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GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 483-500

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 cohomologies and their application to completely integrable Hamiltonian systems are studied.

§ 1. Introduction
If on a smooth Riemannian manifold M n we have a distribution V of
dimension k, which is actually a smooth section of the Grassman fiber bundle Gk (T M n ) → M n adjoint to the tangent fibration T M n to the manifold
M n , 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 M n → V ⊂ T M n be a
natural projection. The operator P defines the mapping P ∗ : Λ∗ (M n ) →
Λ∗ (M n ) ((Λ∗ (M n ), d∗ ) is the de Rham differential complex) by the formula (P ∗ α)(X1 , . . . , Xq ) = α(P X1 , . . . , P Xq ), where α ∈ Λq (M n ) and
X1 , . . . , Xq ∈ S(M n ) are the smooth vector fields on M n .
Denote by ΛV∗ (M n ) all fixed points of the operator P ∗ . In what follows we
shall consider the case where V is integrable, i.e., where M n is partitioned

into leaves and the tangent space to the leaf that passes through the point
x ∈ M n is Vx . Then the pair (Λ∗V (M n ), d∗V ) forms a differential complex
with the differential d∗V = P ∗ ◦ d∗ .
The cohomologies of the complex (ΛV∗ (M n ), d∗V ) are called vertical and
denoted by HV∗ (M n ) ([1]). It has turned out that these cohomologies coincide with those of the classical BRST operator ([2], [3]).
In §2 the vertical cohomologies are defined without fixing the metric on
M n , and some of their functorial properties are studied. The FOL category
1991 Mathematics Subject Classification. 53C12, 70H05.
Key words and phrases. Foliation, vertical cohomology, Hamiltonian systems, isoenergetic surfaces, Liouville torus.

483
c 1998 Plenum Publishing Corporation
1072-947X/98/0900-0483$15.00/0 

484

Z. TEVDORADZE

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. Moreover, for a leaf-to-leaf transforming mapping f : (M n , 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. Finally, a double complex (K ∗∗ , D∗ ) is constructed for the countable covering
U = {uα }α∈A of the manifold M n . It is shown that the cohomologies of
(K ∗∗ , D∗ ) are isomorphic to the vertical cohomologies. A combinatorial
ˇ
definition of vertical comologies in the Cech
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 cohomologies, 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 M n be a smooth ndimensional manifold, and V be a k-dimensional involutive distribution on

M n whose foliation is denoted by F. The bundle of exterior p-forms on V
is denoted by Ap (V ), and the set of smooth sections of the bundle Ap (V )
by ΛpF (M n ). Then ΛpF (M n ) is a module over the algebra of infinitely differentiable functions C ∞ (M n ) on M n .
Let α ∈ ΛpF (M n ), and let X1 , . . . , Xp be the smooth vector fields on M n
which are tangent to the leaves of the foliation F, i.e., they are the smooth
pF
M n . Then the mapping defined by the formula
sections of the bundle V →
j

α(X1 , . . . , Xp ) : x 7−→ α(X1 (x), . . . , Xp (x)),

(2.1)

which on the module of sections S(V ) of V assigns an exterior p-form to an
element α ∈ ΛpF (M n ), is an isomorphism.
Let us now define the operator
p
n
: ΛpF (M n ) → Λp+1

dF
F (M )

VERTICAL COHOMOLOGIES AND THEIR APPLICATION

485

by the relation
(dpF α)(X1 , . . . , Xp+1 ) =
+

X

(−1)

i 1.
Denote by FOL a category whose objects are smooth foliations (M n , F),
and morphisms from (M1n , 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
(M1n , F1 ) and (M2n , F2 ), then h defines the morphism h∗ between the
C ∞ (M2m )-module Λ∗F2 (M2m ) and the C ∞ (M1n )-module Λ∗F1 (M1n ) by the formula
(h∗ α)(X1 , . . . , Xp )(x) = α(h⊤ X1 (x), . . . , h⊤ Xp (x)),

(2.4)

where α ∈ ΛpF2 (M2 ) and X1 , . . . , Xp ∈ S(V1 ), V1 is the distribution associated with F1 , and h⊤ is the tangent mapping to h. We have the commutative diagram
h∗

Λ∗ (M2m ) −−−−→ Λ∗ (M1n )


i∗ .
i∗
y1
y2
h∗

∗

(M1n )
Λ∗F2 (M2m ) −−−−→ ΛF
1

A direct calculation shows that h∗ is a cochain mapping, i.e., d∗F1 ◦ h∗ =
h
◦ d∗F2 . Hence we have
∗+1

VERTICAL COHOMOLOGIES AND THEIR APPLICATION

487

Proposition 2.2. The mapping i∗ is a natural transformation of the de
Rham functor Λ∗ to the functor Λ∗F , where Λ∗ and Λ∗F are the contravariant
functors from the FOL category to the category of differential graded algebras
and their homomorphisms.
2.2. A Homotopy Axiom for Vertical Cohomologies. Let (M n , F )
be a foliation of dimension k. On the manifold M n × R we define naturally
a foliation Fb of dimension k + 1 whose leaves are manifolds Lα × R, α ∈ A,

where Lα , α ∈ A, are the leaves of the foliation F.

