CUBO A Mathematical Journal Vol.19, No¯ 02, (33–48). June 2017 Some geometric properties of η− Ricci solitons and gradient Ricci solitons on (lcs)n−manifolds S. K. Yadav1, S. K. Chaubey2 and D. L. Suthar3 1Department of Mathematics Poornima college of Engineering, ISI-6, RIICO, Institutional Area, Sitapura, Jaipur-302022, Rajasthan, India. prof−sky16@yahoo.com 2Section of Mathematics, Department of Information Technology, Shinas College of Technology, Shinas, P.O. Box 77 Postal Code 324, Oman. sk22−math@yahoo.co.in, sudhakar.chaubey@shct.edu.om 3Department of Mathematics Wollo University, P. O. Box: 1145, Dessie, South Wollo, Amhara Region, Ethiopia. dlsuthar@gmail.com ABSTRACT In the context of para-contact Hausdorff geometry η−Ricci solitons and gradient Ricci solitons are considered on manifolds. We establish that on an (LCS)n−manifold (M, φ, ξ, η, g), the existence of an η−Ricci soliton implies that (M, g) is quasi-Einstein. We find conditions for Ricci solitons on an (LCS)n−manifold (M, φ, ξ, η, g) to be shrinking, steady and expanding. At the end we show examples of such manifolds with η−Ricci solitons. RESUMEN En el contexto de geometŕıa para-contacto Hausdorff, consideramos η−Ricci solitones y Ricci solitones gradientes en variedades. Establecemos que en una (LCS)n−variedad (M, φ, ξ, η, g), la existencia de un η−Ricci solitón implica que (M, g) es casi-Einstein. Encontramos condiciones para que los Ricci solitones en una (LCS)n−variedad (M, φ, ξ, η, g) sean contractivos, estables o expansivos. Al concluir, mostramos ejemp- los de dichas variedades con η−Ricci solitones. Keywords and Phrases: η−Ricci solitons, gradient Ricci solitons, (LCS)n−manifold. 2010 AMS Mathematics Subject Classification: 53C25, 53C15, 53C21. 34 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) 1 Introduction In 2003, Shaikh [34] introduced the notion of Lorentzian concircular structure manifolds (briefly, (LCS)n−manifolds) with an example, which generalize the notion of LP-Sasakian manifolds intro- duced by Matsumoto [27] and also by Mihai and Rosca [26]. Then Shaikh and Baishya ([32], [33]) investigated the application of (LCS)n−manifolds to the general theory of relativity and cosmology. The (LCS)n−manifolds are also studied by Atceken et al. ([1], [2], [21]), D. Narain and S. Yadav [30], S. Yadav et al. ([39]-[42]), Shaikh and his co-authors ([35], [36]) and many others. Ricci solitons represent a natural generalization of Einstein metrics on a Riemannian manifold, being generalized fixed points of Hamilton’s Ricci flow ∂ ∂t g = −2S [24]. The evolution equation defining the Ricci flow is a kind of nonlinear diffusion equation, an analogue of heat equation for metrics. Under Ricci flow, a metric can be improved to evolve into more canonical one by smoothing out its irregularities, depending on the Ricci curvature of the manifold, it will expand in the directions of negative Ricci curvature and shrink in the positive case. Ricci solitons have been studied in many contexts: on Kähler manifolds [18], on contact and Lorentzian manifolds ([3], [15], [25], [31], [38]), on Sasakian ([19], [20]), α−Sasakian [25] and K−contact manifolds [31], on Kenmotsu ([4], [28]) and f−Kenmotsu manifolds [15] and by ([16], [43]) etc. In paracontact geometry, Ricci soliton firstly appeared in the paper of G. Calvoruso and D. Perrone [13]. C. L. Bejan and M. Crasmarean studied Ricci solitons on 3−dimensional normal paracontact manifolds [5]. A more general notion is that of η−Ricci soliton introduced by J. T. Cho and M. Kimura [6], which was treated by C. Calin and M. Crasmareanu on Hopf hypersurfaces in complex space forms [14]. 2 (LCS)n−manifolds (M, φ, ξ, η, g) An n−dimensional Lorentzian manifold M is a smooth connected paracontact Hausdorff manifold with Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type (0, 2) such that for each point p ∈ M the tensor gp : TpM × TpM → Re is a non degenerate inner product of signature (−, +, ..., +), where TpM denotes the tangent space of M at p and Re is the real number space. A non-zero tangent vector v ∈ TpM is said to be timelike (resp., non-spacelike, null and spacelike) if it satisfies gp(v, v) < 0 (resp., < 0, =, > 0) [29]. Definition 2.1. In a Lorentzian manifold (M, g) a vector field ρ defined by g(X, ρ) = A(X), for any X ∈ χ(M) is said to be a concircular vector field [44] if (∇XA) (Y) = α {g(X, Y) + ω(X)A(Y)} , where α is a non-zero scalar and ω is a closed 1−form. Here ∇ denotes the operator of covariant differentiation with respect to the Lorentzian metric g. CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 35 Let M be a Lorentzian manifold admitting a unit timelike concircular vector field ξ, called the generator of the manifold. Then we have g(ξ, ξ) = −1. (2.1) Since ξ is the unit concircular vector field, then there exists a non-zero 1−form η such that for g(X, ξ) = η(X), (2.2) the following equation holds (∇Xη) (Y) = α {g(X, Y) + η(X)η(Y)} (α 6= 0), (2.3) that is, ∇Xξ = α{X + η(X)ξ}, for all vector fields X, Y on M, where α is a non-zero scalar function satisfies ∇Xα = (Xα) = dα(X) = ρη(X), (2.4) ρ being a certain scalar function given by ρ = −(ξα). If we put φX = 1 α ∇Xξ, (2.5) then from (2.3) and (2.5), we have φX = X + η(X)ξ, (2.6) from which it follows that φ is a (1, 1) tensor and called the structure tensor of the manifold. Thus the Lorentzian manifold M together with the unit timelike concircular vector field ξ, its associated 1−form η and (1, 1) tensor field φ, is said to be a Lorentzian concircular structure manifold (briefly (LCS)n−manifold) [34]. Specially, if we take α = 1, then we can obtain the LP-Sasakian structure of Matsumoto [27]. In a (LCS)n−manifold (n > 2), the following relations hold ([32]-[35]): a) η(ξ) = −1, b) φξ = 0, c) φ2X = X + η(X)ξ, (2.7) d) η(φX) = 0, e) g(φX, φY) = g(X, Y) + η(X)η(Y), η(R(X, Y)Z) = (α2 − ρ) {g(Y, Z)η(X) − g(X, Z)η(Y)} , (2.8) R(X, Y)ξ = (α2 − ρ) {η(Y)X − η(X)Y } , (2.9) R(ξ, X)Y = (α2 − ρ) {g(X, Y)ξ − η(Y)X} , (2.10) R(ξ, X)ξ = (α2 − ρ) {η(X)ξ + X} , (2.11) (∇Xφ) (Y) = α {g(X, Y)ξ + 2η(X)η(Y)ξ + η(Y)X} , (2.12) S(X, ξ) = (n − 1)(α2 − ρ)η(X), (2.13) S(φX, φY) = S(X, Y) + (n − 1)(α2 − ρ)η(X)η(Y), (2.14) (Xρ) = dρ(X) = βη(X), (2.15) for any vector fields X, Y, Z on M and β = −(ξρ) is a scalar function, where R is the curvature tensor and S is the Ricci tensor of the manifold. 