CUBO A Mathematical Journal Vol.20, No¯ 01, (17–29). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100017 W2-Curvature Tensor on Generalized Sasakian Space Forms Venkatesha and Shanmukha B. Department of Mathematics, Kuvempu University Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA. vensmath@gmail.com, meshanmukha@gmail.com ABSTRACT In this paper, we study W2-pseudosymmetric, W2-locally symmetric, W2-locally φ- symmetric and W2-φ-recurrent generalized Sasakian space form. Further, illustrative examples are given. RESUMEN En este art́ıculo, estudiamos formas espaciales Sasakianas generalizadas W2-seudosimétricas, W2-localmente φ-simétricas y W2-φ-recurrentes. Ejemplos ilustrativos son dados. Keywords and Phrases: Generalized Sasakian space form, W2-curvature tensor, pseudosym- metric, φ-recurrent, Einstein manifold. 2010 AMS Mathematics Subject Classification: 53C15, 53C25, 53C50. http://dx.doi.org/10.4067/S0719-06462018000100017 Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo 18 Venkatesha and Shanmukha B. CUBO 20, 1 (2018) 1 Introduction The nature of a Riemannian manifold depends on the curvature tensor R of the manifold. It is well known that the sectional curvatures of a manifold determine its curvature tensor completely. A Riemannian manifold with constant sectional curvature c is known as a real space form and its curvature tensor is given by R(X, Y)Z = c{g(Y, Z)X − g(X, Z)Y}. Representation for these spaces are hyperbolic spaces (c < 0), spheres (c > 0) and Euclidean spaces (c = 0). The φ-sectional curvature of a Sasakian space form is defined by Sasakian manifold and it has a specific form of its curvature tensor. Same notion also holds for Kenmotsu and cosymplectic space forms. In order to generalize such space forms in a common frame Alegre, Blair and Carriazo [1] introduced and studied generalized Sasakian space forms. A generalized Sasakian space form is an almost contact metric manifold (M2n+1, φ, ξ, η, g), whose curvature 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}, (1.1) The Riemanian curvature tensor of a generalized Sasakian space form M2n+1(f1, f2, f3) is simply given by R = f1R1 + f2R2 + f3R3, where f1, f2, f3 are differential functions on M 2n+1(f1, f2, f3) and R1(X, Y)Z = g(Y, Z)X − g(X, Z)Y, R2(X, Y)Z = g(X, φZ)φY − g(Y, φZ)φX + 2g(X, φY)φZ, and R3(X, Y)Z = η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ, where f1 = c+3 4 , f2 = f3 = c−1 4 . Here c denotes the constant φ-sectional curvature. The properties of generalized Sasakian space form was studied by many geometers such has [2, 9, 10, 14, 17, 18, 19, 21, 26]. The concept of local symmetry of a Riemanian manifold has been studied by many authors in several ways to a different extent. The locally φ-symmetry of Sasakian manifold was introduce by Takahashi in [28]. De and et al generalize this to the notion of φ-symmetry and then introduced the notion of φ-recurrent Sasakian manifold in [11]. Further φ-recurrent condition was studied on Kenmotsu manifold [8], LP-Sasakian manifold [29] and (LCS)n-manifold [20]. In[16], Pokhariyal and Mishra have defined the W2-curvature tensor, given by W2(X, Y)Z = R(X, Y)Z + 1 2n {g(X, Z)QY − g(Y, Z)QX}, (1.2) CUBO 20, 1 (2018) W2-Curvature Tensor on Generalized Sasakian Space Forms 19 here R and Q are the Riemanian curvature tensor and Ricci