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.
© Kyungpook Mathematical Journal. All rights reserved.

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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
roduction

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)=XYYX[X,Y],R(X,Y)Z=XYZYXZ[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)Xη(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 gt=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
iminaries

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,ηϕ=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)ξη(Y)X)+β(g(ϕX,Y)ξη(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)ξδη(Y)X)+β(g(ϕX,Y)ξδη(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)Xη(X)Y]+2αβ[η(Y)ϕXη(X)ϕY]+δ[(Yα)ϕX(Xα)ϕY+(Yβ)ϕ2X(Xβ)ϕ2Y],R(ξ,Y)=(α2+β2)[δg(X,Y)ξη(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)=[((n1)(α2+β2)(ξβ)]η(X)+δ((ϕX)α)+(n2)δ(Xβ),S(ξ,ξ)=(n1)(α2+β2)δ(n1)(ξβ),Qξ=(δ(n1)(α2+β2)(ξβ)ξ+δϕ(gradα)δ(n2)(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(n2)(gradβ),and 2αβδ(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)Z1(n2)[S(Y,Z)XS(X,Y)Y+g(Y,Z)QXg(X,Z)QY]+r(n1)(n2)[g(Y,Z)Xg(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)Yη(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)Xη(X)Y,¯XYXY=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)Zη(Z)X,Y),g(T(X,Y),Z)=η(X)g(Z,Y)δg(X,Y)g(ξ,Z),T(X,Y)=η(X)Yδg(X,Y)ξ,T(Y,X)=η(Y)Xδg(X,Y)ξ.From (2.23), (2.24),(2.26) and (2.27), we get ¯XY=XY+η(Y)Xδg(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)Xδg(X,Y)ξ.

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

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)Yg(Y,Z)X]+(β+δ)[g(Y,Z)η(X)g(X,Z)η(Y)]ξ(βδ1)[η(Y)Xη(X)Y]η(Z),+α[g(ϕX,Z)Yg(ϕY,Z)ϕXg(X,Z)ϕY+g(Y,Z)ϕX],where R(X,Y)Z=XYZ=YXZ[X,Y]Zis 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)ξδη(Y)X)+β(g(ϕX,Y)ξ(δβ+δ)η(Y)ϕX),¯Xξ=(1+δβ)X(1+δβ)η(X)ξδαϕX,(¯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)Yη(Y)ϕX.In view of (2.7), the last equation gives (¯Xϕ)(Y)=α(g(ϕX,Y)ξδη(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)Xδg(X,ξ)ξ.By using (2.2), (2.4) and (2.8), the above equation gives ¯Xξ=(1+δβ)X(1+δβ)η(X)ξδαϕX.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, thenR¯(X,Y)ξ=(α2+β2δβ)[η(X)Yη(Y)X].+(2αβ+δα)[η(Y)ϕXη(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)Yg(Y,ξ)X]+(β+δ)[g(Y,ξ)η(X)g(X,ξ)η(Y)]ξ(βδ1)[η(Y)Xη(X)Y]η(ξ)+α[g(ϕX,ξ)Yg(ϕY,ξ)ϕXg(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+β2δβ)[η(X)Yη(Y)X]+(2αβ+δα)[η(Y)ϕXη(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+β2δβ)[Xη(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+β2δβ)[η(Y)ξY](2αβ+δα+δ(ξα))[ϕY+δ(ξβ)ϕ2Y].

Remark 3.3

Replace Y = X in (3.8), we get R¯(ξ,X)ξ=(α2+β2δβ)[Xη(X)ξ](2αβ+δα+δ(ξα))[ϕXδ(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)(n2)+β]g(Y,Z)(βδ1)(n2)η(Z)η(Y)α(n2)g(ϕY,Z),where and S are the Ricci tensors of the connections ∇̄ and ∇, respectively on M.

