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VERTICAL COHOMOLOGIES AND THEIR APPLICATION TO COMPLETELY INTEGRABLE HAMILTONIAN

SYSTEMS

Z. TEVDORADZE

Dedicated to the memory of Roin Nadiradze

Abstract. Some functorial and topological properties of vertical co-homologies and their application to completely integrable Hamilto-nian systems are studied.

§ _1. Introduction

If on a smooth Riemannian manifold Mn _{we have a distribution V of}

dimensionk, which is actually a smooth section of the Grassman fiber bun-dleGk(T Mn)→Mn adjoint to the tangent fibrationT Mn to the manifold

Mn_{, then by means of the Riemannian metric we obtain T M}n _{=V ⊕N,}

where N is a normal fiber bundle to V. Let P :T Mn _{→V ⊂T M}n _{be a}

natural projection. The operator P defines the mapping P∗ _{: Λ}∗_(Mn_)→

Λ∗_(Mn_{) ((Λ}∗_(Mn_{), d}∗_{) is the de Rham differential complex) by the} for-mula (P∗α)(X1, . . . , Xq) = α(P X1, . . . , P Xq), where α ∈ Λq(Mn) and

X1, . . . , Xq ∈S(Mn) are the smooth vector fields onMn.

Denote by Λ∗

V(Mn) all fixed points of the operatorP∗. In what follows we

shall consider the case where V is integrable, i.e., whereMn _{is partitioned}

into leaves and the tangent space to the leaf that passes through the point

x∈ Mn _{is V}

x. Then the pair (Λ∗V(Mn), d∗V) forms a differential complex

with the differentiald∗

V =P∗◦d∗.

The cohomologies of the complex (Λ∗

V(Mn), d∗V) are called vertical and

denoted byH∗

V(Mn) ([1]). It has turned out that these cohomologies

coin-cide with those of the classical BRST operator ([2], [3]).

In§2 the vertical cohomologies are defined without fixing the metric on

Mn_{, and some of their functorial properties are studied. The FOL category}

1991Mathematics Subject Classification. 53C12, 70H05.

Key words and phrases. Foliation, vertical cohomology, Hamiltonian systems, isoen-ergetic surfaces, Liouville torus.

483

of smooth foliations and leaf-to-leaf transforming mappings is introduced, and a natural transformation of the de Rham functor Λ∗ _{to the functor} Λ∗

F is constructed (Proposition 2.2). The notion of leaf-to-leaf transforming homotopic mappings is introduced, and the homotopy axiom for vertical cohomologies is proved (Theorem 2.5). The notion of a relative group of vertical cohomologies is introduced by analogy with de Rham’s theory, and the long exact cohomologic sequences (2.6) and (2.7) are derived. More-over, for a leaf-to-leaf transforming mapping f : (Mn_{,F1) → (N}m_,F2),

the cohomology groups H∗_{(f) are constructed and proved (Theorem 2.8)} to be isomorphic for leaf-to-leaf transforming homotopic mappings. Fi-nally, a double complex (K∗∗_{, D}∗_{) is constructed for the countable covering}

U ={uα}α∈A of the manifold Mn. It is shown that the cohomologies of

(K∗∗, D∗) are isomorphic to the vertical cohomologies. A combinatorial definition of vertical comologies in the ˇCech sense (Theorems 2.10 and 2.12) is also given.

In §3 some of the main facts from the topological theory of integrable Hamiltonian systems ([5], [6]) are presented. Using the notion of vertical co-homologies, the groups corresponding to nonresonance Hamiltonian systems are constructed (Theorems 3.3 and 3.4).

In§4 the case of a spherical pendulum is considered as an example.

§_2. Vertical Cohomologies

2.1. Definition of Vertical Cohomologies. Let Mn _{be a smooth n}

-dimensional manifold, andV be ak-dimensional involutive distribution on

Mn _{whose foliation is denoted byF. The bundle of exteriorp-forms onV}

is denoted byAp_{(V), and the set of smooth sections of the bundle A}p_(V)

by Λp_F(Mn_{). Then Λ}p

F(Mn) is a module over the algebra of infinitely dif-ferentiable functionsC∞_(Mn_{) onM}n_.

Letα∈Λp_F(Mn_{), and letX}

1, . . . , Xp be the smooth vector fields onMn

which are tangent to the leaves of the foliationF, i.e., they are the smooth sections of the bundleV pF

→Mn_{. Then the mapping defined by the formula}

α(X1, . . . , Xp) :x j

7−→α(X1(x), . . . , Xp(x)), (2.1)

which on the module of sectionsS(V) ofV assigns an exteriorp-form to an elementα∈Λp_F(Mn), is an isomorphism.

Let us now define the operator

by the relation

(dp_Fα)(X1, . . . , Xp+1) = p+1

X

i=1

(−1)i−1Xiα(X1, . . . ,Xbi, Xi+1, . . . , Xp+1) +

+X

i<j

(−1)i+j_α([X

i, Xj], X1, . . . ,Xbi, . . . ,Xbj, . . . , Xp+1),

whereα∈Λp_F(Mn_),X

i∈S(V),i= 1, p+ 1.

It is easy to verify that the embeddingi:S(V)֒→S(T Mn_{) induces the}

projectioni∗

q :Aq(S(T M))→Aq(S(V)). Indeed, letα∈Aq(S(V)). Define

α∈Aq_{(S(T M)) so thati}∗

qα=α. Any Riemannian metric defines a smooth

sectionP :Mn _{→End(T M}n_{) of the bundle of homomorphisms End(T M}n₎

of the tangent bundle T Mn_{, where P(x) : T}

xMn → Vx ⊂ TxMn is the

orthogonal projection onto a leaf of the foliation F which passes through the pointx∈Mn_{. The elementαcan be defined by the formula}

α(X1, . . . , Xq) =α(P X1, . . . , P Xq), Xi∈S(T Mn), i= 1, q.

