Global Existence of Bi-Hamiltonian Structures on Orientable Three ...

arXiv:1612.06996v1 [math.SG] 21 Dec 2016

Global Existence of Bi-Hamiltonian Structures on Orientable Three Dimensional Manifolds ˙ sim Efe, E. Abado˘glu M. I¸ December 22, 2016 Abstract In this work, we showed 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 Chern class of the normal bundle of the given vector field vanishes. Furthermore, 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.

MATH. SUBJECT CLASSIFICATION: 53D17, 53D35 KEYWORDS: Bi-Hamiltonian systems, Chern class, Bott class.

1

Introduction

An autonomous dynamical system on a manifold M ·

x (t) = v (x (t))

(1)

is determined by a vector field v (x) on a manifold up to time reparametrization. Most basic geometric quantities, I, related to a dynamical system are functions invariant under the flow of the vector field. Lv I = iv dI = v · ∇I = 0

(2)

Given such a function there exists a skew-symmetric linear transformation JI : Λ1 (M ) → X(M )

(3)

v = JI (dI)

(4)

such that The invariants satisfying also the Jacobi identity condition, [JI , JI ]SN = 0

(5)

where [, ]SN is the Schouten-Nijenhuis bracket, are called Hamiltonian functions and (3) is called the Poisson bivector field on M defined by (1). The local structure of such manifolds is first introduced in [1]. Since a dynamical system may 1

admit more than one invariant, there may be more than one Poisson structure related with a dynamical system. In [2], 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 [3], by following [4] we adapt the definitions below: Definition 1 A dynamical system is called bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways: v = J1 (dH2 ) = J2 (dH1 )

(6)

such that J1 and J2 are not multiples of each other. This bi-Hamiltonian structure is compatible if the J1 + J2 is also a Poisson structure. For three dimensional manifolds, which is the simplest nontrivial example without a symplectic structure on it, Poisson structures have a simpler form. Poisson structures of dynamical systems on three manifolds are extensively studied in [5] and also in [6] and [7]. Following the definitions in [5], a Poisson bivector field, which is skew-symmetric, can be associated with a vector field, which is called the Poisson vector field on M by using the Lie algebra isomorphism so (3) ≃ R3 . J = Jk ∂xk (7) and the Jacobi identity has the form J · (∇ × J) = 0

(8)

and the bi-Hamiltonian form (6) becomes v = J1 × ∇H2 = J2 × ∇H1

(9)

Since J1 and J2 are not multiples of each other by definition, J1 × J2 6= 0

(10)

Ji · v = 0

(11)

and by the bi-Hamiltonian form (6), {v, J1 , J2 } form a local frame field on the support of v. This work is focused on bi-Hamiltonian structure of dynamical systems defined by nonvanishing vector fields on an orientable three manifold, or equivalently vector fields on three manifolds whose support is an orientable three dimensional manifold. Since all orientable three dimensional manifolds are parallelizable, there are no topological obstructions for the global existence of a nonvanishing vector field. However, the global existence of the frame field {v, J1 , J2 } is by no means guaranteed. The simplest example is the gradient flow of the S 2 in R3 − {0}. Here, a frame field can not be defined globally since S 2 is not parallelizable. The goal of this paper is to give necessary and sufficient conditions for a nonvanishing vector field on a three manifold to admit a compatible bi-Hamiltonian structure. For this purpose, we will first analyze the local solutions of the equation (8) defining Poisson vectors, which is also studied in [8]. 2

2

Local Existence of bi-Hamiltonian Structure in 3D

Let M be an orientable three dimensional manifold, and v be a non-vanishing vector field. Let v eb1 = (12) kvk and extend this vector field to a local orthonormal frame field {b e1 , eb2 , eb3 }. Define
k the structure functions Cij (x) via the relation k [b ei , ebj ] = Cij (x) ebk

(13)

Now, we have the following proposition.

