Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.


Introduction
Given a vector field v on M , finding a Poisson structure according to which the vector field becomes Hamiltonian may not be an easy task in general. However, if a given dynamical system can be put into Hamiltonian form then, there may be more than one Poisson structure which makes it into a Hamiltonian system. In [9], a bi-Hamiltonian system is introduced for the analysis of certain infinite-dimensional soliton equations. In such a case, there arises the question of the relation between these Poisson structures, which is called compatibility. Although there are at least two different approaches to compatibility [11], by following [10] we adapt the definitions below: Definition 1.1. A dynamical system is called bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways: v = J 1 (dH 2 ) = J 2 (dH 1 ), (1.3) such that J 1 and J 2 are nowhere multiples of each other. This bi-Hamiltonian structure is compatible if J 1 + J 2 is also a Poisson structure.
In this paper we confine ourselves to dynamical systems on three-dimensional orientable manifolds. For three-dimensional manifolds, where there is no symplectic structure for dimensional reasons, Poisson structures have a simple form. Poisson structures of dynamical systems on three manifolds are extensively studied first in [4] and then also in [5] and [8]. Following the definitions in [4], choosing any Riemannian metric g on M , a Poisson bivector field, which is skew-symmetric, can be associated to a vector field by using the Lie algebra isomorphism so(3) R 3 bi-Hamiltonian, {v, J 1 , J 2 } becomes a global frame field on M . For example, for M = R 3 and v = ∂ x 0 we have J i = ∂ x i and {∂ x 0 , ∂ x 1 , ∂ x 2 } forms such a global frame field. However, the global existence of the frame field {v, J 1 , J 2 } is by no means guaranteed. The simplest counterexample is the gradient flow of the S 2 in R 3 \ {0}. Here, a frame field {v, J 1 , J 2 } cannot be defined globally since J 1 , J 2 are sections of the tangent bundle of S 2 which is not trivial and does not admit two nonvanishing linearly independent vector fields.
The goal of this paper is to give necessary and sufficient conditions for a nonvanishing vector field on an orientable three-dimensional manifold to admit a compatible bi-Hamiltonian structure. The paper is organized as follows: In Section 2, the local existence of bi-Hamiltonian systems is investigated in a neighbourhood of a point, possibly refined by the existence conditions of solutions of certain ODE's related with the problem, and it is shown in Theorem 2.7 that it is always possible to find a pair of compatible Poisson structures such that the system defined by the nonvanishing vector field becomes bi-Hamiltonian. In Section 3, obstructions to the global existence of a pair of Poisson structures are studied. In Section 3.2 the primary obstruction for the existence of a global pair of Poisson structures is investigated, and it is shown in Theorem 3.6 that such a pair, which is not necessarily compatible, exists if and only if the first Chern class of the normal bundle vanishes. Finally, the global compatibility of this pair is investigated in Section 3.3 and it is shown in Theorem 3.8 that under the assumption of global existence, the vanishing of the Bott class of the complex codimension one foliation is the necessary and sufficient condition for the global compatibility of the pair of Poisson structures.
Throughout the work, bivectors are denoted by calligraphic and forms are denoted by bold letters.

Local existence of bi-Hamiltonian structure in 3D
For this purpose, we will first analyze the local solutions of the equation (1.4) defining Poisson vectors, which is also studied in [6]. Let M be an orientable three-dimensional manifold with an arbitrary Riemannian metric g, and v be a nonvanishing vector field. Let and extend this vector field to a local orthonormal frame field { e 1 , e 2 , e 3 }. Define the structure functions (C k ij (x)) via the relation Proof . Adopting the frame defined above and using (1.7), we have the Poisson vector field and its curl is given by Now the Jacobi identity (1.4) is obtained by taking the dot product of (2.2) with (2.3), and using triple vector product identities we get If J = 0 then v = 0 and hence v = 0, which contradicts with our assumption that the vector field is nonvanishing. Therefore, we assume J = 0, which means that α = 0 or β = 0. Now we assume α = 0, while the case β = 0 is similar and amounts to a rotation of the frame fields. Defining µ = β α and dividing (2.4) by α 2 , we get whose characteristic curve is the integral curve of (1.1) in arclength parametrization and in the arclength variable s. The Riccati equation (2.6) is equivalent to a linear second order equation and hence, possesses two linearly independent solutions leading to two Poisson vector fields. Since the vector field v is assumed to be nonvanishing, for each x 0 ∈ R 3 it is possible to find a neighborhood foliated by the integral curves of v which are nothing but characteristic curves of (2.5). Therefore, solutions of (2.6) can be extended to a possibly smaller neighborhood on which the Riccati equation has two independent solutions which we call µ i for i = 1, 2. Hence, we have two Poisson vector fields where the coefficients α i are arbitrary.
