On Concircular φ-Recurrent K-Contact Manifold Admitting Semisymmetric Metric Connection Venkatesha

In the present paper, we have studied 𝜙-recurrent and concircular 𝜙-recurrent 𝐾-contact manifold with respect to semisymmetric metric connection and obtained some interesting results.


Introduction
The idea of semisymmetric linear connection on a differentiable manifold was introduced by Friedmann and Schouten 1 .In 2 , Hayden introduced idea of metric connection with torsion on a Riemannian manifold.Further, some properties of semisymmetric metric connection has been studied by Yano 3 .In 4 , Golab defined and studied quarter-symmetric connection on a differentiable manifold with affine connection, which generalizes the idea of semisymmetric connection.Various properties of semisymmetric metric connection and quarter-symmetric metric connection have been studied by many geometers like Sharfuddin and Hussain 5 , Amur and Pujar 6 , Rastogi  The paper is organized as follows.Section 2 is devoted to preliminaries.In Section 3, we study semisymmetric metric connection in a K-contact manifold.In Section 4, it is proved that a φ-recurrent K-contact manifold with respect to semisymmetric metric connection is an International Journal of Mathematics and Mathematical Sciences Einstein manifold.Finally, in Section 5 it is also shown that concircular φ-recurrent K-contact manifold admitting semisymmetric metric connection is an Einstein manifold, and the characteristic vector field ξ and the vector field ρ associated to the 1-form A are codirectional.

Preliminaries
An n-dimensional differentiable 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 a 1-form η on M, respectively, such that, Thus a manifold M equipped with this structure φ, ξ, η is called an almost contact manifold and is denoted by M, φ, ξ, η .If g is a Riemannian metric on an almost contact manifold M such that, where X, 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 an contact metric manifold.
holds, where ∇ denotes the operator of covariant differentiation with respect to g.In a K-contact manifold M, the following relations holds: for all vector fields X, Y , and Z.Here R and S are the Riemannian curvature tensor and the Ricci tensor of M, respectively.
International Journal of Mathematics and Mathematical Sciences 3 Definition 2.1.A K-contact manifold M is said to be φ-recurrent if there exists a nonzero 1form A such that, where A is defined by A W g W, ρ , and ρ is a vector field associated with the 1-form A.
Definition 2.2.A K-contact manifold M is said to be concircular φ-recurrent 12 if there exists a non-zero 1-form A such that, where C is a concircular curvature tensor given by 21 as follows: where R is the Riemannian curvature tensor and r is the scalar curvature.
A linear connection ∇ in an n-dimensional differentiable manifold M is said to be a semisymmetric connection if its torsion tensor T is of the form for all X, Y on TM.A semisymmetric connection ∇ is called semisymmetric metric connection, if it further satisfies ∇g 0.

Semisymmetric Metric Connection in a K-Contact Manifold
A semisymmetric metric connection ∇ in a K-contact manifold can be defined by where ∇ is the Levi-Civita connection on M 3 .
A relation between the curvature tensor of M, with respect to the semisymmetric metric connection ∇ and the Levi-Civita connection, ∇ is given by where R and R are the Riemannian curvatures of the connections ∇ and ∇, respectively.From 3.2 , it follows that where S and S are the Ricci tensors of the connections ∇ and ∇, respectively.
International Journal of Mathematics and Mathematical Sciences Contracting 3.3 , we get where r and r are the scalar curvatures of the connections ∇ and ∇, respectively.

φ-Recurrent K-Contact Manifold with respect to Semisymmetric Metric Connection
A K-contact manifold is called φ-recurrent with respect to the semisymmetric metric connection if its curvature tensor R satisfies the following condition: By virtue of 2.1 and 4.1 , we have Let {e i }, i 1, 2, . . ., n be an orthonormal basis of the tangent space at any point of the manifold.Then putting X U e i in 4.3 and taking summation over i, 1 ≤ i ≤ n, we get

Y Z η e i
A W S Y, Z .

4.4
Put Z ξ, then the second term of 4.4 takes the following form:

4.5
On simplification, we obtain g ∇ W R e i , Y ξ, ξ 0. Now 4.4 implies that We know that Using 3.3 , 2.5 , and 2.7 in the above relation, we get 2 n − 2 g φY, φW .

4.8
In view of 4.6 and 4.8 , we have Again putting Y φY in 4.9 , we get

Concircular φ-Recurrent K-Contact Manifold with respect to Semisymmetric Metric Connection
Let us consider a concircular φ-recurrent K-contact manifold with respect to the semisymmetric metric connection defined by where C is a concircular curvature tensor with respect to the semisymmetric metric connection given by

International Journal of Mathematics and Mathematical Sciences
By virtue of 2.1 and 5.1 , we have where

5.5
Let {e i }, i 1, 2, . . ., n be an orthonormal basis of the tangent space at any point of the manifold.Then putting X U e i in 5.4 and taking summation over i, 1 ≤ i ≤ n, we get

5.6
Replacing Z by ξ in 5.6 , we obtain We know that International Journal of Mathematics and Mathematical Sciences 7 Using 3.3 , 2.5 and 2.7 , the above relation becomes 5.9 In view of 5.7 and 5.9 , we obtain

5.10
Replacing Y by φY in 5.10 , we have Interchanging Y and W in 5.11 , we get Adding 5.11 and 5.12 , which on simplification, we have S Y, W n − 1 g Y, W .

5.13
Thus, we obtain the following theorem.
Theorem 5.1.A Concircular φ-recurrent K-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.

5.15
Putting Y Z e i in 5.15 and taking summation over i, 1 ≤ i ≤ n, one gets A φW η X − A φX η W .

5.16
Replacing X by ξ in 5.16 , one gets for any vector field W. Hence, one states the following.
Theorem 5.2.In a concircular φ-recurrent K-contact manifold admitting semisymmetric metric connection the characteristic vector field ξ and the vector field ρ associated to the 1-form A are codirectional and the 1-form A is given by 5.18 .

7 , 8 ,
Mishra and Pandey 9 , Bagewadi et al. 10-14 , De et al. 15, 16 , 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 17 introduced the notion of local φ-symmetry on a Sasakian manifold.Generalizing the notion of φ-symmetry, De et al. 18 introduced the notion of φ-recurrent Sasakian manifolds.