View of Almost η-Ricci Solitons on the Pseudosymmetric Lorentzian Para-Kenmotsu Manifolds

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https://doi.org/10.34198/ejms.12223.183206

Almost η−Ricci Solitons on the Pseudosymmetric Lorentzian Para-Kenmotsu Manifolds

Tu˘gba Mert1,* and Mehmet At¸ceken2

1 Department of Mathematics, Faculty of Science, University of Sivas Cumhuriyet, 58140, Sivas, Turkey

e-mail: tmert@cumhuriyet.edu.tr

2 Department of Mathematics, Faculty of Art and Science, University of Aksaray, 68100, Aksaray, Turkey

e-mail: mehmet.atceken382@gmail.com

Abstract

In this paper, we consider Lorentzian para-Kenmotsu manifold admitting almost η−Ricci solitons by virtue of some curvature tensors. Ricci pseudosymmetry concepts of Lorentzian para-Kenmotsu manifolds admittingη−Ricci soliton have introduced according to the choice of some curvature tensors such as Riemann, concircular, projective, M−projective, W1 and W2. After then, according to the choice of the curvature tensors, necessary conditions are given for Lorentzian para-Kenmotsu manifold admitting η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are given and classifications have made under the some conditions.

1 Introduction

Para-Kenmotsu and special para-Kenmotsu manifolds, also known as almost paracontact metric manifolds, were defined in 1989 by Sinha and Sai Prasad [1].

Received: February 4, 2023; Accepted: March 20, 2023; Published: March 29, 2023 2020 Mathematics Subject Classification: 53C15, 53C44, 53D10.

Keywords and phrases: Lorentzian manifold, Para-Kenmotsu manifold, pseudoparallel submanifold.

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Sinha and Sai Prasad obtained important characterizations of para-Kenmotsu manifolds. In the following years, para-Kenmotsu manifolds attracted a lot of attention and many authors revealed the important properties of these manifolds. In 2018, Lorentzian para-Kenmotssu manifolds, known as Lorentzian almost paracontact metric manifolds, were introduced [2]. Then, the concept of q−semisymmetry for Lorentzian para-Kenmotsu manifolds is studied [3]. M.

At¸ceken studied invariant submanifolds of Lorentzian para-Kenmotsu manifolds in 2022 and in this study he gave the necessary and sufficient conditions for the an invarıant submanifold of Lorentzian para-Kenmotsu manifold to be total geodesic [4].

The notion of Ricci flow was introduced by Hamilton in 1982.With the help of this concept, Hamilton found the canonical metric on a smooth manifold. Then Ricci flow has become a powerful tool for the study of Riemannian manifolds, especially for those manifolds with positive curvature. Perelman used Ricci flow and it surgery to prove Poincare conjecture in [5], [6]. The Ricci flow is an flow is an evolution equation for metrics on a Riemannian manifold defined as follows:

∂tg(t) =−2S(g(t)).

A Ricci soliton emerges as the limit of the solitons of the Ricci flow. A solution to the Ricci flow is called Ricci soliton if it moves only by a one parameter group of diffeomorphism and scaling.

During the last two decades, the geometry of Ricci solitons has been the focus of attention of many mathematicians. In particular, it has become more important after Perelman applied Ricci solitons to solve the long standing Poincare conjecture posed in 1904. In [7], Sharma studied the Ricci solitons in contact geometry. Thereafter Ricci solitons in contact metric manifolds have been studied by various authors such as Bagewadi et al. in [8–11], Bejan and Crasmareanu in [12], Blaga in [13], Chandra et al. in [14], Chen and Deshmukh in [15], Deshmukh et al. in [16], He and Zhu in [17], At¸ceken et al. in [18], Nagaraja and Premalatta in [19], Tripathi in [20] and many others.

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Motivated by all these studies, we consider Lorentzian para-Kenmotsu manifold admitting almost η−Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentzian para-Kenmotsu manifolds admitting η−Ricci soliton have introduced according to the choice of some curvature tensors such as Riemann, concircular, projective,M−projective,W1and W2.After then, according to the choice of the curvature tensors, necessary conditions are given for Lorentzian para-Kenmotsu manifold admitting η−Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and classifications have made under the some conditions.