The following diagram is commutative:

Aq_{(S(T M}n₎₎ i ∗

q

−−−−→ Aq_F(Mn₎ 

 ydq

  ydqF

Aq+1_{(S(T M}n₎₎ i ∗

q+1

−−−−→ Aq_F+1(Mn₎

,

where byA∗_F(Mn) is denotedA∗(S(V)). Indeed,

(dq_Fi∗_qα)(X1, . . . , Xq+1) = q+1

X

j=1

(−1)j−1(Xj)(i∗qα)(X1, . . . ,Xbj, . . . , Xq+1) +

+X

j<t

(−1)j+t(i∗_qα)([Xj, Xt], X1, . . . ,Xbj, . . . ,Xbt, . . . , Xq+1) =

= (dqα)(X1, . . . , Xq+1) = (iq∗+1dqα)(X1, . . . , Xq+1).

The commutativity dq_F◦i∗q =i∗q+1◦dq impliesdq_F◦dq− 1

F = 0. Thusi∗ is a cochain mapping between the differential complexes (A∗_{(S(T M}n_{), d}∗₎ and (A∗

F(Mn), d∗F).

Definition 2.1. Cohomology groups of the complex (A∗_F(M), d∗_F) are called vertical cohomologies of the foliation (Mn,F), and we denote them byH∗

It is easy to verify that d∗

F is an antidifferentiation of order 1, i.e., if

α∈Aq_F(Mn_{),β ∈A}l

F(Mn), we have

dq_F+l(α∧β) = (dq_Fα)∧β+ (−1)qα∧(dl_Fβ),

where ∧ is an exterior product. This implies that the homomorphism i∗ induces a homomorphism between the de Rham cohomology algebra and the cohomology algebraH∗

F(Mn). We denote this homomorphism by the same symboli∗_{. Since we already know that the cochain mapping i}∗ _{is an} epimorphism, we have a short exact sequence of cochain differential com-plexes

0−→Z_F j ∗

−→A∗(S(T Mn)) i

∗

−→A∗_F(Mn)−→0, (2.2)

where by (Z∗

F, d∗) is denoted the kernel of the mappingi∗. Sequence (2.2) induces a long exact sequence of cohomology groups

0−→He_F0(Mn) j

∗

−→H0(Mn) i

∗

−→H_F0(Mn)−→δ He_F1(Mn) j

∗

−→ · · · . (2.3)

Here byHe∗

F(Mn) are denoted the cohomology groups of the complex (ZF∗, d). SinceH_Fk+m(Mn_{) = 0,m >0, from (2.3) we getHe}m+k

F (Mn)≈Hm+k(Mn),

m >1.

Denote by FOL a category whose objects are smooth foliations (Mn_,F),

and morphisms from (Mn

1,F1) to (M2m,F2) are leaf-to-leaf transforming mappings, i.e., smooth mappings h from M1n to M2n which preserve the foliation structure by performing a leaf-to-leaf transformation.

If h is the leaf-to-leaf transforming mapping between the foliations (Mn

1,F1) and (M2n,F2), then h defines the morphism h∗ between the

C∞_(Mm

2 )-module Λ∗_F2(M

m

2 ) and theC∞(M1n)-module Λ∗_F1(M

n

1) by the for-mula

(h∗α)(X1, . . . , Xp)(x) =α(h⊤X1(x), . . . , h⊤Xp(x)), (2.4)

where α∈Λp_F

2(M2) andX1, . . . , Xp∈S(V1),V1 is the distribution associ-ated withF1, andh⊤ _{is the tangent mapping toh. We have the} commuta-tive diagram

Λ∗_(Mm

2 )

h∗

−−−−→ Λ∗_(Mn

1)

  yi∗

2

  yi∗

1

Λ∗ F2(M

m

2 )

h∗

−−−−→ Λ∗ F1(M

n

1)

.

A direct calculation shows thath∗ is a cochain mapping, i.e.,d∗_F1◦h∗₌

Proposition 2.2. The mapping i∗ _{is a natural transformation of the de}

Rham functorΛ∗_{to the functorΛ}∗

F, whereΛ∗ andΛ∗F are the contravariant

functors from theFOLcategory to the category of differential graded algebras and their homomorphisms.

2.2. A Homotopy Axiom for Vertical Cohomologies. Let (Mn_,F)

be a foliation of dimensionk. On the manifoldMn_{×Rwe define naturally}

a foliationFbof dimensionk+ 1 whose leaves are manifoldsLα×R,α∈A,

whereLα,α∈A, are the leaves of the foliationF.

Lemma 2.3. The projectionπ:Mn_×R→Mn _{defines an isomorphism}

in vertical cohomologies.

Proof. Consider the zero sectionsof the trivial bundleMn_×R−→π _Mn_{, i.e.,}

s(x) = (x,0),x∈Mn_{. Then the mappingsπandsare the leaf-to-leaf}

trans-forming mappings which define the cochain mappingsπ∗: (Λ∗

F(Mn), d∗F)→ (Λ∗

b

F(M

n_{×R), d}∗

b

F) and s ∗ _{: (Λ}∗

b

F(M

n_{×R), d}∗

b

F) → (Λ ∗

F(Mn), d∗F). Since

s∗_◦π∗ _{= 1, π}∗ _{is an monomorphism. We shall show that π}∗ _{induces an} isomorphism at the cohomology level. To this end, we shall construct a cochain equivalence of the mappings 1 andπ∗_◦s∗_.

