Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 3s (2025) 

 

699 
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Some Properties of RICCI Solitons in LP-Kenmotsu Manifolds 

 

Bidyabati Thangjam1, M. S. Devi*2 
1,2

 

Article History: 

Received: 28-09-2024 

Revised: 18-11-2024 

Accepted: 30-11-2024 

Abstract:  

Examining Ricci solitons in Lorentzian para-Kenmotsu manifolds is the goal of this paper. 

We have established that a symmetric parallel second-order covariant tensor in a Lorentzian 

para-Kenmotsu manifold is a constant multiple of the metric tensor. We have shown that if 

L_V g+2S is parallel to the Levi-Civita connection associated with g, where V is a given 

vector field, then (g,V,λ) is a Ricci soliton. We have observed that a Ricci soliton in a W_2-

semi-symmetric Lorentzian para-Kenmotsu manifold is shrinking. Furthermore, certain 

curvature properties of Lorentzian para-Kenmotsu manifolds admitting Ricci solitons are 

studied. Finally, we have provided an example of a 3-dimensional Lorentzian para-Kenmotsu 

manifold. 

Keywords:  Lorentzian Para-Kenmotsu manifolds, Ricci solitons, Symmetric second order 

tensors. 

Mathematics Subject Classification (2020): 53C21, 53C25, 53E20. 

 

Introduction 

Hamilton [5] introduced the concept of Ricci solitons, which is a natural generalization of an Einstein 

metric and is defined on a Riemannian manifold 𝑀. A Ricci soliton is a tripled (𝑔, 𝑉, λ) such that 

  ℒ𝒱 + 2𝑆 + 2𝜆𝑔 = 0, (1.1) 

where 𝑔 is a Riemannian metric, 𝑉 is a vector field, 𝜆 is a real scalar, 𝑆 is a Ricci tensor of 𝑀 and ℒ𝒱 

denotes the Lie derivative operator along the vector field 𝑉. A Ricci soliton is said to be shrinking if 𝜆 

is negative, steady if  𝜆 is zero, and expanding if  𝜆 is positive. 

In recent years, many geometers have studied Ricci solitons. Ingalahalli and Bagewadi [8] studied 

Ricci solitons on 𝛼-Sasakian manifolds. Pokhariyal et al. [12] found some results on Trans-Sasakian 

manifolds. Ayar and Demirhan [1] provided basic information about Ricci solitons on nearly 

Kenmotsu manifolds and obtained some structures on this manifold satisfying a semi-symmetric metric 

connection. Later, Shah [15] studied Ricci solitons in Lorentzian Para-Sasakian manifolds, while Chen 

et al. [2] studied Ricci solitons and certain related metrics on a three-dimensional Trans-Sasakian 

manifolds. Many other geometers had also studied Ricci solitons on various manifolds. 

Sato[14] introduced the concept of an almost para-contact Riemannian manifold. Subsequently, Sinha 

and Prasad [7] defined a specific class of almost para-contact metric manifolds, namely para Kenmotsu 

and special para Kenmotsu manifolds. Another related structure is Lorentzian para-Sasakian manifold, 

which were introduced by Matsumoto [9]. Several other researchers had also studied this manifold 

([4], [10], [16], [18]). Recently, Haseeb and Prasad [6, 7] focused on studying the properties of 

Lorentzian para-Kenmotsu manifolds, particularly in terms of Ricci-pseudosymmetricity and Ricci- 

Department of Mathematics, Mizoram University, Tanhril, Aizawl-796004, India.
                                     devi_saroja@rediffmail.com



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generalized pseudosymmetricity conditions. Pandey et al. [11] investigated the geometric properties 

of 𝜂-Ricci solitons on Lorentzian para-Kenmotsu manifolds. Based on these studies, our motivation is 

to investigate Ricci solitons on  Lorentzian para-Kenmotsu manifolds. 

The paper is structured as follows: Section 1 is the introduction, then there is a preliminaries section. 

In section 3, parallel symmetric second-order tensors and Ricci solitons in Lorentzian para-Kenmotsu 

manifolds are studied. The next section investigates the properties of Ricci solitons in 𝑊2-semi-

symmetric Lorentzian para-Kenmotsu manifolds of dimensions (2𝑛 + 1). In section 5, we study Ricci 

tensor of a Lorentzian para-Kenmotsu manifold admitting a Ricci soliton. Then, in the next section, 

curvature properties of Lorentzian para-Kenmotsu manifolds admitting Ricci solitons are examined. 

The last section provides an example of a Lorentzian para-Kenmotsu manifold. The paper concludes 

with a summary of the findings. 

1. Preliminaries 

A (2𝑛 + 1) −differentiable manifold 𝑀 with a (1,1) tensor field 𝜙, contravariant vector field 𝜉, a 1-

form 𝜂, and a Lorentzian metric 𝑔 is referred as a Lorentzian almost para-contact manifold[9]   if 𝑔 

satisfies the following conditions:    

 𝜙2𝑋1 = 𝑋1 + 𝜂(𝑋1),  𝜙𝜉 = 0,  (2.1) 

 𝑔(𝜙𝑋1, 𝜙𝑋2) = 𝑔(𝑋1, 𝑋2) + 𝜂(𝑋1)𝜂(𝑋2) (2.2) 

and 

 𝜂(𝜉) = −1,  𝑔(𝑋1, 𝜉) = 𝜂(𝑋1),  (2.3) 

 𝜂(𝜙𝑋1) = 0.   (2.4) 

A Lorentzian almost para-contact manifold 𝑀 is a Lorentzian para-Kenmotsu manifold if it satisfies 

[6] 

 (∇𝑋1
 𝜙) = −𝑔(𝜙𝑋1, 𝑋2)𝜉 − 𝜂(𝑋2)𝜙𝑋1,  (2.5) 

for any vector fields 𝑋1 and 𝑋2 on 𝑀 and ∇ is the operator of covariant differentiation with respect to 

the Lorentzian metric 𝑔. 

