1. Introduction

The idea of metric connection with torsion in a Riemannian manifold was introduced by Hayden [1]. Further, some properties of semisymmetric metric connection have been studied by Yano [2]. In [3], Golab defined and studied quarter-symmetric connection on a differentiable manifold with affine connection, which generalizes the idea of semisymmetric connection. Various properties of quarter-symmetric metric connection have been studied by many geometers like Rastogi [4, 5], Mishra and Pandey [6], Yano and Imai [7], De et al. [8, 9], Pradeep Kumar et al. [10], and many others.

The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, Takahashi [11] introduced the notion of local -symmetry on a Sasakian manifold. Generalizing the notion of -symmetry, the authors De et al. [12] introduced the notion of -recurrent Sasakian manifolds.

A linear connection on an -dimensional differentiable manifold is said to be a quarter-symmetric connection [3] if its torsion tensor is of the form where is a 1-form and is a tensor of type . In particular, if we replace by and by , then the quarter-symmetric connection reduces to the semisymmetric connection [13]. Thus, the notion of quarter-symmetric connection generalizes the idea of the semisymmetric connection. And if quarter-symmetric linear connection satisfies the condition for all , where is the Lie algebra of vector fields on the manifold , then is said to be a quarter-symmetric metric connection.

2. Preliminaries

An -dimensional differentiable manifold is called an almost paracontact manifold if it admits an almost paracontact structure consisting of a tensor field , a vector field , and a 1-form satisfying

If is a compatible Riemannian metric with , that is, for all vector fields and on , then becomes a almost paracontact Riemannian manifold equipped with an almost paracontact Riemannian structure .

An almost paracontact Riemannian manifold is called a para-Sasakian manifold if it satisfies where denotes the operator of covariant differentiation. From the above equation it follows that

In an -dimensional para-Sasakian manifold , the following relations hold [14, 15]: for any vector fields , and , where and are the Riemannian curvature tensor and the Ricci tensor of , respectively.

A para-Sasakian manifold is said to be -Einstein if its Ricci tensor is of the form for any vector fields and , where and are some functions on .

Definition 2.1. A para-Sasakian manifold is said to be locally -symmetric if for all vector fields orthogonal to . This notion was introduced for Sasakian manifold by Takahashi [11].

Definition 2.2. A para-Sasakian manifold is said to be locally concircular -symmetric if for all vector fields orthogonal to . Where the concircular curvature tensor is given by [16] where is the Riemannian curvature tensor and is the scalar curvature.

Definition 2.3. A para-Sasakian manifold is said to be -recurrent if there exists a nonzero 1-form such that where is a 1-form and it is defined by and is a vector field associated with the 1-form .

3. Quarter-Symmetric Metric Connection

Let be a linear connection and a Riemannian connection of an almost contact metric manifold such that where is a tensor of type . For to be a quarter-symmetric metric connection in , then we have [3] From (1.1) and (3.3), we get Using (1.1) and (3.4) in (3.2), we obtain Thus a quarter-symmetric metric connection in a para-Sasakian manifold is given by Hence (3.6) is the relation between Riemannian connection and the quarter-symmetric metric connection on a para-Sasakian manifold.

A relation between the curvature tensor of with respect to the quarter-symmetric metric connection and the Riemannian connection is given by where and denote the Riemannian curvatures of the connections and , respectively. From (3.7), it follows that where and are the Ricci tensors of the connections and , respectively.

Contracting (3.8), we get where and are the scalar curvatures of the connections and , respectively.

4. -Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A para-Sasakian manifold is called -recurrent with respect to the quarter-symmetric metric connection if its curvature tensor satisfies the condition By virtue of (2.1) and (4.1), 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 (4.3) and taking summation over ,  , we get The second term of (4.4) by putting takes the form On simplification we obtain Therefore (4.4) can be written in the form Replacing by in the above relation, then using (3.8) and (2.9), we have We know that Using (3.8), (2.6) and (2.9) in the above relation, we get In view of (4.8) and (4.10), we obtain Replacing by in (4.11) and then using (2.3) and (2.10), we have Hence, we can state the following.

Theorem 4.1. If para-Sasakian manifold is -recurrent with respect to quarter-symmetric metric connection then it is an -Einstein manifold with respect to Riemannian connection.

5. Concircular -Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A concircular -recurrent para-Sasakian manifold with respect to the quarter-symmetric metric connection is defined by where is a concircular curvature tensor with respect to the quarter-symmetric metric connection given by By virtue of (2.1) and (5.1), we have from which it follows that where Let be an orthonormal basis of the tangent space at any point of the manifold. Then putting in (5.4) and taking summation over , we get Replacing by in (5.6) and using (2.9), we have We know that Using (2.6) and (2.9) in the above relation, it follows that In view of (5.7) and (5.9), we obtain Replacing by in (5.10) and then using (2.3) and (2.10), we obtain This leads to the following theorem.

Theorem 5.1. If para-Sasakian manifold is concircular -recurrent with respect to quarter-symmetric metric connection then it is an -Einstein manifold with respect to Riemannian connection.

Now from (5.3), we have This gives Now from (5.13) and Bianchi’s second identity, we have By virtue of (2.7), we obtain from (5.14) that Putting in (5.15) and taking summation over , , we get for all vector fields . Replacing by in (5.16), we get for any vector field .

Hence from (5.16) and (5.17), we can state the following.

Theorem 5.2. In a concircular -recurrent para-Sasakian manifold with respect to quarter-symmetric metric connection, the characteristic vector field and the vector field associated to the 1-form are in codirectional and the 1-form is given by (5.17).

Acknowledgments

The authors express their thanks to DST (Department of Science and Technology), Government of India, for providing financial assistance under major research project (no. SR/S4/MS: 482/07). They are also thankful to the referees for valuable suggestions.