Some Results on Generalized Sasakian Space Forms

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 $R (X, Y) Z = c {g (Y, Z) X - g (X, Z) Y} .$ R(X,Y)Z = c\{ g(Y,Z)X - g(X,Z)Y\}.

A Sasakian manifold with constant ϕ-sectional curvature is a Sasakian space form and it has a specific form of its curvature tensor. Similar 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. These space forms are defined as follows:

A generalized Sasakian space form is an almost contact metric manifold (M, ϕ, ξ, η, g), whose curvature tensor is given by (1) $\begin{array}{l} R (X, Y) Z & = f_{1} {g (Y, Z) X - g (X, Z) Y} \\ + f_{2} {g (X, ϕ Z) ϕ Y - g (Y, ϕ Z) ϕ X + 2 g (X, ϕ Y) ϕ Z} \\ + f_{3} {η (X) η (Z) Y - η (Y) η (Z) X + g (X, Z) η (Y) ξ - g (Y, Z) η (X) ξ}, \end{array}$ \matrix{ {R(X,Y)Z} \hfill & { = {f_1}\{ g(Y,Z)X - g(X,Z)Y\} } \hfill \cr {} \hfill & { + {f_2}\{ g(X,\phi Z)\phi Y - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\} } \hfill \cr {} \hfill & { + {f_3}\{ \eta (X)\eta (Z)Y - \eta (Y)\eta (Z)X + g(X,Z)\eta (Y)\xi - g(Y,Z)\eta (X)\xi \},} \hfill}

The Riemanian curvature tensor of a generalized Sasakian space form M²ⁿ⁺¹ (f₁, f₂, f₃) is simply given by (2) $R = f_{1} R_{1} + f_{2} R_{2} + f_{3} R_{3} .$ R = {f_1}{R_1} + {f_2}{R_2} + {f_3}{R_3}. where f₁, f₂, f₃ are differential functions on M²ⁿ⁺¹ (f₁, f₂, f₃) and $\begin{array}{l} R_{1} (X, Y) Z = g (Y, Z) X - g (X, Z) Y, \\ R_{2} (X, Y) Z = g (X, ϕ Z) ϕ Y - g (Y, ϕ Z) ϕ X + 2 g (X, ϕ Y) ϕ Z, \\ R_{3} (X, Y) Z = η (X) η (Z) Y - η (Y) η (Z) X + g (X, Z) η (Y) ξ - g (Y, Z) η (X) ξ, \end{array}$ \matrix{ {{R_1}(X,Y)Z = g(Y,Z)X - g(X,Z)Y,} \hfill \cr {{R_2}(X,Y)Z = g(X,\phi Z)\phi Y - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z,} \hfill \cr {{R_3}(X,Y)Z = \eta (X)\eta (Z)Y - \eta (Y)\eta (Z)X + g(X,Z)\eta (Y)\xi - g(Y,Z)\eta (X)\xi,} \hfill} where $f_{1} = \frac{c + 3}{4}$ {f_1} = {{c + 3} \over 4} , $f_{2} = f_{3} = \frac{c - 1}{4}$ {f_2} = {f_3} = {{c - 1} \over 4} . Where c denotes the constant ϕ-sectional curvature. The properties of generalized Sasakian space form was studied by many geometers such as those mentioned in Refs. [2, 11, 12, 18, 21]. 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 introduced by Takahashi in Ref. [26]. De et.al., generalize the notion of ϕ-symmetry and then introduced the notion of ϕ-recurrent Sasakian manifold in Ref. [13]. Further ϕ-recurrent condition was studied on Kenmotsu manifold [10], LP-Sasakian manifold [27] and (LCS)_n-manifold [22].

