Poisson Structures on Cotangent Bundles

We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poisson structure from M to T*M via connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure on T*M. Finally, for a 2-form on a Riemannian manifold, we study the conditions for some associated 2-forms on T*M to define Poisson structures on cotangent bundles.


Graded Poisson structures on cotangent bundles
Let M be an n−dimensional differentiable manifold and π : T * M −→ M its cotangent bundle. If (x i ), (i = 1, ..., n), are local coordinates on M, we denote by (p i ) the covector coordinates with respect to the cobasis (dx i ). (We assume that everything is C ∞ in this paper).
In this section we discuss graded Poisson structures W on the cotangent bundle T * M obtained as lifts of Poisson structures w on the base manifold M, in the sense that the canonical projection π is a Poisson mapping (see [6]).
Denote by S k (T M) the space of k−contravariant symmetric tensor fields on M and by ⊙ the symmetric tensor product on the algebra S(T M) = Hereafter, by a polynomial on T * M we always mean a fiberwise polynomial. Also, we write f for both f on M, and f • π on T * M. Proof. Any function f on M is a polynomial (f • π) ∈ P 0 (T * M). By (1.3), ∀f, g ∈ C ∞ (M), {f • π, g • π} W ∈ C ∞ (M) and (1.4) {f, g} w := {f • π, g • π} W , defines a Poisson structure w on M.
If the local coordinate expression of the Poisson structure w introduced by Proposition 1.2 is where w, ϕ, η, A, B, C are local functions on M.
The Poisson structure W is completely determined by the brackets {f, g}, {m(X), f } and {m(X), m(Y )}, where f, g ∈ C ∞ (M) and X, Y ∈ χ(M), since the local coordinates x i and p i are functions of this type (p i = m(∂/∂x i )).
where Z X f ∈ C ∞ (M) and γ X f ∈ χ(M). {m(X), . } is a derivation of C ∞ (M). Hence, Z X is a vector field on M, and the mapping γ X : C ∞ (M) −→ χ(M) also is a derivation. Therefore, γ X f depends only on df.
From the Leibniz rule we get that Z hX = hZ X (h ∈ C ∞ (M)) and γ must satisfy The bracket of two affine functions has an expression of the form , the Leibniz rule gives that β is a 2−form on M and Remark that a polynomially graded structure on T * M is graded iff Z X = 0, β = 0 and V = 0. In this case (1.6) reduces to As in [6], a bivector field W on T * M which is locally of the form (1.6) (respectively (1.11)) is called a polynomially graded (respectively graded) bivector field. PROPOSITION 1.4. If W is a graded bivector field on T * M which is π−related with a Poisson structure w on M, there exists a contravariant connection D on the Poisson manifold (M, w) such that Moreover, if W is a graded Poisson structure on T * M then the connection D is flat. Proof. A contravariant connection on (M, w) is a contravariant derivative on T M with respect to the Poisson structure [10].
The required connection is defined by That we really get a connection, which is flat in the Poisson case, follows in exactly the same way as in [6].
The relation (1.12) extends to PROPOSITION 1.5. If Q is a symmetric contravariant tensor field on M andQ is its corresponding polynomial then, for any graded Poisson bivector field W on T * M, one has Proof. D df of (1.14) is extended to S(T M) by where α 1 , ..., α k ∈ Ω 1 (M), and D df α is defined by We put and by a straightforward computation we get for {Q, f } and − (D df Q) the same local coordinate expression. (See [6] for the complete proof in the case of a symmetric covariant tensor field on M.) In order to discuss the next two Jacobi identities, let us make some remarks concerning the operator Ψ of (1.9), which is given in the case of a graded Poisson structure on T * M by With (1.13), the second relation (1.10) becomes and this allows us to derive the local coordinate expression of Ψ. If X = X i (∂/∂x i ) and Y = Y j (∂/∂x j ), we obtain Remark that Ψ : T M × T M −→ ⊙ 2 T M is a bidifferential operator of the first order. PROPOSITION 1.6. If we define an operator D df which acts on Ψ by the Jacobi identity has the equivalent form Proof. Using (1.12), (1.14) and (1.16) for Q = Ψ(X, Y ), (1.20) becomes (1.21).
We also find Concerning the Jacobi identity (putting indices between parentheses denotes that summation is on cyclic permutations of these indices) remark that one must have an operator Θ such that We get the formula and then, the local coordinate expression Using the operator Θ, the Jacobi identity (1.23) becomes and formula (1.21) holds. d) an operator Θ defined by (1.24), satisfying (1.27).
