# On Generalizedϕ-Recurrent (

Authors: C. S. Bagewadi; Dakshayani A. Patil
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

We study generalizedϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifolds. A relation between the associated 1-forms A and B and relation between characteristic vector field ξ and the vector fields ρ
1
,
ρ
2 for a generalized ϕ-recurrent.


---

## Body

## 1.  Introduction

As a generalization of both Sasakian and Kenmotsu manifolds, Oubiña [1] introduced the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kahler manifolds. Trans-Sasakian manifolds of types (
0,0
),(
α
,
0
), and (
0
,
β
) are, respectively, called the cosymplectic, α-Sasakian, and β-Kenmotsu manifolds, with α,β being scalar functions. In particular, ifα
=
0,β
=
1;α
=
1,β
=
0; then a trans-Sasakian manifold will be Kenmotsu and Sasakian manifolds respectively. The local structure of trans-Sasakian manifolds of dimension n
≥
5 has been completely characterized by Marrero [2]. He proved that a trans-Sasakian manifold of dimension n
≥
5 is either cosymplectic or β-Kenmotsu or α-Sasakian manifold.The investigation of manifolds with indefinite metrics was taken by several authors. In 1993, Bejancu and Duggal [3] introduced the concept of (
ϵ
)-Sasakian manifolds and Xufeng and Xiaoli [4] established that these manifolds are real hypersurfaces of indefinite Kahlerian manifolds. Kumar et al. [5] studied the curvature conditions of these manifolds. Tripathi et al. [6] introduced and studied (
ϵ
)-almost paracontact manifolds.The paper is organised as follows: Section2 contains preliminaries of (
ϵ
,
δ
)-trans-Sasakian manifolds. In Sections 3 and 4 we obtain results for generalized ϕ-recurrent and generalized concircular ϕ-recurrent (
ϵ
,
δ
)-Trans-Sasakian Manifolds.


## 2.  Preliminaries

LetM be the connected almost contact metric manifold with an almost contact metric structure (
ϕ
,
ξ
,
η
,
g
); that is, ϕ is a (
1,1
) tensor field, ξ is a vector field, η is a 1-form, and g is compatible Riemannian metric, such that
(1)
ϕ
2
X
=
-
X
+
η
(
X
)
ξ
,
(2)
η
(
ξ
)
=
1
,
ϕ
ξ
=
0
,
η
(
ϕ
X
)
=
0
.
Almost contact metric manifold M is called (
ϵ
)-almost contact metric manifold if
(3)
g
(
ξ
,
ξ
)
=
ϵ
,
η
(
X
)
=
ϵ
g
(
X
,
ξ
)
,
g
(
ϕ
X
,
ϕ
Y
)
=
g
(
X
,
Y
)
-
ϵ
η
(
X
)
η
(
Y
)
,
for all vector fieldsX,YonM, where ϵ
=
g
(
ξ
,
ξ
)
=
±
1.An(
ϵ
)-almost contact metric manifold is called a (
ϵ
,
δ
)-trans-Sasakian manifold if
(4)
(
∇
X
ϕ
)
(
Y
)

