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

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Volume 2012, Article ID 757032,9pages doi:10.1155/2012/757032

Research Article

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

Venkatesha, K. T. Pradeep Kumar, C. S. Bagewadi, and Gurupadavva Ingalahalli

Department of Mathematics, Kuvempu University, Shankaraghatta, Shimoga 577 451, India

Correspondence should be addressed to Venkatesha,[email protected] Received 29 March 2012; Accepted 29 May 2012

Academic Editor: J. Dydak

Copyrightq2012 Venkatesha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

The idea of semisymmetric linear connection on a differentiable manifold was introduced by Friedmann and Schouten1. In2, Hayden introduced idea of metric connection with torsion on a Riemannian manifold. Further, some properties of semisymmetric metric connection has been studied by Yano3. In4, 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 Hussain5, Amur and Pujar6, Rastogi7,8, Mishra and Pandey9, 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, Taka- hashi17introduced the notion of localφ-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.Section 2is devoted to preliminaries. InSection 3, we study semisymmetric metric connection in aK-contact manifold. InSection 4, it is proved that aφ-recurrentK-contact manifold with respect to semisymmetric metric connection is an

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Einstein manifold. Finally, inSection 5it is also shown that concircularφ-recurrentK-contact manifold admitting semisymmetric metric connection is an Einstein manifold, and the characteristic vector fieldξand the vector fieldρassociated to the 1-formAare codirectional.

2. 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,

φ²−I η⊗ξ, ηξ 1, η◦φ0, φξ0. 2.1

Thus a manifoldMequipped with this structureφ, ξ, ηis called an almost contact manifold and is denoted by M, φ, ξ, η. If g is a Riemannian metric on an almost contact manifoldMsuch that,

g

φX, φY

gX, Y−ηXηY, gX, ξ ηX, 2.2

whereX,Y are vector fields defined onM, then,Mis said to have an almost contact metric structureφ, ξ, η, g, andMwith this structure is called an almost contact metric manifold and is denoted byM, φ, ξ, η, g.

If onM, φ, ξ, η, gthe exterior derivative of 1-formηsatisfies dηX, Y g

X, φY

, 2.3

thenφ, ξ, η, gis said to be a contact metric structure, andMequipped with a contact metric structure is called an contact metric manifold.

If moreoverξis killing vector field onM, then,Mis called aK-contact Riemannian manifold19,20. AK-contact Riemannian manifold is called Sasakian19, if the relation

∇Xφ

Y gX, Yξ−ηYX 2.4

holds, where∇denotes the operator of covariant diﬀerentiation with respect tog.

In aK-contact manifoldM, the following relations holds:

∇Xξ−φX, 2.5

gRX, YZ, ξ gY, ZηX−gX, ZηY, 2.6

SX, ξ n−1ηX, 2.7

for all vector fieldsX,Y, andZ. HereRandSare the Riemannian curvature tensor and the Ricci tensor ofM, respectively.

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Definition 2.1. AK-contact manifoldMis said to beφ-recurrent if there exists a nonzero 1- formAsuch that,

φ²∇WRX, YZ AWRX, YZ, 2.8

whereAis defined byAW gW, ρ, andρis a vector field associated with the 1-formA.

Definition 2.2. AK-contact manifoldMis said to be concircularφ-recurrent12if there exists a non-zero 1-formAsuch that,

φ²

∇WC

X, YZ

AWCX, YZ, 2.9

whereCis a concircular curvature tensor given by21as follows:

CX, YZRX, YZ− r nn−1

gY, ZX−gX, ZY

, 2.10

whereRis the Riemannian curvature tensor andris the scalar curvature.

A linear connection∇ in ann-dimensional diﬀerentiable manifoldMis said to be a semisymmetric connection if its torsion tensorTis of the form

TX, Y ∇XY−∇YX−X, Y ηYX−ηXY, 2.11

for allX,Y onTM. A semisymmetric connection∇ is called semisymmetric metric connection, if it further satisfies∇g 0.

