ON A CLASS OF N(k)-CONTACT METRIC MANIFOLDS

ON A CLASS OF N(k)-CONTACT METRIC MANIFOLDS UDAY CHAND DE, AHMET YILDIZ and SUJIT GHOSH Communicated by the fome editoial boad The object of the pesent pape is to study ξ-conciculaly flat and φ-conciculaly flat N(k)-contact metic manifolds. Beside these, we also study N(k)-contact metic manifolds satisfying Z(ξ, X).S = 0. Finally, we constuct an example to veify some esults. AMS 2010 Subject Classification: 53C15, 53C25. Key wods: ξ-conciculaly flat, φ-conciculaly flat, Sasakian manifold, Einstein manifold, η-einstein manifold. 1. INTRODUCTION A tansfomation of a (2n + 1)-dimensional Riemannian manifold M, which tansfoms evey geodesic cicle of M into a geodesic cicle, is called a concicula tansfomation [16, 21]. A concicula tansfomation is always a confomal tansfomation [21]. Hee, geodesic cicle means a cuve in M whose fist cuvatue is constant and whose second cuvatue is identically zeo. Thus, the geomety of concicula tansfomations, i.e., the concicula geomety, is a genealization of invesive geomety in the sense that the change of metic is moe geneal than that induced by a cicle peseving diffeomophism (see also [6]). An inteesing invaiant of a concicula tansfomation is the concicula cuvatue tenso Z. It is defined by [16, 17] (1.1) Z(X, Y )W =R(X, Y )W [g(y, W )X g(x, W )Y ], whee X, Y, W T M and is the scala cuvatue. Riemannian manifolds with vanishing concicula cuvatue tenso ae of constant cuvatue. Thus, the concicula cuvatue tenso is a measue of the failue of a Riemannian manifold to be of constant cuvatue. Let M be an almost contact metic manifold equipped with an almost contact metic stuctue (φ, ξ, η, g). At each point p M, decompose the tangent space T p M into diect sum T p M = φ(t p M) {ξ p }, whee {ξ p } is the MATH. REPORTS 16(66), 2 (2014), 207 217

208 Uday Chand De, Ahmet Yıldız and Sujit Ghosh 2 1-dimensional linea subspace of T p M geneated by {ξ p }. Thus, the confomal cuvatue tenso C is a map C : T p M T p M T p M φ(t p M) {ξ p }, p M. It may be natual to conside the following paticula cases: (1) C : T p (M) T p (M) T p (M) {ξ p }, i.e., the pojection of the image of C in φ(t p (M)) is zeo. (2) C : T p (M) T p (M) T p (M) φ(t p (M)), i.e., the pojection of the image of C in {ξ p } is zeo. This condition is equivalent to (1.2) C(X, Y )ξ = 0. (3) C : φ(t p (M)) φ(t p (M)) φ(t p (M)) {ξ p }, i.e., when C is esticted to φ(t p (M)) φ(t p (M)) φ(t p (M)), the pojection of the image of C in φ(t p (M)) is zeo. This condition is equivalent to (1.3) φ 2 C(φX, φy )φz = 0. An almost contact metic manifold satisfying (1.2) and (1.3) ae called ξ-confomally flat and φ-confomally flat, espectively. Almost contact metic manifolds satisfying the cases (1), (2) and (3) ae consideed in [11], [12] and [13], espectively. In [12], it is poved that a K-contact manifold is ξ-confomally flat if and only if it is an η-einstein Sasakian manifold. In [20], the authos studied ξ-confomally flat N(k)-contact metic manifold. A compact φ-confomally flat K-contact manifold with egula contact vecto field has been studied in [13]. Moeove, in [15] the autho studied φ-confomally flat (k, µ)-contact metic manifolds. Motivated by the above studies, in this pape we study ξ- conciculaly flat and φ-conciculaly flat N(k)-contact metic manifolds. Analogous to the consideations of confomal cuvatue tenso hee we define the following: Definition 1.1. A (2n + 1)-dimensional N(k)-contact metic manifold is said to be ξ-conciculaly flat if (1.4) Z(X, Y )ξ = 0, fo X, Y T M. Definition 1.2. A (2n + 1)-dimensional N(k)-contact metic manifold is said to be φ-conciculaly flat if (1.5) g(z(φx, φy )φw, φu) = 0 fo X, Y, W, U T M. In [7], D.E. Blai et al. stated a study of concicula cuvatue tenso of contact metic manifolds. A (2n + 1)-dimensional N(k)-contact metic manifold satisfying Z(ξ, X).Z = 0, Z(ξ, X).R = 0 and R(ξ, X).Z = 0 have been

