On 3-dimensional f-Kenmotsu manifolds and Ricci solitons
The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisy...
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| author | De, U. C. Turan, M. Yildiz, A. Де, У. К. Туран, М. Ілдиз, А. |
| author_facet | De, U. C. Turan, M. Yildiz, A. Де, У. К. Туран, М. Ілдиз, А. |
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| description | The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Moreover, we investigate an η-parallel Ricci tensor in a 3-dimensional f-Kenmotsu manifold. Finally, we study Ricci solitons in a 3-dimensional f-Kenmotsu manifold. |
| first_indexed | 2026-03-24T02:23:36Z |
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UDC 515.12
A. Yildiz (Dumlupınar Univ., Kütahya, Turkey),
U. C. De (Univ. Calcutta, India),
M. Turan (Dumlupınar Univ., Kütahya, Turkey)
ON 3-DIMENSIONAL f -KENMOTSU MANIFOLDS AND RICCI SOLITONS
ПРО 3-ВИМIРНI f -МНОГОВИДИ КЕНМОЦУ ТА СОЛIТОНИ РIЧЧI
The object of the present paper is to study 3-dimensional f -Kenmotsu manifolds and Ricci solitons. Firstly we give an
example of a 3 -dimensional f -Kenmotsu manifold. Then we consider Ricci-semisymmetric 3-dimensional f -Kenmotsu
manifold and prove that a 3-dimensional f -Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein
manifold. Also η-parallel Ricci tensor in a 3-dimensional f -Kenmotsu manifold have been studied. Finally we study Ricci
solitons in a 3-dimensional f -Kenmotsu manifold.
Метою даної статтi є вивчення 3-вимiрних f -многовидiв Кенмоцу та солiтонiв Рiччi. Спочатку наведено приклад
3-вимiрного f -многовиду Кенмоцу. Потiм розглянуто напiвсиметричний за Рiччi 3-вимiрний f -многовид Кенмоцу i
доведено, що 3-вимiрний f -многовид Кенмоцу є напiвсиметричним за Рiччi тодi i тiльки тодi, коли вiн є многовидом
Ейнштейна. Також дослiджено η-паралельний тензор Рiччi у 3-вимiрному f -многовидi Кенмоцу. Насамкiнець,
дослiджено солiтони Рiччi у 3-вимiрному f -многовидi Кенмоцу.
1. Introduction. Let M be an almost contact manifold, i.e., M is a connected (2n+ 1)-dimensional
differentiable manifold endowed with an almost contact metric structure (φ, ξ, η, g) [1]. As usually,
denote by Φ the fundamental 2-form of M, Φ(X,Y ) = g(X,φY ), X, Y ∈ χ(M), χ(M) being the
Lie algebra of differentiable vector fields on M.
For further use, we recall the following definitions [1, 5, 19]. The manifold M and its structure
(φ, ξ, η, g) is said to be:
(i) normal if the almost complex structure defined on the product manifold M × R is integrable
(equivalently [φ, φ] + 2dη ⊗ ξ = 0),
(ii) almost cosymplectic if dη = 0 and dΦ = 0,
(iii) cosymplectic if it is normal and almost cosymplectic (equivalently, ∇φ = 0, ∇ being
covariant differentiation with respect to the Levi – Civita connection).
The manifold M is called locally conformal cosymplectic (respectively, almost cosymplectic) if
M has an open covering {Ut} endowed with differentiable functions σt : Ut → R such that over
each Ut the almost contact metric structure (φt, ξt, ηt, gt) defined by
φt = φ, ξt = eσtξ, ηt = e−σtη, gt = e−2σtg
is cosymplectic (respectively, almost cosymplectic).
Olszak and Rosca [14] studied normal locally conformal almost cosymplectic manifold. They
gave a geometric interpretation of f -Kenmotsu manifolds and studied some curvature properties.
Among others they proved that a Ricci symmetric f -Kenmotsu manifold is an Einstein manifold.
By an f -Kenmotsu manifold we mean an almost contact metric manifold which is normal and
locally conformal almost cosymplectic.
