A NEW TYPE LORENTZIAN ALMOST PARA CONTACT MANIFOLD

Abstract

The present study initially introduced a new type Lorentzian almost para contact manifold using the generalized symmetric metric connections of type $(\alpha,\beta)$. Later, some results is given about new type Lorentzian almost para contact manifold. \end{abstract}

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Journal of Universal Mathematics
Vol.5 No.2 pp.117-121 (2022)
ISSN-2618-5660
DOI: 10.33773/jum.1134272
A NEW TYPE LORENTZIAN ALMOST PARA CONTACT
MANIFOLD
O˘
GUZHAN BAHADIR
0000-0001-5054-8865
Abstract. The present study initially introduced a new type Lorentzian al-
most para contact manifold using the generalized symmetric metric connec-
tions of type (α, β). Later, some results is given about new type Lorentzian
almost para contact manifold.
1. Introduction
A linear connection ∇on a (semi-)Riemannian manifold Mis suggested to be a
generalized symmetric connection if its torsion tensor Tis presented as follows:
(1.1) T(X, Y ) = α{u(Y)X−u(X)Y}+β{u(Y)ϕX −u(X)ϕY },
for any vector fields Xand Yon M, where αand βare constant functions on M
[2, 3]. ϕcan be viewed as a tensor of type (1,1) and uis regarded as a 1-form
connected with the vector field which has a non-vanishing smooth non-null unit.
Moreover, the connection ∇is called to be metric connection if ∇g= 0, with g
metric tensor [1].
A linear metric connection satisfying the equation (1.1) is called generalized
symmetric metric connections of type (α, β). We remark that generalized symmet-
ric metric connections of type (1,0) is semi-symmetric connection and generalized
symmetric metric connections of type (0,1) is quarter-symmetric connection, re-
spectively [4, 5].
In the present paper, we define a new type Lorentzian almost para contact man-
ifold using the generalized symmetric metric connections of type (α, β)
Date:Received: 2022-06-22; Accepted: 2022-07-27 .
2000 Mathematics Subject Classification. 53C15, 53C25, 53C40.
Key words and phrases. Lorentzian almost para contact manifold, Generalized symmetric met-
ric connection, Parallelized generalized symmetric metric connection.
117

118 O ˘
GUZHAN BAHADIR
2. Lorentzian almost para contact manifold with parallelized
generalized symmetric metric connection
Let Mbe a differentiable manifold endowed with a (1,1) tensor field φ, a con-
travariant vector field ξ, a 1-form ηand Lorentzian metric g, which satisfies
η(ξ) = −1, φ2(X) = X+η(X)ξ,(2.1)
g(φX, φY ) = g(X, Y ) + η(X)η(Y), g(X, ξ ) = η(X),(2.2)
φξ = 0, η(φX)=0.(2.3)
for all vector fields X,Yon M, where ∇is the Levi-Civita connection with respect
to the Lorentzian metric g. Such manifold (M, ξ, η, g) is called Lorentzian almost
para contact manifold. If we mark Φ(X, Y ) = g(φX, Y ) for all vector fields X,Y
on M, then the tensor field Φ is a symmetric (0,2) tensor field [6, 7, 8].
Now, we will give the main characterization theorem.
Theorem 2.1. For an Lorentzian almost para contact manifold, the generalized
symmetric metric connection ∇of type (α, β)is given by
(2.4) ∇XY=∇XY+α{η(Y)X−g(X, Y )ξ}+β{η(Y)φX −g(φX, Y )ξ}.
Proof. There is the following relationship between a linear connection ∇and Levi-
Civita connection ∇
∇XY=∇XY+H(X, Y ),
for all vector field Xand Y. The following is obtained so that ∇is a generalized
symmetric metric connection of ∇, in which His viewed as a tensor of type (1,2);
(2.5) H(X, Y ) = 1
2[T(X, Y ) + T
0(X, Y ) + T
0(Y, X )],
where Tis viewed as the torsion tensor of ∇and
(2.6) g(T
0(X, Y ), W ) = g(T(W, X), Y ).
Thanks to (1.1) and (2.6), we obtain the following;
(2.7) T
0(X, Y ) = α{η(X)Y−g(X, Y )ξ}+β{η(X)φY −g(φX, Y )ξ}.
Using (1.1), (2.5) and (2.7) we obtain
H(X, Y ) = α{η(Y)X−g(X, Y )ξ}+β{η(Y)φX −g(φX, Y )ξ}.
This proves to our assertion. 
Using (2.4), we have
(2.8) ∇Xξ=∇Xξ−α{X+η(X)ξ} − βφX.
If the unit timelike vector field ξis parallel with respect to generalized symmetric
metric connection, that is,
(2.9) ∇Xξ= 0 ⇔ ∇Xξ=α{X+η(X)ξ}+βφX,
then ∇is called generalized symmetric metric ξconnection.
With the help of equations (2.1), (2.2) and (2.4), we obtain
(2.10) (∇Xη)Y= (∇Xη)Y−α{η(X)η(Y) + g(X, Y )} − βg(φX, Y )
and
(2.11)
(∇Xφ)Y= (∇Xφ)Y−α{g(X, φY )ξ+η(Y)φX}−β{g(X, Y )ξ+2η(X)η(Y)ξ+η(Y)X},

