DOI: 10.2478/auom-2014-0011
An. Şt. Univ. Ovidius Constanţa
Vol. 22(1),2014, 127–139
A note on coeffective 1–differentiable
cohomology
Cristian Ida and Sabin Mercheşan
Dedicated to Professor Mirela Ştefănescu at her 70-th anniversary
Abstract
After a brief review of some basic notions concerning 1–differentiable
e
e
cohomology, named here d-cohomology,
we introduce a Lichnerowicz d–
cohomology in a classical way. Next, following the classical study of
coeffective cohomology, a special attention is paid to the study of some
problems concerning coeffective cohomology in the graded algebra of 1–
differentiable forms. Also, the case of an almost contact metric (2n+1)–
dimensional manifold is considered and studied in our context.
1
Introduction
The 1–differentiable cohomology was introduced and intensively studied by A.
Lichnerowicz in [16, 9] in the context of symplectic and contact manifolds and
in [17, 18] in the context of Poisson or Jacobi manifolds. Further signifiant
developments of a such cohomology in the context of Lichnerowicz-Jacobi cohomology are given by M. de León, B. López, J. C. Marrero and E. Padrón,
see for instance [13, 14, 15]. Here we consider the 1–differentiable cohomology of a manifold as follows: for every 1–form η on a smooth manifold M we
e • (M ) = Ω• (M ) ⊕ Ω•−1 (M )
define a coboundary operator de on the complex Ω
e ψ) = (dϕ − dη ∧ ψ, −dψ), where Ω• (M ) = ⊕p≥0 Ωp (M ); Ωp (M ) is
by d(ϕ,
Key Words: 1–differentiable form, coeffective cohomology, almost contact structure.
2010 Mathematics Subject Classification: Primary 58A10, 58A12.; Secondary 53D15.
Received: August 2013
Accepted:November 2013
127
128
Cristian Ida and Sabin Mercheşan
e
the space of p–forms on M . The resulting cohomology is named here d–
cohomology of M . Also, we notice that an harmonic and C–harmonic theory
of 1–differentiable forms on Sasakian manifolds with respect to the operator
de is recently studied in [12].
The coeffective cohomology was introduced by T. Bouché [3] for symplectic
manifolds. Further signifiant developments are given by D. Chinea, M. de León
and J. C. Marrero for cosymplectic manifolds [6] and M. Fernández, R. Ibáñez,
M. de León for contact manifolds [7] and other papers by these authors.
The purpose of this note is to extend the study of coeffective cohomology
in the context of 1–differentiable forms.
The paper is organized as follows. In Section 2, after a briefly review of
some basic notions concerning to 1–differentiable cohomology
e
e
(or d–cohomology
associated
n to an one form η), a vanishing invariant d–class
of order 2p+1, p = 1, . . . , 2 is defined in terms of the 1–form η and following
the general study of Lichnerowicz cohomology (also known as Morse-Novikov
e
cohomology) we define a Lichnerowicz d–cohomology
in the graded algebra of
•
e
e
1–differentiable forms Ω (M ). Also some vanishing Lichnerowicz d–classes
are
given. In Section 3 are given the main results of the paper. Taking into account
the classical construction of coeffective cohomologies which is strongly related
to closed forms, [3, 6, 7, 8], we give some coeffective cohomologies in the graded
e
e
e • (M ), ∧
e ) and some relation with d–cohomology.
algebra (Ω
Since (η, 1) is d–
closed, firstly we define and we study an (η, 1)–coeffective cohomology. Next
e
using the fact that (dη, η) is closed with respect to Lichnerowicz d–differential
e
e
e
d(η,1) = d + (η, 1)∧ we define and we study a (dη, η)–coeffective cohomology.
Also, the case when η is the fundamental 1–form of an almost contact metric (2n + 1)–dimensional manifold is considered and studied. We obtain that
the (dη, η)–coeffective cohomology groups of an almost contact metric manifold of finite type have finite dimension (called the (dη, η)–coeffective numbers
and denoted by e
cp (M, (dη, η))). Also, in this case, we prove that the (dη, η)–
coeffective numbers are bounded by topological numbers depending on the
Betti numbers of the manifold. The methods used here are similarly and
closely related to those used by [6, 7, 8].
