() CUBO A Mathematical Journal Vol.13, No¯ 03, (141–152). October 2011 On Weak concircular Symmetries of Trans-Sasakian manifolds Shyamal Kumar Hui Nikhil Banga Sikshan Mahavidyalaya, Bishnupur, Bankura – 722 122, West Bengal, India. email: shyamal hui@yahoo.co.in ABSTRACT The object of the present paper is to study weakly concircular symmetric and weakly concircular Ricci symmetric trans-Sasakian manifolds. RESUMEN El objeto del presente trabajo es el estudio de variedades simétricas débilmente concir- culares y variedades simétricas trans-Sasakian débilmente concircular de Ricci. Keywords. weakly symmetric manifold, weakly concircular symmetric manifold, weakly Ricci symmetric manifold, concircular Ricci tensor, weakly concircular Ricci symmetric manifold, α- Sasakian manifold, β-Kenmotsu manifold, trans-Sasakian manifold. Mathematics Subject Classification: 53C15, 53C25. 142 Shyamal Kumar Hui CUBO 13, 3 (2011) 1 Introduction The notion of weakly symmetric manifolds was introduced by Tamássy and Binh [9]. A non-flat Riemannian manifold (Mn, g) (n > 2) is called a weakly symmetric manifold if its curvature tensor R of type (0, 4) satisfies the condition (∇XR)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V) (1.1) + H(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + E(V)R(Y, Z, U, X) for all vector fields X, Y, Z, U, V ∈ χ(Mn); χ(M) being the Lie algebra of smooth vector fields of M, where A, B, H, D and E are 1-forms (not simultaneously zero) and ∇ denotes the operator of covariant differentiation with respect to the Riemannian metric g. The 1-forms are called the associated 1-forms of the manifold and an n-dimensional manifold of this kind is denoted by (WS)n. In 1999 De and Bandyopadhyay [3] studied a (WS)n and proved that in such a manifold the associated 1-forms B = H and D = E. Hence (1.1) reduces to the following: (∇XR)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V) (1.2) + B(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + D(V)R(Y, Z, U, X). A transformation of an n-dimensional Riemannian manifold M, which transforms every geodesic circle of M into a geodesic circle, is called a concircular transformation [11]. The interesting in- variant of a concircular transformation is the concircular curvature tensor C̃, which is defined by [11] C̃(Y, Z, U, V) = R(Y, Z, U, V) − r n(n − 1) [ g(Z, U)g(Y, V) − g(Y, U)g(Z, V) ] , (1.3) where r is the scalar curvature of the manifold. Recently Shaikh and Hui [7] introduced the notion of weakly concircular symmetric manifolds. A Riemannian manifold (Mn, g)(n > 2) is called weakly concircular symmetric manifold if its concircular curvature tensor C̃ of type (0, 4) is not identically zero and satisfies the condition (∇XC̃)(Y, Z, U, V) = A(X)C̃(Y, Z, U, V) + B(Y)C̃(X, Z, U, V) (1.4) + H(Z)C̃(Y, X, U, V) + D(U)C̃(Y, Z, X, V) + E(V)C̃(Y, Z, U, X) for all vector fields X, Y, Z, U, V ∈ χ(Mn), where A, B, H, D and E are 1-forms (not simultaneously zero) an n-dimensional manifold of this kind is denoted by (WC̃S)n. Also it is shown that [7], in a (WC̃S)n the associated 1-forms B = H and D = E, and hence the defining condition (1.4) of a (WC̃S)n reduces to the following form: (∇XC̃)(Y, Z, U, V) = A(X)C̃(Y, Z, U, V) + B(Y)C̃(X, Z, U, V) (1.5) + B(Z)C̃(Y, X, U, V) + D(U)C̃(Y, Z, X, V) + D(V)C̃(Y, Z, U, X), CUBO 13, 3 (2011) On Weak concircular Symmetries . . . 