# On 3-Dimensional Contact Metric Generalized(

Authors: D. G. Prakasha; Shyamal Kumar Hui; Kakasab Mirji
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

The present paper deals with a study of 3-dimensional contact metric generalized(
k
,
μ
)-space forms. We obtained necessary and sufficient condition for a 3-dimensional contact metric generalized (
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q to be of constant curvature. We also obtained some conditions of such space forms to be pseudosymmetric and ξ-projectively flat, respectively.


---

## Body

## 1. Introduction

In 1995, Blair et al. [1] introduced the notion of contact metric manifolds with characteristic vector field ξ belonging to the (
k
,
μ
)-nullity distribution and such types of manifolds are called (
k
,
μ
)-contact metric manifolds. They obtained several results and examples of such a manifold. A full classification of this manifold has been given by Boeckx [2]. A contact metric manifold (
M
,
ϕ
,
ξ
,
η
,
g
) is said to be a generalized (
k
,
μ
)-space if its curvature tensor satisfies the condition
(1)
R
(
X
,
Y
)
ξ
=
k
{
η
(
Y
)
X
-
η
(
X
)
Y
}
+
μ
{
η
(
Y
)
h
X
-
η
(
X
)
h
Y
}
for some smooth functions k and μ on M independent choice of vector fields X and Y. If k and μ are constant, the manifold is called a (
k
,
μ
)-space. If a (
k
,
μ
)-space M has a constant ϕ-sectional curvature c and a dimension greater than 3, the curvature tensor of this (
k
,
μ
)-space form is given by [3]
(2)
R
=
c
+
3
4
R
1
+
c
-
1
4
R
2
+
(
c
+
3
4
-
k
)
R
3
+
R
4
+
1
2
R
5
+
(
1
-
μ
)
R
6
,
where R
1, R
2, R
3, R
4, R
5, and R
6 are the tensors defined by
(3)
R
1
(
X
,
Y
)
Z
=
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
,
R
2
(
X
,
Y
)
Z
=
g
(
X
,
ϕ
Z
)
ϕ
Y
-
g
(
Y
,
ϕ
Z
)
ϕ
X
+
2
g
(
X
,
ϕ
Y
)
ϕ
Z
,
R
3
(
X
,
Y
)
Z
=
η
(
X
)
η
(
Z
)
Y
-
η
(
Y
)
η
(
Z
)
X
+
g
(
X
,
Z
)
η
(
Y
)
ξ
-
g
(
Y
,
Z
)
η
(
X
)
ξ
,
R
4
(
X
,
Y
)
Z
=
g
(
Y
,
Z
)
h
X
-
g
(
X
,
Z
)
h
Y
+
g
(
h
Y
,
Z
)
X
-
g
(
h
X
,
Z
)
Y
,
R
5
(
X
,
Y
)
Z
=
g
(
h
Y
,
Z
)
h
X
-
g
(
h
X
,
Z
)
h
Y
+
g
(
ϕ
h
X
,
Z
)
ϕ
h
Y
-
g
(
ϕ
h
Y
,
Z
)
ϕ
h
X
,
R
6
(
X
,
Y
)
Z
=
η
(
X
)
η
(
Z
)
h
Y
-
η
(
Y
)
η
(
Z
)
h
X
+
g
(
h
X
,
Z
)
η
(
Y
)
ξ
-
g
(
h
Y
,
Z
)
η
(
X
)
ξ
for all vector fields X, Y, Z on M, where 2
h
=
L
ξ
ϕ and L is the usual Lie derivative.The notion of generalized Sasakian-space-forms was introduced and studied by Alegre et al. [4] with several examples. A generalized Sasakian-space-form is an almost contact metric manifold (
M
,
ϕ
,
ξ
,
η
,
g
) whose curvature tensor is given by
(4)
R
(
X
,
Y
)
Z
=
f
1
R
1
+
f
2
R
2
+
R
3
f
3
,
where R
1, R
2, and R
3 are the tensors defined above and f
1, f
2, f
3 are differentiable functions on M. In such case we will write the manifold as M
(
f
1
,
f
2
,
f
3
). Generalized Sasakian-space-forms have been studied by several authors, namely, [5–11].By motivating the works on generalized Sasakian-space-forms and(
k
,
μ
)-space forms, Carriazo et al. [12] introduced the concept of generalized (
k
,
μ
)-space forms. A generalized (
k
,
μ
)-space form is an almost contact metric manifold (
M
,
ϕ
,
ξ
,
η
,
g
) whose curvature tensor R is given by
(5)
R
=
f
1
R
1
+
f
2
R
2
+
f
3
R
3
+
f
4
R
4
+
f
5
R
5
+
f
6
R
6
,
where R
1, R
2, R
3, R
4, R
5, R
6 are the tensors defined above and f
1, f
2, f
3, f
4, f
5, f
6 are differentiable functions on M.The object of the paper is to study 3-dimensional contact metric generalized(
k
,
μ
)-space forms. The paper is organized as follows. Section 2 deals with some preliminaries on contact metric manifolds and contact metric generalized (
k
,
μ
)-space forms. Section 3 is concerned with 3-dimensional contact metric generalized (
k
,
μ
)-space forms. Here it is proved that a 3-dimensional contact metric generalized (
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q is of constant curvature if and only if 3
f
2
+
f
3
=
0. In Section 4, it is proved that a 3-dimensional contact metric generalized (
k
,
μ
)-space form is Ricci-semisymmetric; then either f
1
=
f
3 or 3
f
2
+
f
3
=
0. In Sections 5 and 6, we obtained that some equivalent conditions for a 3-dimensional contact metric generalized (
k
,
μ
)-space form are pseudosymmetric and ξ-projectively flat, respectively.