Definition 1

A Riemannian manifold (M²ⁿ⁺¹, g) is called a semi-generalized recurrent manifold if its curvature tensor R satisfies [6, 9] (3) $(\nabla_{X} R) (Y, Z) W = A (X) R (Y, Z) W + B (X) g (Z, W) Y,$ ({\nabla _X}R)(Y,Z)W = A(X)R(Y,Z)W + B(X)g(Z,W)Y, where A and B are two 1-forms, B is non-zero, ρ₁ and ρ₂ are two vector fields such that $g (X, ρ_{1}) = A (X), g (X, ρ_{2}) = B (X),$ g(X,{\rho _1}) = A(X),g(X,{\rho _2}) = B(X), for any vector field X, Y, Z, W and ∇ denotes the operator of covariant differentiation with respect to the metric g.

Definition 2

A Riemannian manifold (M²ⁿ⁺¹, g) is semi generalized Ricci-recurrent if [6, 9] (4) $(\nabla_{X} S) (Y, Z) = A (X) S (Y, Z) + (2 n + 1) B (X) g (Y, Z),$ ({\nabla _X}S)(Y,Z) = A(X)S(Y,Z) + (2n + 1)B(X)g(Y,Z), where A and B are two 1-forms, B is non-zero, ρ₁ and ρ₂ are two vector fields such that $g (X, ρ_{1}) = A (X), g (X, ρ_{2}) = B (X),$ g(X,{\rho _1}) = A(X),g(X,{\rho _2}) = B(X),

Definition 3

A Sasakian manifold (M²ⁿ⁺¹, ϕ, ξ, η, g), n ≥ 1, is said to be an extended generalized ϕ-recurrent Sasakian manifold if its curvature tenor R satisfies the relation (5) $ϕ^{2} (\nabla_{W} R) (X, Y) Z = A (W) ϕ^{2} (R (X, Y) Z) + B (W) ϕ^{2} (G (X, Y) Z)$ {\phi ^2}({\nabla _W}R)(X,Y)Z = A(W){\phi ^2}(R(X,Y)Z) + B(W){\phi ^2}(G(X,Y)Z) for all vector fields X, Y, Z, W, where A and B are two non-vanishing 1-forms such that A(X) = g(X, ρ₁), B(X) = g(X, ρ₂). Here ρ₁ and ρ₂ are vector fields associated with 1-forms A and B respectively.

Definition 4

A generalized Sasakian space form is said to be locally ϕ-symmetric if $ϕ^{2} (\nabla_{W} R) (X, Y) Z = 0$ {\phi ^2}({\nabla _W}R)(X,Y)Z = 0 for all vector fields X, Y, Z orthogonal to ξ. This notion was introduced by T. Takahashi for Sasakian manifolds [26].

In 1940, Yano introduce the concircular curvature tensor. A (2n + 1) dimensional concircular curvature tensor C is given by [30, 31] $C (X, Y) Z = R (X, Y) Z - \frac{r}{2 n (2 n + 1)} {g (Y, Z) X - g (X, Z) Y},$ C(X,Y)Z = R(X,Y)Z - {r \over {2n(2n + 1)}}\{ g(Y,Z)X - g(X,Z)Y\}, where R and r are the Riemannian curvature tensor and scalar curvature tensor, respectively.

Author in Ref. [5] studies the symmetric conditons of generalized Sasakian space forms with concircular curvature tensor such as C(ξ, X) · C = 0, C(ξ, X) · R = 0, C(ξ, X) · S = 0 and C(ξ, X) · P = 0. Recently, researcher in Ref. [28] investigate some symmetric condition on generalized Sasakian space forms with W₂-curvature tensor, such as pseudosymmetric, locally symmetric, locally ϕ-symmetric and ϕ-recurrent. Moreover many geometer’s studied the generalized Sasakian space forms with different conditions such as those mentioned in Refs. [11,12,13, 15, 16].

