CUBO, A Mathematical Journal Vol.22, No¯ 01, (23–37). April 2020 http: // doi. org/ 10. 4067/ S0719-06462020000100023 η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds Sampa Pahan Department of Mathematics, Mrinalini Datta Mahavidyapith Kolkata-700051, India. sampapahan25@gmail.com ABSTRACT In this paper, we study η-Ricci solitons on 3-dimensional trans-Sasakian manifolds. Firstly we give conditions for the existence of these geometric structures and then observe that they provide examples of η-Einstein manifolds. In the case of φ-Ricci symmetric trans-Sasakian manifolds, the η-Ricci soliton condition turns them to Ein- stein manifolds. Afterward, we study the implications in this geometric context of the important tensorial conditions R · S = 0, S · R = 0, W2 · S = 0 and S · W2 = 0. RESUMEN En este art́ıculo estudiamos solitones η-Ricci en variedades trans-Sasakianas tridimen- sionales. En primer lugar damos condiciones para la existencia de estas estructuras geométricas y luego observamos que ellas dan ejemplos de variedades η-Einstein. En el caso de variedades trans-Sasakianas φ-Ricci simétricas, la condición de solitón η-Ricci las convierte en variedades Einstein. A continuación estudiamos las implicancias en este contexto geométrico de las importantes condiciones tensoriales R · S = 0, S · R = 0, W2 · S = 0 y S · W2 = 0. Keywords and Phrases: Trans-Sasakian manifold, η-Ricci solitons. 2010 AMS Mathematics Subject Classification: 53C21, 53C25, 53C44. http://doi.org/10.4067/S0719-06462020000100023 24 Sampa Pahan CUBO 22, 1 (2020) 1 Introduction In 1982, the notion of the Ricci flow was introduced by Hamilton [10] to find a canonical metric on a smooth manifold.The Ricci flow is an evolution equation for Riemannian metric g(t) on a smooth manifold M given by ∂ ∂t g(t) = −2S. A solution to this equation (or a Ricci flow) is a one-parameter family of metrics g(t), parameter- ized by t in a non-degenerate interval I, on a smooth manifold M satisfying the Ricci flow equation. If I has an initial point t0, then (M, g(t0)) is called the initial condition of or the initial metric for the Ricci flow (or of the solution) [14]. Ricci solitons and η-Ricci solitons are natural generalizations of Einstein metrics. A Ricci soli- ton on a Riemannian manifold (M, g) is defined by S + 1 2 LXg = λg where LXg is the Lie derivative along the vector field X, S is the Ricci tensor of the metric and λ is a real constant. If X = ∇f for some function f on M, the Ricci soliton becomes gradient Ricci soliton. Ricci solitons appear as self-similar solutions to Hamiltons’s Ricci flow and often arise as limits of dilations of singularities in the Ricci flow [11]. A soliton is called shrinking, steady and expanding according as λ > 0, λ = 0 and λ < 0 respectively. In 2009, the notion of η-Ricci soliton was introduced by J.C. Cho and M. Kimura [6]. J.C. Cho and M. Kimura proved that a real hypersurface admitting an η-Ricci soliton in a non-flat complex space form is a Hopf-hypersurface [6]. An η-Ricci soliton on a Riemannian manifold (M, g) is defined by the following equation 2S + Lξg + 2λg + 2µη ⊗ η = 0, (1.1) where Lξ is the Lie derivative operator along the vector field ξ, S is the Ricci tensor of the metric and λ, µ are real constants. If µ = 0, then η-Ricci soliton becomes Ricci