## 2. Contact Metric Generalized(
k
,
μ
)-Space Forms

A contact manifold is aC
∞-(2
n
+
1) manifold M
2
n
+
1 equipped with a global 1-form η such that η
∧
(
d
η
)
n
≠
0 everywhere on M
2
n
+
1. Given a contact form η it is well known that there exists a unique vector field ξ called the characteristic vector field of η such that η
(
ξ
)
=
1 and d
η
(
X
,
ξ
)
=
0 for every vector field X on M
2
n
+
1. A Riemannian metric is said to be associated metric if there exists a tensor field ϕ of type (1, 1) such that d
η
(
X
,
Y
)
=
g
(
X
,
ϕ
Y
), η
(
X
)
=
g
(
X
,
ξ
), ϕ
2
X
=
-
X
+
η
(
X
)
ξ, ϕ
ξ
=
0, η
(
ϕ
X
)
=
0, and g
(
ϕ
X
,
ϕ
Y
)
=
g
(
X
,
Y
)
-
η
(
X
)
η
(
Y
) for all vector fields X, Y on M
2
n
+
1. Then the structure (
ϕ
,
ξ
,
η
,
g
) on M
2
n
+
1 is called a contact metric structure, and then manifold M
2
n
+
1 equipped with such a structure is called a contact metric manifold [13].Given a contact metric manifold(
M
2
n
+
1
,
ϕ
,
ξ
,
η
,
g
) we define a (1, 1) tensor field h by 2
h
=
L
ξ
ϕ. Then h is symmetric and satisfies the following relations:
(6)
h
ξ
=
0
,
h
ϕ
=
-
ϕ
h
,
tr
⁡
(
h
)
=
tr
⁡
(
ϕ
h
)
=
0
,
η
·
h
=
0
.
Moreover, if ∇ denotes the Riemannian connection of g, then the following relation holds:
(7)
∇
X
ξ
=
-
ϕ
X
-
ϕ
h
X
.
The vector field ξ is a Killing vector with respect to g if and only if h
=
0. A contact metric manifold (
M
2
n
+
1
,
ϕ
,
ξ
,
η
,
g
) for which ξ is a Killing vector is said to be a K-contact manifold. Therefore, a generalized (
k
,
μ
)-space form with such a structure is actually a generalized Sasakian-space-form.A generalized(
k
,
μ
)-space-form is an almost contact metric manifold (
M
,
ϕ
,
ξ
,
η
,
g
) whose curvature tensor R is given by
(8)
R
(
X
,
Y
)
Z
=
f
1
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
}
+
f
2
{
g
(
X
,
ϕ
Z
)
ϕ
Y
-
g
(
Y
,
ϕ
Z
)
ϕ
X
+
2
g
(
X
,
ϕ
Y
)
ϕ
Z
}
+
f
3
{
η
(
X
)
η
(
Z
)
Y
-
η
(
Y
)
η
(
Z
)
X
+
g
(
X
,
Z
)
η
(
Y
)
ξ
-
g
(
Y
,
Z
)
η
(
X
)
ξ
}
+
f
4
{
g
(
Y
,
Z
)
h
X
-
g
(
X
,
Z
)
h
Y
+
g
(
h
Y
,
Z
)
X
-
g
(
h
X
,
Z
)
Y
}
+
f
5
{
g
(
h
Y
,
Z
)
h
X
-
g
(
h
X
,
Z
)
h
Y
+
g
(
ϕ
h
X
