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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\}.
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
\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
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
\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} = {{c + 3} \over 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]
({\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,{\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]
({\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,{\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
{\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
{\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 - {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]:
{\phi ^2}X = - X + \eta (X)\xi,\eta (\xi ) = 1,{\kern 5pt}\phi \xi = 0,{\kern 5pt}\eta (\phi X) = 0,{\kern 5pt}g(X,\xi ) = \eta (X),g(\phi X,\phi Y) = g(X,Y) - \eta (X)\eta (Y),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({\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])
({\nabla _X}\phi )Y = ({f_1} - {f_3})(g(X,Y)\xi - \eta (Y)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:
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),R(X,Y)\xi = ({f_1} - {f_3})\{ \eta (Y)X - \eta (X)Y\},R(\xi,X)Y = ({f_1} - {f_3})\{ g(X,Y)\xi - \eta (Y)X\},\eta (R(X,Y)Z) = ({f_1} - {f_3})\{ g(Y,Z)\eta (X) - g(X,Z)\eta (Y)\},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
({\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,{\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
({\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
\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
\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
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
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
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
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
{\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,{\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
\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
\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 ei, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = ei in (28) and taking summation over i, 1 ≤ i ≤ (2n + 1), we get
\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)(ei, Y)Z, ξ) = 0. So, by replacing Z by ξ in (29) and with the help of (7) and (12), we get
\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
- 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) = 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
{\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
\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
\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 ei, i = 1, 2,...n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X = U = ei 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
\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,
({\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
\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
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) = 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].
{\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
({\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
{\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
\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
{\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.