CUBO A Mathemati al Journal Vol.15, N o 03, (59�70). O tober 2013 On quasi- onformally �at and quasi- onformally semisymmetri generalized Sasakian-spa e-forms D.G. Prakasha Department of Mathemati s, Karnatak University, Dharwad-580 003 Karnataka State, India. prakashadg�gmail. om H.G. Nagaraja Department of Mathemati s Central College Campus, Bangalore University, Bangalore-560 001, India. hgnraj�yahoo. om ABSTRACT The obje t of the present paper is to study quasi- onformally �at and quasi- onformally semisymmetri generalized Sasakian-spa e-forms. RESUMEN El objeto del artí ulo a tual es estudiar formas de espa io Sasakian uasi- onforma ionales planas y uasi- onforma ionales generalizadas semisimétri as. Keywords and Phrases: Generalized Sasakian-spa e-forms, quasi- onformally �at, quasi- onformally semisymmetri , Einstein manifold, s alar urvature. 2010 AMS Mathemati s Subje t Classi� ation: 53C25, 53D15. 60 D.G. Prakasha & H.G. Nagaraja CUBO 15, 3 (2013) 1 Introdu tion The notion of generalized Sasakian-spa e-forms was introdu ed and studied by Alegre et al [1℄ with several examples. A generalized Sasakian-spa e-form is an almost onta t metri manifold (M, φ, ξ, η, g) whose urvature tensor is given by R(X, Y)Z = f1{g(Y, Z)X − g(X, Z)Y} + f2{g(X, φZ)φY − g(Y, φZ)φX + 2g(X, φY)φZ} + f3{η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ} where f1, f2, f3 are di�erentiable fun tions on M and X, Y, Z are ve tor �elds on M. In su h ase we will write the manifold as M(f1, f2, f3). This kind of manifolds appears as natural generalization of the Sasakian-spa e-forms by taking: f1 = c + 3 4 and f2 = f3 = c − 1 4 , where c denotes onstant φ-se tional urvature. The φ-se tional urvature of generalized Sasakian- spa e-forms M(f1, f2, f3) is f1 + 3f2. Moreover, osymple ti spa e-forms and Kenmotsu spa e- forms are also parti ular ase of generalized Sasakian-spa e-forms. In the re ent paper P. Alegre and A. Carriazo [2℄ studied onta t metri and trans-Sasakian generalized Sasakian-spa e-forms. Generalized Sasakian-spa e-forms have been studied by several authors, viz., [5,6,10,12℄. In Riemannian geometry, many authors have studied urvature properties and to what extent they determined the manifold itself. Two important urvature properties are �atness and symme- try. As a generalization of lo al symmetri spa e, the notion of semisymmetri spa e [13℄ is de�ned as R(X, Y) · R = 0, where R(X, Y) a ts on R as a derivation. In this onne tion, the onformal �atness and lo al symmetry of generalized Sasakian-spa e-forms was studied in [10℄. Also in [6℄, generalized Sasakian-spa e-forms with vanishing proje tive urvature tensor and some symmetry properties have been onsidered. Motivated by these fa ts, in this paper we study the �atness and semisymmetry property of generalized Sasakian-spa e-form regarding the quasi- onformal urva- ture tensor. The notion of the quasi- onformal urvature tensor was given by Yano and Sawaki [14℄. A - ording to them in a (2n + 1)-dimensional (n > 1) almost onta t metri manifold the quasi- onformal urvature tensor C̃ is de�ned by C̃(X, Y)Z = aR(X, Y)Z + b[S(Y, Z)X − S(X, Z)Y + g(Y, Z)QX − g(X, Z)QY] − r 2n + 1 ( a 2n + 2b ) [g(Y, Z)X − g(X, Z)Y] (1) where a and b are onstants and R, S, Q and r are the Riemannian urvature tensor of type (1, 3), the Ri i tensor of type (0, 2), the Ri i operator de�ned by g(QX, Y) = S(X, Y) and the s alar CUBO 15, 3 (2013) On quasi- onformally �at and quasi- onformally semisymmetri . . . 