EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 5743
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Characterization of Three-dimensional Jacobi-Poisson
Manifolds

Mahamane Saminou Ali1, Ibrahima Hamidine2,∗, Abdoulaye Tankary Banao2,
Mouhamadou Hassirou2

1 Département de Mathématiques, Faculté des Sciences et Techniques, Université d’Agadez,
Agadez, Niger
2 Département de Mathématiques et Informatique, Faculté des Sciences et Techniques, Uni-
versité Abdou Moumouni, Niamey, Niger

Abstract. After a reminder of essential notions concerning Jacobi and Poisson manifolds, we
study the relationships between their structures. We also show that for any three-dimensional
Jacobi manifold, we can construct a Poisson structure on the same manifold. The paper concludes
with some examples of such manifolds.

2020 Mathematics Subject Classifications: 53C15, 53C25, 53D10, 53D17, 70G45

Key Words and Phrases: Jacobi structures, Poisson structures, pseudo-Riemannian metric.

1. Introduction

Jacobi and Poisson manifolds are nowadays a research subject in expansion in dif-
ferential geometry. The Jacobi manifolds are introduced by A. Lichnerowicz [1] as a
generalisation of both symplectic, Poisson and contact manifold. A Jacobi bracket is just
a Lie bracket on the algebra of smooth functions given by bilinear first order differential
operator.

The Jacobi and Poisson manifolds formalize the Hamitonnian geometry and are used
to quantify physical systems. This motivates severals studies of such manifolds. In this
context M. Boucetta studies the compatibility between Poisson and pseudo-Riemannian
structures [2–4]. This allows better characterization of symplectic leaves [5, 6]. In the
same way, Y. A. Amrane and A. Zeglaoui study the compatibility between Riemannian
structures and Jacobi structures [7].

To better understand the geometry behind the Jacobi structure, some researchers try
to look at this structure as a generalization of a Poisson structure. So they rewrite notions

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.5743

Email addresses: mahamanesaminou@gmail.com (M. S. Ali),
ibrahima.hamidine@uam.edu.ne (I. Hamidine), tankarybanaou@gmail.com (A. Tankary Banao),
mouhamadou.hassirou@uam.edu.ne (M. Hassirou)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 2 of 14

on Jacobi manifolds so that they generalize the same notions on Poisson manifolds. Thus,
one can ask about the possibility of having a Jacobi structure and a Poisson structure on
the same manifold, and the relationship that can exist between the two structures. The
first answer is given in [8, 9]. They show that from any n-dimensional Jacobi manifold
one can construct a natural Poisson structure on the cone M × R∗

+.
In this paper we try to give an answer to the above question. We study a special

case of a three-dimensional Jacobi manifold. We show that for any three-dimensional
Jacobi manifold we can construct a Poisson structure on the same manifold. We give
some examples of such manifolds.

The organisation of the paper is as follows: in section 2 we begin with a brief review of
the Jacobi structure and its correspondence with Poisson structures. Section 3 is devoted
to the compatibility between Riemannian and Jacobi structures on a manifold as a gen-
eralisation of the compatibility between Riemannian and Poisson structures. In section 4
we give the connection between Jacobi and Poisson manifolds in R3. In addition, we give
the partial compatibility with the Riemannian metric g and the Jacobi structure (π,E);
we also prove that if (M,π,E) is a Jacobi manifold, then (M,P ) is a Poisson manifold
under certain conditions. And some examples of 3-dimensional Jacobi-Poisson manifolds
are given.

2. Jacobi manifolds and Poisson manifolds

Let (M, g, π) be an n-dimensional manifold with a pseudo-Riemannian metric g and a
bivector field π. Let E be a vector field on M . The pair (π,E) defines a Jacobi structure
on M if we have the following relations

[π, π] = 2E ∧ π and [E, π] := LEπ = 0, (1)

where [., .] is the Schouten-Nijenhuis bracket, see [7].
In local coordinates (x1, . . . , xn) the tensor π is determined by the matrix πij(x) =

{xi, xj}. The rank of this matrix is called the rank of π at x.
Note that a be vector π is a Poisson tensor ((M,π) is a Poisson manifold) if [π, π] = 0.

A Poisson structure is called regular if the rank of π is constant on M . If this matrix is
invertible a each x, then π is called non-degenerate or symplectic. We call (M,π,E) a
Jacobi manifold.