Apart from conformal curvature tensor, the projective curvature tensor is another important tensor
from the differential geometric point of view. Let M be a (2n+1)-dimensional Riemannian manifold.
If there exist an one-to-one correspondence between each coordinate neighborhood of M and a
c© A. YILDIZ, U. C. DE, M. TURAN, 2013
620 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON 3-DIMENSIONAL f -KENMOTSU MANIFOLDS AND RICCI SOLITONS 621
domain in Euclidean space such that any geodesic of the Riemannian manifold corresponds to a
straight line in the Euclidean space, then M is said to be locally projectively flat. For n ≥ 1, M is
locally projectively flat if and only if the well known projective curvature tensor P vanishes. Here P
is defined by [12]
P (X,Y )Z = R(X,Y )Z − 1
2n
{S(Y, Z)X − S(X,Z)Y },
for X,Y, Z ∈ χ(M), where R is the curvature tensor and S is the Ricci tensor of M . In fact, M
is projectively flat (that is, P = 0) if and only if the manifold is of constant curvature [21, p. 84,
85]. Thus, the projective curvature tensor is a measure of the failure of a Riemannian manifold to be
of constant curvature. ξ-conformally flat K-contact manifolds have been studied by Zhen, Cabrerizo
and Fernandez [22]. Likewise we define ξ-projectively flat f -Kenmotsu manifold. An f -Kenmotsu
manifold is called ξ-projectively flat if the condition P (X,Y )ξ = 0 holds on M .
A Ricci soliton is a generalization of an Einstein metric. In a Riemannian manifold (M, g), g is
called a Ricci soliton if [7]
£V g + 2S + 2λg = 0, (1.1)
where £ is the Lie derivative, S is the Ricci tensor, V is a complete vector field on M and λ is
a constant. Compact Ricci solitons are the fixed points of the Ricci flow
∂
∂t
g = −2S projected
from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise as
blow-up limits for the Ricci flow on compact manifolds.
The Ricci soliton is said to be shrinking, steady and expanding according as λ is negative, zero
and positive respectively. If the vector field V is the gradient of a potential function −f , then g is
called a gradient Ricci soliton and equation (1.1) assumes the form
∇∇f = S + λg.
A Ricci soliton on a compact manifold has constant curvature in dimension 2 [7] and also in di-
mension 3 [8]. For details we refer to Chow and Knopf [3] and Derdzinski [4]. We also recall the
following significant result of Perelman [15]: A Ricci soliton on a compact manifold is a gradient
Ricci soliton.
In [18], Sharma has started the study of Ricci solitons in K-contact manifolds. In a K-contact
manifold the structure vector field ξ is Killing, that is, £ξg = 0, which is not in general, in an
f -Kenmotsu manifold.
In the present paper we have studied some curvature properties of a 3-dimensional f -Kenmotsu
manifold and Ricci solitons. The paper is organized as follows: After preliminaries we give an ex-
ample of a regular 3-dimensional f -Kenmotsu manifold which is ξ-projectively flat. In Section 4, we
consider Ricci semisymmetric 3-dimensional f -Kenmotsu manifolds and prove that a 3-dimensional
f -Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Section 5
deals with η-parallel Ricci tensor in a 3-dimensional f -Kenmotsu manifold. Finally we study Ricci
solitons in a 3-dimensional f -Kenmotsu manifold. We prove that if in a 3-dimensional f -Kenmotsu
manifold the metric g is a Ricci soliton and V is pointwise collinear with ξ, then g is η-Einstein
and the Ricci soliton is expanding, provided λ = 2(f2 + f ′). We also prove that if the manifold is
η-Einstein, then an f -Kenmotsu manifold admits a Ricci soliton, provided f is constant.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
622 A. YILDIZ, U. C. DE, M. TURAN
2. f -Kenmotsu manifolds. Let M be a real (2n + 1)-dimensional differentiable manifold en-
dowed with an almost contact structure (φ, ξ, η, g) satisfying
φ2 = −I + η ⊗ ξ, η(ξ) = 1,
φξ = 0, η ◦ φ = 0, η(X) = g(X, ξ), (2.1)
g(φX, φY ) = g(X,Y )− η(X)η(Y ),
for any vector fields X,Y ∈ χ(M), where I is the identity of the tangent bundle TM , φ is a tensor
field of (1, 1)-type, η is a 1-form, ξ is a vector field and g is a metric tensor field. We say that
(M,φ, ξ, η, g) is an f -Kenmotsu manifold if the covariant differentiation of φ satisfies [13]:
(∇Xφ)(Y ) = f
(
g(φX, Y )ξ − η(Y )φX
)
, (2.2)
where f ∈ C∞(M) such that df ∧ η = 0. If f = α = constant 6= 0, then the manifold is a
α-Kenmotsu manifold [9]. 1-Kenmotsu manifold is a Kenmotsu manifold [10, 16]. If f = 0, then the
manifold is cosymplectic [9]. An f -Kenmotsu manifold is said to be regular if f2 + f ′ 6= 0, where
f ′ = ξf.
