DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 523–538

REDUCTION OF BIHAMILTONIAN MANIFOLDS

AND RECURSION OPERATORS

Joana Margarida Nunes da Costa and Charles-Michel Marle

Acknowledgements. The authors, particularly C.-M. M., are indebted to Professor Franco Magri for many helpful discussions.

All the manifolds, maps, differential forms, vector and tensor fields, considered in what follows are assumed to be differentiable of class C^{∞}.

1. Poisson manifolds and the Schouten-Nijenhuis bracket

1.1. Some general properties and notations. Let Λ be a bivector, that means a 2-times contravariant skew-symmetric tensor field, on a smooth manifold M. It defines a vector bundle map Λ^{♯} : T^{∗}M → T M, such that, for any x ∈ M, α and β ∈ T_{x}^{∗}M,

⟨β, Λ^{♯}(α)⟩ = Λ_{x}(α, β) .

We can therefore associate with any differential 1-form α, a vector field Λ^{♯}(α). For any pair of smooth functions f and g ∈ C^{∞}(M, R), we set

{f, g} = Λ(df, dg) = i(Λ^{♯}(df))dg = −i(Λ^{♯}(dg))df .

We shall say that (f, g) ↦ {f, g} is the bracket associated with Λ. It satisfies, for all f, g, g_{1} and g_{2} ∈ C^{∞}(M, R),

{f, g} = −{g, f} , {f, g_{1}g_{2}} = {f, g_{1}}g_{2} + g_{1}{f, g_{2}} .

1.2. Definition. A bivector Λ on a smooth manifold M is said to be a Poisson tensor if the bracket associated with it satisfies the Jacobi identity, that is, if for all f, g and h ∈ C^{∞}(M, R),

{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 .

1991 Mathematics Subject Classification. 58F05.

Key words and phrases. Poisson manifolds, bihamiltonian manifolds, reduction.

This paper is in final form and no version of it will be submitted for publication elsewhere.

In that case, the manifold M equipped with Λ is called a Poisson manifold and denoted by (M, Λ); the bracket associated with Λ is called a Poisson bracket and, for any smooth function f on M, the vector field Λ^{♯}(df) is called the Hamiltonian vector field associated with f.

Let us recall the well known result (Lichnerowicz [8]):

1.3. Proposition. Let Λ be a bivector on a smooth manifold M. The following three properties are equivalent:

(i) the bivector Λ is a Poisson tensor; (ii) the Schouten-Nijenhuis bracket [Λ, Λ] of Λ with itself vanishes; (iii) for all f and g ∈ C^{∞}(M, R),

[Λ^{♯}(df), Λ^{♯}(dg)] = Λ^{♯}(d{f, g}) .

When these equivalent properties are satisfied, C^{∞}(M, R), with the Poisson bracket as composition law, is a Lie algebra and the map f ↦ Λ^{♯}(df) is a Lie algebra homomorphism.

1.4. Some properties of the Schouten-Nijenhuis bracket.

Let us recall the definition and some properties of the Schouten-Nijenhuis bracket [15, 13, 14, 5]. We use in the following the conventions of Koszul [5], which may differ by signs and even numerical factors from those used in [8] or [9].

Let M be a smooth manifold. We denote by A(M) = ⊕_{p}A^{p}(M) and by Ω(M) = ⊕_{p}Ω^{p}(M) the graded algebras, repectively, of skew-symmetric contravariant tensor fields, and of differential forms on M.

The Schouten-Nijenhuis bracket is a bilinear composition law on A(M), denoted by (P, Q) ↦ [P, Q], which satisfies the following properties:

(i) if P ∈ A^{p}(M) and Q ∈ A^{q}(M), then [P, Q] ∈ A^{p+q−1}(M); (ii) if X ∈ A^{1}(M) is a vector field on M, then for any Q ∈ A(M), [X, Q] is the Lie

derivative L(X)Q of Q with respect to X; (iii) if P ∈ A^{p}(M) and Q ∈ A^{q}(M), then

[P, Q] = −(−1)^{(p−1)(q−1)}[Q, P] ;

(iv) for any P ∈ A^{p}(M), the map Q ↦ [P, Q] is a derivation of degree p − 1 of the graded exterior algebra A(M) (with the wedge product as composition law); in other words, for any Q_{1} ∈ A^{q_{1}}(M) and Q_{2} ∈ A(M),

[P, Q_{1} ∧ Q_{2}] = [P, Q_{1}] ∧ Q_{2} + (−1)^{(p−1)q_{1}}Q_{1} ∧ [P, Q_{2}] .

These four properties determine the Schouten-Nijenhuis bracket in a unique way.