143 where A, B and D are 1-forms (not simultaneously zero). Again Tamássy and Binh [10] introduced the notion of weakly Ricci symmetric manifolds. A Riemannian manifold (Mn, g) (n > 2) is called weakly Ricci symmetric manifold if its Ricci tensor S of type (0, 2) is not identically zero and satisfies the condition (∇XS)(Y, Z) = A(X)S(Y, Z) + B(Y)S(X, Z) + D(Z)S(Y, X), (1.6) where A, B and D are three non-zero 1-forms, called the associated 1-forms of the manifold, and ∇ denotes the operator of covariant differentiation with respect to the metric tensor g. Such an n-dimensional manifold is denoted by (WRS)n. Let {ei : i = 1, 2, · · · , n} be an orthonormal basis of the tangent space at each point of the manifold and let P(Y, V) = n∑ i=1 C̃(Y, ei, ei, V), (1.7) then from (1.3), we get P(Y, V) = S(Y, V) − r n g(Y, V). (1.8) The tensor P is called the concircular Ricci symmetric tensor [4], which is a symmetric tensor of type (0, 2). In [4] De and Ghosh introduced the notion of weakly concircular Ricci symmetric manifolds. A Riemannian manifold (Mn, g)(n > 2) is called weakly concircular Ricci symmetric manifold [4] if its concircular Ricci tensor P of type (0, 2) is not identically zero and satisfies the condition (∇XP)(Y, Z) = A(X)P(Y, Z) + B(Y)P(X, Z) + D(Z)P(Y, X), (1.9) where A, B and D are three 1-forms (not simultaneously zero). In [5] Oubiña introduced the notion of trans-Sasakian manifolds which contains both the class of Sasakian and cosympletic structures, and are closely related to the locally conformal Kähler man- ifolds. A trans-Sasakian manifold of type (0, 0), (α, 0) and (0, β) are the cosympletic, α-Sasakian and β-Kenmotsu manifold respectively. In particular, if α = 1, β = 0; and α = 0, β = 1, then a trans-Sasakian manifold reduces to a Sasakian and Kenmotsu manifold respectively. Thus trans- Sasakian structures provide a large class of generalized quasi-Sasakian structures. Tamássy and Binh [10] studied weakly symmetric and weakly Ricci symmetric Sasakian manifolds and proved that in such a manifold the sum of the associated 1-forms vanishes everywhere. Again Özgür [6] studied weakly symmetric and weakly Ricci symmetric Kenmotsu manifolds and proved that in such a manifold the sum of the associated 1-forms is zero everywhere and hence such a manifold does not exist unless the sum of the associated 1-forms is everywhere zero. The object of the present paper is to study weakly concircular symmetric and weakly concircu- lar Ricci symmetric trans-Sasakian manifolds. Section 2 deals with preliminaries of trans-Sasakian manifolds. Recently Shaikh and Hui [8] studied weakly symmetric and weakly Ricci symmetric trans-Sasakian manifolds and proved that the sum of the associated 1-forms of a weakly symmetric and also of a weakly Ricci symmetric trans-Sasakian