Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 109-117.

Some Remarks on the Poisson-Nijenhuis and the Jacobi Structures

J. M. Nunes da Costa^{1}

Departamento de Matemática, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal e-mail: jmcosta@mat.uc.pt

Abstract. We study the relationship between two (compatible) Jacobi structures on a manifold M, using their associated homogeneous Poisson structures on ℝ×M, in the case where these Poisson tensors are related by a Nijenhuis tensor. M. S. Classification (1991): 58F05. Keywords: Poisson-Nijenhuis manifold, Jacobi manifold.

Introduction

All objects considered in this paper (manifolds, maps, differential forms, vector and tensor fields) are assumed to be differentiable of class C^{∞}.

Let M be a smooth manifold equipped with a Poisson tensor Λ and a Nijenhuis tensor N, that is, a tensor field of type (1, 1) whose Nijenhuis torsion τ(N) vanishes everywhere. The Nijenhuis torsion τ(N) of N is given by the following formula, where X and Y are vector fields on M,

τ(N)(X, Y) = [NX, NY] − N([NX, Y] + [X, NY] − N[X, Y]).

We denote by Λ^{#} : T^{∗}M → T M the vector bundle map such that, for any x ∈ M, α, β ∈ T_{x}^{∗}M,

< β, Λ^{#}(α) >= Λ_{x}(α, β).

1This work was partially supported by CMUC-FCT and PRAXIS. The paper is in final form and no other version has been submitted for publication elsewhere.

110 J.M. Nunes da Costa: Some Remarks on the Poisson-Nijenhuis and the Jacobi Structures

With Λ and N we can define a tensor field R(Λ, N) of type (2, 1), called the Magri-Morosi concomitant [Ma-Mo 84] of Λ and N, that is defined, for any pair of a 1-form α and a vector field X, by

R(Λ, N)(α, X) = (L_{Λ^{#}(α)}N)X − Λ^{#}(L_{X}(^{t}Nα)) + Λ^{#}(L_{NX}α),  (1)

where ^{t}N is the transpose of N : T M → T M.

Let Λ and N be, respectively, a Poisson tensor and a Nijenhuis tensor on M. The triple (M, Λ, N) is called a Poisson-Nijenhuis manifold [KSch-Ma 90] if

NΛ^{#} = Λ^{#} ^{t}N and R(Λ, N) = 0.

If (M, Λ, N) is a Poisson-Nijenhuis manifold, there exists a sequence (Λ_{k})_{k∈ℕ} of Poisson tensors on M, with Λ_{k} = N^{k}Λ. Moreover, these Poisson tensors are pairwaise compatible, that is, [Λ_{l}, Λ_{k}] = 0, for all l, k ∈ ℕ.

Besides the Poisson-Nijenhuis structure, we are going to use, in what follows, the notion of Jacobi manifold.

A Jacobi manifold [Lich 78] is a triple (M, C, E) where C and E are respectively a bivector and a vector field on M, such that

[E, C] = 0 and [C, C] = 2E ∧ C.

When E = 0, the Jacobi manifold is a Poisson manifold. The Jacobi bracket of f, g ∈ C^{∞}(M, ℝ) is given by

{f, g} = C(df, dg) + f(E.g) − g(E.f),

and it defines a local Lie algebra structure on C^{∞}(M, ℝ).

If (M, C, E) is a Jacobi manifold and h is a nowhere vanishing function on M, then the pair (hC, C^{#}(dh) + hE) = (C_{h}, E_{h}) defines a new Jacobi structure on M, which is said to be conformally equivalent to (C, E).

With each Jacobi manifold (M, C, E), we may associate a homogeneous Poisson structure

(Λ, \frac{∂}{∂t}) on ℝ × M, with Λ given by

Λ = exp(−t)(C + \frac{∂}{∂t} ∧ E),  (2)

where t is the canonical coordinate on ℝ.

This paper is divided into two sections. Section 1 is devoted to the subject of compatible Jacobi manifolds. We show how the compatibility is related with the Lichnerowicz-Jacobi cohomology and we present a way of generating compatible Jacobi structures on a manifold. In Section 2, we establish the conditions on the Poisson-Nijenhuis structure of ℝ × M to ensure the compatibility of the corresponding Jacobi structures on M.