Note that each form from Λ∗

F(Mn×R) can be uniquely represented by linear combinations of the following two types of forms:

(I) (π∗ϕ)·f, ϕ∈Λ∗

F(Mn), f ∈C∞(Mn×R); (II) (π∗ϕ)∧dt·f, ϕ∈Λ∗_F(Mn), f ∈C∞(Mn×R),

where t is the coordinate on the straight lineR. Define the operatorK∗ : Λ∗

b

F(M

n_×R)→Λ∗

b

F −1₍

Mn×R) as follows:

K∗((π∗ϕ)·f) = 0,

K∗((π∗ϕ)∧dt·f) =π∗ϕ·

t Z

0

f dt.

A direct calculation shows that the relation

1−π∗◦s∗= (−1)q−1₍

dq−1

b

F K

q_−Kq+1

dq

b

F) is fulfilled on the forms of types (I) and (II).

Definition 2.4. Two leaf-to-leaf transforming mappingsf, g: (Mn

1,F1) →(Mm

2 ,F2) between the foliations (M1n,F1) and (M2m,F2) are called leaf-to-leaf transforming homotopic if there exists a leaf-leaf-to-leaf transforming mapping

such that ₍

F(x, t) =f(x), t≥1,

F(x, t) =g(x), t≤0, x∈M, t∈R.

Theorem 2.5 (A Homotopy Axiom). Leaf-to-leaf transforming ho-motopic mappings induce identical mappings in vertical cohomologies.

Proof. Let f, g : (M1n,F1) → (M2m,F2) be the leaf-to-leaf transforming homotopic mappings andF : (Mn

1 ×R,F1)b →(M2m,F2) be the homotopy between f and g. Denote by s0 and s1 the sections s0, s1 : (M1n,F1) → (Mn

1 ×R,F1),b s0(x) = (x,0), s1(x) = (x,1), x∈M1n. Thenf =F◦s1 and

g=F◦s0. Hence we havef∗_=s∗

1◦F∗ andg∗=s∗0◦F∗. From the proof of Lemma 2.3 it follows that s∗

1=s∗0 = (π1∗)−1, whereπ1:M1n×R→M1n is the projection. Thereforef∗_=g∗_.

The foliations (Mn

1,F1) and (M2m,F2) will be said to be of the same homotopy type if there are leaf-to-leaf transforming smooth mappings f :

Mn

1 → M2m and g : M2m → M1n such that g◦f and f ◦g are leaf-to-leaf transforming homotopic mappings to the identical mappings of the foliations (Mn

1,F1) and (M2m,F2), respectively. Corollary 2.6. If two foliations(Mn

1,F1)and(M2m,F2)are of the same

homotopy type, then their vertical cohomologies are isomorphic.

2.3. Relative Vertical Cohomologies. Let (Mn,F1) and (Nm,F2) be two foliations, and let f be a leaf-to-leaf transforming smooth mapping

f :Mn_→Nm_{. Define the differential complex}

(Λ∗_{(f), d}∗_{), Λ}∗_{(f) = ⊕}

q≥0Λ

q_(f),

where

Λq_{(f) = Λ}q

F2(N

m_)⊕Λq−1 F1 (M

n_{), d}∗_{(ω, θ) = (−d}∗ F2ω, f

∗_ω+d∗ F1θ).

We easily verify thatd2 _{= 0 and denote the cohomology groups of this} complex by H∗_{(f). Note that the complex (Λ}∗_{(f), d}∗_{) is the cone of the} cochain mappingf∗_{: Λ}∗

F2(N

m_)→Λ∗ F1(M

n_{). If we regraduate the complex}

Λ∗ F1(M

n_{) asΛe}p

F1(M

n

1)≡Λp− 1 F1 (M

n_{), then we obtain an exact sequence of}

differential complexes

0−→Λe∗ F1(M

n₎ α

−→Λ∗_(f)−→β _Λ∗ F2(N

m_{)−→0 (2.5)}

with the obvious mappingsαandβ: α(θ) = (0, θ),β(ω, θ) =ω. From (2.5) we have an exact sequence in cohomologies

· · · −→H_Fq−₁1(Mn) α

∗

−→Hq(f) β

∗

−→H_Fq₂(Nm) δ

∗

It is easily seen thatδ∗_=f∗_{. Letω∈Λ}q

F2(N

m_{) be the closed form, and}

(ω, θ)∈ Λq_{(f). Then d(ω, θ) = (0, f}∗_ω+dq−1

F1 θ), and by the definition of the operator δ∗ _{we have δ}∗_{[ω] = [f}∗_ω+dq−1

F1 θ] = f

∗_{[ω]. Hence we finally} get a long exact sequence

· · · −→H_Fq−1

1 (M

n₎ α∗

−→Hq(f) β

∗

−→H_Fq

2(N

m_)−→f∗ _Hq

F1(M

n₎ α∗

−→ · · · . (2.6)

Corollary 2.7. If the foliations(Mn,F1)and(Nm,F2)are of the p-th andq-th dimension, respectively, then

(i) β∗:Hp+1(f)→H_Fp+1₂ (Nm)is an epimorphism, α∗_:Hq

F1(M

n_)→Hq+1_{(f)is an epimorphism,}

β∗_:Hi_(f)→Hi

F2(N

m_{)is an isomorphism fori > p+ 1,}

α∗_:Hi

F1(M

n_)→Hi+1_{(f)is an isomorphism fori > q;} (ii) Hi_{(f) = 0fori >max{p+ 1, q}.}

Theorem 2.8. If f, g: (Mn_,F1)→(Nm_{,F2)are leaf-to-leaf}

transform-ing homotopic mapptransform-ings, then H∗_{(f) =H}∗_(g).