The Schouten-Nijenhuis bracket has several other remarkable properties. In particular, (v) it satisfies the graded Jacobi identity

\begin{aligned}(−1)^{(p−1)(r−1)}[P, [Q, R]] & + (−1)^{(q−1)(p−1)}[Q, [R, P]] \\ & + (−1)^{(r−1)(q−1)}[R, [P, Q]] = 0 , \end{aligned}

where P ∈ A^{p}(M), Q ∈ A^{q}(M) and R ∈ A^{r}(M); therefore, A(M), with the Schouten-Nijenhuis bracket as composition law, is a graded Lie algebra. There is another important property of the Schouten-Nijenhuis bracket, which

links it with the exterior differentiation operator d. We must first recall some conventions about the exterior algebras A(M) and Ω(M). There exists a bilinear map from Ω(M) × A(M) into C^{∞}(M, R), called the coupling map, denoted by (η, P) ↦ ⟨η, P ⟩. We define it first when η = η_{1}∧⋯∧η_{q} ∈ Ω^{q}(M) is a decomposable q-form (the η_{i} ∈ Ω^{1}(M) being 1-forms on M), and when P = P_{1}∧⋯∧P_{p} ∈ A^{p}(M) is a decomposable p-vector (the P_{i} ∈ A^{1}(M) being vector fields on M). Then

⟨η_{1} ∧ ⋯ ∧ η_{q}, P_{1} ∧ ⋯ ∧ P_{p}⟩ = \{ \begin{matrix}0 & if p ≠ q, \\ det(⟨η_{i}, P_{j}⟩) & if p = q. \end{matrix}

We have denoted by ⟨η_{i}, P_{j}⟩ = i(P_{j})η_{i} the usual coupling of a 1-form η_{i} with a vector field P_{j}.

The coupling map, which is local, can then be extended by bilinearity to the whole of Ω(M) × A(M), since locally any differential q-form and any p-vector are sums of decomposable elements.

We define next a bilinear map from A(M) × Ω(M) into Ω(M), called the inner product of a form by a multivector, and denoted by (P, η) ↦ i(P)η. When P ∈ A^{p}(M) and η ∈ Ω^{q}(M) are homogeneous of degrees p and q, respectively, then i(P)η ∈ Ω^{q−p} is the unique (q − p)-form such that, for any R ∈ A^{q−p}(M),

⟨i(P)η, R⟩ = (−1)^{(p−1)p/2}⟨η, P ∧ R⟩ .

We observe that when p > q, i(P)η vanishes.

The definition of the inner product i(P)η can be readily extended by bilinearity when P and η are not homogeneous.

When P ∈ A^{p}(M) is a fixed homogeneous p-tensor, then i(P) : η ↦ i(P)η is a graded endomorphism of the graded exterior algebra Ω(M), of degree −p.

The numerical coefficient (−1)^{(p−1)p/2} in the above formula may seem unnatural; however it is such that when X ∈ A^{1}(M) is a vector field, then i(X) is the usual inner product by the vector field X; it is also such that, for any P ∈ A^{p}(M) and Q ∈ A^{q}(M),

i(P)i(Q) = i(P ∧ Q) .

Let D_{1} and D_{2} be two graded endomorphisms of the exterior algebra Ω(M), of degrees d_{1} and d_{2}, respectively. We define their bracket, denoted by [D_{1}, D_{2}], as the graded endomorphism of Ω(M), of degree d_{1} + d_{2},

[D_{1}, D_{2}] = D_{1}D_{2} − (−1)^{d_{1}d_{2}}D_{2}D_{1} .

We can now indicate the next property of the Schouten-Nijenhuis bracket: (vi) let P ∈ A^{p}(M), Q ∈ A^{q}(M), [P, Q] their Schouten-Nijenhuis bracket; let i(P),

i(Q) and i([P, Q]) be the inner products by P, Q and [P, Q], respectively, considered as graded endomorphisms of Ω(M); then we have

i([P, Q]) = [[i(P), d], i(Q)] ,

where d denotes the exterior differentiation operator.

The above property determines the Schouten-Nijenhuis bracket in a unique way, and could be used as a definition.

1.5. Applications of the Schouten-Njenhuis bracket to Poisson manifolds.

Let Λ be a bivector on a smooth manifold M. Using the definitions of the bracket associated with Λ, of the coupling map and of the inner product given above we have, for all f and g ∈ C^{∞}(M, R),

{f, g} = i(Λ^{♯}(df))dg = Λ(df, dg) = ⟨df ∧ dg, Λ⟩ = −i(Λ)(df ∧ dg) .

By property 1.4.(vi), we have

i([Λ, f])dg = [[i(Λ), d], i(f)]dg = i(Λ)(df ∧ dg) .

Therefore,

Λ^{♯}(df) = −[Λ, f] = −[f, Λ] , {f, g} = −[[Λ, f], g] = [[Λ, g], f] .

Similarly, for all f, g and h ∈ C^{∞}(M, R),

[Λ, Λ](df ∧ dg ∧ dh) = −i([Λ, Λ])(df ∧ dg ∧ dh) = −[[i(Λ), d], i(Λ)](df ∧ dg ∧ dh)
= −2i(Λ)di(Λ)(df ∧ dg ∧ dh)
= 2i(Λ)(d{f, g} ∧ dh + d{g, h} ∧ df + d{h, f} ∧ dg)
= −2({{f, g}, h} + {{g, h}, f} + {{h, f}, g})
= 2({f, {g, h}} + {g, {h, f}} + {h, {f, g}})
= 2([Λ^{♯}(df), Λ^{♯}(dg)] − Λ^{♯}(d{f, g})).h .

Proposition 1.3 follows easily from this formula.

Let Λ_{0} and Λ_{1} be two bivectors on the smooth manifold M. For all f and g ∈ C^{∞}(M, R), we denote by {f, g}_{0} = Λ_{0}(df, dg) and {f, g}_{1} = Λ_{1}(df, dg) the brackets defined by Λ_{0} and Λ_{1}, respectively. We set

X_{f} = Λ_{0}(df) , Y_{f} = Λ_{1}(df) .