A NEW TYPE LORENTZIAN ALMOST PARA CONTACT MANIFOLD 119
for any vector field X,Yon M.
Now, we suppose that (∇Xη)Y= 0 and (∇Xφ)Y= 0. Then the equations (2.10)
and (2.11) will be as follows:
(2.12) (∇Xη)Y=α{η(X)η(Y) + g(X, Y )}+βg(φX, Y )
and
(2.13) (∇Xφ)Y=α{g(X, φY )ξ+η(Y)φX}+β{g(X, Y )ξ+2η(X)η(Y)ξ+η(Y)X}.
A linear ∇satifying equations (2.12) and (2.13) is called η−parallel generalized
symmetric metric and φ−parallel generalized symmetric metric connection, respec-
tively.
Definition 2.2. Let Mbe a Lorentzian almost para contact manifold. If Mpro-
vide the equations (2.9), (2.12) and (2.13), then Mis called Lorentzian almost
para contact manifold with parallelized generalized symmetric metric connection,
this means that the connection ∇is generalized symmetric metric ξconnection,
η−parallel generalized symmetric metric connection and φ−parallel generalized
symmetric metric connection.
In this new type manifold, using (2.9), (2.12) and (2.13) we have the following
identities.
Proposition 1. In a Lorentzian almost para contact manifold with parallelized
generalized symmetric metric connection, curvature tensor Rand Ricci tensor S
has the following relations
R(X, Y )ξ= (α2+β2)(η(Y)X−η(X)Y) + 2αβ (η(Y)φX −η(X)φY ),(2.14)
η(R(X, Y )Z) = −(α2+β2)(η(Y)g(X , Z)−η(X)g(Y, Z ))
+−2αβ(η(Y)g(φX, Z )−η(X)g(φY, Z )),(2.15)
R(ξ, X )Y= (α2+β2)(g(X, Y )ξ−η(Y)X) + 2αβ(g(φY , X)ξ−η(Y)φX ),
S(Y, ξ ) = (α2+β2)(−n−1) + 2αβ(traceφ)η(Y),(2.16)
for any X, Y, Z ∈χ(M).
In a 3−dimensional manifold, the curvature tensor is given by
R(X, Y )Z=S(Y, Z )X−g(X, Z)QY +g(Y , Z)QX −S(X, Z )Y
−r
2{g(Y, Z )X−g(X, Z)Y},(2.17)
where Qis Ricci operator.
Theorem 2.3. In a 3−dimensional Lorentzian almost para contact manifold with
parallelized generalized symmetric metric connection, scalar and Ricci curvature is
given by the following expressions
r=−8
32(α2+β2)−αβφ,(2.18)
S(Y, Z ) = 2
3nβ2+α2−αβφog(Y , Z) + η(Y)η(Z)
+ 8β2+α2−4αβφη(Y)η(Z)
+ (−2αβ)g(φY , Z).(2.19)

120 O ˘
GUZHAN BAHADIR
Proof. If we write ξinstead of Xin (2.17), we have
η(R(ξ, Y )Z) = −S(Y, Z)−η(Z)S(Y , ξ) + g(Y, Z )S(ξ, ξ)−S(ξ, Z )η(Y)
+r
2{g(Y, Z ) + η(Y)η(Z)}.(2.20)
Using Proposition1 in (2.20), we obtain
S(Y, Z )=(−β2−α2+r
2+ Λ)(g(Y, Z ) + η(Y)η(Z)) + (−2αβ)g(φY, Z )
+ 2Λη(Y)η(Z),(2.21)
where Λ = 4(β2+α2)−2αβφ. Taking an orthanormal frame field in the equation
(2.21) over Xand Y, we have the scalar curvature.
Using (2.21) in (2.18), the expression of the Ricci tensor is obtained. 
We know that
(∇XS)(Y, Z ) = XS(Y, Z )−S(∇XY, Z)−S(Y , ∇XZ).
Using this with (2.21), we have
(∇XS)(Y, Z ) = µ(∇Xη)Y η(Z)+(∇Xη)Z η(Y),(2.22)
where µ=1
3β2+α2−8αβφ. If we use the equation (2.13) in (2.22), we obtain
(∇XS)(Y, Z ) = µ2αη(X)η(Y)η(Z)−βg(φX, Y )η(Z)−βg(φX, Z )η(Y)
+g(X, Y )Z+g(X, Z )Y.(2.23)
The equation (2.23) gives the following result
Proposition 2. In a 3−dimensional Lorentzian almost para contact manifold
with parallelized generalized symmetric metric connection, the manifold is Ricci
symmetric if and only if β2+α2−8αβφ = 0.
3. Conclusion
In this study, we introduced new type Lorentzian almost para contact manifold
using generalized symmetric metric connection. This concept is similar Lorentzian
trans sasakian structure. Thus, all the studies done in this structure can also be
examined for the new structure we introduced.
4. Acknowledgments
The authors would like to thank the reviewers and editors of Journal of Universal
Mathematics.
Funding
The author(s) declared that has no received any financial support for the research,
authorship or publication of this study.
The Declaration of Conflict of Interest/ Common Interest
The author(s) declared that no conflict of interest or common interest
The Declaration of Ethics Committee Approval

A NEW TYPE LORENTZIAN ALMOST PARA CONTACT MANIFOLD 121
This study does not be necessary ethical committee permission or any special per-
mission.
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The author(s) declared that they comply with the scientific, ethical, and citation
rules of Journal of Universal Mathematics in all processes of the study and that
they do not make any falsification on the data collected. Besides, the author(s)
declared that Journal of Universal Mathematics and its editorial board have no re-
sponsibility for any ethical violations that may be encountered and this study has
not been evaluated in any academic publication environment other than Journal of
Universal Mathematics.
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(O˘guzhan Bahadır) Department of Mathematics, Faculty of Arts and Sciences, K.S.U,
Kahramanmaras, TURKEY.
Email address:oguzbaha@gmail.com