In Lorentzian para-Kenmotsu manifolds, the following relations hold [11]: 

 ∇𝑋1
𝜉 = −𝑋1 − 𝜂(𝑋1)𝜉,  (2.6) 

and 

 (∇𝑋1
 𝜂)𝑋2 = −𝑔(𝑋1, 𝑋2) − 𝜂(𝑋1)𝜂(𝑋2).  (2.7) 

In addition to these, the following relations also hold [6]: 

 𝜂(𝑅(𝑋1, 𝑋2 ) 𝑋3) = 𝑔(𝑋2,  𝑋3 )𝜂(𝑋1 ) − 𝑔(𝑋1, 𝑋3)𝜂(𝑋2),  (2.8) 

 𝑅(𝜉, 𝑋1) 𝑋2 = 𝑔(𝑋1, 𝑋2 )𝜉 − 𝜂(𝑋1 ) 𝑋2,  (2.9) 

 𝑅(𝑋1, 𝑋2 )𝜉 = 𝜂(𝑋2) 𝑋1 − 𝜂(𝑋1)𝑋2,  (2.10) 

 𝑆(𝑋1, 𝜉) = 2𝑛𝜂(𝑋1),    𝑄(𝜉) = 2𝑛,  (2.11) 



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 𝑆(𝜙𝑋1, 𝜙𝑋2 ) = 𝑆(𝑋1, 𝑋2 ) + 2𝑛𝜂(𝑋1)𝜂(𝑋2 ),  (2.12) 

for all vector fields 𝑋1 and 𝑋2 on 𝑀. 

Let (𝑔, 𝑉, 𝜆) be a Ricci soliton in a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold 𝑀. 

Then, we have 

 (ℒξ𝑔)(𝑋1, 𝑋2) = 𝑔(∇𝜉 𝑋1, 𝑋2 ) + 𝑔(𝑋1, ∇𝜉𝑋2 ).  (2.13) 

Using [2.6] in [2.13], we have 

 (ℒξ𝑔)(𝑋1, 𝑋2) = −2𝑔(𝑋1, 𝑋2) − 2𝜂(𝑋1)𝜂(𝑋2 ).  (2.14) 

From [1.1] and [2.14], we get 

 𝑆(𝑋1, 𝑋2) = (1 − 𝜆)𝑔(𝑋1, 𝑋2) + 𝜂(𝑋1)𝜂(𝑋2),  (2.15) 

 𝑄𝑋1 = (1 − 𝜆)𝑋1 + 𝜂(𝑋1)𝜉,      𝑟 = 2𝑛 − 𝜆(2𝑛 + 1).   (2.16) 

In view of [2.3] and [2.15], we have 

 𝑆(𝑋1, 𝜉) = −𝜆𝜂(𝑋1),  (2.17) 

 𝑄𝜉 = −𝜆𝜉.  (2.18) 

Definition 2.1 [7]: A (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold 𝑀 is called an 𝜂-

Einstein manifold if its Ricci tensor 𝑆 satisfies the following equation: 

 𝑆(𝑋1, 𝑋2) = 𝛼𝑔(𝑋1, 𝑋2) + 𝛽𝜂(𝑋1)𝜂(𝑋2),  (2.19) 

where 𝛼 and 𝛽 are scalars. 

2. Parallel Symmetric Second Order Tensor and Ricci Solitons in Lorentzian para-

Kenmotsu manifolds 

In this section, we study parallel symmetric second order tensor and Ricci solitons in Lorentzian para-

Kenmotsu manifolds. 

Theorem 3.1: A symmetric parallel second order covariant tensor in a Lorentzian para-Kenmotsu 

manifold is a constant multiple of the metric tensor. 

Proof: Let ℎ be a symmetric tensor field of (0,2)-type which is parallel with respect to ∇ that is ∇ℎ =

0. 

Then by applying the Ricci identity [8], we obtain 

 𝛻2ℎ(𝑋1, 𝑋2; 𝑋3, 𝑋4) − 𝛻2ℎ(𝑋1, 𝑋2; 𝑋4, 𝑋3) = 0,  (3.1) 

which implies that  

 ℎ(𝑅(𝑋1, 𝑋2)𝑋3, 𝑋4) + ℎ(𝑋3, 𝑅(𝑋1, 𝑋2)𝑋4) = 0.  (3.2) 

Replacing 𝑋3 = 𝑋4 = 𝜉 in (3.2) and using (2.10) and the symmetry of ℎ, we get 

 2[𝜂(𝑋2)ℎ(𝑋1, 𝜉) − 𝜂(𝑋1)ℎ(𝑋2, 𝜉)] = 0.  (3.3) 

Putting 𝑋1 = 𝜉 in (3.3), we have 



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 ℎ(𝑋2, 𝜉) = −𝜂(𝑋2)ℎ(𝜉, 𝜉).  (3.4) 