Then by a calculation similar to that above (or by polarizing the above formulas) we have, for all f, g and h ∈ C^{∞}(M, R),

[Λ_{0}, Λ_{1}](df, dg, dh) = {f, {g, h}_{0}}_{1} + {g, {h, f}_{0}}_{1} + {h, {f, g}_{0}}_{1}
+ {f, {g, h}_{1}}_{0} + {g, {h, f}_{1}}_{0} + {h, {f, g}_{1}}_{0}
= ([X_{f}, Y_{g}] + [Y_{f}, X_{g}] − X_{{f,g}_{1}} − Y_{{f,g}_{0}}).h .

We can also write

[Λ_{0}, Λ_{1}](df, dg, dh) = {f, {g, h}_{0}}_{1} + {g, {h, f}_{0}}_{1} + {h, {f, g}_{0}}_{1}
+ {f, {g, h}_{1}}_{0} + {g, {h, f}_{1}}_{0} + {h, {f, g}_{1}}_{0}
= Y_{f}.{g, h}_{0} − {g, Y_{f}.h}_{0} − {Y_{f}.g, h}_{0}
+ X_{f}.{g, h}_{1} − {g, X_{f}.h}_{1} − {X_{f}.g, h}_{1}
= (L(Y_{f})Λ_{0} + L(X_{f})Λ_{1})(dg, dh) .

These formulas will be used in the next section.

2. Bihamiltonian manifolds

2.1. Definition. Two linearly independent Poisson tensors Λ_{0} and Λ_{1} defined on the same smooth manifold M are said to be compatible when Λ_{0} + Λ_{1} is again a Poisson tensor. In such a case the manifold M, equipped with the two Poisson tensors Λ_{0} and Λ_{1}, is called a bihamiltonian manifold, and denoted by (M, Λ_{0}, Λ_{1}).

2.2. Proposition. Let M be a smooth manifold equipped with two linearly independent Poisson tensors Λ_{0} and Λ_{1}. With the notations indicated at the end of the preceding section, the following properties are equivalent:

(i) the Poisson tensors Λ_{0} and Λ_{1} are compatible; (ii) the Schouten-Nijenhuis bracket [Λ_{0}, Λ_{1}] vanishes; (iii) for all f, g and h ∈ C^{∞}(M, R),

\begin{matrix}\{f, \{g, h\}_{0}\}_{1} + \{g, \{h, f\}_{0}\}_{1} + \{h, \{f, g\}_{0}\}_{1} \\ + \{f, \{g, h\}_{1}\}_{0} + \{g, \{h, f\}_{1}\}_{0} + \{h, \{f, g\}_{1}\}_{0} = 0 ; \end{matrix}

(iv) for all f and g ∈ C^{∞}(M, R),

[X_{f}, Y_{g}] + [Y_{f}, X_{g}] − X_{{f,g}_{1}} − Y_{{f,g}_{0}} = 0 ;

(v) for all f ∈ C^{∞}(M, R),

L(Y_{f})Λ_{0} + L(X_{f})Λ_{1} = 0 .

When these properties are satisfied, any two linearly independent linear combinations of Λ_{0} and Λ_{1} are compatible Poisson structures.

Proof. Taking into account the bilinearity of the Schouten-Nijenhuis bracket, it follows immediately from Proposition 1.3 and the formulas in 1.5.

The importance of bihamiltonian manifolds arises from the fact that many completely integrable systems are vector fields on a bihamiltonian manifold, which are Hamiltonian with respect to both Poisson structures. Euler’s equations, which govern the motion of a rigid body with a fixed point, are of that type.

2.3. Example: Euler’s equations for the rigid body. Let us assume for simplicity that the three principal moments of inertia I_{1}, I_{2} and I_{3} of the body, at its fixed point, are strictly positive and distinct. Euler’s equation are

\begin{aligned}\frac{dx_{1}}{dt} & = (I_{2}^{−1} − I_{3}^{−1})x_{2}x_{3} , \\ \frac{dx_{2}}{dt} & = (I_{3}^{−1} − I_{1}^{−1})x_{3}x_{1} , \\ \frac{dx_{3}}{dt} & = (I_{1}^{−1} − I_{2}^{−1})x_{1}x_{2} . \end{aligned}

We have denoted by x_{1}, x_{2}, x_{3} the components of the angular momentum of the body at its fixed point, in the inertial orthogonal frame, attached to the moving body, made by the eigenvectors of the inertia operator. We set:

\begin{aligned}Λ_{0} & = − (x_{1} \frac{∂}{∂x_{2}} ∧ \frac{∂}{∂x_{3}} + x_{2} \frac{∂}{∂x_{3}} ∧ \frac{∂}{∂x_{1}} + x_{3} \frac{∂}{∂x_{1}} ∧ \frac{∂}{∂x_{2}} ) , \\ Λ_{1} & = I_{1}^{−1}x_{1} \frac{∂}{∂x_{2}} ∧ \frac{∂}{∂x_{3}} + I_{2}^{−1}x_{2} \frac{∂}{∂x_{3}} ∧ \frac{∂}{∂x_{1}} + I_{3}^{−1}x_{3} \frac{∂}{∂x_{1}} ∧ \frac{∂}{∂x_{2}} , \\ H & = \frac{1}{2}(I_{1}^{−1}x_{1}^{2} + I_{2}^{−1}x_{2}^{2} + I_{3}^{−1}x_{3}^{2}) , \\ L & = \frac{1}{2}(x_{1}^{2} + x_{2}^{2} + x_{3}^{2}) . \end{aligned}

We observe that

Λ_{0}^{♯}(dH) = Λ_{1}^{♯}(dL) ,

and that the differential equations defined by that vector field, which is Hamiltonian with respect to both Λ_{0} and Λ_{1}, are Euler’s equations. In [4], Holm and Marsden have used the fact that Euler’s equations are Hamiltonian with respect to any nonvanishing linear combination of Λ_{0} and Λ_{1} to derive remarkable properties of that completely integrable system.

