KYUNGPOOK Math. J. 2019; 59(3): 537-562
η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection
Mohd Danish Siddiqi
Department of Mathematics, Jazan University, Faculty of Science, Jazan, Kingdom of Saudi Arabia
e-mails : anallintegral@gmail.com, msiddiqi@jazanu.edu.sa
Received: February 14, 2018; Revised: August 29, 2018; Accepted: October 2, 2018; Published online: September 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The aim of the present paper is to study the δ-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting η-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ϕ-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying R̄.S̄ = 0, P̄.S̄ = 0 is an η-Einstein manifold. More-over, we obtain the conditions for the δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric-metric connection to be conformally flat and ξ-conformally flat.

Keywords: η-Ricci Solitons, δ-Lorentzian trans-Sasakian manifold, semi-symmetric metric connection, curvature tensors, Einstein manifold.
Introduction

In 1924, the idea of a semi-symmetric linear connection on a differentiable manifold was introduced by A. Friedmann and J. A. Schouten [13]. In 1930, Bartolotti [5] gave a geometrical meaning of such a connection. In 1932, H. A. Hayden [16] defined and studied semi-symmetric metric connections. In 1970, K. Yano [42], started a systematic study of semi-symmetric metric connections in a Riemannian manifold and this was further studied by various authors such as Sharfuddin Ahmad and S. I. Hussain [31], M. M. Tripathi [34], I. E. Hirică and L. Nicolescu [17, 18], G. Pathak and U.C. De [27].

Let ∇ be a linear connection in an n-dimensional differentiable manifold M. The torsion tensor T and the curvature tensor R of ∇ are given respectively by $T(X,Y)=∇XY−∇YX−[X,Y],R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z.$

The connection ∇ is said to be symmetric if its torsion tensor T vanishes, otherwise it is non-symmetric. The connection ∇ is said to be a metric connection if there is a Riemannian metric g in M such that ∇g= 0, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if it is the Levi-Civita connection.

A linear connection ∇ is said to be a semi-symmetric connection if its torsion tensor T is of the form $T(X,Y)=η(Y)X−η(X)Y,$where η is a 1-form. Semi-symmetric connections play an important role in the study of Riemannian manifolds. There are various physical problems involving the semi-symmetric metric connection. For example, if a man is moving on the surface of the earth always facing one definite point, say Jaruselam or Mekka or the North pole, then this displacement is semi-symmetric and metric [13].

The study of differentiable manifolds with Lorentizain metric is a natural and interesting topic in differential geometry. In 1996, Ikawa and Erdogan studied Lorentzian Sasakian manifold [20]. Also Lorentzian para contact manifolds were introduced by Matsumoto [24]. Trans Lorentzian para Sasakian manifolds have been used by Gill and Dube [15]. In [41], Yildiz et al. studied Lorentzian α-Sasakian manifold and Lorentzian β-Kenmotsu manifold studied by Funda et al. in [40]. S. S. Pujar and V. J. Khairnar [28] have initiated the study of Lorentzian trans-Sasakian manifolds and studied the some basic results with some of its properties. Earlier to this, S. S. Pujar [29] studied the δ-Lorentzian α-Sasakian manifolds and δ-Lorentzian β-Kenmotsu manifolds.

The study of manifolds with indefinite metrics is of interest from the standpoint of physics and relativity. In 1969, Takahashi [36] has introduced the notion of al-most contact metric manifolds equipped with pseudo Riemannian metric. These indefinite almost contact metric manifolds and indefinite Sasakian manifolds are known as (ε)-almost contact metric manifolds. The concept of (ε)-Sasakian manifolds was initiated by Bejancu and Duggal [6] and further investigation was taken up by X. Xufeng and C. Xiaoli [39]. U. C. De and A. Sarkar [11] studied the notion of (ε)-Kenmotsu manifolds with indefinite metric. S. S. Shukla and D. D. Singh [32] extended with indefinite metric which are natural generalization of both (ε)-Sasakian and (ε)-Kenmotsu manifolds called (ε)-trans-Sasakian manifolds. Siddiqi et al. [33] also studied some properties of Indefinite trans-Sasakian manifolds which is closely related to this topic.

The semi Riemannian manifolds has the index 1 and the structure vector field ξ is always a time like. This motivated Thripathi and others [34] to introduced (ε)-almost paracontact structure where the vector filed ξ is space like or time like according as (ε) = 1 or (ε) = −1.

When M has a Lorentzian metric g, that is a symmetric non-degenerate (0, 2) tensor field of index 1, then M is called a Lorentzian manifold. Since the Lorentzian metric is of index 1, Lorentzian manifold M has not only spacelike vector fields but also timelike and lightlike vector fields. This difference with the Riemannian case gives interesting properties on the Lorentzian manifold. A differentiable manifold M has a Lorentzian metric if and only if M has a 1-dimensional distribution. Hence odd dimensional manifold is able to have a Lorentzian metric. Inspired by the above results in 2014, S. M Bhati [8] introduced the notion of δ-Lorentzian trans Sasakian manifolds.

In 1982, R. S. Hamilton [19] said that the Rici solitons move under the Ricci flow simply by diffeomorphisms of the initial metric that is they are sationary points of the Ricci flow is given by $∂g∂t=−2Ric(g).$

### Definition 1.1

A Ricci soliton (g, V, λ) on a Riemannian manifold is defined by $LVg+2S+2λ=0,$where S is the Ricci tensor, LV is the Lie derivative along the vector field V on M and λ is a real scalar. Ricci soliton is said to be shrinking, steady or expanding according as λ < 0, λ = 0 and λ > 0, respectively.

In 1925, Levy [22] obtained the necessary and sufficient conditions for the existence of such tensors. later, R. Sharma [30] initiated the study of Ricci solitons in contact Riemannian geometry. After that, Tripathi [35], Nagaraja et al. [25] and others like C. S. Bagewadi et al. [4] extensively studied Ricci soliton. In 2009, J. T. Cho and M. Kimura [9] introduced the notion of η-Ricci solitons and gave a classification of real hypersurfaces in non-flat complex space forms admitting η-Ricci solitons. Later η-Ricci solitons in (ε)-almost paracontact metric manifolds have been studied by A. M. Blaga et al. [3]. A. M. Blaga and various others authors also have been studied η-Ricci solitons in different structures (see [1, 2, 10]). Recently in 2017, K. Venu and G. Nagaraja [38] study the η-Ricci solitons in trans-Sasakian manifold. It is natural and interesting to study η-Ricci soliton in δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric metric connection not as real hypersurfaces of complex space forms but a special contact structures. In this paper we derive the condition for a 3 dimensional δ-Lorentzian Trans-Sasakian manifold with a semi-symmetric metric connection as an η-Ricci soliton and derive expression for the scalar curvature.