If E = 0, then [π, π] = 0, which correspond for Poisson structure (M,π). Thus the
Jacobi manifold generalised at once the Poisson manifolds, the contact manifolds and the
locally conformal symplectic manifolds. Furthermore, if g is a Hermitian metric on M,
then (M, g, π,E, λ) is called a pseudo-Riemannian Jacobi manifold, with λ a contact form
[7].

A correspondence between Jacobi and Poisson structures is given by the following fact:
Fact . (π,E) is a Jacobi structure on M if and only if P = e−t(π+ ∂

∂t ∧E) is a Poisson
structure on the cone M = M ×R∗

+
† (more generally on M ×R, and the Poisson structure

†The non-degeneracy of the canonical metric motivates the choice of the cone M × R∗
+.



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 3 of 14

is homogeneous, see [10, 11]).
Let π =

∑
1≤i<j≤4

πij ∂
∂xi ∧ ∂

∂xj be a bivector field on M . Then by definition of the

Schouten-Nijenhuis bracket, one has locally[
πij ∂

∂xi
∧ ∂

∂xj
, πsk ∂

∂xs
∧ ∂

∂xk
]
=

πij ∂π
sk

∂xi
∂

∂xs
∧ ∂

∂xk
∧ ∂

∂xj
− πsk ∂π

ij

∂xs
∂

∂xi
∧ ∂

∂xk
∧ ∂

∂xj

+πsk ∂π
ij

∂xk
∂

∂xs
∧ ∂

∂xi
∧ ∂

∂xj
− πij ∂π

sk

∂xj
∂

∂xi
∧ ∂

∂xs
∧ ∂

∂xk
. (2)

Theorem 1. Let (M,π,E) be a Jacobi manifold. If the pair (π,E) is a Jacobi structure
on M then P = e−t(π + ∂t ∧ E) is a Poisson structure on the cone M = M × R∗

+.

Proof. It’s clear that (π,E) satisfies (1). Now it suffices to prove that [P, P ] = 0, where
P = e−tπ + e−t∂t ∧ E.
By (2), we get

[P, P ] = [e−tπ + e−t∂t ∧ E, e−tπ + e−t∂t ∧ E]

= [e−tπ, e−tπ] + [e−tπ, e−t∂t ∧ E] + [e−t∂t ∧ E, e−tπ] + [e−t∂t ∧ E, e−t∂t ∧ E]

= e−2t[π, π] + 2[e−t∂t ∧ E, e−tπ]

= e−2t[π, π] + 2e−t∂t[E, e−tπ] + 2[e−t∂t, e
−tπ] ∧ E.

Since
[e−tπ, e−t∂t ∧ E] = [e−t∂t ∧ E, e−tπ] and [e−t∂t ∧ E, e−t∂t ∧ E] = 0,

one has

[P, P ] = e−2t[π, π] + 2e−2t∂t ∧ [E, π]− 2e−2tπ ∧ E

= e−2t
(
[π, π]− 2π ∧ E

)
+ 2e−2t∂t[E, π].

So by equation (1);

[P, P ] = 0 ⇐⇒ [π, π]− 2π ∧ E = 0 and [E, π] = 0.

Let’s denote by ♯π : T ∗M → TM the anchor map given by β(♯π(α)) = π(α, β), and by
[, ]π the Koszul bracket given by

[α, β]π = L♯π(α)β − L♯π(β)α− d(π(α, β)), α, β ∈ Ω1(M),

and its inverse ♭g : TM → T ∗M taking X 7→ X♭g := g(X, ·), that for X,Y ∈ TM and
α, β ∈ Ω1(M) satisfies

g(X♭g , Y ♭g) := g(X,Y ) = X♭g(Y ) and g∗(α, β) := g(♯π(α), ♯π(β)) = α(♯π(β)),



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 4 of 14

where g∗ has an associated dual metric g.

Let’s consider the map bundle (see e.g. [7]) ♯π,E : T ∗M → TM , by

♯π,E(α) = ♯π(α) + α(E)E,

and, for λ ∈ Ω1(M) a 1-form, the map [·, ·]λπ,E : Ω1(M)× Ω1(M) → Ω1(M) defined by

[α, β]λπ,E := [α, β]π + α(E)
(
LEβ − β

)
− β(E)

(
LEα− α

)
− π(α, β)λ,

where Ω1(M) denotes the space of differential 1-forms on M .

Recall that for the differential forms α, β, γ ∈ Ω1(M) we have

γ
(
♯π([α, β]π)− [♯π(α), ♯π(β)]

)
=

1

2
[π, π](α, β, γ),

and for any functions f1, f2, f3 ∈ C∞(M) one has

[df1, [df2, df3]π]π + [df2, [df3, df1]π]π + [df3, [df1, df2]π]π

= −1

2
d
(
[π, π])(df1, df2, df3)

)
.