Generalized Sasakian space-forms

A (2n + 1)-dimensional Riemannian manifold is called an almost contact metric manifold if the following result holds [6], [7]: (6) $ϕ^{2} X = - X + η (X) ξ,$ {\phi ^2}X = - X + \eta (X)\xi,(7) $η (ξ) = 1, ϕ ξ = 0, η (ϕ X) = 0, g (X, ξ) = η (X),$ \eta (\xi ) = 1,{\kern 5pt}\phi \xi = 0,{\kern 5pt}\eta (\phi X) = 0,{\kern 5pt}g(X,\xi ) = \eta (X),(8) $g (ϕ X, ϕ Y) = g (X, Y) - η (X) η (Y),$ g(\phi X,\phi Y) = g(X,Y) - \eta (X)\eta (Y),(9) $g (ϕ X, Y) = - g (X, ϕ Y), g (ϕ X, X) = 0$ g(\phi X,Y) = - g(X,\phi Y),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g(\phi X,X) = 0(10) $(\nabla_{X} η) (Y) = g (\nabla_{X} ξ, Y)$ ({\nabla _X}\eta )(Y) = g({\nabla _X}\xi,Y) for all vector field X and Y. On a generalized Sasakian space form M²ⁿ⁺¹ (f₁, f₂, f₃), we have ([1, 15]) (11) $(\nabla_{X} ϕ) Y = (f_{1} - f_{3}) (g (X, Y) ξ - η (Y) X),$ ({\nabla _X}\phi )Y = ({f_1} - {f_3})(g(X,Y)\xi - \eta (Y)X),(12) $\nabla_{X} ξ = - (f_{1} - f_{3}) ϕ X .$ {\nabla _X}\xi = - ({f_1} - {f_3})\phi X.

Again, we know that from Ref. [1], (2n + 1)-dimensional generalized Sasakian space forms holds the following relations: (13) $S (X, Y) = (2 n f_{1} + 3 f_{2} - f_{3}) g (X, Y) - (3 f_{2} + (2 n - 1) f_{3}) η (X) η (Y),$ S(X,Y) = (2n{f_1} + 3{f_2} - {f_3})g(X,Y) - (3{f_2} + (2n - 1){f_3})\eta (X)\eta (Y),(14) $R (X, Y) ξ = (f_{1} - f_{3}) {η (Y) X - η (X) Y},$ R(X,Y)\xi = ({f_1} - {f_3})\{ \eta (Y)X - \eta (X)Y\},(15) $R (ξ, X) Y = (f_{1} - f_{3}) {g (X, Y) ξ - η (Y) X},$ R(\xi,X)Y = ({f_1} - {f_3})\{ g(X,Y)\xi - \eta (Y)X\},(16) $η (R (X, Y) Z) = (f_{1} - f_{3}) {g (Y, Z) η (X) - g (X, Z) η (Y)},$ \eta (R(X,Y)Z) = ({f_1} - {f_3})\{ g(Y,Z)\eta (X) - g(X,Z)\eta (Y)\},(17) $S (X, ξ) = 2 n (f_{1} - f_{3}) η (X) .$ S(X,\xi ) = 2n({f_1} - {f_3})\eta (X).

Semi generalized recurrent generalized Sasakian space forms

Definition 5

A generalized Sasakian space form (M²ⁿ⁺¹, g) is semi-generalized recurrent manifold if (18) $(\nabla_{X} R) (Y, Z) W = A (X) R (Y, Z) W + B (X) g (Z, W) Y,$ ({\nabla _X}R)(Y,Z)W = A(X)R(Y,Z)W + B(X)g(Z,W)Y, here A and B are two 1-forms, B is non-zero, ρ₁ and ρ₂ are two vector fields such that $A (X) = g (X, ρ_{1}) and B (X) = g (X, ρ_{2})$ A(X) = g(X,{\rho _1}){\kern 10pt} and {\kern 10pt} B(X) = g(X,{\rho _2})

Definition 6

A generalized Sasakian space forms (M²ⁿ⁺¹, g) is semi generalized Ricci-recurrent if (19) $(\nabla_{X} S) (Y, Z) = A (X) S (Y, Z) + (2 n + 1) B (X) g (Y, Z) .$ ({\nabla _X}S)(Y,Z) = A(X)S(Y,Z) + (2n + 1)B(X)g(Y,Z).