=

α
{
g
(
X
,
Y
)
ξ
-
ϵ
η
(
Y
)
X
}
+

β
{
g
(
ϕ
X
,
Y
)
ξ
-
δ
η
(
Y
)
ϕ
X
}
holds for some smooth functions α and β on M and ϵ
=
±
1, δ
=
±
1. For β
=
0, α
=
1, (
ϵ
,
δ
)-trans-Sasakian manifold reduces to (
ϵ
)-Sasakian and for α
=
0, β
=
1 it reduces to (
ϵ
)-Kenmotsu manifold. (
ϵ
,
δ
)-Trans-Sasakian manifolds have been studied by authors [7] and they obtained the following results:
(5)
∇
X
ξ
=
-
ϵ
α
ϕ
X
-
δ
β
{
X
-
η
(
X
)
ξ
}
,
(6)
(
∇
X
η
)
(
Y
)
=
δ
β
[
ϵ
g
(
X
,
Y
)
-
η
(
X
)
η
(
Y
)
]
-
α
g
(
ϕ
X
,
Y
)
,
(7)
R
(
X
,
Y
)
ξ
=
(
α
2
-
β
2
)
[
η
(
Y
)
X
-
η
(
X
)
Y
]
-
ϵ
[
(
X
α
)
ϕ
Y
-
(
Y
α
)
ϕ
X
]
-
δ
[
(
X
β
)
ϕ
2
(
Y
)
-
(
Y
β
)
ϕ
2
X
]
+
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
X
,
Y
)
ξ
+
2
ϵ
δ
α
β
[
η
(
Y
)
ϕ
X
-
η
(
X
)
ϕ
Y
]
,
(8)
S
(
X
,
ξ
)
=
-
(
ϕ
X
)
α
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
(
ξ
β
)
]
η
(
X
)
-
(
2
n
-
1
)
X
β
,
(9)
(
ξ
α
)
+
2
δ
α
β
=
0
,
(10)
S
(
ξ
,
ξ
)
=
2
n
(
ϵ
α
2
-
β
2
δ
-
ξ
β
)
,
(11)
ϕ
grad
α
=
grad
β
,
(12)
η
(
R
(
X
,
Y
)
Z
)
=
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
X
,
Y
)
η
(
Z
)
-
(
X
α
)
g
(
Y
,
ϕ
Z
)
+
(
Y
α
)
g
(
X
,
ϕ
Z
)
-
α
β
(
δ
+
ϵ
)
×
[
g
(
Y
,
ϕ
Z
)
η
(
X
)
-
g
(
X
,
ϕ
Z
)
η
(
Y
)
]
-
(
ϵ
α
2
-
δ
β
2
)
×
[
g
(
X
,
Z
)
η
(
Y
)
-
g
(
Y
,
Z
)
η
(
X
)
]
-
(
X
β
)
[
g
(
Y
,
Z
)
-
ϵ
η
(
Y
)
η
(
Z
)
]
+
(
Y
β
)
[
g
(
X
,
Z
)
-
ϵ
η
(
X
)
η
(
Z
)
]
.