manifold of non-vanishing ξ-sectional curva- ture are non-zero everywhere and hence such two structure exists, provided that the manifold is 144 Shyamal Kumar Hui CUBO 13, 3 (2011) of non-vanishing ξ-sectional curvature. However, in section 3 of the paper we have obtained all the 1-forms of a weakly concircular symmetric trans-Sasakian manifold and hence such a structure exist always. Again in section 4 we study weakly concircular Ricci symmetric trans-Sasakian mani- folds and obtained all the 1-forms of a weakly concircular Ricci symmetric trans-Sasakian manifold and consequently such a structure is always exist. Also it is proved that the sum of the associ- ated 1-forms of a weakly concircular Ricci symmetric trans-Sasakian manifold is non-vanishing everywhere. 2 Trans-Sasakian manifolds A (2n + 1)-dimensional smooth manifold M is said to be an almost contact metric manifold [1] if it admits an (1, 1) tensor field φ, a vector field ξ, an 1-form η and a Riemannian metric g, which satisfy φξ = 0, η(φX) = 0, φ2X = −X + η(X)ξ, (2.1) g(φX, Y) = −g(X, φY), η(X) = g(X, ξ), η(ξ) = 1, (2.2) g(φX, φY) = g(X, Y) − η(X)η(Y) (2.3) for all vector fields X, Y on M. An almost contact metric manifold M2n+1(φ, ξ, η, g) is said to be trans-Sasakian manifold [5] if (M × R, J, G) belongs to the class W4 of the Hermitian manifolds, where J is the almost complex structure on M × R defined by J ( Z, f d dt ) = ( φZ − fξ, η(Z) d dt ) for any vector field Z on M and smooth function f on M × R and G is the product metric on M × R. This may be stated by the condition [2] (∇Xφ)(Y) = α{g(X, Y)ξ − η(Y)X} + β{g(φX, Y)ξ − η(Y)φX}, (2.4) where α, β are smooth functions on M and such a structure is said to be the trans-Sasakian structure of type (α, β). From (2.4) it follows that ∇Xξ = −αφX + β{X − η(X)ξ}, (2.5) (∇Xη)(Y) = −αg(φX, Y) + βg(φX, φY). (2.6) In a trans-Sasakian manifold M2n+1(φ, ξ, η, g), the following relations hold: R(X, Y)ξ = (α2 − β2)[η(Y)X − η(X)Y] − (Xα)φY − (Xβ)φ2(Y) (2.7) + 2αβ[η(Y)φX − η(X)φY] + (Yα)φX + (Yβ)φ2(X), CUBO 13, 3 (2011) On Weak concircular Symmetries . . . 145 η(R(X, Y)Z) = (α2 − β2)[g(Y, Z)η(X) − g(X, Z)η(Y)] (2.8) − 2αβ[g(φX, Z)η(Y) − g(φY, Z)η(X)] − (Yα)g(φX, Z) − (Xβ){g(Y, Z) − η(Y)η(Z)} + (Xα)g(φY, Z) + (Yβ){g(X, Z) − η(Z)η(X)}, S(X, ξ) = [2n(α2 − β2) − (ξβ)]η(X) − ((φX)α) − (2n − 1)(Xβ), (2.9) R(ξ, X)ξ = (α2 − β2 − ξβ)[η(X)ξ − X], (2.10) S(ξ, ξ) = 2n(α2 − β2 − ξβ), (2.11) (ξα) + 2αβ = 0, (2.12) Qξ = [2n(α2 − β2) − (ξβ)]ξ + φ(gradα) − (2n − 1)(gradβ), (2.13) where R is the curvature tensor of type (1, 3) of the manifold and Q is the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor S, that is, g(QX, Y) = S(X, Y) for any vector fields X, Y on M. 3 Weakly concircular symmetric trans-sasakian manifolds Definition 3.1. A trans-Sasakian manifold (M2n+1, g)(n > 1) is said to be weakly concircular symmetric if its concircular curvature tensor C̃ of type (0, 4) satisfies (1.5). Setting Y = V = ei in (1.5) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get (∇XS)(Z, U) − dr(X) n g(Z, U) (3.1) = A(X) [ S(Z, U) − r n g(Z, U) ] + B(Z) [ S(X, U) − r n g(X, U) ] +D(U) [ S(X, Z) − r n g(X, Z) ] + B(R(X, Z)U) + D(R(X, U)Z) − r n(n − 1) [ {B(X) + D(X)}g(Z, U) − B(Z)g(X, U) − D(U)g(Z, X) ] . Plugging X = Z = U = ξ in (3.1) and then using (2.7) and (2.11), we obtain A(ξ) + B(ξ) + D(ξ) = 2n2{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) 2n2{α2 − (ξβ) − β2} − r . (3.2) This leads to the following: Theorem 3.1. In a weakly concircular symmetric trans-Sasakian manifold (M2n+1, g) (n > 1), the relation (3.2) holds. 146 Shyamal Kumar Hui CUBO 13, 3 (2011) Next, substituting X and Z by ξ in (3.1) and then using (2.7) and (2.12) we obtain (∇ξS)(ξ, U) − dr(ξ) n η(U) (3.3) = [ A(ξ) + B(ξ) ] [ S(U, ξ) − r n η(U) ] + D(U) [ (2n − 1){α2 − (ξβ) −β2} − n − 2 n(n − 1) r ] + [ α 2 − (ξβ) − β2 − r n(n − 1) ] η(U)D(ξ). From (2.9), we have (∇ξS)(ξ, U) = ∇ξS(ξ, U) − S(∇ξξ, U) − S(ξ, ∇ξU) (3.4) = ∇ξS(ξ, U) − S(ξ, ∇ξU) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ))]η(U) −(2n − 1)(U(ξβ)) − (φU(ξα)). By virtue of (3.3) and (3.4) we obtain from (3.2) that D(U) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ]η(U) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r (3.5) − (2n − 1)(U(ξβ)) + (φU(ξα)) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r + D(ξ) [ (2n − 1){(α2 − β2)η(U) − (Uβ)} − ((φU)α) − n−2 n(n−1) rη(U) (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(U) − (2n − 1)(Uβ) − ((φU)α) ] . Next, setting X = U = ξ in (3.1) and proceeding in a similar manner as above, we get B(Z) = [2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ]η(Z) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r (3.6) − (2n − 1)(Z(ξβ)) + (φZ(ξα)) (2n − 1)[α2 − (ξβ) − β2] − n−2 n(n−1) r + B(ξ) [ (2n − 1){(α2 − β2)η(Z) − (Zβ)} − ((φZ)α) − n−2 n(n−1) rη(Z) (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(Z) − (2n − 1)(Zβ) − ((φZ)α) ] . CUBO 13, 3 (2011) On Weak concircular Symmetries . . . 147 Again, setting Z = U = ξ in (3.1), we get (∇XS)(ξ, ξ) − dr(X) n = A(X) [ S(ξ, ξ) − r n ] + [B(ξ) + D(ξ)][S(X, ξ) (3.7) − n − 2 n(n − 1) rη(X)] + B(R(X, ξ)ξ) + D(R(X, ξ)ξ) − r n(n − 1) [B(X) + D(X)] = [2n{α2 − (ξβ) − β2} − r n ]A(X) + [B(ξ) + D(ξ)] [ S(X, ξ) − { n − 2 n(n − 1) r + α2 − (ξβ) − β2}η(X) ] + [B(X) + D(X)] [ α 2 − (ξβ) − β2 − r n(n − 1) ] . Now we have (∇XS)(ξ, ξ) = ∇XS(ξ, ξ) − 2S(∇Xξ, ξ), which yields by using (2.5) and (2.9) that (∇XS)(ξ, ξ) = 2n[2α(Xα) − 2β(Xβ) − (X(ξβ))] (3.8) + 2α[(Xα) − η(X)(ξα) − (2n − 1)((φX)β)] + 2β[((φX)α) + (2n − 1){(Xβ) − (ξβ)η(X)}]. In view of (3.8), (3.7) yields [ 2n{α 2 − (ξβ) − β2} − r n ] A(X) + [ α 2 − (ξβ) − β2 − r n(n − 1) ][ B(X) + D(X) ] (3.9) = 2n [ 2α(Xα) − 2β(Xβ) − (X(ξβ)) ] + 2α [ (Xα) − η(X)(ξα) −(2n − 1)((φX)β) ] + 2β [ ((φX)α) + (2n − 1){(Xβ) − (ξβ)η(X)} ] − dr(X) n − {B(ξ) + D(ξ)} [ {(2n − 1)(α2 − β2) − n − 2 n(n − 1) r}η(X) − ((φX)α) − (2n − 1)(Xβ) ] . 