Permutating equation (3) twice with respect to X, Y, Z, adding the three equations and using Bianchi second identity, we have (20) $\begin{array}{l} A (X) R (Y, Z) W + B (X) g (Z, W) + A (Y) R (Z, X) W \\ + B (Y) g (X, W) Z + A (Z) R (X, Y) W + B (Z) g (Y, W) = 0 . \end{array}$ \matrix{ {A(X)R(Y,Z)W + B(X)g(Z,W) + A(Y)R(Z,X)W} \hfill \cr { + B(Y)g(X,W)Z + A(Z)R(X,Y)W + B(Z)g(Y,W) = 0.} \hfill}

Contracting (20) with respect to Y, we get (21) $\begin{array}{l} A (X) S (Z, W) + B (X) g (Z, W) - g (R (Z, X) ρ, W) \\ B (Z) g (X, W) - A (Z) S (X, W) + B (Z) g (X, W) = 0 . \end{array}$ \matrix{ {A(X)S(Z,W) + B(X)g(Z,W) - g(R(Z,X)\rho,W)} \hfill \cr {B(Z)g(X,W) - A(Z)S(X,W) + B(Z)g(X,W) = 0.} \hfill}

Setting S(Y, Z) = g(QY, Z) in (21) and factoring off W, we get (22) $A (X) QZ + (2 n + 1) B (X) Z - R (Z, X) ρ + 2 B (Z) X - A (Z) QX = 0 .$ A(X)QZ + (2n + 1)B(X)Z - R(Z,X)\rho + 2B(Z)X - A(Z)QX = 0.

Again contracting with respect to Z and then substitute X = ξ in (22), one can get (23) $r = - \frac{1}{η (ρ_{1})} {{(2 n + 1)}^{2} - 2 η (ρ_{2}) - 2 n (f_{1} - f_{3}) [η (ρ_{2}) + η (ρ_{1})]} .$ r = - {1 \over {\eta ({\rho _1})}}\{ {(2n + 1)^2} - 2\eta ({\rho _2}) - 2n({f_1} - {f_3})[\eta ({\rho _2}) + \eta ({\rho _1})]\}.

Now, we can state the following statement

Theorem 1

The scalar curvature r of a semi-generalized recurrent generalized Sasakian space forms is related in terms of contact forms η(ρ₁) and η(ρ₂) is given in (23).

Next, we prove the semi generalized Ricci-recurrent generalized Sasakian space form, inserting Z = ξ in (19), we have (24) $2 n {(f_{1} - f_{3})}^{2} g (W, ϕ Y) + (f_{1} - f_{3}) S (Y, ϕ W) = A (X) 2 n (f_{1} - f_{3}) η (Y) + (2 n + 1) B (X) η (Y) .$ 2n{({f_1} - {f_3})^2}g(W,\phi Y) + ({f_1} - {f_3})S(Y,\phi W) = A(X)2n({f_1} - {f_3})\eta (Y) + (2n + 1)B(X)\eta (Y).

Again setting Y = ξ in (24), we get (25) $A (X) 2 n (f_{1} - f_{3}) + (2 n + 1) B (X) = 0 .$ A(X)2n({f_1} - {f_3}) + (2n + 1)B(X) = 0.