,
Z
)
ϕ
h
Y
-
g
(
ϕ
h
Y
,
Z
)
ϕ
h
X
}
+
f
6
{
η
(
X
)
η
(
Z
)
h
Y
-
η
(
Y
)
η
(
Z
)
h
X
+
g
(
h
X
,
Z
)
η
(
Y
)
ξ
-
g
(
h
Y
,
Z
)
η
(
X
)
ξ
}
for all vector fields X, Y, Z on M, where f
1, f
2, f
3, f
4, f
5, f
6 are differentiable functions on M, 2
h
=
L
ξ
ϕ, and L is the usual Lie derivative. In such case we will denote the manifold as M
(
f
1
,
…
,
f
6
). This kind of manifold appears as natural generalization of the (
k
,
μ
)-space forms by taking
(9)
f
1
=
c
+
3
4
,
f
2
=
c
-
1
4
,
f
3
=
c
+
3
4
-
k
,
f
4
=
1
,
f
5
=
1
2
,
f
6
=
1
-
μ
as constant. Here c denotes constant ϕ-sectional curvature. The ϕ-sectional curvature of generalized (
k
,
μ
)-space form M
(
f
1
,
…
,
f
6
) is f
1
+
3
f
2
+
f
4
g
(
(
h
-
ϕ
h
ϕ
)
X
,
X
). Generalized Sasakian-space-forms [4] are also examples with f
4
=
f
5
=
f
6
=
0 and f
1, f
2, f
3 not necessarily constant.We inferred the following result from [12].Theorem 1.
LetM
(
f
1
,
…
,
f
6
) be a generalized (
k
,
μ
)-space form. If M is a contact metric manifold with f
3
=
f
1
-
1, then it is a Sasakian manifold.Next, by using the definitions of the tensorsR
1, R
2, R
3, R
4, R
5, R
6 and properties (6) of the tensor h in the formula (8), we obtain that the curvature tensor of a generalized (
k
,
μ
)-space form satisfies
(10)
R
(
X
,
Y
)
ξ
=
(
f
1
-
f
3
)
{
η
(
Y
)
X
-
η
(
X
)
Y
}
+
(
f
4
-
f
6
)
{
η
(
Y
)
h
X
-
η
(
X
)
h
Y
}
for every X, Y. Thus we have the following Theorem.Theorem 2 (see [12]).
IfM
(
f
1
,
…
,
f
6
) is a contact metric generalized (
k
,
μ
)-space form, then it is a generalized (
k
,
μ
)-space, with k
=
f
1
-
f
3 and μ
=
f
4
-
f
6.We know from Theorem1 that M
(
f
1
,
…
,
f
6
) is Sasakian if k
=
f
1
-
f
3
=
1. Under the same hypothesis, we also know that f
2
=
f
3
=
f
1
-
1 and h
=
0, so we may take f
4
=
f
5
=
f
6
=
0. Therefore, in this paper we will consider non-Sasakian generalized (
k
,
μ
)-space forms M
(
f
1
,
…
,
f
6
), that is, those with k
=
f
1
-
f
3
≠
1.A contact metric manifold is said to beη-Einstein manifold [13] if it satisfies
(11)
S
=
a
g
+
b
η
⊗
η
for some smooth functions a and b.