operator of Riemanian manifold respectively. In a generalized Sasakian space forms, the W2-curvature tensor satisfies the condition η(W2(X, Y)Z) = 0. (1.3) Many Geometers studied the W2 curvature tensor studied on different manifolds such has general- ized Sasakian space forms [13], Lorentzian para Sasakian manifolds [30] and Kenmotsu manifolds [25] Motivated by these ideas, we made an attempt to study the properties of generalized Sasakian space form. The present paper is organized as follows: In section 2, we review some preliminary results. In section 3, we study W2-pseudosymmetric generalized Sasakian space form. Section 4, deals with the W2-locally symmetric generalized Sasakian space forms and it is shown that a generalized Sasakian space form of dimension greater than three is W2-locally symmetric if and only if it is conformally flat. Section 5, is devoted to the study of W2-locally φ-symmetric generalized Sasakian space forms. Finally in last section, we discus the W2-φ-recurrent generalized Sasakian space form and found to be Einstein manifold. 2 Generalized Sasakian space-forms The Riemannian manifold M2n+1 is called an almost contact metric manifold if the following result holds [5, 6]: φ2X = −X + η(X)ξ, (2.1) η(ξ) = 1, φξ = 0, η(φX) = 0, g(X, ξ) = η(X), (2.2) g(φX, φY) = g(X, Y) − η(X)η(Y), (2.3) g(φX, Y) = −g(X, φY), g(φX, X) = 0 (2.4) (∇Xη)(Y) = g(∇Xξ, Y), ∀ X, Y ∈ (TpM). (2.5) A almost contact metric manifold is said to be Sasakian if and only if [5, 23] (∇Xφ)Y = g(X, Y)ξ − η(Y)X, (2.6) ∇Xξ = −φX. (2.7) 20 Venkatesha and Shanmukha B. CUBO 20, 1 (2018) Again we know that [1] in (2n + 1)-dimensional generalized Sasakian space form: S(X, Y) = (2nf1 + 3f2 − f3)g(X, Y) − (3f2 + (2n − 1)f3)η(X)η(Y), (2.8) S(φX, φY) = S(X, Y) + 2n(f1 − f3)η(X)η(Y), (2.9) QX = (2nf1 + 3f2 − f3)X − (3f2 + (2n − 1)f3)η(X)ξ, (2.10) r = 2n(2n + 1)f1 + 6nf2 − 4nf3, (2.11) R(X, Y)ξ = (f1 − f3){η(Y)X − η(X)Y}, (2.12) R(ξ, X)Y = (f1 − f3){g(X, Y)ξ − η(Y)X}, (2.13) η(R(X, Y)Z) = (f1 − f3){g(Y, Z)η(X) − g(X, Z)η(Y)}, (2.14) S(X, ξ) = 2n(f1 − f3)η(X). (2.15) Here R, S, Q and r are the Riemannian curvature tensor, Ricci tensor, Ricci operator and scalar curvature tensor of generalized Sasakian space forms in that order. 3 W2-pseudosymmetric generalized Sasakian space forms The concept of a pseudosymmetric manifold was introduced by Chaki [7] and Deszcz [12]. In this article we shall study properties of pseudosymmetric manifold according to Deszcz. Semisymmetric manifolds satisfies the condition R·R = 0 and were categorized by Szabo in [27]. Every pseudosym- metric manifold is semisymmetric but semisymmetric manifold need not be pseudosymmetric. An (2n + 1)-dimensional Riemannian manifold M2n+1 is said to be pseudosymmetric, if (R(X, Y) · R)(U, V)W = LR{((X ∧ Y) · R)(U, V)W)}. (3.1) where LR is some smooth function on UR = {x ∈ M 2n+1|R − r n(n−1) G 6= 0 at x}, where G is the (0, 4)-tensor defined by G(X1, X2, X3, X4) = g((X1 ∧X2)X3, X4) and (X∧Y)Z is the endomorphism and it is defined as, (X ∧ Y)Z = g(Y, Z)X − g(X, Z)Y (3.2) An (2n+1)-dimensional generalized Sasakian space form M2n+1 is said to be W2-pseudosymmetric, if (R(X, Y) · W2)(U, V)Z = LW2{(X ∧ Y) · W2)(U, V)Z}, (3.3) holds on the set UW2 = {x ∈ M 2n+1|W2 6= 0 at x}, where LW2 is some function on UW2. Suppose that generalized Sasakian space form is