Note that, (2.5) determines µ i alone, but not α i . Taking the advantage of the freedom of choosing arbitrary scaling factors we may restrict these factors by imposing compatibility on our Poisson vector fields.
For the Poisson vector fields defined in (2.5), taking the dot product of both sides of (2.3) by J j , leads to Therefore, the compatibility condition (2.9) implies that and hence, we get whose characteristic curve is the solution curve of (1.1) in arclength parametrization By a similar line of reasoning as above, the solutions of (2.12) can also be extended to the whole neighborhood, and the proposition follows.
However, having a pair of Poisson structures obtained in (2.5) and even a compatible pair satisfying (2.11) do not guarantee the existence of Hamiltonian functions even locally.
Proof . For this purpose we first need to write down the equations for the Hamiltonian functions.
The invariance condition of Hamiltonian functions under the flow generated by v implies so the gradients of Hamiltonian functions are linear combinations of e 2 and e 3 . Then, inserting (2.7) into (1.5) we get another condition or by defining (2.15) can be written as Equations (2.14) and (2.16) for Hamiltonian functions are subject to the integrability condition Using the commutation relations given in (2.1) and (2.5), we obtain Applying H j to both sides of (2.17) and using two equations (2.14) and (2.16) for Hamiltonian functions, we get Therefore, our integrability condition for Hamiltonian functions becomes hence, and the proposition follows. Proof . What we need is to show that (2.8) is satisfied. Indeed, writing (2.18) for α i and α j and subtracting the second from the first, the corollary follows.
Note that, for a pair of compatible Poisson structures, J 1 and J 2 , the dilatation symmetry J → f J and the additive symmetry J 1 + J 2 do not imply that J 1 + f J 2 is a Poisson structure. Indeed, if we apply the Jacobi identity condition and using triple vector identity Now we try to describe the relation between the pair of compatible Poisson structures and Hamiltonian functions. But first, we need the following lemma to describe this relation.
Proof . Since Poisson vector fields are linearly independent, one could write Hamiltonians in terms of Poisson vector fields as By using (1.5), we get .
On the other hand, we have Taking the dot product of both sides with J 1 and J 2 , and using the compatibility condition, we obtain Inserting (2.13) into (2.10) and using (2.21), which leads to where e 1 · ∇Ψ i j = 0.
Therefore, we have Inserting (2.22) into (1.5), we get and finally, Note that, For the Hamiltonians to be functionally independent, r.h.s. of (2.23) must not vanish, i.e., 1 + Ψ 1 2 Ψ 2 1 = 0. Now let us define By (1.5), we get Choosing K i 's to be our new Poisson vector fields, the proposition follows.
Consequently, we can write the local existence theorem of bi-Hamiltonian systems in three dimensions.
Theorem 2.7. Any three-dimensional dynamical systeṁ has a pair of compatible Poisson structures in which µ i 's are determined by the equation and α i 's are determined by the equation Furthermore, (2.24) is a locally bi-Hamiltonian system with a pair of local Hamiltonian functions determined by 3 Global existence of compatible bi-Hamiltonian structure in 3D In this section, we investigate the conditions for which the local existence theorem holds globally.
To study the global properties of the vector field v by topological means, we relate the vector field with its normal bundle. Let E be the one-dimensional subbundle of T M generated by v. Let Q = T M/E be the normal bundle of v. By using the cross product with e 1 , we can define a complex structure Λ on the fibers of Q → M , and Q becomes a complex line bundle over M .