148 Shyamal Kumar Hui CUBO 13, 3 (2011) Using (3.5) and (3.6) in (3.9), we obtain [2n{α2 − (ξβ) − β2} − r n ]A(X) (3.10) = 2n[2α(Xα) − 2β(Xβ) − (X(ξβ))] + 2α[(Xα) − η(X)(ξα) −(2n − 1)((φX)β)] + 2β[((φX)α) + (2n − 1){(Xβ) − (ξβ)η(X)}] − dr(X) n − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ {(2n − 1)(α2 −β2) − n − 2 n(n − 1) r}η(X) − ((φX)α) − (2n − 1)(Xβ) ] +A(ξ) 2n{α2 − (ξβ) − β2} − r n (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ {(2n − 1)(α2 −β2) − n − 2 n(n − 1) r}η(X) − ((φX)α) − (2n − 1)(Xβ) ] − 2{α2 − (ξβ) − β2 − r n(n−1) } (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ 2n{2α(ξα) −2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(X) + 2{α2 − (ξβ) − β2 − r n(n−1) } (2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r [ (2n − 1)(X(ξβ)) + (φX(ξα)) ] + 2{α2 − (ξβ) − β2 − r n(n−1) }[2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n ] [2n{α2 − (ξβ) − β2} − r n ][(2n − 1){α2 − (ξβ) − β2} − n−2 n(n−1) r] [ {2n(α2 − β2) − (ξβ) − r n }η(X) − (2n − 1)(Xβ) − ((φX)α) ] . This leads to the following: Theorem 3.2. In a weakly concircular symmetric trans-Sasakian manifold (M2n+1, g) (n > 1), the associated 1-forms D, B and A are given by (3.5), (3.6) and (3.10) respectively. 4 Weakly concircular Ricci symmetric trans-Sasakian man- ifolds Definition 4.1. A trans-Sasakian manifold (M2n+1, g)(n > 1) is said to be weakly concircular Ricci symmetric if its concircular Ricci tensor P of type (0, 2) satisfies (1.9). CUBO 13, 3 (2011) On Weak concircular Symmetries . . . 149 In view of (1.8), (1.9) yields (∇XS)(Y, Z) − dr(X) n g(Y, Z) = A(X) [ S(Y, Z) − r n g(Y, Z) ] (4.1) + B(Y) [ S(X, Z) − r n g(X, Z) ] + D(Z) [ S(X, Y) − r n g(X, Y) ] . Setting X = Y = Z = ξ in (4.1), we get the relation (3.2) and hence we can state the following: Theorem 4.1. In a weakly concircular Ricci symmetric trans-Sasakian manifold (M2n+1, g) (n > 1), the relation (3.2) holds. Next, substituting X and Y by ξ in (4.1), we obtain (∇ξS)(ξ, Z) − dr(ξ) n η(Z) = [A(ξ) + B(ξ)] [ S(ξ, Z) (4.2) − r n η(Z) ] + D(Z) [ S(ξ, ξ) − r n ] . Using (3.2) and (3.4) in (4.2), we get D(Z) = [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(Z) 2n[α2 − (ξβ) − β2] − r n (4.3) − (2n − 1)(Z(ξβ)) + (φZ(ξα)) 2n[α2 − (ξβ) − β2] − r n + D(ξ) [ 2n{(α2 − β2) − (ξβ) − r n }η(Z) − ((φZ)α) − (2n − 1)(Zβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(Z) − (2n − 1)(Zβ) − ((φZ)α) ] for all Z. Again putting X = Z = ξ in (4.1) and proceeding in a similar manner as above we get B(Y) = [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(Y) 2n[α2 − (ξβ) − β2] − r n (4.4) − (2n − 1)(Y(ξβ)) + (φY(ξα)) 2n[α2 − (ξβ) − β2] − r n + B(ξ) [ 2n{(α2 − β2) − (ξβ) − r n }η(Y) − ((φY)α) − (2n − 1)(Yβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(Y) − (2n − 1)(Yβ) − ((φY)α) ] for all Y. 150 Shyamal Kumar Hui CUBO 13, 3 (2011) Again, setting Y = Z = ξ in (4.1) and using (2.9) and (2.11), we get (∇XS)(ξ, ξ) − dr(X) n = [ 2n{α 2 − (ξβ) − β2} − r n ] A(X) (4.5) + [B(ξ) + D(ξ)] [ {2n(α2 − β2) − (ξβ)}η(X) − ((φX)α) − (2n − 1)(Xβ) ] . Using (3.2) and (3.8) in (4.5), we get A(X) = 2n[2α(Xα) − 2β(Xβ) − (X(ξβ))] 2n{α2 − (ξβ) − β2} − r n (4.6) + 2α[(Xα) − η(X)(ξα) − (2n − 1)((φX)β)] 2n{α2 − (ξβ) − β2} − r n + 2β[((φX)α) + (2n − 1){(Xβ) − (ξβ)η(X)}] 2n{α2 − (ξβ) − β2} − r n + A(ξ) [ {2n(α2 − β2) − (ξβ) − r n }η(X) − ((φX)α) − (2n − 1)(Xβ) 2n{α2 − (ξβ) − β2} − r n ] − 