2 Preliminary

Let ˆMn be ann−dimensional Lorentzian metric manifold. This means that it is endowed with a structure (φ, ξ, η, g), where φ is a (1,1)−type tensor field, ξ is a vector field,η is a 1−form on ˆMn and g is a Lorentzian metric tensor satisfying;





φ2ω11+η(ω1)ξ,

g(φω1, φω2) =g(ω1, ω2) +η(ω1)η(ω2),

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η(ξ) =−1, η(ω1) =g(ω1, ξ), (2) for all vector fields ω1, ω2 on ˆMn. Then ˆMn(φ, ξ, η, g) is said to be Lorentzian almost paracontact manifold.

A Lorentzian almost paracontact manifold ˆMn(φ, ξ, η, g) is called Lorentzian para-Kenmotsu manifold if

(∇ω1φ)ω2=−g(φω1, ω2)ξ−η(ω2)φω1, (3) for allω1, ω2∈Γ(TMˆ),where∇and Γ

TMˆ

denote the Levi-Civita connection and differentiable vector fields set on ˆMn,respectively.

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Lemma 1. Let Mˆn(φ, ξ, η, g) be the n−dimensional Lorentzian para-Kenmotsu manifold. The following relations are provided forMˆn(φ, ξ, η, g).

ω1ξ=−φ2ω1 =−ω1−η(ω1)ξ, (4) (∇ω1η)ω2=−g(ω1, ω2)−η(ω1)η(ω2), (5) R(ω1, ω2)ξ=η(ω21−η(ω12, (6) η(R(ω1, ω23) =g(η(ω12−η(ω21, ω3), (7) S(ω1, ξ) = (n−1)η(ω1), (8) where R and S are the Riemann curvature tensor and Ricci curvature tensor of Mˆn(φ, ξ, η, g), respectively.

Example 1. Let us consider the5−dimensional manifold Mˆ5={(x1, x2, x3, x4, z)|z >0},

where (x1, x2, x3, x4, z) denote the standard coordinates of R5. Then let e1, e2, e3, e4, e5 be vector fields on Mˆ5 given by

e1 =z ∂

∂x1, e2 =z ∂

∂x2, e3 = ∂

∂x3, e4 = ∂

∂x4, e5= ∂

∂z

which are linearly independent at each point of Mˆ5 and we define a Lorentzian metric tensorg onMˆ5 as

g(ei, ei) = 1,1≤i≤4 g(ei, ej) = 0,1≤i6=j ≤5

g(e5, e5) =−1.

Let η be the 1−form defined byη(ω1) =g(ω1, e5) for all ω1 ∈Γ

TMˆ

. Now, we define the tensor field (1,1)−type ϕsuch that

ϕe1 =−e2, ϕe3 =−e4, ϕe5 = 0.

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Then for ω1=xiei, ω2 =yjej ∈Γ

TMˆ

,1≤i, j≤5,we can easily see that ϕ2ω11+η(ω1)ξ, ξ =e5, η(ω1) =g(ω1, ξ)

and

g(ϕω1, ϕω2) =g(ω1, ω2) +η(ω1)η(ω2). By direct calculations, only non-vanishing components are

[ei, e5] =−ei,1≤i≤4.

From Kozsul’s formula, we can compute

eie5=−ei,1≤i≤4.

Thus forω1=xiei, ω2 =yjej ∈Γ

TMˆ

, we have 5˜ω1ξ =−ω1−η(ω1)ξ, and

ω

1ϕ

ω2=−g(ϕω1, ω2)ξ−η(ω2)ϕω1, that is, Mˆ5(ϕ, ξ, η, g) is a Lorentzian para-Kenmotsu manifold [4].

