# A Note onLP-Sasakian Manifolds with Almost Quasi-Yamabe Solitons

Authors: Sunil Kumar Yadav; D. L. Suthar; Biniyam Shimelis
Journal: Journal of Mathematics
 (2021)

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## Abstract

We categorize almost quasi-Yamabe solitons onLP-Sasakian manifolds and their CR-submanifolds whose potential vector field is torse-forming, admitting a generalized symmetric metric connection of type α,β. Finally, a nontrivial example is provided to confirm some of our results.


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## Body

## 1. Introduction

Yamabe solitons (YS) are ideas that generate self-similar Yamabe flow (YF) solutions [1]:(1)∂∂tˇg¨tˇ=−δ^tˇg¨tˇ.Di Cerbo and Disconzi were the first to notice them in [2]. Chen and Deshmukh proposed the concept of quasi-Yamabe soliton (QYS) in [3], which we will explore in this study for a more broader situation when the constants are functions.Definition 1.
An almost quasi-Yamabe solitons (AQYS) on Remannian manifoldΘn^,g¨ is a set of data g¨,K,μ,σ^ that fulfill the following equation:(2)12LKg¨=δ^−σ^g¨+μK⋄⊗K⋄,where LK is operator of the Lie derivative in view of K, σ^ and μ are smooth functions on Θn^,g¨, K⋄ is the 1-form of K, and δ^ is the scalar curvature. If σ^<0, σ^=0, or σ^>0, we call an AQYS shrinking, stable, or growing, respectively, using Yamabe solitons nomenclature.
IfK=∇f˜ is a gradient type, (2) yields(3)Hessf^=δ^−σ^g¨+μdf^⊗df^.
It is nothing more than a generalized quasi-Yamabe gradient soliton (GQYGS) (see [4, 5]). Several authors have thoroughly investigated AQYS and QYGS in [6–13].Definition 2.
A vector fieldK on Θn^,g¨ meets the following conditions and is known as a torse-forming vector field [14]:(4)∇P∗K=ψP∗+πP∗K,∀P∗∈χΘ,where π is a 1-form and ψ is in C∞Θ.
We classify such vector field as(i)
It is concircular if the 1-formπ vanishes identically [15](ii)
For concurrent,ψ=1 and π=0 [16](iii)
It is recurrent ifψ=0(iv)
Parallel ifψ=π=0
If the vector fieldK fulfills (4) with πK=0, it called as torqued vector field [17].
The content of the paper is as follows. After the opening remark, Section2 contains the fundamental result of an LP-Sasakian manifold. We discuss the concept of a generalized symmetric metric connection-α,β (GSMC-α,β) in Section 3. With regard to GSMC-α,β, Section 4 is devoted to CR-submanifolds of an LP-Sasakian manifold. With respect to such a connection, we examine QYS in view of a torse-forming vector field on an LP-Sasakian manifold in Section 5. The study of QYS with a torse-forming vector field on CR-submanifolds of an LP-Sasakian manifolds is also covered in Section 6. Finally, in Section 7, we look at AQYGS with a torse-forming vector field by considering the tangential Kt and normal Kn components of such a vector field on CR-submanifolds.