3. Poisson-Nijenhuis manifolds

3.1. The Nijenhuis torsion. Let N be a tensor field of type (1, 1) (one time contravariant and one time covariant) on a smooth manifold M. We shall consider N : T M → T M as a vector bundle map of the tangent bundle T M into itself, and we shall denote by ^{t}N : T^{∗}M → T^{∗}M the transpose map. Nijenhuis [12] has shown that by setting, for any pair (X, Y) of vector fields on M,

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

we define a vector bundle map T (N) : T M ⊕ T M → T M or, in other words, a tensor field T (N) of type (1, 2) (one time contravariant and two-times covariant), called the Nijenhuis torsion of N.

3.2. The recursion operator. Let Λ_{0} be a Poisson tensor and N a tensor field of type (1, 1) on a smooth manifold M. With the same conventions as above, we assume that

N ∘ Λ_{0}^{♯} = Λ_{0}^{♯} ∘ ^{t}N ,

and we set

Λ_{1}^{♯} = N ∘ Λ_{0}^{♯} = Λ_{0}^{♯} ∘ ^{t}N .

The bundle map Λ_{1}^{♯} : T^{∗}M → T M is associated with a 2-times contravariant skew-symmetric tensor field Λ_{1}. For any smooth functions f and g on M, we set

X_{f} = Λ_{0}^{♯}(df) , Y_{f} = Λ_{1}^{♯}(df) = NX_{f} ,

and similarly

{f, g}_{0} = Λ_{0}(df, dg) = X_{f}.g , {f, g}_{1} = Λ_{1}(df, dg) = Y_{f}.g .

As seen in 1.5, Λ_{1} is a Poisson tensor if and only if, for any pair (f, g) of smooth functions on M,

[NX_{f}, NX_{g}] = NX_{{f,g}_{1}} .

Similarly, the Schouten-Nijenhuis bracket [Λ_{1}, Λ_{0}] vanishes if and only if, for any pair (f, g) of smooth functions on M,

[NX_{f}, X_{g}] + [X_{f}, NX_{g}] − X_{{f,g}_{1}} − NX_{{f,g}_{0}} = 0 .

Since Λ_{0} is a Poisson tensor, we have

X_{{f,g}_{0}} = [X_{f}, X_{g}] ,

therefore [Λ_{1}, Λ_{0}] vanishes if and only if, for any pair (f, g) of smooth functions on M,

X_{{f,g}_{1}} = [NX_{f}, X_{g}] + [X_{f}, NX_{g}] − N[X_{f}, X_{g}] .

Therefore, when Λ_{1} is a Poisson tensor compatible with Λ_{0} we have, for any pair (f, g) of smooth functions on M,

[NX_{f}, NX_{g}] − N([NX_{f}, X_{g}] + [X_{f}, NX_{g}] − N[X_{f}, X_{g}]) = 0 .

Using the expression of the Nijenhuis torsion of N, we may state:

3.3. Proposition. Let Λ_{0} be a Poisson tensor and N a tensor field of type (1, 1) on a smooth manifold M, such that

NΛ_{0}^{♯} = Λ_{0}^{♯} ^{t}N .

We assume that the tensor field Λ_{1} associated with the vector bundle map Λ_{1}^{♯} = NΛ_{0}^{♯} satisfies

[Λ_{1}, Λ_{0}] = 0 .

Then Λ_{1} is a Poisson tensor if and only if the Nijenhuis torsion T (N) vanishes on the image Λ_{0}^{♯}(T^{∗}M) of Λ_{0}^{♯}. In such a case, (M, Λ_{0}, Λ_{1} = NΛ_{0}) is a bihamiltonian manifold, called a Poisson-Nijenhuis manifold, and the tensor field N is called its recursion operator.

3.4. The Magri-Morosi concomitant. In what follows, M is a smooth manifold and, for any p-times contravariant skew-symmetric tensor field P ∈ A^{p}(M), and any family (α_{1}, …, α_{p−1}) of p − 1 Pfaff forms on M, we will denote by P (α_{1}, …, α_{p−1}) the unique vector field on M such that, for any Pfaff form α_{p} on M,

⟨α_{p}, P (α_{1}, …, α_{p−1})⟩ = P (α_{1}, …, α_{p−1}, α_{p}) .

In particular, for p = 2, we will write

P (α) = P^{♯}(α) .

Let P ∈ A^{2}(M) be a skew-symmetric, two-times contravariant tensor field, and N a tensor field of type (1, 1) on the smooth manifold M. We assume that

NP^{♯} = P^{♯} ^{t}N .

Magri and Morosi [9] have shown that with P and N one can construct a new tensor field R(P, N) of type (2, 1), by setting, for any pair of a 1-form α and a vector field X,

R(P, N)(α, X) = (L(P^{♯}α)N)X − P^{♯}(L(X)(^{t}Nα)) + P^{♯}(L(NX)α) .

