Abstract
The paper deals with the study on conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric connection, and it is shown that these manifolds are η-Einstein with respect to Levi-Civita connection, and the results are illustrated with examples.
1. Introduction
In 1924, Friedmann and Schouten [1] introduced the idea of semisymmetric linear connection on a differentiable manifold. In 1932, Hayden [2] introduced the idea of metric connection with torsion on a Riemannian manifold. A systematic study of the semisymmetric metric connection on a Riemannian manifold was published by Yano [3] in 1970. After that the properties of semisymmetric metric connection have been studied by many authors like Amur and Pujar [4], Bagewadi [5], Sharfuddin and Hussain [6], De and Pathak [7], and so forth.
A K-contact manifold is a differentiable manifold with a contact metric structure such that is a Killing vector field ([8, 9]). These are studied by many authors like ([8–11]). The notion of Kenmotsu manifolds was defined by Kenmotsu [12]. Kenmotsu proved that a locally Kenmotsu manifold is a warped product of an interval and a Kaehler manifold with warping function , where is a nonzero constant. For example it is hyperbolic space . Kenmotsu manifolds were studied by many authors such as Binh et al. [13], Bagewadi and Venkatesha [14].
In this paper we study conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric connection and obtain results on K-contact and Kenmotsu manifolds with respect to Levi-Civita connection.
2. Preliminaries
An -dimensional differential manifold is said to have an almost contact structure if it carries a tensor field of type , a vector field , and 1-form on , respectively, such that
Thus a manifold equipped with this structure is called an almost contact manifold [9] and is denoted by . If is a Riemannian metric on an almost contact manifold such that, where and are vector fields defined on , then is said to have an almost contact metric structure and with this structure is called an almost contact metric manifold and is denoted by .
If on the exterior derivative of 1-form satisfies, then is said to be a contact metric structure and equipped with a contact metric structure is called contact metric manifold.
If moreover is Killing vector field, then is called a K-contact manifold [8, 9]. A K-contact manifold is called Sasakian [9], if the relation holds, where denotes the operator of covariant connection with respect to .
An almost contact metric manifold, which satisfies the following conditions, where denotes the Riemannian connection of hold, is called a Kenmotsu manifold.
In a K-contact manifold , the following properties hold: where is the Riemannian curvature tensor, is the Ricci tensor and is the Ricci operator of , respectively.
In a Kenmotsu manifold , the following properties hold [12]: where is the Riemannian curvature tensor, is the Ricci tensor, and is the Ricci operator of , respectively.
Let be an -dimensional Riemannian manifold of class with metric tensor and let be the Levi-Civita connection on . A linear connection in an almost contact metric manifold is said to be a semisymmetric connection if the torsion tensor of the connection satisfies where is a 1-form on with as associated vector field, that is, for any differentiable vector field on .
A semisymmetric connection is called semisymmetric metric connection if it further satisfies .
In an almost contact manifold semisymmetric metric connection is defined by identifying the 1-form of (2.14) with the contact-form , that is, with as associated vector field, that is, .
The relation between the semisymmetric metric connection and the Levi-Civita connection of has been obtained by Yano [3], which is given by The above condition satisfies K-contact and Kenmotsu manifolds also.
We denote , , and by curvature tensor, Ricci tensor, and scalar curvature with respect to Levi-Civita connection and correspondingly , , and with respect to semisymmetric metric connection. If denotes C-Bochner curvature tensor [15] with respect to Levi-Civita connection with respect to semisymmetric metric connection is given by where .
Differentiate (2.17) covariantly with respect to and then contracting we get
3. Relation between , , and , , in a K-Contact Manifold
A relation between the curvature tensor and of type of the connections and by using (2.16) is given by From (3.1), it follows that where denotes the Ricci tensor with respect to semisymmetric metric connection and denotes the Ricci tensor. On contacting (3.2), we get where and are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.
In a Riemannian manifold , is a Killing vector field in K-contact manifold, that is, and are invariant under it, that is, where is Lie-derivative. We know that by using (3.5) in (3.7) and by virtue of (2.6), we have Now in a K-contact manifold , , and are with respect to semisymmetric metric connection, that is, This shows that it is not K-contact with respect to a semisymmetric metric connection.