## 3. On 3-Dimensional Contact Metric Generalized(
k
,
μ
)-Space Forms

IfM
3
(
f
1
,
…
,
f
6
) is a contact metric generalized (
k
,
μ
)-space form, then its Ricci tensor S and the scalar curvature r can be written as [12]
(12)
S
(
X
,
Y
)
=
(
2
f
1
+
3
f
2
-
f
3
)
g
(
X
,
Y
)
-
(
3
f
2
+
f
3
)
η
(
X
)
η
(
Y
)
+
(
f
4
-
f
6
)
g
(
h
X
,
Y
)
,
(13)
r
=
2
(
3
f
1
+
3
f
2
-
2
f
3
)
.
It is known that in a 3-dimensional Riemannian manifold (
M
3
,
g
), the curvature tensor R is given by
(14)
R
(
X
,
Y
)
Z
=
S
(
Y
,
Z
)
X
-
S
(
X
,
Z
)
Y
+
g
(
Y
,
Z
)
Q
X
-
g
(
X
,
Z
)
Q
Y
-
r
2
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
for any vector fields X, Y, Z on M. By substituting (12) and (13) in (14) we have
(15)
R
(
X
,
Y
)
Z
=
(
f
1
+
3
f
2
)
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
}
+
(
3
f
2
+
f
3
)
{
η
(
X
)
η
(
Z
)
Y
-
η
(
Y
)
η
(
Z
)
X
+
g
(
X
,
Z
)
η
(
Y
)
ξ
-
g
(
Y
,
Z
)
η
(
X
)
ξ
}
+
(
f
4
-
f
6
)
{
g
(
Y
,
Z
)
h
X
-
g
(
X
,
Z
)
h
Y
+
g
(
h
Y
,
Z
)
X
-
g
(
h
X
,
Z
)
Y
}
.
From (12), we have
(16)
S
(
X
,
ξ
)
=
2
(
f
1
-
f
3
)
η
(
X
)
.
Also, from (15) we obtain
(17)
R
(
ξ
,
Y
)
Z
=
(
f
1
-
f
3
)
{
g
(
Y
,
Z
)
ξ
-
η
(
Z
)
Y
}
+
(
f
4
-
f
6
)
{
g
(
h
Y
,
Z
)
ξ
-
η
(
Z
)
h
Y
}
.IfM
3
(
f
1
,
…
,
f
6
) is a contact metric generalized (
k
,
μ
)-space form, then Q
ϕ
=
ϕ
Q is true if and only if f
4
-
f
6
=
0. In this connection, the following result appears in [12].Lemma 3.
IfM
3
(
f
1
,
…
,
f
6
) is a contact metric generalized (
k
,
μ
)-space form, then the following conditions are equivalent:(i)
M
3 is an η-Einstein,
(ii)
Q
ϕ
=
ϕ
Q, where Q denotes the Ricci tensor,
(iii)
M
3 is a (
f
1
-
f
3
,
0
)-space,
(iv)
f
4
-
f
6
=
0.Now we begin with the following.Lemma 4.
A 3-dimensional contact metric generalized(
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q is of constant curvature if and only if 3
f
2
+
f
3
=
0.Proof.
Let a 3-dimensional contact metric generalized(
k
,
μ
)-space form M
3
(
f
1
,
…
,
f
6
) with Q
ϕ
=
ϕ
Q be a space of constant curvature. Then
(18)
R
(
X
,
Y
)
Z
=
c
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
}
,
where c is constant curvature of the manifold. By using the definition of Ricci curvature and (18) we have
(19)
S
(
X
,
Y
)
=
2
c
g
(
X
,
Y
)
.
If we use (19) in the definition of the scalar curvature we get
(20)
r
=
6
c
.
From (19) and (20) one can easily see that
(21)
S
(
X
,
Y
)
=
r
3
g
(
X
,
Y
)
.
By putting X
=
Y
=
ξ in (12) and using (21) we obtain r
=
6
(
f
1
-
f
3
). By comparing this value of r with (13), we have 3
f
2
+
f
3
=
0.
Conversely, if3
f
2
+
f
3
=
0, then from (15) we can easily see that
(22)
R
(
X
,
Y
)
Z
=
(
f
1
-
f
3
)
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
}
+
(
f
4
-
f
6
)
{
g
(
Y
,
Z
)
h
X
-
g
(
X
,
Z
)
h
Y
+
g
(
h
Y
,
Z
)
X
-
g
(
h
X
,
Z
)
Y
}
.
If M
3
(
f
1
,
…
,
f
6
) satisfies Q
ϕ
=
ϕ
Q, then in view of Lemma 3 we have f
4
-
f
6
=
0. Thus from (22), we have
(23)
R
(
X
,
Y
)
Z
=
(
f
1
-
f
3
)
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
}
.
If M
3
(
f
1
,
…
,
f
6
) satisfies f
4
-
f
6
=
0, then f
1
-
f
3 is constant. Hence, from (23), we conclude that the M
3 is of constant curvature. This completes the proof.Theorem 5.
Every 3-dimensional contact metric generalized(
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q is an N
(
f
1
-
f
3
)-η-Einstein manifold.Proof.
The proof follows from (12), (10), and Lemma 3.


