# On Concircularϕ-Recurrent K-Contact Manifold Admitting Semisymmetric Metric Connection

Authors: Venkatesha; K. T. Pradeep Kumar; C. S. Bagewadi; Gurupadavva Ingalahalli
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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


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## Body

## 1. 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 [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.The paper is organized as follows. Section2 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 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.


## 2. Preliminaries

Ann-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,
(2.1)ϕ2=-I+η⊗ξ,η(ξ)=1,η∘ϕ=0,ϕξ=0.Thus a manifoldM 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,
(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ξ)=η(X),
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
(2.3)dη(X,Y)=g(X,ϕY),
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.If moreoverξ is killing vector field on M, then, M is called a K-contact Riemannian manifold [19, 20]. A K-contact Riemannian manifold is called Sasakian [19], if the relation
(2.4)(∇Xϕ)Y=g(X,Y)ξ-η(Y)X
holds, where ∇ denotes the operator of covariant differentiation with respect to g.In aK-contact manifold M, the following relations holds:
(2.5)∇Xξ=-ϕX,(2.6)g(R(X,Y)Z,ξ)=g(Y,Z)η(X)-g(X,Z)η(Y),(2.7)S(X,ξ)=(n-1)η(X),
for all vector fields X, Y, and Z. Here R and S are the Riemannian curvature tensor and the Ricci tensor of M, respectively.Definition 2.1.
AK-contact manifold M is said to be ϕ-recurrent if there exists a nonzero 1-form A such that,
(2.8)ϕ2((∇WR)(X,Y)Z)=A(W)R(X,Y)Z,
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.
AK-contact manifold M is said to be concircular ϕ-recurrent [12] if there exists a non-zero 1-form A such that,
(2.9)ϕ2((∇WC-)(X,Y)Z)=A(W)C-(X,Y)Z,
where C- is a concircular curvature tensor given by [21] as follows:
(2.10)C-(X,Y)Z=R(X,Y)Z-rn(n-1)[g(Y,Z)X-g(X,Z)Y],
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
(2.11)T(X,Y)=∇~XY-∇~YX-[X,Y]=η(Y)X-η(X)Y,
for all X, Y on TM. A semisymmetric connection ∇~ is called semisymmetric metric connection, if it further satisfies ∇~g=0.


## 3.   Semisymmetric Metric Connection in aK-Contact Manifold

A semisymmetric metric connection∇~ in a K-contact manifold can be defined by
(3.1)∇~XY=∇XY+η(Y)X-g(X,Y)ξ,
where ∇ is the Levi-Civita connection on M [3].A relation between the curvature tensor ofM, with respect to the semisymmetric metric connection ∇~ and the Levi-Civita connection, ∇ is given by
(3.2)R~(X,Y)Z=R(X,Y)Z+[g(ϕY,Z)X-g(ϕX,Z)Y]+[g(Y,Z)ϕX-g(X,Z)ϕY]+[g(ϕX,ϕZ)Y-g(ϕY,ϕZ)X]+[η(X)g(Y,Z)-η(Y)g(X,Z)]ξ,
where R~ and R are the Riemannian curvatures of the connections ∇~ and ∇, respectively.From (3.2), it follows that
(3.3)S~(Y,Z)=S(Y,Z)-(n-2)g(Y,Z)+(n-2)g(ϕY,Z)+(n-2)η(Y)η(Z),
where S~ and S are the Ricci tensors of the connections ∇~ and ∇, respectively.Contracting (3.3), we get
(3.4)r~=r-(n-1)(n-2),
where r~ and r are the scalar curvatures of the connections ∇~ and ∇, respectively.