## 2.LP-Sasakian Manifolds

If a1,1 tensor field ϕ∗, a contravariant vector field ζ^, a 1-form η∗, and the Lorentzian metric g¨ are admitted to a differentiable manifold Θn^; it is termed as an LP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨ (see [18, 19]); then,(5)η∗ζ^=−1,ϕ∗2P∗=P∗+η∗P∗ζ^,g¨ϕ∗P∗,ϕ∗Q∗=g¨P∗,Q∗+η∗P∗η∗Q∗,g¨P∗,ζ^=η∗P∗,∇P∗ζ^=ϕ∗P∗,∇P∗ϕ∗Q∗=g¨P∗,Q∗ζ^+η∗Q∗P∗+2η∗P∗η∗Q∗,where ∇ be the Levi-Civita connection along the metric g¨. In an LP-Sasakian manifold, we yields(6)ϕ∗ζ^=0,η∗ϕ∗P∗=0,rankϕ∗=n^−1.If we write(7)Φ^P∗,Q∗=g¨ϕ∗P∗,Q∗,P∗,Q∗∈Θn^,then Φ^P∗,Q∗ is a symmetric 0,2 tensor field. Now, η∗ is closed on Θn^ϕ∗,ζ^,η∗,g¨ (see [18, 20]); then,(8)∇P∗ηQ∗=Φ^P∗,Q∗,Φ^P∗,ζ^=0.In anLP-Sasakian manifold Θn^,g¨, the following relationships are maintained (see [20, 21]):(9)g¨ℛP∗,Q∗Z∗,ζ^=g¨Q∗,Z∗η∗P∗−g¨P∗,Z∗η∗Q,g¨ℛζ^,P∗Q∗=g¨P∗,Q∗ζ^−η∗Q∗P∗,ℛP∗,Q∗ζ^=η∗Q∗P∗−η∗P∗Q∗,ℛζ^,P∗ζ^=P∗+η∗P∗ζ^,SP∗,ζ^=n^−1η∗P∗,Sϕ∗P∗,ϕ∗Q∗=SP∗,Q∗+n^−1η∗P∗η∗Q∗,for any vector fields P∗, Q∗, and Z∗ on Θn^, where ℛ and S are the curvature tensor and Ricci tensor of Θn^, simultaneously.LetΘ′ be a submanifold of an LP-Sasakian manifold. The Gauss and Weingarten formulas are given by(10)∇P∗Q∗=∇`P∗Q∗+hP∗,Q∗,∀P∗,Q∗∈ΓT∗Θ`,(11)∇P∗N∗=−AN∗P∗+∇P∗⊥N∗,∀N∗∈ΓT∗⊥Θ`,where ∇P∗Q∗ and hP∗,Q∗,∇P∗⊥N∗ belong to ΓTΘ′ and ΓT∗⊥Θ′, respectively.


## 3. Generalized Symmetric Metric Connection of Typeα,β

Let∇˜ and ∇ be a linear and Levi-Civita connection on an LP-Sasakian manifold. Now, we will go through the results that will be used.Lemma 1 (see [22]).
In anLP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, the GSMC ∇˜ of type α,β is given by(12)∇˜P∗Q∗=∇P∗Q∗∗+αη∗Q∗P∗−g¨P∗P∗,Q∗ζ^+βη∗Q∗ϕ∗P∗−g¨ϕ∗P∗,Q∗ζ^,for all P∗ and Q∗ on Θn^.Lemma 2 (see [22]).
The following relations hold on anLP-Sasakian manifold in light of GSMC-α,β:(13)∇˜P∗ϕ∗Q∗=1−βg¨P∗,Q∗+2η∗P∗η∗Q∗−αΦ^P∗,Q∗ζ^+1−βη∗Q∗P∗−αη∗Q∗ϕ∗P∗,∇˜P∗ζ^=1−βϕ∗P∗−αP∗−αη∗P∗ζ^,∇˜P∗η∗Q∗=1−βΦ^P∗,Q∗−αg¨ϕ∗P∗,ϕ∗Q∗,ℛ˜P∗,Q∗ζ^=1−β+β2η∗Q∗P∗−η∗P∗Q∗+α1−βη∗P∗ϕ∗Q∗−η∗Q∗ϕ∗P,ℛ˜ζ^,Q∗ζ^=1−β+β2η∗Q∗ζ^+Q∗∗+αβ−1ϕ∗Q∗,S˜P∗,Q∗=SP∗,Q∗+−αβ+n^−2αβ−α+β2−2βΦ^Φ^P∗,Q∗+−2α2+β−β2+n^α2+αβ−αΦ^g¨P∗,Q∗+−2α2+n^α2+β−β2η∗P∗η∗Q∗,for any P∗, Q∗∈TΘ.