3. Semisymmetric Metric Connection in a K-Contact Manifold

A semisymmetric metric connection∇ in aK-contact manifold can be defined by

∇XY ∇XY ηYX−gX, Yξ, 3.1

where∇is the Levi-Civita connection onM3.

A relation between the curvature tensor of M, with respect to the semisymmetric metric connection∇ and the Levi-Civita connection,∇is given by

RX, YZ RX, YZ g

φY, Z X−g

φX, Z Y

gY, ZφX−gX, ZφY g

φX, φZ Y−g

φY, φZ

X

ηXgY, Z−ηYgX, Z

ξ, 3.2

whereRandRare the Riemannian curvatures of the connections∇ and∇, respectively.

From3.2, it follows that

SY, Z SY, Z−n−2gY, Z n−2g φY, Z

n−2ηYηZ, 3.3

whereSandSare the Ricci tensors of the connections∇ and∇, respectively.

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Contracting3.3, we get

rr−n−1n−2, 3.4

whererandrare the scalar curvatures of the connections∇ and∇, respectively.

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

AK-contact manifold is calledφ-recurrent with respect to the semisymmetric metric connection if its curvature tensorRsatisfies the following condition:

φ²

∇WR

X, YZ

AWRX, Y Z. 4.1

By virtue of2.1and4.1, we have

−

∇WR

X, YZ η

∇WR

X, YZ

ξAWRX, Y Z, 4.2

from which, it follows that

−g

∇WR

X, YZ, U η

∇WR

X, YZ

gξ, U AWg

RX, Y Z, U

. 4.3

Let {ei}, i 1,2, . . . , n be an orthonormal basis of the tangent space at any point of the manifold. Then puttingXUe_iin4.3and taking summation overi, 1≤i≤n, we get

−

∇WS

Y, Z ⁿ

i1

η

∇WR

ei, YZ

ηei AWSY, Z. 4.4

PutZξ, then the second term of4.4takes the following form:

g

∇WR

ei, Yξ, ξ g

∇WRe i, Yξ, ξ

−g R

∇Wei, Y ξ, ξ

−g R

ei,∇WY ξ, ξ

−g

Re i, Y∇Wξ, ξ .

4.5

On simplification, we obtaing∇WRe i, Yξ, ξ 0.

Now4.4implies that

∇WS

Y, ξ −AWSY, ξ. 4.6

We know that

∇WS

Y, ξ ∇WSY, ξ −S

∇WY, ξ

−S

Y,∇Wξ

. 4.7

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Using3.3,2.5, and2.7in the above relation, we get ∇WS

Y, ξ S Y, φW

−SY, W−n−1g Y, φW n−1gY, W 2n−2g

φY, φW .

4.8

In view of4.6and4.8, we have SY, W−S

Y, φW

n−1g Y, φW

−n−1gY, W−2n−2g

φY, φW

n−1AWηY. 4.9

Again puttingY φYin4.9, we get S

φY, W

−S

φY, φW

n−1g

φY, φW

−n−1g φY, W

2n−2g Y, φW

0.

4.10

InterchangingY and W in4.10, we obtain S

φW, Y

−S

φW, φY

n−1g

φW, φY

−n−1g φW, Y

2n−2g W, φY

0.

4.11

Adding4.10and4.11which on simplification, we have

SY, W n−1gY, W. 4.12

Therefore, we can state the following.

Theorem 4.1. Aφ-recurrentK-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.

5. Concircular φ-Recurrent K-Contact Manifold with respect to Semisymmetric Metric Connection

Let us consider a concircularφ-recurrentK-contact manifold with respect to the semisymmetric metric connection defined by

φ²

∇WC

X, YZ

AWCX, Y Z, 5.1

whereC is a concircular curvature tensor with respect to the semisymmetric metric connection given by

CX, YZ RX, Y Z− r nn−1

gY, ZX−gX, ZY

. 5.2

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By virtue of2.1and5.1, we have