Now, differentiating (3.4) covariantly with respect to 𝑋1, we obtain 

(𝛻𝑋1
ℎ)(𝑋2, 𝜉) + ℎ(𝛻𝑋1

𝑋2, 𝜉) + ℎ(𝑋2, 𝛻𝑋1
𝜉) = −[(𝛻𝑋1

𝜂)(𝑋2) + 𝜂(𝛻𝑋1
𝑋2)]ℎ(𝜉, 𝜉) 

                                                                       +𝜂(𝑋2)[(𝛻𝑋1
ℎ)(𝜉, 𝜉) + 2ℎ(𝛻𝑋1

𝜉, 𝜉)].            (3.5) 

 

Using (2.6), (3.4) and the parallel condition ∇ℎ = 0 in (3.5), we get 

 ℎ(𝑋2, 𝛻𝑋1
𝜉) = −(𝛻𝑋1

𝜂)(𝑋2)ℎ(𝜉, 𝜉).  (3.6) 

In consequences of (2.6), (2.7), (3.4) and (3.6), it yields 

 ℎ(𝑋1, 𝑋2) = −𝑔(𝑋1, 𝑋2)ℎ(𝜉, 𝜉).  (3.7) 

From the above (3.7) and (3.4), we can conclude that ℎ(𝜉, 𝜉) is a constant.  

This completes the proof of the theorem. 

Theorem 3.2. Let 𝑀 be a Lorentzian para-Kenmotsu manifold. Assume that a symmetric metric tensor 

field ℎ = ℒ𝒱 𝑔 + 2𝑆 is parallel with respect to the Levi-Civita connection associated with 𝑔. Then 

(𝑔, 𝑉, λ) yields a Ricci-soliton on 𝑀. 

Proof: Let us assume that a symmetric tensor field ℎ = ℒ𝒱𝑔 + 2𝑆 is parallel with respect to the Levi-

Civita connection associated with 𝑔. 

Then, ℎ(𝜉, 𝜉) = 2𝜆, this shows that  𝜆 =
1

2
ℎ(𝜉, 𝜉). 

Now, as ℎ is parallel with respect to 𝑔, then from (3.7) we get 

 𝐻(𝑋1, 𝑋2) = −2𝜆𝑔(𝑋1, 𝑋2),  (3.8) 

for all vector fields 𝑋1 and 𝑋2 on 𝑀, which leads to 

 ℒ𝒱𝑔(𝑋1, 𝑋2) = −2𝜆𝑔(𝑋1, 𝑋2) − 2𝑆(𝑋1, 𝑋2).  (3.9) 

Hence, we complete the proof of the theorem. 

Theorem 3.3. A Ricci semi-symmetric Lorentzian para-Kenmotsu manifold is an Einstein manifold. 

Proof: We consider a Ricci semi-symmetric Lorentzian para-Kenmotsu manifold, i.e., 𝑅  ∘ 𝑆 = 0. 

We have [8], 

 (𝑅(𝑋1, 𝑋2) ∘  𝑆)(𝑋3, 𝑋4) = −𝑆(𝑅(𝑋1, 𝑋2)𝑋3, 𝑋4) − 𝑆(𝑋3, 𝑅(𝑋1, 𝑋2)𝑋4).  (3.10) 

Putting 𝑋1 = 𝜉 and using 𝑅 ∘ 𝑆 = 0 in (3.10), we have 

 𝑆(𝑅(𝜉, 𝑋2)𝑋3, 𝑋4) + 𝑆(𝑋3, 𝑅(𝜉, 𝑋2), 𝑋4) = 0.  (3.11) 

Using (2.9) in (3.11), we obtain 

 −𝑔(𝑋2, 𝑋3)𝑆(𝜉, 𝑋4) − 𝜂(𝑋3)𝑆(𝑋2, 𝑋4) + 𝑆(𝑋3, 𝜉)𝑔(𝑋2, 𝑋4) + 𝜂(𝑋4)𝑆(𝑋3, 𝑋2) = 0. (3.12) 



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Setting 𝑋3 = 𝜉 in (3.12) and by making used of (2.3) and (2.11), we have 

 𝑆(𝑋2, 𝑋4) = 2𝑛𝑔(𝑋2, 𝑋4).  (3.13) 

Thus, we complete the proof. 

Proposition 3.1. If  a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold is an 𝜂-Einstein 

manifold, then the Ricci soliton with constant scalar curvature is shrinking. 

Proof:  

Suppose that  the Lorentzian para-Kenmotsu manifold is an 𝜂-Einstein manifold. Then, we will find 

the values of  𝛼 and 𝛽. 

 Let {𝑒1, 𝑒2, … , 𝑒2𝑛+1} be an orthonormal basis of the tangent at any point of the manifold. Putting 𝑋1 =

𝑋2 = 𝑒𝑖 in (2.19) and taking summation over 𝑖, we get 

 𝑟 = (2𝑛 + 1)𝛼 − 𝛽.  (3.14) 

Again, setting 𝑋1 = 𝑋2 = 𝜉  in (2.19), and using (2.11), we have 

 −2𝑛 = −𝛼 + 𝛽.  (3.15) 

Then from (3.14) and (3.15), we get  

 𝛼 = [
𝑟

2𝑛
− 1] , 𝛽 = [−2𝑛 − 1 +

𝑟

2𝑛
].  (3.16) 

Substituting the value of  𝛼 and 𝛽 in (2.19), we have 

 𝑆(𝑋1, 𝑋2) = [
𝑟

2𝑛
− 1] 𝑔(𝑋1, 𝑋2) + [−2𝑛 − 1 +

𝑟

2𝑛
] 𝜂(𝑋1)𝜂(𝑋2).  (3.17) 