## 3. Generalizedϕ-Recurrent (
ϵ
,
δ
)-Trans-Sasakian Manifolds

Definition 1.
A
(
ϵ
,
δ
)-trans-Sasakian  manifold M
2
n
+
1
(
ϕ
,
ξ
,
η
,
g
) is called generalized ϕ-recurrent if its curvature tensor R satisfies the condition
(13)
ϕ
2
(
(
∇
W
R
)
(
X
,
Y
)
Z
)
=
A
(
W
)
R
(
X
,
Y
)
Z
+
B
(
W
)
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
,
where A and B are two 1-forms, B is nonzero, and these are defined by
(14)
A
(
W
)
=
g
(
W
,
ρ
1
)
,
B
(
W
)
=
g
(
W
,
ρ
2
)
,
and ρ
1, ρ
2 are vector fields associated with 1-forms A and B, respectively.Let us consider a generalizedϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifold. Then by virtue of (1) and (13) we have
(15)
-
(
∇
W
R
)
(
X
,
Y
)
Z
+
η
(
(
∇
W
R
)
(
X
,
Y
)
Z
)
ξ
=
A
(
W
)
R
(
X
,
Y
)
Z
+
B
(
W
)
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
,
from which it follows that
(16)
-
g
(
(
∇
W
R
)
(
X
,
Y
)
Z
,
U
)
+
η
(
(
∇
W
R
)
(
X
,
Y
)
Z
)
η
(
U
)
=
A
(
W
)
g
(
R
(
X
,
Y
)
Z
,
U
)
+
B
(
W
)
[
g
(
Y
,
Z
)
g
(
X
,
U
)
-
g
(
X
,
Z
)
g
(
Y
,
U
)
]
.
Let {
e
i
}, i
=
1,2
,
…
,
(
2
n
+
1
), be an orthonormal basis of the tangent space at any point of the manifold. Then putting X
=
U
=
e
i in (16) and taking summation overi, 1
≤
i
≤
2
n
+
1, we get
(17)
-
(
∇
W
S
)
(
Y
,
Z
)
+
∑
i
=
1
2
n
+
1
‍
η
(
(
∇
W
R
)
(
e
i
,
Y
)
Z
)
η
(
e
i
)
=
A
(
W
)
S
(
Y
,
Z
)
+
2
n
B
(
W
)
g
(
Y
,
Z
)
.
Put Z
=
ξ in (17) and by virtue of (8) and the equation g
(
(
∇
W
R
)
(
e
i
,
Y
)
ξ
,
ξ
)
=
0, we have
(18)
(
∇
W
S
)
(
Y
,
ξ
)
=
-
A
(
W
)
×
[
-
(
ϕ
Y
)
α
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
f
g
g
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
-
Y
β
]
-
2
n
B
(
W
)
η
(
Y
)
ϵ
.
Now we have
(19)
(
∇
W
S
)
(
Y
,
ξ
)
=
∇
W
S
(
Y
,
ξ
)
-
S
(
∇
W
Y
,
ξ
)
-
S
(
Y
,
∇
W
ξ
)
.
Using (5) and (6) in the above relation it follows that
(20)
(
∇
W
S
)
(
Y
,
ξ
)
=
-
W
(
ϕ
Y
)
α
-
ϕ
Y
(
W
α
)
+
[
4
n
β
δ
(
W
β
)
-
W
(
ξ
β
)
4
n
ϵ
α
(
W
α
)
-
4
n
β
δ
(
W
β
)
g
h
-
W
(
ξ
β
)
]
η
(
Y
)
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
×
{
δ
β
[
ϵ
g
(
W
,
Y
)
-
η
(
W
)
η
(
Y
)
]
g
h
h
-
α
g
(
ϕ
W
,
Y
)
}
-
(
2
n
-
1
)
W
(
Y
β
)
+
(
ϕ
∇
W
Y
)
α
+
(
2
n
-
1
)
(
∇
W
Y
)
β
+
ϵ
α
S
(
Y
,
ϕ
W
)
-
δ
β
S
(
Y
,
W
)
+
δ
β
η
(
W
)
×
[
-
(
ϕ
Y
)
α
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
g
h
g
-
(
2
n
-
1
)
Y
β
]
.
From (18) and (20) we obtain
(21)
-
W
(
ϕ
Y
)
α
-
ϕ
Y
(
W
α
)
+
[
4
n
ϵ
α
(
W
α
)
-
4
n
β
δ
(
W
β
)