## 4.CR-Submanifolds of an LP-Sasakian Manifold with GSMC-α,β

Here, we have recall the well-known definition in the following manner.Definition 3 (see [23]).
A Riemannian manifoldΘn^,g¨ of an LP-Sasakian manifold Θn^ϕ∗,ζ^,η,g¨ is called a CR-submanifold Θ′,g¨ if ζ^ is tangent to Θ and there exists on Θ a differentiable distribution D∗:x⟶Dx∗⊂Tx∗Θ such that(i)
D∗ is invariant under ϕ∗, i.e., ϕ∗D∗⊂D∗(ii)
The orthogonal complement distributionD∗⊥:x⟶Dx∗⊥⊂Tx∗Θ of the distribution D∗ on Θ is totally real, i.e., ϕ∗D∗⊥⊂T∗⊥ΘDefinition 4 (see [23]).
If the distributionD∗resp.,D∗⊥ is horizontal (resp., vertical), then the pair D∗,D∗⊥ is called ζ^-horizontal (resp., ζ^-vertical) if ζ^∈ΓD∗ (resp., ζ^∈ΓD∗⊥). The CR-submanifold is also called ζ^-horizontal (resp., ζ^-vertical) if ζ^∈ΓD∗resp.,ζ^∈ΓD∗⊥.
The orthogonal complementϕ∗D∗⊥∈T∗⊥Θ is given by(14)T∗Θ=D∗⊕D∗⊥,T∗⊥Θ=ϕ∗D∗⊥⊕ϱ,where ϕ∗ϱ=ϱ.
LetΘ′ be a CR-submanifold of an LP-Sasakian manifold with a GSMC-α,β. For any P∗∈ΓT∗Θ and N∗∈ΓT∗⊥Θ, we can write(15)P∗=UP∗+VP∗,UP∗∈ΓD∗,VP∗∈ΓD∗⊥,(16)ϕ∗N∗=BN∗+CN∗,BN∗∈ΓD∗⊥,CN∗∈Γϱ.
The Gauss and Weingarten formulas with respect to∇˜ are, respectively, given by(17)∇˜P∗Q∗=∇˜′P∗Q∗+h˜P∗,Q∗,(18)∇˜P∗N∗=−A˜N∗P∗+∇˜P∗⊥N∗,for any P∗,Q∗∈ΓT∗Θ, where ∇˜P∗Q∗, A˜N∗P∗∈ΓT∗Θ. Here, ∇˜′, h˜, and A˜N∗ are called the induced connection on Θ′, the second fundamental form, and the Weingarten mapping with respect to ∇˜, respectively. In view of (10), (12), and (17), we obtain(19)∇′P∗Q∗+h˜P∗,Q∗=∇′P∗Q∗+hP∗,Q∗+αη∗Q∗P∗−g¨P∗,Q∗ζ^+βη∗Q∗ϕ∗P∗−g¨ϕ∗P∗,Q∗ζ^.
Using (15) and (16) in (19), we obtain(20)U∇′P∗Q∗=U∇′P∗Q∗+αη∗Q∗UP∗−αg¨P∗,Q∗Uζ^+βη∗Q∗ϕ∗UP∗−βg¨ϕ∗P∗,Q∗Uζ^,(21)h˜P∗,Q∗=hP∗,Q∗+βη∗Q∗ϕ∗Q∗P∗,(22)V∇′P∗Q∗=V∇′P∗Q∗+αη∗Q∗VP∗−αg¨P∗,Q∗Vζ^−βg¨ϕ∗P∗,Q∗Vζ^,for any P∗,Q∗∈TΘ.


