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A New Type of (𝝐) − Lorentzian Para-Sasakian Manifolds 

 

Shadab Ahmad Khan*1, Toukeer Khan2, Mohd Bilal3, Anis Ahmad4 

1 Assistant Professor, Department of Mathematics & Statistics, Integral University, Lucknow, India- 226026   

sakhan@iul.ac.in* 

2Associate Professor, Department of Liberal Education, Faculty of Science, Era University, Lucknow, India -226003 

3Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al Qura University, Makkah 21955, Saudi 

Arabia 

4Research scholar, Department of Mathematics & Statistics, Integral University, Lucknow, India-226026 

 

Article History: 

Received: 21-08-2024 

Revised: 02-10-2024 

Accepted: 20-10-2024 

Abstract:  

The current investigation commences by introducing a novel category termed 

(ϵ) −Lorentzian para-Sasakian manifolds, employing the generalized symmetric metric 

connection of a specific type(α, β). Several fundamental outcomes concerning with these 

manifolds are derived. Subsequently, we delve into the examination of conformally flat 

and Weyl-semi-symmetric (ϵ) − Lorentzian para-Sasakian manifolds, utilizing the 

generalized symmetric metric connection of the type(α, β). 

Keywords: (ϵ) −Lorentzian para-Sasakian manifolds, generalized symmetric metric 

connection of the type(α, β), Conformally flat, η−Einstein manifold Weyl-

semisymmetric and quasi-constant curvature. 

Mathematics Subject Classification: 2000. 53C15, 53C25, 53C40. 

 

1.  Introduction 

 In [2], the authors introduced and studied the notion of special conformally flat space. Bejancu et. al. 

[1], introduced the concept of (𝜖)-Sasakian manifolds. Also,Xufeng and Xiaoli [4] showed that every 

(𝜖) -Sasakian manifold must be a real hypersurface of some indefinite Kaehler manifold. T. 

Takahashi introduced almost contact manifolds equipped with associated indefinite metrics in 1969 

and studied Sasakian manifolds equipped with an associated indefinite metric. Since the substantial 

role that Sasakian manifolds with indefinite metrics play in physics [5], our inclination naturally lies 

in exploring diverse contact manifolds with indefinite metrics. Recently, in 2009, U.C. De& Sarkar 

[8], studied(𝜖)-Kenmotsu manifolds. K. Matsumoto [7], introduced the notion of Lorentzian Para-

Sasakian manifolds. I. Mihai and R. Rosca [9], defined the same notion independently and several 

authors [10], [11], [12], [14], [16], [17] also studied different structures. 

A linear connection  ∇̅ on a Riemannian manifold 𝑀 is suggested to be a generalized symmetric 

connection if its torsion tensor 𝑇 is defined as: 

𝑇(𝑋, 𝑌)  = 𝛼{𝑢(𝑌)𝑋 − 𝑢(𝑋)𝑌} + 𝛽{𝑢(𝑌)𝜙𝑋 − 𝑢(𝑋)𝜙𝑌}     (1.1) 



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For any vector field 𝑋 and 𝑌 on 𝑀, where 𝛼 and 𝛽 are constant functions on 𝑀[13], 𝜙 can be viewed 

as tensor of type (1, 1) and 𝑢 is regarded as a 1-form connected with the vector field which has a 

non-vanishing smooth non-null unit. A linear metric connection satisfying the equation (1.1) is called 

generalized symmetric metric connection of type(𝛼, 𝛽). 

Moreover, the connection  ∇̅ is said to be metric connection if  ∇̅𝑔 = 0, with 𝑔 as metric tensor [15]. 

This paper is organized as follows: Section I, is introductory. Section II, is devoted to preliminaries. 

In section III, we define (𝜖) −Lorentzian para-Sasakian manifold with generalized symmetric metric 

connection. We also give some basic results of suchtype of manifold in the same section. In Section 

IV, we have studied conformally flat (𝜖) −Lorentzian para Sasakian manifold with generalized 

symmetric metric connection. In section V, we consider Weyl-semi-symmetric (𝜖) −  Lorentzian 

para-Sasakian manifold. 

2.   Preliminaries 

An n-dimensional differential manifold is called an (𝜖) −  Lorentzian para-Sasakian manifold 

i.e. (𝜖)-LP Sasakian manifold, if it admits a (1,1) tensor field 𝜙, a contravariant vector field 𝜉, a 1-

form 𝜂 and a Lorentzian metric 𝑔 which satisfies 

𝜙2𝑋 =  𝑋 + 𝜂(𝑋)𝜉,   𝜂(𝜉) = −1        (2.1) 

𝑔(𝜉, 𝜉) =  −𝜖,   𝜂(𝑋) = 𝜖𝑔(𝑋, 𝜉), 𝜙𝜉 =  0,   𝜂(𝜙𝑋)  =  0     (2.2) 

𝑔(𝜙𝑋, 𝜙𝑌) = 𝑔(𝑋, 𝑌) + 𝜖 𝜂(𝑋) 𝜂(𝑌)             (2.3) 

(∇X𝜙)𝑌 = 𝑔(𝑋, 𝑌)𝜉 + 𝜖𝜂(𝑌)𝑋 + 2𝜖𝜂(𝑋)𝜂(𝑌)𝜉      (2.4) 

∇𝑋𝜉 =  𝜖𝜙𝑋           (2.5) 

(∇𝑋𝜂)𝑋 = 𝑔(𝜙𝑋, 𝑌)           (2.6) 

for arbitrary vector field 𝑋 and 𝑌; where ∇ denotes the operator of covariant differentiation with 

respect to the metric [7], [8] 

On an n- dimensional (𝜖) −  Lorentzian para-Sasakian manifold with structure (𝜙, 𝜉, 𝜂, 𝑔) , the 

following results hold [8]. 

𝑅(𝑋, 𝑌)𝜉 =  𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌         (2.7) 

𝑅(𝜉, 𝑋)𝜉 =  𝜖𝑔(𝑋, 𝑌)𝜉 − 𝜂(𝑋)𝑌         (2.8) 

𝑔(𝑅(𝑋, 𝑌)𝑍, 𝜉)  =  𝑔(𝑌, 𝑍)𝜂(𝑋) − 𝑔(𝑋, 𝑍)𝜂(𝑌)      (2.9) 

𝑆(𝜙𝑋, 𝜙𝑌)  =  𝑆(𝑋, 𝑌) + (𝑛 − 1)𝜂(𝑋)𝜂(𝑌)       (2.10) 

𝑆(𝑋, 𝜉)  = (𝑛 − 1)𝜂(𝑌)          (2.11) 

𝑄𝑋 = 𝜖(𝑛 − 1)𝜉          (2.12) 

for any vector fields 𝑋, 𝑌 and 𝑍; where 𝑅 is the Riemannian curvature tensor, 𝑆 is the Ricci tensor 

and 𝑄 is the Ricci operator given by  

𝑔(𝑄𝑋, 𝑌) = 𝑆(𝑋, 𝑌). 