2
2.1
1–differentiable p–forms and
1–differentiable cohomologies
e
d–cohomology
Let us consider the field Ω0 (M ) = F(M ) of smooth real valued functions
defined on M . For each p = 1, . . . , n = dim M denote by Ωp (M ) the module
of p–forms on M and by Ω(M ) = ⊕p≥0 Ωp (M ) the exterior algebra of M .
A NOTE ON COEFFECTIVE 1–DIFFERENTIABLE COHOMOLOGY
129
e p (M ) = Ωp (M ) ⊕ Ωp−1 (M ) and its elements are pair forms
We denote Ω
(ϕ, ψ) called 1–differentiable p–forms (after a terminology used in [16]). As
in formula (5.5) from [17], we can define a wedge product of 1–differentiable
0
0
e p+p (M ) by:
e p (M ) × Ω
e p (M ) → Ω
e:Ω
forms ∧
0
0
0
0
0
e (ϕ , ψ ) = (ϕ ∧ ϕ , (−1)p ϕ ∧ ψ + ψ ∧ ϕ )
(ϕ, ψ)∧
(2.1)
e • (M ), where (ϕ, ψ) ∈ Ω
e p (M ),
to be the exterior product on the space Ω
0
0
0
p
e (M ). By this definition, we notice that for an 1-differentiable
(ϕ , ψ ) ∈ Ω
0-form (f, 0), where f is a smooth function on M we have (f, 0) · (ϕ, ψ) =
(f ϕ, f ψ). Also, one easily verifies that:
0 0
00
00
0
0
00
00
e (ϕ , ψ ) + (ϕ , ψ ) = (ϕ, ψ)∧
e (ϕ , ψ ) + (ϕ, ψ)∧
e (ϕ , ψ ),
(ϕ, ψ)∧
0 0
00
00
0
0
00
00
e (ϕ , ψ )∧
e (ϕ , ψ ) = (ϕ, ψ)∧
e (ϕ , ψ ) ∧
e (ϕ , ψ )
(ϕ, ψ)∧
and
0
0
0
0
0
e (ϕ , ψ ) = (−1)pp (ϕ , ψ )∧
e (ϕ, ψ),
(ϕ, ψ)∧
e • (M ), ∧
e ) is a graded algebra.
which say that (Ω
e • (M ), ∧
e ):
For any 1–form η on M we define the following operator in (Ω
e ψ) = (dϕ − Lψ, −dψ),
e p (M ) → Ω
e p+1 (M ) , d(ϕ,
de : Ω
(2.2)
where L : Ωp (M ) → Ωp+2 (M ) is given by Lϕ = dη ∧ ϕ.
An easy calculation shows that de2 = e
0, where e
0 := (0, 0).
•
e
e
e • (M ), d)
Denote by H (M ) the cohomology of the differential complex (Ω
e
called 1–differentiable cohomology of M (or d–cohomology of M ).
Remark 2.1. If we replace dη by any 2–closed form, a such differential complex may be defined on every manifold M endowed with a closed 2–form, for
instance symplectic or Kähler manifolds.
We notice that this complex has local cohomology at both p = 0 and p = 1.
e 1 (M )} = {(f, 0) | f = const.} and the
e 0 (M ) → Ω
Specifically, we have ker{de : Ω
cohomology at p = 1 is generated by (η, 1).
e
e 1 (M ).
Proposition 2.1. The d–class
[(η, 1)] is nonzero in H
Proof. If we suppose that [(η, 1)] = e
0 then there exists an 1-differentiable
0
0
e 0) that is
e
zero form (f, 0) ∈ Ω (M ) = Ω (M ) ⊕ {0} such that (η, 1) = d(f,
imposible.
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Cristian Ida and Sabin Mercheşan
e p (M ), α(ϕ) = (ϕ, 0) and
Let us consider now, the mappings α : Ωp (M ) → Ω
e p (M ) → Ωp−1 (M ), β(ϕ, ψ) = ψ for all ϕ ∈ Ωp (M ) and ψ ∈ Ωp−1 (M ),
β :Ω
e • (M ) with
respectively. Then, we have the following result which relates H
•
the de Rham cohomology HdR (M ).