## 5. Quasi-Yamabe Solitons (QYS) with Torse-Forming Vector Field

We classify QYS with torse-forming vector fields on anLP-Sasakian manifold admitting a GSMC-α,β in this section. As a result, we can prove the theorem below.Theorem 1.
AnLP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, n>1, with respect to GSMC-α,β admitting QYS. If K is a torse-forming vector field, then the data g¨,K,σ^,μ is growing, steadying, and contracting in accordance with ψ−δ^˜−1/n^πK+αn^−1η∗K−μτ=><0, unless ψ−δ^˜−1/n^πK+αn^−1η∗K−μτ is constant.Proof.
Let the datag¨,K,σ^,μ be a QYS on Θn^ϕ∗,ζ^,η∗,g¨ in terms of GSMC-α,β. From (3), we have(23)12L˜Kg¨P∗,Q∗=δ^˜−σ^g¨P∗,Q∗+μg¨K⋄,P∗g¨K⋄,Q∗.
From (3) and (12), we obtain(24)L˜Kg¨P∗,Q∗=g¨∇˜P∗K,Q∗+g¨P∗,∇˜Q∗K=2ψg¨P∗,Y+πP∗g¨K,Q∗+πQ∗g¨K,P∗+α2η∗Kg¨P∗,Q∗−g¨P∗,Kη∗Q∗−g¨Q∗,Kη∗P∗+β2η∗Kg¨ϕ∗P∗,Q∗−g¨ϕP∗,Kη∗Q∗−g¨ϕ∗Q∗,Kη∗P∗,for all P∗,Q∗∈χΘ.
With the help of (23) and (24), we obtain(25)ψ−δ^˜−σ^g¨P∗,Q∗=12πP∗g¨K,Q∗+πQ∗g¨K,P∗+αη∗Kg¨P∗,Q∗+βη∗Kg¨ϕ∗P∗,Q∗−α2g¨P∗,Kη∗Q∗+g¨Q∗,Kη∗P∗−β2g¨ϕ∗P∗,Kη∗Q∗+g¨ϕ∗Q∗,Kη∗P∗−μg¨K⋄,P∗g¨K⋄,Q∗.
On contracting (25), we find(26)σ^=ψ−δ^˜−1n^πK+αn^−1η∗K−μτ,where τ =  gK⋄,K⋄=K⋄2.
As a result, Theorem1 is proven.
In this sequel, the corollaries are as follows.Corollary 1.
If (2) defines a QYS on an LP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, n^>1, admitting a GSMC-α,β, then there are the existing relationships representing in Table 1.In Table1, Ω1=ψ−δ^˜−1/n^πK+αn^−1η∗K−μτ, Ω2=ψ−δ^˜−1/n^αn^−1η∗K−μτ, Ω3=1−δ^˜−1/n^αn^−1η∗K−μτ, Ω4=δ^˜−1/n^πK+αn^−1η∗K−μτ, Ω5=δ^˜−1/nαn^−1η∗K−μτ,  and Ω6=ψ−δ^˜−1/n^αn^−1η∗K−μτ.Table 1
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingΩ1=constantΩ1=><0ConcircularΩ2=constantΩ2=><0ConcurrentΩ3=constantΩ3=><0RecurrentΩ4=constantΩ4=><0ParallelΩ5=constantΩ5=><0TorquedΩ6=constantΩ6=><0Corollary 2.
Let datag¨,K,σ^,μ be a QYS on an LP-Sasakian manifold Mn^ϕ∗,ζ^,η∗,g¨, n^>1, with respect to a GSMC-α,β=1,0. If K is torse-forming vector field, then g¨,K,σ^,μ is growing, steadying, and contracting according to ψ−δ^˜−1/n^πK+n^−1η∗K−μτ=><0, unless ψ−δ^˜−1/n^πK+n^−1η∗K−μτ is constant.Corollary 3.
LetΘn^ϕ∗,ζ^,η∗,g¨, n^>1, be an LP-Sasakian manifold endowed with a GSMC-α,β=0,1. If a data g¨,K,σ^,μ be a QYS on Θn^ and K is a torse-forming vector field, then g¨,K,σ^,μ is growing, steadying, and contracting according to ψ−δ^˜−1/n^πK−μτ=><0, unless ψ−δ^˜−1/n^πK−μτ is constant.Corollary 4.
If (2) defines a QYS on an LP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, n^>1, with respect to a GSMC-α,β=0,1, then we obtain the relationship in Table 2.Table 2
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingψ−δ^˜−1/n^πK−μτ=constantψ−δ^˜−1/n^πK−μτ=><0¯Concircularψ−δ^˜−μτ=constantψ−δ^˜−μτ=><0Concurrent1−δ^˜−μτ=constantδ^˜−μτ=><1Recurrentδ^˜−1/n^πK−μτ=constantδ^˜−1/n^πK−μτ=><0Parallelδ^˜−μτ=constantδ^˜−μτ=><0Torquedψ−δ^˜−μτ=constantψ−δ^˜−μτ=><0Corollary 5.
Let a datag¨,K,σ^,μ be a QYS on an LP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, n^>1, with respect to a GSMC-α,β=1,0. Then, the following relationships are maintained in Table 3.In Table3, Ω7=ψ−δ^˜−1/n^πK+n^−1η∗K−μτ, Ω8=ψ−δ^˜−1/n^n^−1η∗K−μτ, Ω9=1−δ^˜−1/n^n^−1η∗K−μτ, Ω10=δ^˜−1/n^πK+n^−1η∗K−μτ, Ω11=δ^˜−1/n^n^−1η∗K−μτ, and Ω12=ψ−δ^˜−1/n^n^−1η∗K−μτ.Table 3
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingΩ7=constantΩ7=><0ConcircularΩ8=constantΩ8=><0ConcurrentΩ9=constantΩ9=><0RecurrentΩ10=constantΩ10=><0ParallelΩ11=constantΩ11=><0TorquedΩ12=constantΩ12=><0