-
W
(
ξ
β
)
]
η
(
Y
)
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
×
{
δ
β
[
ϵ
g
(
W
,
Y
)
-
η
(
W
)
η
(
Y
)
]
-
α
g
(
ϕ
W
,
Y
)
}
-
(
2
n
-
1
)
W
(
Y
β
)
+
(
ϕ
∇
W
Y
)
α
+
(
2
n
-
1
)
(
∇
W
Y
)
β
+
ϵ
α
S
(
Y
,
ϕ
W
)
-
δ
β
S
(
Y
,
W
)
+
δ
β
η
(
W
)
×
[
-
(
ϕ
Y
)
α
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
-
(
2
n
-
1
)
Y
β
g
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
-
(
2
n
-
1
)
Y
β
]
=
-
A
(
W
)
g
×
[
-
(
ϕ
Y
)
α
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
-
(
2
n
-
1
)
Y
β
fg
+
[
2
n
(
ϵ
α
2
-
β
2
δ
)
-
ξ
β
]
η
(
Y
)
-
(
2
n
-
1
)
Y
β
]
-
2
n
B
(
W
)
η
(
Y
)
ϵ
.
Replacing Y
=
ξ in (21) and then using (2) we get
(22)
4
n
ϵ
α
(
W
α
)
-
4
n
β
δ
(
W
β
)
-
2
n
W
(
ξ
β
)
+
[
ϵ
α
(
W
-
η
(
W
)
ξ
)
+
δ
β
ϕ
W
]
α
+
(
2
n
-
1
)
[
-
ϵ
α
ϕ
W
-
δ
β
(
W
-
η
(
W
)
ξ
)
]
β
+
ϵ
α
[
-
(
ϕ
2
W
)
α
-
(
2
n
-
1
)
(
ϕ
W
)
β
]
-
δ
β
[
-
(
ϕ
W
)
α
-
(
2
n
-
1
)
W
β
+
(
2
n
-
1
)
η
(
W
)
ξ
β
]
=
-
A
(
W
)
[
2
n
(
ϵ
α
2
-
β
2
δ
-
ξ
β
)
]
-
2
n
B
(
W
)
ϵ
.Theorem 2.
In a generalizedϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifold, the 1-forms A and B are related as in (22).Ifβ
=
0, then, from (22), we obtain
(23)
-
2
α
(
W
α
)
=
α
2
A
(
W
)
+
B
(
W
)
.Corollary 3.
In a generalizedϕ-recurrent ϵ-α-Sasakian manifold, the 1-forms A and B are related as in (23).(i)
Ifϵ
=
1, then ϵ-α-Sasakian manifold reduces as α-Sasakian manifold.
(ii)
Ifϵ
=
1 and α
=
1, then ϵ-α-Sasakian manifold reduces as Sasakian manifold.Ifα
=
0, then, from (22), we obtain
(24)
(
1
-
2
n
)
δ
β
ϵ
η
(
W
)
(
ξ
β
)
-
4
n
δ
β
ϵ
(
W
β
)
-
2
n
ϵ
W
(
ξ
β
)
=
A
(
W
)
[
2
n
ϵ
(
β
2
δ
+
ξ
β
)
]
-
2
n
B
(
W
)
.Corollary 4.
In a generalizedϕ-recurrent ϵ-β-Kenmotsu manifold, the 1-forms A and B are related as in (24).Ifα
=
β
=
0, then, from (22), we obtain
(25)
B
(
W
)
=
0
.
From definition (13) this leads to the following corollary.Corollary 5.
In cosymplectic manifold, generalizedϕ-recurrent reduces to ϕ-recurrent manifold.Now from (15) we have
(26)
(
∇
W
R
)
(
X
,
Y
)
Z
=
η
(
(
∇
W
R
)
(
X
,
Y
)
Z
)
ξ
-
A
(
W
)
R
(
X
,
Y
)
Z
-
B
(
W
)
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
.
From (26) and second Bianchi identity we get
(27)
A
(
W
)
η
(
R
(
X
,
Y
)
Z
)
+
A
(
X
)
η
(
R
(
Y
,
W
)
Z
)
+
A
(
Y
)
η
(
R
(
W
,
X
)
Z
)
+
B
(
W
)
[
g
(
Y
,
Z
)
η
(
X
)
-
g
(
X
,
Z
)
η
(
Y
)
]
+
B
(
X
)
[
g
(
W
,
Z
)
η
(
Y
)
-
g
(
Y
,
Z
)
η
(
W
)
]
+
B
(
Y
)
[
g
(
X
,
Z
)
η
(
W
)
-
g
(
W
,
Z
)
η
(
X
)
]