Precisely, a Ricci soliton on a Riemannian manifold (M, g) is defined as a triple (g, ξ, λ) onM satisfying

Lξg+ 2S+ 2λg= 0, (9)

where Lξ is the Lie derivative operator along the vector field ξ and λ is a real constant. We note that if ξ is a Killing vector field, then Ricci soliton reduces to an Einstein metric (g, λ). Futhermore, generalization is the notion ofη−Ricci soliton defined by J.T. Cho and M. Kimura as a quadruple (g, ξ, λ, µ) satisfying

Lξg+ 2S+ 2λg+ 2µη⊗η = 0, (10)

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where λand µ are real constants and η is the dual of ξ and S denotes the Ricci tensor of M. Furthermore if λ and µ are smooth functions on M, then it called almost η−Ricci soliton on M.

Suppose the quartet (g, ξ, λ, µ) is almostη−Ricci soliton on manifoldM.Then,

·If λ <0,thenM is shriking.

·If λ= 0,thenM is steady.

·If λ >0,thenM is expanding.

Let M be a Riemannian manifold, T is (0, k)−type tensor field and A is (0,2)−type tensor field. In this case, Tachibana tensor field Q(A, T) is defined as

Q(A, T) (X, ..., Xk1, ω2) =−T((ω1Aω2)X1, ..., Xk)− ...−T(X1, ..., Xk−1,(ω1Aω2)Xk),

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where,

1Aω23 =A(ω2, ω31−A(ω1, ω32, (12)

k≥1, X1, X2, ..., Xk, ω1, ω2∈Γ (T M).

3 Almost η−Ricci Solitons on Ricci Pseudosymmetric and Ricci Semisymmetric of Lorentzian para-Kenmotsu Manifolds

Now let (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold. Then we have

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(Lξg) (ω1, ω2) =Lξg(ω1, ω2)−g(Lξω1, ω2)−g(ω1, Lξω2)

=ξg(ω1, ω2)−g([ξ, ω1], ω2)−g(ω1,[ξ, ω2])

=g(∇ξω1, ω2) +g(ω1,∇ξω2)−g(∇ξω1, ω2) +g(∇ω1ξ, ω2)−g(∇ξω2, ω1) +g(ω1,∇ω2ξ),

for all ω1, ω2 ∈Γ (T M).If we use (4) in the last equation, then we have

(Lξg) (ω1, ω2) =−2g(ω1, ω2)−2η(ω1)η(ω2). (13) Thus, in a Lorentzian para-Kenmotsu manifold, from (10) and (13),we have

S(ω1, ω2) = (1−λ)g(ω1, ω2) + (1−µ)η(ω1)η(ω2). (14) Thus, we can easily give the following result.

Corollary 1. The n−dimensional Lorentz para-Kenmotsu manifold admitting almost η−Ricci soliton

n, g, ξ, λ, µ

is an η−Einstein manifold.

For ω2=ξ in (14),this implies that

S(ξ, ω1) = (µ−λ)η(ω1). (15) Taking into account of (8) and (15),we conclude that

µ−λ=n−1. (16)

Definition 1. LetMˆnbe ann−dimensional Lorentzian para-Kenmotsu manifold.

If R ·S and Q(g, S) are linearly dependent, then the Mˆn is said to be Ricci pseudosymmetric.

In this case, there exists a function h1 on ˆMn such that R·S =h1Q(g, S).

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In particular, ifh1= 0, the manifold ˆMn is said to be Ricci semisymmetric.

Let us now investigate the Ricci pseudosymmetric case of the n−dimensional Lorentzian para-Kenmotsu manifold.

Theorem 1. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn. If Mˆn is a Ricci pseudosymmetric, then Mˆn is either an η−Einstein manifold provided λ= 2−n and µ= 1 or h1 = 1.

Proof. Let us assume that Lorentzian para-Kenmotsu manifold Mˆn be Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold ˆMn. That means

(R(ω1, ω2)·S) (ω4, ω5) =h1Q(g, S) (ω4, ω51, ω2), for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write S(R(ω1, ω24, ω5) +S(ω4, R(ω1, ω25)

=h1{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(17) If we puttingω5=ξ in (17),we get

S(R(ω1, ω24, ξ) +S(ω4, R(ω1, ω2)ξ)

=h1{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω21−η(ω12)}.

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If we make use of (6) and (8) in (18),we have S(ω4, η(ω21−η(ω12) + (n−1)η(R(ω1, ω24)

=h1{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)}.