Moreover, in this paper we introduced the relation between metric connection and semi-symmetric metric connection in an n-dimensional δ-Lorentzian trans-Sasakian manifolds. Also, we have proved some results on curvature tensor, scalar curvature, quasi projective flat, ϕ-projectively flat, . = 0, . = 0, Weyl conformally flat, Weyl ξ-conformally flat receptively in n-dimensional δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric metric connection.

Preliminaries

Let M be a δ-almost contact metric manifold equipped with δ-almost contact metric structure (ϕ, ξ, η, g, δ) [7] consisting of a (1, 1) tensor field ϕ, a vector field ξ, a 1-form η and an indefinite metric g such that $ϕ2=X+η(X)ξ, η∘ϕ=0, ϕξ=0,$$η(ξ)=−1,$$g(ξ,ξ)=−δ,$$η(X)=δg(X,ξ),$$g(ϕX,ϕY)=g(X,Y)+δη(X)η(Y)$for all X, YM, where δ is such that δ2 = 1 so that δ = ± 1. The above structure (ϕ, ξ, η, g, δ) on M is called the δ Lorentzian structure on M. If δ = 1 and this is usual Lorentzian structure [8] on M, the vector field ξ is the time like [42], that is M contains a time like vector field.

In [37], Tanno classified the connected almost contact metric manifold. For such a manifold the sectional curvature of the plane section containing ξ is constant, say c. He showed that they can be divided into three classes. (1) homogeneous normal contact Riemannian manifolds with c > 0. Other two classes can be seen in Tanno [37].

In Grey and Harvella [14] classification of almost Hermitian manifolds, there appears a class W4 of Hermitian manifolds which are closely related to the conformal Kaehler manifolds. The class C6C5 [26] coincides with the class of trans-Sasakian structure of type (α, β). In fact, the local nature of the two sub classes, namely C6 and C5 of trans-Sasakian structures are characterized completely. An almost conatct metric structure [43] on M is called a trans-Sasakian (see [12, 23, 26]) if (M × R, J, G) belongs to the class W4, where J is the almost complex structure on M × R defined by $J(X,fddt)=(ϕ(X)−fξ,η(X)ddt)$for all vector fields X on M and smooth functions f on M × R and G is the product metric on M × R. This may be expressed by the condition $(∇Xϕ)Y=α(g(X,Y)ξ−η(Y)X)+β(g(ϕX,Y)ξ−η(Y)ϕX)$for any vector fields X and Y on M, ∇ denotes the Levi-Civita connection with respect to g, α and β are smooth functions on M. The existence of condition (2.3) is ensure by the above discussion.

With the above literature, we define the δ-Lorentzian trans-Sasakian manifolds [8] as follows:

### Definition 2.1

A δ-Lorentzian manifold with structure (ϕ, ξ, η, g, δ) is said to be δ-Lorentzian trans-Sasakian manifold of type (α, β) if it satisfies the condition $(∇Xϕ)Y=α(g(X,Y)ξ−δη(Y)X)+β(g(ϕX,Y)ξ−δη(Y)ϕX)$for any vector fields X and Y on M.

If δ = 1, then the δ-Lorentzian trans Sasakian manifold is the usual Lorentzian trans Sasakian manifold of type (α, β) [26]. δ-Lorentzian trans Sasakian manifold of type (0, 0), (0, β) (α, 0) are the Lorentzian cosymplectic, Lorentzian β-Kenmotsu and Lorentzian α-Sasakian manifolds respectively. In particular if α = 1, β = 0 and α = 0, β = 1, the δ-Lorentzian trans Sasakian manifolds reduces to δ-Lorentzian Sasakian and δ-Lorentzian Kenmotsu manifolds respectively [21].

Form (2.4), we have $∇Xξ=δ{−αϕ(X)−β(X+η(X)ξ},$and $(∇Xη)Y=αg(ϕX,Y)+β[g(X,Y)+δη(X)η(Y)].$In a δ-Lorentzian trans Sasakian manifold M, we have the following relations: $R(X,Y)ξ=(α2+β2)[η(Y)X−η(X)Y]+2αβ[η(Y)ϕX−η(X)ϕY]​​​​​​​​​​​​​​​​​​ +δ[(Yα)ϕX−(Xα)ϕY+(Yβ)ϕ2X−(Xβ)ϕ2Y],$$R(ξ,Y)=(α2+β2)[δg(X,Y)ξ−η(X)Y] +δ(Xα)ϕY+δg(ϕX,Y)(gradα) +δ(Xβ)(Y+η(Y)ξ)−δg(ϕY,ϕX))(gradβ) +2αβ[δg(ϕX,Y)ξ+η(X)ϕY],$$η(R(X,Y)Z)=δ(α2+β2)[η(X)g(Y,Z)−η(Y)g(X,Z) +2δαβ[−η(X)g(ϕY,Z)+η(Y)g(ϕX,Z)] −[(Yα)g(ϕX,Z)+(Xα)g(Y,ϕZ)] −(Yβ)g(ϕ2X,Z)+(Xβ)g(ϕ2Y,Z)],$$S(X,ξ)=[((n−1)(α2+β2)−(ξβ)]η(X)+δ((ϕX)α)+(n−2)δ(Xβ),$$S(ξ,ξ)=(n−1)(α2+β2)−δ(n−1)(ξβ),$$Qξ=(δ(n−1)(α2+β2)−(ξβ)ξ+δϕ(gradα)−δ(n−2)(gradβ),$where R is curvature tensor, while Q is the Ricci operator given by S(X, Y) = g(QX, Y).

Further in an δ-Lorentzian trans-Sasakian manifold, we have $δϕ(gradα)=δ(n−2)(gradβ),$and $2αβ−δ(ξα)=0.$The ξ-sectional curvature Kξ of M is the sectional curvature of the plane spanned by ξ and a unit vector field X. From (2.11), we have $Kξ=g(R(ξ,X),ξ,X)=(α2+β2)−δ(ξβ).$It follows from (2.17) that ξ-sectional curvature does not depend on X. From (2.11)$g(R(ξ,Y)Z,ξ)=[(α2+β2)−δ(ξβ)]g(Y,Z) +[(ξβ)−δ(α2+β2)]η(Y)η(Z)+[2αβ+δ(δα)]g(ϕY,Z),$$C(X,Y)Z=R(X,Y)Z−1(n−2)[S(Y,Z)X−S(X,Y)Y +g(Y,Z)QX−g(X,Z)QY]+r(n−1)(n−2)[g(Y,Z)X−g(X,Z)Y].$