36 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) 3 η−Ricci solitons on (LCS)n−manifolds Let (M, φ, ξ, η, g) be a (LCS)n−manifold. We follow the equation Lξg + 2S + 2λg + 2µη ⊗ η = 0, (3.1) where Lξ is the Lie-derivative operator along the vector field ξ, S is the Ricci tensor field of the metric g, λ and µ are real constants. We write Lξg in term of the Levi-Civita connection ∇, we obtain (Lξg)(X, Y) = g(∇Yξ, X) + g(Y, ∇Xξ) = 2α[g(X, Y) + η(X)η(Y)]. (3.2) In view of (3.1) and (3.2), we get QX = −(α + λ)X − (α + µ)η(X)ξ, (3.3) r = −nλ − (n − 1)α + µ, (3.4) S(X, Y) = −(α + λ)g(X, Y) − (α + µ)η(X)η(Y), (3.5) S(X, ξ) = S(ξ, X) = (µ − λ)η(X), (3.6) µ − λ = (n − 1)(α2 − ρ), (3.7) for any X, Y ∈ χ(M). Here r is the scalar curvature and Q denotes the Ricci operator corresponding to S, that is, S(X, Y) = g(QX, Y), for all X, Y on M. The structure (g, ξ, λ, µ) that follows the equation (3.1) is said to be an η−Ricci soliton to (M, g) [6]. In particular, if µ = 0, (g, ξ, λ) is a Ricci soliton [24] and it is called shrinking, steady, or expanding according as λ is negative, zero or positive, respectively [12]. Proposition 3.1. On a (LCS)n−manifold (M, φ, ξ, η, g) the following relations hold (i) η(∇Xξ) = 0, (ii) ∇ξξ = 0, (iii) ∇η = α{g + η ⊗ η}, (iv) ∇ξη = 0, (v) Lξφ = 0, (vi) Lξη = 0, (vii) Lξ(η ⊗ η) = 0, (viii) Lξg = 2α(g + η ⊗ η), where ∇ is the Levi-Civita connection associated to g. Also η is closed, the distribution is involutive and tensor field of φ vanishes identically, i. e., the structure is normal. Proof. Since (∇Xφ)(Y) = α{g(X, Y)ξ + 2η(X)η(Y)ξ + η(Y)X}, this indicates that ∇XφY − φ(∇XY) = α{g(X, Y)ξ + 2η(X)η(Y)ξ + η(Y)X}. Taking Y = ξ in above equation, we have φ(∇Xξ) = αφX. Applying φ both sides, we get ∇Xξ + η(∇Xξ) = α{X + η(X)ξ}. CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 37 Since ∇Xξ = αφX and X(g(ξ, ξ)) = 2g(∇Xξ, ξ), therefore η(∇Xξ) = 0 and ∇ξξ = 0. Also (∇Xη)(Y) = αg(Y, φX) = α{g(X, Y) + η(X)η(Y)}, this implies that ∇η = α{g + η ⊗ η}, i.e., ∇ξη = 0. In view of definition of Lie-derivative, we get (Lξφ)(X) = [ξ, φX] − φ([ξ, X]) = ∇ξφX − φ(∇ξX) = (∇ξφ)(X) = 0, i.e., Lξφ = 0. Also, ( Lξη)(X) = ξ(η(X) − η([ξ, X]) = g(X, ∇ξξ) + g(∇Xξ, ξ) = 0, i.e., Lξη = 0. Further we compute (Lξ(η ⊗ η))(X, Y) = ξ(η(X)η(Y)) − η([ξ, X])η(Y) − η(X)η([ξ, Y]), which implies that (Lξ(η ⊗ η))(X, Y) = η(X)g(Y, ∇ξξ) − η(Y)g(X, ∇ξξ) = 0, i.e., Lξ(η ⊗ η) = 0. Again (Lξg)(X, Y) = ξ g(X, Y) − g([ξ, X], Y) − g(X, [ξ, Y]), implies that (Lξg)(X, Y) = α[g(φX, Y) + g(X, φY)]. Using (2.6), we get Lξg = 2α(g + η ⊗ η). At last (dη)(X, Y) = X(η(Y)) − Y(η(X)) − η([X, Y]), that implies (dη)(X, Y) = g(Y, ∇Xξ) − g(X, ∇Yξ) = α{g(Y, X) + η(X)η(Y)} − α{g(X, Y) + η(X)η(Y)} = 0. Finally, Nφ(X, Y) = φ 2[X, Y] + [φX, φY] − φ[φX, Y] − φ[X, φY]. This yields that Nφ(X, Y) = φ 2(∇XY) − φ 2(∇YX) − φ(∇XφY) + φ(∇YφX) +∇φXφY − φ(∇φXY) − ∇φYφX + φ(∇φYX) = 