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We note that, if 𝜖 = 1and the structure vector field 𝜉 is space like, then an (𝜖)-LP Sasakian manifold 

is a usual LP-Sasakian manifold. 

An (𝜖)-LP Sasakian manifold is said to be Einstein manifold if its Ricci tensor 𝑆 is of the form 

𝑆(𝑋, 𝑌)  =  𝜆𝑔(𝑋, 𝑌) 

where 𝜆 is a constant. 

Definition 2.1.An (𝜖)-LP Sasakian manifold will be called a manifold of quasi-constant curvature if 

the curvature tensor �̃� of type (0,4) satisfies the condition 

�̃�(𝑋, 𝑌, 𝑍,𝑊) = 𝑎[𝑔(𝑌, 𝑍)𝑔(𝑋,𝑊) − 𝑔(𝑋, 𝑍)𝑔(𝑌,𝑊)] + 𝑏[𝑇(𝑌)𝑇(𝑍)𝑔(𝑋,𝑊) 

 −𝑇(𝑋)𝑇(𝑍)𝑔(𝑌,𝑊) + 𝑇(𝑋)𝑇(𝑊)𝑔(𝑌, 𝑍) − 𝑇(𝑌)𝑇(𝑊)}𝑔(𝑋, 𝑍)}      (2.13) 

where�̃�(𝑋, 𝑌, 𝑍,𝑊) = 𝑔(𝑅(𝑋, 𝑌)𝑍,𝑊) 

𝑅 is the curvature tensor of type (1,3);𝑎, 𝑏 are scalar functions and 𝜌 is a unit vector field defined by 

𝑔(𝑋, 𝜌) = 𝑇(𝑋)           (2.14) 

The notion of quasi-constant curvature for Riemannian manifolds was given by Chen and Yano [2]. 

Definition 2.2.An (𝜖)-LP Sasakian manifold will be called 𝜂-Einstein manifold if the Ricci tensor 𝑆 

of type (0, 2) satisfies [2] 

𝑆(𝑋, 𝑌) = 𝑎𝑔(𝑋, 𝑌) + 𝑏𝜂(𝑋)𝜂(𝑌) 

where 𝑎 and 𝑏 are scalar functions. 

Definition 2.3.A type of Riemannian manifold whose curvature tensor �̃� of type (0, 4) satisfies the 

condition 

�̃�(𝑋, 𝑌, 𝑍,𝑊) = 𝐹(𝑌, 𝑍)𝐹(𝑋,𝑊) − 𝐹(𝑋, 𝑍)𝐹(𝑌,𝑊)     (2.15) 

is called a special manifold with the associate symmetric tensor 𝐹 of type (0,2) and is denoted by 

𝜓(𝐹)𝑛. 

In 1956, S. S. Chern [3] studied such type of manifolds. These manifolds are important for the 

following reasons. Firstly, for possessing some remarkable properties related to curvature and 

characteristic classes and secondly, for containing a manifold of quasi-constant curvature [2]. 

Definition 2.4.An (𝜖)-LP Sasakian manifold will be called Weyl-semi-symmetric if it satisfies 

(𝑅. (𝑋, 𝑌). 𝐶)(𝑌, 𝑍)𝑊 = 0 

where 𝑅(𝑋, 𝑌) denotes the curvature operator and 𝐶(𝑌, 𝑍)𝑊 is the Weyl-conformal curvature tensor.  

3.   On (𝝐) −Lorentzian Para-Sasakian Manifold with Parallelized Generalized Symmetric 

Metric Connection 

Theorem 3.1. For an (𝜖) −Lorentzian para-Sasakian manifold, the generalized symmetric metric 

connection ∇̅ of type (α, β) is given by 

 ∇̅𝑋𝑌 = ∇𝑋𝑌 + 𝛼{𝜂(𝑌)𝑋 − 𝜖𝑔(𝑋, 𝑌)𝜉} + 𝛽{𝜂(𝑌)𝜙𝑋 − 𝜖𝑔(𝜙𝑋, 𝑌)𝜉}   (3.1) 



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Proof: The relation between a linear connection  ∇̅ and Levi-Civita connection ∇ is given by 

 ∇̅𝑋𝑌 = ∇𝑋𝑌 + 𝐻(𝑋, 𝑌)         (3.2) 

for all vector field 𝑋 and 𝑌. The following equation is such that ∇̅ is a generalized symmetric metric 

connection of∇. In which 𝐻 is viewed as a tensor of type (1, 2), given by 

𝐻(𝑋, 𝑌)  =  
1

2
[𝑇(𝑋, 𝑌)  + 𝑇 ′(𝑋, 𝑌) + 𝑇 ′(𝑌, 𝑋)      (3.3) 

where T is viewed as the torsion tensor of  ∇̅ and 

𝑔(𝑇 ′(𝑋, 𝑌),𝑊)  =  𝑔(𝑇(𝑊,𝑋), 𝑌)        (3.4) 

Replacing 𝑊 by𝜉, using (1.1), (2.1), (2.2) and (2.3) in (3.4), we have 

𝑔(𝑇 ′(𝑋, 𝑌), 𝜉)  = 𝛼𝜂(𝑋)𝑔(𝑌, 𝜉) − 𝜖𝛼𝑔(𝑋, 𝑌)𝑔(𝜉, 𝜉) + 𝛽𝜂(𝑋)𝑔(𝜙𝑌, 𝜉) − 𝜖𝛽𝑔(𝜙𝑋, 𝑌)𝑔(𝜉, 𝜉) 

𝑇 ′(𝑋, 𝑌) = 𝛼𝜂(𝑋)𝑌 − 𝜖𝛼𝑔(𝑋, 𝑌)𝜉 + 𝛽𝜂(𝑋)𝜙𝑌 − 𝜖𝛽𝑔(𝜙𝑋, 𝑌)𝜉    (3.5) 

𝑇 ′(𝑌, 𝑋) = 𝛼𝜂(𝑌)𝑋 − 𝜖𝛼𝑔(𝑋, 𝑌)𝜉 + 𝛽𝜂(𝑌)𝜙𝑋 − 𝜖𝛽𝑔(𝑋, 𝜙𝑌)𝜉    (3.6) 

Using equations (1.1), (2.2), (3.5) and (3.6) in (3.3), we obtained 

𝐻(𝑋, 𝑌)  =  𝛼{𝜂(𝑌)𝑋 − 𝜖𝑔(𝑋, 𝑌)𝜉} + 𝛽{𝜂(𝑌)𝜙𝑋 − 𝜖𝑔(𝜙𝑋, 𝑌)𝜉}    (3.7) 

Using above in (3.2) proves to our assertion. 

