Some Results on Generalized Sasakian Space Forms

In the present frame work, we studied the semi generalized recurrent, semi generalized φ -recurrent, extended generalized φ -recurrent and concircularly locally φ -symmetric on generalized Sasakian space forms.


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 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) + f 2 {g(X, φ Z)φY − g(Y, φ Z)φ X + 2g(X, φY )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 }, The Riemanian curvature tensor of a generalized Sasakian space form M 2n+1 ( f 1 , f 2 , f 3 ) is simply given by (2) where 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].
is called a semi-generalized recurrent manifold if its curvature tensor R satisfies [6,9] ( where A and B are two 1-forms, B is non-zero, ρ 1 and ρ 2 are two vector fields such that g(X, ρ 1 ) = A(X), g(X, ρ 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 4.
A generalized Sasakian space form is said to be locally φ -symmetric if 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] where R and r are the Riemannian curvature tensor and scalar curvature tensor, respectively.

Generalized Sasakian space-forms
A (2n + 1)-dimensional Riemannian manifold is called an almost contact metric manifold if the following result holds [6], [7]: for all vector field X and Y . On a generalized Sasakian space form Again, we know that from Ref. [1], (2n + 1)-dimensional generalized Sasakian space forms holds the following relations: 3 Semi generalized recurrent generalized Sasakian space forms here A and B are two 1-forms, B is non-zero, ρ 1 and ρ 2 are two vector fields such that A(X) = g(X, ρ 1 ) and B(X) = g(X, ρ 2 ) Permutating equation (3) twice with respect to X,Y, Z, adding the three equations and using Bianchi second identity, we have Contracting (20) with respect to Y , we get Setting S(Y, Z) = g(QY, Z) in (21) and factoring off W, we get Again contracting with respect to Z and then substitute X = ξ in (22), one can get 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(ρ 1 ) and η(ρ 2 ) is given in (23).
Next, we prove the semi generalized Ricci-recurrent generalized Sasakian space form, inserting Z = ξ in (19), we have Again setting Y = ξ in (24), we get Now, we can state the following theorem where A and B are two 1-forms, B is non-zero and these are defined by and ρ 1 and ρ 2 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 it follows that 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 The second term of left hand side of (29) by putting Z = ξ takes the form ((∇ W R)(e i ,Y )Z, ξ ) = 0. So, by replacing Z by ξ in (29) and with the help of (7) and (12), we get Inserting Y = ξ in (30) and using (7), we have In view of (31) and replace Y by φY , (30) yields

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 (M 2n+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 for all vector fields X,Y, Z,W , where A and B are two non-vanishing 1-forms such that A(X) = g(X, ρ 1 ), B(X) = g(X, ρ 2 ). Here ρ 1 and ρ 2 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 From which it follows that 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(( It follows that, where Inserting Z = ξ (35) and using (12), (17) and (7), we get Again inserting Y = ξ and using (7), (37) yields By taking the account of (38) in (37) and then replace Y by φY , we get 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 ( 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. 6 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].
for all vector fields X,Y, Z are orthogonal to ξ and an arbitrary vector field W .
Differentiate covariantly with respect W , we have Operate φ 2 on both side, we have In view of (6), and taking the help of relation (1) with X,Y, Z are orthogonal vector field, one can get If the manifold is conformally flat then f 2 = 0. Therefore, (41) yields Hence we can state the following theorem Theorem 6. A generalized Sasakian space forms is concircularly locally φ -symmetric if and only if f 1 and 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 M 2n+1 ( f 1 , f 2 , f 3 ) of dimension greater than three is conformally flat and ξ is Killing, then it is locally symmetric. Moreover, if M 2n+1 ( f 1 , f 2 , f 3 ) is locally symmetric, then f 1 − f 3 is 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 f 1 and 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.