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 (1.1)∂g∂t=−2Ric(g).

Definition 1.1

A Ricci soliton (g, V, λ) on a Riemannian manifold is defined by (1.2)LVg+2S+2λ=0,where S is the Ricci tensor, L_V 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, R̄. S̄ = 0, P̄.S̄ = 0, Weyl conformally flat, Weyl ξ-conformally flat receptively in n-dimensional δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric metric connection.

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 (2.1)ϕ2=X+η(X)ξ,  η∘ϕ=0,  ϕξ=0,(2.2)η(ξ)=−1,(2.3)g(ξ,ξ)=−δ,(2.4)η(X)=δg(X,ξ),(2.5)g(ϕX,ϕY)=g(X,Y)+δη(X)η(Y)for all X, Y ∈ M, 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 C₆ ⊕ C₅ [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 (2.6)(∇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 (2.7)(∇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 (2.8)∇Xξ=δ{−αϕ(X)−β(X+η(X)ξ},and (2.9)(∇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: (2.10)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],(2.11)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],(2.12)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)],(2.13)S(X,ξ)=[((n−1)(α2+β2)−(ξβ)]η(X)+δ((ϕX)α)+(n−2)δ(Xβ),(2.14)S(ξ,ξ)=(n−1)(α2+β2)−δ(n−1)(ξβ),(2.15)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 (2.16)δϕ(gradα)=δ(n−2)(gradβ),and (2.17)2αβ−δ(ξα)=0.The ξ-sectional curvature K_ξ of M is the sectional curvature of the plane spanned by ξ and a unit vector field X. From (2.11), we have (2.18)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)(2.19)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),(2.20)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 (2.21)T¯(X,Y)=∇¯XY−∇¯YX−[X,Y],and (2.22)T¯(X,Y)=η(X)Y−η(Y)X.Moreover, a semi-symmetric connection is called semi-symmetric metric connection if (2.23)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 (2.24)T(X,Y)=η(Y)X−η(X)Y,(2.25)∇¯XY−∇XY=12[T(X,Y)+T′(X,Y)+T′(X,Y)],where (2.26)g(T(Z,X),Y)=g(T′(X,Y),Z).By using (2.4), (2.23) and (2.25), we get (2.27)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,(2.28)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 (2.29)∇¯XY=∇XY+η(Y)X−δg(X,Y)ξ.

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold. The curvature tensor R̄ of M with respect to the semi-symmetric metric connection ∇̄ is defined by (3.1)R¯(X,Y)Z=∇¯X∇¯YZ−∇¯Y∇¯XZ−∇¯[X,Y]Z.By using (2.4), (2.28) and (3.1), we get (3.2)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]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(3.3)(∇¯Xϕ)(Y)=α(g(ϕX,Y)ξ−δη(Y)X)+β(g(ϕX,Y)ξ−(δβ+δ)η(Y)ϕX),(3.4)∇¯Xξ=−(1+δβ)X−(1+δβ)η(X)ξ−δαϕX,(3.5)(∇¯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(3.6)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 (3.7)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 (3.8)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 (3.9)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 (3.10)R¯(X,ξ)ξ=−R¯(ξ,X)ξ.

Now, contracting X in (3.2), we get (3.11)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 S̄ and S are the Ricci tensors of the connections ∇̄ and ∇, respectively on M.

This gives (3.12)Q¯Y=QY−[(δ)(n−2)+β]Y             −(βδ−1)(n−2)η(Y)ξ−α(n−2)ϕY,where Q̄ and Q are Ricci operator with respect to the semi-symmetric metric connection and metric connection respectively and define as g(Q̄Y, Z) = S̄(Y, Z) and g(QY, Z) = S(Y, Z) respectively.

Replace Y = ξ in (3.12) and using (2.15), we get (3.13)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 = e_i and taking summation over i, 1 ≤ i ≤ n − 1 in (3.11), using (2.14) and also the relations r=S(ei,ei)=∑i=1nδiR(ei,ei,ei,ei), we get (3.14)r¯=r−(n−1)[(δ)(n−2)+2β],where r̄ 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(3.15)S¯(ϕY,Z)=−δ(ϕ2Y)α−δ(n−2)(ϕY)β−α(n−2)g(ϕY,ϕZ),(3.16)S¯(Y,ξ)=[(n−1)(α2+β2−δ(ξβ)−δβ(n−1)]η(Y)+δ(n−2)(Yβ)+δ(ϕY)β,(3.17)S¯(ξ,ξ)=[(n−1)(α2+β2−δ(ξβ)−δβ(n−1)]η(Y).

Lemma 3.4

Let M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection, then(3.18)S¯(gradα,ξ)=δ(n−1)(α2+β2(ξβ)−β(n−1)(ξα)−(ξα)(ξβ)+δ(ϕgradα)α+δ(n−2)g(gradα,gradβ),(3.19)S¯(gradβ,ξ)=δ(n−1)(α2+β2(ξβ)−β(n−1)(ξβ)−(ξβ)2+δ(ϕgradβ)α+δ(n−2)g(gradβ)2.

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 P̄ with respect to semi-symmetric metric connection is defined by (4.1)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 (4.2)g(P¯(ϕX,Y)Z,ϕU)=0,where P̄ is the projective curvature tensor with respect to semi-symmetric metric connection.

Now, from (4.1) taking inner product with U, we get (4.3)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 (4.4)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 (4.5)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 (4.6)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.........e_n−1, ξ} be a local orthonormal basis of vector fileds on δ-Lorentzian trans-Sasakian manifold M, then {ϕe₁, ϕe₂........ϕe_n−1, ξ} is also a local orthonormal basis of vector fields on δ-Lorentzian trans-Sasakian manifold M. Now, putting X = U = e_i in equation (4.6) and using (2.2), (2.3),(2.19), (3.11) and (3.16), we have (4.7)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 (4.8)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.

An n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection is said to be ϕ-projectively flat if (5.1)ϕ2(P¯(ϕX,ϕY)ϕZ)=0,where P̄ 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(P̄(ϕ, X, ϕY)ϕZ) = 0 holds if and only if (5.2)g(P¯(ϕX,ϕY)ϕZ,ϕU)=0,for any X, Y, Z, U ∈ TM. Replace Y = ϕY and UϕU in (4.4), we have (5.3)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 (5.4)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 (5.5)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.........e_n−1, ξ} be a local orthonormal basis of vector fileds on δ-Lorentzian trans-Sasakian manifold M, then {ϕe₁, ϕe₂........ϕe_n−1, ξ} is also a local orthonormal basis of vector fields on δ-Lorentzian trans-Sasakian manifold M. Now putting X = U = e_i in equation (5.5) and using (2.1)–(2.5), (2.19), (3.11) and (3.16), we have (5.6)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 (5.7)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.

Now, suppose that M be an n-dimensional δ-Lorentzian trans-Sasakian manifold with a semi-symmetric metric connection satisfying the condition: (6.1)R¯(X,Y).S¯=0.Then, we have (6.2)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 (6.3)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).