Bi-Hamiltonian structure in 3D with dif ferential forms
In order to obtain and express the obstructions to the global existence of bi-Hamiltonian structures on orientable three manifolds by certain cohomology groups and characteristic classes, we will reformulate the problem by using differential forms. For this purpose, let Ω be the volume form associated to the Riemannian metric g of M . Then, there is a local one-form J associated with a local Poisson bivector field J , which is called the local Poisson one-form. The bi-Hamiltonian system (1.5) can be written as Note that, although the l.h.s. of this equality is globally defined, r.h.s. is defined only locally, therefore it holds only locally. Now the Jacobi identity is given by and compatibility amounts to By (2.25), J 1 and J 2 can be chosen to be proportional to dH 1 and dH 2 , respectively, and hence (3.1) takes the form The Jacobi identity for Poisson 1-forms (3.2) implies the existence of 1-forms β i such that for each i = 1, 2. In the next proposition we are going to show that the compatibility of Poisson structures allows us to combine β 1 and β 2 into a single one. Proof . Applying (3.3) to the compatibility condition and therefore, we define Hence β ∧ J i = β i ∧ J i = dJ i , and the proposition follows.
Note that β is a T M -valued 1-form. Namely, ι e 1 β = 0 in general. Now we are going to show that by an appropriate change of Poisson forms, we may reduce it to a connection 1-form on Q.
where φ is the function defined in (2.26).
Proof . For the proof, we carry out the computation with Poisson vector fields, then transform the result to differential forms. The Jacobi identity (1.4) implies that ∇ × J i is orthogonal to J i and therefore, we get By the definition of Poisson vector fields, we have We can rewrite (3.4) in the form Using the compatibility condition (2.9), we obtain

and (3.5) becomes
After a bit of computation it is possible to show that Hence, we have e 1 · ξ = e 1 · ∇ ln φ and defining β = * ι ξ Ω, the lemma follows.
Taking the exterior derivatives of both sides and dividing both sides by φ, Now, by the lemma above, where γ is a connection on Q.

The f irst obstruction: the Chern class of Q
Now we try to find conditions for which a nonvanishing vector field v satisfies for some globally defined functions φ, H 1 and H 2 . For a two-form to be decomposed into the form (3.8), first of all, the two-form must be written as a product of two globally defined, linearly independent nonvanishing factors. However, such a decomposition may not exist globally. Then, the question is to decompose w into a product of two globally defined one forms ρ 1 and ρ 2 Since v is a nonvanishing vector field, w is a 2-form of constant rank 2. If we let S w to be the sub-bundle of T M on which w is of maximal rank, then we have S w ∼ = Q defined above.
The following theorem states the necessary and sufficient conditions for the decomposition of a two-form of constant rank 2s in the large. ii) The representation of its normalization as a map w 1 : M → SO(2s)/U(s) arising from any trivialization of S w lifts to SO(2s) [3].
In our case, when s = 1, since U(1) ∼ = SO(2), then SO(2)/U(1) is a point and it lifts to SO(2) trivially, therefore the second condition in the theorem is satisfied. Hence, the necessary and sufficient condition of decomposition is the triviality of S w ∼ = Q. Since Q is a complex line bundle, it is trivial if and only if c 1 (Q) = 0, or equivalently it has a global section. Since the decomposition of the 2-form w into globally defined 1-forms ρ 1 and ρ 2 is a necessary condition for the existence of a global bi-Hamiltonian structure, the vanishing of the first Chern class of Q becomes a necessary condition.