Now from (3.2), we have Put in (3.10), we have But Put in (3.10), we have
4. K-Contact Manifold Admitting Semisymmetric Metric Connection with
Considering in (2.18) and putting in the equation; using (3.2), (3.3), (3.12), (3.13), and (3.14) and by virtue of (2.1), (2.6), and (2.16), we get In a K-contact manifold, , that is, then the above equation reduces to Interchanging and in the above equation then we have Adding these equation (4.2) and (4.3), we have Then the above equation is written as , where On contracting (4.4), we get Hence we state the following theorem.
Theorem 4.1. If in a K-contact manifold the C-Bochner curvature tensor with respect to semisymmetric metric connection is conservative, then the manifold is -Einstein with respect to Levi-Civita connection and the scalar curvature of such a manifold is given in (4.6).
5. Example for K-Contact Manifold
Consider the 3-dimensional manifold , where are the standard coordinates in . Let be linearly independent at each point of Let be the Riemannian metric defined by where is given by . Let be the vector field, be the 1-form, and be the tensor field defined by The linearity property of and yields that for any vector fields , on . Thus for . The structure defines on . By definition of Lie bracket, we have Let be Levi-Civita connection with respect to the above metric given by Koszul formula, that is Then by Koszula formula, we have Clearly one can see that is a K-contact structure.
The Ricci tensor is given by The nonzero components of , where , 2, 3, and by virtue of (5.7) we have Using these in (5.8), we have This shows that is an -Einstein. This is an example of K-contact manifold which is an -Einstein.
If , in (2.18) and by virtue of (3.2), we obtain . Thus Theorem 4.1 holds true.
However, if , in (2.18) and by virtue of (3.2), we obtain . Hence in general, if , and , then . In this case the converse of Theorem 4.1 does not hold true.
6. Relation between , , and, , in a Kenmotsu Manifold
A relation between the curvature tensor and of type of the connections and by using (2.16) is given by From (6.1), it follows that where denotes the Ricci tensor with respect to semisymmetric metric connection and denotes the Ricci tensor. On contacting (6.2), we get where and are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.
In a Kenmotsu manifold , is a unit vector field in Kenmotsu manifold, then the following properties hold where is a Lie derivative. We know that By using (6.4) in (6.5) and by virtue of (2.10) we have Now in a Kenmotsu manifold , , and are with respect to semisymmetric metric connection, then we define the properties like Now from (6.2), we have Putting in the above equation and by virtue of (6.6) then we have Again putting in (6.9), we have
7. Kenmotsu Manifold Admitting Semisymmetric Metric Connection with
Considering in (2.18) and putting in the equation; using (6.2), (6.3), (6.10), and (6.12) and by virtue of (2.10), (2.16) and on simplification we get In a Kenmotsu manifold , then the above equation reduces to On simplifying the above equation we get Then the above equation is written as , where On contracting (7.3), we get Hence we state the following theorem.
Theorem 7.1. If in a Kenmotsu manifold the C-Bochner curvature tensor with respect to semisymmetric metric connection is conservative, then the manifold is -Einstein with respect to Levi-Civita connection, and the scalar curvature of such a manifold is given in (7.5).
8. Example for Kenmotsu Manifold
Let . Let be linearly independent vector fields given by Let be the Riemannian metric defined by where is given by . Let be the vector field, be the 1-form, and be the tensor field defined by The linearity property of and yields that for any vector fields , on . By definition of Lie bracket, we have Let be the Levi-Civita connection with respect to above metric which is given by Koszula formula (5.6), and by virtue of it we have Clearly one can see that is a Kenmotsu structure.
The nonzero components of , where , 2, 3, and by virtue of (8.6) we have The Ricci tensor is given in (5.8) by virtue of (8.7), and we have This shows that is an -Einstein. This is an example of Kenmotsu manifold which is an -Einstein.
If , in (2.18) and by virtue of (6.2), we obtain . Thus Theorem 7.1 holds true.
However, if , in (2.18) and by virtue of (6.2), we obtain . Hence in general, if , and , then . In this case the converse of Theorem 7.1 does not hold true.
Acknowledgments
The authors express their gratitude 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 grateful to referees for revising the paper.