For a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold the symmetric parallel covariant 

tensor ℎ(𝑋1, 𝑋2) of type (0,2) is given by  

 ℎ(𝑋1, 𝑋2) = (ℒξ 𝑔)(𝑋1, 𝑋2 ) + 2𝑆(𝑋1, 𝑋2 ).  (3.18) 

Using (2.13) and (3.17) in (3.18), we get  

 ℎ(𝑋1, 𝑋2) = [
2𝑟

2𝑛
− 4] 𝑔(𝑋1, 𝑋2) + [−4𝑛 +

2𝑟

2𝑛
− 4] 𝜂(𝑋1)𝜂(𝑋2).  (3.19) 

Taking covariant derivative of (3.19) with respect to 𝑋3, we have 

(𝛻𝑋3
ℎ)(𝑋1, 𝑋2) = [

2(𝛻𝑋3
𝑟)

2𝑛
] [𝑔(𝑋1, 𝑋2) + 𝜂(𝑋1)𝜂(𝑋2)] 

                                        + [
2𝑟

2𝑛
− 4𝑛 − 4] [𝑔(𝑋1, 𝛻𝑋3

𝜉)𝜂(𝑋2) + 𝑔(𝑋2, 𝛻𝑋3
𝜉)𝜂(𝑋1)].  (3.20) 

By putting 𝑋3 = 𝜉  and 𝑋1 = 𝑋2 ∈ (𝑠𝑝𝑎𝑛𝜉)⊥ in (3.20) and by using ∇ℎ = 0, we obtain 

 2𝛻𝜉𝑟 = 0.  (3.21) 

On integrating (3.21), we get 

 𝑟 = 𝑐,  (3.22) 



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where 𝑐 is some integral constant. 

Thus, from (3.22) we have 𝑟 is constant scalar curvature. 

Finally, we will check the nature of the Ricci-soliton. 

From (3.18), we have ℎ(𝑋1, 𝑋2) = −2𝜆𝑔(𝑋1, 𝑋2), then putting 𝑋1 = 𝑋2 = 𝜉, we have 

 ℎ(𝜉, 𝜉) = 2𝜆.  (3.23) 

If we put 𝑋1 = 𝑋2 = 𝜉 in (3.19), that is 

 ℎ(𝜉, 𝜉) = − [
2𝑟

2𝑛
− 4] + [−4𝑛 +

2𝑟

2𝑛
− 4].  (3.24) 

The above equation is reduced as 

 ℎ(𝜉, 𝜉) = −4𝑛.  (3.25) 

Equating (3.23) and (3.25), we obtain 

 𝜆 = −2𝑛 < 0,  (3.26) 

that is the Ricci soliton in a Lorentzian para-Kenmotsu manifold is shrinking. 

Hence, the theorem is proved. 

3. Ricci solitons in a 𝑾𝟐-semisymmetric Lorentzian para-Kenmotsu manifold 

Here, we study the conditions of Ricci solitons in a 𝑊2- semisymmetric Lorentzian para-Kenmotsu 

manifold. 

Definition 4.1. [13] In a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold 𝑀, the 𝑊2-

curvature tensor is defined as 

 𝑊2(𝑋1, 𝑋2)𝑋3 = 𝑅(𝑋1, 𝑋2)𝑋3 +
1

2𝑛
[𝑔(𝑋1, 𝑋3)𝑄𝑋2 − 𝑔(𝑋2, 𝑋3)𝑄𝑋1],  (4.1) 

for all 𝑋1, 𝑋2 and 𝑋3 in 𝑀. 

Theorem 4.1. A Ricci soliton in a 𝑊2-semi symmetric Lorentzian para-Kenmotsu manifold 𝑀 of 

dimension (2𝑛 + 1) is shrinking. 

Proof: Putting 𝑋1 = 𝜉 in (4.1) and using (2.3) and (2.9), we have 

 𝑊2(𝜉, 𝑋2)𝑋3 = 𝑔(𝑋2, 𝑋3)𝜉 − 𝜂(𝑋3)𝑋2 +
1

2𝑛
[𝜂(𝑋3)𝑄𝑋2 − 𝑔(𝑋2, 𝑋3)𝑄𝜉].  (4.2) 

Taking inner product on both sides of (4.1) with respect to 𝜉, we get 

𝜂(𝑊2(𝑋1, 𝑋2)𝑋3) = 𝜂(𝑅(𝑋1, 𝑋2)𝑋3) +
1

2𝑛
[𝑔(𝑋1, 𝑋3)𝑔(𝑄𝑋2, 𝜉) − 𝑔(𝑋2, 𝑋3)𝑔(𝑄𝑋1, 𝜉)].  (4.3) 

Using (2.8) and (2.17) in (4.3), we obtain 

 𝜂(𝑊2(𝑋1, 𝑋2)𝑋3) = (1 +
𝜆

2𝑛
) [𝑔(𝑋2, 𝑋3)𝜂(𝑋1) − 𝑔(𝑋1, 𝑋3)𝜂(𝑋2)].  (4.4) 

Suppose that the condition, 𝑅(𝜉, 𝑋1) ∘ 𝑊2(𝑋2, 𝑋3)𝑋4 = 0 holds in 𝑀. Then by definition, we have 

 𝑅(𝜉, 𝑋1)𝑊2(𝑋2, 𝑋3)𝑋4 − 𝑊2(𝑅(𝜉, 𝑋1)𝑋2, 𝑋3)𝑋4 − 𝑊2(𝑋2, 𝑅(𝜉, 𝑋1)𝑋3)𝑋4 



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−𝑊2(𝑋2, 𝑋3)𝑅(𝜉, 𝑋1)𝑋4 = 0,  (4.5) 

for all vector fields 𝑋1, 𝑋2, 𝑋3 and 𝑋4 on 𝑀. 