− ∇WC

X, YZ η

∇WC

X, YZ

ξAWCX, Y Z, 5.3

from which, it follows that

−g

∇WC

X, YZ, U

η

∇WC

X, YZ

gξ, U AWg

CX, Y Z, U

, 5.4 where

∇WC

X, YZ ∇WRX, YZ 3

gY, WηZX−gX, WηZY 3

gY, ZgW, X−gX, ZgW, Y ξ 2

ηXg φW, Z

Y−ηYg φW, Z

X 2

ηYgX, Z−ηXgY, Z

φW

gY, ZηX−gX, ZηY W

2ηW

ηYgX, Z−ηXgY, Z ξ

2ηZηW

ηXY−ηYX

gZ, W

ηYX−ηXY

−gW, RX, YZξ−ηXRW, YZ

−ηYRX, WZ−ηZRX, YW

− ∇Wr nn−1

gY, ZX−gX, ZY .

5.5 Let {ei}, i 1,2, . . . , n be an orthonormal basis of the tangent space at any point of the manifold. Then puttingXUeiin5.4and taking summation overi, 1≤i≤n, we get

∇WS

Y, Z−∇Wr

n gY, Z − ∇Wr nn−1

gY, Z−ηYηZ

−AW

SY, Z − r ngY, Z

. 5.6

ReplacingZbyξin5.6, we obtain ∇WS

Y, ξ ∇Wr

n ηY−AW

SY, ξ − r nηY

. 5.7

We know that

∇WS

Y, ξ ∇WSY, ξ −S

∇WY, ξ

−S

Y,∇Wξ

. 5.8

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Using3.3,2.5and2.7, the above relation becomes ∇WS

Y, ξS Y, φW

−SY, W−n−1g Y, φW

n−1gY,W 2n−2g

φY,φW . 5.9

In view of5.7and5.9, we obtain S

Y, φW

−SY, W−n−1g Y, φW

n−1gY, W 2n−2gY, W−2n−2ηYηW ∇Wr

n ηY−AW

2n−1²−r

n ηY

.

5.10

ReplacingYbyφYin5.10, we have S

φY, φW

−S φY, W

−n−1g

φY, φW

n−1g φY, W

2n−2g φY, W

0.

5.11

InterchangingY andWin5.11, we get S

φW, φY

−S φW, Y

−n−1g

φW, φY

n−1g φW, Y

2n−2g φW, Y

0.

5.12

Adding5.11and5.12, which on simplification, we have

SY, W n−1gY, W. 5.13

Thus, we obtain the following theorem.

Theorem 5.1. A Concircularφ-recurrentK-contact manifold with respect to semisymmetric metric connection is an Einstein manifold.

Next, from5.3, one has ∇WC

X, YZη

∇WC

X, YZ

ξ−AWCX, YZ. 5.14

Now, using3.2,3.4,5.5, and Bianchi’s identity in5.14, one obtains AWηRX, YZ AXηRY, WZ AYηRW, XZ

−AW g

φY, Z

ηX−g φX, Z

ηY

−AX g

φW, Z

ηY−g φY, Z

ηW

−AY g

φX, Z

ηW−g φW, Z

ηX

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r−n−1n−2

nn−1 AW

gY, ZηX−gX, ZηY r−n−1n−2

nn−1 AX

gW, ZηY−gY, ZηW r−n−1n−2

nn−1 AY

gX, ZηW−gW, ZηX .

5.15

PuttingY Ze_iin5.15and taking summation overi, 1≤i≤n, one gets −nn−1n−2 rn−2−n−1n−2²

nn−1

AXηW

nn−1n−2−rn−2 n−1n−2² nn−1

AWηX A

φW

ηX−A φX

ηW.

5.16

ReplacingXbyξin5.16, one gets

rn−2 2n−1n−2² n²n−1² nn−1rn−2 2n−1n−2

AW−AξηW

0, 5.17

therefore

AW ηWη

ρ

, 5.18

for any vector fieldW.

Hence, one states the following.

Theorem 5.2. In a concircular φ-recurrent K-contact manifold admitting semisymmetric metric connection the characteristic vector fieldξ and the vector fieldρ associated to the 1-formAare co- directional and the 1-formAis given by5.18.

Acknowledgment

The authors express their thanks to DST Department of Science and Technology, Gov- ernment of India for providing financial assistance under major research project no.

SR/S4/MS:482/07.

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