1. Compatible Jacobi manifolds

The notion of compatibility of two Jacobi structures on a manifold was introduced in [NdC 98]. We recall that two Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}) on a manifold M are said to be compatible if (C_{1} + C_{2}, E_{1} + E_{2}) is again a Jacobi structure on M.

Proposition 1.1. [NdC 98] Let M be a manifold endowed with two Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}).Then, (C_{1}, E_{1}) and (C_{2}, E_{2}) are compatible if and only if

[E_{1}, C_{2}] + [E_{2}, C_{1}] = 0 and [C_{1}, C_{2}] = E_{1} ∧ C_{2} + E_{2} ∧ C_{1}.

There are some equivalent ways of expressing the compatibility of two Jacobi structures on a manifold [NdC 98]. But the study of the compatibility of two Jacobi structures on a manifold, can also be done using the Lichnerowicz-Jacobi cohomology. If (M, C, E) is a Jacobi manifold, let us denote by A^{k}(M) the space of skew-symmetric contravariant tensor fields of order k (k-tensors) on M and define the differential operator

σ : A^{k}(M) → A^{k+1}(M), σ(P) = −[C, P] + kE ∧ P.  (3)

The restriction of σ to the subspace

A_{I}^{k}(M) = {P ∈ A^{k}(M) : L_{E}P = 0}

of the invariant k-tensors with respect to the vector field E is a cohomology operator on the Jacobi manifold and the resultant cohomology is called the Lichnerowicz-Jacobi cohomology of M [Le-Ma-Pa 97].

Proposition 1.2. Two Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}) on a manifold M are compatible if and only if

σ_{1}(C_{2}) = −σ_{2}(C_{1}) and σ_{1}(E_{2}) = −σ_{2}(E_{1}),

where σ_{i}, i = 1, 2, are the cohomology operators of the Lichnerowicz-Jacobi cohomology of M, with respect to both Jacobi structures.

Proof. A direct computation using Proposition 1.1 and the definition (3) of the cohomology operators σ_{i}, i = 1, 2, gives the desired result. □

In [NdC 98] it was proved that two conformally equivalent Jacobi structures on M are compatible. Another way of obtaining compatible Jacobi structures uses the Lie derivative on the direction of some vector field.

Proposition 1.3. Let X be a vector field on the Jacobi manifold (M, C, E) such that

L_{X}(L_{X}C) = 0 and L_{X}(L_{X}E) = 0.

Then the pair (C_{1}, E_{1}) = (L_{X}C, L_{X}E) defines a new Jacobi structure on M which is compatible with (C, E).

Proof. With E_{1} = L_{X}E = [X, E], we have L_{E_{1}}C = −L_{E}(L_{X}C), that is

L_{E_{1}}C + L_{E}C_{1} = 0.  (4)

So,

L_{E_{1}}C_{1} = L_{X}(L_{E_{1}}C)
= −L_{X}(L_{E}C_{1})
= −L_{[X,E]}(L_{X}C)
= −L_{E_{1}}C_{1},

and

L_{E_{1}}C_{1} = 0.  (5)

On the other hand,

L_{X}([C, C]) = [L_{X}C, C] + [C, L_{X}C]
= 2[C_{1}, C]

and

L_{X}(E ∧ C) = E_{1} ∧ C + E ∧ C_{1};

so,

[C_{1}, C] = E_{1} ∧ C + E ∧ C_{1}.  (6)

Because L_{X}E_{1} = L_{X}C_{1} = 0, if we take the Lie derivative on the direction of X of both members of (6), we obtain

[C_{1}, C_{1}] = 2E_{1} ∧ C_{1}.  (7)

The equalities (5) and (7) prove that (C_{1}, E_{1}) is a Jacobi structure, while (4) and (6) show the compatibility. □

We can also prove the following result.