61 urvature of the manifold respe tively. If a = 1 and b = − 1 2n−1 , then (1) takes the form C(X, Y)Z = R(X, Y)Z − 1 2n − 1 [S(Y, Z)X − S(X, Z)Y + g(Y, Z)QX − g(X, Z)QY] + r (2n)(2n − 1) [g(Y, Z)X − g(X, Z)Y] = C(X, Y)Z (2) where C is the onformal urvature tensor [8℄. Thus onformal urvature tensor is a parti ular ase of quasi- onformal urvature tensor. A manifold (M, φ, ξ, η, g) shall be alled quasi- onformally �at if the quasi- onformal urva- ture tensor C̃ = 0. It is known that the quasi- onformally �at manifold is either onformally �at if a 6= 0 or Einstein if a = 0 and b 6= 0 [3℄. If the manifold (M, φ, ξ, η, g) satis�es R(X, Y).C̃ = 0, then the manifold is said to be quasi- onformally semisymmetri manifold. A manifold (M, φ, ξ, η, g) is said to be Ri i symmetri if R · S = 0 holds on M [7℄. The lass of Ri i semisymmetri manifolds in ludes the set of Ri i symmetri manifolds (∇S = 0) as a proper subset. Every semisymmetri manifold is Ri i symmetri . The onverse is not true. In the present paper quasi- onformally �at and quasi- onformally semisymmetri generalized Sasakian-spa e-forms are studied. The paper is organized as follows: Se tion 2 of this paper on- tains some preliminary results on generalized Sasakian-spa e-forms. In se tion 3, we study quasi- onformally �at generalized Sasakian-spa e-forms and obtain ne essary and su� ient onditions for a generlized Sasakian-spa e-form to be quasi- onformally �at. Also, we onsider quasi- onformally Ri i tensor and quasi- onformally Ri i symmetri generalized Sasakian spa e-forms. In the next se tion, we deal with quasi- onformally semisymmetri generalized Sasakian-spa e-forms and it is proved that a generalized Sasakian-spa e-form is quasi- onformally semisymmetri if and only if the spa e-form is quasi- onformally �at and f1 = f3. 2 Preliminaries An odd-dimensional manifold M2n+1 is said to admit an almost onta t stru ture (φ, ξ, η), if it arries a tensor �eld φ of type (1, 1), a ve tor �eld ξ and a 1-form η satisfying φ2 = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (3) If g is a ompatible Riemannian metri with (φ, ξ, η) su h that g(φX, φY) = g(X, Y) − η(X)η(Y) (4) or equivalently, g(X, ξ) = η(X), g(X, φY) = −g(φX, Y) (5) for all ve tor �elds X, Y on M2n+1, then M2n+1 be omes an almost onta t metri manifold with an almost onta t metri stru ture (φ, ξ, η, g). An almost onta t metri stru ture is alled a onta t metri stru ture if g(X, φY) = dη(X, Y) 62 D.G. Prakasha & H.G. Nagaraja CUBO 15, 3 (2013) An almost onta t metri manifold is Sasakian if and only if (∇Xφ)Y = g(X, Y)ξ − η(Y)X (6) for all ve tor �elds X, Y on M2n+1. For a (2n + 1)-dimensional generalized Sasakian-spa e-form we have [1℄ R(X, Y)Z = f1{g(Y, Z)X − g(X, Z)Y} + f2{g(X, φZ)φY − g(Y, φZ)φX + 2g(X, φY)φZ} + f3{η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ}, (7) QX = (2nf1 + 3f2 − f3)X − (3f2 + (2n − 1)f3)η(X)ξ, (8) S(X, Y) = (2nf1 + 3f2 − f3)g(X, Y) − (3f2 + (2n − 1)f3)η(X)η(Y), (9) r = 2n(2n + 1)f1 + 6nf2 − 4nf3 (10) for all ve tor �elds X, Y, Z. By virtue of equations(7) and (9), we have η(R(X, Y)Z) = (f1 − f3){g(Y, Z)η(X) − g(X, Z)η(Y)}, (11) R(X, Y)ξ = (f1 − f3){η(Y)X − η(X)Y}, (12) R(ξ, X)Y = (f1 − f3){g(X, Y)ξ − η(Y)X}, (13) S(X, ξ) = 2n(f1 − f3)η(X), (14) S(ξ, ξ) = 2n(f1 − f3). (15) The above results will be used in the next se tions. Now we would like to re olle t some of the examples of generalized Sasakian-spa e-forms. Example 1: ( [11℄)A osymple ti -spa e-form, i.e., a osymple ti manifold with onstant φ- se tional urvature c, is a generalized Sasakian-spa e-form with f1 = f2 = f3 = c/4. Example 2:( [9℄)A Kenmotsu-spa e-form, i.e., a Kenmotsu manifolds with onstant φ-se tional urvature c, is a generalized Sasakian-spa e-form with f1 = (c − 3)/4 and f2 = f3 = (c + 1)/4. Example 3: ( [1℄)Let N(F1, F2) be a generalized omplex-spa e-form. Then, the warped produ t M = R ×f N, endowed with the almost onta t metri stru ture (φ, ξ, η, gf), is a generalized Sasakian-spa e-form M(f1; f2; f3) with fun tions: f1 = (F1 ◦ π) − f ′2 f2 , f2 = F2 ◦ π f2 , f3 = (F1 ◦ π) − f ′2 f2 + f′′ f . In parti ular if N(c) is a omplex-spa e-form, we obtain the generalized Sasakian-spa e-form M ( c − 4f′2 4f2 , c 4f2 , c − 4f′2 4f2 + f" f ) . Hen e, the warped produ ts R ×f C n , R ×f CP n(4) and R ×f CH n(−4) are generalized Sasakian- spa e-forms. CUBO 15, 3 (2013) On quasi- onformally �at and quasi- onformally semisymmetri . . . 63 Example 4: R ×f C m is a generalized Sasakian-spa e-form with f1 = − (f′)2 f2 , f2 = 0, f3 = − (f′1)2 f2 + f′′ f , where f = f(t). 3 Quasi- onformally �at generalized Sasakian-spa e-forms If the generalized Sasakian-spa e-form M(f1, f2, f3) under onsideration is quasi- onformally �at, then we have from (1) R(X, Y, Z, W) = b a [S(X, Z)g(Y, W) − S(Y, Z)g(X, W) (16) +S(Y, W)g(X, Z) − S(X, W)g(Y, Z)] + r (2n + 1)a [ a 2n + 2b][g(Y, Z)g(X, W) − g(X, Z)g(Y, W)], where a and b are onstants and R(X, Y, Z, W) = g(R(X, Y)Z, W). Now putting Z = ξ in (16) and using (4), (12) and (14) we get (f1 − f3)[g(X, W)η(Y) − g(Y, W)η(X)] (17) = 2n(f1 − f3) b a [g(Y, W)η(X) − g(X, W)η(Y) +S(Y, W)η(X) − S(X, W)η(Y)] + r (2n + 1)a [ a 2n + 2b][g(X, W)η(Y) − g(Y, W)η(X)]. Again putting X = ξ in (17) and using (4) and (14) it follows that S(Y, W) = Ag(Y, W) + Bη(Y)η(W), (18) where A = [− a b (f1 − f3) − 2n(f1 − f3) + r (2n + 1)b ( a 2n + 2b)] (19) and B = [ a b (f1 − f3) + 4n(f1 − f3) − r (2n + 1)b ( a 2n + 2b)]. (20) Here A+B = 2n(f1 −f3). In the equation (18) putting Y = W = {ei}, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold and taking summation over i, 1 ≤ i ≤ 2n+1, we get r = (2n + 1)A + B. (21) Now with the help of (19) and (20) the equation (21) gives [a + (2n − 1)b][ r 2n + 1 − 2n(f1 − f3)] = 0. (22) 64 D.G. Prakasha & H.G. Nagaraja CUBO 15, 3 (2013) If a + (2n − 1)b = 0 and a 6= 0 6= b. Then from (1) it follows that C̃(X, Y)Z = aC(X, Y)Z, (23) where C(X, Y)Z denotes the Weyl onformal urvature tensor. But, under the onsideration C̃ = 0. So