0. Thus the structure is normal. In [17], S. Chandra et al. proved that a second order parallel symmetric tensor on a (LCS)n− manifold with α2 − ρ 6= 0, is a constant multiple of the Ricci tensor. Thus we apply this concept for η−Ricci soliton and we prove the following result: 38 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) Theorem 3.2. Let (M, φ, ξ, η, g) be a (LCS)n−manifold. Assume that the symmetric (0, 2) tensor field h = Lξg + 2S + 2µη ⊗ η is parallel with respect to the Levi-Civita connection associated to g, then (g, ξ, λ) yields an η−Ricci soliton on M. Proof. Since h(ξ, ξ) = (Lξg)(ξ, ξ) + 2S(ξ, ξ) + 2µη(ξ)η(ξ) = 2λ, implies that λ = 1 2 h(ξ, ξ). (3.8) In [17], we have h(X, Y) = −h(ξ, ξ)g(X, Y), X, Y ∈ χ(M). (3.9) Therefore, Lξg + 2S + 2µη ⊗ η = 2λg. Our theorem is proved. If µ = 0, it follows that Lξg + 2S + 2(n − 1)(α 2 − ρ)g = 0. Thus we conclude that Corollary 3.3. On a (LCS)n−manifold (M, φ, ξ, η, g) with the property that the symmetric (0, 2) tensor field h = Lξg+2S is parallel with respect to the Levi-Civita connection associated to g, then the equation (3.1), for µ = 0 and λ = −[(n − 1)(α2 − ρ)], defines a Ricci soliton. As a consequence of the existence of η−Ricci soliton on a (LCS)n−manifold. From (3.1), we state that Corollary 3.4. If the equation (3.1) define an η−Ricci soliton on a (LCS)n− manifold, then (M, g) is quasi-Einstein. Since the manifold is quasi-Einstein, if the Ricci tensor field S is a linear combination (with real scalar λ and µ, respectively, with µ 6= 0) of g and the tensor product of a non-zero 1−form η satisfying (2.2) and for an Einstein if S is co-linear with g ([13], [23]). Theorem 3.5. If (M, φ, ξ, η, g) be a (LCS)n−manifold and equation (3.1) define an η−Ricci soliton on (M, g), then (i) Q ◦ φ = φ ◦ Q, (ii) Q and S are parallel along ξ. Proof. The prove of (i) follows by direct computation. For (ii) we have (∇ξQ)X = ∇ξQX − Q(∇ξX) and (∇ξS)(X, Y) = ξ(S(X, Y)) − S(∇ξX, Y) − S(X, ∇ξY). In view of (3.3) and (3.5), above equation leads the result. In a particular case if the manifold is φ−Ricci symmetric, then φ2 ◦ ∇Q = 0, therefore we state the following proposition as: CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 39 Proposition 3.6. If a (LCS)n−manifold (M, φ, ξ, η, g) is φ−Ricci symmetric and equation (3.1) leads to η−Ricci soliton, then µ = −α, λ = −[(n − 1)(α2 − ρ) + α] and the manifold reduces to Einstein. Proof. We compute (∇XQ)Y = ∇XQY − Q(∇XY). In view of (3.3) above equation takes the form (µ + α){X + η(X)ξ}η(Y) = 0, for X, Y ∈ χ(M). From above it follows that µ = −α, λ = −[(n − 1)(α2 − ρ) + α], and S = (n − 1)(α2 − ρ)g. As a weaker version of local symmetry, the notion of local-symmetric Sasakian manifold was introduced by Takahashi [37]. Chaubey ([7]- [11]) studied the properties of symmetric spaces in different extent. Shaikh et al. ([35], [36]) studied locally φ−symmetric and locally φ−recurrent (LCS)n−manifolds. Hui [22] studied φ−pseudosymmetric (LCS)n−manifolds and obtained the form of Ricci tensor S as S(X, Y) = { α(n − 1)(α2 − ρ) α + A(ξ) } g(X, Y) + { (n − 1)(α2 − ρ)A(ξ) α + A(ξ) } η(X)µ(Y), (3.10) provided α + A(ξ) 6= 0. Theorem 3.7. If the tensor field Lξg + 2S on a φ−pseudo Ricci symmetric (LCS)n− manifold with α2 − ρ 6= 0 is parallel with respect to Levi-Civita connection associated to g, then for µ = 0 the Ricci soliton (g, ξ, λ) is shrinking, steady and expanding according as (α 2 −ρ){A(ξ)−α} α+A(ξ) < 0, A(ξ) = α and (α 2 −ρ){A(ξ)−α} α+A(ξ) > 0 respectively. Proof. Let h is a (0, 2) symmetric parallel tensor field on (LCS)n−manifold. In view of (3.1), we obtain h(X, Y) = (Lξg)(X, Y) + 2S(X, Y). (3.11) Using (3.2) and (3.10), equation (3.11) reduces to h(X, Y) = 2α[g(X, Y) + η(X)η(Y)] + 2 { α(n−1)(α 2 −ρ) α+A(ξ) } g(X, Y) +2 { (n−1)(α 2 −ρ)A(ξ) α+A(ξ) } η(X)µ(Y). (3.12) Replacing X = Y = ξ in (3.12), we get h(ξ, ξ) = { 2(n − 1)(α2 − ρ){A(ξ) − α} α + A(ξ) } . (3.13) In view of (3.8) and (3.13), we obtain λ = { (n − 1)(α2 − ρ){A(ξ) − α} α + A(ξ) } . 40 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) Since n > 1, α2 − ρ 6= 0 and α + A(ξ) 6= 0 , we conclude that λ > 0 if (α 2 −ρ){A(ξ)−α} α+A(ξ) > 0, λ = 0 if A(ξ) = α and λ < 0 if (α 2 −ρ){A(ξ)−α} α+A(ξ) < 0. Our theorem is proved. Corollary 3.8. If the tensor field Lξg + 2S on a φ−pseudo Riccisymmetric (LCS)n− manifold with α2 −ρ 6= 0 is parallel with respect to Levi-Civita connection associated to g, then for µ = 0 the Ricci soliton (g, ξ, λ)is shrinking and expanding according as α2−ρ > 0 and α2−ρ < 0 respectively. Let (LCS)n−manifold admits a Ricci soliton defined by (3.1) for µ = 0. It is known that ∇g = 0. We consider λ constant, so ∇λg = 0. Thus LVg + 2S is parallel. Hence LVg + 2S is a constant multiple of metric tensors g, i.e. LVg + 2S = ag, where a is constant. Thus LVg + 2S + 2λg reduces to (a + 2λ)g, we get λ = − a 2 . In view of above statement we state the result as the proposition. Proposition 3.9. In (LCS)n−manifold the Ricci soliton (V, ξ, λ) is shrinking or expanding ac- cording as a is positive or negative. Theorem 3.10. If in a (LCS)n− manifold, the metric g is a Ricci soliton and V is a point-wise co-linear with ξ, then V is a constant multiple of g provided λ = −(n − 1)(α2 − ρ). Proof. Suppose that V is pointwise colinear with ξ, i.e., V = c ξ, where c is a smooth function on (M, g). Then (LVg + 2S + 2λg)(X, Y) = 0 implies that cg(∇Xξ, Y) + (Xc)η(Y) + cg(∇Yξ, X) + (Yc)η(X) +2S(X, Y) + 2λg(X, Y) = 0. With the help of (2.5), the above equation takes the form cαg(φX, Y) + (Xc)η(Y) + cαg(φY, X) + (Yc)η(X) +2S(X, Y) + 2λg(X, Y) = 0. (3.14) Substituting Y = ξ in (3.14) and using (2.7) and (2.13) in it, we get (Xc) = 2 [ λ + (n − 1)(α2 − ρ) ] η(X). (3.15) Since η is closed, i. e., dη = 0 on (LCS)n−manifold. From (3.15) we yield Xc = 0, provided λ = −(n − 1)(α2 − ρ). Our theorem is proved Theorem 3.11. If in a LP-Sasakian manifold the metric g is a Ricci soliton and V is a point-wise co-linear with ξ, then the manifold is an η−Einstein manifold provided c 6= −1. Proof. Particularly if V = ξ, then in view of that equation (LVg + 2S + 2λg)(X, Y) = 0, we have αg(φX, Y) + S(X, Y) + λg(X, Y) = 0. (3.16) Putting