An affine connection ∇̄ in M is called semi-symmetric connection [13], if its torsion tensor satisfies the following relations $T¯(X,Y)=∇¯XY−∇¯YX−[X,Y],$and $T¯(X,Y)=η(X)Y−η(Y)X.$Moreover, a semi-symmetric connection is called semi-symmetric metric connection if $g¯(X,Y)=0.$If ∇ is metric connection and ∇̄ is the semi-symmetric metric connection with non-vanishing torsion tensor T in M, then we have $T(X,Y)=η(Y)X−η(X)Y,$$∇¯XY−∇XY=12[T(X,Y)+T′(X,Y)+T′(X,Y)],$where $g(T(Z,X),Y)=g(T′(X,Y),Z).$By using (2.4), (2.23) and (2.25), we get $g(T′(X,Y),Z)=g(η(X)Z−η(Z)X,Y),g(T′(X,Y),Z)=η(X)g(Z,Y)−δg(X,Y)g(ξ,Z),T′(X,Y)=η(X)Y−δg(X,Y)ξ,$$T′(Y,X)=η(Y)X−δg(X,Y)ξ.$From (2.23), (2.24),(2.26) and (2.27), we get $∇¯XY=∇XY+η(Y)X−δg(X,Y)ξ.$

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold and ∇ be the metric connection on M. The relation between the semi-symmetric metric connection ∇̄ and the metric connection ∇ on M is given by $∇¯XY=∇XY+η(Y)X−δg(X,Y)ξ.$

Curvature Tensor on δ-Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold. The curvature tensor of M with respect to the semi-symmetric metric connection ∇̄ is defined by $R¯(X,Y)Z=∇¯X∇¯YZ−∇¯Y∇¯XZ−∇¯[X,Y]Z.$By using (2.4), (2.28) and (3.1), we get $R¯(X,Y)Z=R(X,Y)Z+(δ)[g(X,Z)Y−g(Y,Z)X] +(β+δ)[g(Y,Z)η(X)−g(X,Z)η(Y)]ξ −(βδ−1)[η(Y)X−η(X)Y]η(Z), +α[g(ϕX,Z)Y−g(ϕY,Z)ϕX−g(X,Z)ϕY+g(Y,Z)ϕX],$where $R(X,Y)Z=∇X∇YZ=∇Y∇XZ−∇[X,Y]Z$is the Riemannian curvature tensor of connection ∇.

### Lemma 3.1

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, then$(∇¯Xϕ)(Y)=α(g(ϕX,Y)ξ−δη(Y)X)+β(g(ϕX,Y)ξ−(δβ+δ)η(Y)ϕX),$$∇¯Xξ=−(1+δβ)X−(1+δβ)η(X)ξ−δαϕX,$$(∇¯Xη)Y=αg(ϕX,Y)+(β+δ)g(X,Y)−(1+βδ)η(X)η(Y).$

Proof

By the covariant differentiation of ϕY with respect to X, we have $∇¯XϕY=(∇¯Xϕ)+ϕ(∇¯XY).$By using (2.1) and (2.28), we have $(∇¯Xϕ)Y=(∇¯Xϕ)Y−η(Y)ϕX.$In view of (2.7), the last equation gives $(∇¯Xϕ)(Y)=α(g(ϕX,Y)ξ−δη(Y)X)+β(g(ϕX,Y)ξ−(δβ+δ)η(Y)ϕX).$To prove (3.4), we replace Y = ξ in (2.28) and we have $∇¯Xξ=∇¯Xξ+η(ξ)X−δg(X,ξ)ξ.$By using (2.2), (2.4) and (2.8), the above equation gives $∇¯Xξ=−(1+δβ)X−(1+δβ)η(X)ξ−δαϕX.$In order to prove (3.5), we differentiate η(Y) covariantly with respect to X and using (2.28), we have $∇¯Xη(Y)=(∇Xη)Y+g(X,Y)−η(X)η(Y).$Using (2.9) in above equation, we get $(∇¯Xη)Y=αg(ϕX,Y)+(β+δ)g(X,Y)−(1+βδ)η(X)η(Y).$

### Lemma 3.2

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, then$R¯(X,Y)ξ=(α2+β2−δβ)[η(X)Y−η(Y)X]. +(2αβ+δα)[η(Y)ϕX−η(X)ϕY] +δ[(Yα)ϕX−(−Xα)ϕY−(Xβ)ϕ2Y+(Yβ)ϕ2X].$

Proof

By replacing Z = ξ in (3.2), we have $R¯(X,Y)ξ=R(X,Y)ξ+(δ)[g(X,ξ)Y−g(Y,ξ)X]+(β+δ)[g(Y,ξ)η(X)−g(X,ξ)η(Y)]ξ−(βδ−1)[η(Y)X−η(X)Y]η(ξ)+α[g(ϕX,ξ)Y−g(ϕY,ξ)ϕX−g(X,ξ)ϕY+g(Y,ξ)ϕX]$In view of (2.2), (2.4) and (2.10), the above equation reduces to $R¯(X,Y)ξ=(α2+β2−δβ)[η(X)Y−η(Y)X] +(2αβ+δα)[η(Y)ϕX−η(X)ϕY] +δ[(Yα)ϕX−(Xα)ϕY−(Xβ)ϕ2Y+(Yβ)ϕ2X].$

### Remark 3.1

Replace Y = ξ and using (3.2), (2.11), (2.2) and (2.4), we obtain $R¯(X,ξ)ξ=(α2+β2−δβ)[−X−η(X)Y]+(2αβ+δα+δ(ξα))[ϕX+δ(ξβ)ϕ2X].$

### Remark 3.2

Now, again replace X = ξ in (3.6), using (2.1), (2.2) and (2.4), we obtain $R¯(ξ,Y)ξ=(α2+β2−δβ)[−η(Y)ξ−Y]−(2αβ+δα+δ(ξα))[ϕY+δ(ξβ)ϕ2Y].$

### Remark 3.3

Replace Y = X in (3.8), we get $R¯(ξ,X)ξ=−(α2+β2−δβ)[−X−η(X)ξ]−(2αβ+δα+δ(ξα))[ϕX−δ(ξβ)ϕ2X].$From (3.7) and (3.9), we obtain $R¯(X,ξ)ξ=−R¯(ξ,X)ξ.$

Now, contracting X in (3.2), we get $S¯(Y,Z)=S(Y,Z)−[(δ)(n−2)+β]g(Y,Z) −(βδ−1)(n−2)η(Z)η(Y)−α(n−2)g(ϕY,Z),$where and S are the Ricci tensors of the connections ∇̄ and ∇, respectively on M.

This gives $Q¯Y=QY−[(δ)(n−2)+β]Y −(βδ−1)(n−2)η(Y)ξ−α(n−2)ϕY,$where and Q are Ricci operator with respect to the semi-symmetric metric connection and metric connection respectively and define as g(Q̄Y, Z) = (Y, Z) and g(QY, Z) = S(Y, Z) respectively.

Replace Y = ξ in (3.12) and using (2.15), we get $Q¯ξ=δ(n−1)(α2+β2)ξ−(ξβ)ξ−2δ(n−2)ξ +δϕ(gradα)−δ(n−2)(gradβ)−β(n−1)ξ.$Putting Y = Z = ei and taking summation over i, 1 ≤ in − 1 in (3.11), using (2.14) and also the relations $r=S(ei,ei)=∑i=1nδiR(ei,ei,ei,ei)$, we get $r¯=r−(n−1)[(δ)(n−2)+2β],$where and r are the scalar curvatures of the connections ∇̄ and ∇, respectively on M.