## 6. Quasi-Yamabe Solitons (QYS) with Potential Vector Field is Torse-Forming onCR-Submanifold

We investigate QYS in relating to a torse-forming vector field onCR-submanifolds of an LP-Sasakian manifold with regard to the induced connection ∇˜′ of type α,β in this section. The following is our theorem.Theorem 2.
Let aCR-submanifold Θ′ of an LP-Sasakian manifold be Θn^ϕ∗,ζ^,η∗,g¨, n^>1, admitting a GSMC ∇′ is ζ^-horizontal (resp. ζ^-vertical) and D∗ is parallel with respect to ∇′. If data g¨,K,σ^,μ is a QYS on Θ′ and K is a torse-forming vector field, then g¨,K,σ^,μ is growing, steadying, or contracting according to ψ−δ^˜−1/n^πK+αn^−1η∗K−μτ=><0, unless ψ−δ^˜−1/n^πK+αn^−1η∗K−μτ is constant.Proof.
IfΘ is ζ^-horizontal for all P∗,Q∗∈ΓD∗ and D∗ is parallel in relation to ∇˜′, then there is, from (20),(27)∇′P∗Q∗=∇′P∗Q∗+αηQ∗P∗−gP∗,Q∗ζ^+βη∗Q∗ϕ∗P∗−gϕ∗P∗,Q∗ζ^.
With the help of Lemma1, the induced connection ∇′ is also a GSMC-α,β. This leads to the statement of Theorem 2.
In this sequel, we write the following corollaries.Corollary 6.
Let aCR-submanifold Θ′ of an LP-Sasakian manifold be Θn^ϕ∗,ζ^,η∗,g¨, n^>1, admitting a GSMC ∇˜′ is ζ^-horizontal (resp. ζ^-vertical) and D∗ is parallel in terms of ∇˜′. If (2) defines a QYS on Θ′ and K is a torse-forming vector field, then the results hold in Table 4.Table 4
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingΩ1=constantΩ1=><0ConcircularΩ2=constantΩ2=><0ConcurrentΩ3=constantΩ3=><0RecurrentΩ4=constantΩ4=><0ParallelΩ5=constantΩ5=><0TorquedΩ6=constantΩ6=><0Corollary 7.
Let aCR-submanifold Θ′ of an LP-Sasakian manifold be Θn^ϕ∗,ζ^,η∗,g¨, n^>1, admitting a GSMC ∇˜′ is ζ^-horizontal (resp. ζ^-vertical) and D∗ is parallel in term of ∇˜′ of type α,β=0,1. If data g¨,K,σ^,μ is a QYS on Θ′ and K is a torse-forming vector field, then, in Table 5, relationships must be true.Table 5
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingΩ7=constantΩ7=><0ConcircularΩ8=constantΩ8=><0ConcurrentΩ9=constantΩ9=><0RecurrentΩ10=constantΩ10=><0ParallelΩ11=constantΩ11=><0TorquedΩ12=constantΩ12=><0Corollary 8.
Let aCR-submanifold Θ′ of an LP-Sasakian manifold be Θn^ϕ∗,ζ^,η∗,g¨, n^>1, admitting a GSMC ∇˜′ of type α,β=1,0 is ζ^-horizontal (resp. ζ^-vertical) and D∗ is parallel with respect to ∇˜′. If g¨,K,σ^,μ is a QYS on Θ′ and K is a torse-forming vector field, then g¨,K,σ^,μ is growing, steadying, or contracting according to ψ−δ^˜−1/n^πK+n^−1η∗K−μτ=><0, unless ψ−δ^˜−1/n^πK+n^−1η∗K−μτ is constant.Corollary 9.
Let aCR-submanifold Θ′ of an LP-Sasakian manifold be Θn^ϕ∗,ζ^,η∗,g¨, n^>1, in relation to a GSMC-α,β=0,1 which is ζ^-horizontal (resp. ζ^-vertical) and D∗ which is parallel in view og ∇˜′. If g¨,K,σ^,μ is a QYS on Θ′ and K is a torse-forming vector field, then g¨,K,σ^,μ is growing, steadying, or shrinking according as ψ−δ^˜−1/n^πK−μτ=><0, unless ψ−δ^˜−1/n^πK−μτ is constant.Corollary 10.
If (2) defines a QYS on an LP-Sasakian manifold Θn^ϕ∗,ζ^,η∗,g¨, n^>1, concerning a GSMC-α,β=0,1, then, in Table 6, relationships are true.Table 6
Relationship betweenK, existence condition, and nature of solitons.
KExistence conditionNature of solitonsTorse-formingψ−δ^˜−1/n^πK−μτ=constantψ−δ^˜−1/n^πK−μτ=><0Concircularψ−δ^˜−μτ=constantψ−δ^˜−μτ=><0Concurrent1−δ^˜−μτ=constantδ^˜−μτ=><1Recurrentδ^˜−1/n^πK−μτ=constantδ^˜−1/n^πK−μτ=><0Parallelδ^˜−μτ=constantδ^˜−μτ=><0Torquedψ−δ^˜−μτ=constantψ−δ^˜−μτ=><0