=
0
.
By virtue of (12) we obtain from (27) that
(28)
A
(
W
)
[
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
X
,
Y
)
η
(
Z
)
-
(
X
α
)
g
(
Y
,
ϕ
Z
)
gghhhh
+
(
Y
α
)
g
(
X
,
ϕ
Z
)
-
α
β
(
δ
+
ϵ
)
gghhhh
×
[
g
(
Y
,
ϕ
Z
)
η
(
X
)
-
g
(
X
,
ϕ
Z
)
η
(
Y
)
]
-
(
ϵ
α
2
-
δ
β
2
)
ghghhh
×
[
g
(
X
,
Z
)
η
(
Y
)
-
g
(
Y
,
Z
)
η
(
X
)
]
hhhggh
-
(
X
β
)
[
g
(
Y
,
Z
)
-
ϵ
η
(
Y
)
η
(
Z
)
]
hghghh
+
(
Y
β
)
[
g
(
X
,
Z
)
-
ϵ
η
(
X
)
η
(
Z
)
]
]
+
A
(
X
)
[
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
Y
,
W
)
η
(
Z
)
-
(
Y
α
)
g
(
W
,
ϕ
Z
)
ggghgghhh
+
(
W
α
)
g
(
Y
,
ϕ
Z
)
-
α
β
(
δ
+
ϵ
)
ggghgghhh
×
[
g
(
W
,
ϕ
Z
)
η
(
Y
)
-
g
(
Y
,
ϕ
Z
)
η
(
W
)
]
ghggghhgh
-
(
ϵ
α
2
-
δ
β
2
)
[
g
(
Y
,
Z
)
η
(
W
)
-
g
(
W
,
Z
)
η
(
Y
)
]
ghggghhgh
-
(
Y
β
)
[
g
(
W
,
Z
)
-
ϵ
η
(
W
)
η
(
Z
)
]
ghggghghh
+
(
W
β
)
[
g
(
Y
,
Z
)
-
ϵ
η
(
Y
)
η
(
Z
)
]
]
+
A
(
Y
)
[
[
g
(
W
,
Z
)
-
ϵ
η
(
W
)
η
(
Z
)
]
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
W
,
X
)
η
(
Z
)
-
(
W
α
)
g
(
X
,
ϕ
Z
)
gggghghhh
+
(
X
α
)
g
(
W
,
ϕ
Z
)
-
α
β
(
δ
+
ϵ
)
gggghghhh
×
[
g
(
X
,
ϕ
Z
)
η
(
W
)
-
g
(
W
,
ϕ
Z
)
η
(
X
)
]
ghgggghhh
-
(
ϵ
α
2
-
δ
β
2
)
ghgggghhh
×
[
g
(
W
,
Z
)
η
(
X
)
-
g
(
X
,
Z
)
η
(
W
)
]
ghgggghhh
-
(
W
β
)
[
g
(
X
,
Z
)
-
ϵ
η
(
X
)
η
(
Z
)
]
ghghghggh
+
(
X
β
)
[
g
(
W
,
Z
)
-
ϵ
η
(
W
)
η
(
Z
)
]
]
+
B
(
X
)
[
g
(
W
,
Z
)
η
(
Y
)
-
g
(
Y
,
Z
)
η
(
W
)
]
+
B
(
Y
)
[
g
(
X
,
Z
)
η
(
W
)
-
g
(
W
,
Z
)
η
(
X
)
]
=
0
.
Putting Y
=
Z
=
e
iin (28) and taking summation over 1
≤
i
≤
3, we get
(29)
A
(
W
)
[
{
ϵ
α
2
-
δ
β
2
-
ϵ
(
ξ
β
)
}
η
(
X
)
-
(
ϕ
X
)
α
]
+
A
(
X
)
[
{
-
(
ϵ
α
2
-
δ
β
2
)
+
ϵ
(
ξ
β
)
}
η
(
W
)
+
(
ϕ
W
)
α
]
+
A
(
ξ
)
×
[
2
α
β
(
δ
-
ϵ
)
g
(
ϕ
W
,
X
)
+
ϵ
(
W
β
)
η
(
X
)
-
ϵ
(
X
β
)
η
(
W
)
]
+
A
(
ϕ
X
)
[
(
W
α
)
+
α
β
(
δ
+
ϵ
)
η
(
W
)
]
+
A
(
ϕ
W
)
[
-
(
X
α
)
-
α
β
(
δ
+
ϵ
)
η
(
X
)
]
+
B
(
W
)
η
(
X
)
-
B
(
X
)
η
(
W
)
=
0
.
If β
=
0, then, from (29), we obtain
(30)
ϵ
α
2
A
(
W
)
η
(
X
)
=
ϵ
α
2
A
(
X
)
η
(
W
)
,
B
(
W
)
η
(
X
)
=
B
(
X
)
η
(
W
)
for all vector fields X,W. Replacing X by ξ in (30), we get
(31)
A
(
W
)
=
η
(
W
)
η
(
ρ
1
)
,
B
(
W
)
=
η
(
W
)
η
(
ρ
2
)
for all vector field W.From (30) and (31), we can state the following.Theorem 6.
A generalizedϕ-recurrent ϵ-α-Sasakian manifold, the characteristic vector field ξ, and the vector fields ρ
1,ρ
2 associated with the 1-forms A,B, respectively, are in the same direction and the 1-forms A,B are given by (31).