(19)

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If we use (7) in the (19), we get

(n−1)g(η(ω12−η(ω21, ω4) +S(η(ω21−η(ω12, ω4)

=h1{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)}.

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In same way, we use (14) in the (20), we can write [(n−1) + (λ−1)] [1−h1]× g(η(ω12−η(ω21, ω4) = 0.

(21)

It is clear from (21),

h1 = 1 or λ= 2−n.

This completes the proof.

We can give the results obtained from this theorem as follows.

Corollary 2. Let Mˆn be Lorentz para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn. If Mˆn is a Ricci semisymmetric, then Mˆn is anη−Einstein manifold providedλ= 2−nand µ= 1.

Corollary 3. Let Mˆn be Lorentz para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn. If Mˆn is a Ricci semisymmetric, then Mˆn is always shriking.

For an n−dimensional semi-Riemann manifold M, the concircular curvature tensor is defined as

C(ω1, ω23 =R(ω1, ω23− r

n(n−1)[g(ω2, ω31−g(ω1, ω32]. (22)

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For an n−dimensional Lorentzian para-Kenmotsu manifold, if we choose ω3 =ξ in (22),we can write

C(ω1, ω2)ξ =

1− r

n(n−1)

[η(ω21−η(ω12], (23) and similarly if we take the inner product of both sides of (22) byξ, we get

η(C(ω1, ω23) =

1− r

n(n−1)

g(η(ω12−η(ω21, ω3). (24) Definition 2. Let Mˆn be an n−dimensional Lorentz para-Kenmotsu manifold.

If C · S and Q(g, S) are linearly dependent, then the manifold is said to be concircular Ricci pseudosymmetric.

In this case, there exists a function h2 on ˆMn such that C·S =h2Q(g, S).

In particular, if h2 = 0, the manifold Mˆn is said to be concircular Ricci semisymmetric.

Thus we have the following theorem.

Theorem 2. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn. If Mˆn is a concircular Ricci pseudosymmetric, then we have

h2 = (n+λ−2) [n(n−1)−r]

n(n−1)2+ [n(n−1)−r] (λ−1).

Proof. Let us assume that Lorentzian para-Kenmotsu manifold ˆMnbe concircular Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold ˆMn.That means

(C(ω1, ω2)·S) (ω4, ω5) =h2Q(g, S) (ω4, ω51, ω2),

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for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write S(C(ω1, ω24, ω5) +S(ω4, C(ω1, ω25)

=h2{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(25)

If we chooseω5 =ξ in (25),we get

S(C(ω1, ω24, ξ) +S(ω4, C(ω1, ω2)ξ)

=h2{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω21−η(ω12)}.

(26)

By using of (8) and (23) in (26),we have

S(ω4, A[η(ω21−η(ω12]) + (n−1)η(C(ω1, ω24)

=h2{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)},

(27)

whereA= 1−n(n−1)r .Substituting (24) into (27), we have A(n−1)g(η(ω12−η(ω21, ω4) +AS(η(ω21−η(ω12, ω4)

=h2{(n−1)g(η(ω12−η(ω21, ω4) +S(η(ω21−η(ω12, ω4)}.

(28)

If we use (14) in the (28), we can write

{A(n+λ−2)−h2[(n−1) +A(λ−1)]}g(η(ω12−η(ω21, ω4) = 0. (29)

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It is clear from (29),

h2 = (n+λ−2) [n(n−1)−r]

n(n−1)2+ [n(n−1)−r] (λ−1). This completes the proof.

We can give the results obtained from this theorem as follows.

Corollary 4. Let Mˆn be Lorentz para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆn is a concircular Ricci semisymmetric, then Mˆn is either manifold with scalar curvaturer=n(n−1)or η−Einstein manifold providedλ= 2−n andµ= 1.

For an n−dimensional semi-Riemann manifold M, the projective curvature tensor is defined as

P(ω1, ω23 =R(ω1, ω23− 1

n−1[S(ω2, ω31−S(ω1, ω32]. (30) For ann−dimensional Lorentzian para-Kenmotsu manifold, if we chooseω3= ξ in (30),we can write

P(ω1, ω2)ξ = 0, (31)

and similarly if we take the inner product of both sides of (30) byξ, we get η(P(ω1, ω23) = 0. (32) Definition 3. LetMˆnbe ann−dimensional Lorentzian para-Kenmotsu manifold.