Proposition 1. Let α ∈ Ω1(M). Then

LE(♯π,Eα) = ♯π,E(LEα).

Proof. By definition of ♯π,E , we have

LE(♯π,Eα) = LE(♯πα+ α(E)E) = LE(♯πα) + LE(α(E)E).

Likewise
♯π,E(LEα) = ♯π(LEα) + (LEα)(E)E.

Then a straightforward calculation yields, for any β ∈ Ω1(M),

β
(
LE(♯π,Eα)− ♯π,E(LEα)

)
= β

(
LE(♯πα) + LE(α(E)E)− ♯π(LEα)− (LEα(E))E

)
= β

(
LE(♯πα)− ♯π(LEα) + [LE(α(E)E)− (LEα)(E)E]

)
=

(
LE(β(♯πα))− (LEβ)(♯πα)

)
− π(LEα, β)

= LEπ(α, β)− (LEβ)
(
♯π(α)

)
− π(LEα, β)

= −(LEβ)
(
♯π(α)

)
− π(LEα, β)

= −π(α,LEβ)− π(LEα, β)

= (LEπ)(α, β)

= 0.



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 5 of 14

3. Riemann-Poisson and Riemann-Jacobi manifolds

Let D be the Levi-Civita contravariant connection associated with (P, g∗). (M, g∗, P )
is said to be a pseudo-Riemannian-Poisson manifold, if

DαP (β, γ) := ♯P (α) · P (β, γ)− P (Dαβ, γ)− P (β,Dαγ) = 0,

where α, β, γ ∈ Ω1(M). D is characterized by

2g∗(Dαβ, γ) = ♯P (α)g
∗(β, γ) + ♯P (β)g

∗(α, γ)− ♯P (γ)g
∗(α, β)

+g∗([α, β]P , γ) + g∗([γ, α], β) + g∗([γ, β]P , α). (3)

Moreover, once we introduce the bundle map J∗ : T ∗M → T ∗M , which behaves like a
contravariant (almost) complex structure [12], such that DαJ

∗β = J∗Dαβ, for all α, β, γ ∈
Ω1(M) the space of 1-forms on M . We denote by ♭g : TM → T ∗M , bundle maps called
musical isomorphisms such that ♭g(X)(Y ) = g(X,Y ) and ♯g its inverse, by g∗ the dual of
g, such that g∗(α, β) = g(♯g(α), ♯g(β)) and such that

g(J♯g(α), ♯g(β)) = π(α, β), and g∗(α, J∗β) = π(α, β), (4)

so J∗ = ♭g ◦ J ◦ ♯g.

Definition 1. A triple (π, g, J) of structures on a tangent space TM is called compatible
if for each α, β ∈ Ω1(M) the relation π(α, β) = g∗(α, J∗β) is satisfied.

To the triple (π,E, g) we associate the differential 1-form λ defined by λ = g(E,E)♭g(E)−
♭g(JE).

We call the contravariant Levi-Civita derivative associated with the triplet (π,E, g) the
unique derivative D, symmetric with respect to the bracket [·, ·]π,E and compatible with
the metric g∗. It is characterised by the formula

2g∗(Dαβ, γ) = ♯π,E(α)g
∗(β, γ) + ♯π,E(β)g

∗(α, γ)− ♯π,E(γ)g
∗(α, β)

+g∗([α, β]π,E , γ) + g∗([γ, α]π,E , β) + g∗([γ, β]π,E , α)

Proposition 2. If (π,E) is a Jacobi structure on M , then

♯π,E(DαJ
∗β) = J♯π,E(Dαβ)

for all α, β ∈ Ω1(M).

Proof. By [7, Proposition 3.1 and 2.6], we have ♯π,E(Dαβ) = ∇♯π,E(α)♯π,E(β), where ∇
is the Levi-Civita connection (covariant) associated with g. Since any Hermitian manifold
is a Kähler manifold if ∇J = 0 which is true (see [13]). Then (M, g, π) is a Kähler-Poisson
manifold and consequently, we have ∇XJY = J∇XY , for all X,Y ∈ TM . If we set
X = ♯π,E(α) and Y = ♯π,E(β), then

♯π,E(DαJ
∗β) = ∇♯π,E(α)J♯π,E(β) = J∇♯π,E(α)♯π,E(β) = J♯π,E(Dαβ).

