A Study on Conservative C-Bochner Curvature Tensor in K-Contact and Kenmotsu Manifolds Admitting Semisymmetric Metric Connection

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.


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 ξ 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 I × f N of an interval I and a Kaehler manifold N with warping function f t se t , where s is a nonzero constant.For example it is hyperbolic space −1 .Kenmotsu manifolds were studied by many authors such as Binh et al. 13 , Bagewadi and Venkatesha 14 .

Preliminaries
An n-dimensional differential manifold M is said to have an almost contact structure φ, ξ, η if it carries a tensor field φ of type 1, 1 , a vector field ξ, and 1-form η on M, respectively, such that Thus a manifold M equipped with this structure is called an almost contact manifold 9 and is denoted by M, φ, ξ, η .If g is a Riemannian metric on an almost contact manifold M such that, where X and Y are vector fields defined on M, then M is said to have an almost contact metric structure φ, ξ, η, g and M with this structure is called an almost contact metric manifold and is denoted by M, φ, ξ, η, g .If on M, φ, ξ, η, g the exterior derivative of 1-form η satisfies, dη X, Y g X, φY , 2.3 then φ, ξ, η, g is said to be a contact metric structure and M equipped with a contact metric structure is called contact metric manifold.If moreover ξ is Killing vector field, then M 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 g.An almost contact metric manifold, which satisfies the following conditions, where ∇ denotes the Riemannian connection of g hold, M, φ, ξ, η, g is called a Kenmotsu manifold.
In a K-contact manifold M, the following properties hold: where R is the Riemannian curvature tensor, S is the Ricci tensor and Q is the Ricci operator of M, respectively.In a Kenmotsu manifold M, the following properties hold 12 : where R is the Riemannian curvature tensor, S is the Ricci tensor, and Q is the Ricci operator of M, respectively.Let M be an n-dimensional Riemannian manifold of class C ∞ with metric tensor g and let ∇ be the Levi-Civita connection on M. A linear connection ∇ in an almost contact metric manifold M is said to be a semisymmetric connection if the torsion tensor T of the connection where π is a 1-form on M with ρ as associated vector field, that is, π X g X, ρ for any differentiable vector field X on M.
A semisymmetric connection ∇ is called semisymmetric metric connection if it further satisfies ∇g 0.
In an almost contact manifold semisymmetric metric connection is defined by identifying the 1-form of 2.14 with the contact-form η, that is, with ξ as associated vector field, that is, g X, ξ η X .The relation between the semisymmetric metric connection ∇ and the Levi-Civita connection ∇ of M has been obtained by Yano 3 , which is given by

2.16
The above condition satisfies K-contact and Kenmotsu manifolds also.

ISRN Geometry
We denote R, S, and r by curvature tensor, Ricci tensor, and scalar curvature with respect to Levi-Civita connection and correspondingly R, S, and r with respect to semisymmetric metric connection.If B denotes C-Bochner curvature tensor 15 with respect to Levi-Civita connection B with respect to semisymmetric metric connection is given by

2.17
where D n − 1 r / n 1 .Differentiate 2.17 covariantly with respect to ∇ and then contracting we get 2.18

Relation between R, S, r and R, S, r in a K-Contact Manifold
A relation between the curvature tensor R and R of type 1, 3 of the connections ∇ and ∇ by using 2.16 is given by where S denotes the Ricci tensor with respect to semisymmetric metric connection and S denotes the Ricci tensor.On contacting 3.2 , we get where r and r are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.
In a Riemannian manifold M, ξ is a Killing vector field in K-contact manifold, that is, S and r are invariant under it, that is, where L is Lie-derivative.We know that 3.7 by using 3.5 in 3.7 and by virtue of 2.6 , we have Now in a K-contact manifold L, S, and r are with respect to semisymmetric metric connection, that is,

3.9
This shows that it is not K-contact with respect to a semisymmetric metric connection.Now from 3.2 , we have

3.10
Put W ξ in 3.10 , we have

K-Contact Manifold Admitting Semisymmetric Metric Connection with Div • B 0
Considering Div • B 0 in 2.18 and putting X ξ 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

4.1
In a K-contact manifold, ξ r 0, that is, ∇ ξ r 0 then the above equation reduces to

4.2
Interchanging Y and Z in the above equation then we have

4.3
Adding these equation 4.2 and 4.3 , we have Then the above equation is written as S Y, Z αg Y, Z βη Y η Z , 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 η-Einstein with respect to Levi-Civita connection and the scalar curvature of such a manifold is given in 4.6 .