If P · S and Q(g, S) are linearly dependent, then the manifold is said to be projective Ricci pseudosymmetric.

In this case, there exists a function h3 on ˆMn such that P·S =h3Q(g, S).

In particular, if h3 = 0, the manifold Mˆn is said to be projective Ricci semisymmetric.

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Let us now investigate the projective Ricci pseudosymmetric case of the Lorentzian para-Kenmotsu manifold.

Theorem 3. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almostη−Ricci soliton onMˆn.If Mˆn is a projective Ricci pseudosymmetric, then Mˆn is either projective Ricci semisymmetric or η−Einstein manifold such that λ= 2−n and µ= 1.

Proof. Let us assume that Lorentzian para-Kenmotsu manifold ˆMn be projective Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold ˆMn.Then we have

(P(ω1, ω2)·S) (ω4, ω5) =h3Q(g, S) (ω4, ω51, ω2), for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write S(P(ω1, ω24, ω5) +S(ω4, P(ω1, ω25)

=h3{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(33)

If we chooseω5 =ξ in (33),we get

S(P(ω1, ω24, ξ) +S(ω4, P(ω1, ω2)ξ)

=h3{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω21−η(ω12)}.

(34)

If we make use of (8) and (31) in (34),we have (n−1)η(P(ω1, ω24)

=h3{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)}.

(35)

(14)

If we use (32) in the (35), we get

h3{(n−1)g(η(ω12−η(ω21, ω4) +S(η(ω21−η(ω12, ω4)}= 0.

(36)

If we use (14) in the (36), we can write

h3(λ+n−2)g(η(ω12−η(ω21, ω4) = 0. (37) It is clear from (37),

h3 = 0 or λ= 2−n.

This completes the proof.

Corollary 5. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almostη−Ricci soliton onMˆn.If Mˆn is a projective Ricci pseudosymmetric, then Mˆn is always shriking providedh36= 0.

For an n−dimensional semi-Riemann manifold M, the M−projective curvature tensor is defined as

M(ω1, ω23 =R(ω1, ω232(n−1)1 [S(ω2, ω31−S(ω1, ω32 +g(ω2, ω3)Qω1−g(ω1, ω3)Qω2]

(38)

For ann−dimensional Lorentzian para-Kenmotsu manifold, if we chooseω3= ξ in (38),we obtain

M(ω1, ω2)ξ = 1

2[η(ω21−η(ω12]− 1

2 (n−1)[η(ω2)Qω1−η(ω1)Qω2], (39) and similarly if we take the inner product of both of sides of (38) by ξ, we get

η(M(ω1, ω23) =1

2g(η(ω12−η(ω21, ω3)− 1

2 (n−1)S(η(ω12

−η(ω21, ω3. (40)

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Definition 4. LetMˆnbe ann−dimensional Lorentzian para-Kenmotsu manifold.

If M ·S and Q(g, S) are linearly dependent, then it is said to beM−projective Ricci pseudosymmetric.

In this case, there exists a function h4 on ˆMn such that M ·S=h4Q(g, S).

In particular, if h4 = 0, the manifold ˆMn is said to be M−projective Ricci semisymmetric.

Let us now investigate the M−projective Ricci pseudosymmetric case of the Lorentzian para-Kenmotsu manifold.

Theorem 4. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆn is a M−projective Ricci pseudosymmetric providedn6= 1 and λ6= 2−n, then we have

h4 = λ+n−2 2 (n−1).

Proof. Let us assume that Lorentzian para-Kenmotsu manifold ˆMn be projective M−projective Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold ˆMn.That means

(M(ω1, ω2)·S) (ω4, ω5) =h4Q(g, S) (ω4, ω51, ω2), for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write

S(M(ω1, ω24, ω5) +S(ω4,M(ω1, ω25)

=h4{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(41)

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If we chooseω5 =ξ in (41),we get

S(M(ω1, ω24, ξ) +S(ω4,M(ω1, ω2)ξ)

=h4{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω21−η(ω12)}.