Let ω ∈ Ω2(M) be a non-degenerate 2-form and θ ∈ Ω1(M). Suppose that the pair
(π,E) is associated with the pair (ω, θ).



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 6 of 14

Definition 2. We say that the Riemannian metric g is associated with the pair (ω, θ) if
♯ω,θ := ♯π,E is an isometry, i.e. if

g
(
♯ω,θ(α), ♯ω,θ(β)

)
= g∗(α, β),

for all α, β ∈ Ω1(M).

If J and J∗ are the endomorphism fields defined by the formulae (4), then

g
(
♯ω,θ(α), ♯ω,θ(β)

)
= g

(
♯ω(α), ♯ω(β)

)
= g∗

(
♭g(♯ω(α)), ♭g(♯ω(β)

)
= g∗(J∗α, J∗β),

for all α, β ∈ Ω1(M).

Theorem 2. An almost Jacobi manifold (M,π, g∗, J∗) with non-degenerate Poisson bivec-
tor field π is a Jacobi manifold if and only if (M,ω, g, J) is a symplectic manifold, where
ω := π−1.

Proof. We know that if g is positive definite, the pair (ω, g) is an almost Hermitian
structure on M and that J is the associated almost complex structure, i.e. we have

g(JX, JY ) = g(X,Y ) and ω(X,Y ) = g(X, JY ),

for all X,Y ∈ TM .
From the equivalence J∗ = ♭g ◦ J ◦ ♯g ⇐⇒ ♯g ◦ J∗ ◦ ♭g = J , we get J∗ ◦ ♭g = ♭g ◦ J and
ω♭ ◦π♯ = Id, and therefore (π, g∗, J∗) is Jacobian if and only if (ω, g, J) is symplectic. The
conclusion follows from [7, Corollary 2.2], to see that

Dπ(α, β, γ) = ∇ω(♯ω,θ(α), ♯ω,θ(β), ♯ω,θ(γ)).

.

Theorem 3. Assume that (π,E) is a Jacobi structure and α, β, γ ∈ Ω1(M). We have the
following property

(L♯π,E(α)π)(β, γ) = L♯π(α)π(β, γ)−
{
γ(E)π

(
β, dα(E)

)
− β(E)π

(
γ, dα(E)

)}
.

Proof. Since ♯π,E(α) = ♯π(α) + α(E)E, we have

(L♯π,E(α)π)(β, γ) = ♯π(α)π(β, γ) + α(E)E · π(β, γ)
−π(β,L♯π(α)+α(E)Eγ)− π(L♯π(α)+α(E)Eβ, γ). (5)

Moreover, we know that, for all vector fields X and Y on TM , α ∈ Ω1(M) and f a smooth
function on M , we have

LX+fY α = LXα+ fLY α+ α(Y )df.



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 7 of 14

So by setting X = ♯π(α), Y = E and f = α(E), we have

π(β,L♯π(α)+α(E)Eγ) = π(β,L♯π(α)γ) + π(β, α(E)LEγ + γ(E)d(α(E)))

= π(β,L♯π(α)γ) + π(β, α(E)LEγ) + π(β, γ(E)d(α(E)))

π(L♯π(α)+α(E)Eβ, γ) = π(L♯π(α)β, γ) + π(α(E)LEβ + β(E)d(α(E)), γ)

= π(L♯π(α)β, γ) + π(α(E)LEβ, γ) + π(β(E)d(α(E)), γ).

Then (5), become

(L♯π,E(α)π)(β, γ) = ♯π(α)π(β, γ) + α(E)E · π(β, γ)

−
(
π(β,L♯π(α)γ) + π(β, α(E)LEγ) + π(β, γ(E)d(α(E)))

)
−
(
π(L♯π(α)β, γ) + π(α(E)LEβ, γ) + π(β(E)d(α(E)), γ)

)
= ♯π(α)π(β, γ)− π(β,L♯π(α)γ)− π(L♯π(α)β, γ)

+α(E)
(
E · π(β, γ)− π(β,LEγ)− π(LEβ, γ)

)
−
(
π(β, γ(E)dα(E)) + π(β(E)dα(E), γ)

)
= (L♯π(α)π)(β, γ) + α(E)LEπ(β, γ)

−
(
π(β, γ(E)dα(E)) + π(β(E)dα(E), γ)

)
= (L♯π(α)π)(β, γ)−

{
γ(E)π

(
β, dα(E)

)
− β(E)π

(
γ, dα(E)

)}
.