This gives Q¯Y=QY[(δ)(n2)+β]Y(βδ1)(n2)η(Y)ξα(n2)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(n1)(α2+β2)ξ(ξβ)ξ2δ(n2)ξ+δϕ(gradα)δ(n2)(gradβ)β(n1)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(n1)[(δ)(n2)+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, thenS¯(ϕY,Z)=δ(ϕ2Y)αδ(n2)(ϕY)βα(n2)g(ϕY,ϕZ),S¯(Y,ξ)=[(n1)(α2+β2δ(ξβ)δβ(n1)]η(Y)+δ(n2)(Yβ)+δ(ϕY)β,S¯(ξ,ξ)=[(n1)(α2+β2δ(ξβ)δβ(n1)]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, thenS¯(gradα,ξ)=δ(n1)(α2+β2(ξβ)β(n1)(ξα)(ξα)(ξβ)+δ(ϕgradα)α+δ(n2)g(gradα,gradβ),S¯(gradβ,ξ)=δ(n1)(α2+β2(ξβ)β(n1)(ξβ)(ξβ)2+δ(ϕgradβ)α+δ(n2)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+β2δ(ξβ)[η(Y)Xη(X)Y],S¯(X,Y)=[(n1)(α2+β2δ(ξβ)δβ(n1)]η(Y),Q¯ξ=δ(n1)(α2+β2ξδ(ξβ)ξ2δ(n2)β(n1)ξ.

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

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)Z1(n1)[S¯(Y,Z)XS¯(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(n1)[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(n1)[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(n1)[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(n1)[S¯(Y,Z)g(ϕX,ϕU)S¯(ϕX,Z)g(Y,ϕU)(δ+β)(n1)g(ϕX,Z)g(Y,ϕU)+(δ+β)(n1)g(Y,Z)g(ϕX,ϕU)(δβ1)(n1)η(Y)η(Z)g(ϕX,ϕU)+(δα)(n1)η(X)η(Z)g(ϕX,ϕU)α(n1)g(X,Z)g(Y,ϕU)α(n1)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)=[(n2)(β+δ)+δ(n1)(α2+β2)(n1)(ξβ)]g(Y,Z)+[δ(n2)(ξβ)+(n2)(βδ1)]η(Y)η(Z)[2δ(n1)αβ+(n1)(ξα)α]g(ϕY,Z)δη(Y)(ϕZ)αδ(n2)(ξβ)η(Y).

If α = 0 and β = constant in (4.7), we get S(Y,Z)=[(n2)(β+δ)+(n1)δβ2]g(Y,Z)+(βδ1)(2n)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
onnection