Now, we can state the following theorem

Theorem 2

A semi-generalized Ricci-recurrent generalized Sasakian space forms, the 1-form A and B holds (25)

Semi generalized ϕ-recurrent generalized Sasakian space forms

Definition 7

A generalized Sasakian space form (M²ⁿ⁺¹, g) is called semi-generalized ϕ-recurrent if its curvature tensor R satisfies the condition (26) $ϕ^{2} (\nabla_{W} R) (X, Y) Z = A (W) R (X, Y) Z + B (W) g (Y, Z) X$ {\phi ^2}({\nabla _W}R)(X,Y)Z = A(W)R(X,Y)Z + B(W)g(Y,Z)X where A and B are two 1-forms, B is non-zero and these are defined by $A (W) = (W, ρ_{1}), B (W) = (W, ρ_{2})$ A(W) = (W,{\rho _1}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} B(W) = (W,{\rho _2}) and ρ₁ and ρ₂ are vector fields associated with 1-forms A and B respectively.

Let us consider a semi-generalized ϕ-recurrent generalized Sasakian space forms. Then by virtue of (6) and (26), we have (27) $\begin{array}{l} - (\nabla_{W} R) (X, Y) Z + η ((\nabla_{W} R) (X, Y) Z) ξ \\ = A (W) R (X, Y) Z + B (W) g (Y, Z) X . \end{array}$ \matrix{ { - ({\nabla _W}R)(X,Y)Z + \eta (({\nabla _W}R)(X,Y)Z)\xi } \hfill \cr { = A(W)R(X,Y)Z + B(W)g(Y,Z)X.} \hfill} it follows that (28) $\begin{array}{l} - g ((\nabla_{W} R) (X, Y) Z, U) + η ((\nabla_{W} R) (X, Y) Z) η (U) \\ = A (W) g (R (X, Y) Z, U) + B (W) g (Y, Z) g (X, U) . \end{array}$ \matrix{ { - g(({\nabla _W}R)(X,Y)Z,U) + \eta (({\nabla _W}R)(X,Y)Z)\eta (U)} \hfill \cr { = A(W)g(R(X,Y)Z,U) + B(W)g(Y,Z)g(X,U).} \hfill}

Let e_i, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = e_i in (28) and taking summation over i, 1 ≤ i ≤ (2n + 1), we get (29) $\begin{array}{l} - (\nabla_{W} S) (Y, Z) + \sum_{i = 1}^{2 n} η (\nabla_{W} R) (e_{i}, Y) Z) η (e_{i}) \\ = A (W) 2 n (f_{1} - f_{3}) S (Y, Z) + B (W) (2 n + 1) g (Y, Z) . \end{array}$ \matrix{ { - ({\nabla _W}S)(Y,Z) + \sum\limits_{i = 1}^{2n} \eta ({\nabla _W}R)({e_i},Y)Z)\eta ({e_i})} \hfill \cr { = A(W)2n({f_1} - {f_3})S(Y,Z) + B(W)(2n + 1)g(Y,Z).} \hfill}

The second term of left hand side of (29) by putting Z = ξ takes the form ((∇_WR)(e_i, Y)Z, ξ) = 0. So, by replacing Z by ξ in (29) and with the help of (7) and (12), we get (30) $\begin{array}{l} - 2 n {(f_{1} - f_{3})}^{2} g (W, ϕ Y) + (f_{1} - f_{3}) S (Y, ϕ W) \\ = A (W) 2 n (f_{1} - f_{3}) η (Y) + B (W) (2 n + 1) g (Y, Z) . \end{array}$ \matrix{ { - 2n{{({f_1} - {f_3})}^2}g(W,\phi Y) + ({f_1} - {f_3})S(Y,\phi W)} \hfill \cr { = A(W)2n({f_1} - {f_3})\eta (Y) + B(W)(2n + 1)g(Y,Z).} \hfill}

Inserting Y = ξ in (30) and using (7), we have (31) $- 2 n (f_{1} - f_{3}) A (W) = (2 n + 1) B (W) .$ - 2n({f_1} - {f_3})A(W) = (2n + 1)B(W).

In view of (31) and replace Y by ϕY, (30) yields $S (Y, W) = 2 n (f_{1} - f_{3}) g (Y, W) .$ S(Y,W) = 2n({f_1} - {f_3})g(Y,W).