Now, we have the following lemmas.

### Lemma 3.3

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, then$S¯(ϕY,Z)=−δ(ϕ2Y)α−δ(n−2)(ϕY)β−α(n−2)g(ϕY,ϕZ),$$S¯(Y,ξ)=[(n−1)(α2+β2−δ(ξβ)−δβ(n−1)]η(Y)+δ(n−2)(Yβ)+δ(ϕY)β,$$S¯(ξ,ξ)=[(n−1)(α2+β2−δ(ξβ)−δβ(n−1)]η(Y).$

Proof

By replacing Y = ϕY in equation (3.11) and using (2.13) and (2.5), we have (3.15). Taking Y = ξ in (3.11) and using (2.13) we get (3.16). (3.17) follows from considering Y = ξ in (3.16) we get (3.17).

### Lemma 3.4

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, then$S¯(gradα,ξ)=δ(n−1)(α2+β2(ξβ)−β(n−1)(ξα)−(ξα)(ξβ)+δ(ϕgradα)α+δ(n−2)g(gradα,gradβ),$$S¯(gradβ,ξ)=δ(n−1)(α2+β2(ξβ)−β(n−1)(ξβ)−(ξβ)2+δ(ϕgradβ)α+δ(n−2)g(gradβ)2.$

Proof

From equation (3.11) and (3.16) and using Y = gradα we have (3.18).

Similarly taking ξ = gradβ in (3.11) and using (3.16), we get (3.19). Using (3.6), (3.13) and (3.16), for constant α and β, we have $R¯(X,Y)ξ=(α2+β2−δ(ξβ)[η(Y)X−η(X)Y],$$S¯(X,Y)=[(n−1)(α2+β2−δ(ξβ)−δβ(n−1)]η(Y),$$Q¯ξ=δ(n−1)(α2+β2ξ−δ(ξβ)ξ−2δ(n−2)−β(n−1)ξ.$

Quasi-projectively flat δ-Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold. If there exists a one to one correspondence between each co-ordinate neighborhood of M and a domain in Euclidean space such that any geodesic of δ-Lorentzian trans-Sasakian manifold corresponds to a straight line in the Euclidean space, then M is said to be locally projectively flat. The projective curvature tensor with respect to semi-symmetric metric connection is defined by $P¯(X,Y)Z=R¯(X,Y)Z−1(n−1)[S¯(Y,Z)X−S¯(X,Z)Y].$

### Definition 4.1

A δ-Lorentzian trans-Sasakian manifold M is said to be quasi-projectively flat with respect to semi-symmetric metric connection, if $g(P¯(ϕX,Y)Z,ϕU)=0,$where is the projective curvature tensor with respect to semi-symmetric metric connection.

Now, from (4.1) taking inner product with U, we get $g(P¯(X,Y)Z,U)=g(R¯(X,Y)Z,U)−1(n−1) [S¯(Y,Z)g(X,U)−S¯(X,Z)g(Y,U)].$Replace X = ϕX and U = ϕU in (4.3), we get $g(P¯(ϕX,Y)Z,ϕU)=g(R¯(ϕX,Y)Z,ϕU)−1(n−1) [S¯(Y,Z)g(ϕX,ϕU)−S¯(ϕX,Z)g(Y,ϕU)].$From (4.2) and (4.4), we have $g(R¯(ϕX,Y)Z,ϕU)=1(n−1)[S¯(Y,Z)g(ϕX,ϕU)−S¯(ϕX,Z)g(Y,ϕU)].$Now, using equations (2.1), (2.4), (3.11) and (3.15) in equation (4.5), we have $g(R¯(ϕX,Y)Z,ϕU)=1(n−1)[S¯(Y,Z)g(ϕX,ϕU)−S¯(ϕX,Z)g(Y,ϕU) −(δ+β)(n−1)g(ϕX,Z)g(Y,ϕU)+(δ+β)(n−1)g(Y,Z)g(ϕX,ϕU) −(δβ−1)(n−1)η(Y)η(Z)g(ϕX,ϕU)+(δα)(n−1)η(X)η(Z)g(ϕX,ϕU) −α(n−1)g(X,Z)g(Y,ϕU)−α(n−1)g(ϕY,Z)g(ϕX,ϕU) +αg(Y,Z)g(X,ϕU)+αg(ϕX,Z)g(ϕX,ϕU).$Let {e1, e2.........en−1, ξ} be a local orthonormal basis of vector fileds on δ-Lorentzian trans-Sasakian manifold M, then {ϕe1, ϕe2........ϕen−1, ξ} is also a local orthonormal basis of vector fields on δ-Lorentzian trans-Sasakian manifold M. Now, putting X = U = ei in equation (4.6) and using (2.2), (2.3),(2.19), (3.11) and (3.16), we have $S(Y,Z)=[(n−2)(β+δ)+δ(n−1)(α2+β2)−(n−1)(ξβ)]g(Y,Z) +[δ(n−2)(ξβ)+(n−2)(βδ−1)]η(Y)η(Z) −[2δ(n−1)αβ+(n−1)(ξα)−α]g(ϕY,Z) −δη(Y)(ϕZ)α−δ(n−2)(ξβ)η(Y).$

If α = 0 and β = constant in (4.7), we get $S(Y,Z)=[(n−2)(β+δ)+(n−1)δβ2]g(Y,Z)+(βδ−1)(2−n)η(Y)η(Z).$Therefore, we have $S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),$where a = (n − 2) (β + δ) + (n − 1)δβ2 and b = (βδ −1) (2 − n).

These results shows that the manifold under the consideration is an η-Einstein manifold. Thus we can state the following theorem:

### Theorem 4.1

An n-dimensional quasi projectively flat δ-Lorentzian trans-Sasakian manifold M with respect to a semi-symmetric metric connection is an η-Einstein manifold if α = 0 and β = constant.