X = ξ in (3.16), we get λ = −(n−1)(α2 −ρ). Since n > 1, α2 −ρ 6= 0. Therefore the Ricci soliton is shrinking or expanding as α2 < ρ or α2 > ρ respectively. Specially, if we take α = 1, CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 41 then (M, g) reduces to a LP-Sasakian structure of Matsumoto [27]. Then in view of (3.14) and (3.15), equation (3.16) reduces to S(X, Y) = ( 2λ 1 + c ) g(X, Y) + ( 2 1 − c ) (−λ − (n − 1)(1 − ρ)η(X)η(Y), (3.17) provided c 6= −1. Our theorem is proved. Corollary 3.12. If in a (LCS)n−manifold, the metric g is a Ricci soliton (V, ξ, λ) and V is a point-wise co-linear with ξ, then the Ricci solution (V, ξ, λ) is shrinking or expanding according as α2 − ρ > 0 or α2 − ρ < 0. In [31], Sharma proved that a compact Ricci soliton of constant scalar curvature is Einstein. On contracting (3.17), we get r = ( 2 1+b ) [λ(n + 1) + (1 − ρ)] =constant. Thus we state the result as: Corollary 3.13. A LP-Sasakian manifold equipped with a compact Ricci soliton is an Einstein manifold. Theorem 3.14. If (LCS)n−manifold is η−Einstein of the form S = δ g + γ η ⊗ η with δ, γ = constant, then the manifold is equipped a Ricci soliton (g, ξ, −(δ + α)). Proof. Let (M, g) be an η−Einstein (LCS)n−manifold, then S(X, Y) = δg(X, Y) + γη(X)η(Y), (3.18) where δ, γ = constants. Taking V = ξ in (3.1) (for µ = 0) and using (3.18), we get (Lξg)(X, Y) + 2S(X, Y) + 2λg(X, Y) = 2(α + δ + λ) + 2(α + γ)η(X)η(Y). (3.19) It is clear from (3.19) that (M, g) admits a Ricci soliton (g, ξ, λ) if α + δ + λ = 0 and α + γ = 0 it implies that γ = −α = constant. Also from (3.18) we have δ = −α + (n − 1)(α2 − 1) = constant. Thus λ = −(α + δ) = constant. Our theorem is proved. Corollary 3.15. If an η−Einstein (LCS)n−manifold with the form S = δ g + γ η ⊗ η admits a compact Ricci soliton (g, ξ, −(δ + α)) then it leads to an Einstein. 4 Gradient Ricci solitons In this section we consider gradient Ricci soliton on (LCS)n− manifold and prove the following results Theorem 4.1. If an η−Einstein (LCS)n−manifold equipped with a gradient Ricci soliton then manifold reduces to an Einstein provided λ = (n − 1)(α2 − ξ) within the frame field ξf = 0. 42 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) Proof. Let the vector field V be the gradient of a potential function f, is called gradient Ricci soliton. Thus (3.1) takes the form ∇∇f = S + λg, (4.1) that implies ∇YDf = QY + λY, (4.2) where ∇ is the gradient operator of g. From above we notice that g(R(ξ, Y)Df, ξ) = g((∇ξQ)Y, ξ) − g((∇YQ)ξ, ξ). (4.3) In view of (2.5) and (3.18), equation (4.3) yields g(R(ξ, Y)Df, ξ) = 0. (4.4) Using (2.10) in (4.4), we get Df = (α2 − ρ){−η(Df)ξ} = −(α2 − ρ)(g(Df, ξ)ξ) = −(α2 − ρ)(ξf)ξ. (4.5) From (4.2) and (4.5), we obtain S(X, Y) + λg(X, Y) = −Y(ξf)(α2 − ρ)η(X) − (ξf)(α2 − ρ)g(φY, X). (4.6) Replacing X = ξ in (4.6) and using (3.15), we yield Y(ξf)(α2 − ρ) = {λ − (n − 1)(α2 − ρ)}η(Y). It implies that if λ = (n − 1)(α2 − ρ) then ξf =constant and therefore from (4.5), we have Df = −(α2 − ρ)(ξf)ξ = ωξ, ω = −(α2 − ρ)(ξf). If we consider ξf = 0, then (4.5) implies that f = constant. Thus (4.1) yields that S = (n−1)(α2 − ρ)g. Our theorem is proved. 