2. Preliminaries
Let
M be an odd-dimensional
manifold. An almost contact metric structure on
M is defined by a
-type tensor field
, a vector field
, a 1-form
and a metric
g on
M such that
for any vector fields
X,
Y on
M. A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. The fundamental 2-form
of the almost contact metric manifold
M is defined by
for all vector fields
on
M, and the form
satisfies
, where dimension of
M is
. Hence, an almost contact metric manifold is orientable. In addition, the structure group of an almost contact metric manifold reduces to the group
.
According to properties of the Levi–Civita covariant derivative of the fundamental 2-form
, there is a classification of almost contact metric manifolds in [
5]. A space having the same symmetries as the covariant derivative of the fundamental 2-form was written. This space is
for all
. The space
was decomposed into 12
irreducible components, denoted by
as shown in
Table 1. There exist
invariant subspaces, each corresponding to a class of almost contact metric manifolds. For example, the trivial class corresponds to the class of cosymplectic (called co-Kähler by some authors) manifolds,
is the class of nearly K-cosymplectic manifolds,
is the class of quasi-sasakian manifolds,
is the class of
-Kenmotsu manifolds etc [
5]. Also, a similar classification was made by [
6]. In this work, we will use the definitions of some other classes in the context by using the notation in [
5,
6]. According to this classification; some special classes of almost contact metric manifolds coincide with a suitable sum of some classes of
.
When the dimension of the manifold is 3, then
[
5].
Let
be an almost contact metric manifold. If we take
where
a and
b are positive functions on
M, one can easily check that
is an almost contact metric manifold too. This deformation is called a generalized D-conformal deformation [
4].
After this deformation, the derivation of the new fundamental 2-form
is
3. Generalized D-Conformal Deformations of Nearly K-Cosymplectic Manifolds
Let
be a nearly K-cosymplectic manifold (that is, belongs to class
). Defining relations of this class are
or equivalently
To calculate the new Levi–Civita covariant derivative of a nearly K-cosymplectic manifold after applying a generalized D-conformal deformation, we need only to consider the property that is parallel. Hence, we state the following lemma.
Lemma 1. Let be an almost contact metric manifold such that characteristic vector field ξ is parallel. If a generalized D-conformal deformation is applied, then the new covariant derivative of the new metric is obtained as Proof. Using Kozsul’s formula and that
is parallel, one can get
Take
in the Equation (
2), to obtain
Hence, we write the new Levi–Civita covariant derivative of the new metric
as in (
1). □
Now we show that under some restrictions, it is possible to obtain a nearly K-cosymplectic structure from an old one by a generalized D-conformal deformation.
Theorem 1. Let be a nearly K-cosymplectic manifold and consider a generalized D-conformal deformation on M with positive functions a and b. is a nearly K-cosymplectic manifold if and only if and b is a constant.
Proof. If we take
in (
1), since
is parallel,
is obtained. Also, we have
if and only if
and
. On the other hand, since
, the relation
is equivalent to
, or by polarization
Now it is easy to see that if and only if and b is a constant. □
We give the following example on nearly K-cosymplectic structures.
Example 1. Let be an almost Hermitian manifold, . Consider the almost contact metric structure on wherewhere f and g are functions on , X, Y are any vector fields on M. Then, is nearly K-cosymplectic if and only if M is nearly Kaehlerian [5]. Consider a generalized D-conformal deformation with a function , and a positive constant b. Since the function a depends only on t, . Then, by Theorem 1, the new almost contact metric structure on is also nearly K-cosymplectic.
Let
be an almost contact metic manifold and
be a local orthonormal frame for the metric
g on
M. Then, after a generalized D-conformal deformation,
is a local orthonormal frame for the metric
, where
and
, for
. If
is a nearly K-cosymplectic manifold,
b is a positive constant and
, then the new almost contact metric structure is nearly K-cosymplectic and the new Levi–Civita covariant derivative is written as
By direct calculation, one can get the new curvature tensor
, Ric operator
and scalar curvature
as
Thus, one can obtain the following theorem.
Theorem 2. Let be a nearly K-cosymplectic manifold whose scalar curvature S is constant. If and , then new nearly K-cosymplectic manifold is locally isometric to the sphere. In addition, if and a is a function such that , the manifolds and are locally isometric.
In addition, since
and
, one can express the new coderivation of the new fundamental 1-form
and 2-form
as
Hence, one can conclude that, if is nearly K-cosymplectic, then is a semi-cosymplectic manifold if and only if and .
4. Generalized D-Conformal Deformations of Quasi-Sasakian Manifolds
In this case, we consider generalized D-conformal deformations of quasi-Sasakian manifolds. An almost contact metric manifold is called quasi-Sasakian if it is normal and its fundamental 2-form
is closed, that is,
for all vector field
X and
Y. Quasi-Sasakian manifolds are the class
. The most important feature of a quasi-Sasakian manifold is that the fundamental vector field
is a Killing vector field [
7].
Let be an almost contact metric manifold such that is Killing. The Levi–Civita covariant derivative of is calculated using Kozsul’s formula, only by considering that is Killing.