Proposition 2.2. Let M be a n–dimensional smooth manifold. Then:
(i) The mappings α and β induce an exact sequence of complexes
β
α
e −→
e • (M ), d)
0 −→ (Ω• (M ), d) −→ (Ω
(Ω•−1 (M ), −d) −→ 0.
(ii) This exact sequence induces a long exact cohomology sequence
α∗
β∗
∗
δp−1
p
e p (M ) −→ H p−1 (M ) −→ H p+1 (M ) −→ . . . ,
. . . −→ HdR
(M ) −→ H
dR
dR
(2.3)
∗
where the connecting homomorphism δp−1
is defined by
p−1
∗
δp−1
[ψ] = [−Lψ] = 0, for any [ψ] ∈ HdR
(M ).
(2.4)
From above proposition, one gets
Corollary 2.1. Let M be a n–dimensional smooth manifold. Then, for all p,
we have
p
p−1
e p (M ) ∼
H
(2.5)
= HdR (M ) ⊕ HdR (M ).
e p (M ) = bp (M ) + bp−1 (M ), where bp (M ) is the p–
Consequently, dim H
e p (M ) is a topological
th Betti number of M . In particular, eb(M ) := dim H
invariant of M , for all p. Also, by applying the Poincaré duality for the de
•
Rham cohomology HdR
(M ) in (2.5) we obtain the following Poincaré duality
for our cohomology:
∗
e c2n+2−p (M ) ,
e p (M ) ∼
(2.6)
H
= H
where the index ”c” denotes the cohomology with compact support. e ∧ (dη)p , (dη)p ) = (0, 0), p = 1, . . . , n and
Also, it is easy to see that d(η
2
e
so we have an invariant d–class
of order 2p + 1 of the n–dimensional smooth
manifold M
h i
e 2p+1 (M ), p = 1, . . . , n .
[(η ∧ (dη)p , (dη)p )] ∈ H
(2.7)
2
p
e
By direct calculus we have (η ∧ (dη)p , (dη)p ) = d((dη)
, −η ∧ (dη)p−1 ) which
e
say that the d–class
[(η ∧ (dη)p , (dη)p )] vanish.
A NOTE ON COEFFECTIVE 1–DIFFERENTIABLE COHOMOLOGY
2.2
131
e
Lichnerowicz d–cohomology
e
As well as we seen the 1–differentiable 1–form (η, 1) is d–closed.
Thus, as in the
classical Lichnerowicz cohomology (also known as Morse-Novikov cohomology)
e • (M ), ∧
e ):
we define the following operator in the graded algebra (Ω
e p (M ) → Ω
e p+1 (M ) , de(η,1) = de + (η, 1)∧
e.
de(η,1) : Ω
(2.8)
By direct calculus we obtain de2(η,1) = e
0, hence we get a differential complex
e • (M ), de(η,1) called the Lichnerowicz complex of 1–differentiable forms; its
Ω
e
e • (M ) is called Lichnerowicz d–cohomology
cohomology H
of 1–differentiable
(η,1)
e
forms on M . Note that d(η,1) does not satisfy the Leibniz property, since for
0
e p (M ) and (ϕ0 , ψ 0 ) ∈ Ω
e p (M ) we have
any (ϕ, ψ) ∈ Ω
0
0
0
0
0
0
e ψ)∧
e (ϕ , ψ ) = d(ϕ,
e (ϕ , ψ )+(−1)p (ϕ, ψ)∧
e de(η,1) (ϕ , ψ ). (2.9)
de(η,1) (ϕ, ψ)∧
e
e • (M ) does not have a ring strucThus the Lichnerowicz d–cohomology
H
(η,1)
e • (M ) is a H
e • (M )–module.
ture. The formula (2.9) also implies that H
(η,1)
We have
e
e
Proposition 2.3. The Lichnerowicz d–cohomology
depends only on the d–
p
p
e
e
class of (η, 1). In fact, we have the isomorphism H
(M ) ≈ H(η,1) (M ).
e
(η,1)+d(f,0)
Proof. Since
de(η,1) (ef , 0) · (ϕ, ψ) = (ef , 0)de(η,1)+d(f,0)
(ϕ, ψ)
e
it results that the map [(ϕ, ψ)] 7→ [(ef , 0) · (ϕ, ψ)] is an isomorphism between
ep
e p (M ).