W2-pseudosymmetric. Now the left- hand side of (3.3) is R(ξ, Y)W2(U, V)Z − W2(R(ξ, Y)U, V)Z − W2(U, R(ξ, Y)V)Z − W2(U, V)R(ξ, Y)Z = 0. (3.4) CUBO 20, 1 (2018) W2-Curvature Tensor on Generalized Sasakian Space Forms 21 In the view of (2.12) the above expression becomes (f1 − f3){g(Y, W2(U, V)Z)ξ − η(W2(U, V)Z)Y − g(Y, U)W2(ξ, V)Z + η(U)W2(Y, V)Z − g(Y, V)W2(U, ξ)Z + η(V)W2(U, Y)Z − g(Y, Z)W2(U, V)ξ + η(Z)W2(U, V)Y} = 0. (3.5) Next the right hand side of (3.3) is LW2{(ξ ∧ Y)W2(U, V)Z − W2((ξ ∧ Y)U, V)Z − W2(U, (ξ ∧ Y)V)Z − W2(U, V)(ξ ∧ Y)Z} = 0. (3.6) By virtue of (3.2), (3.6) becomes LW2{g(Y, W2(U, V)Z)ξ − η(W2(U, V)Z)Y − g(Y, U)W2(ξ, V)Z + η(U)W2(Y, V)Z − g(Y, V)W2(U, ξ)Z + η(V)W2(U, Y)Z − g(Y, Z)W2(U, V)ξ + η(Z)W2(U, V)Y} = 0. (3.7) Using the expressions (3.5) and (3.7) in (3.3) and taking inner product with ξ, we obtain {LW2 − (f1 − f3)}{W2(U, V, Z, Y) − η(W2(U, V)Z)η(Y) − g(Y, U)η(W2(ξ, V)Z) + η(U)η(W2(Y, V)Z) − g(Y, V)η(W2(U, ξ)Z) + η(V)η(W2(U, V)Z) − g(Y, Z)η(W2(U, V)ξ) + η(Z)η(W2(U, V)Z)} = 0, (3.8) where W2(U, V, Z, Y) = g(Y, W2(U, V)Z) and using(1.3) we get either LW2 = (f1 − f3) or W2(U, V, Z, Y) = 0. (3.9) Thus we have following: Theorem 3.1. If M2n+1(f1, f2, f3) is W2-pseudosymmetric generalized Sasakian space form, then M2n+1(f1, f2, f3) is either W2-flat, or LW2 = (f1 − f3) if (f1 6= f3). Also in a generalized Sasakian space form, Singh and Pandey [24] proved the following, Theorem 3.2. A (2n+1)-dimensional (n > 1) generalized Sasakian space form satisfying W2 = 0 is an η-Einstein manifolds. In view of theorem (3.1) and theorem (3.2) we can state the following corollary. Corolary 1. If M2n+1(f1, f2, f3) is a W2-pseudosymmetric generalized Sasakian space forms then M2n+1 is either η-Einstein manifold or LW2 = (f1 − f3) if (f1 6= f3). 22 Venkatesha and Shanmukha B. CUBO 20, 1 (2018) 4 W2-locally symmetric generalized Sasakian space forms Definition 1. A (2n+1) dimensional (n > 1) generalized Sasakian space form is called projectively locally symmetric if it satisfies [18]. (∇WP)(X, Y)Z = 0. for all vector fields X, Y, Z orthogonal to ξ and an arbitrary vector field W. Analogous to this definition, we define a (2n + 1) dimensional (n > 1) W2-locally symmetric generalized Sasakian space form if (∇WW2)(X, Y)Z = 0, for all vector fields X, Y, Z orthogonal to ξ and an arbitrary vector field W. From (1.1) and (1.2), we have W2(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} + 1 2n {g(X, Z)QY − g(Y, Z)QX}. (4.1) Taking covariant differentiation of (4.1) with respect to an arbitrary vector field W, we get (∇WW2)(X, Y)Z = df1(W){g(Y, Z)X − g(X, Z)Y} + df2(W){g(X, φZ)φY − g(Y, φZ)φX + 2g(X, φY)φZ} + f2{g(X, φZ)(∇Wφ)Y + g(X, (∇Wφ)Z)φY − g(Y, φZ)(∇Wφ)X − g(Y, (∇Wφ)Z)φX + 2g(X, φY)(∇Wφ)Z + 2g(X, (∇Wφ)Y)φZ} + df3(W){η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ} + f3{(∇Wη)(X)η(Z)Y + η(X)(∇Wη)(Z)Y − (∇Wη)(Y)η(Z)X − η(Y)(∇Wη)η(Z)X + g(X, Z)(∇Wη)(Y)ξ + g(X, Z)η(Y)∇Wξ − g(Y, Z)(∇Wη)(X)ξ − g(Y, Z)η(X)∇Wξ} + 1 2n {g(X, Z)(∇WQ)(Y) − g(Y, Z)(∇WQ)(X)}. (4.2) where ∇ denotes the Riemannian connection on the manifold. Differentiating (2.10) covariantly with respect to a W, one can get (∇WQ)(Y) = d(2nf1 + 3f2 − f3)(W)Y − d(3f2 + (2n − 1)f3)(W)η(Y)ξ − (3f2 + (2n − 1)f3)[(∇Wη)(Y)ξ + η(Y)(∇Wξ)]. (4.3) CUBO 20, 1 (2018) W2-Curvature Tensor on Generalized Sasakian Space Forms 23 In view of (4.3) and (4.2), it follows that (∇WW2)(X, Y)Z = df1(W){g(Y, Z)X − g(X, Z)Y} + df2(W){g(X, φZ)φY − g(Y, φZ)φX + 2g(X, φY)φZ} + f2{g(X, φZ)(∇Wφ)Y + g(X, (∇Wφ)Z)φY − g(Y, φZ)(∇W φ)X − g(Y, (∇Wφ)Z)φX + 2g(X, φY)(∇Wφ)Z + 2g(X, (∇Wφ)Y)φZ} + df3(W){η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ} + f3{(∇Wη)(X)η(Z)Y + η(X)(∇Wη)(Z)Y − (∇Wη)(Y)η(Z)X − η(Y)(∇Wη)η(Z)X + g(X, Z)(∇Wη)(Y)ξ + g(X, Z)η(Y)∇Wξ − g(Y, Z)(∇Wη)(X)ξ − g(Y, Z)η(X)∇Wξ} + 1 2n [g(X, Z){d(2nf1 + 3f2 − f3)(W)Y − d(3f2 + (2n − 1)f3)(W)η(Y)ξ − (3f2 + (2n − 1)f3)[(∇Wη)(Y)ξ + η(Y)(∇Wξ)]} − g(Y, Z){d(2nf1 + 3f2 − f3)(W)X − d(3f2 + (2n − 1)f3)(W)η(X)ξ − (3f2 + (2n − 1)f3)[(∇Wη)(X)ξ + η(X)(∇Wξ)]}]. (4.4) Taking X, Y, Z orthogonal to ξ in (4.4) and then taking the inner product of the resultant equation with V, followed by setting V = Z = ei in the above equation, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, i = 1, 2, ......, 2n + 1, we get f2{−g(φX, (∇Wφ)Y) + n∑ i=1 g(X, (∇Wφ)ei)g(φY, ei) + g(φY, (∇Wφ)X) − n∑ i=1 g(Y, (∇Wφ)ei)g(φX, ei) + 2 n∑ i=1 g(X, φY)g((∇Wφ)ei, ei)} = 0. (4.5) For Levi Civita connection ∇, (∇Wg)(X, Y) = 0, which gives (∇Wg)(X, Y) − g(∇WX, Y) − g(X, ∇WY) = 0. Putting X = ei and Y = φei in the above equation, we obtain − g(∇Wei, φei) − g(ei, (∇Wφ)ei) = 0, 24 Venkatesha and Shanmukha B. CUBO 20, 1 (2018) which can be written as g(ei, φ(∇Wei)) − g(ei, (∇Wφ)ei) = 0. Thus we have g(ei, (∇Wφ)ei) = 0. (4.6) By the virtue of (4.5) and (4.6) takes the form f2{−g(φX, (∇Wφ)Y) + ∑ i=1 g(X, (∇Wφ)ei)g(φY, ei) + g(φY, (∇Wφ)X) − ∑ i=1 g(Y, (∇Wφ)ei)g(φX, ei)} = 0. (4.7) The above equation yields f2 = 0. It is known that a generalized Sasakian space form of dimen- sion greater than three is conformally flat if and only if f2 = 0 [14]. Hence the manifold under consideration is conformally flat. Conversely, suppose that the manifold is conformally flat. Then f2 = 0. In addition, if we consider X, Y, Z orthogonal to ξ then (1.1) yields R(X, Y)Z = f1{g(Y, Z)X − g(X, Z)Y}. The above equation gives, r = 2n(2n + 1)f1. (4.8) In view of (2.11) and (4.8), we obtain f3 = 0. Hence from (4.4), we get (∇WW2)(X, Y)Z = 0. Therefore, the manifold is W2-locally symmetric. Thus we have the following assertion. Theorem 4.1. A (2n + 1) dimensional (n > 1) generalized Sasakian space form is W2-locally symmetric if and only if it is conformally flat. or Theorem 4.2. A (2n + 1) dimensional (n > 1) generalized Sasakian space form is W2-locally symmetric if and only if f1 is constant. 5 W2-Locally φ-symmetric generalized Sasakian space forms Definition 2. A generalized Sasakian space form M2n+1(f1, f2, f3) of dimension greater than three is called W2-locally φ-symmetric if it satisfies φ2((∇WW2)(X, Y)Z) = 0, (5.1) for all vector fields X, Y, Z orthogonal to ξ on M2n+1. Let us consider a W2-locally φ-symmetric generalized Sasakian space form of dimension greater than three. Then from the definition and (2.1), we have CUBO 20, 1 (2018) W2-Curvature Tensor on Generalized Sasakian Space Forms 25 − ((∇WW2)(X, Y)Z) + η(∇WW2)(X, Y)Z)ξ = 0, (5.2) Taking the inner product g in both sides of the above equation with respect to W, we get − g((∇WW2)(X, Y)Z, W) + η(∇WW2)(X, Y)Z)η(W) = 0, (5.3) If we take orthogonal to W, then the above equation yields, g((∇WW2)(X, Y)Z, W) = 0, (5.4) The above equation is true for all W orthogonal to ξ. If we choose W 6= 0 and not