## 7. Almost Quasi-Yamabe Solitons (AQYS) whose Potential Vector Field is Torse-Forming onCR-Submanifold

In this section, we classify AQYS whose potential field is torse-forming onCR-submanifold of an LP-Sasakian manifold with respect to a GSMC-α,β. At this stage, we denote Kt and Kn as tangential and normal components of such vector field. To begin, we will prove the outcome.Theorem 3.
An almost quasi-Yamabe solitong¨,Kt,σ^,μ on CR-submanifold of an LP-Sasakian manifold in relation to a GSMC-α,β satisfies(28)δ^−σ^−ψ+η∗Kng¨P∗,Q∗+μg¨K⋄,P∗g¨K⋄,Q∗=g¨AKnP∗,Q∗+12πP∗g¨K,Q∗+πQ∗g¨P∗,K+β2g¨Kn,ϕ∗P∗η∗Q∗+g¨ϕ∗Q∗,Knη∗P∗,any type of vector field P∗,Q on Θ′.Proof.
In light of (3), (12), (17), and (18), we have(29)ψP∗+πP∗K=∇˜P∗K=∇˜P∗Kt+Kn=∇′P∗Kt+hP∗,Kt+βη∗Ktϕ∗Q∗P∗−AKnP∗+∇P∗⊥Kn+αη∗KnP∗+βη∗Knϕ∗P∗−βg¨ϕ∗P∗,Knζ^.
We obtain the following equation when we compare the tangential and normal components of (29):(30)∇′P∗Kt=ψP∗+πP∗K+AKnP∗−αηKnP∗−βη∗Knϕ∗P∗+βg¨ϕ∗P∗,Knζ^,(31)hP∗,Kt=−∇P∗⊥Kn−βη∗Knϕ∗Q∗P∗.
We may deduce from the concept of Lie derivative and (31) that(32)LKtg¨P∗,Q∗=2ψg¨P∗,Q∗+2g¨AKnP∗,Q∗−2η∗Kng¨P∗,Q∗+πP∗g¨K,Q∗+πQ∗g¨P∗,K+βg¨Kn,ϕ∗P∗η∗Q∗+g¨ϕQ∗,Knη∗P∗.
Using (32) in (2), we yield(33)δ^−σ^−ψ+η∗Kng¨P∗,Q∗+μg^K⋄,P∗g¨K⋄,Q∗=g¨AKnP∗,Q∗+12πP∗g¨K,Q∗+πQ∗g¨P∗,K+β2g¨Kn,ϕ∗P∗η∗Q∗+g¨ϕ∗Q∗,Knη∗P∗.
This completed our assertion.
As a result, the following corollaries are stated.Corollary 11.
If (2) defines AQYS on CR-submanifold of an LP-Sasakian manifold in relation to a GSMC-α,β which is minimal, consequently, the following relationship holds:(34)δ^−σ^−ψ+η∗Knn^+μτ=πK.Corollary 12.
If (2) defines AQYS on CR-submanifold of an LP-Sasakian manifold and the distribution is ζ^-horizontal (resp. ζ^-vertical), P∗,Q∗∈ΓD∗, where D∗ is parallel with induced connection ∇′ of type α,β, then we have(35)δ^−σ^−ψ+η∗Kng¨P∗,Q∗+μg¨K⋄,P∗g¨K⋄,Q∗=g¨AKnP∗,Q∗+12πP∗g¨K,Q∗+πQ∗g¨P∗,K+β2g¨Kn,ϕ∗P∗η∗Q∗+g¨ϕ∗Q∗,Knη∗P∗,in all vector fields P∗,Q∗ on Θ′.Corollary 13.
If datag¨,Kt,σ^,μ is a AQYS on CR-submanifold of an LP-Sasakian manifold and the distribution is ζ^-horizontal (resp. ζ^-vertical), P∗,Q∗∈ΓD∗, D∗ is parallel with induced connection ∇′ of type α,β is minimal, then the relation holds:(36)δ^−σ^−ψ+η∗Knn^+μτ=πK.