H
(M ) and H
e
(η,1)
(η,1)+d(f,0)
By straightforward caculus we obtain
0
e p (M ) and (ϕ0 , ψ 0 ) ∈ Ω
e p (M ) we have
Proposition 2.4. For any (ϕ, ψ) ∈ Ω
0
0
0
0
0
0
e (ϕ , ψ ) = de(η,1) (ϕ, ψ)∧
e (ϕ , ψ ) + (−1)p (ϕ, ψ)∧
e de−(η,1) (ϕ , ψ ).
de (ϕ, ψ)∧
(2.10)
Consequently
0
0
0
0
e de−(η,1) (ϕ , ψ ) = de (ϕ, ψ)∧
e de−(η,1) (ϕ , ψ ) .
de(η,1) (ϕ, ψ)∧
Formula (2.10) yields the induced map
e • (M ) × H
e•
e•
H
(η,1)
−(η,1) (M ) → H (M ).
(2.11)
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Cristian Ida and Sabin Mercheşan
Now, using (2.1), (2.8) and (2.9), a straightforward calculus leads to
de(η,1) (dη)p , η ∧ (dη)p−1 = e
0 , de(η,1) η ∧ (dη)p−1 , (dη)p−1 = e
0.
(2.12)
Thus, we can define two cohomology classes
e 2p (M ) , η ∧ (dη)p−1 , (dη)p−1 ∈ H
e 2p−1 (M )
(dη)p , η ∧ (dη)p−1 ∈ H
(η,1)
(η,1)
(2.13)
e
called the Lichnerowicz d–classes of M .
Also, it is easy to see that
(dη)p−1
η ∧ (dη)p−1
p
p−1
e
,−
(dη) , η ∧ (dη)
= d(η,1)
2
2
and
p−1
η ∧ (dη)
p−1
, (dη)
= de(η,1)
η ∧ (dη)p−2
(dη)p−1
,−
2
2
e
which say that the Lichnerowicz d–classes
from (2.13) vanishes.
e
The operator d(η,1) will be an important tool in the study of a (dη, η)–
coeffective cohomology in the next section.
3
e
Coeffective d–cohomology
Let us consider again η a differential one form on M . Taking into account
the classical construction of coeffective cohomologies which is strongly related
to closed forms, see for instance [7, 8], the aim of this section is to give some
e • (M ), ∧
e ) and some relation
coeffective cohomologies in the graded algebra (Ω
e
e
with d–cohomology. Since (η, 1) is d–closed, firstly we define and we study an
(η, 1)–coeffective cohomology. Next using the fact that (dη, η) is closed with
respect to Lichnerowicz differential de(η,1) we define and we study a (dη, η)coeffective cohomology. Also, the case when η is a contact form of an almost
contact metric (2n + 1)–dimensional manifold is considered and studied.
3.1
e
(η, 1)–coeffective d–cohomology
We define the operator
e (η,1) : Ω
e p (M ) → Ω
e p+1 (M ) , L
e (η,1) (ϕ, ψ) = (ϕ, ψ)∧
e (η, 1).
L
The space
n
o
ep (M ) = ker L
e (η,1) : Ω
e p (M ) → Ω
e p+1 (M )
A
(η,1)
(3.1)
133
A NOTE ON COEFFECTIVE 1–DIFFERENTIABLE COHOMOLOGY
is called the subspace of (η, 1)–coeffective 1–differentiable forms on the smooth
e
e
e
manifold
M . Since (η, 1) is d–closed, L(η,1) and d commute, which implies
e• (M ), de is a differential subcomplex of the differential complex
that A
(η,1)
•
e(η,1) (M ) is called (η, 1)–coeffective d–
e
e (M ), de . Its cohomology H
ep A
Ω
cohomology of M . If this cohomology
is finite, we
define the (η, 1)–coeffective
p
e
e
numbers by e
cp (M, (η, 1)) = dim H A(η,1) (M ) .
e
e
In the following we relate the (η, 1)–coeffective d–cohomology
with the d–
cohomology by means of a long exact sequence in cohomology. Consider the
following natural short exact sequence for any degree p:
e
L
(η,1)
i ep
ep (M ) −→
e (η,1) −→ e
e (η,1) = A
e
0.