Proposition 2 A non-vanishing vector field v defines two local Poisson structures on M . Proof. Adopting the frames defined above and using (11) we have the Poisson vector field J = αb e2 + βb e3 (14) and its curl is given by ∇ × J = ∇α × b e2 + α∇ × eb2 + ∇β × eb3 + β∇ × eb3

(15)

Now, the Jacobi identity (8) obtained by taking the dot product of (14) with (15), and using triple vector product identities we get
3 2 2 3 =0 (16) − β 2 C12 − αβ C31 + C12 βb e1 · ∇α − αb e1 · ∇β − α2 C31

If J = 0 then kvk = 0 and hence v = 0 which contradicts with our assumption that the vector field is non-vanishing, therefore we assume J 6= 0

(17)

which means that α 6= 0 or β 6= 0. Now we assume α 6= 0, while the case β 6= 0 is similar and amounts to rotation of the frame fields. Defining µ=

β α

(18)

and dividing (16) by α2 we get
3 2 3 2 − µ2 C12 e1 · ∇µ = −C31 b − µ C31 + C12

(19)

whose characteristic curve is the integral curve of (1) in arclenght parametrization and
dµ 3 2 3 2 − µ2 C12 (20) = −C31 − µ C31 + C12 ds 3

in the arclenght variable s. The Riccati equation (20) is equivalent to a linear second order equation and hence possesses two linearly independent solutions leading to two Poisson vector fields for dynamical system under consideration. Since the vector field v is assumed to be non-vanishing, then for each x0 ∈ R3 it is possible to find a neighborhood foliated by the integral curves of v which are nothing but characteristic curves of (19). Therefore the solutions of (20) can be extended to a possibly smaller neighborhood on which the Riccati equation has two independent solutions which we call µi for i = 1, 2. Hence we have two Poisson vector fields Ji = αi (b e2 + µi eb3 ) (21) where the coefficients αi are arbitrary. Note that (19) determines µi alone, but not αi . Taking the advantage of the freedom of choosing arbitrary scaling factors we may restrict these factors by imposing compatibility on our Poisson vector fields. Proposition 3 Poisson structures obtained in (19) are compatible if e1 · ∇ ln b

Proof. Let


αi 3 µi − µj = C12 αj

(22)

J = J1 + J2

(23)

(∇ × J) · J = (∇ × J2 ) · J1 + (∇ × J1 ) · J2 = 0

(24)

Using (8) for J1 , J2 and J

For the Poisson vector fields defined in (19), taking the dot product of both sides of (15) by Jj leads to

2 3 (∇ × Ji ) · Jj = αi αj µi − µj C12 (25) + C12 µi − eb1 · ∇ ln αi Therefore the compatibility condition (24) implies that

2 3 2 3 C12 + C12 µi − eb1 · ∇ ln αi = C12 + C12 µj − eb1 · ∇ ln αj

and hence we get

e1 · ∇ ln b


αi 3 µi − µj = C12 αj

(26)

(27)

whose characteristic curve is the solution curve of (1) in arclenght parametrization
d αi 3 = C12 ln µi − µj (28) ds αj then, by a similar line of reasoning as above the solutions of (28) can also be extended to the whole neighborhood, and the proposition follows. However, having a pair of Poisson structures obtained in (19) and even a compatible pair satisfying (27) as well do not guarantee the existence of Hamiltonian functions even locally. 4

Proposition 4 The dynamical system (1) is locally bi-Hamiltonian with a pair of Poisson structures obtained in (21) if and only if eb1 · ∇ ln

αi 3 3 = C31 + µi C12 kvk

(29)

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 eb1 · ∇Hi = 0

(30)

so the gradients of Hamiltonian functions is a linear combination of eb2 and eb3 Then, inserting (21) into (9) we get another condition eb3 · ∇Hj − µi eb2 · ∇Hj =

or by defining (31) can be written as

kvk αi

(31)

ui = −µi eb2 + eb3

(32)

ui · ∇Hj =

(33)

kvk αi

Two equations (30) and (33) for Hamiltonian functions are subject to the integrability condition eb1 (ui (Hj )) − ui (b e1 (Hj )) = [b e1 , ui ] (Hj )

Using the commutation relation given in (13) and (19) we obtain

1 1 3 3 [b e1 , ui ] = − C31 + µi C12 eb1 − C31 + µi C12 ui

(34)

(35)

and applying Hj to both sides of (35), and using two equations (30) and (33) for Hamiltonian functions we get 3 3 [b e1 , ui ] (Hj ) = − C31 + µi C12


kvk αi

(36)

Therefore our integrability condition for Hamiltonian functions becomes,  
kvk kvk 3 3 = − C31 + µi C12 eb1 · ∇ (37) αi αi

hence

eb1 · ∇ ln



αi kvk



3 3 = µi C12 + C31

(38)

and the proposition follows.