## 8. Example

A 4-dimensional differentiable manifold is taken into consideration, that is,Θ4=p,q,z,t∈ℜ4:p,q,z,t≠0, where p,q,z,t is the standard coordinate in ℜ4. At each point along Θ4, e⌣1,e⌣2,e⌣3,e⌣4 is a set of linearly independent vector fields and is described as(37)e⌣1=ep−t∂∂p,e⌣2=eq−t∂∂q,e⌣3=ez−t∂∂z,e⌣4=∂∂t.Also, the Lie bracket’s nonvanishing components are as follows:(38)e⌣1,e⌣4=e⌣1,e⌣2,e⌣4=e⌣2,e⌣3,e⌣4=e⌣3.Letg¨ on Θ be the Lorentzian metric as(39)g¨ij=g¨e⌣i,e⌣j=0,ifi≠j,−1,ifi=j=4,1,otherwise.Letη∗ be the 1-form corresponding to the Lorentzian metric g¨:(40)η∗P∗=g¨P∗,e⌣4,for any P∗∈ΓΘ. If ϕ∗ is defined as the 1,1-tensor field,(41)ϕ∗e⌣1=e⌣1,ϕ∗e⌣2=e⌣2,ϕ∗e⌣3=e⌣3,ϕ∗e⌣4=0.We can readily prove this using the linearity characteristics ofϕ∗ and g¨:(42)η∗e⌣4=−1,ϕ∗2P∗=P∗+η∗P∗e⌣4,g¨ϕ∗P∗,ϕ∗Q∗=g¨P∗,Q∗+η∗P∗η∗Q∗,for any P∗,Q∗∈ΓTΘ. Thus, for e⌣4=ζ^, the frame ϕ∗,ζ^,η∗,g¨ leads to an LP-contact skeleton, which is known as the LP-contact manifold of dimension 4. Now, for e⌣4=ζ^, Koszul’s formula gives the nonvanishing component:(43)∇e⌣1e⌣1=e⌣4,∇e⌣1e⌣4=e⌣1,∇e⌣2e⌣2=e⌣4,∇e⌣2e⌣4=e⌣2,∇e⌣3e⌣3=e⌣4,∇e⌣3e⌣4=e3.Using the above equation, it can be easily verified that∇P∗e⌣4=ϕ∗P∗ holds for each P∗∈χΘ. Thus, an LP-contact manifold is a 4-dimensional LP-Sasakian manifold. From (12), we calculate as follows:(44)∇˜e⌣1e⌣1=1−α−βe⌣4,∇˜e⌣1e⌣4=1−α−βe⌣1,∇˜e⌣2e⌣2=1−α−βe⌣4,∇˜e⌣2e⌣4=1−α−βe⌣2,∇˜e⌣3e⌣3=1−α−βe⌣4,∇˜e⌣3e⌣4=1−α−βe⌣3.It is clear from (12) that ∇˜P∗e⌣4=1−βϕ∗P∗−αP∗−αη∗P∗e⌣4 holds for each P∗∈χΘ. So, an LP-Sasakian manifold admitted a GSMC-α,β.The nonvanishing components of the curvature tensor using the aforementioned formulas are(45)ℛ˜e⌣1,e⌣2e⌣1=−1−α−β2e⌣2,ℛ˜e⌣1,e⌣3e⌣1=−1−α−β2e⌣3,ℛ˜e⌣1,e⌣4e⌣1=−1−α−βe⌣4,ℛ˜e⌣1,e⌣2e⌣2=1−α−β2e⌣1,ℛ˜e⌣2,e⌣3e⌣2=−1−α−β2e⌣3,ℛ˜e⌣2,e⌣4e⌣2=−1−α−βe⌣4,ℛ˜e⌣1,e⌣3e⌣3=1−α−β2e⌣1,ℛ˜e⌣2,e⌣3e⌣3=1−α−β2e⌣2,ℛ˜e⌣3,e⌣4e⌣3=−1−α−βe⌣4,ℛ˜e⌣1,e⌣4e⌣4=−1−α−βe⌣1,ℛ˜e⌣2,e⌣4e⌣4=−1−α−βe⌣2,ℛ˜e⌣3,e⌣4e⌣4=−1−α−βe⌣3.The Ricci tensor ofΘ is defined as