Lemma 2. Let be an almost contact metric manifold such that characteristic vector field ξ is Killing. If a generalized D-conformal deformation is applied, then the new covariant derivative of the new metric is obtained as Now our aim is to obtain a quasi-Sasakian manifold after applying a generalized D-conformal deformation to a quasi-Sasakian manifold. First we give the condition for to be Killing with respect to .
Lemma 3. is Killing vector field if and only if and .
Proof. Let
be a Killing vector field. Then
for all vector fields
X,
Y. Since
if we take
in (
9), we obtain
hence
In addition, if we take
in (
9), we get
. Converse of the lemma is trivial. □
We can obtain quasi-Sasakian manifolds by deforming the old ones as follows.
Theorem 3. Let be a quasi-Sasakian manifold and consider a generalized D-conformal deformation with positive functions a and b. Then, the new almost contact metric manifold obtained by generalized D-conformal deformation is quasi-Sasakian if and only if b is a constant and a depends only on the direction of ξ.
Proof. Consider a generalized D-conformal deformation on
M with positive functions
a and
b. Let
be a quasi-Sasakian manifold. Then,
is a Killing vector field. Thus from Lemma 3, we get
Conversely, let
b be a constant and
. By Lemma 3,
is Killig. In addition, since a quasi-Sasakian manifold is normal, we have
Since
, the equation
is satisfied. Also, for a quasi-Sasakian manifold
, thus we get
Since the function b is a constant, we obtain . As a result is a quasi-Sasakian manifold. □
In addition, one can obtain the following corollary:
Corollary 1. Let be a quasi-Sasakian manifold. If a is a positive function such that and b is a positive constant, then the new almost contact metric manifold is normal.
If
is a quasi-Sasakian manifold and
a and
b are positive functions such that
(
b need not be constant) and
, then in new almost contact metric manifold
, one can compute directly coderivation of
and
as follows:
and
Let
be a quasi-Sasakian manifold,
b be a positive constant and
a be a function such that
. After a generalized D-conformal deformation, the new covariant derivative is
Moreover, by direct calculation, one can get
and
Hence, we obtain new Ricci operator
and scalar curvature
as
Example 2. Let M be a seven dimensional 3-Sasakian manifold. Since this manifold is Sasakian (), it is in particular quasi-Sasakian (). It is known that its scalar curvature is 42 and also for all vector fields X. For definition and properties of 3-Sasakian manifolds, see [8]. Let be one of the three Sasakian structures on a seven dimensional 3-Sasakian manifold and assume that a generalized D-conformal deformation is applied to this structure. Note that by Theorem [3], the deformed structure is also quasi-Sasakian. Now we calculate the new scalar curvature of the deformed manifold. Sincefrom the Equation (18), Let a and b be positive constants satisfying or . Then, the Equation (19) implies or . Thus the new quasi-Sasakian manifold is locally isometric to the sphere or the hyperbolic space, respectively. On the other hand, if positive constants a and b are chosen as , then and the new quasi-Sasakian manifold has zero scalar curvature.
5. Generalized D-Conformal Deformations of -Kenmotsu Manifolds
An almost contact metric manifold
is called
-Kenmotsu manifold, if the relation
is satisfied, where
is a smooth function on
M. It is known that if
is a
-Kenmotsu manifold, then the equation
is satisfied.
Lemma 4. Consider a generalized D-conformal deformation of an almost contact metric structure such that , where a and b positive functions. After a generalized D-conformal deformation, the new Levi–Civita covariant derivative is Then, we obtain the following lemma:
Lemma 5. Let be a β-Kenmotsu manifold. Consider a generalized D-conformal deformation on M where a and b are positive functions. Thenif and only if the function a depends only on the direction of ξ, that is . Proof. First by taking
in (
21), we get
and
Then, by the definition of the generalized D-conformal deformation and the Equations (
22) and (
23), we have
The Equation (
24) implies
Take
in (
25) to obtain
that is
Conversely, let
. Then, the term
of the Equation (
24) vanishes and thus
is obtained. □
For any
-Kenmotsu manifold, we know that
. After deformation, derivation of
is obtained as:
The new Levi–Civita covariant derivative of
is
Note that the following theorem can be deduced from Lemma 4.1 of [
4]. In [
4], first the new Levi–Civita covariant derivative is calculated under the restriction that
a and
b are positive functions depending on the direction of
and then Lemma 4.1 in [
4] is stated for trans-Sasakian manifolds by using this covariant derivative. In our study, however, we obtain the new Levi–Civita covariant derivative only by assuming that
(equivalently
) in Lemma 5 and then we state the following theorem.
Theorem 4. Let be a β-Kenmotsu manifold, and consider a generalized D-conformal deformation with a and b positive functions. If and , then is a -Kenmotsu manifold, where Proof. Since
for all vector fields
X and
Y, taking
, then
□
Let
be a
-Kenmotsu manifold. Then, if
is
-Kenmotsu, then from Lemma 5, we obtain
. In addition, since
M is
-Kenmotsu, we have
Take
in (
27), then we obtain
. We have been unable to find any restriction on the function
b.
If
is a
dimensional
-Kenmotsu manifold,
a and
b positive functions such that
,
, after a generalized D-conformal deformation, we have
and coderivations of
and
are calculated as
In addition, by long direct calculation, the new scalar curvature
is