However, this may not be sufficient since the existence of a decomposition in the form (3.9) may not imply that the factors ρ i satisfy In order to determine the effect of vanishing Chern class on the constructions made so far, we are going to investigate the equation (2.5) defining the Poisson one-forms. Since our Poisson one-forms and related integrability conditions are determined by the local solutions of (2.5), they are defined locally on each chart. Let J p i and J q i be the Poisson vector fields in charts (U p , x p ) and (U q , x q ) around points p ∈ M and q ∈ M , respectively. Around the point p ∈ M , the Poisson vectors J p i are determined by µ p i , α p i and the local frame e p 2 , e p 3 . Given the local frame, we can write (2.5) whose solutions are µ p i 's, and using µ p i 's we can determine α p i 's by the equation (2.13). Now, if c 1 (Q) = 0, which is a necessary condition for the existence of global bi-Hamiltonian structure, then we have a global section of Q, i.e., global vector fields normal to v. By using the metric on M , normalize this global section of Q and take it as e 2 , then define e 3 = e 1 × e 2 . So we have the global orthonormal frame field { e 1 , e 2 , e 3 }. In order to understand the relation between local Poisson one-forms obtained in two different coordinate neighborhoods, we first need the following lemmas: Lemma 3.4. If two solutions µ 1 (s) and µ 2 (s) of the Riccati equation are known, then the general solution µ(s) is given by where K is an arbitrary constant [7]. for r = p, q. Therefore, on U p ∩ U q , µ p i and µ q i are four solutions of the same Riccati equation for i = 1, 2. By the lemma above we have Now, using the compatibility condition (2.8), where e 1 · ∇K pq i = 0. (3.12) Multiplying both sides by α p 1 α q i in (3.11), gives Rearranging (3.13), we obtain (3.12) and the compatibility, we can take to be our new Poisson vector fields on the neighborhood U p , and obtain J q i × J p i = 0. By compatibility these new Poisson vector fields J p i produce functionally dependent Hamiltonians and therefore, for the simplicity of notation, we will assume without restriction of generality that J p i = J p i and the lemma follows.
Then, we have the following result: Theorem 3.6. There exist two linearly independent global sections j i of Q satisfying if and only if c 1 (Q) = 0.
Proof . The forward part is trivial since the existence of a global section of the complex line bundle Q implies that Q is trivial, and hence c 1 (Q) vanishes. For the converse, we define and the lemma implies that j p i = j q i on U p ∩ U q and the theorem follows.
The lemma above states the reason why one may fail to extend a local pair of compatible Poisson vector fields into a global one, even if c 1 (Q) = 0. In order to do so one should have J q i = J p i on U p ∩ U q . However, not the Poisson vector fields but their unit vector fields can be globalized. Since in general, they may not lead to a pair of compatible Poisson structures. Now we take j 1 as our first global Poisson vector field, and check whether we can find another global Poisson vector field compatible with this one by rescaling j 2 .

Second obstruction: Bott class of the complex codimension 1 foliation
Since v is a nonvanishing vector field on M , it defines a real codimension two foliation on M by orbits of v. Since Q = T M/E is a complex line bundle on M , this foliation has complex codimension one. Now, by assuming our primary obstruction which is the vanishing of the Chern class, we compute the Bott class of the complex codimension one foliation as defined in [2], which is studied in detail in [1], and then show that the system admits two globally defined compatible Poisson structures if and only if the Bott Class is trivial. For the rest of our work, we will assume that Q and its dual Q * are trivial bundles. By (3.14), Q * has two global sections j i = ( * ı j i Ω) satisfying for globally defined Γ i 's. These j i 's are related with the local Poisson one-forms J p i by Redefining Γ i 's if necessary, comparing (3.15) with (3.18), we get Proof . Since j 1 and j 2 may not be compatible, we introduce a local Poisson form j p defined on the coordinate neighborhood U p of p ∈ M , which is compatible with j 1 and parallel to j 2 i.e., and Now (3.20) implies that Putting (3.15) and (3.22) into (3.21) and using (3.20), we get Our aim here is to find the obstruction to extending f p to M , or for (3.23) to hold globally. For this purpose, we consider the connections on Q defined by Γ i 's. By (3.19), we define the curvature of these connections to be κ = dΓ i = dγ p .
Taking the exterior derivative of (3.17) and using (3.16), we get dγ p ∧ J p i = dγ p ∧ j i = 0, which leads to κ = dγ p = ϕ j 1 ∧ j 2 . Since h 1 is globally defined, on U p ∩ U q we have h 1 = γ p + d ln h p = γ q + d ln h q and γ p − γ q = d ln h q h p .