Now, substituting  𝑌 = 𝜉 in equation (3.1), we obtained 

 ∇̅𝑋𝜉 =  ∇𝑋𝜉 − 𝛼{𝑋 + 𝜂(𝑋)𝜉} − 𝛽(𝜙𝑋) 

If the vector field 𝜉 representing a unit of time like is aligned in parallel according to a generalized 

symmetric metric connection, that is ∇̅𝑋𝜉 = 0, we have 

∇𝑋𝜉 = 𝛼{𝑋 + 𝜂(𝑋)𝜉} + 𝛽(𝜙𝑋)        (3.8) 

then  ∇̅ is called generalized symmetric metric 𝜉 connection. 

Using equation (2.2) and (2.3) and replacing 𝑌 by 𝜙𝑌 in equation (3.1), we have 

 ∇̅𝑋𝜙𝑌 =  ∇𝑋𝜙𝑌 − 𝜖𝛼𝑔(𝑋, 𝜙𝑌)𝜉 − 𝜖𝛽{𝑔(𝑋, 𝑌) + 𝜖 𝜂(𝑋) 𝜂(𝑌)}𝜉 

Using covariant differentiation in above, we have 

(∇̅𝑋𝜙)𝑌 = (∇𝑋𝜙)𝑌 − 𝛼{𝜂(𝑌)𝜙𝑋 + 𝜖𝑔(𝑋, 𝜙𝑌)𝜉} − 𝛽{𝜂(𝑌)𝑋 + 𝜖𝑔(𝑋, 𝑌)𝜉 + 𝜂(𝑋)𝜂(𝑌)𝜉} (3.9) 

Now, as we know that 

( ∇̅𝑋𝜂)𝑌 =   ∇̅𝑋𝜂𝑌 + 𝜂( ∇̅𝑋𝑌) 

Using equation (3.1) in above, we obtained 

( ∇̅𝑋𝜂)𝑌 = (∇𝑋𝜂)𝑌 − 𝛼{𝜖𝑔(𝑋, 𝑌) + 𝜂(𝑋)𝜂(𝑌)} − 𝜖𝛽𝑔(𝜙𝑋, 𝑌)    (3.10) 

for any vector field 𝑋 and 𝑌 on 𝑀. 

Now, we suppose that 



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( ∇̅𝑋𝜙)𝑌 = 0 and ( ∇̅𝑋𝜂)𝑌 = 0.  

Then the equation (3.9) and (3.10) will be as follows 

(∇𝑋𝜙)𝑌 = 𝛼{𝜂(𝑌)𝜙𝑋 + 𝜖𝑔(𝑋, 𝜙𝑌)𝜉} + 𝛽{𝜂(𝑌) + 𝜖𝑔(𝑋, 𝑌)𝜉 + 2𝜂(𝑋) 𝜂(𝑌)𝜉}  (3.11) 

and  

(∇𝑋𝜂)𝑌 = 𝛼{𝜖𝑔(𝑋, 𝑌) + 𝜂(𝑋)𝜂(𝑌)} + 𝜖𝛽𝑔(𝜙𝑋, 𝑌)      (3.12) 

A linear connection  ∇̅ satisfying equation (3.11) and 

(3.12) is called 𝜙 −parallel generalized symmetric connection and 𝜂 −parallel generalized 

symmetric connection respectively. 

Definition 3.2.Let 𝑀 be a (𝜖) −Lorentzian para-Sasakianmanifold. If 𝑀 satisfies the equations (3.8), 

(3.11) and (3.12), then 𝑀 is called (𝜖) −Lorentzian para-Sasakian manifold with parallelized 

generalized symmetric metric connection, this means that the connection ∇̅ is generalized symmetric 

metric 𝜉 connection, 𝜙 −parallel generalized symmetric connection and 𝜂 −parallel generalized 

symmetric connection 

Proposition 3.3. Let  𝑀 be a (𝜖) −Lorentzian para-Sasakian manifold with parallelized generalized 

symmetric metric connection then following relation holds 

( ∇̅𝑋𝜙)𝑌 = (𝜖 − 𝛽) 𝜂(𝑌)𝑋 + 2(𝜖 + 1)𝜂(𝑋) 𝜂(𝑌)𝜉 + (1− ϵ𝛽)𝑔(𝑋, 𝑌)𝜉 − 𝛼𝜂(𝑌)𝜙𝑋 − ϵα𝑔(𝑋, 𝜙𝑌)𝜉 

            (3.13) 

for all vector field 𝑋 and 𝑌 on 𝑀. 

Proposition 3.4. In a (𝜖) − Lorentzian para-Sasakian manifold with parallelized generalized 

symmetric metric connection, curvature tensor 𝑅 and Ricci tensor 𝑆 and Ricci operator 𝑄 has the 

following relations 

𝑅(𝑋, 𝑌)𝜉 = (𝛼2 + 𝛽2){𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌} + 2𝛼𝛽{𝜂(𝑌)𝜙𝑋 − 𝜂(𝑋)𝜙𝑌} + (𝑋𝛽)𝜙𝑌 − (𝑌𝛽)𝜙𝑋 

+(𝑋𝛼)𝜙2𝑌 − (𝑌𝛼)𝜙2𝑋                 (3.14) 

𝜂(𝑅(𝑋, 𝑌)𝑍 = 𝜖(𝛼2 + 𝛽2){𝜂(𝑋)𝑔(𝑌, 𝑍) − 𝜂(𝑌)𝑔(𝑋, 𝑍)}     (3.15) 

𝑅(𝜉, 𝑋)𝑌 = (𝛼2 + 𝛽2){𝜖𝑔(𝑋, 𝑌)𝜉 − 𝜂(𝑌)𝑋} + {𝜉𝛽 − 2𝛼𝛽𝜂(𝑌)}𝜙𝑋 + (𝜉𝛼)𝑋 

 +𝜖𝛼𝜉𝑔(𝑋, 𝑌)𝑌 − (𝑋𝛼)𝜉 − (𝑋𝛼)𝜂(𝑌)𝑌      (3.16) 

𝑅(𝜉, 𝑋)𝜉 = (𝛼2 + 𝛽2 + 𝜉𝛼)𝜙2𝑋 + (2𝛼𝛽 + 𝜉𝛽)𝜙𝑋      (3.17) 

𝑆(𝑋, 𝜉) = (𝑛 − 1)(𝛼2 + 𝛽2)𝜂(𝑋)        (3.18) 

𝑄𝜉 = 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)𝜉         (3.19) 

for any 𝑋, 𝑌, 𝑍 ∈ 𝜒(𝑀). 