ϕ-Projectively flat δ-Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection

An n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection is said to be ϕ-projectively flat if $ϕ2(P¯(ϕX,ϕY)ϕZ)=0,$where is the projective curvature tensor of M n-dimensional δ-Lorentzian trans-Sasakian manifold with respect to a semi-symmetric metric connection. Suppose M be ϕ-projectively flat δ-Lorentzian trans-Sasakian manifold with respect to a semi-symmetric metric connection. It is know that ϕ2((ϕ, X, ϕY)ϕZ) = 0 holds if and only if $g(P¯(ϕX,ϕY)ϕZ,ϕU)=0,$for any X, Y, Z, UTM. Replace Y = ϕY and UϕU in (4.4), we have $g(P¯(ϕX,ϕY)ϕZ,ϕU)=g(R¯(ϕX,ϕY)ϕZ,ϕU)−1(n−1) [S¯(ϕY,ϕZ)g(ϕX,ϕU)−S¯(ϕX,ϕZ)g(ϕY,ϕU)].$From (5.2) and (5.3), we have $g(R¯(ϕX,ϕY)ϕZ,ϕU)=1(n−1)[S¯(ϕY,ϕZ)g(ϕX,ϕU) −S¯(ϕX,ϕZ)g(ϕY,ϕU)].$Now, using (2.1),(2.2),(2.4),(2.5), (3.2) and (3.11) in equation (5.4), we have $g(R¯(ϕX,ϕY)ϕZ,ϕU)=1(n−1)[S¯(ϕY,ϕZ)g(ϕX,ϕU)−S¯(ϕX,ϕZ)g(ϕY,ϕU)] −(δ+β)(n−1)g(ϕY,ϕZ)g(ϕX,ϕU)+(δ+β)(n−1)g(ϕX,ϕZ)g(ϕY,ϕU) −α(n−1)g(Y,ϕZ)g(ϕX,ϕU)−α(n−1)g(X,ϕYZ)g(ϕX,ϕU) +αg(ϕY,ϕZ)g(X,ϕU)−αg(ϕX,ϕZ)g(Y,ϕU).$Let {e1, e2.........en−1, ξ} be a local orthonormal basis of vector fileds on δ-Lorentzian trans-Sasakian manifold M, then {ϕe1, ϕe2........ϕen−1, ξ} is also a local orthonormal basis of vector fields on δ-Lorentzian trans-Sasakian manifold M. Now putting X = U = ei in equation (5.5) and using (2.1)(2.5), (2.19), (3.11) and (3.16), we have $S(Y,Z)=[(n−2)(β+α)+δ(n−1)(α2+β2)−(n−1)(ξβ)]g(X,Z) +[2δ(n−2)(ξβ)+(n−2)(βδ−1)]η(Y)η(Z) +[α−2δαβ(n−1)−(n−1)(ξα)]g(ϕY,Z) −[δ(ϕZ)α+δ(n−2)(Zβ)]η(Y)−[δ(ϕY)α+δ(n−2)(Yβ)]η(Z)$If α = 0 and β = constant in (5.6), we get $S(Y,Z)=[(n−2)(β+δ)+(n−1)δβ2]g(Y,Z)+(βδ−1)(2−n)η(Y)η(Z).$Therefore, $S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),$where a = (n − 2)(β + δ) + (n − 1)δβ2 and b = (βδ − 1)(2 − n).

This result shows that the manifold under the consideration is an η-Einstein manifold. Thus we can state the following theorem:

### Theorem 5.1

An n-dimensional ϕ-projectively flat δ-Lorentzian trans-Sasakian manifold M with respect to a semi-symmetric metric connection is an η-Einstein manifold if α = 0 and β = constant.

δ-Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfying R̄.S̄ = 0

Now, suppose that M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection satisfying the condition: $R¯(X,Y).S¯=0.$Then, we have $S¯(R¯(X,Y)Z,U)+S¯(Z,R¯(X,Y)U)=0.$Now, we replace X = ξ in equation (6.2), using equations (2.11) and (6.2), we have $δ(α2+β2)g(Y,Z)S¯(ξ,U)−(α2+β2)η(Z)S¯(Y,U)−2δαβg(ϕY,Z)S¯(ξ,U)+2αβη(Z)S¯(ϕY,U)+δ(Zα)S¯(ϕY,U)−δg(ϕY,Z)S¯(gradα,U)−δg(ϕY,ϕZ)S¯(gradβ,U)+δ(Zβ)S¯(Y,U)−δ(Zβ)η(Y)S¯(ξ,U)−δg(Y,Z)S¯(ξ,U)+δη(Z)S¯(Y,U)+αg(ϕY,Z)S¯(ξ,U)−δαη(Z)S¯(ϕY,U)+δ(α2+β2)g(Y,U)S¯(ξ,Z)−(α2+β2)η(U)S¯(Y,Z)−2δαβg(ϕY,U)S¯(ξ,Z)+2αβη(U)S¯(ϕY,Z)+δ(Uα)S¯(ϕY,Z)−δg(Y,U)S¯(gradα,Z)−δg(ϕY,ϕU)S¯(gradβ,Z)+δ(Uβ)S¯(Y,Z)−δ(Uβ)η(Y)S¯(ξ,Z)−δg(Y,U)S¯(ξ,Z)+δη(U)S¯(Y,Z)+αg(ϕY,U)S¯(ξ,Z)−δαη(U)S¯(ϕY,Z)=0.$Using equations (2.1)(2.5), (2.13), (2.14), (3.11) and (3.15)(3.19) in equation (6.3)$[(α2+β2)−δ(ξβ)−δβ]S(Y,Z)=[δ(n−1)(α2+β2)−2β(n−1)(α2+β2)−2(n−1)(α2+β2)(ξβ)+2δβ(n−1)(ξβ)−δ(ξβ)2+(ϕgradβ)α+(n−2)(gradβ)2+δβ2(n−2)+δ(n−2)(α2+β2)+β(α2+β2)−2α2β(n−2)−δα(ξα)−(n−2)(ξβ)−δβ(ξβ)−β(n−2)+δα2(n−2)]g(Y,Z)+[−δ(ϕgradβ)α−δ(n−2)(gradβ)2+(n−2)(βδ−1)(α2+β2)+2δα2β(n−2)+α(n−2)(ξα)+(β+δ)(n−2)(ξβ)+β(β+δ)(n−2)−α2(n−2)]η(Y)η(Y)η(Z)+[−2δαβ(n−1)(α2+β2)+2(n−2)αβ2+2αβ(n−2)(ξβ)−(n−1)(α2+β2)(ξα)+δβ(n−2)(ξα)+δ(ξα)(ξβ)+(ϕgradα)α+(n−2)(g(gradα,gradβ)+α(α2+β2)−δα(ξβ)−2αβ(n−2)(δ)−(n−2)(δα)+α(n−2)]g(ϕY,Z)+[δ(ξα)+2αβ−δα]S(ϕY,Z)+[(n−2)(ξβ)(Zβ)+[δ(α2+β2)(ϕZ)α−δ(n−2)(α2+β2)(Zβ)+(ξβ)(ϕZ)αβ(ϕZ)α+β(n−2)(Zβ)]η(Y)+[δ(α2+β2)(ϕY)α+δ(n−2)(α2+β2)(Yβ)−2δαβ(ϕ2Y)α−2δαβ(n−2)(ϕYβ)−β(ϕY)α−β(n−2)(Yβ)+α(ϕ2Y)α+α(n−2)(ϕYβ)]η(Z)−(n−2)(Yβ)(Zβ)+(n−2)(Zβ)(ξβ).$

If α = 0 and β = constant in (5.6), we get $S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),$where $a=−[(n−1)δβ4+(n−2)(gradβ)2+(n−2)δβ2+(n−2)δβ2−(n−2)β+(2n−3)β3(β+δ)β]$ and $b=−[(n−2)(βδ−1)β2+(n−2)(β+δ)β−(n−2)δ(gradβ2(β+δ)β]$. This shows that M is an η-Einstein manifold. Thus, we can state the following theorem:

### Theorem 6.1

An n-dimensional δ-Lorentzian trans-Sasakian manifold M with respect to a semi-symmetric metric connection ∇̄ satisfying R̄.S̄ = 0, then δ-Lorentzian trans-Sasakian manifold M is an η-Einstein manifold if α = 0 and β = constant.

