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

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1 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),

2 208 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 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 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.

4 210 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 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.

6 212 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 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.

8 214 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 9 On a class of N(k)-contact metic manifolds 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

10 216 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), [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, [4] D.E. Blai, Riemannian geomety of contact and symplectic manifolds. Pog. Math. 203, Bikhause Boston, Inc., Boston, MA, [5] D.E. Blai, Two emaks on contact metic stuctues. Tohoku Math. J. 29 (1977), [6] D.E. Blai, Invesion theoy and confomal mapping. Stud. Math. Lib. 9, Ame. Math. Soc., [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),

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