## 4.ϕ-Recurrent K-Contact Manifold with respect to Semisymmetric Metric Connection

AK-contact manifold is called ϕ-recurrent with respect to the semisymmetric metric connection if its curvature tensor R~ satisfies the following condition:
(4.1)ϕ2((∇~WR~)(X,Y)Z)=A(W)R~(X,Y)Z.
By virtue of (2.1) and (4.1), we have
(4.2)-(∇~WR~)(X,Y)Z+η((∇~WR~)(X,Y)Z)ξ=A(W)R~(X,Y)Z,
from which, it follows that
(4.3)-g((∇~WR~)(X,Y)Z,U)+η((∇~WR~)(X,Y)Z)g(ξ,U)=A(W)g(R~(X,Y)Z,U).
Let {ei}, i=1,2,…,n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X=U=ei in (4.3) and taking summation over i, 1≤i≤n, we get
(4.4)-(∇~WS~)(Y,Z)+∑i=1n‍η((∇~WR~)(ei,Y)Z)η(ei)=A(W)S~(Y,Z).
Put Z=ξ, then the second term of (4.4) takes the following form:
(4.5)g((∇~WR~)(ei,Y)ξ,ξ)=g(∇~WR~(ei,Y)ξ,ξ)-g(R~(∇~Wei,Y)ξ,ξ)-g(R~(ei,∇~WY)ξ,ξ)-g(R~(ei,Y)∇~Wξ,ξ).
On simplification, we obtain g((∇~WR~)(ei,Y)ξ,ξ)=0.Now (4.4) implies that
(4.6)(∇~WS~)(Y,ξ)=-A(W)S~(Y,ξ).
We know that
(4.7)(∇~WS~)(Y,ξ)=∇~WS~(Y,ξ)-S~(∇~WY,ξ)-S~(Y,∇~Wξ).
Using (3.3), (2.5), and (2.7) in the above relation, we get
(4.8)(∇~WS~)(Y,ξ)=S(Y,ϕW)-S(Y,W)-(n-1)g(Y,ϕW)+(n-1)g(Y,W)+2(n-2)g(ϕY,ϕW).
In view of (4.6) and (4.8), we have
(4.9)S(Y,W)-S(Y,ϕW)+(n-1)g(Y,ϕW)-(n-1)g(Y,W)-2(n-2)g(ϕY,ϕW)=(n-1)A(W)η(Y).
Again putting Y=ϕY in (4.9), we get
(4.10)S(ϕY,W)-S(ϕY,ϕW)+(n-1)g(ϕY,ϕW)-(n-1)g(ϕY,W)+2(n-2)g(Y,ϕW)=0.
Interchanging Y and W in (4.10), we obtain
(4.11)S(ϕW,Y)-S(ϕW,ϕY)+(n-1)g(ϕW,ϕY)-(n-1)g(ϕW,Y)+2(n-2)g(W,ϕY)=0.
Adding (4.10) and (4.11) which on simplification, we have
(4.12)S(Y,W)=(n-1)g(Y,W).
Therefore, we can state the following.Theorem 4.1.
Aϕ-recurrent K-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.