We will call R(P, N) the Magri-Morosi concomitant of P and N. It satisfies the following properties [9], which link it with the Nijenhuis torsion and the Schouten bracket:

(i) Let NP be the skew-symmetric two-times contravariant tensor field associated with the vector bundle map NP^{♯}. It satisfies N(NP)^{♯} = (NP)^{♯} ^{t}N, and for any 1-form α and any vector field X,

R(NP, N)(α, X) = NR(P, N)(α, X) + T (N)(X, P^{♯}α) .

(ii) For any pair (α, β) of 1-forms on M,

[NP, NP](α, β) = N([P, P](^{t}Nα, β) − 2R(P, N)(α, P β)) + 2T (N)(P α, P β) .

(iii) Let P and Q ∈ A^{2}(M) be two 2-times contravariant skew-symmetric tensor fields and N a tensor field of type (1, 1) such that

NP^{♯} = P^{♯} ^{t}N , NQ^{♯} = Q^{♯}^{t}N .

Then, for any pair (α, β) of Pfaff forms on M,

\begin{aligned}[NP, Q](α, β) & = N[P, Q](α, β) − QR^{∗}(P, N)(α, β) \\ & − R(Q, N)(β, P α) + R(Q, N)(α, P β) , \end{aligned}

where the transpose R^{∗}(P, N) of R(P, N) is defined by the formula, in which α and β are any Pfaff forms and X any vector field,

⟨R^{∗}(P, N)(α, β), X⟩ = ⟨β, R(P, N)(α, X)⟩ .

3.5. Remark. Let us indicate another useful formula involving the Schouten-Nijenhuis bracket and the Nijenhuis torsion, but not the Magri-Morosi concomitant. We shall use the same conventions as in Section 3.4. Let P ∈ A^{2}(M) be a two-times contravariant skew-symmetric tensor field and N a tensor field of type (1, 1) such that

NP^{♯} = P^{♯} ^{t}N .

Then

[NP, NP] − 2N[NP, P] − N^{2}[P, P] = 2T (N) ∘ (P, P) ,

where T (N) ∘ (P, P) is the vector bundle map such that, for any pair (α, β) of Pfaff forms,

T (N) ∘ (P, P)(α, β) = T (N)(P α, P β) .

This formula can be easily proven by using results indicated in 1.5.

By using the above formulas, we can easily prove the following result, which is a slight generalization of a theorem due to Magri and Morosi [9]:

3.6. Theorem. Let (M, Λ_{0}, Λ_{1} = NΛ_{0}) be a Poisson-Nijenhuis manifold. For any integer k ≥ 1, we set

Λ_{k}^{♯} = N^{k} ∘ Λ_{0}^{♯} ,

and we denote by Λ_{k} the two-times contravariant skew-symmetric tensor field associated with the vector bundle map Λ_{k}^{♯}. Then, for all k ∈ N, the Λ_{k} are Poisson tensors and are pairwise compatible. In other words, for all k and l ∈ N,

[Λ_{k}, Λ_{l}] = 0 .

4. Reduction of bihamiltonian manifolds

The reduction theorem of Marsden and Ratiu for Poisson manifolds [11] can be easily extended to bihamiltonian manifolds, as was shown independently by F. Magri (personal communication) and one of the authors (J. M. N. da C.):

4.1. Theorem. Let (M, Λ_{0}, Λ_{1}) be a bihamiltonian manifold, S a submanifold of M, and D a vector sub-bundle of T_{S}M, which satisfy the following conditions:

(i) For any pair (F, G) of smooth functions on M whose differentials dF and dG, restricted to S, vanish on the sub-bundle D, the differentials d{F, G}_{0} and d{F, G}_{1} of the Poisson brackets of F and G, for each Poisson structure Λ_{0} and Λ_{1}, restricted to S, vanish on D.

(ii) Let D^{0} ⊂ T_{S}^{∗}M be the annihilator of D. Then

Λ_{0}^{♯}(D^{0}) ⊂ T S + D and Λ_{1}^{♯}(D^{0}) ⊂ T S + D .

(iii) The distribution T S ∩ D on S is completely integrable; the set Ṡ of leaves of the foliation defined on S by that distribution is a smooth manifold, and the canonical projection π : S → Ṡ is a submersion.

Then there exists on Ṡ a unique pair (Λ̇_{0}, Λ̇_{1}) of compatible Poisson structures, such that, for any pair (f, g) of smooth functions on Ṡ and any smooth extensions (to a neighbourhood of S in M) F of f ∘ π and G of g ∘ π whose differentials dF and dG, restricted to S, vanish on D,

{F, G}_{0} |_{S}= {f, g}_{0} ∘ π , {F, G}_{1} |_{S}= {f, g}_{1} ∘ π .

We will say that (Ṡ, Λ̇_{0}, Λ̇_{1}) is the reduced bihamiltonian manifold defined by S and D.

4.2. Remarks.

1. When (M, Λ_{0}, Λ_{1} = NΛ_{0}) is a Poisson-Nijenhuis manifold, condition (ii) of Theorem 4.1 may be replaced by the slightly more restrictive condition

Λ_{0}^{♯}(D^{0}) ⊂ T S + D and N(T S + D) ⊂ T S + D .

But the reduced bihamiltonian manifold (Ṡ, Λ̇_{0}, Λ̇_{1}) is not in general a Poisson-Nijenhuis manifold. However, in many cases the higher order Poisson tensors Λ_{k} = N^{k}Λ_{0} on M (k ≥ 1) induce on Ṡ higher order Poisson tensors Λ̇_{k}, which are pairwise compatible, although they are not deduced from Λ̇_{0} by repeated application of a recursion operator. We shall see an example of that construction in the next section.