To give examples, we consider the following situation, similar to [6]. Let (M, w) be an n−dimensional Poisson manifold and suppose that its symplectic foliation S is contained in a regular foliation F on M, such that T F is a foliated bundle i.e., there are local bases {Y u } (u = 1, ..., p, p = rank F ) of T F with transition functions constant along the leaves of F . Consider a decomposition where νF is a complementary subbundle of T F , and F −adapted local coordinates (x a , y u ) (a = 1, ..., n − p) on M [9]. The Poisson bivector w has the form then their transition functions are constant along the leaves of F . Now, ∀α ∈ T * M, α = ζ a dx a + ε u β u and we may consider (x a , y u , ζ a , ε u ) as distinguished local coordinates on T * M. The transition function are To prove that W is graded, we also consider natural coordinates and show that the expression of W with respect to these coordinates becomes of the form (1.11) (see [6]).
There are some interesting particular cases of Proposition 1.8: a) w is a regular Poisson structure, and the bundle T S is a foliated bundle; in this case we may take F = S. b) S is contained in a regular foliation F which admits adapted local coordinates (x a , y u ) with local transition functions (F is a leaf-wise, locally affine, regular foliation.) In this case (∂/∂y u ) = v a v u (x)(∂/∂ỹ v ) and we may use the local vector fields Y u = ∂/∂y u . c) There exists a flat linear connection ∇ (possibly with torsion) on the Poisson manifold (M, w). In this case we may consider as leaves of F the connected components of M, and the local ∇−parallel vector fields have constant transition functions along these leaves. Therefore, we may take them as Y i (i = 1, ..., n).
In particular, we have the result of c) for a locally affine manifold M (where ∇ has no torsion), using as Y i local ∇−parallel vector fields, and also for a parallelizable manifold M (where we have global vector fields Y i ).
As a consequence, Proposition 2.8 holds for the Lie-Poisson structure [10] of any dual G * of a Lie algebra G, the graded Poisson structure being defined on T * G * = G * × G.

Graded bivector fields on cotangent bundles
In this section we will discuss graded bivector fields on a cotangent bundle T * M, which may be seen as lifts of a given Poisson structure w on M, that satisfy less restrictive existence conditions than in the case of graded Poisson structures.
Recall the following definition from [6]. Let F be an arbitrary regular foliation, with p−dimensional leaves, on an n−dimensional manifold N. We denote by C ∞ f ol (N) the space of foliated functions (the functions on N which are constant along the leaves of F ). A transversal Poisson structure of (N, F ) is a bivector field w on N such that is a Lie algebra bracket on C ∞ f ol (N). A bivector field w on N defines a transversal Poisson structure of (N, F ) iff [6] ( The cotangent bundle T * M of any manifold M has the vertical foliation F by fibers with the tangent distribution V := T F . Obviously, the set of foliated functions on T * M may be identified with C ∞ (M). PROPOSITION 2.1. Any polynomially graded bivector field W on T * M, which is π related with a Poisson structure of M is a transversal Poisson structure of (T * M, V ).
Proof. The local coordinate expression of W is of the form (1.6), and W is π−related with the bivector field w defined on M by the first term of (1.6). Then, (2.2) holds, because w is a Poisson bivector on M. In what follows, we will discuss some interesting classes of graded semi-Poisson structures of T * M. Then, we give a method to construct all the graded semi-Poisson bivector fields on T * M which induce the same Poisson structure w on the base manifold M.
Proof. If the local coordinate expression of w is (1.5), using (2.7) and the properties of <, > [1, 6] we get REMARK 2.5. The relation (2.10) provides us the expression of the operator Ψ W 1 associated to W 1 (see (1.16)): Now, instead of D we consider a linear connection ∇ on a Poisson manifold (M, w) and define the vector field K on T * M by where ♯ w : T * M −→ T M is defined by β(α ♯ ) = w(α, β), ∀β ∈ Ω 1 (M), and the upper index H denotes the horizontal lift with respect to ∇ ( [5,11]). In local coordinates we get (2.14) On T * M we have the canonical symplectic form ω = dλ = dp i ∧ dx i where λ = p i dx i is the Liouville form, and the vector bundle isomorphism and locally one has defines a graded semi-Poisson structure on T * M which is π−related with w. Proof. We get where ∇ j w ai are the components of the (2, 1)−tensor field on M defined by X → ∇ X w, X ∈ χ(M).
We will say that W 2 of (2.17) is the graded ∇−lift of the Poisson structure w of M.
