The object of present paper is to study 3-dimensional $(\varepsilon,\delta)$-trans-Sasakian manifold admitting Ricci solitons and $K$-torse forming vector fields. We prove the conditions for the Ricci solitons to be shrinking, expanding and steady. Further, we have obtained a condition for the vector field $\xi$ in a generalized recurrent 3-dimensional $(\varepsilon,\delta)$-trans-Sasakian manifold to be co-symplectic. We have also shown that such a manifold $M$ is of constant scalar curvatur

Maralabhavi, Y.B.

Shivaprasanna, G. S.

Somashekhara, G.

ePrints@Bangalore University

International Journal of Pure and Applied MathematicsVolume 87 No. 3 2013, 405-413ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: http://dx.doi.org/10.12732/ijpam.v87i3.4PAijpam.euON 3-DIMENSIONAL (ǫ,δ)-TRANS-SASAKIAN STRUCTUREY.B. Maralabhavi1 §, G.S. Shivaprasanna2, G. Somashekhara31,2,3Department of MathematicsCentral College CampusBangalore UniversityBengaluru, 560 001, Karnataka, INDIAAbstract: The object of present paper is to study 3-dimensional (ε, δ)-trans-Sasakian manifold admitting Ricci solitons and K-torse forming vector fields.We prove the conditions for the Ricci solitons to be shrinking, expanding andsteady. Further, we have obtained a condition for the vector field ξ in a general-ized recurrent 3-dimensional (ε, δ)-trans-Sasakian manifold to be co-symplectic.We have also shown that such a manifold M is of constant scalar curvature.AMS Subject Classification: 53D10, 53D15Key Words: (ǫ, δ)trans-Sasakian manifold, Ricci soliton, shrinking, steady,expanding, K-torseforming vector field1. IntroductionThe concept of (ǫ)-Sasakian manifolds was introduced by A. Bejancu and K.L.Duggal [1] and further investigation was taken up by Xufend and Xiaoli[13]and Rakesh kumar et al [7]. De and Sarkar [3] introduced and studied con-formally flat,Weyl semisymmetric, ϕ-recurrent (ε)-Kenmotsu manifolds. In[1], the authors obtained Riemannian curvature tensor of (ǫ)-Sasakian mani-folds and established relations among different curvatures. H.G. Nagraja etal [4] have studied (ǫ, δ)-trans-Sasakian structures which generalizes both (ǫ)-manifolds and (ǫ)-Kenmotsu manifolds.Received: April 13, 2013 c© 2013 Academic Publications, Ltd.url: www.acadpubl.eu§Correspondence author406 Y.B. Maralabhavi, G.S. Shivaprasanna, G. Somashekhara2. PreliminariesLet (M,g) be an almost contact metric manifold of dimension n equipped withan almost contact metric structure (ϕ, ξ, η, g) consisting of a (1,1) tensor fieldϕ, a vector field ξ, a 1-form η and a Riemannian metric g satisfyingϕ2 = −I + η ⊗ ξ, (2.1)η(ξ) = 1, (2.2)ϕξ = 0, η ◦ ϕ = 0. (2.3)An almost contact metric manifoldM is called an (ǫ)-almost contact metricmanifold ifg(ξ, ξ) = ε, (2.4)η(X) = εg(X, ξ), (2.5)g(φX,φY ) = g(X,Y )− εη(X)η(Y ),∀X,Y ∈ TM, (2.6)where ε = g(ξ, ξ) = ±1. An (ε)-almost contact metric manifold is called an(ε, δ)-trans-Sasakian manifold if(∇Xφ)Y = α[g(X,Y )ξ − εη(Y )X] + β[g(φX, Y )ξ − δη(Y )φX], (2.7)holds for some smooth functions α and β on M and ε = ±1, δ = ±1. Forβ = 0,α = 1, an (ε, δ)-trans-Sasakian manifold reduces to an (ε)-Sasakian andfor α = 0, β = 1 it reduces to a (δ)-Kenmotsu