3 On a class of N(k)-contact metic manifolds 209 consideed in [7]. B.J. Papantoniou [1] and D. Peone [9] included the studies of contact metic manifolds satisfying R(ξ, X).S = 0, whee S is the Ricci tenso. Motivated by these studies, we continue the study of the pape [7] and classify N(k)-contact manifolds with concicula cuvatue tenso Z satisfying Z(ξ, X).S = 0. The pesent pape is oganized as follows. Afte peliminaies in Section 3, we study ξ-conciculaly flat N(k)-contact metic manifolds and pove that a (2n + 1)-dimensional, n > 1, ξ-conciculaly flatn(k)-contact metic manifold is locally isometic to Example 2.1. Section 4 deals with the study of φ-conciculaly flat N(k)-contact metic manifolds. In this section, we pove that a φ-conciculaly flat (2n + 1)-dimensional, n 1, N(k)-contact metic manifold is a Sasakian manifold. Section 5 is devoted to study a (2n + 1)- dimensional, n 2, N(k)-contact metic manifold satisfying Z(ξ, X).S = 0 and pove that the manifold satisfies Z(ξ.X).S = 0 if and only if it is an Einstein-Sasakian manifold. Finally, in Section 6, we constuct an example of a 3-dimensional N(k)-contact metic manifold which veifies some esults of Section 3. 2. PRELIMINARIES A (2n + 1)-dimensional diffeentiable manifold M is said to admit an almost contact stuctue if it admits a tenso field φ of type (1, 1), a vecto field ξ and a 1-fom η satisfying [3, 4] (2.1) φ 2 X = X + η(x)ξ, η(ξ) = 1, φξ = 0 and η φ = 0. An almost contact metic stuctue is said to be nomal if the induced almost complex stuctue J on the poduct manifold M R defined by J(X, f d dt ) = (φx fξ, η(x) d dt ) is integable, whee X is tangent to M, t is the coodinate of R and f is a smooth function on M R. Let g be the compatible Riemannian metic with almost contact stuctue (φ, ξ, η), i.e., (2.2) g(φx, φy ) = g(x, Y ) η(x)η(y ). Then M becomes an almost contact metic manifold equipped with an almost contact metic stuctue (φ, ξ, η, g). Fom (2.1) it can be easily seen that (2.3) g(x, φy ) = g(φx, Y ), g(x, ξ) = η(x), fo any vecto fields X, Y. An almost contact metic stuctue becomes a contact metic stuctue if g(x, φy ) = dη(x, Y ), fo all vecto fields X, Y.

210 Uday Chand De, Ahmet Yıldız and Sujit Ghosh 4 A contact metic manifold is said to be Einstein if S(X, Y ) = λg(x, Y ), whee λ is a constant and η-einstein if S(X, Y ) = αg(x, Y ) + βη(x)η(y ), whee α and β ae smooth functions. A nomal contact metic manifold is a Sasakian manifold. An almost contact metic manifold is Sasakian if and only if (2.4) ( X φ)y = g(x, Y )ξ η(y )X, X, Y T M, whee is the Levi-Civita connection of the Riemannian metic g. A contact metic manifold M 2n+1 (φ, ξ, η, g) fo which ξ is a Killing vecto field is said to be a K-contact metic manifold. A Sasakian manifold is K- contact but not convesely. Howeve, a 3-dimensional K-contact manifold is Sasakian [14]. It is well known that the tangent sphee bundle of a flat Riemannian manifold admits a contact metic stuctue stisfying R(X, Y )ξ = 0 [5]. Again on a Sasakian manifold [18] we have R(X, Y )ξ = η(y )X η(x)y. As a genealization of both R(X, Y )ξ = 0 and the Sasakian case: D.E. Blai, T. Koufogiogos and B.J. Papantoniou [8] intoduced the (k, µ)- nullity distibution on a contact metic manifold and gave seveal easons fo studying it. The (k, µ)-nullity distibution N(k, µ) [8] of a contact metic manifold M is defined by N(k, µ) : p N p (k, µ) = {W T p M : R(X, Y )W = (ki + µh)(g(y, W )X g(x, W )Y )}, fo all X, Y T M, whee (k, µ) R 2. A contact metic manifold M with ξ N(k, µ) is called a (k, µ)-contact metic manifold. If µ = 0, the (k, µ)-nullity distibution educes to k-nullity distibution [19]. The k-nullity distibution N(k) of a Riemannian manifold is defined by [19] N(k) : p N p (k) = {Z T p M : R(X, Y )Z = k[g(y, Z)X g(x, Z)Y ]}, k being a constant. If the chaacteistic vecto field ξ N(k), then we call a contact metic manifold as N(k)-contact metic manifold [7]. If k = 1, then the manifold is Sasakian and if k = 0, then the manifold is locally isometic to the poduct E n+1 (0) S n (4) fo n > 1 and flat fo n = 1 [5]. Howeve, fo a N(k)-contact metic manifold M of dimension (2n + 1), we have [7] (2.5) ( X φ)y = g(x + hx, Y )ξ η(y )(X + hx), whee h = 1 2 ξφ. (2.6) h 2 = (k 1)φ 2.