Lemma 2.3. The projection π : M n × R → M n defines an isomorphism
in vertical cohomologies.
π

Proof. Consider the zero section s of the trivial bundle M n × R −
→ M n , i.e.,
n
s(x) = (x, 0), x ∈ M . Then the mappings π and s are the leaf-to-leaf transforming mappings which define the cochain mappings π ∗ : (Λ∗F (M n ), d∗F ) →
(Λ∗ (M n × R), d∗ ) and s∗ : (Λ∗ (M n × R), d∗ ) → (Λ∗F (M n ), d∗F ). Since
b
b
b
b
F
F
F
F
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 and π ∗ ◦ s∗ .
∗
(M n × R) can be uniquely represented by
Note that each form from ΛF
linear combinations of the following two types of forms:
(I) (π ∗ ϕ) · f, ϕ ∈ Λ∗F (M n ), f ∈ C ∞ (M n × R);
(II) (π ∗ ϕ) ∧ dt · f, ϕ ∈ Λ∗F (M n ), f ∈ C ∞ (M n × R),
where t is the coordinate on the straight line R. Define the operator K ∗ :
Λ∗ (M n × R) → Λ∗ −1 (M n × R) as follows:
b
b
F
F
K ∗ ((π ∗ ϕ) · f ) = 0,
∗

∗

∗

K ((π ϕ) ∧ dt · f ) = π ϕ ·

Zt

f dt.

0

A direct calculation shows that the relation
1 − π ∗ ◦ s∗ = (−1)q−1 (dq−1 K q − K q+1 dq )
b
b
F
F

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

Definition 2.4. Two leaf-to-leaf transforming mappings f, g : (M1n , F1 )
→ (M2m , F2 ) between the foliations (M1n , F1 ) and (M2m , F2 ) are called leafto-leaf transforming homotopic if there exists a leaf-to-leaf transforming

mapping
F : (M1n × R, Fb1 ) → (M2m , F2 )

488

Z. TEVDORADZE

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 homotopic mappings induce identical mappings in vertical cohomologies.
Proof. Let f, g : (M1n , F1 ) → (M2m , F2 ) be the leaf-to-leaf transforming
homotopic mappings and F : (M1n × R, Fb1 ) → (M2m , F2 ) be the homotopy
between f and g. Denote by s0 and s1 the sections s0 , s1 : (M1n , F1 ) →
(M1n × R, Fb1 ), s0 (x) = (x, 0), s1 (x) = (x, 1), x ∈ M1n . Then f = F ◦ s1 and

g = F ◦ s0 . Hence we have f ∗ = s∗1 ◦ F ∗ and g ∗ = s∗0 ◦ F ∗ . From the proof
of Lemma 2.3 it follows that s∗1 = s∗0 = (π1∗ )−1 , where π1 : M1n × R → M1n
is the projection. Therefore f ∗ = g ∗ .
The foliations (M1n , F1 ) and (M2m , F2 ) will be said to be of the same
homotopy type if there are leaf-to-leaf transforming smooth mappings f :
M1n → 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
(M1n , F1 ) and (M2m , F2 ), respectively.
Corollary 2.6. If two foliations (M1n , F1 ) and (M2m , F2 ) are of the same
homotopy type, then their vertical cohomologies are isomorphic.
2.3. Relative Vertical Cohomologies. Let (M n , F1 ) and (N m , F2 ) be
two foliations, and let f be a leaf-to-leaf transforming smooth mapping
f : M n → N m . Define the differential complex
(Λ∗ (f ), d∗ ),

Λ∗ (f ) = ⊕ Λq (f ),
q≥0

where
q
n
Λq (f ) = ΛF
(N m ) ⊕ Λq−1
F1 (M ),
2

d∗ (ω, θ) = (−d∗F2 ω, f ∗ ω + d∗F1 θ).