Proposition 1.4. Let X_{1} and X_{2} be two vector fields on the Jacobi manifold (M, C, E) such that

[X_{1}, X_{2}] = 0, [X_{1}, L_{X_{2}}C] = 0, [X_{1}, L_{X_{2}}E] = 0

and

L_{X_{i}}(L_{X_{i}}C) = 0, L_{X_{i}}(L_{X_{i}}E) = 0, i = 1, 2.

Then

(C_{1}, E_{1}) = (L_{X_{1}}C, L_{X_{1}}E) and (C_{2}, E_{2}) = (L_{X_{2}}C, L_{X_{2}}E)

are compatible Jacobi structures on M.

Proof. We only have to check the compatibility. By Proposition 1.3, we know that the Jacobi structures (C, E) and (C_{2}, E_{2}) are compatible. Then,

[C, C_{2}] = E ∧ C_{2} + E_{2} ∧ C and [E, C_{2}] + [E_{2}, C] = 0.  (8)

If we take the Lie derivative, on the direction of X_{1}, of both members of equalities (8), we obtain

[C_{1}, C_{2}] = E_{1} ∧ C_{2} + E_{2} ∧ C_{1} and [E_{1}, C_{2}] + [E_{2}, C_{1}] = 0.  (9)
□

2. Homogeneous Poisson-Nijenhuis manifolds

Let M be a manifold endowed with two Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}). Take the

corresponding homogeneous Poisson structures on ℝ × M, (Λ_{i} = exp(−t)(C_{i} + \frac{∂}{∂t} ∧ E_{i}), \frac{∂}{∂t}), i = 1, 2.

Proposition 2.1. [NdC 98] The Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}) on M are compatible if and only if Λ_{1} and Λ_{2} are compatible Poisson tensors on ℝ × M.

Using this result, we want to study the relationship between two compatible Jacobi structures on M and their associated (compatible) homogeneous Poisson strucutres on ℝ×M, in the case where these Poisson structures are related by a Nijenhuis tensor.

Let N̄ be a (1, 1)-tensor on ℝ × M such that L_{\frac{∂}{∂t}} N̄ = 0. Then, N̄ is given by

N̄ = N + Y ⊗ dt + \frac{∂}{∂t} ⊗ γ + g \frac{∂}{∂t} ⊗ dt,  (10)

where N is a (1, 1)-tensor on M, Y is a vector field on M, γ is a 1-form on M and g ∈ C^{∞}(M, ℝ). Reciprocally, if N̄ is given by (10), then L_{\frac{∂}{∂t}} N̄ = 0.

Remarks

1. The image of a vector field on ℝ × M of the form X̄ = exp(−t)(f \frac{∂}{∂t} + X), where X is a vector field on M and f ∈ C^{∞}(M, ℝ), by the (1, 1)-tensor given by (10) is a vector field on ℝ × M of the same type of X̄.

2. If Λ is the homogeneous Poisson tensor on ℝ × M, given by (2), then N̄Λ is a homogeneous bivector on ℝ × M: L_{\frac{∂}{∂t}} (N̄Λ) = −N̄Λ

Proposition 2.2. Let N̄ be a (1, 1)-tensor on ℝ × M given by (10). Then N̄ is a Nijenhuis tensor on ℝ × M if and only if

i) τ(N) = Y ⊗ dγ;
ii) L_{N}γ = gdγ;
iii) L_{Y} N = −Y ⊗ dg;
iv) ^{t}N(dg) = L_{Y} γ + gdg.

Proof. First, remark that if X is a vector field on M, then N̄X = NX+ < γ, X > \frac{∂}{∂t} and

that N̄(\frac{∂}{∂t}) = Y + g \frac{∂}{∂t}. If X_{1} and X_{2} are vector fields on M, then

τ(N̄)(X_{1}, X_{2}) = τ(N)(X_{1}, X_{2}) + (X_{2}. < γ, X_{1} > −X_{1}. < γ, X_{2} >

+ <γ, [X_{1}, X_{2}]>)N̄(\frac{∂}{∂t}) + ((NX_{1}).<γ, X_{2} > −(NX_{2}).<γ, X_{1} >

− < γ, [NX_{1}, X_{2}] > − < γ, [X_{1}, NX_{2}] > − < γ, N[X_{1}, X_{2}] >)\frac{∂}{∂t}

= τ(N)(X_{1}, X_{2}) − dγ(X_{1}, X_{2})Y − (gdγ(X_{1}, X_{2}) − L_{N}γ(X_{1}, X_{2}))\frac{∂}{∂t}.