In view of (2.9) and (4.5), we get 

𝑔(𝑋1, 𝑊2(𝑋2, 𝑋3)𝑋4)𝜉 − 𝜂(𝑊2(𝑋2, 𝑋3)𝑋4)𝑋1 − 𝑔(𝑋1, 𝑋2)𝑊2(𝜉, 𝑋3)𝑋4 

                  +𝜂(𝑋2)𝑊2(𝑋1, 𝑋3)𝑋4 − 𝑔(𝑋1, 𝑋3)𝑊2(𝑋2, 𝜉)𝑋4 + 𝜂(𝑋3)𝑊2(𝑋2, 𝑋1)𝑋4 

                  −𝑔(𝑋1, 𝑋4)𝑊2(𝑋2, 𝑋3)𝜉 + 𝜂(𝑋4)𝑊2(𝑋2, 𝑋3)𝑋1 = 0.  (4.6) 

Again, taking inner product on both sides of (4.6) with 𝜉 and using (2.3), we have 

−𝑔(𝑋1, 𝑊2(𝑋2, 𝑋3)𝑋4) − 𝜂(𝑊2(𝑋2, 𝑋3)𝑋4)𝜂(𝑋1) − 𝑔(𝑋1, 𝑋2)𝜂(𝑊2(𝜉, 𝑋3)𝑋4) 

 +𝜂(𝑋2)𝜂(𝑊2(𝑋1, 𝑋3)𝑋4) − 𝑔(𝑋1, 𝑋3)𝜂(𝑊2(𝑋2, 𝜉)𝑋4) + 𝜂(𝑋3)𝜂(𝑊2(𝑋2, 𝑋1)𝑋4) 

−𝑔(𝑋1, 𝑋4)𝜂(𝑊2(𝑋2, 𝑋3)𝜉) + 𝜂(𝑋4)𝜂(𝑊2(𝑋2, 𝑋3)𝑋1) = 0.  (4.7) 

In consequence of (4.2), (4.4) and (4.7), it yields 

 𝑔(𝑅(𝑋2, 𝑋3)𝑋4, 𝑋1) +
1

2𝑛
[𝑔(𝑋2, 𝑋4)𝑆(𝑋1, 𝑋3) − 𝑔(𝑋3, 𝑋4)𝑆(𝑋1, 𝑋2)] 

                                                     − (1 +
𝜆

2𝑛
) [𝜂(𝑋1){𝑔(𝑋4, 𝑋3)𝜂(𝑋2) − 𝑔(𝑋2, 𝑋4)𝜂(𝑋3)}  

                                                     +𝑔(𝑋1, 𝑋2){𝜂(𝑋4)𝜂(𝑋3) + 𝑔(𝑋3, 𝑋4)} 

                                  +𝜂(𝑋2){𝑔(𝑋3, 𝑋4)𝜂(𝑋1) − 𝑔(𝑋1, 𝑋4)𝜂(𝑋3)} 

                                  − 𝑔(𝑋1, 𝑋3){𝜂(𝑋4)𝜂(𝑋2) + 𝑔(𝑋2, 𝑋4)}             

                                  +𝜂(𝑋3){𝑔(𝑋1, 𝑋4)𝜂(𝑋2) − 𝑔(𝑋2, 𝑋4)𝜂(𝑋1)} 

                                  +𝜂(𝑋4){𝑔(𝑋3, 𝑋1)𝜂(𝑋2) − 𝑔(𝑋2, 𝑋1)𝜂(𝑋3)}] = 0.  (4.8) 

Let {𝑒1, 𝑒2, … ,  𝑒2𝑛+1} be an orthonormal basis. Putting 𝑋1 = 𝑋2 = 𝑒𝑖 in (4.8) and taking summation 

over 𝑖, where 1 ≤ 𝑖 ≤ (2𝑛 + 1), we have 

 [
2𝑛+1

2𝑛
] 𝑆(𝑋3, 𝑋4) =

𝑟

2𝑛
𝑔(𝑋3, 𝑋4) − 2 (1 +

𝜆

2𝑛
) [𝑛𝑔(𝑋3, 𝑋4) 

                                                    −(2𝑛 + 1)𝜂(𝑋3)𝜂(𝑋4)].  (4.9) 

Again, taking orthonormal frame field over 𝑋3 and 𝑋4, we get 

 𝜆 = −2𝑛 < 0, 

 which implies that the soliton is shrinking. 

Hence, the proof is completed.  

Theorem 4.2. Let 𝑀 be a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold and (𝑔, 𝑉, λ) be 

a Ricci soliton satisfying the condition 𝑊2(𝜉, 𝑋1) ∘ 𝑆 = 0 in 𝑀, then the Ricci soliton is steady. 

Proof: Let 𝑀 be a (2𝑛 + 1)-dimensional Lorentzian para-Kenmotsu manifold and (𝑔, 𝑉, λ) be a Ricci 

soliton in 𝑀. 