Proof. Let F : (Mn_{×R,F1)b → (N}m_{,F2) be the homotopy mapping}

be-tweenf andg. Lets0ands1be the zero and the unit section, respectively, of the trivial bundle Mn×R−→π Mn. Then F ◦s0 = g and F◦s1 = f. Hence we have a homomorphism between the short exact sequences

0 −−−−→ Λe∗

b

F1

(Mn_{×R) −−−−→}α′ _Λ∗_(F) β

′

−−−−→ Λ∗ F2(N

m_{) −−−−→ 0} 

 ys∗

1

  yid⊕s∗

1

  yid

0 −−−−→ Λe∗ F1(M

n_{) −−−−→ Λ}∗_{(f) −−−−→ Λ}∗ F2(N

m_{) −−−−→ 0}

.

This homomorphism defines a homomorphism between the corresponding long cohomologic sequences

··· →Hq

F

2(N

m

)→Hq

b F₁

(Mn

×R)→Hq+1

(F)→Hq+1

F

2 (N

m

)→Hq+1

b F₁

(Mn

×R)→ ···

↓id ↓s∗

1 ↓γ ↓id ↓s

∗

1 ··· →Hq

F₂(N

m

)→ Hq

F₁(M

n

) →Hq+1

(f)→Hq+1

F₂ (N

m

)→ Hq+1

F₁ (M

n

) → ···

,

where γ is the mapping induced by id⊕s∗

1. Since s∗1 is an isomorphism (Lemma 2.3), by virtue of the lemma on five homomorphisms we conclude thatγis also an isomorphism, i.e.,H∗_(f)≈H∗_{(F). By a similar reasoning} we can conclude thatH∗_(g)≈H∗_(F).

If (Mn_{,F1) is a subfoliation of the foliation (W}m_{,F2), i.e., the embedding}

Mn_֒→j _Wm_{is simultaneously a leaf-to-leaf transforming mapping, then the}

cohomology algebraH∗_{(j) will be said to be the algebra of relative vertical} cohomologies. Denote it byH∗

Now sequence (2.6) can be rewritten as

· · · −→H_Fq−1

1 (M)

α∗

−→H_Fq

2,F1(W;M)

β∗

−→H_Fq

2(W)

j∗

−→

j∗

−→H_Fq

1(M)

α∗

−→ · · · . (2.7)

Note that if we forget the structure of the foliation, then, as is known, the embedding Mn ֒→j Wm defines a long exact cohomological sequence of the pair (Wm_{, M}n_{) in de Rham’s theory. One can easily verify that}

the homomorphism i∗ _{from Proposition 2.2 defines a morphism between} the long exact cohomological sequence of the pair (Wm_{, M}n_{) in de Rham’s}

theory and sequence (2.7).

2.4. The Generalized Mayer–Vietoris Principle for Vertical Coho-mologies. A Combinatorial Definition of Vertical CohoCoho-mologies. Let (Mn_{,F) be a smooth foliation of dimensionk; letU ={u}

α}α∈A be an

open countable covering of the manifoldMn_{. Similarly to ˇCech-de Rham’s}

theory, we define a double complex which will be used to calculate vertical cohomologies of the foliation (Mn_,F).

Denote byuα0···αp the intersection of open setsu0, . . . , up and by

`_the

disjunctive union. Then we have a sequence of open sets

Mn ←− a

α0

uα0

∂0 ←− ←−

∂1

a

α0<α1

uα0α1

∂0 ←−_∂ 1 ←− ←− ∂2 a

α0<α1<α2

uα0α1α2 ←− ←− ←− ←− · · · ,

where ∂i is the embedding ∂i(uα0···αp) = u_α0···b_αi···αp. This sequence of

embeddings induces a sequence of restriction mappings of vertical forms

Λ∗ F(Mn)

r∗

−→ Y

α0 Λ∗

F(uα0) δ0 −→ −→

δ1

Y

α0<α1 Λ∗

F(uα0α1)

δ0 −→_δ 1 −→ −→ δ2 δ0 −→_δ1 −→ −→

δ2

Y

α0<α1<α2 Λ∗

F(uα0α1α2) −→ −→ −→ −→ · · · ,

whereδi is induced by the imbedding∂i, i.e.,δi= (∂i)∗.

Let us define the difference operatorδby the following rule: ifωα0···αp∈

Λq_F(uα0···αp) denotes the components of the elementω∈

Q α0<···<αp

Λq_F(uα0···αp),

then

(δω)α0···αp+1=

p+1

X

i=0 (−1)i_ω

α0···bαi···αp+1, (2.8)

whereω_α

0···bαi···αp+1 ≡δi(ωα0···αi−1αi+1···αp+1).

Consider now the double complex

Kp,q≡Cp(U,Λq_F) = Y

α0<···<αp

Λq_F(uα0···αp). (2.9)

whose horizontal mappings are the operators δ∗ _{and whose vertical} map-pings are the operatorsd∗

F. As is known, this double complex can be reduced to an ordinary differential complex (K∗_{, D}∗_):

Kn= ⊕

p+q=nK

p,q_{, D}n _=δp_{+ (−1)}p_dq

F on Kp,q.

Lemma 2.9. The sequence

· · · −→Λ∗ F(Mn)

r∗

−→K0,∗_−→δ0 _K1,∗_−→δ1 _K2,∗_{−→ · · ·}δ2

is exact.

Proof. Letq ≥0 be an integer number. Clearly, Λq_F(Mn_{) is the kernel of}

the first operatorδ0_{, since an element from}Q

α0 Λq_(u

α0) is a global form on

Mn _{if and only if its components are consistent at the intersections.}

Let{θα}α∈Abe the partitioning of unity subordinate to the coveringU =

{uα}α∈A. Ifω∈Kp,q is the cocycle of the operatorδp, then we can assign

to itp−1 cochainsKωby the formula (Kω)α0···αp−1 =

P α

θαωαα0···αp−1 (it

is assumed here that ω...,α,...,β,... =−ω...,β,...,α,..., if α > β; clearly, this is

consistent with the operation δ, i.e., (δω)...,β,...,α,... =−(δω)...,α,...,β,...). In

that case

(δp−1Kω)α0···αp=

X

i

(−1)i_(Kω)

α0···bαi···αp=

X

i,α

(−1)i_θ αω_αα

0···bαi···αp=

=X

α

θα[ωα0···,αp−(δω)αα0···αp] =

X

α

θαωα0···αp=ωα0···αp.