Using local coordinates and the notation of (1.2) we get where D is the contravariant derivative induced by the linear connection ∇, defined by D df = ∇ (df ) ♯ (see [10]). From (2.17) we have (see [1,6]) we get the explicit formula PROPOSITION 2.7. The graded ∇−lift W 2 of w is characterized by: i) the Poisson structure induced on M by W 2 is w, i.e.
iii) for any vector fields X and Y of M we have Proof. i) If f ∈ C ∞ (M) then Df = −X w f and from (2.20), (2.21) and the formula ii) As W 2 is graded, the bracket {m(X), f } W 2 must be of the form (2.24). DenotingD (Γ i jk are the coefficients of the linear connection ∇) and hence (2.25).
iii) (2.26) is a direct consequence of (2.22). Notice from (2.26) that the operator Ψ W 2 associated to W 2 has the same expression as Ψ W 2 of (2.12), but in the case of W 1 the contravariant derivative D is induced by a linear connection ∇ on M.
PROPOSITION 2.8. If the graded semi-Poisson structure W 1 is defined by a linear connection on (M, w) then it coincides with W 2 iff w is ∇−parallel.
We will prove now PROPOSITION 2.9. Let (M, w) be a Poisson manifold and π : T * M −→ M its cotangent bundle. The graded semi-Poisson structures W on T * M which are π−related with w are defined by the relations where D is an arbitrary contravariant connection of (M, w) and the operator Ψ is given by where Ψ 0 is the operator Ψ of a fixed graded semi-Poisson structure, A : T M ×T M −→ ⊙ 2 T M is a skew-symmetric, first order, bidifferential operator such that where τ is a (2, 1)−tensor field on M, and T is a (2, 2)−tensor field on M with the properties T (Y, Proof. If two graded semi-Poisson bivector fields, π−related with w, have associated the same contravariant connection D, it follows from (1.17) that the difference Ψ ′ −Ψ is a tensor field T, as in Proposition. To change D means to pass to a contravariant connection D ′ = D + τ, where τ is a (2, 1)−tensor field on M and from (1.17) again, it follows that A = Ψ ′ − Ψ becomes a bidifferential operator with the property (2.29).

Horizontal lifts of Poisson structures
In this section we define and study an interesting class of semi-Poisson structures on T * M which are produced by a process of horizontal lifting of Poisson structures from M to T * M via connections.
On T * M we distinguish the vertical distribution V, tangent to the fibers of the projection π and, by complementing V by a distribution H, called horizontal, we define a nonlinear connection on T * M [7,8].
We have (adapted) bases of the form and N ij are the coefficients of the connection defined by H. Equivalently, a nonlinear connection may be seen as an almost product structure Γ on T * M such that the eigendistribution corresponding to the eigenvalue −1 is the vertical distribution V [7].
We assume that the nonlinear connection above is symmetric, i.e., N ji = N ij . This condition is independent [7] on the local coordinates.
The complete integrability of H, in the sense of the Frobenius theorem, is equivalent to the vanishing of the curvature tensor field For a later utilization, we also notice the formulas [7,8] Let w be a bivector on M, with the local coordinate expression (1.5). DEFINITION 3.1. The horizontal lift of w to the cotangent bundle T * M is the (global) bivector field w H defined by Proof. In this case the coefficients of Γ are where Γ k ij are the coefficients of ∇ and, with respect to the bases {∂/∂x i , ∂/∂p j }, the local expression of w H becomes R where X H f denotes the usual horizontal lift [5,11], from M to T * M, of the w−Hamiltonian vector field X f on M.
In this case, the projection π : (T * M, w H ) −→ (M, w) is a Poisson mapping.
Proof. We compute the bracket [w H , w H ] with respect to the bases (3.1) and get that the Poisson condition [w H , w H ] = 0 is equivalent with the pair of conditions (Putting indices between parentheses denotes that summation is on cyclic permutations of these indices.) The first condition (3.8) is equivalent to [w, w] = 0 and the second is the local coordinate expression of (3.7). Notice that the condition (3.7) has the equivalent form  (or, equivalently This is equivalent to C D = 0. In the case where w H is a Poisson bivector, it is interesting to study its compatibility with the canonical Poisson structure W 0 of (2.15). PROPOSITION 3.6. If w H is a Poisson bivector, then it is compatible with W 0 iff Proof. By a straightforward computation we get that the compatibility condition [w H , W ] = 0 is equivalent to (3.12).
The Bianchi identity [7] (3.13) shows that the second relation (3.12) implies (3.7). Then COROLLARY 3.7. If (M, w) is a Poisson manifold and the cotangent bundle T * M is endowed with a symmetric nonlinear connection, then w H is a Poisson bivector on T * M compatible with W 0 iff conditions (3.12) hold. REMARK 3.8. Considering the isomorphism where u ∈ T * M and H * u is the dual space of H u , the second condition (3.12) may be written in the equivalent form (3.14) [Ψ(R(X, Y ))](♯ w α) H = 0 , ∀X, Y ∈ χ(T * M), ∀α ∈ Ω 1 (M) .