2. Magri and Morosi [9, 10] considered the following situation: (M, Λ_{0}, Λ_{1}) is a bihamiltonian manifold, and S a symplectic leaf (see for example Weinstein [17], Vaisman [16] or [7]) of the Poisson manifold (M, Λ_{1}). The sub-bundle D of T_{S}M chosen here is

D = Λ_{0}^{♯}(ker Λ_{1}^{♯}) |_{S} .

Assuming that D ∩ T S is of constant rank, one can easily prove that D ∩ T S is completely integrable, and that conditions (i) and (ii) of Theorem 4.1 are satisfied. If in addition the set of leaves of the foliation defined on S by D ∩ T S is a smooth manifold, and if the canonical projection π : S → Ṡ is a submersion, Theorem 4.1 can be applied.

4.3. A simple case. In view of its application to the Toda lattice, we consider the following simple case. Let (M, Λ_{0}) be an 2n-dimensional Poisson manifold whose Poisson tensor Λ_{0} is everywhere of rank 2n; in other words, M is a symplectic manifold, equipped with the Poisson structure canonically associated with its symplectic structure. Let Λ_{1} be another Poisson tensor on M, which may not be compatible with Λ_{0}. Since Λ_{0}^{♯} : T^{∗}M → T M is an isomorphism, we may set

N = Λ_{1}^{♯} ∘ (Λ_{0}^{♯})^{−1} .

Then N is a tensor field of type (1, 1) which satisfies

Λ_{1}^{♯} = NΛ_{0}^{♯} = Λ_{0}^{♯} ^{t}N .

For any integer k ≥ 1, we set

Λ_{k}^{♯} = N^{k}Λ_{0}^{♯} ,

and we denote by Λ_{k} the 2-times contravariant, skew-symmetric tensor field associated with the vector bundle map Λ_{k}^{♯}. Let Z be a nowhere vanishing vector field on M such that

L(Z)Λ_{0} = 0 , L(Z)Λ_{1} = 0 .

It satisfies also

L(Z)N = 0 .

We assume that the set Ṁ of integral curves of Z is a smooth manifold and that the canonical projection π : M → Ṁ is a submersion. We can easily prove that there exists on Ṁ a pair (Λ̇_{0}, Λ̇_{1}) of Poisson structures, such that π : M → Ṁ is a Poisson map from (M, Λ_{0}) onto (Ṁ, Λ̇_{0}), and also from (M, Λ_{1}) onto (Ṁ, Λ̇_{1}).

Similarly, for any integer k ≥ 1, there exists on Ṁ a 2-times contravariant, skewsymmetric tensor field Λ̇_{k}, which is the direct image π_{∗}Λ_{k} of Λ_{k} by the map π.

4.4. Proposition. With the same notations and assumptions as in 4.3, let T (N) be the Nijenhuis torsion of N. We assume that, for any pair (α, β) of Pfaff forms on M,

T (N)(Λ_{1}^{♯}α, Λ_{1}^{♯}β) = 0 .

Then for all k ≥ 1, the Λ_{k} are Poisson tensors on M, and are pairwise compatible; in other words, for all k ≥ 1 and l ≥ 1, [Λ_{k}, Λ_{l}] = 0.

Proof. Let α be a Pfaff form and X a vector field on M. We use 3.4.(ii), with P = Λ_{0} and β = (Λ_{0}^{♯})^{−1}X. Since [Λ_{0}, Λ_{0}] = 0 and [Λ_{1}, Λ_{1}] = 0, we obtain

NR(Λ_{0}, N)(α, X) = T (N)(Λ_{0}^{♯}α, X) .  (∗)

Now we use 3.4.(iii) with P = Q = Λ_{1}. Since [Λ_{1}, Λ_{1}] = 0, we obtain for any pair (α, β) of Pfaff forms,

[Λ_{2}, Λ_{1}](α, β) = −Λ_{1}R^{∗}(Λ_{1}, N)(α, β) − R(Λ_{1}, N)(β, Λ_{1}^{♯}α) + R(Λ_{1}, N)(α, Λ_{1}^{♯}β) .

But by 3.4(i) and the formula (∗) above, since T (N)(X, Y) = −T (N)(Y, X),

R(Λ_{1}, N)(α, X) = NR(Λ_{0}, N)(α, X) + T (N)(X, Λ_{0}^{♯}α) = 0 .

Similarly,

⟨R^{∗}(Λ_{1}, N)(α, β), X⟩ = ⟨β, R(Λ_{1}, N)(α, X)⟩ = 0 .

Therefore [Λ_{2}, Λ_{1}] = 0. Now we use Proposition 3.3, in which we replace Λ_{0} by Λ_{1} and Λ_{1} by Λ_{2}, and we obtain the stated results.

4.5. Remarks.

1. Under the assumptions of Proposition 4.4, for all k ≥ 1, the images Λ̇_{k} = π_{∗}Λ_{k} of the Poisson tensors Λ_{k} by the submersion π are Poisson tensors on Ṁ, and they are pairwise compatible. This result remains true under assumptions slightly weaker than those made in that Proposition.