Henceδp−1_{(Kω) =ω.}

Theorem 2.10. The cohomologies of the double complex Kp,q_{, i.e., the}

cohomologies of the complex (K∗_{, D}∗_{), are isomorphic to the vertical}

co-homologies H_F∗(Mn). This isomorphism is obtained by the mapping of the restrictionr∗.

Proof. SinceD∗_r∗_{= (δ}0_+d∗

F)r∗=δ0r∗+d∗Fr∗=d∗Fr∗=r∗+1d∗F, r∗ is a cochain mapping, it induces the mapping in cohomologiesr_∗.

Letϕ∈Km_{be the cocycle, i.e.,D}m_{ϕ= 0. We can representϕas a sum}

ϕ=ϕ0,m_+ϕ1,m−1_{+· · ·+ϕ}p,m−p_{, whereϕ}i,j_∈Kij _andϕp,m−p_{6= 0. Then}

δp_ϕp,m−p _{= 0. Hence by Lemma 2.9 we find that there exists an element}

ϕ′∈Kp−1,m−p such thatδp−1ϕ′=ϕp,m−p. Then the elementϕ−Dm−1ϕ′

repeating this procedure several times, we obtain an element ϕe ∈ K0,m

which is cohomologic to ϕ. For this element we have dm

Fϕe= 0, δ0ϕe = 0, i.e.,ϕedefines the global vertical closed form onMn _{which by means of r}m

transforms to ϕe. This shows that the mappingrm

∗ is epimorphic. Let us now show that the mappingrm

∗ is monomorphic. Letϕ∈Λm

F(Mn) anddmFϕ= 0. Then ifrm(ϕ) =Dm−1ϕ′,ϕ′ ∈Km−1,

ϕ′=ϕ′0,m−1+· · ·+ϕ′p,m−p−1, whereϕ′p,m−p−16= 0, thenδpϕ′p,m−p−1= 0 (becauseDm−1ϕ′ has only one component inK0,m).

Proceeding as above, we can find an elementϕ′′_{such thatϕ}′′_∈K0,m−1 and ϕ′ _−ϕ′′ _{∈ D}m−2_(Km−2_{). Then r}m_{(ϕ) = d}m−1

F ϕ′′ and δ0ϕ′′ = 0. Thereforeϕ′′_{defines a global formϕe}′′_onMn _{such that d}m−1

F ϕe′′=ϕ.

By analogy with de Rham’s theory Theorem 2.10 can be called the gen-eralized Mayer–Vietoris principle.

From the lower row of the double complexK∗,∗_{let us choose a} subcom-plex, namely, a kernel of the differentiald0

F. We get a sequence

C_F0(U) δ 0

−→C_F1(U) δ 1

−→C_F2(U) δ 2

−→ · · · , (2.10)

where C_Fi(U) ≡ kerd0_F ⊂ Q

α0<···<αi

Λ0

F(uα0···αi), and δ∗ is the difference

operator defined above. The cohomologies of complex (2.10) will be called the ˇCech cohomologies of the foliation (Mn_{,F) for the coveringU. They}

are a purely combinatorial object and will be denoted by H∗

F(U). Let the covering U = {uα}α∈A consist of foliated open sets such that all finite

nonempty intersections are contractible. Any manifold is known to have such a covering. Then, in view of the fact that Poincar´e’s lemma ([I]) is valid for vertical cohomologies, we obtain

Lemma 2.11. The sequence

0−→C_Fp(U) j

p

−→ Y

α0<···<αp

Λ0_F(uα0···αp)

d0

F

−−→ Y

α0<···<αp

Λ1(uα0···αp)

d1

F

−−→ · · ·

is exact,p≥0, wherejp _{is the embedding.}

By Lemma 2.11 and the same reasoning as we used in proving Theorem 2.10 we can prove

Theorem 2.12. The cochain mapping j∗ _{defines an isomorphism}

be-tween the ˇCech cohomologiesH∗

F(U)and cohomologies of the double complex

K∗,∗_{. Hence the ˇCech cohomologies and vertical ones are isomorphic:}

Corollary 2.13. A zero-dimensional vertical cohomology H0

F(Mn) is

isomorphic to a group of smooth functions on the foliated manifold Mn_,

which are constant on the leaves.

§_3. Completely Integrable Hamiltonian Systems and Vertical Cohomologies

3.1. Topology of Constant Energy Surfaces of Completely Inte-grable Hamiltonian Systems. In this subsection we shall briefly recall the basic facts from the topological theory of integrable Hamiltonian sys-tems ([5]).

Let v = sgradH be a Hamiltonian system on the symplectic manifold (M2n_{, ω), whereω is a symplectic 2-form on M}n_{, and let v be integrable.}

Thus there existnindependent (almost everywhere) smooth integralsf1=

H, f2, . . . , fn in involution, i.e., {fi, fj} = 0, i, j = 1, n, where { , } is

the Poisson bracket. Let F : M2n _{→ R}n _{be the moment mapping which}

corresponds to these integrals, i.e., F(x) = (f1(x), . . . , fn(x)), x ∈ M2n.

LetN be the set of critical points of the moment mapping, and Σ =F(N) be the set of all critical values which is called the bifurcation diagram.