## 4.  Generalized Concircularϕ-Recurrent (
ϵ
,
δ
)-Trans-Sasakian Manifolds

In this section we study three-dimensional generalized concircularϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifold.Definition 7.
(
ϵ
,
δ
)-Trans-Sasakian manifold (
M
3
,
g
) is called generalized concircular ϕ-recurrent if its concircular curvature tensor C
¯
(32)
C
¯
(
X
,
Y
)
Z
=
R
(
X
,
Y
)
Z
-
r
6
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]satisfies the condition(33)
ϕ
2
(
(
∇
W
C
¯
)
(
X
,
Y
)
Z
)
=
A
(
W
)
C
¯
(
X
,
Y
)
Z
+
B
(
W
)
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
,
where A
(
W
) and B
(
W
) are defined as in (14) and r is the scalar curvature of the manifold (
M
3
,
g
).Let us consider a generalized concircularϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifold. Then by virtue of (1) we have
(34)
-
(
∇
W
C
¯
)
(
X
,
Y
)
Z
+
η
(
(
∇
W
C
¯
)
(
X
,
Y
)
Z
)
ξ
=
A
(
W
)
C
¯
(
X
,
Y
)
Z
+
B
(
W
)
[
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
]
,
from which it follows that
(35)
-
g
(
(
∇
W
C
¯
)
(
X
,
Y
)
Z
,
U
)
+
η
(
(
∇
W
C
¯
)
(
X
,
Y
)
Z
)
η
(
U
)
=
A
(
W
)
g
(
C
¯
(
X
,
Y
)
Z
,
U
)
+
B
(
W
)
[
g
(
Y
,
Z
)
g
(
X
,
U
)
-
g
(
X
,
Z
)
g
(
Y
,
U
)
]
.
Let e
1,e
2,e
3 be a local orthonormal basis of the tangent space at any point of the manifold. Then putting Y
=
Z
=
e
i in (35) and taking summation overi, 1
≤
i
≤
3, we get
(36)
-
(
∇
W
S
)
(
X
,
U
)
+
∇
W
r
3
g
(
X
,
U
)
+
(
∇
W
S
)
(
X
,
ξ
)
η
(
U
)
-
∇
W
r
3
η
(
X
)
η
(
U
)
=
A
(
W
)
[
S
(
X
,
U
)
-
r
3
g
(
X
,
U
)
]
+
2
B
(
W
)
g
(
X
,
U
)
.
Putting X
=
U
=
ξ in (36) and using (11) we have
(37)
A
(
W
)
[
2
(
β
2
δ
-
ϵ
α
2
+
ξ
β
)
+
r
ϵ
3
]
-
2
ϵ
B
(
W
)
=
0
.
This leads to the following.Theorem 8.
In a generalized concircularϕ-recurrent (
ϵ
,
δ
)-trans-Sasakian manifold M
3
(
ϕ
,
ξ
,
η
,
g
), the 1-forms A and B are related as in (37).



---

*Source: 736846-2014-01-30.xml*