Proof:  As we know that, curvature tensor is  

𝑅(𝑋, 𝑌)𝜉 = ∇𝑋∇𝑌𝜉 − ∇𝑌∇𝑋𝜉 − ∇[𝑋,𝑌]𝜉       (3.20) 



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Using equation (3.8) and (3.12), we obtained 

∇𝑋∇𝑌𝜉 = (𝑋𝛼)(𝜙
2𝑌) + 𝛼(∇𝑋𝑌) + 𝜖𝛼

2𝑔(𝑋, 𝑌 + 𝛼2𝜂(𝑋)𝜂(𝑌)𝜉 + 𝜖𝛼𝛽𝑔(𝜙𝑋, 𝑌)𝜉 

+𝛼2𝜂(𝑌)𝑋 + 𝛼2𝜂(𝑋)𝜂(𝑌)𝜉 + 𝛼𝛽𝜂(𝑌)𝜙𝑋 + (𝑋𝛽)(𝜙𝑌) + 𝛽(∇𝑋𝜙𝑌)   (3.21) 

Similarly, 

∇𝑌∇𝑋𝜉 = (𝑌𝛼)(𝜙
2𝑋) + 𝛼(∇𝑌𝑋) + 𝜖𝛼

2𝑔(𝑋, 𝑌)𝜉 + 𝛼2𝜂(𝑋)𝜂(𝑌)𝜉 + 𝜖𝛼𝛽𝑔(𝑋, 𝜙𝑌)𝜉 

+𝛼2𝜂(𝑋)𝑌 + 𝛼2𝜂(𝑋)𝜂(𝑌)𝜉 + 𝛼𝛽𝜂(𝑋)𝜙𝑌 + (𝑌𝛽)(𝜙𝑋) + 𝛽(∇𝑌𝜙𝑋)   (3.22) 

Using equation (3.8) and (3.11), we can write as 

𝛻[𝑋,𝑌]𝜉 =  𝛼{[𝑋, 𝑌] + 𝜂([𝑋, 𝑌])𝜉} + 𝛽𝜙[𝑋, 𝑌] 

∇[𝑋,𝑌]𝜉 =  𝛼(∇𝑋𝑌) − 𝛼(∇𝑌𝑋) + 𝛽∇𝑋𝜙𝑌 − 𝛽∇𝑌𝜙𝑋 − 𝛼𝛽𝜂(𝑌)𝜙𝑋 + 𝛼𝛽𝜂(𝑋)𝜙𝑌 

+𝛽2𝜂(𝑋)𝑌 − 𝛽2𝜂(𝑌)𝑋         (3.23) 

Using equation (3.20), (3.21), (3.22) and (3.23), we obtained 

𝑅(𝑋, 𝑌)𝜉 = (𝛼2 + 𝛽2){𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌 + 2𝛼𝛽{𝜂(𝑌)𝜙𝑋 − 𝜂(𝑋)𝜙𝑌} + (𝑋𝛽)(𝜙𝑌) 

−(𝑌𝛽)(𝜙𝑋) + (𝑋𝛼)(𝜙2𝑌) − (𝑌𝛼)(𝜙2𝑋)        (3.24) 

Interchanging 𝑍 and 𝜉 in (3.14), we have 

𝑔(𝑅(𝑋, 𝑌)𝑍, 𝜉) = (𝛼2 + 𝛽2){𝜖𝑔(𝑌, 𝑍)𝑔(𝑋, 𝜉) − 𝜖𝑔(𝑋, 𝑍)𝑔(𝑌, 𝜉)} 

                                         +2𝛼𝛽{𝜖𝑔(𝑌, 𝑍)𝑔(𝜙𝑋, 𝜉) − 𝜖𝑔(𝑋, 𝑍)𝑔(𝜙𝑌, 𝜉)} + (𝑋𝛽)𝑔(𝜙𝑌, 𝜉) 

                                                −(𝑌𝛽)𝑔(𝜙𝑋, 𝜉) + (𝑋𝛼)𝑔(𝜙2𝑌, 𝜉) − (𝑌𝛼)𝑔(𝜙2𝑋, 𝜉) 

𝜂(𝑅(𝑋, 𝑌)𝑍) = 𝜖(𝛼2 + 𝛽2){𝜂(𝑋)𝑔(𝑌, 𝑍) − 𝜂(𝑌)𝑔(𝑋, 𝑍)}      (3.25) 

From equation (3.14), we have 

𝑅(𝜉, 𝑋)𝑌 = (𝛼2 + 𝛽2){𝜖𝑔(𝑋, 𝑌)𝜉 − 𝜂(𝑌)𝑋} + {𝜉𝛽 − 2𝛼𝛽𝜂(𝑌)}𝜙𝑋 + (𝜉𝛼)𝑋 

+𝜖𝛼𝜉𝑔(𝑋, 𝑌)𝑌 − (𝑋𝛼)𝜉 − (𝑋𝛼)𝜂(𝑌)𝑌          (3.26) 

Putting 𝑌for 𝜉 in above equation, we have 

𝑅(𝜉, 𝑋)𝜉 = (𝛼2 + 𝛽2){𝑋 + 𝜂(𝑋)𝜉} + {𝜉𝛽 + 2𝛼𝛽}𝜙𝑋 + 𝜉𝛼{𝑋 + 𝜂(𝑋)𝜉} 

𝑅(𝜉, 𝑋)𝜉 = (𝛼2 + 𝛽2 + 𝜉𝛼)𝜙2𝑋 + (𝜉𝛽 + 2𝛼𝛽)𝜙𝑋      (3.27) 

Now, from equation (3.15), we have 

𝑔(𝑅(𝑋, 𝑌)𝑍, 𝜉) = (𝛼2 + 𝛽2){𝜖𝑔(𝑋, 𝜉)𝑔(𝑌, 𝑍) − 𝜖𝑔(𝑌, 𝜉)𝑔(𝑋, 𝑍)} 

Putting  𝑌 = 𝑍 = 𝑒𝑖 , where 𝑒𝑖  is an orthonormal basis of the tangent space at each point of the 

manifold and taking summation over 𝑖 and 𝑖 = 1,2, . . . . . , 𝑛 then, we get 

𝑔(𝑅(𝑋, 𝑒𝑖)𝑒𝑖, 𝜉) = (𝛼
2 + 𝛽2){𝜖𝑔(𝑋, 𝜉)𝑔(𝑒𝑖, 𝑒𝑖) − 𝜖𝑔(𝑒𝑖, 𝜉)𝑔(𝑋, 𝑒𝑖)} 

𝑆(𝑋, 𝜉) = (𝑛 − 1)(𝛼2 + 𝛽2)𝜂(𝑋)        (3.28) 