For an f -Kenmotsu manifold from (2.2) it follows that
∇Xξ = f{X − η(X)ξ}. (2.3)
The condition df ∧ η = 0 holds if dimM ≥ 5. In general this does not hold if dimM = 3 [14].
In a 3-dimensional Riemannian manifold, we always have
R(X,Y )Z = g(Y, Z)QX − g(X,Z)QY + S(Y,Z)X − S(X,Z)Y−
−τ
2
{
g(Y,Z)X − g(X,Z)Y
}
. (2.4)
In a 3-dimensional f -Kenmotsu manifold we have [14]
R(X,Y )Z =
(τ
2
+ 2f2 + 2f ′
)
(X ∧ Y )Z−
−
(τ
2
+ 3f2 + 3f ′
){
η(X)(ξ ∧ Y )Z + η(Y )(X ∧ ξ)Z
}
, (2.5)
S(X,Y ) =
(τ
2
+ f2 + f ′
)
g(X,Y )−
(τ
2
+ 3f2 + 3f ′
)
η(X)η(Y ), (2.6)
where τ is the scalar curvature of M and f ′ = ξ(f).
From (2.5), we obtain
R(X,Y )ξ = −(f2 + f ′)[η(Y )X − η(X)Y ], (2.7)
and (2.6) yields
S(X, ξ) = −2(f2 + f ′)η(X). (2.8)
From equations (2.7) and (2.8) it can be easily verified that P (X,Y )ξ = 0.
Remark 2.1. A 3-dimensional f -Kenmotsu manifold is always ξ-projectively flat.
In the next section we verify the remark by an example.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON 3-DIMENSIONAL f -KENMOTSU MANIFOLDS AND RICCI SOLITONS 623
3. Example of a 3-dimensional f -Kenmotsu manifold. We consider the three-dimensional
manifold M = {(x, y, z) ∈ R3, z 6= 0}, where (x, y, z) are the standard coordinates in R3. The
vector fields
e1 = z2
∂
∂x
, e2 = z2
∂
∂y
, e3 =
∂
∂z
are linearly independent at each point of M . Let g be the Riemannian metric defined by
g(e1, e3) = g(e2, e3) = g(e1, e2) = 0,
g(e1, e1) = g(e2, e2) = g(e3, e3) = 1.
Let η be the 1-form defined by η(Z) = g(Z, e3) for any Z ∈ χ(M). Let φ be the (1, 1) tensor
field defined by φ(e1) = −e2, φ(e2) = e1, φ(e3) = 0.
Then using linearity of φ and g we have
η(e3) = 1, φ2Z = −Z + η(Z)e3,
g(φZ, φW ) = g(Z,W )− η(Z)η(W ),
for any Z,W ∈ χ(M). Now, by direct computations we obtain
[e1, e2] = 0, [e2, e3] = −2
z
e2, [e1, e3] = −2
z
e1.
The Riemannian connection ∇ of the metric tensor g is given by the Koszul’s formula which is
2g(∇XY,Z) = Xg(Y,Z) + Y g(Z,X)− Zg(X,Y )−
−g
(
X, [Y,Z])− g(Y, [X,Z]) + g(Z, [X,Y ]
)
. (3.1)
Using (3.1) we have
2g(∇e1e3, e1) = 2g
(
−2
z
e1, e1
)
,
2g(∇e1e3, e2) = 0 and 2g(∇e1e3, e3) = 0.