2. By using the formula indicated in 3.5, we can easily prove that the assumption of Proposition 4.4, that for any pair (α, β) of Pfaff forms on M,

T (N)(Λ_{1}^{♯}α, Λ_{1}^{♯}β) = 0 ,

is equivalent to the following assumption: for any triple (α, β, γ) of Pfaff forms on M,

[Λ_{1}, Λ_{0}](^{t}Nα, ^{t}Nβ, ^{t}Nγ) = 0 .

Proposition 4.4 does not say anything about the compatibility of the Λ_{k} with Λ_{0}, nor with the compatibility of the Λ̇_{k} with Λ̇_{0}. The next Proposition deals with that question.

4.6. Proposition. Under the assumptions of Proposition 4.4, we assume in addition that for any pair (α̇, β̇) of Pfaff forms on Ṁ and any vector field X on M,

\begin{aligned}[Λ_{1}, Λ_{0}](π^{∗}α̇, π^{∗}β̇) & = 0 , \\ R(Λ_{0}, N)(π^{∗}α̇, X) & = 0 , \\ T (N)(X, Λ_{1}^{♯}π^{∗}α̇) & = 0 . \end{aligned}

Then for any k ≥ 1, the Poisson tensor Λ̇_{k} on Ṁ is compatible with Λ̇_{0}.

Proof. Let (α̇, β̇) be a pair of Pfaff forms on Ṁ. We have seen in the proof of 4.4 that R(Λ_{1}, N) = 0. We can easily prove by induction on k that for all k ≥ 1, first (by using 3.4.(i)) that R(Λ_{k}, N)(π^{∗}α̇, X) = 0, second (by using 3.4.(iii)) that [Λ_{k}, Λ_{0}](π^{∗}α̇, π^{∗}β̇) = 0. This result implies that for all k ≥ 1, [Λ̇_{k}, Λ̇_{0}] = 0.

5. Application to the Toda lattice

5.1. The Toda lattice and the Flaschka transformation. On R^{2n} (coordinates q^{1}, …, q^{n}, p_{1}, …, p_{n}), let

Λ_{0} = ∑_{i=1}^{n} \frac{∂}{∂p_{i}} ∧ \frac{∂}{∂q^{i}}

be the canonical Poisson tensor. The Toda lattice is the Hamiltonian system (relative to Λ_{0}) with Hamiltonian

H = \frac{1}{2} ∑_{i=1}^{n} p_{i}^{2} + ∑_{i=1}^{n−1} exp(q_{i} − q_{i+1}) .

We set, for 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n,

a^{i} = \frac{1}{2} exp (\frac{1}{2}(q^{i} − q^{i+1})) , b_{j} = −\frac{1}{2}p_{j} .

The map π : R^{2n} → R^{2n−1}, (q^{1}, …, q^{n}, p_{1}, …, p_{n}) ↦ (a^{1}, …, a^{n−1}, b_{1}, …, b_{n}), is the Flaschka transformation [3]. The fibers of π are integral curves of the vector field

Z = ∑_{i=1}^{n} \frac{∂}{∂q^{i}} = Λ_{0}^{♯} (d ∑_{i=1}^{n} p_{i}) .

Since L(Z)Λ_{0} = 0, there is on the image of π a unique Poisson tensor Λ̇_{0} such that π is a Poisson map. This Poisson tensor, which extends to the whole R^{2n−1}, is

Λ̇_{0} = \frac{1}{4} ∑_{i=1}^{n−1} a^{i} \frac{∂}{∂a^{i}} ∧ (\frac{∂}{∂b_{i}} − \frac{∂}{∂b_{i+1}} ) .

The Hamiltonian H of the Toda lattice factors through π: it can be expressed as H = Ḣ ∘ π, with

Ḣ = 2 ∑_{i=1}^{n} b_{i}^{2} + 4 ∑_{i=1}^{n−1}(a^{i})^{2} ,

and the Toda vector field projects by π onto the Hamiltonian vector field T = Λ̇_{0}^{♯}(dḢ).

5.2. Higher order Poisson structures for the Toda lattice. The projection T of the Toda vector field on R^{2n−1} is Hamiltonian with respect to Λ̇_{0}; it is Hamiltonian for another Poisson tensor Λ̇_{1}, obtained by Adler [1]:

\begin{aligned}Λ̇_{1} & = −2 ∑_{i=1}^{n−1}(a^{i})^{2} \frac{∂}{∂b_{i}} ∧ \frac{∂}{∂b_{i+1}} \\ & + \frac{1}{2} ∑_{i=1}^{n−1} a^{i} \frac{∂}{∂a^{i}} ∧ (a^{i+1} \frac{∂}{∂a^{i+1}} + 2b_{i+1} \frac{∂}{∂b_{i+1}} − 2b_{i} \frac{∂}{∂b_{i}} ) , \end{aligned}

with, by convention, a^{n} = 0. We have indeed

T = Λ̇_{0}^{♯}(dḢ) = Λ̇_{1}^{♯}(dL̇) , with L̇ = ∑_{i=1}^{n} b_{i} .

Moreover, the Poisson tensors Λ̇_{1} and Λ̇_{0} are compatible: [Λ̇_{1}, Λ̇_{0}] = 0.

Kupershmidt [6] has found a third Poisson tensor Λ̇_{2} on R^{2n−1}, whose coefficients are polynomials of degree 3 in the coordinates a^{i} and b_{j}, compatible with Λ̇_{0} and Λ̇_{1}.