Corollary 5 The pair of Poisson structures Ji = αi (b e 2 + µi b e3 ) where αi is defined by (38) and µi is defined by (19) is compatible. 5

Proof. What we need is to show that (22) is satisfied. Indeed, writing (38) for αi and αj and subtracting the second from the first, the corollary follows. Note that, for a pair of compatible Poisson structures, J1 and J2 , the dilatation symmetry J → f J and the additive symmetry J1 + J2 do not imply that J1 + f J2 is a Poisson structure. Indeed if we apply the Jacobi identity condition and using triple vector identity (J1 + f J2 ) · ∇ × (J1 + f J2 ) = −∇f · (J1 × J2 ) = 0

(39)

which implies that eb1 · ∇f = 0

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. Lemma 6 For the bi-Hamiltonian system with a pair of compatible Poisson structures defined above, α1 α2 (µ2 − µ1 )

∇ · eb1 = eb1 · ∇ ln

kvk

2

(40)

Proof. Adding the equations for integrability conditions of Hamiltonian functions (38) for i = 1, 2 we get   3 3 + (µ1 + µ2 ) C12 (41) eb1 · ∇ ln (α1 α2 ) = eb1 · ∇ ln kvk2 + 2C31

On the other hand, subtracting the equations (19) satisfied by µ1 and µ2 , and dividing by (µ2 − µ1 ) gives
3 2 3 − (µ1 + µ2 ) C12 (42) eb1 · ∇ ln (µ2 − µ1 ) = − C31 + C12 Adding (41) to (42) and using

we get

i ∇ · eb1 = Ci1

  eb1 · ∇ ln (α1 α2 (µ2 − µ1 )) = eb1 · ∇ ln kvk2 + ∇ · eb1

(43)

(44)

and the lemma follows.

Proposition 7 Given a bi-Hamiltonian system with a pair of compatible Poisson structures, then there exists a canonical pair of compatible Poisson structures K1 , K2 with the same Hamiltonian functions H1 , H2 such that i+1

Ki = (−1) where φ=

φ∇Hi

(45)

α1 α2 (µ2 − µ1 ) kvk

(46)

6

Proof. Since Poisson vector fields are linearly independent one could write Hamiltonians in terms of Poisson vector fields ∇Hi = σ ji Jj

(47)

By using (9) we get σ 22 = −σ 11 =

kvk α1 α2 (µ2 − µ1 )

(48)

On the other hand we have ∇ × ∇Hi = ∇σ ji × Jj + σ ji ∇ × Jj = 0

(49)

Taking the dot product of both sides with J1 and then with J2 , and using the compatibility condition we obtain eb1 · ∇ ln σ ij =

J1 · (∇ × J2 ) α1 α2 (µ2 − µ1 )

(50)

Inserting (29) into (25), and using (50) eb1 · ∇ ln σ ij = −b e1 · ∇ ln φ

which leads to

σ ij =

(51)

Ψij φ

(52)

where Therefore we have

eb1 · ∇Ψij = 0

(53)

∇H1

=

1 φ

Ψ11 J1 + Ψ21 J2

∇H2

=

1 φ

Ψ12 J1 − Ψ11 J2

Inserting (54) into (9) we get




(54)




Ψ11 = −1

(55)

and finally ∇H1

=

∇H2

=

− α1 α2kvk (µ −µ 2

1)

kvk α1 α2 (µ2 −µ1 )

Note that ∇H1 × ∇H2 = − 1 + Ψ12 Ψ21




J1 − Ψ21 J2 Ψ12 J1 + J2







(56)

2

kvk eb1 α1 α2 (µ2 − µ1 )

(57)

For the Hamiltonians to be functionally independent RHS of (57) must not vanish, i.e. 1 + Ψ12 Ψ21 6= 0 (58)