Theorem 3

A semi generalized ϕ-recurrent generalized Sasakian space forms (M²ⁿ⁺¹, g) is an Einstein manifold and moreover; the 1-forms A and B are related as −2n(f₁ − f₃)A(W) = (2n + 1)B(W).

Extended generalized ϕ-recurrent generalized Sasakian space forms

According to the definition of extended generalized ϕ-recurrent Sasakian manifolds, we will define the Extended generalized ϕ-recurrent generalized Sasakian space forms

Definition 8

A generalized Sasakian space forms (M²ⁿ⁺¹, ϕ, ξ, η, g), n ≥ 1, is said to be an extended generalized ϕ-recurrent generalized Sasakian space forms if its curvature tenor R satisfies the relation (32) $ϕ^{2} (\nabla_{W} R) (X, Y) Z = A (W) ϕ^{2} (R (X, Y) Z) + B (W) ϕ^{2} (G (X, Y) Z)$ {\phi ^2}({\nabla _W}R)(X,Y)Z = A(W){\phi ^2}(R(X,Y)Z) + B(W){\phi ^2}(G(X,Y)Z) for all vector fields X, Y, Z, W, where A and B are two non-vanishing 1-forms such that A(X) = g(X, ρ₁), B(X) = g(X, ρ₂). Here ρ₁ and ρ₂ are vector fields associated with 1-forms A and B respectively.

Let us consider an extended generalized ϕ-recurrent generalized Sasakian space forms. Then by virtue of (6), we have (33) $\begin{array}{l} - (\nabla_{W} R) (X, Y) Z + η ((\nabla_{W} R) (X, Y) Z) ξ \\ = A (W) {- R (X, Y) Z + η (R (X, Y) Z)} \\ + B (W) {- G (X, Y) Z + η (G (X, Y) Z)} . \end{array}$ \matrix{ { - ({\nabla _W}R)(X,Y)Z + \eta (({\nabla _W}R)(X,Y)Z)\xi } \hfill \cr { = A(W)\{ - R(X,Y)Z + \eta (R(X,Y)Z)\} } \hfill \cr { + B(W)\{ - G(X,Y)Z + \eta (G(X,Y)Z)\}.} \hfill}

From which it follows that (34) $\begin{array}{l} - g ((\nabla_{W} R) (X, Y) Z, U) + η ((\nabla_{W} R) (X, Y) Z) η (U) \\ = A (W) {- g (R (X, Y) Z, U) + η (R (X, Y) Z) η (U)} \\ + B (W) {- g (G (X, Y) Z, U) + η (G (X, Y) Z) η (U)} . \end{array}$ \matrix{ { - g(({\nabla _W}R)(X,Y)Z,U) + \eta (({\nabla _W}R)(X,Y)Z)\eta (U)} \hfill \cr { = A(W)\{ - g(R(X,Y)Z,U) + \eta (R(X,Y)Z)\eta (U)\} } \hfill \cr { + B(W)\{ - g(G(X,Y)Z,U) + \eta (G(X,Y)Z)\eta (U)\}.} \hfill}

Let e_i, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = e_i in (34) and taking summation over i, 1 ≤ i ≤ (2n + 1), and the relation g((∇_WR)(X, Y)Z, U) = −g((∇_WR)(X, Y)U, Z), we get (35) $\begin{array}{l} - (\nabla_{W} S) (Y, Z) = A (W) {- S (Y, Z) + η (R (ξ, Y) Z)} \\ + B (W) {- (2 n - 1) g (Y, Z) - η (Y) η (Z)} . \end{array}$ \matrix{ { - ({\nabla _W}S)(Y,Z) = A(W)\{ - S(Y,Z) + \eta (R(\xi,Y)Z)\} } \hfill \cr { + B(W)\{ - (2n - 1)g(Y,Z) - \eta (Y)\eta (Z)\}.} \hfill}

It follows that, (36) $(\nabla_{W} S) (Y, Z) = A \otimes S (Y, Z) + Kg (Y, Z) + μ η (Y) (Z) .$ ({\nabla _W}S)(Y,Z) = A \otimes S(Y,Z) + Kg(Y,Z) + \mu \eta (Y)(Z). where K = [(2n − 1)B(W) − A(W)(f₁ − f₃)] and μ = [(f₁ − f₃)A(W) + B(W)].