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(n1)[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(n1)[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(n1)[S¯(ϕY,ϕZ)g(ϕX,ϕU)S¯(ϕX,ϕZ)g(ϕY,ϕU)](δ+β)(n1)g(ϕY,ϕZ)g(ϕX,ϕU)+(δ+β)(n1)g(ϕX,ϕZ)g(ϕY,ϕU)α(n1)g(Y,ϕZ)g(ϕX,ϕU)α(n1)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)=[(n2)(β+α)+δ(n1)(α2+β2)(n1)(ξβ)]g(X,Z)+[2δ(n2)(ξβ)+(n2)(βδ1)]η(Y)η(Z)+[α2δαβ(n1)(n1)(ξα)]g(ϕY,Z)[δ(ϕZ)α+δ(n2)(Zβ)]η(Y)[δ(ϕY)α+δ(n2)(Yβ)]η(Z)If α = 0 and β = constant in (5.6), we get S(Y,Z)=[(n2)(β+δ)+(n1)δβ2]g(Y,Z)+(βδ1)(2n)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
ic#!# = 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(n1)(α2+β2)2β(n1)(α2+β2)2(n1)(α2+β2)(ξβ)+2δβ(n1)(ξβ)δ(ξβ)2+(ϕgradβ)α+(n2)(gradβ)2+δβ2(n2)+δ(n2)(α2+β2)+β(α2+β2)2α2β(n2)δα(ξα)(n2)(ξβ)δβ(ξβ)β(n2)+δα2(n2)]g(Y,Z)+[δ(ϕgradβ)αδ(n2)(gradβ)2+(n2)(βδ1)(α2+β2)+2δα2β(n2)+α(n2)(ξα)+(β+δ)(n2)(ξβ)+β(β+δ)(n2)α2(n2)]η(Y)η(Y)η(Z)+[2δαβ(n1)(α2+β2)+2(n2)αβ2+2αβ(n2)(ξβ)(n1)(α2+β2)(ξα)+δβ(n2)(ξα)+δ(ξα)(ξβ)+(ϕgradα)α+(n2)(g(gradα,gradβ)+α(α2+β2)δα(ξβ)2αβ(n2)(δ)(n2)(δα)+α(n2)]g(ϕY,Z)+[δ(ξα)+2αβδα]S(ϕY,Z)+[(n2)(ξβ)(Zβ)+[δ(α2+β2)(ϕZ)αδ(n2)(α2+β2)(Zβ)+(ξβ)(ϕZ)αβ(ϕZ)α+β(n2)(Zβ)]η(Y)+[δ(α2+β2)(ϕY)α+δ(n2)(α2+β2)(Yβ)2δαβ(ϕ2Y)α2δαβ(n2)(ϕYβ)β(ϕY)αβ(n2)(Yβ)+α(ϕ2Y)α+α(n2)(ϕYβ)]η(Z)(n2)(Yβ)(Zβ)+(n2)(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=[(n1)δβ4+(n2)(gradβ)2+(n2)δβ2+(n2)δβ2(n2)β+(2n3)β3(β+δ)β] and b=[(n2)(βδ1)β2+(n2)(β+δ)β(n2)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
ic#!# = 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)Z1(n1)[S¯(Y,Z)ξS¯(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)(n1)+(β+δ)(n2)(n1)g(Y,Z)S¯(ξ,U)1(n1)S(Y,Z)S¯(ξ,U)(n2)(n1)(βδ1)η(Y)η(Z)S¯(ξ,U)+α2δαβ(n1)(n1)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(n1)δ(ξβ)η(Z)S¯(Y,U)(n2)(n1)δ(Zβ)S¯(Y,U)1(n1)δ(ϕZ)αS¯(Y,U)δ(α2+β2)(n1)+(β+δ)(n2)(n1)g(Y,U)S¯(ξ,Z)1(n1)S(Y,U)S¯(ξ,Z)(n2)(n1)(βδ1)η(Y)η(U)S¯(ξ,Z)+α2δαβ(n1)(n1)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(n1)δ(ξβ)η(Z)S¯(Y,Z)(n2)(n1)δ(Uβ)S¯(Y,Z)1(n1)δ(ϕU)αS¯(Y,Z)=0Putting 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(n1)(α2+β2)+(n2)(βδ)(α2+β2)β(n1)(α2+β2)δ(n2)(βδ1)2(n1)(ξβ)(α2+β2)(n2)(βδ1)(ξβ)2α2β(n2)δα(n2)(ξα)+δα2(n2)+δβ(n1)+δ(ξβ)2+(ϕgradα)α+(n2)(gradβ)2]g(Y,Z)+(n2)β(β+δ)(n2)(α2+β2)+2(n2)δα2β+α(n2)(ξα)+(n2)(β+δ)(ξβ)α2(n2)δ(n2)(gradβ)2δ(ϕgradβ)α]η(Y)(η)(Z)+[α(α2+β2)2δαβ(α2+β2)(n1)2αβ2nδ(ξβ)δβ(ξα)+2αβ(ξβ)2δαβ(n2)(n1)(ξα)+α(n2)(n1)(α2+β2)(ξα)+(n1)δβ(ξα)+δ(ξα)(ξβ)+(ϕgradα)α+)n2)g(gradα,gradβ)]g(ϕY,Z)+[δα+δ(ξα)δα]S(ϕY,Z)+[δ(n+3)(α2+β2)(Zβ)+β(n2)(Zβ)delta(α2+β2)(ϕZ)α+(n1)β(ϕZ)α+(ξβ)(ϕZ)α)]η(Y)+[2δαβ(ϕ2Y)α2δαβ(n2)(ϕYβ)+α(ϕ2Y)α+α(n2)(ϕYβ)+δ(α2+β2)(ϕY)α+δ(n2)(α2+β2)(Yβ)β(ϕY)αβ(n2)(Yβ)]η(Z)(Zα)(ϕ2Y)α(n2)(Zβ)(ϕYβ)(Zβ)(ϕY)αβ(n2)(Yβ).If α = 0 and β = constant in (7.6), we get S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),where a=[(n1)β4+(n2)β2(βδ)+(n1)β3(n2)β(βδ1)+(n1)δβ+(n2)(gradβ)2β(βδ)] and b=[(n2)β(β+δ)+(n2)β2(n2)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
onnection