Damianou [2] has shown the existence of an infinite sequence (Λ̇_{k}), k ∈ N, of Poisson tensors on R^{2n−1}, the coefficients of Λ̇_{k} being polynomials of degree k + 1 in the coordinates a^{i}, b_{j}, the first terms Λ̇_{0}, Λ̇_{1}, Λ̇_{2} of that sequence being the Poisson tensors already obtained by Flaschka, Adler and Kupershmidt. Moreover the terms of that sequence are pairwise compatible. These Poisson tensors are not obtained from each other by repeated application of a recursion operator, and Damianou does not indicate an explicit formula for obtaining them.

Such an explicit formula can be obtained by going back to R^{2n} and using Proposition 4.4. First we set

z = \frac{1}{n} ∑_{i=1}^{n} q^{i} .

Then z, a^{1}, …, a^{n−1}, b_{1}, …, b_{n} can be used as (nonlinear) coordinates on R^{2n}. The Poisson tensors Λ̇_{0} and Λ̇_{1} on R^{2n−1} can now be considered as Poisson tensors on R^{2n}, whose expressions, in terms of these coordinates, are those given above,

which do not contain z nor \frac{∂}{∂z}. We have

Λ_{0} = Λ̇_{0} + Z ∧ Y ,

where Y is a vector field and Z the vector field already defined. Their expressions, in the new coordinates z, a^{i}, b_{j}, are

Y = \frac{1}{2n} ∑_{i=1}^{n} \frac{∂}{∂b_{i}} , Z = \frac{∂}{∂z} .

Although Λ_{0} and Λ̇_{1}, considered as Poisson tensors on R^{2n}, are not compatible, we can define a recursion operator N = Λ̇_{1}^{♯} ∘(Λ_{0}^{♯})^{−1}, and set, for any integer k ≥ 1, Λ_{k}^{♯} = N^{k} ∘ Λ_{0}^{♯}. Let Λ_{k} be the corresponding 2-times contravariant skew-symmetric tensor fields.

By using 4.5.2, we can easily prove that the torsion T (N) of the recursion operator N satisfies the condition of Proposition 4.4. Therefore, the Λ_{k}, for k ≥ 1, are pairwise compatible Poisson tensors. Their projections Λ̇_{k} by π are Poisson tensors on R^{2n−1}; the first terms Λ̇_{1} and Λ̇_{2} are equal to those found by Adler and Kupershmidt. We conjecture that the whole sequence (Λ̇_{k}), k ∈ N, obtained by that method, is the same as that obtained by Damianou (the non uniqueness of the terms in that sequence observed by Damianou corresponding to the various possible choices of a Poisson tensor Λ_{1} on R^{2n} which projects by π onto Λ̇_{1}).

6. References

[1] Adler, M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg de-Vries type equations, Invent. Math., 50, 1979, 219–248.

[2] Damianou, P. A., Master symmetries and R-matrices for the Toda lattice, Letters in Mathematical Physics, 20, 1990, 101–112.

[3] Flaschka, H., On the Toda lattice, Phys. Rev. B 9, 1974, 1924–1925.

[4] Holm, D. D., and Marsden, J. E., The rotor and the pendulum, in Symplectic geometry and mathematical physics, (P. Donato et al., eds), Birkhäuser, Boston, 1991, 189–203.

[5] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie, in Élie Cartan et les mathématiques d’aujourd’hui, Astérisque, numéro hors série, 1985, 257-271.

[6] Kupershmidt, B., Discrete Lax equatioons and differential difference calculus, Astérisque 123, 1985.

[7] Libermann, P., and Marle, C.-M., Symplectic geometry and analytical mechanics, D. Reidel Publishing Company, Dordrecht, 1987.

[8] Lichnerowicz, A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry, 12, 1977, 253–300.

[9] Magri, F., and Morosi, C., A geometric characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S 19 , 1984, Università di Milano.

[10] Magri, F., Geometry and soliton equations, in La “Mécanique analytique” de Lagrange et son héritage, Acta Academiae Scientiarum Taurinensis, Torino 1990, 181–209.

[11] Marsden, J. E., and Ratiu, T., Reduction of Poisson manifolds, Letters in Mathematical Physics, 11, 1986, 161–169.

[12] Nijenhuis, A., X_{n+1}-forming sets of eigenvectors, Proc. Kon. Ned. Akad. Wet. Amsterdam, A 54, 1951, 200–212.

[13] Nijenhuis, A., Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Math., 17, 1955, 390–403.

[14] Ouzilou, R., Hamiltonian actions on Poisson manifolds, in Symplectic geometry (A. Crumeyrolle and J. Grifone, eds), Research notes in mathematics 80, Pitman, Boston, 1983, 172–183.

[15] Schouten, J. A., On the differential operators of the first order in tensor calculus, in Convegno Int. Geom. Diff. Italia, 1953. Ed. Cremonese, Roma, 1954, 1–7.

[16] Vaisman, I., Lectures on the geometry of Poisson manifolds, Birkhäuser, Boston, 1994.

[17] Weinstein, A., The local structure of Poisson manifolds, J. Differential Geometry, 18, 1983, 523–557 and 22, 1985, 255.

Universidade de Coimbra, Departamento de Matemática Apartado 3008, 3000 Coimbra, Portugal

Université Pierre et Marie Curie, Institut de Mathématiques 4, Place Jussieu, 75252 Paris cedex 05, France

E–mails : jmcosta@mat.uc.pt, marle@mathp6.jussieu.fr