soliton. In the last few years, many authors have worked on Ricci solitons and their generalizations in different Contact metric manfolds in [1], [7], [8], [9], [12] etc. In 2014, B. Y. Chen and S. Desh- mukh have established the characterizations of compact shrinking trivial Ricci solitons in [5]. Also, in [2], A. Bhattacharyya, T. Dutta, and S. Pahan studied the torqued vector field and established some applications of torqued vector field on Ricci soliton and conformal Ricci soliton. A.M. Blaga [3], D. G. Prakasha and B. S. Hadimani [17] observed η-Ricci solitons on different contact metric manifolds satisfying some certain curvature conditions. CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 25 In this paper we study the existence of η-Ricci soliton on 3-dimensional trans-Sasakian manifold. Next we show that η-Ricci soliton on 3-dimensional trans-Sasakian manifolds becomes η-Einstein Manifold under some conditions. Next we prove that φ-Ricci symmetric trans-Sasakian manifold (M, g) manifold satisfying an η-Ricci soliton becomes an Einstein manifold. Next we give an ex- ample of an η-Ricci soliton on 3-dimensional trans-Sasaian manifold with λ = −2 and µ = 6. Later we obtain some different types of curvature tensors and their properties under certain conditions. 2 Preliminaries The product M̄ = M×R has a natural almost complex structure J with the product metric G being Hermitian metric. The geometry of the almost Hermitian manifold (M̄, J, G) gives the geometry of the almost contact metric manifold (M, φ, ξ, η, g). Sixteen different types of structures on M like Sasakian manifold, Kenmotsu manifold etc are given by the almost Hermitian manifold (M̄, J, G) . The notion of trans-Sasakian manifolds was introduced by Oubina [15] in 1985. Then J. C. Mar- rero [13] have studied the local structure of trans-Sasakian manifolds. In general a trans-Sasakian manifold (M, φ, ξ, η, g, α, β) is called a trans-Sasakian manifold of type (α, β). An n (= 2m + 1) dimensional Riemannian manifold (M, g) is called an almost contact manifold if there exists a (1,1) tensor field φ, a vector field ξ and a 1-form η on M such that φ2(X) = −X + η(X)ξ, (2.1) η(ξ) = 1, η(φX) = 0, (2.2) φξ = 0, (2.3) η(X) = g(X, ξ), (2.4) g(φX, φY) = g(X, Y) − η(X)η(Y), (2.5) g(X, φY) + g(Y, φX) = 0, (2.6) for any vector fields X, Y on M. A 3-dimensional almost contact metric manifold M is called a trans-Sasakian manifold if it satisfies the following condition (∇Xφ)(Y) = α{g(X, Y)ξ − η(Y)X} + β{g(φX, Y)ξ − η(Y)φX}, (2.7) for some smooth functions α, β on M and we say that the trans-Sasakian structure is of type (α, β). For 3-dimensional trans-Sasakian manifold, from (2.7) we have, ∇Xξ = −αφX + β(X − η(X)ξ), (2.8) 26 Sampa Pahan CUBO 22, 1 (2020) (∇Xη)(Y) = −αg(φX, Y) + βg(φX, φY). (2.9) In a 3-dimensional trans-Sasakian manifold, we have R(X, Y)Z = [ r 2 − 2(α2 − β2 − ξβ)][g(Y, Z)X − g(X, Z)Y] − [ r 2 − 3(α2 − β2) + ξβ][g(Y, Z)η(X) − g(X, Z)η(Y)]ξ + [g(Y, Z)η(X) − g(X, Z)η(Y)][φ grad α − grad β] − [ r 2 − 3(α2 − β2) + ξβ]η(Z)[η(Y)X − η(X)Y] − [Zβ + (φZ)α]η(Z)[η(Y)X − η(X)Y] − [Xβ + (φX)α][g(Y, Z)ξ − η(Z)Y] − [Yβ + (φY)α][g(X, Z)ξ − η(Z)X], S(X, Y) = [ r 2 − (α2 − β2 − ξβ)]g(X, Y) − [ r 2 − 3(α2 − β2) + ξβ]η(X)η(Y) − [Yβ + (φY)α]η(X) − [Xβ + (φX)α]η(Y). When α and β are constants the above equations