Hence ∇e1e3 = −2
z
e1. Similarly, ∇e2e3 = −2
z
e2 and ∇e3e3 = 0. (3.1) further yields
∇e1e2 = 0, ∇e1e1 =
2
z
e3,
∇e2e2 =
2
z
e3, ∇e2e1 = 0,
∇e3e2 = 0, ∇e3e1 = 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
624 A. YILDIZ, U. C. DE, M. TURAN
From the above it follows that the manifold satisfies ∇Xξ = f{X − η(X)ξ} for ξ = e3, where
f = −2
z
. Hence we conclude that M is an f -Kenmotsu manifold. Also f2 + f ′ 6= 0. Hence M is a
regular f -Kenmotsu manifold.
It is known that
R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z. (3.2)
With the help of the above formula and using (3.2) it can be easily verified that
R(e1, e2)e3 = 0, R(e2, e3)e3 = − 6
z2
e2, R(e1, e3)e3 = − 6
z2
e1,
R(e1, e2)e2 = − 4
z2
e1, R(e3, e2)e2 = − 6
z2
e3, R(e1, e3)e2 = 0,
R(e2, e1)e1 = − 4
z2
e2, R(e2, e3)e1 = 0, R(e3, e1)e1 = − 6
z2
e3.
From the above expressions of the curvature tensor we obtain
S(e1, e1) = g(R(e1, e2)e2, e1) + g(R(e1, e3)e3, e1) = −10
z2
.
Similarly, we have
S(e2, e2) = −10
z2
and S(e3, e3) = −12
z2
.
Now from the expressions of curvature tensors and Ricci tensor we have
P (e1, e2)e3 = P (e1, e3)e3 = P (e2, e3)e3 = 0,
that is, M is ξ-projectively flat. Hence the Remark 2.1 is verified.
4. Ricci-semisymmetric 3-dimensional f -Kenmotsu manifolds. An f -Kenmotsu manifold is
called Ricci-semisymmetric if R(X,Y ) ·S = 0 [20], where R(X,Y ) is treated as a derivation of the
tensor algebra for any tangent vectors X, Y. Then
S(R(X,Y )U, V ) + S(U,R(X,Y )V ) = 0. (4.1)
Putting X = U = ξ in (4.1) we have
S(R(ξ, Y )ξ, V ) + S(ξ,R(ξ, Y )V ) = 0. (4.2)
Then using (2.7) in (4.2) we obtain
(f2 + f ′)
[
η(Y )S(ξ, V )− S(Y, V )
]
+ (f2 + f ′)
[
g(Y, V )S(ξ, ξ)− η(V )S(ξ, Y )
]
= 0.
Now using (2.8) in the above equation yields
S(Y, V ) = −2
(
f2 + f ′
)
g(Y, V ),
on the set of M on which f 6= 0. Thus we obtain the following:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON 3-DIMENSIONAL f -KENMOTSU MANIFOLDS AND RICCI SOLITONS 625
Proposition 4.1. Let M be a Ricci-semisymmetric 3-dimensional regular f -Kenmotsu mani-
fold. Then M is an Einstein manifold.
Since Ricci symmetric manifold implies Ricci-semisymmetric, therefore Proposition 4.1 general-
izes the result of Olszak and Rosca [14].
Now we prove the following:
Theorem 4.1. Let M be a 3-dimensional non-cosymplectic f -Kenmotsu manifold. The follow-
ing conditions are equivalent:
(i) M is an Einstein manifold,
(ii) the Ricci tensor S of M is parallel (∇S = 0),
(iii) M is Ricci-semisymmetric.
Proof. (i) ⇒ (ii) and (ii)⇒ (iii) are clear. (iii)⇒ (i) by Proposition 4.1.
Theorem 4.1 is proved.
Remark 4.1. It is obvious that by the formula (2.4) the conditions (i) – (iii) in Theorem 4.1 can
be replaced by the following conditions:
(i) M is of constant curvature,
(ii) M is locally symmetric (∇R = 0),
(iii) M is semisymmetric.