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Using equation (3.28), we have 

𝑄𝜉 = 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)𝜉         (3.29) 

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)𝑄 − 𝑆(𝑋, 𝑍)𝑌 −
𝑟

2
{𝑔(𝑌, 𝑍)𝑋 − 𝑔{𝑋, 𝑍)𝑌}  (3.30) 

Theorem 3.5:  In a 3 dimensional  (𝜖) −Lorentzian para-Sasakian manifold with parallelized 

generalized symmetric metric connection, the Ricci operator 𝑄 is given by 

𝑄𝑋 = {
𝑟

2
− 2𝜖(𝛼2 + 𝛽2) + 𝜖𝜉𝛼}𝑋 + {

𝑟

2
− 4𝜖(𝛼2 + 𝛽2) + 𝜖𝜉𝛼}𝜂(𝑋)𝜉 + 𝜖{𝜉𝛽 + 2𝛼𝛽} 

 (3.31) 

for any 𝑋, 𝑌, 𝑍 ∈ 𝜒(𝑀). 

Proof:In a 3-dimensional (𝜖) −Lorentzian para-Sasakian manifold with parallelized generalized 

symmetric metric connection, the curvature tensor is 

𝑅(𝑋, 𝑌)𝜉 = 𝑆(𝑌, 𝜉)𝑋 − 𝑆(𝑋, 𝜉)𝑌 − 𝑔(𝑋, 𝜉)𝑄𝑌 + 𝑔(𝑌, 𝜉)𝑄𝑋 −
𝑟

2
{𝑔(𝑌, 𝜉)𝑋 − 𝑔(𝑋, 𝜉)𝑌} 

Using equation (3.14) and (3.18) in above equation, then we have  

{
r

2𝜖
− 2(α2 + β

2)} {𝜂(𝑌)𝑋 − 𝜂(𝑋)𝑌} + 2𝛼𝛽{𝜂(𝑌)𝜙𝑋 − 𝜂(𝑋)𝜙𝑌} 

+(𝑋𝛽)𝜙𝑌 − (𝑌𝛽)𝜙𝑋 + (𝑋𝛼)𝜙2𝑌 − (𝑌𝛼)𝜙2𝑋 =
1

𝜖
𝜂(𝑌)𝑄𝑋 −

1

𝜖
𝜂(𝑋)𝑄𝑌 

Putting 𝑌 = 𝜉 in above equation, we obtained 

𝑄𝑋 = {
r

2
− 2𝜖(α2 + β

2) + 𝜖𝜉𝛼}{𝑋 + 𝜂(𝑋)𝜉} + 𝜖(2𝛼𝛽 + 𝜉𝛽))𝜙𝑋 − 2𝜖(α2 + β
2)𝜂(𝑋)𝜉 

Thus, from above we have result (3.31). 

Theorem 3.6:  In a 3 dimensional (𝜖) −Lorentzian Para-Sasakian manifold with parallelized 

generalized symmetric metric connection, scalar and Ricci curvature are given by the following 

expressions 

𝑟 = 2 (1+
8

𝜖−1
) (𝛼2 + 𝛽2)         (3.32) 

𝑆(𝑌, 𝑍) = 2(𝛼2 + 𝛽2) {(1+
4

𝜖−1
)𝑔(𝑌, 𝑍) + 2𝜂(𝑌)𝜂(𝑍)}     (3.33) 

for any 𝑋, 𝑌, 𝑍 ∈ 𝜒(𝑀). 

Proof: On putting 𝜉 in place of 𝑋 in equation (3.30), we have the equation as 

𝑅(𝜉, 𝑌)𝑍 = 𝑆(𝑌, 𝑍)𝜉 − 𝑔(𝜉, 𝑍)𝑄𝑌 + 𝑔(𝑌, 𝑍)𝑄𝜉 − 𝑆(𝜉, 𝑍)𝑌 −
𝑟

2
{𝑔(𝑌, 𝑍)𝜉 − 𝑔{𝜉, 𝑍)𝑌} 

𝜂(𝑅(𝜉, 𝑌)𝑍) = −𝑆(𝑌, 𝑍) − 𝜂(𝑍)𝑆(𝑌, 𝜉) + 𝜖𝑆(𝜉, 𝜉)𝑔(𝑌, 𝑍) − 𝜂(𝑌)𝑆(𝜉, 𝑍) +
𝑟

2
{𝑔(𝑌, 𝑍)

+ 𝜖𝜂(𝑌)𝜂(𝑍)} 



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Using equation (3.15), (3.18) in above, we have 

𝜖(𝛼2 + 𝛽2){𝜂(𝑋)𝑔(𝑌, 𝑍) − 𝜂(𝑌)𝑔(𝑋, 𝑍)} = −𝑆(𝑌, 𝑍) − 𝜂(𝑍){2(𝛼2 + 𝛽2)𝜂(𝑌)} 

+𝜖{2𝜖(𝛼2 + 𝛽2)𝑔(𝜉, 𝜉)}𝑔(𝑌, 𝑍) 

−𝜂(𝑌){2(𝛼2 + 𝛽2)𝜂(𝑌)} +
𝑟

2
{𝑔(𝑌, 𝑍) + 𝜖𝜂(𝑌)𝜂(𝑍)} 

𝑆(𝑌, 𝑍) = {
𝑟

2
− 𝜖(𝛼2 + 𝛽2) + Λ} {𝑔(𝑌, 𝑍) + 𝜖𝜂(𝑌)𝜂(𝑍)} − Λ𝑔(𝑌, 𝑍)  − 2𝜖Λ𝜂(𝑌)𝜂(𝑌) (3.34) 

where Λ = 2ϵ(α2 + β
2), now taking an orthonormal frame filed in the above equation over 𝑌 and 𝑍, 

we have the scalar curvature.  

Using equations (3.32) and (3.34), the expression of the Ricci tensor is obtained. 

4. Conformally Flat (𝝐) −Lorentzian Para-Sasakian Manifold with Parallelized Generalized 

Symmetric Metric Connection 

The Weyl conformal curvature tensor 𝐶 of type (1, 3) of an n-dimensional Riemannian manifold is 

given by 

𝐶(𝑋, 𝑌)𝑍 =  𝑅(𝑋, 𝑌)𝑍 −
1

(𝑛 − 2)
[𝑆(𝑌, 𝑍)𝑋 − 𝑆(𝑋, 𝑍)𝑌 + 𝑔(𝑌, 𝑍)𝑄𝑋 − 𝑔(𝑋, 𝑍)𝑄𝑌] 

+
𝑟

(𝑛−1)(𝑛−2)
[𝑔(𝑌, 𝑍)𝑋 − 𝑔(𝑋, 𝑍)𝑌]        (4.1) 

where 𝑄 is the Ricci operator defined by g(QX,Y) = S(X,Y)and 𝑟 is the scalar curvature.  