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)Z1(n2)[S¯(Y,Z)XS¯(X,Z)Y+g(Y,Z)Q¯Xg(X,Z)Q¯Y]+r¯(n1)(n2)[g(Y,Z)Xg(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(n2)[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¯(n1)(n2)[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(n2)[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(n1)(n2)[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(n2)[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(n1)(n2)[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
onnection

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(n2)[S¯(Y,Z)XS¯(X,Z)Y+g(Y,Z)Q¯Xg(X,Z)Q¯Y]+r¯(n1)(n2)[g(Y,Z)Xg(X,Z)Y],Now, taking the inner product of equation (9.1) with U. then we get g(R¯(X,Y)Z,U)=1(n2)[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¯(n1)(n2)[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(n2)[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(n1)(n2)[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(n2)[δS(Y,Z)δη(Y)S(ξ,Z)+g(Y,Z)S(ξ,ξ)δη(Z)S(Y,ξ)]r(n1)(n2)[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(n1)]g(Y,Z)+[δ(n4)(ξβ)+n(α2+β2)δr(n1)]η(Y)η(Z)[2δαβ(n2)+(n2)(ξα)]g(ϕY,Z)[δ(ϕZ)α+δ(Zβ)(n2)]η(Y)[δ(ϕY)α+δ(n2)(Yβ)]η(Z).If α = 0 and d β = constant in (7.6), we get S(Y,Z)=[δβ2+r(n1)]g(Y,Z)+[nβ2δr(n1)]η(Y)η(Z).Therefore S(Y,Z)=ag(Y,Z)+bη(Y)η(Z),where a=[δβ2+r(n1)] and b=[nβ2δr(n1)]. 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)Z1(n2)[S¯(Y,Z)XS¯(X,Z)Y+g(Y,Z)Q¯Xg(X,Z)Q¯Y]+r¯(n1)(n2)[g(Y,Z)Xg(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)Yg(Y,Z)X]+(δ+β)[η(X)g(Y,Z)η(Y)g(X,Z)]ξ(βδ1)η(Z)[η(Y)Xη(X)Y]+α[g(ϕX,Z)Yg(ϕ,Z)Xg(Y,Z)ϕX+g(X,Z)ϕY]+1(n2)(βδ1)(n2)η(Y)η(Z)((δ)(n2)+β)g(Y,Z)X+α(n2)g(ϕY,Z)X+((δ)(n2)+β)g(X,Z)Y+(βδ1)(n2)η(X)η(Z)Yα(n2)g(ϕX,Z)Y((δ)(n2)+β)g(Y,Z)X+(β+δ)(n2)g(Y,Z)η(X)ξα(n2)g(Y,Z)ϕX+((δ)(n2)+β)g(X,Z)Y(β+δ)(n2)g(X,Z)η(Y)ξα(n2)g(X,Z)ϕY]β+δ+(n2)(n2)[g(Y,Z)Xg(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
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+2μηη=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)ξδαϕX,Y]+(Xb)η(Y)+bg[1(1+δβ)Y(1+δβ)η(Y)ξδαϕY,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+β2δ(ξβ))2δβ]η(X)+2λη(X)+2μη(X)=0.or (Xb)+(ξb)η(X)+[4b(1+δβ)+2(2(α2+β2δ(ξβ))2δβ+2λ+2μ]η(X)=0.Again replacing X = ξ in (10.7), we obtain ξb=[2b(1+δβ)+(2(α2+β2δ(ξβ))δβ+λ+μ]Putting (10.8) in (10.7), we obtain db=[2b(1+δβ)(2(α2+β2δ(ξβ))δβλμ]η.By applying d on (10.9), we get [2b(1+δβ)(2(α2+β2δ(ξβ))δβλμ]dη=0.Since ≠ 0 from (10.10), we have [2b(1+δβ)(2(α2+β2δ(ξβ))δβλμ]=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 negativeλ=[2b(1+δβ)(2(α2+β2δ(ξβ))δβμ].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 toλ=[2b(1+δβ)(2(α2+β2δ(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 fieldse1=zx,e2=zy,e3=zz,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)=1andg(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(Xη(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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