## 5. Concircularϕ-Recurrent K-Contact Manifold with respect to Semisymmetric Metric Connection

Let us consider a concircularϕ-recurrent K-contact manifold with respect to the semisymmetric metric connection defined by
(5.1)ϕ2((∇~WC-~)(X,Y)Z)=A(W)C-~(X,Y)Z,
where C-~ is a concircular curvature tensor with respect to the semisymmetric metric connection given by
(5.2)C-~(X,Y)Z=R~(X,Y)Z-r~n(n-1)[g(Y,Z)X-g(X,Z)Y].
By virtue of (2.1) and (5.1), we have
(5.3)-(∇~WC-~)(X,Y)Z+η((∇~WC-~)(X,Y)Z)ξ=A(W)C-~(X,Y)Z,
from which, it follows that
(5.4)-g((∇~WC-~)(X,Y)Z,U)+η((∇~WC-~)(X,Y)Z)g(ξ,U)=A(W)g(C-~(X,Y)Z,U),
where
(5.5)(∇~WC-~)(X,Y)Z=((∇WR)(X,Y)Z)+3[g(Y,W)η(Z)X-g(X,W)η(Z)Y]+3[g(Y,Z)g(W,X)-g(X,Z)g(W,Y)]ξ+2[η(X)g(ϕW,Z)Y-η(Y)g(ϕW,Z)X]+2[η(Y)g(X,Z)-η(X)g(Y,Z)]ϕW+[g(Y,Z)η(X)-g(X,Z)η(Y)]W+2η(W)[η(Y)g(X,Z)-η(X)g(Y,Z)]ξ+2η(Z)η(W)[η(X)Y-η(Y)X]+g(Z,W)[η(Y)X-η(X)Y]-g(W,R(X,Y)Z)ξ-η(X)R(W,Y)Z-η(Y)R(X,W)Z-η(Z)R(X,Y)W-∇Wrn(n-1)[g(Y,Z)X-g(X,Z)Y].
Let {ei}, i=1,2,…,n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X=U=ei in (5.4) and taking summation over i, 1≤i≤n, we get
(5.6)(∇~WS~)(Y,Z)-∇~Wr~ng(Y,Z)=-∇~Wr~n(n-1)[g(Y,Z)-η(Y)η(Z)]-A(W)[S~(Y,Z)-r~ng(Y,Z)].
Replacing Z by ξ in (5.6), we obtain
(5.7)(∇~WS~)(Y,ξ)=∇~Wr~nη(Y)-A(W)[S~(Y,ξ)-r~nη(Y)].
We know that
(5.8)(∇~WS~)(Y,ξ)=∇~WS~(Y,ξ)-S~(∇~WY,ξ)-S~(Y,∇~Wξ).
Using (3.3), (2.5) and (2.7), the above relation becomes
(5.9)(∇~WS~)(Y,ξ)=S(Y,ϕW)-S(Y,W)-(n-1)g(Y,ϕW)+(n-1)g(Y,W)+2(n-2)g(ϕY,ϕW).
In view of (5.7) and (5.9), we obtain
(5.10)S(Y,ϕW)-S(Y,W)-(n-1)g(Y,ϕW)+(n-1)g(Y,W)+2(n-2)g(Y,W)-2(n-2)η(Y)η(W)=∇Wrnη(Y)-A(W)[2(n-1)2-rnη(Y)].
Replacing Y by ϕY in (5.10), we have
(5.11)S(ϕY,ϕW)-S(ϕY,W)-(n-1)g(ϕY,ϕW)+(n-1)g(ϕY,W)+2(n-2)g(ϕY,W)=0.
Interchanging Y and W in (5.11), we get
(5.12)S(ϕW,ϕY)-S(ϕW,Y)-(n-1)g(ϕW,ϕY)+(n-1)g(ϕW,Y)+2(n-2)g(ϕW,Y)=0.
Adding (5.11) and (5.12), which on simplification, we have
(5.13)S(Y,W)=(n-1)g(Y,W).
Thus, we obtain the following theorem.Theorem 5.1.
A Concircularϕ-recurrent K-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.Next, from (5.3), one has
(5.14)(∇~WC-~)(X,Y)Z=η((∇~WC-~)(X,Y)Z)ξ-A(W)C-~(X,Y)Z.
Now, using (3.2), (3.4), (5.5), and Bianchi's identity in (5.14), one obtains
(5.15)A(W)η(R(X,Y)Z)+A(X)η(R(Y,W)Z)+A(Y)η(R(W,X)Z)=-A(W)[g(ϕY,Z)η(X)-g(ϕX,Z)η(Y)]-A(X)[g(ϕW,Z)η(Y)-g(ϕY,Z)η(W)]-A(Y)[g(ϕX,Z)η(W)-g(ϕW,Z)η(X)]+r-(n-1)(n-2)n(n-1)A(W)[g(Y,Z)η(X)-g(X,Z)η(Y)]+r-(n-1)(n-2)n(n-1)A(X)[g(W,Z)η(Y)-g(Y,Z)η(W)]+r-(n-1)(n-2)n(n-1)A(Y)[g(X,Z)η(W)-g(W,Z)η(X)].
Putting Y=Z=ei in (5.15) and taking summation over i, 1≤i≤n, one gets
(5.16)[-n(n-1)(n-2)+r(n-2)-(n-1)(n-2)2n(n-1)]A(X)η(W)+[n(n-1)(n-2)-r(n-2)+(n-1)(n-2)2n(n-1)]A(W)η(X)=A(ϕW)η(X)-A(ϕX)η(W).
Replacing X by ξ in (5.16), one gets
(5.17)[[r(n-2)+2(n-1)(n-2)]2+n2(n-1)2n(n-1)[r(n-2)+2(n-1)(n-2)]][A(W)-A(ξ)η(W)]=0,
therefore
(5.18)A(W)=η(W)η(ρ),
for any vector field W.Hence, one states the following.Theorem 5.2.
In a concircularϕ-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 co-directional and the 1-form A is given by (5.18).



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