7

Now let us define K1

=

J1 −Ψ21 J2 1+Ψ12 Ψ21

=

J2 +Ψ12 J1 1+Ψ12 Ψ21

=

1 α2 (µ2 −µ1 ) − α1+Ψ 1 Ψ2 kvk ∇H1 ( 2 1)

=

α1 α2 (µ2 −µ1 ) ∇H2 (1+Ψ12 Ψ21 )kvk

(59) K2 By (9) and (59) we get K1 × ∇H1

= K2 × ∇H2

=

0

K2 × ∇H1

= K1 × ∇H2

=

v

(60) Choosing Ki ’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 8 Any three dimensional dynamical system ·

x (t) = v (x (t))

(61)

has a pair of compatible Poisson structures Ji = αi (b e2 + µi eb3 )

(62)

in which µi ’s are determined by the equation


3 3 2 2 − µ2i C12 + C12 eb1 · ∇µi = −C31 − µi C31

and αi ’s are determined by the equation αi 3 3 eb1 · ∇ ln = C31 + µi C12 kvk

(63)

(64)

Furthermore (61) is a locally bi-Hamiltonian system with a pair of local Hamiltonian functions determined by i+1

Ji = (−1) where φ=

3

φ∇Hi

(65)

α1 α2 (µ2 − µ1 ) kvk

(66)

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 will 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 eb1 we can define a complex structure Λ on the fibers of Q → M , and Q becomes a complex line bundle over M . 8

3.1

Bi-Hamiltonian Structure in 3D with Differential Forms

In order to obtain and express the obstructions to the global existence of biHamiltonian 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 of M . Then, there is a one-form J associated with a Poisson bivector field Λ, J = ıΛ Ω

(67)

which is called the Poisson one-form. The bi-Hamiltonian system (9) can be written as ιv Ω = J1 ∧ dH2 = J2 ∧ dH1 (68) and the Jacobi identity is given by Ji ∧ dJi = 0, for i = 1, 2

(69)

and compatibility amounts to J1 ∧ dJ2 = −J2 ∧ dJ1

(70)

By (65) J1 and J2 can be chosen to be proportional to dH1 and dH2 , respectively, and hence (68) takes the form ιv Ω = φdH1 ∧ dH2 .

(71)

The Jacobi identity for Poisson 1-forms (69) implies the existence of 1-forms β i such that dJi = β i ∧ Ji (72) for each i = 1, 2. In the next proposition we are going to show that compatibility of Poisson structures allows us to combine β 1 and β 2 into a single one. Proposition 9 There is a 1-form β such that dJi = β ∧ Ji

(73)

for each i = 1, 2. Proof. Applying (72) to the compatibility condition J1 ∧ dJ2 + J2 ∧ dJ1 = 0

(74)

(β 1 − β 2 ) ∧ J1 ∧ J2 = 0

(75)

β 1 − β 2 = b 1 J1 + b 2 J2

(76)

β = β 1 − b 1 J1 = β 2 + b 2 J2

(77)

we get which implies that and therefore we can define

9

Hence β ∧ Ji = β i ∧ Ji = dJi

(78)

and the proposition follows. Note that β is a T M -valued 1-form. Namely, ιbe1 β 6= 0

(79)

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. Lemma 10 ιbe1 β = ιeb1 (d ln φ)

(80)

where φ is the function defined in (66) Proof. For the proof, we carry out the computation with Poisson vector fields, then map them to differential forms. The Jacobi identity (8) implies that ∇× Ji is orthogonal to Ji and therefore we get ∇ × Ji = ai1 eb1 + ai2 eb1 × Ji

(81)

J1 × J2 = φ kvk eb1

(82)

By the definition of Poisson vector fields we have

we can rewrite (81) in the form ∇ × Ji =

ai1 J1 × J2 + ai2 eb1 × Ji φ kvk

(83)

Using the compatibility condition (24) we obtain ai1

= (∇ × Ji ) · eb1

(84)

a21 J1 − a11 J2 + ((∇ × J1 ) · J2 ) eb1 → φ k− vk

(85)

ai2

=

(∇×J1 )·J2 φkvk

Now, we define, ξ= and (83) becomes

∇ × J i = ξ × Ji After a bit of computation it is possible to show that   [b e1 × J1 , eb1 × J2 ] ξ = ∇ ln φ + eb1 × − eb1 × ∇ ln kvk φ kvk

