equipped with contact structures ω and ω respectively. Let γ : M −→ M be a bijective map such that for any map f : Proof. We use the Taken's theorem (see Takens, 1979) and the same arguments as in (Banyaga, 1986). Proof. Note that, these groups are γ-transitive. There exists a homeomorphism γ : M −→ M such that τ( f ) = γ f γ −1 , for any f ∈ Di f f ω (M). Useing the above theorem, we have γ * ω = ω.