5 On a class of N(k)-contact metic manifolds 211 (2.7) R(X, Y )ξ = k[η(y )X η(x)y ]. (2.8) R(ξ, X)Y = k[g(x, Y )ξ η(y )X]. (2.9) S(X, Y ) = 2(n 1)g(X, Y ) + 2(n 1)g(hX, Y ) +[2nk 2(n 1)]η(X)η(Y ), n 1. (2.10) S(Y, ξ) = 2nkη(X). (2.11) = 2n(2n 2 + k). Given a non-sasakian (k, µ)-contact manifold M, E. Boeckx [10] intoduced an invaiant I M = 1 µ 2 1 k and showed that fo two non-sasakian (k, µ)-manifolds M 1 and M 2, we have I M1 = I M2 if and only if up to a D-homothetic defomation, the two manifolds ae locally isometic as contact metic manifolds. Thus, we see that fom all non-sasakian (k, µ)-manifolds of dimension (2n + 1) and fo evey possible value of the invaiant I M, one (k, µ)-manifold M can be obtained. Fo I M > 1 such examples may be found fom the standad contact metic stuctue on the tangent sphee bundle of a manifold of constant cuvatue c whee we have I M = 1+c 1 c. Boeckx also gives a Lie algeba constuction fo any odd dimension and value of I M < 1. Example 2.1. Using this invaiant, D.E. Blai, J-S. Kim and M.M. Tipathi [7] constucted an example of a (2n + 1)-dimensional N(1 1 n )-contact metic manifold, n > 1. The example is given in the following: Since the Boeckx invaiant fo a (1 1 n, 0)-manifold is n > 1, we conside the tangent sphee bundle of an (n + 1)-dimensional manifold of constant cuvatue c so chosen that the esulting D-homothetic defomation will be a (1 1 n, 0)-manifold. That is, fo k = c(2 c) and µ = 2c we solve fo a and c. The esult is 1 1 n = k + a2 1 a 2, 0 = µ + 2a 2 a c = n ± 1 n 1, a = 1 + c and taking c and a to be these values we obtain N(1 1 n )-contact metic manifold.

212 Uday Chand De, Ahmet Yıldız and Sujit Ghosh 6 The above example will be used in Theoem 3.1. Hee, we state some Lemmas which will be neccessay to pove ou main esults. Lemma 2.1 ([5]). A contact metic manifold M 2n+1 satisfying the condition R(X, Y )ξ = 0 fo all X, Y is locally isometic to the Riemannian poduct of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of positive cuvatue 4, i.e., E n+1 (0) S n (4) fo n > 1 and flat fo n = 1. Lemma 2.2 ([2]). Let M be an η-einstein manifold of dimension (2n + 1) (n 1). If ξ belongs to the k-nullity distibution, then k = 1 and the stuctue is Sasakian. Lemma 2.3 ([19]). Let M be an Einstein manifold of dimension (2n + 1) (n 2). If ξ belongs to the k-nullity distibution, then k = 1 and the stuctue is Sasakian. 3. ξ-concircularly FLAT N(k)-CONTACT METRIC MANIFOLDS In this section, we study ξ-concicully flat N(k)-contact metic manifolds. Let M be a (2n + 1)-dimensional, n 1, ξ-concicully flat N(k)- contact metic manifold. Putting W = ξ in (1.1) and applying (1.4) and g(x, ξ) = η(x), we have (3.1) R(X, Y )ξ = [η(y )X η(x)y ]. Using (2.7) in (3.1), we obtain (3.2) (k )[η(y )X η(x)y ] = 0. Now [η(y )X η(x)y ] 0 in a contact metic manifold, in geneal. Theefoe, (3.2) gives (3.3) k =. Using (2.11) in (3.3) yields (3.4) k = 1 1 n. Hence, we can state the following: Theoem 3.1. If a (2n + 1)-dimensional (n > 1) N(k)-contact metic manifold is ξ-conciculaly flat, then it is locally isometic to Example 2.1.