We can conclude by [13, section 3, definition and properties, iii)], since LEπ = 0.Gives us
a necessary condition so that

(
R3, g∗, P

)
is a Riemann-Poisson manifold using divergence,

where g∗ is the canonical metric.

Remark 1. If ♯π(α) = E, then we have a Hermitian-Poisson manifold, since LEπ = 0.

The characterization of finite dimensional Riemann-Poisson manifolds started with [4].
The following theorem gives another method for the characterization obtained in [4].

Theorem 4. If the bivector field P is compatible with g∗ then there exists a differential
function µ on R3 such that

P = −
(
∂µ

∂z

)
∂

∂x
∧ ∂

∂y
+

(
∂µ

∂y

)
∂

∂x
∧ ∂

∂z
−
(
∂µ

∂x

)
∂

∂y
∧ ∂

∂z

Proof. According to the work of M. Boucetta [4], the compatibility between P and g∗

implies that the application being C∞ (
R3

)
-bilinear

(α, β) 7−→ LHh
g∗ (α, β) = g∗

(
DP

α dh, β
)
+ g∗

(
α,DP

β dh
)



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 8 of 14

which is anti-symmetric for any differentiable function h on R3, where Hh is the Hamilto-
nian field of h, consequently the vanishing of modular vector field given by

ϕ (h) := g∗
(
DP

dxdh, dx
)
+ g∗

(
DP

dydh, dy
)
+ g∗

(
DP

dzdh, dz
)

∀ h ∈ C∞ (
R3

)
.

We calculate ϕ (h), as

g∗
(
DP

dxdh, dx
)
=

∂h

∂x
g∗

(
DP

dxdx, dx
)
+

∂h

∂y
g∗

(
DP

dxdy, dx
)
+

∂h

∂z
g∗

(
DP

dxdz, dx
)
+ ♯P (dx) ·

(
∂h

∂x

)
=

∂h

∂x
· Γ11

1 +
∂h

∂y
· Γ12

1 +
∂h

∂z
· Γ13

1 +

(
∂P12

∂y
+

∂P13

∂z

)(
∂h

∂x

)
=

∂h

∂y
· ∂P12

∂x
+

∂h

∂z
· ∂P13

∂x
+ P12

∂2h

∂y∂x
+ P13

∂2h

∂z∂x
,

So

g∗
(
DP

dydh, dy
)
=

∂h

∂x
· Γ21

2 +
∂h

∂y
· Γ22

2 +
∂h

∂z
· Γ23

2 +

(
−∂P12

∂x
+

∂P23

∂z

)(
∂h

∂y

)
= −∂h

∂x
· ∂P12

∂y
+

∂h

∂z
· ∂P23

∂y
− P12

∂2h

∂x∂y
+ P23

∂2h

∂z∂y

and

g∗
(
DP

dzdh, dz
)
=

∂h

∂x
· Γ31

3 +
∂h

∂y
· Γ32

3 +
∂h

∂z
· Γ33

3 +

(
−∂P13

∂x
− ∂P23

∂y

)(
∂h

∂z

)
= −∂h

∂x
· ∂P13

∂z
− ∂h

∂y
· ∂P23

∂z
− P13

∂2h

∂x∂z
− P23

∂2h

∂y∂z
.

Now, let us compute the contravariant connection D. We will use the Christoffel symbols
Γij
k . For example, Ddxdx = Γ11