114 J.M. Nunes da Costa: Some Remarks on the Poisson-Nijenhuis and the Jacobi Structures

So, τ(N̄)(X_{1}, X_{2}) = 0 if and only if

τ(N)(X_{1}, X_{2}) = (Y ⊗ dγ)(X_{1}, X_{2})  (11)

and

L_{N}γ(X_{1}, X_{2}) = gdγ(X_{1}, X_{2}).  (12)

Let X be a vector field on M. Then,

τ(N̄)(X, \frac{∂}{∂t}) = [NX, Y] + ((NX).g)\frac{∂}{∂t} − (Y. < γ, X >)\frac{∂}{∂t}

−N[X, Y]− < γ, [X, Y] > \frac{∂}{∂t} − (X.g)(Y + g \frac{∂}{∂t})

= −(L_{Y} N)(X) − (Y ⊗ dg)(X)

+(< ^{t}Ndg, X > − < L_{Y} γ, X > − < gdg, X >)\frac{∂}{∂t}.

So, τ(N̄)(X, \frac{∂}{∂t}) = 0 if and only if

(L_{Y} N)(X) = −(Y ⊗ dg)(X)  (13)

and

< ^{t}Ndg − L_{Y} γ − gdg, X >= 0.  (14)

Equalities (11), (12), (13) and (14) end the proof. □

Let us take the homogeneous Poisson tensor Λ = exp(−t)(C + \frac{∂}{∂t} ∧ E) on (ℝ × M). Lemma 2.1. With the notations of Proposition 2.2, N̄Λ = Λ^{t}N̄ if and only if

i) NE = C^{#}(γ) + gE;
ii) NC − C^{t}N = E ⊗ Y + Y ⊗ E;
iii) < γ, E >= 0.

Proof. Let α be any 1-form on M. Then,

N̄(Λ^{#}(α)) = exp(−t)(N̄(C^{#}(α)− < α, E > \frac{∂}{∂t}))
= exp(−t)(N(C^{#}(α))− < α, E > Y − (< α, C^{#}(γ) > + < α, E > g)\frac{∂}{∂t})

and, taking account that

tN̄(α) = ^{t}Nα+ < α, Y > dt,

we compute

Λ^{#}(^{t}N̄α) = exp(−t)(C^{#}(^{t}Nα)+ < α, Y > E− < α, NE > \frac{∂}{∂t})

and we conclude that N̄Λ^{#}(α) = Λ^{#}(^{t}Nα) if and only if

< α, NE >=< α, C^{#}(γ) + gE >  (15)

and

(NC^{#} − Y ⊗ E)α = (C^{#} ^{t}N + E ⊗ Y)α.  (16)

Equalities (15) and (16) give conditions i) and ii).

On the other hand,

N̄(Λ^{#}(dt)) = exp(−t)(NE+ < γ, E > \frac{∂}{∂t})  (17)

and, because ^{t}N̄(dt) = γ + gdt,

Λ^{#}(^{t}N̄(dt)) = exp(−t)(C^{#}(γ) + gE).  (18)

Once the right members of (17) and (18) are equal, we obtain condition iii). □

Let us take the Magri-Morosi concomitant R(Λ, N̄) of Λ and N̄, (1).

Lemma 2.2. With the notations of Proposition 2.2, R(Λ, N̄) = 0 if and only if

i) NE + L_{Y} E = gE − C^{#}(dg)
ii) L_{Y} C = gC − NC + Y ⊗ E
iii) (L_{E}N)X+ < γ, X > E = C^{#}(i_{X}dγ) + (X.g)E
iv) R(C, N)(df, X) = (X.(Y.f))E − (X.(E.f))Y − < γ, X > C^{#}(df),

where X is any vector field on M and f is any function of C^{∞}(M, ℝ).