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We assume that the condition 𝑊2(𝜉, 𝑋1) ∘ 𝑆 = 0 holds in 𝑀, then we have 

 𝑆(𝑊2(𝜉, 𝑋1)𝑋2, 𝑋3) + 𝑆(𝑋2, 𝑊2(𝜉, 𝑋1)𝑋3) = 0.  (4.10) 

Using (2.17), (4.2) and (4.10), we obtain 

 −𝜆𝑔(𝑋1, 𝑋2)𝜂(𝑋3) − 𝜆𝑔(𝑋1, 𝑋3)𝜂(𝑋2) +
1

2𝑛
[𝑆(𝑄𝑋1, 𝑋3)𝜂(𝑋2) + 𝑆(𝑄𝑋1, 𝑋2)𝜂(𝑋3)] 

                                 −𝑆(𝑋1, 𝑋3)𝜂(𝑋2) − 𝑆(𝑋1, 𝑋2)𝜂(𝑋3) −
1

2𝑛
[𝑔(𝑋1, 𝑋2)𝑆(𝑄𝜉, 𝑋3) 

                                 +𝑔(𝑋1, 𝑋3)𝑆(𝑄𝜉, 𝑋2)] = 0.             (4.11) 

Setting 𝑋3 = 𝜉  in (4.11) and using (2.1), (2.3) and (2.17), we get 

 (𝜆 +
1

2𝑛
𝜆2) 𝑔(𝑋1, 𝑋2) + 𝑆(𝑋1, 𝑋2) +

1

2𝑛
[𝜆2𝜂(𝑋1)𝜂(𝑋2) − 𝑆(𝑄𝑋1, 𝑋2)] 

                                 −
1

2𝑛
𝜂(𝑋1)𝑆(𝑄𝜉, 𝑋2) = 0.  (4.12) 

Again, putting 𝑋2 = 𝜉  in (4.12) and using (2.1) and (2.17), we have 

𝜆 = 0. 

The above equation implies that the Ricci soliton is steady. 

Hence, we complete the proof. 

4. Ricci tensor of a Lorentzian para-Kenmotsu manifold admitting a Ricci soliton 

In this section, we study Ricci tensor of a Lorentzian para-Kenmotsu manifold admitting a Ricci 

soliton. 

Theorem 5.1. Let 𝑀 be a Lorentzian para-Kenmotsu manifold admitting a Ricci soliton (𝑔, 𝑉, λ). If 

the Ricci tensor 𝑆 of the manifold is 𝜂-recurrent, then the Ricci soliton is steady. 

Proof: Suppose that the Ricci tensor of the Lorentzian para-Kenmotsu manifold is 𝜂-recurrent, i.e., 

 (𝛻𝑋1
𝑆)(𝑋2, 𝑋3) = 𝜂(𝑋1)𝑆(𝑋2, 𝑋3),  (5.1) 

for all vector fields 𝑋1, 𝑋2, 𝑋3 on 𝑀. 

Then, by using (2.15), we have 

 (𝛻𝑋1
𝑆)(𝑋2, 𝑋3) = −2𝜂(𝑋1)𝜂(𝑋2)𝜂(𝑋3) − 𝑔(𝑋1, 𝑋2)𝜂(𝑋3) − 𝑔(𝑋1, 𝑋3)𝜂(𝑋2).  (5.2) 

Using (2.15) in (5.1) and comparing with (5.2), we get 

−𝑔(𝑋1, 𝑋2)𝜂(𝑋3) − 𝑔(𝑋1, 𝑋3)𝜂(𝑋2) − (1 − 𝜆)𝑔(𝑋2, 𝑋3)𝜂(𝑋1) = 3𝜂(𝑋1)𝜂(𝑋2)𝜂(𝑋3).  (5.3) 

Setting 𝑋2 = 𝑋3 = 𝜉  in (5.3), we obtain 

 −𝜆𝜂(𝑋1) = 0.  (5.4) 

Since 𝜂(𝑋1) ≠ 0, we have 𝜆 = 0. 

 Therefore, the Ricci soliton is steady. 



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Theorem 5.2. Let 𝑀 be an Lorentzian para-Kenmotsu manifold, admitting a Ricci soliton (𝑔, 𝑉, λ). 

Then 𝑄 and 𝑆 are parallel along 𝜉, where 𝑄 is the Ricci operator, defined by 𝑆(𝑋1, 𝑋2) = 𝑔(𝑄𝑋1, 𝑋2) 

and 𝑆 is the Ricci tensor of 𝑀.   

Proof: We can express the equations for the Ricci operator and Ricci tensor along 𝜉 as follows, 

 (∇𝜉𝑄)𝑋1 = ∇𝜉𝑄(𝑋1) − 𝑄(∇𝜉𝑋1)  (5.5) 

and 

 (∇𝜉 𝑆)(𝑋1, 𝑋2) = ∇𝜉𝑆(𝑋1, 𝑋2) − 𝑆(∇𝜉𝑋1, 𝑋2) − 𝑆(𝑋1, ∇𝜉𝑋2).  (5.6) 

Using (2.16) in (5.5), we can simplify to obtain 

 (∇𝜉𝑄)𝑋1 = 0.  (5.7) 

Similarly, applying (2.15) to (5.6), we get 

 (∇𝜉 𝑆)(𝑋1, 𝑋2) = 0.  (5.8) 

Therefore, from (5.7) and (5.8), we can conclude that 𝑄 and 𝑆 are parallel along 𝜉, which completes 

the proof. 