the quasi- onformal �atness and onformally �atness are equivalent. This implies that C = 0. If a + (2n − 1)b 6= 0 and a 6= 0. Then from (22). r = 2n(2n + 1)(f1 − f3). (24) So by omparing (10) and (24) we have 3f2 + (2n − 1)f3 = 0. (25) By taking a ount of (25) in (9), we get S(X, Y) = 2n(f1 − f3)g(X, Y). (26) This shows that, M(f1, f2, f3) is an Einstein. Thus we state the following: Theorem 1. A quasi- onformally �at generalized Sasakian-spa e-from is either onformally �at or an Einstein manifold with s alar urvature r = 2n(2n + 1)(f1 − f3). In the above theorem we have seen if a + (2n − 1)b = 0 and a 6= 0 6= b, then it follows that a quasi- onformally �at generalized Sasakian-spa e-form is onformally �at. But, it is known that [10℄ a (2n + 1)-dimensional (n > 1) generalized Sasakian-spa e-form M(f1, f2, f3) is onformally �at if and only if f2 = 0. So in this ase M(f1, f2, f3) is quasi- onformally �at if and only if f2 = 0. On the other hand, if a + (2n − 1)b 6= 0 and a 6= 0 then we have (24). By omparing the equations (10) and (24), one an get (25). Conversely, suppose that (25) holds. Then in view of (7), (9) and (25), we an write the equation (1) as C̃(X, Y, Z, W) = a 1 − 2n f2[g(Y, Z)g(X, W) − g(X, Z)g(Y, W)] +af2[g(X, φZ)g(φY, W) − g(Y, φZ)g(φX, W) + 2g(X, φY)g(φZ, W)] + 3a 1 − 2n f2[g(Y, W)η(X)η(Z) − g(X, W)η(Y)η(Z) +g(X, Z)η(Y)η(W) − g(Y, Z)η(X)η(W)], (27) where C̃(X, Y, Z, W) = g(C̃(X, Y)Z, W). Repla ing X by φX and Y by φY in (27) we get C̃(φX, φY, Z, W) = a 1 − 2n f2[g(φY, Z)g(φX, W) − g(φX, Z)g(φY, W)] +af2[g(φX, φZ)g(φ 2Y, W) − g(φY, φZ)g(φ2X, W) + 2g(φX, φ2Y)g(φZ, W)]. (28) CUBO 15, 3 (2013) On quasi- onformally �at and quasi- onformally semisymmetri . . . 65 Putting Y = W = ei, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold, and taking summation over i, (1 ≤ i ≤ 2n + 1), we get 2n+1 ∑ i=1 C̃(φX, φei, Z, ei) = a 2n − 1 f2g(φX, φZ) + af2[−g(φX, φZ)g(φei, φei) + 3g(φ 2X, φ2Z)]. (29) Again putting X = Z = ei, where {ei} is an orthonormal basis of the tangent spa e at ea h point of the manifold, and taking summation over i, (1 ≤ i ≤ 2n + 1), we get after simpli� ation f2 = 0 with a 6= 0. Then in view of (25), we get f3 = 0. Therefore, we obtain from (7) that R(X, Y)Z = f1{g(Y, Z)X − g(X, Z)Y}. (30) From (30) we have S(X, Y) = 2nf1g(X, Y) and r = 2n(2n + 1)f1. Hen e in view of (1), we have C̃(X, Y)Z = 0. This leads to the following: Theorem 2. Let M(f1, f2, f3) be a (2n+1)-dimensional (n > 1) generalized Sasakian-spa e-form. Then M(f1, f2, f3) is quasi- onformally �at if and only if one of the following statements is true: (i) a + (2n − 1)b = 0, a 6= 0 6= b and f2 = 0. (ii) a + (2n − 1)b 6= 0, a 6= 0 and 3f2 + (2n − 1)f3 = 0. In a (2n + 1)-dimensional (n > 1) manifold (M, φ, ξ, η, g), let {ei}, i = 1, 2, ..., 2n + 1 be a lo al orthonormal basis. Then the quasi- onformal urvature tensor C̃(X, Y)Z de�ned as in (1), we an de�ne a symmetri tensor of type (0, 2) alled as quasi- onformal Ri i tensor and whi h is denoted by S C̃ (X, Y) = 2n+1 ∑ i=1 C̃(ei, X, Y, ei), (31) where ∑ 2n+1 i=1 C̃(ei, X, Y, ei) = ∑ 2n+1 i=1 g(C̃(ei, X)Y, ei). From (31)and (1), we have S C̃ (X, Y) = {a + (2n − 