Inserting Z = ξ(35) and using (12), (17) and (7), we get (37) $\begin{array}{l} 2 n {(f_{1} - f_{3})}^{2} g (W, ϕ Y) + (f_{1} - f_{3}) S (Y, ϕ W) \\ = {2 n (f_{1} - f_{3}) A (W) + 2 nB (W)} η (Y) . \end{array}$ \matrix{{\kern 15pt} {2n{{({f_1} - {f_3})}^2}g(W,\phi Y) + ({f_1} - {f_3})S(Y,\phi W)} \hfill \cr { = \{ 2n({f_1} - {f_3})A(W) + 2nB(W)\} \eta (Y).} \hfill}

Again inserting Y = ξ and using (7), (37) yields (38) $2 n (f_{1} - f_{3}) A (W) + 2 nB (W) = 0 .$ 2n({f_1} - {f_3})A(W) + 2nB(W) = 0.

By taking the account of (38) in (37) and then replace Y by ϕY, we get $S (Y, W) = 2 n (f_{1} - f_{3}) g (Y, W) .$ S(Y,W) = 2n({f_1} - {f_3})g(Y,W).

Thus we have the following assertion

Theorem 4

An extended generalized ϕ-recurrent generalized Sasakian space forms is an Einstein manifold and moreover the associated 1-forms A and B are related by (f₁ − f₃)A + B = 0.

It is known that a generalized Sasakian space form is Ricci-semisymmetric if and only if it is an Einstein manifold. In fact, by Theorem 4, we have the following:

Corollary 5

An extended generalized ϕ-recurrent generalized Sasakian space forms is Ricci-semisymmetric.

Concircularly locally ϕ-symmetric generalized Sasakian space forms

Definition 9

A (2n + 1) dimensional (n > 1) generalized Sasakian space form is called concircularly locally ϕ-symmetric if it satisfies [12]. $ϕ^{2} (\nabla_{W} C) (X, Y) Z = 0 .$ {\phi ^2}({\nabla _W}C)(X,Y)Z = 0. for all vector fields X, Y, Z are orthogonal to ξ and an arbitrary vector field W.

Differentiate covariantly with respect W, we have (39) $(\nabla_{W} C) (X, Y) Z = (\nabla_{W} R) (X, Y) Z - \frac{dr (W)}{2 n (2 n + 1)} {g (Y, Z) X - g (X, Z) Y} .$ ({\nabla _W}C)(X,Y)Z = ({\nabla _W}R)(X,Y)Z - {{dr(W)} \over {2n(2n + 1)}}\{ g(Y,Z)X - g(X,Z)Y\}.

Operate ϕ² on both side, we have (40) $ϕ^{2} ((\nabla_{W} C) (X, Y) Z) = ϕ^{2} ((\nabla_{W} R) (X, Y) Z) - \frac{dr (W)}{2 n (2 n + 1)} {g (Y, Z) ϕ^{2} X - g (X, Z) ϕ^{2} Y} .$ {\phi ^2}(({\nabla _W}C)(X,Y)Z) = {\phi ^2}(({\nabla _W}R)(X,Y)Z) - {{dr(W)} \over {2n(2n + 1)}}\{ g(Y,Z){\phi ^2}X - g(X,Z){\phi ^2}Y\}.