Let us suppose that the manifold is conformally flat. Then from the above equation, we have 

𝑔(𝑅(𝑋, 𝑌)𝑍,𝑊)  =
1

(𝑛 − 2)
[𝑆(𝑌, 𝑍)𝑔(𝑋,𝑊) − 𝑆(𝑋, 𝑍)𝑔(𝑌,𝑊) + 𝑔(𝑌, 𝑍)𝑆(𝑋,𝑊) 

−𝑔(𝑋, 𝑍)𝑆(𝑌,𝑊)]  −
𝑟

(𝑛−1)(𝑛−2)
[𝑔(𝑌, 𝑍)𝑔(𝑋,𝑊) − 𝑔(𝑋, 𝑍)𝑔(𝑌,𝑊)]        (4.2) 

Putting 𝑊 = 𝜉 in (4.2), we get 

𝑔(𝑅(𝑋, 𝑌)𝑍, 𝜉) =
1

(𝑛 − 2)
[𝑆(𝑌, 𝑍)𝑔(𝑋, 𝜉) − 𝑆(𝑋, 𝑍)𝑔(𝑌, 𝜉) + 𝑔(𝑌, 𝑍)𝑆(𝑋, 𝜉) − 𝑔(𝑋, 𝑍)𝑆(𝑌, 𝜉)] 

−
𝑟

(𝑛 − 1)(𝑛 − 2)
[𝑔(𝑌, 𝑍)𝑔(𝑋, 𝜉) − 𝑔(𝑋, 𝑍)𝑔(𝑌, 𝜉)] 

𝜖𝜂(𝑅(𝑋, 𝑌)𝑍)  =
1

(𝑛 − 2)
[𝜖𝑆(𝑌, 𝑍)𝜂(𝑋) − 𝜖𝑆(𝑋, 𝑍)𝜂(𝑌) + 𝑔(𝑌, 𝑍)𝑆(𝑋, 𝜉) − 𝑔(𝑋, 𝑍)𝑆(𝑌, 𝜉)] 

−
𝑟

(𝑛−1)(𝑛−2)
[𝜖𝑔(𝑌, 𝑍)𝜂(𝑋) − 𝜖𝑔(𝑋, 𝑍)𝜂(𝑌)] (4.3) 

Using equation (3.15), (3.18) and (4.3), we obtained 

(𝛼2 + 𝛽2){𝜂(𝑋)𝑔(𝑌, 𝑍) − 𝜂(𝑌)𝑔(𝑋, 𝑍)}  

=
𝜖

(𝑛 − 2)
[𝑆(𝑌, 𝑍)𝜂(𝑋) − 𝑆(𝑋, 𝑍)𝜂(𝑌) + (𝑛 − 1)(𝛼2 + 𝛽2)𝑔(𝑌, 𝑍)𝜂(𝑋) 



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                −(𝑛 − 1)(𝛼2 + 𝛽2)𝑔(𝑋, 𝑍)𝜂(𝑌)] −
𝜖𝑟

(𝑛 − 1)(𝑛 − 2)
[𝑔(𝑌, 𝑍)𝜂(𝑋) − 𝑔(𝑋, 𝑍)𝜂(𝑌)] 

𝑆(𝑌, 𝑍)𝜂(𝑋) = 𝑆(𝑋, 𝑍)𝜂(𝑌) + {
𝑟

(𝑛−1)
− 𝜖(𝛼2 + 𝛽2)} {𝑔(𝑌, 𝑍)𝜂(𝑋) − 𝑔(𝑋, 𝑍)𝜂(𝑌)} (4.4) 

Using equation (3.18) and replacing  𝑋 = 𝜉 in (4.4), we obtained 

𝑆(𝑌, 𝑍) = {
𝑟

(𝑛−1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑌, 𝑍) + {

𝑟

(𝑛−1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑌)𝜂(𝑍)  (4.5) 

Hence, we can state the following using definition (2.2). 

Theorem 4.1.An 𝑛 −dimensional(𝑛 > 1) conformally flat (𝜖) − Lorentzian para-Sasakian manifold 

with parallelized generalized symmetric metric connection is an  𝜂 −Einstein manifold. 

Using equation (4.5) in (4.2), we get 

𝑔(𝑅(𝑋, 𝑌)𝑍,𝑊)  =
1

(𝑛 − 2)
[{{

𝑟

(𝑛 − 1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑌, 𝑍) 

       + {
𝑟

(𝑛 − 1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑌)𝜂(𝑍)}𝑔(𝑋,𝑊) 

−{{
𝑟

(𝑛 − 1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑋, 𝑍) 

    + {
𝑟

(𝑛 − 1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑋)𝜂(𝑍)}𝑔(𝑌,𝑊) 

+{{
𝑟

(𝑛 − 1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑋,𝑊) 

  + {
𝑟

(𝑛 − 1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑋)𝜂(𝑊)}𝑔(𝑌, 𝑍) 

{
𝑟

(𝑛 − 1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑌,𝑊) 

     + {
𝑟

(𝑛 − 1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋, 𝑍)]  

−
𝑟

(𝑛 − 1)(𝑛 − 2)
[𝑔(𝑌, 𝑍)𝑔(𝑋,𝑊) − 𝑔(𝑋, 𝑍)𝑔(𝑌,𝑊)] 

 

𝑔(𝑅(𝑋, 𝑌)𝑍,𝑊)   = {
𝑟 − 2𝜖(𝑛 − 1)(𝛼2 + 𝛽2)

(𝑛 − 1)(𝑛 − 2)
} {

𝑔(𝑌, 𝑍)𝑔(𝑋,𝑊)

− 𝑔(𝑋, 𝑍)𝑔(𝑌,𝑊)
} 

                 + {
𝑟 − 𝜖𝑛(𝑛 − 1)(𝛼2 + 𝛽2)

(𝑛 − 1)(𝑛 − 2)
}

{
 

 
𝜂(𝑌)𝜂(𝑍)𝑔(𝑋,𝑊)

−𝜂(𝑋)𝜂(𝑍)𝑔(𝑌,𝑊)

+𝜂(𝑋)𝜂(𝑊)𝑔(𝑌, 𝑍)

−𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋, 𝑍)}
 

 
 

In view of definition (2.1) and above relation, we have the following. 