manifold.Let (M,g) be a (ε, δ)-trans-Sasakian manifold. Then from (2.7), it is easyto see that(∇Xξ) = −εαφX − βδφ2X, (2.8)(∇Xη)Y = −αg(Y, φX) + εδβg(φX,φY ), (2.9)ξα+ 2αβ = 0. (2.10)In a 3-dimensional (ε, δ)-trans-Sasakian manifold, the curvature tensor Rand Ricci tensor S are given by[5]R(X,Y )Z =(2A−r2)(g(Y,Z)X − g(X,Z)Y ) +B(g(Y,Z)η(X) − g(X,Z)η(Y ))ξ+Bη(Z)(η(Y )X − η(X)Y ),(2.11)R(X,Y )ξ = ε[(α2 − β2) + (2− ε2)r][η(Y )X − η(X)Y ], (2.12)ON 3-DIMENSIONAL (ǫ,δ)-TRANS-SASAKIAN STRUCTURE 407S(X,Y ) = Ag(X,Y ) +Bη(X)η(Y ), (2.13)where εδ = 1, A = ( r2 − (α2 − β2)), B = (3(α2 − β2)− ε r2 ) and r is the scalarcurvature.Let (M,g) be a Riemannian manifold with metric g. The metric g is calleda Ricci soliton if[8]LV g + 2S + 2λg = 0, (2.14)where L is the Lie derivative, S is a Ricci tensor, V is a complete vector field onM and λ is a constant. So Ricci soliton is a generalization of Einstein metric.In[9],Ramesh sharma started the study of the Ricci solitons in contact geometry.Later Mukutmani Tripathi[11], Cornelia Livia Bejan and Mircea Crasmareanu[2] and others extensively studied Ricci solitons in contact metric manifolds.The Ricci soliton is said to be shrinking, steady and expanding according as λis positive, zero and negative respectively.We also study K-torse forming vector fields in 3-dimensional (ε, δ)-trans-Sasakian manifold. Torse forming vector fields were introduced by K.Yano [6],∇Xρ = aX + π(X)ρ, (2.15)where ρ is a vector field and π is a non-zero 1-form. Further, a complex analogueof a torse forming vector field is called K-torse forming vector field and it wasintroduced by S.Yamaguchi[10],∇Xρ = aX + bϕX +B(X)ρ+D(X)ϕρ, (2.16)where ρ is a vector field, a and b are functions, B(X) and D(X) are 1-forms.A 3-dimensional (ε, δ)-trans-Sasakian manifold (M,g) is called generalizedrecurrent[12], if its curvature tensor R satisfies the condition(∇XR)(Y,Z)(W ) = A(X)R(Y,Z)W +B(X)[g(Z,W )Y − g(Y,W )Z)], (2.17)where A and B are two 1-forms and B is non zero.3. Ricci SolitonLetM be a 3-dimensional (ε, δ)-trans-Sasakian manifold with metric g. A Riccisoliton is a generalization of an Einstein metric and defined on a Riemannianmanifold (M,g) by (2.14).Let V be pointwise collinear with ξ ie.,V = bξ,Then (2.14) impliesg(∇Xbξ, Y ) + g(∇Y bξ,X) + 2S(X,Y ) + 2λg(X,Y ) = 0, (3.1)408 Y.B. Maralabhavi, G.S. Shivaprasanna, G. Somashekharawhich is reduced tobg(∇Xξ, Y )+(Xb)η(Y )+bg(∇Y ξ,X)+(Y b)η(X)+2S(X,Y )+2λg(X,Y ) = 0.(3.2)Using (2.8) in (3.2), we obtain2bβδg(X,Y )−2bβη(X)η(Y )+(Xb)η(Y )+(Y b)η(X)+2S(X,Y )+2λg(X,Y ) = 0.(3.3)Replace Y by ξ in (3.3) and use (2.13) to get2bβδεη(X)− 2bβη(X)+ (Xb)+ (ξb)η(X)+2Aεη(X)+2Bη(X)+2λεη(X) = 0.(3.4)Again putting X = ξ in (3.4), we obtainξb = −bβδε+ bβ −Aε−B − λε. (3.5)By using (3.5) in (3.4), we getXb = [−bβδε + bβ −Aε−B − λε]η(X), (3.6)ordb = [−bβδε + bβ −Aε−B − λε]η, (3.7)where we have put ∇X = d.Applying d on (3.7), we get[bβδε − bβ +Aε+B + λε]dη = 0. (3.8)Since dη 6= 0 we havebβδε− bβ +Aε+B + λε = 0. (3.9)Using (3.9) in (3.7) then db = 0 and hence b is a constant. Therefore from (3.3)it followsS(X,Y ) = −(λ+ bβδ)g(X,Y ) + bβη(X)η(Y ), (3.10)which implies that M is of constant scalar curvature