reduce to, R(ξ, X)ξ = (α2 − β2)(η(X)ξ − X), (2.10) S(X, ξ) = 2(α2 − β2)η(X), (2.11) R(ξ, X)Y = (α2 − β2)(g(X, Y)ξ − η(Y)X). (2.12) R(X, Y)ξ = (α2 − β2)(η(Y)X − η(X)Y). (2.13) Definition 2.1. A trans-Sasakian manifold M3 is said to be η-Einstein manifold if its Ricci tensor S is of the form S(X, Y) = ag(X, Y) + bη(X)η(Y), where a, b are smooth functions. CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 27 3 η-Ricci solitons on trans-Sasakian manifolds To study the existence conditions of η-Ricci solitons on 3-dimensional trans-Sasakian manifolds, we prove the following theorem. Theorem 3.1: Let (M, g, φ, η, ξ, α, β) be a 3-dimensional trans-Sasakian manifold with α, β constants (β 6= 0). If the symmetric (0, 2) tensor field h satisfying the condition βh(X, Y) − α 2 [h(φX, Y) + h(X, φY)] = Lξg(X, Y) + 2S(X, Y) + 2µη(X)η(Y) is parallel with respect to the Levi- Civita connection associated to g. Then (g, ξ, µ) becomes an η-Ricci soliton. Proof: We consider a symmetric (0, 2)-tensor field h which is parallel with respect to the Levi- Civita connection (∇h = 0). Then it follows that h(R(X, Y)Z, W) + h(R(X, Y)Z, W) = 0, (3.1) for an arbitary vector field W, X, Y, Z on M. Put X = Z = W = ξ we get h(R(X, Y)ξ, ξ) = 0, (3.2) for any X, Y ∈ χ(M) By using the equation (2.13) h(Y, ξ) = g(Y, ξ)h(ξ, ξ), (3.3) for any Y ∈ χ(M). Differentiating the equation (3.3) covariantly with respect to the vector field X ∈ χ(M) we have h(∇XY, ξ) + h(Y, ∇Xξ) = g(∇XY, ξ)h(ξ, ξ) + g(Y, ∇Xξ)h(ξ, ξ), (3.4) Using the equation (2.8) we have βh(X, Y) − αh(φX, Y) = −αg(φX, Y)h(ξ, ξ) + βh(ξ, ξ)g(X, Y). (3.5) Interchanging X by Y we have βh(X, Y) − αh(X, φY) = −αg(X, φY)h(ξ, ξ) + βh(ξ, ξ)g(X, Y). (3.6) Then adding the above two equations we get βh(X, Y) − α 2 [h(φX, Y) + h(X, φY)] = βh(ξ, ξ)g(X, Y). (3.7) We see that βh(X, Y)− α 2 [h(φX, Y)+h(X, φY)] is a symmetric tensor of type (0, 2). Since Lξg(X, Y), S(X, Y), η(X) = g(X, ξ) and η(Y) = g(Y, ξ) are symmetric tensors of type (0, 2) and λ, µ are real constants, the sum Lξg(X, Y) + 2S(X, Y) + 2µη(X)η(Y) is a symmetric tensor of type (0, 2). 28 Sampa Pahan CUBO 22, 1 (2020) Therefore, we can take the sum as an another symmetric tensor field of type (0, 2). Hence for we can assume that βh(X, Y) − α 2 [h(φX, Y) + h(X, φY)] = Lξg(X, Y) + 2S(X, Y) + 2µη(X)η(Y). Then we compute βh(ξ, ξ)g(X, Y) = Lξg(X, Y) + 2λg(X, Y) + 2µη(X)η(Y). As h is parallel so, h(ξ, ξ) is constant. Hence, we can write h(ξ, ξ) = − 2 β λ where β is constant and β 6= 0. So, from the equation (3.7) we have βh(X, Y) − α 2 [h(φX, Y) + h(X, φY)] = −2λg(X, Y), (3.8) for any X, Y ∈ χ(M). Therefore Lξg(X, Y) + 2S(X, Y) + 2µη(X)η(Y) = −2λg(X, Y) and so (g, ξ, µ) becomes an η-Ricci soliton. Corollary 3.2: Let (M, g, φ, η, ξ, α, β) be a 3-dimensional trans-Sasakian manifold with α, β constants (β 6= 0). If the symmetric (0, 2) tensor field h admitting the condition βh(X, Y) − α 2 [h(φX, Y) + h(X, φY)] = Lξg(X, Y) + 2S(X, Y) is parallel with respect to the Levi-Civita connec- tion associated to g with λ = 2n. Then (g, ξ) becomes a Ricci soliton. Next theorem shows the necessary condition for the existence of η-Ricci soliton on 3-dimensional trans-Sasakian manifolds. Theorem 3.3: If 3-dimensional trans-Sasakian manifold satisfies an η-Ricci soliton then the man- ifold becomes η-Einstein manifold with