Proof. Letf be any function of C^{∞}(M, ℝ). The component of the vector field R(Λ,N̄)(df,\frac{∂}{∂t})

on \frac{∂}{∂t} is

exp(−t)(< df, C^{#}(dg) + L_{Y} E + NE − gE >).  (19)

The other components (without \frac{∂}{∂t}) of R(Λ, N̄)(df, \frac{∂}{∂t}), obtained from the computation of

< dh, R(Λ, N̄)(df, \frac{∂}{∂t}) >, for any h ∈ C^{∞}(M, ℝ), are

exp(−t)(< dh, (−L_{Y} C)^{#}(df) + gC^{#}(df) − N(C^{#}(df)) + (E.f)Y >).  (20)

From (19) and (20), we conclude that R(Λ, N̄)(df, \frac{∂}{∂t}) = 0 gives conditions i) and ii) of the Lemma.

On the other hand, if X is any vector field on M, the component of the vector field

R(Λ, N̄)(df, X) on \frac{∂}{∂t} is

exp(−t)(< df, −(L_{E}N)X− < γ, X > E + C^{#}(i_{X}dγ) + (X.g)E >,  (21)

while the components without \frac{∂}{∂t}, obtained from the computation of

< dh, R(Λ, N̄)(df, X) >, for any h ∈ C^{∞}(M, ℝ), are

exp(−t)(< dh, R(C, N)(df, X)+ < γ, X > C^{#}(df) + (X.(E.f))Y − (X.(Y.f))E > . (22)

From (21) and (22), we conclude that R(Λ, N̄)(df, X) = 0 gives conditions iii) and iv) of the Lemma. □

Theorem 2.1. Let Λ = exp(−t)(C + \frac{∂}{∂t} ∧ E) and N̄ = N + Y ⊗ dt + \frac{∂}{∂t} ⊗ γ + g \frac{∂}{∂t} ⊗ dt, be respectively the homogeneous Poisson tensor and the Nijenhuis tensor on ℝ × M. Then, the triple (ℝ × M, Λ, N̄) is a Poisson-Nijenhuis manifold if and only if conditions i), ii) and iii) of Lemma 2.1 and i) – iv) of Lemma 2.2 hold. Moreover, the homogeneous Poisson tensor

N̄Λ is given by exp(−t)(C_{1} + \frac{∂}{∂t} ∧ E_{1}), where

(C_{1}, E_{1}) = (gC − L_{Y} C, gE − C^{#}(dg) − L_{Y} E)  (23)

is a Jacobi structure on M, compatible with (C, E).

Proof. We only have to find the expressions of C_{1} and E_{1}. For any pair (f, h) of functions on M,

< dh, (N̄Λ)^{#}(df) >= exp(−t) < dh, NC^{#}(df) − (Y ⊗ E)(df) >  (24)

and

< dh, (Λ^{#}(^{t}N̄)(df) >= exp(−t) < dh, C^{#}(^{t}N(df)) + (E ⊗ Y)(df) > .  (25)

Since N̄Λ = \frac{1}{2}(N̄Λ + Λ^{t}N̄), we obtain from (24) and (25), using conditions ii) of Lemma 2.1 and ii) of Lemma 2.2,

(N̄Λ)(df, dh) = exp(−t)(NC − Y ⊗ E)(df, dh)
= exp(−t)(gC − L_{Y} C)(df, dh).  (26)

Also,

< dh, N̄(Λ^{#}(dt)) >= exp(−t) < dh, NE >  (27)

and

< dh, Λ^{#}(^{t}N̄(dt)) >= exp(−t) < dh, C^{#}(γ) + gE > .  (28)

From (27) and (28), and taking into account conditions i) of Lemma 2.1 and i) of Lemma 2.2, we conclude that

(N̄Λ)(dt, dh) = exp(−t)NE(dt, dh)
= exp(−t)(gE − C^{#}(dg) − L_{Y} E)(dt, dh).  (29)
□

Acknowledgements. The author wishes to thank Professors C.-M. Marle and J. Monterde for helpful discussions.

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