provided β = constant.This leads to the following:Theorem 3.1. Let M be a 3-dimensional (ε, δ)-trans-Sasakian manifoldin which the metric tensor g is Ricci soliton with the vector field V as pointwisecollinear with ξ, then M is a space of constant scalar curvature provided β is aconstant.ON 3-DIMENSIONAL (ǫ,δ)-TRANS-SASAKIAN STRUCTURE 409Now let V = ξ. Then the (2.14) becomesLξg + 2S + 2λg = 0. (3.11)Using (2.8), we getLξg(X,Y ) = 2βδg(X,Y )− 2βδεη(X)η(Y ), (3.12)and therefore,(Lξg + 2S)(X,Y ) = 2βδ[g(X,Y )− εη(X)η(Y )]− 2[(r2− (α2 − β2))g(X,Y )+(3(α2 − β2)− εr2)η(X)η(Y )].(3.13)Using (3.13) in (3.11), we obtain2[−r2+ βδ + (α2 − β2) + λ]g(X,Y )−2[βδε + 3(α2 − β2)− εr2]η(X)η(Y ) = 0.(3.14)Take X = Y = ξ in (3.14). We getλ =(β2 − α2)(ε+ 3)− r(ε− 1)2ε. (3.15)Hence we state the followingTheorem 3.2. Let M be a 3-dimensional (ε, δ)-trans-Sasakian manifold.Then a Ricci soliton (g, ξ, λ) in (M,g) is:(i) shrinking for (β2 − α2) (ε+ 3) > r(ε-1),(ii) expanding for (β2 − α2) (ε+ 3) < r(ε-1).Corollary 3.1. In a 3-dimensional (ε, δ)-trans-Sasakian manifold M Riccisoliton is:(i) shrinking if β > α and ε = 1,(ii) expanding if β < α and ε = 1,(iii) steady if α = β and ε = 1.Suppose (M,g) is a 3-dimensional (ε, δ)-trans-Sasakian manifold and (g, V, λ)is a Ricci soliton in (M,g). If V is a conformal killing vector field, thenLV g = ρg, (3.16)410 Y.B. Maralabhavi, G.S. Shivaprasanna, G. Somashekharafor some scalar function ρ. From (2.14)S(X,Y ) = −[λg(X,Y ) +12LV g(X,Y )]. (3.17)Now from (3.16) it follows thatS(X,Y ) = −(λ+ρ2)g(X,Y ), (3.18)QX = −(λ+ρ2)X, (3.19)r = −3(λ+ρ2). (3.20)Put X = Z = ξ and use (3.20) in (2.11) to obtainR(ξ, Y )ξ = (α2 − β2)(3 − 2ε)[η(Y )ξ − Y ]. (3.21)Put X = ξ in (2.12) to obtainR(ξ, Y )ξ = ε[(α2 − β2) +(2− ε2)r][η(Y )ξ − Y ]. (3.22)Comparing (3.21)and (3.22), we obtainr =6(1− ε)(α2 − β2)(2ε − 1). (3.23)By using (3.20) and (3.23), we getλ =2(ε − 1)(α2 − β2)(2ε− 1)−ρ2. (3.24)Theorem 3.3. Let M be a 3-dimensional (ε, δ)-trans-Sasakian manifoldadmitting Ricci soliton (g, V, λ), where V is conformal killing vector field. Then(g, V, λ) is:i) expanding for ρ > 4(ε−1)(α2−β2)(2ε−1) ,ii) shrinking for ρ < 4(ε−1)(α2−β2)(2ε−1)iii) steady for ρ = 4(ε−1)(α2−β2)(2ε−1) .ON 3-DIMENSIONAL (ǫ,δ)-TRANS-SASAKIAN STRUCTURE 4114. K-Torse Forming Vector FieldIn 3-dimensional (ε, δ)-trans-Sasakian manifold, ξ is always a K-torse formingvector field. Take ρ = ξ in (2.16) and compare the result with (2.8) to obtaina = βδ, b = −εα, B(X) = −βη(X) and D(X)=0. Then it implies∇Xξ = βδX − εαϕX − βη(X)ξ, (4.1)(∇Xη)Y = βδg(X,Y )− εαg(ϕX,Y )− εβη(X)η(Y ), (4.2)R(X,Y )ξ = (aβ + bα)[η(X)Y − η(Y )X], (4.3)R(ξ,X)Y = (aβ + bα)[η(Y )X − εg(X,Y )ξ], (4.4)andS(Y, ξ) = −2(aβ + bα)η(Y ), (4.5)where α and β are constants.Taking Y =W = ξ in (2.17),we obtain(∇XR)(ξ, Z)(ξ) = A(X)R(ξ, Z)ξ +B(X)[εη(Z)ξ − εZ). (4.6)By the definition of covariant derivative, we have(∇XR)(ξ, Z)(ξ) = ∇XR(ξ, Z)ξ −R(∇Xξ, Z)ξ −R(ξ,∇XZ)ξ −R(ξ, Z)∇Xξ.(4.7)Using (4.1),(4.2) and (4.4) in (4.7), we get(∇XR)(ξ, Z)(ξ) = d(aβ + bα)(X)[Z − η(Z)ξ] + (aβ + bα)[(1 − δ)βg(X,Z)ξ+ (ε− 1)αg(ϕX,Z)ξ + (ε− 1)βη(X)η(Z)ξ + 2η(∇XZ)ξ − 2βη(X)Z + 2εβη(X)Z].