7 On a class of N(k)-contact metic manifolds 213 Let us conside a 3-dimensional ξ-conciculaly flat N(k)-contact metic manifold. Then n = 1 and in that case we have k = 0. Hence, in view of Lemma 2.1, we can state the following: Coollay 3.1. A 3-dimensional N(k)-contact metic manifold is ξ- conciculaly flat if and only if the manifold is flat. 4. φ-concircularly FLAT N(k)-CONTACT METRIC MANIFOLDS This section is devoted to study φ-conciculaly flat N(k)-contact metic manifolds. Let M be a (2n + 1)-dimensional φ-conciculaly flat N(k)-contact metic manifold. Using (1.5) in (1.1) we obtain (4.1) g(r(φx, φy )φw, φv ) = [g(φy, φw )g(φx, φv ) g(φx, φw )g(φy, φv )]. Let {e 1, e 2,..., e n, φe 1, φe 2,..., φe n, ξ} be an othonomal φ-basis of the tangent space. Putting X = V = e i in (4.1) and taking summation ove i = 1 to 2n and using (2.7), we obtain (2n 1) (4.2) S(φY, φw ) = [ + k]g(φy, φw ). Replacing Y and W by φy and φw in (4.2) and using (2.1), (2.2), (2.10) and (2.11) yields (4.3) S(Y, W ) = Ag(Y, W ) + Bη(Y )η(w ), whee A and B ae given by the following elations: A = (2n 1)(2n 2) + 4nk 2n + 1 and B = In view of the equation (4.3) we state the following: 2(2n 1){n(k 1) 1}. 2n + 1 Poposition 4.1. A (2n+1)-dimensional φ-conciculaly flat N(k)-contact metic manifold is an η-einstein manifold. Using the Lemma 2.2 we have the following: Theoem 4.1. A (2n + 1)-dimensional φ-conciculaly flat N(k)-contact metic manifold is a Sasakian manifold.

214 Uday Chand De, Ahmet Yıldız and Sujit Ghosh 8 5. N(k)-CONTACT METRIC MANIFOLD SATISFYING Z(ξ, X).S = 0 This section deals with a (2n+1)-dimensional, n 2, N(k)-contact metic manifold satisfying Z(ξ, X).S = 0. Now, the elation Z(ξ, X).S = 0 implies (5.1) S(Z(ξ, X)Y, W ) + S(Y, Z(ξ, X)W ) = 0. (5.2) (5.3) Using (1.1) in (5.1), we have S(R(ξ, X)Y, W ) + S(Y, R(ξ, X)W ) [g(x, Y )S(ξ, W ) η(y )S(X, W ) + g(x, W )S(Y, ξ) η(w )S(X, Y )] = 0. Using (2.8) and (2.10) in (5.2) yields 2nk[g(X, Y )η(w ) + g(x, W )η(y )] [S(X, W )η(y ) +S(X, Y )η(w )] [2nk{g(X, Y )η(w ) +g(x, W )η(y )} {S(X, W )η(y ) + S(X, Y )η(w )}] = 0. Putting W = ξ in (5.3) and using g(x, ξ) = η(x) and (2.10), we obtain (5.4) { 1}[S(X, Y ) 2nkg(X, Y )] = 0. Putting the value of fom (2.11) in =, we get k = 3. We know that k 1 in a N(k)-contact metic manifold. Theefoe and hence, fom (5.4) we have (5.5) S(X, Y ) = 2nkg(X, Y ). Theefoe, a N(k)-contact metic manifold satisfying Z(ξ, X).S = 0 is an Einstein manifold. Theefoe, in view of Lemma 2.3, we have the manifold is a Sasakian manifold. Convesely, let the manifold be an Einstein-Sasakian manifold. Then we have k = 1 and S(X, Y ) = 2ng(X, Y ). Theefoe, we have (5.6) (Z(ξ, X).S)(Y, W ) = S(Z(ξ, X)Y, W ) + S(Y, Z(ξ, X)W ) = 2n[g((Z(ξ, X)Y, W ) + g(y, Z(ξ, X)W )]. Using (1.1), (2.8) and (2.11) in (5.6) we easily obtain (5.7) (Z(ξ, X).S)(Y, W ) = 0. In view of the above discussions, we state the following: Theoem 5.1. A (2n+1)-dimensional, n 2, N(k)-contact metic manifold satisfies Z(ξ, X).S = 0 if and only if the manifold is an Einstein-Sasakian manifold.