5. η-parallel Ricci tensor.
Definition 5.1. The Ricci tensor S of a 3-dimensional f -Kenmotsu manifold is called η-parallel
if it satisfies
(∇WS)(φX, φY ) = 0
for all vector fields W, X and Y.
The notion of η-parallel Ricci tensor for Sasakian manifolds was introduced by Kon [11].
Let us consider a 3-dimensional f -Kenmotsu manifold. Replacing X by φX and Y by φY in
(2.6), we have
S(φX, φY ) =
(τ
2
+ f2 + f ′
)
g(φX, φY ). (5.1)
Using (2.1) in (5.1) we get
S(φX, φY ) =
(τ
2
+ f2 + f ′
){
g(X,Y )− η(X)η(Y )
}
. (5.2)
Differentiating the equation (5.2) covariantly with respect to W, we obtain
(∇WS)(φX, φY ) =
(
dτ(W )
2
+ 2f(Wf) + (Wf ′)
){
g(X,Y )− η(X)η(Y )
}
+
+
(τ
2
+ f2 + f ′
){
−fg(W,X)η(Y ) + fg(W,Y )η(X)
}
.
Suppose the Ricci tensor is η-parallel. Then, we get from above(
dτ(W )
2
+ 2f(Wf) + (Wf ′)
){
g(X,Y )− η(X)η(Y )
}
+
+
(τ
2
+ f2 + f ′
){
−fg(W,X)η(Y ) + fg(W,Y )η(X)
}
= 0. (5.3)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
626 A. YILDIZ, U. C. DE, M. TURAN
Then putting X = Y = ei in (5.3), where {ei} is an orthonormal basis of the tangent space at each
point of the manifold, and taking summation over i, 1 ≤ i ≤ 3, we obtain
dτ(W )
2
+ 2f(Wf) + (Wf ′) = 0.
Thus we have the following:
Theorem 5.1. If in a 3-dimensional f -Kenmotsu manifold the Ricci tensor S is η-parallel, then
the scalar curvature is constant, provided f is a constant.
6. Ricci solitons. Suppose that a 3-dimensional f -Kenmotsu manifold admits a Ricci soliton. It
is well known that ∇g = 0. Since λ in the Ricci soliton equation (1.1) is a constant, so ∇λg = 0.
Thus £V g + 2S is parallel.
In [2] Calin and Crasmareanu proved that a symmetric parallel second order tensor in an f -
Kenmotsu manifold is a constant multiple of the metric tensor on the set on which f 6= 0. Hence we
conclude that £V g + 2S = β (say) is a constant multiple of the metric tensor g, that is,
(£V g + 2S))(X,Y ) = β(X,Y ) = β(ξ, ξ)g(X,Y ),
where β(ξ, ξ) is given by
β(ξ, ξ) = (£V g + 2S))(ξ, ξ) = −4(f2 + f ′).
So £V g + 2S + 2λg reduces to (−4(f2 + f ′) + 2λ)g. Using (1.1) we get λ = 2(f2 + f ′). Thus
we have the following:
Proposition 6.1. In a 3-dimensional regular f -Kenmotsu manifold the Ricci soliton (g, λ, V )
is shrinking or expanding according as (f2 + f ′) is negative or positive.
In particular, let V be pointwise collinear with ξ, i.e., V = bξ, where b is a function on the
3-dimensional f -Kenmotsu manifold. Then
(£V g + 2S + 2λg)(X,Y ) = 0,
which implies that
g(∇Xbξ, Y ) + g(∇Y bξ,X) + 2S(X,Y ) + 2λg(X,Y ) = 0,
or,
bg(∇Xξ, Y ) + (Xb)η(Y ) + bg(∇Y ξ,X) + (Y b)η(X) + 2S(X,Y ) + 2λg(X,Y ) = 0. (6.1)
Using (2.3) in (6.1), we obtain
2fbg(X,Y )− 2fbη(X)η(Y ) + (Xb)η(Y ) + (Y b)η(X) + 2S(X,Y ) + 2λg(X,Y ) = 0. (6.2)
In (6.2) replacing Y by ξ it follows that
Xb+ (ξb)η(X)− 4(f2 + f ′)η(X) + 2λη(X) = 0. (6.3)
Putting X = ξ in (6.3), we get
ξb = 2(f2 + f ′)− λ.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
ON 3-DIMENSIONAL f -KENMOTSU MANIFOLDS AND RICCI SOLITONS 627
Putting this value in (6.3), we obtain
Xb+ (2(f2 + f ′)− λ)η(X)− 4(f2 + f ′)η(X) + 2λη(X) = 0,
or,
db = {2(f2 + f ′)− λ}η. (6.4)
Applying d on (6.4), we get
{λ− 2(f2 + f ′)}dη = 0.