1 dx+ Γ11
2 dy + Γ11

3 dz. From (3), one has

Γ11
1 = 0, Γ11

2 = − ∂2µ

∂x∂z
, Γ11

3 =
∂2µ

∂x∂y
,

Γ12
1 =

∂2µ

∂x∂z
, Γ12

2 = 0, Γ12
3 =

1

2

(
− ∂2µ

∂x2
+

∂2µ

∂y2
+

∂2µ

∂z2

)
Γ21
1 = 0, Γ21

2 = − ∂2µ

∂y∂z
, Γ21

3 =
1

2

(
− ∂2µ

∂x2
+

∂2µ

∂y2
− ∂2µ

∂z2

)
Γ13
1 = − ∂2µ

∂x∂y
, Γ13

2 =
1

2

(∂2µ

∂x2
− ∂2µ

∂y2
− ∂2µ

∂z2

)
, Γ13

3 = 0

Γ31
1 = 0, Γ31

2 =
1

2

(∂2µ

∂x2
+

∂2µ

∂y2
− ∂2µ

∂z2

)
, Γ31

3 =
∂2µ

∂y∂z
,

Γ22
1 =

∂2µ

∂y∂z
, Γ22

2 = 0, Γ22
3 = − ∂2µ

∂x∂y
,

Γ23
1 =

1

2

(∂2µ

∂x2
− ∂2µ

∂y2
+

∂2µ

∂z2

)
, Γ23

2 =
∂2µ

∂x∂y
, Γ23

3 = 0,



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 9 of 14

Γ32
1 =

1

2

(
− ∂2µ

∂x2
− ∂2µ

∂y2
+

∂2µ

∂z2

)
, Γ32

2 = 0, Γ32
3 = − ∂2µ

∂x∂z
,

Γ33
1 = − ∂2µ

∂y∂z
, Γ33

2 =
∂2µ

∂x∂z
, Γ33

3 = 0.

Therefore

ϕ (h) =
∂h

∂y
· ∂P12

∂x
+

∂h

∂z
· ∂P13

∂x
− ∂h

∂x
· ∂P12

∂y
+

∂h

∂z
· ∂P23

∂y
− ∂h

∂x
· ∂P13

∂z

=

(
−∂P13

∂z
− ∂P12

∂y

)
∂h

∂x
+

(
∂P12

∂x
− ∂P23

∂z

)
∂h

∂y
+

(
∂P23

∂y
+

∂P13

∂x

)
∂h

∂z

=

{(
−∂P13

∂z
− ∂P12

∂y

)
∂

∂x
+

(
∂P12

∂x
− ∂P23

∂z

)
∂

∂y
+

(
∂P23

∂y
+

∂P13

∂x

)
∂

∂z

}
· h;

since

{
∂

∂x
,
∂

∂y
,
∂

∂z

}
is a basis of C∞ (

R3
)
-free module of A

(
R3

)
then, ϕ (h) = 0 ∀ h ∈

C∞ (
R3

)
is equivalent to the system of equations

−∂P13

∂z
− ∂P12

∂y
= 0

∂P12

∂x
− ∂P23

∂z
= 0

∂P23

∂y
+

∂P13

∂x
= 0,

which is equivalent to:

∂

∂x
(P12 + P13) +

∂

∂y
(−P12 + P23) +

∂

∂z
(−P13 − P23) = 0.

By setting

a = −P13 − P23, b = −P12 + P23, c = P12 + P13,

one has

∂c

∂x
+

∂b

∂y
+

∂a

∂z
= 0 ⇔ div · v = 0 where v = (c, b, a) .

We can say that the 2-form α = adx ∧ dy + bdx ∧ dz + cdy ∧ dz on R3 is closed. Since R3

is simply connected, then α is exact, there are three differentiable functions η, µ, ν and
an 1-form β = ηdx+ µdy + νdz on R3 such that α = dβ

dβ =

(
∂µ

∂x
− ∂η

∂y

)
dx ∧ dy +

(
∂ν

∂x
− ∂η

∂z

)
dx ∧ dz +

(
∂η

∂y
− ∂µ

∂z

)
dy ∧ dz ⇒



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 10 of 14
a = −P13 − P23 =

∂µ

∂x
− ∂η

∂y

b = −P12 + P23 =
∂η

∂z
− ∂ν

∂x

c = P12 + P13 =
∂ν

∂y
− ∂µ

∂z

⇒ a+b+c=0 (6)

which is equivalent to

∂

∂x
(µ− ν) +

∂

∂y
(ν − η) +

∂

∂z
(η − µ) = 0

which is also equivalent to 
η = ν
η = µ
µ = ν

⇔ η = µ = ν.

Therefore the system (6) becomes
a = −P13 − P23 =

∂µ

∂x
− ∂µ

∂y

b = −P12 + P23 =
∂µ

∂z
− ∂µ

∂x

c = P12 + P13 =
∂µ

∂y
− ∂µ

∂z
.

By identification we obtain

P12 = −∂µ

∂z
, P13 =

∂µ

∂y
, P23 = −∂µ

∂x
.

Which gives the desired result.

Remark 2. By setting f = −µ we find the results of M. Boucetta [4].

4. Link between Jacobi structures and Poisson structures

Contrary to Theorem 1, here we show the existence of a Poisson structure from a
Jacobi structure on a manifold of dimension 3.
Let (M,π,E) and (M,P ) respectively a Jacobi manifold and Poisson manifold, where

π = π12∂x ∧ ∂y + π13∂x ∧ ∂z + π23∂y ∧ ∂z, E = E1∂x + E2∂y + E3∂z.