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Theorem 4.2.An 𝑛 −dimensional (𝑛 > 1) conformally flat (𝜖) − Lorentzian para-Sasakian manifold 

with parallelized generalized symmetric metric connection is of quasi-constant curvature. 

It is already proved that 𝜓(𝐹)𝑛 contains a manifold of quasi-constant curvature as a subclass. 

Let us suppose 

𝐹(𝑋, 𝑌) = 𝑝𝑔(𝑋, 𝑌) + 𝑞 (𝑋)𝜂(𝑌)        (4.6) 

where, 

𝑝 = √{
𝑟 − 2𝜖(𝑛 − 1)(𝛼2 + 𝛽2)

(𝑛 − 1)(𝑛 − 2)
} 

and  

𝑞 = {
𝑟 − 𝜖𝑛(𝑛 − 1)(𝛼2 + 𝛽2)

(𝑛 − 1)(𝑛 − 2)
}√{

(𝑛 − 1)(𝑛 − 2)

𝑟 − 2𝜖(𝑛 − 1)(𝛼2 + 𝛽2)
} 

Now, from the equation (2.13), we have 

 �̃�(𝑋, 𝑌, 𝑍,𝑊) = 𝐹(𝑌, 𝑍)𝐹(𝑋,𝑊) − 𝐹(𝑋, 𝑍)𝐹(𝑌,𝑊) 

Therefore, the manifold of quasi-contact curvature is a 𝜓(𝐹)𝑛. 

From the above equation & Theorem 4.2, we have the following result 

Theorem4.3.A conformally flat (𝜖) − Lorentzian para-Sasakian manifold with generalized 

symmetric metric connection is a 𝜓(𝐹)𝑛. 

5. Weyl-Semi-symmetric (𝝐) − Lorentzian Para-Sasakian Manifold with Parallelized 

generalized Symmetric Metric Connection 

An (𝜖) − Lorentzian para-Sasakian manifold is said to be Weyl-semi-symmetric if 

𝑅. 𝐶 = 0           (5.1) 

From (4.1), we have 

𝜂(𝐶(𝑋, 𝑌)𝑍)  =  
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
 – 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} {𝑔(𝑌, 𝑍)𝜂(𝑋)  − 𝑔(𝑋, 𝑍)𝜂(𝑌) 

−𝑆(𝑌, 𝑍)𝜂(𝑋) + 𝑆(𝑋, 𝑍)𝜂(𝑌)}]        (5.2) 

Putting 𝑍 = 𝜉 in above equation, we get 

𝜂(𝐶(𝑋, 𝑌)𝜉)  =  0          (5.3) 

Again putting 𝑋 = 𝜉 in equation (5.2), we get 

𝜂(𝐶(𝜉, 𝑌)𝑍)  =  
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
 − 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} +  𝑆(𝑌, 𝑍) 

   +(𝑛 − 1)(𝛼2 + 𝛽2)𝜂(𝑋)𝜂(𝑌) − 𝑔(𝑋, 𝑍) − 𝜖𝜂(𝑋)𝜂(𝑌)}]    (5.4) 



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If the manifold is Weyl-semi-symmetric then, we have 

𝑔[𝑅(𝜉, 𝑌)𝐶(𝑈, 𝑉)𝑊, 𝜉]  −  𝑔[𝐶(𝑅(𝜉, 𝑌)𝑈, 𝑉)𝑊, 𝜉] −  𝑔[𝐶(𝑈, 𝑅(𝜉, 𝑌)𝑊, 𝜉] 

− 𝑔[𝐶(𝑈, 𝑉)𝑅(𝜉, 𝑌)𝑊, 𝜉]   = 0         (5.5) 

From equation (3.16), we obtained 

𝑔(𝑅(𝜉, 𝑋)𝑌, 𝜉) = −(𝛼2 + 𝛽2){𝑔(𝑋, 𝑌) + 𝜖𝜂(𝑋)𝜂(𝑌)} + 𝜖(𝜉𝛼)𝜂(𝑋) + (𝛼𝜉)𝑔(𝑋, 𝑌)𝜂(𝑌) 

+𝜖(𝑋𝛼) − 𝜖(𝑋𝛼)𝜂(𝑌)𝜂(𝑌)         (5.6) 

Using equation (5.5) and (5.6), we obtained 

(𝛼2 + 𝛽2){𝐶′(𝑈, 𝑉,𝑊, 𝑌) + 𝜖𝜂(𝑌)𝜂(𝐶(𝑈, 𝑉)𝑊)} − 𝜖(𝜉𝛼)𝜂(𝑌) − (𝛼𝜉)𝐶 ′(𝑈, 𝑉,𝑊,𝑈)𝜂(𝐶(𝑈, 𝑉)𝑊)

− 𝜖(𝑌𝛼) + 𝜖(𝑌𝛼)𝜂(𝐶(𝑈, 𝑉)𝑊)𝜂(𝐶(𝑈, 𝑉)𝑊) + 𝑔[𝐶((𝛼2 + 𝛽2)(𝜖𝑔(𝑌, 𝑈)𝜉

− 𝜂(𝑈)𝑌) + (𝜉𝛽 − 2𝛼𝛽𝜂(𝑈))𝜙𝑌 + (𝜉𝛼)𝑌 + 𝜖𝛼𝜉𝑔(𝑌, 𝑈)𝑈 − (𝑌𝛼)𝜉 

− (𝑌𝛼)𝜂(𝑈)𝑈, 𝑉)𝑊, 𝜉] + 𝑔[𝐶(𝑈, (𝛼2 + 𝛽2)(𝜖𝑔(𝑌, 𝑉)𝜉 − 𝜂(𝑉)𝑌) + (𝜉𝛽

− 2𝛼𝛽𝜂(𝑉))𝜙𝑌 + (𝜉𝛼)𝑌 + 𝜖𝛼𝜉𝑔(𝑌, 𝑉)𝑉 − (𝑌𝛼)𝜉 − (𝑌𝛼)𝜂(𝑉)𝑉,𝑊), 𝜉]

+ 𝑔[𝐶(𝑈, 𝑉)((𝛼2 + 𝛽2)(𝜖𝑔(𝑌,𝑊)𝜉 − 𝜂(𝑊)𝑌) + (𝜉𝛽 − 2𝛼𝛽𝜂(𝑊))𝜙𝑌 + (𝜉𝛼)𝑌 

+ 𝜖𝛼𝜉𝑔(𝑌,𝑊)𝑊  

−(𝑌𝛼)𝜉 − (𝑌𝛼)𝜂(𝑊)𝑊), 𝜉] = 0        (5.7) 

where 𝐶 ′(𝑈, 𝑉,𝑊, 𝑌) = 𝑔(𝐶(𝑈, 𝑉)𝑊, 𝑌) 