(4.8)From (4.6) and (4.8), we haved(aβ + bα)(X)[Z − η(Z)ξ] + (aβ + bα)[(1 − δ)βg(X,Z)ξ + (ε− 1)αg(ϕX,Z)ξ+ (ε− 1)βη(X)η(Z)ξ + 2η(∇XZ)ξ − 2βη(X)Z + 2εβη(X)Z]= A(X)[(aβ + bα)(Z − η(Z)ξ)] +B(X)[εη(Z)ξ − εZ].(4.9)Put Z = ξ and use (2.2),(2.4),(2.5) in (4.9) to obtain(aβ + bα)[2(ε − 1)βη(X)ξ + η(∇Xξ)ξ] = 0. (4.10)If (aβ + bα) 6= 0 and ε = 1, then (4.10) is∇Xξ = 0. (4.11)Thus we have412 Y.B. Maralabhavi, G.S. Shivaprasanna, G. SomashekharaTheorem 4.4. In a generalized recurrent 3-dimensional (ε, δ)-trans-SasakianmanifoldM the vector field ξ is co-symplectic provided (aβ+bα) 6= 0 and ε = 1.Suppose the Ricci tensor S is semi conjugate in 3-dimensional (ε, δ)-trans-Sasakian manifoldM with respect to the vector field ξ, which isK-torse formingvector field in M . So R(X, ξ).S(Y,Z) = 0.Then we haveS(R(X, ξ)Y,Z) + S(Y,R(X, ξ)Z) = 0. (4.12)Using (4.4) and (4.5) in (4.12), we get(aβ + bα)[S(X,Z)η(Y ) + 2ε(aβ + bα)g(Y,X)η(Z) + S(Y,X)η(Z)+2ε(aβ + bα)g(X,Z)η(Y )] = 0.(4.13)Put Z = ξ and use (2.5),(4.5) in (4.13) to obtainS(X,Y ) = −2ε(aβ + bα)g(X,Y ), (4.14)which givesr = −6ε(aβ + bα), (4.15)where a and b are constants. Hence r is constant.Theorem 4.5. Suppose the ricci tensor S in a 3-dimensional (ε, δ)-trans-Sasakian manifold M is semi-conjugate with respect the vector field ξ.Then Mis a space of constant scalar curvature.References[1] Bejancu A. and Duggal K.L.,Real hypersurfaces of idefinite Kahler mani-folds, Int.J.Math.Math.sci.,16,No.3(1993),545-556.[2] Cornelia Livia Benjan and Mircea Crasmareanu, Ricci Solitonsin manifolds with Quasi-constant curvature, Publ.Math.Debrecen,78,No.1(2011),235-243.[3] De.U.C. and Sarkar A.,On (ε)-Kenmotsu manifolds, Hadronic journal, 32(2009),231-242.[4] Halammanavar G.Nagaraja, Rangaswami C.Premalatha, Ganganna So-mashekara, On an (ε, δ)-trans-Sasakian structure, Proceedings of the Es-tonian Academy of sciences,61,No.1(2012),20-28.ON 3-DIMENSIONAL (ǫ,δ)-TRANS-SASAKIAN STRUCTURE 413[5] H.G.Nagaraja and Somashekhara,Special Trans-Sasakian manifolds andcurvature conditions, International journal of pure and applied Mathe-matics,81, 3 (2012),411-420.[6] K.Yano, On the torse forming direction in Riemannian spaces,Proc.Imp.Acd.Tokyo., 20 (1944),340-345.[7] R.Kumar, R.Rani and R.K.Nagaich,On sectional curvature of (ε)-Sasakianmanifolds, Int.J.Math.Math.sci.,(2007),ID.93562.[8] R.S.Hamilton, The Ricci flow on surfaces, Mathematics and gen-eral relativity(Santa Cruz, CA,1986),Contemp.Math.,71,237-262,AmericanMath.soc.,(1988).[9] Sinha B.B. and Ramesh sharma,On Para-A-Einstein manifolds, Publica-tions De L’Institute Mathematique.Nouvelle Serie,34,No.48(1983),211-215.[10] S.Yamaguchi, On Kaehlerian torse forming vector fields, KodaiMath.J.,2(1979),103-115.[11] Tripathi M.M., Ricci solitons in contact metric manifolds,arXiv:0801.4222v1,[math.DG],28(2008).[12] U.C.De, N.Guha and D.Kamilya, On generalized Ricci-recurrent mani-folds, Tensor,N.S.,56(1995),312-317.[13] X.Xufeng, C.Xiaoli, Two theorems on ε-Sasakian manifolds,Int.J.Math.Math.sci.,21,No.2(1998),249-254.414

On 3-Dimensional (ε,δ)-Trans-Sasakian Structure

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On 3-Dimensional (ε,δ)-Trans-Sasakian Structure

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