On Generalized 𝜙-Recurrent (𝜖,𝛿)-Trans-Sasakian Manifolds

Abstract

We study generalized -recurrent -trans-Sasakian manifolds. A relation between the associated 1-forms and and relation between characteristic vector field ξ and the vector fields for a generalized -recurrent.

1. Introduction

As a generalization of both Sasakian and Kenmotsu manifolds, Oubiña [1] introduced the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kahler manifolds. Trans-Sasakian manifolds of types ,, and are, respectively, called the cosymplectic, -Sasakian, and -Kenmotsu manifolds, with , being scalar functions. In particular, if,;,; then a trans-Sasakian manifold will be Kenmotsu and Sasakian manifolds respectively. The local structure of trans-Sasakian manifolds of dimension has been completely characterized by Marrero [2]. He proved that a trans-Sasakian manifold of dimension is either cosymplectic or -Kenmotsu or -Sasakian manifold.

The investigation of manifolds with indefinite metrics was taken by several authors. In 1993, Bejancu and Duggal [3] introduced the concept of -Sasakian manifolds and Xufeng and Xiaoli [4] established that these manifolds are real hypersurfaces of indefinite Kahlerian manifolds. Kumar et al. [5] studied the curvature conditions of these manifolds. Tripathi et al. [6] introduced and studied -almost paracontact manifolds.

The paper is organised as follows: Section 2 contains preliminaries of -trans-Sasakian manifolds. In Sections 3 and 4 we obtain results for generalized -recurrent and generalized concircular -recurrent -Trans-Sasakian Manifolds.

2. Preliminaries

Let be the connected almost contact metric manifold with an almost contact metric structure ; that is, is a tensor field, is a vector field, is a -form, and is compatible Riemannian metric, such that Almost contact metric manifold is called -almost contact metric manifold if for all vector fields,on, where .

An -almost contact metric manifold is called a -trans-Sasakian manifold if holds for some smooth functions and on and , . For , , -trans-Sasakian manifold reduces to -Sasakian and for , it reduces to -Kenmotsu manifold. -Trans-Sasakian manifolds have been studied by authors [7] and they obtained the following results:

3. Generalized -Recurrent -Trans-Sasakian Manifolds

Definition 1. -trans-Sasakian  manifold is called generalized -recurrent if its curvature tensor satisfies the condition where and are two -forms, is nonzero, and these are defined by and , are vector fields associated with -forms and , respectively.

Let us consider a generalized -recurrent -trans-Sasakian manifold. Then by virtue of (1) and (13) we have from which it follows that Let , , be an orthonormal basis of the tangent space at any point of the manifold. Then putting in (16) and taking summation over, , we get Put in (17) and by virtue of (8) and the equation , we have Now we have Using (5) and (6) in the above relation it follows that From (18) and (20) we obtain Replacing in (21) and then using (2) we get

Theorem 2. In a generalized -recurrent -trans-Sasakian manifold, the -forms and are related as in (22).

If , then, from (22), we obtain

Corollary 3. In a generalized -recurrent --Sasakian manifold, the -forms and are related as in (23).(i)If , then --Sasakian manifold reduces as -Sasakian manifold.(ii)If and , then --Sasakian manifold reduces as Sasakian manifold.

If , then, from (22), we obtain

Corollary 4. In a generalized -recurrent --Kenmotsu manifold, the -forms and are related as in (24).

If , then, from (22), we obtain From definition (13) this leads to the following corollary.

Corollary 5. In cosymplectic manifold, generalized -recurrent reduces to -recurrent manifold.

Now from (15) we have From (26) and second Bianchi identity we get By virtue of (12) we obtain from (27) that Putting in (28) and taking summation over , we get If , then, from (29), we obtain for all vector fields ,. Replacing by in (30), we get for all vector field .

From (30) and (31), we can state the following.

Theorem 6. A generalized -recurrent --Sasakian manifold, the characteristic vector field , and the vector fields , associated with the -forms ,, respectively, are in the same direction and the -forms , are given by (31).

4. Generalized Concircular -Recurrent -Trans-Sasakian Manifolds

In this section we study three-dimensional generalized concircular -recurrent -trans-Sasakian manifold.

Definition 7. -Trans-Sasakian manifold is called generalized concircular -recurrent if its concircular curvature tensor

satisfies the condition where and are defined as in (14) and is the scalar curvature of the manifold .

Let us consider a generalized concircular -recurrent -trans-Sasakian manifold. Then by virtue of (1) we have from which it follows that Let ,, be a local orthonormal basis of the tangent space at any point of the manifold. Then putting in (35) and taking summation over, , we get Putting in (36) and using (11) we have This leads to the following.

Theorem 8. In a generalized concircular -recurrent -trans-Sasakian manifold , the -forms and are related as in (37).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.