## 4. Ricci-Semisymmetric 3-Dimensional Contact Metric Generalized(
k
,
μ
)-Space Forms

A contact metric generalized(
k
,
μ
)-space form is said to be Ricci-semisymmetric if its Ricci tensor S satisfies the condition
(24)
R
(
X
,
Y
)
·
S
=
0
,
X
,
Y
∈
χ
(
M
)
,
where R
(
X
,
Y
) acts as a derivation on S. This notion was introduced by Mirzoyan [14] for Riemannian spaces.From (24) we have
(25)
S
(
R
(
X
,
Y
)
U
,
V
)
+
S
(
U
,
R
(
X
,
Y
)
V
)
=
0
.
If we substitute X
=
ξ in (25) and then using (17), we get
(26)
(
f
1
-
f
3
)
[
g
(
Y
,
U
)
S
(
ξ
,
V
)
-
η
(
U
)
S
(
Y
,
V
)
+
g
(
Y
,
V
)
S
(
ξ
,
U
)
-
η
(
V
)
S
(
Y
,
U
)
]
+
(
f
4
-
f
6
)
[
g
(
h
Y
,
U
)
S
(
ξ
,
V
)
-
η
(
U
)
S
(
h
Y
,
V
)
+
g
(
h
Y
,
V
)
S
(
ξ
,
U
)
-
η
(
V
)
S
(
h
Y
,
U
)
]
=
0
.
By using (16) in (26) we obtain
(27)
2
(
f
1
-
f
3
)
[
(
f
1
-
f
3
)
{
g
(
Y
,
U
)
η
(
V
)
+
g
(
Y
,
V
)
η
(
U
)
}
+
(
f
4
-
f
6
)
{
g
(
h
Y
,
U
)
η
(
V
)
+
g
(
h
Y
,
V
)
η
(
U
)
}
]
-
(
f
1
-
f
3
)
[
S
(
Y
,
V
)
η
(
U
)
+
S
(
Y
,
U
)
η
(
V
)
]
-
(
f
4
-
f
6
)
[
S
(
h
Y
,
V
)
η
(
U
)
+
S
(
h
Y
,
U
)
η
(
V
)
]
=
0
.
Consider that {
e
1
,
e
2
,
e
3
} is an orthonormal basis of the T
p
M, p
∈
M. Then by putting Y
=
U
=
e
i in (27) and taking the summation for 1
≤
i
≤
3, we have
(28)
(
f
1
-
f
3
)
[
8
(
f
1
-
f
3
)
η
(
V
)
-
S
(
V
,
ξ
)
-
r
η
(
V
)
]
=
0
.
Again by using (16) in (28), we get
(29)
(
f
1
-
f
3
)
[
6
(
f
1
-
f
3
)
-
r
]
η
(
V
)
=
0
,
which gives either f
1
=
f
3 or r
=
6
(
f
1
-
f
3
). For the second case, that is, r
=
6
(
f
1
-
f
3
), we have 3
f
2
+
f
3
=
0. Thus we can state the following.Theorem 6.
If a 3-dimensional contact metric generalized(
k
,
μ
)-space form is Ricci-semisymmetric then either f
1
=
f
3 or 3
f
2
+
f
3
=
0.