Theorem 5. Let (M,π,E) be a Jacobi manifold. Then there exists a vector field Z on
TM different from E such that the bivector P defined by P = π + E ∧ Z, is a Poisson



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 11 of 14

tensor on M. There is a natural one-to-one correspondence between Jacobi manifold and
Poisson manifold ([P, P ] = 0), which is, if and only if

−2E1π13 + 2E2π13 − 2E3π12 =
(
2π12 −A

)∂B
∂x

+
(
2π13 − C

)∂A
∂y

+A
(2∂π13

∂x
− ∂C

∂y

)
+B

(
2
∂π12
∂x

− ∂C

∂z

)
+ C

(
2
∂π12
∂y

− ∂B

∂z

)
+ 2π13

∂A

∂x
, (7)

wehere

π = π12∂x ∧ ∂y + π13∂x∧∂z + π23∂y ∧ ∂z

E = = E1∂x + E2∂y + E3∂z

Z = Z1∂x + Z2∂y + Z3∂z.

A = E1Z2 − E2Z1, B = E1Z3 − E3Z1 and C = E2Z3 − E3Z2.

Moreover

P = (π12+E1Z2−E2Z1)∂x∧∂y+(π13+E1Z3−E3Z1)∂x∧∂z+(π23+E2Z3−E3Z2)∂y∧∂z.

Proof. To obtain the result it is enough to show that, by using equation (2),

[P, P ] = [π + E ∧ Z, π + E ∧ Z]

= [π, π]π + [π,E ∧ Z] + [E ∧ Z, π] + [E ∧ Z,E ∧ Z]

= 0

We can assume that E ̸= Z. Then one has

E ∧ Z =
(
E1∂x + E2∂y + E3∂z

)
∧
(
Z1∂x + Z2∂y + Z3∂z

)
=

(
E1Z2 − E2Z1

)
∂x ∧ ∂y +

(
E1Z3 − E3Z1

)
∂x ∧ ∂z +

(
E2Z3 − E3Z2

)
∂y ∧ ∂z

= A∂x ∧ ∂y +B∂x ∧ ∂z + C∂y ∧ ∂z.

A straightforward computation show that

[E ∧ Z,E ∧ Z] =
(
−A

∂B

∂x
− C

∂A

∂y
−A

∂C

∂y
− C

∂B

∂z
−B

∂C

∂z

)
∂x ∧ ∂y ∧ ∂z. (8)

[π, π] = 2E ∧ π

= 2
(
E1π23 − E2π13 + E3π12

)
∂x ∧ ∂y ∧ ∂z (9)

[π,E ∧ Z] + [E ∧ Z, π] = 2
(
B
∂π12
∂x

− π12
∂B

∂x
+ C

∂π12
∂y

+A
∂π13
∂x

+π13
∂A

∂x
+ π23

∂A

∂y

)
∂x ∧ ∂y ∧ ∂z (10)



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 12 of 14

So equation (8) by (9)-(10), becomes

[P, P ] =
(
2E1π23 − 2E2π13 + 2E3π12 + (2π12 −A)

∂B

∂x
+ (2π23 − C)

∂A

∂y

+A(
2∂π13
∂x

− ∂C

∂y
) +B(2

∂π12
∂x

− ∂C

∂z
) + C(2

∂π12
∂y

− ∂B

∂z
) + 2π13

∂A

∂x

)
∂x ∧ ∂y ∧ ∂z.

We deduce the result (7).

Example 1. Let us equip R3 with the bivector field π defined by

π = (2y + 3y2)∂x ∧ ∂y + (2 + 3y)∂y ∧ ∂z

and the vector field E defined by E = −2∂x − 3∂z.
A direct calculation shows that

[π, π] = 2E ∧ π = 2(−9y2 − 4− 12y)∂x ∧ ∂y ∧ ∂z and [E, π] = 0.

By choosing Z1 = f(x) = −x2 − 1, Z2 = g(z), Z3 = h(y) = y2 + 1, in this example, we
have A = −2Z2, B = −2Z3 + 3Z1, C = 3Z2 and by (7)

[P, P ] =
(
− 8− 24y − 18y2 + (−2h(y) + 3f(x))(−∂g

∂z
) + (4 + 12y)(3g(z))

)
∂x ∧ ∂y ∧ ∂z.