δ-Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfying P̄.S̄ = 0

Now, we consider δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection satisfying $(P¯(X,Y).S¯)(Z,U)=0,$where is the projective curvature tensor and is the Ricci tensor with a semi-symmetric metric connection. Then, we have $S¯(P¯(X,Y)Z,U)+S¯(Z,P¯(X,Y)U)=0.$Replace X = ξ in the equation (7.2), we get $S¯(P¯(ξ,Y)Z,U)+S¯(Z,P¯(ξ,Y)U)=0.$Putting X = ξ in (4.1), we get $P¯(ξ,Y)Z=R¯(ξ,Y)Z−1(n−1)[S¯(Y,Z)ξ−S¯(ξ,Z)Y].$Using (2.1), (2.2), (2.4), (2.11), (3.2), (3.11), (3.17) and (7.4) in (7.3), we get $δ(α2+β2)(n−1)+(β+δ)(n−2)(n−1)g(Y,Z)S¯(ξ,U)−1(n−1)S(Y,Z)S¯(ξ,U)−(n−2)(n−1)(βδ−1)η(Y)η(Z)S¯(ξ,U)+α−2δαβ(n−1)(n−1)g(ϕY,Z)S¯(ξ,U)−δg(ϕY,Z)S¯(gradα,U)−δg(ϕY,ϕZ)S¯(gradβ,U)+2αβη(Z)S¯(ϕY,U)+δ(Zα)S¯(ϕY,U)+δ(Zβ)S¯(Y,U)−δ(Zβ)η(Y)S¯(ξ,U)−δαη(Z)S¯(ϕY,U)−1(n−1)δ(ξβ)η(Z)S¯(Y,U)(n−2)(n−1)δ(Zβ)S¯(Y,U)−1(n−1)δ(ϕZ)αS¯(Y,U)δ(α2+β2)(n−1)+(β+δ)(n−2)(n−1)g(Y,U)S¯(ξ,Z)−1(n−1)S(Y,U)S¯(ξ,Z)−(n−2)(n−1)(βδ−1)η(Y)η(U)S¯(ξ,Z)+α−2δαβ(n−1)(n−1)g(ϕY,U)S¯(ξ,Z)−δg(ϕY,U)S¯(gradα,Z)−δg(ϕY,ϕU)S¯(gradβ,Z)+2αβη(U)S¯(ϕY,Z)+δ(Uα)S¯(ϕY,Z)+δ(Zβ)S¯(Y,Z)−δ(Uβ)η(Y)S¯(ξ,Z)−δαη(U)S¯(ϕY,Z)−1(n−1)δ(ξβ)η(Z)S¯(Y,Z)(n−2)(n−1)δ(Uβ)S¯(Y,Z)−1(n−1)δ(ϕU)αS¯(Y,Z)=0$Putting U = ξ and Using (2.1)(2.5), (3.11) and (3.15)(3.20) in (7.5), we get $[(α2+β2)−δ(ξβ)−δβ]S(Y,Z)=[δ(n−1)(α2+β2)+(n−2)(βδ)(α2+β2)−β(n−1)(α2+β2)−δ(n−2)(βδ−1)−2(n−1)(ξβ)(α2+β2)−(n−2)(βδ−1)(ξβ)−2α2β(n−2)δα(n−2)(ξα)+δα2(n−2)+δβ(n−1)+δ(ξβ)2+(ϕgradα)α+(n−2)(gradβ)2]g(Y,Z)+(n−2)β(β+δ)−(n−2)(α2+β2)+2(n−2)δα2β+α(n−2)(ξα)+(n−2)(β+δ)(ξβ)−α2(n−2)−δ(n−2)(gradβ)2−δ(ϕgradβ)α]η(Y)(η)(Z)+[α(α2+β2)−2δαβ(α2+β2)(n−1)−2αβ2n−δ(ξβ)−δβ(ξα)+2αβ(ξβ)−2δαβ(n−2)−(n−1)(ξα)+α(n−2)−(n−1)(α2+β2)(ξα)+(n−1)δβ(ξα)+δ(ξα)(ξβ)+(ϕgradα)α+)n−2)g(gradα,gradβ)]g(ϕY,Z)+[δα+δ(ξα)−δα]S(ϕY,Z)+[δ(n+3)(α2+β2)(Zβ)+β(n−2)(Zβ)−delta(α2+β2)(ϕZ)α+(n−1)β(ϕZ)α+(ξβ)(ϕZ)α)]η(Y)+[−2δαβ(ϕ2Y)α−2δαβ(n−2)(ϕYβ)+α(ϕ2Y)α+α(n−2)(ϕYβ)+δ(α2+β2)(ϕY)α+δ(n−2)(α2+β2)(Yβ)−β(ϕY)α−β(n−2)(Yβ)]η(Z)−(Zα)(ϕ2Y)α−(n−2)(Zβ)(ϕYβ)−(Zβ)(ϕY)α−β(n−2)(Yβ).$If α = 0 and β = constant in (7.6), we get $S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),$where $a=−[(n−1)β4+(n−2)β2(βδ)+(n−1)β3−(n−2)β(βδ−1)+(n−1)δβ+(n−2)(gradβ)2β(βδ)]$ and $b=−[(n−2)β(β+δ)+(n−2)β2−(n−2)δ(gradβ)2β(β+δ)]$.

This result show that the manifold under the consideration is an η-Einstein manifold. Thus we have the following theorem:

### Theorem 7.1

An n-dimensional δ-Lorentzian trans-Sasakian manifold M with respect to a semi-symmetric metric connection ∇̄ satisfying P̄.S̄ = 0, then δ-Lorentzian trans-Sasakian manifold M is an η-Einstein manifold if α = 0 and β = constant.

Weyl Conformal Curvature Tensoron δ-Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection

The Weyl conformal curvature tensor of type (1, 3) of M an n-dimensional δ-Lorentzian trans-Sasakian manifold a with semi-symmetric metric connection ∇̄ is given by [16] $C¯(X,Y)Z=R¯(X,Y)Z−1(n−2)[S¯(Y,Z)X−S¯(X,Z)Y+g(Y,Z)Q¯X−g(X,Z)Q¯Y]+r¯(n−1)(n−2)[g(Y,Z)X−g(X,Z)Y],$where is the Ricci operator with respect to the semi-symmetric metric connection ∇̄. Let M ba an n-dimensional δ-Lorentzian trans-Sasakian manifold. The Weyl conformal curvature tensor of M with respect to the semi-symmetric metric connection ∇̄ is defined in equation (8.1).