(42)

If we make use of (8) and (39) in (42),we have (n−1)η(M(ω1, ω24)

+S

ω4,12[η(ω21−η(ω12]−2(n−1)1 [η(ω2)Qω1−η(ω1)Qω2]

=h4{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)}.

(43)

By using (40) in the (43), we get

(n−1)

2 g(η(ω12−η(ω21, ω4)−S(η(ω12−η(ω21, ω4)

2(n−1)1 S(η(ω2)Qω1−η(ω1)Qω2, ω4)

=h4{(n−1)g(η(ω12−η(ω21, ω4) +S(η(ω21−η(ω12, ω4)}.

(44)

If we put (14) in (44), we can write

1

2(n−1)g(η(ω12−η(ω21, ω4) + (λ−1)g(η(ω12−η(ω21, ω4) +2(n−1)(λ−1) S(η(ω21−η(ω12, ω4)

=h4[λ+n−2]g(η(ω12−η(ω21, ω4)

(45)

(17)

Again, if we use (14) in the (45), we obtain n 1

2(n−1)(λ−1)2+ (λ−1) + 12(n−1)−h4[λ+n−2]o

×

g(η(ω12−η(ω21, ω4) = 0.

(46)

It is clear from (46),

h4 = λ+n−2 2 (n−1). This completes the proof.

We can give the following corollary.

Corollary 6. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn. If Mˆn is a M−projective Ricci semisymmetric, thenλ= 2−nthat is Mˆn is always shriking.

For an n−dimensional semi-Riemann manifold M, the W1−curvature tensor is defined as

W11, ω23=R(ω1, ω23+ 1

n−1[S(ω2, ω31−S(ω1, ω32]. (47) For an n−dimensional Lorenrzian para-Kenmotsu manifold ˆMn, if we choose ω3 =ξ in (47),we can write

W11, ω2)ξ = 2 [η(ω21−η(ω12], (48) and similarly if we take the inner product of both of sides of (47) by ξ, we get

η(W11, ω23) = 2g(η(ω12−η(ω21, ω3). (49) Definition 5. LetMˆnbe ann−dimensional Lorentzian para-Kenmotsu. IfW1·S and Q(g, S) are linearly dependent, then the manifold is said to be W1−Ricci pseudosymmetric.

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In this case, there exists a function h5 on ˆMn such that W1·S =h5Q(g, S).

In particular, if h5 = 0, the manifold Mˆn is said to be W1−Ricci semisymmetric.

Let us now investigate theW1−Ricci pseudosymmetric case of the Lorentzian para-Kenmotsu manifold.

Theorem 5. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton onMˆn.If MˆnisW1−Ricci pseudosymmetric, thenMˆnis either an η−Einstein manifold provided λ= 2−n and µ= 1 or h5 = 2.

Proof. Let us assume that Lorentzian para-Kenmotsu manifold ˆMnbeW1−Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold ˆMn. That means

(W11, ω2)·S) (ω4, ω5) =h5Q(g, S) (ω4, ω51, ω2), for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write S(W11, ω24, ω5) +S(ω4, W11, ω25)

=H5{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(50)

If we chooseω5 =ξ in (50),we get

S(W11, ω24, ξ) +S(ω4, W11, ω2)ξ)

=h5{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω12−η(ω21)}.

(51)

(19)

If we make use of (8) and (48) in (51),we have 2S(ω4, η(ω21−η(ω12) + (n−1)η(W11, ω24)

=h5{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω21−η(ω12)}.

(52)

If we use (49) in the (52), we get

2 (n−1)g(η(ω12−η(ω21, ω4) +2S(η(ω21−η(ω12, ω4)

=h5{(n−1)g(η(ω12−η(ω21, ω4) +S(η(ω21−η(ω12, ω4)}.

(53)

If we use (14) in the (53), we can write

[n+λ−2] [2−h5]g(η(ω12−η(ω21, ω4) = 0. (54) It is clear from (54),

h5 = 2 or λ= 2−n.

This completes the proof.

We can give the following corollaries.