0 −→ ker L
Ω (M ) −→ Imp+1 L
(η,1)
e
(3.2)
e (η,1) and de commute, (3.2) becomes a short exact sequence of differenSince L
tial complexes.
Therefore, we can consider the associated long exact sequence in cohomology:
e∗
e∗
L
(η,1)
i
e(η,1) (M ) −→
ep A
e p (M ) −→
. . . −→ H
H
δe∗
p
e(η,1) (M ) −→ . . .
e p+1 ImL
e (η,1) −→
e p+1 A
H
H
(3.3)
e∗
e
where ei∗ and L
(η,1) are the homomorphisms induced in cohomology by i and
e (η,1) , respectively, and δe∗ is the connecting homomorphism defined by
L
p
h
i
e 0 , ψ0 )
δep∗ [(ϕ, ψ)] = d(ϕ
(3.4)
0
0
e p (M ) such that L
e (η,1) (ϕ0 , ψ 0 ) = (ϕ, ψ).
for any (ϕ , ψ ) ∈ Ω
If η is an 1–form without zeros we have
e(η,1) (M ) ∼
e0 A
0}.
H
= {e
e (η, 1) = (0, 0) implies (ϕ, ψ) = (ω, θ)∧
e (η, 1) we
Moreover, since (ϕ, ψ)∧
deduce that
e (η,1) : Ω
e p (M ) → Ω
e p+1 (M )} = Im {L
e (η,1) : Ω
e p−1 (M ) → Ω
e p (M )}.
ker{L
Now decompose the long exact sequence (3.3) in the following short exact
sequences:
e∗
L
(η,1)
i
e∗
ep
e∗
e
e
0 −→ Im ei∗ = ker L
(η,1) −→ H (M ) −→ Im L(η,1) −→ 0.
134
Cristian Ida and Sabin Mercheşan
Then we deduce the formula:
ebp (M ) = dim ker L
e∗
e∗
(η,1) + dim Im L(η,1) .
(3.5)
From (3.5) we obtain the following result:
Proposition 3.1. Let M be a n–dimensional smooth manifold. Asume that η
e
is an 1–form without zeros on M and (η, 1)–coeffective d–cohomology
is finite.
Then we have
ebp (M ) ≤ e
cp (M, (η, 1)) + e
cp+1 (M, (η, 1))
(3.6)
for all p.
3.2
e
(dη, η)–coeffective d-cohomology
As in the previous subsection, the main purpose of this subsection is to cone
e
struct a (dη, η)–coeffective d–cohomology.
Althouhg (dη, η) is not d–closed
e
it is de(η,1) –closed and this fact allow to construct an associated coeffective d–
cohomology. The case when M is an almost contact manifold is also considered
and studied in the next subsection.
Let us define the operator
e (dη,η) : Ω
e p (M ) → Ω
e p+2 (M ) , L
e (dη,η) (ϕ, ψ) = (ϕ, ψ)∧
e (dη, η).
L
(3.7)
The space
n
o
ep
e (dη,η) : Ω
e p (M ) → Ω
e p+2 (M )
A
(M
)
=
ker
L
(dη,η)
is called the subspace of (dη, η)–coeffective 1–differentiable forms on M .
Taking into acount that de(η,1) (dη, η) = e
0 the relation (2.9) say that
e ψ)∧
e (dη, η) = d(ϕ,
e (dη, η)
de(η,1) (ϕ, ψ)∧
or, equivalently
e
e (dη,η) = L
e (dη,η) d.
de(η,1) L
(3.8)
The identity (3.8) suggests us to consider a family of operators de(kη,k) , k ∈ R,
which we abbreviate as dek if no misunderstanding occurs. We get immediatly
from (3.8)
e
ep
ep
dek L
(3.9)
(dη,η) = L(dη,η) dk−p , ∀ p ≥ 0,
e0
where L
e
(dη,η) = Id|Ω(M
).