Clearly, we have two cases: (a) dim Σ< n−1 and (b) dim Σ =n−1. In the case (a) the set Σ does not separate the space Rn _{and therefore all}

nonsingular leavesBa=F−1(a) are diffeomorphic to one another (it is well

known that if they are compact, then they are diffeomorphic to the toriTn, and if they are noncompact, then they are diffeomorphic to the cylinders

Tk ×Rn−k). The case (b) is more difficult. Below we shall consider a theorem from [5].

Suppose that the restriction f =f1|Xn+1 to a joint compact nonsingu-lar surface of the level of the rest of the n−1 integrals Xn+1 = {x ∈

M2n|fi(x) =ci, i= 2, n} is a Bott function, i.e., all critical points of this

restriction are organized into nondegenerate critical submanifolds (a critical submanifold Lk _⊂Xn+1 _{is nondegenerate if the restriction of the function}

f to every normal planePn+1−k _{has a nondegenerate Morse singularity at}

the pointPn+1−k_∩Lk_).

Let c1 be a critical value of the function f on the surface Xn+1, and let c = (c1, . . . , cn), i.e., c = (c1, c2, . . . , cn) ∈ Σ. Let Bc = F−1(c) be a

critical fiber of the moment mapping. Thus Bc = {f1 =c1} is a critical level surface off1onXn+1.

Theorem 3.1 (A. T. Fomenko). Each connected compact component B_c0 of the critical fiber Bc is homeomorphic to a set which is one of the

following four types: (1) a torus Tn_{; (2) the nonorientable manifolds K}n

0

andKn

1;(3)a torusTn−1; or(4)a cell complexT1n∪T2n obtained by

remov-ingn−1-dimensional toriT₁n−1fromT1n andT2n−1 fromT2n, which realize

and glueingTn

1 andT2ntogether by identifying only the toriT1n−1andT2n−1 (by means of a diffeomorphism). In cases (1)–(3) the critical fibers consist entirely of critical points of the functionf on which a maximum or a mini-mum is attained. In case (4)the critical points of f in the critical fiberB0

c

forms a torusTn−1_{(the result of glueing together the toriT}n−1

1 andT2n−1)

which is a “saddle” for the function f.

The manifolds K1n and K2n from Theorem 3.1 are the factor-sets of the torusTngenerated by the action of some group onTnwithout fixed points ([5]).

It is important to investigate a particular case of the integrable Hamil-tonian system v = sgradH on the four-dimensional symplectic manifold (M4, ω). For mechanical and physical reasons, it is useful to study the inte-grability effect on an individual isoenergetic surfaceQ3⊂M4 given by the equationH(x) =h, wherehis a regular value of the HamiltonianH. The restriction of the systemvonQ3 will be denoted by the same symbolv. In what followsQ3_{will be assumed to be the closed manifold. We also assume} that an additional integralf is the Bott function onQ3_{. Then to the critical} fibers from Theorem 3.1 there correspond the following manifolds:

(1) the torusT2; (2) Klein’s bottleK2; (3) the circleS1_;

(4) a piecewise smooth two-dimensional polyhedron with a singularity of the type of ”a fourfold line” (a transversal intersection of two planes).

The Hamiltonian H is called the nonresonance one onQ3 if the set of Liouville tori with irrational windings is everywhere dense in Q3.

We say that the Hamiltonian systemv is integrable in the Bott sense if among Bott functions there is an additional first integral ofv.

Definition 3.2 ([6]). Two integrable, in the Bott sense, nonresonance Hamiltonian systemsv1, v2on the oriented manifolds Q31, Q32are said to be topologically equivalent if there exists an orientation preserving diffeomor-phismg:Q3

1→Q32 which transforms the Liouville tori of the systemv1 to those of the systemv2 (critical tori are included in the number of Liouville tori), and the isolated critical circles to the isolated critical circles with the same orientation which is by the fieldv.

The partitioning of the manifold Q3 into Liouville tori and critical level surfaces of the Bott integralf is called the Liouville foliation onQ3 (for a given integrable nonresonance Hamiltonian systemv). Obviously, the diffeo-morphismg, appearing in Definition 3.2, is a leaf-to-leaf mapping between Liouville foliated manifolds.

An analogous definition can be introduced for the multidimensional case as well. Two completely integrable, in Bott sense, nonresonance Hamilto-nian systems v1 andv2 on the nonsingular level surfaces X1n+1 and X2n+1 are said to be topologically equivalent if there exists a diffeomorphism

h : X₁n+1 → X₂n+1 which transforms the Liouville tori of the system v1 to those of the system v2 (critical tori Tn are included in the number of Liouville tori), and critical fibersBc of the systemv1to those of the system

v2.

Note that the Liouville foliation is not a foliation in the sense of§2. Let v = sgradH be a completely integrable, in the Bott sense, nonres-onance Hamiltonian system on the isoenergetic surface Q3. As we already know, this system defines the Liouville foliation onQ3_{. Denote it byF.}

If fromQ3 we remove all critical (i.e., maximal, minimal and ”saddle”-type) circles S1, then on the resulting open manifoldQ′3 the foliation F induces a true foliationF′_{. Thus we have defined the vertical cohomology} groupsH_Fi′(Q

′

3_{),i= 0,1,2. Similarly, one can derive the vertical} cohomol-ogy groupsHi

F′(X ′

n+1_{) in the multidimensional case,i= 0,1, . . . , m.} If nowv1 andv2 are assumed to be the topologically equivalent systems on the manifolds Q31 and Q32, respectively, then the diffeomorphism g :

Q31→Q32which preserves the Liouville foliation obviously induces the leaf-to-leaf transforming diffeomorphismg′ _:Q′

3 1 →Q

′

3

2. Therefore the following theorem is valid.

Theorem 3.3. The cohomology groups H_Fi′(Q ′

3_{) are a topological}

in-variant of completely integrable, in the Bott sense, nonresonance Hamilton-ian systemsv= sgradH on the isoenergetic surfaceQ3_.