α and β constants. Proof: From the equation (1.1) we get 2S(X, Y) = −g(∇Xξ, Y) − g(X, ∇Yξ) − 2λg(X, Y) − 2µη(X)η(Y). (3.9) By using the equation (2.8) we get S(X, Y) = −(β + λ)g(X, Y) + (β − µ)η(X)η(Y) (3.10) and S(X, ξ) = −(λ + µ)η(X). (3.11) Also from (2.11) we have λ + µ = 2(β2 − α2). (3.12) The Ricci operator Q is defined by g(QX, Y) = S(X, Y). Then we get QX = (µ − β + 2(α2 − β2))X + (β − µ)η(X)ξ. (3.13) CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 29 Then we can easily see that the manifold is an η-Einstein manifold. We know a manifold is φ-Ricci symmetric if φ2 ◦ ∇Q = 0. Now we prove the next theorem. Theorem 3.4: If a φ-Ricci symmetric trans-Sasakian manifold (M, g) satisfies an η-Ricci soliton then µ = β, λ = 2(β2 − α2) − β and (M, g) is an Einstein manifold. Proof: From the equation (3.13) we have (∇XQ)Y = ∇XQY − Q(∇XY) = −α(β − µ)η(Y)φX + β(β − µ)η(Y)X − (β − µ)η(Y)η(X)ξ +(β − µ)[−αg(φX, Y) + βg(φX, φY)]ξ. Now applying φ2 both sides we have µ = β, λ = 2(β2 −α2)−β and (M, g) is an Einstein manifold. We construct an example of η-Ricci soliton on 3-dimensional trans-Sasakian manifolds in the The next section. 4 Example of η-Ricci solitons on 3-dimensional trans-Sasakian manifolds We consider the three dimensional manifold M = {(x, y, z) ∈ R3 : y 6= 0} where (x, y, z) are the standard coordinates in R3. The vector fields e1 = e 2z ∂ ∂x , e2 = e 2z ∂ ∂y , e3 = ∂ ∂z are linearly independent at each point of M. Let g be the Riemannian metric defined by gij = { 1 for i = j, 0 for i 6= j. Let η be the 1-form defined by η(Z) = g(Z, e3) for any Z ∈ χ(M 3). Let φ be the (1, 1) tensor field defined by φ(e1) = e2, φ(e2) = −e1, φ(e3) = 0. Then using the linearity property of φ and g we have η(e2) = 1, φ 2(Z) = −Z + η(Z)e2, g(φZ, φW) = g(Z, W) − η(Z)η(W), for any Z, W ∈ χ(M3). Thus for e2 = ξ, (φ, ξ, η, g) defines an almost contact metric structure on M. Now, after some calculation we have, 30 Sampa Pahan CUBO 22, 1 (2020) [e1, e3] = −2e1, [e2, e3] = −2e2, [e1, e2] = 0. The Riemannian connection ∇ of the metric is given by the Koszul’s formula which is 2g(∇XY, Z) = Xg(Y, Z) + Yg(Z, X) − Zg(X, Y) − g(X, [Y, Z]) − g(Y, [X, Z]) + g(Z, [X, Y]). By Koszul’s formula we get, ∇e1e1 = 2e3, ∇e2e1 = 0, ∇e3 e1 = 0, ∇e1e2 = 0, ∇e2e2 = 2e3, ∇e3e2 = 0, ∇e1e3 = −2e1, ∇e2e3 = −2e2, ∇e3e3 = 0. From the above it can be easily shown that M3(φ, ξ, η, g) is a trans-Sasakian manifold of type (0, −2). Here R(e1, e2)e2 = −4e1, R(e3, e2)e2 = 4e2, R(e1, e3)e3 = −4e1, R(e2, e3)e3 = −4e2, R(e3, e1)e1 = −4e2, R(e2, e1)e1 = 4e3. So, we have S(e1, e1) = 0, S(e2, e2) = 0, , S(e3, e3) = −8. (4.1) From the equation (1.1) we get λ = −2 and µ = 6. Therefore, (g, ξ, λ, µ) is an η-Ricci soliton on M3(φ, ξ, η, g). In the next sections we consider η-Ricci Solitons on 3-dimensional trans-Sasakian manifolds satis- fying some curvature conditions. 