Now, taking inner product with U in (8.1), we get $g(C¯(X,Y)Z,U)=g(R¯(X,Y)Z,U)−1(n−2)[S¯(Y,Z)g(X,U)]−S¯(X,Z)g(Y,U)+g(Y,Z)g(Q¯X,U)−g(X,Z)g(Q¯Y,U)+r¯(n−1)(n−2)[g(Y,Z)g(X,U)−g(X,Z)g(Y,U)].$Using (2.4), (3.2), (3.11), (3.12) and (3.14) in (8.2), we get $C¯(X,Y,Z,U)=g(R¯(X,Y)Z,U)−1(n−2)[S(Y,Z)g(X,U)]−S(X,Z)g(Y,U)+g(Y,Z)g(QX,U)−g(X,Z)g(QY,U)+r(n−1)(n−2)[g(Y,Z)g(X,U)−g(X,Z)g(Y,U)],$where g((X, Y)Z, U) = (X, Y, Z, U) and R(X, Y)Z, U) = C(X, Y, Z, U) are Weyl curvature tensor with respect to the semi-symmetric metric connection respectively, we have $C¯(X,Y,Z,U)=C(X,Y,Z,U),$where $C(X,Y,Z,U)=g(R¯(X,Y)Z,U)−1(n−2)[S(Y,Z)g(X,U)]−S(X,Z)g(Y,U)+g(Y,Z)g(QX,U)−g(X,Z)g(QY,U)]+r(n−1)(n−2)[g(Y,Z)g(X,U)−g(X,Z)g(Y,U)].$

### Theorem 8.1

The Weyl conformal curvature tensor of a δ-Lorentzian trans-Sasakian manifold M with respect to a metric connection is equal to the Weyl curvature of δ-Lorentzian trans-Sasakian manifold with respect to the semi-symmetric metric connection.

δ-Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connection

Let us consider that the δ-Lorentzian trans-Sasakian manifold M with respect to the semi-symmetric metric connection is Weyl conformally flat, that is = 0. Then from equation (8.1), we get $R¯(X,Y)Z=1(n−2)[S¯(Y,Z)X−S¯(X,Z)Y+g(Y,Z)Q¯X−g(X,Z)Q¯Y]+r¯(n−1)(n−2)[g(Y,Z)X−g(X,Z)Y],$Now, taking the inner product of equation (9.1) with U. then we get $g(R¯(X,Y)Z,U)=1(n−2)[S¯(Y,Z)g(X,U)−S¯(X,Z)g(Y,U)+g(Y,Z)g(Q¯X,U)−g(X,Z)g(Q¯,U)]−r¯(n−1)(n−2)[g(Y,Z)g(X,U)−g(X,Z)g(Y,U)].$Using equations (2.4), (3.2), (3.11), (3.12) and (3.14) in equation (9.2), we get $g(R(X,Y)Z,U)=1(n−2)[S(Y,Z)g(X,U)−S(X,Z)g(Y,U)+g(Y,Z)g(QX,U)−g(X,Z)g(QY,U)]−r(n−1)(n−2)[g(Y,Z)g(X,U)−g(X,Z)g(Y,U)].$Putting X = U = ξ in equation (9.3) and using (2.2), (2.3) and (2.4), we get $g(R(ξ,Y)Z,ξ)=1(n−2)[δS(Y,Z)−δη(Y)S(ξ,Z)+g(Y,Z)S(ξ,ξ)−δη(Z)S(Y,ξ)]−r(n−1)(n−2)[δg(Y,Z)−η(Y)η(Z)],$where g(QY, Z) = S(Y, Z).

Now, using equations (2.13), (2.14) and (2.16), we get $S(Y,Z)=[(δ(α2+β2)−(ξβ)]+r(n−1)]g(Y,Z)+[δ(n−4)(ξβ)+n(α2+β2)−δr(n−1)]η(Y)η(Z)−[2δαβ(n−2)+(n−2)(ξα)]g(ϕY,Z)−[δ(ϕZ)α+δ(Zβ)(n−2)]η(Y)−[δ(ϕY)α+δ(n−2)(Yβ)]η(Z).$If α = 0 and d β = constant in (7.6), we get $S(Y,Z)=[δβ2+r(n−1)]g(Y,Z)+[nβ2−δr(n−1)]η(Y)η(Z).$Therefore $S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),$where $a=[δβ2+r(n−1)]$ and $b=[nβ2−δr(n−1)]$. This shows that M is an η-Einstein manifold. Thus we can state the following theorem:

Let M ba an n-dimensional Weyl conformally flat δ-Lorentzian trans-Sasakian manifold with respect to the semi-symmetric metric connection ∇̄ is an η-Einstein manifold if α = 0 and β =constant. Now, taking equation (8.1)$C¯(X,Y)Z=R¯(X,Y)Z−1(n−2)[S¯(Y,Z)X−S¯(X,Z)Y+g(Y,Z)Q¯X−g(X,Z)Q¯Y]+r¯(n−1)(n−2)[g(Y,Z)X−g(X,Z)Y].$Using (2.20), (3.2), (3.11), (3.12) and (3.14) in equation (9.7), we get $C¯(X,Y)Z=C(X,Y)Z+δ[g(X,Y)Y−g(Y,Z)X]+(δ+β)[η(X)g(Y,Z)−η(Y)g(X,Z)]ξ−(βδ−1)η(Z)[η(Y)X−η(X)Y]+α[g(ϕX,Z)Y−g(ϕ,Z)X−g(Y,Z)ϕX+g(X,Z)ϕY]+1(n−2)(βδ−1)(n−2)η(Y)η(Z)−((δ)(n−2)+β)g(Y,Z)X+α(n−2)g(ϕY,Z)X+((δ)(n−2)+β)g(X,Z)Y+(βδ−1)(n−2)η(X)η(Z)Y−α(n−2)g(ϕX,Z)Y−((δ)(n−2)+β)g(Y,Z)X+(β+δ)(n−2)g(Y,Z)η(X)ξα(n−2)g(Y,Z)ϕX+((δ)(n−2)+β)g(X,Z)Y−(β+δ)(n−2)g(X,Z)η(Y)ξ−α(n−2)g(X,Z)ϕY]−β+δ+(n−2)(n−2)[g(Y,Z)X−g(X,Z)Y].$Let X and Y are orthogonal basis to ξ. Putting Z = ξ and using (2.1), (2.2) and (2.4) in (9.8), we get $C¯(X,Y)ξ=C(X,Y)ξ.$

### Theorem 9.1

An n-dimensinal δ-Lorentzian trans-Sasakian manifold M is Weyl ξ-conformally flat with respect to the semi-symmetric metric connection if and only if the manifold is also Weyl ξ-conformally flat with respect to the metric connection provided that the vector fields are horizontal vector fields.