Corollary 7. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆnis aW1−Ricci semisymmetric, thenMˆn is anη−Einstein manifold providedλ= 2−nand µ= 1.

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Corollary 8. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆnis aW1−Ricci semisymmetric, thenMˆn is always shriking.

For an n−dimensional semi-Riemann manifold M, the W2−curvature tensor is defined as

W21, ω23 =R(ω1, ω23− 1

n[g(ω2, ω3)Qω1−g(ω1, ω3)Qω2]. (55) For ann−dimensional Lorentzian para-Kenmotsu manifold ˆMn, if we chooseω3= ξ in (55),we can write

W21, ω2)ξ = [η(ω21−η(ω12]

(n−1)1 [η(ω1)Qω2−η(ω2)Qω1],

(56)

and similarly if we take the inner product of both sides of (56) byξ, we get η(W21, ω23) =g(η(ω12−η(ω21, ω3)

+(n−1)1 S(η(ω12−η(ω21, ω3).

(57)

Definition 6. LetMˆnbe ann−dimensional Lorentzian para-Kenmotsu manifold.

If W2 ·S and Q(g, S) are linearly dependent, then the manifold is said to be W2−Ricci pseudosymmetric.

In this case, there exists a function h6 on ˆMn such that W2·S =h6Q(g, S).

In particular, if h6 = 0, the manifold Mˆn is said to be W2−Ricci semisymmetric.

Let us now investigate theW2−Ricci pseudosymmetric case of the Lorentzian para-Kenmotsu manifold.

(21)

Theorem 6. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆn is a W2−Ricci pseudosymmetric, then

h6 = n+λ−2 n−1 , providedn6= 1.

Proof. Let us assume that Lorentzian para-Kenmotsu manifold be W2−Ricci pseudosymmetric and (g, ξ, λ, µ) be almost η−Ricci soliton on Lorentzian para-Kenmotsu manifold. That means

(W21, ω2)·S) (ω4, ω5) =h6Q(g, S) (ω4, ω51, ω2), for all ω1, ω2, ω4, ω5 ∈Γ

TMˆn

.From the last equation, we can easily write S(W21, ω24, ω5) +S(ω4, W21, ω25)

=h6{S((ω1gω24, ω5) +S(ω4,(ω1gω25)}.

(58) If we chooseω5 =ξ in (58),we get

S(W21, ω24, ξ) +S(ω4, W21, ω2)ξ)

=h6{S(g(ω2, ω41−g(ω1, ω42, ξ) +S(ω4, η(ω21−η(ω12)}.

(59)

If we make use of (8) and (56) in (59),we have (n−1)η(W21, ω24) +S(ω4,[η(ω21−η(ω12]

(n−1)1 [η(ω1)Qω2−η(ω2)Qω1]

=h6{(n−1)g(η(ω12−η(ω21, ω4) +S(ω4, η(ω12−η(ω21)}.

(60)

(22)

If we use (57) in the (60), we get

(n−1)g(η(ω12−η(ω21, ω4)

(n−1)1 S(ω4, η(ω1)Qω2−η(ω2)Qω1)

=H6{S(ω4, η(ω21−η(ω12)

+2n(f1−f3)g(η(ω12−η(ω21, ω4)}.

(61)

If we use (14) in the (61), we have

(n−1)g(η(ω12−η(ω21, ω4)

1−λn−1S(η(ω12−η(ω21, ω4)

=h6[n−λ]g(η(ω12−η(ω21, ω4)

(62)

Again, if we use (14) in (62),we obtain n

(n−1)−(1−λ)n−12 −h6(n−λ)o

×

g(η(ω12−η(ω21, ω4) = 0.

(63)

It is clear from (63),

h6 = n+λ−2 n−1 . This completes the proof.

We can give the results obtained from this theorem as follows.

Corollary 9. Let Mˆn be Lorentzian para-Kenmotsu manifold and (g, ξ, λ, µ) be almost η−Ricci soliton on Mˆn.If Mˆnis aW2−Ricci semisymmetric, thenMˆn is anη−Einstein manifold providedλ= 2−nand µ= 1 or λ=n, µ= 2n−1 and it is always shriking.

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