[P, P ] = 0 ⇐⇒
(
2h(y)− 3f(x)

)
g′(x) + (12 + 36y)g(z) = 18y2 + 24y + 8

⇐⇒ αg′(z) + βg(z) = γ

We get

g(z) = Ke−
β
α
z +

αγ

β
, ∀K ∈ R,

where α = 2y2 + 3x2 + 5, β = 12 + 36y, γ = 18y2 + 24y + 8 with y ̸= −1
3 .

Thus we obtain
Z = −(x2 + 1)∂x + Z2∂y + (y2 + 1)∂z

with

Z2 = K exp
( −12− 36y)

3x2 + 2y2 + 5
z
)
+

(9x2 + 6y2 + 15)(y + 2
3)

2

6y + 2
, ∀K ∈ R.

Corollary 1. Let us assume that E = f∂z where f ̸≡ 0. Let (π,E) a Jacobi structure on
M . P = π + E ∧ Z, (Z = ∂x + ∂y + ∂z) is a Poisson structure on M if and ally if

∂π12
∂z

= 0, (11)

π12
∂f

∂y
+ π13

∂f

∂z
+ f

∂π13
∂z

= 0, (12)

π12
∂f

∂x
+ π23

∂f

∂z
+ f

∂π23
∂z

= 0, (13)

2f
(∂π12

∂x
+

∂π12
∂y

+
1

2

∂π13
∂y

+
1

2

∂π13
∂z

)
= π13

(∂f
∂y

− ∂f

∂z

)
+ 2π12

(
f +

∂f

∂x

)
. (14)



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 13 of 14

Proof. Since
E ∧ Z = −f∂x ∧ ∂z − f∂y ∧ ∂z,

one has:

[P, P ] = [π + E ∧ Z, π + E ∧ Z]

= [π, π] + [π,E ∧ Z] + [E ∧ Z, π] + [E ∧ Z,E ∧ Z]

= 2E ∧ π + π13

(∂f
∂y

− ∂f

∂z

)
∂x ∧ ∂y ∧ ∂z + 2π12

∂f

∂x
∂x ∧ ∂y ∧ ∂z

−f
(
2
∂π12
∂x

+ 2
∂π12
∂y

+
∂π13
∂y

+
∂π13
∂z

)
∂x ∧ ∂y ∧ ∂z

= −2f
(∂π12

∂x
+

∂π12
∂y

+
1

2

∂π13
∂y

+
1

2

∂π13
∂z

)
∂x ∧ ∂y ∧ ∂z

π13

(∂f
∂y

− ∂f

∂z

)
∂x ∧ ∂y ∧ ∂z + 2π12

(
f +

∂f

∂x

)
∂x ∧ ∂y ∧ ∂z.

So [P, P ] = 0 if and only if

2f
(∂π12

∂x
+

∂π12
∂y

+
1

2

∂π13
∂y

+
1

2

∂π13
∂z

)
= π13

(∂f
∂y

− ∂f

∂z

)
+ 2π12

(
f +

∂f

∂x

)
which give (14).

[E, π] = f
∂π12
∂z

∂x ∧ ∂y +
(
π12

∂f

∂x
+ f

∂π23
∂z

+ π23
∂f

∂z

)
)∂y ∧ ∂z

+
(
π12

∂f

∂y
+ f

∂π13
∂z

+ π13
∂f

∂z

)
∂x ∧ ∂z

= 0.

Since f ̸= 0, one has 
∂π12
∂z

= 0

π12
∂f
∂x

+ f ∂π23
∂z

+ π23
∂f
∂z

= 0

π12
∂f
∂y

+ f ∂π13
∂z

+ π13
∂f
∂z

= 0.

Thus we obtain (11),(12) and (13).

Example 2. Let
π = (x4 + y4)∂x ∧ ∂y − x∂x ∧ ∂z + y∂y ∧ ∂z

and E = f∂z with f is smooth function.

Then a straightforward calculation yields
(x4 + y4)∂f∂x + y ∂f

∂z = 0,

(x4 + y4)∂f∂y − x∂f
∂z = 0,

2f(x4 + y4 + 4x3 + 4y3) + 2(x4 + y4)∂f∂x − x∂f
∂y

+ x∂f
∂z = 0.

On can obtain
f(x, y, z) = exp

(
zg(x, y) + k(x, y)

)
.



M. S. Ali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5743 14 of 14

Acknowledgements

The authors would like to sincerely thank the reviewers for their careful reading and
many enlightening comments that significantly improve the quality of the article.

This works was supported by the EDP-MC Network with finance support from the
International Science Program (ISP Grants).

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