9 On a class of N(k)-contact metic manifolds 215 6. EXAMPLE In this section, we constuct an example of a N(k)-contact metic manifold which is ξ-conciculaly flat. We conside 3-dimensional manifold M = {(x, y, z) R 3 }, whee (x, y, z) ae the standad coodinate in R 3. Let e 1, e 2, e 3 ae thee vecto fields in R 3 which satisfies [e 1, e 2 ] = (1 + λ)e 3, [e 2, e 3 ] = 2e 1 and [e 3, e 1 ] = (1 λ)e 2, whee λ is a eal numbe. Let g be the Riemannian metic defined by g(e 1, e 3 ) = g(e 2, e 3 ) = g(e 1, e 2 ) = 0, g(e 1, e 1 ) = g(e 2, e 2 ) = g(e 3, e 3 ) = 1. Let η be the 1-fom defined by η(u) = g(u, e 1 ) fo any U χ(m). Let φ be the (1, 1)-tenso field defined by φe 1 = 0, φe 2 = e 3, φe 3 = e 2. Using the lineaity of φ and g we have η(e 1 ) = 1, φ 2 (U) = U + η(u)e 1 and g(φu, φw ) = g(u, W ) η(u)η(w ) fo any U, W χ(m). Moeove, he 1 = 0, he 2 = λe 2 and he 3 = λe 3. The Riemannian connection of the metic tenso g is given by Koszul s fomula which is given by, 2g( X Y, Z) = Xg(Y, Z) + Y g(z, X) Zg(X, Y ) g(x, [Y, Z]) g(y, [X, Z]) + g(z, [X, Y ]). Using Koszul s fomula we get the following: e1 e 1 = 0, e1 e 2 = 0, e1 e 3 = 0, e2 e 1 = (1 + λ)e 3, e2 e 2 = 0, e2 e 3 = (1 + λ)e 1, e3 e 1 = (1 λ)e 2, e3 e 2 = (1 λ)e 1, e3 e 3 = 0. In view of the above elations we have X ξ = φx φhx, fo e 1 = ξ