(86)

(87)

Hence, we have

eb1 · ξ = b e1 · ∇ ln φ 10

(88)

and defining β = ∗ιξ Ω

(89)

the lemma follows. Now we define new Poisson 1-forms Ki Ji = φKi

(90)

Taking the exterior derivatives of both sides dJi = dφ ∧ Ki + φdKi = β ∧ φKi

(91)

and dividing both sides by φ dKi = (β − d ln φ) ∧ Ki

(92)

γ = β − d ln φ

(93)

ιeb1 γ = ιeb1 β − ιeb1 (d ln φ) = 0

(94)

dKi = γ ∧ Ki

(95)

Let Now, by the lemma above

therefore where γ is a connection on Q.

3.2

The First Obstruction: Chern Class of Q

Now, we try to find conditions for which a non-vanishing vector field v satisfies w = ιv Ω = φdH1 ∧ dH2

(96)

for some globally defined functions φ, H1 and H2 . For a two-form to be decomposed into the form (96), first of all, the two-form must be written as a product of two globally defined, linearly independent non-vanishing 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 ρ1 and ρ2. w = ρ1 ∧ ρ2

(97)

Since v is a non-vanishing vector field then w is a 2-form of constant rank 2. If we let Sw to be the sub-bundle of T M on which w is of maximal rank, then we have Sw ∼ = Q defined before. The following theorem states the necessary and sufficient conditions for the decomposition of a two-form of constant rank 2s in the large. Theorem 11 [9] Let Σ be an Rn -bundle over a connected base space M . Let w be a 2-form on Σ of constant rank 2s. Let Sw be the subbundle of Σ on which w is of maximal rank. w decomposes if and only if 11

i. Sw is a trivial bundle. ii. The representation of its normalization as a map w1 : M → SO(2s)/U (s) arising from any trivialization of Sw lifts to SO(2s). 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. Therefore the necessary and sufficient condition of decomposition is the triviality of Sw ∼ = Q. Since Q is a complex line bundle, it is trivial if and only if c1 (Q) = 0, or equivalently it has a global section. Since the decomposition of the 2-form w into globally defined 1-forms ρ1 and ρ2 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, the decomposition may not imply that the factors ρi satisfy the Jacobi identity ρi ∧ dρi = 0 (98) In order to determine the effect of a vanishing Chern class condition on the the constructions made so far, we are going to investigate the equation (19) defining the Poisson one-forms. Since our Poisson one-forms and related integrability conditions are determined by the local solutions of the (19), they are defined locally on each chart. Let {Jip } and {Jiq } be the Poisson vector fields in charts (Up , xp ) and (Uq , xq ) around points p ∈ M and q ∈ M , respectively. Around point p ∈ M , the Poisson vectors {Jip } are determined by µpi , αpi and the local frame {b ep2 , ebp3 }. Given the local frame, we can write (19) whose solutions are µpi ’s, and using µpi ’s we can determine αpi ’s by the equation (29). Now, if c1 (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 field normal to v. By using the metric on M , normalizing this global section of Q and take it as eb2 , then define eb3 = eb1 × eb2 . So we have the global orthonormal frame field {b e1 , eb2 , eb3 }. In order to understand the relation between local Poisson one-forms obtained in two different coordinate neighborhoods, we first need the following lemmas. Lemma 12 If two solutions µ1 (s) and µ2 (s) of the Riccati equation
dµi 3 3 2 2 − µ2i C12 + C12 = −C31 − µi C31 ds

(99)

are known, then the general solution µ (s) is given by µ − µ1 = K (µ − µ2 ) e

R

3 C12 (µ2 −µ1 )ds

(100)

where K is an arbitrary constant. Lemma 13 If c1 (Q) = 0, then two pairs of compatible Poisson vector fields {Jip } and {Jiq } on Up and Uq respectively, are related on Up ∩ Uq by Jiq Jip = kJiq k kJip k 12

(101)