We recall that a symmetric linear connection ∇ on a Poisson manifold (M, w) is called a Poisson connection if ∇w = 0. Such connections exists iff w is regular, i.e. rank w = const (see [10]). PROPOSITION 3.9. Let (M, w) be a regular Poisson manifold with a Poisson connection ∇. Then, the bivector w H , defined with respect to ∇, is a Poisson structure on T * M compatible with the canonical Poisson structure W 0 iff the 2−form vanishes for every Pfaff form α on M.
Proof. With (3.5), the first condition (3.12) becomes ∇w = 0, which we took as a hypotheses. The second condition (3.12) becomes w ih R l hjk = 0 , and we get the required conditions. REMARK 3.10. If w is defined by a symplectic structure of M then (3.15) means R ∇ = 0.

Poisson structures derived from differential forms
If ω is a 2−form on a Riemannian manifold (M, g) we associate with it a 2−form Θ(ω) on the cotangent bundle π : T * M −→ M, and considering (pseudo-) Riemannian metrics on T * M related to g, we study the conditions for Θ(ω) to produce a Poisson structure on this bundle. Let (M, g) be a n−dimensional manifold and ∇ its Levi-Civita connection. If Γ k ij are the local coefficients of ∇, a connection Γ with the coefficients (3.5) is obtained on T * M.
The components of the curvature form are given by (3.2). Since the connection is symmetric, the Bianchi identity (3.13) holds. The elements Φ k ij of (3.3) are The Riemannian metric g provides the "musical" isomorphism ♯ g : T * M −→ T M and the codifferential and (g st ) are the entries of the inverse of the matrix (g ij ) [10].
where λ is the Liouville form, is said to be the associated 2−form of ω.
With respect to the cobases (dx i , δp i ) we get Now, we consider two (pseodo-) Riemannian metrics G 1 and G 2 on T * M and study the conditions for the bivectors W i = ♯ G i Θ(ω), (i = 1, 2) to define Poisson structures on T * M. The Poisson condition [W i , W i ] = 0, i = 1, 2 is equivalent to [10] (4.6) First, consider [7,8] the pseudo-Riemannian metric G 1 of signature (n, n) To find the condition which ensure that (4.6) holds, we need the local expression of the codifferential δ G 1 of G 1 . Denote by∇ the Levi-Civita connection of G 1 , and for simplicity we put PROPOSITION 4.2. The bivector ♯ G 1 Θ(ω) defines a Poisson structure on the cotangent bundle T * M iff ω is a closed 2−form on M and Γ a ai = 0, ∀i = 1, ..., n. In this case Θ(ω) is a symplectic form.
Let us consider now the Riemannian metric of Sasaki type (4.13) G 2 = g ij dx i ⊙ dx j + g ij δp i ⊙ δp j (see [2] for the Sasaki metric). LEMMA 4.3. The local coordinate expression of the Levi-Civita connec-tion∇ of G 2 is where we used again the notations of (4.8) and R jk i (also R i k j ) are obtained from R kij by the operation of lifting the indices, i.e. R jk i = g ja g kb R abi , R i k j = g ia g kb R ajb .
Proof. The result is proved by a straightforward computation. PROPOSITION 4.4. The bivector δ G 2 Θ(ω) defines a Poisson structure on the cotangent bundle T * M iff (4.15) ∇ω = 0 , g ab R k abi = 0 , ω ab R k iab = 0 , where ω ab = g ai g bj ω ij are the components of the bivector w = ♯ g ω on M.
Identifying the coefficients, the Poisson conditions (4.6) for W 2 becomes:  Let us remark that the conditions (4.17) implies (4.16), because, if ∇ω = 0 then ∇ a ω ij = 0, and g ab R k abi = 0 implies g ab ω ij R h abk = 0. REMARK 4.5. If the bivector ♯ G 2 Θ(ω) defines a Poisson structure on T * M then w = ♯ g ω defines a Poisson structure on M, as the second condition (4.16) is equivalent to the Poisson condition [10] (i,j,k) w ia ∇ a w jk = 0 .
(The local coordinate expression of w is (1.5).) COROLLARY 4.6. If ♯ G 2 Θ(ω) is a Poisson bivector on T * M, then the scalar curvature r of (M, g) vanishes.
Proof: The expression of r is r = g ab R ab , where R ba = R k akb = R ab are the components of the Ricci tensor, and if we make the contraction k = i in the second relation (4.15) we get g ab R k akb = 0, and whence r = 0.