216 Uday Chand De, Ahmet Yıldız and Sujit Ghosh 10 Theefoe, the manifold is a contact metic manifold with the contact stuctue (φ, ξ, η, g). Now, we find the cuvatue tensos as follows: R(e 1, e 2 )e 2 = (1 λ 2 )e 1, R(e 3, e 2 )e 2 = (1 λ 2 )e 3, R(e 1, e 3 )e 3 = (1 λ 2 )e 1, R(e 2, e 3 )e 3 = (1 λ 2 )e 2, R(e 2, e 3 )e 1 = 0, R(e 1, e 2 )e 1 = (1 λ 2 )e 2, R(e 3, e 1 )e 1 = (1 λ 2 )e 3. In view of the expessions of the cuvatue tensos we conclude that the manifold is a N(1 λ 2 )-contact metic manifold. Using the expessions of the cuvatue tenso we find the values of the Ricci tensos as follows: S(e 1, e 1 ) = 2(1 λ 2 ), S(e 2, e 2 ) = 0, S(e 3, e 3 ) = 0. Hence, = S(e 1, e 1 ) + S(e 2, e 2 ) + S(e 3, e 3 ) = 2(1 λ 2 ). Let X and Y ae any two vecto fields given by X = a 1 e 1 + a 2 e 2 + a 3 e 3 and Y = b 1 e 1 + b 2 e 2 + b 3 e 3. Using (1.1) we get Z(X, Y )e 1 = 2(1 λ2 ) [(a 2 b 1 a 1 b 2 )e 2 + (a 3 b 1 a 1 b 3 )e 3 ]. 3 Theefoe, the manifold will be ξ-conciculaly flat if λ = 1 and in that case the manifold will be a N(0)-contact metic manifold with = 0 which veifies the esult of Section 3. Acknowledgments.The authos ae thankful to the efeees fo thei valuable suggestions in the impovement of this pape. REFERENCES [1] B.J. Papantoniou, Contact Riemannian manifolds satisfying R(ξ, X).R = 0 and ξ (k, µ)-nullity distibution. Yokohama Math. J. 40 (1993), 149 161. [2] C. Baikoussis and T. Koufogiogos, On a type of contact manifolds. J. Geomety 46 (1993), 1 9. [3] D.E. Blai, Contact manifolds in Riemannian geomety. Lectue Notes in Math. 509, Spinge-Velag, 1976. [4] D.E. Blai, Riemannian geomety of contact and symplectic manifolds. Pog. Math. 203, Bikhause Boston, Inc., Boston, MA, 2002. [5] D.E. Blai, Two emaks on contact metic stuctues. Tohoku Math. J. 29 (1977), 319 324. [6] D.E. Blai, Invesion theoy and confomal mapping. Stud. Math. Lib. 9, Ame. Math. Soc., 2000. [7] D.E. Blai, J-S. Kim and M.M. Tipathi, On the concicula cuvatue tenso of a contact metic manifold. J. Koean Math. Soc. 42 (2005), 883 892.

11 On a class of N(k)-contact metic manifolds 217 [8] D.E. Blai, T. Koufogiogos and B.J. Papantoniou, Contact metic manifolds satisfying a nullity condition. Isael J. Math. 91 (1995), 189 214. [9] D. Peone, Contact Riemannian manifolds satisfying R(X, ξ).r = 0. Yokohama Math. J. 39 (1992), 141 149. [10] E. Boeckx, A full classification of contact metic (k, µ)-spaces. Illinois J. Math. 44 (2000), 1, 212 219. [11] G. Zhen, On confomal symmetic K-contact manifolds. Chinese Quat. J. Math. 7 (1992), 5 10. [12] G. Zhen, J.L. Cabeizo, L.M. Fenández and M. Fenández, On ξ-confomally flat K- contact metic manifolds. Indian J. Pue Appl. Math. 28 (1997), 725 734. [13] J.L. Cabeizo, L.M. Fenández and M. Fenández, The stuctue of a class of K-contact manifolds. Acta Math. Hunga. 82 (1999), 331 340. [14] J-B. Jun and U.K. Kim, On 3-dimensional almost contact metic manifolds. Kyungpook Math. J. 34 (1994), 2, 293 301. [15] K. Aslan, C. Muathan and C. Özgü, On φ-confomally flat contact metic manifolds. Balkan J. Geom. Appl., 5 (2000), 1 7. [16] K. Yano, Concicula geomety I. concicula tansfomations. Poc. Imp. Acad. Tokyo 16 (1940), 195 200. [17] K. Yano and S. Bochne, Cuvatue and Betti numbes. Ann. Math. Stud. 32, Pinceton Univ. Pess, 1953. [18] S. Sasaki, Lectue notes on almost contact manifolds. Tohoku Univesity I, 1965. [19] S. Tanno, Ricci cuvatue of contact Riemannian manifolds. Tohoku Math. J. 40 (1988), 441 448. [20] U.C. De and S. Biswas, A note on ξ-confomally flat contact manifolds. Bull. Malays. Math. Soc. 29 (2006), 51 57. [21] W. Kuhnel, Confomal tansfomations between Einstein spaces. Confom. Geomety. Aspects Math. 12 (1988), Vieweg, Baunschweig, 105 146. Received 10 Januay 2012 Calcutta Univesity, Depatment of Pue Mathematics, 35, Ballygaunge Cicula Road, Kolkata 700019, West Bengal, India uc de@yahoo.com Inonu Univesity, Education Faculty, 44280, Malatya Tukey ayildiz44@yahoo.com a.yildiz@inonu.edu.t Madanpu K. A. Vidyalaya (H.S.), Vill+P.O. Madanpu, Dist. Nadia, Pin. 741245, W. B., India ghosh.sujit6@gmail.com