Proof. Given the global frame field {b e2 , eb3 } defined both on coordinate neighborhoods Up and Uq , Riccati equations for µi ’s can be written as 2

e1 · ∇µri = (∇ × eb2 ) · eb2 + µri ((∇ × eb2 ) · b b e3 + (∇ × eb3 ) · eb2 ) + (µri ) (∇ × eb3 ) · eb3 (102) for r = p, q. Therefore, on Up ∩ Uq , µpi and µqi are four solutions of the same Riccati equation for i = 1, 2. By the lemma above we have µqi − µp1 = Kipq (µqi − µp2 ) e

R

3 C12 (µp2 −µp1 )ds

(103)

Now, using the compatibility condition (22)

(103) becomes

3 C12 (µp2 − µp1 ) = b e1 · ∇ ln

µqi − µp1 = Kipq (µqi − µp2 ) where Multiplying both sides by

αp2 αp1

(104)

αp2 αp1

(105)

e1 · ∇Kipq = 0 b

(106)

Jiq × J1p = Kipq Jiq × J2p

(107)

αp1 αqi

in (105) gives

Rearranging (107) we obtain Jiq × (J1p − Kipq J2p ) = 0

(108)

Using (106) and the compatibility, we can take Jeip = J1p − Kipq J2p

(109)

Jiq × Jeip = 0

(110)

Jeip = Jip

(111)

to be our new Poisson vector fields on the neighborhood Up , and obtain

By compatibility this new Poisson vector fields Jeip produce functionally dependent Hamiltonians and therefore, for the simplicity of notation, we will assume without restriction of generality that

and the lemma follows. Then, we have the following result.

Theorem 14 There exist two linearly independent global sections b ji of Q satisfying   b ji · ∇ × b ji = 0 (112)

if and only if c1 (Q) = 0.

13

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 c1 (Q) vanishes. For the converse, we define Jp b jip = ip (113) kJi k

and the lemma implies that jip = jiq on Up ∩ Uq and the theorem follows. The lemma above states clearly the reason one may fail to extend a local pair of compatible Poisson vector fields into a global one. Even if c1 (Q) = 0. In order to do so one should have Jiq = Jip on Up ∩ Uq . However not the Poisson vector fields but their unit vector fields can be globalized. However, since e1 · ∇ b

kJ2p k 6= 0 kJ1p k

(114)

in general, they may not lead to a pair of compatible Poisson structures. Now, we take b j1 as our first global Poisson vector field, and then going to check if we can find another global Poisson vector field compatible with this one by rescaling b j2 .

3.3

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 . 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 will compute the Bott class of the complex codimension one foliation as defined in [10], which is studied in detail in [11], 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 (112) it has two global sections b ji satisfying db ji = Γi ∧ b ji

(115)

dJip = γ p ∧ Jip

(117)

for globally defined Γi ’s. These b ji ’s are related with the local Poisson vector fields Jip by Jip = kJip k b ji (116)

By (95) , we have

Inserting (116) and (117) into (115) we also have db ji = (γ p − d ln kJip k) ∧ b ji

(118)

Γi = γ p − d ln kJip k

(119)

Redefining Γi ’s if necessary, comparing (115) with (118) we get

14

Proposition 15 Let κ be the curvature of Q. There exists a a compatible pair of global Poisson structures if and only if Ξ = (Γ1 − Γ2 ) ∧ κ

(120)

is exact. Proof. Since, b j1 and b j2 are not compatible we introduce a local Poisson form p j defined on the coordinate neighborhood Up of p ∈ M , which is compatible with b j1 and parallel to b j2 i.e., j p = f pb j2 (121) and

Now, (121) implies that

b j1 ∧ dj p + j p ∧ db j1 = 0

(122)

dj p = (Γ2 + d ln f p ) ∧ j p

(123)

Putting (115) and (123) into (122) and using (121) we get

we find

(Γ1 − Γ2 − d ln f p ) ∧ b j1 ∧ j p = 0

(124)

(Γ1 − Γ2 ) ∧ b j1 ∧ b j2 = d ln f p ∧ b j1 ∧ b j2

(125)

κ = dΓi = dγ p

(126)