S˜P∗,Q∗=∑i=14εig¨ℛ˜e⌣i,P∗Q∗,e⌣i, where εi=g¨e⌣i,e⌣i, and is given by(46)S˜e⌣i,e⌣j=31−α−β2000031−α−β2000031−α−β20000−31−α−β2.Also, the scalar curvatureδ^˜ = ∑i=14S˜e⌣i,e⌣j=61−α−β2.Let any vector fieldsP∗,Q∗, and Z∗∈χΘ4; it is possible to write(47)P∗=a1e⌣1+b1e⌣2+c1e⌣3+d1e⌣4,Q∗=a2e⌣1+b2e⌣2+c2e⌣3+d2e⌣4,Z=a3e⌣1+b3e⌣2+c3e⌣3+d3e⌣4,where ai,bi,ci,di∈ℜ+,i=1,2,3,4, in order for(48)β22c2−d2a1a3+b1b3+c1c3−c3−d3a1a2+b1b2+c1c2−c1−d1a2a3+b2b3+c2c3a1a3+b1b3+c1c3−d1d3≠0.If we consider the 1-formπ by πP∗=g¨P∗,1−α−βe⌣4, for any P∗∈χΘ and considering ψ∈C∞Θ as(49)ψ=β22c2−d2a1a3+b1b3+c1c3−c3−d3a1a2+b1b2+c1c2−c1−d1a2a3+b2b3+c2c3a1a3+b1b3+c1c3−d1d3,therefore, the relation,(50)∇P∗Q∗=ψP∗+πP∗Q∗,holds. As per these consequences, from (24), we obtain(51)L˜Q∗g¨P∗,Z∗=g¨∇˜P∗Q∗,Z∗+g¨P∗,∇˜Z∗Q∗=2ψg¨P∗,Z∗+πP∗g¨Q∗,Z∗+πZ∗g¨Q∗,P∗+α2η∗Q∗g¨P∗,Z∗−g¨P∗,Q∗η∗Z∗−g¨Z∗,Q∗η∗P∗+β2η∗Q∗g¨ϕ∗P∗,Z∗−g¨ϕ∗P∗,Q∗η∗Z∗−g¨ϕ∗Z∗,Q∗η∗P∗.Also, we calculate(52)g¨P∗,Z∗=a1a3+b1b3+c1c3−d1d3,g¨Q∗,Z∗=a2a3+b2b3+c2c3−d2d3,g¨P∗,Q∗=a1a2+b1b2+c1c2−d1d2.Also,(53)πP∗=−1−α−βd1,πQ∗=−1−α−βd2,πZ∗=−1−α−βd3.With the help of (52) and (53), equation (51) reduces(54)12LQ∗g¨P∗,Z∗=ψa1a3+b1b3+c1c3−d1d3−121−α−βd1a2a3+b2b3+c2c3−d2d3+d3a1a2+b1b2+c1c2−d1d2+α22c2−d2a1a3+b1b3+c1c3−d1d3−c3−d3a1a2+b1b2+c1c2−d1d2−c1−d1a2a3+b2b3+c2c3−d2d3+β22c2−d2a1a3+b1b3+c1c3−c3−d3a1a2+b1b2+c1c2−c1−d1a2a3+b2b3+c2c3.Also,(55)δ^˜−σ^g¨P∗,Z∗=61−α−β2−σ^a1a3+b1b3+c1c3−d1d3.We consider thata1a3+b1b3+c1c3−d1d3≠0, 2d1a2a3+b2b3+c2c3−d2d3+2d3a1a2+b1b2+c1c2−d1d2+d2a1a3+b1b3+c1c3−d1d3=0, and −2c3−d3a1a2+b1b2+c1c2−d1d2−2c1−d1a2a3+b2b3+c2c3−d2d3−3c2−d2a1a3+b1b3+c1c3−d1d3=0.Thus, we getg¨,Q∗,σ^ is a Yamabe soliton, that is, 1/2LQ∗g¨P∗,Z∗=δ^˜−σ^g¨P∗,Z∗ holds, unless(56)σ^=ψ−61−α−β2−14−1−α−βd2+3αc2−d2=ψ−δ^˜−14πQ∗+3αη∗Q∗=constant.As a result, the existence of the YSg¨,Q∗,σ^ on a 4-dimensional LP-Sasakian manifold admitting a GSMC-α,β with potential vector field Q∗ as torse-forming is justified. Then, Theorems 1 and 2 are verified.



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