Since dη 6= 0, we have
2(f2 + f ′)− λ = 0. (6.5)
Using (6.5) in (6.4) yields b is a constant. Therefore from (6.2) it follows
S(X,Y ) = −(λ+ fb)g(X,Y ) + fbη(X)η(Y ), (6.6)
which implies that M is an η-Einstein manifold. This leads to the following:
Theorem 6.1. If in a 3-dimensional f -Kenmotsu manifold the metric g is a Ricci soliton and
V is pointwise collinear with ξ, then V is a constant multiple of ξ and g is an η-Einstein manifold
of the form (6.6) and the Ricci soliton is expanding provided λ = 2(f2 + f ′).
The converse of the above theorem is not true, in general. However if we take f = constant,
i.e., if we consider a 3-dimensional η-Einstein f -Kenmotsu manifold, then it admits a Ricci soliton.
This can be proved as follows:
Let M be a 3-dimensional η-Einstein f -Kenmotsu manifold and V = ξ. Then
S(X,Y ) = γg(X,Y ) + δη(X)η(Y ), (6.7)
where γ and δ are certain scalars.
Now using (2.3)
(£ξg)(X,Y ) = g(∇Xξ, Y ) + g(∇Y ξ,X) = 2f
{
g(X,Y )− η(X)η(Y )
}
.
Therefore
(£ξg)(X,Y ) + 2S(X,Y ) + 2λg(X,Y ) = 2(f + γ + λ)g(X,Y )− 2(f − δ)η(X)η(Y ). (6.8)
From equation (6.8) it follows that M admits a Ricci soliton (g, ξ, λ) if f + γ + λ = 0 and δ =
= f = constant. From (6.7) we have using (2.8), −2f2 = γ+ δ. Hence γ = −2f2− f = constant.
Therefore λ = −(γ + δ) = constant. So we have the following:
Theorem 6.2. If a 3-dimensional f -Kenmotsu manifold is η-Einstein of the form S = γg +
+ δη ⊗ η, then the manifold admits a Ricci soliton (g, ξ,−(γ + δ)).
1. Blair D. E. Contact manifolds in Riemannian geometry // Lect. Notes Math. – 1976. – 509.
2. Calin C., Crasmareanu. From the Eisenhart problem to the Ricci solitons in f -Kenmotsu manifolds // Bull. Malays.
Math. Soc. – 2010. – 33.
3. Chow B., Knopf D. The Ricci flow: An introduction // Math. Surv. and Monogr. – 2004. – 110.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
628 A. YILDIZ, U. C. DE, M. TURAN
4. Derdzinski A. Compact Ricci solitons (preprint).
5. Goldberg S. I., Yano K. Integrability of almost cosymplectic structures // Pacif. J. Math. – 1969. – 31. – P. 373 – 382.
6. Friedan D. Nonlinear models in 2 + ε dimensions // Ann. Phys. – 1985. – 163. – P. 318 – 419.
7. Hamilton R. S. The Ricci flow on surfaces // Contemp. Math. – 1988. – 71.
8. Ivey T. Ricci solitons on compact 3-manifolds // Different. Geom. Appl. – 1993. – 3. – P. 301 – 307.
9. Janssens D., Vanhecke L. Almost contact structures and curvature tensors // Kodai Math. J. – 1981. – 4, № 1. –
P. 1 – 27.
10. Kenmotsu K. A class of almost contact Riemannian manifolds // Tohoku Math. J. – 1972. – 24. – P. 93 – 103.
11. Kon M. Invariant submanifolds in Sasakian manifolds // Math. Ann. – 1976. – 219. – P. 227 – 290.
12. Mishra R. S. Structures on differentiable manifold and their applications. – Allahabad: Chandrama Prakasana, 1984.
13. Olszak Z. Locally conformal almost cosymplectic manifolds // Colloq. math. – 1989. – 57. – P. 73 – 87.
14. Olszak Z., Rosca R. Normal locally conformal almost cosymplectic manifolds // Publ. Math. Debrecen. – 1991. – 39.
– P. 315 – 323.
