On -Recurrent Para-Sasakian Manifold Admitting
Quarter-Symmetric Metric Connection
K. T. Pradeep Kumar,1 Venkatesha,1and C. S. Bagewadi1
Academic Editor: R. Vรกzquez-Lorenzo, T. Friedrich, M. Korkmaz, O. Mokhov
Received03 Nov 2011
Accepted06 Dec 2011
Published16 Feb 2012
Abstract
We obtained the relation between the Riemannian connection
and the quarter-symmetric metric connection on a para-Sasakian manifold.
Further, we study -recurrent and concircular -recurrent para-Sasakian
manifolds with respect to quarter-symmetric metric connection.
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 ,โโ, 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 Bianchiโs 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.
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