Note that in the multi-dimensional case the groupsH_Fi′(X ′

n+1_{),i= 0, n,} are also topological invariants.

Thus to determine a topological invariant we had to deal with the open manifoldQ′3. To avoid this, we shall slightly modify the definition of coho-mology groups.

The isoenergetic surfaceQ3_{can be considered as the set of Liouville tori,} circlesS1(which are maximal, minimal and ”saddle”-type) and rings of the typeS1×(0,1) (which are obtained from a hyperbolic critical fiber of type (4) (Theorem 3.1) by removing hyberbolic critical circles of the integralf). We denote this partitioning of the manifoldQ3 byP. ByS(Q3) we denote a Lie algebra of smooth vector fields onQ3 and byS(Q3) a Lie subalgebra of the algebra S(Q3) consisting of smooth vector fields on Q3 tangent to the manifolds of the partitioning P. The algebraC∞(Q3) is, obviously, a module over S(Q3) with respect to the product x·g = xg, x ∈ S(Q3),

By a k-dimensional cochain of the algebra S(Q3_{) with coefficients in}

C∞_(Q3_{) we shall mean an element of the spaceA}k_(S(Q3_{)) which is a} skew-symmetrick-linear functional onS(Q3_{) with values inC}∞_(Q3_{). The} differ-entialdk_:Ak_(S(Q3_))→Ak+1_(S(Q3_{)) is defined by the formula}

dkα(X1, . . . , Xk+1) = k+1

X

i=1

(−1)i−1Xiα(X1, . . . ,Xbi, Xi+1, . . . , Xk+1) +

+X

i<j

(−1)i+j_α([X

i, Xj], X1, . . . ,Xbi, . . . ,Xbj, . . . , Xk+1),

whereα∈Ak(S(Q3));X1, . . . , Xk+1∈S(Q3).

It is easy to verify thatdk◦dk−1= 0. Thus (A∗(S(Q3)), d∗) is a cochain complex. We denote the cohomologies of the complex (A∗(S(Q3)), d∗) by

H_P∗(Q3). Obviously,H_Pi(Q3) = 0, i >2.

We have also defined the cohomology groupsH_P∗(Xn+1) for the multidi-mensional case. Note that hereH_Pi(Xn+1) = 0, i > n.

Obviously, A∗(S(Q3)) is a graduated ring, and the differentiald is the antidifferentiation. Hence the cohomologies H_P∗(Q3) also acquire a ring structure.

If nowv1andv2are, as above, the topologically equivalent systems on the manifoldsQ3

1 and Q32, respectively, then the diffeomorphism g :Q31 →Q32 defines an isomorphism between the differential complexes (A∗_(S(Q3

1)), d∗) and (Ak_(S(Q3

2)), d∗). Therefore we have

Theorem 3.4. Letv= sgradH be an integrable, in the Bott sense, non-resonance Hamiltonian system on the isoenergetic surfaceQ3, and letP be the partitioning of the manifoldQ3 into Liouville tori, critical circles of the additional Bott integral f, and rings of the type S1_{×(0,1). Then the}

co-homology groupsH∗

P(Q3)are topological invariants of the systemv, i.e., to

the topologically equivalent systems there correspond the isomorphic groups.

An analogous theorem holds in the multidimensional case as well. Here the partitioning P consists of Liouville tori Tn, (n−1)-dimensional tori

Tn−1_{, and rings of the typeT}n−1_×(0,1). SinceA0(S(Xn+1)) =C∞(Xn+1), we have

H_P0(Xn+1) = ker(d0:C∞(Xn+1)→A1(S(Xn+1)) =

=ˆg∈C∞(Xn+1)|Xg= 0, ∀X ∈S(Xn+1‰= Inv_SC∞(Xn+1),

where Inv_SC∞(Xn+1) denotes the set of smooth functions onXn+1 whose restrictions are constant functions on the elements of the partitioningP.

the set of continuous functions onGwhose liftings toXn+1_{by the natural} projectionXn+1_{→Gare smooth functions.}

Remark 3.5. For the four-dimensional case, a complete topological invari-ant was introduced in [6]. This is the so-called labelled molecule consisting of a graph with edges to which are attached rational numbers from [0,1) or ∞. Note that in the four-dimensional case the above-mentioned G co-incides with the graph-molecule from [6]. Therefore the zero-dimensional groupsH_P0(Xn+1_{) already pick up “nonzero” information on the topological} equivalence of integrable Hamiltonian systems.

§ _4. The Case of a Spherical Pendulum

For a spherical pendulum the phase space is a cotangent bundle to the two-dimensional sphere T∗_S2_{, where S}2 _{= {x∈ R}3 _{: x}2

1+x22+x23 = 1}. Using the Riemannian metric onS2, we can identifyT∗_S2_{with the tangent} bundleT S2and define the Hamiltonian of the system by the energy function

E(x, v) =1

2hv, vi+x3, x∈S

2_{, v∈T}

xS2. (4.1)

A kinetic moment with respect to thex3-axis has the form

I(x, v) =x1v2−v1x2. (4.2)

The critical points of the moment mapping F = (E, I) : T S2 → R2

are the points x=±(0,0,1), v = 0 and v =α(−x2, x1,0), 1 +α2x3 = 0,

x36=±1. The corresponding singular values of the moment mapping are

F = (±1,0) and F=1 2α

2₋3 2α

−2_{, α−α}−3‘_{, where |α|>1.(4.3)}

Whenαtends to±1, the point of the curve defined by formula (4.3) tends to (−1,0). For I 6= 0, i.e., forx 6=±(0,0,1), we can introduce the polar coordinates

x1= sinϕcosθ, x2= sinϕsinθ, x3= cosϕ,

whereθ∈[0.2π],ϕ∈]0, π[.