orthogonal to (∇WW2)(X, Y)Z, then it follows that (∇WW2)(X, Y)Z = 0 (5.5) Hence, the manifold is W2-locally symmetric and hence by theorem 4.3, it is conformally flat. Conversely, let the manifold is conformally flat and hence f2 6= 0. Again, for X, Y, Z orthogonal to ξ, we have applying φ2 on both side to equation (4.4), one can get φ2(∇WW2)(X, Y)Z = −df2(W){g(X, φZ)φX − g(Y, φZ) + 2g(X, φY)φZ} − 1 2n {d(3f2 − f3)(W)[g(X, Z)Y − g(Y, Z)X]}. (5.6) if f2 = f3 = 0, the above equation yields φ2(∇WW2)(X, Y)Z = 0 for all X, Y, Z are orthogonal to ξ, therefore the manifold is W2-locally φ-symmetric. Now we are in a position to state the following statement, Theorem 5.1. A (2n + 1)-dimensional (n > 1) generalized Sasakian space form M2n+1 is W2- locally φ-symmetric if and only if it is conformally flat. 6 W2-φ-recurrent generalized Sasakian Space form Definition 3. A generalized Sasakian space form is said to be φ-recurrent if there exists a non-zero 1-form A such that,(see[11]) φ2((∇WR)(X, Y)Z) = A(W)R(X, Y)Z, for arbitrary vector fields X, Y, Z, W. If the 1-form A vanishes, then the manifold reduces to a φ-symmetric manifold. 26 Venkatesha and Shanmukha B. CUBO 20, 1 (2018) According to the definition of φ-recurrent generalized Sasakian space form, we define W2-φ- recurrent generalized sasakian space form by φ2((∇WW2)(X, Y)Z) = A(W)W2(X, Y)Z. (6.1) Then by (2.1) and (6.1), we have − (∇WW2)(X, Y)Z + η((∇WW2)(X, Y)Z)ξ = A(W)W2(X, Y)Z, (6.2) for arbitrary vector fields X, Y, Z, W. From the above equation it follows that − g((∇WW2)(X, Y)Z, U) + η((∇WW2)(X, Y)Z)η(U) = A(W)g(W2(X, Y)Z, U). (6.3) Let {ei}, i = 1, 2, ......2n + 1, be an orthogonal basis of the tangent space at any point of the manifold. Then putting X = U = ei in (6.3) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get − (∇WS)(Y, Z) − 1 2n [(∇WS(Y, Z)) − g(Y, Z)dr(W)] + 2n+1∑ i=1 η((∇WW2)(ei, Y)Z)η(ei) = A(W){(∇WS)(Y, Z) − 1 2n [(∇WS)(Y, Z) − g(Y, Z)dr(W)]}. (6.4) Setting Z = ξ in (6.4) then using (2.5), (2.13) and (2.7) and then replace Y by φY in (6.4), we get S(Y, W) = 2n(f1 − f3)g(Y, W). (6.5) Hence we can state following theorem: Theorem 6.1. Let generalized Sasakian space forms M2n+1is W2-φ-recurrent, then it is an Ein- stein manifold, provided (f1 − f3) 6= 0. 7 Example In [1], generalized complex space-form of dimension two is N(a, b) and the warped product M = R×N endowed with the almost contact metric structure is a three dimensional generalized Sasakian- space-form whose smooth functions f1 = a−(f ′ ) 2 f2 , f2 = b f2 and f3 = a−(f ′ ) 2 f2 + f ′′ f . Here f = f(t), t ∈ R and f ′ indicates the derivative of f with respect to t. Suppose we set a = 2, b = 0 and f(t) = t with t 6= 0, then f1 = 1 t2 , f2 = 0 and f3 = 1 t2 , we have from (1.2) W2(X, Y)Z = 1 t2 {g(Y, Z)X − g(X, Z)Y + η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ} + 1 2t2 {g(X, Z)Y − g(Y, Z)X − g(X, Z)η(Y)ξ + g(Y, Z)η(X)ξ}. 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Sumangala, on M-projective curvature tensor of generalised Sasakian space form. Acta Math. Univ. Comenianae, 2 (2013), 209–217. Introduction Generalized Sasakian space-forms W2-pseudosymmetric generalized Sasakian space forms W2-locally symmetric generalized Sasakian space forms W2-Locally -symmetric generalized Sasakian space forms W2–recurrent generalized Sasakian Space form Example