In view of (6), and taking the help of relation (1) with X, Y, Z are orthogonal vector field, one can get (41) $\begin{array}{l} ϕ^{2} ((\nabla_{W} C) (X, Y) Z) & = d f_{1} (W) {g (Y, Z) X - g (X, Z) Y} \\ + d f_{2} (W) {g (X, ϕ Z) ϕ Y - g (Y, ϕ Z) ϕ X + 2 g (X, ϕ Y) ϕ Z} \\ + f_{2} {g (X, ϕ Z) (\nabla_{W} ϕ) Y + g (X, (\nabla_{W} ϕ) Z) ϕ Y \\ - g (Y, ϕ Z) (\nabla_{W} ϕ) X - g (Y, (\nabla_{W} ϕ) Z) ϕ X \\ + 2 g (X, ϕ Y) (\nabla_{W} ϕ) Z + 2 g (X, (\nabla_{W} ϕ) Y) ϕ Z} \\ + \frac{dr (W)}{2 n (2 n + 1)} {g (Y, Z) X - g (X, Z) Y} . \end{array}$ \matrix{ {{\phi ^2}(({\nabla _W}C)(X,Y)Z)} \hfill & { = d{f_1}(W)\{ g(Y,Z)X - g(X,Z)Y\} } \hfill \cr {} \hfill & { + d{f_2}(W)\{ g(X,\phi Z)\phi Y - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\} } \hfill \cr {} \hfill & { + {f_2}\{ g(X,\phi Z)({\nabla _W}\phi )Y + g(X,({\nabla _W}\phi )Z)\phi Y} \hfill \cr {} \hfill & { - g(Y,\phi Z)({\nabla _W}\phi )X - g(Y,({\nabla _W}\phi )Z)\phi X} \hfill \cr {} \hfill & { + 2g(X,\phi Y)({\nabla _W}\phi )Z + 2g(X,({\nabla _W}\phi )Y)\phi Z\} } \hfill \cr {} \hfill & { + {{dr(W)} \over {2n(2n + 1)}}\{ g(Y,Z)X - g(X,Z)Y\}.} \hfill}

If the manifold is conformally flat then f₂ = 0. Therefore, (41) yields $ϕ^{2} ((\nabla_{W} C) (X, Y) Z) = {d f_{1} (W) + \frac{dr (W)}{2 n (2 n + 1)}} {g (Y, Z) X - g (X, Z) Y} .$ {\phi ^2}(({\nabla _W}C)(X,Y)Z) = \left\{ {d{f_1}(W) + {{dr(W)} \over {2n(2n + 1)}}} \right\}\{ g(Y,Z)X - g(X,Z)Y\}.

Hence we can state the following theorem

Theorem 6

A generalized Sasakian space forms is concircularly locally ϕ-symmetric if and only if f₁and the scalar curvature are constant

Note 7.In [18], U. K. Kim studied generalized Sasakian space forms and proved that if a generalized Sasakian space forms M²ⁿ⁺¹ (f₁, f₂, f₃) of dimension greater than three is conformally flat and ξ is Killing, then it is locally symmetric. Moreover, if M²ⁿ⁺¹ (f₁, f₂, f₃) is locally symmetric, then f₁ − f₃is constant. In the above theorem it is shown that a conformally flat generalized Sasakian space form of dimension greater than 3 is locally ϕ-symmetric if and only if f₁and scalar curvature is constant. Thus, we observe the difference between locally symmetric generalized Sasakian space forms and concircularly locally ϕ-symmetric generalized Sasakian space forms.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Some Results on Generalized Sasakian Space Forms

Published Online: Mar 30, 2020

Page range: 85 - 92

Received: May 28, 2019

Accepted: Jul 29, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00009

KeywordsGeneralized Sasakian space forms, extended generalized -recurrent, Einstein manifold

© 2020 Shanmukha B. Venkatesha, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Generalized Sasakian space forms, extended generalized -recurrent, Einstein manifold