Our aim here is to find the obstruction for the extending f p to M , or for (125) to hold globally. For this purpose we consider the connections on Q defined by Γi ’s. By (119) we define the curvature of these connections to be

Taking the exterior derivative of (117) and using (116) we get

which leads to

dγ p ∧ Jip = dγ p ∧ b ji = 0

(127)

κ = dγ p = ϕb j1 ∧ b j2

(128)

(Γ1 − Γ2 ) ∧ κ = d ln f p ∧ κ = d ((ln f p ) κ)

(129)

Now multiplying both sides of (125) with ϕ

and the proposition follows. Now we are going to show that the cohomology class of Ξ vanishes if and only if the Bott class of the complex codimension 1 foliation vanishes. Since Q is a complex line bundle we have c1 (Q) = [κ]

15

(130)

and the vanishing of c1 (Q) is a necessary condition c1 = dh1

(131)

c1 = κ = dγ p

(132)

h1 = γ p + d ln hp

(133)

h1 ∧ c1 = (γ p + d ln hp ) ∧ dγ p = d ln hp ∧ κ + γ p ∧ dγ p

(134)

So we have which implies that on Up Then the Bott class [10]

Now by (94) and (128) we have γ p ∧ dγ p = 0

(135)

h1 ∧ c1 = d ((ln hp ) κ)

(136)

and therefore Since h1 is globally defined, on Up ∩ Uq we have h1 = γ p + d ln hp = γ q + d ln hq and γ p − γ q = d ln

hq hp

(137)

(138)

Now, we have the following theorem. Theorem 16 The cohomology class of Ξ vanishes if and only if the Bott class of the complex codimension one foliation defined by the nonvanishing vector field vanishes. Proof. If the Bott class vanishes, then we have a globally defined function h such that d ((ln h) κ) = 0 (139) Then choosing f = h leads to a compatible pair of global Poisson structures. Conversely if there is a pair of globally defined compatible Poisson structures then γ becomes a global form and by (138) we have d ln on Up ∩ Uq . Therefore where c

qp

hq =0 hp

ln hq − ln hp = cqp

(140)

(141)

is a constant on Up ∩ Uq . Now fixing a point x0 ∈ Up ∩ Uq cqp = ln hq (x0 ) − ln hp (x0 ) = ln cq − ln cp 16

(142)

we obtain

hp hq = =h cp cq where h is a globally defined function. and

(143)

d ln h = d ln hp

(144)

[h1 ∧ c1 ] = [d ((ln h) κ)] = 0

(145)

Therefore and the theorem follows.

References [1] A. Weinstein, ”The local structure of Poisson manifolds”. J. Diff. Geom. 18 (3): 523–557, (1983). [2] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (5) , pp. 1156–1162, (1978). [3] M. Santoprete, On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems, SIGMA 11 (2015) [4] P. J. Olver, Canonical forms and integrability of bi-Hamiltonian systems, Phys. Lett. A, volume 148 number 3,4, p. 177-187 [5] H. G¨ umral and Y. Nutku, Poisson structure of dynamical systems with three degrees of freedom, J.Math.Phys. 34 5691-5723, (1993). [6] F. Haas and J. Goedert, On the Generalized Hamiltonian Structure of 3D Dynamical Systems, Phys. Lett. A 199 173-179, (1995). [7] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures, vol. 347 in Grundlehren der mathematischen Wissenschaften. SpringerVerlag, Heidelberg, Berlin, New York, (2013). [8] B. Hernandez-Bermejo, New solutions of the Jacobi equations for three dimensional Poisson structures, J. Math. Phys. 42 4984-4996, (2001). [9] I. Dibag,Decomposition in the large of two-forms of constant rank, Annales de l’institut Fourier , Volume: 24, Issue: 3, page 317-335, (1974). [10] R. Bott, Lectures on characteristic classes and foliations. Notes by Lawrence Conlon, with two appendices by J.Stasheff. In Lectures on algebraic and differential topology, Lecture Notes in Math., Vol. 279, Springer, Berlin, 1-94, (1972). [11] T. Asuke, A remark on the Bott class, Ann. Fac. Sci. Toulouse Math (6), 10:5, 21, (2001)

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