5. Ricci soliton in a Lorentzian para-Kenmotsu manifold and its curvature properties 

In this section, we explore some curvature properties of a Lorentzian para-Kenmotsu manifold 

admitting a Ricci soliton. 

Definition 6.1. [19] In a Lorentzian para-Kenmotsu manifold 𝑀, the projective curvature 𝑃 of the 

manifold is defined as 

 𝑃(𝑋1, 𝑋2)𝑋3 = 𝑅(𝑋1, 𝑋2)𝑋3 −
1

2𝑛
[𝑆(𝑋2, 𝑋3) 𝑋1 − 𝑆(𝑋1, 𝑋3 ) 𝑋2 ],  (6.1) 

for all vector fields 𝑋1, 𝑋2 and 𝑋3 on 𝑀. 

Proposition 6.1. A Lorentzian para-Kenmotsu manifold 𝑀, admitting a Ricci soliton (𝑔, 𝑉, λ) is 𝜉-

projectively flat iff the soliton is shrinking. 

Proof: Putting 𝑋3 = 𝜉 in (6.1) and by using (2.10) and (2.11), we get 

 𝑃(𝑋1, 𝑋2)𝜉 = [
2𝑛+𝜆

2𝑛
] [𝜂(𝑋2) 𝑋1 − 𝜂(𝑋1 ) 𝑋2 ].  (6.2) 

This implies that 𝑃(𝑋1, 𝑋2)𝜉 = 0 if and only if  𝜆 = −2𝑛, which proves the proposition. 

Definition 6.2. [13] In a Lorentzian para-Kenmotsu manifold 𝑀, the concircular curvature 𝐶 of the 

manifold is defined as 

 𝐶(𝑋1, 𝑋2)𝑋3 = 𝑅(𝑋1, 𝑋2)𝑋3 −
𝑟

2𝑛(2𝑛+1)
[𝑔(𝑋2, 𝑋3)𝑋1 − 𝑔(𝑋1, 𝑋3)𝑋2].  (6.3) 

Proposition 6.2. A Lorentzian para-Kenmotsu manifold 𝑀, admitting a Ricci soliton (𝑔, 𝑉, λ) is 𝜉-

concircularly flat if the soliton is shrinking. 

Proof: Setting 𝑋3 = 𝜉 in (6.3) and using the equations (2.3) and (2.16), we get 



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 𝐶(𝑋1, 𝑋2)𝑋3 = [
4𝑛2+𝜆(2𝑛+1)

2𝑛(2𝑛+1)
] [𝜂(𝑋2 ) 𝑋1 − 𝜂(𝑋1 )𝑋2 ].  (6.4) 

This shows that 𝐶(𝑋1, 𝑋2)𝜉 = 0 if and only if  𝜆 = −
4𝑛2

2𝑛+1
. 

Hence, we prove the proposition.    

Definition 6.3. [9] In a Lorentzian para-Kenmotsu manifold 𝑀, the conharmonic curvature tensor 𝐻 

is defined as 

 𝐻(𝑋1, 𝑋2)𝑋3 = 𝑅 (𝑋1, 𝑋2)𝑋3 −
1

2𝑛−1
[𝑔(𝑋2, 𝑋3 )𝑄𝑋1 − 𝑔(𝑋1, 𝑋3 )𝑄𝑋2 ] 

 +𝑆(𝑋2, 𝑋3) 𝑋1 − 𝑆(𝑋1, 𝑋3) 𝑋2 ].  (6.5) 

Proposition 6.3. A Lorentzian para-Kenmotsu manifold 𝑀, admitting a Ricci soliton (𝑔, 𝑉, λ) is 𝜉-

conharmonically flat if the soliton is shrinking. 

Proof: Putting 𝑋3 = 𝜉  in (6.5), we obtain 

 𝐻(𝑋1, 𝑋2)𝜉 = 𝑅(𝑋1, 𝑋2)𝜉 −
1

2𝑛−1
[𝑔(𝑋2, 𝜉)𝑄𝑋1 − 𝑔(𝑋1, 𝜉)𝑄𝑋2  

                             +𝑆(𝑋2, 𝜉)𝑋1 − 𝑆(𝑋1, 𝜉)𝑋2 ].  (6.6) 

Using (2.3), (2.16) and (2.17) in (6.6), we have 

 𝐻(𝑋1, 𝑋2)𝜉 =
2𝑛−2+2𝜆

2𝑛−1
[𝜂(𝑋2)𝑋1 − 𝜂(𝑋1)𝑋2].  (6.7) 

Thus, 𝐻(𝑋1, 𝑋2)𝜉 = 0 if and only if   λ= −(𝑛 − 1). 

Hence, the proof is completed. 

Definition 6.4. [3] In a Lorentzian para-Kenmotsu manifold 𝑀, the Weyl conformal curvature tensor 

𝑊 is defined as 

 𝑊(𝑋1, 𝑋2)𝑋3 = 𝑅(𝑋1, 𝑋2)𝑋3 −
1

2n−1
[𝑔(𝑋2, 𝑋3)𝑄𝑋1 − 𝑔(𝑋1, 𝑋3)𝑄𝑋2 

  +𝑆(𝑋2, 𝑋3)𝑋1 − 𝑆(𝑋1, 𝑋3)𝑋2]  

 +
𝑟

2𝑛(2𝑛−1)
[𝑔(𝑋2, 𝑋3)𝑋1 − 𝑔(𝑋1, 𝑋3)𝑋2].  (6.8) 

Proposition 6.4. A Lorentzian para-Kenmotsu manifold 𝑀, admitting a Ricci soliton (𝑔, 𝑉, λ) is 𝜉-

conformally flat if the soliton is shrinking. 