5 Examples of an η−Ricci solition on (LCS)n−manifolds Example 5.1. Let M = { (x, y, z) ∈ Re3 : z 6= 0 } be a 3−dimensional smooth manifold, where (x, y, z) are the standard coordinates in Re3. Let {E1, E2, E3} be linearly independent global frame on M given by E1 = e z ( x ∂ ∂x + y ∂ ∂y ) , E2 = e z ∂ ∂y , E3 = e 2z ∂ ∂z . Let g be the Lorentzian metric defined by g(E1, E3) = g(E2, E3) = g(E1, E2) = 0, g(E1, E1) = g(E2, E2) = 1, g(E3, E3) = −1. CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 43 and η be the 1-form defined by η(V) = g(V, E3) for any V ∈ χ(M). Let φ be the (1, 1) tensor field defined by φE1 = E1, φE2 = E2, φE3 = 0. Then using the linearity of φ and g we have η(E3) = −1, φV = V + η(V)E3, g(φV, φW) = g(V, W) + η(V)η(W), for any V, W ∈ χ(M). Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor of g. Then we obtain [E1, E2] = −e zE2, [E1, E3] = −e 2zE1, [E2, E3] = −e 2zE2. Taking E3 = ξ and using Koszul’s formula for the Lorentzian metric g, we have ∇E1E3 = −e 2zE1, ∇E1E1 = −e 2zE3, ∇E1E2 = 0, ∇E2E3 = −e 2zE2, ∇E3E2 = 0, ∇E2E1 = −e 2zE2, ∇E3E3 = 0, ∇E2E2 = −e 2zE3 − e zE1, ∇E3E1 = 0. From the above it can be easily see that E3 = ξ is a unit timelike concircular vector field and hence (φ, ξ, η, g) is a (LCS)3-structure on M. Consequently M 3(φ, ξ, η, g) is a (LCS)3-manifold with α = −e2z 6= 0 such that (Xα) = ρη(X) where ρ = 2e4z. Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor R and Ricci tensor S as follows: R(E2, E3)E3 = e 4zE2, R(E1, E3)E3 = e 4zE1, R(E1, E2)E2 = {e 4z − e2z}E1, R(E2, E3)E2 = e 4zE3, R(E1, E3)E1 = e 4zE3, R(E1, E2)E1 = {−e 4z − e2z}E2, S(E1, E1) = 0, S(E2, E2) = 0, S(E3, E3) = 2e 4z. Also from (3.5), we calculated that S(E1, E1) = −(α + λ), S(E2, E2) = −(α + λ), S(E3, E3) = (λ − µ). Thus we conclude that from (3.5) for α = −e2z, λ = e2z and µ = e2z −e4z, the structure (g, ξ, λ, µ) is an η−Ricci soliton on M3(φ, ξ, η, g). Example 5.2. Let a 3−dimensional manifold M = { (x, y, z) ∈ Re3 : z 6= 0 } , where (x, y, z) are the standard coordinates in Re3 . Let{E1, E2, E3} be linearly independent global frame on M given by E1 = e −z ( x ∂ ∂x + y ∂ ∂y ) , E2 = e −z ∂ ∂y , E3 = e −2z ∂ ∂z . Let g be the Lorentzian metric defined by g(E1, E3) = g(E2, E3) = g(E1, E2) = 0, g(E1, E1) = g(E2, E2) = 1, g(E3, E3) = −1. Let η be the 1−form defined by η(V) = g(V, E3) for any V ∈ χ(M). Let φ be the (1, 1) tensor field defined by φE1 = E1, φE2 = E2, φE3 = 0. Then using the linearity of φ and g we have η(E3) = −1, φV = V + η(V)E3, g(φV, φW) = g(V, W) + η(V)η(W), 44 S. K. Yadav, S. K. Chaubey & D. L. Suthar CUBO 19, 2 (2017) for any V, W ∈ χ(M). Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor of g. Then we obtain [E1, E2] = −e −zE2, [E1, E3] = −e −2zE1, [E2, E3] = −e −2zE2. Taking E3 = ξ and using Koszul’s formula for the Lorentzian metric g, we have ∇E1E3 = e −2zE1, ∇E1E1 = e −2zE3, ∇E1E2 = 0, ∇E2E3 = e −2zE2, ∇E3E2 = 0, ∇E2E1 = e −2zE2, ∇E3E3 = 0, ∇E3E2 = e −2zE3 − e −zE1, ∇E3E1 = 0. From the above it can be easily seen that E3 = ξ is a unit timelike concircular vector field and hence (φ, ξ, η, g) is a (LCS)3-structure on M. Consequently M 3(φ, ξ, η, g) is a (LCS)3-manifolds with α = e−2z 6= 0 such that (Xα) = ρη(X) where ρ = 2e−4z Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor R and Ricci tensor S as follows: R(E2, E3)E3 = e −4zE2, R(E1, E3)E3 = e −4zE1, R(E1, E2)E2 = {e −4z − e−2z}E1, R(E2, E3)E2 = e −4zE3, R(E1, E3)E1 = e −4zE3, R(E1, E2)E1 = {−e −4z − e−2z}E2, S(E1, E1) = 2e −4z − e−2z, S(E2, E2) = 2e −4z − e−2z, S(E3, E3) = 2e −4Z. Also from (3.5), we calculated that S(E1, E1) = −(α + λ), S(E2, E2) = −(α + λ), S(E3, E3) = (λ − µ). We summaries that from (3.5) for α = e2z, λ = −2e−4z and µ = −4e−4z, the data (g, ξ, λ, µ) admits an η−Ricci soliton on M3(φ, ξ, η, g). Example 5.3. We consider the 4−dimensional manifold M = { (x, y, z, u) ∈ Re4 : u 6= 0 } , where (x, y, z, u) are the standard coordinates in Re4 . Let{E1, E2, E3, E4} be linearly independent global frame on M given by E1 = u ( x ∂ ∂x + y ∂ ∂y ) , E2 = u ∂ ∂y , E3 = u ( ∂ ∂y + ∂ ∂z ) , E4 = (u) 3 ∂ ∂u . Let g be the Lorentzian metric defined by g(E1, E1) = g(E2, E2) = g(E3, E3) = 1, g(E4, E4) = −1, g(Ei, Ej) = 0, i 6= j, i, j = 1, 2, 3, 4. Let η be the 1−form defined by η(V) = g(V, E4), ξ = (u) 4 ∂ ∂u for any V ∈ χ(M). Let φ be the (1, 1) tensor field defined by φE1 = E1, φE2 = E2, φE3 = E3, φE4 = 0. Then using the linearity of φ and g we have η(E4) = −1, φV = V + η(V)E3, g(φV, φW) = g(V, W) + η(V)η(W), CUBO 19, 2 (2017) Some geometric properties of η− Ricci solitons ... 45 for any V, W ∈ χ(M). Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor of g. Then we obtain [E1, E2] = −uE2, [E1, E4] = −(u) 4E1, [E2, E4] = −(u) 4E2, [E3, E4] = −(u) 4E3. Taking E4 = ξ and using Koszul’s formula for the Lorentzian metric g, we have ∇E1E4 = −(u) 2E1, ∇E2E1 = uE3, ∇E1E1 = −(u) 4E4, ∇E3E4 = −(u) 4E3, ∇E3E3 = −(u) 4E4, ∇E2E2 = −(u) 2E4 − uE1. From the above it can be easily seen that the structure (φ, ξ, η, g) is an (LCS)4− structure on M. Consequently M4(φ, ξ, η, g) is an(LCS)4-manifold with α = −(u) 4 6= 0 such that (Xα) = ρη(X) where ρ = 2(u)4. Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor R and Ricci tensor S as follows: R(E1, E4)E1 = (u) 4E4, R(E2, E4)E2 = (u) 4E4, R(E3, E4)E3 = (u) 4E4, R(E1, E3)E3 = (u) 4E1, R(E1, E3)E1 = −(u) 4E3, R(E2, E3)E2 = −(u) 4E3, R(E1, E4)E4 = (u) 4E1, R(E2, E4)E4 = (u) 4E2, R(E1, E2)E2 = [(u) 4 − (u)2]E1, R(E2, E3)E3 = (u) 4E2, R(E3, E4)E4 = (u) 4E3, R(E1, E2)E1 = −[(u) 4 − (u)2]E2, S(E1, E1) = 3(u) 4 − (u)2, S(E2, E2) = 3(u) 4 − (u)2, S(E3, E3) = 3(u) 4, S(E4, E4) = 3(u) 4. Also from (3.5), we calculated that S(E1, E1) = −(α + λ), S(E2, E2) = −(α + λ), S(E3, E3) = (λ − µ), S(E4, E4) = (λ − µ). We conclude that from (3.5) for α = −(u)4, λ = −3(u)4 + 2(u)2 and µ = −6(u)4 + 2(u)2 the data (g, ξ, λ, µ) admits an η−Ricci soliton on M4(φ, ξ, η, g). References [1] M. Atceken, On geometry of submanifold of (LCS)n−manifolds, Int. J. Math. Sci., (2012), Art. ID304647. [2] M. Atceken and S. K. 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