## 5. Pseudosymmetric 3-Dimensional Contact Metric Generalized(
k
,
μ
)-Space Forms

Let(
M
,
g
) be a Riemannian manifold with the Riemannian metric ∇. A tensor field F
:
χ
(
M
)
×
χ
(
M
)
×
χ
(
M
)
→
χ
(
M
) of type (1, 3) is said to be curvature-like if it has the properties of R. For example, the tensor R given by
(30)
R
(
X
,
Y
)
Z
=
(
X
∧
Y
)
Z
=
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
,
00
00
000000000000000000000000
X
,
Y
,
Z
∈
χ
(
M
)
defines a curvature-like tensor field on M. For example, if R
(
X
,
Y
)
Z
=
k
(
(
X
∧
Y
)
Z
)
=
k
(
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
), then the manifold is of constant curvature k.It is well known that every curvature-like tensor fieldF acts on the algebra 𝔗
s
1
(
M
) of all tensor fields on M of type (
1
,
s
) as a derivation [15]
(31)
(
F
·
P
)
(
X
1
,
…
,
X
s
;
Y
,
X
)
=
F
(
X
,
Y
)
{
P
(
X
1
,
…
,
X
s
)
}
-
∑
j
=
1
s
‍
P
(
X
1
,
…
,
F
(
X
,
Y
)
X
j
,
…
,
X
s
)
for all X
1
,
…
,
X
s
∈
χ
(
M
) and P
∈
𝔗
s
1
(
M
). The derivation F
·
P of P by F is a tensor field of type (
1
,
s
+
2
). For a tensor field P of type (
1
,
s
), we define the derivative of P with respect to the curvature-like tensor defined by (30) as
(32)
Q
(
g
,
P
)
(
X
1
,
…
,
X
s
;
Y
,
X
)
=
(
X
∧
Y
)
P
(
X
1
,
…
,
X
s
)
-
∑
j
=
1
s
‍
P
(
X
1
,
…
,
(
X
∧
Y
)
X
j
,
…
,
X
s
)
.
As a generalization of locally symmetric (
∇
R
=
0
) manifolds, the notion of a semisymmetric (
R
·
R
=
0
) manifold is introduced by Szabó [16]. As a proper generalization of a semisymmetric manifold, the notion of pseudosymmetric manifold is introduced by Deszcz [17]. A Riemannian manifold (
M
,
g
) is said to be pseudosymmetric if there exists a real valued smooth function L
:
M
→
ℝ such that R
·
R
=
L
(
g
,
R
). In particular, if L is constant, then M is said to be a pseudosymmetric manifold of constant type [17]. A pseudosymmetric manifold is said to be proper if it is not semisymmetric.For 3-dimensional Riemannian spaces, the following characterization of pseudosymmetry is known (cf. [18, 19]).Theorem 7.
A 3-dimensional Riemannian space is pseudosymmetric if and only if it isη-Einstein.We also know that a 3-dimensional contact metric generalized(
k
,
μ
)-space form is η-Einstein if and only if f
4
-
f
6
=
0. Hence, in view of Theorem 7, we can state the following.Theorem 8.
A 3-dimensional contact metric generalized(
k
,
μ
)-space form is pseudosymmetric if and only if f
4
-
f
6
=
0.From Lemma3 and Theorem 8, we have following.Lemma 9.
A 3-dimensional contact metric generalized(
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q is pseudosymmetric.


## 6.ξ-Projectively Flat 3-Dimensional Contact Metric Generalized (
k
,
μ
)-Space Forms

In a 3-dimensional contact metric generalized(
k
,
μ
)-space form M
3
(
f
1
,
…
,
f
6
) the projective curvature tensor 𝒫 is defined as
(33)
𝒫
(
X
,
Y
)
Z
=
R
(
X
,
Y
)
Z
-
1
2
[
S
(
Y
,
Z
)
X
-
S
(
X
,
Z
)
Y
]
for all X
,
Y
,
Z
∈
T
M.By pluggingZ
=
ξ in (33) and using (16) and (10), we get
(34)
𝒫
(
X
,
Y
)
ξ
=
(
f
4
-
f
6
)
[
η
(
Y
)
h
X
-
η
(
X
)
h
Y
]
.
If f
4
-
f
6
=
0, then from (34) one can get
(35)
𝒫
(
X
,
Y
)
ξ
=
0
;
that is, M
3
(
f
1
,
…
,
f
6
) is ξ-projectively flat.Conversely, suppose thatM
3
(
f
1
,
…
,
f
6
) is ξ-projectively flat. Then from (34) we obtain
(36)
(
f
4
-
f
6
)
[
η
(
Y
)
h
X
-
η
(
X
)
h
Y
]
=
0
.
This implies that f
4
-
f
6
=
0. Thus we can state the following.Theorem 10.
A 3-dimensional contact metric generalized(
k
,
μ
)-space form is ξ-projectively flat if and only if f
4
-
f
6
=
0.From Lemma3 and Theorem 10, we obtain the following.Lemma 11.
A 3-dimensional contact metric generalized(
k
,
μ
)-space form with Q
ϕ
=
ϕ
Q is ξ-projectively flat.By combining Lemma3, Theorem 5, Theorem 8, and Theorem 10 the following result is obtained.Theorem 12.
IfM
3
(
f
1
,
…
,
f
6
) is a contact metric generalized (
k
,
μ
)-space form, then the following conditions are equivalent to each other:(i)
M
3 is N
(
f
1
-
f
3
)-η-Einstein,
(ii)
Q
ϕ
=
ϕ
Q, where Q denotes the Ricci operator,
(iii)
f
4
-
f
6
=
0,
(iv)
M
3 is pseudosymmetric,
(v)
M
3 is ξ-projectively flat.



---

*Source: 797162-2014-04-15.xml*