Proof: Putting 𝑋3 = 𝜉  in (6.8), we obtain  

 𝑊(𝑋1, 𝑋2)𝜉 = 𝑅(𝑋1, 𝑋2)𝜉 −
𝑟

2𝑛−1
[𝑔(𝑋2, 𝜉)𝑄𝑋1 

                                                       −𝑔(𝑋1, 𝜉)𝑄𝑋2 + 𝑆(𝑋2, 𝜉)𝑋1 − 𝑆(𝑋1, 𝜉)𝑋2] 

                            +
𝑟

2𝑛(2𝑛−1)
[𝑔(𝑋2, 𝜉) 𝑋1 − 𝑔(𝑋1, 𝜉) 𝑋2 ].  (6.9) 

Using (2.3), (2.16) and (2.17) in (6.9), we have 



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 𝑊(𝑋1, 𝑋2)𝜉 =
2𝑛+λ

2𝑛
[𝜂(𝑋2)𝑋1 − 𝜂(𝑋1)𝑋2].  (6.10) 

This implies that 𝑊(𝑋1, 𝑋2)𝜉 = 0 if and only if  λ = −2𝑛. 

This completes the proof of the proposition. 

6. Example of a Lorentzian para-Kenmotsu manifold 

In this section we establish an example of a Lorentzian para-Kenmotsu manifold. We consider the 3-

dimensional manifold 𝑀 = {(𝑥1, 𝑥2, 𝑥3) ∈ 𝑅3: 𝑥3 ≠ 0}, where (𝑥1, 𝑥2, 𝑥3) are the standard coordinates 

in 𝑅3. Let 𝐸1, 𝐸2 and 𝐸3 be a linearly independent vector fields in 𝑀 which satisfy 

  [𝐸1, 𝐸2] = 𝐸2,  [𝐸2, 𝐸3] = 0,   [𝐸1, 𝐸3] = 𝐸3. 

Let 𝑔 be the Lorentzian metric defined by 

𝑔(𝐸1, 𝐸1) = −1, 𝑔(𝐸2, 𝐸2) = 𝑔(𝐸3, 𝐸3) = 1, 

𝑔(𝐸1, 𝐸2) = 𝑔(𝐸2, 𝐸3) = 𝑔(𝐸1, 𝐸3) = 0. 

Let 𝜂 be the 1-form defined by 𝜂(𝑋1) = 𝑔(𝑋1, 𝐸3), for any vector field 𝑋1. 

Let 𝜙 be (1,1)-tensor field defined by 

𝜙𝐸1 = 0,  𝜙𝐸2 = 𝐸3,  𝜙𝐸3 = 𝐸2. 

Then we have  

𝜂(𝐸1) = −1, 𝜙2(𝑋1) = 𝑋1 + 𝜂(𝑋1)𝐸1 

and 

𝑔(𝜙𝑋1, 𝜙𝑋2 ) = 𝑔(𝑋1, 𝑋2) + 𝜂(𝑋1)𝜂(𝑋2). 

Thus for 𝜉 = 𝐸1, (𝜙, 𝜉, 𝜂, 𝑔) defines a Lorentzian almost paracontact metric structure on 𝑀. Let ∇ be 

the Levi-Civita connection of the Lorentzian metric 𝑔. Then using Koszul's formula, we obtain 

∇𝐸1
𝐸1 = 0, ∇𝐸1

𝐸2 = 0, ∇𝐸1
𝐸3 = 0, 

∇𝐸2
𝐸1 = −𝐸2, ∇𝐸2

𝐸2 = −𝐸1, ∇𝐸2
𝐸3 = 0, 

∇𝐸3
𝐸1 = −𝐸3, ∇𝐸3

𝐸2 = 0, ∇𝐸3
𝐸3 = −𝐸1. 

From the above calculation, one can easily verify that 

 ∇𝑋1
𝜉 = −𝜙2𝑋1, 

and  

 (∇𝑋1
𝜙)𝑋2 = −𝑔(𝜙X1, X2) − 𝜂(𝑋2)𝜙𝑋1. 

Therefore, the manifold (𝑀, 𝑔, 𝜉, 𝜙, 𝜂) is a Lorentzian para-Kenmotsu manifold. 

 On this manifold (𝑀, 𝑔, 𝜉, 𝜙, 𝜂), we can easily verify our results. 

 



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Conclusion: The study suggests that in a Lorentzian para-Kenmotsu manifold, a symmetric parallel 

second-order covariant tensor is proportional to the metric tensor. Additionally, if ℒ𝒱𝑔 + 2𝑆 is 

parallel, where 𝑉 is a vector field, then (𝑔, 𝑉, λ) is a Ricci soliton. The study further states that a Ricci 

soliton in a 𝑊2-semi-symmetric Lorentzian para-Kenmotsu manifold is shrinking, whereas a Ricci 

soliton satisfying the condition 𝑊2(𝜉, 𝑋1) ∘ 𝑆 = 0 in a Lorentzian para-Kenmotsu manifold is steady. 

The study concludes by analyzing certain curvature properties of Lorentzian para-Kenmotsu manifolds 

admitting Ricci soliton and provides an example of a 3-dimensional Lorentzian para-Kenmotsu 

manifold. 

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