Example for K-Contact Manifold
Consider the 3-dimensional manifold M { x, y, z : x, y, z ∈ R 3 }, where x, y, z are the standard coordinates in R 3 .Let E 1 , E 2 , E 3 be linearly independent at each point of Let g be the Riemannian metric defined by where g is given by g 1 − y 2 dx ⊗ dx 1 − x 2 dy ⊗ dy dz ⊗ dz .Let ξ be the vector field, η be the 1-form, and φ be the 1, 1 tensor field defined by

5.3
The linearity property of φ and g yields that for any vector fields U, W on M. Thus for E 3 ξ.The structure φ, ξ, η, g defines on M. By definition of Lie bracket, we have Let ∇ be Levi-Civita connection with respect to the above metric g given by Koszul formula, that is

5.6
Then by Koszula formula, we have

5.7
Clearly one can see that φ, ξ, η, g is a K-contact structure.
The Ricci tensor S X, Y is given by

5.8
The nonzero components of R X, E i E i , where i 1, 2, 3, and by virtue of 5.7 we have 5.9 Using these in 5.8 , we have

5.10
This shows that R 3 is an η-Einstein.This is an example of K-contact manifold which is an η-Einstein.
18 and by virtue of 3.2 , we obtain Div • B 0. Thus Theorem 4.1 holds true.
However, if X / Y / Z E i , in 2.18 and by virtue of 3.2 , we obtain Div • B / 0. Hence in general, if In this case the converse of Theorem 4.1 does not hold true. 10 ISRN Geometry

Relation between R, S, r and R, S, r in a Kenmotsu Manifold
A relation between the curvature tensor R and R of type 1, 3 of the connections ∇ and ∇ by using 2.16 is given by where S denotes the Ricci tensor with respect to semisymmetric metric connection and S denotes the Ricci tensor.On contacting 6.2 , we get where r and r are scalar curvatures with respect to semisymmetric metric connection and Levi-Civita connection.In a Kenmotsu manifold M, ξ is a unit vector field in Kenmotsu manifold, then the following properties hold where L is a Lie derivative.We know that

6.5
By using 6.4 in 6.5 and by virtue of 2.10 we have Now in a Kenmotsu manifold L, S, and r are with respect to semisymmetric metric connection, then we define the properties like L ξ r Div ξ r.

6.9
Putting X ξ in the above equation and by virtue of 6.6 then we have ∇ ξ S Y, Z 0, 6.10 ∇ ξ r 0.

ISRN Geometry
In a Kenmotsu manifold M, ∇ ξ r 0 then the above equation reduces to

7.2
On simplifying the above equation we get

7.3
Then the above equation is written as S Y, Z αg Y, Z βη Y η Z , 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 η-Einstein with respect to Levi-Civita connection, and the scalar curvature of such a manifold is given in 7.5 .

Example for Kenmotsu Manifold
Let M { x, y, z ∈ R 3 }.Let E 1 , E 2 , E 3 be linearly independent vector fields given by Let g be the Riemannian metric defined by where g is given by g e 2z /2 dx ⊗ dx dy ⊗ dy dz ⊗ dz.Let ξ be the vector field, η be the 1-form, and φ be the 1, 1 tensor field defined by The linearity property of φ and g yields that g φU, φW g U, W − η U η W , 8.4 for any vector fields U, W on M. By definition of Lie bracket, we have Let ∇ be the Levi-Civita connection with respect to above metric g which is given by Koszula formula 5.6 , and by virtue of it we have 8.6 Clearly one can see that φ, ξ, η, g is a Kenmotsu structure.The nonzero components of R X, E i E i , where i 1, 2, 3, and by virtue of 8.6 we have

8.7
The Ricci tensor S X, Y is given in 5.8 by virtue of 8.

3 i 1
b i E i and Z 3 i 1 c i E i , then Div • B X, Y Z / 0.In this case the converse of Theorem 7.1 does not hold true.
This shows that R 3 is an η-Einstein.This is an example of Kenmotsu manifold which is an η-Einstein.
i , in 2.18 and by virtue of 6.2 , we obtain Div • B / 0. Hence in general, ifX 3 i 1 a i E i , Y