15. Perelman G. The entopy formula for the Ricci flow and its geometric applications. – Preprint //
http://arxiv.org/abs/math.DG/02111159.
16. Pitis G. A remark on Kenmotsu manifolds // Bul. Univ. Brasov Ser. C. – 1988. – 30. – P. 31 – 32.
17. Rosca R. Exterior concurrent vector fields on a conformal cosymplectic manifold endowed with a Sasakian structure
// Libertos Math. – 1986. – 6. – P. 167 – 171.
18. Sharma R. Certain results on K-contact and (k, µ)-contact manifolds // J. Geometry. – 2008. – 89. – P. 138 – 147.
19. Sasaki S., Hatakeyama Y. On differentiable manifolds with certain structures which are closely related to almost
contact structures II // Tohoku Math. J. – 1961. – 13. – P. 281 – 294.
20. Verstraellen L. Comments on pseudo-symmetry in the sense of R. Deszcz // Geometry and Topology of Submanifolds. –
1993. – 6. – P. 199 – 209.
21. Yano K., Bochner S. Curvature and Betti numbers // Ann. Math. Stud. – 1953. – 32.
22. Zhen G., Cabrerizo J. L., Fernandez L. M., Fernandez M. On ξ-conformally flat contact metric manifolds // Indian
J. Pure and Appl. Math. – 1997. – 28. – P. 725 – 734.
Received 18.10.11
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
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| id | umjimathkievua-article-2446 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:36Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/32/02e1d190f2eb4760a2fd5401224a1e32.pdf |
| spelling | umjimathkievua-article-24462020-03-18T19:15:53Z On 3-dimensional f-Kenmotsu manifolds and Ricci solitons Про 3-вимірні $f$-многовиди Кенмоцу та солітони Річчі De, U. C. Turan, M. Yildiz, A. Де, У. К. Туран, М. Ілдиз, А. The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Moreover, we investigate an η-parallel Ricci tensor in a 3-dimensional f-Kenmotsu manifold. Finally, we study Ricci solitons in a 3-dimensional f-Kenmotsu manifold. Метою даної статті є вивчення 3-вимірних $f$-многовидів Кенмоцу та солітонів Річчі. Спочатку наведено приклад 3-вимірного $f$-многовиду Кенмоцу. Потім розглянуто напівсиметричний за Річчі 3-вимірний $f$-многовид Кенмоцу i доведено, що 3-вимірний $f$-многовид Кенмоцу є напівсиметричним за Річчі тоді i тільки тоді, коли він є многовидом Ейнштейна. Також досліджено n-паралельний тензор Річчі у 3-вимірному $f$-многовиді Кенмоцу. Насамкінець, досліджено солітони Річчі у 3-вимірному $f$-многовиді Кенмоцу. Institute of Mathematics, NAS of Ukraine 2013-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2446 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 5 (2013); 620–628 Український математичний журнал; Том 65 № 5 (2013); 620–628 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2446/1658 https://umj.imath.kiev.ua/index.php/umj/article/view/2446/1659 Copyright (c) 2013 De U. C.; Turan M.; Yildiz A. |
| spellingShingle | De, U. C. Turan, M. Yildiz, A. Де, У. К. Туран, М. Ілдиз, А. On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title | On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title_alt | Про 3-вимірні $f$-многовиди Кенмоцу та солітони Річчі |
| title_full | On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title_fullStr | On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title_full_unstemmed | On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title_short | On 3-dimensional f-Kenmotsu manifolds and Ricci solitons |
| title_sort | on 3-dimensional f-kenmotsu manifolds and ricci solitons |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2446 |
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