Putting 𝑌 = 𝑈 in (5.7), we have 

(𝛼2 + 𝛽2){𝐶′(𝑈, 𝑉,𝑊,𝑈) + 𝜖𝜂(𝑈)𝜂(𝐶(𝑈, 𝑉)𝑊)} − 𝜖(𝜉𝛼)𝜂(𝑈) 

                          −(𝛼𝜉)𝐶 ′(𝑈, 𝑉,𝑊,𝑈)𝜂(𝐶(𝑈, 𝑉)𝑊) − 𝜖(𝑈𝛼)

+ 𝜖(𝑈𝛼)𝜂(𝐶(𝑈, 𝑉)𝑊)𝜂(𝐶(𝑈, 𝑉)𝑊)                         + 𝑔[𝐶((𝛼2 + 𝛽2)(𝜖𝑔(𝑈, 𝑈)𝜉

− 𝜂(𝑈)𝑈) + (𝜉𝛽 − 2𝛼𝛽𝜂(𝑈))𝜙𝑈 + (𝜉𝛼)𝑈 

+𝜖𝛼𝜉𝑔(𝑈, 𝑈)𝑈 − (𝑈𝛼)𝜉 − (𝑈𝛼)𝜂(𝑈)𝑈, 𝑉)𝑊, 𝜉] 

+𝑔[𝐶(𝑈, (𝛼2 + 𝛽2)(𝜖𝑔(𝑈, 𝑉)𝜉 − 𝜂(𝑉)𝑈) 

 +(𝜉𝛽 − 2𝛼𝛽𝜂(𝑉))𝜙𝑈 + (𝜉𝛼)𝑈 + 𝜖𝛼𝜉𝑔(𝑈, 𝑉)𝑉 − (𝑈𝛼)𝜉 − (𝑈𝛼)𝜂(𝑉)𝑉,𝑊), 𝜉] 

+𝑔 [𝐶(𝑈, 𝑉) ((𝛼2 + 𝛽2)(𝜖𝑔(𝑈,𝑊)𝜉 − 𝜂(𝑊)𝑈) + (𝜉𝛽 − 2𝛼𝛽𝜂(𝑊))𝜙𝑈 +                         (𝜉𝛼)𝑈 +

𝜖𝛼𝜉𝑔(𝑈,𝑊)𝑊 − (𝑈𝛼)𝜉 − (𝑈𝛼)𝜂(𝑊)𝑊) , 𝜉] = 0  (5.8) 

Taking an orthogonal frame in the equation (5.8) over 𝑈, we obtained  

∑ 𝐶 ′(𝑒𝑖, 𝑉,𝑊, 𝑒𝑖)
𝑛
𝑖=1 = 0         (5.9) 

and using equation (5.3) in (5.8), we have 

𝜂(𝐶(𝜉, 𝑉)𝑊)  =  0          (5.10) 

Using equation (5.3) and (5.8) we have 

𝐶 ′(𝑈, 𝑉,𝑊, 𝑌) + 𝜂(𝑌)𝜂(𝐶(𝑈, 𝑉)𝑊) + 𝜖𝜂(𝑈)𝜂(𝐶(𝑌, 𝑉)𝑊) + 𝜖𝜂(𝑉)𝜂(𝐶(𝑈, 𝑌)𝑊)  



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                                                                  +𝜖𝜂(𝑊)𝜂(𝐶(𝑈, 𝑉)𝑌) = 0   (5.11) 

Using equation (5.2) and (5.11), we have 

𝐶′(𝑈, 𝑉,𝑊, 𝑌) + 𝜂(𝑌)
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
– 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} {𝑔(𝑉,𝑊)𝜂(𝑈) − 𝑔(𝑈,𝑊)𝜂(𝑉)

− 𝑆(𝑉,𝑊)𝜂(𝑈) + 𝑆(𝑈,𝑊)𝜂(𝑉)}]

+ 𝜖𝜂(𝑈)
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
– 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} {𝑔(𝑉,𝑊)𝜂(𝑌) − 𝑔(𝑌,𝑊)𝜂(𝑉)

− 𝑆(𝑉,𝑊)𝜂(𝑌) + 𝑆(𝑌,𝑊)𝜂(𝑉)}]

+ 𝜖𝜂(𝑉)
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
– 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} {𝑔(𝑉,𝑊)𝜂(𝑈) − 𝑔(𝑈,𝑊)𝜂(𝑉)

− 𝑆(𝑉,𝑊)𝜂(𝑈) + 𝑆(𝑈,𝑊)𝜂(𝑉)}]

+ 𝜖𝜂(𝑊)
1

(𝑛 − 2)
[{

𝑟

(𝑛 − 1)
– 𝜖(𝑛 − 1)(𝛼2 + 𝛽2)} {𝑔(𝑉, 𝑌)𝜂(𝑈) − 𝑔(𝑈, 𝑌)𝜂(𝑉)

− 𝑆(𝑉, 𝑌)𝜂(𝑈) + 𝑆(𝑈, 𝑌)𝜂(𝑉)}] = 0                                                                    (5.12) 

From equation (5.4) and (5.10), we have 

𝑆(𝑌, 𝑍) = {
𝑟

(𝑛−1)
− 𝜖(𝛼2 + 𝛽2)} 𝑔(𝑌, 𝑍) + {

𝑟

(𝑛−1)
− 𝜖𝑛(𝛼2 + 𝛽2)} 𝜂(𝑌)𝜂(𝑍)  (5.13) 

Using equation (5.12) and (5.11), we have 

𝐶 ′(𝑈, 𝑉,𝑊, 𝑌) = 0          (5.14) 

From the above equation we can see that 𝑅. 𝐶 = 0, this implies that 𝐶 = 0. 

Hence this condition with the help of Theorem 4.2, gives the following results: 

Theorem 5.1.An𝑛 −dimensional Weyl-semi-symmetric  (𝜖) − Lorentzian para-Sasakian manifolds 

with parallelized generalized symmetric metric connection is of quasi-constant curvature. 

Theorem 4.3 and equation (5.14) leads to the following result: 

Corollary 5.2.An𝑛 −dimensional Weyl-semi-symmetric  (𝜖) − Lorentzian para-Sasakian manifolds 

with parallelized generalized symmetric metric connection is a 𝜓(𝐹)𝑛. 

Application. (𝜖) −  Lorentzian para-Sasakian manifolds with parallelized generalized symmetric 

metric connection are used in the Newtons law of gravitational field and theory of Relativity. 

Acknowledgement. The authors are thankful to Integral University, Lucknow for giving manuscript 

communication number IU/R&D/2024-MCN0002708. 

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