A NOTE ON COEFFECTIVE 1–DIFFERENTIABLE COHOMOLOGY
135
ep
e
Now, the relation (3.8) say that if (ϕ, ψ) ∈ A
(dη,η) (M ) then d(ϕ, ψ) ∈
p
p+1
e
e
e
A
(dη,η) (M ), hence A(dη,η) (M ), d is a differential subcomplex of the differen
e(dη,η) (M ) is called (dη, η)–
e p (M ), de . Its cohomology H
ep A
tial complex Ω
e
coeffective d–cohomology
of M . If this cohomology is finite,
the
then we define
e(dη,η) (M ) .
ep A
(dη, η)–coeffective numbers by e
cp (M, (dη, η)) = dim H
e
As in the case of (η, 1)–coeffective d–cohomology,
in the sequel we relate the
e
e
(dη, η)–coeffective d–cohomology with the d–cohomology
by means of a long
exact sequence in cohomology. Consider the following natural short exact
sequence for any degree p:
L(dη,η)
e
i ep
p+2 e
ep
e (dη,η) = A
e
L(dη,η) −→ e
0. (3.10)
0 −→ ker L
(dη,η) (M ) −→ Ω (M ) −→ Im
e
e (dη,η) = L
e (dη,η) de−(η,1)
By means of (3.9), for k = 0 and p = 1, we obtain deL
p+1
p e
e ψ) ∈ Im
e (dη,η) , hence
L
which say that if (ϕ, ψ) ∈ Im L(dη,η) then d(ϕ,
p e
p
e (M ), de . Thus,
Im L(dη,η) , de is a subcomplex of the differential complex Ω
(3.10) becomes a short exact sequence of differential complexes.
Therefore, we can consider the associated long exact sequence in cohomology:
e∗
e∗
L
(dη,η)
i
e(dη,η) (M ) −→
ep A
e p (M ) −→
. . . −→ H
H
∆
e∗
p+2
e(dη,η) (M ) −→ . . .
e p+2 ImL
e p+1 A
e (dη,η) −→
H
H
(3.11)
e∗
e
where ei∗ and L
(dη,η) are the homomorphisms induced in cohomology by i and
∗
e
e
L(dη,η) , respectively, and ∆
is the connecting homomorphism.
p+2
3.3
The almost contact case
In this subsection we consider that η is the almost contact 1–form of a (2n+1)–
dimensional almost contact manifold M .
Let us recall the following fundamental result due to [5]:
Proposition 3.2. Let (M, F, ξ, η, g) be a (2n + 1)–dimensional almost contact
metric manifold, (for necessary definitions see for instance [1, 4, 20]). Then
the map
L : Ωp (M ) → Ωp+2 (M ) , Lϕ = ϕ ∧ dη
is injective for p ≤ n − 1, and surjective for p ≥ n.
Using the above proposition, we have
136
Cristian Ida and Sabin Mercheşan
Proposition 3.3. Let (M, F, ξ, η, g) be a (2n + 1)–dimensional almost contact
e (dη,η) given in (3.7) is injective for p ≤ n−1,
metric manifold. Then the map L
and surjective for p ≥ n + 1.
e (dη,η) (ϕ, ψ) = (Lϕ, η ∧ ϕ + Lψ). Now, from
Proof. Using (2.1) we have L
e (dη,η) (ϕ1 , ψ1 ) = L
e (dη,η) (ϕ2 , ψ2 ) we obtain
L
Lϕ1 = Lϕ2 , η ∧ ϕ1 + Lψ1 = η ∧ ϕ2 + Lψ2
(3.12)
and by Proposition 3.2 if p ≤ n − 1, L is injective and from the first relation of
(3.12) it results that ϕ1 = ϕ2 . Replacing in the second relation of (3.12) we
e (dη,η) is injective
obtain ψ1 = ψ2 , and so (ϕ1 , ψ1 ) = (ϕ2 , ψ2 ) which say that L
for p ≤ n − 1.
Taking into account that for any p ≥ n, L is surjective we obtain that
0
0
for any ϕ ∈ Ωp+2 (M ) there is ϕ ∈ Ωp (M ) such that ϕ = Lϕ. Also, for ϕ
0
as above, if p − 1 ≥ n then for any ψ ∈ Ωp+1 (M ) there is ψ ∈ Ωp−1 such
0
that ψ − η ∧ ϕ = Lψ, and so we conclude that if p ≥ n + 1 then for any
0
0
e p+2 (M ) there exists (ϕ, ψ) ∈ Ω
e p (M ) such that L
e (dη,η) (ϕ, ψ) =
(ϕ , ψ ) ∈ Ω
0
0
e (dη,η) is surjective for p ≥ n + 1.