η-Ricci Solitons and Ricci Solitons in δ-Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection

Let M be 3-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection and V be pointwise collinear with ξ i.e. V = , where b is a function. Then consider the equation [9] $LVg+2S¯+2λg+2μη⊗η=0,$where LV is the Lie derivative operator along the vector field V, is the Ricci curvature tensor field of the metric g and λ and µ are real constants. Then equation (10.1) implies, $g(∇¯Xbξ,Y)+g(∇¯Ybξ,X)+2S¯(X,Y)+2λg(X,Y)+2μη(X)η(Y)=0,$or $bg(∇¯Xξ,Y)+(Xb)η(Y)+bg(∇¯Yξ,X)+(Yb)η(X) +2S¯(X,Y)+2λg(X,Y)+2μη(X)η(Y)=0.$

Using (3.4) in (10.3), we get $bg[−(1+δβ)X−(1+δβ)η(X)ξ−δαϕX,Y]+(Xb)η(Y)+bg[1−(1+δβ)Y−(1+δβ)η(Y)ξ−δαϕY,X]+(Yb)η(X)+2S¯(X,Y)+2λg(X,Y)+2μη(X)η(Y)=0.$$−2b(1+δβ)g(X,Y)−2b(1+δβ)η(Y)η(X)+(Xb)η(Y)+(Yb)η(X)+2S¯(X,Y)+2λg(X,Y)+2μη(X)η(Y)=0.$With the substitution of Y with ξ in (10.5) and using (3.21) for constants α and β, it follows that $(Xb)+(ξb)η(X)−4b(1+δβ)η(X)+2[2(α2+β2−δ(ξβ))−2δβ]η(X)+2λη(X)+2μη(X)=0.$or $(Xb)+(ξb)η(X)+[−4b(1+δβ)+2(2(α2+β2−δ(ξβ))−2δβ+2λ+2μ]η(X)=0.$Again replacing X = ξ in (10.7), we obtain $ξb=−[−2b(1+δβ)+(2(α2+β2−δ(ξβ))−δβ+λ+μ]$Putting (10.8) in (10.7), we obtain $db=[2b(1+δβ)−(2(α2+β2−δ(ξβ))−δβ−λ−μ]η.$By applying d on (10.9), we get $[2b(1+δβ)−(2(α2+β2−δ(ξβ))−δβ−λ−μ]dη=0.$Since ≠ 0 from (10.10), we have $[2b(1+δβ)−(2(α2+β2−δ(ξβ))−δβ−λ−μ]=0.$By using (10.9) and (10.11), we obtain that b is a constant. Hence from (10.5) it is verified $S¯(X,Y)=[b(1+δβ)−λ]g(X,Y)+[b(1+δβ)−μ]η(X)η(Y).$which implies that M is an η-Einstien manifold. This lead to the following:

### Theorem 10.1

In a 3-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, the metric g is an η-Ricci soliton and V is a positive collinear with ξ, then V is a constant multiple of ξ and g is an η-Einstien manifold of the form (10.12) and η-Ricci soliton is expanding or shrinking according as the following relation is positive and negative$λ=−[2b(1+δβ)−(2(α2+β2−δ(ξβ))−δβ−μ].$For µ = 0, we deduce equation (10.12)$S¯(X,Y)=[b(1+δβ)−λ]g(X,Y)+[b(1+δβ)]η(X)η(Y).$Now, we have the following corollary:

### Corollary 10.1

In a 3-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, the metric g is a Ricci soliton and V is a positive collinear with ξ, then V is a constant multiple of ξ and g is an η-Einstien manifold and Ricci soliton is shrinking according as the following relation is negative. For µ = 0, (10.13) reduce to$λ=−[2b(1+δβ)−(2(α2+β2−δ(ξβ))−δβ].$

Here is an example of η-Ricci soliton on δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection.

### Example 10.1

We consider the three dimensional manifold M = [(x, y, z) ∈ ℝ3| z ≠ 0], where (x, y, z) are the Cartesian coordinates in3. Choosing the vector fields$e1=z∂∂x, e2=z∂∂y, e3=−z∂∂z,$which are linearly independent at each point of M. Let g be the Riemannian metric define by $g(e1,e3)=g(e2,e3)=g(e2,e2)=0, g(e1,e1)=g(e2,e2)=g(e3,e3)=δ,$where δ = ±1. Let η be the 1-form defined by η(Z) = ∊g(Z, e3) for any vector field Z on TM. Let ϕ be the (1, 1) tensor field defined by ϕ(e1) = −e2, ϕ(e2) = e1, ϕ(e3) = 0. Then by the linearity property of ϕ and g, we have $ϕ2Z=Z+η(Z)e3, η(e3)=1 and g(ϕZ,ϕW)=g(Z,W)−δη(Z)η(W)$for any vector fields Z, W on M.

Let ∇ be the Levi-Civita connection with respect to the metric g. Then we have $[e1,e2]=0, [e1,e3]=δe1, [e2,e3]=δe2.$

The Riemannian connection ∇ with respect to the metric g is given by $2g(∇XY,Z)=Xg(Y,Z)+Yg(Z,X)−Zg(X,Y) +g([X,Y],Z)−g([Y,Z],X)+g([Z,X],Y).$From above equation which is known as Koszul’s formula, we have $∇e1e3=δe1,∇e2e3=δe2,∇e3e3=0,∇e1e2=0,∇e2e2=−δe3,∇e3e2=0,∇e1e1=−δe3,∇e2e1=0,∇e3e1=0.$Using the above relations, for any vector field X on M, we have $∇Xξ=δ(X−η(X)ξ)$for ξe3, α = 0 and β = 1. Hence the manifold M under consideration is an δ-Lorentzian trans Sasakian of type (0, 1) manifold of dimension three.

Now, we consider this example for semi-symmetric metric connection from (2.9) and (10.14), we obtain: $∇¯e1e3=(1+δ)e1,∇¯e2e3=(1+δ)e2,∇¯e3e3=0,∇¯e1e2=0,∇¯e2e2=−(1+δ)e3,∇¯e3e2=0,∇¯e1e1=−(1+δ)e3,∇¯e2e1=0,∇¯e3e1=0.$Then the Riemannian and the Ricci curvature tensor fields with respect to the semi-symmetric metric connection are given by: $R¯(e1,e2)e2=−(1+δ)2e1,R¯(e1,e3)e3=−δ(1+δ)e2,R¯(e2,e1)e1=−(1+δ)2e2,R¯(e2,e3)e3=−δ(1+δ)e2,R¯(e3,e1)e1=δ(1+δ)e3,R¯(e3,e2)e2=−δ(1+δ)e3, S¯(e1,e1)= S¯(e2,e2)=−(1+δ)(1+2δ), S¯(e3,e3,)=2δ(1+δ).$From (10.14), for $λ=(1+δ)2δ$ and µ = −(1 + δ)(1 + 3δ), the data (g, ξ, λ, µ) is an η-Ricci soliton on (M, ϕ, ξ, η, g) which is expanding.

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