1)b}{S(X, Y) − r 2n + 1 g(X, Y)}. (32) We �rst assume that a (2n + 1)-dimensional generalized Sasakian-spa e-form M(f1, f2, f3) is Ri i semisymmetri . That is, (R(X, Y).S)(Z, W) = −S(R(X, Y)Z, W) − S(Z, R(X, Y)W) = 0. Now, sin e the urvature tensor R of type (0, 4), de�ned by g(R(X, Y)Z, W) = R(X, Y, Z, W) 66 D.G. Prakasha & H.G. Nagaraja CUBO 15, 3 (2013) is skew-symmetri where R is the urvature tensor of type (1, 3), we get from (33) and (11) by taking a ount that a + (2n − 1)b 6= 0 S C̃ (R(X, Y)Z, W) + S C̃ (Z, R(X, Y)W) = 0 (33) whi h implies that (R(X, Y).S C̃ )(Z, W) = 0. So the spa e-form M(f1, f2, f3) is quasi- onformally Ri i semisymmetri . Again, let us suppose that the spa e-form is quasi- onformally Ri i semisymmetri , that is, R.S C̃ = 0 holds in M(f1, f2, f3). Then (33) holds. Now using (33), and the skew-symmetri properties of R we get after simpli� ation R.S = 0, whi h implies that the spa e-form is Ri i semisymmetri . Hen e the following theorem holds: Theorem 3. A (2n + 1)-dimensional (n > 1) generalized Sasakian-spa e-form M(f1, f2, f3) is Ri i semisymmetri if and only if it is quasi- onformally Ri i semisymmetri provided that a + (2n − 1)b 6= 0. 4 Quasi- onformally semisymmetri generalized Sasakian- spa e-forms In this se tion we onsider a generalized Sasakian-spa e-form M(f1, f2, f3) satisfying the ondition R(X, Y) · C̃ = 0. (34) Then we obtain from (1) by using (4), (12) and (14) η(C̃(X, Y)Z) = { (a + 2nb)(f1 − f3) − r (2n + 1) { a 2n + 2b } } [g(Y, Z)η(X) −g(X, Z)η(Y)] + b[S(Y, Z)η(X) − S(X, Z)η(Y)]. (35) On taking Z = ξ in the equation (35), we get η(C̃(X, Y)ξ) = 0. (36) Again putting X = ξ in the equation (35), we have η(C̃(ξ, Y)Z) = { (a + 2nb)(f1 − f3) − r (2n + 1) { a 2n + 2b } } [g(Y, Z) −η(Y)η(Z)] + b[S(Y, Z) − 2n(f1 − f3)η(Y)η(Z)]. (37) In virtue of (34) we get R(X, Y)C̃(U, V)W − C̃(R(X, Y)U, V)W −C̃(U, R(X, Y)V)W − C̃(U, V)R(X, Y)W = 0. (38) CUBO 15, 3 (2013) On quasi- onformally �at and quasi- onformally semisymmetri . . . 67 Whi h implies that (f1 − f3){C̃(U, V, W, Y) − η(Y)η(C̃(U, V)W) +η(U)η(C̃(Y, V)W) + η(V)η(C̃(U, Y)W) +η(W)η(C̃(U, V)Y) − g(Y, U)η(C̃(ξ, V)W) −g(Y, V)η(C̃(U, ξ)W) − g(Y, W)η(C̃(U, V)ξ)} = 0. (39) Putting U = Y in (39) and with the help of (35) and (36) we get either f1 = f3 (40) or {C̃(Y, V, W, Y) + η(W)η(C̃(Y, V)Y) −g(Y, Y)η(C̃(ξ, V)W) − g(Y, V)η(C̃(Y, ξ)W)} = 0. (41) Let {e1, e2, ..., e2n+1} is an orthonormal basis of the tangent spa e at ea h point of the manifold. Putting Y = ei in (41) and taking summation over i, (1 ≤ i ≤ 2n + 1), and using (35), (37) we get S(V, W) = A′g(V, W) + B′η(V)η(W)} (42) where A′ = 2n(a + 2nb)(f1 − f3) − rb a − b (43) and B′ = −2n(2n + 1)b(f1 − f3) + rb a − b . (44) Here A′ + B′ = 2n(f1 − f3). Now ontra ting (42) we get r = (2n + 1)A′ + B′. (45) By (43) and (44) the equation (45) gives (a + (2n − 1)b)(r − 2n(2n + 1)(f1 − f3)) = 0. (46) Therefore, either a + (2n − 1)b = 0 or r = 2n(2n + 1)(f1 − f3). (47) From (43) and (47) we obtain A′ = 2n(f1 − f3). (48) By (44) and (47) we get B′ = 0. (49) So, from (42), (48) and (49) we have S(V, W) = 2n(f1 − f3)g(V, W). (50) 68 D.G. Prakasha & H.G. Nagaraja CUBO 15, 3 (2013) Therefore, M(f1, f2, f3) is an Einstein manifold. Now with the help of (47) and (50) the equations (35) and (37) imply that η(C̃(X, Y)Z) = 0 (51) and η(C̃(ξ, Y)Z) = 0 (52) respe tively. So using (36),(51) and (52) in (39) we get C̃(U, V, W, Y) = 0. (53) Therefore, by taking a ount of (40) and (53), we have either f1 = f3 or M(f1, f2, f3) is quasi- onformally �at. Conversely, if f1 = f3 then from (13) R(ξ, U) = 0. Then obviously the ondition R(ξ, U) · C̃ = 0, that is, quasi- onformally semisymmetri ondition is satis�ed. Again if the spa e-form is quasi- onformally �at, then learly it is quasi- onformally semisymmetri . Hen e we on lude the following: Theorem 4. A (2n+1)-dimensional (n > 1) generalized Sasakian-spa e-form is quasi- onformally semisymmetri if and only if either the spa e-form is quasi- onformally �at or f1 = f3. By ombining the Theorem2 and Theorem 4, we an state the following orollary: Theorem 5. Let M(f1, f2, f3) be a (2n+1)-dimensional (n > 1) generalized Sasakian-spa e-form. Then M(f1, f2, f3) is quasi- onformally semisymmetri if and only if f1 = f3 or one of the following statements is true: (i) a + (2n − 1)b = 0, a 6= 0 6= b and f2 = 0. (ii) a + (2n − 1)b 6= 0, a 6= 0 and 3f2 + (2n − 1)f3 = 0. It an be easily seen that ∇P = 0 implies R.P = 0. Hen e by virtue of Theorem 4 we get Corollary 4.1. A (2n+1)-dimensional (n > 1) quasi- onformally symmetri generalized Sasakian- spa e-form is either quasi- onformally �at or f1 = f3. A Riemannian manifold is said to be quasi- onformally re urrent if ∇P = A ⊗ P, where A is a non-zero 1-form. It an be easily shown that a quasi- onformally re urrent manifold satis�es R · P = 0. Hen e we immediately get the following: Corollary 4.2. A (2n+1)-dimensional (n > 1) quasi- onformally re urrent generalized Sasakian- spa e-form is either quasi- onformally �at or f1 = f3. In parti ular, for Sasakian-spa e-form f1 = c+3 4 and f3 = c−1 4 . So, f1 6= f3. Hen e we an have the following orollary: Corollary 4.3. A (2n+1)-dimensional (n > 1) Sasakian-spa e-form is quasi- onformally semisym- metri if and only if it is quasi- onformally �at. CUBO 15, 3 (2013) On quasi- onformally �at and quasi- onformally semisymmetri . . . 69 Remark: If we take f(t) = et in Example 4, we have f1 = −1, f2 = 0 and f3 = 0. Therefore, the ondition 3f2 + (2n − 1)f3 = 0 and f2 = 0 holds. Hen e from Theorem 2, generalized sasakian- spa e-form R ×f C m with f(t) = et is quasi- onformally �at. Similarly from Theorem 5, generalized Sasakian-spa e-form R ×f C m with f(t) = et is quasi- onformally semisymmetri . A knowledgement: The �rst author (DGP) is thankful to University Grants Commission, New Delhi, India for �nan ial support in the form of Major Resear h Proje t F. No. 39-30/2010 (SR), dated: 23-12-2010. Re eived: May 2012. A epted: September 2013. Referen es [1℄ P. Alegre, D. Blair and A. Carriazo, Generalized Sasakian-spa e-forms, Israel J. Math. 14 (2004), 157-183. [2℄ P. Alegre and A. Carriazo, Stru tures on generalized Sasakian-spa e-form, Di�erential Geom. and its appli ation 26 (2008), 656-666. doi. 10.1016/j difgeo. [3℄ K. 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