(ϕ , ψ ), which say that L
ep
e
Corollary 3.1. A
(dη,η) (M ) = {0}, for p ≤ n − 1, and as a consequence
e(dη,η) (M ) = {0} , ∀ p = 0, 1, . . . , n − 1,
ep A
H
or equivalently e
cp (M, (dη, η)) = 0, for any p = 0, 1, . . . , n − 1.
e (dη,η) = Ω
e p+2 (M ), for p ≥ n + 1.
By Proposition 3.3 we have that Imp+2 L
As a consequence, we have
e p+2 (M ) , ∀ p ≥ n + 2.
e p+2 Im L
e (dη,η) = H
H
Furthermore, for p ≥ n + 2, the long exact sequence in cohomology (3.11) may
be expressed as
e∗
e∗
L
(dη,η)
i
e(dη,η) (M ) −→
ep A
e p (M ) −→
. . . −→ H
H
e∗
∆
p+2
e(dη,η) (M ) −→ . . . .
e p+2 (M ) −→
e p+1 A
H
H
(3.13)
Now, we shall decompose the long exact sequence (3.13) in 5–terms exact
sequences:
e∗
e∗
L
(dη,η)
i
i
e(dη,η) (M ) −→
ep A
e p (M ) −→
e ∗p+1 −→
0 → Im ∆
H
H
A NOTE ON COEFFECTIVE 1–DIFFERENTIABLE COHOMOLOGY
e∗
∆
p+2
e p+2 (M ) −→
e ∗p+2 −→ 0,
H
Im ∆
137
(3.14)
e ∗ = ker ei∗ .
where Im ∆
p+1
If M is of finite type, the de Rham cohomology groups have finite
dimene∗ ≤
sion, and so ebp (M ) = bp (M ) + bp−1 (M ) is finite. Since 0 ≤ dim Im ∆
p
ebp (M ), for p ≥ n + 4 we have the following result:
Proposition 3.4. Let (M, F, ξ, η, g) be a (2n + 1)–dimensional almost contact
e
metric
manifold of finite type, then the (dη, η)–coeffective d–cohomology group
p
e(dη,η) (M ) has finite dimension, for p ≥ n + 3.
e A
H
From (3.14), we have
e(dη,η) (M ) +dim Im ∆
ebp+2 (M )−ebp (M ) = dim Im ∆
e ∗p+1 −dim H
ep A
e ∗p+2 ,
for p ≥ n + 3, from which we deduce
e ∗p+1 +ebp (M )−ebp+2 (M )+dim Im ∆
e ∗p+2 . (3.15)
e
cp (M, (dη, η)) = dim Im ∆
Now, as a consequence of (3.15), we obtain
Theorem 3.1. For p ≥ n + 3, we have
ebp (M ) − ebp+2 (M ) ≤ e
cp (M, (dη, η)) ≤ ebp (M ) + ebp+1 (M ).
(3.16)
Now, using Proposition 3.1 we obtain
Corollary 3.2. Let (M, F, ξ, η, g) be a (2n + 1)–dimensional almost contact
metric manifold of finite type, then
e
cp (M, (dη, η)) ≤ e
cp (M, (η, 1)) + 2e
cp+1 (M, (η, 1)) + e
cp+2 (M, (η, 1))
(3.17)
for every p ≥ n + 3.
Acknowledgments
The authors wishes to express their deep gratitude to the Organizing Committee of The International Conference A new approach in theoretical and applied
methods in algebra and analysis, Constanta, 4-6 April 2013, CNCS-UEFISCDI
grant PN-II-ID-WE-2012-4-169. The first author is supported by the Sectorial
Operational Program Human Resources Development (SOP HRD), financed
from the European Social Fund and by the Romanian Government under the
Project number POSDRU/ID134378 .
138
Cristian Ida and Sabin Mercheşan
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Cristian IDA,
Department of Mathematics and Computer Science,
Transilvania University of Braşov,
Braşov 500091, Str. Iuliu Maniu 50, România
Email: cristian.ida@unitbv.ro
Sabin MERCHEŞAN,
Department of Mathematics and Computer Science,
Transilvania University of Braşov,
Braşov 500091, Str. Iuliu Maniu 50, România
Email: s.merchesan@unitbv.ro
140
Cristian Ida and Sabin Mercheşan