5 η-Ricci solitons on 3-dimensional trans-Sasakian mani- folds satisfying R(ξ, X) · S = 0 First we suppose that 3-dimensional trans-Sasakian manifolds with η-Ricci solitons satisfy the con- dition R(ξ, X) · S = 0. Then we have S(R(ξ, X)Y, Z) + S(Y, R(ξ, X)Z) = 0 CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 31 for any X, Y, Z ∈ χ(M). Using the equations (2.12), (3.10), (3.11) we get (β − µ)g(X, Y)η(Z) + (β − µ)g(X, Z)η(Y) − 2(β − µ)η(X)η(Y)η(Z) = 0. Put Z = ξ we have (β − µ)g(X, Y) − (β − µ)η(X)η(Y) = 0. Setting X = φX and Y = φY in the above equation we get (β − µ)g(φX, φY) = 0. Again using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β. Also we can easily see that M is an Einstein manifold. So we have the following theorem. Theorem 5.1: If a 3-dimensional trans-Sasakian manifold (M, g, φ, η, ξ, α, β) with α, β constants admitting an η-Ricci soliton satisfies the condition R(ξ, X) · S = 0 then µ = β, λ = 2(β2 − α2) − β and M is an Einstein manifold. Corollary 5.2: A 3-dimensional trans-Sasakian manifold with α, β constants satisfies the condi- tion R(ξ, X) · S = 0, there is no Ricci soliton with the potential vector field ξ. 6 η-Ricci solitons on 3-dimensional trans-Sasakian mani- folds satisfying S(ξ, X) · R = 0 We consider 3-dimensional trans-Sasakian manifolds with η-Ricci solitons satisfying the condition S(ξ, X) · R = 0. 32 Sampa Pahan CUBO 22, 1 (2020) So we have S(X, R(Y, Z)W)ξ − S(ξ, R(Y, Z)W)X + S(X, Y)R(ξ, Z)W − S(ξ, Y)R(X, Z)W +S(X, Z)R(Y, ξ)W − S(ξ, Z)R(Y, X)W + S(X, W)R(Y, Z)ξ − S(ξ, W)R(Y, Z)X = 0. Taking inner product with ξ then the above equation becomes S(X, R(Y, Z)W) − S(ξ, R(Y, Z)W)η(X) + S(X, Y)η(R(ξ, Z)W) −S(ξ, Y)η(R(X, Z)W) + S(X, Z)η(R(Y, ξ)W) − S(ξ, Z)η(R(Y, X)W) + S(X, W)η(R(Y, Z)ξ) − S(ξ, W)η(R(Y, Z)X) = 0. (6.1) Put W = ξ and using the equations (2.10), (2.12), (3.10), (3.11) we get − (β + λ)g(X, R(Y, Z)ξ) + (λ + µ)η(R(Y, Z)X) = 0. (6.2) Also we have η(R(Y, Z)X) = −g(X, R(Y, Z)ξ). So from the equation (6.2) we get (β + 2λ + µ)g(X, R(Y, Z)ξ) = 0. Again using the equation (3.12) we have µ = β + 4(β2 − α2), λ = −[2(β2 − α2) + β]. So we have the following theorem. Theorem 6.1: If a 3-dimensional trans-Sasakian manifold (M, g, φ, η, ξ, α, β) with α, β constants admitting an η-Ricci soliton satisfies the condition S(ξ, X) · R = 0 then µ = β + 4(β2 − α2), λ = −[2(β2 − α2) + β]. CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 33 Corollary 6.2: A 3-dimensional trans-Sasakian manifold with α, β constants satisfies the condi- tion S(ξ, X) · R = 0, there is no Ricci soliton with the potential vector field ξ. 7 η-Ricci solitons on 3-dimensional trans-Sasakian mani- folds satisfying W2(ξ, X) · S = 0 Definition 7.1. Let M be 3-dimensional trans-Sasakian manifold with respect to semi-Symmetric metric connection. The W2-curvature tensor of M is defined by [16] W2(X, Y)Z = R(X, Y)Z + 1 2 (g(X, Z)QY − g(Y, Z)QX). (7.1) We assume 3-dimensional trans-Sasakian manifolds with η-Ricci solitons satisfying the condition W2(ξ, X) · S = 0. Then we have S(W2(ξ, X)Y, Z) + S(Y, W2(ξ, X)Z) = 0 for any X, Y, Z ∈ χ(M). Using the equations (2.12), (3.10), (3.11), (7.1) we get [− (β + λ) 2 (λ + µ) + (β + λ)2 2 + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(X, Y)η(Z) +[ (β + λ)2 2 − (β + λ) 2 (λ + µ) + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(X, Z)η(Y) +[−(β + λ)(β − µ) − 2(β − µ)(α2 − β2) − (β − µ)(λ + µ)]η(X)η(Y)η(Z) = 0. Put Z = ξ in the above equation we get [− (β + λ) 2 (λ + µ) + (β + λ)2 2 + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 ]g(X, Y) +[ (β + λ)2 2 − (β + λ) 2 (λ + µ) + (β − µ)(α2 − β2) + (λ + µ) (β − µ) 2 34 Sampa Pahan CUBO 22, 1 (2020) −(β + λ)(β − µ) − 2(β − µ)(α2 − β2) − (β − µ)(λ + µ)]η(X)η(Y) = 0. Setting X = φX and Y = φY in the above equation we get (β − µ)( (β + 2λ + µ + 2(α2 − β2)) 2 )g(φX, φY) = 0. Again using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. So we have the following theorem. Theorem 7.1: If a 3-dimensional trans-Sasakian manifold (M, g, φ, η, ξ, α, β) with α, β constants admitting an η-Ricci soliton satisfies the condition W2(ξ, X)·S = 0 then µ = β, λ = 2(β 2 −α2)−β or µ = 2(β2 − α2) + β, λ = −β. Corollary 7.2: A 3-dimensional trans-Sasakian manifold with α, β constants satisfies the condi- tion W2(ξ, X) · S = 0, there is no Ricci soliton with the potential vector field ξ. 8 η-Ricci solitons on 3-dimensional trans-Sasakian mani- folds satisfying S(ξ, X) · W2 = 0 Suppose that 3-dimensional trans-Sasakian manifolds with η-Ricci solitons satisfy the condition S(ξ, X) · W2 = 0. So we have S(X, W2(Y, Z)V)ξ − S(ξ, W2(Y, Z)V)X + S(X, Y)W2(ξ, Z)V − S(ξ, Y)W2(X, Z)V +S(X, Z)W2(Y, ξ)V − S(ξ, Z)W2(Y, X)V + S(X, V)W2(Y, Z)ξ − S(ξ, V)W2(Y, Z)X = 0. CUBO 22, 1 (2020) η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds 35 Taking inner product with ξ then the above equation becomes S(X, W2(Y, Z)V) − S(ξ, W2(Y, Z)V)η(X) + S(X, Y)η(W2(ξ, Z)V) −S(ξ, Y)η(W2(X, Z)V) + S(X, Z)η(W2(Y, ξ)V) − S(ξ, Z)η(W2(Y, X)V) + S(X, V)η(W2(Y, Z)ξ) − S(ξ, V)η(W2(Y, Z)X) = 0. (8.1) Put V = ξ and using the equations (2.10), (2.12), (3.10), (3.11), (7.1) we get − (β + λ)g(X, W2(Y, Z)ξ) + (λ + µ)η(W2(Y, Z)X) = 0. (8.2) Using the equations (3.10), (3.11), (7.1) then the equation (8.2) becomes [(β + λ)2 + (λ + µ)2 + 2(α2 − β2)(β + 2λ + µ)]g(X, R(Y, Z)ξ) = 0. Using the equation (3.12) we have µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. So we have the following theorem. Theorem 8.1: If Let a 3-dimensional trans-Sasakian manifold (M, g, φ, η, ξ, α, β) with α, β constants admitting an η-Ricci soliton satisfies the condition S(ξ, X) · W2 = 0 then µ = β, λ = 2(β2 − α2) − β or µ = 2(β2 − α2) + β, λ = −β. Corollary 8.2: A 3-dimensional trans-Sasakian manifold with α, β constants satisfies the condi- tion S(ξ, X) · W2 = 0, there is no Ricci soliton with the potential vector field ξ. Acknowledgement: The author wish to express her sincere thanks and gratitude to the ref- eree for valuable suggestions towards the improvement of the paper. 36 Sampa Pahan CUBO 22, 1 (2020) References [1] C. S. Bagewadi, G. Ingalahalli, S. R. Ashoka, A stuy on Ricci solitons in Kenmotsu Manifolds, ISRN Geometry, (2013), Article ID 412593, 6 pages. [2] A. Bhattacharyya, T. Dutta, and S. Pahan, Ricci Soliton, Conformal Ricci Soliton And Torqued Vector Fields, Bulletin of the Transilvania University of Brasov Series III: Mathematics, Infor- matics, Physics,, Vol 10(59), No. 1 (2017), 39-52. [3] A. M. Blaga, Eta-Ricci solitons on para-Kenmotsu manifolds, Balkan Journal of Geometry and Its Applications, Vol.20, No.1, 2015, pp. 1-13. [4] C. Călin, M. 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Introduction Preliminaries -Ricci solitons on trans-Sasakian manifolds Example of -Ricci solitons on 3-dimensional trans-Sasakian manifolds -Ricci solitons on 3-dimensional trans-Sasakian manifolds satisfying R(, X)S=0 -Ricci solitons on 3-dimensional trans-Sasakian manifolds satisfying S(, X)R=0 -Ricci solitons on 3-dimensional trans-Sasakian manifolds satisfying W2(, X)S=0 -Ricci solitons on 3-dimensional trans-Sasakian manifolds satisfying S(, X)W2=0