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) − (ξβ) − r n }η(X) − (2n − 1)(Xβ) − ((φX)α) ] for all X. This leads to the following: Theorem 4.2. In a weakly concircular Ricci symmetric trans-Sasakian manifold (M2n+1, g) (n > 1), the associated 1-forms D, B and A are given by (4.3), (4.4) and (4.6) respectively. Adding (4.3), (4.4) and (4.6) and using (3.2), we get A(X) + B(X) + D(X) (4.7) = 2n[2α(Xα) − 2β(Xβ) − (X(ξβ))] 2n{α2 − (ξβ) − β2} − r n + 2α[(Xα) − η(X)(ξα) − (2n − 1)((φX)β)] 2n{α2 − (ξβ) − β2} − r n + 2β[((φX)α) + (2n − 1){(Xβ) − (ξβ)η(X)}] 2n{α2 − (ξβ) − β2} − r n 2 [ 2n{2α(ξα) − 2β(ξβ)} − (ξ(ξβ)) − dr(ξ) n ] η(X) 2n{α2 − (ξβ) − β2} − r n − (2n − 1)(X(ξβ)) + (φX(ξα)) n{α2 − (ξβ) − β2} − r 2n − 2 [ 2n{2α(ξα) − 2β(ξβ) − (ξ(ξβ))} − dr(ξ) n ] [2n{α2 − (ξβ) − β2} − r n ]2 [ {2n(α2 − β2) −(ξβ) − r n }η(X) − (2n − 1)(Xβ) − ((φX)α) ] CUBO 13, 3 (2011) On Weak concircular Symmetries . . . 151 for any vector field X. This leads to the following: Theorem 4.3. In a weakly concircular Ricci symmetric trans-Sasakian manifold (M2n+1, g) (n > 1), the sum of the associated 1-forms is given by (4.7). In particular, if φ(grad α) = grad β, then (ξβ) = 0 and hence the relation (4.7) reduces to the following form A(X) + B(X) + D(X) (4.8) = 2n[2α(Xα) − 2β(Xβ)] 2n(α2 − β2) − r n + 2α{(Xα) − η(X)(ξα) − (2n − 1)((φX)β)} 2n(α2 − β2) − r n + 2β{((φX)α) + (2n − 1)(Xβ)} + 2{4nα(ξα) − dr(ξ) n }η(X) − 2(φX(ξα)) 2n(α2 − β2) − r n − 2[4nα(ξα) − dr(ξ) n ] [2n(α2 − β2) − r n ]2 [ {2n(α2 − β2) − r n }η(X) − ((φX)α) − (2n − 1)(Xβ) ] . for any vector field X. This leads to the following: Corollary 4.1. If a weakly concircular Ricci symmetric trans-Sasakian manifold (M2n+1, g) (n > 1) satisfies the condition φ(grad α) = grad β, then the sum of the associated 1-forms is given by (4.8). If β = 0 and α = 1, then (4.7) yields A(X) + B(X) + D(X) = 0 for all X and hence we can state the following: Corollary 4.2. There is no weakly concircular Ricci symmetric Sasakian manifold M2n+1(n > 1), unless the sum of the 1-forms is everywhere zero. Corollary 4.3. If an α-Sasakian manifold is weakly concircular Ricci symmetric, then the sum of the 1-forms, i.e., A + B + D is given by A(X) + B(X) + D(X) = 2α[(2n + 1)(Xα) − η(X)(ξα)] − 2(φX(ξα)) 2nα2 − r n + 2[4nα(ξα) − dr(ξ) n ]((φX)α) (2nα2 − r n )2 . Again, if α = 0 and β = 1, then (4.7) yields A(X) + B(X) + D(X) = 0 for all X. This leads to the following: Corollary 4.4. There is no weakly concircular Ricci symmetric Kenmotsu manifold M2n+1(n > 1), unless the sum of the 1-forms is everywhere zero. Corollary 4.5. If a β-Kenmotsu manifold is weakly concircular Ricci symmetric, then the sum of 152 Shyamal Kumar Hui CUBO 13, 3 (2011) the 1-forms, i.e., A + B + D is given by A(X) + B(X) + D(X) = 2n{2β(Xβ) + (X(ξβ))} − 2(2n − 1)β{(Xβ) − (ξβ)η(X)} 2n{(ξβ) + β2} + r n + 2[{4nβ(ξβ) + (ξ(ξβ) + dr(ξ) n }η(X) + (2n − 1)(X(ξβ))] 2n{(ξβ) + β2} + r n − 2[2n{2β(ξβ) + (ξ(ξβ))} + dr(ξ) n ][{2nβ2 + (ξβ) + r n }η(X) + (2n − 1)(Xβ)] [2n{(ξβ) + β2} + r n ]2 . Received: April 2010. Revised: September 2010. 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