We easily verify that d2 = 0 and denote the cohomology groups of this
complex by H ∗ (f ). Note that the complex (Λ∗ (f ), d∗ ) is the cone of the
∗
∗
(N m ) → ΛF
(M n ). If we regraduate the complex
cochain mapping f ∗ : ΛF
2
1
p
p−1
e (M n ) ≡ Λ (M n ), then we obtain an exact sequence of
Λ∗F1 (M n ) as Λ
1
F1
F1
differential complexes
β

α

e ∗ (M n ) −
→Λ
0−
→ Λ∗F2 (N m ) −
→ Λ∗ (f ) −
→0
F1

(2.5)

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

β∗

δ∗

→ ··· .
··· −
→ HFq−1
(M n ) −→ H q (f ) −→ HFq 2 (N m ) −→ HFq 1 (M n ) −
1

VERTICAL COHOMOLOGIES AND THEIR APPLICATION

489

q
It is easily seen that δ ∗ = f ∗ . Let ω ∈ ΛF
(N m ) be the closed form, and
2
q−1
∗
q
(ω, θ) ∈ Λ (f ). Then d(ω, θ) = (0, f ω + dF1 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
β∗

α∗

f∗

α∗

··· −
→ HFq−1
(M n ) −→ H q (f ) −→ HFq 2 (N m ) −→ HFq 1 (M n ) −→ · · · . (2.6)
1
Corollary 2.7. If the foliations (M n , F1 ) and (N m , F2 ) are of the p-th
and q-th dimension, respectively, then
(N m ) is an epimorphism,
(i) β ∗ : H p+1 (f ) → HFp+1
2
q
α∗ : HF1 (M n ) → H q+1 (f ) is an epimorphism,
β ∗ : H i (f ) → HFi 2 (N m ) is an isomorphism for i > p + 1,
α∗ : HFi 1 (M n ) → H i+1 (f ) is an isomorphism for i > q;
(ii) H i (f ) = 0 for i > max{p + 1, q}.
Theorem 2.8. If f, g : (M n , F1 ) → (N m , F2 ) are leaf-to-leaf transforming homotopic mappings, then H ∗ (f ) = H ∗ (g).
Proof. Let F : (M n × R, Fb1 ) → (N m , F2 ) be the homotopy mapping between f and g. Let s0 and s1 be the zero and the unit section, respectively,
π
→ M n . Then F ◦ s0 = g and F ◦ s1 = f .
of the trivial bundle M n × R −
Hence we have a homomorphism between the short exact sequences
′

′

β
α
e ∗ (M n × R) −−−
0 −−−−→ Λ
−→ Λ∗ (F ) −−−−→ Λ∗F2 (N m ) −−−−→ 0
b1 
F


s∗
id⊕s∗

.
y1
y
yid
1

0 −−−−→

e ∗ (M n )
Λ
F1

−−−−→ Λ∗ (f ) −−−−→ Λ∗F2 (N m ) −−−−→ 0

This homomorphism defines a homomorphism between the corresponding
long cohomologic sequences
q+1
q
(N m ) → H q+1 (M n ×R) → ···
··· → HF
(N m ) → H q (M n ×R) → H q+1 (F ) → HF
2

q
··· → HF

2

b

2

F1

↓id
(N m ) →

q
HF

1

↓s∗
1
(M n )

2

s∗1 .

b

F1

↓γ
↓id
q+1
→ H q+1 (f ) → HF
(N m ) →

s∗1

↓s1∗
q+1
(M n )
HF
1

,

→ ···

Since
is an isomorphism
where γ is the mapping induced by id ⊕
(Lemma 2.3), by virtue of the lemma on five homomorphisms we conclude
that γ is also an isomorphism, i.e., H ∗ (f ) ≈ H ∗ (F ). By a similar reasoning
we can conclude that H ∗ (g) ≈ H ∗ (F ).
If (M n , F1 ) is a subfoliation of the foliation (W m , F2 ), i.e., the embedding
j

M n ֒→ W m is simultaneously a leaf-to-leaf transforming mapping, then the
cohomology algebra H ∗ (j) will be said to be the algebra of relative vertical
cohomologies. Denote it by HF∗ 2 ,F1 (W ; M ).

490

Z. TEVDORADZE

Now sequence (2.6) can be rewritten as
j∗

β∗

α∗

··· −
→ HFq−1
(M ) −→ HFq 2 ,F1 (W ; M ) −→ HFq 2 (W ) −→
1
j∗

α∗

−→ HFq 1 (M ) −→ · · · .

(2.7)

Note that if we forget the structure of the foliation, then, as is known,
j

the embedding M n ֒→ W m defines a long exact cohomological sequence
of the pair (W m , 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 (W m , M n ) in de Rham’s
theory and sequence (2.7).
2.4. The Generalized Mayer–Vietoris Principle for Vertical Cohomologies. A Combinatorial Definition of Vertical Cohomologies.
Let (M n , F) be a smooth foliation of dimension k; let U = {uα }α∈A be an
ˇ
open countable covering of the manifold M n . Similarly to Cech-de
Rham’s
theory, we define a double complex which will be used to calculate vertical
cohomologies of the foliation (M n , F).
`
the
Denote by uα0 ···αp the intersection of open sets u0 , . . . , up and by
disjunctive union. Then we have a sequence of open sets
M n ←−

a
α0

∂0
uα0 ←−
←−
∂
1

a

α0