The functionsE andI in terms of these coordinates are written as

E= 1 2ϕ˙

2_+V

I(ϕ), I= (sin2ϕ) ˙θ, (4.4)

whereVI(ϕ) = 12(sin

2_{ϕ) ˙θ}2_{+ cosϕis the effective potential, and (ϕ, θ,ϕ,˙ θ˙)} define the local coordinate system on T S2. The image F is given by the relationI =α−α−3_{,E ≥} 1

2α 2₋3

2α−

2_{, |α| ≥ 1. For regular values of the} mapping F we haveE > 1₂α2−3

2α−

✲ ✻

s

s (0,−1) (0,1)

E

I T S2

[image:16.612.195.364.68.260.2]

✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘_✘✘✘✘✘✘✘✘✘ ✘✘_✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘✘✘✘✘✘ ✘✘✘_✘✘✘✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘✘✘✘✘✘ ✘✘✘_✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘✘✘✘✘✘ ✘✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘_✘✘✘✘✘✘_✘✘✘ ✘ ✘ ✘✘✘✘✘ Fig. 1

Let us consider the functionIon the isoenergetic surfaceQ3={E= 1₂}. If (ϕ, θ,ϕ,˙ θ˙) is a critical point of the function I onQ3_{, then at that point} gradE and gradI are collinear. Hence we obtain ϕ0 = arccos(1−

√ 13 6 ), ˙

θ1 =

q√

13+1

2 , ˙θ2 =−

q√

13+1

2 . Thus gives us two isolated critical circles

S1={(ϕ0, θ,ϕ˙ = 0,θ˙1)|θ∈[0,2π]}andS2={(ϕ0, θ,ϕ˙ = 0,θ˙2)|θ∈[0,2π]}. Using the method of constant Lagrange multipliers, we can conclude that

S1 is the maximal critical circle, andS2 is the minimal one.

By Morse’s lemma it follows that the compact isoenergetic surface cor-responding to the values of energy E close to −1 is the three-dimensional sphere S3. Therefore Q3 is also diffeomorphic toS3, since the points ±1 are critical values of the integral E. The isoenergetic surface Q3 can be represented as two solid toriS1

1×D2 andS21×D2 glued together along the boundary tori by means of a diffeomorphism which transforms the parallel to the meridian, and vice versa (here D2 _{is a two-dimensional disk, and} the central circles of the solid toriS1

1× {0} andS21× {0}coincide with the critical circles of the integralI); the toriS1

i ×Sr,r∈(0,1],i= 1,2, where

Sr ={x∈ D2| kxk = r}, are the usual Liouville tori of the Hamiltonian

systemv=sgradE. In that case, using the notation from [6], the labelled molecule has the form

A•————

r=0 •A, and the factor setGcoincides with the segment.

Now consider the functionIon the isoenergetic surfaceQ3={E=α >

1}. A direct calculation shows that, as in the preceding case, for the integral

I we have two isolated critical circles S1 ={(ϕ0, θ,0,θ˙1)|θ ∈ [0,2π]} and

S2 = {(ϕ0, θ,0,θ˙2)|θ ∈ [0,2π]}, where ϕ0 = arccosα− √

α2₊₁₂

3 and ˙θ1 = −q_√ 3

α2_+12−α, ˙θ2=

q

3 √

In terms of the coordinates (ϕ, θ,ϕ,˙ θ˙) the Euclidean metric is written as

ds2_=dϕ2_{+ sin}2_ϕdθ2 _{so that the the equationE=αtakes the form}

kξk2+ 2 cosϕ= 2α,

where ξ = ( ˙ϕ,θ˙) is the tangent vector at the point (ϕ, θ) ∈ S2_{, and kξk} denotes the norm of ξ. Hence it follows that the norm of the vector ξ is a function ofϕ, and kξk 6= 0. Therefore Q3 is diffeomorphic to the space

T1S2, where T1S2 ={(a, ξ)|x ∈S2, kξk = 1}. As is wellknown, T1S2 is diffeomorphic to the three-dimensional projective space RP3 so thatQ3≈

RP3. As above,Q3can be represented as two solid toriS11×D2andS21×D2 glued together along the boundary tori by means of the diffeomorphism

h whose corresponding induced mapping h_∗ : H1(T2;Z) → H1(T2;Z) is

given by the matrix

’

1 2 1 1

“

. Thus for the integrable Hamiltonian system

v= sgradE onQ3_{the labelled molecule has the form}

A•————

r=1 2

•A,

and the factor set again coincides with the segment. This example shows that vertical cohomologies are not a complete topological invariant of inte-grable Hamiltonian systems.

Acknowledgments

The author expresses his gratitude to Dr. Z. Giunashvili for helpful discussions.

This research was partly carried out under Grant No. RJV200 from the Intenational Science Foundation.

References

1. Jos´e M. Figueroa-O’Farrill, A topological characterization of classical BRST cohomology. Commun. Math. Phys. 127(1990), 181–186.

2. M. Henneaux and C. Teitelboim, BRST cohomology in classical me-chanics. Commun. Math. Phys. 115(1988), 213–230.

3. J. Fisch, M. Henneaux, J. Stasheff, and C. Teitelboim, Existence, uniqueness, and cohomology of the classical BRST charge with ghosts of ghosts. Preprint,1988.

6. A. V. Bolsinov, S. V. Matveev, and A. T. Fomenko, Topological classification of integrable Hamiltonian systems with two degrees of freedom. A list of systems of small complexity. (Translated from Russian) Math. Surveys45(1990), No. 2, 49–77.

(Received 1.03.1996)

Author’s address: